Deterministic Chaos in One-Dimensional Continuous Systems 2015035631, 9789814719698, 9814719692

Table of contents :
0_front-matter-2016
1_bifurcational-and-chaotic-dynamics-of-simple-structural-members--2016
2_introduction-to-fractal-dynamics-2016
3_introduction-to-chaos-and-wavelets-2016
4_simple-chaotic-models-2016
5_discrete-and-continuous-dissipative-systems-2016
6_eulerbernoulli-beams-2016
7_timoshenko-and-sheremetevpelekh-beams-2016
8_panels-2016
9_plates-and-shells-2016
10_back-matter-2016

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Deterministic Chaos in One-Dimensional Continuous Systems J. Awrejcewicz, V. A. Krysko, I. V. Papkova & A. V. Krysko

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Series Editor: Leon O. Chua

Deterministic Chaos in One-Dimensional Continuous Systems Jan Awrejcewicz Lodz University of Technology, Poland

Vadim A Krysko Irina V Papkova Saratov State Technical University, Russia

Anton V Krysko Saratov State Technical University, Russia Cybernetic Institute, National Research Tomsk Polytechnic University, Russia

International Institute for Applied Systems Analysis

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Library of Congress Cataloging-in-Publication Data Names: Awrejcewicz, J. (Jan) Title: Deterministic chaos in one dimensional continuous systems / Jan Awrejcewicz [and three others]. Description: New Jersey : World Scientific, 2015. | Series: World Scientific series on nonlinear science. Series A ; vol. 90 | Includes bibliographical references and index. Identifiers: LCCN 2015035631| ISBN 9789814719698 (hardcover : alk. paper) | ISBN 9814719692 (hardcover : alk. paper) Subjects: LCSH: Engineering mathematics. | Buildings--Vibration. | Strength of materials--Mathematics. | Building materials. Classification: LCC TA342 .D48 2015 | DDC 624.1/7011857--dc23 LC record available at http://lccn.loc.gov/2015035631

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Preface There is no hope, at least in the coming future, to solve in full the problem devoted to vibrations of 3D thermo-elastic structural elements, which are widely applied in mechanical and civil engineering (bridges, silos, oil platforms), civil and military industries (aircrafts, space launchers, missiles, tanks, satellites), biomechanics and biomedical engineering (stents, surgical devices), maritime and offshore engineering (ships, boats, sea platforms, tubes and pipelines), cars and motorcycle factories, as well as MEMS engineering. In general, the proposed book deals with nonlinear vibrations of structural members (beams, plates, panels, shells), where the dynamical problems can be reduced to that of one spatial variable and time, and therefore they are further named as 1D systems. Our aim is not to synthesize and overview the general literature devoted to this problem as it is very easy (using Google, Wikipedia, and so on) nowadays to find a lot of material and descriptions including the history as well as state-of-the art information about the mechanics/dynamics of structural members. The roots of structural mechanics come from Cauchy (1828), Poisson (1829), Kirchhoff (1850) and von K´ arm´an (1910). The firstorder shear deformation utilized by the Reissner–Mindlin theory was extended by Reddy (1990), who developed the higher order shear deformation theory including cubic terms. On the other hand, the Donnell–Mushtari–Vlasov nonlinear shallow shell theory has been proposed, which is validated for very thin shells by neglecting the in-plane inertia, transverse shear deformation and rotary inertia. Nonlinear theories for moderate and large deformations of thin elastic shells have been presented by Mushtari and Galimov (1957), Vlasov v

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(1944) and Vorovich (1999). The tensor notation has been applied by Sanders (1968) and Koiter (1966) yielding the Sanders–Koiter equations governing the nonlinear shells vibrations. Another track in the development shell theory was added by the Fl¨ ugge–Lur’e–Byrne’s modification, which was further generalized by Novozhilov (1953). Additional important contributions to shell theories have been given by Naghdi and Nordgren (1963), Librescu (1987), and Libai and Simmonds (1988). As it has already been mentioned, investigation of stability, vibrations and buckling of mechanical structures should be considered as a 3D time-dependent nonlinear problem taking into account various nonlinear factors such as geometrical nonlinearity, physical nonlinearity, elastic-plastic properties, cyclic loadings, damping and material rheological properties, relaxation and hysteresis phenomena, fatigue resistance among others. Note that the mentioned complex problem cannot be solved fully in spite of the recent remarkable development of numerical approaches. The main barrier stopping the investigation is associated with a so-called curse of dimension, since three independent spatial variables and time are involved in the studied dynamical process. On the other hand, engineering and physical processes require reliable results obtained in engineering-accepted simulation time intervals, which requires many novel proposals of modeling of particular problems of dynamics devoted to simple structural members like beams, strings, plates, rods, panels, shells, as well as their dynamical interactions. Nowadays, engineering constructions can be viewed as a collection of interacting sets consisting of the mentioned simple structures usually modelled either as 1D or 2D time-dependent spatial problems. The mathematical theory of elasticity takes into account various aspects of static and dynamic problems of deflection of beams and shells in a 3D formulation. The same problem, from an engineering approach, can be approximated via reduction of sets of 3D PDEs to that of 1D and 2D problems. The carried out reduction is based on physical motivations and imaginations being analogous to a formal mathematical approach aimed at reduction of problems with infinite dimension to finite one, or equivalently, the employed reduction

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process includes the transition from governing PDEs to a set of finite (relatively large) number of ODEs, or finally the reduction is aimed at decreasing of an order of the governing input PDEs. Therefore, in the case of beams and rods two coordinates can be removed while bending requires the introduction of numerous hypotheses. The mentioned approaches have been initiated by Bernoulli, Euler, Rayleigh, Timoshenko, Sheremetev, Pelekh, Ambartsumian, Grigoliuk and others. In the case of shell-type structures, two main classes of problems can be studied: (i) 2D spatial problems, where the generalized beam hypotheses can be applied; (ii) axially symmetric shells. It is observed that the mentioned modeling yields the counterpart 1D problems but on the other hand, approximates the shell-type constructions. In what follows, we show how the problem regarding thickness of the shell-type constructions can be overcome. Recall that in classical theories of Euler and Bernoulli developed in the XVIII Century [Euler (1757)] the following main assumptions were introduced: (1) transversal cross sections of a beam that are flat and perpendicular to the beam axis before deformation remain plain and perpendicular to the beam axis during the deformation process in time; (2) Normal stresses on the fibers located in parallel to the rod axis are small and can be neglected, i.e. the longitudinal cross sections induce resistance against deflection independently, without any interaction between them; (3) inertial effects of rotation of rod elements are omitted within the deflection process. In Rayleigh’s theory proposed in 1873 [Strutt (1899)], inertial effects introduced by the beam element rotation while a rod undergoes simulteneous bending have been taken into account, which implies a reconstruction of the form of the beam kinetic energy. In addition, in the theory proposed by Timoshenko in 1921 [Timoshenko (1921a)] it is assumed that the transversal cross sections remain flat but not perpendicular to the beam axis. The latter hypothesis implies an occurrence of additional terms in the formula governing the rod potential energy. In the published works of B.L. Pelekh and M.P. Sheremetev in 1964 [Pelekh (1973, 1978); Pelekh and Teters (1968); Sheremetev and Pelekh (1964)] the following generalizations of the Timoshenko

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theory have been proposed: (i) a transversal cross section being perpendicular to the beam axis before deformation is not only perpendicular to the beam axis after the rod deformation process but also curved. This hypothesis implies the occurrence of a principally novel set of PDEs governing rod dynamics. Around 17 years after publishing the results by Pelekh and Sheremetev, similar conclusions have been published by Levinson and Reddy [Levinson (1981); Reddy, (1984a,1984b)]. The mathematical model introduced by Pelekh–Sheremetiev has been applied and developed in the works of V.A. Krysko and his co-workers [Krysko (1976)] (we cite only the monograph while the remaining papers have been published in Russian). This brief description given in the above explains why beam-type structures and axially symmetric shells are interpreted as 1D spatial structures in this book. The book is organized in the following manner — it consists of nine chapters. Each chapter begins with a short introductory overview of the chapter contents. Chapter 1 presents a short literature critical overview emphasizing on the recent results devoted to analysis and control of nonlinear dynamics of beams, plates, panels and shells, and their interplay with nonlinear problems of stability, buckling, bifurcation and chaos. The first few chapters concentrate on introduction to fractal dynamics (Chapter 2), definitions of chaos and scenarios of transition from regular to chaotic dynamics as well as an introduction of the classical Fourier analysis (FFT) versus wavelet transform approaches (Chapter 3). Simple chaotic systems are briefly displayed and commented in Chapter 4, and the illustrated examples of strange chaotic attractors are then revisited in the remaining chapters which are purely devoted to the study of structural members. Modeling of dissipative systems and factors met in real-world processes for both discrete (lumped mechanical systems) and continuous structural members are briefly introduced in Chapter 5. Chaotic dynamics of Euler–Bernoulli beams including geometric and physical nonlinearities, taking into account thermal effects and elasticplastic deformations, has been modeled, illustrated and discussed in Chapter 6. Chapter 7 presents a study of the Timoshenko and

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the Pelekh–Sheremetev beams along with numerous examples of rich nonlinear dynamics with application of wavelet transforms and computations of Lyapunov exponents. Bifurcational dynamics as well as chaos, hyperchaos, hyper–hyper chaos and deep chaos exhibited by the rectangular plate-strips and the cylindrical panels are investigated in Chapter 8. The interplay of the obtained results with the Sharkovsky series and his theorem is illustrated among others, and the realibility of the obtained results is addressed. Chapter 9 presents numerous novel bifurcation and chaotic phenomena exhibited by spherical and conical shells with constant and variable thickness emphasizing on spatio-temporal dynamics and control of chaotic vibrations with the help of the wavelet-based analysis. The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with nonlinear dynamical systems, as well as theoretically inclined mechanical and civil engineers. The book has the following unique and original features which distinguishes it from other books existing in the market: (i) The state-of-the-art nonlinear dynamics of structural members is briefly addressed allowing a reader to follow the main book track, which spans across various research topics such as vibrations of beams, plates, panels and shells with nonlinearity, bifurcation and chaos. (ii) Novel methods versus classical approaches are presented to study nonlinear phenomena exhibited by continuous systems with the help of Lyapunov exponents (exponents up to four are estimated), the wavelet-based analysis as well as neural network approaches to achieve realiable numerical results in a faster way. (iii) Our approach, contrary to majority of analyses (see the literature overview given in Chapter 1), relies on truncation of nonlinear PDEs governing the dynamics of structural members in a way that gives reliable and validated results. In other words, the approximated set contains a relatively large number of nonlinear ODEs modelling real objects with infinite degrees of freedom.

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(iv) All the presented results devoted to structural members include numerous novel nonlinear phenomena and are based either on previously/recently published or unpublished authors’ results. (v) Applied mathematicians and physicists should be attracted by the fascinating and rich dynamics exhibited by simple structural members and by the solution properties of the governing 1D nonlinear PDEs. On the other hand, the engineeringoriented researchers and graduate students will sometimes find unexpected nonlinear phenomena, which are a waiting experimental validation. (vi) Our book is based only on numerical computations, but a reader may find other analytical or semi-analytical approaches based on asymptotic theories in the following supplementary books [Andrianov et al. (2004, 2014); Awrejcewicz et al. (1998)]; (vii) The book endeavours to utilize and extend our earlier results presented in the monographs [Awrejcewicz and Krysko (2003, 2008); Awrejcewicz et al. (2007, 2004)]. (viii) The book covers a wide variety of the studied PDEs, the way of their validated reduction to ODEs, classical and non-classical methods of analysis, influence of various boundary conditions and control parameters, as well as the illustrative presentation of the obtained results in the form of 2D and 3D figures and vibration type charts and scales. (ix) The book contains originally discovered, illustrated and discussed novel and/or modified classical scenarios of transition from regular to chaotic dynamics exhibited by 1D structural members. It shows a way to control chaotic and bifurcational dynamics as well as gives directions to study other dynamical systems modelled by chains of nonlinear oscillators. (x) The book presents numerous challenges for civil/mechanical engineers showing that more sophisticated tools are required to understand and predict real structural responses.

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The authors wish to express their thanks to M. Ka´zmierczak, R. Kepi´ nski and O. Szymanowska for their help in the book preparation. Finally, J. Awrejcewicz acknowledges the financial support of National Science Centre of Poland under the grant MAESTRO 2, No. 2012/04/A/ST8/00738, during 2013–2016. Lodz and Saratov, 2015

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Contents Preface 1.

Bifurcational and Chaotic Dynamics of Simple Structural Members 1.1 1.2 1.3 1.4

2.

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Cantor Set and Cantor Koch Snowflake . . . . 1D Maps . . . . . . . Sharkovsky’s Theorem Julia Set . . . . . . . . Mandelbrot’s Set . . .

Dust . . . . . . . . . . . . . . . . . . . . .

Introduction to Chaos and Wavelets 3.1 3.2

4.

Beams Plates Panels Shells

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. 2 . 5 . 9 . 12

Introduction to Fractal Dynamics 2.1 2.2 2.3 2.4 2.5 2.6

3.

v

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14 17 20 23 25 27 31

Routes to Chaos . . . . . . . . . . . . . . . . . . . . 31 Quantifying Chaotic Dynamics . . . . . . . . . . . . 46

Simple Chaotic Models 4.1 4.2 4.3

14

86

Introduction . . . . . . . . . . . . . . . . . . . . . . . 86 Autonomous Systems . . . . . . . . . . . . . . . . . 87 Non-Autonomous Systems . . . . . . . . . . . . . . . 102 xiii

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5.

6.5 6.6 6.7

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Introduction . . . . . . . . . . . . . . . . . . . . . . Planar Beams . . . . . . . . . . . . . . . . . . . . . Linear Planar Beams and Stationary Temperature Fields . . . . . . . . . . . . . . . . . . . . . . . . . Curvilinear Planar Beams and Stationary Temperature Fields . . . . . . . . . . . . . . . . . . Flexible Curvilinear Beam in Stationary Temperature and Electrical Fields . . . . . . . . . Beams with Elasto-Plastic Deformations . . . . . . Multi-Layer Beams . . . . . . . . . . . . . . . . . .

Timoshenko and Sheremetev–Pelekh Beams 7.1 7.2 7.3

105 106 109 116 117 118 129

. 130 . 140 . 169 . 207 . 233 . 238 . 269 307

340 Infinite Length Panels . . . . . . . . . . . . . . . . . 341 Cylindrical Panels of Infinite Length . . . . . . . . . 438

Plates and Shells 9.1 9.2

. . . . . .

The Timoshenko Beams . . . . . . . . . . . . . . . . 307 The Sheremetev–Pelekh Beams . . . . . . . . . . . . 324 Concluding Remarks . . . . . . . . . . . . . . . . . . 339

Panels 8.1 8.2

9.

Introduction . . . . . . . . . . . . . . Linear Friction . . . . . . . . . . . . Nonlinear Friction . . . . . . . . . . Hysteretic Friction . . . . . . . . . . Impact Damping . . . . . . . . . . . Damping in Continuous 1D Systems

105

Euler–Bernoulli Beams

6.4

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Discrete and Continuous Dissipative Systems

6.1 6.2 6.3

7.

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5.1 5.2 5.3 5.4 5.5 5.6 6.

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459

Plates with Initial Imperfections . . . . . . . . . . . 460 Flexible Axially-Symmetric Shells . . . . . . . . . . . 491

Bibliography

530

Index

551

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Chapter 1

Bifurcational and Chaotic Dynamics of Simple Structural Members: Literature Review There are numerous papers and books devoted to dynamics of structural members, i.e. beams, plates, panels and shells. The aim of this chapter is to overview the literature and state-of-the art research devoted to dynamics of the aforementioned structural members and their interplay with bifurcation and chaotic phenomena. This is a novel challenging research track, and hence not many papers and books are published in this topic. In general, from a mathematical point of view, the book deals with nonlinear PDEs and the developed methods for analysis of their solutions with respect to stability, bifurcations, buckling, as well as regular and chaotic dynamics of the modeled continuous objects. On the other hand, the problem is always reduced (though by different ways) to a study of a set of large amount of nonlinear ODEs. In the case of a few first-order nonlinear ODEs, they may govern dynamics of simple nonlinear autonomous and non-autonomous oscillators (two or three first-order ODEs) or dynamics of coupled oscillators. This is why in the first four chapters a background about bifurcational and chaotic dynamics of difference and differential equations has been given to introduce the reader with basic knowledge devoted to dynamics of lumped mechanical systems and beyond. This effort allows for a smooth transition from nonlinear dynamics exhibited by simple dynamical systems to that of simple continuous systems, which can be also understood as 1D or 2D chains of nonlinear oscillators.

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Beams

Though Galileo Galilei and Leonardo da Vinci are among the first scientists who considered modeling of a beam, their theories have not been completed due to incorrect assumptions and hypotheses. Jacob Bernoulli (1654–1705) observed that the elastic beam curvature at any of its point is proportional to the bending moment at that point. This idea has been extended by Daniel Bernoulli (1700–1782), who derived the partial differential equation governing beam dynamics. This theoretical background has been used and extended by Leonhard Euler (1707–1783), who rigorously investigated elastic beams subjected to various loads. From an engineering point of view, a beam is a mechanical (civil engineering) structure with one of its dimensions significantly larger than the remaining two dimensions. The so far introduced rough definition of a beam can be directly applied to beam-like structures such as shafts, manipulator arms, airplane wings, long-span bridges, flexible satellites, fuselages, etc. Nowadays, there exist several beam theories originating from solid mechanics. However, the concepts introduced by Bernoulli and Euler are simple and acceptable in engineering, and this theory is commonly known as Euler– Bernoulli beam theory, the classical beam theory, Euler beam theory, Bernoulli beam theory and Bernoulli–Euler beam theory. The Euler–Bernoulli beam theory relies on the main assumption that the cross-section of the beam is infinitely rigid in its own plane (in-plane displacement field is composed of two rigid body translations and one rigid body rotation). The second assumption says that the cross-section of a beam remains plane after deformation. The third assumption is that the cross-section remains normal to the deformed axis of the beam. The given assumptions are known as the Euler–Bernoulli assumptions for beams or as kinematic assumptions for Euler–Bernoulli beams. The Euler–Bernoulli beam can be extended to include an analysis of curved beams, buckling beam phenomena, composite beams, geometrically nonlinear beams, 3D transverse loading beams, as well as beams under viscoelastic/plastic deformations. In spite of that the

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Bifurcational and Chaotic Dynamics of Simple Structural Members

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3

original Euler–Bernoulli theory holds for infinitesimal strains and small rotations and can be extended to fit problems of moderately large rotations with inclusion of the von K´arm´an strains. There exists a vast literature devoted to the Euler–Bernoulli beam dynamics and its state-of-the-art that will be omitted here. In what follows, our review of the literature is mainly aimed at the bifurcational and chaotic dynamics of the Euler–Bernoulli beams. Abhyankar et al. [Abhyankar et al. (1993)] have applied an explicit finite difference scheme to study partial differential equations governing the Euler–Bernoulli beam dynamics, aimed at analysis of the beam chaotic vibrations. The space–time spectral method has been employed to solve a simply supported modified Euler–Bernoulli nonlinear beam exhibiting lateral forced vibrations [Bar–Yoseph et al. (1996)]. The authors used a generalized Galerkin method for the temporal discretization as well as a discontinuous mixed Galerkin method for the temporal discretization. The obtained solutions have been compared visually with the reference solution of the Duffing equation. The classical Euler–Bernoulli theory has been employed to study microtubules, i.e. proteins organized in a network [Civalek and Demir (2011)]. Eringen’s non-local elasticity theory has been used to include the size effect, but only static analysis has been presented. Nonlinear responses of a clamped–clamped buckled Euler– Bernoulli beam governed by a PDE with a cubic nonlinearity has been studied by Barari et al. [Barari et al. (2011)]. However, the problem has been strongly truncated and reduced to that of the Duffing equation. The nonlinear transverse vibrations of a simply supported Euler–Bernoulli beam under both principal parametric and internal resonances have been analyzed in reference [Sahoo et al. (2013)]. Periodic, quasiperiodic and chaotic dynamics matched with resonances have been carried out using the method of multiple scales. Recently, attention has been paid to microbeams widely applied in micro-electro-mechanical systems (MEMS). Avsec [Avsec (2011)] has studied vibrations of microbeams and nanotubes, whereas Batra et al. [Batra et al. (2008)] analyzed vibrations of narrow microbeams initially predeformed by an electric field. Pull-in instability of

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structural members including microbeams has been analyzed in references [Kacem et al. (2012); Krylov (2007); Moghimi and Ahmadian (2009, 2010); Vyasarayani et al. (2011)]. Chaotic electrostatic microbeam oscillator has been studied by Towfighian et al. [Towfighian et al. (2010)], and bifurcation diagrams in the plane voltage amplitude-frequency have been constructed. A micro Euler–Bernoulli beam, under electro-statically actuated voltage has been analyzed in reference [Sedighi and Shirazi (2013)] taking into account the von K´ arm´an nonlinearity, and the pull-in instability behavior has been investigated. However, only the first mode approximation has been taken into account while applying the Bubnov–Galerkin procedure. The Euler–Bernoulli simply supported beam taking into account the von K´ arm´an geometric nonlinearity under an extremal excitation has been studied in reference [Dai and Sun (2014)]. They carried out a comparison of a chaotic and a multi-dimensional model and concluded that in order to control chaotic beam dynamics the contribution of high order vibrations should be taken into account. In spite of Euler–Bernoulli beam theory, there exists the widely applied Timoshenko beam theory [Rosinger and Ritchie (1977); Timoshenko (1921a, 1922, 1932)] developed by S. Timoshenko (1878– 1972). The Timoshenko model is particularly suitable for analysis of a short beam, composite beams or beams under high-frequency excitation, since it takes into account rotational inertia effects and shear deformation. The Timoshenko beam model can be viewed as more general with respect to the Euler–Bernoulli model because if the shear modulus of the beam material tends to infinity (beam is rigid in shear) and if rotational inertia phenomena can be neglected, the Timoshenko model takes the form of the Euler–Bernoulli model. Since the Euler–Bernoulli beam does not include a shear deformation yielding rotation, it is stiffer compared to the Timoshenko beam. However, if a real beam has large length and small thickness, then a difference between two models is small. It should be noted that the effect of rotary inertia was proposed by Rayleigh in 1894, and in literature there exists the Rayleigh beam theory. Recently, the Euler–Bernoulli and Timoshenko models have been reconsidered with respect to the two-dimensional elasticity model

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[Labuschagne et al. (2009)]. Using the example of a cantilever beam, the authors conclude that the Timoshenko model is close to that constructed with the help of the 2D theory for models of practical use. The Euler–Bernoulli beam theory corresponds to the first approximation, whereas the Timoshenko theory stands for the refined beam theory. Li [Li (2008)], using examples of functionally graded beams with the rotary inertia and shear deformation included, showed that the Euler–Bernoulli and Rayleigh beam theories can be derived from the Timoshenko beam theory. Nonlinear flexural waves and chaotic vibrations of a Timoshenko beam have been analyzed by Zhang and Liu [Zhang and Liu (2010)]. The finite-deflection and the axial inertia are included into the derived PDEs governing flexural waves in the beam. Then the problem has been strongly reduced into a nonlinear ordinary equation. The latter non-autonomous equation has been analyzed following the classical Melnikov method to define the threshold condition of the occurrence of a transverse heteroclinic point, and to predict the deterministic chaos. He’s homotopy perturbation method has been applied to study nonlinear free vibrations of clamped-clamped and clamped-free Timoshenko microbeams by Moeenfard et al. [Moeenfard et al. (2011)]. Again, the original problem of infinite dimension has been strongly truncated to consideration of only one nonlinear ordinary differential equation. 1.2

Plates

The study of vibrations of plates has a long history in mechanics and applied mathematics and numerous books and papers have been published dedicated to this issue. In this brief state-of-the-art review, we aim only at recent results matching nonlinear vibrations of plates with bifurcation and chaotic phenomena. Both local and global bifurcations of parametrically excited nearly square plates have been studied by Yang and Sethna [Yang and Sethna (1991)], putting emphasis on occurrence of the Smale horseshoe manifold. This approach based mainly on the

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averaging procedure has been extended to analyze vibrations of nearly square plates to the anti-symmetric case [Yang and Sethna (1992)]. Shilnikov-type homoclinic orbits, bifurcations and chaotic vibrations of thin plates subjected to parametric excitation have been analyzed in reference [Feng and Sethna (1993)] using the so-called global perturbation method. The multiple scales method has been applied to study modal interaction of nonlinear clamped plates and harmonic excitation by Hadian and Nayfeh [Hadian and Nayfeh (1990)]. A double mode approach has been employed by Shu et al. [Shu et al. (1999)], and chaotic vibrations have been detected using the Melnikov method. The global perturbation method has been applied in references [Samoylenko and Lee (2007); Yeo and Lee (2006)] to study global vibrations of an imperfect circular plate for the case of 1:1 internal resonance, and the chaotic orbits. A similar analytical technique has been used by Yu and Chen [Yu and Chen (2010)] to study global bifurcations of a simply supported rectangular metallic plate under a transverse harmonic excitation. Both Melnikov’s approach and averaging procedure have been employed by Zhang and Li [Zhang and Li (2010)] to detect resonant chaotic vibrations of a simply supported rectangular plate excited externally and periodically. The extended Melnikov-type analytical approach has been applied by Zhang et al. [Zhang et al. (2008, 2010)] to study the global bifurcations and multipulse chaotic vibrations of a buckled thin plate as well as a laminated composite piezoelectric rectangular plate. The so-called energy-phase method has been utilized by Yao and Zhang [Yao and Zhang (2007)] to analyze the Shilnikov-type multi-pulse heteroclinic orbits and chaotic vibrations of a parametrically and externally excited rectangular thin plate. The similar-like approach has been used to examine the Shilnikov-type multi-pulse homoclinic orbits exhibited by a circular plate excited harmonically [Yu and Chen (2010b)]. More recently, Yao and Zhang [Yao and Zhang (2014)] have investigated multi-pulse global heteroclinic bifurcations and chaotic dynamics of a simply supported rectangular thin plate in the resonant case using the extended Melnikov method.

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The so far reported references deal with a strong truncation of the studied PDEs governing dynamics of plates, and hence many nonlinear phenomena have been omitted during those studies. There are also papers mainly based on numerical simulations, which are closer to the investigation topic of this book. The fractal dimensions, the maximum Lyapunov exponents and bifurcation diagrams have been employed to study bifurcational and chaotic vibrations of a simply supported thermoelastic plate with variable thickness in reference [Yeh et al. (2003)]. The global bifurcation and chaotic dynamics of a rectangular thin plate have been studied by Zhang [Zhang (2001)]. Touz´e et al. [Touz´e et al. (2011)] have studied the transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates based on the von K´ arm´an PDEs for thin plates including geometric nonlinearity. The obtained numerical results have been confirmed with experimental investigations. High dimensional chaos versus the framework of wave turbulence have been illustrated and discussed. Recently, a challenging interest coming from an industry has been devoted to new functionally graded materials (FGMs), being inhomogeneous composites made of a mixture of metals and ceramics. FGM plates have been widely used in various branches of the industry since their material properties allow for a smooth and continuous change from one surface to another, and hence the problem of thermal stress concentrations can be withdrawn. The nonlinear transient thermoelastic analysis of FGM plates under pressure loading and temperature fields has been carried out by Parveen and Reddy [Parveen and Reddy (1998)]. Yang and Shen [Yang and Shen (2002)] studied free and forced vibrations of FGM plates in a thermal environment, and the material parameters have been temperature-dependent. Dynamics of the pre-stressed graded layer and two surface-mounted piezoelectric actuator layer have been examined by Yang et al. [Yang et al. (2003)]. Nonlinear dynamics of FGM plates embedded in thermal environment including heat conduction and temperature-variable material properties have been studied by Huang and Shen [Huang and Shen (2004)]. The effect of transverse shear strains and rotary inertia have been taken into

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account while analyzing nonlinear vibrations of FGM plates with initial stress by Chen [Chen (2005)]. The nonlinear dynamics of a simply supported FGM rectangular plate under both transversal and in-plane excitation in a thermal environment has been investigated by Hao et al. [Hao et al. (2008)]. The problem has been reduced to the study of a two-degree-of-freedom system with the quadratic and cubic nonlinear terms. The cases of internal and principal parametric resonances have been analyzed using asymptotic perturbation methods. Periodic, quasi-periodic and chaotic vibrations have been studied. A 3D-exact solutions for free and forced vibration of simply supported FGM rectangular plates has been derived by Vel and Batra [Vel and Batra (2004)]. Nonlinear vibrations of FGM plates with randomness of the material properties and plates with geometric imperfections have been considered by Kitipornchai et al. [Kitipornchai et al. (2004, 2006)]. Yang and Huang [Yang and Huang (2007)] dealt with a semianalytical investigation of geometrical imperfections on the nonlinear vibrations of simply supported FGM plates. Nonlinear vibration, bifurcation and chaos of viscoelastic cracked plates have been investigated by Hu and Fu [Hu and Fu (2007)]. The von K´ arm´ an plate theory and the linear isotropic constitutive theory have been employed to a nonlinear integral-partial differential equation for a rectangular plate with an all-over part-through crack. In particular, the effects of the depth and the position of the crack and the viscoelastic material parameters’ impact on the bifurcational and chaotic dynamics with movable simply-supported boundary conditions have been analyzed. Onozato et al. [Onozato et al. (2009)] carried out laboratory experiments on chaotic vibrations of a rectangular plate under in-plane elastic constraint at clamped edges. Chaotic responses have been examined using the Fourier spectra, the Poincar´e projections, the maximum Lyapunov exponents and the Karhunen–Lo´eve method. Zhang et al. [Zhang et al. (2010)] investigated nonlinear vibrations and chaos of a FGM rectangular plate in thermal environment and under parametric and external excitations. The governing PDEs are derived based on the Reddy third-order shear deformation plate theory and the Hamilton’s principle. Application of

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the Galerkin procedure yielded a three-degree-of-freedom nonlinear systems, which has been further studied via multiple scales and classical numerical approaches. Bifurcation and chaotic phenomena exhibited by an axially moving plate under external and parametric excitations have been examined by Liu et al. [Liu et al. (2012)]. The derived coupled PDEs of transverse deflection and stress have been reduced to a set of ODEs, which has been studied numerically. A relevance between the onset of chaos with the corresponding linear instability range has been illustrated and discussed. The global bifurcations and multiple chaotic vibrations of a simply supported laminated composite piezoelectric rectangular thin plate subjected to parametric and transverse excitations have been investigated by Zhang and Zhang [Zhang and Zhang (2011)]. Both von K´ arm´an plate theory and the first-order piston theory are used to derive PDEs governing nonlinear dynamics of cantilever plate in supersonic flow in reference [Xie et al. (2014)]. The Rayleigh–Ritz procedure has been employed to get ODEs being then solved numerically. Chaotic, prechaotic and postchaotic regimes have been identified, and the routes to chaos have been studied. 1.3

Panels

In this section, we proceed to a brief review of the published papers contributing to bifurcational and chaotic vibrations of panels. Maestrello et al. (1992) studied the dynamic response of an aircraft panel forced at resonance and off-resonance by plane acoustic waves. Period doubling bifurcations and chaotic panel vibrations have been detected, when the sound pressure level of the excitation increased. Good agreement between the experimental and numerical results has been obtained. Yamaguchi and Nagai [Yamaguchi and Nagai (1997)] have presented numerical results devoted to chaotic vibrations of a shallow cylindrical shell-panel subjected to harmonic lateral excitation. Based on the Donnell–Mushtari–Vlasov theory, the problem has been reduced to that of a multiple-degree-of freedom system by the Galerkin procedure. Chaotic vibrations combined with a dynamic

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snap-through have been monitored with the help of Lyapunov exponents and Poincar´e maps. The effect of the in-plane elastic constraint on the chaos of the shell-panel has been illustrated and discussed. Chaotic vibrations of a panel forced by turbulent boundary layer and sound have been studied by Maestrello [Maestrello (1999)]. The panel response combined with period-doubling bifurcations makes a transition to chaos when forced by the boundary layer, which has been associated with quasi-periodic dynamics as the wave loses the spatial homogeneity. Nonlinear bifurcations and chaotic dynamics of fluttering panel in post-critical domain have been analyzed by Bolotin et al. [Bolotin et al., (1998a,1998b)]. Chaotic vibrations of a panel forced by buffeting aerodynamic loads have been demonstrated and investigated by Epureanu et al. [Epureanu et al. (2004)]. The finite difference method has been employed to study coherent structures of the panel nonlinear dynamics, which have been identified by a proper orthogonal decomposition. Nagai et al. [Nagai et al. (2004)] have presented a shallow cylindrical shell-panel with a concentrated mass under periodic excitation. Furthermore, Nagai et al. [Nagai et al. (2007)] carried out the detailed experimental and analytical analyses on chaotic vibrations of a shallow cylindrical shell-panel under gravity and periodic excitation. The detected chaotic dynamics has been monitored and quantified by Fourier spectra, Poincar´e maps, maximum Lyapunov exponents and Lyapunov dimension. It has been shown that the dominant chaotic dynamics is yielded by the responses of the sub-harmonic resonance of 1/2 order and of the ultra-sub-harmonic resonance of 2/3 order. The similar-like approach has been extended to study a modal interaction in chaotic vibrations of a shallow double-curved shell-panel by Maruyama et al. [Maruyama et al. (2008)]. The shell-panel with square boundary and initial geometric imperfection has been simply supported, whereas the in-plane displacement at the boundary has been constrained elastically. Similar to the case of the plates, the FGM panel has been recently extensively studied. It has been shown that owing to the aero-thermoelastic interactions at high Mach numbers, skin panels may exhibit divergence and flutter instability behavior. The combined action of

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thermal and aerodynamic loads on both static and dynamic stabilities of FGM panels have been analyzed by Sohn and Kim [Sohn and Kim (2008)]. Ibrahim et al. [Ibrahim et al. (2008)] investigated the nonlinear flutter and thermal buckling of a FGM panel using FEM (Finite Element Method). The aero-thermo-elastic post-critical and vibration characteristics of temperature-dependent FGM panels in a supersonic flow have been studied by Hosseini and Fazelzadeh [Hosseini and Fazelzadeh (2010)]. The von K´ arm´ an theory has been employed, whereas the material properties have been considered as temperature-dependent and graded in the thickness direction. PDEs have been converted to ODEs and then classical numerical tools have been implemented to study chaotic vibrations. Hosseini et al. [Hosseini et al. (2011)] investigated chaotic and bifurcation dynamics of FGM curved panels subjected to aero-thermal loads. The panel has been infinitely long and simply supported, and the material properties have been taken as temperature-dependent and varying through the thickness direction. The governing PDE has been truncated to a set of nonlinear ODEs solved numerically. Regular and chaotic vibrations regimes have been detected and monitored via Poincar´e maps, time histories, frequency spectra and Lyapunov exponents. The effect of the geometrically imperfect curved skin panel parameters on the flutter behavior has been investigated by Abbas et al. [Abbas et al. (2011)]. The thermal degradation and Kelvin’s model of structural damping have been included into the analysis. Aginsky and Gottlieb [Aginsky and Gottlieb (2012)] studied a nonlinear bifurcation structure of panels under periodic acoustic fluid-structure interaction. In particular, an intricate bifurcation structure near the fifth-mode panel resonance including coexisting symmetric and asymmetric periodic solutions has been detected. In addition, the emergence of a non-stationary spatio-temporal chaotic solution has been found. Li et al. [Li et al. (2012)] addressed the problem of the aero-elastic stability and bifurcation structure of subsonic nonlinear thin panels subjected to external excitation. The von K´arm´an large deflection model and Kelvin’s structural damping have been utilized while

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deriving the governing equation of the simply supported 2D-panel. The problem has been reduced to that of finite number of nonlinear ODEs, which then have been solved numerically using the fourth order Runge–Kutta method. It has been shown that: (i) the panel lost its stability by divergence; (ii) periodic and chaotic zones appear alternately; (iii) a route to chaos is realized via period-doubling bifurcation. The similar-like approach has been extended to study bifurcation structure and scaling properties of a subsonic periodically driven panel with geometric nonlinearity by Li et al. [Li et al. (2015)]. In particular, the scaling properties of the bifurcation structure are discussed in terms of discrete mapping and based on linear approximation. However, only one-mode reduction has been employed during the studies. 1.4

Shells

In the case of shells chaotic and bifurcational dynamics, the number of reports is rather limited. Period-doubling bifurcations of an infinitely long cylindrical shell under the condition of internal resonance has been studied by Nayfeh and Raouf [Nayfeh and Raouf (1987)]. Chaotic vibrations of non-shallow arches loaded at their crown by a vertical harmonic force have been investigated by Thomsen [Thomsen (1992)]. The quasi-periodic break-up, intermittency and long transient behavior have been detected as routes to chaotic dynamics. Chaotic energy pumping through auto-parametric resonance in cylindrical shell has been investigated by Popov et al. [Popov et al. (2001)]. Transient and steady-state instabilities matched with chaotic vibrations exhibited by pressure-loaded shallow spherical shells have been analyzed in reference [Soliman and Goncalves (2003)]. Amabili [Amabili (2005)] studied chaotic vibrations of doubly curved shallow shells. Nagai and Yamaguchi [Nagai and Yamaguchi (1995)] analyzed chaotic vibrations of a shallow cylindrical shell with rectangular boundary under cyclic excitation. Sheng and Wang [Sheng and Wang (2011)] have investigated a nonlinear response of FGM cylindrical shells under both mechanical and thermal loads within the von K´ arm´ an nonlinear theory.

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The influence of temperature change, fraction exponent of FGM and geometry parameters has been addressed. Krasnopolskaya et al. [Krasnopolskaya et al. (2013)] introduced two new mathematical models of cross-wave generation in fluid free surface between two cylindrical shells. Two eigenmodes approximations have been employed, and periodic, quasi-periodic and chaotic regimes have been detected and illustrated. Alijani and Amabili [Alijani and Amabili (2012)] introduced sub-harmonic, quasi-periodic and chaotic dynamics of FGM doubly curved shells under concentrated harmonic load. Gear’s backward differentiation formula has been applied to get bifurcation diagrams, Poincar´e maps and time histories, as well as the Lyapunov spectrum has been computed.

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Chapter 2

Introduction to Fractal Dynamics In this chapter, in order to get a deeper understanding of nonlinear dynamical phenomena covered by this book, the fundamental concepts of one-dimensional (1D) maps and fractal sets are briefly reviewed and illustrated. First notions of Cantor’s set, Cantor’s dust and Koch’s snowflake are presented. Then 1D maps are considered putting emphasis on their regular and chaotic dynamics, the cobweb diagrams and the period doubling bifurcation routes to chaos including estimation of the Feigenbaum constant are also briefly revisited. The Sharkovsky theorem is addressed with presentation of its advantages and limitations. Next, the Julia, Fabout and Mandebrot sets are shortly described and illustrated. 2.1

Cantor Set and Cantor Dust

The term Cantor dust was introduced by German mathematician Georg Cantor in 1883 [Cantor (1883)], though it was discovered earlier by H. J. Smith [Smith (1874)]. This set has been considered as a set of zero Lebesque measure. Fractal properties of the Cantor set have important meaning since many of the known functions are similar families of this set. The Cantor ternary set is constructed by repeatedly removing the open middle part of the unit interval [0, 1], assuming that it is divided into three parts. We begin (the first step) by deleting the open middle third (1 /3 , 2 /3 ) leaving two line segments [0, 1 /3 ] ∪ [2 /3 , 1]. Proceeding in

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Construction of the Cantor set.

a similar way with all the remaining parts, we obtain (see Fig. 2.1) C0 = [0, 1] ,  
C1 = 0, 1 /3 ∪ 2 /3 , 1 ,
 


C2 = 0, 1 /9 ∪ 2 /9 , 1 /3 ∪ 2 /3 , 7 /9 ∪ 8 /9 , 1 ,   2 cn−1 Cn−1 ∪ + . Cn = 3 3 3


(2.1)

The obtained remaining segments/points can be presented via the following general formula    ∞ 3m −1    3k + 1 3k + 2 ,1 , (2.2) 0, ∪ C= 3m 3m m=1 k=0

which has been proved by Soltanifar [Soltanifar (2006)]. Limiting set C, which contains all of the non-deleted sets Cn , n = 0, 1, 2, . . . , is called the classical Cantor dust. In what follows, we discuss a few important properties of the Cantor set. 1. Cantor set is a self-similar fractal of dimension d = log(2) /log(3) ≈ 0.6309,

(2.3)

where N r d = 1 is satisfied for N = 2 and r = 1 /3 ; N is the number of equal parts; 1 /2 — decrease of 1 /2 times; d — fractal (nonlog N integer) dimension or dimension similarity d = log 1 / . Logarithm 2 can be taken with any basis, for instance, using e ≈ 2, 7183 . . . . The Hausdorf dimension of the Cantor set is ln 2/ ln 3 = log3 2, whereas its Lebesque measure is zero. 2. Cantor set does not include intervals of integer length. 3. Sum of removed intervals while constructing the set C is equal to 1.

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This statement can be proved in the following way. Length of the first interval is 1 /3 . In order to get C2 , we remove two intervals of length 1 /32 . In the next step, we remove 22 intervals, each of length 1 /33 , and so on. Therefore, the sum of removed intervals is S=

2 22 2n−1 1 + 2 + 3 + ... + n + ..., 3 3 3 3

or equivalently   2  3 

S = 1 /3 · 1 + 2 /3 + 2 /3 + 2 /3 + . . . .

(2.4)

(2.5)

Let us recall the geometric series 1 = 1 + x + x2 + x3 + . . . 1−x

|x| < 1,

(2.6)

and hence we get S=

1 /3 ·

1 = 1. 1 − 2 /3

(2.7)

4. Comparison of the Cantor set with the interval [0, 1] shows that powers of these two sets are equal, i.e. power of the Cantor set C is equal to the power of continuum [0, 1]. 5. Classical Cantor dust presents the example of a compact fully discontinuous set. The Cantor set cannot have any interval of non-zero length. Since the subsequent subsets of the Cantor set construction do not remove end points, the Cantor set is not empty and contains an uncountably infinite number of points. Another observation is that all algebraic irrational numbers are normal, whereas the Cantor set members are either rational or transversal (they are not normal). The Cantor set can serve as the archetype of a fractal. It exhibits the left and right self-similarity transformations fl (x) = x/3 and fr (x) = (2 + x)/3, which acting on the Cantor set yield fl (C) ∼ = ∼ fr (C) = C. It means that the Cantor set is invariant with respect to the introduced homeomorphism. The set {fl , fr } together with the function composition creates the dyadic monoid. On the other hand, the Cantor dust is a multi-dimensional Cantor set, i.e. it is

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constructed as a finite Cartesian product of the Cantor set with itself by yielding a Cantor space also with zero measure. The method of subdividing a shape into copies of itself simultaneously removing a few of such copies has been extended to 2D objects. In spite of the Cantor dust, one may construct the Sierpi´ nski triangle discovered by W. Sierpi´ nski in 1916 [Sierpi´ nski (1916)] (an equilateral triangle is subdivided into four equilateral triangles removing the middle triangle) and the Menger sponge reported by K. Menger in 1928 [Menger (1928)] (a cube is taken; each face is divided into nine squares; the smaller cube in the middle of each face is removed and also the smaller cube in the center of the large cube is removed to get a void cube; the last two steps are repeated for the remaining smaller cubes and the described iterations are repeated to infinity). It is remarkable that each face of the Menger sponge is a Sierpi´ nski carpet, and its Lebesque measure is zero (uncountable set). However, the Lebesque covering dimension of the Menger sponge is one. It can be shown that every curve is homeomorphic to a subset of the Menger sponge including trees and graphs, vertices and closed loops, etc. giving the right to understand the Menger sponge as a universal curve. On the other hand, the Sierpi´ nski carpet can be viewed as a universal curve for all planar curves, i.e. projection of the Menger universal curves onto a plane yields the Sierpi´ nski universal curves. The Hansdorff dimension of the Sierpi´ nski carpet is log 8/ log 3 ∼ = 1.893, whereas the Hansdorff dimension of the Menger sponge is log 20/ log 3 ∼ = 2.727. 2.2

Koch Snowflake

In 1904, the Swedish mathematician Helge von Koch presented a way to construct the so-called Koch snowflake, known as the Koch island and Koch star, as a mathematical curve being a prototype for a fractal [Peitgen et al. (1926)]. Boundary of Koch’s snowflake is described by a curve consisting of three similar fractals of dimension d ≈ 1.2618. Each third part of the snowflake is constructed in an iterative way, beginning from one side of an equilateral triangle.

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Fig. 2.2

Subsequent step of construction of Koch’s curve.

Let K0 be the initial part. We remove the third middle part and we add two new parts of the same length, as shown in Fig. 2.2. The obtained Koch’s curve has an infinite length, since each iteration adds one-third of the previous length. In addition, it is continuous everywhere but differentiable nowhere. The obtained set is called K1 . We repeat the described procedure a few times changing in each step the middle third part by new parts. Denoting by Kn the figure obtained from the nth step for each of the triangle side, we get a Koch snowflake (three Koch’s curves contribute to the snowflake), see Fig. 2.3. The fractal dimension of the Koch curve is d = log 4 /log 3 ≈ 1.2618.

(2.8)

Intuitively, the series {Kn }∞ n=1 → K. If we take a copy of K, reduced by three times (r = 1/3), then the whole set K can be composed of N = 4 such copies. One of the most important properties of the Koch’s snowflake is its infinite length, i.e. the length of the curve Kn = 4n /3n and its limit lim (4n /3n ) = ∞. n→∞ Since the number of new triangles yielded by n iterations is tn = 3 · 4n−1 =

3 n ·4 , 4

and the area of each new triangle is s0 sn−1 = n, sn = 9 9 the total new area obtained in iteration n follows   3 4 n n s , s n tn = 4 9

(2.9)

(2.10)

(2.11)

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Fig. 2.3

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Koch’s snowflake.

and finally the total area of the snowflake is    n   n   4 3 3 4 k 1− = s0 1 + sn = s0 1 + 4 9 5 9 k=1   n  4 s0 8−3 , = 5 9

(2.12)

where s0 is the area of the starting triangle. The snowflake perimeter after n iterations follows  n 4 , (2.13) Ln = 3 · l · 3 where l denotes the length of each triangle sides. In the limiting case n → ∞, one gets 8 lim sn = s0 . (2.14) n→∞ n→∞ 5 Nowadays, various variants of the Koch curves have been proposed by taking into account various angles (quadratic types 1 and 2 Koch curve and Ces´ aro fractal), quadratic type 1 and type 2 Koch surfaces, sphere Koch flake and Koch cube, etc. lim Ln = ∞,

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1D Maps

1D maps, also known as iterated maps or recursion relations or difference equations, are mathematical systems governing the dynamics of a single variable x over discrete time values. Let us assume that a simple discrete dynamical system consists of an initial  point (n) (x ) ∞ = f x0 and the function f (x). Sequence {xn }∞ n n=0 is n=0  (n) called an orbit f (x) = f (f . . . (f (x))) of the initial point x0 . We take x0 as a real, and f as an elementary function, for instance f (x) = x2 + c, f (x) = cx (1 − x), f (x) = cos x. Here, the map f is called a compressing one. Therefore, the theorem of a fixed point cannot be applied, and furthermore we cannot get any conclusion for the convergence of the series {xn }∞ n=0 . Investigating a chaotic dynamics we consider a nonlinear (non-affine) function which cannot be presented in the form f (x) = ax+b, since in either linear or affine cases, chaos is not observed. Let ∃ |f  (x)|. If x is a fixed point and |f  (x)| < 1, then x is an attracting point. If |f  (x)| > 1, then it is a repiller, and if |f  (x)| = 1, then any conclusion regarding point x cannot be made. An orbit is called periodic of period p, if xn + p = xn for n = 0, 1, 2, . . . , and p is the smallest integer number. If the equation of periodicity xn + p = xn becomes true only after a certain finite number of steps, say, n ≥ n0 , then the orbit is also periodic after a few iterative steps. The socalled cobweb diagram is often used to follow an orbit of the real function f . We consider function f (x) = x2 (Fig. 2.4). The shown diagonal lines are plots of y = x (y/x denotes vertical/horizontal axis), whereas the curved line is a plot of the map f (x). The remaining series of lines with arrows are cobweb elements. How to construct a cobweb diagram? Let us take the map y = xn+1 = f (xn ). The initial point x0 is mapped to a new point x1 , which is found by drawing a vertical line from the axis x to the curve f (x). Then we move horizontally to the line y = x to find the next x-coordinate xn+1 . A fixed point of the map is defined as an intersection of the graph y = f (x) and the line y = x.

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Fig. 2.4 Cobweb diagram of the map f (x) = x2 for the following initial conditions: (a) x0 = 1.1, (b) x0 = −0.9.

Another case of iterations is a limit-cycle. It represents a periodic dynamics, since the orbit tends to a series of points that repeats itself (a number of iterations needed to repeat itself is called a period of the orbit). In our case x0 = 0 and x0 = 1 are fixed points of the map y = x2 . If x0 > 1, then the orbit tends to + ∞, if 0 < x0 < 1 or −1 < x0 < 0, then the orbit tends to a fixed point 0. If x0 = 1, then the orbit is [−1, 1 1 1 . . .], i.e. it is periodic, and finally if x0 < −1, then xn → ∞ and the orbit diverges. In the given case, the fixed point 0 is attractive, whereas the fixed point 1 is repelling. The functions f (x) = x2 and f (x) = x2 − 1 are particular cases of the map f (x) = x2 + c, and they are widely used in the theory of dynamical systems [Sharkovsky et al. (1997)]. Although f (x) = x2 +c is only a quadratic function, it is widely applied. We consider a real case, i.e. when x and √c are real numbers. We √ 1+ 1−4c 1− 1−4c 2 ;η= . We solve equation x + c = x, and we get ξ = 2 2 deal with a fixed point for 1 − 4c ≥ 0 (c ≤ 14 ), if −ξ < η < ξ, and besides, f (−ξ) = ξ. Orbits for x0 > ξ and x0 < −ξ tend to + ∞. For −3 /4 < c < 1 /4 , the fixed point η is attractive, i.e. |f  (x)| < 1 and all orbits tend to η. For c < −3 /4 |f  (x)| > 1, i.e. η becomes a repeller. When c transits over the value −3 /4 , the system exhibits a period doubling bifurcation. A second period doubling bifurcation takes place for c = −5 /4 . If c < −5 /4 , then an attractive periodic orbit of period 4 appears. Decreasing c, attractive orbits of length 8, 16 and 32 are observed, i.e. after period doubling bifurcations chaos

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occurs. For c = −2, there exist periodic orbits of f (x) with periods 2, 3, 4, . . . . Feigenbaum studied intervals between period doubling bifurcation of the square function f (x) = cx (1 − x), also known as the logistic map [Feigenbaum (1990); Thunberg (2001)]. The obtained diagram of orbits is similar to that of the function f (x) = x2 + c. The fundamental meaning of the Feigenbaum analysis relies on the detection of the mechanism’s universality of a route to chaos via period doubling bifurcation, which is exhibited not only by the logistic map f (x) = cx (1 − x), but also by the maps f (x) = x2 + c, f (x) = c sin (πx) , f (x) = cx2 sin (πx) defined on appropriate intervals. The aforementioned class includes functions f (x), defined on the interval [0, 1] and achieving a maximum at point xm ∈ (0, 1) for f  (xm ) = 0, whereas f (x) on [0, xm ] and [xm , 1], as well



2


(x) (x) − 32 ff
(x) is negative as its Schwartz derivative Sf (x) = ff
(x) ∀ x ∈ [0, 1]. We denote by c0 , c1 , c2 , . . . , bifurcation points on the orbits’ diagram (Fig. 2.5), i.e. those points, where iterations f (x) = x2 + c change the attracting orbit of period 2n into attracting orbit 2n+1 . One may check that c∞ = lim = −1.401155 . . . , i.e. we deal with n→∞

Fig. 2.5 x2 + c.

Diagram exhibiting period doubling bifurcation of the map f (x) =

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the Feigenbaum type process. For c = −1.7548777 . . . period-3 orbit exists, which corresponds to the more lighted part of the diagram. −cn−1 Further, we have d = lim ccnn+1 −cn = 4.669162 . . . , which defines the n→∞ Feigenbaum constant. It should be noted that it is, in general, difficult to find analytically bifurcation points cn for any given function, for instance x2 + c, and hence to define the Feigenbaum constant. However, there exists another approach. For each pair of bifurcation points cn and cn+1 there exists a point c∗n , which corresponds to an orbit with period 2n . For this value of c, the critical point x0 of the function fc satisfies (n) the equation fc = x0 . Then, the Feigenbaum constant d can be found through the following equation c∗n − c∗n−1 , ∗ n→∞ c∗ n+1 − cn

d = lim

(2.15)

where c∗n is found numerically with the help of the Newton method. The stretching and folding exhibited by the logistic map produces sequences of iterates, which finally yield an exponential divergence validated by the Lyapunov exponent computation. The logistic map correlation dimension is 0.5, and its Hausdorff dimension is 0.538. The logistic map as well as other chaotic maps have found applications in developing image encryption schemes [Pareek et al. (2006); Wong et al. (2008)]. 2.4

Sharkovsky’s Theorem

Diagram of orbits shown in Fig. 2.5 presents attracting periodic orbits for fc (x). For 1/4 < c < −3/4, there is an attracting orbit of period 1. For −3/4 < c < −9/4, we deal with the attracting orbit of period 2, which bifurcates into attracting period-4 orbit, for c = −5/4. On some intervals, the diagram exhibits windows. For instance, for c ≈ −1, 75 we have a white zone with period-3 orbits. A natural question appears: Do other periodic orbits exist? Those orbits should be repellers, since in the diagram we observe only attracting

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orbits. It is shown that the occurrence of a period-3 orbit implies the occurrence of orbits with periods n = 1, 2, 3, . . . . In 1964, A. N. Sharkovsky published a general theorem, regarding a map onto itself valid for the real functions, which had an important impact on the development of the theory of nonlinear dynamical systems [Sharkovsky (1964)]. A brief description of Sharkovsky’s theorem follows. Let I be finite or infinite interval in R. We assume the map f : I → I is continuous. We define the number x as a periodic point of period m if f m(x) = x and f k (x) = x for all 0 < k < m. Let us construct the following ordering of the positive integers 3, 5, 7, 9, . . . 2 · 3, 2 · 5, 2 · 7, 2 · 9, . . . 22 · 3 , 22 · 5 , 22 · 7 , 22 · 9 , . . . 23 · 3 , 23 · 5 , 23 · 7 , 23 · 9 , . . . ......... n . . . , 2 , . . . , 23 , 22 , 21 , 1.

(2.16)

We start with the construction of the odd numbers in the increasing order, followed by 2 times the odds, 22 times the odds, 23 times the odds, and so on, whereas at the end we take powers of two in the decreasing order. Then, if f has a periodic point of least period m and m precedes in the ordering (2.16), f also has a periodic point of least period n. Assuming now that f has only a finite number of periodic points, then those points have periods numbers of which are powers of 2. If there exists a periodic point of period-3, then there exist also periodic points of all other periods. It should be emphasized that the Sharkovsky theorem does not concern with the stability of periodic orbits but rather their existence. In other words, if one takes the logistic map, for a wide range of the control parameter only one period-3 orbit is detected, which is an attractor. The remaining orbits of all periods are not visible, since they are unstable and cannot be detected through the cobweb construction described so far. The Sharkovsky Theorem is valid only for a real function defined on a real interval. If, for instance, function f is given by a rotation of each point lying on a circle about the angle 2/n, then the orbits

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of each point have the same period n. In this case, there is lack of any additional periods, and Sharkovsky’s theorem is not applicable. On the other hand, if there exists an orbit of the odd period larger than one, then a number of different periods tends to infinity. 2.5

Julia Set

We denote by C a set of complex numbers a + bi = z, where a and b are respectively Re (z), √ real and imaginary parts of z, i.e. a = ∞ 2 2 b = Im(z), |z| = a + b . Series of complex numbers {zn }n=1 → lim zn = ∞, which means that ∀ M > 0, ∃ N > 0, ∀ n > N : n→∞

|zn | > M, i.e. all points zn lie inside a circle of radius M for sufficiently large n and it is necessary that absolute magnitudes increase to infinity. Let f (z) = an z n + an−1 z n−1 + . . . + a1 z + a0 , an = 0, be a polynomial of order n ≥ 2, and its coefficients an , an−1 , , . . . , a1 , a0 are complex numbers (in particular case, they are real). The Julia set of a function f (known as J(f ) set) is the limit of a set of points z, approaching infinity while iterating f (z). This set honors the French mathematician Gaston Julia [Julia (1918)] (1893–1978), who simultaneously with Pierre Fabout [Fatou (1917)] (1878–1929) in 1917–1919 wrote the fundamental papers devoted to the iteration of the functions of a complex variable. The factor set of the function exhibits a property that all nearby values behave in a similar manner under the action of repeated iterations of functions. On the other hand, the Julia set exhibits values in which perturbations yield drastic changes in the sequence of iterated function values. A simple example of Julia’s set is: f (z) = z 2 . In what follows, we consider the following Julia set fc (z) = z 2 + c, where c is constant in C. In Figs. 2.6 and 2.7, a few Julia sets are presented. In order to proceed with a more rigorous statement, we consider a complex rational function f (z) = p(z)/g(z) from the plane onto itself, where p(z), g(z) are complex polynominals. There exists a finite number of open sets F1 , F2 , . . . that are invariant owing to the action of f (z) having two properties: (i) a set of Fi is dense in the plane; (ii) f (z) behaves regularly on each of the sets Fi . One may

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Fig. 2.6 The Julia set for f (z) = z 2 − 1, where z is a complex number (adapted from [Douady (1986)]).

Fig. 2.7 The Julia set for f (z) = z 2 − 0.20 + 0.75i (adapted from [Sierpi´ nski (1916)]).

distinguish between attracting cycles and neutral cycles. The sets Fi are Fatou domains of f (z), and each of the domains has at least one critical point of f (z). The complement of F (f ) (Fatou set) is the Julia set J(f ). The latter one is invariant by f (z), and the iteration is repelling in a neighborhood of z, which yields chaotic deterministic iterations of f (z) on the Julia set. The Julia set may include a finite number of regular points, i.e. those whose sequence of operation is finite. There exist various equivalent descriptions of the Julia set. It can be also proved that the Julia set and the Fatou set of f are completely invariant under iterations of the holomorphic function f ,

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i.e.

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f −1 (J(f )) = f (J(f )) = J(f ),

f −1 (F (f )) = f (F (f )) = F (f ). (2.17) More detailed description of Fatou and Julia sets can be found in references [Barnsley (1988); Douady (1986); Lauwerier (1991); Paitgen and Richter (1986)]. There are also a few proposals for the direct application of Julia/Fatou set theories to pure and applied sciences. The Newton’s-secants method for finding numerical solutions to the nonlinear equation F (z) = z 2 − C = 0 has been proposed and then extended to solve Newton’s-secants and Tchebishev’s-secants imaginary problems based on the Julia set theory by Tomova [Tomova (2001)]. Another application of Julia sets to switched dynamical processes has been presented by Lakhatakia [Lakhtakia (1991)]. The Julia set theory has been employed to transient chaos detection in process control systems [Russell and Alpigini (1996)]. The quaternion Julia set has been applied to generate real-time-based symmetric keys for cryptography in the reference [Rubesh Anand et al. (2009)]. 2.6

Mandelbrot’s Set

Mandelbrot extended investigations of French mathematicians Fatou and Julia by studying the parameter space of quadratic polynomials in 1980 [Mandelbrot (1980)]. The Mandelbrot set is governed by the quadratic recurrence equation zn+1 = zn2 + z0 ,

(2.18)

where the orbit zn does not tend to infinity (remains bounded). Namely, a complex number z0 = c belongs to the Mandelbrot set M , when the absolute value of zn remains bounded. For example, for z0 = 1 we get the sequence {0, −1, 0, −1, 0, . . . } being bounded, whereas for z0 = 1 we obtain the sequence {0, 1, 2, 5, 26, . . . }, which is unbounded. The Mandelbrot set boundary also exhibits a smaller version of the main shape (see Fig. 2.8), and hence the fractal property of self-similarity is applied to the entire set. A zone of period-3 Mandelbrot’s set is shown in Fig. 2.9.

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Fig. 2.8 Mandelbrot’s set fc (z) = z 2 + c (a) and the window (b) of Mandelbrot’s set M around point c = −1.75 + 0i (points belonging to set M are colored black); adapted from [Mandelbrot (1980)].

More precisely, the Mandelbrot set can be defined in the following way. Set M for fc (z) = z 2 +c is defined as {c} ∈ C, for which the orbit (n) of point 0 is bounded, i.e.: (i) M = {c ∈ C : {fc (0)}∞ n=0 bounded} (n) or equivalently (ii) M = {c ∈ C : fc (0) → ∞ for n → ∞}. Equivalence of definition 1 and definition 2 is yielded by the 2 observation that lim z z+c = ∞, i.e. ∃ R > 0 we have |z| > R → z→∞

(n )

(n )

|fc (c)| > 2|z|. If ∀n0 ∃|fc 0 (0)| > R, ∀n > n0 : |fc 0 (0)| > 2n−n0 R, (n) i.e. fc (0) → ∞. Point zero is the only one point where the derivative is equal to zero. Problems regarding boundaries of the orbits should be justified using the following theorem.

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Fig. 2.9 Area of period-3 of Mandelbrot’s set (it should be considered simultaneously with Figs. 2.8 and 2.9); adapted from [Mandelbrot (1980)].

If |c| > 2 and |z| ≥ c, then on orbit z → ∞. In particular, this implies that points c ∈ / M . Point c = −2 is the only one point of the circle |c| = 2, which belongs to Mandelbrot’s set. In Fig. 2.10, certain parts of the Mandelbrot’s set corresponding to the existence of the attractive periodic orbits of different orbits are presented. The presented diagram of the orbits shows what happens on the real axis of the Mandelbrot set. Each bifurcation corresponds to a new frame, which intersects the abscissa axis, and the period corresponds to a number of branches of the orbital diagram. A value of c, for which periodic attracting points of period-2 exist in the Julia set lie inside the circle |c + 1| = 1 /4 . It has been impossible so far to find an analytical dependence between a period and a frame for periods larger than two. The Hausdorff dimension of the Mandelbrot set boundary is equal to two [Shishikura (1998)]. There is a strong correspondence between the geometry of the Mandelbrot set at a given point and

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Fig. 2.10 (1980)].

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Periods of frames (Mandelbrot’s set); adapted from [Mandelbrot

the associated Julia set. Properties and various peculiarities of the Mandelbrot set are described and illustrated in numerous publications including [Lei (1990, 2000)]. The real-world applications of fractal and Mandelbrot sets can be found in physics, mechanics, economics, biomechanics and financial mathematics.

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Chapter 3

Introduction to Chaos and Wavelets This chapter is devoted to definitions of chaos and description of various routes to chaos including the Landau–Hopf scenario, the Ruelle–Takens–Newhouse scenario, the Feigenbaum scenario and the Pomeau–Manneville scenario. The synchronization of chaos is briefly addressed. Quantification of chaotic dynamics via Fourier analysis, Poincar´e maps, Lyapunov exponents is described. The Melnikov method to detect strange chaotic attractors is presented. The remaining chapter part is focused on advantages of the wavelet analysis versus the classical Fourier analysis. Properties of different wavelets are illustrated and discussed. 3.1

Routes to Chaos

3.1.1

Introduction

There is no rigorous definition of chaos. Encyclopedia Britanica refers to the Greek word “χαoζ”. Poet Ovidius in his “Metamorphoses” writes: Before the seas, and this terrestrial ball, And Heav’n’s high canopy, that covers all, One was the face of Nature; if a face: Rather a rude and indigested mass: A lifeless lump, unfashion’d, and unfram’d, Of jarring seeds; and justly Chaos nam’d. No sun was lighted up, the world to view; No moon did yet her blunted horns renew: Nor yet was Earth suspended in the sky, Nor pois’d, did on her own foundations lye: Nor seas about the shores their arms had thrown. 31

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Therefore, Ovidius understood chaos as the infinite space existing before the appearance of all other matters. Romes understood chaos as the initial disordered matter, where a Creator introduced order and harmony, i.e. the same description for chaos holds for Romes and Greeks. Nothing has been changed up till now: G. Schuster, for instance, describes chaos as a state of disorder and irregularity [Schuster (1998)]. Nowadays, more attention is paid to the investigation of the structural stochasticity (chaos is born from order), which can be exhibited by the deterministic dynamics of structural members in their pre-critical states. Stochasticity is implied by a complex intrinsic system dynamics and is not given as a result of the noise or fluctuation input. It can be treated, in some cases, as the occurrence of a turbulent behavior. It is well known that there exists relatively large amount of the scenarios of transitions into turbulent behavior being (in majority of cases) governed by the Navier–Stokes equations, describing dynamics of the uncompressed fluid, which have the following form 1 ∂u + (u · ∇) u − ν∇2 u = − ∇p + f, (3.1) ∂t ρ div u = 0, (3.2) u = 0 on D,

(3.3)

where u = u(xi , t), i = 1, 2, 3, p is pressure, D is boundary of the space containing a fluid, ρ is fluid density, f is external force, ν is kinematic viscosity. Energy dissipation is governed by the term ν∇2 u. Equation (3.1) presents a 3D partial differential equation regarding u (velocities) with respect to a fixed system of coordinates (Euler’s approach), whereas Eq. (3.2) is responsible for the condition of the uncompressed fluid behavior, and Eq. (3.3) governs the boundary conditions. Note that there is lack of proof of the turbulent solution existence to Eqs. (3.1)–(3.3), when time goes to infinity. However, there is a proof of the turbulent solutions existence for 2D equations. On the other hand, physical properties described by the Navier– Stokes equations are relatively well investigated and understood. The first work in this field was published by Reynolds [Reynolds

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(1883)]. He introduced the following non-dimensional parameters: Reynold’s number R = UνL , u/U — non-dimensional velocities, xi /L — non-dimensional coordinates, t = L/U — non-dimensional time, p = p/ρU 2 — non-dimensional pressure, and he considered the following partial differential equation 1 ∂u + (u · ∇) u − ∇2 u = −∇p. (3.4) ∂t R Reynolds showed that an increase in R may change the fluid motion qualitatively from a regular (laminar) to disordered chaotic (turbulent) flow. In hydrodynamics, the word “turbulence” is used as a description of the spatio-temporal chaos. It implies that chaos in a fluid is exhibited in all scales in space and time. However, the mathematical description of this state belongs to one of the most tedious and complicated problems to be solved. Up to now, it is not clear how a stochastic attractor of the turbulent flow should be constructed. On the other hand, more important results have been recently obtained for the investigation of simple dynamical systems yielded, for example, by truncation of the original continuous system, and being governed by ordinary differential equations as well as maps (difference equations). However, truncated systems include only timing chaos. It allows, in the first approximation, to analyze a turbulence birth, i.e. the case when the velocity field began to fluctuate in time in a disordered manner. One of the aims of the book is to illustrate and discuss various scenarios of transition of the space structural members as beams into spatio-temporal chaos for different mathematical models taken into account. In what follows, we address a few examples of transitions from regular to chaotic dynamics exhibited by simple dynamical systems, in order to introduce the reader briefly to the topics covered by the book contents. 3.1.2

On chaos definitions

In the beginning, we introduce a few definitions of chaos exhibited by dynamical systems.

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(i) Devaney’s definition [Devaney (1989); Eckmann and Ruelle (1985); Stewart (1989)]. The Devaney three features of chaos are formulated for a continuous map f : X → X, where X is a metric space. First (1) Devaney’s condition states that for all non-empty open subsets U and V of X there exists k (natural number) such that f k (U ) ∩ V is non-empty, which defines f as transitive. Second (2) Devaney’s condition exhibits the so-called “element of regularity”, where there are periodic points of f , forming a dense subset of X. Third (3) Devaney’s condition states, that f satisfies the property of sensitive dependence on initial conditions, which means that if there is a positive real number δ such that for every x in X and every neighborhood N of X there exists a point y in N and n (nonnegative integer) such that the iterates f n (x), f n (y) are more than distance δ apart. Hence, the fundamental characteristics of chaos require three conditions: essential dependence on the initial conditions; mixing caused by transitivity; regularity condition implied by density of periodic points. There is a theorem saying that if f : X → X is transitive and has dense periodic points then f has sensitive dependence on initial conditions, which has been proved in the reference [Banks et al. (1992)]. Let us present Devaney’s definition in a more rigorous way: let there be a metric space (x, d). Now we explain the meaning of the already introduced three conditions and the introduced metric space (X, d). 1. Let x ∈ X and U be an open set containing x. Map f depends essentially on the initial conditions if (∀δ > 0), (∃n > 0), (y ∈ U ): d(f (n) (x), f (n) (y)) > δ. (3.5) 2. Let f is transitive, then ∀(u, v) of open sets (∃n > 0) f (n) (u)Πv = 0.

(3.6)

3. Density of the periodic points means that at any arbitrary neighbourhood of any point in X there exists at least one periodic point.

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(ii) Baker’s map ([Driebe (1999); Fox (1997); Hasegawa and Saphir (1992)]). A map from the unit square onto itself is called the Baker map, which is a chaotic map. The name comes from the bakers’ operation. Namely, kneading is a process applied by bakers in the making of the dough. During kneading, the dough is folded in half and compressed. The horseshoe map is topologically conjugate to the baker’s map. It preserves the 2D Lebesgue measure, it is strong topological mixing and it can be understood as a two-sided operator of the symbolic dynamics of a 1D lattice. (iii) One may apply the following simple properties of chaos, i.e. there either exists the essential sensitivity on the initial conditions or one of the Lyapunov exponents measuring divergence of the neighborhood trajectories is positive. 3.1.3

Landau–Hopf (LH) scenario

The first scenario was proposed by Landau [Landau (1944)] in 1944, and then by Hopf [Hopf (1948)] in 1948. When Reynold’s number R regarding parameters characterizing flow activity achieves the critical value R, the previously stationary flow loses its stability. For R → ∞, the velocity of occurrence of new frequencies increases (Pn 0; 1), and the solution can be presented in the following form: u(x, t) =

∞


Am (x)i

m (ω t+δ)

,

(3.7)

n=1

where ω = {ω1 , ω2 , . . . , ωn }; n → ∞; R → ∞. Although the frequency ratio is irrational, the spectrum becomes a broad band and similar to chaotic one, i.e. an infinite quasi-periodic process of “turbulence” is observed, which has not been verified by experiments (the Couette flow and the Rayleigh–Bernard convection). The so far described LH scenario (Fig. 3.1) is associated with Hopf theory of bifurcations, which we will briefly present now. Let a motion be described by ordinary differential equations: dx = Fp (x), x = x1 , . . . , xk , (3.8) dt

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Fig. 3.1

The LH scenario.

Fig. 3.2 (a) Spiral type trajectories approaching a stable critical point; (b) Spiral type trajectories approaching a stable limit cycle.

where p is a system parameter (for example, p can be an amplitude of excitation). Critical points of equation (3.8) are the points x = xc , where dxc = 0 , i.e. Fp (xc ) = 0. (3.9) dt Stability of the points defined by (3.9) is estimated via the associated linearized equations and their characteristic values λ = λ(p). Assuming that λ lies in the left half-plane, i.e. they have negative real parts, the investigated critical point is stable [Fig. 3.2(a)]. The Hopf bifurcation takes place under the condition that a complex conjugated pair of the eigenvalues moves from the left-hand side of the complex plane to its right-hand side. For the critical values of

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p = p+ (such p, for which λ moves to the right-hand side) additional conditions should be taken into account including the first, second and third derivatives of Fp (xc ). As a result of this bifurcation, the previously stable critical point [Fig. 3.2(a)] is transformed into a stable periodic orbit (limit cycle) as shown in [Fig. 3.2(b)]. The transition from a stable critical point to a stable limit cycle can be illustrated using an example of a particle motion in the potential well, where the second minimum occurs while increasing p (Fig. 3.3). There exists also another situation, shown in Fig. 3.4, when for p = p+ and for p > p+ there do not exist of any attractive (stable) solutions.

Fig. 3.3 Normal Hopf bifurcation of a particle moving in a potential well: (a) stable critical point, (b) unstable limit cycle.

Fig. 3.4 Inverse Hopf bifurcation of a particle moving in a potential well: (a) stable limit cycle, (b) unstable critical point.

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Ruelle–Takens–Newhouse (RTN) scenario

RTN scenario plays an important role since it yields an alternative proposal beyond that of the LH scenario. In 1971, Ruelle and Takens [Ruelle and Takens (1971)] have shown that, on the contrary to the LH scenario, only a few bifurcations may cause chaotic dynamics. Firstly, they showed that after three Hopf bifurcations a torus T 3 may lose its stability and change its geometric structure to that of a strange attractor with properties of chaotic dynamics. Further, in 1978, Ruelle, Takens and Newhouse [Newhouse et al. (1978)] proved the theorem according to which a strange chaotic attractor appears after two successive Hopf bifurcations (Fig. 3.5). In fact, Ruelle and Takens assumed that after two Hopf bifurcations the particle motion is bounded by manifolds, which are not smooth and possess rather complicated topology. Those manifolds have been named after a strange attractor. They do not have integer dimension, which represents a transitional set between a surface and volume. Fractal dimension has been studied in detail by Mandelbrodt [Mandelbrodt (1982)] in the context of fractals. Attractor, yielded by the RTN scenario, should satisfy the additional conditions which are applied to the so-called attractors of “axiom A”, and hence the motion within this attractor is chaotic (practically, this class of attractors is rather limited). This type of motion is very sensitive to a change in initial conditions. Two groups of researchers, Feigenbaum et al. [Feigenbaum et al. (1982)] and Rand et al. [Rand et al. (1982)] considered the problem of how a quasi-periodic motion with two independent frequencies ω1 and ω2 on a torus could exhibit the phase locking phenomena after the introduction of a perturbation (Fig. 3.6).

Fig. 3.5

The RTN scenario.

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Motion on a torus.

For rational values ω1 /ω2 = p/q, a trajectory closes after q cycles (modes synchronization). For irrational ratio ω1 /ω2 , the motion is quasi-periodic and trajectory is never closed and covers the whole torus. RTN scenario has been validated experimentally (Bernard instability and Taylor instability). In the case of the hydrodynamic turbulence the RTN scenario is weak, i.e. it is associated with resonances of higher harmonics. In the latter case, two fundamental processes are realized: stretching, protecting sensitivity to the initial conditions and protecting properties of attraction. Smale horseshoe may serve as an example of the processes so far described [Wiggins (2000)]. 3.1.5

Feigenbaum scenario (F)

In 1978, Feigenbaum [Feigenbaum (1983)] proposed the universal mechanism of transition into chaos associated with an infinite process of period doubling of the initial periodic motion. This behavior is exhibited by simple nonlinear maps, including the following onedimensional map: xn+1 = f (xn ).

(3.10)

However, function f (x) should satisfy certain conditions. One widely used example of this map is the following one xn+1 = 4λxn (1 − xn ),

0 < x < 1,

(3.11)

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where λ is a parameter. The map (3.11) has been initially applied in biology as a simple model of population dynamics. Feigenbaum discovered the geometric convergence of the sequence of period doubling bifurcation by defining the quantity λn+1 − λn = 4.6692016 . . . , λn+2 − λn+1

(3.12)

i.e. he got the universal number, which has been detected in the models of Lorenz and Henon, and has been also validated by numerous experiments. In what follows, we present dynamics of the map (3.11) for various λ. Table 3.1 shows the results obtained in the following way: for each value λ the iterations are carried out until the transitional processes do not vanish and the trajectory reaches its asymptotic position (here, we have 2-cycle, 4-cycle, 2n -cycle, ..., 6- or 3-cycle or an aperiodic attractor, and so on). The following essential properties should be emphasized: 1. 2n cycles (n = 1, 2, . . . ) in λ space: they become more and more compressed. 2. For λ > λ∞ , chaotic zones appear. 3. 3-cycles or other odd cycles appear in a chaotic regime. Li and Yorke pointed out that period 3 implies chaos [Li and Yorke (1975)]. Similar observation can be validated also for differential equations. 4. λ = 1 chaotic dynamics is developed in full. Table 3.1

Period doubling scenario.

Cycle type 1. 2. 3. 4. 5. .. . ∞

2-cycle 4-cycle 8-cycle 16-cycle 32-cycle ... aperiodic attractor

λn birth of a cycle 0.75 0.86237 0.88602 0.89218 0.8924728 ... 0.892486418

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Beyond the period doubling limiting point, the obtained structure is extremely rich in dynamics. Grebogi et al. [Grebogi et al. (1983)] introduced new notations describing the dynamics. Namely, chaotic zones narrowing and widening have been called subduction and internal crisis, respectively, whereas the final extension for λ = 1 has been named crisis. In the Rayleigh–Benard experiment for mercury in the magnetic field, four period doublings have been detected and the Feigenbaum constant has been estimated with the error of 5%. The following important remark can be formulated. One of the important factors while increasing λ is the occurrence of 3-cycle or other odd cycles. They are treated as points of tangent bifurcations. Each of the fixed points of a third cycle is transformed into a pair of fixed points, one stable and one unstable, which implies the so-called saddle-node bifurcation. Note that they differ from a period doubling bifurcation, where an unstable fixed point bifurcates into a pair of stable points. 3.1.6

Pomeau–Manneville (PM) scenario

The fourth scenario of a transition into chaos has been proposed by Pomeau and Manneville [Pomeau and Manneville (1980)]. At that time, numerous results devoted to dynamical chaos have been already reported and, in particular, it has been illustrated that a transition from periodicity to chaos can be realized through a jump via only one bifurcation. This transition is understood as the stiff route to chaos and it is associated with the intermittency phenomenon. In simple words, by intermittency we understand a signal where, in a stochastic way, long regular oscillations interleave and relatively short irregular bursts occur. Now, with the increase in a control parameter, the yields increase in the chaotic bursts unless the fully developed chaos occurs. This phenomenon has been detected by Pomeau and Manneville while solving the differential equations governing the Lorenz model. For the control parameter values less than the critical values, an associated Poincar´e map presents a stable fixed point. When the control parameter passes through its critical value, this point becomes unstable. Transition into the unstable state can be realized

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via the so far mentioned scenarios, and hence the classification of the I, II and III type intermittencies has been introduced. The so far introduced intermittency types correspond to a saddle-node bifurcation, a subcritical Hopf bifurcation or an inverse period doubling bifurcation, respectively. Note that all three types of intermittency moduli of the eigenvalues of the corresponding linearized Poincar´e maps are larger than one. In Table 3.2, a few characteristics (characteristic behavior and Poincar´e maps, eigenvalues and signal forms) are reported. In the mentioned table, parameter ε controls the critical system behavior. For the I type of intermittency (ε = 0) a tangent bifurcation takes place. Both vertical and horizontal Poincar´e sections (Lamerey’s diagrams) present a two-asymptotic trajectory of a saddle-node point. For ε > 0, in the vicinity of the vanished fixed point, the graph of the function exhibits the so-called channel, in which the trajectory point moves relatively far, and this behavior is associated with the laminar phase of the intermittency. However, when the point leaves the channel, a turbulent dynamics is observed. In the case of the II type intermittency (ε = 0), we deal with the subcritical Andronov–Hopf bifurcation, whereas for the III type intermittency (ε = 0) the associated one-dimensional map exhibits a subcritical period doubling bifurcation of the first cycle. In reference [Klimaszewska and Zebrowski (2009)], a recurrence plot method has been applied to the analysis of short time series through which the intermittency type was identified. In particular, this approach was successfuly used to study experimental data regarding heart rate variability recordings, and the obtained results confirmed the occurrence of I intermittency type. Since type-I intermittency is associated with an inverse tangent bifurcation in nonlinear iterated maps, the horizontal visibility algorithm has been applied to trace chaotic trajectories of the unimodal (quadratic logistic) maps. It has been shown in the reference [N´ unez et al. (2013)] how the alternation of laminar episodes and chaotic bursts are inherited by the variance of the graph degree distribution. In other words, since a stable fixed point and unstable limit cycle transit into a single unstable fixed point, the laminar phase occurs

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ε 0

Laminar signal

Real eigenvalues intersect the unit circle at point +1 xn+1 = ε + xn + ux2n Monotonous increase

Second

Two complex-conjugated eigenvalues simultaneously intersect the unit circle rn+1 = (1 + ε)rn + urn θn+1 = θn + Ω

Real eigenvalues intersect the unit circle at point −1

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Transition

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First

Intermittency types versus control parameter ε.

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when the system trajectory is in the neighborhood of an unstable point. Then, the trajectory moves outward from the fixed point, the chaotic dynamics occurs ending with the trajectory return into the phase space part between the fixed point and unstable limit cycle. Experimental evidence of type-II intermittency in a hydrodynamic system has been reported by Ringuet et al. [Ringuet et al. (1993)] as well as in a coupled nonlinear oscillator in the reference [Huang and Kim (1987)]. An experimental evidence of type-III intermittency has been reported by Kreisberg et al. [Kreisberg et al. (1991)] while studying the Belousov–Zhabotinsky reaction in a well-stirred flow reactor as well as in a simple electronic circuit detected and illustrated by Kye et al. [Kye et al. (2003)]. Nowadays, in spite of the classical Pomeau–Manneville three types of intermittencies, there exist type-X and type-V intermittencies [He (1989); Price and Mullin (1991)], as well as the one-off chaos–chaos intermittency [Platt et al. (1993)] and the in–out chaos–chaos intermittency [Ashwin et al. (2001)]. 3.1.7

Synchronization of chaos

Synchronization has a Greek root and means a correlation of different processes in time. Synchronization phenomena play a crucial role in physics and biology, and in engineering sciences including mechanics, mechatronics and electrical engineering. Mostly, a phase synchronization is exhibited by engineering processes, where a few cyclic signals after a transitional dynamics start to oscillate with identical phase angles per consecutive cycles. The synchronized signals have integer relation between their frequencies and exhibit mode/phase locking quantified by Arnold tongues. In spite of synchronization regarding regular signals, the chaotic systems can be also synchronized yielding eventually locking of phases of the chaotic signals. In physics, two main couplings between subsystem can be distinguished, i.e. undirectional coupling and bidirectional coupling. In the first case, a so-called master–slave dynamics

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takes place, where one of the subsystem dynamics transmits energy to other subsystems (master ) and it is not influenced by the remaining subsystems (slaves). In this case, chaotic dynamics of the master energy is transmitted into slave subsystems. In the language of mechanics, it may correspond to a chaotic motor with high energy, and its output serves as excitation to the coupled either lumped or continous mechanical oscillators. However, dynamics of the oscillators does not influence the motor dynamics. In the second case, one deals with a so-called bidirectional coupling between oscillators, where all oscillators (subsystem) interact with each other. In the language of mechanics/control, the oscillating subsystems satisfy the third Newton’s law of action and reactions or obey feedback interactions. Recently, many different approaches have been applied to study complete/identical synchronization [Fujisaka and Yamada (1983); Pecora and Carroll (1990)], generalized synchronization [Rulkov et al. (1992)], lag [Rosenblum et al. (1997)] and phase [Rosa et al. (1998)] synchronization, imperfect phase synchronization [Zaks et al. (1999)], and almost synchronization [Femat and Solis–Perales (1999)]. This topic has been extensively reviewed in the reference [Boccaletti et al. (2002)], where an impact of the mentioned ideas on nonlinear optics, fluid dynamics and physiology has been outlined. There are also numerous works which concern experimental validation of various synchronization phenomena, including the classical work, where it has been demonstrated experimentally that a transition from a quasi-periodicity to chaos is associated with synchronization (phase locking) of frequencies ω1 , ω2 (see Fig. 3.7). A universal transition from quasi-periodicity to chaos is illustrated and discussed by Ostlund et al. [Ostlund et al. (1983)]. Finally, a recent tendency to match intermittency phenomenon and phase synchronization has been observed. The reference [Kiss and Hudson (2001)] is devoted to phase synchronization and suppression of chaos through intermittency in a electromechanical oscillator via laboratory experiment. Coupled discrete and continuous in time systems have been studied regarding phase synchronization

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Fig. 3.7 Frequency synchronization in the Benard experiment [Gollub and Benson (1979)].

effects of chaotic type-I intermittent oscillations by Ivanchenko et al. [Ivanchenko et al. (2004)]. In our book, we aim at studying synchronization phenomena in continuous (spatially extended) systems which approximate the infinite dimensional systems. Although there are a few papers dealing with PDEs and including 1D complex Ginzburg–Landau equations [Boccaletti et al. (1999); Bragard et al. (2000)], there is a gap in results concerning synchronization of chaotic dynamics regarding spatial structural members like beams, plates and shells. 3.2 3.2.1

Quantifying Chaotic Dynamics Introduction

Investigation of regimes of dynamical chaos consists of four directions. The first one is associated with rigorous mathematical background of the hyperbolic systems. The second one relies on analysis of dynamics of mathematical models of real physical systems. The third one, perhaps the most important from the point of view of nonlinear dynamics, includes a study and investigation of relatively simple basic models exhibiting the fundamental properties of chaotic systems. Finally, the fourth direction is oriented toward the investigation of continuous systems governed by partial differential

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equations. As the latter direction begins to develop, we aim to contribute to this topic on the basis of an investigation of mechanical continuous systems, i.e. systems with infinite number of degrees of freedom. It should be noted that while investigating mechanical continuous systems, a wide spectrum of approaches known and used during the analysis of simple basic models for chaos detection and observation are applied. The so far presented directions create the fundamental problem of theory of vibrations as a science investigating basic phenomena using examples of dynamics of simple basic models associated with different nonlinear phenomena. Discovery of the deterministic chaos implied the need to formulate a novel chapter of theory of nonlinear vibrations aimed at analyzing the basic models of chaos. In computer experiments with mathematical models of various complexity, chaotic effects are usually studied through analysis of signals in time and space, phase portraits, Poincar´e maps, time series and power spectra using the Fourier transformation (FT or Fast Fourier Transform (FFT)) and/or autocorrelation functions. It will be shown further that investigation of continuous mechanical systems requires also the application of the wavelet analysis. 3.2.2

Fourier Transformation (FT)

Time series is a function f (t) like, for instance, that describes beam deflections versus time. Monitoring of the time series on the sufficiently large time intervals allows for estimation of the vibration character consisting of three possible regimes: periodicity, quasiperiodicity and chaos. An important role while studying the time series plays a frequency distribution displayed by a power spectrum S(ω) governed by the FT. FT of a function f (t) yields the new function  ∞ 1 f (t)e−iωt dt, (3.13) F (ω) = 2π −∞ and



∞

f (t) = −∞

F (ω)eiωt dω.

(3.14)

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Note that F (ω) can be complex. Square of its modules S(ω) = |F (ω)|2 is called the power spectrum. If f (t) has a period T , then the power spectrum exhibits the main peak on the frequency values ω = T1 , and possibly other peaks regarding the frequency rational with ω. Function f (t) is called quasi-periodic if it can be presented as a sum of periodic functions, but with irrational periods. If in the time series f (t) neither long term regularity nor periodicity/quasiperiodicity is observed, the oscillations are referred as aperiodic or chaotic ones. Numerical simulation of oscillating processes requires application of the discrete analog of FT, since direct computation of the power spectrum for the discrete FT needs long computational time. Therefore, in a commercially proposed algorithm a so-called FFT is applied. For integer k, the FFT yields F (k) =

n


f (j)e−2πi

(j−1)(k−1) n

.

(3.15)

j=1

FFT is of a particular advantage, when n = 2m . 3.2.3

Types of Poincar´ e maps

While investigating dynamical systems, a so-called map (transformation/section) is introduced, which is defined through the series {w(t1 ), w(t2 ), . . . , w(tn ), . . . , w(tN )}, where wn = w(tn ). In simple deterministic transformation the quantity wn+1 is defined via wn , which is described by the form wn+1 = f (wn ). In fact, this formula presents a difference equation. In order to detect a chaotic quantity through time series analysis, the following two criteria should be satisfied: (i) points produced by wn+1 and wn should be grouped by a certain functional dependence; (ii) function f (w) should exhibit either maximum or minimum. Assuming that the mentioned criteria are satisfied, it is recommended to choose a polynomial approximation of the obtained points and apply the defined map to carry out further numerical experiments in a way similar to that presented for the square map.

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Examples of functions f (w) of mechanical systems can be found in the reference [Combes et al. (1989)]. Existence of such function implies the possibility (in some cases) to detect, even in complex dynamical systems, mathematical properties exhibited by 1D mapping such as period doubling phenomena, the Feigenbaum constant or various routes to chaos. In the case when there is an excitation force with period T , one may introduce the following time sequence tn = nT + τ0 to produce a Poincar´e map. This allows to distinguish periodic and non-periodic motions. In the book, we mainly apply the dependence wt (wt+T ), where T stands for a period of the driven force. In what follows, we consider certain types of the maps. (a) Poincar´ e map Consider a certain regime of motion of a system governed by differential equations characterized by the trajectory Γ in the phase space Rn . Let us introduce a certain hyperplane S  of dimension N − 1. We assume that the phase trajectory Γ successively and transversally (with non-zero angle) intersects this surface. Surface S is called the Poincar´e section regarding the phase trajectory Γ. Trajectory Γ generates a certain unique transformation (but not mutually unique) → on the Poincar´e plane, which maps − w (k) through intersections of Γ → − with S into a successive point w (k + 1). Sequence of transformation is defined via intersections of Γ with S in one direction. The obtained → discrete set {− w (k)}, where k = 0, 1, 2, . . . on S is called the Poincar´e map for the trajectory Γ. (b) Successive cross-section In order to get a cross-section we consider the following intersecting plane w − w˙ = 0 in the phase space (w, w). ˙ In this plane, three points or equivalently three equilibria may occur. In order to keep the uniqueness of the transformation, an initial point should belong to the intersecting plane and located in the vicinity of the right equilibrium point with the additional condition w¨ < 0. One may consider the following 1D transformation. We take a plane transformation of the point (x, y) into (x0 , y0 ) in such a way that the radius beginning at this point and arbitrarily directed should

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not intersect the mapping curve more than once. Thus, the transformation θn → θn+1 describes an angle between directions on the curve edge and nth point of the successive transformation. Applying the normalization procedure, the problem is reduced to mapping the unit interval [0, 1] onto itself, which is denoted by Ln+1 = ϕ(Ln ). The obtained curve fully describes the system dynamic properties. (c) Poincar´ e pseudo-map In this work, we often apply the Poincar´e pseudo-map, where the dependence wt (wt+T ) is constructed and T is the period of an exciting force. 3.2.4

Lyapunov characteristic exponents

Lyapunov exponents play a crucial role in the theory of Hamiltonian and dissipative systems. They allow to quantify the system stochasticity. Besides, there exists a link between Lyapunov exponents and other characteristics like the Kolmogorov entropy or the fractal dimension. Lyapunov exponents characterize the averaged velocity of the exponential divergence of the neighborhood trajectories. The characteristics of stochasticity have been introduced by Henon and Heiles in their numerical experiments [Daubechies (1988)]. Earlier, Poincar´e had pointed out the link between chaotization and the exponential instability [Farge (1992)]. Let us assume that after application of one of the methods reducing the problem of infinite dimension to that of a finite one, we get a system of the first-order ordinary differential equations of the form x˙ = F (x, M ),

(3.16)

which, in general, cannot be solved analytically. The method of computation of Lyapunov exponents has been introduced by Benettin et al. [Benettin et al. (1976, 1980); Staszewski and Worden (1999)], and we briefly describe main ideas of this approach. We consider two neighbourhood trajectories with the initial conditions x0 and x0 + ∆x0 . Their time evolution defines the tangent vector ∆x(x0 , t)

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d(x0 , t) = ∆x(x0 , t).

(3.17)

Dynamics of ∆x can be defined by the linearized equation (3.16), i.e. we get ∂F ∆x = · ∆x, (3.18) dt ∂x where ∂F ∂x is the Jacobi matrix F(x, M ). We introduce the averaged exponential divergence of the neighborhood trajectories in the following way 1 d(x0 , t) ln . (3.19) λ(x0 , ∆x0 ) = lim t→∞ t d(x0 , 0) d(0)→0

It can be proved that a limit of λ exists and is bounded. Furthermore, there exists a full system of M fundamental solutions (ei ) of equations (3.18), and each of the equations possesses the value λi (x0 ) = λ(x0 , ei ),

(3.20)

which is called a characteristic Lyapunov exponent. The Lyapunov exponents do not depend on the choice of the phase space metric G and they can be ordered in the following way: λ1 ≥ λ2 ≥ λ3 ≥ · · · ≥ λM . In the particular case of a periodic trajectory, Eq. (3.18) define a certain linear transformation with the period T , which can be governed by the following map ∆xn+1 = A · ∆xn .

(3.21)

Matrix A has M eigenvalues αM , in general complex ones, which can be ordered in the following way |α1 | ≥ |α2 | ≥ · · · ≥ |αM |.

(3.22)

We denote the corresponding eigenvalues by ei , and (3.18) for ∆x0 = ei yields ∆xn = αni ei . Hence, owing to (3.19), we get 1 λi (ei ) = ln |αi | = λi . T

(3.23)

(3.24)

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Formula (3.23) implies that for ∆x0 = c1 e1 + · · · + cM eM =

M


ci ei ,

(3.25)

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i=1

dynamics of the vector ∆xn is defined by the first zero-value coefficient ci . Each of the Lyapunov exponents defines a velocity λ in a certain subspace with dimension of one less than previous one. Consequently, almost for all ∆x we get the value λ = λ1 . This property is illustrated in Fig. 3.8. While studying a dynamical system, the so-called Poincar´e map is widely used and its concept is schematically presented in Figs. 3.8 to 3.10. Oseledec [Oseledec (1968)] generalized the concept of eigenvalues and eigenvectors onto periodic orbits. The main idea of this approach is that any non-periodic trajectory can be approximated

Fig. 3.8

Fig. 3.9

Scheme of a Poincar´e map construction.

Sketch of the successive cross-section construction.

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Sketch of a Poincar´e map construction.

Fig. 3.11 Two neighborhood diverging trajectories 1 and 2, where ∆x(t) is a x tangent vector (a); ∀ vector ∆ x non-parallel to e2 , λ(∆x0 ) = λ1 , whereas for ∆ parallel to e2 , λ(∆x0 ) = λ2 < λ1 (b).

by a periodic one with a sufficiently large period. For any continuous trajectory governed by differential equations (3.16) at least one of the Lyapunov exponents associated with eigenvector tangent to the trajectory is zero. Lyapunov exponents for vectors ∆x are named also as the first-order exponents. Oseledec generalized this notion in order to describe the average velocity of the exponential increase in a p-dimensional velocity Vp constructed with the help of the vectors ∆x1 , ∆x2 , . . . , ∆xn (p ≤ M ). Therefore, 1 Vp (x0 , t) ln t→∞ t Vp (x0 , 0)

λp (x0 , Vp ) = lim

(3.26)

defines the Lyapunov exponent of order p. Osedelec [Oseledec (1968)] and Benettin [Benettin and Galgani (1979); Benettin et al. (1976, 1980)], showed that λp (x0 , Vp ) is expressed by a sum p of the first

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order. Analogously, since for almost all ∆x the following formula holds λ(x0 , ∆x) = λ1 (x0 ), then for almost all Vp we have

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λp = λ1 + λ2 + · · · + λp =

p


λi .

(3.27)

i=1

This relation is used for numerical estimation of all Lyapunov exponents. For p = M , the following average velocity of the exponential increase in the phase volume is obtained λ

M

=

M


λi (x0 ).

(3.28)

i=1

Largest exponent is often used as a criterion for chaos occurrence. As we have already mentioned, λ(x0 , ∆x0 ) = λ1 (x) holds for almost all tangent vectors ∆x0 . Therefore, in order to compute λ1 the initial vector ∆x0 can be taken arbitrarily. Integrating simultaneously equation (3.19), we find d(t) = |∆x(t)|, and we can take further d0 = d(0) = 1. Problems may occur in the case, when |∆x| increases exponentially, which introduces errors. The use of the linearized equation has an important advantage, because its solution does not depend on |∆x|. However, in some cases it is convenient to integrate two trajectories (compare Figs. 3.11 and 3.12). Let us choose the time interval τ and let us normalize |∆x| into a unit at the end of each taken interval, i.e. the vector length |∆x|(|∆x|) → d0 = 1 is conserved while keeping its same direction.

Fig. 3.12 Scheme of computation of the largest Lyapunov exponent [Benettin et al. (1976)].

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We introduce the notation y = x + ∆x, where τ denotes the final time interval. Therefore, we can successfully compute

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dk = |∆xk−1 (τ )|, ∆xk (0) =

∆xk−1 (τ ) , dk

(3.29) (3.30)

where ∆xk−1 (τ ) is obtained by integration of (3.16), (3.19) on the interval τ with the initial conditions x(kτ ), ∆xk (0). If we introduce the quantity λn =

n 1
di , nτ

(3.31)

i=1

then (3.19) yields λ1 = lim λn . n→∞

(3.32)

For a regular dynamics λ1 ≤ 0, whereas the occurrence of the chaotic component yields λ1 > 0, and it does not depend on x. This method is either applied for continuous trajectories or for discrete maps. Further, we apply it to investigate continuous trajectories of beams, plates and shells. Benettin’s algorithm [Benettin et al. (1976)] yields the possibility of computing the full spectrum of the Lyapunov exponents in the M -dimensional phase space. Let us choose the initial basis from p orthonormal tangent vectors and let us numerically estimate the p-dimensional volume Vp (t) defined by these vectors. One may find the Lyapunov exponent λp1 of order p (see (3.26)). Carrying out this procedure for p = 1, 2, . . . , M , the formula (3.27) allows to define all exponents λ1 , λ2 , . . . , λM . Here, another difficulty occurs. Namely, in the process of motion angles between tangent vectors exponentially decrease, whereas the numerical errors increase. Therefore, in addition to the renormalization of the vector’s ∆x length it is necessary to carry out the renormalization of the vectors. New vectors should lie in the same space as the old ones. The latter procedure can be carried out by the Gramm–Schmidt method. Let ∆xk−1 (τ ) be a tangent vector at time instant starting from ∆xk−1 (0). Let us

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compute for each time interval the following quantities (1)

(1)

dk = |∆xk−1 (τ )|,

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(1) ∆xk

(3.33)

(1)

∆xk−1

. (1) dk Here, for j = 2, . . . , M we find the following quantities (i) uk−1 (τ )

=

(j) ∆xk−1 (τ )

−

d−1


=

(i)

(j)

(3.34)

(i)

(∆xk (0) · ∆xk−1 (τ ) · ∆xk (0)), (3.35)

i=1 (j)

(j)

dk = |uk−1 (τ )|, (j) ∆xk (0)

(3.36)

(j)

=

uk−1 (τ ) (j)

.

(3.37)

dk

Then, in (k − 1)th interval of τ the volume Vp increases by (p) dk times. Therefore, (3.26) yields 1 (1) (2) (p) ln(di di ... di ). (3.38) λ1 = lim n→∞ nτ

(1) (2) dk dk . . .

(p−1)

(p)

from λ1 Computing λ1 exponent is defined as follows

and using (3.27), the pth Lyapunov

n 1
(p) ln di . n→∞ nτ

λp = lim

(3.39)

i=1

3.2.5

Melnikov’s method

In 1962, Melnikov [Melnikov (1962, 1963)] proposed a method to investigate a motion in the vicinity of the system separatrix, where a studied system is close to the integrable system. This method allows to achieve a criterion of chaos occurrence in the neighborhood of the separatrix with dissipation. Taking dissipation as the control parameter, one may derive criterion for occurrence of chaos. Morozov [Morozov (1976)] and Holmes [Holmes (1979)] investigated the Duffing equation, and we report here the results given by Holmes. Consider the equation (3.40) x˙ = f0 (x) + εf1 (x, t),

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 0 (saddle closed Fig. 3.13 Input and output separatrices of the hyperbolic point X orbit (a); stable and unstable manifolds of the saddle (b,c); intersection of stable and unstable manifolds (d)).

where x = (x1 , x2 ), f1 is periodic with respect to t with period T . 0 For ε = 0, the system is integrable and has the hyperbolic point X  0 . It has been shown with the unique separatrix x0 (t) and x0 (t) = X schematically in Fig 3.13(a), where in the phase space (x1 , x2 ) the integrable system is represented by input xs (t) and output xu (t) separatrices. There exist elliptic points inside of the homoclinic loop. When the excitation is included, the phase space becomes threedimensional (x1 , x2 , t), and therefore we follow the system motion on the intersection surface t = const(mod T ), where T denotes the excitation period. In a dissipative system, three cases are possible: Cases 1 and 2 — separatrices never intersect x1 , and one of them can overlap the second one [Fig. 3.13(b)], i.e. the output separatrix xu (t) surrounds the input separatrix xs (t), Fig. 3.13(c) — input separatrix xs (t) surrounds the output one xu (t)); case 3 — separatrices intersect each other in an infinite number of points [Fig. 3.13(d)]. The Melnikov method is aimed at finding the first intersection point of

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the separatrices, i.e. stable and unstable manifolds associated with a saddle. Let us apply the theory of perturbation to compute the distance D between the separatrices in a certain time instant t0 [Awrejcewicz and Krysko (2006); Nayfeh (2004)]. Observe that, in the first case [Fig. 3.13(b)] we have D < 0, in the second case [Fig. 3.13(c)] we have D > 0 for ∀ t0 . In the third case [Fig. 3.13(d)], the quantity D changes its sign for each intersection of separatrices, and hence this stands for the criterion of chaos occurrence. All these three situations have been studied by Melnikov [Melnikov (1962, 1963)]. In order to compute D, it is sufficient to know two separatrices xs (t) and xu (t) with the approximations truncated to the first form of ε. Let xs,u (t, t0 ) = x0 (t − t0 ) + εxs,u 1 (t, t0 ),

(3.41)

where t0 is the arbitrary initial time, and x0 is the separatrix being perturbed (homoclinic loop). Substitution of (3.41) into (3.40) yields dxs,u 1  x0 (t − t0 ), t), = M (x0 ) · xs,u 1 + ε f1 ( dt where

 ∂f01 M (x0 ) =

∂x1

∂f01 ∂x2

∂f02 ∂x1

∂f02 ∂x2

(3.42)

 .

(3.43)

The Jacobi matrix of the vector f0 is computed along the nonperturbed trajectory x0 (t − t0 ). We need the solution xs (t) of (3.42) for t > t0 and for xu (t) for t < t0 such that  p, (3.44) lim xs = lim xu = X t→+∞

t→−∞

 p denotes the perturbed location of the hyperbolic point. where X Both solutions differ by the vector (ε = 1) α  (t, t0 ) = xs (t, t0 ) − xu (t, t0 ) = xs1 (t, t0 ) − xu1 (t, t0 ).

(3.45) The Melnikov distance D(t, t0 ) is defined as a projection d onto  to the non-perturbed separatrice x0 of the time the normal N instant t0 :   · d. (3.46) D(t, t0 ) = N

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 · d (dashed curve presents the unperturbed Fig. 3.14 Melnikov’s distance D = N  p under perturbation). separatrice of the hyperbolic point  x0 , which moves into X

 (t, t0 ) Applying (3.41) (for ε = 0), we define the normal vector N as follows   −f ( x ) 02 0  (t, t0 ) = . (3.47) N f01 (x0 ) We introduce the operator (3.48) x ∧ y = x1 y2 − x2 y1 , and its action can be generated to the form (3.49) x ∧ y = εik xi yk , where εik is a unit anti-symmetric matrix (ε12 = 1). Then the scalar product (3.46) can be presented in the following form  (3.50) D(t, t0 ) = f0 ∧ d. Let (3.51) D = Ds − Du, where (3.52) D s,u (t, t0 ) = f0 ∧ xs,u 1 . s Differentiating (3.52) and D with respect to time, we get (3.53) D˙ s = f˙0 ∧ xs1 + f0 ∧ x˙ s1 = (M (x0 ) · x0 ) ∧ xs1 + f0 ∧ x˙ s1 . Using (3.51) and x0 = f0 , we get D˙ s = (M (x0 ) · f0 ) ∧ xs1 + f0 ∧ x˙ s1 ∧ (M (x0 ) · xs1 ) + f0 ∧ f1 (3.54) = Sp M (x0 ) f0 ∧ xs1 + f0 ∧ f1 = Sp M (x0 ) Ds + f0 + f1 , where (3.55) Sp M = div f0 . Next, (3.55) should be integrated in order to get Ds . D˙ u should be obtained in a similar way and then integrated. The obtained result

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should be substituted into (3.51), and finally the Melnikov distance D will be found. Let us consider the particular case of a Hamilton system, i.e. when the dissipation is equal to zero, i.e. Sp M = 0. Integrating (3.51) from t0 to ∞, and taking into account the asymptotic condition D s (∞, t0 ) = f0 (x0 ) ∧ x1 = 0, we get  ∞ D s (t0 , t0 ) = − f0 ∧ f1 dt. (3.56) t0

Likely, we have  D (t0 , t0 ) = − u

t0

∞

f0 ∧ f1 dt,

and hence



D (t0 , t0 ) = D (t0 , t0 ) + D (t0 , t0 ) = − s

u

∞ −∞

(3.57)

f0 ∧ f1 dt,

(3.58)

which defines the motion kind. As it has been already mentioned, if D(t0 ) changes, then owing to Melnikov the motion is chaotic. In what follows, we apply Melnikov’s method to define the conditions of transitions into chaos for the following Duffing equation [Holmes (1979)]: x ¨ − x + x3 = −εδx˙ + εγcos(ωt).

(3.59)

This equation governs nonlinear oscillation of one degree of freedom system with small damping εδ and harmonically excited with the frequency ω and amplitude εγ. We reduce Eq. (3.54) to a system x˙ = v, 3

v˙ = x + x + ε(γcos(ωt) − δv).

(3.60)

Curves of constant energy correspond to an unperturbed Hamiltonian (δ = 0) of the form 1 1 1 H0 = v 2 − x2 + x4 , 2 2 4

(3.61)

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Fig. 3.15

61

Curves of constant energy of the unperturbed Hamiltonian.

and they are shown in Fig. 3.15. Point x = v = 0 is a unique hyperbolic point associated with one sepratrix for H0 = 0. In order to find the solution of the separatrix for H0 = 0, from (3.56) we find v and then we substitute it into the first equation of system (3.60), and we get  1/2 x2 dx = −x 1 − , dt 2

(3.62)

hence √

2 , coth t √ 2 sh t . v0 (t) = − coth2 t x0 (t) =

(3.63) (3.64)

Comparing (3.60) and (3.61), we get f01 = v,

f11 = 0,

f02 = x − x3 ,

f12 = γcos(ωt) − δv, (3.65)

and therefore, f0 ∧ f1 = v0 [γcos(ω t) − δv] .

(3.66)

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Substituting (3.63) into (3.56) and (3.57) and carrying out summation, we finally find the Melnikov distance  +∞   (3.67) γv0 (t − t0 ) cos (ω t) − δv02 (t − t0 ) dt. D=− Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

−∞

Substituting (3.67) into (3.64) and changing the integration variable τ = t − t0 , we get  +∞ √ sinh τ dτ sin(ω τ ) D = 2 γ sin(ω t0 ) coth2 τ −∞  +∞ tanh2 τ (3.68) dτ + 2δ 2 −∞ coth τ √ sin(ω t0 ) 4 πω + δ. = 2πγω 3 coth 2 Chaotic motion in the vicinity of the separatrix (D changes its sign) is √ 3 2πγω . (3.69) δ < δc = 2 ch πω 2 Numerical simulation shows that the occurrence of strange chaotic attractor for the Duffing equation is associated with the cascade of bifurcations of two focuses x = ±1 , v = 0 (Fig. 3.16). Nowadays, there are numerous examples of direct application of the Melnikov technique to predict and study 1DOF dynamical systems (see, for example, references [Awrejcewicz and Holicke (1999, 2006); Awrejcewicz and Pyryev (2003)]). An extension of the Melnikov method of multi-DOFs systems with Coulomb type friction firstly proposed by Gruendler [Gruendler (1985)] is widely described in the monograph [Awrejcewicz and Holicke (2007)].

Fig. 3.16 Scenario of oscillation of the Duffing equation versus the damping coefficient γ.

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3.2.6

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Wavelet analysis Introduction

In general, wavelets are devoted to analyzing the signals (time histories) of various real world processes. They are used to get hidden information in different kinds of data via the so-called wavelet-based compression/decompression. The word “wavelet” has been applied for the first time by Morlet and Grossmann as continuous wavelet transforms in 1980 (the equivalent French word is ondelette, which means “small wave”) [Grossmann and Morlet (1984)]. The wavelet theory is strongly linked with the continuous of Str¨ omberg wavelets [Str¨ omberg (1983)], Daubechies [Daubechies (1992); Hazewinkel (2001)] (orthogonal wavelets with compact support), Mallat [Mallat (1989)] (multiresolution-based wavelet transforms), Delprat et al. [Delprat et al. (1992)] (interpretation of continuous wavelets transform) and Newland [Newland (1993)] (harmonic wavelet transform). A wavelet means a small oscillation (or a few oscillations) begining at zero, then increasing, and finally ending at zero. The wavelet transforms allow to get the timefrequency characteristic for time-continuous signals. They are separated into three classes: continuous, discrete and multiresolutionbased wavelet transforms. In spite of the given geometrical description, a wavelet can be viewed as a function applied for the separation of another function being analyzed into different scale components. On the other hand, a wavelet transform allows to represent the given function by the applied wavelets. In spite of the already mentioned continuous (analog) wavelet transforms, discrete wavelet transforms are also available. Contrary to the classical Fourier transforms, the wavelet transforms are particularly useful while studying chaotic and transient dynamics. The discrete wavelets mainly use a chosen subset of scale and translation values and this class includes Coiflet wavelets [Beylkin et al. (1991)], Daubechies wavelet [Daubechies (1992); Hazewinkel (2001)], Cohen–Daubechies–Feauveau wavelet [Cohen et al. (1992); Daubechies (1992)], Haar wavelet [Chui (1992); Haar (1910)],

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Legendre wavelet [Banifatemi et al. (2007); Shang and Han (2007)], and Symlets [Kumari and Vijay (2012)]. The continuous wavelet transforms have no limitations in using scales and translations, and this class includes real valued (Beta, Hermitian, Meyer, Mexican hat, Shannon and Str¨ omberg) and complex valued (Mexican hat, Morlet, Shannon) wavelets [Blatter (2002); Louis et al. (1997)]. Among many existing signal transformations, FT certainly belongs to the most rigorously established and the mostly used. The majority of signals met in practice is presented in the time domain, i.e. a signal is the function of time. On a graph, we take the time axis as the horizontal one (independent) and the amplitude (system response) axis as the vertical one (dependent). However, in many cases the amplitude–time signal representation does not belong to a suitable choice. It may happen that the most important information is hidden in the frequency domain. A frequency spectrum contains a set of frequency (spectral) components. It exhibits occurrence and magnitude of various frequencies in a studied signal. Frequency is measured in Hertz (Hz) or in a number of periods per second. Frequency of a signal is yielded by FT. Given a signal in time domain we get its spectral representation. Instead of time, we take frequencies on the horizontal axis, whereas the vertical coordinate presents an amplitude of a frequency withdrawn from the signal. In what follows, we give a background related to wavelet theory and application, though we are mainly focused on definitions, properties and their consequences using the wavelet transformation of 1D functions. However, the consideration can be extended in a similar way for a study of multi-dimensional cases. (i) From Fourier to wavelet transformation Integral FT as well as the Fourier series play a fundamental role in harmonic analysis. As a result, the obtained Fourier coefficients have relatively simple physical interpretation and they contain many important information regarding the character type of a signal being investigated. Application of the integral transformation and Fourier series is very clear and simple. All properties and formulas refer

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only to two real functions sin(t), cos(t) or one complex exp(it) = cos(t) + i sin(t), i = (−1)1/2 in the form of a sinusoidal wave, and all associated transformations can be relatively easily proved. On the other hand, the wavelet transformation appeared rather late, and its mathematical tools are still under development. Therefore, in order to avoid the rigorous mathematical presentation, we follow [Holschneider (1988)] and we introduce necessary definitions associated with the wavelet analysis in comparison to the known Fourier analysis, where the latter one is widely known and accepted. (ii) Fourier series We recall here a few fundamental statements being used further. Let L2 (0, 2π) be the space of squared integrable functions with finite energy. The formula  2π |f (t)|2 dt < ∞, t ∈ (0, 2π), (3.70) 0

defines a piecewise continuous function f (t). It can be extended in a periodic manner and defined on the whole axis R(−∞, ∞) such that f (t) = f (t − 2π) for all t from R.

(3.71)

Any arbitrary f (t) taken from the space of 2π-periodic and square integrable functions can be presented in the form of the Fourier series f (t) =

∞


cn exp(int),

(3.72)

f (t)exp(−int)dt,

(3.73)

−∞

with the following coefficients  −1 cn = (2π)

2π 0

and the series is uniformly convergent with regard to f (t), i.e.  2π N
|f (t) − cn exp(int)|2 dt = 0. (3.74) lim M,N →∞ 0

M

Observe that wn (t) = exp(int),

n = . . . , 0, 1, . . .

(3.75)

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is the orthonormalized basis of the space L2 (0, 2π), constructed with the help of a scale transformation of the unique function w(t) = exp(it), in such a way that wn (t) = w(nt). In other words, each 2π-periodic square integrable function can be obtained by a superposition of the scale transformations of the basic function w(t) = exp(it) = cos(t) + i sin(t), i.e. it is a composition of sinusoidal curves with various frequencies. The coefficients depend on a harmonic (frequency) number. Recall that for the Fourier coefficients the following Parseval’s formula holds:  2π ∞
−1 2 |f (t)| dt = |cn |2 . (3.76) (2π) 0

−∞

(iii) Wavelet series Consider the space L2 (R) of function f (t) with finite energy defined on the whole real axis R(−∞, ∞)  2π |f (t)|2 dt < ∞. (3.77) Ef = 0 2 L (0, 2π)

and L2 (R) are essentially different. Functional spaces In particular, local average values of each of functions from L2 (R) should tend to zero on (±∞). Since a sinusoidal wave does not belong to L2 (R), a family of sinusoidal waves wn cannot serve as a basis for the functional space L2 (R). Therefore, we are going to find relatively simple functions allowing a construction of the basis of space L2 (R). Waves creating the space L2 (R) should tend to zero on ±∞ approaching zero relatively fast, which is important in practice. Let us take wavelets as basic functions, which are well-localized solitontype small waves. Similar to the case of the space L2 (0, 2π), which has been described in full with the help of one basic function w(t), we are going to construct the functional space L2 (R) also with the help of only one wavelet ψ(t). A question arises: How to cover the whole axis R(−∞, ∞) using only one localized function ψ(t) rapidly tending to zero? It can be done by applying a system of shifts along the axis. For simplicity, we consider integer shifts, i.e. we take ψ(t−k). In what follows, we introduce an analog of a sinusoidal frequency. In order to keep simplicity

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of this introduction and to obtain directly the main wavelet properties, we introduce the multiplier 2: ψ(2j t − k), where j, k are integers (j, k ∈ I). Therefore, owing to the help from scale transformations (1/2j ) and shifts (k/2j ) we can describe all frequencies, and we can cover the whole axis using only one unique basic wavelet  ∞ψ(t). Recall definition of a norm p 2 = < p, p >1/2 , < p, q >= −∞ p(t)q(t)dt, where a bar denotes a complex conjugated quantity. Consequently, we have ψ(2j t − k) 2 = 2−j/2 ψ(t) 2 . Therefore, if the wavelet ψ(t) ∈ L2 (R) has a unique norm, then also all wavelets of the family {ψjk } of the form ψjk (t) = 2j/2 ψ(2j t − k)

(j, k) ∈ I

(3.78)

are also normalized into one: ψjk 2 = ψ 2 = 1. We say that the wavelet ψ ∈ L2 (R) is orthogonal, if the family {ψjk } defined by (3.78) presents the orthonormalized basis of the functional space L2 (R): < ψjk , ψlm > = δlk δkm , and each f ∈ L2 (R) can be presented in a form of the following series f (t) =

∞


cjk ψjk (t),

(3.79)

j,k=−∞

which is uniformly convergent in L2 (R) N2
N1
cjk ψjk f− lim = 0. M1 ,N1 ,M2 ,N2 →∞ −M2 −M1

(3.80)

2

One of the simple examples of the orthogonal wavelet in the HAAR wavelet is defined as follows   1, 0 ≤ t < 1/2, (3.81) ψ H (t) = −1, 1/2 ≤ t < 1,  0, t < 0, t ≥ 1. H , ψ H , yielded by this One may check that any two functions ψjk im wavelet through formula (3.78) and with the help of scale transformations (1/2j ), (1/2l ) and shifts (k/2j ), (m/2l ) are orthogonal and have a unique norm. Let us construct a basis of the functional space L2 (R) with the help of scale transformations and shifts of the wavelets with

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arbitrary values of the basic parameters, i.e. scale coefficient a and shift parameters b:

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ψab (t) = |a|−1/2 ψ((t − b)/a),

a, b ∈ R,

ψ ∈ L2 (R).

(3.82)

Now, we can define the following integral wavelet-transformation  ∞  ∞ −1/2 ¯ f (t)ψ((t − b)/a)dt = f (t)ψ¯ab (t)dt. [Wψ f ](a, b) = |a| −∞

−∞

(3.83) Extending further the analogy with respect to FT, we can define the coefficients cjk = of the series (3.79) of the function f into a series of wavelets in the following wavelet transformation: cjk = [Wψ f ](1/2j , k/2j ). Instead of [Wψ f ](a, b) for the wavelettransform coefficients (amplitudes), we apply (sometimes) further notation as W (a, b) or Wψ f or W [f ]. Therefore, each function from L2 (R) can be constructed via superposition of the scale transformations and shifts of the basic wavelet. In other words, there is a composition of small waves with coefficients depending on wave number, frequency, scale factor and shift parameter (time). Application of the discrete-wavelet transformation (discrete frequency–time space in the form of shifts and stretching regarding powers of 2) allows to prove majority of the results of the theory of wavelets regarding compactness and orthogonality of the wavelet basis, series convergence, etc. [Combes et al. (1989); Daubechies (1988); Farge (1992); Staszewski and Worden (1999)]. Proofs of those results are sometimes necessary, for instance, in the case of information compression or in numerous problems of modeling, i.e. in the cases, where it is important to carry out the transformations and to get an exact form of the inverse transformation. In order to analyze signals, a continuous wavelet transformation is widely applied. However, its discrete counterpart allows to exhibit the hidden information in the studied data more efficiently and rigorously. (iv) Inverse wavelet transformation A sinusoidal wave creates the orthonormalized basis of the functional space L2 (0, 2π) and the inverse FT does not introduce any problems.

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On the contrary, orthonormalization of the bases of space L2 (R), constructed through wavelets are defined by a choice of the basis wavelet and a way of the basis construction (values of the basis parameters, i.e. scale and shift coefficients). A wavelet can be considered as the basis function L2 (R) only if the constructed (with its help) basis is orthonormalized and the inverse transformation does exist. However, rigorous proofs of compactness and orthogonal properties are not easy [Combes et al. (1989); Daubechies (1988); Farge (1992); Staszewski and Worden (1999)]. Besides, for practical reasons, in many cases the used bases are not rigorously orthogonal, and the description of quasi-basis is introduced and applied. Although we omit further detailed and rigorous statements/analysis, we consider the inverse transformation for the two cases already mentioned: for the basis (3.78), allowing stretching and shifts (1/2j , 1/2j ), j, k ∈ I, and basis (3.82), constructed for arbitrary values of (a, b), a, b ∈ R. For the basis parameters (a, b), a, b ∈ R the inverse wavelet transformation is defined via the same basis as for the direct wavelet transformation, i.e.  f (t) = Cψ−1

[Wψ f ](a, b)ψab (t)dadb/a2 ,

(3.84)

where Cψ is the normalized coefficient (analogous to the coef(2π)1/2 of the normalized Fourier transform Cψ = ficient ∞ 2 −1 ∧ ∧ we denote the Fourier transform). −∞ |ψ (ω)| |ω| dω < ∞ (by Condition of finite Cψ bounds the class of function ψ(t) ∈ L2 (R), which can be taken as the basis wavelets. In particular, it is evident that the Fourier transform ψ ∧ should be equal to zero in the coordinates origin ω = 0, and consequently, its zero moment should also be equal to zero 

∞

ψ(t)dt = 0.

ψ(t) :

(3.85)

−∞

In many practical situations, it is sufficient to consider only positive frequencies, i.e. a > 0. The wavelet, respectively, should satisfy

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the following condition  ∞   ∧ 2 −1   ψ (ω) ω dω = 2 Cψ = 2 0

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0

 ψ ∧ (−ω)2 ω −1 dω < ∞. (3.86)

In the case of discrete wavelet transformation, a stable basis is defined in the following way. Function ψ(t) ∈ L2 (R) is called the R function if the basis {ψjk }, defined by (1) is the Riesz basis in the following sense: there exist two constants A and B, 0 < A ≤ B < ∞, for which the following relation holds 2 ∞ ∞

2 2 A {cjk } 2 ≤ cjk ψjk (3.87) ≤ B {cjk } 2 . j=−∞ k=−∞ 2

In addition, for any bounded series cjk we have

{cjk } 22 ≤

∞


∞


| cjk |2 < ∞.

(3.88)

j=−∞ k=−∞

For any R function, there exists the basis {ψ jk } being the counterpart of the basis {ψjk } in the sense that < ψjk , ψ lm >= δlk δkm . It helps to construct the following formula ∞
ψ jk (t). (3.89) f (t) = j,k=∞

Now, if ψ is the orthogonal wavelet and {ψjk } stands for the orthonormal basis then {ψ jk } and {ψjk } coincide and formula (3.90) is the formula of the inverse transformation. On the contrary, if ψ is not orthogonal wavelet, but a dyadic wavelet R then it has the dyad {ψ ∗ }, which allows for construction of the dyad of the family {ψjk } similar to basis (1): ψ jk (t) = ψ ∗ (t) = 2j/2 ψ ∗ (t) = (2j t − k),

j, k ∈ I.

(3.90)

In the general case, the reconstructing formula (3.90) does not necessarily present a wavelet series in the sense that ψ∗ is not a wavelet and {ψ jk } may not have the dyad-basis constructed via the way of (3.82). (v) Frequency–time localization Fourier transform and Fourier series are remarkable mathematical tools and they are widely described in numerous applications in order

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to get physical interpretation of various processes while studying different kind of signals. A real signal always belongs to the space L2 (R). Fourier transform of the signal f (t) with a finite energy defined by the norm f 2 presents the spectrum of this signal f ∧ (ω) =



∞

f (t)e−iωt dt.

(3.91)

−∞

In some cases, a physical interpretation yielded by this formula is difficult. Therefore, in order to get a spectral information for a chosen frequency, it is necessary to have the whole past and future information. However, the formula does not include the possibility of frequency evaluation in time. The Fourier transform, for example, does not distinguish the signal composed of two harmonics from the signal composed of these harmonics, when one harmonic is included into the second one. Besides, it is known that the frequency of a signal is inversely proportional to its length. Therefore, in order to get the information about high frequencies with the required accuracy, the latter should be obtained rather from small time intervals instead of the whole time interval. And vice versa, information about low frequencies should be yielded with the help of relatively wide time intervals. A part of the mentioned problems can be solved using the window FT. However, continuously oscillating basis function (sinusoidal wave) does not allow to get the required localized information. On the other hand, an element of the wavelet basis belongs to the well-localized function rapidly approaching zero outside of the small interval, which does not allow to carry out a proper “localized spectral analysis”. In other words, the wavelet transform is automatically equipped in the moveable frequency–time window being narrow (wide) on small (large) scales. Which parameters characterize a frequency–time window of the wavelet transformation? Since the wavelet ψ and its Fourier counterpart ψ ∧ rapidly decrease, they can be used as a window-function with the centre and the wideness defined in the following way. For the non-trivial window function z(t) ∈ L2 (R) it is necessary that tz(t) should also belong to L2 (R), its centre t∗ and radius

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∞ ∆z are defined by formulas t∗ = (1/ z 22 ) −∞ t|z(t)|2 dt and ∆z = ∞ 1/2 (1/ z 22 ){ −∞ (t − t∗ )2 |z(t)|2 dt} , whereas a width of the window function is ∆z . Let t∗, ∆ψ , ω ∗ , ∆ψ∧ be centres and radii of the wavelet ψ and its Fourier–Transform ψ ∧ , respectively. Then, the integral wavelet transformation (5) is bounded by the following time window [wint ] = [b + at∗ − 2at∗ − 2a∆ψ , b + at∗ + 2a∆ψ ].

(3.92)

It means that the time localization with the window centre at b + at∗ and the window width 2a∆ψ takes place. Let us introduce the function η(ω) = ψ ∧ (ω + ω ∗ ), which is also a window-type function with the center in zero and with the radius ∆ψ∧ . Using the formula = /2π, one may find the integral wavelet transformation (5) for the Fourier–picture f ∧ of the following form  ∞ f ∧ (ω)eibω η(a(ω − ω ∗ /a))dω. (3.93) W (a, b) = |a|1/2 −∞

Omitting the meaning of the phase shift and constants, it is evident that transformation (3.93) yields the localized information about spectrum f ∧ (ω) of signal f (t) with the following frequency window [winω ] = [ω ∗ /a − ∆ψ∧ a, ω ∗ /a + ∆ψ∧ /a].

(3.94)

Frequency localization takes place in the vicinity of the window center at ω ∗ /a with the width of 2∆ψ∧ /a. Observe that the ratio of the central frequency and the window width (ω ∗ /a)/(2∆ψ∧ /a) = ω ∗ /(2∆ψ∧ )

(3.95)

do not depend on the position of the central frequency, whereas the frequency–time window [wint ] × [winω ], with the surface 4∆ψ ∆ψ∧ is compressing for a high central frequency ω ∗ /a and is stretching for a low central frequency [Fig. 3.17(a)]. In Fig. 3.17, localization in the frequency–time space using the Fourier [Fig. 3.17(b)] and Shannon [Fig. 3.17(c)] transforms is reported. It is known that FT of the series of data with uniform discretization regarding time ∆t cannot be more discretized regarding frequency using the principle of ∆ω = ∆t/2 (the Nyquist

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Fig. 3.17 Frequency–time localization of transformations: (a) wavelets, (b) Fourier harmonics, (c) Shannon functions.

frequency presents a particular case of the principle of indeterminability between time and frequency localization). Analogous limitations between time and frequency localization for the wavelet transform can be described with the ratio ∆t∆ω ≥ 1/(4π). Figure 3.17 shows that FT may localize a frequency well. Shannon’s transformation does not have the frequency localization. Wavelet transformation has a moveable window, localized around the chosen time instant and the window can be extended with the scale increase, which is the most suitable property while getting a spectral information. Let us compare the wavelet transformation (3.83) with a shorttime FT  (3.96) F (ω, b) = f (t)z(t − b)eiωt dt, where the signal f (t) is multiplied by the window-type function z. Therefore, F (ω, b) stands for the signal development of the family of functions z(t − b)eiωt , composed of a unique function z(t) with the help of shifts ω in a frequency domain. On the other hand, the wavelet transform W (a, b) is the signal development of the family ψ((t−b)/a) composed of the unique function ψ(t) with the help of shift b in time and extension a also in time. Wavelet transformation presents a continous set of window FTs with different windows associated

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with each of the frequencies. Therefore, basis functions of the window FT possess the same resolution in time and frequency domains (z(t), z ∧ (ω)) for all points of the transformation plane, whereas the basis functions of the wavelet transformation exhibit a decreased resolution ψ(t/a) regarding scale a or increased resolution regarding ψ ∧ (aω) with respect to the frequency. The mentioned property gives remarkable advantage while analyzing signals, since the high-frequency characteristics are usually well localized, and slowly changing characteristics require application of a good low frequency resolution. It is evident that the so far described wavelet advantages are useful also while solving differential equations. 3.2.7

Basic functions

We have used “wavelet” to describe a certain soliton-like function without a rigorous definition. We have also introduced a few fundamental notions and we have described their certain properties. It is difficult so far to achieve a unique and commonly accepted definition for the wavelet. It should be mentioned that the majority of the applied constraints on wavelets deals with a requirement to have the inversed transformation or a reconstruction formula. In what follows, we describe the wavelet features. For a proper application, it is required to know the properties of a function called wavelet. We formulate and discuss the wavelet features, and then we consider a few well-known functions and their correspondence to the previously given features. (i) Localization Wavelet transform, contrary to FT, uses a localized basic function. A wavelet should be localized in both time and frequency domain. (ii) Zeroth average



∞

ψ(t)dt = 0.

(3.97)

−∞

In many cases, it is necessary that not only zero-order moment but also all first m-moments should be equal to zero (wavelet

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of the mth order):



∞

tm ψ(t)dt = 0.

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(3.98)

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−∞

Those wavelets allow, ignoring the most regular polynomial signal components, to study small scale signal fluctuations as well as highorder peculiarities. (iii) Boundaries



|ψ(t)|2 dt < ∞.

(3.99)

Requirement regarding good localization and boundaries of a wavelet can be written in the following way: |ψ(t)| < 1/(1 + |t|n ) or |ψ ∧ (ω)| < 1/(1 + |k − ω0 |n ), where ω0 denotes the dominating wavelet frequency, and n should be possibly large. (iv) Self-similarity A characteristic feature of the wavelet transform is its self-similarity: All wavelets of the family ψab (t) have the same number of oscillations as the basic wavelet ψ(t) since they are yielded by a basic wavelet via scale transformations and shifts. We illustrate the so far introduced background on the examples [Staszewski and Worden (1999)]. Among the illustrated functions there are wavelets (three last examples), and functions, which are not wavelets (three first examples). In Fig. 3.18, there are time-dependent functions (upper series) and their Fourier transforms. The δ-function being localized in t, space does not exhibit this property in k-space [Fig. 3.18(a)]. Sinus is well localized in k-space, but is not localized in t-space [Fig. 3.18(b)]. Gabor’s function [Fig. 3.18(c)] is defined through the formula G(t) = exp[iΩ(t − t0 ) − iv] exp[−(t − t0 )2 /2σ 2 ]/[σ(2π)1/2 ] and represents a modulated Gauss function with the following four parameters: shift t0 , the standard averaged squared deviation σ, frequency modulation Ω and the phase shift v. Development of the Gabor functions stands for the development on the piecewise modulated sinusoid. Since the piecewise length is constant for all frequencies, it gives different number of oscillations for different harmonics. Good localization observed in both t

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Fig. 3.18 Functions often applied (time dependence — first row, Fourier transform — second row): (a) Shannon wavelet, (b) Fourier wavelet, (c) Gabor wavelet, (d) HAAR-wavelet, and (e) LP-wavelet.

and k spaces of the Gabor functions does not have the self-similarity property. HAAR-wavelet [see Fig. 3.18(d)] presents an example of the orthogonal discrete wavelet with orthonormalized basis. The main disadvantage of this wavelet is the lack of smoothness. Sharp borders appear in t-space, which imply the occurrence of infinite tails (decreasing as k−1 ) in k-space as well as a lack of symmetry. However, these drawbacks do not play (sometimes) an important role, and in some applications this property can be treated as an advantage. Often a symmetric FHAT wavelet known as French hat is applied:  1, |t| ≤ 1/3,  ψ(t) = −1/2, 1/3 < |t| ≤ 1, ψ ∧ (k) = 3H(k)(sin k/k−sin 3k/3k),  0, |t| > 1, (3.100) where H(k) denotes the Heaviside function. In Fig. 3.18(e), the Littlewood–Paley (LP) FHAT wavelet exhibits irregularities in time domain and it is not sufficiently decreasing fast in the frequency domain. On the contrary, the LP wavelet has clearly exhibited boundaries in Fourier space, and it is rather badly decreasing in time space. Therefore, these two wavelets can be treated as limiting ones, where in between more suitable wavelets exist.

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We give now a few examples of wavelet-shaped functions. Since a wavelet transform is a scalar product of the analyzing wavelet and the analyzed signal, coefficients of the wavelet transformation (they are denoted by W (a, b)), where a is the scale coefficient and b is the shift parameter) consist of the combined information about the wavelet and the signal (similar to the case of coefficients of FT which hold information about a signal and a sinusoidal wave). A choice of the analyzing wavelet depends on which information should be deleted from a signal. Each wavelet possesses its own characteristic features in time and frequency spaces, and hence through application of different wavelets it is possible to detect various properties of the analyzed signal. Sometimes, the wavelet analysis is compared to a peculiar mathematical microscope. Namely, the shift parameter b controls the “mathematical microscope” focus position, the scale coefficient a is responsible for the magnitude, and finally a choice of the basic wavelet ψ is defined by the “optical microscope properties”. Real bases are often constructed using derivatives of the Gauss functions    ψm (t) = (−1)m ∂tm exp −|t|2 /2 , (3.101)    ∧ m 2 ψ m (k) = m(ik) exp −|k| /2 , where ∂tm = ∂ m [ ]/∂tm . Higher order derivatives have more zero order moments and allow to withdraw information about signal peculiarities of a higher order hidden in the signal. Figures 3.19(a) and 3.19(b) show wavelets obtained by m = 1 and m = 2, respectively. The first one is called the WAVE-wavelet, whereas the second one is named MHAT wavelet or “Mexican hat”. MHAT-wavelet having a narrow energy spectrum and two zero moments is particularly suitable to study complex signals. The Gauss functions allow to study the well-known DOG-wavelet property (difference of Gaussians)     ψ(t) = exp −|t|2 /2 − 0.5 exp −|t|2 /8 , (3.102)      ∧ 1 ψ (k) = (2π)− 2 exp −|k|2 /2 − exp −2|k|2 .

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Fig. 3.19 Examples of often used wavelets: (a) WAVE, (b) MHAT, (c) Morlet, b (d) Paul, (e) LMB, (f) Doubechies (ψ(t) — time dependent signal, ψ(k) — corresponding Fourier transform).

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Examples of complex wavelets are shown in Fig. 3.19 (their real components are reported). Mostly used complex basis is constructed on well localized k- and t-space Morlet wavelet. The formulas   ψ(t) = exp (ik0 t) exp −|t|2 /2 , (3.103)   ∧ ψ (k) = H(k) exp −[k − k0 ]2 /2 , report a plane wave modulated by a Gaussian of unit width. In Fig. 3.19(c), the Morlet wavelet is shown for k0 = 6. Increasing k0 implies an increase in the angle basis sensitivity, but the space sensitivity is getting worse. In quantum mechanics often the so-called Paul wavelet of the following form ψm (t) = [(m + 1)im /1 − it]m+1 , ∧

ψ m (k) = H(k)km exp (−k) ,

(3.104)

is applied, which is shown in Fig. 3.19 for m = 4. Increasing m implies an increase in zero moments of the wavelet. In Fig. 3.19, examples of wavelets are given, which are often used for construction of the orthogonal discrete bases of the form ψjk (t) = 2j/2 ψ(2j t − k)

(3.105)

with the norm ψjk 2 = ψ 2 = 1. Wavelet ψ ∈ L2 (R) is called orthogonal if the earlier defined family {ψjk } presents an orthonormalized basis of the functional space L2 (R): ψjk , ψlm δjl δkm and ∀f ∈

L2 (R)

(3.106)

can be recast in the following series form f (t) =

∞


Cjk ψjk (t).

(3.107)

j,k=−∞

The uniform convergence of the series in L2 (R) N2
N1
=0 C ψ f − lim jk jk M1 ,N1 ,M2 ,N2 →∞ −M2 −M1

(3.108)

2

is carried out with a help of the Mall procedure: LMB-wavelet and one of the Daubechies wavelets. The so far presented complex

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wavelets are progressive, since their Fourier coefficients are equal to zero. These wavelets guarantee time direction and they do not yield any wrong information between past and future. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

3.2.8

Properties and advantages of wavelets

Contrary to 1D FT giving 1D information on the relative contribution (amplitudes) of various time scales (frequencies), a 1D wavelettransform yields a 2D massive of the amplitudes of the wavelet transformation, i.e. values of the coefficient W (a, b). Position of their values in the space (a, b) = (time scale, time localization) yields information about the frequency evolution (similarly to the amplitudes spectrum or Fourier coefficients) and the coefficients of the wavelet transformation, frequency spectrum or scale–time spectrum or wavelet spectrum. (i) Presentation of the results Spectrum W (a, b) of a 1D signal presents a surface in a 3D space. There are various ways of visualization of this information. Besides the mentioned surface, projections onto the plane (a, b) are used with isoclines or color figures allowing for monitoring the wavelet transform amplitude on various scales in time, as well as curves of local extreme of these surfaces (scaleograms) yielding essential properties of the studied process. The term skeleton describes well the physical meaning of the curves with local extrema. In cases where it is necessary to show a wide interval of scales, it is recommended to use the coordinates (log a, b). (ii) Properties of the wavelet transforms Coefficients of the wavelet transform include the combined information on the analyzing wavelet and the analyzed signal. It should be emphasized that the wavelet transformation does not depend on the analyzing wavelet. Independence of the analyzed object plays an important role in studying various signal parameters/characteristics. In what follows, we list fundamental elementary properties of the wavelet transformation of the function f (t). The following notation is further used [Wψ f ](a, b) = W [f ] = W (a, b).

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(a) Linearity W [αf1 (t) + βf2 (t)] = αW [f1 ] + βW [f2 ] = αW1 (a, b) + βW2 (a, b). (3.109) Owing to this formula, the wavelet transformation of a vectorial function yields a vector with components where each of them undergoes a wavelet transformation separately. (b) Invariance regarding time shift W [f (t − b0 )] = W (a, b − b0 ).

(3.110)

This property implies commutations of the differentiation and in particular ∂t W [f ] = W [∂t f ], where ∂t = ∂/∂t . (c) Invariance regarding stretching (compressing) W [f (t/a0 )] = (1/a0 )W (a/a0 , b/a0 ).

(3.111)

This formula allows, in particular, to define an occurrence and a character of particularities of the analyzing function (see the next point). Besides the three mentioned elementary properties being independent of the choice of the analyzing wavelet-type, the wavelet transform possesses also a few more important properties listed below. 1. Frequency–time localization and occurrence of the frequency–time window and angle influence. 2. Differentiation W [∂tm f ]

 m

∞

= (−1)

−∞

f (t)∂tm [ψ ab (t)]dt.

(3.112)

Therefore, in order to integrate large scale polynomial components and analyze higher order peculiarities or small scale variations of the function f , the scale result is obtained when the analyzing wavelet/studied function is differentiated a few times. This property is of a particular importance assuming that the function f is represented by a series of numbers, whereas a wavelet is given by an explicit formula.

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3. Wavelet transform follows the Parseval’s theorem in the sense that the following formula is satisfied    (3.113) W1 (a, b)W 2 (a, b)dadb/a2 . f1 (t)f 2 (t)dt = Cψ−1 It means that the signal energy can be computed through amplitudes (coefficients) of the wavelet transform in a way similar to that of the Fourier amplitudes estimation   2 (3.114) E = f (t)dt = |A(ω) − iB(ω)|2 dω. Definitions and properties of a 1D continuous wavelet transformation are generalized into a multi-dimensional as well as discrete cases. Each of them possesses its own peculiarities. We omit here the detailed discussion regarding those phenomena, since we further use continuous wavelet transformation applied to 1D functions. 4. Advantages of the wavelet-based analysis. Having wavelet spectra, one may compute required characteristics of a studied process and analyze their properties. In what follows, we present possibilities given by the analysis of the signal peculiarities and its energetic characteristics. (a) Analysis of local regularity [Combes et al. (1989)]. We consider a few implications of the scale invariance 4. If f ∈ C m (t0 ), i.e. the analyzed function at point t0 is continuously differentiated up to an arbitrary order, then the coefficients of the wavelet transformation for t = t0 should satisfy the inequality W (a, t0 ) ≤ am+1 a1/2 for a → 0. Multiplier a1/2 appears, since due to the scale invariance 4 the skeleton properties of the function f should be studied on the basis of the L1 normalized coefficients. Recall that L1 and L2 are normalized coefficients coupled by a simple estimation W (a, t0 ) ≤ a−1/2 W (a, b). If f ∈ Λα (t0 ), i.e. the analyzing function belongs to a space of the Holder’s function with exponent α (it means that f is continuous though not necessarily differentiable in t0 , but such that |f (t + t0 ) − f (t)| = c|t0 |α , α < 1, c = const > 0), then coefficients of its wavelet transform for t = t0 should satisfy the relation W (a, t0 ) ≈ caα a1/2 for a → 0.

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Wavelet transformation is constructed in a way that W (a, t) is regular even if f (t) is irregular. The whole information of possible peculiarities f (t) is hidden in the asymptotic coefficients of W (a, t0 ) for small a (localization t0 , intensity c, exponent α). If the coefficients of small scales are divergent, then f function has a singularity, where the latter is characterized by the α coefficient governing the slope between log |W (a, t0 )| and log a at point t0 . On the contrary, if these coefficients are close to zero in the neighborhood of t0 in small scales, then function f is regular at point t0 . The so far described property is often and successfully applied while analyzing fractal and multi-fractal signals [Arneodo et al. (1988); Holschneider (1988)]. Typical property of the fractal sets is their asymptotic self-similarity. Namely, considering f in vicinity of the point t0 with various amplification on each scales we see practically the same: f (λt+λt0 )−f (λt) ≈ λa(t0 ) (f (t+t0 )−f (t)). The basis of the transformation is self-similar. One may easily show the coefficients of the transformation are also scaled with the same exponent as the analyzing function W (λa, t0 + λb) ≈ λa(t0 ) W (a, t0 ). This allows to get the scaling exponent α(t0 ), which is associated with the fractal set dimension. Analysis of the multi-fractal set allows to define the spectrum of exponents and the spectrum of dimensions. Let us note that the analysis of a local regularity is universal, since it does not depend on a wavelet choice. (b) Energetic characteristics [Combes et al. (1989)]: We consider some implications of the property 3. Analog of Parseval’s formula implies that in a space of real function, full energy of the signal f can be presented via amplitudes of the wavelet transformation in the   following form Ef =

f 2 (t)dt = Cψ−1

W 2 (a, b)dadb/a2 .

(3.115)

Density of the signal energy EW (a, b) = W 2 (a, b) characterizes the energetic levels (levels of excitation) of the investigated signal f (t) in the space (a, b) = (scale, time). (iii) Local energy spectrum One of the fundamental differences of the wavelet-transform in comparison to other approaches is the possibility of getting the localized

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characteristics and studying local properties of the processes. Knowing the density of energy EW (a, b) one may define local energy density at point b0 (or t0 ) with the help of the window  (3.116) Eξ (a, t0 ) = EW (a, b)ξ((b − t0 )/a)db. Window  function ξ supports the interval around t0 and satisfies formula ξ(b)db = 1. If as ξ a Dirac function is taken, then the local energy spectrum takes the form Eδ (a, t0 ) = W 2 (a, t0 ).

(3.117)

The latter characteristic allows to study dynamics of energy transition of the process along scales, i.e. energy exchange between process components of different scales in an arbitrary time instant. (iv) Global energy spectrum Full energy distributed along scales is associated with the global spectrum of energy of the coefficients  (3.118) EW (a) = W 2 (a, b)db. The global energy spectrum is also referred to as a scalogram or wavelet variance. Expressing energy spectrum EW (a) via signal energy spectrum in the Fourier space EF (ω) = |f ∧ (ω)|2 , we get  (3.119) EW (a) = EF (ω)|ψ ∧ (aω)|2 dω. One may conclude that the wavelet spectrum of the signal energy corresponds to the Fourier spectrum of the signal energy being smoothened on each scale by the Fourier spectrum of the analyzing spectra. Signal energy is defined via energy spectrum in the following way  (3.120) Ef = Cψ−1 EW (a)da/a2 . Therefore, Ef is proportional to the surface under the curve EW (a)/a2 , and the scalogram exhibits a relative contribution of different scales into full energy and presents energy distribution of the process over scales. The analyzed function has a time energy,

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whereas the analyzed wavelet has an average zero value, and hence the energy spectrum EW (a) should tend to zero on both scale ends and should have at least one maximum. Localization of the similar maxima (peaks) of the Fourier spectrum EF (ω) is usually associated with frequencies and the corresponding characteristic modes of the analyzed signal carrying out the fundamental energy of the process. Maximum values of the energetic spectrum EW (a) can be interpreted in a similar way: they define the fundamental contribution into full energy Ef . In further chapters of this book, we use a few advantages of the wavelet-based analysis. Owing to the mentioned multi-resolution properties, the wavelets allow for better signal representation. In particular, they do not exhibit problems related to the so-called frequency–time resolution trade-off. On the contrary, they do so to the often applied short-time Fourier transform. The wavelet multi-resolutional properties yield large temporal supports for lower frequencies keeping short temporal widths for higher frequencies. Finally, the further-applied wavelets allow to follow the frequency evolutions in time including their birth and death.

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Chapter 4

Simple Chaotic Models This chapter presents a short study of simple autonomous and nonautonomous systems exhibiting strange chaotic attractors. In particular, the Lorenz and R¨ ossler equations are discussed and the geometrical properties of the chaotic attractors governed by these equations are described. Other examples of autonomous chaotic systems include modified generators with inertial nonlinearity, Chua circuit, systems with quadratic nonlinearities, labyrynth chaos, jerk equations, two-scroll and three-scroll attractors, and Rikitake chaotic attractor. Examples of simple non-autonomous systems generating chaos include forced van der Pol equation, Rayleigh equation, Duffing oscillator and single-well oscillator, and externally and parametrically excited oscillators. The background given in this chapter is useful while exploring the next chapters since many of the presented and discussed chaotic attractors can also be found in the systems with infinite dimension. 4.1

Introduction

Nowadays, it is well known that numerous dynamical systems modelling phenomena met in our universe are essentially nonlinear and coupled, and cannot be presented and explained by traditional methods of mathematical analysis. It is impossible to obtain the algebraic description for solutions to the mentioned problems in closed forms, even though infinite series as well as special functions are applied. However, desirable success can be achieved with the help of numerical methods. 86

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In many cases met in practice, the so-called transitional processes are characterized by non-stationary and nonlinear phenomena, including occurrence of large displacements of a studied system and a complex and rich mechanism of the interaction between coupled subsystems, energy pumping and dissipation. They include, for instance, problems of fluid dynamics, such as the trans-sound phenomena associated with the moving body and its interaction with the flow, and other. All of the so far described phenomena may exhibit turbulent behavior. This is why solutions to the so far mentioned problems require computers with very large memories, and with the possibility of realizing a few billions of operations per second. The following three directions of research are associated with the qualitative analysis of nonlinear dynamical systems: (i) Rigorous mathematical proofs and analyses of properties of hyperbolic and parabolic systems. (ii) Construction of mathematical models of dynamics aimed at purely scientific investigations, and their qualitative analysis. (iii) Construction and investigation of simple archetypical models exhibiting fundamental properties of chaotic systems. The last direction of investigation attracted a lot of interest by numerous researchers, what yielded the fundamental results in the field of deterministic chaos. 4.2 4.2.1

Autonomous Systems Lorenz mathematical model (LMM)

In what follows, we give a remarkable example demonstrating chaotic dynamics. In 1959, E. Lorenz began his numerical investigation of some meteorological problems including modeling and numerical simulations of convective flows in the atmosphere. The LMM has been published in 1963, i.e. before the definition of a strange attractor [Lorenz (1963)]. Lorenz aimed at a weather forecast for a longer period. Then, about thirty years later he came back to his accidently discovered sensitive dependence on initial conditions of the derived

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Fig. 4.1

The Lorenz set (a photo from [www.giport.ru]).

differential equations describing atmospheric convection at the Massachusetts Institute of Technology (MIT) (see [Lorenz (1991, 1993)]). The Lorenz strange attractor is shown in Fig. 4.1. Let us consider a fluid layer of constant thickness H, subjected to action of the temperature gradient ∆T . If all motions are parallel to the plane (x − z) and they are homogenous in direction of y, then the governing equations take the following form δψ δ δΨ δ δθ δ (∇2 ψ) = (∇2 ψ) − (∇2 ψ + ν∇2 (∇2 ψ)) + gα , δt δz δx δx δz δx δθ δψ δθ δψ ∆T δψ δθ = − + k∇2 θ + , δt δz δx δx δz H δx

(4.1)

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where ψ is the function of the 2D flow, i.e. the velocity (u = (u, w)) is defined by the following formulas δψ δψ , W = − , (4.2) δz δx where θ is the temperature field characterizing deviation from the equilibrium state; g is the Earth acceleration, α is the coefficient of the temperature extension; ν is the kinematic viscosity, k is the heat transfer coefficient. A solution to this problem is known since Rayleigh, and it reads
πz 
πz 
παx 
παx  sin , θ = θ0 cos sin . (4.3) ψ = ψ0 sin H H H H The given solution starts to increase if the Rayleigh number Rα = gαH 3 ∆T exceeds the following critical value γk

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u=

3

π 4 (1 − α2 ) = , (4.4) α2 27π 1 = 657.511 for α2 = . (4.5) min Rα+ = 4 2 One may solve system (4.1) in higher order approximations, when instead of the simple Rayleigh approximation (4.3), the following one is applied ∞  ∞
nπx 
mπαx   sin , ψmn (t) sin ψ(x, z, t) = H H m=1 n=1 (4.6) ∞  ∞
nπx 
mπαx   sin . θmn (t) cos θ(x, z, t) = H H m=1 n=1 Rα+

In this case, periodic boundary conditions on both directions are formulated. Substituting (4.6) into (4.1), we get an infinite number of ordinary differential equations (ODEs). Lorenz [Lorenz (1963)] considered the following strong truncation of the problem: ψ11 = X, θ11 = Y and θ02 = Z. In this case, with the help of scaling transformations, the input system of equations has been reduced to three ODEs of the form: .

x = σ(y − z),

.

y = −xz + rx − y,

.

z = xy − bz,

(4.7)

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where σ = ν/k is the Prandtl number, r = Rα /Ra+ is the Rayleigh normalized number, b = 4(1 + a2 ) is the geometric factor (time variable is rescaled to the form τ = π 2 (1 + a2 )kt/H 2 ). Equations (4.7) are known as Lorenz equations. We have the following variables: x is the convection intensity, y is the temperature difference between input and output flows, z is the deviation of the vertical temperature profile with respect to the linear one. In what follows, we report a few properties of equations (4.7). 1. Divergence .

.

.

∂y ∂z ∂x + + = −(b + σ + 1) (4.8) D= ∂x ∂y ∂z is negative, since b > 0, σ > 0 . Let us denote the volume element of the phase space by Γ(t), and hence a flow compression can be presented in the following form Γ(t) = Γ(0)e−(b+σ+1)t .

(4.9)

It is observed that all trajectories are bounded by a certain limiting manifold. 2. Critical points Condition x = y˙ = z˙ = 0 is satisfied by the following points: (a) x = y = z = 0— pure heat transfer without convection; (b) X = Y = ± b(r − 1), Z = (r − 1) — stationary convection, which is possible for r > 1. 3. Stability We recast the linearized equations associated with (4.7) to the following matrix form      δX −σ σ 0 δX d  (4.10) δY  =  (r − z) −1 −x   δY  . dt δZ y x −b δZ Linear ODEs (4.10) allow to get some conclusions regarding stability of critical points:

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(a) Point (X, Y, Z) = (0, 0, 0) is stable for r < 1, i.e. all eigenvalues have negative real part; for r > 1, a real part of one of the eigenvalues becomes positive, i.e. the critical point becomes unstable and hence the infinite small perturbation may imply convection. Note that the stability of a critical point depends only on the Rayleigh number.   (b) Point (X, Y, Z) = (± b(r − 1); ± b(r − 1); r − 1) for r > 1 exhibits eigenvalues consisting of one real negative root and a pair of complex conjugated roots. This pair of critical points loses its stability for r =

σ(σ + b + 1) . σ−b−1

(4.11)

If r > 0, the mentioned conditions are satisfied only in the case if σ > (b + 1). Remark 1 It occurs that the stability of the studied critical points does not depend only on the Rayleigh number. Lorenz [Lorenz (1963)] chose the following parameters: b = 8/3, σ = 10. They yielded stability loss of the convective state for r = 470/19 ≈ 24.74 . . . , D = −13.67. Let us divide {r} into four intervals: (1) 0 > r > 1, (2) 1 < r < 24.74, (3) r ∼ = 24.74, (4) r > 24.74, and let us analyze a solution of Lorenz equations. 1. r ∈ (0; 1). All trajectories associated with arbitrarily chosen initial conditions move along a spiral into the coordinates origin, i.e. we have a globally attractive stationary solution. 2. r ∈ (1; 24.74). Origin loses its stability via bifurcation and is transformed into stationary   a pair of locally attracting   solutions: c1 ( b(r − 1), b(r − 1), r − 1), c2 (− b(r − 1), − b(r − 1), r − 1). All trajectories tend either to point c1 or c2 , possibly besides a set of trajectories of zero measure remaining in the origin neighborhood. 3. r ≈ 24.74. We deal with a critical value, for which c1 and c2 lose their stability, but c1 and c2 do not tend to the limit cycles, since for large r an inversed bifurcation takes place.

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Fig. 4.2

Solution to Lorenz equations for r = 28, σ = 10, b = 8/3, z = 27.

Fig. 4.3

Power spectrum of the Lorenz equations.

4. r > 24.74. The motion is of an essential disorder. It evolves in a spiral manner in the neighborhood of one of the fixed points (c1 and c2 ) within a certain time interval and then jumps into the neighborhood of the second fixed point in an unpredictible manner, and so on. This implies occurence of stretching and folding and creation of a complex manifold called a strange chaotic attractor. Typical trajectories of this attractor are reported in Fig. 4.2, whereas the associated power spectrum is shown in Fig. 4.3.

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Fig. 4.4

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Time history of the Lorenz equations.

Lorenz carried out long-term computations and stopped his numerical experiment. He observed a peculiar numerical behavior. Assuming that the computations will be repeated with exactly the same initial conditions, the numerical results should also be repeated. Lorenz slightly changed initial conditions, decreasing a number of less meaningful digits. Errors introduced by those changes were small. In the beginning, the obtained results coincided with the previous ones. However, the increase in the computational time yielded a new solution completely different from the previous one (Fig. 4.4). Lorenz repeated the computations many times to be sure of the obtained results. However, what has been later recognized, he experienced the high solution sensitivity to the introduced initial conditions, i.e. the fundamental property of chaotic dynamics. The obtained dependence is referred to as a butterfly effect, which has been illustrated and discussed in his work published in 1972 with the provoking title “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” [Lorenz (1972)]. It means that the long-term weather forecast may fail due to the sensitivity of the initial conditions (small changes may result in qualitatively different response).

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Remark 2 1. For r ≈ 24.74, the dynamics becomes chaotic, but a sequence of the associated events implies chaos does not include any periodic regimes, i.e. it does not coincide with the Ruelle–Takens scenario initiating the occurence of a turbulent flow [Ruelle and Takens (1971)]. 2. For r = 145−148, the strange chaotic attractor is transformed into a periodic limit cycle. 3. Further increase in r forces the limit cycle to vanish, and again strange attractor is born, but for r = 210−234 the occurence of another limit cycle is observed. 4. While moving between a limit cycle and a strange attractor, new phenomena have been observed referred as intermittency of various types. 5. Since the studied Lorenz system is truncated, another challenging research direction concerns the increase in the number of modes in equations (4.6) to be taken into account during numerical simulations. Remark 3 1. One of the main disadvantages of the Lorenz analysis relies on the rigorously introduced truncation of modes. Therefore, it is of interest to know what happens when the number of modes increases. The increase in the number of terms in (4.6) by one implies a series of different regimes. Though it was possible to find a strange chaotic attractor, the most sensitive parameter has been associated with a number of modes. This means that truncation of PDEs by keeping only a few modes cannot guarantee a real system behaviour. 2. In spite of the criticism introduced so far, simple models such as the Lorenz model are of interest since they may exhibit very rich nonlinear dynamics. 3. An important role is played also by the investigation of convergence of the solution series (4.1).

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4. Another challenging direction of investigations of PDEs is associated with the proof of existence of attractors of finite dimensions. Investigation of the Lorenz equations opened a challenging approach aimed at finding an even simpler set of ODEs exhibiting chaotic strange attractors. Namely, it has been observed that rescaling the Lorenz Equations (4.7) in the following way: (x, y, z) → (σx, σy, σz + r), t → t/σ, and taking into account r, σ → ∞ while r ∗ = br/σ 2 remains finite, the so-called diffusionless Lorenz systems have been obtained x˙ = y − x,

z˙ = xy − r ∗ ,

y˙ = −xz,

(4.12)

where for wide range of only one parameter r ∗ a strange chaotic attractor has been detected (for instance, for r ∗ = 1 the largest Lyapunov exponent λmax = 0.21). 4.2.2

R¨ ossler mathematical model (RMM)

In 1976, R¨ ossler [R¨ossler (1976)] proposed the model governing a chemical reaction of the following form .

x = −y − z,

.

y = x + ay,

.

z = b − cz + xz,

(4.13)

where a, b, c are the system parameters. Lorenz in book [Lorenz (1993)] has pointed out that O. R¨ ossler has discovered a simpler system of differential equations with chaotic solutions. The system (4.13) is recast in the following way: first equation is differentiated with respect to time, and after removing the second equation we get: ..

.

x − ax + (1 + z)x = (a + c)z − b,

.

z = b − cz + xz.

(4.14)

In this case, the R¨ossler system can be interpreted as an oscillator with a parametric and external excitation, and their input depends on the value of the amplitude of oscillations. Increase in the parameter c yields a series of period doubling bifurcations, and then a strange chaotic attractor appears (Fig. 4.5). After achieving a critical point, a series of bifurcations matching stripes of a chaotic attractor

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 4.5 (a) Trajectories of the R¨ ossler equations projected into the (x, y) plane and the corresponding power spectra a = b = 0.1, c = 2.6, (b) c = 3.5, (c) c = 4.1, (d) c = 4.18, (e) c = 4.21, (f) c = 4.23, (g) c = 4.3, (h) c = 4.6.

appeared. The R¨ossler system allows for an approximation of a 3D flow into one-dimensional map [Lichtenberg and Lieberman (1984)], which is defined by the strong contraction of the phase volume into one of the eigenvectors. Apparently, R¨ ossler detected a simpler system exhibiting chaotic dynamics governed by the following equations [R¨ ossler (1979)]: x˙ = −y − z,

y˙ = x,

z˙ = a(y − y 2 ) − bz.

(4.15)

The R¨ossler prototype-4 type (4.15) displays strange chaotic attractors for a = b = 0.5 (in this case, λmax = 0.094).

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Modified generator with inertial nonlinearity (MGIN)

4.2.3

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97

The MGIN is described by the following equation [Anishchenko et al. (1999)] (similar to these have already been discussed in the first and second sections of this chapter): .

x = mx + y − xz,

.

y = −x,

.

z = −gz + zΦ(x),

(4.16)

where Φ(x) = x2 (for x ≥ 0) and Φ(x) = 0 (for x < 0), m, g are the system parameters, and m is proportional to a difference between the input and output energies, g is the relative relaxation time of a thermistor. The given model is based on the classical schemes of generators, and in asymptotic cases it describes the Van der Pol generator. In a classical generator with an internal nonlinearity [Teodorchik (1964)], the self-oscillations are guaranteed by introducing a thermo-resistor into the oscillation contour, whose properties depend on the electrical loop current. Model of the generator with the inertial nonlinearity illustrates various mechanisms of occurrence of the deterministic chaos in the system with a homoclinic trajectory of the saddle-focus separatrix type. Figure 4.6 presents (a) phase portrait, (b) time history, and (c) power spectrum characterizing the period doubling scenario. It has been verified experimentally that in all dynamical systems where chaos occurs through the period doubling bifurcations, a chaotic attractor has a fractal dimension 2 < d < 3, and its crosssection has a horseshoe shape. Chua circuit

4.2.4

Mathematical model of the Chua generator [Chua (1992); Chua et al. (1982, 1986)] is more complex than described in Section 4.2.2, since we have three equilibria, a symmetry and more complex types of the trajectories. Equations governing system dynamics follow .

x = α[y − h(x)],

.

y = x − y + z,

.

z = −βy − γz, (4.17)

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Fig. 4.6 (a) Series of period doubling bifurcations: phase portrait, (b) time history, (c) power spectrum.

where: h(x) = bx + 0.5(a − b)(|x + 1| − |x − 1|), and a, b, α, β are the non-dimensional system coefficients. They can be recast to the following form of equations α. .. . . z + z + βz = βx, x = − z − αh(x). (4.18) β In the system (4.18), owing to the symmetry, a doubled loop of the saddle-focus separatrix is constructed which is responsible for the

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Simple Chaotic Models

Fig. 4.7

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Chaotic “double scroll” attractor (a = 0.1, b = 0.1, α = 15, β = 23).

occurrence of a more complex type of the chaotic attractor, and is called a “doubled scroll” attractor. Its projection is shown in Fig. 4.7. Chua circuit presents not only typical properties, being similar to the generator with MGIN model and to the R¨ ossler model, but also exhibits a series of specific properties due to the symmetry. In the reference [Awrejcewicz and Calvisi (2002)] the proposals to build mechanical models/prototypes of the Chua circuit are given. A gallery of Chua’s type attractors has been reported in the book by Bilotta and Pantano [Bilotta and Pantano (2008)], where nearly about 900 strange attractors have been detected, illustrated and discussed. The original Chua’s system (4.17) containing five parameters α, β, γ, a, b can be further simplified to get a system with only one parameter of the form x˙ = ay − x + (x + 1) − (x − 1),

y˙ = z − x,

z˙ = y.

(4.19)

The system (4.19) possesses a strange chaotic attractor for a = 0.5 (λmax = 0.11). 4.2.5

Conservative and other systems

The autonomous conservative systems in the language of mechanics refer to the lack of damping and gyroscopic forces. Systems without

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damping and friction are rather unlikely to be found in nature, since they are indifferent, i.e. neither attractors nor repellers. In this and the next section, we do not present figures of strange chaotic attractors. On the contrary, we present only the differential equations and the values of associated parameters responsible for the occurence of a chaotic dynamics. A 3D autonomous system with quadratic nonlinearities governed by the following first order ODEs x˙ = y,

y˙ = yz − x,

z˙ = α − y 2 ,

(4.20)

exhibits chaos for α = 1 (λmax = 0.014), and it has been studied by Nos´e and Hoover [Hoover (1995); Nos´e (1991)]. Thomas [Thomas (1999)] has discovered the so-called labyrinth chaos governed by the following ODEs x˙ = sin y,

y˙ = sin z,

z˙ = sin x,

(4.21)

where the state space is partitioned into 3D cells, and a trajectory wanders between the cells via a pseudorandom way. The word...jerk defines the derivative of acceleration and it is → → denoted by − r , where − r is the position of a particle of mass m. In England, the word jolt is used instead of the word jerk. It is recommended to control the jerk of public transportation vehicles to avoid discomfort to passengers (it should be less than 2 ms−3 ) or to avoid damage to transported fragile objects, like eggs. In addition, in the aerospace industry a sensor measuring jerk is used, and it is named a jerkometer. Since the equation ... → − → → → ¨r , − r = f (− r˙ , − r) (4.22) can be projected onto one of the introduced rectangular co-ordinates, i.e. x, y or z, it yields a third-order differential ODE which can exhibit chaotic dynamics. Though a general approach to transform any three first-order ODEs to one “jerk” equation can be found elsewhere, one may observe that both Lorenz [Eq. (4.23)] and R¨ ossler [Eq. (4.24)] systems have their counter part jerk forms as follows ... x + (1 − σ + b − x/x)¨ ˙ x + [b(1 + σ + x2 ) − (1 + σ)x/x] ˙ x˙ − bσ(r − 1 − x2 )x = 0,

(4.23)

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... y + (c − a)¨ y + (1 − ac)y˙ + cy − b

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− (y˙ − ay)(¨ y − ay˙ + y) = 0.

(4.24)

The simplest quadratic jerk equation yielding chaos has been proposed by Sprott [Sprott (1997)], and it has the following form ... x = −a¨ (4.25) x + x˙ 2 − x. For a = 2.02, it gives a chaotic trajectory x(t), and its largest Lyapunov exponent λmax = 0.05. The simplest cubic case of the jerk equation has been presented by Malasoma [Malasoma (2000)], and it reads ... x = −a¨ (4.26) x + xx˙ 2 − x, which exhibits chaos for a = 2.03 (λmax = 0.08). Another jerk equation studied by Malasoma [Malasoma (2002)] has the following form ... x = −α(x + x ¨) + xx˙ + x¨ ˙ x/x, (4.27) which has been yielded by the systems y˙ = −αy + z,

x˙ = z,

z˙ = −x + xy.

(4.28)

The system governed by (4.27) exhibits chaos for 10.28 < α < 10.37, and its Kaplan–Yorke dimension Dky = 2.003. Linz and Sprott [Linz and Sprott (1999)] presented the chaotic attractor of the jerk equation ... x = −a¨ x − x˙ + |x| − 1, (4.29) with an absolute value nonlinearity. For a = 0.6, the estimated λmax = 0.036. The already illustrated Lorenz attractor (Fig. 4.7) is known as a two-scroll attractor. The three-scroll system governed by equations x˙ = x − yz,

y˙ = −y + xz,

z˙ = −3z + xy,

(4.30)

possesses also a chaotic attractor of λmax = 0.378. Finally, a fourscroll attractor can be exhibited by the following ODEs x˙ = x − yz,

y˙ = x − y + xz,

z˙ = −3z + xy,

where the estimated largest λmax = 0.248.

(4.31)

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The last example of three autonomous first-order ODEs exhibiting chaos deals with the so-called Rikitake dynamics [Rikitake (1958)]. The ODEs are as follows Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

x˙ = −µx + yz,

y˙ = −µy + x(z − α),

z˙ = 1 − xy,

(4.32)

where the parameter α is responsible for the difference in the angular velocities of two disks, and the parameter α displays resistive dissipation. For α = µ = 1, Eq. (4.32) exhibits the two-scroll attractor with λmax = 0.13. 4.3

Non-Autonomous Systems

This section is devoted to simple nonlinear non-autonomous systems, which in the language of mechanics are simple stationary or nonstationary (parametric) oscillators usually harmonically excited. We begin with the van der Pol externally excited oscillator governed by the second-order non-homogeneous differential equation of the form x ¨ + b(x2 − 1)x˙ + x = F sin ωt.

(4.33)

This equation has been studied by Cartwright and Littlewood [Cartwright and Littlewood (1945)], Levinson [Levinson (1949)], and more recently by Levi [Levi (1981)]. Strange attractor yielded by Eq. (4.33) can be obtained for b = 1, F = 1, ω = 0.45, and the largest Lyapunov exponent λmax = 0.04. Another equivalent example of a strange chaotic attractor is exhibited by the Rayleigh equation x ¨ + (x˙ 2 − b)x˙ + x = F sin ωt

(4.34)

for fixed b = 4, F = 5, ω = 4 (λmax = 0.15). One of the simplest second-order non-autonomous ODEs is that of the following form x ¨ + xx˙ 2 = sin ωt,

(4.35)

which exhibits chaos for ω = 4 (λmax = 0.014). Certainly, the non-autonomous Duffing equation [Duffing (1918)] belongs to the

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mostly revisited equations in mechanics, since numerous mechanical/physical systems can be modelled by it. It has the following form

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x ¨ + cx˙ + α1 x + α2 x3 = F sin ωt,

(4.36)

where chaos has been detected by Ueda [Ueda (1979)] and Moon and Holmes [Moon and Holmes (1979)]. In the mechanical language, the case α1 > 0, α2 > 0 corresponds to the stiffening elasticity of the system, whereas the case of α1 > 0, α2 < 0 describes the softening elasticity of the system. The latter one is also named as the Duffing two-well oscillator. The two-well oscillator with damping governed by the ODE x ¨ + x˙ − x + x3 = sin ωt,

(4.37)

displays a strange attractor for ω = 0.8 with λmax = 0.12. Another example governed by equation x ¨ + x3 = sin 2t

(4.38)

concerns a single-well oscillator exhibiting strange chaotic attractor with λmax = 0.09 [Gottlieb and Sprott (2001)]. Instead of a geometric-type nonlinearity one may take into account a piecewiselinear system of the form x ¨ + |x˙ − x| + x − 2 = sin t,

(4.39)

which possesses a strange chaotic attractor with λmax = 0.08. Two other examples deal with the so-called conservative signum oscillator x ¨ + sgnx = sin t

(4.40)

and the exponential oscillator x ¨ + x˙ + ex − 1 = 21 sin t,

(4.41)

where both of them have chaotic response. There also exists a documented research devoted to the analysis of simple oscillators parametrically and externally excited, governed by the following non-dimensional differential equation x ¨ + αx˙ + ω02 (1 + h cos(νt))x + βx2 + ξx3 = γ cos ωt,

(4.42)

and this case has been extensively studied both analytically and numerically by Belhaq and Houssni [Belhaq and Houssni (1999)],

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assuming that the difference between frequencies ω − ω0 is small. In particular, it has been shown how the original quasi-periodic system, i.e. the parametric and external excitations having incommensurable frequencies, can be reduced to a periodically driven system. Chaotic behavior of this oscillator has already been reported in reference [Nayfeh (1983)] for ξ = 0. The approximate analytical criteria of chaos occurrence have been proposed by Szempli´ nska–Stupnicka for ξ = 0 [Szempli´ nska– Stupnicka et al. (1989)]. Luo [Luo (2004)] gave analytical predictions of resonant separatrix bands and studied chaotic oscillations of the Mathieu–Duffing oscillator with a twin-well potential. Shen et al. [Shen et al. (2008)] investigated analytically the bifurcation route to chaos also in the Mathieu–Duffing oscillator. Luo and Yu [Luo and Yu (2014)] studied the case of Eq. (4.42) for ξ = 0, γ = 0. A route from periodicity to chaos has been illustrated and discussed via harmonic amplitudes varying with excitation amplitude in the finite term Fourier series solution. The case of equation (4.42) for h < 0, β = 0 for negative and positive spring constants putting emphasis to MEMS applications has been revisited in reference [Jin et al. (2014)]. Both Melnikov and Galerkin methods as well as the computation of the maximum Lyapunov exponents have been used to unveil the controllability of chaotic vibrations of the system driven by parametric pumps. The Melnikov analytical and numerical techniques have been applied to study chaotic motions of the Duffing–Van der Pol oscillator with external and parametric excitations governed by the following equation x ¨ + px(1 ˙ − x2 ) − αx + βx3 = f (1 + δx) cos ωt,

(4.43)

in reference [Zhou and Chen (2014)]. The prediction of chaotic dynamics based on the Melnikov approach has been verified numerically.

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Chapter 5

Discrete and Continuous Dissipative Systems 5.1

Introduction

In equations governing chaotic dynamics the dissipative properties of a system are usually represented as resistance forces proportional to a velocity. It is well known that there exist various models taking into account the dissipative properties of mechanical systems. It is known from an undergraduate course of theory of vibrations that there are linear and nonlinear resistance forces, and the nonlinear dependence can be approximated by various analytical formulas. However, when a body is cyclically deformed, a certain violation of the Hook principle may occur, which is represented by the hysteresis loop. The surface bounded by a hysteresis loop defines energy lost per one cycle of vibration in a unit material volume. It has been proved that the hysteretic loop surface, for majority of the construction materials, practically does not depend on the deformation frequency, but depends on the deformation amplitude. We briefly revisit typical dissipation processes which are widely met while studying vibrations of the mechanical systems [Panovko (1991)].

105

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5.2

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Deterministic Chaos in One-Dimensional Continuous Systems

Linear Friction

In order to study a 1-DOF system with a linear friction (damping), one may apply the following Lagrange equation
 ∂T ∂Π d ∂T − =− + Q∗ , (5.1) dt ∂ q˙ ∂q ∂q where Q∗ is the generalized force of a linear friction. In order to define it, we assume that in each point of the system we deal with linear dissipation Ri = −βi vi ,

(5.2)

where βi is the friction coefficient. Since a general force n  ∂ri , Ri Q= ∂q

(5.3)

i=1

and since ∂vi ∂ri = , ∂q ∂ q˙ we have Q∗ = −

n  i=1

(5.4)

 ∂ri ∂vi =− . βi vi βi vi ∂q ∂ q˙ n

(5.5)

i=1

From the following formula 1 ∂ 1 ∂νi2 ∂vi = (vi vi ) = , vi ∂ q˙ 2 ∂ q˙ 2 ∂ q˙ we get n n  ∂  βi νi2 βi ∂νi2 vi =− . Q∗ = − 2 ∂ q˙ ∂ q˙ 2 i=1

(5.6)

(5.7)

i=1

The calculated sum Φ=

n  βi ν 2 i

i=1

2

,

(5.8)

formally coincides with the kinetic energy and its associated dissipative Rayleigh function. We transform (5.8) into a more compact form 1 (5.9) Φ = bq 2 , 2 where b is the generalized coefficient of a viscosity.

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Finally, we get the following formula for the generalized friction force/viscous damping force ∂Φ = −bq. ˙ (5.10) Q∗ = − ∂ q˙ Since the kinetic energy T = 12 aq 2 and the potential energy Π = 1 2 2 cq , the Lagrange Equation (5.1) takes the following form

dq d2 q + b + cq = 0. (5.11) 2 dt dt For relatively small values of the viscous damping coefficient, √ when b < 2 ac, the general solution to the second order differential Equation (5.11) is a

q = e−ht (C1 sin k∗ t + C2 cos k∗ t),

(5.12)

where

 b , k∗ = h2 − k2 , (5.13) 2a and constants C1 and C2 are defined by the following initial condition dq0 dq (0) = , (5.14) q(0) = q0 , dt dt which means that dq0 + hq0 , C2 = q0 . (5.15) C1 = dt k∗ Another solution form follows h=

q = Ae−ht sin(k∗ t + α), where

(5.16)



√ 2 0 ( dq q0 h2 − k2 dt + hq0 ) 2 + q0 , tan α = . (5.17) A= h2 − k2 q˙0 + hq0 As it can be seen from (5.12) and (5.16), the motion can be viewed as the damped vibration with a constant frequency, but with successively decreasing amplitudes, and the full process is characterized by a monotonous amplitude dissipation. Envelopes of the damped oscillations are described by the function A = ±A0 e−ht , where A0 stands for the initial envelope coordinate.

(5.18)

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Frequency of the free damped vibrations is defined by the formula √  b2 − 4ac 2 2 , (5.19) k∗ = h − k = 2a Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

and time length of one cycle is T∗ =

2π 4πa . =√ 2 k∗ b − 4ac

(5.20)

Since the influence of the damping force on the eigenfrequency of the oscillating process is small, k∗ ≈ k, T∗ ≈ T . Sequence of maximal deviations fits the geometric progression rule, since owing to (5.18) the ratio of two successive maximum deviations A(t) and A(t + T∗ ), separated by time interval T∗ stands for a constant value equal to ehT∗ . Natural logarithm of that ratio is called the logarithmic decrement, which is defined as follows 2πb πb ≈√ . Λ = hT∗ = √ 2 ac 4ac − b

(5.21)

The logarithmic decrement measures a way of damping of the free vibrations. For essentially large values of the damping coefficient, when √ b > 2 ac, the general solution to (5.11) instead of (5.12), takes the following form q = C1 es1 t + C2 es2 t , where s1,2 =

−b ±

(5.22)

√

b2 − 4ac . 2a

(5.23)

Constants of integration are expressed through the initial conditions and take the following form C1 =

−C2 es2 t + q0 , es1 t

C2 =

0 s1 t −s2 q0 + dq dt e . s2 es2 t es1 t − s1 es2 t

(5.24)

Motion governed by (5.22) does not have vibration character: for arbitrary initial conditions, the values q and q˙ asymptotically tend to zero.

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√ The case when b = 2 ac (critical damping), the solution to the differential Equation (5.11) is as follows

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q = e−kt [C1 e−kt + tC2 e−kt ],

(5.25)

−kt , C = ( dq0 )/e−kt . 0 where C1 = (q0 − t dq 2 dt )/e dt

5.3

Nonlinear Friction

In the case when amplitudes decrease is different from that of the so far illustrated geometric progression, we deal with a nonlinear friction. Nonlinear dependence of the friction forces versus velocity can be approximated by different analytical formulas. We assume that the generalized friction force Q∗ is proportional to the nth power of velocity, whereas the power exponent n = 1 depends on the actual properties of the friction force. Thus, dependence can be cast to the following form ˙ n−1 q. Q∗ = −b|q| The governing fundamental equation takes the form  n−1  ∂q  ∂q ∂2q + cq = 0. a 2 + b  ∂t ∂t ∂t

(5.26)

(5.27)

Exact solution of the nonlinear Equation (5.27) in the form of elementary functions is not known, and hence in order to find q = q(t) one may apply various types of approximating methods. (a) Method of energy balance Let the solution being sought be close to harmonic one and be characterized by the frequency k, corresponding to the conservative system without friction. Now, considering an arbitrary vibration cycle and beginning time measurement with time instant where the deflection achieves maximum, the motion is described by the function q = A(t) cos kt,

(5.28)

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where A(t) is the slowly changing function in time, i.e. AT  A, A  Ak. Then in the generalized velocity

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q = −Ak sin kt + A cos kt,

(5.29)

one may omit the second term, and hence q = −Ak sin kt.

(5.30)

Owing to (5.26), the following generalized friction force is defined as Q∗ = −b(Ak)n |sin kt|m−1 sin kt.

(5.31)

Work of the friction force in the considered cycle is  T  T n+1 Q∗ qdt ˙ = −bk [A |sin kt| ]m−1 dt. U= 0

(5.32)

0

We may assume here approximately that in the considered period the quantity A is constant, hence  T /4  π/2 m+1 m+1 m+1 n sin ktdt = −4bA k sinm+1 ϕ dϕ. U = −4b(Ak) 0

0

(5.33) The integral appearing in (5.33) will be denoted by I, and it is approximated by the Gamma function in the following way:   π/2 2n−2 m2 Γ n2 m+1 . (5.34) sin ϕ dϕ = I= m(m + 1)Γ(m) 0 Now, owing to formula (5.34), one may compute following values versus the exponent m (see Table 5.1). It is easy to find that U = −4bAkm+1 km+1 I(n).

(5.35)

Formula (5.35) presents the system energy change within the considered cycle. Since at the beginning and at the end of the considered Table 5.1

m versus I [see (5.34)].

m

0

0.5

1.0

1.5

2.0

2.5

3.0

I

1.000

7/8 = 0.875

π/4 = 0.785

0.718

2/3 = 0.667

0.624

0.589

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cycle the kinetic energy is equal to zero, the change of the full system energy is defined by the change of the potential energy, i.e. it is necessary to take into account A(0) and A(T ). At the beginning of the cycle, we have Π(0) =

1 2 cA (0), 2

(5.36)

whereas at the end, we have 1 2 cA (T ). (5.37) 2 Consequently, the increase (negative) in the system potential energy is Π(T ) =

1 1 c[A2 (T ) − A2 (0)] = c[A(T ) + A(0)][A(T ) − A(0)]. (5.38) 2 2 After a few transformations, the following finite difference equation is obtained as ∆Π =

∆A = −

4b(Ak)m I(n) . c

(5.39)

This equation matches the amplitude increase (negative) per one cycle with the amplitude value at the cycle beginning, i.e. it defines the shape of the upper envelope. Considering this envelope as a continuous curve governed by time function A = A(t), the following approximating formula holds ∆A = T

2π dA dA = . dt k dt

(5.40)

Therefore, the equation of infinite differences (5.39) takes the form of the following differential equation for the envelope: 2bkm+1 I(m) n dA =− A . (5.41) dt πc Integrating this equation requires considering two cases: m = 1 and m = 1. In the case m = 1 (linear damping), owing to the definition in Table 5.1 I = π/4, Eq. (5.41) takes the following form dA = −hA, dt

(5.42)

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where b bk2 = . 2c 2a A solution to the linear Equation (5.49) is as follows Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

h=

A = A0 e−ht ,

(5.43)

(5.44)

where A0 stands for the initial coordinate of the envelope. Therefore, for m = 1 we get the previous exact result (5.44). Although this matching holds only for the envelope (due to the difference between k and k∗ the graphs would be different), it supports the idea of application of the method of energetic balance. In the case m = 1 Equation (5.41) is nonlinear, but we may find its solution, since the variables can be separated. 2bkm+1 I(m) dA dt. (5.45) = − Am πc Integrating (5.45) and taking into account the initial condition A(0) = A0 , the following dependence is derived A0

. (5.46) A= m−1 2b(m−1)k m+1 I(m)Am−1 0 1+ t πc A specific form of this formula depends on m. In the case, when m = 2 (squared friction), formula (5.46) yields A0 , (5.47) A= 4bk 3 A 1 + 3πc 0 t which means that the envelope is described by a hyperbola. Applying solution (5.46) we may also get the envelope for another important case, when m = 0. Owing to (5.26), this case is associated with the value q˙ (5.48) Q∗ = −b , |q| ˙ defining the Coulomb friction, the volume of which does not depend on the velocity magnitude. Substituting m = 0 into the general solution (5.46), we get 2bk t, (5.49) A = A0 − πc

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i.e. amplitudes decrease via a linear rule and the amplitudes fit the arithmetic progression. The latter result corresponds also to the exact solution. For m = 1 the ratio of two neighborhood largest deviations is not constant. It means that the logarithmic decrement will depend on the amplitude Λ = ln

Ai , Ai+1

(5.50)

where i is the number of the considered cycle. If, as it has been assumed earlier, the difference ∆Ai = Ai+1 − Ai is small in comparison to Ai , then
 ∆Ai ∆A Ai+1 − ∆Ai . (5.51) = ln 1 − ≈− Λ = ln Ai+1 Ai+1 A Substituting (6.5) into the last formula, we find the dependence of the logarithmic decrement versus the amplitude of the form 4bkm I(m) m−1 A . (5.52) c Consequently, only for m = 1 the logarithmic decrement does not depend on the vibration amplitude and is constant within the vibration regime. For m = 2, in the process of damped vibrations, the logarithmic decrement decreases simultaneously with the amplitude decrease, whereas for m = 0 (Coulomb friction), it increases while the amplitude decreases. Λ=

(b) Method of slowly changing amplitudes This approximating method has been proposed by Van der Pol for a wide class of systems with weak nonlinearity, when the differential equation can be presented in the following form 
∂q ∂2q 2 , (5.53) + k0 q = f q, ∂t2 ∂t where f (q, ∂q ∂t ) is the function containing relatively small nonlinear terms. Solution to the differential Equation (5.49) takes the following

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form

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q = A cos(k0 t − ϕ),

(5.54)

where it is assumed that A and ϕ are functions of time. Depending on the properties of the introduced functions A(t) and ϕ(t), formula (5.54) may fit well or wrong with the harmonic vibration of frequency k0 . For constant A and ϕ formula (5.54) fits the harmonic vibration in the exact manner. In the case when A and ϕ are “almost constant”, i.e. they exhibit slow changes in time, formula (5.54) describes vibrations with slowly changing amplitude and phase. This behaviour is typical for system dynamics with a weak nonlinearity. If we substitute formula (5.54) into the fundamental Equation (5.53), we get an equation with two unknown functions A and ϕ. In order to carry out properly the change of one function q by two functions A and ϕ, we need to add one more additional relation between them. Van der Pol proposed the following one ∂ϕ ∂A cos(k0 t − ϕ) + A sin(k0 t − ϕ) = 0. ∂t ∂t

(5.55)

If one integrates formula (5.53) then, taking into account (5.55), the following simple formula for velocity is obtained: ∂q = −k0 A sin(k0 t − ϕ), ∂t

(5.56)

which is similar to that of constant A and ϕ. Therefore, formulas for accelerations will be relatively simple and they will not contain the 2 ∂2ϕ second-order derivative ∂∂tA 2 and ∂t2 : ∂A ∂ϕ ∂2q k0 sin(k0 t − ϕ) − Ak02 cos(k0 t − ϕ) + Ak0 cos(k0 t − ϕ). =− ∂t2 ∂t ∂t (5.57) Substituting formulas (5.54)–(5.56) into the given Equation (5.53), we get the following first-order differential equation −

∂ϕ ∂A k0 sin ψ + Ak0 cos ψ = f [A cos ψ − Ak0 sin ψ], ∂t ∂t

where ψ = k0 t − ϕ.

(5.58)

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Relations (5.55) and (5.58) yield the following derivatives:

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1 ∂A = − f [A cos ψ − Ak0 sin ψ] sin ψ, ∂t k0 1 ∂ϕ = f [A cos ψ − Ak0 sin ψ] cos ψ. ∂t Ak0

(5.59)

Assuming that the studied system is close to the linear one, we may suspect that A and ϕ are not able to get relatively large increase ∂ϕ per one cycle 2π/k0 and that derivatives ∂A ∂t and ∂t are constant within a period of an arbitrary but one cycle. Therefore, although the derivatives are expressed by rather complicated nonlinear functions (5.59), we do not introduce a large error by taking their averaged values over the period 2π/k0 :  2π 1 ∂A =− f (A cos ψ − Ak0 sin ψ) sin ψdψ, ∂t 2πk0 0  2π 1 ∂ϕ = f (A cos ψ − Ak0 sin ψ) cos ψdψ. ∂t 2πAk0 0

(5.60)

Note that carrying the integration in (5.58) we keep A as a constant value. In fact, this averaging procedure plays a key role in the described method of slowly changing amplitudes. Equation (5.60) can be rewritten in a more compact way Φ(A) ∂A = , ∂t 2πk0

Ψ(A) ∂ϕ = , ∂t 2πAk0

(5.61)

(short Van der Pol equation), where  Φ(A) = −  Ψ(A) =

2π

0 2π 0

f (A cos ψ − Ak0 sin ψ) sin ψdψ, (5.62)

f (A cos ψ − Ak0 sin ψ) cos ψdψ.

Therefore, first of all, the integrals (5.62) should be computed assuming that A is constant. Next, differential Equations (5.61) are integrated, but without an assumption of constant A.

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Fig. 5.1

5.4

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Hysteretic loop sketch.

Hysteretic Friction

When cycling deformation of the elastic bodies takes place, even for small stresses the Hook’s law is violated. In Fig. 5.1, a hysteresis loop is presented in the following coordinates: stress σ versus deformation ε. The surface located in the inside of the loop defines energy amount per one vibration cycle measured in a unit material volume. Since a distance between loop branches are usually small, getting exact hysteretic form via laboratory experiment is difficult. It has been found that the hysteretic surface of majority of design materials does not depend on the frequency of the deformation process but rather on the deformation amplitude, which is expressed by the following formula (5.63) Ω = αAm+1 , where α and m are constants defined via experiments. This dependence differs principally from the formula (5.35), where the power exponent of order m + 1 appears. Although formula (5.35) also includes frequency k, it does not depend on the coefficient α in (5.63). In order to define a pattern describing the damped vibration with the hysteretic friction, the equation of energetic balance will be applied. Namely, we compare the dissipated energy (it is taken with the minus sign) and energy increase governed by (5.38) per one cycle (period): (5.64) −αAm+1 = cA∆A. This approach yields the equation in the form of finite differences α (5.65) ∆A = − Am , c

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which, as in (5.39), can be recast to the following differential equation

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αk m dA =− A . dt 2πc

(5.66)

Integration of Eq. (5.66) under the initial condition A(0) = A0 yields A=

A0



m−1

1+

α(m−1)kAm−1 0 2πc

t.

(5.67)

Note that for the hysteretic friction one may get the exponential dependence (if m = 1), which is typical for the case of linear viscous friction. Observe also that the Coulomb friction can be treated as the particular case of the nonlinear friction (5.26), but also as the particular case of the applied fundamental formula (5.66). In both aforementioned cases, it is characterized by the value m = 0. 5.5

Impact Damping

In certain engineering systems, a key role of energy dissipation is played by frequent impacts instead of the so far discussed continuous action of friction forces. Let us, for simplicity, consider the case when impacts take place in time instants corresponding to the system transitions through the equilibrium position assuming that the sudden system energy loss is proportional to the system energy before the impact. In this case, the instantaneous energy loss can be measured through the system velocity before the impact Ω = bν 2 ,

(5.68)

where b is a certain constant coefficient of mass dimension. Let us consider a half-cycle of the vibration, which begins with the largest deviation A(0). During the first quarter of the cycle, the system moves with the constant energy 12 cA2 (0), and at the end of this quarter-cycle we have ν 2 = Ac A2 (0). Then, an impact takes place implying a sudden energy loss of the value of (5.68). Next, the system begins to move with the

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 2b 1− , a

(5.69)

which is constant within the second quarter-cycle of vibration. Therefore, at the time instant corresponding to the end of this quartercycle, the system potential energy is equal to that of (5.69):
 cA2 (0) 2b cA2 (T /2) = 1− . (5.70) 2 2 a Thus, we can define a ratio of displacements at the beginning and the end of the first half-cycle as follows 1 A(0) . = A(T /2) 1 − 2b/a Similarly, for the second half-cycle we obtain 1 A(0) . = A(T ) 1 − 2b/a

(5.71)

(5.72)

Comparison of the largest displacements A(0) and A(T ) yields 1 A(0) . = A(T ) 1 − 2b/a

(5.73)

In other words, the ratio of successive largest displacements is constant. Therefore, the envelope of the curve of the damped vibrations is characterized by the following exponent A = A0 e−ht ,

(5.74)

which corresponds to the logarithmic decrement of the following form 1 . (5.75) Λ = hT = ln 1 − 2b/a For small ratio 2b/a, we get Λ ≈ 2b/a. 5.6

(5.76)

Damping in Continuous 1D Systems

In this section, we consider only linear continuous systems. Since internal processes of energy loss of vibrating continuous 1D systems

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are very complex and require deep modeling including atomic structural patterns of a material, there is no hope, at least in the coming future, that a proper modeling of energy loss will be achieved. 5.6.1

Free vibrations

Let us apply a classical approach. Stress–strain relation of a linear continuous one-dimensional structural member (string, rod, beam) has the following form 
∂ε , (5.77) σ =E ε+µ ∂f where E is Young’s modulus, µ = E  /E, E  stands for viscous damping coefficient, σ(ε) is the stress (strain) function in an arbitrary cross-section of the analyzed mechanical object. Considering a rod, its longitudinal force is derived using the following form 
∂2u ∂u +µ , (5.78) Sσ = SE ∂x ∂x∂t where u(x, t) is the longitudinal rod displacement and S denotes the area of the rod cross-section. Dynamical equilibrium configuration of the rod infinitely small element defined by intersection of two parallel cross-sections governed by coordinates x and x + dx yields the following equation 
 ∂u ∂2u ∂ ∂2u ES + , (5.79) ρS 2 = ∂t ∂x ∂x ∂x∂t where ρ is the rod material density. Assuming that ES = const we arrive at the following PDE  2 ∂3u ∂2u 2 ∂ u , (5.80) = a + µ ∂t2 ∂x2 ∂ 2 u∂t where a2 ≡ a2r = E/ρ. The same equation is derived for the case of either transversal string vibrations or rotational rod vibrations (in this case µ = G /G, where G is the shear modulus, and G stands for the viscous damping associated with a shear processes, but with different coefficient a).

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The following form of solution of (5.80) is assumed

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u(x, t) = U (x)T (t), and hence we get T¨(t)U (x) = a2 [T (t)U  (x) + µT˙ (t)U  (x)] = 0,

(5.81)

(5.82)

or equivalently U  (x) T¨(t) = a2 = −α2 , U (x) µT˙ (t) + T (t)

(5.83)

where the constant value has been denoted by −α2 (here and further  = d/dx, ˙ = d/dt). The problem has been reduced to solve the following separated two second-order differential equations U  (x) +

α2 U (x) = 0, a2

T¨(t) + α2 µT˙ (t) + α2 T (t) = 0.

(5.84) (5.85)

Solutions of these equations are widely described in the books devoted to vibrations, and more recently in [Osi´ nski (1998)] (see also Section 4.2). In order to solve the problem uniquely, we need to define boundary and initial conditions. The introduced boundary conditions allow to define the associated infinite number of eigenfunctions and the corresponding eigenvalues αn , n = 1, ..., ∞ [see (5.84)]. Equation (5.85), after substitution of T (t) = ert , yields the following characteristic equation r 2 + α2n µr + α2n = 0,

(5.86)

µα2 r1,2 = − n + αn λn , 2

µ2 α2n − 1. λn = 4

(5.87)

and hence

2 2

The damped vibrations occur, when µ 4αn < 1, and  

−µα2 µ2 α2n µ2 α2n n t + Bn sin 1 − t e 2 t. Tn (t) = An cos 1 − 4 4 (5.88)

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The upper critical damping yields the following solution Tn (t) = (An ch λn t + Bn sh λn t) e

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In the case of critical damping

(µ2 α2n

−µα2 nt 2

.

(5.89)

= 4), we get

Tn (t) = (An + Bn t) e

−µα2 nt 2

.

(5.90)

It is worth mentioning that the damping depends on frequency αn , and by increasing n the term αn µ increases. It means that even for constant µ, the successive modes of vibrations are more strongly damped [the case governed by (5.88)]. In the remaining two cases [Eqs. (5.89) and (5.90)], the system movement is aperiodic, perhaps with the occurrence of only one vibration. Free vibrations of our mechanical object are described by the following infinite series ∞  u(x, t) = Tn (t)Un (x). (5.91) n=1

Assuming the following initial conditions u(x, 0) = u0 (x), v(x, 0) = v0 (x),

(5.92)

we get ∞  n=1 ∞ 

Tn (0)Un (x) = u0 (x), (5.93) T˙n (0)Un (x) = v0 (x).

n=1

Since the eigenfunctions (modes) are orthogonal and are defined by the boundary conditions, we obtain l u0 (x)Un (x)dx , Tn (0) = 0  l 2 0 Un (x)dx (5.94) l v (x)U (x)dx 0 n , T˙n (0) = 0  l 2 0 Un (x)dx where l stands for the length of the structural member. Equations (5.94) define the initial conditions for Eq. (5.85). Using one

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Fig. 5.2 dx.

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Forces Q(x, t) and torques M (x, t) action on the beam element of length

of Equations (5.88)–(5.90), depending on the damping, one finds the associated An , Bn . Let us consider beam vibrations now. In this simple model of transversal beam vibrations y(x, t) influence of transversal forces and influence of the rotational motion of beam cross-section are neglected. Using the introduced rectangular co-ordinates (axis OX is horizontal and it coincides with the beam middle line, whereas the axis OY goes down). Equation of motion of the beam element dx can be derived by taking into account Fig. 5.2. Projection of the forces onto axis OX yields ρSdx

∂ 2 y(x, t) = −Q(x, t) + Q(x, t) + dQ(x, t), ∂t2

(5.95)

where Q(x, t) =

∂M (x, t) . ∂x

(5.96)

In the above, E is the longitudinal elasticity modulus, I is the moment of inertia of the beam cross-section regarding the middle beam axis, and R denotes the radius of the beam curvature. In what follows, we assume that the beam material is made of a viscous and elastic material, and the stress–strain relation in the beam fibers has the following form 
∂ε , (5.97) σ =E ε+µ ∂t and µ = E  /E, where E  is the viscous damping coefficient.

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Since

 1 ∂ y M =− +µ , EI R ∂t R

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(5.98)

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for a small beam deflection we have ∂ 2 y(x, t) 1 = . R(x, t) ∂x2 We finally derive the bending torque 
2 ∂ 3 y(x, t) ∂ y(x, t) , +µ M (x, t) = −EI ∂x2 ∂x2 ∂t

(5.99)

(5.100)

and the transversal force

 ∂ ∂ 2 y(x, t) ∂ 3 y(x, t) ∂M = −EI , +µ Q(x, t) = ∂x ∂x ∂x2 ∂x2 ∂t

and the increment of the transversal force  ∂ ∂ 2 y(x, t) ∂ 3 y(x, t) ∂Q dx = −EI 2 dx. +µ dQ = ∂x ∂x ∂x2 ∂x2 ∂t

(5.101)

(5.102)

Taking into account (5.102) in (5.95), we get 2 ∂ 5 y(x, t) ∂ 4 y(x, t) 2 ∂ y(x, t) + a + µ = 0, b ∂x4 ∂x4 ∂t ∂t2

(5.103)

where a2b = EI/ρS. Equation (5.103) governs free beam transversal vibrations with viscous internal damping. Assuming the solution in the form (5.81), we get   (5.104) U IV (x) T (t) + µT˙ (t) + a2b U (x)T¨(t) = 0, and hence −

T¨(t) U IV (x) = −k4 . = a2b U (x) T (t) + µT˙ (t)

(5.105)

Finally, the problem is reduced to the study of the following two independent linear ODEs U IV (x) − k4 U (x) = 0, T¨ + µα2 T˙ (t) + α2 T (t) = 0,

(5.106) (5.107)

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where α2 = k4 /a2b . It should be emphasized that Eq. (5.106) defines the eigenfunctions (modes) and they do not depend on the internal viscous damping. We solve linear ODE (5.106), i.e. assuming U (x) = erx we obtain the following characteristic equation r 4 − k4 = 0,

(5.108)

and hence r1 = k, r2 = −k, r3 = ik, r4 = −ik. It means that a general solution of equation (5.106) is U (x) = C1 ekx + C2 e−kx + C3 eikx + C4 e−ikx

(5.109)

or equivalently U (x) = A sin kx + B cos kx + C sh kx + D ch kx,

(5.110)

where four constant values A, B, C, D are defined via the boundary conditions. We restrict our further analysis only to the case of free-free beam support. For the beam of length l the boundary conditions follow y(0, t) = 0,  ∂ 2 y(x, t)  ∂x2 

= 0, x=0

y(l, t) = 0,  ∂ 2 y(x, t)  = 0. ∂x2 x=l

(5.111) (5.112)

Substituting (5.81), (5.110) into (5.111), (5.112) yields B + D = 0,

−Bk2 + Dk2 = 0,

A sin kl + B cos kl + C sh kl + D ch kl = 0,

(5.113)

−Ak2 sin kl − Bk2 cos kl + Ck2 sh kl + Dk2 ch kl = 0. The characteristic equation is defined by     0 1 0 1   2 2   0 k 0 −k =0   sin kl cos kl sh kl ch kl   −k2 sin kl −k2 cos kl k2 sh kl k2 ch kl

(5.114)

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or equivalently k4 sh kl sin kl = 0.

(5.115)

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Since sh kl = 0 for kl = 0, we get sin kl = 0,

(5.116)

which means that kn l = nπ,

n = 1, 2, . . . .

(5.117)

Each kn defines Un (x) = An sin kn x + Bn cos kn x + Cn sh kn x + Dn ch kn x. (5.118) Each of the infinite number of eigenfunctions should satisfy the boundary conditions. Substitution of (5.117) into algebraic Equations (5.113) yields Dn = 0, Bn = 0, sin kn l sin nπ An = − An = 0, Cn = − sh kn l sh nπ

(5.119)

x Un (x) = An sin kn x = sin nπ . l

(5.120)

and hence

Equation (5.107) is the same as Eq. (5.86), and its solution has been studied earlier. 5.6.2

Excited vibrations

In this section, we consider 1D structural members (strings, rods, beams) subjected to external load action. The load q = q(x, t) is continuously distributed along a structural member length per its unit length and depends on time. In the case of a longitudinally vibrating rod, the load (force) is distributed along its length continuously and in parallel to the rod axis x, whereas in the case of the rod vibrating torsionally the torque is distributed along its length. In the case of the string transversal vibrations, the load (torque) is continuously distributed along its length, and is perpendicular to the string axis.

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One may show, proceeding in the way described in Section 5.6.1, that in all cases the governing equation has the following form  2 ∂ 3 u(x, t) ∂ 2 u(x, t) 2 ∂ u(x, t) −a +µ = bq(x, t), (5.121) ∂t2 ∂x2 ∂t∂x2 where b = 1/(ρS). In the case of a string a2 = T /(ρS), (T is the string tension, ρ is the string material density, S is the area of string cross-section). In the case of the rod longitudinal (torsional) vibrations a2 = E/ρ, b = 1/(ρS) (a2 = G/ρ, b = 1/(ρI0 )), where G is the shear modulus, I0 is the moment of inertia of the rod cross-section, and ES = const, GI0 = const. The studied PDE Eq. (5.121) is linear and non-homogeneous. Its solution consists of a sum of a general solution of the homogeneous PDE (q = 0) and a particular solution of the non-homogeneous PDE. We have already shown how to find the general solution to the homogeneous equation and how to determine constants satisfying boundary and initial conditions. In the case of a particular solution to the non-homogeneous equation it should satisfy the boundary conditions, whereas the initial conditions follow u(x, 0) = 0,  ∂u(x, t)  = 0. ∂t t=0

(5.122)

We apply the following approximation q(x, t) =

∞ 

Qn (t)Un (x),

(5.123)

n=1

and the solution is assumed to be of the following form u(x, t) =

∞  n=1

ξn (t)Un (x).

(5.124)

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In order to find Qn (t), we multiply both sides of Eq. (5.123) by Um (x), and compute an integral from 0 to l to get  l  l ∞  q(x, t)Um (x)dx = Qn (t) Un (x)Um (x)dx. (5.125) 0

0

n=1

Since Un and Um are mutually orthogonal, then  l 1 q(x, t)Un (x)dx, Qn (t) = 2 γn 0  l Un (x)dx. γn2 =

(5.126)

0

We substitute (5.123) and (5.124) into (5.121) and we obtain ∞ 

ξ¨n (t)Un (x)−

n=1

∞ 

a2 [ξn (t)Un (x)+µξ˙n (t)Un (x)] = b

n=1

∞  n=1

Qn (t)Un (x), (5.127)

which yields U  (x) ξ¨n (t) − bQn (t) = a2 n = −ωn2 . Un (x) ξn (t) + µξ˙n (t)

(5.128)

Finally, we separate time and space dependent functions to get Un (x) +

ωn2 Un (x) = 0, a2

ξ¨n (t) + µωn2 ξ˙n (t) + ωn2 ξn (t) = bQn (t).

(5.129) (5.130)

Equation (5.130) governs oscillations of damped non-autonomous linear oscillators. Taking into account our earlier considerations corresponding to (5.130), we have the following solutions. (i) Undercritical damping (4 > ωn2 µ2 )  t 1 2 1 Qn (τ )e− 2 ωn µ(t−τ ) sin cn (t − τ )dτ, ξn (t) = bcn 0

ω2 µ cn = ωn 1 − n . 4

(5.131)

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(ii) Supercritical damping (4 < ωn2 µ2 )  t 1 2 1 Qn (τ )e− 2 ωn µ(t−τ ) sh cs (t − τ )dτ, ξn (t) = bcs 0 (5.132)

ωn2 µ − 1. cs = ωn 4 (iii) Critical damping  1 2 1 t (5.133) Qn (τ )e− 2 ωn µ(t−τ ) (t − τ )dτ. ξn (t) = b 0 Finally, the equation governing transversal beam vibrations, where the beam is continuously transversally loaded by the load q = q(x, t) per beam unit length, has the following form [see Eq. (5.103)] ∂ 5 y(x, t) q(x, t) ∂ 2 y(x, t) ∂ 4 y(x, t) + a2b . (5.134) +µ = 4 4 ∂x ∂x ∂t ∂t2 EI Substituting ∞  ξn (t)Un (x), (5.135) y(x, t) = n=1

q(x, t) =

∞ 

Qn (t)Un (x),

(5.136)

n=1

into (5.134) we get ξn (t)UnIV (x) + µξ˙n (t)UnIV (x) + a2b ξ¨n (t)Un (x) = Qn (t)Un (x) (5.137) or equivalently 1 a2b ξ¨n (t) − ρF Qn (t) UnIV (x) = −kn4 . = (5.138) Un (x) ξn (t) + µξ˙n (t) Equations (5.138) yield two separated sets of ordinary differential equations

−

(5.139) UnIV (x) − kn4 Un (x) = 0, 1 Qn (t), (5.140) ξ¨n (t) + ωn2 ξ˙n (t) + ωn2 ξn (t) = ρS where ωn2 = kn4 /a2b , a2b = ρE/EI. Solutions of the obtained Eqs. (5.139) and (5.140) have already been discussed.

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Chapter 6

Euler–Bernoulli Beams This chapter is devoted to studying chaotic dynamics of the Euler– Bernoulli beams. At first, numerical approaches being further applied including both Benettin and Wolf algorithms, neural networks and classical and modified Gramm–Schmidt orthonormalization procedures are described and validated using the standard maps. Then the planar beams are investigated in a standard way, i.e. the governing PDEs are reduced to nonlinear ODEs, and the latter are analyzed using time histories, Fourier frequency spectra, phase curves, 3D beam deflections, as well as the first four Lyapunov exponents (LES) and Morlet wavelets. Numerous novel examples of beam chaotic and hyperchaotic dynamics are illustrated and discussed. Chaotic dynamics of linear (straight line) beams embedded in a stationary temperature field has been analyzed in Section 6.3. The beam governing PDEs with thermal effect are derived first and the associated mechanical boundary conditions (BCs) are defined. The 2D heat transfer PDE has been added, and four kinds of thermal (BCs) have been briefly discussed. A way of numerically solving the heat transfer PDEs via Finite Difference Method (FDM) has been illustrated and validated. Then the full problem governed by PDEs has been reduced to ODEs and solved numerically including the discussion of the reliability of the obtained results. Next, numerous novel examples of the beam thermal chaotic dynamics have been presented and discussed. A similar way of modeling and analysis has been employed to study chaotic dynamics of curvilinear planar beams embedded in stationary temperature field. In particular, the influence of BCs on beam chaotic dynamics and scenarios of transitions from regular to 129

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chaotic beam vibrations has been reported and discussed putting emphasis on thermal effects, including influence of heat boundary conditions (HBCs) and geometric beam parameters. Similar theoretical and numerical backgrounds have been used to study flexible curvilinear beams embedded in both temperature and electric fields (Section 6.5). Section 6.6 is aimed at investigating chaotic vibrations of the Euler–Bernoulli beams taking into account the elastic-plastic deformations. Contrary to previous sections, here chaotic vibrations of physically nonlinear beams with and without deflection constrains have been analyzed. The last section is focused on the analysis of multilayer beams with gaps, i.e. hybrid-type chaotic beams dynamics are studied including the geometric and physical beam nonlinearity. Both chaos and synchronization phenomena have been studied as well as the hybrid contact/no contact beams chaotic vibrations exhibiting the spatiotemporal distribution of the contact pressure have been analyzed. 6.1 6.1.1

Introduction On the generalized Benettin’s algorithm

We begin with consideration of the generalized Benettin’s algorithm to find LEs [Benettin et al. (1976)]. Let point x0 belong to the attractor A of a dynamical system. This point belongs to unperturbed (real) trajectory. We take a positive quantity ε essentially smaller than the attractor dimension. Further, we take an arbitrary perx0 − x0 || = ε. Now, turbed point x ˜0 in a way to satisfy the relation ||˜ ˜0 within the we consider the evolution of the chosen points x0 and x ˜1 , time interval T , and we denote the obtained points by x1 and x ˜1 − x1 will be called the excitation respectively. The vector ∆x1 = x vector. We aim to estimate the value of λ in the following way ˜ 1 = 1 ln ||∆x1 || . (6.1) λ T ε Time interval T can be chosen in a way that the excitation amplitude should be less than the linear dimensions of the nonhomogeneity properties of the phase space and the dimension of the

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attractor. Consider the normalized vector of the perturbation ∆x
1 = ε∆x1 /||∆x1 ||, and the associated new perturbation x˜
1 = x1 + ∆x
1 . We extend the so far described procedure by considering points x1 and x ˜1 instead of x0 and x ˜0 , respectively. Repeating the so far described procedure of M times, one may ˜l estimate λ as the averaged arithmetic value of the quantities λ obtained in each of the computational steps. 6.1.2

Wolf ’s algorithm

A concept of Lyapunov’s exponent belongs to the quantitative characteristics, allowing to estimate an order of the system chaotization. Number of LEs depends on a number of equations governing the studied system’s dynamics. Each of the equations corresponds to one of the LEs. If the LE is negative, then the associated dynamic process is periodic. If the LE is positive, then a chaotic component occurs in the trajectory of motion. The method proposed by Wolf et al. [Wolf et al. (1985)] allows to estimate the largest Lyapunov exponent via the choice of only one coordinate, and it is of particular use, when the system equations are not known, and the remaining phase coordinates are not known. Let us have a time series x(t), t = 1, . . . , N of a measurement at one coordinate of the chaotic process, carried out via equal time intervals. The method of mutual information defines the time delay τ , whereas via monitoring of the neighborhood events the embedding space dimension is measured. As a result of the construction carried out, the following set of points of the space Rm is obtained xi = (x(i), x(i − τ ), . . . , x(i − (m − 1) · τ )) = (x1 (i), x2 (i), . . . , xm (i)), (6.2) where i = ((m − 1)τ + 1), . . . , N . Let us choose a point x0 from the series (6.2). Considering the series (6.2), we find point x ˜0 , such that the following norm ||˜ x0 − x0 || = ε0 < ε is satisfied, where ε is the fixed quantity essentially smaller than the dimension of an attractor being reconstructed. ˜0 are isolated in time. Next, evoIt is necessary that points x0 and x lution of these points is traced on the reconstructed attractor, unless

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the distance between them achieves the given quantity εmax . Let us ˜1 , the distance between them denote the obtained points by x1 and x
by ε0 , and the time interval of the evolution by T1 . After that, we consider again the series (6.2) and we find the point
x
1 − x1 || = ε1 < ε x ˜1 , located closely to x1 such that the formula ||˜
˜1 − x1 should have, possibly, the same holds. Vectors x ˜1 − x1 and x ˜
1 . direction. The same procedure is repeated for points x1 and x Repeating the so far described procedure M times, the largest Lyapunov exponent can be estimated by the following formula λ∼ =

M −1


ln(ε
k /εk )/

k=0

6.1.3

M


Tk .

(6.3)

k=1

Neural network approach

In the work [Golovko (2005)], the method of LE computation via choice of only one monitored coordinate and using neural networks approach is proposed. The idea of the method relies on the computation (with the help of the neural mesh) of the divergence of two neighborhood trajectories into n steps ahead. The constructed mesh is shown in Fig. 6.1. Elements of the hidden layer of the neural mesh have the sigmoidal activation function shown in Fig. 6.2. It is a monotonously increasing and almost everywhere differentiable S-shape nonlinear function with saturation. Sigmoid allows to amplify the weak signals and resist

1

aij bi

2

xn

xn

Fig. 6.1

Single-layer neural network of direct distribution.

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Euler–Bernoulli Beams

Fig. 6.2

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Sigmoidal function of activation.

against saturation of the strong signals. As an example, a sigmoid function may serve the following logistic function: 1 , (6.4) σ(x) = 1 + exp(−tx) where t is the parameter defining the function curvature. For t → ∞, the function tends to a threshold function. For t = 0, the sigmoid becomes a constant function of the value 0.5. Area of the function values covers the interval [0, 1] and its derivative follows dσ(x) = tf (x)(1 − f (x)). (6.5) dx In this book, we propose a modification of the neural network method. Namely, our algorithm is based on the following steps: 1. 2. 3. 4. 5.

Computation with a proper choice of the time delay. Computation of dimension of the embedding space. Pseudo-phase reconstruction of a trajectory via time delays. Construction of approximations of the neural network. Teaching of the neural network to compute a successive vector of the series. 6. Computation of a spectrum using the generalized Benettin algorithm [Benettin et al. (1976, 1980)].

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In order to realize the proposed algorithm, the neural network should be constructed which can be classified by the following criterions: (i) Owing to the input information, the network is analog one (information is presented in the form of real numbers). (ii) The mesh is self-organized (output solutions are constructed on the basis of input solutions). (iii) Mesh exhibits a direct expansion due to the character of couplings (all couplings are directed from input to output neurons). (iv) Mesh has dynamic couplings due to the character of the adjusted synapses (in the teaching process the synaptic couplings are properly adjusted (dW/dt = 0), where W denotes the weight mesh coefficients). The scheme of the used neural network is presented in Fig. 6.3. There exists a hidden layer of networks, which has the hyperbolic tangent playing the key role in the activation function: exp(Ax) − exp(−Ax) . (6.6) htan(Ax) = exp(Ax) + exp(−Ax) A derivative of the hyperbolic tangent, similar to the logistic function, is expressed via a quadratic function. However, contrary to the

Fig. 6.3

Function of activation of a hyperbolic tangent.

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logistic function, values of the hyperbolic tangent are located in the interval of (−1; 1). It allows to achieve faster convergence, contrary to properties of a standard logistic function. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

6.1.4

Classical Gramm–Schmidt orthogonalization

Let there be linearly independent vectors a1 , . . . , aN . Vector of projection is defined in the following way: projb a =

a, b b, a, b

(6.7)

where a, b is the scalar product of vectors a and b. The operator projects vector a onto vector b in the orthogonal way. The process being applied is carried out in the following way (see Fig. 6.4): b1 = a1 , b2 = a2 − projb1 a2 , b3 = a3 − projb1 a3 − projb2 a3 , b4 = a4 − projb1 a4 − projb2 a4 − projb3 a4 , .. . N −1
projbj aN . bN = aN −

(6.8)

j=1

5

a3

b3 projb1a3

b1

4

1 2 projb2a3

b3

3

projb1a3 + projb2a3

b2 Fig. 6.4

Geometric representation of the classical Gramm–Schmidt process.

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Using vector bj (j = 1 . . . N ), the following normalized vectors b ej = ||bjj || are constructed. As a result of this process, either a set of orthogonal vectors b1 , . . . , bN or unit vectors e1 , . . . , eN is obtained. 6.1.5

Modified Gramm–Schmidt orthogonalization

The normalization procedure is carried out using the modified Gramm–Schmidt method. The applied procedure allows to construct, on the basis of linearly independent vectors a1 , . . . , aN , a set of orthogonal vectors b1 , . . . , bN or unit vectors e1 , . . . , eN such that each of them can be expressed via a linear combination of a1 , . . . , aN . Resultant vector bj is computed with the help of formulas (1)

aj = aj − projb1 aj , (2)

(1)

(6.9)

(1)

aj = aj − projb2 aj ,

(6.10)

.. . (j−2)

aj

(j−3)

− projbj−2 aj

(j−2)

− projbj−2 aj

= aj

bj = aj

(j−3)

,

(j−2)

.

(6.11) (6.12)

In formula (6.9), from vector aj , a component in direction of vec(1) tor b1 is removed. The obtained vector aj does not include this component, and hence it is orthogonal to vector b1 . In the next step, from formula (6.10), the vector component in (2) direction of vector b2 is removed. The obtained vector aj does not include the component in direction b2 and is orthogonal to vector b1 . As a result of further removing the appropriate components, vector bj is obtained, which does not include components in directions b1 , . . . , bj−1 being orthogonal to vectors b1 , . . . , bj−1 (see Fig. 6.5). 6.1.6

Results comparison

In order to compare the results of the method proposed in the reference [Golovko (2005)] with our method, we use the H´enon map [H´enon (1976)]. The H´enon’s transformation maps the point (Xn , Yn )

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Fig. 6.5

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Geometric representation of the modified Gramm–Schmidt process.

into a new one through the following operation Xn+1 = 1 − aXn2 + Yn ,

(6.13)

Yn+1 = bXn . Usually, the following standard parameters are taken a = 1.4, b = 0.3. Exact values of the LEs are: 0.418, 1.622, and they have been computed by Sprott [Rowlands and Sprott (1992)] with the help of the Benettin’s method. In order to compare both results 70 elements have been chosen, and the modeling includes one step. Results are obtained for 1,000 iterations of the network learning. The results reported in the reference [Golovko (2005)] are: largest Lyapunov exponent: 0.43; mean squared error: 5.92 · 10−5 . Our results yield LEs: 0.4259215243; −1.6230484826; mean squared error: 1.326 · 10−7 . 6.1.7

Hyperchaos

The developed numerical method has been applied to analyze the generalized hyperchaotic H´enon map [Bayer and Klein (1990)].

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Fig. 6.6

The LEs versus T (a) and M (b).

Table 6.1 Generalized H´enon’s map: signal (a), Morlet wavelet (b) and phase portrait (c).

The first two of the LEs are positive, which validates the hyperchaotic attractor of H´enon’s map. Our results yield: Kaplan–Yorke dimension: 2.15358; KS entropy: 0.42534; phase space compression: −2.34408 (see also Table 6.2).

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Table 6.2

139

Estimation of the LEs (H´enon’s generalized map).

Lyapunov’s exponent number

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First Second Third

Benettin’s method [H´enon (1976)] 0.276 0.257 −4.040

Our method 0.2204903426 0.2048540485 −2.7694287583

The governing equations have the following form: x1 (k + 1) = 1.76 − x22 (k) − 0.1x3 (k), x2 (k + 1) = x1 (k),

(6.14)

x3 (k + 1) = x2 (k). Spectrum of the LEs has been reported by Bayer and Klein in 1990 [Bayer and Klein (1990)]. In what follows, we briefly describe a few results from the area of electric engineering for circuit self-excited systems with aperiodic elements reported by Dmitriev and Kislova [Dmitriev and Kislova (1989)], where the development of strange attractors with an increase in super-critical parameter, and chaos–hyperchaos phenomena regarding the phase transitions have been illustrated and discussed. In this section, we consider transitions into chaos versus LEs monitoring. We study a transition from the strange chaotic attractor with one positive Lyapunov exponent into that with two positive LEs, i.e. a transition abbreviated as chaos–hyperchaos. Four LEs are computed, and their time evolution λi (T ) as well as the dependencies λi (M ), where M is a constant, are shown in Fig. 6.6 (T denotes time constant associated with the aperiodic elements). In Fig. 6.7, the development of the Poincar´e map versus the coefficient M has been reported. It should be mentioned that moving along the straight line T = 2 we follow a transition from chaos to hyperchaos associated with a sequence of bifurcations, and the transition from chaos–hyperchaos is exhibited by the qualitative change of the physical characteristics and processes.

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Fig. 6.7 Evolution of the Poincar´e map with the increase in M (a) resonance orbit (M = 6.05) (b) orbit before stability loss (M = 6.5) with one positive LE (M = 8) (c) beginning of the attractor reconstruction with second positive LE (M = 9.5) (d) beginning of strange attractor collapse (e),(f) development of the hyperchaos (M = 10(e), 15(f )).

The second LE becomes positive for T = 0.8, and then it starts to increase smoothly [Fig. 6.7(a)] which, in turn, is accompanied by a smooth Poincar´e map transformation which validates a weak transition from chaos to hyperchaos. Moving along the line T = 2, a transition from chaos to hyperchos is associated with a series of bifurcations, and it implies the qualitative change in physical characteristics and processes. 6.2 6.2.1

Planar Beams Hyperchaos

We apply the modified neural network method for Lyapunov spectrum computation while analyzing vibrations of the flexible

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q x

h

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l z Fig. 6.8

The studied flexible Euler–Bernoulli beam.

Euler–Bernoulli beams. Two-dimensional beam space defined by a rectangular coordinates has the form Ω = {(x, z) ∈ [0, l] × [−h/2, h/2]} (Fig. 6.8). The beam mathematical model is based on the following hypotheses: (i) Any transversal section being normal to the middle beam curve before deformation remains normal to the middle beam curve, 2 and the cross-section height is not changed: εx,i = −z ∂∂xw2 . (ii) Nonlinear dependence between deformations and displacement 1 ∂w 2 follows the K´ arm´ an form: εx = ∂u ∂x + 2 ( ∂x ) , where: u(x, t) — displacement of the beam middle line along axis x; w(x, t) — beam deflection. (iii) We take isotropic and elastic beam material: εzx = εx + εx,i . (iv) Normal stresses on layers parallel to the middle beam line are small in comparison to other stresses and they are neglected. Using the D’Alembert principle, we consider dynamic equilibrium state taking into account given forces, dynamic reactions, inertial forces and dissipative forces, and the following equations regarding displacements govern the beam element motion:   2 γ ∂2u ∂ u h + L (w, w) − = 0, (6.15) Eh 3 ∂x2 g ∂t2   2 4 γ ∂ 2 w γ ∂w h ∂ w +q = 0, + L1 (w, u) + L2 (w, w) − h 2 − hε Eh − 4 12 ∂x g ∂t g ∂t (6.16)

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Table 6.3 Time histories (a), Fourier spectra (b), phase curves (c) and (d) beam deflection w(x, t) for different beam partition n.

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  2 ∂ 2 w ∂u 3 ∂ 2 w ∂w 2 where: L1 (w, u) = ∂∂xu2 ∂w ∂x + ∂x2 ∂x ; L2 (w, w) = 2 ∂x2 ∂x ; 2 L3 (w, w) = ∂∂xw2 ∂w ∂x ; γ is the beam material unit weight; g is the Earth acceleration; t is time; ε is the damping coeficient. The following non-dimensional parameters are introduced: l w ul x l4 w = ; u = 2; x = ; λ = ; q = q 4 ; h h l h h E  l t εl Eg ,ε = . t= ; τ = ; c= τ c γ c Taking into account the introduced variables, Eqs. (6.15) and (6.16) take the following form (bars over non-dimensional parameters are already omitted): ∂2u ∂2u + L (w, w) − = 0, (6.17) 3 ∂x2 ∂t2   1 ∂4w ∂2w ∂w 1 = 0. (w, u) + L (w, w) − + q − −ε L 1 2 2 4 2 λ 12 ∂x ∂t ∂t (6.18) Equations (6.17) and (6.18) should be supplemented with the BCs. Let both beam ends have the simple and non-moveable support: w(0, t) = w(1, t) = u(0, t) = u(1, t) = Mx (0, t) = Mx (1, t) = 0, (6.19) and the initial conditions are as follows w(x, 0) = w(x, ˙ 0) = u(x, 0) = u(x, ˙ 0) = 0.

(6.20)

As it is well known, LEs characterize the stability of the motion of the dynamical system in a phase space. The mathematical background for their existence is given by the Oseledec multiplicative ergodic theorem [Oseledec (1966)] for discrete lumped mass systems. However, the Euler–Bernoulli beam, as a continuous structural member, has an infinite number of degrees of freedom. In the reference [Schaumloffel (1991)] the theorem is generalized to include the case of a phase space of infinite dimension. In order to get an approximated solution, we may reduce the problem to that of finite dimension using either finite elements method or finite difference method,

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Table 6.4 Vibration characteristics for different load amplitude q0 (see text for more information).

and reduce PDEs to ODEs. Then, LEs can be easily determined. The constructed mathematical Euler–Bernoulli model governs the dynamics of a dissipative continuous system, which exhibits the compression of a phase volume during its evolution. In other words, an attractor can be embedded in the so-called inertial manifold of a

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Continued

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Table 6.4

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finite dimension [Schaumloffel (1991)]. It means that the dynamics of such a system can be described with the help of ODEs, which again validates our computations of LEs of a continuous system via a set of ODEs.

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In order to reduce a system of PDEs to a system of ODEs regarding time coordinate, we apply the finite difference approximations by using the time series development in the vicinity of a point xi . Namely, we take the mesh GN = {0 ≤ xi ≤ 1, xi = i/N, i = 0 . . . N }. Partial differential Eqs (6.17) and (6.18) are reduced to a system of the second-order ODEs by using difference operators. In each mesh node, we get a system of equations i = 0, . . . , n, where n denotes the number of space partition, Λx (·i ); Λx2 (·i ); Λx4 (·i ) are difference operators associated with approximation O(∆2 ), where ∆ stands for a computational step regarding the spatial coordinate. u ¨t = Λx2 (ui ) + Λx (wi )Λ2x (wi ), 1 1 w ¨t + εw˙ t = 2 − Λx4 (wi ) + Λx2 (ui )Λx (wi ) λ 12

3 2 +Λx2 (wi )Λx (ui ) + (Λx (wi )) Λx2 (wi ) + q , 2 (6.21)

and Λx , Λx2 , Λx4 are difference operators. Boundary and initial conditions are also presented by the finite differences with approximation 0(∆2 ). A natural question arises: How many intervals, denoted by n, should be used to divide the interval x ∈ [0; 1], where n1 = ∆? Consider ODEs (6.17) and (6.18) with the BCs (6.19) and zero initial conditions (6.20) for the following fixed parameters ε = 1; ωp = 5, q0 = 35 · 103 , where the transversal load is uniformly distributed in the following harmonic form q = q0 sin(ωp t). In order to investigate the convergence of a solution of the equations of the beam motion, we carry out the analysis for q0 = 35 · 103 . For the so far fixed parameters, the system vibrations exhibit three frequencies, and two of them are linearly independent, i.e. ωp and ω1 , whereas the third one ω2 is linearly dependent (see Table 6.3). We are going to study the signal/time history (a) versus a partition number regarding the spatial coordinate. For this purpose, we construct frequency power spectra (b) for x = 0.5, phase trajectories (c) for x = 0.5, and the space–time surface of beam deflection w(x, t). The obtained Cauchy problem is solved with the fourth-order Runge– Kutta method.

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The results are reported for n = 60, 80, 100, the partition number of the spatial coordinate x ∈ [0, 1] while solving the problem through FDM. Time step has been chosen applying the Runge stability condition. Analysis of the obtained results shows that increasing n we may achieve the solution convergence not only for the Fourier power spectrum, but also via other used numerical characteristics. Owing to the obtained results we may conclude that they are true and validated and that they refer to the real behavior of the studied continuous system with an infinite number of degrees of freedom, since a further increase in the interval x ∈ [0, 1] partition does not yield any new effects. The cells of Table 6.4 present the following characteristics: (a) signal t ∈ [1836; 1855]; (b) Fourier power spectrum of the signal; (c) phase portrait approximated by the neural network; (d) 2D Morlet wavelet [Grossmann and Morlet (1984)]; (e) 3D Morlet wavelet [Grossmann and Morlet (1984)]; (f) LEs with the values of KS entropy, and a strength of the space compression. The following observations are yielded by our computational process.

1. For the load amplitude q0 = 4·103 , the first dependent frequency ω2 = 5.64 occurs, which satisfies the formula ω2 = ωp − 2ω1 . This frequency can be observed on the Morlet wavelets, but its amplitude is small. First Lyapunov exponent is positive, the Kaplan– Yorke dimension is positive as well as the KS entropy values (the latter is equal to the value of the first Lyapunov exponent). Phase portrait implies that the attractor consists of a few rings. The values of compression of the phase volume are reduced. Further increasing the load forces the system to shift into chaos. The associated LEs are: λ1 = 0.0002833; λ2 = −0.0016836; λ3 = −0.0038560; λ4 = −0.0038560. 2. For the load q0 = 5 · 103 , we have new frequency ω3 = 1.89 which is defined by the ratio ω3 /ω1 = 3. Wavelets exhibit the increase in the amplitude of the frequency ω2 , but frequency ω3 cannot be observed. Orbits of the phase map are remarkably different. From

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this time instant the two first LEs are positive, and the Kaplan– Yorke dimension increases. The value of KS entropy achieves its maximum for the second Lyapunov exponent having its values larger than for the load q0 = 4 · 103 . Phase portrait reports a torus. The values of compression of the phase volume increase, but continue to remain negative. The LEs are equal to: λ1 = 0.0007082; λ2 = −0.0005367; λ3 = −0.000661; λ4 = −0.0022174. It means that chaos is more developed, and the system begins to transit into chaos–hyperchaos regime. 3. After achieving the value q0 = 6 · 103 , a new frequency ω4 = ω2 − 2ω1 = 4.38 occurs. Orbits reported on the phase plots have different diameters. Both LEs and the Kaplan–Yorke dimensions increased compared to previous load values. The KS entropy value and the magnitude of the phase volume compression increase, whereas the latter one became positive. On the Morlet wavelets, the frequencies ω3 and ω4 are visible though their amplitudes are small, which confirms the plane ring structure, but now with a collapse. From this time instant the system exhibits pulsation, which can be observed on the phase portrait and on the history of LEs (λ1 = 0.0009791; λ2 = 0.0005991; λ3 = −0.0004654; λ4 = −0.0008653). It means that the system is in the chaos–hyperchaos regime. 4. Next, the load increased up to q0 = 10 · 103 yields linearly dependent frequencies ω6 = 2ω5 = 5.02; ω5 = 2ω7 = 2.5; ω8 = 2ω3 = 3.78. The Morlet wavelet becomes more noisy. Now we have only one positive Lyapunov exponent, but its magnitude is twice larger in comparison to the previous load values. The KS entropy is equal to the value of the first Lyapunov exponent. The LEs are: λ1 = 0.0192312; λ2 = −0.5624457; λ3 = −0.6462801; λ4 = −0.9727997. Orbits of the system exhibit large convergence, and the attractor structure is not clearly shown (see the phase portrait). Time series possesses a large asymmetry. In what follows, we study transitions into chaos and hyperchaos of the continuous mechanical structure in the form of a simply supported flexible Euler–Bernoulli beam. The mentioned beam has been

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loaded periodically q = q0 sin(ωp t), and we have fixed λ = l/h = 50, ωp = 6.9. The problem aimed at tracing a scenario of transition from periodic to chaotic vibrations has been solved. Figures 6.9–6.19 report the following characteristics: (a) signal for t ∈ [1836; 1855]; (b) Fourier power spectrum; (c) phase portrait; (d) deflections from the time instant t = 1836; (e) time history w(x, t), x ∈ [0; 1], t ∈ [1836; 1855]; (f) pseudo Poincar´e map; (g) phase portrait (neural network); (h) LEs; (i) 2D Haar wavelet; (j) 2D Morlet wavelet [Grossmann and Morlet (1984)]; (k) 3D Haar wavelet; (l) 3D Morlet wavelet [Grossmann and Morlet (1984)]. Let us analyze results reported in Figs 6.9–6.19. 1. For the load q0 = 0.125 · 103 , only one excitation frequency ωp = 6.9 is exhibited in the Fourier spectrum. Elliptic shape of the phase portrait implies the occurrence of a periodic vibration. Increasing the load up to q0 = 0.5 · 103 results in occurrence of the second independent frequency ω1 = 0.63, which is also visible on the wavelet, though its amplitude is small. All LEs are negative,

Fig. 6.9

Dynamics of simply supported Euler–Bernoulli beam (q0 = 0.125 · 103 ).

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150

Fig. 6.10

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 0.5 · 103 ).

and the Kaplan–Yorke dimension is equal to zero, whereas the KS entropy (h) is negative. Time history (signal) and the beam vibrations are symmetric at a fixed time instant. All of the mentioned characteristics approve the periodicity of the signal. Magnitude of the phase volume compression (d) has a relatively large negative value. 2. The increase in the load amplitude up to q0 = 4 · 103 , yields the occurrence of the first dependent frequency ω2 = 5.64, which satisfies the relation ω2 = ωp − 2ω1 . Though this frequency is visible on Morlet wavelets, it is not visible on the Haar wavelets (it has small amplitude). One may conclude that the appearance of the dependent frequency allowed to distinguish similarity of the waves on the time histories. The first LE as well as the Kaplan–Yorke dimension become positive. The same holds for the KS entropy magnitude which is equal to the first LE. The signal exhibits a small symmetry as well as beam vibrations in a fixed time instant.

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 4 · 103 ).

The attractor is composed of a few rings (see the phase portrait). Magnitude of the phase volume compression decreases, and further increase in the excitation amplitude shifts the system into chaos. 3. For the load q0 = 5 · 103 , a new frequency ω3 = 1.89 appears, and the following relation holds ω3 /ω1 = 3. Wavelets exhibit an increase in the amplitude associated with frequency ω2 , but frequency ω3 cannot be distinguished. Orbits of the phase transform begin to diverge. From this time instant the two first LEs have a positive sign, and the Kaplan–Yorke dimension is increased. KS entropy achieves the maximum for the second LE, and it is larger than in the case of the load q0 = 4 · 103 . Phase portrait exhibits a torus. Beam vibrations for a fixed time instant become more asymmetric. Magnitude of the phase volume compression increases, though it remains negative. In other words, the chaotic state of the system transits into the chaos-hyperchaos state.

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Fig. 6.12

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 5 · 103 ).

4. After achieving the excitation amplitude q0 = 6·103 , the frequency ω4 = ω2 − 2ω1 = 4.38 appears. Orbits shown on phase transformations differ from each other in diameter. Magnitude of LEs together with the Kaplan–Yorke dimension increased. Magnitude of the KS entropy and the magnitude of the phase volume compression become positive. On the Haar and Morlet wavelets, the frequencies ω3 and ω4 are distinguishable, but have small amplitudes, which are also seen on the Fourier spectrum. Phase portrait has also the plane ring structure but with a collapse. Thus, the system exhibits pulsation which can be further shown on phase portraits and LEs. System is in a chaos–hyperchaos state. 5. For the load q0 = 7·103 , and q0 = 8·103 , chaotization is on the same level. Magnitudes of LEs slightly decrease for the load q0 = 7 · 103 , and then again increase up to the first level. The magnitudes of the RS entropy and phase volume compression are also changed

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 6 · 103 ).

(for the load q0 = 7 · 103 they decrease, whereas for q0 = 8 · 103 they increase again to the previous values). Kaplan–Yorke dimension as well as the amplitudes of frequencies ω3 and ω4 (on the power spectrum and on Haar and Morlet wavelets) are increased. Asymmetry of the signal also increases. Phase portrait loses its flat ring structure, although the attractor becomes more ordered. System prolongs its stay in the chaotic–hyperchaotic regime. 6. For the load q0 = 9 · 103 , the phase portrait returns to the previously cut ring structure. Magnitudes of LEs become smaller in comparison to the previous case. Amplitudes of frequencies ω3 and ω4 continue to increase. Then the system transits from the chaos–hyperchaos state to the chaotic state. This transition is accompanied by the essential change of the attractor as shown in the Poincar´e map. 7. The successive increase in the load up to q0 = 10 · 103 yields the occurrence of linearly dependent frequencies ω6 = 2ω5 = 5.02; ω5 = 2ω7 = 2.5; ω8 = 2ω3 = 3.78. The Haar and Morlet

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154

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 7 · 103 ).

wavelets become more noisy. In the Haar wavelets, the separated frequencies cannot be distinguished, but their general tendency can be followed. For the given load, only the first Lyapunov exponent becomes positive, but its value is two times larger in comparison to the previous loads. KS entropy is equal to the value of the first Lyapunov exponent. Orbits of the system have very large divergence, and the attractor is not clearly exhibited. The beam vibrations exhibit large asymmetry in a fixed time instant. 8. However, beginning with q0 = 50 · 103 the system starts to move back into a less chaotic intensity state, which is exhibited by the phase portrait. The signal asymmetry decreases. The first two LEs become positive. However, the magnitude of the largest one is of an order less than that corresponding to the load q0 = 10 · 103 , as well as the Kaplan–Yorke dimension is less than that of the loads q0 = 6·103 –q0 = 9·103 . KS entropy value corresponds to the second Lyapunov exponent and remains arbitrarily high. The Fourier and

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 8 · 103 ).

wavelet spectra do not contain frequencies ω7 , ω8 and ω2 anymore. The signal again becomes symmetric, whereas the beam vibrations in the fixed time instant also have less asymmetry. In other words, the system went out from the deep chaotic regime. 9. Now we deal with the system’s, deep chaotic state which is reported for the load q0 = 100 · 103 . The beam vibrations exhibit the largest asymmetry for a fixed time instant. The LEs have their largest values comparable to the previous ones. The phase portrait exhibits almost undistinguishable structure, i.e. the occurrence of the full mixture of the attractor. Wavelets along the time axis have a non-homogenous structure for all frequencies. In Table 6.5, the following quantities versus q0 are reported: (a) Kaplan–Yorke dimension, (b) LEs, (c) entropy, (d) phase space compression, whereas in Table 6.6 there are given values of the quantities shown in Table 6.5.

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Fig. 6.16

Dynamics of simply supported Euler–Bernoulli beam (q0 = 9 · 103 ).

In order to compare the results obtained via the method of neural network with the results of other methods we have computed the largest LE using the Wolf method. The obtained results are shown in Fig. 6.20. 6.2.2

Frequency analysis

Henri Poincar´e observed that in the investigation of nonlinear dynamical systems, a knowledge of how those systems behave depending on the chosen control parameters plays a key role. This is why we have used this idea while studying nonlinear dynamics of flexible beams with the help of developed dedicated algorithms and programs for numerical analysis. They allow to construct and analyze signals (w(0.5, t), w(x, t)x ∈ [0, 1]) of the frequency power spectrum (with the help of Fast Fourier Transform (FFT)). In addition, this approach allows to detect different types of vibrations (periodicity, period doubling bifurcations, quasi-periodicity, chaos, etc.).

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 104 ).

In particular, an important role in both theory and applications is played by the charts of the vibration regimes versus the control parameters q0 , ωp . The charts of the vibration regimes contain main features exhibited by nonlinear dynamic processes. In Fig. 6.21, an exemplary chart of the vibration regimes for a flexible beam has been reported for the following parameters: n = 120, λ = 50, ε = 1, ωp = [0; 10.35), q0 = [0; 2·105 ), and the BCs (6.19). The vibration regime has been defined with the help of the frequency power spectrum obtained via FFT. Application of the chart of vibration regimes allows to get a global picture of the system dynamical behavior. 6.2.3

Solution to the Cauchy problem

The following question arises: Is the fourth order Runge–Kutta method realizable while solving the Cauchy problem? For this

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Fig. 6.18

Dynamics of simply supported Euler–Bernoulli beam (q0 = 5 · 104 ).

purpose, we apply the sixth order Runge–Kutta method to solve the same problem (Butcher’s method [Hairer et al. (1993)]). In Fig. 6.22, the obtained results regarding the fourth- and sixthRunge–Kutta methods are compared for the following fixed parameters: n = 80, λ = 50, ε = 1, ωp = [0; 10.35), q0 = [0; 2 · 105 ). From the reported charts, one may conclude that the obtained results are the same, but the time duration required to compute the chart via sixth order Runge–Kutta method is 1.5–2 times larger than that of the fourth order Runge–Kutta method. Therefore, in further computations the fourth order Runge–Kutta method is used. 6.2.4

Axially symmetric solutions

It should be noted that an investigation of nonlinear dynamics of flexible Euler–Bernoulli beams subjected to harmonic transversal excitation yields a non-symmetric solution, though the applied BCs

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Fig. 6.19

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Dynamics of simply supported Euler–Bernoulli beam (q0 = 105 ).

are symmetric. However, the occurrence of asymmetric vibration is possible only if the investigated beam has an initial imperfection, and hence it is suspected that small errors appear while solving the Cauchy problem. Since in our investigations we fully study symmetric conditions, the non-symmetric solutions may appear only due to errors introduced by the applied Runge–Kutta method. In particular, this behavior has often been observed in chaotic regimes, and the obtained result may be interpreted as a non-symmetric solution. Note that neither increase in the partition points of the interval [0; 1] nor application of the sixth order Runge–Kutta method removes the stated problem. This is why the occurred asymmetry should be removed on each computation step. An analogous effect can be achieved through an artificial partition of the symmetric loads with respect to a center into two symmetric parts, assuming that a negligible small difference in the

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Table 6.5 The Kaplan–Yorke dimension (a), the LEs (b), entropy (c) and phase space compresion (d) versus q0 .

symmetry is introduced. In other words, in order to get reliable numerical results both solutions for left- and right-hand sides of the investigated loading intervals should be the same. This can be achieved via transformations of one half of loading intervals into the second half with the use of appropriate signs. In what follows, we report the numerical experimental results for the beam for the interval x ∈ [0; 1] partitioned into n = 80 parts, and for the following fixed parameters: ε = 1, ωp = 9, q0 = 5 · 104 . The obtained results are shown in Table 6.9. Analysis of the obtained results allows to conclude that the power spectra constructed via FFT coincide regarding frequencies, but differ regarding their powers. The following frequency ratios are detected: ωp is the nondependent frequency, ω1 = 1/3 ωp , ω14 = 1/3 ωp , ω2 = 3/18 ωp , ω3 = 4/18 ωp , ω4 = 6/18 ωp , ω5 = 8/18 ωp , ω6 = 9/18 ωp ,

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λ1 λ2 λ3 λ4 Dkj h d

125

500

4 × 103

−0.0006 −0.0005 −0.0911 −2.0561 0 −0.0007 −2.1483

−0.0003 −0.0001 −1.3026 −2.0799 0 −0.0004 −3.3831

0.00028 −0.0016 −0.0039 −0.0051 1.168 0.0003 −0.0103

5 × 103

6 × 103

7 × 103

8 × 103

9 × 103

0.00070 0.00097 0.00065 0.00098 0.00061 −0.0005 0.00059 0.00055 0.00068 0.00045 −0.0007 −0.0005 −0.0004 −0.0009 −0.0005 −0.0022 −0.0009 −0.0010 −0.0018 −0.0009 2.594 2.681 3.788 3.627 3.638 0.00071 0.00158 0.00121 0.00167 0.00107 −0.0027 0.00024 −0.0002 −0.0010 −0.0003

104

5 × 104

105

0.01923 0.00901 0.01382 −0.5624 0.00273 0.00363 −0.6463 −0.0306 −0.0114 −0.9728 −0.0322 −2.7302 1.824 2.645 3.002 0.01923 0.01174 0.01746 −2.1622 −0.0510 −2.7241

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q0

Values of the characteristics reported in Table 6.5. Euler–Bernoulli Beams

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Table 6.6

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Fig. 6.20 Largest LE versus amplitude of external load estimated via our (neural network) and Wolf algorithms.

Fig. 6.21 details).

Chart of vibration regimes of the flexible beam (see text for more

ω7 = 17/18 ωp , ω8 = 14/18 ωp , ω9 = 1/18 ωp , ω10 = 2/18 ωp , ω11 = 5/18 ωp , ω12 = 7/18 ωp , ω13 = 10/18 ωp , ω15 = 13/18 ωp , ω16 = 15/18 ωp , ω17 = 16/18 ωp . The data of Table 6.7 imply that although the results regarding frequencies almost coincide for both cases, i.e. with and without corrections, the energy distribution exhibits an essential difference. In Table 6.8, beam deflections at the time instant t = 1836 are shown. From the results given in Table 6.8, one may conclude that the solutions shown in cases (b) and (c) coincide fully and are symmetric,

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Fig. 6.22 Table 6.7 spectra).

163

Charts of vibration types obtained via different Runge–Kutta methods. Dynamics of the beam for different modeling (frequency power

Table 6.8

Dynamics of the beam for different modeling (time histories).

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whereas the solution presented in the case (a) is non-symmetric. In order to clarify the open question, if the occured error is introduced by the improper beam partition of the space coordinate (to small partitions number), in Table 6.9 charts of the vibration regimes for the following set of parameters: λ = 50, ε = 1, ωp = [0; 10.35), q0 = [0; 2 · 105 ) and for symmetric and non-symmetric stability loss mode with different number of partitions (n = 40, n = 60, n = 80, n = 120) are reported. 6.2.5

Influence of BCs and relative beam thickness on chaos

In Table 6.10, charts of vibration regimes for the following control parameters: n = 120, ωp = [0; 10.35), q0 = [0; 2 · 105 ) and different types of BCs: (a) both sides are simply supported; (b) clamping on both ends; (c) simple support-clamping, and for different beam thicknesses (λ = 50, 100) are shown. Results shown in Table 6.10 imply that for symmetric BCs, in the case of small and average frequencies, vibration regimes coincide. In the case of non-symmetric mixed BCs a coincidence with symmetric BCs is observed only for small frequencies. Furthermore, the nonsymmetric BCs imply the increase in the chaotic vibration regimes. The increase in beam thickness yields the increase in periodic zones. 6.2.6

Charts of LEs

The chart of LEs is one of the novel tools to globally investigate the nonlinear dynamics. In general, in order to estimate the LEs, two algorithms are often used, i.e. Bennetin’s and Wolf’s algorithms. Both of them require monitoring of the associated linearized system of equations. It is clear that in general it is difficult to get an analytical form of the linearized equations, whereas a direct application of the numerical approaches requires very long computational time. On the other hand, Wolf’s algorithm is associated only with searching solutions to a system of differential equations. Considering those remarks, Wolf’s algorithm has been chosen. We take the same parameters as in the previous case, whereas the applied BCs are now mixed, i.e. we deal with the simple support-clamping.

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Table 6.9 Charts of beam vibrations for symmetric and non-symmetric governing equations and for different beam partitions number n.

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Table 6.10 Charts of beam vibrations: simple support (a), clamping (b) and simple support-clamping (c) for different beam thicknesses λ.

Since the trajectory of a beam point is described by a system of four differential equations, we consider four LEs. In Table 6.11, both the charts of vibration regimes and LEs are shown for comparison purpose. In Table 6.12, color notation for this chart is also given. It is clear that the increase in the number of positive LEs implies the increase in the strength of the chaotic dynamics. Comparison of the chart of vibration regimes with the chart of the LEs shows efficiency of both methods of analysis for frequency

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Table 6.11

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Chart of vibration regimes (λ) versus chart of LEs.

spectrum and the chaotic dynamics quantified by LEs. Practically, all detected periodic vibrations are associated with negative LEs, whereas vibrations with the combination of independent frequencies appear only in zones with one positive LE. Although the chaotic behavior has been reported relatively long time ago, the chaos– hyperchaos state has been illustrated and discussed in 1989 in the reference [Dmitriev and Kislova (1989)]. On the other hand, qualitative results describing chaos, chaos–hyperchaos as well as the deep chaos have been reported in the reference [Awrejcewicz et al. (2012)]. In Table 6.12, the zones associated with each studied vibration regime separately as well as all results are collected together in the form of the LE chart. We have also detected chaotic vibrations with four positive LE and the latter phenomenon has been called deep chaos. The number of LE depends on the number of equations governing the studied system dynamics. Each equation is associated with one LE. If the LE is negative then we have a periodic process. If at least one Lyapunov exponent is positive, then chaos occurs. The analysis carried so far allows to present the following brief summary of the obtained results. In this section, we have proposed the modification of the neural network method for computation of the LE spectrum. The method is illustrated using an example of the generalized H´enon system and the flexible Euler–Bernoulli beam. Further development of the Wolf method has been proposed, which allows to study the four positive LE, and then to detect and discuss the phenomena of chaos–hyperchaos, chaos–hyper-hyperchaos.

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Table 6.12

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Chart of all LEs and their separated parts.

Scenarios of transition from periodic to chaotic vibrations in flexible Euler–Bernoulli beams have been detected with the help of the Fourier transformation and charts of the four largest LEs have been computed via the Wolf algorithm. Results of the nonlinear analysis of flexible Euler–Bernoulli beams are validated and are found to be reliable. Namely, spatial coordinates have been partitioned by the finite difference method with approximation of the secondorder, with respect to power spectrum and also regarding the signal

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Euler–Bernoulli Beams

169

coincidence. It means that the studied beams have been treated as systems with infinite degrees of freedom. The Cauchy problem has been solved with the fourth- and sixth order Runge–Kutta methods. Results of both methods coincide, and hence the further computation with the help of the fourth order Runge–Kutta method has been carried out. Various approaches for the study of spatial chaos have been applied including modal portraits, which for each beam point present the rotation angle ( ∂w ∂x ) and the approximate value of the ∂2w curvature ( ∂x2 ). The constructed charts allow to control the chaotic vibrations of the Euler–Bernoulli beams. 6.3 6.3.1

Linear Planar Beams and Stationary Temperature Fields Problem statement

The applied beam mathematical model is based on the following hypotheses: (i) Any transversal cross-section normal to the middle beam curve after deformation remains a straight line normal to the middle beam curve, and the cross-section height is not changed  ∂2w εx,u = −z ∂x2 . (ii) Nonlinear dependence between deformations and displacements is taken in the form proposed by Von Karm´ an [Von K´ arm´an   ∂u 1 ∂w 2 (1910)]: εx = ∂x + 2 ∂x , where u(x, t) is the middle beam line displacement along axis x, and w(x, t) denotes the beam deflection. (iii) Beam material is isotropic, elastic and satisfies the Duhamel– Neuman rule: εzx = εx + εx,u + αT (x, z), where α is the linear extension coefficient, and T (x, z) denotes the beam temperature. (iv) There is no restriction to the temperature field, and it is governed by a 2D heat transfer equation. We consider physical materials which depend on the temperature. (v) Normal stresses present on the layer parallel to the middle beam curve are neglected (they are small compared to other stresses).

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The problem is reduced to the quasi-static one, i.e. the temperature field does not depend on time. Taking into account the action of both temperature field T (x, z) and Hook’s law, the integral relations for beam stresses and torques take the following form 

2 

h/2 1 ∂w ∂u + εzx dz = Eh − NxT, (6.22) Nx = E ∂x 2 ∂x −h/2

h3 ∂ 2 w − MxT, 2 12 ∂x −h/2   h/2 h/2 where NxT = αE −h/2 T dz, MxT = aE −h/2 T dz. Mx = E

h/2

εzx zdz = −E

(6.23)

Applying the D’Alembert principle, we add dynamical relations, inertial forces and a dissipative force to the given forces, and hence the beam element movement is described by the following equations for displacements:   2 γ ∂2u ∂NxT ∂ u + h 2 = 0, + L (w, w) − (6.24) Eh 3 2 ∂x ∂x g ∂t   2 4 ∂2w T ∂w ∂NxT h ∂ w + + L (w, u) + L (w, w) − N Eh − 1 2 12 ∂x4 ∂x ∂x ∂x2 x −

∂ 2 MxT γ ∂ 2 w γ ∂w h + q = 0, − − hε ∂x2 g ∂t2 g ∂t

(6.25)

  2 ∂ 2 w ∂u 3 ∂ 2 w ∂w 2 where: L1 (w, u) = ∂∂xu2 ∂w ∂x + ∂x2 ∂x ; L2 (w, w) = 2 ∂x2 ∂x ; 2 L3 (w, w) = ∂∂xw2 ∂w ∂x ; γ is the volume beam material weight; g is the Earth acceleration; t is time; ε is the damping coefficient. The studied beam is shown in Fig. 6.23.

Fig. 6.23

The studied linear beam.

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Equations (6.24) and (6.25) should be supplemented with one of the following BCs:

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1. Two beam ends are clamped: w(0, t) = w(l, t) = u(0, t) = u(l, t) = w
x (0, t) = w
x (l, t) = 0. (6.26) 2. Two beam ends are simply supported: w(0, t) = w(l, t) = u(0, t) = u(l, t) = Mx (0, t) = Mx (l, t) = 0. (6.27) 3. One of the beam ends is clamped, and other is simply supported w(0, t) = w(l, t) = u(0, t) = u(l, t) = w
x (0, t) = Mx (l, t) = 0. (6.28) Initial conditions follow: w(x, 0) = w(x, ˙ 0) = u(x, 0) = u(x, ˙ 0) = 0.

(6.29)

In order to analyze the obtained system, we use the theory of similarity and dimension [Awrejcewicz (2014)], and we introduce the non-dimensional variables and parameters w ¯= t t¯ = , τ

w , h

u ¯=

l τ= , c

ul x , x ¯= , 2 h  l

c=

Eg , γ

MxT =

MxT l , Eh2

ε¯ =

λ= εl , c

l , h

q¯ = q

NxT =

l4 , h4 E

NxT l2 , Eh3

(6.30)

T¯ = αT λ2 ,

where l is the beam length, h is the beam thickness, E is Young modulus, w(x, t) is the beam deflection, u(x, t) is the beam displacement along axis x. Taking into account the introduced non-dimensional quantities, Eqs. (6.24) and (6.25) are recast to the following form (bars over the

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non-dimensional quantities are omitted): ∂2u ∂NxT ∂2u − t = 0, + L3 (w, w) − 2 ∂x ∂x ∂t   1 ∂4w 1 +q L1 (w, u) + L2 (w, w) − λ2 12 ∂x4

∂2w ∂w ∂ 2 MxT ∂ ∂w T Nx − = 0. − −ε − 2 2 ∂x ∂x ∂x ∂t ∂t

(6.31)

(6.32)

Both boundary and initial conditions (6.26)–(6.29) are also recast to the non-dimensional counterpart forms. In order to investigate the stress–strain relations of the objects working in conditions of the nonhomogeneous heating, a temperature field should be defined first. In this regard, it is necessary to formulate methods devoted to solutions of the associated heat transfer problems for thin walled structural elements. In this work, in order to define a temperature field in beams, the following 2D heat transfer equation should be solved without the internal heat source (W0 = 0): 1 ∂2T ∂2T + = 0. ∂x2 λ ∂z 2 To find a solution to the stationary heat transfer equation we need to introduce the BCs. Four kinds of BCs are known, as described in the following paragraph (see [Carlaw and Jeager (1986)]). I. The first kind of BCs. The temperature distribution on the surface S of a body is given as a function of the coordinates: Ts = g(x, z),

x, z ∈ S.

(6.33)

These kinds of problems include heating and cooling of the system to a given temperature on the boundary or rather intensive heat exchange on the surface where the beam surface temperature is close to the medium temperature. However, these problems are rather limited and conditions (6.33) are used rather only for development of the mathematical models of the BC and validation of the obtained numerical results.

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II. The second kind of BCs. The density of a heat stream on the body surface as a function of coordinates is given as

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Ws = θ(x, z),

x, z ∈ S.

(6.34)

Owing to the Fourier law, the condition (6.34) takes the following form [Carlaw and Jeager (1986)]:

∂T = θ(x, z), x, z ∈ S. (6.35) −λ ∂n S In the case when the heat stream density on the body surface is constant, we have WS = W0 = const. Such heat exchange conditions can appear when the bodies are heated by high temperature heat sources, and when the heat exchange is carried out mainly by the Stefan–Boltzmann rule as well as when the body temperature is essentially smaller than the temperature produced by the radiation of a surface. In the case when  ∂T  = 0, (6.36) WS = − ∂n S we have the so-called heat isolation. III. The third kind of BCs. On the boundaries of the body surface, a dependence of the heat stream density regarding the heat transfer from the body side on the temperature TS of the body surface and the surrounding medium temperature T0 is given. In the case when the body is cooling TS > T0 WS = ξ (TS − T0 ) ,

(6.37)

where ξ is the proportionality coefficient called the heat exchange coefficient. Equation ∂T = ξ (TS − T0 ) (6.38) −λ ∂n presents an analytical form of the BC of the third kind, which is widely applied in the investigations of heat transfer in the stream/gas fluid interaction on the boundaries between different types of continuous objects.

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BCs of the third kind may yield BCs of the first and second kind as the particular cases. Therefore, if λξ → ∞ (ξ → ∞ for λ = const, or λ → ∞ for ξ = const), then we obtain the BCs of the first kind

  1 ∂T , (6.39) TS − T0 = limξ/λ→∞ ξ/λ ∂n S i.e. we have TS = T0 . In the case when ξ → 0, we get the particular case of the BCs of the second kind

∂T −λ = 0. (6.40) ∂n S IV. The fourth kind of BCs correspond to the heat transfer of the body surface with the neighborhood medium (convective heat transfer of the body and the surrounding fluid) or heat exchange of the contacting rigid bodies when the temperature of the contacting surfaces is the same −λ1

T1S = T2S ,



∂T1 ∂T2 = λ2 . ∂n S ∂n S

(6.41) (6.42)

Formula (6.41) presents the condition of the temperature field continuity, whereas formula (6.42) presents the energy conservation on the surface of the contacting two bodies. 6.3.2

Solution of the heat transfer equation

The temperature field is described by the following PDE and BCs: ∇2 T =

∂2T 1 ∂2T + =0 ∂x2 λ2 ∂z 2

T = f (x, z) on Γ, F =

∂T =q ∂n

on Γ2 ,

∈Ω

(6.43) (6.44) (6.45)

where T is the partial function of a potential, f , F are given values, Γ = Γ1 + Γ2 is the contour of the space Ω, and n stands for the unit vector of the external normal.

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Equivalent integral equation can be derived using the Green formula, and it takes the following form:

∗ (6.46) T (ξ) = F (x)T (ξ, x)d(x) − T (x)F ∗ (ξ, x)d(x). Function T ∗ is a Green function computed for a free space, and it is a fundamental solution of the Laplace equation. Here, it satisfies the following PDE with a discrete singularity of the form ∇2 T ∗ (ξ, x) = −∆(ξ, x),

(6.47)

where ∆ is the Dirac delta function, and ξ and x are the points of ∗ (ξ,x) the source and the field, respectively. Function F ∗ (ξ, x) = ∂f∂n(x) is understood as the derivative of f ∗ with respect to an external normal. In the case of 2D problems, we have  1 /2 1 1 1 ln , F ∗ = − rn , r = x2 + z 2 , T∗ = 2π r 2πr rn =

1 ∂r(ξ, x) = − (χ1 xn + χ2 zn ), ∂n(x) r

χi = xi (ξ) − zi (x),

(6.48)

i = 1, 2,

where xi,n is the direction cosine of the external normal at point x. The obtained system of equations allows to compute the values of the potential in the internal body points if all values of T and F are known along the boundary. In the properly defined problems at each boundary point either T or F are defined, and then an additional equation is used to compute the remaining unknowns. For this purpose, we shift the source point to the boundary and we take into account the violation of continuity in the second integral. Finally, we get the relation

(6.49) C(ξ)T (ξ) = F (x)T ∗ (ξ, x)d(x) − T (x)F ∗ (ξ, x)d(x), in which the second integral is computed with the help of the main Cauchy value. The coefficient C, which is the function of the internal source at the point ξ, can be computed either analytically or, alternatively, by defining of a constant potential. Now, we discritize

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Eq. (6.49) via partition of the boundary Γ onto a chain of boundary elements. Furthermore, we assume that the values of T and F in a point located inside of the element are linked with the values of nodes via certain interpolating functions. In this work, this is done with the help of linear elements using linear interpolating functions. Then, Eq. (6.49) takes the following form Ci Ti =

m
j=1

(ψ1 F1 + ψ2 F2 )u∗ d

m


j

j

j=1

1 1 ψ1 = (1 − η), ψ2 = (1 + η) 2 2

(ψ1 T1 + ψ2 T2 )F ∗ d, (6.50)

i = 1, . . . , N,

where η = 2x /l is the non-dimensional coordinate, l is the element length, x denotes the local coordinate. Denoting Fijk =   ∗ k ∗ j ψk T d; hij = j ψk F d, and taking into account an input of two closing elements (j − 1) and (j) into one term, equation (6.50) can be recast to the following form Ci Ti =

N


Gij Fj −

j=1

N


ˆ ij Tj , H

(6.51)

j=1

where N is the general number of boundary nodes, and each of the node elements is equal to Fij2 of the element (j − 1) plus term Fij1 of the element (j), assuming that the numeration of nodes is carried k ). out in the counter-clockwise way (the same is applicable also to Hij Equation (6.51) can be presented in a more compact way N
j=1

Hij Tj =

N


Gij qj ,

(6.52)

j=1

ˆ ij + δij Ci , and δij is the Kronecker delta. Note that where Hij = H equation (6.52) can be applied, using the method of collocation, to all nodal points. Finally, we get a system of algebraic equations of order N × N , which have the following matrix representation HT = GQ.

(6.53)

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Taking into account BCs, the equation is transformed to a suitable form to get the following system of equations

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KY = F,

(6.54)

where K is a fully composed matrix and Y is the vector containing all boundary unknowns. From a computational point of view, the coefficients Hij (i = j) and Gij (i = j) can be located in matrix K in such a way that Eq. (6.53) is not needed. In the method of boundary elements we use the fundamental solution obtained earlier. 6.3.3

Numerical versus analytical solution

A solution to the heat transfer equation is obtained using the boundary element method with different steps ∆x and ∆z. In order to check the validity of the results, the convergence of numerical and analytical solutions has been investigated first. The analytical solution to a stationary equation of the heat transfer equation for a 2D body is reported in [Vasilenko et al. (1976)] and it has the form ∞ 4
sin[(2n + 1)πx] sinh[(1/2 − z)(2n + 1)π]   . (6.55) T = π (2n + 1) sinh (2n + 1) π2 n=1 The problem is solved for a body with a rectangular cross-section 0 < x < 1, −1/2 < z < 1/2 and with the BCs given in Table 6.13. Boundary surface z = −1/2 has a constant temperature T = 1, whereas the remaining surfaces have zero temperature. In the given case, we deal with the first kind of BCs which have the forms given in Table 6.13. Figure 6.24 shows a graphic representation of both solutions, i.e. these are obtained analytically and numerically for the mesh applied on beam space Ω with the numbers of nodes nx × nz , where T and N T in Eqs. (6.53 nx = 50 and nz = 10. In order to compute MX X Table 6.13 T (x, z) = 1 T (x, z) = 0 T (x, z) = 0 T (x, z) = 0

First kind HBCs.

z = −1/2 z = 1/2 x=1 x=0

0≤x≤1 0≤x≤1 −1/2 ≤ z ≤ 1/2 −1/2 ≤ z ≤ 1/2

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Fig. 6.24

Fig. 6.25

Analytical versus numerical solution of the heat transfer equation.

Comparison of numerical and analytical values of NxT and MxT .

and 6.54), the Simpson numerical technique is applied. Convergence of the numerical and analytical solutions is monitored using the beam partition ∆x = 1/50 and ∆z = 1/10. Results of convergence investigation are shown in Fig. 6.25. The presented computational results reported in Fig. 6.25 indicate high accuracy of the applied numerical approach. 6.3.4

Solving heat transfer equation via FDM

We use FDM to solve the heat transfer equation. The process of finding a solution consists of two fundamental steps: (i) at first, the governing PDE and BCs are transformed to difference equations; (ii) a solution to the difference equations is found. The first step is associated with a question about the accuracy of the applied approximation. Here we apply the fourth-order approximation, which is of particular importance while solving 3D equations.

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z

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1

0

Fig. 6.26

1

x

Unit square applied to solve the Dirichlet problem.

The second step concerns the following question: Which method, i.e. direct or indirect, should be applied to solve the difference equations. The iteration methods are used since the system of difference equations is of high order, and it is impossible to solve it via a direct method. There are various iteration methods to solve our problem. In practice, a method which requires short computational time, small computational memory and is relatively simple to realize is usually chosen. An experimental comparison of different methods allows us to choose the most appropriate method with regard to arithmetic operations and computational time for the whole problem. This problem has been discussed in works [Andreev (1965); Dyakonov (1974)], among others. The problem of efficiency of differential iteration methods applied to solve the Dirichlet problem for the Laplace equation describing a squared space (Fig. 6.26) has been solved in [Vakhlaeva and Misnik (1975)]. We consider iteration methods based on a successive Seidel relaxation algorithm, a triangular method with Chebyshev’s acceleration and an implicit method of variable directions. Two ways of approximation of partial derivatives are applied, i.e. the fifthand the ninth-order scheme, with the approximations of O(h2 ), O(h4 ). The mentioned methods were compared with respect to the convergence velocity, the volume of a required computer memory and the simplicity of its realization. The problem of a proper choice of the parameters for increasing the computational speed was also addressed. For the Laplace equation ∆T = 0, the Dirichlet problem

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with BCs applied on the unit square sides  e3x , z = 0,    cos(3z), x = 0, T |Γ = e3x cos(3), z = 0,    3 e cos(3z), x = 0,

was studied first: 0 ≤ x ≤ 1, 0 ≤ z ≤ 1, 0 ≤ x ≤ 1, 0 ≤ z ≤ 1.

(6.56)

An exact solution to the stated problem is T (x, z) = e3x cos(3z). Let us define the system of difference equations approximations ∆T = −f with BCs T |Γ = ϕ(x, z) in the following form AT = F,

(6.57)

where A = K + D + N , D is the diagonal, matrix K is the upper triangle, matrix N is the lower triangle matrix. The method of upper relaxation for system (6.57) is reduced to the form [Marchuk (1982)]:   (n) (n−1) (n) (n−1) = DT − ω KT + (D + N )T −F . (6.58) DT For ω = 1, it yields Seidel method. For the five-point approximation, the difference equation takes the form  (n) (1) (n) (2) (n) (3) (n−1) (4) (n−1) Tij = ω aij Ti−1j + aij Tij−1 + aij Ti+1j + aij Tij+1 + f (n−1)

+(1 − ω)Tij (1)

(3)

aij = aij =

,

h2 1 · 2 2 2, 2 h1 + h2

(2)

(4)

aij = aij =

h2 1 · 2 1 2, 2 h1 + h2

(6.59)

where h1 , h2 — denotes the length of cells in direction x and z, respectively. In the case of nine-point approximation, we have  (n) (1) (n) (2) (n) (3) (n−1) (4) (n−1) Tij = ω aij Ti−1j + aij Tij−1 + aij Ti+1j + aij Tij+1 (5)

(n−1)

(6)

(n)

(7)

(n)

+aij Ti−1j+1 + aij Ti−1j−1 + aij Ti+1j−1 (8) (n−1) (n−1) +aij Ti+1j+1 + f + (1 − ω)Tij ,

(6.60)

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(3)

where aij = aij =

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(7)

(8)

2

2

1 5h2 −h1 10 · h21 +h22 ,

(2)

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(4)

aij = aij =

2

2

1 5h1 −h2 10 · h21 +h22 ,

(5)

(6)

aij = aij =

1 aij = aij = 20 The iteration multiplier is yielded by the known formula [Marchuk (1982)]: 2  , (6.61) ωopt = 1 + 1 − λ21

where λ1 is the matrix spectral radius corresponding to Seidel’s method. For the rectangular space the value of λ1 is known, either for five-point or nine-point approximations:



h22 1 h21 1 πh1 πh2 (5) + · 2 , (6.62) cos cos λ1 = · 2 2 h1 + h22 a 2 h1 + h22 b (9) λ1

1 = 5





cos

πh1 a



5h2 − h22 cos + 21 h1 + h22



cos

πh2 b





πh2 b



5h2 − h21 + 22 cos h1 + h22

,



πh1 a



(6.63)

where a and b are the sides of the rectangle. For an arbitrary space, and for ωopt obtained in (6.61), one may use the approximate value of λ1 . There is a formula [Marchuk (1982)] coupling the spectral matrix radius of the iteration process (6.58) µ1 and ω: (µ1 + ω − 1)2 = λ21 . (6.64) ω 2 µ1 Therefore, one may estimate the approximate value of the relation multiplier ωopt via the following formula 2



ωm+1 = 1+

1−

(m)

µ1

+ωm −1

(m)

µ1

.

(6.65)

2 ωm

Approximating value µ1 is yielded by the Lusternik method [Marchuk (1982)], and it reads   (n+1)    (n+1) T − T (n)  − T (n)  i,j T (n)    µ1 = limn→∞  T (n) − T (n−1)  ≈  T (n) − T (n−1)  , (6.66) i,j

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where m stands for a minimum value satisfying the inequality     µ(m)   1 − 1 (6.67)  ≤ εµ .  (m−1)  µ Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

1

Here, εµ is the small given quantity and ω0 is the given quantity (for instance, we can take ω0 = 1). Carrying out m1 iterations via 1 Seidel’s method we find µm 1 . However, the speed of convergence can be increased. It can be chosen with the accuracy of 0.1 and 0.2. The latter can be achieved using the method of an equivalent rectangular. It relies on seeking the matrix spectral radius in the Gauss–Seidel method on the basis of an equivalent rectangle whose surface is equal to the given space. The latter one can be obtained in the following way. Its width is determined by the diameter of the largest circle which can be drawn between the boundaries of the space boundary (the circle can neither intersect the boundary curve nor include the space). Its length is defined by the space surface divided by its width, whereas ωopt can be taken equal to its asymptotic value  2πh1 h2 1 1 + , (6.68) ωopt = 2 −  2 2 b2 h1 + h22 a for the case of the five-point scheme, and 

1 1 2h21 + 25h22 2    + , ωopt = 2 − 62πh1 h2  2 a2 b2 h1 + 10h22 h21 + h22 20h22 + h21 (6.69) for the nine-point scheme, where h1 , h2 , a, b are the same as in (6.62). It should be emphasized that the number of iterations can be reduced if ωopt can be defined via indirect ways either by formulas (6.68), (6.69) or by an equivalent rectangle. Formula (6.67) yields dependence of n and εµ . The experiment has shown that for the applied square, the optimal interval for εµ is [0.001; 0.005] for h1 = h2 = 0, 0625 in the case of the fifth-order approximation, whereas in the case of the ninth-order approximation εµ ∈ [0.005; 0.01], and for the five-point scheme on the S-shaped space εµ ∈ [0.001; 0.01]. A decreasing step implies a decrease of εµ . Among all explicit methods of variable directions, the triangle methods exhibit a high

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convergence speed keeping economical choice of the computational steps since triangular matrices are easily manipulated. Furthermore, the choice of computational steps is realized via explicit methods. For instance, if we divide the input matrix (6.60) into two triangular matrices (1)

(2)

(3)

(4)

(A1 T )i,j = a(ij) Ti−1j − 12 Tij + a(ij) Tij−1 , (A2 T )i,j = a(ij) Ti+1j − 12 Tij + a(ij) Tij+1 ,

(6.70)

the corresponding explicit method of variable directions is governed by the following formulas  1 1 1 (n+1) (n) (1) (n+ ) (2) (n+ ) = + τ a(ij) Ti−1j 2 + a(ij) Tij−1 2 T Tij ij τ 1+ 2 

1 (n) (3) (n) (4) (n) +a(ij) Ti+1j + a(ij) Tij+1 − Tij , (6.71) 2 i = 1, 2, . . . , N1 − 1, j = 1, 2, . . . , N2 − 1,  1 1 (n+ 12 ) (n+1) (3) (n+1) (2) (n+ ) ˆ = + τ a(ij) Tˆi+1j + a(ij) Tij−1 2 Tij Tij τ 1+ 2 

1 1 (n) (4) ˆ (n+1) (2) (n+ 2 ) +a(ij) Tij+1 + a(ij) Tij−1 − Tij , (6.72) 2 i = (N1 − 1) , . . . , 2, 1,

j = N2 − 1, . . . , 2, 1.

We have detected experimentally that multiplier τ accelerates the convergence in (6.71), (6.72) if it is equal to τcp =

2 . h1 + h2

(6.73)

In order to improve the convergence speed of the applied triangular method [Marchuk (1982)], we use the Chebyshev algorithm. The iteration process is defined in the following way: the iteration formulas (6.71) and (6.72) are supplemented by the following formula  (n+1) (n) (n+1) (n) = Tij + αn Tˆij − Tij , (6.74) Tij

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where αn are certain parameters improving the convergence of the iteration method, and they read

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αn =

µ 2n−1 2 (1 + cos( 2N π) . 1 − µ2 (1 + cos( 2n−1 2N π)

(6.75)

The maximum value M of the iteration process matrix eigenvalue can be practically computed in the approximated way. For this purpose, one may apply the Lusternik algorithm (6.54). The n1 iterations are carried using (6.71), (6.72), i.e. with the Chebyshev economization method, and its efficiency can be estimated by the parameter    (n+1) (n)  − T i,j  i,j Ti,j (n ) , (6.76) λ1 1 =   (n) (n−1)  − T T  i,j i,j i,j which tends to λ1 = M for n → ∞. If the latter one undergoes slight changes from one iteration to the next one, this λ1 can be taken as M . Next, αn are computed and the iterations are carried out with the Chebyshev acceleration. Among the series of iteration schemes of the variable iterations, we consider the Peaceman–Rachford Alternating Direction Implicit (ADI) method. This method requires transition from one iteration to the other one via two steps [Samarsky and Nikolaev (1978)]:  1  T (n+ 2 )  = ϕ, Γ  n+ 12 ) (n+ 12 ) (2) (n+1) (n+1)  ( =T − τn (A1 T ), T + A2 T  = ϕ, 1

1

T (n+ 2 ) = T (n) − τn(1) (A1 T (n+ 2 ) + A2 T (n) ), T (n+1)

Γ

(6.77) (1)

(2)

where τn and τn are the chosen parameters improving the acceleration of the iteration steps. The first formula in (6.77) is implicit with respect to the horizontal line, whereas the second one is associated with the vertical line. Here we have: Aα T = Λα T = Tx¯α xα ,

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(1)

(−A1 T )ij = aij Ti−1j −

2 aij 2

185

(3)

Tij + aij Ti+1j ,

(0)

(2)

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aij

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(6.78)

(4)

Tij + aij Ti+1j ,

(k)

where aij are defined in (6.60). In the case of the scheme of the improved accuracy order, we have A = A1 + A2 −

h21 + h22 A1 A2 . 2

(6.79)

The basic idea of the method of variable directions consists in the transition from one iteration to the successive solution along the rows and columns of one-dimensional problems which are solved with the method of relaxation. The algorithm of the solution to the system of algebraic equations (6.77) is reduced to that of the successive solution for rows of the following equation 1

1

T (n+ 2 ) + τn (1) A1 T (n+ 2 ) = Fn ,  Fn = T (n) + τn (1) A2 T (n) , T (n+1) Γ = ϕ;

(6.80)

and for the columns of the following equation T (n+1) + τn (2) A2 T (n+1) = Fn+ 1 , 2

Fn+ 1 = T 2

(n+ 12 )

+

1 τn (2) A1 T (n+ 2 ) .

(6.81)

In the case of the increased order accuracy, the formulas analogous to (6.80) and (6.81) can be derived in the following form [Samarsky (1977)]:   1 h2 E + τn (1) − 121 A1 T (n+ 2 ) = Φn ,   h2 (6.82) Φn = E − τn (1) + 122 A2 T (n) ,  1  h2 +h2 T (n+ 2 )  = ϕ − 112 2 A2 ϕ, Γ

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 E + τn (2) −

Φn+ 1 2

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h22 12



A2 T (n+1) = Φn+ 1 ,

  = E − τn (2) +

h21



12



2

1

A1 T (n+ 2 ) ,

(6.83)

 T (n+1) Γ = ϕ.

Improvement of the convergence acceleration of the iteration process can be achieved by an appropriate choice of parameters τn (1) and τn (2) . Owing to the reference [Samarsky (1977)], the optimal parameters τn (1) and τn (2) for problems (6.80) and (6.81) are defined by the following formulas sωn + r sωn − r , τn (2) = , (6.84) τn (1) = 1 + ωn p 1 − ωn p where (∆1 − δ1 )∆2 ae − ξ , χ= , ae + ξ (∆2 − δ1 )∆1      ∆1 − ∆2 + ∆1 + ∆2 p ∆1 − δ1 ∆2 − δ2  , , ξ=  r= 2∆1 ∆2 ∆1 + δ2 ∆2 + δ1 s=r+

1−p , ∆1

p=

2n − 1 (1 + 2θ) (1 + 2θ σ ) , σ= , n = 1, 2, . . . , v, σ/2 1−σ 1+σ 2n 2θ (1 + θ +θ )



4 1 4 1 1−ξ 1 ln , θ = η2 1 + η2 , η = , v ≈ 2 ln π ε η 16 2 1+ξ

ωn =

where δ1 and δ2 are the minimal eigenvalues of operators A1 and A2 , respectively; ∆1 and ∆2 are the maximum eigenvalues of the same operators (they are assumed to be known); ε is responsible for the required accuracy of the iteration process. In the case of the improved accuracy order of (6.82), (6.83), the computation of the iteration parameters can be realized via formulas ˆ α , which are coupled by the (6.84) by changing δα , ∆α through δˆα , ∆ relation ¯α δα ∆ ˆα = , ∆ (6.85) δˆα = ¯α, 1 − aeα δα 1 − aeα ∆ where: α = 1, 2, aeα =

h2α 12 .

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187

In the case of the rectangular space, the improved order accuracy scheme with the choice of the optimal sequence of the parameter is followed by formulas (6.84), (6.85). The results are reported in Table 6.14. They show that the method is highly convergent, and after 12 iterations the required accuracy of order 10−6 is achieved. Besides, for the schemes (6.82)–(6.84) the memory volume to keep (n+ 1 ) values Tijn , Tij 2 is equal to 2N , where N denotes the number of space points. In all the considered iteration methods, the computation is (n)

extended until maxi,j

(n−1)

|Tij −Tij

|

(n) Tij

< εit. , where εit. is the given small

quantity. In Table 6.14, the relation of the number of iterations in the relaxation and Seidel’s methods versus the approximation points (five or nine) are shown. The nine-point approximation has a higher order convergence in comparison to the five-point approximation. Besides, the nine-point approximation allows us to choose a more robust mesh to achieve the same accuracy as the five-point approximation, which finally results in the decrease of the computational time while achieving a solution with required accuracy. Table 6.14 reports the methods of upper Seidel relaxation as well as that of

Table 6.14 methods.

Comparison of advantages/disadvantages of the applied numerical

Method of variable direction Method

Seidel Fivepoints

Alternatingtriangle Implicit

Five- Ninepoints points

Fivepoints

Ninepoints

Iterational parameter 1 1 1.65 1.65 Memory volume N N N N Number of iterations 247 225 89 49 Error in point(0.5; 0.5) 0.001 0.0006 0.0006 0.00009 π 2 h2 1, 2π 2 h2 2πh 2.09πh Convergence speed 2

20 N 80 0.0001 2πh

τk , τ k 2N 12 0.00003

Scheme

Fivepoints

Upper relaxation

(1)

(2)

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Table 6.15

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Convergence of the FDM.

Finite differences m×n

Temperature T (0.5; 0.5)

Number of elements

Temperature T (0.5; 0.5)

4×4 8×8 16 × 16 32 × 32 64 × 64 128 × 128

0.37 0.30 0.27 0.26 0.25 0.25

4 8 16 32 64 128

0.26 0.25 0.24 0.25 0.25 0.25

z

T=1

1 (0.5; 0.5)

T=0

T=0

0

Fig. 6.27

T=0

1

x

The tested squared plane.

algebraic direction with five- and nine-point approximations in the application to the problem (6.56) for the squared space. Results reported in Table 6.14 imply that for the squared space the method of variable accelerations is the most efficient. The next one is that of upper relaxation with the nine-point approximation. The method of upper relaxation can be applied to an arbitrary space since the optimal iteration parameter is computed in a computational step which makes it more universal in comparison to other methods. Besides, the method of upper relaxation is simple to realize and requires a minimum computer memory value (one working field N ). On the basis of numerical experiments, one may conclude that for the considered type of problems the most economical one is the method of upper relaxation h1 = h2 = h = 0.0625, εit = 10−6 . Next, we also consider the convergence of the second-order FDM and the FEM for the squared space shown in Fig. 6.27. The data

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Table 6.16

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HBCs.

reported in Table 6.15 allow us to conclude that the convergence is observed for partition n × m = 64 × 64, whereas in the FEM we require partition into 32 elements. In this work, the temperature field is defined by the FDM of the second-order accuracy, i.e. the approximation regarding the spectral coordinate x ∈ [0; 1] in the Euler–Bernoulli beam equations is realized via the FDM with the second-order accuracy. We consider vibration of a flexible Euler–Bernoulli beam embedded in a temperature field with three kinds of BCs of the heat transfer equations, which are reported in Table 6.16. 6.3.5

Reduction of PDEs to ODEs

In order to reduce our PDEs to a system of ODEs regarding the time coordinate, the finite difference approximations are used applying Taylor series development in the neighborhood of xi . Let us consider the mesh space GN = {0 ≤ xi ≤ 1, xi = i/N, i = 0, . . . , N }. PDEs (6.31)–(6.32) are reduced to the second-order ODEs with respect to time. In each mesh node, we get the system of equations i = 0, . . . , n,

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where n denotes the number of partitions regarding a spatial coordinate: Λx (·i ); Λx2 (·i ); Λx4 (·i ) are the difference operators of order O(Λ2 ), where ∆ is the step of the spatial coordinate. The ODEs are as follows
 u ¨t = Λx2 (ui ) + Λx (wi )Λx2 (wi ) − Λx NiT ,  1 1 w ¨t + εw˙ t = 2 − Λx4 (wi ) + Λx2 (ui )Λx (wi ) + Λx2 (wi )Λx (ui ) λ 12
 3 + (Λx (wi ))2 Λx2 (wi ) − Λx2 (wi )NiT + Λx (wi )Λx NiT 2  −Λx2 (MiT ) + q . (6.86) Both boundaries and initial conditions are reduced to their finite difference counterpart forms. In the given system of equations, the temperature terms are provisionally computed depending on the kind of BCs for the heat transfer equations. The obtained system of the second-order ODEs is reduced to ODEs of the firstorder, and hence they are solved using the fourth-order Runge–Kutta method. 6.3.6

Reliability of results

Reliability of the obtained results for the flexible beam vibration without the temperature field has been discussed in the first part of this book. It is necessary to consider the reliability and validity of results regarding beam vibrations within the stationary temperature field (see Table 6.17). The applied BCs are of the first kind for the heat transfer equation with the temperature intensity of T = 300 (Table 6.16). Charts of vibration regimes are reported in Table 6.17. They present characters of nonlinear dynamic processes obtained with the help of FFT. An important question arises: How to define the optimal number (n) of the intervals of partitions of frequency ωp and amplitude q0 of the excitation load? One may observe that the chart resolutions do not play a sufficient role in the visual analysis of charts with 200 × 200 pixels. However, the charts of high resolutions allow to define boundaries between

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Charts of the beam vibration type.

different vibration regimes more precisely. Therefore, further analysis is carried out with the resolution of 200 × 200. 6.3.7

Chaos versus heat excitation

In order to investigate the influence of the temperature field intensity, the following parameters are taken: chart resolution: 200 × 200; number of nodes: n = 80; time step: ∆t = 0, 00390625 = 1/256; time interval: t ∈ [1836; 2348); load interval: q0 ∈ [0; 200000]; frequency interval: ωp ∈ [0; 10]; dissipation factor: ε = 1; temperature field computed for the BCs of the first type (Table 6.16). We investigate function wmax (q0 ) depending on temperature magnitude and maximum beam deflection wmax in its center with respect to transversal beam axis Z versus q0 for fixed frequency ωp = 5 (Fig. 6.28). The color diagram reported in the figures corresponds to that of Table 6.17. The occurrence of a chaotic regime (white color) yields a sudden increase in beam deflection. Observe that the maximum amplitude has been computed on the whole time interval. The influence of temperature T = 100 does not yield any change in the vibration amplitude. Furthermore, it does not change the vibrational regime. This is why in our further investigations the temperature influence is not considered. Charts showing the influence of temperature on the vibration regime for the first kind of BC of the heat transfer equation are reported in Table 6.18.

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6 T=0 T=100 T=200 T=300

4 w(0.5)

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5

3

2

1

0

0

2

4

6

8 q

Fig. 6.28

6.3.8

10

12

14 4

x 10

Deflection of the beam center for different temperatures.

Influence of BCs

In order to study the influence of BCs on beam vibrations the following relations are monitored: a color palette of the temperature field vibration in the beam space with respect to the beam length and thickness; the relations and scales of the vibration character versus intensity of the external load reported with the help of the power spectrum (HSA) and by tracing of the first four Lyapunov exponents (LPN). We fix the frequency of excitation ωp = 5.

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Table 6.18

6.3.9

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The influence of temperature on vibration charts.

LEs

In order to compute the LEs for flexible Euler–Bernoulli beams embedded in a stationary temperature field, Wolf’s algorithm is applied [Wolf et al. (1985)]. Tables 6.19–6.21 show computational results for the flexible beams with the BCs ∂w (0, t) = Mx (1, t) = 0, w (0, t) = w (1, t) = u (0, t) = u (1, t) = ∂x (6.87) and for three types of the temperature field (BCs for the heat transfer equation are shown in Table 6.16). In these figures, the following notation is introduced: the first row color palette presents  the temperature field change of coordinates (x ∈ [0; 1]), z ∈ − 12 ; 12 , the second row shows changes wmax (0, 5) - q0 for three values of the temperature field intensity (T = 0; 200; 300) as well as the scales of the vibrations’ character frequency spectrum (FFT). Besides, the scales of all four LEs for the following five particular cases are presented: 1. All four LEs are negative (periodic vibrations). 2. One of the LEs is positive, whereas the second one is negative (chaos).

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Table 6.19

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The influence of the first type of BCs on beam dynamics.

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Table 6.20

The influence of the second type of BCs on beam dynamics.

195

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Table 6.21

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The influence of the third type of BCs on beam dynamics.

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3. Two LEs are positive (chaos–hyperchaos). 4. Three LEs are positive (chaos–chaos–hyperchaos). 5. Four LEs are positive (deep chaos). The scales are constructed for T = 0; 200; 300. In the case of the deep chaos the temperature field damps the system chaotic regime. Beam zones of periodic vibrations for the first and second kinds of BCs increase essentially. Table 6.22 reports charts of the vibration regimes (simply supported beams) in the stationary temperature field of the first kind and for the temperature intensity T = 0; 200; 300. Charts shown in Table 6.22 allow us to construct a full picture of beam vibration regimes versus control parameters {ωp ; q0 } and temperature excitation T . Analyzing the charts, one may conclude that an increase in the temperature excitation implies an increase in the periodic zones. It means that chaotic zones decrease. This phenomenon was observed in the zone of the maximum investigated frequency as well as in the zone of the maximum excitation force. In other words, the increase in temperature allows us to keep the system working in a regular regime for large values of the control parameters. 6.3.10

Scenarios of transition into chaos (first kind BCs)

In order to analyze the influence of temperature field on beam vibrations more deeply, the following characteristics are taken into account: signal (w(0.5, t), w(x, t), x ∈ [0, 1]), power spectrum obtained via FFT, phase trajectories (w(w, ˙ w) ¨ for the given point of space coordinate x), Poincar´e section (w[w(t), w(t + T )]) as well as the amplitude characteristics in the space–time coordinates for characteristic interval {q0 } for fixed frequency ωp = 5. In what follows, we compare (a) signal, (b) Fourier power spectra, (c) phase trajectories, (d) Poincar´e maps, and (e) spatial-temporal surfaces. A comparison was made for different values of control parameter q0 and for frequency ω0 = 5. Tables 6.23–6.29 present the analysis of results of the solution for simple beam supports, for

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Table 6.22

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Beam vibration charts for different temperature values.

temperature intensity T = 0, T = 200, T = 300 and for three types of HBCs (see Table 6.16). The following conclusions can be formulated on the basis of the obtained results of computations (i) For q0 = 500, a periodic signal is observed. ωp = 5 dominates in the frequency power spectrum, phase portraits tend to a stable orbit, the Poincar´e map exhibits one point. In spatial-temporal domain, the signal repeats itself. However, for BC of kind II (T = 300) and for BC of kind III (T = 200 and T = 300) other vibration regimes are exhibited. New frequencies ω1 and

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Table 6.23 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 500).

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Table 6.24 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 6 · 103 ).

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Table 6.25 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 24 · 103 ).

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Table 6.26 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 30 · 103 ).

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Table 6.27 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 57 · 103 ).

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Table 6.28 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 80 · 103 ).

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Table 6.29 Time histories, FFT, 3D phase portraits, Poincar´e maps and beam deflection shapes for different BCs and temperatures (q0 = 95 · 103 ).

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(ii)

(iii)

(iv)

(v)

(vi)

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ω2 appear in the frequency spectra. Increasing q0 to 6·103 implies the occurrence of new frequencies, but they have a negligible effect on either the amplitude of vibrations or on the global system behavior; For q0 = 24 · 103 , the regularity is observed for T = 200 and T = 300 for BC of kind II and for BC of kind III. For T = 0, the system exhibits chaotic dynamics which is characterized by a set of frequencies visible in the Fourier spectrum as well as by the occurrence of non-periodic waves on the spatial-temporal surface; An increase in q0 up to q0 = 30 · 103 yields evident differences in the frequency spectrum. For BC of kind II we deal with the periodic behavior. For T = 0, BC of kind I (T = 200 ÷ 300) and BC of kind II (T = 200) the Poincar´e maps present two crossing orbits; For q0 = 57·103 , differences in the mentioned characteristics are more evident and similar characteristics are changed. Now the similarities exhibit BC of kind I and BC of kind II for T = 200 and 300, respectively; For q0 = 80 · 103 for all BC types and temperature values, excluding BC of kind I (T = 200), the Hopf bifurcation ω1 = 1/2ωp = 2.5 is detected; The amount of q0 equal 95 · 103 shifts the system dynamics to a chaotic regime. One may distinguish ω1 = 1/2ωp = 2.5 in the frequency spectra corresponding to the first Hopf bifurcation. However, for BC of kind II (T = 300) again a periodic dynamics appears.

Although temperature has a negligible effect on the magnitude of beam vibrations, it changes the vibration regime. Our investigations show that by changing the system parameters like BCs and the magnitude of temperature field one may control the system regular/chaotic dynamics. It has been illustrated and discussed how an increase in the temperature effect decreases the magnitude of chaotic zones. We have also shown that the temperature effect may have an influence on the scenarios of transition from regular to

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chaotic system dynamics. Now, changing the system parameters in the way indicated by the reported charts one may achieve the required system dynamics even though the beam is dynamically and thermally externally loaded. We have also detected, illustrated and discussed various aspects of chaotic dynamics of the flexible beam in the stationary temperature field, and novel dynamic phenomena are reported including: chaos–hyperchaos (two LEs are positive), hyper–hyperchaos (three LEs are positive) and deep chaos (four LEs are positive). The occurrence of the stationary temperature field essentially changes both magnitude and zones of the above mentioned characteristics. Finally, the occurrence of chaos– hyperchaos dynamics yields the birth of spatial-temporal chaotic dynamics. 6.3.11

Conclusions

We rigorously discussed the problem of the choice of the most suitable method to find solutions to the Laplace equations. Temperature field and its type essentially influence the character of vibrations of flexible Euler–Bernoulli beams for the considered type of boundary and initial beam conditions. One may also apply the temperature field to control the beam vibrations. 6.4 6.4.1

Curvilinear Planar Beams and Stationary Temperature Fields Introduction

We consider a flexible one layer thin curvilinear beam of length l, height h and geometric curvature kx = 1/Rx , where Rx is the beam curvature radius. The beam is loaded continuously and uniformly via load q(x, t) acting in the direction normal to the middle beam surface (Fig. 6.29). The mathematical model of the beam is based on the hypotheses presented in the first and second chapters of the book taking into account the hypothesis that for shallow spherical shells the shallow threshold is defined by the ratio of the shell arrow’s rise f to the

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h

l Rx

z Fig. 6.29

Rx

The studied curvilinear beam model.

smallest shell dimension l: owing to Reissner [Reissner (1946)] f /l ≤ 1/8, owing to Vlasov f /l ≤ 1/5, assuming that the geometry in space coincides with shell geometry in plane [Vlasov (1949)]. Mathematical model of the curvilinear beam is governed by the system of non-linear PDEs. They describe a motion of the beam element taking into account energy dissipation, and they are written in the non-dimensional form (regarding displacements [Volmir (1956)]) ∂2u ∂NxT ∂w ∂2u + L − − k (w, w) − = 0, x 3 ∂x2 ∂x ∂x ∂t2     1 ∂w 2 ∂2w 1 ∂4w ∂u 1 − kx w − + kx −w 2 − λ2 12 ∂x4 ∂x 2 ∂x ∂x 

 ∂ ∂w T N L1 (u, w) + L2 (w, w) + q − ∂x ∂x x −

∂2w ∂w ∂ 2 MxT = 0, − − ε1 2 2 ∂x ∂t ∂t

∂ 2 u ∂w ∂u ∂ 2 w + , L1 (u, w) = ∂x2 ∂x ∂x ∂x2 L3 (w, w) =

3 L2 (w, w) = 2 ∂w ∂ 2 w , ∂x ∂x2



∂w ∂x

2

∂2w , ∂x2

(6.88) where L1 (u, w), L2 (w, w), L3 (w, w) are nonlinear operators, w(x, t) is the beam normal deflection, u(x, t) is the beam longitudinal displacement, ε1 is the damping coefficient, E denotes Young modulus, h is the height of the transversal beam cross-section, γ is the beam

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unit weight density, g is the Earth acceleration, kx is the beam curvature, t is time, q = q0 sin(ωp t), where q0 is the amplitude of the external harmonic load, and ωp is the frequency of the external load. Non-dimensional parameters are introduced in the following way w ua t x a ¯= , u ¯ = 2, x ¯ = , t¯ = , λ= , w h h h a τ  a qa4 Eg a , ε¯ = , q¯ = 4 , τ = , p= (6.89) p γ p h E T 2 T kx a ¯ T = Nx l , M ¯ T = Mx l , T¯ = λT λ2 . , N k¯x = x x λ Eh3 Eh2 Bars over the non-dimensional parameters in Eq. (6.88) are omitted. The following BCs are introduced: one side has simple support (x = 0), whereas the second is clamped (x = a)

w(0, t) = w(a, t) = u(0, t) = u(a, t) = w
x (a, t) = w

xx (0, t) = 0, (6.90) and initial condition follows: w(x, 0) = w(x, ˙ 0) = u(x, 0) = u(x, ˙ 0) = 0.

(6.91)

For the curvilinear beams made of the isotropic material, the following heat transfer equation has been derived in the reference [Podstrigatch and Shvetz (1978)]: ∆T + 2kx ∂T ∂z = 0. Solution to the Laplace equation ∆T = 0 has been described in Section 6.3.2. In further investigations, owing to the introduced shallow conditions, we consider heat transfer equations of the Laplace type with boundary conditions (reported in Table 6.30). PDEs (6.88) are reduced to the Cauchy problem of the finite differences of the second-order accuracy, and they follow u ¨t = Λx2 (ui ) − kx Λx (wi ) + Λx (wi )Λx2 (wi ), 

1 1 1 w ¨t + εw˙ t = 2 − Λx4 (wi ) + kx Λx (ui ) − kx Λx (wi ) − Λx2 (wi ) λ 12 2 −Λx2 (ui )Λx (wi ) + Λx2 (wi )Λx (ui )

3 + (Λx (wi ))2 Λx2 (wi ) + q . 2

(6.92)

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Types of the HBCs.

The obtained second-order ODEs (6.92) with the associated boundary and initial conditions formulated with the help of the finite difference method are solved with the fourth-order Runge– Kutta method. Validity of the chosen Runge–Kutta method has been discussed in detail in Section 6.2.3. Since the equations governing dynamics of the flexible curvilinear beam differ from the linear beam only by the term including the beam curvature, we apply the method of solution presented in Section 6.3.1 for validity and reliability of

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Table 6.31

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Vibration charts for different kx (FFT).

the obtained results. At first, we consider vibration regimes of the curvilinear beams without the temperature field. Interval x ∈ [0; 1] is divided into 120 parts, λ = 100, ε1 = 1, with curvature kx = 0, 12, 48 with the help of the FDM. In Table 6.31, changes of the vibrations character for the set {q0 , ωp } are presented. Results are obtained using the FFT. The meaning of colour vibration characteristics is given in the table. In addition, the analysis has been supported by the LE charts. As in the charts given in Section 6.2.6, here the LE charts are constructed depending on the control parameters {ωp , q0 }. In Table 6.32, LE charts obtained using Wolf’s algorithm are presented, and the first four LEs are computed. Comparison of Table 6.31 with Table 6.32 shows a remarkable convergence of various types of analysis. Furthermore, each chart exhibits its own aspects of the dynamical process, whereas they all contribute to the full dynamical picture of the system. Owing to the charts analysis, an increase in the beam curvature yields an extension

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Charts of LEs for different kx .

of the regular zones (four LE are negative). After achieving stability loss, the vibration process is transited into deep chaos. The latter is the novel result, which cannot be found in the vibrations of beams without curvature or with a small curvature value. 6.4.2

BCs versus chaotic vibrations

We consider three types of beam BCs with the following curvature parameter kx = 48 and with the parameter λ = 100. 1. Two beam ends are clamped: w(0, t) = w(l, t) = u(0, t) = u(l, t) = wx
(0, t) = wx
(l, t) = 0. (6.93) 2. Two beam ends are simply supported: w(0, t) = w(l, t) = u(0, t) = u(l, t) = Mx (0, t) = Mx (l, t) = 0. (6.94)

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Table 6.33 BCs.

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Periodic and chaotic beam dynamics yielded by FFT for different

3. One beam side is clamped and the other one is simply supported: w(0, t) = w(l, t) = u(0, t) = u(l, t) = wx
(0, t) = Mx (l, t) = 0. (6.95) The remaining parameters are the same as in the previous case. The presented data indicate similar dependence as for the linear beam. Clamping is associated with the regular beam dynamics, whereas simple–simple support has less periodic zones than in the mentioned case, and finally the non-symmetric support yields the most strongly exhibited chaotic vibrations. Charts reported in Table 6.33 are constructed with the help of the frequency spectrum (FFT). 6.4.3

Scenarios of transitions into chaos

We have considered the beam with the curvature coefficient kx = 48 and the parameters n = 120, λ = 100, ε1 = 1. Analysis has been carried out for the fixed frequency (ωp = 5.7615) and various amplitudes of the harmonics excitation. In Tables 6.34–6.50, their cells describe the following characteristics: (1) frequency Fourier spectrum (t ∈ [1836, 2348]; x = 0.5, i.e. beam center), (2) pseudoPoincar´e map, (3) phase portrait, (4) modal portrait, (5) autocorrelation function, (6) beam displacement form in time instant t = 1836, (7) beam displacement form in time intervals t ∈ [1836, 1852], (8) 2D Morlet wavelet, (9) 3D Morlet wavelet. Analysis of the data reported in Tables 6.34–6.50 allows to formulate the following conclusions. For the amplitude of excitation q0 = 57,500, periodic vibrations appear. In the modal portrait the

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Table 6.34

Beam characteristics for q0 = 57,500, ωp = 5.7615.

Table 6.35

Beam characteristics for q0 = 62,500, ωp = 5.7615.

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Table 6.36

Beam characteristics for q0 = 64,500, ωp = 5.7615.

Table 6.37

Beam characteristics for q0 = 83,000, ωp = 5.7615.

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Table 6.38

Beam characteristics for q0 = 85,000, ωp = 5.7615.

Table 6.39

Beam characteristics for q0 = 86,000, ωp = 5.7615.

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Table 6.40

Beam characteristics for q0 = 86,500, ωp = 5.7615.

Table 6.41

Beam characteristics for q0 = 87,000, ωp = 5.7615.

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Table 6.42

Beam characteristics for q0 = 87,500, ωp = 5.7615.

Table 6.43

Beam characteristics for q0 = 88,500, ωp = 5.7615.

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Table 6.44

Beam characteristics for q0 = 94,500, ωp = 5.7615.

Table 6.45

Beam characteristics for q0 = 95,000, ωp = 5.7615.

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Table 6.46

Beam characteristics for q0 = 95,500, ωp = 5.7615.

Table 6.47

Beam characteristics for q0 = 95,500, ωp = 5.7615.

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Table 6.48

Beam characteristics for q0 = 103,000, ωp = 5.7615.

Table 6.49

Beam characteristics for q0 = 104,000, ωp = 5.7615.

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Beam characteristics for q0 = 198000, ωp = 5.7615.

attractor is exhibited, but its magnitude is so small that it is not sufficiently presented via both phase portrait and Fourier power spectrum. Relatively small increase in the loading amplitude yields the occurrence of the independent frequency as well as a set of dependent frequencies. The phase portrait and Poincar´e map exhibit an attractor. For the amplitude interval q0 = 64,500 − 83,000, the situation does not change globally, though some changes in Poincar´e maps as well as in the energy distribution along the frequencies are observed. For the amplitude q0 = 85,000, the collapse of Poincar´e maps and phase portraits are visible, whereas the jump-type system transition regarding the amplitude q0 = 86,000 is shown. However, the so far described state is not robust, and already in interval q0 = 86,500 − 87,000 a transition into chaos appears (2D Marlet wavelets exhibit frequencies intermittency, whereas the 3D wavelet shows the energy transition into low frequencies, the Poincar´e map presents the attractor, which is also illustrated by phase and modal

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portraits, the autocorrelation function rapidly increases). It should be mentioned that the so far illustrated changes are not only associated with vibrations character, but also with characteristics such as the maximum deflection. For amplitude q0 = 86,500 it achieves the value 1.75, for q0 = 86,500 it is 4.25, and for q0 = 87,500 it is equal to 10.5 (this is visible on the beam deflection form). Deflection maximum is moved to the beam center along coordinate x, and the system loses its sensitiveness to the non-symmetric BC. In practice, the thin curvature beam buckles and loses ability to resist to the external load, which is well seen on the constructed beam deflection form. Further analysis has rather only a theoretical meaning, since the curvilinear beam deflection is outside the required boundaries of the earlier introduced hypotheses. For all amplitudes of the external load reported further, phase portraits exhibit the Lorenz attractor. Attractor of the Poincar´e section is consequently destroyed to achieve a cloud of points. Autocorrelation function suddenly decreases. The Fourier frequency spectrum becomes noisy step by step, and in the last tables we observe the broadband spectrum and the deep chaos. For the amplitude of excitation load less than q0 = 57,500, harmonic vibrations with excitation frequency ωp = 5.7615 appear. Increasing the loading amplitude yields the birth of ω1 = 2.614, which is irrational with respect to the fundamental frequency, and in practice simultaneously the linearly dependent frequencies appear ω2 = ωp − ω1 . The process is further developed, and the Rayleigh– Takens–Newhouse scenario is observed. It is clearly demonstrated for q0 = 62,500. The scenario is robust up to q0 = 86,000 and after that the system returns into a harmonic regime. The increase in q0 = 86,500 up to q0 = 87,000 implies the intermittency phenomenon (it is visible on the 2D wavelet). Strong noise perturbation is observed in the spectrum, but the following frequencies ω1 = 2.405, ω2 = ωp − ω1 are distinguishable, and the same Ruelle–Takens–Newhouse scenario is followed. This scenario takes place up to q0 = 87,500. After that, the system exhibits two scenarios simultaneously. Namely, for q0 = 88,500 we observe the fundamental frequency ωp , two frequencies ω1 and ω2 = ωp − ω1 from the previous Ruelle–Takens–Newhouse scenario, and the frequencies ω3 = ωp /2 = 2.884, ω4 = ωp /4 = 1.424,

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ω5 = 3/4ωp = 4.308. As it can be seen from frequencies ω3 , ω4 , ω5 , this is the second period doubling bifurcation scenario of transition into chaos. The system prolongs to stay in this doubled regime. However, for amplitude q0 = 95,500 the system undergoes a sudden change, and we observe the frequency ω1 = ωp /2 = 2.884 as well as frequencies ω2 = 1.387, ω3 = 1.497, ω4 = 4.271, ω5 = 4.369. It is remarkable that frequencies ω2 , ω3 are localized in the neighborhood of ωp /4 = 1.436, whereas frequencies ω4 , ω5 are gathered in the neighborhood of 3/4ωp = 4.307, i.e. Hopf bifurcations take place in both explicit and implicit forms, and the damped Ruelle–Takens– Newhouse scenario is observed. A more detailed analysis of vibrations of the curvilinear beam from q0 = 95,000 to q0 = 96,000 shows that the system jumps a few times between two dynamical states. The doubled system dynamics is robust also for high values of q0 , until deep chaos is reached with the broadband chaotic foundation of the Fourier spectrum. 6.4.4

Chaos versus temperature fields

Type 1 problem Tables 6.51–6.56 present the analysis of the stationary temperature field for HBCs of types 1–3 (BCs of the temperature field are given in Table 6.30), for a beam with curvatures kx = 12; 24. The first graph presents the color palette of the temperature field along the beam thickness and length. Further, wmax (0, 5)(q0 ) and scales yielded by FFT and LEs for T = 0; 200; 300 are reported. The scales exhibit periodic vibrations (all LEs are negative), chaos (one LE is positive), hyperchaos (two LEs are positive), hyper-hyperchaos (three LEs are positive) and deep chaos (four LEs are positive). Analysis of the obtained results indicates that the increase in the temperature implies the increase in the periodic zones. Zones of different types of chaos are simultaneously decreased. Parameter kx essentially influences the character of vibrations of curvilinear beams embedded in the stationary temperature field. The increase in the temperature field intensity causes the death of the deep chaos. The chaos–hyper– hyperchaos also almost vanishes while q0 is increased.

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Table 6.51 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T1 , kx = 12).

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Type 2 problem

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Table 6.52 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T2 , kx = 12).

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Table 6.53 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T3 , kx = 12).

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6.4.5

Chaos versus beam curvature

Type 1 problem

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Table 6.54 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T1 , kx = 24).

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Type 2 problem

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Table 6.55 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T2 , kx = 24).

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Type 3 problem

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Table 6.56 Dependence wmax (q0 ) and scales of HSA and LPE for different T values (T3 , kx = 24).

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Influence of HBC type and geometric parameter kx

In Table 6.57, graphs of changes of wmax (0.5)(q0 ) for three types of HBCs (see Table 6.16) versus kx = 0; 12; 24 for the given initial deflection, which are taken as initial conditions while solving the Cauchy problem, are reported. Initial system configuration is defined via the set-up method presented in the reference [Feodosev (1963)]. The increase in a parameter yields the increase in the beam initial bending in the temperature field. All presented diagrams consider the curvature parameter kx = 0. 6.4.7

Solution to heat transfer equation (influence of kx)

In this section, in order to define the temperature field for a curvature beam, a 3D dimensional heat transfer equation is solved directly. For beams made of an isotropic material, the stationary heat transfer equation takes the following form [Podstrigatch and Shvetz (1978)]: W0 ∂2T ∂T ∂2T =− , + + 2kx 2 2 ∂x ∂z ∂x λ Table 6.57

Function wmax (q0 ) for different Ti (i = 1, 2, 3) and kx .

(6.96)

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Table 6.58 Temperature intensity in the beam centre for different kx and beam partitions.

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Number of elements 32 64 128

kx = 12

kx = 24

Finite differences m×n

kx = 12

kx = 24

0.28 0.28 0.28

0.35 0.35 0.35

32 × 32 64 × 64 128 × 128

0.32 0.28 0.28

0.36 0.31 0.31

Fig. 6.30

Functions wmax (q0 ) for different kx .

where λ is the heat transfer coefficient of an isotropic body. Since in the considered problems we do not take into account the internal heat source, we take W0 = 0 in Eq. (6.96), which eventually takes the following form: ∇2 T + 2kx

∂T = 0. ∂x

(6.97)

In Table 6.58, the values of the temperature in the curvilinear beam center S = {0 ≤ x ≤ 1; 0 ≤ z ≤ 1} and T (0.5; 0.5) are defined for the BC given in Table 6.31 and Fig. 6.30. Temperature is obtained through the method of BCs (numbers of elements and finite differences are shown in Table 6.58). Number of partitions should be increased while increasing kx in order to get reliable results from the finite differences method.

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Euler–Bernoulli Beams

233

In Fig. 6.30, the dependencies wmax (0.5)(q0 ) for type 1 problem (Table 6.30) for the values of kx = 12 and kx = 24 are reported. Dotted (solid) curves are associated with lack (existence) of kx . Analysis of the mentioned dependencies yields the result that the increase in the curvature kx changes essentially wmax (0.5)(q0 ), which is used to estimate the stability of the beam versus the temperature field. 6.5 6.5.1

Flexible Curvilinear Beam in Stationary Temperature and Electrical Fields Problem treatment

The so far used mathematical model has been extended to derive curvilinear beam vibrations in both temperature and electric fields. The beam surface is covered with electricity conducting polarized layer (Fig. 6.31). We study the curvilinear beam of length a (along the axis Ox), having thickness h (along axis Oz) and the unit width. Beam material properties are described in the frame of the linear theory of piezo-effects. The beam is fixed, and subjected to the transversal distributed load action q(x, t). On the surfaces z = ±h, the electric action of the potentials difference V (t) is applied. Surfaces x = 0, x = a are not covered with the electrodes. We have uz = u − z

∂w , ∂x

wz = w,

−h ≤ z ≤ h,

(6.98)

where u = u(x, t), w = w(x, t) are longitudinal displacement and deflection of the beam middle line, respectively. q=q0sinωpt x

h

a z

Fig. 6.31

Rx

Rx

Flexible curvilinear beam with the electricity conducting layer.

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Taking into account (6.5.1), we obtain
 ∂u 1 ∂w 2 ∂2w + −z 2, εxx = ∂x 2 ∂x ∂x Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

9in x 6in

εzx = 0.

(6.99)

Vectorial electric field characteristics are: D = D(x, z, t) denotes induction, E = E(x, z, t) denotes intensity. State equations, taking into account linear equations of a direct and inverse piezo-effect and pyro-electric-effects, have the following form σxx = cE 11 (εxx − αT T ) − e31 Ez , Dx = εS11 Ex ,

(6.100)

Dz = εS33 Ez + e31 εxx + gpyr T. State equations include physical constants of the beam material: cE 11 is the elasticity modulus for a constant electric field, e31 is the piezo-electric coefficient, εS11 , εS33 denotes dielectric permeability for constant deformation, αT is the coefficient of the linear heat extension, T = Θ(x, z, t) − T0 is the temperature with respect to the temperature of the surrounding state T0 ; gpyr is the pyro-electric coefficient; gpyr = (2 . . . 3) · 10−3 in the direction along the prepolarization and gpyr = 0 for other directions. State equations (6.100) are derived for the case when the beam material is initially polarized along beam thickness. 6.5.2

Equations of both electrostatics and motion

Equations of the introduced electrostatics follow: ∂Dx ∂Dz + = 0, ∂x ∂z

Ex = −

∂ψ , ∂x

Ez = −

∂ψ , ∂z

(6.101)

where ψ = ψ(x, z, t) is the electric potential. Taking into account (6.99) and (6.100), the first of equations (6.101) is recast to the following form: e31 ∂ 2 w εS11 ∂ 2 ψ ∂ 2 ψ gpyr ∂T = 0. + S + − S ∂z 2 eS33 ∂x2 ε33 ∂x2 ε33 ∂z

(6.102)

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Euler–Bernoulli Beams

235

Now equations of motion will be derived taking into account a dissipation. We introduce the following mechanical stresses,  h  h ∂ ∂Mx = σxx dz, Qx = σxx zdz, (6.103) Nx = ∂x ∂x −h −h and after a few transformations we get  
2  h 1 ∂w e α ∂u 31 T + + V (t) − T dz , Nx = 2hcE 11 ∂x 2 ∂x 2h −h 2hcE 11  h 2h3 E ∂ 3 w ∂T E c11 3 − c11 αT zdz. Qx = − 3 ∂x −h ∂x (6.104) Equation of motion projected onto the axis Ox has the following form ˙ = 2hρ(¨ u + ε1 u)

∂Nx , ∂x

(6.105)

and taking into account (6.104), it is recast to the following form αT ∂2u + L3 (w, w) − ∂x2 2h



h

−h

ρ ∂T dz = E (¨ u + ε1 u). ˙ ∂x c11

(6.106)

Equation of motion projected onto axis Oz is: ∂ 2 w ∂Nx ∂w ∂Qx + Nx 2 + + q = (2hρ)(w¨ + ε1 w), ˙ ∂x ∂x ∂x ∂x

(6.107)

which taking into account (6.101), is recast to the following form
 h2 ∂ 4 w q e31 + + L1 (u, w) + L2 (w, w) − 3 ∂x4 2hcE 2hcE 11 11
  h h ∂2w ∂2T ∂ 2 w αT zdz + T dz × V (t) 2 − (6.108) 2 ∂x 2h ∂x2 −h −h ∂x 
  ρ ∂w h ∂T dz = ˙ (w ¨ + ε2 w). + ∂x −h ∂x cE 11

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Equations (6.102), (6.104), (6.108) are given in non-dimensional forms. We denote dimensional quantities with the wave (∼), whereas without thewave the non-dimensional quantities are denoted as Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

follows: c =

cE ˜ = ax, 11 /ρ is the velocity, d31 is the piezo-modulus, x 2

a ˜ = 2hw, λ = 2h , t˜ = ac t, ε˜1,2 = ac ε1,2 , z˜ = 2hz, u ˜ = (2h) a u, w 2h 4 1 2h 1 2h ˜ ˜ ˜ q˜ = cE 11 ( a ) q, V = λ2 ( d31 )V , ψ = λ2 ( d31 )ψ, T = T0 T . Equations (6.102), (6.104), (6.108) are derived and they have the following form

1 εS11 ∂ 2 ψ ∂ 2 ψ ∂2w ∂T 2 = 0, (6.109) + + − kpyr λ2 ∂x2 λ2 εS33 ∂x2 ∂z 2 ∂z  1/2 ∂T ∂2u 2 dz = u ¨ + ε1 u, + L3 (w, w) − λ (αT T0 ) ˙ 2 ∂x −1/2 ∂x

k12

(6.110) 1 λ2




1 ∂4w ∂2w 2 + q + k V (t) L1 (u, w) + L2 (w, w) − 2 12 ∂x4 ∂x2   1/2 2 ∂ 2 w 1/2 ∂ T zdz + T dz (αT T0 ) 2 ∂x2 −1/2 −1/2 ∂x   ∂w 1/2 ∂T dz = w ¨ + ε2 w. ˙ + ∂x −1/2 ∂x



(6.111)

In Eqs. (6.109)–(6.111), k12 = e31 d31 /εS33 and k22 = e31 /(cE 11 d31 ) are non-dimensional coefficients of the electro-chemical coupling and 2 = gpyr T0 d31 /εS33 is the non-dimensional coefficient of the pyrokpyr electric coupling, whereas ∂u ∂ 2 w ∂ 2 u ∂w , + 2 ∂x ∂x2 ∂x ∂x 
3 ∂ 2 w ∂w 2 , L2 (w, w) = 2 ∂x2 ∂x L1 (u, w) =

L3 (w, w) =

∂w ∂ 2 w 1 = L1 (w, w). 2 ∂x ∂x 2

(6.112)

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Table 6.59

6.5.3

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Chart of beam vibrations for different V .

Initial and BCs for the electric potential

For ψ = ψ(x, z, t), BCs in the non-dimensional quantities ψ(x, −1/2, t) = −V (t)/2, ψ(x, 1/2, t) = V (t)/2 (0 ≤ x ≤ 1, t > 0) ∂ψ ∂x = 0 for x = 0, x = 1 (−1/2 ≤ z ≤ 1/2, t > 0) are taken. Initial conditions expressed via the non-dimensional quantities are: ψ = 0, ∂ψ ∂t = 0 for t = 0 (0 ≤ x ≤ 1, −1/2 ≤ z ≤ 1/2). Modeling parameters are taken to be the same as in the previous section. As it has been seen from the data reported in Table 6.59, for the uncoupled problem, the influence of the electric field on the beam chaotic dynamics is negligible. 6.5.4

Concluding remarks

Investigation of the flexible curvilinear beam embedded in the stationary field supports our investigations carried out in previous chapters. For a properly defined temperature field, the chaotic dynamics can be shifted into a required regime. Temperature field initiates the occurrence of the beam curvature, which causes an increase in the beam stiffness. Therefore, the temperature can be used as an important parameter to control chaotic beam vibrations. However, in the frame of the assumed model, the temperature does not yield plastic deformations in the beam material and does not decrease its elasticity. In the stationary field, the chaotic dynamics is practically not changed, when the electric field is applied.

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6.6

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6.6.1

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Beams with Elasto-Plastic Deformations Mathematical model

In this section, we study vibrations of Euler–Bernoulli beams taking into account the elasto-plastic deformations. The rectangular system of coordinates has been introduced in the following way: axis OX goes along the beam axis, whereas axes OY and OZ coincide with the main axes of the beam’s cross-sections. We consider the beam of length l with the rectangular transversal cross-section of height h subjected to the transversally applied load per unit length p(x, t), acting in the direction of the axis OZ and changing in time t. Beam vibrations are analyzed using the Euler–Bernoulli hypothesis, i.e. we assume that the transversal cross-sections, normal to the beam axis, in the pre-bending state remain in plane and they are normal to the bended axis as well as are not deformed in the plane. Besides, we consider only small elasto-plastic deformations, and we do not apply any external forces in the direction of the OX axis. However, we consider the case of a physically non-homogeneous beam, i.e. we take into account that the Young modulus E and Poisson’s coefficient ν depend on the deformable system state in a given point. They are coupled with the shear modulus G and the value deformation K via the following formulas E=

9KG , 3K + G

ν=

1 3K − 2G · , 2 3K + G

(6.113)

where K = K0 = const, and the shear modulus is defined via the formula G=

1 σi (ei ) · , 3 ei

(6.114)

where is the σi is the stress intensity, ei is the strain intensity, which taking into account (2.2.9), takes the following form ei =

2 |εx |. 3

(6.115)

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Besides, we have taken into account the so-called physical nonlinearity, i.e. the dependence σi (ei ) is defined in the following three ways [Krysko (1976)]: Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

1. Ideally elasto-plastic body: σi = 3G0 ei for ei < es , σi = σs for ei ≥ es ,

(6.116)

where es stands for the beam material deformation. 2. Elastic-plastic body with linear reinforcement: σi = 3G0 ei for ei < es , σi = 3G0 es + 3G1 (ei − es )

for ei ≥ es .

3. We consider a diagram for the pure aluminum: 
 ei , σi = σs 1 − exp − es

(6.117)

(6.118)

though we can take into account other materials too. Relation σi (ei ) can be arbitrarily defined including even a table form, which is obtained through investigation of the standard probes of the stress– strain tests. The governing equation of the beam longitudinal vibrations has the following form [Volmir (1972)] γ ∂2w ∂w ∂ 2 Mx h 2 −ε = 0, + p − 2 ∂x g ∂t ∂t

(6.119)

where (with the help of introduction of the dissipation coefficient) we take into account also an internal friction. If one assumes that the Young modulus is constant, then the bending torque in the transversal cross-section takes the form  ∂2w ∂2w Ez 2 dydz = − 2 EJ. (6.120) Mx = − 2 ∂x ∂x S 2

4

Consequently, ∂∂xM2x = −EJ ∂∂xw4 , Eq. (6.120) is linear and takes the form of (6.120). In order to convert Eq. (6.120) into its

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non-dimensional counter-part form, the following non-dimensional relations are introduced

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w = w · h,

x=x ¯ · l,

z = zh,

E = E0 · E,

t=

p=p·

1 l2 ·

1/2 · t¯, h E0 g γ

l2 h2 E0

(6.121)

,

where E0 is a constant value. Taking into account (6.121) and assuming that the beam has a rectangular cross-section, the beam height is h, and the beam thickness can be taken as a unit, the following non-dimensional relations are obtained εx =

h2 εx , l2

E0 h2 E0 h4 σx , Mx = 2 Mx , 2 l l  h/2  1/2 Ez 2 dz = E0 h3 E1 (x) = σx =

−h/2



×

−1/2

1/2

−1/2



1/2 −1/2

(6.122)

Ez 2 dz = E0 h3 E1 (x) ,

and hence the non-dimensional equation (6.119) is ∂2w ∂w ∂ 2 Mx = 0. + p − −ε· 2 2 ∂x ∂t ∂t

(6.123)

Since we further apply only the non-dimensional quantities, the bars are omitted. In order to find a solution it is more convenient to recast (6.123) to the following form ∂2 ∂2w = p − ∂t2 ∂x2




∂2w E1 (x) · ∂x2

 −ε·

∂w . ∂t

(6.124)

Equation (6.124) should be supplemented with one of the earlier introduced BCs (6.26)–(6.28), and we take the initial conditions defined by formula (6.29).

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6.6.2

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Fundamental hypotheses and mathematical model

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We consider beams in the frame of the following fundamental hypotheses: (1) Longitudinal beam dimension is larger than the beam crosssectional dimensions. (2) Beam axis presents a straight line, and planes of main axes of the beam cross-section as well as the beam axis are mutually perpendicular. The applied rectangular system of coordinates is introduced in the following way: axis OX coincides with the beam axis, whereas axes OY and OZ coincide with the main axes of the beam cross section. Contrary to the previous cases, in this chapter the system behavior is studied by taking into account the so-called geometrical nonlinearity, i.e. we take into consideration the middle surface deformation as well as a torsional torque. This means that the considered stress–strain relations are not limited to the small elastoplastic deformations. Consequently, the first relation of (6.122) takes the following form εx = εx,cr + εx,u


 ∂u 1 ∂w 2 ∂2w + = −z 2. ∂x 2 ∂x ∂x

(6.125)

It is assumed that the considered system is physically linear, i.e. the Young modulus E and Poisson’s coefficient ν do not depend on the system deformable state in a beam point (they are constant). Intensity of the external loads has been taken into account with regard to the normal direction, and it is defined by the transversal load value per the beam length unit. We introduce the longitudinal force Nx in the transversal cross-section associated with the normal stress σx :  σx dydz. (6.126) Nx = S

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In order to derive the beam governing equations we use the principle of virtual work [Volmir (1972)], which takes the following form  l  σx δεx dxdydz − pδwdx Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

V

0



− S

 → → → X− n δu + Y− n δv + Z− n δw dv = 0.

(6.127)

The last term of the left-hand side of Eq. (6.127) takes various forms depending on the BCs, which will be considered further. The first two terms can be transformed by taking into account (6.125) as well as formulas (6.120) and (6.126). Owing to the integration carried out by parts, formula (6.127) takes the following form  l    Nx δu + w δw − Mx δw − pδw dx 0 (6.128)   → → → X− − n δu + Y− n δv + Z− n δw dv = 0. S

Then, after a few additional transformations, the following system of equations is obtained M  x + (Nx w ) + p = 0, N  x = 0.

(6.129)

System (6.129) governs a static case of the considered problem. Now, we proceed to the dynamical case taking into account the internal damping, and (6.129) takes the form [Volmir (1972)]:
 γ ∂2w ∂ ∂w ∂w ∂ 2 Mx N + p − h 2 −ε = 0, + x 2 ∂x ∂x ∂x g ∂t ∂t (6.130) ∂Nx γ ∂ 2 u − h 2 = 0. ∂x g ∂t Validity of (6.130) can be approved assuming εx,cr = 0, and hence 4 ∂ 2 Mx = −EJ ∂∂xw4 , and ∂x2  h/2   1/2  h/2  ∂ 2 w 1/2 z 2

σx dzdy = −E 2 dy = 0, Nx = ∂x −1/2 2 −h/2 −1/2 −h/2 (6.131) and the system of equations (6.130) takes the linear form.

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Equations (6.130) are transformed to their counterpart form being more suitable for computation purposes. Namely, we introduce the following transformation (observe the integration along the beam  1/2 1/2 thickness yields −1/2 dy = y|−1/2 = 1): 

(1)

Mx =



h/2

−h/2

σx zdz =

−h/2

Eεx zdz


 ∂2w ∂u 1 ∂w 2 + − z 2 zdz =E 2 ∂x ∂x −h/2 ∂x 
   h/2 ∂u 1 ∂w 2 + zdz =E ∂x 2 ∂x −h/2 

(2)

h/2



h/2

 Eh3 ∂ 2 w ∂ 2 w h/2 2 · z dz = − . −E 2 ∂x −h/2 12 ∂x2  h/2  h/2 σx dz = Eεx dz Nx = −h/2

(6.132)

−h/2


 ∂2w ∂u 1 ∂w 2 + − z 2 dz =E 2 ∂x ∂x −h/2 ∂x  
  h/2 ∂u 1 ∂w 2 + dz =E ∂x 2 ∂x −h/2 

h/2

∂2w −E 2 ∂x





h/2 −h/2

zdz = Ehεx,cr .

Then, system (6.130) can be recast to the following form:
 ∂w Eh3 ∂ 4 w ∂ ∂w γ ∂2w h 2 =p− ε −ε , + Eh x,cr g ∂t 12 ∂x4 ∂x ∂x ∂t ∂εx,cr γ ∂2u . =E g ∂t2 ∂x

(6.133)

(6.134)

System of equations (6.134) should be transformed to the associated non-dimensional form. We have used relations (6.121), where 2 we have added u = uhl and formulas (6.122), whereas instead of εx ,

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we have applied

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εx,cr


 ∂u 1 ∂w 2 l2 + = = 2 εx,cr . ∂x 2 ∂x h

Then, system (6.134) takes the following form
 E0 ∂ 4 w ∂w ∂ ∂w ∂2w , −ε = p − + E ε 0 x,cp 2 4 12 ∂x ∂x ∂x ∂t ∂t ∂2u

E0 l2 ∂εx,cp 2 = h2 · ∂x . ∂t

(6.135)

(6.136)

The bars over the non-dimensional quantities will be omitted. The mentioned equations should be supplemented by boundary and initial conditions which should include boundary and initial conditions for the function u. We have taken u = 0 for x = 0, x = 1, which allows to study simple and clamped support depending on the condition applied to function w. Then, BCs will be given in one of the following ways. 1. Clamping on both beam sides ∂w = u = 0 for x = 0, ∂x 2. Simple support on both sides w=

x = 1.

(6.137)

∂2w = u = 0 for x = 0, x = 1. (6.138) ∂x2 3. One beam end is clamped, whereas the other is simply supported w=

w=

∂2w = u = 0 for x = 0, ∂x2

w=

∂w = u = 0 for x = 1. ∂x (6.139)

Initial conditions, in the general form, are as follows ∂u = g2 (x) for t = 0. ∂t (6.140) PDEs (6.136) are transformed to a system of four ODEs for each node regarding x [Samarsky and Nikolaev (1978)]. Introducing the w = f1 (x);

∂w = f2 (x); ∂t

u = g1 (x);

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following notation: y1 = w, y2 = (1977)], we get

∂w ∂t ,

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y3 = u and y4 =

∂u ∂t

[Samarsky

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dy1i = y2i , dt E0 y1i+2 − 4y1i+1 + 6y1i − 4y1i−1 + y1i−2 dy2i =p− · dt 12 h4x

∂ ∂y1 εx,cr · + − ε · y2i , ∂x ∂x i

(6.141)

dy3i = y4i , dt
2 i+1 εx,cr − εi−1 l dy4i x,cr = , dt h 2hx where

i−1 i+1 εi+1 ∂y1 − y1i−1 ∂ x,cr − εx,cr y1 εx,cr · = · ∂x ∂x 2hx 2hx i y i+1 − 2y1i + y1i−1 + εix,cr · 1 . h2x

(6.142)

The system (6.141) is supplemented with one of the following counterpart BCs: 1. Clamping on both beam sides y10 = y1n = 0;

y1n+1 = y1n−1 ;

y1−1 = y11 ;

y30 = y3n = 0. (6.143)

2. Simple support on both sides y10 = y1n = 0;

y30 = y3n = 0. (6.144) 3. One beam end is clamped, whereas the other is simply supported y10 = y1n = 0;

y1−1 = −y11 ,

y1−1 = −y11 ,

y1n+1 = −y1n−1 ;

y1n+1 = y1n−1 ;

y30 = y3n = 0 (6.145)

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and the following initial conditions:





0,n 0,n = f1 x0,n ; y2 = f2 x0,n ; y1 t=0 t=0







= g1 x0,n ; y40,n = g2 x0,n . y30,n t=0

(6.146)

t=0

In fact, the given system is composed of 4(n + 1) ODEs. It has four groups consisting of n + 1 same type of equations, i.e. it can be considered through matrix layers, and in each layer the Cauchy problem should be solved for four ODEs. Its solution can be obtained with the help of the fourth-order Runge–Kutta method [Samarsky (1982)], and after that we may proceed to the next layer. In what follows, we study the problem of beam vibrations with the rectangular cross-section and with simply supported both ends. The beam in the initial time instant has been in the equilibrium position, i.e. initial conditions for w and u follow:

dw

du

= 0; u|t=0 = 0; = 0. (6.147) w|t=0 = 0; dt t=0 dt t=0 Recall that though we have assumed that the beam is physically homogeneous and linear, we have taken into account the geometric nonlinearity. Note that the transversal load per beam length unit is defined by p = p0 sin(ωt). Finally, our problem has been reduced to the following one dy1i = y2i , dt

 E0 y1i+2 − 4y1i+1 + 6y1i − 4y1i−1 + y1i−2 dy2i = p0 sin (ωt) − · dt 12 h4x

∂ ∂y1 εx,cr · + − ε · y2i , (6.148) ∂x ∂x i dy3i = y4i , dt
2 i+1 εx,cr − εi−1 l dy4i x,cr = dt h 2hx

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with the following BCs

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y10 = y1n = 0;

y1−1 = −y11 ,

and initial conditions



= y20,n y10,n t=0

t=0

y30 = y3n = 0 (6.149)

y1n+1 = −y1n−1 ;

= y30,n

t=0

= y40,n

t=0

= 0.

(6.150)

As a result of solution of (6.148)–(6.150), time histories of the beam deflection have been obtained for all beam partition points along the beam length. The obtained dependencies have been applied to find the beam vibrations spectra and beam phase portraits. 6.6.3

FDM

In order to reduce the system to ODEs, we have applied the FDM of the derivatives with respect to x with the accuracy of O(h2x ). With its help, the fourth-order derivative for x has been supplemented by its difference analog, i.e. the whole interval x ∈ [0, 1] has been divided into n intervals with the given step hx . The second-order ODEs have been transformed to first-order ODEs by the introduction of additional functions y1 = w and y2 = ∂w ∂t [Samarsky (1977)], and as a result the following equations have been obtained dy1i = y2i , dt


 ∂2 ∂ 2 y1 dy2i = p − 2 E1 · − ε · y2i , dt ∂x ∂x2 i

(6.151)

where
 ∂ 2 y1 E1i+1 − 2E1i + E1i−1 y1i+1 − 2y1i + y1i−1 ∂2 · = · E 1 ∂x2 ∂x2 i h2x h2x +2 ·

E1i+1 − E1i−1 y1i+2 − 2y1i+1 + 2y1i−1 − y1i−2 · hx h3x

+ E1i ·

y1i+2 − 4y1i+1 + 6y1i − 4y1i−1 + y1i−2 . h2x (6.152)

System (6.151) should be supplemented with BCs (6.137)–(6.139) and initial conditions (6.140). In fact, the given system of 2(n + 1)

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ODEs includes two pairs of n + 1 same type of equations. The latter observation allows to consider them as independent layers (each of them is defined by a point of beam partition, and each of the layers has its own boundary and initial conditions). Therefore, a solution to the problem can be finally obtained via the fourth-order Runge–Kutta method [Samarsky (1982)] taking into account the latter observation. 6.6.4

Vibrations of physically nonlinear beams with deflection constraints

We consider the problem of vibrations of a beam with the rectangular cross-section being simply supported. The beam in the initial time instant is in an equilibrium state. We take into account the physical non-homogeneity and nonlinearity, and the beam has been subjected to harmonic load p = p0 sin(ωt) per its length unit. The value of the first-order physical-geometric parameter es = 0, 098, which is associated with material properties and with the geometrically non-dimensional beam parameter (l/h), has been taken for pure aluminum. Diagram σi (ei ) is taken for the elastic-plastic body with a linear reinforcement in the form of (6.117). It has been considered for two cases, when the diagram and the horizontal line have an angle of 300◦ and 150◦ for ei ≥ es . Diagram of the stress intensity exhibits the following feature: the elastic-plastic model with the linear reinforcement, when the diagram constitutes the angle of 15◦ with a horizontal line for ei ≥ es cannot be validated for large values of the excitation amplitude. Namely, a negligible increase in the amplitude implies a jump in the maximum deflection, and the process is out of boundaries guaranteed by the theory of small elastic-plastic deformations and cannot be analyzed by the solution of Eqs. (6.141), (6.142), (6.144) and (6.146). Therefore, further investigations have been carried out for the angle of 30◦ . As a result, solutions of (6.141), (6.142), (6.144) and (6.146) yield time histories of beam deflection and its velocity for all beam points along its length. The obtained dependencies have been used for finding vibrations spectrum and phase portraits in different beam points. We have considered the cases of different values of dissipation. A decrease in the dissipation coefficient makes the beam robust to

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small loading. An increase in the damping factor pushes the beam to its equilibrium position, and the internal friction plays a more important role. Although for all values of damping ε the beam behavior tendency is unchanged, for small values of the parameter ε the initial period of beam vibrations plays a crucial role. Processes realized in this period define the whole possible beam behavior, and therefore we have taken ε = 0.1 as the most representative beam damping coefficient. We have studied the influence of different values of frequency ωp on the beam dynamics. Although in general beam dynamics behavior versus the increase in the excitation amplitude does not change qualitatively, there are a few exceptional cases. In Figs. 6.32–6.34, the fundamental characteristics of the beam central point versus the different values of the excitation amplitude for three chosen frequencies have been reported. In Fig. 6.32, we have ωp < ω0 , where ω0 is the natural beam frequency. Figure 6.33 corresponds to the case ωp = ω0 , and finally Fig. 6.34 deals with the case ωp > ω0 . In all the three tables, beam deflections versus time, phase portraits, Poincar´e maps and the power spectra are shown. For ωp < ω0 , the following scenario of transition from quasi-periodic vibrations to deterministic chaos while increasing the excitation amplitude (Fig. 6.32) has been detected. For small values of the load the twofrequency quasi-periodic vibrations appear. Both phase portrait and Poincar´e section show a limit cycle. One of two frequencies (ω2 ) has been related to the excitation frequency, whereas ω1 being defined by the system parameters remains unchanged. In Fig. 6.32, one may see that the beam time histories (for p0 = 3.0 and p0 = 3.2) are split into two parts, with large and small amplitudes. In the beginning (p0 = 2.2), the beam phase portrait exhibits a large distribution of the phase curves, but after long transitional process the limiting cycle in the form of an ellipse is visible in both phase portrait and Poincar´e map. The latter one indicates a quasiperiodic orbit with two incommensurable frequencies ω1 and ω2 . Then, a third frequency ω3 appears (p0 = 3.0), which is not coupled with the two previous frequencies by linear relations. Therefore, owing to the Ruelle–Takens–Newhouse scenario, there is a possibility

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250 w

0,4 w' t

0,2

0,2

0,00

0,0

0,0

-0,02

-0,2

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0,02

-0,04

30

45

t 75

60

-0,4 -0,04 -0,02 0,00

0,1 0,0 -0,1 -0,2 20

40

0,02

w 0,04

60

80

t 100

ω1

w -0,04 -0,02 0,00 0,02 0,04

0,0 -0,1

1

0,016 |Ck/Cmax|

0,4

0,4

0,012

0,0

0,0

0,008

-0,4

-0,4

0,004

-0,2

-0,1

0,0

0,1

0,2

w

-0,8 -0,2 -0,1 0,0 0,1 0,2

0,8 w't

0,4

0,4

0,0

0,0

-0,4

-0,4

3

4

w

ω2

ω3 ω1

ω

0,000 0,0

0,3

(g)

0,8 w 't

2

(d)

0,8 w't

(f)

0,1

0

0,8 w't

(e) w 0,2

ω

0,0000

(c)

-0,8

ω2

0,0005

(b)

w 0,2

|Ck/Cmax|

0,0010

-0,2

(a)

0

b2304-ch06

Deterministic Chaos in One-Dimensional Continuous Systems 0,4 w't

0,04

9in x 6in

0,6

3

(h) |Ck/Cmax| 0,016

ω3

0,012

ω '3

ω2

ω '1 ω1 ω '2

0,008 0,004

-0,2 0

30

60

90

120

t 150

-0,8

-0,2 -0,1

0,0

0,1

0,2

w

-0,8 -0,2

-0,1

0,0

0,1

0,2 w

0,000 0,0

ω 0,3

(k)

(j)

(i) 0,3 w

4

1,0 w 't

1,0 w't

0,5

0,5

0,0

0,0

-0,5

-0,5

w -1,0 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3

w -1,0 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3

(n)

(o)

0,6

3

4

3

4

(l) |Ck/Cmax| 0,06

0,2 0,1

0,04

0,0 -0,1

0,02

-0,2 -0,3

0

20

40

(m)

60

80

t 100

0,00 0,0

ω 0,3

0,6

(p)

Fig. 6.32 Fundamental characteristics of the beam center (ωp < ω0 ) for p0 = 2.2 (a–d), p0 = 3.0 (e–h), p0 = 3.2 (i–l), p0 = 3.8 (m–p).

of transition from quasi-periodic to chaotic vibration via the birth of a third frequency. Further increase in the excitation amplitude validates this hypothesis, but the route to chaos is slightly different. The so-called period of steady-state periodic vibrations, i.e. those corresponding to small amplitude vibrations, increases, the phase portrait

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Euler–Bernoulli Beams w

0,4

w't

251

0,4 w't

6 ω1

0,0010 |Ck/Cmax|

0,10 0,05

0,2

0,2

0,00

0,0

0,0

-0,2

-0,2

-0,05

ω1

0,0005

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-0,10 0

15

30

45

t 60

-0,4 -0,1

0,1 w

0,0

(a)

-0,4 -0,1

(b)

w

0,1 w

0,0

0,0000

0,1

0,3

0,0

0,0

0,0

0,08

-0,1

-0,3

-0,3

0,04

20

40

60

80

t 100

-0,6 -0,2

-0,1

0,0

w 0,2

0,1

-0,6 -0,2

0,16

w

ω2

w -0,1

0,0

0,1

0,2

0,00

ω1

6ω 1 ω

0

1

(g)

0,6 w't

0,6 w't

0,3

0,3

2

3

0,12 |Ck/Cmax|

0,08

0,0

0,0

0,0

-0,3

-0,3

4

(h) 6ω1

0,2 0,1

4

0,12

(f)

(e)

3

(d)

0,6

0

2

|Ck/Cmax|

0,3

-0,2

1

(c) w't

0,6 w't

0,2

ω 0

ω2

ω1

0,04

-0,1 -0,2

0

15

30

45

t 60

-0,6 -0,2

-0,1

(i)

0,0

0,1

0,2

w

-0,6 -0,2

w -0,1

1,0

0,2

0,1

0,2

ω

0,00 0

1

(k)

(j)

w

0,0

0,5

0,5

0,0

0,0

-0,5

-0,5

3

4

(l)

1,0 w't

w't

2

0,15

|Ck/Cmax|

0,10

0,0

-0,2 150

165

(m)

180

t 195

-1,0

-1,0 -0,2

0,0

(n)

0,2

w

0,05

-0,2

0,0

(o)

0,2

w

0,00 0,0

ω 0,5

1,0

3

4

(p)

Fig. 6.33 Fundamental characteristics of the beam center (ωp = ω0 ) for p0 = 2.0 (a–d), p0 = 2.4 (e–h), p0 = 2.6 (i–l), p0 = 3.2 (m–p).

consists of the broadband parts, and the Poincar´e map shows an ellipse. Then, in the frequency spectrum a bifurcation occurs associated with two low frequencies ω1 and ω3 (p0 = 3.2). Between them, another frequency (ω1
= (ω1 + ω3 )/2) occurs. Besides, from the right of ω1 and from the left of ω3 two more frequencies ω2
and ω3


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252

w't

w't

0,3

0,3

0,2

0,2

0,02

0,1

0,1

0,00

0,0

0,0

-0,1

-0,1

-0,2

-0,2

w -0,3 -0,04 -0,02 0,00 0,02 0,04 0,06

w -0,3 -0,04 -0,02 0,00 0,02 0,04 0,06

0,04

-0,02 -0,04

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

10

20

30

40

t 50

(a) 0,6

0,20

|Ck/Cmax|

0,04

ω2

ω1

0,02

(b)

w

0,25

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Deterministic Chaos in One-Dimensional Continuous Systems

0,06 w

0

9in x 6in

ω

0,00 0

1

(c)

0,3

0,0

0,0

ω2

ω1 0,012

0,15

ω3

0,10

-0,3

4

0,018 |CK/Cmax|

0,6

0,3

3

(d)

w't

w't

2

0,006

-0,3

0,05 0,00

0

15

30

45

60

t 75

-0,6

-0,6

0,0

0,1

0,2

w

0,0

0,1

(f)

(e) 0,2 w

0,4

0,1

0,2

0,2

w

ω

0,000 0

1

(g)

w't

0,4

2

3

4

(h)

w't

0,12 |Ck/Cmax|

ω3

0,2

0,08 0,0

0,0

-0,2

-0,2

0,04

-0,1 -0,2

0,0

0

15

30

45

60

t 75

-0,4 -0,2

-0,1

0,0

0,1

w 0,2

-0,4 -0,2

(j)

(i) 0,4 w't

0,4 w't

0,1

0,2

0,2

0,0

0,0

0,0

-0,1

-0,2

-0,2

-0,2 30

60

90

120

t 150

-0,4 -0,2

-0,1

(m)

0,0

0,1

w 0,2

ω

0,00 0

1

0,1

w 0,2

2

3

4

(l) 0,10 |Ck/Cmax|

ω3 0,05 ω

4

ω2

-0,4 -0,2

-0,1

0,0

0,1

w 0,2

0,00 0,0

ω 0,5

(o)

(n)

w

0,0

(k)

0,2 w

0

-0,1

ω2

0,6 w't

0,6 w't

0,3

0,3

0,0

0,0

-0,3

-0,3

1,0

3

4

3

4

(p) 0,03

|Ck/Cmax|

0,2 0,1

0,02

0,0 -0,1 -0,2 30

60

90

(r)

120

t 150

-0,6 -0,2

-0,1

0,0

(s)

0,1

0,2 w

0,01

-0,6 -0,2

-0,1

0,0

(t)

0,1

0,2 w

0,00 0,0

ω 0,5

1,0

(u)

Fig. 6.34 Fundamental characteristics of the beam center (ωp > ω0 ) for p0 = 1.6 (a–d), p0 = 2.0 (e–h), p0 = 2.6 (i–l), p0 = 2.8 (m–p), p0 = 3.2 (r–u).

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occur, respectively, located at a distance equal to half of the interval length between them. A successive change of the contribution of three fundamental frequencies is observed; in the beginning, the most dominating is ω2 , and then its contribution is compared with that of ω3 . The successive increase in the load amplitude pushes the system to vibrate chaotically. The following factors validate chaos: deflection in time does not exhibit regular vibrations, and the introduced steady-state period tends to infinity (in the figure, only a short interval of time is shown); phase portrait exhibits chaotic wandering of the trajectory; Poincar´e sections are composed of infinite numbers of geometrically disordered points; power spectrum with broadband part on the interval [0; 0, 7] is shown, and the frequency ω2 has rather negligible contribution to beam dynamics (its level achieves ≈ 1/3 of the maximum value). Therefore, increase in the excitation amplitude acting on the beam with physical nonlinearity for ωp < ω0 implies the transition from quasi-periodic vibrations to the chaotic one after the birth of the third frequency via one bifurcation, and it is associated with the occurrence of intermittency, i.e. the Pomeau–Manneville scenario takes place [Pomeau and Manneville (1980)]. In the case when ωp = ω0 , a transition scenario to the deterministic chaos is associated with a few peculiarities in the initial part of this process. In Table 6.31, the fundamental system characteristics for different amplitudes of the load p0 = 2.0, p0 = 2.4, p0 = 2.6, p0 = 3.2 are given. For p0 = 2.0, two frequencies are distinguished in the power spectrum, as in the case of ωp < ω0 . One of them is associated with ωp , whereas the other is defined by the beam parameters. Therefore, when the excitation frequency ωp = ω0 the frequencies of the spectrum are rational with the coefficient k = 6 (ω1 and 6ω1 ). It means that we have periodic vibrations with two rational frequencies exhibited in the spectrum (phase portrait and Poincar´e map show the limit cycle in the form of an ellipse). The increase in the load up to p0 = 2.4 yields changes in the beam deflection history. If earlier a steady-state vibration regime of small amplitude has been observed (p0 = 2.0), then now the steady-state vibration regime is large and achieves the order of 0.2 of the beam

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thickness. Vibrations in the vicinity of the upper and lower edge positions have a complex form, since in those positions the fundamental concentration of the phase curves takes place (see the phase portrait and Poincar´e map). On the left-hand side of the frequency spectrum, a new low frequency ω2 appears which is not coupled via any linear relation with two previous frequencies. Further increase in the excitation load implies the occurrence of more frequencies between ω1 and ω2 which divide the distance between them into three equal parts. Phase portrait is composed of dense dark spots, and the Poincar´e map has irregularly distributed points. The power spectrum demonstrates the occurrence of a new frequency first, and then period 3 bifurcation takes place (not shown). However, further increase in the excitation amplitude (p0 = 2.6) does not show chaotic vibrations. The beam vibrations are small, phase portrait and Poincar´e map demonstrate a quasi-periodic solution. Further increase in the load is similar to the case considered earlier for ωp < ω0 , i.e. in the frequency spectrum one more frequency ω3 appears which is not coupled with the other ones (not shown here). The time history exhibits intermittency, and after that vibrations are transmitted to chaotic regime, i.e. the Pomeau–Manneville scenario takes place. Observe that in the considered case the vibrations character has been also qualitatively changed. In the case when ωp < ω0 (p0 = 3.2), the beam in the chaotic regime vibrates practically in a uniform way in relation to the equilibrium configuration, i.e. it exhibits a symmetry of origin. However, in the considered case chaotic vibrations do not have any line of symmetry. Furtheremore, it has been observed that the chaotic vibrations character is the same for any of the chosen exciting frequency. To conclude, for ω = ωp , we have reported a transition from periodic to quasi-periodic vibrations, while during the transition period the qualitative changes in the vibration character as well as the period tripling bifurcation have been observed. Then, after occurrence of a third frequency, and via only one bifurcation accompanied by the intermittency phenomena, the Pomeau–Manneville scenario has been demonstrated. Chaotic vibrations exhibit the symmetry breaking of the origin (the symmetry has been observed for ωp < ω0 ).

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In the case when ωp > ω0 , we have observed a few peculiarities regarding the transition into chaos as it has happened for ωp = ω0 . In Fig. 6.34, the fundamental characteristics of the system behavior for p0 = 1.6, p0 = 2.0, p0 = 2.6, p0 = 2.8 and p0 = 3.2 have been reported. For relatively small loads, i.e. p0 = 1.6, quasi-periodic vibrations are detected. In the power spectrum, two frequencies ω1 and ω2 dominate. Phase portrait and Poincar´e map show the form of a broadband ellipse. Increasing the load implies the beam jump, and vibrations take place around the new equilibrium position. These phenomena were observed earlier only when the impulse type excitation was applied [Lorenz (1981)]. Increasing the load up to p0 = 2.0, the beam from the beginning moves into the upper half plane and starts to vibrate around the new equilibrium center located on the level of ≈ 0.13 of its thickness. The maximum beam deflection is less than the allowed value of 0.25 of the beam thickness. Both phase portrait and Poincar´e map are changed. Power spectrum exhibits the occurrence of a third frequency ω3 = ω1 /2. Increasing further the amplitude p0 = 2.6 implies a stabilization of vibrations. They exhibit again a symmetry regarding the origin, but the transitional state is different. Beam vibrations are not converged to steady-state vibration with a small amplitude, but they achieve the steady-state with the maximum deflection value of the order of 0.2 beam thickness. In the beam deflection, the steadystate period can be distinguished, which is rather negligible for those loads. Phase portrait and Poincar´e map exhibit the regular attractor of a complex form. Beam vibrations are spanned into two frequencies, i.e. ω3 = 0.5ω1 and ω2 . A successive increase in the load up to p0 = 2.8 implies an increase in the steady-state period, phase portrait trajectory and curves in the Poincar´e map are broadband. Within the phase space and in the frequency spectrum ω4 appears, which is not associated with any existing earlier frequency. Then the transition into chaos follows the Pomeau–Manneville scenario. Here, we briefly address the changed character of the beam chaotic vibrations for p0 = 3.2. Contrary to the previously illustrated two cases, for ωp > ω0 , the beam vibrating chaotically has vibration parts associated with the two new symmetry centers. In

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other words, the beam vibrates between the two new centers, hence it demonstrates a series of unpredictable jumps. To conclude, the increase in the exciting amplitude of the transversal load for ωp < ω0 yields the beam vibrations transition from two-frequency to threefrequency quasi-periodic vibrations, and the position of the center of vibrations symmetry is changed. Then, the beam is again shifted into a two-frequency regime, while in the transitional process the qualitative changes of vibrations and the change of vibration frequencies are observed. Increasing further the excitation amplitude implies the increase in the steady-state period and the third independent frequency is born, and next the system is transited into a chaotic regime via the Pomeau–Manneville scenario. Chaotic vibrations occur around a novel beam equilibrium configuration. All of the so far described investigations have been carried out with the linear loading–release phenomena, i.e. we have assumed that the beam material has only linear elastic properties and the loading/release processes are realized via the same rule. Figure 6.35 presents dependencies of the deflection versus time, deformation intensity versus time and the space trajectory σi (t, ei ) for two different values of the excitation amplitude: p0 = 2.2 and p0 = 3.8 and for ωp < ω0 . The following conclusions can be formulated based on the computational results and the reported graphs in Figure 6.35. 1. The buckling and loading/unloading, as demonstrated in the intervals AB, BC, CD, DE, EF and FG, can be treated as independent ones. 2. In the case when regular vibrations are considered (p0 = 2.2), the intensity of stresses and strains are in a frame of linear dependencies, i.e. ei < es and we do not need to take into account the secondary plastic deformations. 3. In the case of chaotic vibrations (p0 = 3.8), there are three release parts in the unit time interval (AB, CD, EF), the influence of the final deformations should only be taken into account in the neighborhood of point F, where the curve σi (t, ei ) is tangent to ei . In the remaining parts, the final deformations are rather negligible.

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0,02

w

257

ei 0,10

es

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0,00

0,05

-0,02 45

46

47

48

49

t 50

0,00 45

46

47

(a)

48

49

t 50

(b)

0,3 w

1,2 ei

0,2 0,1 0,0 -0,1 -0,2 -0,3 45

0,8

А C B DF E G 46

47

A

t 48

es 0,0 45

G

C

0,4

B 46

E D F 47

(c)

(d)

(e)

(f)

t 48

Fig. 6.35 Deflection (a,c) and deformation intensity (b,d) versus time; stress intensity and deformation intensity versus time (e,f) for different values of the excitation amplitudes: a,b,e–p0 = 2.2; c,d,f–p0 = 3.8.

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Even if we have at least one point on each unit time interval where the final plastic deformations should be taken into account, their influence do not change the qualitative beam behavior. We have solved the problem of vibrations of a beam with the rectangular cross-section taking into account the physical nonhomogeneity and nonlinearity. We deal with simply supported beam ends, and the beam is subjected to an action of the transversal p = p0 sin(ω · t) load per unit length. The problem is reduced to a solution of the system (6.141), (6.142) taking into account boundary and initial conditions of the form (6.144) and (6.146). Pure aluminum is served as the beam material. Diagram σi (ei ) has been taken as that for the elastic-plastic body with the linear reinforcement in the form (6.117). The case of reinforcement has been included, when the diagram creates the angle 30◦ with a horizontal line for ei ≥ es . We have studied different frequencies of excitations ωp , where ωp < ω0 , ωp = ω0 , ωp > ω0 , and ω0 is the natural beam frequency. We have introduced different constraints on deflections in various beam points. When the barrier is moved over the middle of the beam, the problem is similar to that of considering only the first half of the beam due to the exhibited symmetry. Therefore, we consider one half of the beam only. In the case when the barrier is located at the center of the beam, the beam vibrations are symmetric with respect to the beam center, i.e. axial symmetry occurs. As a result of solutions of system (6.141), (6.142), (6.144), (6.146) and taking into account the introduced constraints, we have obtained the values of beam deflections and velocities for all points of the beam partition along its length. The obtained dependencies have been used to find a spectrum of vibrations and phase portraits in different points of the partition. Contrary to the linear theory introduced in Chapter 3, dynamics of all partition points is independent on the barrier location. This observation is true except for the beam point where a barrier is located, but this case will be studied separately. This is why in all the given figures we present only vibration characteristics of one beam point, i.e. that having the largest amplitude for a given barrier

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position. Observe that the latter point coincides with the beam center or lies in its close vicinity. We have observed that independently of the constraint position, the beam can carry large loads, i.e. for large values of the excitation amplitude the deflection values are within the assumed theory of elastic-plastic deformations. However, zones of vibrations with buckling appear, i.e. intervals of amplitudes of exciting loads, where the maximum deflection values exceed 0.25h, appear. In Fig. 6.36, we show the case of the barrier position in point 0.3, and the beam deflection versus time is presented, at point 0.7. In this case for p0 ≤ 2.6, the beam deflections are predicted with the theory of small elastic-plastic deformations (p0 = 2.6), for 2.6 < p0 < 3.4 the deflection values exceed 0.25h of the beam (p0 = 3.0), whereas for p0 ≥ 3.4 the beam deflections again become sufficiently small to be within the validity of the mentioned theory of small elastic-plastic deformations (p0 = 3.4). Besides, the beam vibration character is changed which, in particular, is exhibited by a ratio between maximum deflections on the transitional interval and amplitude of the steady-state vibrations. This phenomenon has been monitored for all values of the excitation frequency and for the arbitrary barrier position. Figure 6.36 corresponds to the case when ωp = 34 ω0 and the increase in ωp zones of splashes are shifted to small amplitudes of vibrations. For all barriers position the effect of beam penetration into the barrier is observed (Fig. 6.37). For small values of the excitation load, in those points we have vibrations with a cut upper half-period (p0 = 0, 4), whereas increasing the excitation successively smoothens vibrations (p0 = 1, 0), and after certain critical values the beam starts to drive in the barrier, besides the cases when a sudden beam deflection in a direction contrary to the beam position (p0 = 1, 8) is observed. This phenomenon is observed along the whole beam length and for all frequency of excitations. The reason is that the beam is in contact with the barrier, it harvests the energy, and its velocity suddenly increases, which is visible on the phase portraits. This causes sudden jumps of the beam deflection, and then the velocity is decreased to the value of zero.

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0,0

-0,1

30

40

t 50

30

40

t 50

30

40

t 50

-0,2 0

10

20

(a) 0,2 w

0,0

-0,2

-0,4 0

10

20

(b) 0,08

w

0,04 0,00 -0,04 -0,08

0

10

20

(c)

Fig. 6.36 Beam deflection versus time for different values of the excitation amplitudes: p0 = 2.6 (a) p0 = 3.0 (b) p0 = 3.4 (c).

In the case of barriers, localization in points 0.1, 0.2 and 0.5, the beam vibrates around a new configuration state. In Fig. 6.38, the diagrams of dependence of new localization centers versus the excitation amplitude for different beam partition points are reported (in the case of the barrier position in the beam center we have symmetric vibrations). In Fig. 6.38, the case ωp = 34 ω0 has been studied.

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w't

w 0,001

0,4 0,3

0,000

0,2

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-0,001

0,1 0,0

-0,002 -0,003 0

3

6

9

t 12

-0,1 -0,003

-0,002

-0,001

0,000

w 0,001

(b)

(a) w

1,6

0,00

w't

1,2 0,8

-0,04

0,4 0,0 -0,08

0

20

40

60

80

t 100

-0,08

-0,04

(c)

w 0,00

(d)

w

3,0

0,00

w't

2,5 -0,05

2,0

-0,10

1,5 1,0

-0,15

0,5 -0,20

0,0

-0,25 0

20

40

(e)

60

80

t 100

-0,5 -0,2

-0,1

w 0,0

(f)

Fig. 6.37 Deflection versus time and phase portrait in the point of barrier location for different values of the excitation amplitudes: p0 = 0.4 (a,b) p0 = 1.0 (c,d) p0 = 1.8 (e,f).

It should be emphasized that when the barrier is located in the beam partition point 0.1, the centers of distances between the barrier and the remaining beam ends have different deflections. New equilibrium positions have large values for the points being measured

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wo

x=0,3 x=0,5 x=0,6 x=0,8

0,10

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9in x 6in

0,1 0,3

0,05

0,00

6

7

8

9

0,8

po

(a) w0

x=0,1 x=0,3 x=0,5 x=0,6 x=0,7 x=0,9

0,025 0,020 0,015

0,6

0,2

0,010 0,005 0,000 1

2

3

4

5

p0

6

(b) w0

x=0,1; 0,9 x=0,4; 0,6

0,020 0,015

0,1

0,4

0,010 0,005 0,000 0

2

4

6

8

10

p0

(c) Fig. 6.38 Shifted equilibrium state (w0 ) versus amplitude of excitation (p0 ) in beam points, where a barrier is located: 0.1 (a) 0.2 (b) 0.5 (c).

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from the beams end. At the beam end, the deflection value is zero, whereas in the point of the barrier location the maximum upper beam deflection is equal to the gap between the beam and the barrier (the minimum lower beam deflection does not have any constraint). For this barrier position, a shift in the equilibrium states takes place for the exciting amplitudes larger than those corresponding to buckling zones. When the barrier is located in point 0.2, then the values of new equilibrium positions for points 0.3 and 0.9, 0.5 and 0.7 are the same. It can be explained in the following way. Occurrence of the left beam part, i.e. the part between the left beam end and the barrier yields an effect of symmetry of vibrations of larger beam part regarding the new center located in point 0.6 (dotted curve line). The shift of the equilibrium state takes place for the excitation amplitudes less than those corresponding to the splashes zone. In the case when vibrations are bounded in the beam center, there exist amplitudes of excitations such that vibrations are shifted with respect to the origin. Besides, the ratio between the shifted equilibrium positions having equal distances between the beam centers and its ends is the same as in point 0.1. For the studied case of the barrier location, two buckling zones are observed (p0 ∈ (3; 4, 5] ∪ (7; 8)). The analogous tendencies have been observed for all studied frequencies of excitations, but for large values of the exciting frequency the corresponding equilibrium positions have large values. For all the considered cases of barrier positions, the increase in the amplitude of excitation yields either one or a few buckling zones. Periodic vibrations exhibit rational frequencies and then, after a series of period doubling bifurcations, a chaotic attractor occurs. The described dynamics is shown in Fig. 6.39, where the following characteristics are given: beam deflection time history, phase portrait and power spectrum. We consider the case, when ωp < ω0 , the barrier is located in point 0.2, and the results are reported for point 0.6. For small amplitudes of excitation the beam vibrates periodically, which is well demonstrated for p0 = 1.2. The increase in the excitation amplitude (p0 = 2.2; p0 = 3.8) yields a shift in the center of the

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Deterministic Chaos in One-Dimensional Continuous Systems w

0,06

0,4

0,04

w't

|Ck/Cmax| 0,012

ωB

0,2

0,008

0,02 0,0

0,00

-0,04

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

0,004

-0,2

-0,02 0

3

6

9

12

t 15

-0,4 -0,04

-0,02

0,00

0,02

0,04

w 0,06

0,000

ω 0

4

0,4 0,2

0,00

0,0

-0,02

16

0,02 |Ck/Cmax|

w't

0,02

12

(c)

(b)

(a) w 0,04

8

ωB

3ωB 2ω B

5 ωB

0,01

4ωB

-0,2

-0,04 0

5

10

15

t 20

-0,4 -0,04

-0,02

(d)

0,02

0,04

w

0,00

ω 0

4

( e)

w

0,15

0,00

8

(f) ωB

3ωB

0,3

4ωB 5ω B

2ωB

0,05

0,0

0,00 -0,05

16

0,01 |Ck/Cmax|

0.6 w't

0,10

12

0,005

-0,3

-0,10 -0,15

0

4

8

12

16

t 20

-0,6 -0,1

(g)

0,0

ω

0,00

0,1

0

4

(h)

ω B 2ω B

3ω B

0,5 0,0

0,0

16

(i)

1,0

0,1

12

0,01 |Ck/Cmax|

w't

0,2 w

8

4ω B

5ω B

0,005

-0,5

-0,1

-1,0

0

10

20

30

40

t 50

-0,1

0,0

0,1

w 0,2

0,00

ω 0

4

12

16

12

16

(l)

(k)

(j) 0,2 w

8

1,5 w' t

|Ck/Cmax| 0,006

1,0 0,1

0,5

0,004

0,0

0,0

-0,5

0,002

-0,1

-1,0 -0,2 0

35

70

105

(m)

140

t 175

-1,5 -0,2

ω

w -0,1

0,0

(n)

0,1

0,2

0,000

0

4

8

(o)

Fig. 6.39 Characteristics of the beam dynamics in partition point 0.6, when the barrier is situated in point 0.2: p0 = 1.2 (a–c) p0 = 3.8 (d–f) p0 = 6.0 (g–i) p0 = 6.62 (j–l) p0 = 6.5 (m–o).

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vibrations. In the reported figures, for p0 = 3.8, the new equilibrium position is represented by a line parallel to the time axis. The successive increase in the load (p0 = 6.62) implies the occurrence of a splash zone, which takes place for p0 ∈ (6; 6.5], and then vibrations take place about the coordinates origin. The increase in the excitation amplitude yields a complexity of the phase portrait, since the trajectories occupy its large part. Power spectrum exhibits rational frequencies: ωp : 2ωp , 3ωp , 4ωp , 5ωp (Fig. 6.39). Though this behavior is similar to that observed in the linear problem, here, on the contrary, a route to chaos is observed due to the occurrence of the physical nonlinearity. Namely, period doubling bifurcations appear (the first and the second bifurcations are shown), and then in between each of the two neighborhood rational subharmonic frequencies appear being rational with ω1 /2m . Subharmonics occurring after an even bifurcation are well distinguished with respect to their level. The following bifurcation sequence of p0 is obtained: p1 = 3.8, p2 = 6.62, p3 = 7.22, p4 = 7.345, . . ., tending to the critical value pcr ≈ 7.46. The latter sequence allowed to compute the Feigenbaum constant δ ≈ 4.8, which yielded an error of 2.8% with respect to the exact (theoretical) value. For p0 > pcr , the discrete lines in the spectrum collapsed in an inversed manner to their occurrence, and the broadband zones of power spectrum appear. Deflection time histories are chaotic, phase portrait becomes a dark part of the plane, i.e. the full system dynamics is chaotic. To sum up, the system transits from periodic to chaotic vibrations in the following way: the increase in the excitation amplitude yields an occurrence of rational frequencies, and then the period doubling bifurcations take place, i.e. the Feigenbaum scenario is exhibited [Feigenbaum (1978)]. Analogous scenario is observed for all studied frequencies of excitation. 6.6.5

Geometrically nonlinear beam with/without constraints

Three cases of periodic load action on the beam have been studied (ωp < ω0 , ωp = ω0 , ωp > ω0 ). For small loads, when the beam

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deflection is less than 0.25, the beam deflection is well approximated by the linear theory, and hence deformation of the beam middle surface does not introduce any important contribution. The beam vibrates periodically, which is validated by the deflection time histories and power spectrum (Fig. 6.40). All dynamical features are reported in Figs. 6.40 and 6.41 (fundamental vibration characteristics are shown for the central beam point for different values of the excitation amplitude). In Fig. 6.40, beam deflection versus time, phase portrait and power spectrum are shown. Furthermore, for each frequency spectrum the frequencies occurring in the given spectrum are listed, and their ratio is given with respect to frequency ω1 , which is the excitation frequency, i.e. ω1 = ωp . The increase in the load amplitude (p0 = 0.5, p0 = 5.0, p0 = 13.0, p0 = 20.0) yields the increase in beam deflections, and hence a contribution of the middle surface deformation also increases. Beam vibrations have more complex character, the phase portrait shows subharmonic behavior, the power spectrum has rational components, which are exhibited to the left and to the right of the first frequency. The novel rational frequencies are well distinguished with respect to their level. Each of the two neighborhood frequencies is located in a distance of 3ω1 . Therefore, the increase in the load contributes to the increase in the teeth number of the saw tooth spectrum shape, whose teeth are symmetrically located with respect to the central tooth, and finally the “frequency saw” consists of seven teeth. In Fig. 6.41 as well as in Fig. 6.40, the columns present beam deflection versus time, phase portrait and power spectrum, corresponding to large values of the load amplitude (p0 = 20.2, p0 = 23.2, p0 = 25.0). In this case, an increase in the load does not imply an increase in the saw teeth width. Between each of the teeth, the subharmonics appear with frequencies being rational with 3ω1 /2m . In addition, subharmonics occurring after an odd bifurcation are well distinguished with respect to their level. Further increase in the load yields the collapse of the discrete spectral lines, and zones of the broadband spectrum appear. Simultaneously, the beam deflection versus time loses its regularity, phase portraits contain a wash out

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0,02

w't

|Ck/Cmax|

0,10 0,01

0,05

0,00

0,00

267

1

0,003

1.

28=28

1

1. 2. 3.

28=28

1

25=25

1

31=31

1

1. 2. 3. 4. 5.

28=28

1

25=25

1

31=31

1

22=22

1

34=34

1

1. 2. 3. 4. 5. 6. 7.

28=28

1

25=25

1

31=31

1

22=22

1

34=34

1

19=19

1

37=37

1

0,002

0,001

-0,05

-0,01

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-0,10 -0,02 10

12

14

16

18

t 20

-0,02

-0,01

(a)

0,00

0,01

w 0,02

0,000 0,0

ω 0,2

0,6

0,8

(c)

(b) w't

0,2 w

0,4

1

0,03 |C/C | k max

1,0 0,1

0,5

0,0

0,02

0,0 -0,5

-0,1 -0,2 10

12

14

16

18

t 20

-1,0 -0,2

-0,1

(d) 0,4

0,0

0,1

w 0,2

0,00 0,0

ω 0,2

0,4

(e)

w

3

2

0,01

0,6

0,8

(f) 0,03 |C/C | k max

3 w't

1

2 0,2

0,02

1

2 3

0

0,0

0,01

-1

4

-0,2

5

-2 -0,4 10

12

14

16

18

t 20

-3

w -0,4

-0,2

(g)

0,0

0,2

0,4

0,00 0,0

ω 0,2

(h) 0,03 |Ck/Cmax|

2

4

0,4

2

0,0

0

-0,4

-2

-0,8 10

12

14

(j)

16

18

t 20

0,6

0,02

0,01

13

4

5 7

6

-4 -0,8

-0,4

0,0

(k)

0,8

(i)

w't

0,8 w

0,4

0,4

w 0,8

0,00 0,0

ω 0,2

0,4

0,6

0,8

(l)

Fig. 6.40 System behavior characteristics for different values of the excitation amplitude, when the tooth saw frequencies shape appears.

trajectory. It means that the system dynamics is shifted into a chaotic regime. In Fig. 6.41, frequency spectra for the first, second, third and fourth bifurcations are shown. Figures 6.41(j) and 6.41(k) include

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Deterministic Chaos in One-Dimensional Continuous Systems w't

0,8 w

0,03 |Ck/Cmax|

4

0,4

2

0,0

0,02

0

0,01

-2

-0,4

1

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-4

-0,8 10

15

20

25

t 30

-0,8

-0,4

0,0

ω

w 0,8

0,2

0,4

6 w't

w

0,4

3

0,0

0

-0,4

-3

0,6

0,8

0,6

0,8

(c)

(b)

(a) 0,8

0,4

0,03 |C /C | k max

0,02

0,01 2

-0,8 10

15

20

25

t 30

-6 -0,8

-0,4

0,0

0,4

0,8

w

ω

0,00 0,2

0,4

(e)

(d)

(f)

6 w't

w

0,04 |C /C | k max

1,0

3

0,5

0

0,0 -0,5

0,02

-3

-1,0 10

15

20

25

t 30

-6 -0,5

0,0

(g) 0,03

0,03

0,02

|Ck/Cmax|

0,01

0,45

0,50

(j)

0,00 0,0

ω 0,2

ω 0,55

0,00

2 ω 0,45

0,50

(k)

0,55

0,4

0,6

0,8

1,0

(i) 0,03

0,02

0,01

0,00

w 1,0

(h)

|Ck/Cmax|

1

0,5

|Ck/Cmax|

0,03

0,02

0,02

0,01

0,01

0,00

ω 0,45

0,50

(l)

0,55

0,00

|Ck/Cmax|

ω 0,45

0,50

0,55

(m)

Fig. 6.41 System behavior characteristics for different values of the excitation amplitudes, when period doubling bifurcations occur.

parts 1 and 2 taken from Figs. 6.41(c) and 6.41(d), respectively. We have detected the following series of bifurcation with respect to the control parameter p0 : p1 = 20, 2, p2 = 23, 2, p3 = 23, 8, p4 = 23, 925, . . ., tending to the critical value of pcr ≈ 23, 958. Hence, the estimated Feigenbaum constant is δ ≈ 4, 8, which yields the error of 2, 8% in comparison to the exact value.

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The following main conclusions yielded by our study can be withdrawn. The increase in the excitation amplitude implies richer beam dynamics, and the frequency spectrum consists of rational frequencies. The latter ones are located symmetrically with respect to the initial frequency, and the distance of each neighborhood frequencies is the same. Further increase in the load implies period doubling bifurcations, i.e. the Feigenbaum scenario occurs [Feigenbaum (1978)]. Analogous scenario has been detected for all studied frequencies of excitation. However for ωp < ω0 , the mentioned scenario is developed slower, i.e. vibrations are periodic even for large loads. If ωp > ω0 , then the collapse of periodic vibrations occurs for small loads. Finally for ωp = ω0 , the behavior exhibits the resonance phenomenon. The fundamental part of the power spectrum belongs to the first spectrum “tooth” and the remaining part is of less meaning, but the general picture of a route from periodic to chaotic vibrations is the same as in the two previous cases. 6.7 6.7.1

Multi-Layer Beams Problem formulation

The classical approach devoted to the analysis of interaction of beams, plates and shells can be reduced to the following differential equation ¨ + εU˙ + L(U ) = F − qk ψM, (6.153) U with the BCs GU (A) = H(A), initial conditions U (t = 0) = Φ1 and  +U  ) ≥ 0, A ∈ U˙ (t = 0) = Φ2 , and the geometrical conditions f (R  are vector function ω, where ω denotes a contact zone, U and U and vector of displacements of a point on the surface Ω; L, G, H are differential operators, F is the vector function of the external / ω) = 0; M is a column, and its elements load; ψ(ω) = 1, ψ(A ∈ corresponding to the equilibrium equation projected on a normal to the surface Ω is equal to one, and the remaining elements are equal to zero; fk = 0 stands for an equation of the contact surface (when a point is out of the contact surface we take f > 0). Conditions

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pertaining to lack of penetration between the contacting bodies are usually presented in the following vectorial equation

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 +U  ∇f  (R)  ≥ 0, f (R)

A ∈ ω.

(6.154)

A solution to problems (6.153) and (6.154) is composed of a vector function, a contact zone and a contact pressure. If the surface ω ¯ is not given, we get a design-type nonlinear problem, when in the process of searching for solution the computational scheme is changed. It should be mentioned that in many cases the contact reaction on the border Ω may include a discrete resultant transversal force and a resultant torque being tangent to the border of ω ¯. Another approach often used to study the static problems is based on the problem reduction to the Fredholm type integral equation of the first order, i.e.
G (A, B)qk (B) d¯ ω = g (A) , A∈ω ¯, (6.155) ω

where G(A, B) is the Green function defining deflection in a point A, whereas in a point B a discrete force being normal to the surface beam acts; g(A) is the sum, describing the form of the beam surface taking part in the contact, and the function describing the displacement of the penetrating body treated as the rigid one. If Eq. (6.155) is obtained via the Euler–Bernoulli hypothesis, then it possesses only a general solution; contact reaction presents discrete forces on the contact zone border (if a zone is not a priori given) and a torque for the unknown space ω [Galin (1976); Popov and Tolkachev (1980)]. The contact problem of theory of beams can be formulated as mathematically correct via one of the regularization methods and then a transition from equation (6.155) is carried out into the following Fredholm equation of the second type
G (A, B)qk (B) d¯ ω = g (A) , A∈ω ¯. (6.156) ω

Methods of mathematical regularization are based on a definition of the regularization operators [Popov and Tolkachev (1980); Tichonov and Arsenin (1979)]. It has been shown that the most

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efficient and mathematically approved method of the choice of regularization parameter is that of the error minimization introduced by Tikhonov [Lavrentev et al. (1980); Tichonov and Arsenin (1979)]. In this case, though it is enough to keep a minimum of the a priori information about a solution, there is a need to solve the additional problem of finding the associated functional minimum. There exist also other approaches to the problem of regularization of the integral equations of the considered contact problems. They are based on the accuracy improvement of the physical problem statement, and they are known as methods of physical regularization. In a contact zone, an elastic layer is introduced [Grigolyuk and Tolmachev (1975)], which takes into account the properties of the micro-geometry of the contact surfaces and it has the following properties: 1. Deformations of the contact layer have a local character and do not depend on the displacements of the contacting bodies. 2. Mutual influence of the contact deformations of the contacting bodies, i.e. the displacement of an arbitrary part of the contact layer does not depend on the displacements of other parts. 3. Deformations of the contact layer are proportional to the mutual contact forces. The second type of Fredholm integral equation is defined using a condition of the contact in the integral form. The coefficient K plays the role of regularization parameter, quantitatively characterizing properties of contacting surfaces and it is estimated experimentally [Aleksandrov and Romalis (1986); Demkin (1970); Levina and Reshetov (1971)]. One of the ways to define K in Eq. (6.156) relies on an estimation of displacement of a half-space border subjected to the contact pressure. In references [Aleksandrov (1962); Aleksandrov and Mhitaryan (1983); Bohatyrenko et al. (1982)] the relation for normal displacements of a layer with small thickness has been matched with undeformed foundation [Vorovich et al. (1974)], and it has been shown that the decrease in the layer thickness approximates the Winkler foundation. In references [Kantor (1983, 1990); Kantor and Bogatyrenko (1986)] the approach to solve the contact problems of nonlinear theory

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of shells is based on removing the contact pressure qk from the unknown functions using Winkler’s relation. This type of the beam contact is equivalent to the statement (6.156), but allows to omit the construction of a Green function numerically and implicitly define a solution from the equations of equilibrium configurations (6.153). 6.7.2

Contact pressure versus transversal beam band

We show that in order to model the transversal beam contact Winkler’s hypothesis can be applied. In order to study this approach, we begin with a solution to the problem of a contact layer of thickness 2l1 with a rigid body. In a zone of length 2l, let the layer be subjected to pressure q symmetrically located regarding the axes x, z (Fig. 6.42). Considering only the plane problem, the pressure is approximated by the following  mπx infinite trigonometric series q(x) = ∞ m=0 Am cos l . The solutions are as follows [Strelbitskaya et al. (1971)] σx = 2 ± Fm



− (z) cos βx, Am Fm

σz = 2



+ (z) cos βx, Am Fm

(βl1 chβl1 ± shβl1 ) chβz − βz shβz shβl1 = , sh2βl1 + 2βl1

(6.157)

q

2l1

x

2l Fig. 6.42 Contact layer of thickness 2l1 subjected to transversal load q of width 2l symmetrically located.

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where β = mπ l . We take q(x) = −σz (x, l1 ). In the case of the deformed plane state, we have  1  ∂uz (x, z) = (q − υ 2 )σz − υ (1 + υ) σz . (6.158) εz = ∂z E It should be emphasized that this modelling of the transversal band in the contact zone can be applied within the frame of the classical kinematic Euler–Bernoulli hypothesis [Volmir (1972)]. Namely, the normal load and the contact pressure satisfy the same requirement. Both are less than normal stresses in the transversal shell crosssections, and they can be omitted in the Green relations.  l1 Integrating (6.158) with respect to z, and taking into account 0 σx dz = 0, since there is a lack of external load along the axis x, we get
  1 − υ 2 l1 σz dz = 2 1 − υ 2 uz (x, l1 ) = E 0  ch2βl1 − 1 cos βx. (6.159) Am β (sh2βl1 + 2βl1 ) m Developing nominator and denominator into a series with respect to βl1 ≡ c, we obtain 2

4

1 + c3 + 2c 1 − υ2  45 + · · · cos βx l1 Am uz (x, l1 ) = 2 4 c 3c E 1 + 3 + 45 + · · · m

c4 1 − υ2  l1 + · · · cos βx, Am 1 − = E 45 m  1 ¯ 4 IV 1 − υ2 l1 q(¯ x) + · · · , x) − (l1 ) qx (¯ = E 45

(6.160)

l1 x , ¯l1 = . l l Introducing 2l1 = h, we get the relation matching the transversal x): band ∆ = 2uz (x, t) with function q(¯

1 h 4 IV 1 E ∆ = q(¯ x ) − qx¯ (¯ x,
t) + · · · (6.161) 1 − υ2 h 720 l x ¯=

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z

x

2l Fig. 6.43

Clamping of the contact layer by two punches.

We consider clamping of the same layer by two smooth pressing dies (punches) symmetrically located (Fig. 6.43). A solution to the contact problem, where uz (x, l1 ) is the function given in interval −l ≤ x ≤ l which satisfies the following equation


∞ x−ξ q(ξ) K (6.162) dξ = πs0 uz (x, l1 ), l1 −∞ where in the interval |x| ≤ l for |z| = l1 the following condition is satisfied σz = τ = 0, where
∞
∞ ch2u − 1 cos up L(u) du ≡ cos up du, K(p) = sh2u + 2u u u 0 0 (6.163) E . s0 = 2(1 − υ 2 ) Solution to Eq. (6.162) in the form


∞
∞ s0 u cos pu ∗ x−ξ ∗ du uz (ξ, l1 ) K dξ, K (p) = q(x) = 2 l L(u) πl1 −∞ 1 0 (6.164) is reported in the reference [Korn and Korn (1968)], where using the  u (2i) (p), the following relation has been = A1 + ∞ series L(u) i=1 Bi u  i π (2i) (p), where δ (2i) is a δ0 (p) + π ∞ derived K ∗ (p) = A 0 i=1 (−1) Bi δ derivative of the 2ith order.

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In what follows, we find a solution in the explicit form using the already mentioned dependence from [Korn and Korn (1968)]:

x−ξ = l1 δ0 (x − ξ), δ0 (x − ξ) = −δ0 (ξ − x), δ0 (p) = δ0 l1
l f (ξ)δ0 (x − ξ) dξ = f (x),


−l

l

−l

(2i)

f (ξ)δ0 (x − ξ) dξ = f (2i) (x),

|x| ≤ l.

(6.165)

∗ x−ξ Since K ∗ (p) is odd, then K ∗ ( ξ−x l1 ) = K ( l1 ). Substituting a series of u/L(u) into the solution of Eq. (6.162), integrating it in the interval −l, l and taking into account relations for the generalized functions, we get

q(x) =

 E 1 2 4 IV (x, l ) − B l u + B l u − · · · . (6.166) u z 1 1 z 2 1 1 z 2(1 − υ 2 ) l1

u 1 4 = u(sh2u+2u) = 2(1 + 45 u + · · · ), then owing to the Since L(u) ch2u−1 earlier introduced results we get 1/A = 2, B1 = 0, B2 = 2/45, and finally



1 h 4 IV 1 E ∆+ ∆ + ··· . (6.167) q(¯ x) = 1 − υ2 h 720 l

One may check that relations (6.161) and (6.167) are mutually inversed. They imply that in the case of lack of bending deformations and extension of the middle surface of beams into so far considered problems for the layer, the relation between clamping and contact pressure differs from the Winkler model by terms such as derivatives of the fourth and higher orders multiplied by small coefficients. Even for h/l = 1, where l is the characteristic contact zone dimension, the first coefficient is equal to 1/720, and for the beams contact we have usually h  l. This validates our assumption that Winkler’s model (6.167) is the appropriate one and can be used in our future investigations. The same result has been obtained in the reference [Aleksandrov and Mhitaryan (1983)], where the problem

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has been analyzed asymptotically, as well as in the reference [Kantor (1990)], where a wide class of contact problems has been solved for asymmetric shells. When we take into account tangential deformation implied by bending and extension deformations of the contacting beams, the coefficient ∆ is changed, and a small additional term appears. The formula governing clamping for plates yields [Pelekh and Suhorol3(1−υ) E 2 skiy (1980)]: qk = 4(1+υ)(1−2υ) h [∆ + o(h ∆0 w)] (for beams, when

Poisson’s coefficient ν = 0, qk = 34 Eh [∆ + o(h2 ∆0 w)]). Analogous formulas are obtained in [Bloh (1977)]. They differ only in the coefficient K standing by E/h, which in all formulas is close to one. Taking into account the given estimations, as well as the observation that while solving contact problems of theory of Kirchhoff shells the Winkler relation model can be applied, the contact pressure between the Euler–Bernoulli beam and the punch can be estimated via difference ∆ = w − a, where w denotes the normal displacement of the middle surface [Aleksandrov and Mhitaryan (1983); Galin (1976); Grigolyuk and Tolmachev (1975); Pelekh and Suhorolskiy (1980)]: E (6.168) qk = K (w − a) , a > 0. h For the contact problem of two beams, formula (6.168) takes the following form E (6.169) qk = K (w1 − a − w2 ). h If the contact takes place, then qk > 0. As it has been already mentioned, K ≈ 1. 6.7.3

Geometric and physical nonlinearities

We consider a multi-layer beams package (Fig. 6.42). The following notation is introduced: hl , h0l refer to thickness of lth beam and thickness measured in beam center, respectively; bl , b0l denote beam width and beam width in its center, respectively (it means that a beam may have a non-constant transversal cross-section); wl (x, t) is the beam deflection; ul (x, t) is the displacement in a middle surface;

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277

El is Young’s modulus; t is time; εl is the damping coefficient; a is the beam length; ρl is the unit material mass density; G0l is the shear modulus; νl is Poisson’s coefficient; eil and σil are deformations and stresses intensity, respectively; esl and σsl denote intensity of deformation and stress of plasticity, respectively; Kl is the volume elasticity modulus, ε0l is the volume deformation; K is Winkler’s proportionality coefficient. Let beams occupy in R2 the spaces Ωl = {(x, z)| 0 ≤ x ≤ a, αl ≤ z ≤ βl }, where αl1 = −hl1 /2, αl2 = (hl1 /2 + δ), βl1 = hl1 /2, βl2 = (hl1 /2 + δ + h2l ) (see Fig. 6.42). The coordinates’ origin is located on the left side of the upper beam in its middle line, whereas a clearance between beams is denoted by δ. Axis Oz goes down, and the length of the beams package is a (Fig. 6.42). We assume beam’s material is isotropic, but non-homogeneous, such that Young’s modulus is denoted by El , shear modulus by Gl , coefficients of volume deformation by Kl , coefficient of the transversal deformation by γl , and plasticity limit by σs l = σs l (x, z, ε0 , ei ). In order to define E = E(x, z, ε0 , ei ), Hencky’s small elasto-plastic deformations are used [Birger (1951); Budiansky (1959); Ilyushin (1948); Rabotnov (1966)]. The following non-dimensional parameters are introduced ¯ l h0l , wl = w ¯l G0l , ¯l h0l , El = E x=x ¯a, z = z¯ h0l , hl = h bl = ¯bl b0l ,

4 2 ¯ h0 b0 , t = t¯ a pl = p¯l h0l , K = K a4 h0l  a2 ρl , l = 1, n. εl = ε¯l 2 h0l G0l b0l



ρl , G0l b0l

(6.170)

Taking into account the Euler–Bernoulli hypothesis, the nonlinear relation between deformations and displacements, and the method of non-constant elasticity parameters [Krysko (1976)], equations of motion of the beam in non-dimensional parameters have the following form (bars over the non-dimensional quantities are already omitted): 

∂ 1 ∂ 2 ul 2 E0l u
l + (w
l ) − E1l w

l , (6.171) bl hl 2 = ∂t ∂x 2

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∂2 1
2 ∂ 2 wl ∂wl ∗


= ql + 2 E1l u l + (w l ) − E2l w l bl hl 2 + εl ∂t ∂t ∂x 2  

 1
2 ∂



w l E0l u l + (w l ) − E1l w l , + ∂x 2 (l = 1, 2) (6.172) where


Eil = b

βl αl

El z i dz,

ql∗ = ql + qkl

(i = 0, 1, 2).

(6.173)

Stress ql∗ , acting on the beam, is defined by the sum of external periodic load ql and contact stresses qkl . The jump effect of the beam interaction is included. The contact stresses have the following form

E1 h02 w1 − δ − w2 ψ, (l = 1, 2), (6.174) qk l = (−1)l K h1 h01 where K is the proportionality coefficient between the contact pressure and clamping. Function ψ defines the dimension of the contact zone, and has the form  h0 2 1 1 + sign(w1 − δ − w2 ) . (6.175) ψ= 2 h01 It should be emphasized that the occurrence of the term ψ in the equations governing beams dynamics transits these equations to a new type, and the problem becomes constructively nonlinear. By the latter nonlinearity we mean the changes in the computational algorithm in the deformation process. BCs of the beams support can be taken arbitrarily. In order to include the physical material beam nonlinearity we apply theory of plasticity and the method of non-constant elasticity parameters [Krysko (1976)]. The diagram of beam material deformations σli (eil ) can be taken arbitrarily. In this work we consider beams, for which the following relation governing deformation holds 

eil . (6.176) σli = σls 1 − exp − esl

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Gaps, geometric and physical nonlinearities

Integration of Eqs. (6.171) and (6.172) with arbitrary initial and BCs is carried out via FDM. For this purpose, the space D = {(x, t)|0 ≤ x ≤ 1, 0 ≤ t ≤ T } has been covered by a rectangular grid xi = ihx , tj = jht (i = 0, 1, 2 . . . n; j = 0, 1, 2, . . .), where xi = xi+1 − xi = hx = 1/nx (nx integer) and ht = tj+1 − tj , hz = 1.0/nz . Differential Eqs. (6.171 and 6.172) can be approximated on the grid with corresponding finite-difference relations. In order to increase their accuracy, symmetric formulas for derivatives are given. We get 

1 εl ht h2t 2wli,j + − 1 wli,j−1 + Ali,j , wli,j+1 = (1 + εl ht /2bl hl ) 2hl bl hl 

h2t ∂E0l 1
2
u + (w ) + E0l (u

+ w
w

) uij+1 = bh ∂x 2 ∂E1l




w − E1l w + 2uij − uij−1 , − ∂x ij (6.177) where 

∂2 1
2


E1l u l + (w l ) − E2l w Ali,j = ∂x2 2 (6.178) 

1 ∂ 2 w
l E0l u
l + (w
l ) − E1l w

. − ∂x 2 i,j Applying the FDM, boundary and initial conditions are written in the corresponding appropriate manner. In this case, the governing equations are described by the three-layer scheme. In the beginning of computation, wl (x, t) and ul (x, t) on the layer (j +1) are computed using values wl (x, t) and ul (x, t) from the two previous layers (j − 2) and (j − 1). In order to begin the computations, the values of wl (x, t) and ul (x, t) have been computed using a dummy layer with number j = −1. In order to apply the method of non-constant elasticity parameters [Birger (1951)], the beam has been divided into nz layers for its thickness. Further, for each time step for node xj , the deformation intensity (6.176) layer by layer is found. Formulas (6.177) and (6.178) define the elasticity modulus and the integrals (6.173)

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are computed using the Simpson method. The transversal load has the following harmonic form

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ql = q0l cos(ωl t).

(6.179)

The described algorithm of a solution to the governing equations of the beams dynamics has been investigated from the point of view of the convergence of results regarding the spatial and timing grid for stationary problems. The relaxation method [Feodosev (1963)] has been applied to find a deflection in the central point of the simply supported beam subjected to uniformly distributed load along the beam length (the load does not depend on the time and ε = εcr , where critical damping εcr implies fast damping of the vibrations). The analysis of the obtained results shows that the deflection does not depend on the step magnitudes for 2 · 10−5 ≤ ht ≤ 10−3 . On the other hand, a change in the partitions number of space coordinate from nx = 28 to nx = 10 yields an error of 1%. Taking ht = 2 · 10−5 and nx = 28, we have obtained the deflection of 0.02346 in the beam center, whereas the analytical method of statics yields its counterpart value of 0.02343 [Bernshtein (1961)]. For the same values of ht and nx as in the case of a static problem, a dynamical problem has been solved. The simply supported beam subjected to the transversal load (6.179) has been studied, for which the trans for q10 = 1.5 and ω = 0.5. Change in the beam partition number regarding its length from nx = 30 to nx = 10 yields the error of the amplitude of natural frequency estimation of only 1.6%. Finally, we have taken nx = 30, ht = 2 · 10−5 . A change in number of layers regarding beam thickness from 12 to 20 yields an error only of 0.2%. Therefore, we have taken nz = 12. 6.7.5

Wavelet analysis

We consider the nonlinear system consisting of two beams coupled via BCs. The transversal harmonic load acts only on one beam, whereas the second beam vibrations are excited only via a contact with the first beam. Frequency of excitations ω1 = 6.28, whereas the gap between beam is changed from δ = 0.025 to δ = 0.1, and ε = 1.4.

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Depending on the beam deflection magnitude and the beam material type, the following problems can be considered: A — Both beams are elastic (beam material is physically linear); B — The first beam is made of physically nonlinear material, whereas the second is made of elastic material; C — The first beam is elastic, and the second is made of physically nonlinear material; D — Both beams are made of physically nonlinear material. Below, the results obtained for solutions to the problems of type A–D for two beams coupled via BCs with the clearance of δ = 0.05 have been reported. For other clearances, similar beam behavior is observed, but it is associated with application of large loads. In Figs. 6.44 and 6.45 vibrations of the middle point of the 1st and 2nd beams are shown: (a) and (c) denote vibration of the 1st and 2nd beams, respectively; (b) and (d) denote time-frequency wavelet spectra; (e) is the frequency–time phase shift distribution of beams in order to trace the phase synchronization [Osipov et al. (1997); Pikovsky et al. (2001)]; f is the spatio-temporal contact pressure distribution; (g) is the comparison of deflections of the middle point of the first and second beams. Problems A. We consider a series of problems of elastic beams, i.e. without taking into account the physical nonlinearity. For the value of q01 = 2.25 (small), the upper beam vibrations are small (w < 0.25h, and the geometrical nonlinearity influence is not exhibited). Owing to the wavelet spectra [Figs. 6.44(b) and (d)] vibrations are harmonic, and phase synchronization [Fig. 6.44(e)] is not observed. Comparison of the deflection magnitudes validates this observation. For q01 = 7.5 (Fig. 6.45), vibrations are more complex and the wavelet spectrum consists of two-frequency vibrations of both beams, but the phase synchronization takes place only on the excitation frequency. For q01 = 23.0 (Fig. 6.46), we may observe the phase chaotic synchronization: both beams exhibit chaotic vibrations, but their phases coincide. Furthermore, owing to the developed numerical algorithms we may follow how a vibration regime (regular or chaotic) defined

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Fig. 6.44 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial–temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 2.25).

through the wavelet-spectra influences the spatio-temporal contact pressure. For the amplitude of excitations q1 = 2.25, 7.50, 23.00, the wavelet spectrum of vibrations of both beams implies a transition from harmonic (q1 = 2.25) to chaotic (q1 = 23.00) vibrations. In the mentioned series of the load, the spatio-temporal distribution of the contact pressure is transited from periodic to chaotic one [Figs. 6.44(f)–6.46(f)]. Problems D. Here we illustrate a transition from asynchronous to synchronous vibrations, where the chart of phase difference implies

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Fig. 6.45 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial–temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 7.5).

synchronization on frequency ω = 4.0, instead of the excitation frequency. In Figs. 6.47(f)–(n), the deflection magnitude allows to trace the process of synchronization in time. In problems related to beams made of a physically nonlinear material one may also observe the interesting behavior of the dynamical stability loss of the first beam. As a result, it starts to vibrate around a new equilibrium position and does not touch the second beam (q10 = 20.75). For ε = 1.4, at the end of the observation time the second beam does not move. By decreasing the dissipation coefficient for the second beam (ε = 0.7, 0.3, 0.0), a dynamical stability loss of the first beam is also observed. In the

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Fig. 6.46 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial-temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 13.0).

latter case, the second beam’s vibration is damped within the larger time interval compared with the previous case (Figs. 6.48–6.51). As an example, we take two beams coupled via the simple BCs: ∂wl2 (0, t) ∂wl2 (0, t) = = 0; ul (1, t) = ul (l, t) = wl (1, t) = wl (l, t) = ∂x2 ∂x2 l = 1, 2. (6.180) The initial conditions follow ∂wl (x, 0) ∂ul (x, 0) = ul (x, 0) = = 0; wl (x, 0) = ∂t ∂t

l = 1, 2, (6.181)

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Fig. 6.47 Time histories of the first (a) and second (c) beams, ω(t) of the first (b) and second (d) beams, beams phase difference ϕ1 −ϕ2 (e) and beam center deflections wi (t) for different observation time intervals of 5–15 (f), 15–25 (g), 25–35 (h), 35–45 (i), 45–54 (j), 55–65 (k), 65–75 (l), 75–85 (m), and 85–95 (n) for Problems D (q01 = 2.5).

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Fig. 6.48 Time histories of the first (a) and second (c) beam centers, 2D Morlet wavelets of the first (b) and second (d) beams, beams phase difference ϕ1 −ϕ2 , and beam center deflections wi (f ) for ε = 1.4 (Problems D).

Fig. 6.49 Time histories of the first (a) and second (c) beam centers, 2D Morlet wavelets of the first (b) and second (d) beams, beams phase difference ϕ1 −ϕ2 , and beam center deflections wi (f ) for ε = 0.7 (Problems D).

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Fig. 6.50 Time histories of the first (a) and second (c) beam centers, 2D Morlet wavelets of the first (b) and second (d) beams, beams phase difference ϕ1 −ϕ2 , and beam center deflections wi (f ) for ε = 0.3 (Problems D).

Fig. 6.51 Time histories of the first (a) and second (c) beam centers, 2D Morlet wavelets of the first (b) and second (d) beams, beams phase difference ϕ1 −ϕ2 , and beam center deflections wi (f ) for ε = 0.0 (Problems D).

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and the transversal load is defined as

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ql = q0l sin (ωlp t).

(6.182)

The harmonic load (6.182) excites only one (upper) beam, whereas the second (lower) can be excited via a contact with the first beam (ω1p = 6.28, δ = 0.05, ε = 1.4). Depending on the beam deflection magnitude and the beam material, the following problems will be studied: A — both beams are elastic (material is physically linear); B — both beams are made of physically nonlinear material. In Figs. 6.52–6.55, vibrations of both beam centers (x = 0.5) are reported, and the following notation is given: a, c are vibrations of each beam; b, d denote frequency-temporal wavelet spectrum; e is the frequency-temporal distribution of the phase difference in order

Fig. 6.52 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial–temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 2.25).

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Fig. 6.53 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial–temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 7.5).

to follow the phase synchronization; f is the space–temporal distribution of the contact pressure; g are the simultaneous vibrations of beam centers in given time intervals. Problems A. For q01 = 2.25, the upper beam exhibits small deflections (w < 0.25h, geometric nonlinearity does not play any essential influence on the beam dynamics). Owing to the wavelet spectra (b), (d) vibrations are harmonic with the excitation frequency acting on the first beam, and the phase synchronization (e) is not observed. Comparison of the deflection magnitude g validates the fact that the vibrations are not chaotic. For q01 = 7.5 (Fig. 6.53), vibrations are more complex, and we can observe two-frequency vibrations of the first and second beams

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Fig. 6.54 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial-temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems A q01 = 23.0).

on the wavelet-spectra, but the phase synchronization takes place only in the neighborhood of the excitation frequency. Analysis of the contact pressure shows its chaotic distribution, and on the contact zone border we have sharp peaks being characteristics of the kinematic Euler–Bernoulli model and the contact interaction of beams. Vibrations of the two beams (g) show phase synchronization, and amplitudes of both signals are mutually coupled, i.e. we have the full synchronization for both phases and signals. The increase in the excitation amplitude up to q01 = 23.0 (Fig. 6.54) implies chaotic vibrations of upper and lower beams, and the phase synchronization on the whole interval of the frequencies being analyzed (the phase synchronization history (e) shows its different character in time). Frequencies playing a key role in synchronization essentially change but frequencies of synchronization in the

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Fig. 6.55 Time histories of first (a) and second (c) beam vibrations, time frequency wavelet spectra of first (b) and second (d) beam, and frequency–time phase shift distribution of beams (e), spatial–temporal contact distribution of beams (f) and deflections of middle beam points (g) (Problems B q01 = 2.5).

vicinity of the excitation frequency remain unaffected. Contact pressure (f ) exhibits chaotic character. Deflections in the upper and lower beams (g) are close to each other on the whole time interval, i.e. full synchronization of frequency and signal takes place. Problems B. Results given in Fig. 6.55 illustrate the phenomenon of transition from asynchronous to synchronous vibrations, whereas the chart of phase difference shows that synchronization takes place in the interval of frequencies [6; 2.5]. In Fig. 6.55(f)–(h) we may follow how synchronization develops in time. The beginning of phase synchronization shown in Fig. 6.55 in time instant for t > 40, and for ω = 4.0. For t > 46, we have the fully developed phase synchronization in the wide interval of the investigated frequencies.

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Fig. 6.56 Time histories of the first (a) and second (c) beam centers, and 2D wavelet spectrum of the first (b) and second (d) beam centers.

In problems with inclusion of the material physical nonlinearity we have observed the first beam dynamical stability loss (Fig. 6.56). The beam begins to vibrate around a new equilibrium state and does not have any contact with the second beam. For ε = 1.4, the second beam vibrations are damped. It should be emphasized that it is possible to control chaotic vibrations with the help of a gap between beams, and the choice of the beams material. 6.7.6

Contact interaction of beams with and without the physical nonlinearity

In what follows, we report numerical results for two-layer beams ωp = ω0 = 6.28 and excitation amplitude q1 = 2.25, 7.50, 23.0. We consider the influence of clearance between beams on their vibrations. In particular, we aim at studying the character of initial simultaneous beam vibration, i.e. beginning from their movement

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with a contact between beams. The investigations have been carried out for the following beam clearances δ ∈ {0.025; 0.05; 0.1; 0.2}. In Figs. 6.57 and 6.58, the results for δ = 0.025 are reported, whereas in Figs. 6.58, 6.59 for δ = 0.05, show the most influenced parameters of the carried out analysis. Results of Figs. 6.57 and 6.59 correspond to the initial simultaneous vibrations of two-layer beams, and the results reported in Figs. 6.58 and 6.60 refer to the simultaneous chaotic vibrations. Numerical results are given in figures in the following way: (a) signal w(t); (b) phase portrait w(w); ˙ (c) power spectrum (FFT); (d) and (e) 2D and 3D wavelet spectrum, respectively; (f) phase difference; (g) signal of simultaneous vibrations of two beams; (h) contact pressure versus time; (i) largest LE versus time. Analysis of the numerical results showed that before the contact between both beams, the upper beam vibrations were harmonic with the excitation frequency. In the time instant when a contact between beams takes place the vibrations change qualitatively, since the funω damental frequencies are ωp , 23 ωp , 3p , i.e. period tripling takes place. On the other hand, the wavelet analysis shows that the whole energy ω of the second beam is concentrated on the frequency 3p , where frequencies ωp and 23 ωp also appear. Synchronization (dark areas on graph (f)) takes place in the frequency intervals 1 < ω < 1.5 and 4.2 < ω < 5.2 and has a local-temporal character. Contact pressure (h) possesses the almost synchronous character. The successive increase in the excitation amplitude implies period doubling bifurcation, and hence the two-beam system is transited into chaotic dynamics via the modified Feigenbaum scenario [Feigenbaum (1978)]. In chaotic regimes (Figs. 6.58 and 6.59), a dynamical stability loss occurs, i.e. harmonic excitation of the upper beam implies the increase in its deflection. The power spectrum is broadband and the contact pressure distribution is chaotic. Chaotic history is phase synchronized, but the largest synchronization is exhibited on low frequencies ω < 4, and for ω ∈ [5; 8]. Time interval associated with the increase in deflections exhibits synchronization on the whole frequency interval. As it is seen on the graph (g) there is locking of the upper beam by the lower beam (Fig. 6.60).

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Fig. 6.57 Vibrational characteristics of the first and second beams separately, and of two beams (δ = 0.025, q01 = 0.5) — see text for more details.

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(Continued)

Let us briefly summarize the results obtained so far. First, the mathematical model of simultaneous vibrations of the multi-layer beams is constructed (beams are coupled via BCs, taking into account geometric, physical and constructive nonlinearities). Numerical algorithm to solve the stated problems has been proposed, and the beams are considered as mechanical systems with infinite degrees of freedom. It should be emphasized that traditional approaches to detect phase synchronization regimes based on the introduction of the phase of chaotic signal are correct only for the time series, which have sharp representation of the fundamental frequency ω0 in the Fourier-power spectrum. Otherwise [see Figs. 6.58 and 6.59(c)], traditional methods using the phase of chaotic signal may yield incorrect results. 6.7.7

Influence of BCs (two-layer isotropic beams)

We consider the system consisting two isotropic beams of the same thickness h1 = h2 = h and material E1 = E2 = E, coupled through

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Fig. 6.58

(Continued)

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Fig. 6.58 Vibrational characteristics of the first and second beams separately, and of two beams (δ = 0.025, q01 = 0.8) — see text for more details.

BCs. Clearance between the beams δ = 0.05h, and the upper beam is subjected to harmonic excitation of the form q = q0 sin ωp t,

(6.183)

where q0 = 0.5 is the amplitude of external load, ωp = 6.28 is the excitation frequency. BCs include three types of the two-layer beams: 1st type — two cantilever beams wl (0, t) = ul (0, t) = w
lx (0, t) = 0, Mx (1, t) = 0,

Nx (1, t) = 0,

Qx (1, t) = 0.

(6.184) (l = 1, 2);

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Fig. 6.59 Vibrational characteristics of the first and second beams separately, and of two beams (δ = 0.5, q01 = 0.9) — see text for more details.

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299

(Continued)

2nd type — one cantilever beam, and one simply supported w1 (0, t) = u1 (0, t) = w
1x (0, t) = 0, Mx (1, t) = 0,

Nx (1, t) = 0,

Qx (1, t) = 0,

∂ 2 w2 (1, t) ∂ 2 w2 (0, t) = w (0, t) = = w2 (1, t) = 0 2 ∂x2 ∂x2 u2 (0) = u2 (1) = 0;

(6.185)

3rd type — one cantilever beam, and one with fixed support w1 (0, t) = u1 (0, t) = w
1x (0, t) = 0, Mx (1, t) = 0,

Nx (1, t) = 0,

Qx (1, t) = 0,

∂w2 (1, t) ∂w2 (0, t) = w2 (0, t) = = w2 (1, t) = 0, ∂x ∂x

u2 (0) = u2 (1) = 0.

(6.186) In Figs. 6.61–6.62, the following characteristics of the dynamical process are shown: (a), (b) deflection wl [1, t], l = 1, 2 for the 1st type

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Fig. 6.60 Vibrational characteristics of the first and second beams separately, and of two beams (δ = 0.5, q01 = 2.5) — see text for more details.

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(Continued)

BC; wl [0.5, t], l = 1, 2 for 2nd and 3rd type BC; (c) wavelet analysis of the phase difference in time t ∈ [20, 100]; (d) deflections on time interval t ∈ [45, 55]. We consider the influence of BCs of the second beam on the vibrations character of both beams and their synchronization. Before a contact between beams, even for small load non-periodic regular vibrations are observed. In the beginning, chaotic vibrations of geometrically nonlinear two-layer conservative beams are observed (Fig. 6.61). Power spectrum of the first beam exhibits ωp /3 though other frequencies also appear. The second beam vibrations exhibit frequencies depending on the BCs: 1st type — ωp /3; 2nd type — ≈ 4.5; 3rd type — ≈ 8. Phase locking of the frequencies is observed

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Fig. 6.61 Beam center deflections (a, b), phase difference ϕ1 −ϕ2 (c), and windows of beam center deflections (a) for three types of BCs (geometric and physical nonlinearities for two beams are taken).

for the 1st type of BCs, whereas for two other types of BCs the phase locking phenomenon does not appear [Fig. 6.61(d)]. BCs also play an important role in phase synchronization. It occurs for the first type of BCs in the interval of frequencies [0; 3], though areas of synchronization also occur in other frequencies. Synchronization areas increase with time. BCs of the second-type imply synchronization of frequencies in the interval [1, 5; 5], as well as in other frequencies there are separated zones of synchronization. Synchronization is changed via splashes in time (for t = 20, 50, 70, 85). In the case of the third-type BCs, synchronization is observed on the

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Fig. 6.62 Beam center deflections (a, b), phase difference ϕ1 − ϕ2 (c), and windows of beam center deflections (a) for three types of BCs (geometric and physical nonlinearities for two beams are taken).

whole frequency domain, but it is not uniformly distributed in time [Fig. 6.61(c)]. If we take into account geometric and physical nonlinearities, then vibrations of our beams with second and third BCs do not change. Two-layer cantilever system exhibits a stability loss (in this case a sudden increase in deflection, and a change of the vibration type occur) for initially initiated simultaneous beam vibrations. Physical nonlinearity of the second beam does not have any influence on the synchronization of two-layer beams.

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Let us consider now two-layer systems, where the first beam exhibits the physical and geometrical nonlinearities, and the second beam exhibits only geometric nonlinearity (Fig. 6.62). Stability loss of the initially simultaneous vibration of the conservative beams is observed [Figs. 6.62(a) and (b)]. Vibrations are chaotic with the same frequencies as in the case of geometrically nonlinear systems studied earlier. Occurrence of physical and geometric nonlinearities implies that in a certain time instant there is practically lack of synchronization for all types of BCs [Fig. 6.62(c)]. For the cantilever system (condition 1), the synchronization takes place for t > 40 on the frequency 1.5. Area of synchronization increases with time, but it is localized in low frequency zone. For the BC of the 2nd and 3rd types, synchronization occurs for t > 70. For the 2nd type BC, synchronization takes place on the frequencies 1.5 and 3.5 with splashes of frequencies observed in time. Influence of the physical nonlinearity of the upper beam essentially changes the character of the vibrations of the beams system, and hence the synchronization zone. In the case, when both beams are physically and geometrically nonlinear, and their vibrations do not depend on the type of the BCs of the lower beam, vibrations and phase synchronization are analogous to the previous case. 6.7.8

Three beams coupled via BCs

We consider a phase synchronization of the mechanical structure composed of three Euler–Bernoulli beams (Fig. 6.63). The governing equations of its nonlinear vibrations taking into account their geometric and physical nonlinearities and contact interaction within Winkler’s theory are reported in the reference [Kantor (1990)]. For q01 = 0.9, a contact between the second and the third beam occurs. In Fig. 6.64, for each of the beams, the following are reported: power spectra (FFT); 3D wavelets; phase difference between the first and second, second and third, and first and third beams in time, respectively; largest LE versus time. Time histories w(0.5; t), t ∈ [50; 65] for the transversal load acting on the upper beam δ = 0.025 hl , hl = h (beam thickness), l = 3, ε = 1 are shown (black color corresponds to synchronization zones).

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t)

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x

z Fig. 6.63

Three Euler–Bernoulli beams with clearance.

Fig. 6.64 Dynamical characteristics of the three layers Euler–Bernoulli beam: power spectra (a–c), largest LEs (d), 3D wavelets (e–g), phase differences (h–k).

6.7.9

Conclusions

The main results follow: (i) A novel dynamical phenomenon of interacting beams has been detected. Namely, when one of the beams is subjected to periodic load, the phase synchronization of beam vibrations takes place not only on the excitation frequency, but also on the independent frequency. Besides, after synchronization interval a transition of

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the second beam from chaos into equilibrium configuration has been observed. (ii) Spatio-temporal distribution of the contact pressure in the system of uncoupled beams can be used as a tool for chaos identification.

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Chapter 7

Timoshenko and Sheremetev–Pelekh Beams This chapter deals with the Timoshenko and Sheremetev–Pelekh beam theories and the chaotic vibrations of the beams within the mentioned theories. In both cases the applied assumptions and hypotheses are formulated first, and then the governing PDEs are studied. Numerous examples of chaotic dynamics with the help of classical tools of numerical analysis supported by wavelet transforms and Lyapunov exponents (LEs) computations are reported. 7.1 7.1.1

The Timoshenko Beams Hypotheses

The following hypotheses are introduced in the case of Timoshenko beam model: (i) Beam material is isotropic and homogenous; (ii) We consider an elastic material satisfying Hooke’s law (plastic and rheological deformations are not taken into account); (iii) Normal stress σzz in layers parallel to middle surface is considered as small in comparison to the stresses σxx , σyy in the normal layers; (iv) u = u(x, t), w = w(x, t) are components of displacements vector (z = 0); uz = u(x, z, t), wz = w(x, z, t) denote displacements of an arbitrary beam point; (v) Beam is subjected only to action of the transversal load q(t) = q0 sin ωp t; 307

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q x,t

a

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0

x

2h

x ∂w ∂x

z

+

Fig. 7.1

∂w γ ∂x

The applied Timoshenko beam model.

(vi) Nonlinear dependencies between deformations and displacements are taken in the von K´arm´an form. For a given model, we take the hypothesis stating that tangential displacements uz , wz are distributed along beam thickness {−h ≤ z ≤ h} owing to the following linear rule: uz = u + zγx ,

wz = w,

(7.1)

where γx = γx (x, t) is the angle of rotation of a normal to the line z = 0 (Fig. 7.1). Then, taking into account (7.1) yields
 ∂u 1 ∂w 2 ∂γx ∂w z + , ezxz = γx + . (7.2) +z exx = ∂x 2 ∂x ∂x ∂x Now, denoting tangential deformation of the middle line by ε11 , x bending deformation by H11 = ∂γ ∂x , and shear deformation by ε13 = ∂w γx + ∂x , the formulas for deformations can be restricted to include only linear terms of the series development regarding z: ezxx = ε11 + z H11 ,

ezxz = ε13 ,

z ∈ (δi − ∆, δi+1 − ∆).

(7.3)

The shear deformation is εz13 =

1 Q1 f (z), 2hG13

(7.4)

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where Q1 is the transversal force, G13 is the shear modulus, f (z) is the function characterizing the rule of tangential stresses along beam thickness. This function is even, i.e. f (−z) = f (z),

(7.5)

which corresponds to the symmetric distribution of tangential stresses regarding the middle line. This implies two conditions:  h  h 1 f (z) · z dz = 0, f (z) dz = 1. (7.6) 2h −h −h In order to define transversal forces, we apply the following relations:  1 h 1 Q1 ε13 = σ13 εz13 dz. (7.7) 2 2 −h After a few transformations, we get h

Q1 = 2hG13 k2 ε13 ,

(7.8)

1 2 where k12 = 2h −h f (z)dz. Function f (z) satisfying the conditions (7.6) may have a different form; we can take for instance   1  z 2 − . (7.9) f (z) = 6 4 2h

The quantity k2 for this function yields 5/6. The latter value has been obtained by Ambartsumian [Ambartsumian (1987)] and Reissener [Reissner (1946)]. Timoshenko in the reference [Timoshenko (1921)] considered two values: 5/6 and 8/9, and the latter one has been validated experimentally. Hooke’s law for each ith orthotropic layer takes the form: eixx =

1 i νi i σxx − 13i σzz , i E1 E1 eixz =

eizz = i σxz , Gi13

i −ν31 1 i i σxx + i σzz , i E3 E3

(7.10)

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and hence

i

E1i ν31 E1i i i = ¯ i exx + ¯ i ezz , ∆ ∆ i

E3i ν13 E3i i i i i i = e + xx ¯i ¯ i ezz , σxz = G13 exz , ∆ ∆

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i σxx

i σzz

(7.11)

i , ν i are Poisson’s coefficients, where E1i , E3i are elasticity moduli, ν13 31 ¯ i = 1 − ν i ν i . For a Gi13 is the shear modulus in the ith layer, ∆ 31 13 chosen orthotropic layer form, the following relations hold [Ambartsumian (1987)]: i i = E11 ν31 , E3i ν13

which implies E1i =

(7.12)

i E3i ν13 . i ν31

i = 0, we find a forTaking into account the static hypothesis σzz mula for deformation eizz , and we substitute it to two first relations of (7.10). Then, the components of the stress tensor in each beam layer take the form i = E1i eixx , σxx

i σxz = Gi13 eixz .

(7.13)

Using notation ϕi1 = E1i ,

ϕi2 = 0,

(7.14)

and taking into account (7.3), we may present the stresses in each layer in the following convenient form i = ϕi1 ε11 + zϕi1 H11 , σxx

i σxz = Gi13 ε13 .

(7.15)

Denoting n + m − 1 = k, δi − ∆ = ai , δi+1 − ∆ = ai+1 , we define (in a way analogous to one layer beam) the internal stresses T11 ; Q1 are shear forces; M11 are bending moments, i.e. we have k  ai+1 k  ai+1   i 1 σxx dz, Q1 = σxz dz, T11 = i=0

ai

M11 =

i=0 k   i=0

ai+1

ai

ai

1 σxx · z dz,

(7.16)

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311

and finally we obtain    δV = (T11 δε11 ) dX + (M11 δH11 ) dX + (Q1 δε13 ) dX. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

(7.17) Substituting T11 =

i σxx

k  

and

ϕi1 dz

Q1 =

M11 =

ε11 +

αi+1 αi

k   i=0

k  

αi+1 αi

i=0

k   i=0

into (7.11), we get

αi+1

αi

i=0

i σxz

ϕi1 zdz

ε11 +

ϕi1 zdz

αi

H11 ,

Gi13 dz



αi+1

(7.18)

ε13 ,

k  

αi+1

αi

i=0

ϕi1 z 2 dz

H11 ,

and after defining the following quantities C11 =

k   i=0

D11 =

ai+1 ai

k   i=0

K11 =

k  

ϕij

2

· z dz,

A44 =

ai+1 ai

i=0

ai+1 ai

ϕij dz,

k   i=0

j = 1, 2,

ϕij · z dz,

ai+1

ai

Gi13 dz,

(7.19)

l = 2, 1,

relations (7.18) are cast to the following form: T11 = C11 ε11 + K11 H11 ,

7.1.2

Q1 = A44 ε13 ,

M11 = K11 ε11 + D11 H11 . (7.20)

Fundamental equations

The Hamilton principle yields 

t1

δS = δ t0

(K − Π) dt = 0,

(7.21)

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where Π = ΠC + ΠI . Energy of the middle    ∂U 1 1 + T11 ε11 dx = T11 ΠC = 2 e 2 e ∂x Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

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surface is
  1 ∂w 2 dx, 2 ∂x

whereas bending energy ΠI has the following form 
  k  1 ∂w dx, γ = γi. M11 H11 + Q11 γ + ΠI = 2 e ∂x

(7.22)

(7.23)

i=0

Kinetic energy is  

2   
∂w 2 ∂γ ∂U 2 1 γ1 2h0 + +b dx, K= 2 g ∂t ∂t ∂t e k k  αi+1   i γ, b= z 2 dz, γ= i=0

i=0

(7.24)

αi

whereas energy of the external forces follows 
δ w = (P11 δU + qδw)dx.

(7.25)

Substituting (7.22)–(7.25) into (7.21), and through a variation procedure, the following equations are obtained γ1 ∂2u ∂T11 + Pxx − (2h0 ) 2 = 0, ∂x g ∂t
 ∂ γ1 ∂2w ∂w ∂Q11 + T11 +q− (2h0 ) 2 = 0, ∂x ∂x ∂x g ∂t

(7.26)

γ1 ∂ 2 γ ∂M11 − Q11 − b 2 = 0. ∂x g ∂t Substituting (7.18) into (7.26), the following equations regarding displacements of the Timoshenko model are obtained   
  ∂γ ∂u 1 ∂w 2 ∂ C11 + + K11 ∂x ∂x 2 ∂x ∂x +Pxx −

γ1 ∂2u (2h0 ) 2 = 0, g ∂t

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∂ ∂w A44 γ + ∂x ∂x   
  ∂γ ∂w ∂u 1 ∂w 2 ∂w ∂ C11 + + K11 + ∂x ∂x 2 ∂x ∂x ∂x ∂x ∂2w γ1 (2h0 ) 2 = 0, g ∂t   
  
∂γ ∂w ∂u 1 ∂w 2 ∂ K11 + − A44 γ + + D11 ∂x ∂x 2 ∂x ∂x ∂x +q −

−

γ1 ∂ 2 γ b = 0. g ∂t2

(7.27)

System (7.27) should be supplemented with the initial and boundary conditions. The angle of rotation of a beam normal element around the axis Oy and the bending moment read γ = γ0,

0 M11 = M11 ,

(7.28)

respectively. Normal displacement of contour points and the magnitude of the transversal stress are as follows ∂w = Q011 , Q11 + T11 (7.29) w = w0 , ∂x respectively. Displacement of the contour points of the middle surface in a direction of axis x and the magnitude of an external compression stress follow u = u0 ,

0 T11 = T11 .

(7.30)

Owing to the Euler–Bernoulli hypothesis, a position of the normal element in the boundary section after a deformation is defined through four parameters. At each beam end, four boundary conditions are formulated, respectively. From the mathematical point of view and owing to the Timoshenko hypothesis, a number of degrees of freedom of the normal element is increased up to five due to independent rotation along axis x (on a boundary, we have x = const). ∂ δw = −δγ, If we neglect the transversal shear effect, i.e. ∂x then removing δγ in the variational equation with the help of

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these relations, we obtain the equations corresponding to the Euler– Bernoulli hypotheses instead of Eq. (7.27).

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7.1.3

Validity of numerical results

Similar to the Euler–Bernoulli beams, in order to validate numerical solutions of the equations governing the dynamics of Timoshenko beams, both FDM (Finite Difference Method) and FEM (Finite Element Method) are used. For this purpose, the following input data are taken: ε1 = 1, ε2 = 0, ε3 = 0, ω = 6.9, λ = a/2h = 50, whereas the amplitude  of the harmonic excitation is taken from the interval 4 q0 ∈ 0; 6 · 10 . Spatial step c and timing step ∆t are chosen considering stability conditions of the obtained solutions due to the Runge principle. The problem has been solved using n = 40, c = 1/40, and time step ∆t = 3.9052 · 10−3 . The choice motivation of optimal step values regarding time and spatial coordinate is described in the reference [Awrejcewicz et al. (2008)]. Reliability and validity of the obtained results are confirmed by comparison of the scales presenting the dependencies of the beam vibrations character versus the excitation amplitude, which are computed only for one value ω, as well as the graphs of the maximum beam deflection versus the excitation amplitude wmax (q0 ), obtained via both FDM and FEM (Fig. 7.2). Graphs of the maximum beam deflection versus the excitation amplitude are obtained via different methods. They fully coincide perhaps in spite of chaotic zones, which proves the reliability of the obtained numerical results. Besides, computations carried out either by FDM or FEM allowed to observe the loss in dynamic stability subjected to the action of the transversal harmonic load, which is characterized by a sudden change of the maximum deflection while changing the excitation amplitude slightly. In Fig. 7.2, the dynamic stability loss is clearly exhibited during transition from point A to point B, as well as from point C to point D. On the associated scales these transitions correspond to chaotic zones. On the contrary, transition from point E to point F is associated with a route from chaotic to periodic vibrations, which is also exhibited by corresponding scales (the values of deflections are decreased).

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315

2.5 МКР МКЭ

E

2

D

maxw

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1.5

1

A

0.5

0

Fig. 7.2

7.1.4

C

B

0

1

2

F

3 q0

4

5

6 4

x 10

Dependence wmax (q0 ) and the associated vibration scales.

Wavelet analysis

The investigation allows to conclude that the change in chaotic vibrations character of the studied continuous mechanical system depends essentially on the control parameters. Below, the results illustrating peculiarities of the system transition from regular to chaotic regimes while increasing the excitation load amplitude for three values of the frequency are reported. We denote points (ω, q0 ) ∈ {ω, q0 } of the chosen parameters in the control parameters space. Point (4.9; 10,000) (Fig. 7.3) is associated with the harmonic vibrations with the excitation frequency. The wavelet spectrum (1) implies that the vibrations character is constant. In this case, the traditional analysis can be applied, i.e. signal (2), its phase portrait (3) and Fourier power spectrum (4) are constructed. Therefore, if the vibrations character remains unchanged (in the considered case, we deal with harmonic vibrations), then both traditional methods as well as wavelet-based analysis give the same results. Increasing the load to (4.0; 10,600) (Fig. 7.4), the wavelet spectrum (1) shows that the second maximum appears beginning from the time instant t ≥ 1,000, which is manifested by the Fourier spectrum (3) showing

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Deterministic Chaos in One-Dimensional Continuous Systems w__4.000_10000_ 0.25

w

0.2 0.15 0.1 0.05

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w

0 -0.05 -0.1 -0.15 -0.2 -0.25 1102

1104

1106

1108

(1)

1112

1114

1116

1118

1120

3.5

4

4.5

5

(2) fourier_w__4.000_10000_

w__4.000_10000_

1.5

1110 t

4 3.5

1

3 0.5

dw

2.5 F(ω)

0

2

1.5 1

-0.5

0.5 -1

-1.5 -0.4

0 -0.5 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.5

1

1.5

2

2.5

w

3

ω

(3)

(4)

Fig. 7.3 Timoshenko beam vibration characteristics: (1) 2D wavelet, (2) time history, (3) phase portrait, and (4) FFT (ω = 4.0, q0 = 10,000, t ∈ [0; 1,300]).

w__4.000_10600_

2

fourier_w__4.000_10600_ 4

1.5

dw

3.5

1

3

0.5

2.5

0

F(ω) 2 1.5

-0.5

1

-1

0.5

-1.5 -2 -0.4

(1)

0 -0.5 -0.3

-0.2

-0.1

0 w

(2)

0.1

0.2

0.3

0.4

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(3)

Fig. 7.4 Timoshenko beam vibration characteristics: (1) 2D wavelet, (2) phase portrait, and (3) FFT (ω = 4.0, q0 = 10,600, t ∈ [0; 1,300]).

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w__4.000_10600_

2

0.3

317 fourier_w__4.000_10600_

w

3.5 1.5 3

0.2

1

0.1

2.5

0.5 dw

w 0

2 F(ω) 1.5

0

-0.5

1

-1

0.5

-0.1

-0.2

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200

202

204

206

208

(1)

Fig. 7.5

0

-1.5

210 t

212

214

216

-2 -0.3

218

-0.5 -0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

1

1.5

2

w

(2)

2.5

3

3.5

4

4.5

5

3.5

4

4.5

5

ω

(3)

(4)

Same as in Fig. 7.3 (ω = 4.0, q0 = 10,600, t ∈ [200; 400]).

w__4.000_10600_

w__4.000_10600_

2

0.3

fourier_w__4.000_10600_

w

3.5 1.5 3

0.2 1 0.1 w

2.5

0.5 dw

0

-0.1

2 F(ω) 1.5

0

-0.5

1

-1

0.5

-0.2 0

-1.5 -0.3 1102

(1)

Fig. 7.6

1104

1106

1108

1110 t

1112

(2)

1114

1116

1118

1120

-2 -0.4

-0.5 -0.3

-0.2

-0.1

0 w

0.1

(3)

0.2

0.3

0.4

0.5

1

1.5

2

2.5

3

ω

(4)

Same as in Fig. 7.3 (ω = 4.0, q0 = 10,600, t ∈ [1100; 1300]).

two closely located frequencies. Wavelet spectrum constructed for t ∈ [200; 400] (1), phase portrait (3) and power spectrum (4), exhibits periodic vibrations with one frequency on the Fourier spectrum (Fig. 7.5). Monitoring the wavelet spectrum, where the second frequency maximum begins with t ≥ 1,000, the wavelet spectrum (1), signal (2), phase portrait (3) and Fourier spectrum (4) for t ∈ [1,100; 1,300] are constructed and shown in Fig. 7.6. Now, the Fourier spectrum (4) really exhibits two additional frequencies (besides the excitation frequency). Comparison of the Fourier spectra and phase portraits in Figs. 7.4–7.6 yields the conclusion that the Fourier-transformation gives only an integral-type of general picture of the studied vibrations. In Fig. 7.4, one may see how the additional time interval t < 1,000, as well as more delayed time instants contribute to the vibrations (see Fig. 7.6). Therefore, general conclusion regarding vibrations drawn only on the basis of the Fourier spectrum can yield erroneous results unless they are not validated by the frequency-temporal wavelet spectrum approach.

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Deterministic Chaos in One-Dimensional Continuous Systems fourier_w__4.000_12200_

w__4.000_12200_

2

4 1.5

3.5

1

3

0.5

2.5

0

F(ω) 2

dw

1.5

-0.5

1

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-1

0.5

-1.5

0

-2 -0.5

-0.4

-0.3

-0.2

-0.1

(1)

Fig. 7.7

0 w

0.1

0.2

0.3

0.4

-0.5

0.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(2)

(3)

Same as in Fig. 7.4 (ω = 4.0, q0 = 12,200, t ∈ [0; 1300]).

w__4.000_12200_

w__4.000_12200_

2

fourier_w__4.000_12200_

w 0.3

3.5

1.5

0.2

3

1

2.5 0.1

0.5 2

w

dw

0

1.5

-0.5

-0.1

1

-1

-0.2

0.5

-1.5

-0.3 200

202

204

206

(1)

Fig. 7.8

F(ω)

0

208

210 t

212

214

216

0

-2 -0.4

218

-0.3

-0.2

-0.1

(2)

0 w

0.1

0.2

0.3

0.4

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(4)

Same as in Fig. 7.3 (ω = 4.0, q0 = 12,200, t ∈ [200; 400]). w__4.000_12200_

fourier_w__4.000_12200_ 3.5

1.5

3 1

0.2

2.5 0.5

0.1 w

w__4.000_12200_

2 w

0.3

dw

0 -0.1 -0.2 -0.3

2 F(ω) 1.5

0

-0.5

1

-1

0.5 0

-1.5

-0.4 600

Fig. 7.9

0.5

(3)

0.4

(1)

-0.5

602

604

606

608

610 t

612

(2)

614

616

618

-2 -0.5

-0.5 -0.4

-0.3

-0.2

-0.1

0 w

0.1

(3)

0.2

0.3

0.4

0.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(4)

Same as in Fig. 7.3 (ω = 4.0, q0 = 12,200, t ∈ [600; 800]).

For ω = 4.0, q0 = 12,200 (Fig. 7.7), only three frequencies are visible, whereas the occurrence of additional frequencies takes place already for t ≈ 450. For smaller loads (Fig. 7.4), it happened for t ≈ 1,000. Carrying out similar analysis to that of ω = 4.0, q0 = 10,600, the wavelet spectrum, signal, phase portrait and Fourier spectrum for t ∈ [200; 400] (Fig. 7.8) and for t ∈ [600; 800] (Fig. 7.9) are constructed. The picture presenting the occurrence of new frequencies is similar to that of Figs. 7.4–7.6.

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w__4.000_14200_

3

fourier_w__4.000_14200_ 4

2

3.5 3

1

2.5 dw

F(ω) 2

0

1.5

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-1

1 0.5

-2

0 -3 -0.8

-0.6

-0.4

-0.2

0 w

(1)

Fig. 7.10

0.4

0.6

0.8

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(2)

(3)

Same as in Fig. 7.4 (ω = 4.0, q0 = 14,200, t ∈ [0; 450]). w__4.000_14200_ w

fourier_w__4.000_14200_

2

0.4

3

1.5

0.3

2.5

1

0.2 w

w__4.000_14200_

2.5

0.5

2

0.5

0.1

dw

0

F(ω)1.5

0

-0.5

-0.1

1

-1

-0.2

0.5

-1.5

-0.3

0

-2

-0.4 100

102

104

106

108

(1)

Fig. 7.11

0.2

110

112

114

116

-0.5

-2.5 -0.8

118

-0.6

-0.4

(2)

-0.2

0

0.2

0.4

0.6

0.5

1

1.5

2

2.5

(3)

3

3.5

4

4.5

5

(4)

Same as in Fig. 7.3 (ω = 4.0, q0 = 14,200, t ∈ [100; 150]). w__4.000_14200_

w__4.000_14200_

3

fourier_w__4.000_14200_ 3.5

w 0.4

3

2

2.5

0.2 1

2

0 w

0

F(ω)1.5

-1

0.5

dw

1

-0.2

-0.4

0

-2

-0.5

-0.6 380

(1)

Fig. 7.12

382

384

386

388

390 t

(2)

392

394

396

398

-3 -0.8

-0.6

-0.4

-0.2

0 w

(3)

0.2

0.4

0.6

0.8

-1

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ω

(4)

Same as in (ω = 4.0, q0 = 14,200, t ∈ [300; 450]).

Further increase in the load to q0 = 14,200 (Fig. 7.10) forces the system to transit from periodic vibrations into a chaotic regime (it takes place for t = 300). If we construct wavelet spectrum, signal, phase-portrait and Fourier spectrum for time intervals of t ∈ [100; 150] (Fig. 7.11) and for t ∈ [300; 450] (Fig. 7.12), then one may conclude that the phase portrait and Fourier spectrum imply regular dynamics (Fig. 7.11), whereas the drawings of Fig. 7.12 imply chaos. These characteristics for the whole time interval (Fig. 7.10),

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without additional information given by the wavelet spectrum, can lead to wrong conclusions regarding the vibrations character. Comparison of the wavelet spectrum and Fourier spectra reported in the above given figures allows to conclude that together with maximum amplitudes associated with fundamental frequencies exhibited by both spectra, the Fourier spectrum includes a certain number of less exhibited maxima, which are not validated by the wavelet spectra. In other words, the wavelet transform reports only the fundamental frequencies, i.e. those transmitting the fundamental energy of the vibrating process. For the frequency ω = 7.0 (Fig. 7.13), different phenomena are observed. The increase in the loading amplitude from q0 = 2,000 (1,2) to q0 = 5,000 (3,4) yields only one additional frequency (besides of the excitation frequency). For q0 = 27,000 (5,6), signal chaotization is observed suddenly (contrary to the frequency ω = 4, chaos begins practically with the vibrations). For q0 = 27,400 (7,8), elements of chaotization appear in both wavelet spectrum and Fourier spectrum in comparison to the previous results. Further increase in the load up to q0 = 29,000 (9,10) and q0 = 29,200 (11,12) leads to the occurrence of chaos again. However, for the given values of the control parameters the frequency-temporal signal characteristics given by the wavelet transform allows to detect and monitor windows of quasiperiodicity (they refer to time periods, where on the wavelet spectrum one may monitor the energy distribution regarding different frequencies). On (9,10) those “windows” are visible for t ∈ (100; 150), t ∈ (175; 200). For q0 = 29,200 (11,12) those “windows” are observed in time instants t = 150 and t = 400 and they have short duration. For q0 = 29,400 (13,14), we see the relatively long “window” t ∈ (130; 350). Note that in all three cases shown in (9–14), the Fourier spectrum indicates only that the signal is chaotic, but no information regarding signal time evolution is given. Only the wavelet spectrum allows to solve the question regarding the time character exactly. The vibrations type for q0 = 29,400 for different time intervals coincides with the dynamics of the largest LE (Fig. 7.14). Since for q0 = 29,400 the transitional processes are reported, we also use this case for the illustration of the vibration regimes given by

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(1) q0

= 2000

(2)q0 = 2000

321

(3)q0 = 5000

(4)q0 = 5000

fourier_w__7.000_27000_

3

fourier_w__7.000_27400_ 3

2.5

2.5

2

2

1.5

1.5

1 F(ω) 1

F(ω) 0.5

0.5

0

0

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-0.5 -1

-0.5

-1.5

-1 1

2

3

4

5

6

7

8

1

2

3

ω

(5) q0 = 27000

4

5

6

7

8

ω

(6) q0 = 27000

(7) q0 = 27400

(8) q0 = 27400 fourier_w__7.000_29200_

fourier_w__7.000_29000_ 3 2.5

2.5 2

2

1.5

1.5 F(ω) 1

F(ω)

1

0.5

0.5

0 -0.5

0

-1

-0.5

-1.5 1

2

3

4

5

6

7

8

1

ω

(9) q0 = 29000

2

3

4

5

6

7

8

ω

(10) q0 = 29000

(11) q0 = 29200

(12) q0 = 29200

fourier_w__7.000_29400_ 3 2.5 2 1.5 F(ω)

1

0.5 0 -0.5 -1 -1.5 1

2

3

4

5

6

7

8

ω

(13) q0 = 29400

Fig. 7.13

(14) q0 = 29400

Wavelet and Fourier spectra for different q0 and ω = 7.0.

Fig. 7.14 The largest LE versus time for ωp = 7.0, q0 = 29,400, T = {[60; 70], [210; 220], [360; 380]}.

different wavelets. In Fig. 7.15, the results of the wavelet transform using Gauss-1, Mexican hat and Gauss-8, respectively, are presented. One may conclude that only the Gauss-8 wavelet allows to exhibit frequencies, on which the fundamental energy of vibrations is concentrated. However, this wavelet does not yield an adequate picture

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Deterministic Chaos in One-Dimensional Continuous Systems

(1)

(2)

(3)

Fig. 7.15 Wavelet spectra for Gauss-1 (1), Mexican hat (2) and Gauss (3) for ω = 7.0, q0 = 29,400, t ∈ [0; 450]. w__7.000_29400_

w__7.000_29400_

3

fourier_w__7.000_29400_

w

0.4

2.5 2

0.3 0.2

2

1

1.5

0

F(ω) 1

0.1 w

dw

0 -0.1

0.5

-1

-0.2

0

-0.3

-2

-0.4 200

(1)

-0.5 202

204

206

208

210 t

212

(2)

214

216

218

-3 -0.5

-0.4

-0.3

-0.2

-0.1

0 w

0.1

(3)

0.2

0.3

0.4

0.5

1

2

3

4

5

6

7

8

ω

(4)

Fig. 7.16 2D wavelet spectrum (1), time history (2), phase portrait (3) and FFT (4) (ω = 7.0, q0 = 29,400, t ∈ [200; 220]).

of qualitative changes of the vibration character (in particular, recall the occurrence of the mentioned windows). For q0 = 29,400, windows of quasi-periodicity have a large zone of attraction. Using traditional methods of analysis through signals, phase portraits and power spectrum on the time interval t ∈ [200; 220], it is not difficult to conclude that the fundamental energy is concentrated on three frequencies (Fig. 7.16). In this case, the FFT can be used as the tool to control observations yielded by wavelet spectra. The leading role while investigating the nonstationary signals certainly belongs to the wavelet-based analysis. Besides, in the situations shown in Fig. 7.13 (9,12), a construction of the power spectrum for the quasi-periodicity windows is difficult due to their small time duration. However, the sufficient exact localization of the windows with the help of wavelet analysis allows to analyze their structure. For example, in Fig. 7.13 (9,10) (ω = 7.0, q0 = 29,000), i.e. on the separated time intervals we have also

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Timoshenko and Sheremetev–Pelekh Beams w__7.000_29000_

0.5

323 fourier_w__7.000_29000_

w__7.000_29000_

3 w

0.4

2.5 2

0.3 0.2

2 1.5

1

0.1 w

dw

0

F(ω)

0

-1

-0.2 -0.3

0 -0.5

-2

-0.4

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

-0.5 110

-1 112

114

116

(1)

Fig. 7.17

1

0.5

-0.1

118

120 t

122

124

126

-3 -0.8

128

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

1

2

3

4

w

(2)

5

6

7

8

7

8

ω

(3)

(4)

Same as in Fig. 7.16 (ω = 7.0, q0 = 29,000, t ∈ [110; 130]).

w__7.000_29200_

w__7.000_29200_

3

fourier_w__7.000_29200_ 2.5

w

0.4

2

0.3

2

0.2 1

1.5

0

F(ω) 1

0.1 w

dw

0 -0.1

0.5

-1

-0.2

0

-0.3

-2

-0.4

-0.5 146

148

150

152

154 t

(1)

Fig. 7.18

(2)

156

158

-3 -0.5

-0.4

-0.3

-0.2

-0.1

0 w

0.1

(3)

0.2

0.3

0.4

0.5

1

2

3

4

5

6

ω

(4)

Same as in Fig. 7.16 (ω = 7.0, q0 = 29,200, t ∈ [145; 160]).

windows. In Fig. 7.17, results of application of classical vibration approaches are presented: wavelet spectrum, signal, phase portrait and power spectrum for the time interval t ∈ [110; 130]. The latter example also shows that the fundamental energy of the vibrational process is distributed along three frequencies. Analogous constructions for q0 = 29,200, t ∈ [145; 160] are shown in Fig. 7.18. Comparison of the results shown in Figs. 7.16–7.18 yields the conclusion that the structure of quasi-periodicity windows obtained for different loads is practically the same. Namely, in the mentioned time intervals the fundamental energy of vibrations is mainly distributed along three frequencies. The change of vibrations character for the frequency ω = 8.0 also possesses its own peculiarities: with increase in the load, new frequencies appear, but for other loads the reported windows do not appear. In Fig. 7.19, wavelet spectra and FFT power spectra are reported related to the load increase: q0 = 2,000; 4,000; 7,200; 14,000; 21,600. It should be emphasized that for all three considered frequencies ω = {4.0; 7.0; 8.0} the wavelet transform allows for monitoring

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Deterministic Chaos in One-Dimensional Continuous Systems fourier_w__8.000_2000_

fourier_w__8.000_4000_ 3

2.5

2.5

2

2

1.5

1.5

1 F(ω)0.5

ω)

1

F(

0.5

0

0

-0.5

-0.5

-1

-1

-1.5

-1.5

-2 1

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

b2304-ch07

2

3

4

5

6

7

8

9

ω

(2) q0 = 2000

(1) q0 = 2000

1

2

4

5

6

7

8

9

8

9

(4) q0 = 4000

fourier_w__8.000_7200_

fourier_w__8.000_14000_ 3.5

3

3

2.5

2.5

2

2

1.5 F(ω)

3

ω

(3) q0 = 4000

ω)1.5

1

F(

1

0.5

0.5

0

0

-0.5

-0.5 -1 1

2

3

4

5

6

7

8

-1

9

ω

(5) q0 = 7200

1

2

3

4

5

6

7

ω

(6) q0 = 7200

(7) q0 = 14000

(8) q0 = 14000

fourier_w__8.000_21600_ 3.5 3 2.5 2 F(ω) 1.5 1 0.5 0 -0.5 1

(9) q0 = 21600

2

3

4

5

6

7

8

9

ω

(10) q0 = 21600

Fig. 7.19

Same as in Fig. 7.13 (ω = 8.0).

of change in vibration character in time though all the remaining parameters are fixed. 7.2

The Sheremetev–Pelekh Beams

The Sheremetev–Pelekh models of beams are not widely known in the Western literature. Therefore, we begin with a short background description of this topic. 7.2.1

Hypotheses

The following hypotheses are introduced while constructing the Sheremetev–Pelekh beam model: (i) Beam material is homogenous and isotropic; (ii) Beam material is elastic and the Hooke’s principle holds, i.e. plastic and rheological deformations are not taken into account;

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Timoshenko and Sheremetev–Pelekh Beams

325

(iii) Normal stresses σzz in layers parallel to the middle surface are small in comparison to stresses σxx , σyy in the normal layers; (iv) u = u(x, t), w = w(x, t) are components of the displacement points of the middle line (z = 0), whereas uz = u(x, z, t), wz = w(x, z, t) are displacements of an arbitrary beam point; (v) Beam is subjected only to the transversal harmonic load q(t) = q0 sin ωp t action; (vi) Nonlinear dependencies between deformations and displacements are taken in the K´ arm´an form: ezxx


 1 ∂w 2 ∂uz + = , ∂x 2 ∂x

ezxz =

∂wz ∂uz + . ∂z ∂x

(7.31)

Following [Sheremetev and Pelekh (1964)], displacement field (see Fig. 7.20 and observe rotation and curvature of the normal) for a multi-layer beam can be formulated in the following form uz = u + zγ + z 2 uT + z 3 γ T ,

wz = w.

(7.32)

Note that (7.32), similar to the Timoshenko model, allows to satisfy the condition of compatibility of displacements on a border of

q x, t

a

0

x

2h

x ∂w ∂x

z

+ Fig. 7.20

∂w γ ∂x

The Sheremetev–Pelekh beam model.

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contacting layers. In order to define unknown functions uT (x, y), γxT (x, y), we use additional relations on the external beam surfaces:

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 n+m  = 0, σxz δn+m −∆

 1  σxz = 0, δ0 −∆

(7.33)

which, taking into account (7.32), yield the following system of equations (n + m − 1 = k, δi − ∆ = ai ) ∂w + γx + 2ak uT + 3a2k γxT = 0, ∂x ∂w + γx + 2a0 uT + 3a20 γxT = 0, ∂x

(7.34)

and they have the following solutions


∂w + γx ∂x


z

u = u + z γx +







 ak + a0 = u = · − . 2ak a0 (7.35) Owing to the introduced notation, we have: δn+m = 2h0 , δ0 = 0, 2h0 −2∆ 1 k +a0 = 2(2h , 3ak1a0 = − 3(2h0 −∆)·∆ . Then, relations (7.2.1) − a2a 0 −∆)·∆ k a0 take the following form γxT

1 , 3ak a0

∂w + γx ∂x

T




∂w + γx ∂x

 h0 − ∆ 1 3 −z , z (2h0 − ∆) ∆ 3∆ (2h0 − ∆) 2

wz = w (7.36) or ∂w +ϕ· u =u−z ∂x z




 h0 − ∆ 1 3 z+z −z , (2h0 − ∆) ∆ 3∆ (2h0 − ∆) 2

wz = w, (7.37) ∂w +γ. Function ϕ(x) = +γ describes the beam shear where ϕ = ∂w x ∂x ∂x in plane XOZ. Now we apply relation (7.36), considering rotation

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Timoshenko and Sheremetev–Pelekh Beams

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327

angles of the normal separately: ∂∂ wx or in the form (7.37), and we introduce the unknown shear function ϕ. Further, we apply (7.36), and using the known procedure (see [Sheremetev and Pelekh (1964)]), the shear deformation in an arbitrary beam point is obtained:
 ∂w + γx , (7.38) ezxz = f (z) · ∂x where f (z) is the function of the stress distribution along beam thickness [Sheremetev and Pelekh (1964)]: f (z) = 1 + z ·

z2 2h0 − 2∆ − . (2h0 − ∆) · ∆ (2h0 − ∆) · ∆

(7.39)

Note that in the case of Timoshenko model, we have f (z) ≡ 1. 0 ·∆ , h2 = − 3∆ (2h10 −∆) , the remaining compoDenoting h1 = (2hh0 −∆)·∆ nents of the deformation take the following form



z = ε11 + z + h1 z 2 + h2 z 3 H11 + h1 z 2 + h2 z 3 H111 + αz11 Θ, ezxx lxx (7.40) 2

where H111 = ∂∂ xw2 . For the symmetric layers, we get δn+m − ∆ = h, δ1 − ∆ = −h, f (z) = 1 − (z/h)2 , uT ≡ 0, h1 ≡ 0, h2 = −(1/3h2 ). Now, in a way similar to Timoshenko and Euler–Bernoulli models, we get stresses in each of the layers in the form



i = ϕi1 ε11 + z + h1 z 2 + h2 z 3 ϕi1 H11 σxx

+ (h1 z 2 + h2 z 3 ) ϕi1 H111 − ϕi1 (αi11 )Θ, (7.41) i = Gi13 ε13 · f (z). σxz

In order to obtain 1D equation of a multi-layered beam, the following Lagrange functional is taken on the interval x ∈ [0, e]  n+m−1   δi+1 −∆ i (σxx (δε11 + t11 δH11 + δH11 ) e

i=0

δi −∆

i f (z)δε13 )dz + σxz

 qδwdx,

= e

dx

(7.42)

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where t11 = (z + h1 z 2 + h2 z 3 ), r11 = (h1 z 2 + h2 z 3 ). Owing to the analogy with the Timoshenko and Euler–Bernoulli models, we introduce integral characteristics of the beam internal stresses regarding bending moments and transversal forces Mij , Qj , (i, j = 1), which have the same order as the stresses in the Timoshenko model. However, here we take into account higher order moments and tangential forces, which depend on the transversal shear: M112 = h1

k−1   i=0

ai+1

ai

i σxx z 2 dz,

M113 = h2

k−1   i=0

(1)

Q1 =

k−1   i=0

ai+1

ai

ai+1

ai

i σxx z 3 dz,

i σxz f (z) dz.

(7.43) Relations of elasticity of a multi-layer beam can be derived by coupling the internal stresses (7.43) with the deformation of middle surface, i.e. we substitute (7.43) into (7.41) to get (1)

Q1 = A1313 ε13 , M11 = K11 ε11 + (D11 + K1113 + D114 ) H11 + (K1113 + D114 ) H111 − (KP11 + KP12 ) M112

M113

M223

0 0 H11 + D111 H111 − (KP11 + KP12 ) , = K11 ε11 + D11 1

1 1 = K112 ε11 + K1123 + K1124 + K1125 H11 1

1 + K1124 + K1125 H111 + M P112 0 0 = K112 ε11 + K1121 H11 + K1123 H111 + M P112 , 1

1 1 = K113 ε11 + D1134 + K1135 + D1136 H11 1

1 + K1135 + D1136 H111 + M P113

(7.44)

0 0 = K113 ε11 + D1131 H11 + D1133 H111 + M P113 , 1

1 1 = K213 ε11 + D2234 + K2235 + D2236 H11 1

1 + K2235 + D2236 H111 + M P223 0 0 = K213 ε11 + D2231 H11 + D2233 H111 + M P223 .

Coefficients of the series are different in comparison to the analogous ones corresponding to the Euler–Bernoulli model, and they

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have the following form: k  ai+1  ϕi1 z 2 dz, K112 = h1

A1313 =

i=1

ai

k  

ai+1

ai

i=1

=

h21

k   i=1

1 = h2 D1134

2 Gi13 f(z) dz,

ai+1

ai+1

ai

k   i=1

i=1

ai

i=1 1 K1124

k  

K1113 = h1

ai+1

ai

k  

ϕi1 z 4 dz,

1 K1123 = h1

k   i=1

ϕi1 z 4 dz,

1 K1125

= h1 h2

ai+1

ai+1 ai

k  

ai

ϕi1 z 4 dz,

M P112 = −h1

1 D1136 = h22

k   i=1

k   i=1

M P113 = −h2

ai+1

ai

k   i=1

ϕi1 z 3 dz,

ai+1

ai

ϕi1 z 5 dz,

ϕi1 z 6 dz, (7.45)



ϕi1 αi11 z 2 Θ dz ,

ai+1

ai

ϕi1 z 3 dz,

ai+1

ai

i=1 ai+1

ϕi1 z 3 dz,

ai

i=1

k  

D114 = h2

K113 = h2

329

ϕi1 αi11 z 3 Θ dz

.

Equations governing the dynamics regarding displacements for the one-layer beam of the kinematic Sheremetov–Pelekh model have the following form: ∂u ∂ 2 u ∂w ∂ 2 w ∂ 2 u = 0, + − 2 − ε1 ∂x2 ∂x ∂x2 ∂t ∂t  3    4 ∂ γx 1 ∂ 4 w ∂γx ∂ 2 w 1 2 + − D + k γ λ2 63 5 ∂x3 4 ∂x4 ∂x ∂x2 1 ∂2w ∂w 1 = 0, [L (w, w) + L (w, u)] + q − − ε2 2 1 2 2 2 λ λ ∂t ∂t   ∂ 2 γx 48 ∂ 3 w ∂w ∂γx 204 ∂ 2 γx 2 2 − = 0. − − 12λ k D γ + − ε3 x 2 3 2 315 ∂x 315 ∂x ∂x ∂t ∂t (7.46) +

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Bars over the non-dimensional parameters are omitted, Dγ = GE13 , 1 and the operators L1 (w, u), L2 (w, w), L3 (w, u) have the following form 
∂ 2 u ∂w ∂u ∂ 2 w 3 ∂ 2 w ∂w 2 L1 (u, w) = + , L2 (w, w) = , ∂x2 ∂x ∂x ∂x2 2 ∂x2 ∂x L3 (w, w) =

∂ 2 w ∂w . ∂x2 ∂x

(7.47) System (7.46) should be supplemented by boundary conditions (here we consider the simply supported beam only): w(0, t) = w(1, t) = 0,

u(0, t) = u(1, t) = 0,

(γx )
x (0, t) = 0,

(γx )
x (1, t) = 0,

w

xx (0, t) = 0,

w

xx (1, t) = 0,

and initial conditions: w(x, t)|t=0 = u(x, t)|t=0 = γ(x, t)|t=0 = 0,    ∂u(x, t)  ∂γx (x, t)  ∂w(x, t)  = = = 0.  ∂t t=0 ∂t t=0 ∂t t=0 7.2.2

(7.48)

(7.49)

Reliability of numerical results

As in the case of Euler–Bernoulli beams, in order to verify the reliability of the numerical solutions of equations governing the dynamics of Sheremetov–Pelekh beams, two numerical approaches have been applied, i.e. FDM and FEM. The following input data are used: ε1 = 1, ε2 = 0, ε3 = 0, ω = 6.9 (frequency of excitation), a = 50 (relative beam length), λ = 2h   and the amplitude of transver4 sal harmonic load q0 ∈ 0; 6 · 10 . A comparison of the numerical solutions of (7.46) obtained through FDM and FEM including investigation of the obtained results convergence, influence of both the partition of the spatial coordinate and time step has been carried out. The obtained results via FDM and FEM practically coincide in full, but FDM has been further chosen due to less computational time needed to get reliable results. Beam partition of n = 40, which defines spatial step

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Fig. 7.21 Maximum beam deflection versus q0 (a) and scales of beam vibrations character (b).

c = 1/40, and time step of ∆t = 0.0039 have been chosen to preserve the required computational accuracy with a simultaneous short computational time. Comparison of the results obtained via FDM and FEM is illustrated in Fig. 7.21, where drawings of the maximum beam deflection and scales of the vibrations character are shown. By a scale of vibrations character we mean a graphical representation of the vibrations type (periodic, quasi-periodic, chaotic, etc.) versus the applied load. It is clear that both methods correctly exhibit characteristic jumps of the maximum deflection. These jumps indicate the loss in beam stability in time instants associated with transition from periodic to chaotic vibrations. We report also time history and power spectra for three different beam partitions regarding spatial coordinates for n = 20, n = 40

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332 1

n=20 n=40 n=60

0 -0.5 -1

3

3

2.5

2.5

2

2

F(ω)1.5

1.5 F(ω) 1

1

-1.5

0.5

0.5

-2

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Deterministic Chaos in One-Dimensional Continuous Systems

0.5

w

9in x 6in

0

0

-2.5 1016

1017

1018 t

1019

1020

1

2

3

(a)

ω

4

5

6

n=40 n=60

-0.5

n=20

-0.5

1

7

2

(b)

3

ω

4

5

6

7

(c)

Fig. 7.22 Time histories (a) and power spectra (b, c) for different beam partitions n (periodicity). n=20 n=40 n=60

2 1

2

w0

F(ω) 1

-1

0

-2 -3 1010

3

n=20 n=40 n=60

-1 1011

1012

t

1013

1014

(a)

1

2

3

ω

4

5

6

7

(b)

Fig. 7.23 Time histories (a) and power spectra (b) for different beam partitions n (chaos).

and n = 60. In Fig. 7.22, the convergence is illustrated for the case of periodic vibrations: (a) time history convergence, (b) (n = 20) and (c) (n = 40, 60) — power spectrum convergence. For n = 40 and n = 60, the convergence regarding time history as well as the power spectrum is observed. In the case of chaotic vibrations, the convergence regarding time series cannot be achieved, and hence the coincidence in signals [Fig. 7.23(a)] is not observed. Power spectra [Fig. 7.23(b)] have broad-band character for all used partitions. 7.2.3

Wavelet-based analysis

We investigate a character of nonlinear vibrations of the Sheremetev– Pelekh beam subjected to the transversal harmonic excitation

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w__6.000_5000_

0.15

fourier_w__6.000_5000_ 3

0.04

2

0.05

0.02 0.01 w

1.5 1

0 dw

0

F(ω) 0.5

-0.05

-0.01 -0.02

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0.1

0.03

333

0 -0.5

-0.1

-1

-0.03

-0.15

-1.5

-0.04 501

502

503

504

505 t

506

507

508

509

-0.2 -0.05

510

-0.04

-0.03

-0.02

-0.01

(a)

0 w

0.01

0.02

0.03

0.04

0.05

-2

1

2

3

4

5

6

7

ω

(b)

(c) _6.000_5000_sp_500_800_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

(e)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.24 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 5,000 and t ∈ [500; 510]. fourier_w__6.000_10800_

w__6.000_10800_

0.8

3.5

0.6

3 2.5

0.4

2 0.2 w

1.5 F(ω)

0

1

0.5

-0.2

0 -0.4

-0.5 -1

-0.6 -0.8 -0.4

-1.5 -0.3

-0.2

-0.1

0 w

a)

0.1

0.2

0.3

0.4

1

2

3

4

5

6

7

ω

b)

c)

d)

Fig. 7.25 Phase portrait (a), FFT (b), 2D wavelet (c), 3D wavelet (d) for ωp = 6.0, q0 = 10,800, t ∈ [0; 1,324].

q = q0 sin(ωp t) for the excitation frequency ωp = 6.0 and for a series of the excitation amplitude q0 . In Fig. 7.24, the case of periodic vibrations with ωp = 6.0, q0 = 5,000 is illustrated. The wavelet spectrum ((e) is the wavelet spectrum surface in space, (d) is the projection of a given surface onto plane (t, ω)), as well as the Fourier spectrum (c) demonstrate the occurrence of only one frequency being equal to the excitation frequency. Signal/time history (a), phase portrait (b) and the Poincar´e map (f) also confirm the harmonic character of vibrations.

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w__6.000_10800_

w__6.000_10800_

0.3

fourier_w__6.000_10800_

0.1

3.5

0.2

0.08

3

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0.1

2.5

0.04

0

0.02 w

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dw

0

F(ω)

-0.1

2

1.5

-0.02

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

1

-0.2

-0.04 -0.06

0.5

-0.3

-0.08 -0.1 150

0 151

152

153

154

155 t

(a)

156

157

158

159

160

-0.4 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

1

w

2

3

4

5

6

7

ω

(b)

(c) _6.000_10800_sp_190_200_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

(e)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.26 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 10,800, t ∈ [10; 200].

The increase in the excitation amplitude (ωp = 6.0) up to q0 = 10,800 (Fig. 7.25) implies more complex vibrations. Frequencytemporal wavelet spectrum (c, d) shows that in time instant t ≈ 500 a transition from one-frequency to multi-frequency vibrations takes place. Traditional tools of analysis like the (a) phase portrait and (b) Fourier spectrum indicate the occurrence of many frequencies. However, without implementing the wavelet spectrum one may yield a wrong conclusion that the multi-frequency character of vibrations takes place for all t ∈ [0; 1,324]. In fact, we can distinguish three time intervals: [10; 200] — one frequency vibrations, [200; 800] — transitional process with a birth of new frequencies, [800; 1,200] — multi-frequency vibrations. In Fig. 7.26, the phase portrait (b), signal (a), Fourier spectrum (c) and the Poincar´e map (f) are presented for time interval t ∈ [10; 200]. The same is shown in Fig. 7.27 for t ∈ [800; 1200]. In the given case, Fourier spectra confirm change in vibrations character exhibited also on the wavelet spectrum (Fig. 7.25). Fourier-spectrum in Fig. 7.26 reports one frequency, whereas

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w__6.000_10800_

fourier_w__6.000_10800_

w__6.000_10800_

0.8

335

3

0.3

0.6

0.25 0.2 0.15 0.1 w

dw

0.05 0

0.4

2

0.2

1.5 F(ω)

0

1

0.5

-0.2

-0.05

Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

2.5

-0.4

0

-0.6

-0.5

-0.1 -0.15 -0.2 901

902

903

904

905 t

(a)

906

907

908

909

910

-0.8 -0.25

-0.2

-0.15

-0.1

-0.05

0 w

0.05

0.1

0.15

0.2

0.25

-1

1

2

3

4

5

6

7

ω

(b)

(c) _6.000_10800_sp_800_900_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

(e)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.27 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 10,800, t ∈ [800; 1200].

in Fig. 7.27 two fundamental frequencies (they are visible also on the wavelet spectrum) are shown. Therefore, in the wavelet spectrum only fundamental scales are well distinguished, which are associated with the main system energy. The wavelet spectrum allows for adding important information to the traditional vibration analysis. For instance, a comparison of Figs. 7.26 and 7.27(c) yields the conclusion that in spite of vibrations with frequency ωp the part of the energy is transmitted via frequency ωp /2, which indicates the occurrence of Hopf bifurcation. Besides, there exists one more low independent frequency. Increasing the excitation amplitude up to q0 = 13,600 for ωp = 6.0 (Fig. 7.28) the frequency-temporal wavelet spectrum shows that frequency maxima appear (besides of the excitation frequency) in time. In Figs. 7.29–7.31, time histories (a), phase portraits (b), Fourier spectra (c), Poincar´e maps (f), as well as 2D (d) and 3D (e) wavelet surfaces for time intervals t ∈ [0; 200], [600; 800], [1,000; 1,200] are reported. It is illustrated that besides the fundamental frequency

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w__6.000_13600_

2.5

3

2

2.5

1.5

2 1 dw

1.5 F(ω)

0.5 0

1

0.5 0

-0.5

-0.5

-1

-1

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-1.5 -0.4

-0.2

0

0.2

w

0.4

0.6

0.8

1

1

2

3

4

5

6

7

ω

(a)

(b)

(c)

(d)

Fig. 7.28 Phase portrait (a), FFT (b), 2D wavelet (c), 3D wavelet (d) for ωp = 6.0, q0 = 13,600, t ∈ [0; 1,324]. w__6.000_13600_

0.15

w__6.000_13600_

0.4

fourier_w__6.000_13600_

3

0.3

2.5

0.1

0.05 w

dw

0.2

2

0.1

1.5 F(ω)

0

1

0 -0.1

-0.05

0.5

-0.2

0

-0.3

-0.5

-0.1 180

181

182

183

184

185 t

186

(a)

187

188

189

190

-0.4 -0.05

0

0.05

0.1

0.15

0.2

0.25

1

0.3

2

3

w

4

5

6

7

ω

(b)

(c) _6.000_13600_sp_200_300_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

(e)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.29 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 13,600, t ∈ [0; 200].

ωp = 6.0, ωp /2 = 3.0 and ωp /4 = 1.5 appear. The choice of control parameters ωp = 6.0, q0 = 17,000 (Fig. 7.32) allows to observe the multi-frequency vibrations on the whole time interval without a change in the frequency set, where at ωp = 6.0 the frequencies ωp /2 = 3.0 and ωp /4 = 1.5 appear. For the control parameters ωp = 6.0, q0 = 33,400 (Fig. 7.33) we may observe chaos in the whole time interval t ∈ [0; 1,324], which is confirmed also by the frequencytemporal wavelet spectrum.

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fourier_w__6.000_13600_

w__6.000_13600_

1.5

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337

3

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1

2.5

0.5

2

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0.4 w

dw

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1 0.2

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0

-1

0 -0.1

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702

703

704

705 t

706

707

708

709

-1.5 -0.5

710

-0.4

-0.3

-0.2

-0.1 w

(a)

0

0.1

0.2

1

0.3

2

3

4

5

6

7

ω

(b)

(c) _6.000_13600_sp_600_700_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

0

0.1

0.2

0.3

0.4

(e)

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.30 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 13,600, t ∈ [600; 800]. w__6.000_13600_

fourier_w__6.000_13600_

w__6.000_13600_

2.5

0.7

3.5

0.6

2

0.5

3

1.5

2.5

0.4

2

1

0.3 w

dw

0.2

F(ω)1.5

0.5

1

0.1

0

0.5

0 -0.5

-0.1 -0.2

0 -0.5

-1

-0.3 1101

1102

1103

1104

1105 t

(a)

1106

1107

1108

1109

1110

-1.5 -0.4

-1 -0.2

0

0.2

0.4

0.6

0.8

1

1

2

3

4

w

5

6

7

ω

(b)

(c) _6.000_13600_sp_1000_1100_ 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.1

(d)

(e)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

(f)

Fig. 7.31 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 13,600, t ∈ [1,000; 1,200].

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w__6.000_17000_

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fourier_w__6.000_17000_ 3.5

1.3

3

1.5

1.2

2.5

1.1

1

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dw

w 0.9

F(ω)1.5

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0

-0.5

-0.5

0.5 610

620

630

640

650 t

660

670

680

690

-1 0.4

700

0.5

0.6

0.7

0.8

0.9 w

(a)

1

1.1

1.2

1.3

1

2

3

4

5

6

7

ω

(b)

(c) _6.000_17000_sp_600_800_

1.5

1

0.5

0 0

(d)

0.5

1

(e)

1.5

(f)

Fig. 7.32 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 17,000, t ∈ [0; 1,324]. w__6.000_33400_

w__6.000_33400_

6

2

fourier_w__6.000_33400_ 4.5

1.5

4

4 3.5

1 2

3

0

2.5 F(ω) 2

0.5 w

dw

0

1.5

-0.5

-2

1

-1

0.5

-4

0

-1.5 400

405

410

415

420

425 t

430

(a)

435

440

445

450

-6 -2

-0.5

-1.5

-1

-0.5

0

w

0.5

1

1.5

2

2.5

1

2

3

4

5

6

7

ω

(b)

(c) _6.000_33400_sp_200_600_

3 2.5 2 1.5 1 0.5 0 -0.5 -0.5

(d)

(e)

0

0.5

1

1.5

2

2.5

3

(f)

Fig. 7.33 Time history (a), phase portrait (b), FFT (c), 2D wavelet (d), 3D wavelet (e), Poincar´e map (f) for ωp = 6.0, q0 = 33,400, t ∈ [0; 1,324].

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Concluding Remarks

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In what follows, we briefly summarize the obtained results: (i) General theories and mathematical models of chaotic vibrations of nonlinear continuous mechanical systems (beams) with the help of the kinematic Euler–Bernoulli, Timoshenko and Sheremetev–Pelekh hypotheses are constructed; (ii) Good coincidence between the wavelet spectrum and the LEs have been achieved while defining the vibrations character; (iii) Advantages of the application of Morlet’s wavelet to study chaotic vibrations of various beam models, i.e. the mentioned wavelet exhibiting the largest possibility of frequency exposition, are illustrated; (iv) The numerical experiments imply that the wavelet transformation exhibits only the fundamental frequencies, which are associated with the fundamental part of the vibration energy transmission; (v) The wavelet analysis allowed to detect the modified Ruelle– Takens–Newhouse scenario for the Bernoulli–Euler, Timoshenko and Sheremetev–Pelekh beams, when three linearly independent frequencies persist in the spectrum, but they exhibit the frequency intermittency phenomena; (vi) Wavelet-based analysis yielded a transition into chaos via intermittency for the Timoshenko beam; comparison of the results reported for different frequencies and amplitudes of the loads allows to conclude that a structure of the observed windows is practically the same, i.e. in the investigated time intervals, the vibration energy is localized on three frequencies.

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Chapter 8

Panels In Section 8.1, the rectangular plate-strip under uniformly distributed harmonic load taking into account physical and elastic-plastic deformations is studied. Various stress–strain relations while cyclic loading are applied. Then a computational algorithm including reduction of PDEs to ODEs via either the Bubnov–Galerkin Method (BGM) or the Finite Difference Method (FDM) is described. Different numerical methods for computation of ODEs are reviewed and discused. Estimation of Lyapunov exponents (LEs) via analytical and numerical approaches is illustrated, and the problem of stability of the studied system is analyzed. Charts of vibrations’ regimes for longitudinal and transversal vibrations are reported. Convergence and reliability of the obtained numerical results have been discussed. Different transition scenarios from regular to chaotic dynamics have been detected and discussed. Then the Sharkovsky’s bifurcation series has been reported. Chaos–hyperchaos and hyper–hyperchaos phase transitions have been studied with the help of LEs and Lyapunov dimension (LD). Reliability of the obtained chaotic zones has been addressed. In Section 8.2, cylindrical panels of infinite length are studied. At first, the governing PDEs are formulated with boundary and initial conditions. Reliability of the solutions obtained via FDM is studied. The method of LEs computations is presented, and the chaos– hyperchaos transition as well as the Sharkovsky’s series are detected and discussed.

340

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Panels

Fig. 8.1

8.1 8.1.1

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Rectangular plate under periodic load.

Infinite Length Panels Mathematical models

We consider nonlinear vibrations of a rectangular plate subjected to the action of uniformly distributed periodic load (Fig. 8.1). We take into account two types of nonlinearities: physical and elastic-plastic deformations (assuming that loading and relief processes lie on the same curve, then we deal only with the physical nonlinearity). We consider a long plate (a
b), and we study the plate parts in the neighborhood of its short sides, whereas on the remaining lengths the plate bending is located on a cylindrical surface. We consider a beam-strip of a unit width and having length a. This approach allows to reduce a PDE to that of only one spatial coordinate x. The bended plate is supported by a set of elastic ribs with the same stiffness which are located in parallel to the plate short side a, and the distance between ribs is c. We additionally assume that the ribs may be only compressed.

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Surface of the transversal section of each rib is F , i.e. the separated beam-strip has the width Fc = F/c. The deformations in the middle beam-strip surface are Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

exx = ε11 + zχ11 , ∂2w , ∂x2
 ∂u 1 ∂w 2 + = + kx w. ∂x 2 ∂x χ11 = −

ε11

Hooke’s law has the following form E [exx − εTxx ], σxx = 1 − ν2

(8.1)

(8.2)

where εTxx = αT T + ε1p xx . The following notation is further used: αT is the coefficient of linear material heat extension; T is the temperature; w is the deflection of the middle beam surface; u is the displacement in the direction of the axis Ox; kz = 1/Rx is the curvature regarding x; h is the beam thickness; ν is the Poisson’s coefficient; g is the Earth acceleration; γ is the specific beam material weight; ε1p xx is the plastic component per loading moment. Owing to the Birger method of variable elasticity parameters in the form (8.2), we get a relation between stresses and deformations. We assume that the Young’s modulus and Poisson’s coefficient are not fixed but they depend on the deformed state, i.e. E = E(x, z, e0 , eis , T ), ν = ν(x, z, e0 , eis , T ), where e0 is volume deformation, eis is plastic deformation. Substitution of (8.1) into (8.2), yields E [ε11 + zχ11 − εTxz ]. (8.3) σxz = 1 − ν2 Integration of (8.3) regarding z allows to get forces in the middle plate-strip surface:  h/2  h/2 E σxx dz = ε11 dz T11 = 1 − ν2 −h/2 −h/2 (8.4)  h/2  h/2 Ez E dz − εT dz. + χ11 2 2 xx −h/2 1 − ν −h/2 1 − ν

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Introducing notation  h/2  h/2 Ez i Ez i T dz, b = ε dz, ai = i 2 2 xx −h/2 1 − ν −h/2 1 − ν

i = 0, 1, 2,

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(8.5) we get T11 = a0 ε11 + a1 χ11 − b0 .

(8.6)

Note that the coefficient b0 contains components of the temperature and the remaining plastic deformations. Solving (8.6) with respect to tangential deformations we get ε11 =

1 a1 b0 T11 − χ11 + . a0 a0 a0

(8.7)

Further, we multiply both sides of Eq. (8.3) by z and carry out the integration of the plate-strip along its thickness. We get the bending moment  h/2 σxx zdz = a1 ε11 + a2 χ11 − b1 . (8.8) M11 = −h/2

We substitute deformation of the middle surface (8.7) into (8.8) to get
 a1 b0 1 T11 − χ11 + + a2 χ11 − b1 Mxx = a1 a0 a0 b1 (8.9)

  a1 a1 a1 b0 = T11 + a2 − − b1 . χ11 + a0 a0 a0 Taking into account the following notation A1 =

a1 , a0

A2 = a2 −

a21 , a0

0 Mxz =

a1 b0 − b1 , a0

(8.10)

Eq. (8.9) can be cast to the following form 0 . Mxx = A1 T11 + A2 χ11 + Mxx

The equilibrium equation is as follows
 hγ ∂ 2 w ∂w ∂w ∂ 2 Mxx ∂ T +kx T11 +q− = 0. + −εh 11 2 ∂x ∂x ∂x g ∂t2 ∂t

(8.11)

(8.12)

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Equation of compatibility of deformations is not applied here, since the beam deflections depend only on x. The forces T11 occurred in the beam middle surface are generated by reactions coming from reinforced ribs, and they will be further denoted by T . They are constant along the whole length of the beam-strip, and Eq. (8.12) can be rewritten in the following form hγ ∂ 2 w ∂w ∂ 2 Mxx ∂ 2 w = 0. + + T k + q − − εh x 2 2 2 ∂x ∂x g ∂t ∂t

(8.13)

We derive the equation of compatibility of deformation of the beam-strip and the stiff rib. We define the joint support displacement generated by the beam-strip deformation. For this purpose we write equation of coupling between deformations of the middle surface with displacements
 ∂u 1 ∂w 2 = ε11 + kx w − . (8.14) ∂x 2 ∂x In the latter relation we substitute ε11 from the Hook law for the middle surface (8.7), and we obtain
 1 a1 b0 1 ∂w 2 + kx − . (8.15) ε11 = T11 − χ11 + a0 a0 a0 2 ∂x Therefore, we may compute the full displacement of support ∆ (we call it positive, when supports approach each other):   
 a
 a 1 a ∂w 2 a1 b0 ∂u 1 dx = dx − T11 − χ11 + dx ∆=− 2 0 ∂x a0 a0 b1 0 ∂x 0  
 a  a 1 a ∂w 2 wdx = dx − T ∗ − kx wdx. (8.16) −kx 2 0 ∂x 0 0 On the other hand, the quantity ∆ is described as shortening of the reinforced beam   σp 1 − νp2 a. (8.17) ∆= Ep The following notation has been applied: σp is the compressing stressin a rib,  Ep is the constant elasticity material modulus. Multi2 plier 1 − νp is introduced because the separated stiff rib plays the

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role of a thin beam of a rib. Comparison of expressions (8.16) and (8.17) yields    
 a σp 1 − νp2 1 a ∂w 2 ∗ a+T = dx − kx wdx. (8.18) Ep 2 0 ∂x 0 We have introduced a series of assumptions so far. In particular, we have assumed that the deflections of the beam-strip are small in comparison to dimension a, and forces T11 acting on the ends of the elementary element dx should be equal, i.e. we have applied here the theory of average bending. Equilibrium condition of a movable beam-strip end yields σp =

T∗ , Fe

and substituting it to (8.18) yields      
 a 2 1 − ν σ 1 a ∂w 2 p p ∗ a+1 = dx − kx wdx, T Ep Fe 2 0 ∂x 0

(8.19)

(8.20)

and hence ∗

T =

1 2

 a  ∂w 2 a 0
∂x dx − kx  0 wdx . 2 σp (1−νp ) Ep F e a + 1

(8.21)

In what follows, while applying the iteration procedure, the value T ∗ will be given from a previous iterational step, i.e.   a
a1 b0 1 ∗ T11 − χ11 + dx T = a0 a0 b1 0  (8.22)  a
 a a1 1 b0 dx − − χ11 dx = T11 β1 + β0 . = T11 a0 a0 0 a0 0 Therefore, using (8.21) we obtain

  2 a 1 1 a ∂w dx − k wdx x 0 β1 2 0 ∂x β0 
− . T11 = 2 β1 σp (1−νp ) Ep F e a + 1

(8.23)

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In the case when both ends of the beam-strip are clamped we can take Fe = ∞ and the formula is essentially simplified. Let us come back to Eq. (8.12). The values of T11 are found through the iteration process via formula (8.23), whereas the value of Mxx is yielded by 2 formula (8.11), assuming that χ = − ∂∂xw2 . Therefore, we have got the algorithm of computation of infinite panels taking into account geometric nonlinearities and elasto-plastic deformation as well as the external load action. 8.1.1.1

Relief process and secondary plastic deformations

We consider the dependence between strains and stresses for an isotropic material in the following way 1 (8.24) exx = σxx + ε1p xx . E The Mises flow material criterion is applied [Hill (1956)], since it is validated experimentally by soft materials like aluminum, copper, iron and steel. In the theory of small elastic-plastic deformations, the properties of the dependence between stresses and deformations are defined by the function σi = f (ei ),

(8.25)

where σi is the stress intensity, ei is the strain intensity, G0 is the characteristic value of the shear modulus in the non-deformed state, σs is the plastic flow threshold, ep1 is the residual deformation in a studied element in the loading instants, ep2 is the begining of a secondary loading (Fig. 8.2). It is assumed that formula (8.25) does not depend on the stress state, and it can be found experimentally while stretching– compressing a cylindrical sample. Below, we give a few analytical forms of dependence σi = f (ei ): 1. Ideally elastic-plastic body:  3G0 ei , for ei < es , σi = for ≥ es . σs ,

(8.26)

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Dependence of the intensity of the deformation of the stress intensity.

2. Elastic-plastic body with a linear strain hardening:  3G0 ei , for ei < es , σi = 3G0 ei + 3G1 (ei − es ) , for ei ≥ es .

(8.27)

3. Pure aluminum model: σi = σs [1 − exp (−ei /es )].

(8.28)

4. Fractional exponent dependence: σi = Aem i ,

0 ≤ m ≤ 1.

(8.29)

where A and m are defined experimentally. 5. Cubic dependence: σi = Eei − me3i .

(8.30)

where E and m are material constants. 6. Fifth degree polynomial: σi = Eei − me3i − m2 e5i . where E, m1 and m2 are material constants. 7. Square formula: Aei σi =  ei 2 . 1+ m

(8.31)

(8.32)

8. Ramberg–Osgood formula [Ramberg and Osgood (1943)]: σi = Eei − Aem i .

(8.33)

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Besides the mentioned formulas σi (ei ) one may find more examples of the stress–strain relations in the literature. However, for our algorithm the dependence σi (ei ) can be defined arbitrarily, even in the form of a table obtained from standard experimental results. In what follows, we consider σi (ei ) presented in Fig. 8.2. In the elastic part, we have σi = 3G0 ei ,

(8.34)

where G0 is the characteristic value of the shear modulus in the nondeformed state. After achieving an elasticity threshold, the loading process (dei > 0) σi (ei ) is described by (8.25). For lightening (dei ≤ 0), owing to the reference [Schevchenko (1970)], σi (ei ) is defined by relations: σi = 3G0 e i ,

e i = ei − ep1 ,

(8.35)

where ep1 is the remaining deformation in an element in the lightening time instant. Formula (8.35) defines σi (ei ) in the case, when in the element (while lightening) the secondary plastic flow deformation does not appear. In a domain of the secondary plastic deformation (ei > es ), their intensity is defined by the following formula σi = f1 (ei ).

(8.36)

Functions f1 (ei ) and f (ei ) do not depend on the stress material state and they are defined experimentally by a stress–strain investigation of cylindrical samples. Owing to the geometrical interpretation of the deformation process in the plane σi − σs , the quantity e i can be presented in the following form e i = ei − ep1 ,

(8.37)

i.e. in the way as in (8.35), but in this case e i and σi are negative quantities. The so far given representation of e i corresponds to system coordinates (e i ,σi ), and the directions of axes coincide with the directions of axes (ei ,σi ). If after the lightening process a body is loaded by the forces with opposite signs with respect to the firstly initiated loading, then it is more convenient to apply the system coordinates (e1i ,σi1 ) with the axes “e” having directions opposite to the axes of

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ei and σi . In the latter case e i = ep1 − ei for ei ≥ ep1 , and for ei < ep1 the following formula holds: e i = ei − ep1 . The corresponding changes should be carried out in relations (8.35), (8.36) and all successive relations. In both cases, the origin of estimation of stress–strain intensity is related to the body element state, which has only plastic components of deformation ε1p ij , occurring in this element up to the beginning of the lightening. Further, we apply the coordinates (e i ,σi ). For material exhibiting the ideal Bauschinger effect [Kadashevich and Novozhilov (1958)], the equation of curve (8.36) in the space of secondary plastic deformations can be written applying the transformed function σi = f1 (ei ) in the following form [Schevchenko (1970)]:
 σ1 − σs p − ei , (8.38) σi = σ1 − σs − f e1 + 3G0 where σ1 is a stress corresponding to the beginning of lightening. Function (8.38) in plane (ei ,σi ) represents a curve which can be obtained via a parallel displacement of the curve σi = f1 (ei ) on the 1 −σs , and then the rotation by value of σi = σ1 − σs , ei = ep1 + σ3G 0 an amount of radians around the point with the given coordinates. Therefore, for ei = e0s we have σi = σ1 − 2σs . Relation (8.38) defines the dependence between intensities of σi and ei in a domain of elastic lightening, if the function f in (8.38) for ei > e0s is written in the following form 

σ1 − σs σ1 − σs p p − ei = 3G0 e1 + − ei . (8.39) f e1 + 3G0 3G0 If in the separated body elements, after the lightening with secondary plastic deformations, the process of the next loading dei > 0, the dependence σi = f1 (ei ) will be described by the formula (8.35), where e i = ei − ep2 , and ep2 corresponds to the beginning of the secondary loading. If the secondary loading process is associated with the change in plastic deformations, then the intensity of stresses is not governed by (8.35), but rather by the following formula σi = f2 (e i ).

(8.40)

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The formula (8.40), similar to functions (8.25) and (8.36), does not depend on the stress state character, and it is defined experimentally. Functions (8.40) define in plane (ei ,σi ) a curve, a straight line (8.35) (elastic deformation), under the condition that ei is defined by the following equation e i = ei − ep2 .

(8.41)

For a material with the ideal Bauschinger effect [Kadashevich and Novozhilov (1958)], the function (8.40) is defined in the following way [analogously to (8.38) and (8.39)]
 σ2 − σs p , (8.42) σi = σ2 + σs + f ei − e2 + 3G0 and the elastic part of the deformation curve is approximated by the following formula 

σ2 − σs σ2 − σs p p = 3G0 ei − e2 + , (8.43) f ei − e2 + 3G0 3G0 where σ2 is the stress intensity defined by the formula (8.38) in time instant associated with the beginning of the secondary loading. It should be emphasized that the given process takes place in all sample points and in all time instants. 8.1.1.2

Mathematical model and computational algorithm

We consider nonlinear vibrations of a rectangular plate subjected to the longitudinal harmonic load under the condition that one of the plate dimensions is larger than the second one, i.e. a
b (Fig. 8.3). We are interested in beam-strip parts adjacent to short edges, and we assume that on the remaining parts the plate is bent along a cylindrical surface. Then, it is sufficient to investigate vibrations of a beam-strip of length a and width equal to 1. Therefore, it is necessary to solve Eq. (8.13) satisfying the following assumptions: lack of curE vature kx = 0; intensity of the stress–state state σi = 1−ν 2 ei , where E (Young’s modulus) is constant; temperature influence is neglected αT = 0.

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h

a

0

x

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P1 = P0 sin(ωpt)

z

q = q2 sin(ω4t) Fig. 8.3

Computational scheme.

The so far given assumptions essentially simplify the computational algorithm given in the previous subsection. Namely, we get Eh Eh3 , b1 = 0, , a = 0, a = 1 2 1 − ν2 12(1 − ν 2 )    α
 α
Eh ∂w 2 ∂w 2 1 dx = dx, T = 2β1 0 ∂x 2α(1 − ν 2 ) 0 ∂x a0 =

Mxx = a2 χ11 = −

(8.44)

∂2w Eh3 . 12(1 − ν 2 ) ∂x2

In this way, we have omitted the iteration procedure, and we have got the following 1D variant of the von K´ arm´an equations:


  a ∂4w Eh ∂2w ∂w 2 hγ w ¨ + hεw˙ = −D 4 + dx g ∂x 2a(1 − ν 2 ) ∂x ∂x2 0 − Px (t)

∂2w + q (t, x) , ∂x2

(8.45) where w(x, t) is the deflection function, x is the spatial coordinate, t is time, a is the dimension of beam-strip, h is the thickness, Px (t), q(x, t) are longitudinal/parametric and transversal loads, respecEh3 tively, E is Young’s modulus, D = 12(1−ν 2 ) is the cylindrical stiffness, ν is Poisson’s coefficient, γ is volume specific material weight, g is the Earth acceleration, ε is the damping coefficient. We recast the governing equations to the counterpart non-dimensional form by

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introducing the following non-dimensional parameters (with bars):

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x = a¯ x,

Eh3 Eh4 q = 4 q¯, w = hw, ¯ Px (t) = 2 Px (t), a a  1 γ a2 ε¯, λ = . t= 2 h Eh2 g 12 (1 − ν 2 )

Finally, Eq. (8.45) takes the following non-dimensional form (bars are omitted)


  1 ∂4w ∂2w ∂2w ∂w 2 w ¨ + εw˙ = −λ 4 + 6λ dx − P (t) + q(t, x). x ∂x ∂x ∂x2 ∂x2 0 (8.46) In Eq. (8.46), dissipative properties being proportional to the motion velocity are taken into account. It should be emphasized that different models of dissipative properties are used (see Chapter 5 and the reference [Bolotin (1961)] for more details). As it is known from a linear theory of vibrations, there exists linear and nonlinear frictions (see [Golovin (2012)]). The latter one can be approximated by various analytical formulas. When the bodies are cyclically loaded and deformed, the Hooke’s law violation is observed (even for loading) exhibited by the occurrence of hysteretic loops. The surface of a hysteresis loop defines energy dissipated per one cycle of vibrations and per unit of material volume. It is already known that hysteresis loop surface, for majority of the used materials in industrial applications, does not depend on the deformation frequency process, but it rather depends on the deformation amplitude. Equation (8.46) should be supplemented with one of the following boundary conditions: (i) Simple–simple support: w = w x = 0,

for x = 0, 1;

(8.47)

for x = 0, 1;

(8.48)

w (1) = w x (1) = 0,

(8.49)

(ii) Clamping–clamping support: w = w x = 0, (iii) Simple–clamping support: w (0) = w x (0) = 0,

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and one of the following initial conditions associated with the mentioned boundary conditions: (i) w (x) |t=0 = w0 sin (πx) ,

w˙ (x) |t=0 = 0;

(8.50)

w (x) |t=0 = w0 (1 − cos (2πx)) ,

w˙ (x) |t=0 = 0;

(8.51)

(ii)

(iii) w (x) |t=0 = f (x) ,

w˙ (x) |t=0 = 0.

(8.52)

Function f (x) describes a deflection defined by the set-up method with respect to a small value of the transversal external load action [Feodosev (1963)]. In this chapter, we consider nonlinear vibrations with linear friction. It should be emphasized that when a constant load is applied, then the problem is reduced to the solution of a static problem. Namely, a so-called set-up problem can be applied, when ε = εcr (critical damping). Then the vibrations of the investigated plate rapidly fast decrease, and the solution tends to its stationary counterpart for {qi } → {wi }. The set-up method was firstly applied by Feodosev, who considered a stability problem of a spherical dome subjected to the transversal load. It should be noted that dissipative terms are introduced artificially even to solve large sets of algebraic equations, and then the iterational process is obtained having a lot of advantages in comparison to the widely used Newton’s method. For time dependent loads Px (t), q(x, t), the parameter ε essentially influences the character of vibrations of a mechanical system as well as the location of chaotic zones. 8.1.2 8.1.2.1

Reduction of PDEs to ODEs The BGM

The BGM [Volmir (1972)] allows to reduce the initial-boundary problem governed by a PDE (problem of infinite dimension) to a set of truncated ODEs (problem of finite dimension). Although this method has a series of advantages, it also possesses drawbacks. From one side

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it allows to get reliable qualitative results even if rough approximation is used, but from the other side the method strongly depends on the introduced boundary conditions. A numerical experiment shows that the mentioned method allows to achieve convergence of the iteractions already using 2–3 series terms, which means that the system of ODEs with low dimension can be applied. Let us introduce a solution in the following form

w(x, t) =

N 

Ai (t)wi (x),

(8.53)

i=0

where w(x, t) should satisfy the introduced boundary conditions. After application of the BGM to Eq. (8.46), the following system of ODEs is obtained for Ai (t): N 

N N

  N ˙ ¨ Ai bir + 6λL({Ai }1 ) Ai cik Ai + εAi aik = −λ

i=0

i=0

i=0

− Px (t)

N 

(8.54)

Ai cik + Qk (t),

i=0

where k = 0, 1, . . .. In the above, the following notation for the coefficients is applied  aik =

0

 cik =

1 0



1

wi (x)wk (x) dx, wi (x)wk (x)dx,

 N L {Ai }1 =

1 0

bik =

1

0

wiIV (x)wk (x) dx,



Qk (t) =

N  i=0

0

1

q(x, t)wk (x) dx,

Ai (t)w i (x)

2 dx.

(8.55)

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The system of second-order ODEs can be reduced to that of firstorder ODEs, which can be directly solved with one of the widely applied Runge–Kutta methods. We analyze the BGM applied vibrations of an infinite panel with a geometric nonlinearity. We consider a solution to Eq. (8.46) with simple support (8.47). We take {sin π(2i + 1)x}i=1...N as basic functions of BGM. Since they satisfy the boundary conditions, we take the solution in the following form N  Ai (t) sin(π(2i + 1)x). (8.56) w(x, t) = i=1

Substituting (8.56) into (8.46), we get N

 A¨i + εA˙ i sin(π(2i + 1)x) i=1

= −λ

N 



Ai (π(2i + 1))4 sin(π(2i + 1)x) − (6λL {Ai }N 1

i=1

−Px (t))

N 

Ai (π(2i + 1))2 sin(π(2i + 1)x) + q(x, t),

i=1

(8.57) where

L {Ai }N 1





1

= 0

=

N 

2 Ai (π(2i + 1)) cos(π(2i + 1)x)

N  N   i=1 j=1

0

1

{Ai Aj π 2 (2i + 1)(2j + 1)

× cos(π(2i + 1)x) cos(π(2j + 1)x)dx =

N   i=1

=

dx

i=1

1 0

{Ai π(2i + 1) cos(π(2i + 1)x)}2 dx

N π2  (Ai (2i + 1))2 . 2 i=1

(8.58)

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We multiply both sides of (8.57) by sin(π(2j + 1)x), and then we integrate both sides of the new equations for x from 0 to 1 to get N

 1  ˙ ¨ sin(π(2i + 1)x) sin(π(2j + 1)x)dx Ai + εAi 0

i=1

= −λ

N 

Ai (πi)

i=1

4



1

sin(π(2i + 1)x) sin(π(2j + 1)x)dx 0

+ (6λL {Ai }N 1 − Px (t))

N  i=1  1

Ai (πi)

(8.59) 2



1

sin(πix) sin(πjx)dx 0

sin(πjx)dx.

+ q(x, t) 0

Since functions {sin(πix)}N 1 are mutually orthogonal in the interval [0; 1], the following equations are finally obtained   N  4 2 2 (Ai i) − Px (t) Aj (πj)2 A¨j + εA˙ j = −λAj (πj) − 3λπ i=1

4 q(x, t). (8.60) πj The N second-order ODEs are transformed to a system of 2N first-order differential equations of the following form A˙ j (t) = A j (t),   N  4   2 2 (Ai i) − Px (t) Aj (πj)2 A j = −εA j − λAj (πj) − 3λπ +

i=1

4 q(x, t), (8.61) πj which are solved numerically using one of the Runge–Kutta methods. We consider the problem separately for lack of transversal excitation (q(x, t) = 0). It is obvious that in the case when the initial system deflection does not exist, the solution to the problem is trivial (zero). This is why we introduce the initial conditions in the form of (8.50), +

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which corresponds to the introduction of an artificial small deflection of the plate. Owing to the structure of Eq. (8.61) one may conclude that the initial perturbation in the form of a first harmonic does not interact with higher order harmonics (the coefficients standing by higher harmonics are equal to zero). In the latter case, the system of Eq. (8.60) is reduced to the well-known Duffing equation [Nayfeh and Mook (1995)] of the form A¨1 + εA˙ 1 = −(λπ 2 (1 + 3A21 ) − Px (t))A1 π 2 .

(8.62)

We consider a solution of Eq. (8.46) with boundary conditions (8.48). As basic functions of the Bubnov–Galerkin approach, we take: {cos π2ix − 1}i=0...N , or equivalently, we assume the following solution form N 

w(x, t) =

Ai (t)(cos(2πix) − 1).

(8.63)

i=1

Substituting (8.63) into Eq. (8.46), we get N

 A¨i + εA˙ i (cos(2πix) − 1) i=1

= −λ

N 



Ai (2πi)4 cos(2πix) − (6λL {Ai }N 1

(8.64)

i=1

− Px (t))

N 

Ai (2πi)2 cos(2πix) + q(x, t),

i=1

where

 N L {Ai }1 =

0

=

1

N 

2 Ai 2πi sin(2πix)

dx

i=1

N  N   i=1 j=1

0

1

 Ai Aj 4π 2 sin(2πix) sin(2πjx) dx

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=

1 0

i=1

{Ai 2πi sin(2πix)}2 dx = 2π 2

N 

(Ai i)2 .

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(8.65) Multiplying both sides of Eq. (8.64) by cos(2πjx), and carrying out the integration procedure from 0 to 1, we obtain N

  A¨i + εA˙ i i=1 N 

= −λ

1 0

(cos(2πx) − 1) cos(2πjx)dx

Ai (2πi)

i=1



4



1

cos(2πix) cos(2πjx)dx 0

1

(8.66)

cos(2πjx)dx

+ q(x, t) 0

N

 (t)) Ai (2πi)2 − P − (6λL {Ai }N x 1

 ×

i=1

1

cos(2πix) cos(2πjx)dx. 0

Since the functions {cos(2πix)}N 1 are mutually orthogonal on interval [0 : 1], the following system of equations is obtained   N  4 2 2 (Ai i) − Px (t) Aj (2πj)2 A¨j + εA˙ j = −λAj (2πj) − 12λπ i=1

−

N 

(A¨i + εA˙ i ) = q(x, t),

(8.67)

i=1

which is reduced to the following first-order differential equations A˙ j (t) = A j (t),   N  (Ai i)2 − Px (t) Aj (2πj)2 . A˙  j = −εA j − λAj (2πj)4 − 12λπ 2 i=1

(8.68)

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FDM

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Explicit and implicit schemes In order to reduce PDEs (8.46) to ODEs, we apply the FDM to the spatial coordinate x. In the mesh GN = {0 ≤ xi ≤ 1, xi = i/N , i = 0, . . . , N },

(8.69)

we substitute the partial derivatives by their difference counterparts, and we get  1  2 Λx (wi ) dx Λx2 (wi ) w ¨i + εw˙ i = −λΛx4 (wi ) + 6λ 0 (8.70) − Px (t)Λx2 (wi ) + q (ih, t) . Integral occurring in (8.70) can be found numerically, for example, via the Simpson formula. The following formulas hold for the boundary conditions (8.48),(8.47): 1. Simply supported edge (·)−i = −(·)i . 2. Clamped edge (·)−i = (·)i . Introduction of the change in variables w˙ i = w i

(8.71)

reduces second-order ODEs (8.70) to the first-order ODEs regarding deflections wi and velocities wi of the following form  1    2 Λx (wi ) dx Λx2 (wi ) w˙ i + εw i = −λΛx4 (wi ) + 6λ 0 (8.72) − Px (t)Λx2 (wi ) + q (ih, t) . In order to construct an implicit scheme, we take a regular mesh GN . On this mesh, we define Eq. (8.46) in the form of three-layer difference scheme with weights (the Krenck–Nicholson method). We have  1 2  j Λx (wi ) dx, αj = 0

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Zik = −λΛkx4 (wi ) + {6λα − Px (t)} Λkx2 (wi ),

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wij+1 − wij−1 wij+1 − 2wij + wij−1 + ε τ2 2τ = σZij + (1 − σ) Zij+1 + q(ih, t).

(8.73)

Superscripts (subscripts) correspond to layers in time  (space). Time derivatives are approximated with accuracy of O τ 2 , whereas  4 2 space  6 derivatives are estimated with accuracy of O h , O h , O h . Computation of the coefficient α as well as boundary conditions is similar to the case of the explicit method. It should be noted that contrary to the explicit method, the integration in time is carried out not by the Runge–Kutta method, but rather via a solution to the linear equations. The system matrix is of a band type and   its 4 width depends on the approximation order of the derivatives O h     and O h6 . For approximation O h2 , the associated matrix has 5, 7 and 9 diagonals, respectively. Computation of difference derivatives Operators Λx , Λx2 , Λx4 can be estimated with various steps of applied approximation. In order to get formulas for difference derivatives, we apply the following program for symbolic computation in Maple environment. Here we use the classical approach to find the difference derivatives. We define the dimension of a pattern required for interpolation of a given derivative with required accuracy. Then we construct the interpolating polynomial, and in the next step the obtained polynom is differentiated by the required number of times. Finally, the polynomial value in the center of the pattern is computed. The following procedure is applied f d := proc(n, m) locall, df ; l := f loor(m/2) + f loor((n − 1)/2); interp([seq(k ∗ h, k = −l..l)], [seq(y(k ∗ h), k = −l..l)], x);

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df := simplify(subs(x = 0, diff (%, x$n))); print(df ); print(series(series(df, h, m + n), h, m)) Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

endproc Here n denotes the order of a derivative, whereas m stands for the approximation order. It is clear that the dimension of the difference derivative is coupled with m and n in the following way l = [m/2] + [(n − 1)/2].

(8.74)

A given maple-function not only yields a formula for the difference derivative, but also shows that approximation of the difference derivative corresponds to the given parameters. Application of symbolic computations essentially simplifies the process of routine type computations, and also allows to omit possible errors introduced by hand-made derivations. We demonstrate this approach  6  regarding the difference derivative computation of order O h . In what follows, the formulas for different orders  2  of approximations are given. For O h , we have 
 1  ∂ (·) ≈ (·)i+1 − (·)i−1 = Λx (·) + O(h2 ), (8.75) ∂x 2h
2   1  ∂ (·) ≈ 2 (·)i+1 − 2(·)i + (·)i−1 = Λx2 (·) + O(h2 ), (8.76) 2 ∂x i h
4   1  ∂ (·) ≈ 4 (·)i+2 − 4(·)i+1 + 6(·)i − 4(·)i−1 + (·)i−2 , 4 ∂x i h = Λx4 (·) + O(h2 ).

(8.77)

For O(h4 ), we have 
 1  ∂ (·) (·)i−2 − 8(·)i−1 + 8(·)i+1 − (·)i+2 ≈ ∂x i 12h = Λx (·) + O(h4 ),

(8.78)

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∂ 2 (·) ∂x2

∂ 4 (·) ∂x4

 ≈ i

 1  (·)i+2 + 16(·)i+1 − 30(·)i + 16(·)i−1 + (·)i−2 2 12h

= Λx2 (·) + O(h4 ),



≈ i

(8.79)

1  −(·)i+3 + 12(·)i+2 − 39(·)i+1 + 56(·)i − 39(·)i−1 h4  + 12(·)i−2 − (·)i−3 = Λx4 (·) + O(h4 ). (8.80)

  For O h6 we have








∂ (·) ∂x

∂ 2 (·) ∂x2

∂ 4 (·) ∂x4

 ≈ i

1 [(·) − 9(·)i+2 + 45(·)i+1 − 45(·)i−1 60h i+3 + 9(·)i−2 − (·)i−3 ] = Λx (·) + O(h6 ),

 ≈ i

(8.81)

1  2(·)i+3 − 27(·)i+2 + 270(·)i+1 − 490(·)i 180h2  + 270(·)i−1 + 27(·)i−2 + 2(·)i−3 = Λx2 (·) + O(h6 ), (8.82)

 ≈ i

1  7(·)i+4 − 96(·)i+3 + 676(·)i+2 − 1952(·)i+1 240h4 +2730(·)i − 1952(·)i−1 + 676(·)i−2 − 96(·)i−3 + 7(·)i−4

= Λx4 (·) + O(h6 ).



(8.83)

As it can be seen from the reported formulas, the number of outcontour increases order increase: 1 —   4with  the approximation  6  2points for O h , 2 — for O h , 3 — for O h . In order to compute the partial derivatives, we may also apply a more general approach. Consider the following relation b0 f  (x − h) + b1 f  (x) + b2 f  (x + h) ≈ c0 f (x − h) + c1 f (x) + c2 f (x + h).

(8.84)

We aim at finding coefficients b0 , b1 , b2 , c0 , c1 , c2 such that the formula (8.84) is satisfied with the highest accuracy order.

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d

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s n Fig. 8.4 n = 7).

Illustration of a Pad´e approximation parameters (s = 3/2, d = 3,

Let f (x) = eiωh , then we get −ω 2 (b0 e−iωh + b1 + b2 eiωh )eiωh ≈ (c0 e−iωh + c1 + c2 eiωh )eiωh . (8.85) We are going to find the mentioned coefficients in the neighborhood of h = 0. Taking into account ξ = eiωh (ln ξ = iωh), we get   ln ξ 2 c0 + c1 ξ + c2 ξ 2 ≈ . (8.86) h b0 + b1 ξ + b2 ξ 2 Formula (8.86) implies that the best approximation in the neighborhood of ξ = 1 is yielded by the Pad´e approximation of the function  2 ln ξ (see Fig. 8.4) h   12 − 24ξ + 12ξ 2 (ξ − 1)2 ln ξ 2 = . ≈ 1 h h2 (1 + 10ξ + ξ 2 ) h2 (1 + (ξ − 1) + 12 (ξ − 1)2 ) (8.87) Parameters of Pad´ e approximations (s = 3/2, d = 3 and n = 7). The coefficients of implicit difference derivatives are defined by the following Maple-function: with(numapprox); implicid f d := proc(n, m, s, d) localt; t := pade(xs ∗ ln(x)m , x = 1, [n, d]) : print(coef f (expand(denom(t)), x, i)$i = 0..d); print(coef f (expand(numer(t)), x, i)/hm $i = 0..n); end proc

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Here, m is the order of the derivative approximation; s is the pattern shift of derivatives; d is the dimension of the pattern derivatives (without 1); n is the dimension of the pattern functions (without 1). For the explicit difference derivatives, the nominator of the Pad´e approximation has zero order (d = 0) and we may apply a simple Taylor series development. In other words, we have got one more way to estimate the coefficients of the difference derivatives. This is realized by the following Maple-function: f d := proc(n, m) locall, df ; l := f loor(m/2) + f loor((n − 1)/2); expand(convert(taylor(xl ∗ ln(x)n , x = 1, 2 ∗ l + 1), polynom)); df := simplif y(sum(coef f (%, x, i) ∗ y((i − l) ∗ h)/hn , i = 0..2 ∗ l)); print(df ); print(series(series(df, h, m + n), h, m)); end proc The latter function coincides in full with the classical approach function, and it may also serve for checking of the obtained difference derivative of given parameters. The problem of computations of partial derivatives can be studied from five different points of view: (i) Interpolating approach. Operator of a partial derivative is the exact derivative of an interpolating polynomial constructed on a certain pattern. This approach is widely used by spectral methods devoted to computations of partial derivatives. (ii) Approximation of differential operators. The operator of a partial derivative is defined by the finite difference approximation of a differential operator. This approach has been used for checking the results presented in earlier examples. (iii) Correlation approach. The differential operator can be considered as a correlating filter from the position of frequency filters with the appropriately chosen coefficients to approximate a partial derivative.

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(iv) Multiplication by a Toeplitz matrix. Computation of a partial derivative can be viewed algebraically, as the multiplication by a band matrix (Toeplitz matrix). This approach plays the fundamental role while constructing implicit schemes. (v) Spectral methods. This method is based on the Fourier series. Operator for a partial derivative is presented as an inverse Fourier transformation regarding a direct discrete Fourier transformation and the properly defined coefficients. This approach can be generalized into an arbitrary basis of a space of continuous functions. However, introduced approximations to continuous differential equations by the difference equations imply a modification of the physical properties of an investigated system. From the point of view of dispersive and diffusion properties, the continuous and discrete solutions are not fully equivalent. This is why we require adequate information on the modification of solutions introduced by discretization. This problem can be solved with the theory of digit filters. Let us consider the difference approximations of space derivatives from the point of view of the digital filters. They can be classified as linear stationary non-recursive filters. One of the important characteristics of a digital filter is the transition function characterizing the ratio of an output complex signal amplitude and an input complex signal amplitude. In our case for differentiation of a spatial derivative, we deal with the relation of discrete and continuous dispersion. The continuous rule of dispersion should be negligibly perturbed in order to keep possibly “flat” transition function. We consider an action of the difference operator regarding the fourth derivative using different approximation orders on a harmonic signal. Substituting the harmonic input w(x) = eiωx into Eqs. (8.77), (8.80), (8.83), after trigonometric transformations, and division by the dispersion term ω 4 eiωx , the following transition functions corresponding to the applied operators of digital filters are obtained η2 = 4

cos2 (πϕ) − 2 cos (πϕ) + 1 , π 4 ϕ4

(8.88)

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η4 = − η6 =

Deterministic Chaos in One-Dimensional Continuous Systems

4 cos3 (πϕ) − 6cos2 (πϕ) + 9 cos (πϕ) − 4 , 3 π 4 ϕ4

(8.89)

1 15 ×

7cos4 (πϕ) − 48cos3 (πϕ) + 1626cos 2 (πϕ) − 208 cos(πϕ) + 87 . π 4 ϕ4 (8.90)

The latter relations have been obtained with the help of the following Maple-function: f d f ilter := proc(n, m) locall, df ; l := f loor(1/2 ∗ m) + f loor(1/2 ∗ n − 1/2); expand(convert(taylor(xl ∗ ln(x)n , x = 1, 2 ∗ l + 1), polynom)); simplify(sum(coeff (%, x, i) ∗ exp(I ∗ (i − l) ∗ h)/hn , i = 0..2 ∗ l))/I n ; subs(h = P i ∗ phi, %) endproc In the above n is the derivative order, whereas m denotes the approximation order. In Fig. 8.5, graphs of transition functions of digital filters corresponding to operators (8.77), (8.80), (8.83) are reported. Drawings are given in interval [0, 1], where 0 corresponds to zero frequency, and 1 is the Nyquist frequency ω = π /h . Figure 8.5 shows that for all approximations the transition function is strongly damped in high frequencies interval. However, the transition function of the operator L4 possesses more “flat” profile than L2 , i.e. it perturbs less the continuous dispersion rule, in particular on low frequencies, where the fundamental energy localization is expected. It is clear that the operator L6 differs from L4 rather marginally, and hence in many applications we may use L4 .

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Fig. 8.5 Transition functions for the fourth difference derivative with different approximations.

Approximation of functions and their derivatives on a mesh From a historical point of view, the fundamental method of investigation of partial differential equations is associated with the use of explicit difference schemes of low orders. These methods are universal and have simple realization. Even today they are used to a general picture of the PDEs behavior, to find the direction of further investigations that emphasize on requirements regarding accuracy, stability and computational speed. Difference schemes are simple in realizations and they can be easily adapted to various forms of the initial and boundary conditions. The latter property is of a particular importance in comparison to the Bubnov–Galerkin approach which depends essentially on the applied boundary conditions. Since the computation of the difference derivatives is carried on a certain narrow mesh interval, the computed values of the derivatives depend only on the values of the neighborhood mesh nodes. It is tempting to expect that the increase in the approximation order and the increase in the difference pattern should solve the problem of localization, but it leads to other problems related to computation requirements. Errors associated with digital accuracy increase, as well as the occurence of computational complexity.

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The validity, reliability and quality estimation of the numerical solution requires a rigorous definition of errors in the computational methods. In the case when we look for a scalar quantity, the estimation of absolute and relative errors allows for adequate monitoring of the evaluation of the quality of the applied algorithm. However, in the case when a function is the solution, there exist a few alternative methods of error computations, based on various norms. If the error on inteval [a, b] is defined in the following way e(x) = w(x) ˜ − w(x),

(8.91)

where w(x) is the exact solution, and w(x) ˜ is the numerical solution, then in practice the following definitions of norms are applied e∞ = max |e(x)| , a≤x≤b



e1 =

b

|e(x)| dx,

(8.92) (8.93)

a





e2 =

b

e(x)2 dx.

(8.94)

a

All those definitions are particular cases of the so-called p-norm defined as follows   b p |e(x)|p dx. (8.95) ep = a

In the case when the numerical solution is in the form of a mesh function, we must compare it with the exact solution. The following vector of errors is introduced ei = wi − w(xi ),

(8.96)

which allows for numerical error estimation with respect to an arbitrarily chosen norm. However, an arbitrary vector norm may increase while increasing the number of mesh points, which can yield erroneous results regarding estimation of the error order.

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In order to remove the mentioned drawback one may discretize one of the norms (8.92)–(8.94), for example, in the following way e1 = h

b 

|e(x)|.

(8.97)

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a

Writing programs realizing one of the numerical methods devoted to finding a solution to PDEs requires verification and validation of the applied algorithm and its realization. There are a few approaches to solve the mentioned problem. In the case when the exact solution of a given problem is known for some initial and boundary conditions, this solution allows not only for validations of the correctness of the written program, but it also allows to follow asymptotic evolution of the error, and even to define a constant of the known asymptotics. Let us assume that we know the exact solution w(x) to a problem. Carrying out the numerical solution on the introduced mesh with step h, we find an approximating solution w ˜h . The following error function is introduced: E(h) = w ˜h − wh .

(8.98)

If the used method has an approximation order p, we may expect that E(h) = Chp + O(hp ).

(8.99)

Then we decrease the step twice to get Chp (8.100) E(h/2 ) = p , 2 and hence E(h) = 2p , (8.101) E(h/2 ) E(h) . (8.102) p = log2 E(h/2 ) Knowing p, one may define the constant of asymptotics of the method: C = E(h)/hp .

(8.103)

Therefore, having in hand the exact solution, two series of computations allow to define both order and constant of the asymptotic

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method. Even if we monitor a “good behavior” of the numerical solution it is worthy to carry out other computations on a more dense mesh, in order to follow a proper asymptotic error evolution. The latter method allows to exhibit certain small errors introduced while writing a program code. In the case when the exact solution is not known, one may try to simplify the problem, and in the beginning to solve the counterpart simplified mathematical model, for which the exact solution is known. The latter approach, though not fully sufficient, allows to discover introduced errors in the initial step of the algorithm. Besides, introduction of more dense mesh yields a good estimation of computational errors. Let us assume that we have carried out two series of computations with steps h and h/2. Then, taking the solution associated with the dense mesh as exactly one, we get           h   ˜ ˜ h  + w ˜ − wh/2  E(h) = wh − wh/2  = wh − w

p  h . (8.104) = E(h) + O 2 In other words, the latter approach yields very good approximation of the errors introduced by the applied method. It should be noted, however, that the obtained numerical solution may converge to a function which, in principle, has no relation to the real searched solution. 8.1.3

Solving ODEs in time

Consider a problem with initial conditions y  (t) = f (t, y(t)),

y(t0 ) = y0 ,

(8.105)

where function f : [t0 , ∞) × d → d , and y0 ∈ d is a given vector. The so far stated problem is called the Cauchy problem and, owing to the theorem on existence and uniqueness of the solution to an ordinary differential equation, the solution does exist and it is unique [Hairer et al. (1993)]. In order to solve numerically the formulated problem, a series of methods have been developed, but the more popular and useful are the Runge–Kutta methods. There exist

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boundary values problems, where the solution values are given in a few points, and they require special methods like shooting, finite differences and collocation. One may say that it is sufficient to solve differential equations of the first-order. Equations of higher order can be reduced to the first-order ODEs by introduction of additional variables. For instance, the second-order equation y  = −y can be presented in the form of two equations of the first-order: y  = z and z  = −y. 8.1.3.1

Euler’s method

In Eq. (8.105), we substitute y  by the following difference approximation y(t + h) − y(t) . (8.106) y  (t) ≈ h Hence, the following formula is obtained y(t + h) ≈ y(t) + hf (t, y(t)).

(8.107)

The given formula is applied in the following way. We take the integration step h and we consider a series of time instants t0 , t1 = t0 + h, t2 = t0 + 2h, . . . . By yn , we denote numerical approximation to the exact solution y(tn ). Owing to formula (8.107), we compute successive approximations to the exact solution via the following recursive scheme yn+1 = yn + hf (tn , yn ).

(8.108)

The obtained equation presents the Euler method, proposed in 1798 by L. Euler [Hairer et al. (1993)]. Observe that Eq. (8.106) can be also presented in the following way y(t) − y(t − h) . (8.109) h In this case, we deal with the so-called implicit Euler method y  (t) ≈

yn+1 = yn + hf (tn+1 , yn+1 ).

(8.110)

In the latter approach, it is necessary to solve the equation of type (8.110) on each computation step, and this method is implicit. It can be solved also by the Newton method [Hairer and Wanner (1996)]. It

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is obvious that in this case each computational step requires essentially more time in comparison to the explicit method. In many cases, the Euler method does not have high accuracy, and mathematicians have developed methods of higher order accuracy. It is clear that in order to compute successive values, one may use not only one value from the previous computational step, but also the whole series of the previously obtained values. In practice, all multistep methods belong to a family of linear multi-step methods, and can be presented in the following form αk yn+k + αk−4 yn+k−4 + · · · + α0 yn = h(βk f (tn+k , yn+k ) + βk f (tn+k−1 , yn+k−1 )

(8.111)

+ · · · + βk f (tn , yn )) where αk , βk are certain constants. The latter approach yields a series of the Runge–Kutta methods in memory of Karl Runge and Martin Kutta [Hairer and Wanner (1996)]. Mostly known and used is obviously the fourth-order Runge–Kutta (RK4) method. 8.1.3.2

Runge–Kutta methods

We are looking for a solution to problem (8.105) in points {ti }N 0 . Then the following formula holds  yn+1 = yi +

ti+1

f (t, y(t))dt.

(8.112)

ti

t We compute the integral using the trapezoid method: tii+1 f (t, y(t))dt = 12 h(f (ti , yi )+ f (ti+1 , yi+1 )). Owing to Euler’s formula, we have: yi+1 = yi + hf (ti , yi ). Therefore, we get 1 yi+1 = yi + h(f (ti , yi ) + f (ti+1 , yi + hf (ti , yi ))). 2

(8.113)

The method based on formula (8.113) is called the modified Euler method. However, in fact this is the second-order Runge–Kutta (RK2) method. It can be verified describing a solution y(t) in the

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vicinity of yi in the form of the Taylor series up to the terms of order O(h3 ): 1 yi+1 = yi + hy  i + h2 y  i + O(h3 ) 2 1 = yi + hf (ti , yi ) + h2 f  (ti , yi ) + O(h3 ) 2 1 = yi + hf (ti , yi ) + h2 (f  x (ti , yi ) + f  y (ti , yi )y  i ) + O(h3 ) 2 = yi + hf (ti , yi ) 1 + h2 (f  x (ti , yi ) + f  y (ti , yi )f (ti , yi )) + O(h3 ). 2

(8.114)

The right-hand side of (8.113) is also developed into Taylor’s series: 1 yi+1 = yi + h [f (ti , yi ) + f (ti+1 , yi + hf (ti , yi ))] 2 1 = yi + h[f (ti , yi ) + f (ti , yi ) + hf  x (ti , yi ) 2 + hf  y (ti , yi )f (ti , yi ) + O(h2 )] = yi + hf (ti , yi ) +

h2  (f x (ti , yi ) 2

+ f  y (ti , yi )f (ti , yi )) + O(h3 ).

(8.115)

Comparison of right-hand side of Eqs. (8.114) and (8.115) proves that the modified Euler’s method is the RK2. In Eq. (8.112), we compute the integral by the method of rectt angulars tii+1 f (t, y(t))dt = 12 h(f (ti + h2 , y(ti + h2 ))). Owing to the Euler formula, we have y(ti + h2 ) = yi + h2 f (ti , yi ), and hence
 1 1 (8.116) yi+1 = yi + hf ti + h, yi + hf (ti , yi ) . 2 2 The method based on formula (8.116) is known as the improved Euler method. We show that the latter method also coincides with the Taylor series (8.114) up to the terms of the second-order. We

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have 1 1 1 yi+1 = yi + hf (ti + h, yi + hf (ti , yi )) 2 2 2  1 = yi + h f (ti , yi ) + hf  x (ti , yi ) 2  1  2 + hf y (ti , yi )f (ti , yi ) + O(h ) 2  1  = yi + hf (ti , yi ) + h2 f  x (ti , yi ) + f  y (ti , yi )f (ti , yi ) + O(h3 ), 2 (8.117) and hence we have proved that in fact this is the RK2 method. In general, the following RK2 method and the following generalized formula are used 1 yi+1 = yi + h(f (ti , yi ) + f (ti+1 , yi + hf (ti , yi ))), 2
 1 1 yi+1 = yi + hf ti + h, yi + hf (ti , yi ) , 2 2

(8.118)

which yields yi+1 = yi + αhf (ti , yi ) + bhf (ti + αh, yi + βhf (ti , yi )),

(8.119)

and it can be reduced either to the modified Euler method (a = 0, b = 1, α = 12 , β = 12 ) or to the improved Euler method (a = 12 , b = 12 , α = 1, β = 1). The latter formula can be generalized for higher order approximations:  k1 (h) = hf (ti , yi ) ,      k2 (h) = hf (ti + α2 h, yi + β21 k1 (h)) ,      ... kq (h) = hf (ti + αq h, yi + βq1 k1 (h) + · · · + βqq−1 kq−1 (h)) ,     q       = y + pi ki (h). y i  i+1 i=1

(8.120)

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All Runge-Kutta methods are described by formulas (8.120). However, not all formulas yielded by (8.120) define the Runge-Kutta methods. Important question arises: How to choose parameters α, β and p to get a Runge-Kutta method of the given order s? The stated question will be solved using an example of the third Runge-Kutta method. Taking q = s = 3, the following formula is obtained   k (h) = hf (ti , yi ) ,   1   k (h) = hf (t + α h, y + β k (h)) , 2 i 2 i 21 1 (8.121)  (h) = hf (t + α h, y + β k (h)+ k (h)) , k 3 i 3 i 31 1 32 2    y = y + p k (h) + p k (h) + p k (h). i+1

i

1 1

2 2

3 3

We develop all terms of the right hand part of (8.121) into the Taylor series up to h3 order, and we get  k1 (h) = hf (ti , yi )         k2 (h) = hf (ti + α2 h, yi + β21 k1 (h)) = h[f + h f  x α2 + f  y β21 f        1   2 2  f + O(h3 ) + h2 f  xx α22 + f  xy α2 β21 f + f  yy β21    2!        k3 (h) = hf (ti + α3 h, yi + β31 k1 (h)+32 k2 (h)) + h[f + f x α3 h                       

+ f  y {β31 hf + β32 h(f + f  x α2 h + f  y β21 hf )} 1 + h2 f  xx α23 + 2f  xy α3 (β31 + β32 )f 2!

+f  yy (β31 + β32 )2 f 2 + O(h3 ) yi+1 = yi + p1 k1 + p2 k2 + p3 k3 .

(8.122) On the other hand, developing y(t) in a vicinity of point ti into Taylor series up to terms of order h3 , the following formula is obtained 1 1 yi+1 = yi + hy  i + h2 (f  x + f  y f ) + h3 (f  x + f  y f ) + O(h4 ) 2 6

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1 1 = yi + f h + h2 (f  x + f  y f ) + h3 (f  xx + 2f  xy f + f  yy f 2 2 6 2

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+ f  x f  y + (f  y ) f ) + O(h4 ).

(8.123)

We compare the coefficients standing by h, h2 , h3 , and we get h: p1 f + p2 f + p3 f = f,     h2 : p2 f  x α2 + f  y β21 f + p3 f  x α3 + f  y (β31 + β32 )f = h3 :

 1  f x + f y f , 2

 1   2 2 p2 f xx α22 + 2f  xy α2 β21 f + f  yy β21 f 2    + p3 f  y β32 f  x α2 + f  y β21 f 1   f xx α23 + 2f  xy α3 (β31 + β32 )f 2  2 2  +f yy (β31 + β32 ) f

+

1  2 (f xx + 2f  xy f + f  yy f 2 + f  x f  y + (f  y ) f ). 6 (8.124) We carry out the same operation, but this time with respect to other coefficients: =

f : p1 + p2 + p3 = 1, 1 f  x : p2 α2 + p3 α3 = , 2 1 f  y f : p2 β21 + p3 (β31 + β32 ) = , 2 1 f  y f : p2 β21 + p3 (β31 + β32 ) = , 2 1 1 1 f  xx : p2 α22 + p3 α23 = , 2 2 6

(8.125)

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1 1 f  yy f : p2 β21 + p3 (β31 + β32 ) = , 2 3 1 1 1 2 f  yy f 2 : p2 β21 + p3 (β31 + β32 )2 = , 2 2 6 1 1 f  x f  y : p3 α2 β32 = , 2 6 1 1 2 (f  y ) f : p3 β32 β21 = . 2 6 We have got eight equations, but only six of them are linearly independent. We take β21 and β32 as parameters, and the remaining parameters are derived in the following form  
1 α2 1 α2 α22 ± − − , α2 = β21 , α3 = 2 4 p3 2 3
 1 1 1 − p2 β21 − β32 , p3 = , p1 = 1 − (p2 + p3 ), p2 = p3 2 6β32 β21 (8.126) Therefore, an infinite set of solutions depending on the choice of free variables β21 and β32 is obtained. It is suitable to take rational coefficients, therefore, the most popular RK3 method is that with β21 = 12 and β32 = 2. Hence, α2 = 12 , p3 = 16 , α3 = 1, p2 = 23 , β3 = −1, p1 = 16 . It means that the following one-step formula is obtained: k1 = f (ti , yi ),
 1 1 k2 = hf ti + h, yi + k1 , 2 2 k3 = hf (ti + h, yi − k1 + 2k2 ), 1 2 1 yi+1 = yi + k1 + k2 + k3 + O(h4 ). 6 3 6

(8.127)

The so far described technique can be applied to find formulas with higher order approximations. We rewrite the most popular

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and widely used Runge–Kutta methods [Abramowitz and Stegun (1978)]: Alternative RK3 method k1 = f (ti , yi ),
1 k2 = hf ti + h, yi + 3
2 k3 = hf ti + h, yi + 3

 1 k1 , 3  2 k2 , 3

(8.128)

1 3 yi+1 = yi + k1 + k3 + O(h4 ). 4 4 Classical RK4 method k1 = f (ti , yi ),
1 k2 = hf ti + h, yi + 2
1 k3 = hf ti + h, yi + 2

 1 k1 , 2  1 k2 , 2

(8.129)

k4 = hf (ti + h, yi + k3 ), 1 1 1 1 yi+1 = yi + k1 + k2 + k3 + k4 + O(h5 ). 6 3 3 6 Improved RK4 method k1 = f (ti , yi ),
k2 = hf ti +
k3 = hf ti +

1 h, yi + 3 2 h, yi − 3

 1 k1 , 3  1 k1 + k2 , 3

k4 = hf (ti + h, yi + k1 − k2 + k3 ), 1 3 3 1 yi+1 = yi + k1 + k2 + k3 + k4 + O(h5 ). 8 8 8 8 Runge–Kutta Cash–Karp method [RKCK] k1 = f (ti , yi ),

(8.130)

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k3 = hf
k4 = hf
k5 = hf


 1 1 ti + h, yi + k1 , 5 5

379

(8.131)

 3 3 9 ti + h, yi + k1 + k2 , 10 40 40

 3 3 9 6 ti + h, yi + k1 − k2 + k3 , 5 10 10 5

 11 5 70 35 ti + h, yi − k1 + k2 − k3 + k4 , 54 2 27 27

7 1631 175 575 k1 + k2 + k3 ti + h, yi + 8 55296 512 13824  44275 253 k4 + k5 , + 110592 4096

k6 = hf

yi+1 = yi +

37 250 125 512 k1 + k3 + k4 + k6 + O(h6 ). 378 621 594 1771

Fehlberg method [RKF45] k1 = f (ti , yi ),
 1 1 k2 = hf ti + h, yi + k1 , 4 4
 3 3 9 k3 = hf ti + h, yi + k1 + k2 , 8 32 32
 12 1932 7200 7296 k1 − k2 + k3 , k4 = hf ti + h, yi + 13 2197 2197 2197
 8341 32832 29440 845 k1 − k2 + k3 − k4 , k5 = hf ti + h, yi + 4104 4104 4104 4104
1 6080 41040 28352 k1 + k2 − k3 k6 = hf ti + h, yi − 2 20520 20520 20520  9295 5643 k4 − k5 , + 20520 20520

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1 (902880k1 + 3953664k3 + 3855735k4 7618050

− 1371249k5 + 277020k5 ) + O(h6 ). (8.132) It should be emphasized that there exist even higher order Runge– Kutta methods. For example, the Prince–Dormand method is nothing but the RK8PD method [Bader and Deuflhard (1983)]. For this method, q = 13 and computational complexity of one-step in time is much higher than while applying low order Runge–Kutta methods. The increase in the integration step does not play a crucial role, and hence in majority of applications the methods with high order approximations attract attention rather from a purely scientific side, since they do not give any essential benefits from the point of view of computations. 8.1.3.3

Results obtained by different order Runge–Kutta methods

Since we have a set of various Runge–Kutta methods, a question appears as to which one among them is the most effective to solve our problem. In order to get an answer, we compare the results obtained by different methods with different integration steps in time. We study a solution to problem (8.46) with boundary initial conditions (8.47) solved with the FDM with difference derivatives of order O(h4 ). We compute parametric vibrations of the beam-strip subjected to harmonic excitation Px (t) = P0 sin(ω0 t), where ω0 = 3 is the frequency closely located to the fundamental beam frequency. Amplitude of excitation is chosen in a way that the vibrations are regular P0 = 2.5. As a pattern result, we take that of the RK8PD method with the integration step dt = 2−13 . Since during vibrations a stable regime has been observed, it was possible to get the case when stationary vibrations coincide with the pattern with accuracy up to a sign. Therefore, instead of trajectories, the frequency power spectra, i.e. integral characteristics, should be compared. In Fig. 8.6, a comparison of the beam-strip center vibrations and frequency power spectra for numerical integrations of the RK2 and RK8 method, is given. Stationary vibrations coincide up to the sign,

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-1

RK8 RK2

0.6

-3 -4

0.2

A -5

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RK8 RK2

-2

0.4

-6

-0.2

-7

-0.4

-8

-0.6 -0.8 68

381

-9 69

70

71

72

73

74

-10

0

1

2

T

3

4 ω

5

6

7

Fig. 8.6 Comparison of the beam-strip center point vibrations and frequency power spectra for different Runge–Kutta methods. Table 8.1 Error of integration of simply supported beam via different Runge–Kutta methods in %, formula (8.125). DT

RK2

RK2IMP

RK4

RK4IMP

RKF45

RKCK

2−8 0.054002 0.056899 0.053990 0.053990 ∞ ∞ 0.026166 0.026890 0.026164 0.026164 0.026164 0.026164 2−9 2−10 0.012219 0.012400 0.012219 0.012219 0.012219 0.012219 2−11 0.005240 0.005284 0.005240 0.005240 0.005240 0.005240 2−12 0.001748 0.001759 0.001748 0.001748 0.001748 0.001748

RK8PD 0.053989 0.026164 0.012219 0.005240 0.001748

whereas frequency power spectra coincide with the accuracy up to a constant component (zero frequency). Power spectra have been truncated up to frequency 2.1ω0 , and the relative errors for all methods in comparison to the pattern result have been computed. In Table 8.1 comparison of the relative error for different Runge–Kutta methods and different time steps has been carried out. The following notation has been used: RK2 is the modified Euler method (RK2 second-order Runge–Kutta method); RK2IM P is the improved Euler method (RK2 second-order Runge–Kutta method), formula (8.115); RK4 is the classical fourth-order Runge–Kutta RK4 method, formula (8.129); RK4IM P is the improved RK4 fourth-order Runge–Kutta method, formula (8.130); RKF 45 is the Fehlberg method (fifth-order Runge– Kutta method), formula (8.132); RKCK is the Cash–Karp method (fifth-order Runge–Kutta method), formula (8.131); RK8P D is the Prince–Dormand method (eight-order Runge–Kutta method).

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Table 8.2 Errors introduced by integration of cos via different Runge–Kutta methods in %. DT

RK2

RK2IMP

RK4

RK4IMP

RKF45

RKCK

RK8PD

2−2

9.499220 10.149400 7.148090 1.016780 0.158051 0.042727 0.017180 0.007865 0.03615 0.001543 0.000514

10.164300 11.566600 19.016600 17.917600 9.929830 3.145560 0.850096 0.218204 0.055427 0.014103 0.003573

9.461320 7.976540 0.834543 0.148066 0.067255 0.032769 0.016011 0.007722 0.003598 0.001541 0.000513

9.461320 7.976540 0.834543 0.148066 0.067255 0.032769 0.016011 0.007722 0.003598 0.001541 0.000513

13.477200 0.532819 0.285462 0.137571 0.066866 0.032748 0.016010 0.007722 0.003598 0.001541 0.000513

0.991611 0.604951 0.288984 0.137675 0.066869 0.032748 0.016010 0.007722 0.003598 0.001541 0.000513

1.522040 0.636559 0.289736 0.137695 0.066870 0.032748 0.016010 0.007722 0.003598 0.001541 0.000513

2−3 2−4 2−5 2−6 2−7 2−8 2−9 2−10 2−11 2−12

The carried out numerical experiments show that the numerical scheme is unstable for the time step dt = 2−7 for all methods. Furthermore, the table implies that the numerical algorithm is divergent also for dt = 2−8 for the Fehelberg and Cash–Karp methods. The obtained results show that the obtained error practically does not depend on the method used but it depends on the chosen time step. This is motivated by a complex part of the right-hand side of the integrated differential equations, and in result by the multi-mode vibrational regimes. For comparison purpose, in Table 8.2 a similar comparison of results for the integration of differential Eq. (8.133) is conducted, whose solution is the function y(t) = cos(t): y  (t) = −y(t),

y(0) = 0,

y  (0) = 1.

(8.133)

Right-hand side of this differential equation is very simple and does not introduce any errors into the numerical scheme. Resultant function has very simple form of one-frequency oscillator. Numerical scheme is sufficiently stable and does not diverge even if large time steps are applied. Here, we clearly see the advantage of the Runge– Kutta methods of higher orders. Observe also that the improved Euler’s method is more suitable for computation compared to the modified Euler method though both have the same approximation order. It is clear that the improved Euler method yields reliable

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results for steps less than dt = 2−8 , whereas the modified Euler method exhibits computational error of 1% for the step dt = 2−5 . After getting the required approximation, further increase in accuracy does not introduce any qualitative changes into the results. Therefore, a key role in the choice of a method of integration plays not only on its computational accuracy, but also on its computational complexity, i.e. the time needed to achieve reliable results. In the model problem of integration of Eq. (8.133) via the Cash– Karp method we deal with the smallest error, and the duration of computational results is only 15% slower than the (fastest) modified Euler method. It is obvious that the Cash–Karp method is more suitable to solve this problem since already for the step dt = 2−2 the error is less than 1%. Therefore, owing to the choice of a sufficiently large computational step, the given method essentially overcomes the modified Euler’s method from the point of view of computational complexity. However, the so far obtained results are not applicable to the problem of vibrations of the beam-strip. Errors of all methods are in practice equal for equal steps. The remaining methods give reliable results with an error of 0.5%. For the given problem, the modified Euler method is remarkable owing to its simplicity. Contrary to the studied model problem, in our case the majority of computational time deals with the computation of the right-hand side of the differential equation. In this case, for the same time step, the Runge–Kutta methods of higher order require a few times longer computational time than the modified Euler method. One time step in the modified Euler method requires three times computations of the right-hand side of the equation, whereas computations carried out by the improved Euler method, RK4 method, RK4IMP method, Fehelberg method and Cash–Carp method requires 4, 4, 7, 6 and 13 times, respectively. For the model’s problem, computation of the right-hand side has been carried out practically at once, therefore the higher order methods require typically 15% more computation time. In the case of analysis of the beam-strip vibration, the Cash–Karp method is beaten by the modified Euler method in two times. Reported results of the remaining methods have been

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8.1.4 8.1.4.1

LE Computation of the LEs spectrum

The LEs play an important role in the theory of dissipative dynamical systems. They allow to compute a quantitative measure of chaotization. Besides, there is a link between LEs and other characteristics of the chaotization, like the Kolmogorov entropy and the dynamical dimension. The theory of LEs has been developed by Oseledec [Oseledec (1968)]. Here, we aim at presenting relations between LEs and mechanical vibrations of structural members. Linkage between LEs and Kolmogorov entropy has been studied by Benettin et al. [Benettin et al. (1980)] and rigorously analyzed by Pesin [Pesin (1976)]. In the literature for chaotic oscillations of various dynamical system, the Benettin et al. algorithm [Benettin et al. (1980)] has been widely applied. In what follows, we give a background of the algorithm emphasizing on the problems devoted to direct numerical realization. We take an arbitrary system composed of n-dimensional vector x (n-dimensional phase space), and an input system of differential equations is integrated on a certain interval. Further, we consider the linearized system for the initial point of the integration interval. We get a new system composed of n vectors. Then this system is orthonormalized via the Gramm–Schmidt algorithm, and serves to get a new system of vectors xi for the next computation step. Logarithms of the normalization coefficients for each of the vectors are averaged on sufficiently large number of iterations, and the obtained limits of the digital sequences define a spectrum of LEs. It should be emphasized that the initial system should be in a certain stationary state, and the phase point at the end of each integration step serves as the initial condition for the next computational interval. In the work Benettin et al. [Benettin et al. (1980)], it has been shown that a choice of the initial system of vectors xi can be realized

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in an arbitrary manner, and does not influence the limiting values of LEs. In practice, the numerical integration of the linearized systems should be carried out using the same method as it is done for a nonlinear system. It means that the computational complexity of the algorithm is of n + 1 times larger than for the integration of the original input system. In practice, the orthogonalization of the system should be carried out at each step of the Runge–Kutta method in order to achieve the required accuracy. Small angles between the vectors χi , and the orthonormalization process causes high error rates. It means that the computational complexity of the algorithm increases. For a system with small dimension of the linearized equations, the problem can be solved analytically. For example, for the elastic beam-strip the Jacobi matrix of the system (8.144) has the following form 

0

I(A1 , t) =  −π 2 (λπ 2 (1 + 9A21 ) − Px (t))

1



. −ε

(8.134)

Let χ1 , χ2 be eigenvalues of the matrix (8.134), then a solution to the differential equation dA dt = I × A with the initial condition  A(0) = 

A1 (0) A

 ,

(8.135)

1 (0)

has the following form  1 {[−χ2 A1 (0) + A 1 (0)]eχ1 t   χ1 − χ2    χ2 t   − [−χ A (0) + A (0)]e } 1 1 1   .  A(t) =   1   {χ1 [−χ2 A1 (0) + A 1 (0)]eχ1 t     χ1 − χ2  χ2 t − χ2 [−χ1 A1 (0) + A 1 (0)]e } 

(8.136)

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Observe that in general χ1 , χ2 are complex, and in practice the formula has different representation for complex and real χ. Therefore, solutions of the linearized system can be found directly without applying the numerical integration formula (8.136). It essentially decreases the computational time and increases the obtained accuracy. Comparison of these two ways shows that they are practically equivalent (the difference is of 0.1%). Solution (8.136) character is associated with the second method of determining the LE. The formula implies that the length of vectors xi exponentially increases/decreases according to the real parts of χ1 , χ2 . Therefore, computation of LEs can be carried out by the direct averaging of the real parts of the eigenvalues of the Jacobi matrix along a phase curve. The latter method yields a qualitative picture of LEs contrary to the classical approach aimed at the approximate numerical approach. Therefore, for the beam-strip [from (8.134)], the following characteristic equation is obtained χ(χ + ε) + π 2 (λπ 2 (1 + 9A21 ) − Px (t)) = 0, which yields the eigenvalues  ε ε2 − π 2 (λπ 2 (1 + 9A21 ) − Px (t)). χ1,2 = − ± 2 2

(8.137)

(8.138)

The spectrum of LEs (χ1 , χ2 ) has following properties χ1 + χ2 = −ε and χ2 < 0. It is also relatively easy to formulate a criterion of chaos:    ε 1 t ε2 − π 2 (λπ 2 (1 + 9A2 ) − Px (t)) ≥ . Re χ1 ≥ 0 ⇒ lim t→∞ t 0 2 2 (8.139) From the point of view of computational complexity, this method is the most suitable, since the number of additional actions is minimal, and majority of time is spent in the numerical integration of the nonlinear system (here the problem is solved directly). Furthermore, the speed of convergence of the method is higher compared to methods based on numerical integrations.

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Fig. 8.7 Comparison of two computational methods to estimate the largest LE (the BGM).

However, although both methods give the same qualitative results, they differ with respect to the quantitative results. For instance, Fig. 8.7 shows a comparison of the largest LEs versus the external Px for the fixed frequency u being equal to the fundamental system frequency. The first curve is obtained via the classical Benettin method, the second curve is computed by formula (8.139). The picture consists of the “color scale” defining the vibrations character, and it has been obtained on the basis of analysis of the power spectra of system vibrations. 8.1.4.2

Results reliability and LEs

Numerical experiment shows that the LEs coincide with the “color scale”, i.e. periodic vibrations correspond to negative/positive value at the largest LE, whereas bifurcations correspond to χ1 close to zero. The problem of comparison of different characteristics is important while investigating reliability of the obtained results. This is why we have not limited ourselves to compare the characteristics associated with the change of one control parameter, but we have carried out a multi-scaled numerical experiment. The obtained results

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Fig. 8.8 Comparison of the largest LE and “chart” obtained through analysis of the power spectrum (BGM): (a) character of oscillations; (b) LEs; (c) maximal characteristic exponent depiced as 3D surface; (d) maximal characteristic exponent depiced as gray gradient (minimal LE (white), maximal LE (black).

are reported in Fig. 8.8. Figure 8.8(a) is constructed for ω ∈ [3; 4], P0 ∈ [0; 10]. Remaining three figures refer to three different ways of visualization of the maximum LE. Figure 8.8(c) corresponds to the largest characteristic exponent presented in the form of a 3D surface. The cross-section close to zero level (−0.05) is shown. Output of the surface above the cross-sectional plane denotes a transition of the LE into the space of positive values, i.e. chaotic vibrations. It is interesting to consider this plane in zones where bifurcations appear. For instance, let us study the line of first bifurcation. It is seen how the surface tends to zero level but never crossing it. Right-hand side Fig. 8.8(b) exhibits the largest exponent in the form

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of grey color. The increase in darkness yields the increase in the characteristic exponent magnitude. Minimum value of the LE (−0.5) corresponds to the white color, and the maximum value corresponds to black color. Drops of bifurcation, zones and periodic vibrations in chaotic zones are visible. There are clearly visible curves of stiff bifurcations, whereas within bifurcation zones there is a border between the first and the second period doubling bifurcation. Inside there are drops of periodic vibrations and bifurcations. Furthermore, different zones of chaos have different measures of chaotization. In darker zones, the maximum LE is a little bit larger than in other chaotic zones. The last Fig. 8.8(d) deals with the cross-section of the surface of LEs by the plane close to (−0.05) and boundary of a zone where the exponent tends to zero. This figure is less informative than the previous one, but it allows to distinguish zones of regular and chaotic vibrations. Comparison of the first figure with remaining ones gives the possibility to estimate the validity of the analysis on the basis of the power spectrum. It is clear that “charts” coincide with each other and mutually supplement each other [Awrejcewicz et al. (2002)]. First “chart” yields additional information on the character of onefrequency vibrations, allows to separate zones with vibrations with excitation frequencies and vibrations with frequency ω/2. “Charts” of LEs allow to trace the evolution in transitional zones, give a possibility to estimate the measure of chaotization in different zones of chaos. Coincidence of boundaries of fundamental zones allows to conclude that the analysis of vibrations on the basis of power spectrum yields the results being in agreement with other methods, and can be applied to estimate the general chart of system behavior. 8.1.4.3

Stability versus maximal deflection

In order to describe scenarios of the system behavior with respect to its regular/chaotic dynamics it is useful to consider LEs and scales obtained on the basis of analysis of the power spectra together with purely mechanical characteristics like for instance maximal deflection. The latter quantity plays an important role in monitoring

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Fig. 8.9 Comparison of the maximum deflection, maximum LE and a “scale” constructed using analysis of the power spectrum (the BGM).

mechanical properties of the analyzed system. Figure 8.9 shows all three parameters. Comparison of all three characteristics implies that all of them are coupled and mutually supplement each other. On the graph of maximum deflection, boundaries of the fundamental zones are visible. They are characterized by jump-type changes of the maximum deflection, i.e. the system suffers from a sudden stability loss associated with quantitative changes in mechanical regimes. Zones of chaos differ remarkably from zones of periodic vibrations. In the periodic vibration zones, the maximal deflection increase occurs smoothly contrary to chaotic zones where the jumps appear. In comparison to the scale, the maximum deflection yields additional information for periodic vibrations. For example, for P0 = 4.8 and P0 = 9.4 the maximum deflection exhibits smooth jumps, which are not visible on the scale. Comparison of the maximum deflection with the maximum LE gives a picture of a “strange” system behavior. The LE points tend to zero, i.e. the system is on the threshold of transition into a new regime. These jumps remarkably differ from the changes in the maximum deflection on the boundaries of zones, since they are more smooth and do not change the smoothness of

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the curve. Consideration of all exponents allows to yield wider and deeper conclusions regarding the system behavior. Non-homogeneity of one-frequency zones of vibrations and existence of the internal border are clearly visible on the curves of maximum deflection and maximum LE. It should be emphasized that the obtained dependencies of maximum deflection can be compared with estimation of the maximum LE. In many cases, when a computation of the LE spectrum is difficult owing to the complex form of the right-hand side of the differential equation, one may take into account only the maximum deflection. 8.1.5 8.1.5.1

Vibrations of flexible panels with infinite length The BGM

In order to verify the reliability of the numerical realization of the BGM a series of the numerical experiments has been carried out. At first, we define the fundamental frequency of the system. We consider the free vibration of the beam-strip for ε = 0 and with Px (t) = q(t) = 0 for the initial boundary conditions (8.47)–(8.50) (simple support). We get the particular case of the Duffing equation (8.62) and hence   (8.140) A¯1 = − 1 + 3A21 λπ 4 A1 . For small vibrations, we can neglect the nonlinear term influence, and the fundamental frequency can be defined by the following linearized equation A¯1 = −λπ 4 A1 .

(8.141) √ Its solution is A1 = cos (ω0 t + ϕ0 ), and the frequency ω0 = π 2 λ. The numerical experiment has been carried out for Poisson’s coefficient ν = 0.3. The fundamental frequency is ω0 = 2.9866. The following parameters are fixed within the numerical experiment: ε = 0, amplitude of initial deflection a0 = 0.01, time step dt = 2−7 = 0.0078125, and the integration is carried out by the RK2 method. In Fig. 8.10, time history of the beam-strip central point and the power spectrum are reported. As it can be seen from

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0.008

-6

0.006 0.004

-8

0.002 W

0

A

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-0.002

-10 -12

-0.004 -0.006

-14

-0.008 1

Fig. 8.10

2

T

3

4

5

-16

0

0.5

1

1.5

ω

2

2.5

3

3.5

Free vibrations (ε = 0) of the beam-strip (the BGM).

the figure, the amplitude of the harmonic vibration is 0.01, whereas its non-dimensional, period T = 2.10156 and the angular frequency ω = 2π T = 2.9897. Therefore, the introduced numerical error is less than 1%. This simple experiment verifies the validity of the first approximation of BGM. Additional verification of the result can be yielded by a solution to the static problem via the “set up” method. Static problems of structural members are widely described and investigated. Here, we aim only at the so-called problem of buckling. For this purpose, we consider the problem of longitudinal beam strip loading. Let the longitudinal load Px (t) = P1 be an infinite length impulse. Depending on the value of the external load, the beam-strip will deflect up to a certain amplitude. For small values of the external force the deflection is zero, but after a certain critical value the deflection will be different from zero and starts to increase with the increase in P1 . In order to solve the static problem via the set-up method, we consider the beam-strip with the small initial deflection a0 = 0.01 in the strongly dissipative medium (ε = 10) under the action of the infinite length impulse. As a result, for each value of the external force we can obtain the curve shown in Fig. 8.11(a). The threshold value of the deflection corresponds to a solution of the static problem for the given load (in our example, P1 = 2.5). Therefore, the function Ws (P1 ) is constructed [Fig. 8.11(b)]. Integration has been carried out by the second, RK2 method. Integration has been with time step dt = 2−8 = 0.00390625.

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0.7

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W

0.3

0.4

0.2

0.2

0.1 0

0.6

0

Fig. 8.11 (BGM).

2

4

T

6

8

10

0

0

1

2

P

3

4

5

Buckling effect while solving the static problem via the set-up method

In order to get the analytical form of this dependence, the stationary solution of Eq. (8.62) should be considered (λπ 2 (1 + 3A21 ) − P1 )A1 π 2 = 0, 
 1 P1 −1 . A1 = ± 3 λπ 2

(8.142) (8.143)

  It is clear that in the interval P1 ∈ 0; λπ 2 , solution (8.143) has no physical meaning, and hence zero solution is obtained. In interval P1 ∈ [0; ∞], the results of the numerical experiment coincide in full with the analytical solution. The sign of the numerical solution is defined by a sign of the initial deflection. For a positive (negative) initial deflection, the positive (negative) deflection branch is realized. One of the important questions in the BGM is the choice of an approximation order that would guarantee a sufficient reliability level. The first approximation has a simple character and opens the door for another class of problems. Solution in higher approximations yields a system with a large system of degrees of freedom (DOFs), and a more complex behavior in comparison to the first approximation arises. In this work, we aim at a determination of the order of the series guaranteeing qualitative reliable results. The stream convengence consequence principally cannot be achieved in the problem of nonlinear dynamics owing to the dynamical instability of the input problem.

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Fig. 8.12 (BGM).

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Stream and integral convergence in a zone of harmonic vibration

In Figs. 8.12 and 8.13, different orders of approximation has been applied. The beam-strip subject to harmonic load q(t) = q0 sin(ω0 t) has been studied, where ω0 stands for the system fundamental frequency. We have considered two characteristic points: q0 = 900 — the system is in a harmonic regime (Fig. 8.12) and q0 = 1200 — the system is in a chaotic regime (Fig. 8.13). In each of the pictures, a comparison of vibrations of the central beam point for different orders of approximation and comparison of integral characteristics (power spectra) are carried out. Vibrations in the chosen beam points have sufficiently large amplitudes (order of 5–6 beam thickness), i.e. nonlinear terms play a key role in Eq. (8.61). Integration has been carried out by the RK2 method with the step in time dt = 2−8 = 0.00390625. One of the aims of this work is to investigate scenarios of transition from periodic to chaotic vibrations. It is important to have a general imagination of the system behavior regarding the external excitation parameters. Therefore, it is necessary to construct the

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395

Stream and integral convergence in chaotic vibrations (BGM).

so-called charts of plate vibrations for each type of the boundary conditions and external load excitation. A chart represents the plane (A, ω), where A is the control parameter and P0 stands for the excitation amplitude. Each point of the chart is denoted by a color characterizing the regime of vibrations. Therefore, we get informative visual representation of the system behavior. Character of vibrations is defined in full by the beam center point vibrations (x = 0.5), since all of the remaining points move in a synchronized manner. For each pair of the parameters (A, ω), on the basis of analysis of the Fourier spectrum of the beam center point vibrations, the vibration character is defined. While constructing the chart, the frequency interval ω 1 where ω0 is the plate fundamental frequency for the from 2ω1 0 to 32ω 0 given boundary conditions, is analyzed. Interval of changes of A has been chosen in a way that the maximum beam deflection does not achieve the value of seven beam thickness. Hence, the hypothesis of the average deflection taking into account the geometric nonlinearities has been satisfied.

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Fig. 8.14

Charts of vibrations for the transversal load (BGM).

We have already considered the problem regarding the BGM convergence for fixed characteristic pairs (q0 , ω) corresponding to periodic and chaotic vibrations. We should ensure that the results of different approximations coincide for whole choice of the considered parameters. Therefore, the vibration charts have been computed for the transversal load. In Fig. 8.14, the charts for the first four series of BGM have been presented. The integration has been carried out via the RK2 method with time step dt = 2−8 = 0.00390625. One may distinguish the difference between the first and the second approximation, where further increase in the approximation terms does not bring any quantitative changes in the chart. The obtained results allow to conclude that in order to get qualitative results it is sufficient to consider only three terms of the series. This is enough to separate zones of periodic, chaotic or quasi-periodic dynamics. We have also analyzed the problem of convergence regarding time step and the applied integration method. The given numerical experiments with application of the second-order Runge– Kutta method (RK2IMP), fourth-order Runge–Kutta method (RK4,

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RK4IMP), fourth-order modified Runge–Kutta method (Fehlberg method (RKF45) and Cash–Karp method RKCK), eighth-order Runge–Kutta method (Prince method–Dormand method RK8PD) of the accuracy order and implicit Gear methods (GEAR1 and GEAR2). Steps in time have been chosen from 2−7 to 2−11 uniformly along the logarithmic scale. Principle differences between the results obtained through different methods have not been found, hence in the fundamental numerical experiments the RK2 method with time step dt = 2−8 = 0.00390625 has been used (it is most suitable one from the point of computations). We have also constructed the chart for the transversal load Px (t) = P0 sin(ωt). As it has been mentioned earlier, in this case we may take only the first term of the series and we get the Duffing Eq. (8.62). Then, changing the variables Aj (t) = A˙ j (t) we reduce the problem to the following one A˙ 1 = A 1 ,      A˙ 1 = −ε A 1 − λπ 2 1 + 3A21 − Px (t) A1 π 2 π 2 A1 .

(8.144)

In Fig. 8.15, the chart obtained via BGM is presented for the initial boundary conditions (8.47)–(8.50). Integration has been carried out by the RK2 method with time step dt = 2−8 = 0.00390625.

Fig. 8.15

Chart of vibrations for the longitudinal load (BGM).

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The chart investigation allowed to separate the following characteristic regimes: (i) This vibration character cannot be exhibited while applying the transversal load, and hence the variant of the longitudinal load is of a particular interest. In this case, we can observe a stiff bifurcation, i.e. the jump-type transition from damped to periodic vibrations while negligibly increasing the amplitude of the external load. Two characteristic cases of vibrations have been observed after the stiff bifurcations: stationary one-frequency vibrations with the synchronized frequency equal to excitation frequency, and the subharmonic vibrations with ωp /2. (ii) After the first stiff bifurcation, a further increase in the excitation amplitude shifts the system dynamics into a periodic regime. As it has been already mentioned, either harmonic or a subharmonic motion with ωp /2 has been exhibited. There are also narrow subharmonic zones with frequency equal to ωp /4. (iii) Chaotic zones are of particular interest. Here we apply only frequency power spectrum to measure chaotic dynamics and the vibration zones. (iv) In between zones of different dynamics one may observe small transitional zones associated with soft type bifurcations. Vibrations in those zones are still regular ones, but the trajectories of vibrations are not so strongly stable as it happened in periodic regimes. The associated power spectrum exhibits the occurrence of peaks on subharmonic frequencies. We analyze the chart given in Fig. 8.15 in a more detailed manner. We consider the chart constructed by cross-sections along the axis Px , we change the amplitude of the external excitation but keeping a fixed excitation frequency. This way of analysis is based on the consideration of the whole frequency interval, which can be divided into three characteristic zones with the following magnitude of the frequencies: [1.5; 2.2] — low frequencies, [2.2; 3.8] — average frequencies, [3.8; 4.5] — high frequencies. For small amplitudes of the excitation force the delivered energy is not sufficient to generate undamped vibrations. The bottom part

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of the chart is associated with damped vibrations. The upper border of this zone corresponds to the first stiff bifurcation. In other words, a small increase in the external excitation amplitude changes qualitatively the picture of vibrations, i.e. there is a transition from damped to periodic vibrations. We may distinguish two characteristic cases after the stiff bifurcation: either the vibration frequency is equal to ωp or ωp /2. For zones of low and high frequencies we have periodic ω vibrations with 2p , whereas in a zone of average frequencies we have ωp . It has been observed numerically that the beam vibrations are realized with three frequencies ωp /4, ωp /2, ωp . We consider the scenario of successive bifurcations in a zone of average frequencies. After the stiff bifurcation we have periodic vibrations of frequency ωp . Successive increase in Px implies a series of bifurcations and a transition to chaotic vibrations. Similar scheme of the frequency switching is also characteristic for two other zones, but instead of ωp , we have ωp /2, i.e. subharmonic vibrations appear. On the chart, there are separated zones where there is subharmonic vibration with ωp /4, and zones without bifurcations. Zones with bifurcations exhibit two characteristic cases either a transition from periodic to chaotic vibrations or associated with zones located within periodic zones. In what follows, we consider separately particular cases on boundaries between frequency zones. Here jump-type frequency switching is observed without any series either of bifurcations or a transition through a chaotic regime. The mentioned zones of frequencies are located in intervals [2.1; 2.2] and [3.8; 4.2]. After the first stiff bifurcation periodic, vibrations appear. Further, on a short interval of Px , values of vibrations are again damped. After that, stiff bifurcation takes place again, and a periodic solution with a different frequency occurs. More careful attention allows to distinguish the similarity between the chart’s parts. In general, all three frequency zones are self-similar. This is particularly exhibited in a zone of small amplitude of excitations. Shapes of areas are repeated in three zones; the larger the frequency, the larger is the area. As it has been already mentioned, the neighborhood zones differ from each other by an order of the frequency switching for periodic vibrations. Therefore, similar zones

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may have the same qualitative character of vibrations but they may differ in quantitative characteristics. Since in certain parts a transition into chaos is associated with a series of bifurcations, we have investigated a correspondence of the observed series behavior to the Feigenbaum scenario [Awrejcewicz et al. (2002); Krysko et al. (2002)]. We have carefully studied the first cascade of bifurcations while transiting into chaos in a zone of averaged frequencies. We have detected eight bifurcations. For example, we report here the case of the Feigenbaum constant computation for the fundamental frequency ω0 = 3 (Table 8.3). Values of Ci are obtained from the following relation Ci =

Pxn − Pxn−1 . Pxn+1 − Pxn

(8.145)

These results correspond well to the theoretical values Pxn − Pxn−1 ≈ 4.6692 . . . n→∞ Pxn+1 − P n x

C∞ = lim

(8.146)

In Fig. 8.16, the spectrum for ω = 3, Px = 2.616819 associated with nine bifurcations is shown. It is interesting to follow how the largest LE behaves in the series of Hopf bifurcations. In Fig. 8.17, the largest LE behavior while approaching to the critical point of a transition to chaos is shown. On the axis Ox, we have logarithmic scale and the first four Hopf bifurcations are visible. After achieving the successive bifurcation, the largest LE tends to zero and then decreases again. Process of chaotic vibrations in the parameters plane (Px , ω) begins with a period doubling bifurcation. Series of these bifurcations is easily observed on the power spectrum (see Fig. 8.16). Up to eight bifurcations have been monitored. The process of chaotization Table 8.3

Px Ci

Amplitude of the external load in the first series of bifurcations.

1

2

3

4

5

6

7

8

2.5732

2.6036 3.028

2.6136 3.975

2.61615 4.770

2.616679 4.759

2.616790 4.708

2.616814 4.683

2.616819

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Fig. 8.16 Power spectra with eight bifurcations (BGM) for ωp = 3, Px = 2.616819 (a) and a part of the graph a (b).

LCE 0.1

LCE

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6

0.1

Fig. 8.17

0.01

0.001

0.0001

0.00001 Px-Po

Largest LE exhibited by the series of Hopf bifurcations.

observed via the frequency spectrum is characterized by a smooth noising of the spectrum base. Sharp peaks remain on the fundamental frequencies and their harmonics. A series of the period doubling bifurcations implies occurrence of the infinite cascade of the period doubling bifurcations as the mechanism of transition into chaotic vibrations [Afraimovich et al. (1986);

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Arnold (1979); Astahov and Bezruchko (1987)], and also fits properties of similarity yielded by the theory of universality. For many dynamical systems with continuous time, five or more period doubling bifurcations have been illustrated and studied in references [Franceschini (1980); Franceschini and Tabaldi (1979); Grutchfield and Farmer (1980)]. A route to chaos via period doubling bifurcations and occurrence of chaotic (Smale) attractors (see [Shilnikov (1956)]) yield the socalled Smale horseshoe structure. The obtained attractors are either called Feigenbaum attractors or strange attractors [Arneodo and Collet (1985); Arneodo and Thual (1985)]. Particular feature of the Smale attractor relies on keeping maximum harmonic value associated with the fundamental frequency after the occurrence of the broadband spectrum. In Fig. 8.18, a series of the period doubling bifurcations tending to the Smale attractor is presented. The following time histories w(t), Poincar´e sections w(w) ˙ and frequency power spectra are given. It is also important to trace the evolution of the attractor along parameter P0 for eigenfrequency ω0 = 3. In Fig. 8.19, the most characteristic attractors and the corresponding Poincar´e sections are presented. In what follows, we explain the observed nonlinear phenomena using the qualitative theory of differential equations. When a stable limit cycle is born as a result of the Hopf bifurcation, the previous stable equilibrium point becomes unstable of the saddle-focus type with one-dimensional stable W s and two-dimensional unstable W u manifolds. Then two multiplicators of the limit cycle become complex conjugated. Now W u starts wrapping on the limit cycle creating configurations similar to [Shilnikov (1956)]. Increase in the load over the criticality may lead to the following behavior. W s and W u approach each other up to the occurrence of a homoclinic curve of the saddle-focus type. In this case, the complex structure is created which implies the existence of a chaotic attractor (the name of this attractor is not uniquely defined). In references [Arneodo and Collet (1985); Arneodo and Thual (1985)] it is called the screw-type attractor. In Shilnikov’s work [Shilnikov (1956)], it is called the spiral attractor. The typical

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Fig. 8.18 Time histories, phase portraits and frequency power spectra for different P0 : (a) 2.61, (b) 2.7, (c) 2.77 in the neighborhood of the smale attractor.

situation occurs when period doubling bifurcations take place before the occurrence of Shilnikov’s attractor, and then a transition to the Smale attractor takes place. On the other hand, while moving along the parameter the following situation is possible: there is no loop, but Shilnikov’s attractor does exist. In other words, there is a special type of attractor which can be identified only by its vicinity to the bifurcation manifold with the saddle-focus loop. The existence of an attractor with defined properties being characteristic for the Shilnikov attractor has been illustrated in the reference [Arneodo and Thual (1985)]. Therefore, criterions are needed allowing to distinguish Smale attractors from other attractors located in the vicinity of a saddle-focus loop. One of such criterions is motivated by a fact that in chaos associated with the Smale attractor, time instants for which maxima of oscillations are

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Fig. 8.19

Evolution of attractors and Poincar´e maps.

regularly distributed, whereas within Shilnikov’s type chaotic attractor they constitute a stochastic sequence. Besides the Smale and Shilnikov attractors, we have also R¨ ossler type attractors. We demonstrate R¨ ossler type attractors occurred after a series of period doubling bifurcations. In Fig. 8.20 a scenario yielding its birth for the values P0 = 4; 4.48; 4.5 is reported. The last of the phase portraits is the R¨ossler attractor. In Fig. 8.20 w(t), w(w) ˙ and the power spectrum are shown. 8.1.5.2

Numerical experiment (FDM)

Convergence of the difference scheme We study a convergence of the explicit scheme of problem (8.71) and (8.72) versus an order of approximation of the difference derivatives. As the pattern problem we consider the case of parametric vibrations

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Fig. 8.20 Time histories, phase portraits and frequency power spectra for differossler attractor. ent P0 : (a) 4.0, (b) 4.48, (c) 4.5 in the neighborhood of the R¨

with the external load Px (t) = P0 sin(ω0 t), where ω0 is the fundamental frequency of the system, q(x, t) = 0 and with the boundary/initial condition (8.49)–(8.52). In comparison to the boundary/initial condition (8.47)–(8.50) and (8.48)–(8.51) the considered problem is not symmetric, and hence we analyze the difference scheme convergence. We are aimed at an estimation of the vibrations character versus the excitation amplitude. From a point of view of applied mechanics it is important to define zones of changes of the excitation amplitude where vibrations are chaotic. Therefore, we tried to achieve the integral convergence of the method. For this purpose scales for various partitions, used approximations order and integration methods have been constructed. A “scale” presents a color representation of a vibration character versus the excitation amplitude P0 . These

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scales have been constructed on the basis of analysis of the vibration power spectra of the beam center. Namely, for each value of P0 simulation has been carried out, then the power spectrum of vibrations of the beam center has been obtained, and finally the vibration beam regime has been defined. Each regime is marked with a different color. The following characteristic regimes have been detected: damped vibrations — energy introduced by external force is too small to beat the dissipative forces; harmonic vibrations with frequency ω0 — synchronization between the system and external force; subharmonic vibrations with frequency ω20 ; period doubling bifurcations; chaotic vibrations. In Fig. 8.21 scales regarding the explicit method are reported. For the boundary initial conditions (8.49)–(8.52) the fundamental frequency ω0 = 5, whereas for the interval of P0 variations [0; 50] this corresponds to the boundaries, where assumptions regarding applicability of theory of shallow plates are valid, i.e. deflection of the beam should not be larger than its 6–7 thickness. As a pattern result we consider the scales obtained for n = 32, dt = 2−11 , whereas the numerical integration has been carried out via the eighth-order method RK8PD. Analysis of the obtained results yields a conclusion that the integral convergence is achieved for partition n = 16, dt = 2−8 for the applied RK2 method. Approximation of order O(h2 ) smoothes higher frequencies of the signals, and hence for large values of the excitation amplitude the difference scheme does not properly enough present the real system behavior. On the other hand, in the case of sufficient flat transition, functions of O(h6 ) (Fig. 8.25) amplify high frequencies input, and in result the difference scheme is divergent (n = 16, dt = 2−8 for integration via the RK2 method). Finally, the results obtained by the RK2 method are reliable, and there is no need to apply higher orders Runge–Kutta methods. In Fig. 8.21, for comparison purpose, numerous results obtained via the implicit method with different values of the weight coefficient σ are reported. Implicit difference schemes are more stable, which allows to apply larger steps in time than for explicit schemes for the same partitions of the space coordinates. Stability of the implicit difference scheme is guaranteed by a strong smoothing of

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Convergence of explicit difference methods with various approximation

high frequencies, but simultaneously frequencies associated with the fundamental system energy are also smoothened. This is why for the weight coefficient σ = 0.5, the stability of the difference scheme has not been achieved. The result being close to the pattern ones can be only achieved for the weight coefficient σ ≈ 1, which in fact reduces the problem to that of integration via the Euler method. Besides, the implicit scheme for equivalent partitions has larger computational complexity than the associated explicit scheme. Only for σ = 0 the method yields a computational benefit, since the computation of the operator Z on the previous layer can be avoided. In Fig. 8.22 the

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Fig. 8.22

Comparison with the implicit Cranck–Nicholson scheme.

Table 8.4

Square average errors of different methods.

n

8

12

14

16

18

20

32

O(h2 )

1.14e + 2

5.12e + 1

3.76e + 1

2.88e + 1

2.28e + 1

1.84e + 1

7.17e + 0

O(h2 )

1.18e + 1

2.41e + 0

1.31e + 0

7.69e − 1

4.81e − 1

3.16e − 1

4.82e − 2

O(h2 )

1.29e + 0

1.20e − 1

4.82e − 2

2.18e − 2

1.08e − 2

5.75e − 3

3.45e − 4

PS

1.88e + 0

1.90e − 3

3.20e − 5

2.46e − 7

2.36e − 5

3.11e − 5

7.12e + 1

scales for the implicit method n = 32, dt = 2−11 of approximation order O(h4 ) σ = 0; 0.5; 1, and the pattern result, explicit method of n = 32, dt = 2−11 as well as integration via the RK8PD method with approximation of the spatial derivatives O(h6 ) are reported. A pseudo-spectral method with a Chebyshev mesh strongly differs from the explicit and implicit methods since points of the mesh have an influence on the values of the partial derivatives, not only on the neighboring ones. The latter fact should theoretically increase an accuracy of the being computed derivatives, but the high order interpolating polynomials oscillate on the interval ends (the Gibbs effect). This causes a lack of accuracy of the computed derivatives. In order to demonstrate this effect we consider the model problem devoted to computation of the fourth derivative. For the boundary conditions of the Dirichlet–Neumann problem (8.48) we take the function w(x) = 1 − cos(2πx), which satisfies the boundary conditions. In Table 8.4 the values of the square averaged numerical error of the 4th derivative computed either by the pseudo-spectral method

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1 0.1 0.01 0.001 0.0001 1e-05 1e-06 1e-07

Fig. 8.23

10

20

Square average errors in computation of the fourth derivative.

or via the analytical value wIV (x) = 16π 4 cos(2πx) are given. In this table, also data for difference derivatives of orders O(h2 ), O(h4 ) and O(h6 ) are reported. The data allow to get the following conclusion: the pseudo-spectral method yields higher order accuracy than the difference methods, but the error increases when the number of partitions increases. The table shows that for n = 16 the optimal partition for the pseudo-spectral method is obtained, and the increase in n implies a stability loss of the computational scheme. For the difference method the dependence of the error on a number of partitions yields a linear function in the logarithmic coordinates, and it monotonously decreases with the increase in n (Fig. 8.23). In order to get an imagination of the relative error of the method, it should be emphasized that the interval of the fourth derivative changes [−1560: 1560], i.e. the same average error of order 1 corresponds to quantities of 0.01. For the first and second derivatives, the dependence of the error on n has the same character as in all considered cases. As in the case of the implicit method, in Fig. 8.24 a comparison of the pattern result of the implicit method (n = 32, dt = 2−11 ,

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Fig. 8.24 mesh.

Comparison of the pseudo-spectral method versus the Chebyshev

Chebyshev mesh n=12

(a) Regular mesh n=12

(b)

Fig. 8.25

The Chebyshev mesh (a) versus the regular mesh (b).

with integration of the RK8PD method and approximation of the spatial derivatives O(h6 )) with the results obtained via the pseudospectral method for various partitions is presented. Observe that steps of integration in time have been chosen smaller than in the case of analogous partition and the same integration method regarding the regular mesh. For n = 8, dt = 2−9 (for regular mesh we have dt = 2−8 ). This paradoxal phenomenon can be explained in the following manner: boundary layer points of the Chebyshev mesh are located essentially closer to the edge than in the case of the regular mesh (Fig. 8.25), and these points are most sensitive to the scheme errors, since their absolute values are always small (owing to the edges fixation — Dirichlet condition), whereas, on the contrary, the fourth derivative has the maximum value. Therefore, the error of the derivatives computation implies divergence of the difference scheme. We separately consider the problem of vibrations convergence for the explicit method.

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Fig. 8.26 Integral convergence in chaos: time histories and frequency power spectra for n = 16, dt = 2−8 (a) and n = 32, dt = 2−11 (b).

For example, in Fig. 8.26 vibrations of the beam center for various spatial partitions, as well as the power spectra of these signals (parameters of the external load P0 = 7, ω0 = 5, partitions n = 16, dt = 2−8 and n = 32, dt = 2−11 , integration has been carried out by the RK4 method, spatial derivatives are approximated by order of O(h4 ) are reported. It is obvious that the integral convergence is achieved (it is exhibited by the power spectra), but a full coincidence of the trajectories is not achieved. For both partitions the results qualitatively coincide, which means that the further increase in the spatial partition will not change the so far obtained result, i.e. chaotic zones are reliable and not introduced by the computational errors. The figure clearly shows that the signals coincide in full on a short initial time interval, but then they diverge and after that no correlation exists. The influence of this problem is similar to that of simple supports of the edges considered earlier, and it can be solved with the BGM

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with higher approximations. However, we use only the FDM since it is the most general and does not allow to introduce additional mathematical constructions on the contrary to the Bubnov–Galerkin approach. Fundamental frequency of the studied system essentially depends on the used boundary condition. For instance, for simple support and clamping (8.47)–(8.50) and (8.48)–(8.51) we may derive analytical formulas to define √ √ fundamental frequency value: ω0 =  2 the π 2 λ ≈ 2.98 and ω0 = 32 π 2 λ ≈ 6.705, respectively. The fundamental frequency has been also defined numerically. We take the dissipation term ε = 0, and we introduce the initial buckling a0 = 0.01. Integration parameters follow: spatial partition n = 16, time step dt = 2−8 = 0.00390625. In Fig. 8.27 vibrations of the beam center are shown as well as the power spectrum of these vibrations. Time history indicates the periodic vibrations, and the power spectrum exhibits only one frequency vibrations, i.e. ω = 6.761. The error magnitude introduced by numerical integration is less than 1%, which validates reliability of the obtained results. Initially, the problem of the convergence of the difference scheme has been investigated. The Runge principle has been used to define the minimum partition required to get the reliable results regarding a chart construction. In Fig. 8.28 scales of the vibrations character constructed for different partitions (ω = 6.7, 0 < P0 < 50) are reported.

-5 0.008

-6

0.006

-7

0.004

-8

0.002

-9

0

A -10

-0.002

-11

-0.004

-12

-0.006

-13

W

-14

-0.008 -0.01 0

0.5

1

1.5

2 T

(a)

2.5

3

3.5

4

-15

0

2

4

6

8

ω

10

12

14

(b)

Fig. 8.27 Free vibrations in problem with clamping (FDM): time histories (a) and frequency power spectraum (b).

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Fig. 8.28 Convergence of the explicit difference methods for various approximations for problems with clamping.

Fig. 8.29

Chart of vibrations for the problem with clamping (FDM).

It can been seen that the necessary computational accuracy is achieved for n = 16, dt = 2−7 = 0.0078125. This partition has been used to construct the chart given in Fig. 8.29. The fundamental frequency ω0 = 6.7, and hence the considered interval for ω is [4.0; 10.0]. The general behavior of the system is similar to that previously studied for the simply supported beam, but there exist also

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Fig. 8.30

Period doubling bifurcation (see text for more description).

a few peculiarities. Namely, in spite of period doubling bifurcation also the period tripling bifurcations have been obtained (Fig. 8.30). In Fig. 8.30 the fundamental characteristics are shown: time histories w(t), phase portraits w(w), ˙ power spectrum and Poincar´e maps (ω = 8.75, P0 = 25.8). Unfortunately, it was difficult to estimate borders between zones in the series of period tripling bifurcation, and therefore we were unable to estimate the Feigenbaum constant for this case. As it has been seen from the power spectrum, after the first triple period bifurcation a series of bifurcations appears, but they are different from either period doubling or period tripling bifurcations. This phenomenon requires further investigation. For the studied problem a transition to chaos via period doubling bifurcations through the Feigenbaum scenario has been detected. In Table 8.5 the data used for the Feigenbaum constant computation associated with the fundamental frequency is given. The obtained results coincide well with the theoretical value C∞ ≈ 4.6692 (formula (8.146)). As in previous case, we have observed a series of eight Hopf bifurcations.

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Table 8.5 Px (amplitude of excitation) corresponding to the first series of bifurcations in the problem with clamping (FDM).

Px Ci

1

2

3

4

5

6

7

8

14.6836

14.8441 2.7395

14.9027 4.7368

14.9150 4.5069

14.9178 4.5705

14.9184 4.4971

14.9185 4.6358

14.9186

Fig. 8.31 Two-piece wise linear (a) and exponential (b) stress–strain dependence.

8.1.6

Geometric and physical nonlinearities

We consider the problem with physical nonlinearity for two variants of stress–strain relations: linear dependence of two linear pieces and the exponential transition into a plastic zone. These possibilities are presented in Fig. 8.31. Important role plays here a ratio of the linear beam with the ratio ha = 100. Hence, the following variants of the dependence σ (e) has been considered: (i) Two piece-wise linear dependence:  kei, for e ≤ es , σ (e) = kes, for e > es .

(8.147)

(ii) Exponential dependence



 e . σ(e) = kes 1 − exp − es

(8.148)

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The common point of two straight lines es corresponds to the critical deformation, after which a plastic zone appears. The coefficient of a slope of two piecewise linear characteristics has been chosen in a way that on the linear part of the dependence σ (e) the governing equations should coincide in full with the equations taking into account only the geometric nonlinearity. All numerical experiments have been carried out for the following fixed parameters: k = 0.39 and es = 9.8. For the problems with physical nonlinearity the fundamental experiments have been given in frame of the nonlinear geometrical problem statement: charts of vibrations types, the spectrum of LEs, as well different variants of the beam support. Besides, we have added one more parameter: beam deflection for frozen time instants taking into account plastic zones. Their analysis allows to estimate the plasticity level of the beam, i.e. to define, which part of the beam cross-section moved to a zone of plasticity. In Fig. 8.32 a comparison of the fundamental characteristics (maximum deflection, largest LE and scales) for the excited longitudinal vibrations with the fundamental frequency (problem with simple support (8.47)–(8.50)) is shown. For small amplitudes of excitation

Fig. 8.32 Comparison of four dynamical characteristics for longitudinal excitation with the fundamental frequency (see text for more details).

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amplitude, the deformations do not achieve plastic zones and solutions coincide in full. The increase in the amplitude of excitation yields the increase in the amplitude of vibrations of the beam, and hence its deformation is increased. For deflections of order of two beam thickness a zone of plastic deformation is visible in the central area of the beam. From this time instant solutions of geometrical nonlinear problem do not coincide with the physical solution of nonlinear problem. It should be emphasized that for a physically nonlinear problem we should consider essentially smaller interval of the external amplitude changes. As it can be seen from the figure, after a certain value the beam achieving the maximum deflection finally transits into a chaotic zone, and the maximum deflection rapidly increases. For this deflection magnitude the applied equations are not useful. It is interesting to compare the charts of vibrations for the problem of physical nonlinearity and more simple modeling variant taking into account the geometric nonlinearity. In Fig. 8.33 the charts regarding the simple support of two edges are shown. The left chart presents a solution to geometrically nonlinear problem (a part of the chart (8.26)), whereas the right corresponds to the problem with two piece-wise characteristics (8.26). As it has been seen the charts fully coincide for small amplitudes of the external load, but the increase in the excitation amplitudes introduces the difference, and for large displacements the charts are clearly different. For a comparison, in Fig. 8.34 a chart for the exponential nonlinearity is reported. Obviously

Fig. 8.33

Comparison of vibration charts.

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Fig. 8.34 form.

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Chart of vibration types for a physical nonlinearity in exponential

the nonlinearity influences the chart of vibrations, and for different types of nonlinearities considered we expect an occurrence of new effects. In order to distinguish a difference between the geometric and physical nonlinearities and only the geometric nonlinearity more deeply, it should be observed that the beam transits from a zone of elastic deformations to a zone of plastic deformations. For this reason we have applied beam cross-sections for different time instants to follow beam deflection for different values of the amplitude of the external load for the cases of clamping (Fig. 8.35). In this figure the beam cross-section corresponding to its maximum deflection are presented for P0 ∈ [7; 20] with the step of ∆P0 = 0.2. This variant of the boundary initial conditions is of more interest comparing to the case of the simple beam support, since the beam has three zones of transitions into plastic deformations. The central zone and points of clamping exhibit maximum deformations and therefore beginning with amplitudes ∆P0 = 7 we observe zones of plastic deformations. For the amplitude of the external force P0 = 12, 5 zones of plastic deformations begin contacting with each other, and the beam exhibits the continuous strip of plasticity. One may say that about of 50% of the beam length is passed to the plastic zone. The beam

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Fig. 8.35 Cross-sections of the beam deflection for various amplitudes of the external load.

cross-section can be treated in the figure symbolically, since a ratio of the linear beam dimension and its thickness equals 100. In the problems of simply supported beam the only one plastic zone is located in a neighborhood of the beam center. The increase in the external load amplitude implies the increase in this zone and for P0 = 20 the beam fully transits into the plastic zone. In Fig. 8.36 the vibration chart for the boundary conditions (8.48) and the physical piece-wise linear characteristic is presented. In the upper chart a plasticity order in the form of color intensity of the beam is shown. White color corresponds to plastic deformations, and the chart of vibrations in these zones repeats in full the chart of the problem devoted to only geometric nonlinearities (Fig. 8.28). A chart of the plasticity order gives a possibility of estimating visually the influence of a physical nonlinearity. Plasticity zones play an important role and the given chart can be considered together with the cross-section of the beam deflections shown in Fig. 8.35. As it has been seen from the upper chart, for P0 = 20 the beam practically is

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Fig. 8.36

Chart of vibrations (a) and chart of plasticity order (b).

transited in full to a plastic zone on all frequencies and vibrations for large loading amplitudes are not predicted by the applied theory. 8.1.7

On the Sharkovsky’s periodicity

Investigation of chaotic regimes in models with a 3D phase space began with investigation of the Lorenz system, and nowadays

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includes numerous mathematical models. They include objects from hydro- and aerodynamics, radio-physics, mechanics, chemistry, biology, economy. In addition, each branch of the mentioned sciences generates its own peculiar mathematical model being not coupled with the so far known and investigated ones. Occurrence of novel models exhibits the rich behavior of nonlinear dynamics which require classification of the detected phenomena. In fact, the rules of changes of dynamical regimes regarding local changes of the parameters can be reduced to a few typical bifurcations. If we consider a route to chaos, then the problem is reduced to a few typical scenarios of transitions or a combination of these transitions. On the other hand, global bifurcational structures of the dynamic regimes, even for the same mechanical system, but for different boundary conditions, may differ completely from each other. In other words, analyzing the bifurcation diagrams require a study of the particular mosaic elements which describe local scenarios of the transitions between different of regimes. From the mathematical point of view, local scenarios of transition into chaos correspond to the local bifurcations or sets of such bifurcations. Then, a complex mathematical object is created, composed of hyperbolic non-trivial sets, as well as stable periodic motions. The mentioned sets are called quasi-periodic. In the above we gave an example of transition of the mechanical system into chaotic state via the Feigenbaum scenario, and we have estimated the universal Feigenbaum constants for the boundary conditions (8.48)–(8.51). We consider a scenario of transition of the mechanical system into a chaotic state for the boundary conditions (8.49)–(8.52). We have detected not only period doubling and period tripling bifurcations, but also period quintuple, period septuple, period ninefold bifurcation phenomena. The given results are shown in Figs. 8.37 and 8.38 while partitioning of the space into parts n = 16 and n = 32, respectively. In these figures time histories w(t), phase portraits w(w), ˙ power spectrum, Poincar´e maps wt+T (wt ), where T is a period of excitation (amplitudes of the exciting force are shown in the figures) are presented. In Fig. 8.37(a) in the power spectrum one

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Fig. 8.37 Following Sharkovsky’s periodicity: time histories, phase portraits, frequency power spectra and Poincar´e maps for different P0 : 6.871 (a), 6.904 (b), 6.909 (c), 6.917 (d) (n = 16).

period doubling bifurcation is clearly visible, after which the period fivefold bifurcation takes place. In Fig. 8.37(b) in the power spectrum two period doubling bifurcations are shown (denoted by 1 and 2, respectively), and then the period tripling bifurcation occurs. Figure 8.37(c) shows one period doubling bifurcation with the following sevenfold bifurcation. Finally, in Fig. 8.37(d) after a period doubling bifurcation, the period ninefold bifurcation appears. In the latter case, in the Poincar´e section, two areas of the gathered points have been transited into one area. We consider an analogous chart for the partition n = 32 (Fig. 8.38). Results have been presented for only one excitation frequency ω = 5. Here, after first period doubling bifurcation (it is denoted by 1 in the

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Fig. 8.38 Following Sharkovsky’s periodicity: time histories, phase portraits, frequency power spectra and Poincar´e maps for different P0 : 6.850 (a), 6.857 (b), 6.877 (c), 6.889 (d) (n = 32).

figures), we have period fivefold and period sevenfold bifurcations [Fig. 8.38(d)] as well as a partition of the spectrum into 11 equal parts is observed [Fig. 8.38(c)]. The novel phenomenon, when the spectrum is divided not into either 2 or 3 parts, but also for more parts with creation of a strange attractor (SA), has been reported. This phenomenon is presented in the Poincar´e map in the form showing collapsing of two local spaces. The interval of the parameter variation (with a step of 0.001) has been studied. The following characteristic features have been monitored. During transition from periodic vibrations to chaotic ones, a series of bifurcations appear which has more complicated scenario than that exhibited by occurrence of the amplitude peak

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associated with the frequency ω2 . This space is relatively wide. Further, a sudden transition into chaos occurs, and then a transitional zone into another regime appears. The series of bifurcations occurs before the system transits into chaos, which are of particular interest. The bifurcation interval consists of the series of short zones, and in each of them the further spectrum partition takes place into rather large number of parts. The common phenomenon for all zones is that of existence of the first period doubling bifurcation. The most dominating zones are: spectrum parts from 0 to ω2 and from ω2 to ω are divided into 5 equal parts; spectrum parts from 0 to ω2 and from ω2 to ω are divided into 11 equal parts; spectrum parts from 0 to ω2 and from ω2 to ω are divided into 7 equal parts. In the transitional zones the spectra have non-chaotic structure, but they exhibit peaks, which are not exactly rational with respect to the fundamental frequency. The most dominating characteristic zone is that 5 and 11 located between bifurcations the spectra of this zone ω and 4ω have two peaks associated with 10 10 , i.e. peaks corresponding 3ω to the fivefold bifurcations, but the peaks associated with 2ω 10 and 10 are substituted by pairs of the equally distant peaks [Fig. 8.38(b)]. It should be emphasized that nowadays simple models of chaotic dynamics, i.e. discrete dynamical systems are relatively well studied. For such systems a series of the theorems have been formulated, in particular, the window of period three, which corresponds to the most interesting part of the 1D map f (x) = x2 + c. The following question occurs: Do other non-periodic orbits exist? The answer is given by the Sharkovsky theorem [Sharkovsky (1964)]: Let J be the finite or infinite interval in R. We assume that the map is continuous. If there is a point of period n, then there exist a point of period k, k > n belonging to the following series 3, 5, 7, 9, . . . . . . , 2n , . . . , 22 , 21 , 1. Sharkovsky’s theorem is applied only to the real function given in the real interval. Ten years after the work [Sharkovsky (1964)] another work has been published [Li and Yorke (1975)] for the particular case of the orbit with period-3.

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One of the important achievements of this work refering to flexible beams is detection and illustration of the bifurcations sequence, which follows the Sharkovsky’s rule of order. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

8.1.8

Chaos — hyperchaos–hyper–hyperchaos phase transitions

We consider the evolution of the vibration regime while increasing the amplitude of the exciting force P0 after occurrence of a strange chaotic attractor (SA). Evolution of the strange chaotic attractor with the increase in the parameter P0 inside of a chaotic space and its associated qualitative changes are not fully investigated for mechanical systems. A general tendency of stochasticity development of chaotic set and the increase in the attractor dimension are known. A full description of chaotic dynamics governed by a system of differential equations is bounded by their number. Qualitative changes in structure of chaos exhibited by mechanical systems is not investigated in full. Analysis of the transition “chaos–chaos” does not include bifurcations in chaos associated with the increase in the LD of the analyzed attractor. Intuitively, an occurrence of the qualitative changes of the physical characteristics of an attractor due to occurrence of the positive LE is more realistic way of study than a change induced by the LD successive integer. In fact, in the case of occurrence of additional positive LE, a new unstable direction on the system trajectory appears, which should either change the system dynamics or its physical characteristics. On the other hand, Lyapunov’s dimension of the local volume preserves its quantity, i.e. it is neither compressed nor extended. Note that the LD includes not only positive and zero value but also negative exponents. The latter ones are responsible for stability of various types of motion. They represent the metric properties of an attractor. In the reference [Li and Yorke (1975)] it has been shown that negative LEs are not associated with any physical characteristics of the dynamic regimes, and in general they cannot be estimated via physical experiment. We consider a transition from a SA with one positive LE to chaos with two and three LEs, which will be called as chaos– hyperchaos and “hyperchaos–hyper–hyperchaos” transition. Moving

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Fig. 8.39 Maximum beam deflection, power spectrum of LEs, and the vibrations scale (mixed boundary conditions).

in the hyperchaos space along the parameter P0 two scenarios are possible: a soft transition of “chaos–hyperchaos”, when a smooth increase in the second LE does not imply any jump-type qualitative changes of the chaotic process. On the other hand a stiff transition “chaos–hyperchaos” is associated with the so far mentioned jumptype changes. It is interesting to investigate mutual dependence of the various vibration characteristics. In Fig. 8.39 the whole spectrum of parameters is presented including: wmax (P0 ) — maximum deflection of the beam center, LEs spectrum {χi (P0 )}5i=1 and a scale, characterizing the vibrations character. Analyzing simultaneously curves wmax (P0 ) and {χi (P0 )}5i=1 their mutual correspondence is observed. A change of sign of the maximum LE is associated with a series of stiff bifurcations exhibited by the curve wmax (P0 ) and the associated color change of the attached scales. Vibrations are damped up to P0 = 3.3. It is well seen on the deflection curve wmax (P0 ). All LEs are negative, since the vibrations are stable, and they are approximately equal to − 2ε . While approaching the amplitude load value P0 = 3.3, the maximum LE χ1 (P0 ) begins to increase and it tends to 0. It means that the system is on a border between two regimes, and a small increase in the external amplitude of the load yields periodic vibrations. The system exhibits a first stiff bifurcation, and the maximum LE is equal

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Spectrum of the LEs in the case of clamping.

− 2ε again. Further, an increase in the amplitude P0 of the external load implies occurrence of a zone of periodic vibrations [3.3; 6.9]. On this interval the maximum deflection wmax (P0 ) smoothly increases, and the maximum LE oscillates in a zone of negative values. The latter zone is finished by a series of bifurcations, and then a transition into chaos takes place. Series of bifurcations are visible on the maximum LE χ1 (P0 ). Beginning of a new bifurcation is associated with the increase in χ1 (P0 ) up to 0. Sudden jump on both dependence χ1 (P0 ) and wmax (P0 ) indicates the first transition of the system into a chaotic state. Positive values χ1 (P0 ) in interval [6.9; 7.2] imply the system instability. This interval is presented in the scale. The next interval P0 ∈ [17.2; 29.2] begins with a stiff stability loss, which is well indicated on the curves wmax (P0 ) and χi (P0 ). Two largest LEs are positive, whereas the remaining LEs are negative, and the system exhibits hyperchaos. Finally, the hyper–hyperchaos regime takes place for P0 ∈ [42.5; 50]. Beginning of this zone is associated with a zone of a stiff stability loss. The system transits via a series of stiff bifurcations into a hyper–hyperchaos (χ1 > 0, χ2 > 0, χ3 > 0, χ4 increases, remaining negative, whereas χ5 oscillates around the value of − 2ε ).

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Comparison of all characteristics taken into account so far yields an observation, that the results obtained via analysis of the vibration spectrum corresponds in full to the results obtained via the LEs spectrum. This correspondence has been observed on the whole considered interval. We emphasize once more peculiarities and mutual relations of the considered parameters in the characteristic system points: (i) In zones with stable vibrations (damped and periodic) the maximum LE χi (P0 ) is positive, and the maximum deflection wmax (P0 ) changes smoothly without jumps. (ii) In bifurcation zones (denoted on a scale by black color) the maximum LE χi (P0 ) tends to zero value, but does not achieve a positive quantity. (iii) The change of regime is associated with a jump effect (χi (P0 ) and wmax (P0 )). Points of the system reconstruction are well visible on the vibrations scale. (iv) In zones with chaotic vibrations the maximum LE χi (P0 ) is positive, and the maximum deflection wmax (P0 ) increases via small jumps. (v) High correspondence between the results obtained via FFT (scale) and graphs of the maximum LE χi (P0 ) is evident. Periodic vibrations and damped vibrations correspond to negative χi (P0 ), whereas chaotic zones are associated with positive LE. It should be emphasized that the so far described phase transitions “chaos–hyperchaos–hyper–hyperchaos” have been observed only in the case of the applied non-symmetric boundary conditions [Krysko et al. (2006)]. In Fig. 8.41 dependences χi (P0 ) are presented for the boundary conditions of clamping type. Here a transition from the state of bifurcation into chaotic state takes place, and from the whole LEs only χ1 changes its sign. In the chaotic regime only χ1 and χ2 increase, whereas the remaining LE oscillate around − 2ε . On the borders of transition from regime of bifurcations into chaos and vice versa all LEs oscillate. In dissipative systems with chaotic dynamics versus the control parameters and initial conditions, an

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Fig. 8.41 Time histories, phase portraits, power spectra and Poincar´e showing the following regimes: (a) periodic vibrations, (b) chaos, (c) chaos–hyperchaos, (d) hyper–hyperchaos.

infinite set of attractors are realized with different structures. Fixation of one of the control parameters does not change the picture: there is an infinite set of attractors and their pools of attraction are divided in the phase space by separatrix surfaces. Since in the phase space there exists a set of regular periodic regimes together with the SA, a theoretical description of the dynamics of dissipative systems is difficult. Numerical results show that attracting pools and existence of periodic regimes in the phase space with respect to parameters decrease with the increase in their periods. It is because the cycles of large periods are not registered due to fluctuations, and an attractor does

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not include periodic trajectories and becomes almost hyperbolic one. The latter attractors are called quasi-attractors, whereas the systems are called quasi-hyperbolic systems. It seems that a lack of sufficient number of rigorous results of the stochastic and ergodic theories of the quasi-attractors should be supplemented with numerical and physical experiments. In Fig. 8.41 the following dependencies are shown: time histories w(t) (180 < t < 200), phase portraits, power spectra and Poincar´e sections for the following values of the amplitude of external load: (i) P0 = 5 — periodic vibrations, the first LE χ1 > 0; (ii) P0 = 7 — chaotic vibrations, the first LE χ1 > 0, whereas the remaining are negative (χ2 < 0); (iii) P0 = 27 — chaos–hyperchaos; the first and the second LE is positive (χ1 > 0, χ2 > 0), and the remaining ones are negative (χ3 < 0); (iv) P0 = 47 — chaos–hyperchaos–hyper–hyperchaos; first, second and third LEs are positive (χ1 > 0, χ2 > 0, χ3 > 0); the remaining one is negative (χ4 < 0). Analysis of the mentioned characteristics implies that periodic vibrations are characterized via the Poincar´e map by a point, whereas the Poincar´e cross sections for P0 = 7 shows two independent attractors localized in different pools of the phase space without any intersection. These attractors are robust since they preserve their structure and statistical properties for small changes of the control parameters. The associated power spectrum has a broadband basis with two extrema on the excitation frequency and first subharmonic. The Poincar´e map exhibits a symmetry regarding a diagonal, and the points are concentrated in two fundamental groups. Transition of the system into chaos–hyperchaos state yields a birth of the unified SA composed of both previous chaotic regimes. Independently of the initial condition choice, the unified attractor attracts trajectories of beginning in initial points, which means that the separatrix plane does not exist more. In other words, creation of unified attractor implies unification of the attraction pools of both previously separated attractors. In the Poincar´e map “splashes” appear,

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which means that either a new stochastic set has been born or a development of a new stochastic set in directions previously being compressed. In the power spectrum, we also observe a broadband base, but exhibiting only one extremum associated with the excitation frequency. Movement along the parameter P0 up to P0 = 47 pushes the system into chaos–hyperchaos–hyper–hyperchaos state. Here more dense unification of the SA is presented, and on the power spectrum there is a lack of clearly manifested local extrema. The power spectrum corresponds to the white noise and has no local extrema. Detailed investigation of the time histories w(t) as well as other characteristics shows, that the system stays regularly on two attractors for P0 = 7. Transition of the system into chaos–hyperchaos state (P0 = 27) is realized via splashes of the positive deflections, i.e. the system exhibits the following dynamical state: stiff stability loss for negative deflections, vibrations around the equilibrium configuration and stiff stability loss for positive deflections. While the system transiting into chaos–hyperchaos state (P0 = 47), vibrations are fully chaotic, i.e. the “turbulent” phenomena are observed, and there is a lack of laminar splashes. Besides of the so far mentioned characteristics, each regime can be analysed using the AF (see Fig. 8.42).

Fig. 8.42

Autocorrelation function (AF) for various dynamic regimes.

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In the case of periodic regime the AF is periodic, whereas for chaotic regime the AF is fastly damped and tends to zero. Sets of finite and non-finite dimensions having chaotic dynamics include systems with 1.5 DOFs on one end, and partial differential equation of the Navier–Stokes type on the other end [Landford (1982)]. In between both ends there are dynamical systems with increasing number of DOFs, including also the von K´ arm´ an equations considered in this book. Investigation of the mentioned dynamical systems should yield an answer to the following fundamental problem: Does any border exist in the system chaotic dynamics, where on one side the system dynamics can be called chaotic, whereas from the other side we deal with a real turbulent dynamics? Observe that in the considered paradigm, the real turbulence does not mean any hydrodynamic phenomenon, but rather a common picture of the typical behavior of continuous systems [Landford (1982)]. 8.1.9

Reliability of chaotic zones

One of the important problems regarding existence of either hyperchaotic or hyper–hyperchaotic deals with their verification. We should be sure that the discovered nonlinear effects are not yielded by peculiarity of the difference scheme or by error of the introduced method of computation of the LEs. For the problem of mixed boundary conditions, the following numerical methods have been used: (i) Explicit method with different partitions of n, with different computational steps dt, and with approximation of spatial derivatives of orders O(h2 ), O(h4 ), O(h6 ), as well as the Runge– Kutta methods RK2, RK4, RK9 have been applied. (ii) Implicit method with different partition n regarding space, time steps dt, and with approximation of spatial derivatives of orders O(h2 ), O(h4 ), O(h6 ). (iii) Pseudo-spectral method with different partitions n, with different time step dt, and the Runge–Kutta methods RK2, RK4, RK9 have been used. Algorithm of estimation of the LE spectrum is applicable to the applied method being based on solutions to differential equations,

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and it assumes integration of additional vactors along a phase curve. Therefore, in order to verify the explicit method we have chosen a pseudo-spectral method. Further comparison of the results obtained by these methods for different partitions regarding space and time has been carried out. All results have been obtained via RK2 method with approximation of the space derivatives with accuracy of O(h4 ). The choice of these parameters has been illustrated and discussed earlier, and it seems that we achieved a compromise between the speed and accuracy of the computations. In Fig. 8.43 spectra of LEs have been presented, obtained via explicit and pseudo-spectral methods for different partitions regarding spatial and timing steps. In the figures there are given also scales, obtained via the same numerical method as that used for computation of LEs spectra. Explicit method yielded results A–D, whereas the pseudo-spectral yielded the resulated denoted by E, F, G. Parameters of computations follow: A — explicit method; spatial partition n = 8, time step dt = 2−8 , derivatives approximation O(h4 ). B — explicit method; spatial partition n = 16, time step dt = 2−8 , derivatives approximation O(h4 ). C — explicit method; spatial partition n = 16, time step dt = 2−9 , derivatives approximation O(h4 ). D — explicit method; spatial partition n = 32, time step dt = 2−11 , derivatives approximation O(h4 ). E — pseudo-spectral method; spatial partition n = 8, time step dt = 2−8 . F — pseudo-spectral method; spatial partition n = 32, time step dt = 2−11 . G — pseudo-spectral method; spatial partition n = 16, time step dt = 2−12 . Qualitative correspondence of the results obtained via various methods approves reliability of existence of hyper–hyperchaotic zones and allows to conclude that the observed phenomenon characterizes the dynamical system properly and does not depend on the applied difference scheme. One more important observation follows: maximum deflection curve wmax (P0 ) yields information regarding the

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Fig. 8.43

Comparison of LE spectra for different numerical methods applied.

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vibration character. This characteristic does not need any additional algorithms and it can be obtained in a rather simple manner. Simplicity in construction of a given characteristics allows to apply it from the beginning of the carried out analysis. More detailed picture can be obtained by adding additional “scale” characteristic constructed on the basis of FFT and LE spectrum. We have also compared a chart of vibrations with a chart of chaotic, hyper–chaotic and hyper–hyperchaotic zones, based on the LE spectrum (see Fig. 8.44). Observation of the charts yields

Fig. 8.44 zones.

Charts of vibrations: chaotic, hyperchaotic and hyper–hyperchaotic

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the following conclusion. Chaotic zone is the widest and zones of hyper–chaos and hyper–hyperchaos are located in it. The hyper– hyperchaos phenomenon occurs for large amplitudes of the external force for high excitation frequencies. Comparison of both charts allows to validate the carried out analysis of the basis of FFT, since zones with chaos correspond to zones with positive LE. It is obvious that charts coincide with each other. The first chart brings the additional information on the periodic vibrations, and it allows to divide zones with separate harmonic and subharmonic ω2 zones. Chart of the LE yields more detailed picture of chaotic zones putting emphasis on the chaotization strength that they are the parameters (P0 , ω). Coincidence of the borders of the fundamental zones allows to conclude that the analysis carried out on a basis of FFT yields the results being in agreement with other methods, and can be applied to estimate the general chart of the system dynamics. 8.1.10

Conclusions

This part yields results of the numerical computation of the vibrations of infinitely long flexible panels subjected to the external parametric load. At first, the problem of the reliability of the numerical results has been addressed. For the problem of simply supported beam we have applied the BGM in higher approximations for two physical processes — free vibrations and buckling phenomenon. Coincidence of the experimental and analytical results implies the results reliability. For the BGM it is required to estimate a number of the series terms to get the reliable results. The experience has shown that we may achieve only the integral convergence. Therefore, the whole further analysis is carried out on a basis of the vibrations power spectrum associated with the beam center. Owing to the so far described approach a methodology of the charts construction has been developed. Note that for the problem regarding the parametric transversal load the practical convergence of the BGM is achieved already for two terms of the series. The beam parametric vibrations have served as an example to study the charts of vibrations, the fundamental vibration regimes, and to investigate a transition into chaos and the

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system evolution within a chaotic dynamics. We have detected a transition into chaos through the Feigenbaum scenario, and we have traced eight period doubling bifurcations. In the chaotic zone we have distinguished Smale, Shilnikov and R¨ossler attractors. The BGM has been used to control the validity and reliability of the pseudo-spectral method on Chebyshev’s mesh. In particular, the problems regarding convergence for the different approximations, spatial partitions, and time steps have been addressed. Results yielded by all methods have been compared with each other on a basis of the integral 3D charts. Good correspondence of all results obtained through the applied methods has been presented. Particular attention has been paid to the problem of integral convergence of the difference scheme with respect to the approximation order of the difference derivatives, i.e. O(h2 ), O(h4 ) and O(h6 ). Optimality of the approximation O(h4 ) to solve PDEs governing problems in mechanics has been demonstrated. The problem of a usage of explicit versus implicit schemes has been outlined. FDM has been used to solve problems with the beam clamping. As in the case of the beam simple support, the period doubling route to chaos has been illustrated, and the system dynamical peculiarities in a chaotic regime has been investigated. A period tripling bifurcation has been also demonstrated. Besides the problem of parametric vibrations of the flexible panels of infinite length with the geometric nonlinearity, we have included into considerations also the physical material nonlinearity. We have considered two variants of the stress–strain relation, i.e. in the form of two piece-wise linear characteristic and in the form of the exponential transition into a plastic zone. For both variants the vibration charts for the problem of clamped beam have been constructed. In the case of the simply supported beam wider spectrum of the characteristics has been applied: maximum deflection, maximum LEs and scales. While solving the problems with physical nonlinearities, we have added a novel parameter — plastic zone. Owing to analysis of the plastic zone, a notion of plasticity global order of the beam has been introduced. Namely, we have investigated beam vibrations up to its transitions into a plastic zone.

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In this chapter part novel results of theory of nonlinear mechanical systems have been presented. At first, a route to chaos due to Sharkovsky’s smooth scenario has been illustrated and discussed. On the contrary to the Feigenbaum scenario, our dynamical system transits into chaos via a cascade of the period doubling bifurcation, and via the sequence governed by Sharkovsky’s order. Earlier the similar transitions have been discovered only for discrete maps, and hence the reported result is novel from a point of view of nonlinear dynamics. For the case with non-symmetric boundary conditions, we have detected the unique phenomenon not exhibited by other nonlinear dynamical systems. Namely, three LEs have got positive values (hyper–hyperchaos). Each of the regimes have been analyzed using the fundamental characteristics. Simultaneous analysis of LE, maximum deflection, vibrations of the beam center, phase portraits, power spectra, Poincar´e maps and the AF imply the increase in the chaotization process corresponding to the increase in a number of positive LE. We have paid attention to the reliability of the obtained results. Comparison of the results of the explicit method, with different space partition as well as the pseudospectral method on a Chebyshev’s mesh approves that the chaos– hyperchaos phenomenon is the real property of our studied system, and not the error introduced by the applied numerical method. The simultaneous analysis of the charts of vibrations and charts of chaotic, hyperchaotic and hyper–hyperchaotic zones has been carried out. Numerical experiments showed a larger efficiency of the applied numerical methods and the used algorithms. The obtained results are new from a point of view of nonlinear mechanics and they await experimental approvement.

8.2 8.2.1

Cylindrical Panels of Infinite Length Problem formulation

We consider elastic isotropic shells of infinite length, i.e. a shell material satisfies the Hooke’s law. In addition, we take into account the

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geometric nonlinearity, i.e. the relation between deformations of middle surface and displacements has the following form
 ∂u 1 ∂w 2 − kx w + . (8.149) εx = ∂x 2 ∂x Full deformations of an arbitrary point εzx is a sum of deformations in middle surface εx and bending deformation (εzx = εx + εx,u ), which owing to the Kirchhoff–Love hypothesis, is as follows ∂2w . (8.150) ∂x2 Let us consider a process of shell motion in time interval t0 and t1 . We compare different trajectories of system points between the initial t0 and final t1 positions. Real trajectories are defined by the following relation  t1 (δK − δΠ + δ W )dt = 0. (8.151) εx,u = −z

t0

Here K stands for the system kinetic energy, Π is the potential, whereas δ W is the sum of elementary works of the external forces. In the case when all forces acting on the system have a potential, Eq. (8.151) takes the form  t1 (K − Π)dt = 0, (8.152) δS = δ  t1

t0

where S = t0 (K − Π)∂t is the Hamilton action. After standard transformations, the non-dimensional counterpart form of equations regarding displacements follows ∂2u ∂w ∂w ∂ 2 w ∂2u + − k + p − = 0, x x 2 ∂x2 ∂x ∂x ∂x ∂t2
 ∂w ∂w ∂ 2 w 1 ∂ 4 w ∂w ∂ 2 u + + − kx − 12 ∂x4 ∂x ∂x2 ∂x ∂x ∂x2
  
∂2w ∂w 1 ∂w 2 ∂u − kx + + kx + ∂x2 ∂x ∂x 2 ∂x +q −

∂w 1 ∂2w = 0. −ε 2 12 ∂t ∂t

(8.153)

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The following relation between dimensional and non-dimensional quantities holds

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a 1 , k¯x = Rx λ

E E h λ3 p¯x , q = λ4 q¯, λ = , 1 − ν2 1 − ν2 a  (8.154) (1 − ν 2 )γ ¯ 2 t=a t, u = u ¯λ , x = a¯ x, Eg

px =

where parameters with bars correspond to non-dimensional quantities. In Eq. (8.153) the bars are already omitted. In relations (8.154) the following notation is applied: E is the elasticity modulus, ν is the Poisson’s coefficient, γ is the material density, g is the Earth acceleration, q is the transversal load (function of x and t), h and a are thickness and linear shell dimension, respectively, w and u stand for deflection and displacement of the middle surface, respectively, and kx = R1x is the shell curvature. System of PDEs (8.153) should be supplemented by boundary and initial conditions. Boundary conditions are as follows: 1. Pinned support u=w=

∂2w = 0, ∂x2

for x = 0; 1.

(8.155)

u=w=

∂w = 0, ∂x

for x = 0; 1.

(8.156)

∂2w = 0, ∂x2 ∂w = 0. u=w= ∂x

(8.157)

2. Fixed support

3. Pinned–fixed (mixed) support for x = 0 for x = 1

u=w=

Initial conditions for t = 0 follow: u = f1 (x),

u˙ = f2 (x),

w = f3 (x),

w˙ = f4 (x).

(8.158)

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8.2.2

441

Solution and its reliability

We approximate the partial derivatives regarding x in system (8.153) by difference relations with the error of O(h41 ) using the Taylor series in vicinity of the point xi of order h1 , where h1 is the partition step of x ∈ [0, 1]: Gh1

  1 . = 0 ≤ xi ≤ 1, xi = ih1 , i = 0, . . . , N, h1 = N

(8.159)

In this case PDEs (8.153) are reduced to second-order ODEs with respect to time for an ith point of the interval [0, 1]: u ¨i = Λx2 (ui ) − Λx (wi )(kx − Λx2 (wi )) + p(ih1 , t),
1 2 w ¨i + εw˙ i = λ − Λx4 (wi ) + Λx (wi )(Λx2 (ui ) 12  − Λx (wi )(kx − Λx2 (wi )) + (Λx2 (wi ) + kx )(Λx (ui ) − kx wi + 0.5(Λx (wi ))2 ) + q(ih1 , t).

(8.160)

The following difference operators are introduced: −(·)i+2 + 8(·)i+1 − 8(·)i−1 + (·)i−2 = Λx (·)i = 12h1




∂(·) ∂x



+ O(h41 ),

i

−(·)i+2 + 16(·)i+1 − 30(·)i + 16(·)i−1 − (·)i−2 12h21
2  ∂ (·) = + O(h41 ), ∂x2 i

Λx2 (·)i =

−(·)i+3 + 12(·)i+2 − 39(·)i+1 + 56(·)i − 39(·)i−1 + 12(·)i−2 − (·)i−3 6h41
4  ∂ (·) = + O(h41 ). ∂x4 i

Λx4 (·)i =

(8.161)

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The difference forms of the boundary and initial conditions follow 1. Pinned support

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ui = wi = Λx2 (wi ) = 0,

i = 0; N.

(8.162)

2. Fixed support ui = wi = Λx (wi ) = 0,

i = 0, . . . , N.

(8.163)

3. Mixed support (8.157) is a combination of boundary conditions (8.162) and (8.163). Initial conditions have the form ui = wi = Λx2 (wi ) = 0, i = 0; N (x = 0), ui = wi = Λx (wi ) = 0, i = 0; N (x = 1),

(8.164)

ui = f1 (ih1 ), u˙ i = f2 (ih1 ), wi = f3 (ih1 ), w˙ i = f4 (ih1 ). (8.165) System of second-order ODEs (8.160) is then transformed to a system of first-order ODEs, and then it is solved with the RK4 method. Numerical convergence of the method with respect to the space coordinate x and time t has been investigated. In Table 8.6 a time history w (0.5; t), phase portrait w (w) ˙ and power spectrum S (ω) for different partitions of spatial coordinate x and with respect to time t and action of harmonic transversal load q (x, t) = q0 sin ωq t, q0 = 500, ωq = 0.46 (kx = 48, ε = 0.1) are reported. Analysis of the given results shows that in order to get practically the exact solution it is sufficient to use interval [0, 1] partition with N = 16. Applying the Runge principle and observing the Table 8.6 Time histories, phase portraits and power spectra for different N (periodic vibrations).

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Table 8.7 Time histories, phase portraits and power spectra for different N (chaotic vibrations).

results given in Table 8.6, one may conclude that taking different N = 32; 64; 128 does not introduce any changes to the time histories w(0.5, t), phase portraits w(w) ˙ and power spectra S (ω). A similar investigation is carried out when the system exhibits a chaotic regime. In Table 8.7 the same characteristics as in Table 8.6 for q0 = 3,500 and ωp = 0.46 are presented. It is clear that though the signals for different partitions do not coincide, their integral characteristics, phase portraits and power spectra practically coincide. Therefore, the further numerical computations have been carried out for N = 32. 8.2.3

LEs

In order to detect particularities of the cylindrical panel vibrations we construct the charts on control parameters {q0 , ωp } plane (Figs. 8.45 and 8.46). The method of charts construction has been based on the power spectrum analysis and the largest LE. Figure 8.45 reports the chart in full, whereas in Fig. 8.46 only its part marked as A

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Fig. 8.45

Charts of panel vibrations.

is shown. In order to construct the chart shown in Fig. 8.45 the plane of {q0 , ωp } parameters have been divided by a mesh of steps {4, 0.00115}. Application of the LE computation plays an important role in theory of Hamilton and dissipative dynamical systems, since they allow to measure stochasticity magnitude. In addition, there exist a dependence of LE on other dynamical characteristics like the Kolmogorov entropy or a fractal dimension. It should be emphasized that LEs characterize the averaged velocity of the exponential divergence of the neighborhood trajectories (see [Wolf et al. (1985)]). We follow here the method developed by Benettin et al. [Benettin et al. (1976, 1980)]. We transform Eq. (8.161) to its normal counterpart form: dwi dui = Ui , = Wi , dt dt

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Charts of panel vibrations (window A).

dUi = Λx2 (ui ) − Λx (wi )(kx − Λx2 (wi )) + p(ih1 , t), dt
1 dWi = −εWi + λ2 − Λx4 (wi ) + Λx (wi )(Λx2 (ui ) dt 12  − Λx (wi )(kx − Λx2 (wi )) + (Λx2 (wi ) + kx )(Λx (ui ) − kx wi + 0.5(Λx (wi ))2 ) + q(ih1 , t), i = 1, . . . , N − 1. (8.166) For the given problem the method of finding all 4(N −1) LE relies on the following approach. We take orthonormal system of 4(N − 1) vectors of dimension 4(N − 1):   0  0 (8.167) v , v  = 1, (v 0 , v 0 ) = δij , i, j = 1, . . . , 4(N − 1) . i

i

i

j

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For simplicity, we take the following vectors:

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(1, 0, . . . ), (0, 1, 0, . . . ), (0, 0, 1, . . . ), . . . , (0, 0, 0, . . . , 1).

(8.168)

Along a trajectory of the system motion we compute a differential of the map with a sufficient step, given by Eq. (8.167), regarding each of vectors (8.168). Then, the obtained vectors {¯ vik , i = 1, . . . , 4(N − 1)} are normalized via the Gramm–Schmidt procedure     i−1   k    v ¯  k k 1 k k k k k k v1  , v1 = k , αi =  v¯i − (vj , v¯i )vj  α1 = ¯  , α1   j=1 (8.169) &i−1 k k k k v¯i − j=1 (vj , v¯i )vj , i = 2, . . . , 4(N − 1). vik = αki Finally, LEs are defined by the following formula k 1  ln αji . (8.170) λi = lim k→∞ kδt j=1

In what follows we investigate a problem of dynamical stability loss of the shell subjected to harmonic load and uniformly distributed along the shell surface, whose natural frequency is ω0 = 0.46. In Fig. 8.47 the dependence wmax (q0 ) for the shell center (x = 0.5) — point A and for (x = 0.75) — point B are shown. In addition, two characteristics λ1 (q0 ) for the shell center (x = 0.5) — point A are reported. Besides, in Fig. 8.47, in window E, there is the vibration scale part versus q0 and λi (q0 ), (i = 1, 2, 3) regarding the interval 4,000 ≤ q0 ≤ 5,000. In the scale showing the vibration type versus q0 zones a, b, c, d, e, f are distinguished, which should be analyzed in a more detailed manner (we deal with so-called peculiar zones). In the interval 0 ≤ q0 ≤ 1, 247.3223 the shell exhibits periodic vibrations with ω0 and ω0 /2, transiting into a zone of bifurcations. Period doubling bifurcations have been monitored up to q0 = 1,247.3223 (see Table 8.8). Increasing q0 up to 5 · 10−5 the mechanical system is transited into chaos owing to the Ruelle–Takens–Newhouse scenario on the two frequencies for q0 = 1,247.32235 (see point A). Owing to the Ruelle–Takens–Newhouse theorem, chaos appears after two Hopf bifurcations, and then a SA exhibiting a complex topology and bounded by non-smooth manifolds occurs.

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Dependence wmax (q0 ) and λi (q0 ), i = 1, 2, 3.

In zone a, the first-order discontinuity in λ1 (q0 ) is observed, and λ1 > 0, λ2 < 0, λ3 < 0. However, a sudden increase in beam deflections is not observed. Further increase in q0 pushes the system into a deep chaos for q0 = 1,256 — signal w (t) exhibits chaotic character, power spectrum has a broadband base, whereas the phase portrait presents a black spot. The change of q0 on the increment of 0.615 (q0 = 1,256.615) pushes the shell vibrations into an ordered deterministic chaos, which is built via thirteen-fold period doubling bifurcation (λ1 > 0). The change of q0 on 5 · 10−3 (q0 = 1,256.62) yields more ordered vibrations exhibited by the signal, phase portrait and power spectrum. A particular feature of the system exhibited by a narrow window of bifurcations of the mentioned type (λi (q0 ) < 0, (i = 1, 2, 3)) is reported. Further increase in q0 on the amount of 0.4 causes the occurrence of the Hopf bifurcation (q0 = 1,260, λi (q0 ) < 0, i = 1, 2, 3), and hence the Sharkovsky order of 2·13 takes place. SA illustrated by Poincar´e map begin to increase, and their number increase too. For q0 = 1,260.5 (i.e. increasing q0 an amount of 0.5), the shell transits again into the chaotic regime on the frequencies of the last Hopf bifurcation, i.e. Sharkovsky’s order of

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Table 8.8 Time histories, phase portraits, Poincar´e maps and frequency power spectra of the shell centre (zones a, b) for different q0 .

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2·13 is exhibited. Step by step the chaos is transited into deep chaos, which is manifested by signal w (t), dark spot in the phase portrait as well analogous picture of the power spectrum as it happened for q0 = 1,256 (λ1 > 0, λ2 < 0, λ3 < 0). The so far given description is related to zone b (see Table 8.7). Maximum shell deflection in zones a and b is practically the same, i.e. no sudden increase in shell deflection is observed in comparison to that regarding the static + = 1,256 estimates the occurrence of the stability loss. The value q0,1 first dynamical critical load. 8.2.4

Chaos–hyperchaos transition

We investigate the evolution of the vibrational regime of the shell increasing amplitude q0 of the harmonic load. It should be mentioned that the problem of chaotic attractor evolution with the increase in the parameter q0 is rather rarely investigated. Our investigations allowed to discover rich nonlinear dynamical features of the investigated shell, including that the smoothing of the power spectrum is bounded due to the system characteristics, in particular, due to a number of governing differential equations, the attractor dimension computation, as well as synchronization processes. It is known that qualitative changes in a chaotic structure can already appear in systems with 1.5 DOFs. In the theory of shells, though DL (LD) [Awrejcewicz et al. (2004)] is computed via LPE, there is no unique relation between the signature of the LPE spectrum and the DL of a SA. Intuitively, occurrence of qualitative changes in the physical characteristics of an attractor through the birth of an additional LE is more realistic than the analogous reconstruction using the concept of DL . In fact, in the case of occurrence of an additional positive LE there is a new unstable direction on the system trajectory, which implies the qualitative new system dynamics. On the other hand, the LD defines the averaged local volume preserving its quantity, i.e. neither compressed nor extended. Observe that the LD includes not only positive and zero exponents, but also negative ones. Though the latter ones may be responsible for stability of any motion type, for the steady-state motion they do not influence the system dynamics,

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since they rather define only metric properties of the attractor. It has been shown in the reference [Wolf et al. (1985)], that in general negative LE cannot be estimated through physical experiments approving our remark that they do not influence the physical characteristics of dynamical regimes. In what follows, we consider transitions into chaos while changing patterns of the LE spectrum, as well as transitions of system dynamics with one positive LE into chaos with two positive LEs, i.e. into hyperchaos. The latter transitions within chaotic regimes are referred as chaos–hyperchaos transitions. The so far described phenomenon for the case of flexible plates subjected to harmonic load has been reported in reference. In Table 8.9, time history w(t), w(w), ˙ Poincar´e map wt (wt+T ) and the power spectrum of the central shell point driven harmonically by q0 sin(0.46t) are reported. The mentioned characteristics are (1) (2) (3) given for three values of q0 = 4,648, q0 = 4,652 and q0 = 4,656, and the mentioned loads are denoted by area C on the dependence λ (q0 ) in Fig. 8.47. For q01 : λ1 > 0, λ2 < 0, q02 : λ1 > 0, λ2 > 0, q03 : λ1 > 0, λ2 < 0, i.e. when we deal here with three equilibrium configurations. Namely, for q01 and q03 the system is in a chaotic regime, whereas for q02 the unification of both attractors takes place Table 8.9 Time histories, phase portraits, Poincar´e maps and frequency power spectra of the shell center (chaotic vibrations) for different q0 .

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in the center of wt (wt+T ). In the chaos–hyperchaos regime, λ1 and λ2 achieve their maximum positive values, whereas λ3 practically remains non-affected. Deterministic Chaos in One-Dimensional Continuous Systems Downloaded from www.worldscientific.com by CHINESE UNIVERSITY OF HONG KONG on 07/02/16. For personal use only.

8.2.5

On the Sharkovsky’s periodicity

One of the key problems in the theory of turbulence reduces to the following one: How to predict the beginning of its occurrence from a condition of stability and equilibrium? Feigenbaum begun his investigations from the analysis of intervals between period doubling bifurcations of the square map. This map was first investigated in 1845 by P. Verhulst, who studied ecosystems, and hence the diagram y = x2 +c is named as Verhulst’s diagram. The fundamental result obtained by Feigenbaum refers to its universal meaning. The analyzed mechanism called a route to chaos via period doubling bifurcations occurs not only for iterations cx (1 − x), but also, in the case of mappings into itself: x2 +c, c sin (πx) and cx2 sin πx defined on certain intervals. Diagram of orbits shown in Fig. 8.48 allows to distinguish the attractive periodic orbits for the functions fc (x) = x2 + c. Observe that in certain parts the diagram is cut off. For instance, for c ≈ 1.75 a white strip is visible, and the attracting orbits are of period-3. The natural question appears: Do other periodic orbits exist? The latter ones should be repellers, since the diagram presents only attractive orbits. It has been detected that the occurrence of the orbits with period-3 implies an occurrence of orbits with periods n = 1, 2, 3, . . . . In 1975, Lie and Yorke [Li and Yorke (1975)] considered orbits with period-3, but it happened that they studied only a particular case of Sharkovsky’s theorem published in 1964 (see Section 2.4). Let us compare two maps fc (x) = x2 + c and fc (x) = z 2 + c, i.e. the logistic map in real and complex plane, and constructed LE λ1 (c) for the map fc (x) = x2 + c (see Figs. 8.48, 2.9 and 2.10). The diagrams of the orbits for two functions have attracting, repelling and neutral points, which correspond to stable, unstable and neutral equilibrium states. When starting from the vicinity of a fixed point, if we approach it via the infinite interations process, this point is called an attractor.

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Fig. 8.48 Bifurcation diagram of the map fc (x) = x2 + c, and zone of period-3 of the Mandelbrodt’s set.

Now, starting from a close neighborhood of the repelling fixed point, we go away from it. Neutral fixed point is characterized by the following: starting from its vicinity, we will remain in its neighborhood neither approaching nor getting away from it. In order to define stability of a fixed point z¯ of the mapf (z), we z )| < 1, then z¯ is attractive, if |f  (¯ z )| > 1 need to compute f  (z): if |f  (¯ z )| = 1, then the point is neutral. then it is a repeller, and if |f  (¯ For the rational maps, owing to the Sharkovsky theorem, there are cycles with all orders n = 2, 3, 4 . . . . A point (or points) of a complex plane is called an attractor, when the process of iterations zn+1 = f (zn ) , n → ∞ takes place. In some cases, a few attractors may exist, or they may constitute infinite number of points and they may present a continuous curve or other set, for instance Cantor set. Mandelbrot set given in Figs. 2.8–2.10 has a particularly complex structure. Each complex number c either belongs to the Mandelbrot set (M ) or it does not. For ∀c ∈ M , the set is compact.

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In Fig. 8.48, certain parts of the Mandelbrot set fc (z) = z 2 + c > 0 are shown. Diagram fc (x) = x2 + c [Fig. 8.48(a)] exhibits points evolution on the real axis of a Mandelbrot’s set. Each bifurcation corresponds to new zones, which intersect axis x, and a period in this case corresponds to the number of orbit branches. Point C∞ is called Feigenbaum point. In the diagram, between C = 1/4 and C∞ the period doubling process takes place when C → C∞ . For C > C∞ , we have chaotic zone, period-3 window occurs for C = −1/7548777 . . .. Occurrence of period-3 implies occurrence of orbits with periods n = 1, 2, 3, . . . In Fig. 8.48(b), the dependence of LE λ1 for the logistic map fC (x) = x2 + C is presented. Analysis of the mentioned figures shows that the chaotic regime is interrupted, where the sequence |fCn (x)| appears again in the periodic interval, which corresponds to λ1 < 0. The so far carried out analysis implies that construction of 1D maps for simple dynamical systems can exhibit similar bifurcational phenomena for transition of mechanical systems into a chaotic regime, as it has been found for our system with infinite number of DOFs. Let us study the application of Sharkovsky theorem to predict the occurence of periodic orbits for the Marger–Vlasov Eqs. (8.153), (8.154) with parameters kx = 48, ε = 0.01 under harmonic load excitation q = q0 sin ω0 t for the simple support along the shell contour (8.155) and for zero initial conditions (8.160). We take shell material as elastic, homogeneous and isotropic ν = 0.3; excitation frequency is equal to shell natural frequency ω0 = 0.46. We report time histories from the shell center and centers of its quadrants, phase portraits, power spectra and Poincar´e maps for the orbits predicted by the Sharkovsky’s theorem. In Table 8.10, Sharkovsky’s orders 3, 5, 7, 9, 11, 13, in Table 8.11 — 2·3, 2·5, 2·7, 2·9, 2·11, 2·13 and finally in Table 8.12 — 21 , 22 , 23 , 24 , 25 , 26 are reported. Besides the given dependencies w (0.5; t), w (w), ˙ wt (wt+T ) and S (w), in Tables 8.10–8.12 modal characteristics w (∂w/∂x ) for the shell point x = 0.375, surfaces and Sharkovsky’s exponents (SE) versus q0 are shown. It should be emphasized that the so-called Sharkovsky’s orders do not appear step by step in our investigated shell, but they can be appropriately withdrawn from the whole plane of control parameters {q0 , ω0 }

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Table 8.10

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Table 8.11

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Table 8.12

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Table 8.12

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(Tables 8.6 and 8.7). The following nonlinear phenomena have been detected: 1. In the case of period tripling, time history exhibits period partition into three equal parts, for 5–5 equal parts, and so on. Poincar´e map consists of 3; 5; 7; 9; 11; 13 points. For orders 2 · 3, 2 · 5, 2 · 7, 2 · 9, 2 · 11, 2 · 13 we have groups consisting of 2 points each. Phase portrait shows period doubling. Observe that localization of points of Poincar´e map is ordered for orbits with periods 3; 2 · 3; 9; 2 · 9; 13; 2 · 13, whereas for orders 5; 7; 11 and 2 · 5; 2 · 7; 2 · 11 a reconstruction takes place, and the points are located on the phase space in an arbitrary manner. The given orbits present windows of periodicity within chaos, and their structure is the same in the whole set of the control parameters {q0 , ωq }. All windows of periodicity have negative LEs λi < 0, i = 1, 2, 3. 2. For each orbit described so far (point 1), the changes in shell deflections in time w(x, t) (0 ≤ x ≤ 1; 127,500 ≤ t ≤ 128,000) are shown, which allows to study a chart of the shell deformation depending on the orbit type 3; 5; 7; 9; 11. The increase in the period implies a transition into spatio–temporal chaos of the shell.

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Chapter 9

Plates and Shells Section 9.1 deals with shallow shells and plates with initial imperfection. The mathematical model is derived, and governing PDEs are reduced to ODEs. Reliability of the obtained numerical results is studied. Chaotic vibrations of spherical and conical shells with constant and variable thicknesses have been examined. In particular, the spatio-temporal chaotic dynamics and control of chaos have been analyzed. Vibrations of flexible axially-symmetric shells are studied in Section 9.2. The mathematical model has been derived, and supplemented with boundary and initial conditions. The FDM and computational algorithm are presented. The method of relaxation has been described and then applied to study the shell vibrations. Problems devoted to dynamical stability loss including stability criteria, and scenarios of transition from periodic to chaotic vibrations are analyzed. Then, the shell vibrations under non-uniform harmonic excitation are studied including discussion regarding the existence of Sharkovsky’s periodicity series. Control of shell chaotic vibrations using the continuous harmonic local force and the harmonic torque is addressed. Then, the wavelet-based analysis of chaotic shell vibrations is presented. Numerous novel chaotic phenomena are detected and studied.

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9.1.1

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Plates with Initial Imperfections Mathematical model and solutions algorithm

In this chapter, we consider a shallow elastic shell (Fig. 9.1), which can be treated as a plate with initial imperfection (this will be discussed later), located in a closed 3D space of R3 with curvilinear system of coordinates α, β, γ [Kantor (1971)]. We assume that Lam´e parameters A, B and radii R1
, R12 , R2
of the shell mean surface are continuous with their first derivatives regarding the functions α, β. In the given coordinates, the shell as a part of the 3D space is defined as follows: Ω = {α, β, γ/(α, β, γ) ∈ [0, a]×[0, b]×[−h /2 , h /2 ]}. The derived PDEs governing nonlinear dynamics of shells (Fig. 9.1) are obtained on the basis of the following hypotheses: shell fibres are one-layer, made of an isotropic, homogeneous and elastic material, and the Kirchhoff–Love hypothesis holds. The associated variational equation has the following form [Kantor (1971)]:

   D (∆w) ¯ 2 − (1 − ν)L(w, ¯ w) ¯ δ 2 S¯    1 ¯ ¯ ¯+w ¯0, F w ¯ − ∆k F + L w 2

Fig. 9.1

Computational scheme of a shallow shell.

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 1  ¯ 2 ¯ ¯ ∆F − (1 + ν) L F , F − ds 2Eh 

 h ¯ ¯ (w ¨ + εw) ˙ δwd¯ ¯ s = 0. q− − γg S¯

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(9.1)

where   ∂ B ∂ ∂ A ∂ 1 , + ∆= AB ∂ α ¯ A ∂α ¯ ∂ β¯ B ∂ β¯  ∂ 1 ∂ ∂ 1 B ∂ ∂ 1 ∂ 1 + ¯ + ∆k =
¯ AB ∂ α ¯ R 1 A ∂α ¯ ∂α ¯ R12 ∂ β ∂ β R12 ∂ α ¯  ∂ 1 A ∂ , + ¯
∂ β R 1 B ∂ β¯ ∂ F¯ 2 ∂2w ∂2w ¯ ∂ 2 F¯ ¯ ¯ ∂ 2 F¯ ∂2w · + · − 2 · . L(w, ¯ F¯ ) = ∂α ¯ 2 ∂ β¯2 ¯2 ∂α ¯ ∂ β¯ ∂ α ¯ ∂ β¯ ∂ β¯2 ∂ α In order to solve Eq. (9.1), in which the deflection function w ¯ and stress function F¯ are independently variated, we cannot apply the Ritz procedure directly (equation does not have the form of functional variation being equal to zero). In order to find the approximated value of elements w ¯ and F¯ , we take the coordinate sequence wi (α, β) and ϕi (α, β), satisfying the same requirements as Eq. (9.1). In order to find w ¯ and F¯ , the systems of functions {ϕij (x, y), ψij (x, y)}, i, j = 0, 1, 2 . . . , should satisfy the following five requirements: 1. ϕij (x, y) ∈ HA , ψij (x, y) ∈ HA , where HA is a Hilbert space, which is called the energetic space; 2. ∀i, j functions ϕij (x, y) and ψij (x, y) are linearly independent, continuous with their partial derivatives to fourth-order inclusively in the space Ω; 3. ϕij (x, y) and ψij (x, y) satisfy the boundary conditions; 4. ϕij (x, y) and ψij (x, y) are compact in HA ;

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5. ϕij (x, y) and ψij (x, y) should represent M first elements of the full system of the functions:

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w ¯=

Mx i=0

¯ x ¯i (t¯)wi (¯ α, β),

F¯ =

Mx

¯ y¯i (t¯)ϕi (¯ α, β).

(9.2)

i=0

The approximating solutions have coefficients x ¯i (t¯) and y¯i (t¯), which are time-dependent functions. Substituting relations (9.2) into Eq. (9.1), carrying out the variational operation, and comparing to zero terms standing by δ¯ xi , δy¯i , we get the following system of ODEs ¯ ¯ for x ¯i (t) and y¯i (t): xk + εx˙ k ) + Bik xk + Cip yp + Dikp xk yp = Qi q0 , Kik (¨ 1 Cpi xi + Epj yj + Dpki xk xi = 0, 2 i, k = 1, 2, . . . , n; p, j = 1, 2, . . . , m.

(9.3)

In the polar coordinates with axial symmetry w = w(r), ϕ = ϕ(r), α = r, β = θ, ds = 2πrdr, and the operators take the form ∆=

1 d d2 , + 2 d¯ r r¯ dr

d2 w 1 dw ¯ d2 F¯ ¯ 1 dF¯ L(w, ¯ F¯ ) = + · 2. ·
d¯ r 2 r d¯ r r d¯ r d¯ r

(9.4)

Substituting r¯ by a¯ ρ in operators (9.4), and carrying out the standard transformations (and after division by 2πEh50 /a4 ), the system is transformed to its counterpart non-dimensional form. In order to reduce Eq. (9.3) to the non-dimensional forms, the following quanϕ y¯i tities are introduced: w ¯ = w/h, x ¯i = xi /h, ϕ¯ = Eh 3 , yi = Eh3 ,

2 2 ¯ ¯ ¯ = ε/τ , τ = a a γ , h = h(ρ) ¯ , h = h(0), F = Eh F , t = tτ , ε h Eh40 q a4 ,

h0

Eg

where w is the deflection, F is the stress function, t is q¯ = time, ε is the damping coefficient, a is the dimension of the square shell, respectively; h is the thickness of the shell, g is the Earth acceleration, γ is the material weight density, ν is Poisson’s coefficient for the isotropic material (ν = 0.3), E is elasticity modulus, w0 is the initial imperfection. Next, bars over the non-dimensional quantities are omitted.

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In the case of axially symmetric deformation of the shallow rotation shell of thickness h = h0 (1 + cρ), the coefficients of system (9.3) take the following form
1 (1 + cρ) wi wk ρdρ, Kik = Nik =

1 12 (1 − ν 2 )

0




1

0

(1 + cρ)3 [∆wi ∆wk − (1 − ν) L (wi , wk )] ρdρ,


Cip = −
Dikp = −
Ejp = −

1 0

1 0

[∆k φp + L (w0 , φp )] wi ρdρ,


1

0

wi L(wk , wp )ρdρ,

Qi =

1 0

wi ρdρ,

1 [∆ϕj ∆ϕp − (1 + ν) L (ϕj , ϕp )] ρdρ. 1 + cρ (9.5)

Solving the second equation of system (9.3) for yi , we get    1  −1 −1 E Dpi xi xs . yi = Ejp Cps + 2 jp s

(9.6)

Multiplying by K −1 the first equation of (9.3) and using notation x˙ i = ri , the problem is reduced to the first-order ODEs of the form 

 −1 Cij + A−1 D x · yj r˙i = −ε ri + Kik s ks ik j −1 −1 −Kik Bks xs + q0 (t )Kik Qk ,

(9.7)

x˙ i = ri , i, k, s = 1, 2, . . . , n;

p, j = 1, 2, . . . , m.

The so far introduced transformation has been possible since −1 −1 and Ejp exist if the coordinate functions are linmatrices Kik early independent. Equations (9.7) with the initial conditions xi = 0, x˙ i = 0 for t = 0, have been solved with the fourth-order Runge–Kutta method. We consider further the axially symmetric deformation of closed shallow rotational shells and circled plates subjected to uniformly distributed load being normal to the mean shell surface. In polar coordinates and in the case of axial symmetry we have: w = w(ρ),

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Fig. 9.2

Surfaces of structural members: (a) plate; (b) cone; (c) sphere.

F = F (ρ), α = ρ, β = θ, ds = 2πρdρ, and the thickness is defined by the function h(ρ) = h0 (1 + cρ). The mean shell surface is defined by the initial deflection w0 = −hR 1 − c1 ρ − c2 ρ2 , K = hH0 , where H is the full shell height over a plane (see Fig. 9.2). For c1 = 1, c2 = 0 we get a conical shell [Fig. 9.2(b)]; c1 = 0, c2 = 1 corresponds to a sphere [Fig. 9.2(c)]; for k = 0 we deal with a plate [Fig. 9.2(a)]. Approximating function for four types of boundary conditions are shown in Table 9.1. In order to investigate vibrations of a shallow conical shell, we consider it as a plate (∆k ϕ ≡ 0) with initial deflection: w0 = −k(1 − ρ ), k = H /ho , and we apply the coordinate functions given in Table 9.1. Each of the formulas of (9.5) can be presented by a sum of integrals of the following form


1

I(x, y) = 0

ρx (1 − ρ2 ) dρ = y

(2y)!!(x − 1)!! , (x + 2y + 1)!!

(9.8)

and for four types of the boundary conditions applied, the coefficients of the system (9.7) take the form:

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Table 9.1

465

System of approximating functions for boundary conditions.

1. Unmovable clamping (4 + 2i + 2k)!! 1 (1) +c , Kik = 6 + 2i + 2k (7 + 2i + 2k)!!   1 ik 4(i + 1)(k + 1) (1) Nik = 2 3(1 − ν ) i + k + 1 (i + k)(i + k + 1)   6ik 1+ν 3c2 − + 2(i + k + 2) (i + k)(i + k − 1) 2 +

 3c(2i + 2k − 4)!! [15ik − (1 + ν)(i + k)(i + k − 1)] , (2i + 2k + 3)!!

(1)

Cip = −2(p + 1)  
1
1

H 2 i+p+1 2 2 i+p (1 − r ) dr + 2p r 1−r dr , − · h0 0 0 (1)

Ejp = 4ip [(j + p − 1) (1 + ν) − 2jp]   c c2 1 − , + × j + p − 1 j + p − 1 /2 j + p (1)

Dikp = 4 (i + 1) (k + 1) p Qi =

1 . 2(i + 2)

p! (i + k + 1) . . . (i + k + p + 1) (9.9)

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2. Movable clamping (2)

(1)

(2)

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(2)

Cip

(2)

Ejp

(1)

Nik = Nik ,
1 H i+1 (2) (1) = 2 · p(2p − 1) · r 2p−2 · (1 − r 2 ) dr Q i = Qi , h0 0  pj = −16 (j + 1) (p + 1) (j + p + 1) (j + p) (j + p − 1)   c2 6jp 1−ν + − 2 (j + p + 1) (j + p + 2) (j + p) (j + p − 1) 2  (2j + 2p − 4)!! [15jp − (1 − ν) (j + p) (j + p − 1)] , −c (2j + 2p + 3)!!

(2)

Dikp = −

4 (i + 1) (k + 1) (p + 1) . (i + k + p + 1) (i + k + p + 2) (9.10)

3. Unmovable simple support (3)

(1)

Kik = Ki−1,k−1

(3)

Nik

(3)

 2 + c (4 + 3c)   , k = i = 1,     6 (1 − ν) c (4 + 5c) = , i = 1, k = 2; i = 2, k = 1, −   15 (1 − ν)     (1) Ni−1,k−1 ,

(1)

Cip = Ci−1,p ,

(3)

(1)

(3)

Dikp = Di−1,k−1,p ,

Qi

(1)

(3)

= Qi−1 ,

(1)

Ejp = Ejp . (9.11)

4. Movable simple support (4)

(1)

(4)

(2)

Kik = Ki−1,k−1 , Dikp = Di−1,k−1,p ,

(4)

(3)

Nik = Nik , (4)

Qi

(1)

= Qi−1 ,

(4)

(2)

Cip = Ci−1,p , (4)

(2)

(9.12)

Ejp = Ejp .

As it has been already mentioned, once we investigate the spherical shell we treat it as a plate with the initial deflection w0 = −k(1 − r 2 ). For four types of boundary conditions shown in Table

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9.1, the coefficients (9.7) differ from the case of the conical shell only by Cip : (1)

(1+i)!p! . 1. Unmovable clamping: Cip = −4 hH0 p (i+p+1)!

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(2)

(i+p)! . 2. Movable clamping: Cip = 4 hH0 (i + 1) (p + 1) (i+p+2)! (3)

(1)

3. Unmovable simple support: Cip = Ci−1,p . (4)

(2)

4. Movable simple support: Cip = Ci−1,p . The transversal uniformly distributed harmonic load is q = q0 sin(ωp t), where q0 is the amplitude of the harmonic excitation and ωp is a frequency of the excitation. 9.1.2

Results reliability

In order to study PDEs governing dynamics of the mentioned structural members, we introduce mode shape functions and obtain ODEs of infinite dimension. In order to get correct interpretation of the obtained results, the following remarks should be taken into account. When we investigate any continuous system, instead of infinite set of ODEs, we take a truncated system of finite dimension. It is assumed that by increasing a number of equations, we find a threshold beginning from which a further increase in the number of equations does not yield anything new in the system behaviour. This approach is also motivated by an occurrence of finite dimension of the system attractor. However, it may happen that an improper choice of basis functions, which serve to reduce PDEs to ODEs, effects the corresponding system of ODEs, which may have attractors different from that of the original system. This feature may occur, for instance, in the case of a 2D equation governing dynamics of heat convection. The Lorenz system [Lorenz (1963)], presenting a three-mode truncation of the derived approximated PDE, demonstrated complex dynamics including chaos. However, an increase in the number of modes yields first an irregular increase in chaos, and hence its decrease. For sufficiently large number of modes, chaos vanishes. In the work [Curry et al. (1984)] it has been shown that for large Prandtl numbers δ in the considered 2D

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Boissinesque convection there are critical values of the Rayleigh number Ra implying two- and three-mode vibrational motion, whereas with further increase in Ra, the system exhibits a periodic onefrequency convection. The illustrated example shows that in order to get qualitatively true correspondence between the original and truncated dynamics obtained by the use of either Bubnov–Galerkin or Ritz approaches, we need to take into account a sufficient number of modes. Let us investigate the problem of estimating the number of modes in the Ritz procedure using an example of vibrations of spherical and conical shallow shells being geometrically nonlinear and having a constant or non-constant thickness, and being bounded by their contour. The applied load (uniformly distributed along the shell surface) has the following form q = q0 sin(ωp t).

(9.13)

We consider the vibration charts associated with shells of k = 3 and k = 5 (Figs. 9.3 and 9.4, respectively) depending on the magnitude of control parameters {q0 , ωp } for different number of

Fig. 9.3 Charts of control parameters {q0 , ωp } of the conical shell for k = 3 and for different n.

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Fig. 9.4 Charts of control parameters {q0 , ωp } of the conical shell for k = 5 and for different n.

partition terms n = 1 − 7. Further increase in the number n in (9.3) has not changed the charts {q0 , ωp } qualitatively. For n = 1 (Fig. 9.3), the chart differs from the remaining ones, i.e. it presents only bifurcation zones and harmonic and sub-harmonic vibrations with ωp and ωp /2, without any chaotic zones. Increasing n yields new zones of bifurcation and chaos. Similarly, the chart (k = 5) for n = 1 strongly differs from the remaining ones, since an increase in n yields different zones becoming similar, i.e. the converging sequence of vibration character is observed. For instance, the sub-harmonic zone is the same for all n ≥ 2, but for n = 2 it is shifted to the right. Chaotic zones become smaller while increasing n, but the separated parts do not change starting from n = 4. Both cases of k = 5 and k = 3 exhibit better convergence for high frequencies than low frequencies, and for frequencies located in the neighborhood of the natural frequency. We consider two points: the first one for k = 3, n = 6, 7 is located in a bifurcation zone (Fig. 9.5); the second one, for k = 5, n = 6, 7 is

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Fig. 9.5 Dependence w(0, t), t ∈ [50; 53] and S(ω) versus n for the conical shell (k = 3) with constant thickness (movable clamping).

in a chaotic zone (Fig. 9.6). In Fig. 9.5, w (0, t) for 50 ≤ t ≤ 53 and power spectra (S(ω)) are reported. Analysis of the given results in Fig. 9.5 shows that beginning from n ≥ 4 dependences w(0.5 : t) are close to each other, whereas power spectra coincide in full. Results obtained for n = 1, 2, 3 differ essentially from those of n ≥ 4. Hence, we may conclude that beginning from n ≥ 4, the process of bifurcations is reliable for k ≤ 3, i.e. there exists a convergent sequence, which can be modelled in the following way     n n xi (t)wi (ρ) = min , ϕ0 − yi (t)wi (ρ) = min , w0 − i=1

t∈[50,53]

i=1

t∈[50,53]

(9.14)

and this is the best approximation to w0 and ϕ0 in the metric HA .

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Fig. 9.6 Dependence w(0, t), t ∈ [50; 53] and S(ω) versus n for the conical shell (k = 5) with constant thickness (movable clamping).

For n = 4, 5, 6, 7, the power spectrum exhibits a period tripling bifurcation, and the attracting orbits have period-3. Occurrence of period-3 orbits yields the occurrence of orbits with periods n = 1, 2, 3 . . . . The mentioned remarks are applicable to both real functions and maps of an interval into itself. Here, we aim at an analysis of other orbits, which are exhibited by dynamics of the flexible conical shells. In Fig. 9.6, the same characteristics as in Fig. 9.5 are shown. However, in this case we do not observe the previously exhibited uniform convergence. For n = 1, we have periodic vibrations; n = 4 corresponds to the period tripling bifurcation (period-3), where period-1 is defined by 2π/ω; for n = 2, 3, 5, 6, 7 though we have determined chaotic vibrations, we deal with different types of chaos. Namely, for n = 2

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we have chaos associated with excitation frequency. For n = 3, 5 we have chaos exhibiting mainly frequency 3ω, whereas for n = 6, 7 we have chaos associated with frequency 7ω. Owing to this discussion, we may conclude that a convergence of the Ritz procedure versus a number of the terms of the series (9.3) essentially depends on initial deflection parameter k, and on the dynamical regime of the system. In what follows, we investigate convergence of the Ritz procedure versus boundary condition type and a shell geometry using an example of conical shells supported along their edges and having constant (Fig. 9.7) and non-constant (Fig. 9.8) thicknesses (h = h0 (1 + cρ)) for c = 0.1, k = 5. We consider a point, which for n = 6, 7 is in a chaotic zone for the following fixed parameters: q0 = 2.4, ωp = 3.5.

Fig. 9.7 Dependence w(0, t), t ∈ [50; 53] and S(ω) versus n for the conical shell (k = 5) with constant thickness (movable simple clamping).

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Fig. 9.8 Dependence w(0, t), t ∈ [50; 53] and S(ω) versus n for the conical shell (k = 5) with the non-constant thickness (movable simple clamping).

In Fig. 9.7, signals (w (0; t), 150 ≤ t ≤ 156) and power spectra are reported. For n = 2, harmonic vibrations with ωp occur, n = 3 yields the first period doubling bifurcation, whereas for n = 1, 4, 5, 6, 7 we have chaos associated with the fundamental shell frequency. Next, we consider the shell with variable thickness and with the following fixed parameters: q0 = 2.4, ωp = 3.57. In Fig. 9.8, signals (w (0; t), 150 ≤ t ≤ 156) and power spectra are reported. For n = 2 and n = 3, periodic vibrations with the frequency ωp are shown; n = 1 yields sub-harmonic vibrations of ωp /5, i.e. the first approximation of (9.3) yields period-5 vibration, whereas for n = 4, 5, 6 chaos

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Fig. 9.9

Charts {q0 , ωp } of the spherical shell for different n.

occurs associated with natural frequency. In other words, beginning from n = 4, a convergent sequence is observed. Further, all results are reported for n = 6. 9.1.3

Spherical shells of constant thickness

We consider vibrations of spherical shallow shells with constant thickness, supported on their edges. In Fig. 9.9, charts of control parameters {q0 , ωp } for the shell with k = 1, 1.5, 2, 3, 4, 5, are reported. The chart showing a transition into chaos exhibits rich dynamics while increasing coefficient k. Analysis of dependence {q0 , ωp } versus k shows that for the plate (k = 0), {q0 , ωp } exhibits only periodic vibrations under the constraints w (0) ≤ 5 and q0 ≤ 100. Zones of chaos and bifurcations increase with the increase in k, and for k = 1 only two chaotic narrow islands within the harmonic zones appear. Increasing k ≥ 1.5 yields new zones of bifurcation and chaos.

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Considering deterministic vibrations of spherical and conical shells with constant and variable thicknesses for arbitrary boundary conditions and k parameter, none of the known models in its pure form can describe a transition to chaos of the studied continuous mechanical systems. The so far mentioned parameters, i.e. q0 , ωp , k play a key role in the mechanisms of transition of the mechanical system into chaos. For spherical moveably supported shells a new scenario of transition from periodic to chaotic vibrations has been detected, which exist also in the case of conic shells of both constant and non-constant thickness for the same boundary conditions. There appears a new linearly independent frequency and a transition into chaos is carried out via the series of linear combination of two frequencies and the successive Hopf bifurcations. We consider this scenario in detail on an example of a spherical shell with parameter k = 3. The fundamental . characteristic signal w (0, t), phase portrait w(w), power spectrum S(ωp ), Poincar´e section w(w(t + T )) are reported. In Table 9.2, the following notation is used: wi = w(t), wi+1 = w(t + T ), where T is the period of excitation. The reported q0 values are called threshold values, since in between the mentioned q0 values, the chart practically does not change. In what follows, we describe the dynamical phenomena reported in Table 9.2. 1. Vibrations are carried out with the frequency of excitation a1 and they are periodic. Phase portrait presents on invariant set of onerotational cycle (q0 = 15). 2. Further increase in the parameter q0 up to q0 = 15.92 implies an occurrence of the independent frequency b1 , i.e. there is twofrequency motion with frequencies a1 and b1 . Motion is not synchronized, i.e. ab11 = m n = 8.859 . . . is irrational. 3. Increase in q0 up to q0 = 17 yields the series of linearly dependent frequencies bn = n · b1 and an = a1 − (n − 1) b1 , and this process continues up to the moment, when the frequencies ak and bk ∈ [b1 , a1 ] start to approach each other. After that, in the spectrum, a third type of the dependent frequency is exhibited: cn = x ± c2 , c2 = a7 − b6 , xn = an , bn .

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Table 9.2

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(Continued)

4. Then, we observe mutual approach of further frequencies, and the third type of the dependent frequency vanishes, and period vibrations, with period-21 occurs. 5. For the parameter q0 = 16.4, a period doubling bifurcation for frequency b1 takes place. 6. Further increase in q0 = 17.382 implies chaos. Change of q0 on amount of 9 · 10−5 , i.e. for q0 = 17.383, pushes the shell into stiff stability loss (deflections increase suddenly, approximately in two times), and the system again vibrates periodically with the frequency of excitation a1 . Here, we may treat this process as a dynamical stability loss of conical shells subjected to periodic time loading, which can be understood as a novel criterion for the dynamical stability loss. Analysis of the existing dynamical criteria of stability loss of shells is carried out in the reference [Krysko (1976)]. The reported process of the dynamical stability loss is considered as the most general, and its properties before stability loss allow to control the given process. The spherical shells exhibit narrow zones embeded into chaos, where the Feigenbaum scenario [Feigenbaum (1983)] has been

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Fig. 9.10 Dependence wmax (q0 ) and other dynamical characteristics for the spherical shell (k = 3, ωp = 5).

detected. However, only three period doubling bifurcations have occurred, and further sequence of bifurcations has not been detected numerically. In order to illustrate vibrations of spherical shells, in Figs. 9.10 and 9.11, the characteristics wmax monitored in the shell top versus q0 for ωp = 5, k = 3 and for ωp = 8, k = 5 respectively are shown. The Lyapunov exponents play an important role in investigations of dynamical systems. They give computational qualitative measure of the stochasticity order. The proposed and developed idea of computation of a series of vibration character of dynamical system [Awrejcewicz et al. (2002)] based on an analysis of the power spectrum S(ω) reveals a good coincidence with the evolution of the largest characteristic exponent (LE) named here as λ1 (q0 ). In this work, in order to compute LE the Benettin method is applied [Bennetin et al. (1978, 1979)]. In a chaotic state λ1 > 0, and white color corresponds to chaotic zones in the reported scales/charts. In Fig. 9.10, window L is presented in the form of a scale, for which the dependence λ1 (q0 ) (15 ≤ q0 ≤ 19) is shown, and where four zones

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Fig. 9.11 Dependence wmax (q0 ) and other dynamical characteristics for the spherical shell (k = 5, ωp = 8).

of chaos are observed: a (16.1 ≤ q0 ≤ 16.5); b (16.6 < q0 < 16.8); c (16.9 ≤ q0 ≤ 17.3) and d (17.9 < q0 < 18.1), with λ1 > 0, which well coincides with the scale of vibrations. Observe that in zone d, local buckling is observed on the corresponding dependence wmax (q0 ). . Figures include also w(t), w(w), power spectra and space forms of vibrations for three points A, B, C for k = 5 (point A corresponds to periodic vibrations, point B refers to bifurcations, point C refers to chaos) and two points A, B (point A refers to periodic vibrations, B is the linear combination of two independent frequencies a1 and b1 with successive Hopf bifurcations) for k = 3, which are shown in dependencies wmax (q0 ). Point A corresponds to harmonic vibrations, point B to linear combination of two independent frequencies, and point C to chaos. Analysis of these results in either two points w(0, t) or in space w(ρ, t) for 50 ≤ t ≤ 54 allows to conclude that the dynamical phenomena exhibited by the shell do not depend on k in the case of periodic vibrations, whereas observed bifurcations and chaos essentially depend on parameter k.

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Fig. 9.12

9.1.4

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Charts {q0 , ωp } of the conical shell (movable clamping) for different n.

Conical shells of constant thickness

We consider vibrations of conical shallow shells of constant thickness and with the moveable clamping. In Fig. 9.12, the charts {q0 , ωp } for shells with constant thickness and for k = 1, 1.5, 2, 3, 4, 5 are shown. Charts of vibrations character change qualitatively with the increase in k. Let us compare the charts {q0 , ωp } regarding spherical and conical shells. Analysis of the dependence {q0 , ωp } versus k shows that an influence of the shell geometry on the character of vibration increases with the increase in k. For k = 1, in both conical and spherical shells narrow zones of chaos are located between ωp = 5 and ωp = 6. For k = 1.5, both charts display bifurcation zones, which are qualitatively similar. For k = 2, zones of rational frequencies are added, but they have a different organization. For k ≥ 3, an influence of the shell geometry implies essential differences in the charts for spherical and conical shells. Change in the shell geometry has an impact on scenarios of transition into chaos. The conical shell with k = 5 exhibits also zones associated with the Feigenbaum scenarios.

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In Table 9.3, dependencies of the signal w (0, t), 50 ≤ t ≤ 60, the . phase portrait w(w ), the power spectrum S(ωp ) and the Poincar´e map w(w(t + T )) for the conical shell for k = 5 are reported. Values of the sequence q0,n and the sequence d are given in Table 9.4, which yield the following convergent sequence q0,n − q0,n−1 = 4.66830065. dn = q0,n+1 − q0,n

Theoretical value obtained for the function f = 1 − cx2 is equal to d = 4.66916224 . . . . Difference of theoretical computations with the numerical experiments for the conical shell is of 0.018%. Next, we consider the dependence wmax (0) versus q0 for ωp = 5.61 and scales of bifurcations for conical shells k = 3; 5, as well as vibrations of the shell’s surface in time, signals w (0, t) 51.25 ≤ t ≤ 53.75, phase portraits for periodic vibrations, vibrations after the series of bifurcations, and finally chaotic vibrations (Fig. 9.13). In what follows, we investigate an influence of boundary conditions on the vibrations character with the example of the conical shell with k = 5 taking into account boundary conditions of movable clamping and movable simple support (Fig. 9.14). Analysis of the results shows that the vibrations corresponding to the movable clamping are more complex than that of the movable simple support. In the case of simply supported shells, the interesting phenomenon of signal intermittency has been detected. Table 9.5, gives the sig. nal w(t), the phase portrait w(w ), the power spectrum S(w), the Poincar´e map w(w(t + T )), where T stands for the excitation period (ωp = 3.5). In this scenario, two period doubling bifurcations are obtained, and next the intermittency behavior is observed, which transits the system into chaos. 9.1.5

Control of chaos

We investigate the influence of the shell thickness change versus boundary conditions and the shell geometry. We consider shallow conical shells with constant and non-constant thicknesses h = (1 + cρ), considering them as plates with initial deflection

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Table 9.3

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Vibrational characteristics of the conical shell (k = 5) for different q0 .

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q0,n dn

483

Feigenbaum scenarios detected numerically.

1

2

3

4

5

9.605846

11.098 2.19579722

11.77755 4.75708785

11.9204 4.66830065

11.951

Fig. 9.13 Dependence wmax (q0 ) and other dynamical characteristics for the spherical shell (ωp = 8).

w0 = −k(1 − ρ ) , where k = H/h0 . For the given type of boundary conditions, the approximating functions have the following form wi (ρ) = (1 − ρ2 ) , i

ϕj (ρ) = (1 − ρ2 )

i+1

.

(9.15)

We take the load q = q0 sin ωp t and zero value initial conditions. In Fig. 9.15, {q0 , ωp } charts are reported for constant and nonconstant shell thicknesses for c = 0.1, −0.1, 0.2 (k = 5). Here the influence of thickness changes on the system state differs essentially from the previously studied case.

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Fig. 9.14 (a) Charts {q0 , ωp } of the conical shell with movable simple support and (b) with movable clamping.

For c = −0.1, a chaotic zone associated with low frequencies (about 2.5) appears, which does not exist for c = 0.1, c = 0.2 and c = 0, as well as a chaotic zone associated with high frequencies (about 5.5) occurs, which exists for c = 0 and does not exist for c = 0.1, 0.2. It can be seen that the influence of shell thickness change on its dynamics depends essentially on the shell geometry and initial conditions. In Fig. 9.16, charts of control parameters {q0 , ωp } for the conical shell (k = 5) movably clamped with constant (c = 0) and nonconstant (c = 0.1, −0.1) thicknesses are shown. An increase in thickness in the shell center (c = −0.1) yielded new zones of chaos associated with high frequencies and frequencies close to natural frequency for q0 > 35, as well as an increase in zones associated with independent frequencies is observed. For c = 0.1, on the contrary, zones of chaos and bifurcations essentially decreased. In what follows, we analyze spherical shells with k = 5. At first, we consider spherical shells with the boundary condition of movable clamping and with k = 5 of constant and non-constant thicknesses for c = 0.1. In Fig. 9.17, charts of control parameters {q0 , ωp } are given. Here, the influence of thickness on the shell dynamics is more visible than in the previous case. The charts imply that shells with variable

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Table 9.5 ent q0 .

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The vibration characteristics for the conical shell (k = 5) for differ-

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Fig. 9.15 (a) Charts {q0 , ωp } for conical shells of constant and (b,c,d) nonconstant thickness (k = 5).

thickness have smaller zones of bifurcations and chaotic vibrations. The carried out results allow to conclude that by changing the shape of a transversal shell cross-section, and properly choosing parameters q0 and ωp we may control nonlinear vibrations of the studied continuous systems. In order to investigate the vibration character of conical shells versus the parameter q0 we construct the dependence wmax (q0 ). In Fig. 9.18, functions wmax (q0 ) in the shell top are given for k = 5 and c = 0, 0.1, −0.1 for ωp = 3.5, 3.57 and 3.38, respectively. In the dependence wmax (q0 ) for q0 = 1, the first stiff stability loss occurs. In the neighborhood of q0 = 2, there is a zone of the second stiff stability loss. Critical loads for constant thickness are within the mentioned

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Charts of control parameters {q0 , ωp } for spherical shells for different c.

interval. The first critical load occurs for the first Hopf bifurcation (c = −0.1) and for the second independent frequency (for c = 0, 0.1), which are depicted in the scales by vertical lines in the vicinity of q0 = 1. 9.1.6

Spatio-temporal chaos

In this section, we investigate the birth of a spatio-temporal chaos in time. In Figs. 9.19 and Fig. 9.20, the signal w(0, t) and the deflection of the mean shell surface in certain time instants for periodic (Fig. 9.19) and chaotic (Fig. 9.20) vibrations for k = 5 and ωp = 3.3 are shown. Besides, there are given dependencies w(ρ, t) for t ∈ [151; 154.5] [Fig. 9.19(a)] and t ∈ [167.5; 169] [Fig. 9.20(a)].

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Fig. 9.17 (a) Charts of control parameters {q0 , ωp } for spherical shells of constant and (b) non-constant thickness (k = 5).

Fig. 9.18 Dependencies wmax (q0 ) and vibration scales for different c of the conical shells.

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Fig. 9.19

Functions w(ρ, t), w(t) and w(ρ) for periodic regime.

Fig. 9.20

Functions w(ρ, t), w(t) and w(ρ) for chaotic regime.

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Simultaneous study of both figures allows to carry out an analysis of signal changes in the spatio-temporal domain. Point number in Figs. 9.19(b) and 9.20(b) corresponds to a curve with the same number shown in Figs. 9.19(c)–(e) and 9.20(c)–(e), respectively. We are going to analyze changes of deflection in time separately for periodic and chaotic vibrations. Curve 1 in Fig. 9.19(b) characterizes the maximum deflection of the shell center, whereas the shell quadrants go down [Fig. 9.20(c)]. When the shell center goes down along the curve w(t) we reach points 2, 3 [Fig. 9.19(b)], whereas when the shell center moves down the shell quadrants begin to move up (curves 2, 3 in [Fig. 9.19(c)]. Further, periodic vibrations appear around the neutral equilibrium states (curves 4–9 in Figs. 9.19(d) and 9.19(e)). In Fig. 9.20, a similar picture is obtained, but here we have a large number of half-waves (Figs. 9.20(d) and 9.20(e)). Finally, we discuss the problem of stability of our dissipative dynamical systems. For q0 corresponding to periodic vibrations as well as for q0 corresponding to chaotic vibrations we study phase portraits w( ∂w ∂t ) (their analysis allows to study chaos evolution in 2 time), and modal portraits w( ∂∂ρw2 ) (their study allows to investigate spatial chaos). Hence, a spatio-temporal chaos can be studied by taking into account both characteristics. In Table 9.6, the phase and modal portraits for a spherical shell with the boundary condition of movable simple support (k = 5) subjected to harmonic load q = q0 · sin ωp t, where ωp = 5 is equal to the natural shell Table 9.6 Phase and modal portraits of periodic (q0 = 2) and chaotic (q0 = 6.5) spherical shell vibrations.

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frequency, are reported for two vibration regimes, i.e. periodic and chaotic regimes. As can be seen from Table 9.6, in the case of periodic vibra∂2w tion both characteristics w( ∂w ∂t ) and w( ∂ρ2 ) present one-rotational limit cycle. In chaos, both characteristics yield a dense spot, and for q0 = 6.5 we observe a spatio-temporal chaos. It should be emphasized that temporal chaos and spatial chaos appear simultaneously in the considered problems. In the case of an infinite impulse in time for the same boundary conditions and parameter k in all states the 2 dependence w( ∂∂ρw2 ) presents a dense spot. In the latter case, another type of analysis should be carried out. 9.2 9.2.1

Flexible Axially-Symmetric Shells Mathematical model

We consider a flexible spherical circular shell governed by the following equations   ∂2w ∂4w 2 ∂3w 1 ∂2w 1 ∂w Φ


− 1− + 2 2 − 3 w + εw = − 4 − ∂r r ∂r 3 r ∂r r ∂r r ∂r 2   1 ∂w ∂Φ 1− + 4q, − ∂r r ∂r   1 ∂Φ ∂w 1 ∂w ∂F ∂ 2 Φ 1 ∂Φ − = 1 − , Φ= . + ∂r 2 r ∂r r 2 ∂r ∂r 2r ∂r ∂r (9.16) Equation (9.16) requires boundary and initial conditions as well as conditions regarding the shell top. Boundary Conditions (BC) 1. Simple movable support in a meridian direction (a) homogeneous BC ∂2w = 0, for r = b; Φ = w = 0, ∂r 2 (b) non-homogeneous BC ∂2w = M0 sin(ωp t) , for r = b. Φ = w = 0, ∂r 2

(9.17)

(9.18)

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2. Pinned support of the shell contour (a) homogeneous BC

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Φ ∂Φ − ν = 0, ∂r b

∂ 2 w ν ∂w = 0, + ∂r 2 r ∂r

w = 0,

for r = b; (9.19)

(b) non-homogeneous BC Φ ∂Φ − ν = 0, ∂r b

w = 0,

∂ 2 w ν ∂w = M0 sin(ωp t), + ∂r 2 r ∂r

for r = b. (9.20)

3. Loosely clamped shell contour Φ = w = 0,

∂w = 0, ∂r

for r = b.

(9.21)

4. Rigidly clamped shell contour Φ ∂Φ − ν = 0, ∂r b

w = 0,

∂w = 0, ∂r

for r = b.

(9.22)

0 ≤ t < ∞.

(9.23)

w

≈ 2C;

w


≈ 0. (9.24)

Initial conditions w = f1 (r, 0) = 0,

w
= f2 (r, 0) = 0,

Top shell conditions Φ ≈ Ar; Φ
≈ A; w ≈ B + Cr 2 ; w
≈ 2Cr; FDM and computational algorithm In order to reduce a continuous system (9.16) to a system with lumped parameters, FDM with approximation O (∆2 ) will be applied. Equation (9.16), in the form of finite-difference relations regarding the spatial coordinate, has the following form   wi+1 − wi−1 1 Φi+1 − Φi−1


− w i + εw i = − 2∆ 2ri ∆ ri3   1 Φi+1 − Φi−1 Φi wi+1 − 2wi + wi−1 − Φi + − + ri ∆2 ri 2∆ ri

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wi+2 − 4wi+1 + 6wi − 4wi−1 + wi−2 ∆4 wi+2 − 2wi+1 + 2wi−1 − wi−2 − − 4qi , ri ∆3       1 1 1 1 2 1 + Φi + + Φi−1 − 2 + Φi+1 − 2 − ∆ 2ri ∆ ∆2 ri2 ∆ 2ri ∆   wi+1 − wi−1 wi+1 − wi−1 1− , =− 2∆ 4ri ∆ (9.25)

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−

where ∆ = b/n, and n denotes a number of partition of the shell radius. The corresponding boundary conditions are as follows. 1. Simple movable support in a meridian direction (a) homogeneous BC Φ = 0,

wn+1 = −wn−1 ,

wn = 0 for rn = b;

(9.26)

(b) non-homogeneous BC Φ = 0,

wi+1 = M0 sin(ωp t) − wi−1 ;

wn = 0,

for rn = b. (9.27)

2. Pinned support of the shell contour (a) homogeneous BC 2∆ν ν∆ − 2 b Φi ; wi+1 = wi−1 ; Φi+1 = Φi−1 + b 2 b + ν∆ wn = 0 for rn = b; (b) non-homogeneous BC Φi+1

2∆ν = Φi−1 + i; b

wi+1

M0 sin(ωp t) − 1 = ∆2



1 ∆2

+

−

ν 2b∆

ν

2 b ∆

(9.28)

wi−1

;

wn = 0 for rn = b. 3. Loosely clamped shell contour

(9.29)

(9.30) Φn = 0; wn+1 = wn−1 ; wn = 0 for rn = b. 4. Rigidly clamped shell contour 2∆ν wn+1 = wn−1 ; wn = 0 for rn = b. Φi+1 = i−1 + i; b (9.31)

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Initial conditions follow wn = f1 (rk , 0) = 0,

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(0 ≤ k ≤ n),

w
n = f2 (rk , 0) = 0, 0 ≤ t < ∞.

(9.32)

If we neglect small terms and change the differential operators by the central finite-difference once for r = ∆, the following conditions in the shell top are obtained: Φ0 = Φ2 − 2Φ1 ;

9.2.2

w0 =

4 1 w1 − w2 ; 3 3

8 8 w−1 = w1 − w2 + w3 . 3 3 (9.33)

Method of relaxation

The idea of finding a solution to stationary problems via nonstationary problems have been used for the first time in the 1930’s of the previous century by A. N. Tichonov [Tichonov and Samarskiy (1977)]. For the problems of stationary supersonic flows acting on the bodies, the relaxation method was applied by Godunov, Zabrodin and Prokhorov in 1961 [Godunov et al. (1961)]. In 1947, Lyusternik, in order to solve the Poisson equation using the relaxation method, applied FDM [Lusternik (1947)]. This method has been applied to nonlinear problems of shells by Feodosev [Feodosev (1963)]. In this section, we propose a modification to this method to solve Poisson equation for the Dirichlet problem. In what follows, we illustrate the application of this method with various modifications to nonlinear problems of the theory of circle shallow shells within the Kirchhoff–Love model. 9.2.2.1

Main Idea

The main idea of the relaxation method for solutions to numerous problems in mathematical physics relies on the consideration of an unsteady process generated by initial conditions in the limit reaching a steady state, which yields a solution of the associated equilibrium system configurations. The results are obtained fast and in a simpler way than using the method based on direct computation of the equilibrium states. Let us consider the Dirichlet D = {0 ≤ x1, x2 ≤ 1}

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problem of the Poisson equation on the square defined as follows

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∂2u ∂2u + = ϕ(x1 , x2 ), 0 ≤ x1 , x2 ≤ 1 ∂x1 2 ∂x2 2 u|∂ D = ψ(s),

(9.34)

having the boundary ∂D, where s is the arc length along the border ∂ D, and the functions φ(x1, x2 ) and ψ(s) are a priori given. We formulate the problem (9.34) in the counterpart form λx1 x2 um n + λx2 x2 um n = φ (x1, m , x2, n ),

m , n = 1, 2, . . . , M − 1,

um n |∂ D = ψ(sm n ), (9.35) where λx1 1 , λx2 2 are differences of a second-order. Let us give a physical interpretation of Eq. (9.34). Solution u(x1 , x2 ) to the problem (9.34) can be interpreted as a temperature (not dependent on time) in a point (x1 , x2 ) of the plate in a heat transfer equilibrium. Functions ϕ(x1 , x2 ) and ψ(s) govern the temperature distribution of the heat source in the plate, and the temperature distribution along its border, respectively. We consider the following supplemented non-stationary problem regarding the heat distribution

U |∂ D

∂U = ∇2 U − ϕ(x1 , x2 ), ∂t = ψ(s); U (x1 , x2 , 0) = ψ0 (x1 , x2 ),

(9.36)

where ϕ and ψ are the same as in problem (9.34), ψ0 (x1 , x2 ) is arbitrary given in time instant t = 0, i.e. we define the initial con2 ∂ 2 (·) ditions, and ∇2 (·) = ∂∂x(·) 2 + ∂x2 is the Laplace operator. Since the 1 2 heat source ϕ(x1 , x2 ) and the temperature on the border ψ(s) do not depend on time, we may expect that the solution U (x1 , x2 , t) will be changed more slowly in time. The temperature distribution U (x1 , x2 , t) for t → ∞ tends to the stationary temperature distribution u (x1 , x2 ) governed by problem (9.34), i.e. in the case of nonstationary parabolic problems the information can be transmitted only in direction of time increase. Therefore, instead of the stationary problem (9.34) we may solve the associated non-stationary problem (9.36) in time t using the relaxation method until the solution stops

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to change in a priori given interval of stationary problems. This is the main idea of finding solutions of required accuracy. Owing to this observation, instead of problem (9.34) we solve problem (9.36) and instead of the difference scheme (9.35) we consider and define three different schemes to solve problem (9.36). We are mainly oriented toward the following simple explicit difference scheme: 1 p+1 (u − upm n ) = λx1 x1 upm n + λx2 x2 upm n − ϕ(x1,m n , x2,m n ), τ mn u0m n = ψ0 (x1, m x2, n ). up+1 m n |∂ D = ψ (sm n ), (9.37) In parallel, we define also the following simple implicit difference scheme 1 p+1 (u − upm n ) = λx1 x1 up+1 m n − ϕ(x1,m , x2,n ), τ mn (9.38) p+1 0 um n = ψ0 (x1,m , x2, n ), um n |∂ D = ψ(sm n ), as well as the following scheme of variable directions  1 1 0 p ˜mn + Λx x Umn (U mn − Umn Λx1 ,x2 U )= − ϕ(x x ) 1,m, 2,n , 2 2 τ 2  1 1 p+1 ˜ p+1 ˜mn + Λx x Umn (Umn − Umn ) = Λx1 x1 U − ϕ(x x ) , 1,m 2,n 2 2 τ 2   p+1  ˜mn  ∂D = Ψ(Smn ), U 0 = Ψ0 (x1,m , x2,n ). Umn ∂D = U mn (9.39) We assume that Ψ0 (x1,m , x2,n ) is defined in a way to satisfy the following relation on the border Ψ0 | ∂D = Ψ(Smn ).

(9.40)

p+1 p } on already known U p = {Umn } Computation of U p+1 = {Umn via the scheme (9.37) is carried out using the explicit formulas. Comp+1 } via scheme (9.38) requires solution of the putation U p+1 = {Umn following problem

1 p+1 1 p p+1 p+1 + Λx2 x2 Umn − Umn = ϕ(x1,m x2,n ) − Umn , Λ11 Umn τ τ p+1 |∂D Umn

= Ψ(Smn ).

(9.41)

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However, this problem is not simple with respect to the initial problem (9.35). Therefore, there is no rational motivation to apply the implicit scheme to carry out the approximate computation. p+1 p } based on known U p = {Umn } Finally, computation of U p+1 = {Umn via scheme (9.39) can be realized in few stages. At first, they are realized in the direction of axis ox1 to compute solutions {U˜mn } of one-dimensional problems for each fixed n, and then the similar-like computational process is repeated in the direction of axis ox2 in order p+1 } of one-dimensional problems for each to compute solutions {Umn fixed m. A number of arithmetic actions is proportional to a number of unknowns. Consider the difference p − Umn , ε3bm ≡ Umn

(9.42)

p {Umn }

and an exact solution U = between the mesh function U p = {Umn } of problem (9.35). Let us compute a condition for which the p of the non-stationary problems tends to error ε3bm of solution Umn zero with the increase in p, as well as the character of this tendency goes to zero. Let us take an optimal step τ and estimate a volume of computational work required to decrease the norm of the initial error ε0bm = Ψ0 (x1,m , x2,n ),

(9.43)

for a given number of times. 9.2.2.2

Explicit Scheme of Relaxation

Solution {Umn } of the problem (9.35) satisfies the equations 1 (Umn − Umn ) = Λx1 x2 Umn + Λx2 x2 Umn − ϕ(x1,m , x2,n ), τ Umn = Umn1 . (9.44) Umn |∂D = Ψ(Smn ), Taking into account the so far given equations, error εpmn can be estimated in Eq. (9.37) step by step via the following difference problem, where also Eq. (9.42) is taken into account: 1 p+1 (ε − εpmn ) = Λx1 x2 εpmn + Λx2 x2 εpmn , τ mn (9.45) εp+1 mn |∂D = 0,

ε0mn = Ψ0 (x1,m , x2,n ) − Umn .

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Observe that the mesh function ε0mn for each p1 (p = 0, 1, . . . , ) takes zero value on the border. It can be treated as an element of a linear space of functions defined on the mesh (x1,m , x2,n ) = (mh, nh, m, n = 0, 1, . . . , N ), and taking zero values in points on ∂D. We define the following norm in this space:  1 2 |εpmn |2 . (9.46)

εp = mn

In order to solve (Eq. 9.45), we apply the following finite Fourier series (crs λprs )Ψ(r,s) , (9.47) εp = rs

where crs are the coefficients of the series of initial error ε0 = {ε0mn } of the finite Fourier series, and the numbers λrs are defined in the following way 4τ  rπ sπ  + sin2 . (9.48) λrs = 1 − 2 sin2 h 2M 2M Numbers cprs = crs λprs are coefficients of the development of error εp = {εpmx } into the Fourier series via the orthogonal basis Ψ(rs) . Therefore,   1    1 2 2 |crs λprs |2 , ε0  = (9.49) |crs |2 .

εp = Finally, we get

εp

≤ {max |λrs|}p .

εo

(9.50)





Note that ε0 can be given to satisfy the equation ε0 = Ψ(r ,s ) , where (r
, s
) is a chosen pair for which max |λrs| = |λr
s
| ,

(9.51)

rs

  i.e. lim εp / ε0  = 0, and max |λrs | =< 1. Maximum decrease p→∞

rs

for τ , where max |λrs | − min, is estimated from Eq. (9.48): rs

λleft = 1 −

8τ π , cos2 2 h 2M

λright = 1 −

8τ 2 π . sin h2 2M

(9.52)

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λright

λleft -1

0

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1

A scheme of τ estimation.

Increasing τ , beginning from τ = 0, we initiate the movement of these points into left (see Fig. 9.21). This process ends when −λleft = λright . is yielded by Eq. (9.49): 2M 2 1 ≈ . p≥− π π2 ln(1 − 2sin2 2M )

Optimal value τ =

9.2.2.3

(9.53)

h2 /4

(9.54)

Method Limits

The so far described methodology and error estimation are applicable to study difference schemes approximating other boundary value problems for elliptic equations with variable coefficients and with curvilinear boundaries. It is necessary that the operator −Λh ≡ −(Λx1 x1 + Λx2 x2 ) occurring in scheme (3.1.2) should be self-jointed, and its eigenvalues µj should have the same sign: 0 < µ min < µj < µ max . 9.2.2.4

(9.55)

Shells with Finite Deflections

The developed algorithm and programs package allow to solve numerous problems of static and dynamics of axially-symmetric shells. The problems of statics is treated from the point of view of dynamics, and the Feodosev method [Feodosev (1963)] has been applied being a variant of the relaxation method. The method is based on the following approach: for ε = εcr the dependencies {qm , wm (t)} are constructed, where m = 1, 2, . . . denotes numbers of the load values for which the solution via the mentioned relaxation method has been obtained. This allows for computation of q(w) and analysis of the stress–strain state of shells. We follow here the reference [Valishvili (1976)], where the iterative relaxation method has been illustrated and applied. The functions q(w) constructed via the mentioned algorithm (curves a) and via the method proposed by Valishvili (curves b)

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Fig. 9.22

Shell with fixed contour (q = q0 , ε = 1).

[Valishvili (1976)] are shown in Figs. 9.22–9.25. Observe that the solution proposed by Valishvili (Fig. 9.23) does not yield either upper or lower critical load values, since both load and deflection are not defined uniquely. On the contrary, the proposed dynamical approach allows to estimate the critical loads accurately, and hence to predict buckling of shells. The occurred loops have the following origin. The given characteristic q0 (w) is constructed for the shell center, whereas the remaining points of the shell radius behave on its own way, i.e. the shell stability loss occurs not in the shell center, but in its quadrants, hence the shell center versus the load describes the mentioned loops. The reported results not only validate the high efficiency of this method, but also authenticate its wide applications to solve problems of statics. 9.2.3 9.2.3.1

Dynamical stability loss Criterions

The proposed algorithm and the method of computations allow to investigate the stress–strain states and the stability of the already mentioned wide class of static/dynamic problems of shells. It is well

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Fig. 9.23

Fig. 9.24

Shell with pinned contour (q = q0 , ε = 1).

Shell with loosely clamped contour (q = q0 , ε = 1).

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Fig. 9.25

Shell with simple movable contour (q = q0 , ε = 1).

known that shell structures subjected to transversal load of different type suffer stability loss. For the dynamical problems of shells we have a series of criteria of dynamical stability loss, and their analysis is carried out in the reference [Krysko (1976)]. We consider shells with parameter b = 4 subjected to uniform harmonic transversal load q = q0 sin(ωp t). In Fig. 9.26, the dependence wmax (q0 ) for the shell with a simple-movable contour subjected to the mentioned load action for ωp = 0.521 is shown. In the mentioned figure, points a, b, c corresponding to periodic, chaotic and post-critical periodic vibration are reported, respectively. Point a — q0 = 0.16212, point b — q0 = 0.16213, i.e. the change of q0 on magnitude of 1 · 10−5 implies a sudden increase in shell’s deflection (we deal here with a stiff stability loss). The similar-like investigations have been also carried out for the remaining types of the boundary conditions, and the similar results have been obtained in the reference [Krysko (1976)]. While investigating chaotic vibrations, we do not present diagrams about orbits, as it is usually done during investigations of a wide class of two-valued maps into itself, but the vibration scale is constructed instead. Dependence wmax (q0 ), scales of vibrations

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Fig. 9.26 Dependence wmax (q0 ) and other dynamic characteristics for the shell with simple movable contour.

character and Lyapunov exponents well coincide with each other approving the validity and reliability of obtained results. When the largest Lyapunov exponent becomes positive, then chaos appears, which is also validated by the scale of bifurcations and wmax (q0 ) [Awrejcewicz et al. (2006)], where a series of stiff bifurcations is observed (dynamical stability loss). For the given points on the functions wmax (q0 ) a surface w(r, t), the time history w(0, t), the phase portraits w(w
) and the power spectrum S(ωp ) for the shell center are reported (remaining shell points have analogous characteristics). The shell, from its previous chaotic state, is transited into periodic vibrations by a sudden buckling jump. The so far described scenario of stiff stability loss can be understood as the criterion of dynamical stability loss for the case of harmonic transversal excitation. This is a novel criterion of stability loss of shells subjected to periodic loads. 9.2.3.2

From Periodic to Chaotic Vibrations

Analysis of nonlinear dynamics of structures plays an important role in the investigation of vibrations of plates and shells periodically

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loaded taking into account energy dissipation from the point of view of possible scenarios of transition into chaotic dynamics. In particular, this concerns investigations of axially-symmetric spherical shells subjected to periodic loads and dissipative factors [Agamirov (1990); Goldenveizer et al. (1979); Grigoluk and Kabanov (1978); Krysko (2003); Nie and Liu (1994); Palmov (1976); Saliy (2001); Schimmels and Palaiotto (1994); Slawianowska (1996); Soric (1994); Teregulov and Timergalev (1998)]. In problems of theory of shells, some of the known classical scenarios have been detected, like Feigenbaum’s scenario, but also their numerous modifications including: modified Ruelle–Takens– Newhouse scenario, Ruelle–Takens–Newhouse–Feigenbaum scenario, modified Pomeau–Manneville scenario, as well as novel scenarios of transition from periodic to chaotic vibrations. Feigenbaum’s scenario We consider vibration of shells subjected to an action of periodic load q = q0 sin(ωp t) uniformly distributed on their surfaces. We study possible scenarios of transition of those mechanical systems from periodic to chaotic vibrations, and we briefly discuss a few hypotheses regarding the mechanisms of transition of the regular/laminar flow to the hydro-dynamic turbulence. In spite of the earlier proposed Landau hypothesis [Landau (1944)], all other mechanisms are associated with models of finite dimensions including the Ruelle– Takens–Newhouse, Feigenbaum and Pomeau–Manneville scenarios. It should be mentioned that even now there is no general and unique mechanism of transition into turbulence. We here illustrate and describe mechanisms of occurrence of weak turbulence exhibited by transversal vibrations of flexible axially symmetric shells. Transition from regular to chaotic vibrations via period doubling bifurcations is well known and validated by many simple mathematical models. It is also known that period doubling bifurcations are well described in the R¨ ossler attractor and many other simple models. The mentioned phenomenon has been also detected in our problem of the shell with a simple moveable contour. In Figure 9.27 for the central shell point, and for the boundary condition (Fig. 9.26) the

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Fig. 9.27 Shell with simple moveable contour: phase portraits, power spectra and Poincar´e maps for different q0 .

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q0, n dn

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Table 9.7 n

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1 bifurcation

2 bifurcation

3 bifurcation

4 bifurcation

5 bifurcation

0.1335

0.13522 4.19512

0.13563 4.659091

0.135718 4.656084

0.1357369

following characteristics for the limiting values of q0 are reported: phase portrait w(w
), power spectrum S, db (ωp ) and Poincar´e maps wt (wt + T ), where T is the period of excitation. Table 9.7 allows to get the following convergent series regarding the estimation of the Feigenbaum constant dn =

q0,n − q0,n−1 = 4.65608466, q0,n+1 − q0,n

n = 5,

where the theoretical value of d = 4.66916224. The difference between theoretical and numerical value is 0.28%. Values of series q0, n and series dn are shown in Table 9.7. As a result of the vibrational process of the axially symmetric shell with the simple moveable contour for b = 4, we have detected the Smale chaotic attractors [Smale (1962)]. The latter ones are also known as attractors of Feigenbaum’s type or strange attractors (SA). Ruelle–Takens–Newhouse scenario For a spherical shell with a simply supported contour (Fig. 9.20), as well as with moving (Fig. 9.23) and stiff clamping (Fig. 9.24) the classical Ruelle–Takens–Newhouse scenario has been detected, which we study in more detail here. The fundamental characteristics: time . history w (0, t), phase portrait w(w ), power spectrum S, db (ωp ) and Poincar´e maps wt (wt + T ) versus the limiting values of q0 are collected in Fig. 9.28. The values of q0 are referred to as limiting, since between two successive values of q0 the qualitative picture of dynamical state remains unchanged. We consider this scenario using an example of the rigidly clamped shell contour. 1. Vibrations take place on the fundamental frequency a1 of excitation and they are periodic. Phase portrait presents a onerotational limit cycle (q0 = 0.68).

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Fig. 9.28 Dynamic characteristics of the shell exhibiting Ruelle–Takens– Newhouse scenario versus different q0 .

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(Continued)

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2. Further movement along the parameter up to q0 = 0.699 implies the occurrence of independent frequency b1 , i.e. two-frequency motion with two frequencies a1 and b1 occurs. The motion is not synchronized, i.e. ab11 = m n = 3.169 is irrational number. 3. The increase in q0 up to q0 = 0.7 produces a series of linearly dependent frequencies bn = n · b1 and an = a1 − (n − 1) b1 , and this process takes place until both frequencies ak and bk ∈ [b1 , a1 ] approach each other (q0 = 0.706); then one more series of linear combination occurs, i.e. c2 = b2 − a2 , cn = x ± c2 , (x = an , bn ) (q0 = 0.73418). The increase in q0 to q0 = 0.73418 shifts the system into a chaotic regime. The change of q0 on the magnitude 2·10−5 , i.e. for q0 = 0.7342 causes the shell stability loss, and the system exhibits again periodic vibrations with excitation frequency a1 . This process can be treated as the dynamic stability loss of spherical shells subjected to a periodic load action. The obtained scenario is similar to that of Ruelle–Takens, where in the power spectrum the first and then the second independent frequency appear. The mentioned criterion served to study vortices in a fluid between two cylinders, as well as the Rayleigh–Bernard convection. However, the model of transition into turbulence observed by us essentially differs from the Ruelle–Takens model due to the occurrence of a fractal structure of labyrinth shape after the occurrence of the second independent frequency observed in Poincar´e map. Then, a further increase in q0 yields a series of linear combinations of the mentioned independent frequencies, and the reconstruction of the earlier detected fractal structure. The increase in q0 up to q0 = 0.73418 yields SA, whereas the increase in q0 magnitude of only 2 · 10−5 yields a collapse of SA, and its transition into periodic vibrations. 9.2.4

Non-uniform harmonic excitation

We study a simply supported shell subjected to two types of loads: 1. The load q = q0 sin(ωp t) is acting on five points 8 ≤ i ≤ 12, where 0 ≤ i ≤ n; i, n ∈ N in a neighborhood of quadrants, and in

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Fig. 9.29 (a) Dependence wmax (q0 ) of a shell with five and (b) four harmonically excited points.

remaining points q = 0; 2. The load q = q0 sin(ωp t) is acting in five points 0 ≤ i ≤ 4, where 0 ≤ i ≤ n, i, n ∈ N in a neighborhood of the center, and in remaining points q = 0. Dependencies wmax (q0 ) for the first and the second type loads are shown in Fig. 9.29(a) and 9.29(b), respectively. The first case is associated with two stiff bifurcations. The first stiff stability loss occurs while transiting from harmonic vibration to first Hopf bifurcation. The second stiff stability loss appears during transition from chaotic to harmonic vibrations, which is approved by the scale of signal types exhibiting period doubling bifurcations. In this case, the dependence wmax (q0 ) is smoother, there is a first-order discontinuity as in the previous case. Scale of signal types exhibits a small zone of soft bifurcations and there is a lack of chaotic zones. Five period doubling bifurcations have been detected, and the numerically estimated Feigenbaum constant is 4.67784 . . . , and its difference in comparison with the theoretical value is 0.168%. For two types of local loads of the shell, the charts in the plane {q0 , ωp } are reported (the first type is shown in Fig. 9.30(a), whereas the second has been presented in Fig. 9.30(b)). In the chart of the control parameter plane {q0 , ωp } a large chaotic zone on the high frequencies is observed, whereas small zones of Hopf bifurcations are located on low frequencies (there are drops of independent frequencies and their linear combination). Transition of the local load into the neighborhood of the shell center yields large zones of regular vibrations, whereas a small zone

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Fig. 9.30 (a) Charts of shell vibrations on the control parameters {q0 , ωp } plane with harmonic excitations in five and (b) four points.

of chaotic vibrations has been shifted into a space of low frequencies. In this case, drops of independent frequencies and their linear combinations do not exist. 9.2.5

Sharkovsky’s periodicity

We investigate vibrations of the spherical axially-symmetric shell with simple movable support and the shallow parameter b = 4, with damping coefficient ε = 0.1, and subjected to harmonic load q = q0 sin ω0 t. Within the chaotic region, windows of periodic vibrations following the Sharkovsky order have been detected. We study time histories/signals in the shell center, phase portraits, power spectra and Poincar´e maps and we aim to trace the behavior of periodic orbits described in the Sharkovsky theorem. In Fig. 9.31, Sharkovsky order 2 · 3; 2 · 5 is exhibited. It should be mentioned that the so-called Sharkovsky’s orders are not associated with each other, but they are rather detected separately in the whole plane {q0 , ω0 }. The following dynamical features have been illustrated: in the Poincar´e map a number of points is equal to the number of maxima in the power spectrum; order 2·3 — Poincar´e map is divided into two subsets having 3 points each; order 2 · 5 — Poincar´e map consists of 10 points, and the power spectrum exhibits 10 maxima. The phase portrait has loops,

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Fig. 9.31 Vibrations of shell with a simple movable support exhibiting Sharkovsky’s orders of 2 · 3 and 2 · 5.

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(Continued)

and their number coincides with the number of points of the Poincar´e map. Those orbits present periodic windows in chaos and their structure is the same in the whole analyzed set. The mentioned orbits can be also traced on the phase and modal portraits, constructed for three phase variables. 9.2.6

Control of chaos

We have carried out control of the circle shell loaded by the uniform periodic load q = q0 sin(ωp t) by using two additional types of periodic excitations: (1) Local transversal harmonic excitation is applied to five points 8 ≤ i ≤ 12, where 0 ≤ i ≤ n; i, n ∈ Z; (2) Harmonic torque is applied. Two types of excitations have been studied: fixed frequency and synchronization of frequencies. We study the system behavior, when the local harmonic load is applied. The system response has

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Table 9.8

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Applied surface and local shell loads.

Problem

Surface load

1. 2. 3. 4. 5.

q = q0 · sin (ωp · t) — q = q0 · sin (ωp · t) q = q0 · sin (ωp · t) q = q0 · sin (ωp · t)

Local load — q = q0 · sin (ωp · t) q1 = 0.6 sin(0.6 t) q1 = 0.6 sin(0.725 t) q1 = 0.6 sin(0.886 t)

Fig. 9.32 Dependencies wmax (q0 ) and vibrations character scales associated with five problems (see Table 9.8).

been monitored on the basis of numerical results and reported in a graphical form, where also the dependencies of the maximum shell deflection versus excitation amplitude as well as the charts of the vibration character versus control parameters have been shown using a color notation. Identification of the system vibration regime has been carried out with the help of the power spectrum analysis as well as the computation of Lyapunov exponents. We consider the function wmax (q0 ), when two types of the load act on the system: local and continuously distributed (Table 9.8). Five numbers of curves shown in Fig. 9.32 are identified by the corresponding number of problems given in Table 9.8.

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Natural frequency of the shell is ω0 = 0.644. One may conclude that two-frequency excitation does not improve the vibration regime is seen in the figures (in the last three cases, the system works in harmonic regimes less than two first cases considered). In order to investigate the shell behavior under the action of two exciting loads with different frequencies (problems 3–5 of Table 9.8), the mathematical model has been constructed and the charts of the vibrations character for the control parameters {q0 , ωp } are reported in Fig. 9.33. All charts exhibit large zones of chaotic vibrations, while increasing the excitation frequency of the local load implies

Fig. 9.33 Charts of vibration type versus control parameters {q0 , ωp } and the associated number of solved problem.

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Fig. 9.34

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Dependencies wmax (q0 ) and the scales of solved problems for ωp = 0.6.

an increase in chaotic zones. As we have already mentioned, in the case of two-frequency excitation, the system is transited into periodic vibrations only if excitation frequencies are close to each other. In this case, on the charts of vibrations (Fig. 9.33) zones of chaos are located in the vicinity of the frequency of the local load, and this case has been more deeply studied with the help of dependencies wmax (q0 ). Namely, in Figs. 9.34–9.36, wmax (q0 ) have been reported for all five loading cases. In Fig. 9.34, the wmax (q0 ) curves for problems 1, 2, 3 (ωp = 0.6), in Fig. 9.35 for problems 1, 2, 4 (ωp = 0.725), and in Fig. 9.36 for problems 1, 2, 5 (ωp = 0.886) are shown. In all cases, the coincidence of two frequencies of the external load decreases a zone of chaotic vibrations, both independent frequencies and bifurcation do not appear and the dependence wmax (q0 ) becomes smooth. Area of chaotic zones and bifurcations and independent frequencies decrease, whereas a zone of periodic vibration increases (Fig. 9.35). We study the system behavior, when the second type of the exciting load is applied, i.e. the periodic load. In Fig. 9.37, the dependencies wmax (M0 ) and wmax (q0 ) as well as the scales characterizing the signal type for seven problems associated with the mentioned loading types are reported in Table 9.9.

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Fig. 9.35 0.725.

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Dependencies wmax (q0 ) and the scales of solved problems for ωp =

Fig. 9.36

Dependencies wmax (q0 ) and vibration scales (ωp = 0.886).

The values of ωp and M0 have been chosen to illustrate the shell behavior in cases: the first point is taken on the boundary between chaos and bifurcations; the second is taken in deep chaos regime; the third is taken on the border of chaos and periodic vibrations, and the last in a zone of periodic vibrations. Let us analyze the case of

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Fig. 9.37

Dependencies wmax (M0 ) and wmax (q0 ) and vibration scales. Table 9.9

Shell loading types.

No

Continuous load

1. 2. 3. 4. 5. 6. 7.

q = q0 · sin (ωp · t) — q = q0 · sin (ωp · t) q = q0 · sin (ωp · t) q = q0 · sin (ωp · t) q = q0 · sin (ωp · t) q = q0 · sin (ωp · t)

Resistance torque — M = M0 · sin (ωp · t) M1 = 0.1 · sin (0.859 · t) M1 = 3.4 · sin (0.859 · t) M1 = 4.2 · sin (0.859 · t) M1 = 5.5 · sin (0.859 · t) M1 = 9.6 · sin (0.859 · t)

chaotic zones (in Fig. 9.38, the following characteristics are shown: signal w(0, t), where 1000 ≤ t ≤ 1100, phase portrait w(w
), power spectrum S(ωp ), Poincar´e map wt (wt+T )). In the beginning, we consider the shell dynamics in regimes where the shell is subjected to action of only one component of excitation, i.e. either the continuous periodic load or the periodic torque. Frequency of exciting load is the same in all cases (ω1 = 0.859). When the shell is subjected to action of the continuous load, a transition from periodic to chaotic vibration is carried out via a stiff stability loss. The analyzed power spectrum (in a chaotic zone) exhibits the local maximum on frequency ω1 /2, in spite of the global maximum

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Fig. 9.38 Power spectrum, time histories/signals, phase portraits and Poincar´e maps of the shell subjected to continuous load and torque actions.

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on frequency ω1 . In order to control chaos, two exciting forces have been applied. We consider the case when the exciting torque has amplitudes M0 = 0.1; 3.4; 4.2 and the frequency coincides with the frequency of the harmonic continuous load. Since the type of vibrations for all these cases is similar, we consider the influence of M0 for the mentioned three cases simultaneously. The Poincar´e map exhibits one attractor, whereas the phase portrait presents a continuous spot. We consider the influence of both exciting load and torque with M0 = 5.5. In this case, in the chaotic zone there are drops of periodic vibrations. A transition from chaotic to periodic vibrations takes place via the stiff stability loss. Contrary to the earlier studied zone, in this case the chaotic vibrations are changed: phase portrait represents an attractor of the shape of eight, whereas the Poincar´e map consists of two attractors. In the latter case, the excitation torque frequency ωp = 0.859, whereas its amplitude changes in the interval of (0.1; 9.6). In all cases reported in Fig. 9.37, besides the cases of the periodic load and torque (q0 = 9.6, ωp = 0.859), the maximum shell deflection curve exhibits the first-order discontinuity implying the stiff stability loss. Its occurrence is approved by the vibration character scales, since this time instant is associated with a change of the vibration character. In order to study changes in the system’s reaction on the external load, the charts of vibration types in the control parameters plane {qo , ωp } (problem 3–7, Table 9.9) and Fig. 9.39 have been constructed. Analysis of the obtained results implies that action on the shell of the continuous periodic load and the periodic torque with amplitude M0 = 0.1 with the same frequency of both excitations, the detected chaotic zones are similar to those already shown in Fig. 9.40(b) for the case of the periodic load. It means that the action of additional periodic torque with its small amplitude does not change essentially the shell vibrations. Increasing M0 implies non-unique change in the chaotic region with respect to the global chart surface. Namely, in the beginning it starts to increase monotonously and it practically covers the whole chart area, then chaotic zone starts to decrease simultaneously shifting

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Fig. 9.39 Charts of vibration types of the shell subjected to continous load and resistance torque (problems 3–7, Table 9.9).

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Fig. 9.40 Charts of the shell vibration types versus the control parameters {M0 , ωp }, {q0 , ωp }: (a) distributed load q = q0 sin(ωp t), (b) local load q1 = q0 sin(ωp t), (c) support moment M = M0 sin(ωp t), (d) distributed load q = q0 sin(ωp t) and local load q1 = 0.6 sin(ωp t), (e) distributed load q = q0 sin(ωp t) and support moment M1 = 9.6 sin(ωp t).

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Table 9.10 Controlling vibrations of the shell using the continuous harmonic local force and the harmonic torque. No

Continuous load

1. 2. 3. 4. 5.

q = q0 sin (ωp t) — — q = q0 sin (ωp t) q = q0 sin (ωp t)

Local load — q = q0 sin (ωp · t) — q1 = 0.6 sin (ωp t) —

Torque — — M = M0 sin (ωp t) — M1 = 9.6 sin (ωp t)

into an area of higher values of q0 . Areas of periodic zones appear only if the frequencies are rational, then the chart exhibits a large area of bifurcations. Otherwise, an area of independent frequencies and their linear combinations appear. The increase in the excitation amplitude (ωp = 0.859) implies the increase in the periodic zone while the parameter q0 is increased. Analysis of the charts of control parameters for the first type of problems (3–5, Table 9.8) and the second type of problems (3–7, Table 9.9) allows to conclude that in the case of similar frequency ωp = 0.859 of the load and the torque, there exists a vertical zone, ∀q0 ∈ [0, 1], corresponding to periodic vibrations. This motivated us to consider the problems when the shell is subjected to uniformly distributed periodic load and the local load or the periodic torque while keeping the synchronized changes if the frequencies of both excitations (Table 9.10). In Fig. 9.40, charts of the vibration types for the control parameters {q0 , ωp }, {M0 , ωp } are reported. Analysis of the obtained results allows to conclude that the synchronization of external loads shifts the system to qualitatively different type of vibrations. Almost all chaotic zones occurring in the charts {q0 , ωp }, {M0 , ωp } [Figs. 9.40(a), 9.40(b) and 9.40(c)] have been cancelled. In the case of the fourth type of the load [Fig. 9.40(d)] the vibrations are periodic. Only a small chaotic zone has been preserved as well as bifurcation on low and high frequencies. In the case of the fifth type of loading [Fig. 9.40(e)] there exist zones of bifurcations as well as small zones of chaos, but only on low frequencies.

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Fig. 9.41

Charts of vibration types {q0 , ωp }, {M0 , ωp }.

Similar investigations have been carried out for the shell with its contour and the shallow parameter b = 4. Results of the given investigations are shown in Fig. 9.41. Taking into account the earlier obtained results, our further analysis has not been carried out so deeply as it was done for the case of the shell with its simple moveable contour. We have investigated the methods of chaos control, which have earlier given positive results and they have been applied to the shell with the simply supported contour. As in the previous case, we have constructed the charts of the control parameters for the case of uniformly distributed periodic load [Fig. 9.41(a)], and for the case of periodic torque [Fig. 9.41(b)] which have been also reported earlier. In addition, charts describing shell behavior subjected to periodic continuous and local load and torque actions have been given. All applied loads have been shown in Table 9.11 (observe that the phase

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Table 9.11 Controlling vibrations of the shell using the continuous harmonic local force and the harmonic torque.

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Problem 1. 2. 3. 4. 5.

Continuous load q = q0 sin (ωp t) — — q = q0 sin (ωp t) q = q0 sin (ωp t)

Local load — q = q0 sin (ωp t) — q1 = sin (ωp t − π) —

Torque — — M = M0 sin (ωp t) — M1 = 0.5 sin (ωp t)

shift of the local load on the magnitude of π means that the local load works in the anti-phase with respect to the continuous load). Action of two additional loads yields a benefit, since after their introduction a decrease in chaotic zones and an increase in periodic zones take place. Influence of the additional local load is more effective with respect to energy dissipation as well as from the point of view of chaos control. The most important observation is that the methods of control of chaotic vibrations are the same for simple support, moveable and non-moveable support. 9.2.7

Wavelet based analysis

It has been shown earlier that a transition from periodic to chaotic vibrations of our mechanical system has been realized via the Feigenbaum scenario [Feigenbaum (1978)]. The latter scenario can be validated by numerical experiments of simple mathematical models. We have studied the shell with a simply supported moving resistance contour and the shell parameter b = 4. The Feigenbaum constant has been estimated for the control parameter q0 , where the bifurcations take place. The obtained numerical value q0,n − q0,n−1 = 4.65608466, n=5 αn = q0,n+1 − q0,n well coincides with its theoretical counterpart α = 4.66916224 [Schimmels and Palaiotto (1994)], and the difference is of 0.28%. In order to study bifurcational and chaotic vibrations of the flexible axially symmetric shallow shells the wavelet transforms on the basis of Gauss wavelets of order from m = 1 to m = 8 as well as

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real and complex Morlet wavelets have been applied. For further research, we have chosen the Morlet and Gauss of m = 1 and m = 8 wavelets as the most representative and reliable. In order to estimate the Feigenbaum constant α, we have used the one-dimensional (1D) Fourier transform, yielding also a 1D information on the relative input (amplitude) of various time scales (frequencies). Contrary to the 1D Fourier transform, the wavelet transform on 1D series yields 2D set of coefficients of the wavelet transform W (t, ω). Distribution of these coefficients values in the space (t, ω) (time scales, frequency localization) gives information of the evolution of relative input of different scales in time, and it is called a spectrum of coefficients of wavelet transform or equivalently a wavelet spectrum. In Figs. 9.42(a)–9.43(c) and 9.44 (a, b) for different values of the parameter q0 , the wavelet spectra of 1D signal (i.e. magnitudes of |W (t, ω)|) as well as surfaces in a 3D space and their projections on the surface (t, ω) in the form of 2D and 3D wavelets, Poincar´e maps, power spectra and phase portraits are given. The charts reported in Figs 9.42–9.44 allow to follow the change of amplitudes of wavelet transformation regarding various scales and in time, as well as pictures of curves of local extrema on those surfaces (the so-called skeleton diagrams). In Figs. 9.42–9.44, the abscissa axis denotes time, whereas the ordinate axis presents the time scale. Light chart regions correspond to large, whereas dark regions correspond to small values of the energy density |W (t, ω)|. Grey color of charts (Figs. 9.42–9.44) exhibits regions of wavelet transform values for small time intervals, which are gathered in the second column of each of the reported Figs. 9.42 and 9.43 in the enlarged window scale. In the case of the complex Morlet wavelet (Fig. 9.44) such a window denotes the time interval, for which in the second column a surface of arguments corresponding to the coefficients of the wavelet-transformation, is shown (the latter coefficients are complex numbers). Analysis of chaotic vibrations of flexible axially symmetric shallow shells under action of the transversal periodic load with the help of the wavelets Gauss 1 and Gauss 8 yields approximated qualitative results, and does not describe the whole complex picture of

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Fig. 9.42 row (a) Shell’s periodic vibrations: 2D wavelet spectrum (Gauss 1), its enlarged window, and 3D wavelet spectrum (Gauss 1); row (b) 2D wavelet spectrum (Gauss 8), its enlarged window, and 3D wavelet spectrum (Gauss 8); row (c) 2D wavelet spectrum (Morlet), its enlarged window, and 3D wavelet spectrum (Morlet); row (d) Poincar´e map, frequency power spectrum and phase portrait.

vibrations. The validated description of chaotic vibrations may yield only Morlet wavelets both real and complex. Our experience shows that in the case of the studied shallow rotational shells the most suitable for a reliable study are the complex Morlet wavelets. They

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Fig. 9.43 row (a) Shell’s chaotic vibrations: 2D wavelet spectrum (Gauss 1), its enlarged window, and 3D wavelet spectrum (Gauss 1); row (b) 2D wavelet spectrum (Gauss 8), its enlarged window, and 3D wavelet spectrum (Gauss 8); row (c) 2D wavelet spectrum (Morlet), its enlarged window, and 3D wavelet spectrum (Morlet); row (d) Poincar´e map, frequency power spectrum and phase portrait.

possess good properties of the real Morlet wavelet, i.e. good frequency localization from one side, whereas from the other side the values of arguments of the corresponding wavelet-coefficients yield additional information allowing to visualize the properties of self-similarity on different scales monitored in a chaotic signal (see Fig. 9.44(b) for

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Fig. 9.44 Chaotic vibrations of the shell using different scales (rows (a), (b) for q0 = 0.14: 2D wavelet spectrum (complex Morlet), its enlarged window, and 3D wavelet spectrum (complex Morlet).

q0 = 0.14). The mentioned benefit of the Morlet wavelet allows to win with other wavelets. A Morlet wavelet has the largest number of zero moments compared with other wavelets, which allows for reliable and validated description of chaotic vibrations of the flexible shallow shells. The complex Morlet wavelet can be also applied in those cases when we aim at observing possible violations of the uncertainty principle. It seems that the study shows that a choice of the wavelet type depends on the vibration regime, and has its own peculiarity. Though in quantum mechanics the Pauli wavelets have been mainly applied, it seems that the complex Morlet wavelet is most suitable for the problems of theory of flexible structural members including shells.

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Xie, D., Xu, M., Dai, H. and Dowell, E.H. (2014). Observation and evolution of chaos for a cantilever plate in supersonic flow. J. Fluids Struct., 50, pp. 271–291. Yamaguchi, T. and Nagai, K.-I. (1997). Chaotic vibrations of a cylindrical shellpanel with an in-plane elastic-support at boundary. Nonlin. Dynam., 13(3), pp. 259–277. Yang, X.L. and Sethna, P.R. (1991). Local and global bifurcations in parametrically excited vibrations of nearly square plates. Int. J. Nonlin. Mech., 26, pp. 199–220. Yang, X.L. and Sethna, P.R. (1992). Non-linear phenomena in forced vibrations of a nearly square plate: antisymmetric case. J. Sound Vib., 155, pp. 413–441. Yang, J. and Huang, X.L. (2007). Nonlinear transient response of functionally graded plates with general imperfections in thermal environments. Comp. Meth. App. Mech. Eng., 196, pp. 2619–2630. Yang, J. and Shen, H.-S. (2002). Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments. J. Sound Vib., 255(3), pp. 579–602. Yang, J., Kitipornchai, S. and Liew, K.M. (2003). Large amplitude vibration of thermo-electro-mechanically stressed FGM laminated plates. Comp. Meth. App. Mech. Eng., 192(35–36), pp. 3861–3885. Yao, M.-H. and Zhang, W. (2007). Shilnikov-type multipulse orbits and chaotic dynamics of a parametrically and externally excited rectangular thin plate. Int. J. Bif. Chaos, 17(3), pp. 851–875. Yao, M.-H. and Zhang, W. (2014). Using the extended Melnikov method to study multi-pulse chaotic motions of a rectangular thin plate. Int. J. Dynam. Control, 2, pp. 365–385. Yeh, T.L, Chen, C.K. and Lai, H.Y. (2003). Chaotic and bifurcation dynamics of a simply supported thermo-elastic circular plate with variable thickness in large deflection. Chaos Solit. Fract., 15, pp. 811–829. Yeo, M.H. and Lee, W.K. (2006). Evidences of global bifurcations of imperfect circular plate. J. Sound Vib., 293, pp. 138–155. Yu, W.Q. and Chen, F.Q. (2010a). Global bifurcations and chaos in externally excited cyclic systems. Commun. Nonlin. Sci. Num. Sim., 15(12), pp. 4007– 4019. Yu, W.Q. and Chen, F.Q. (2010b). Orbits homoclinic to resonances in a harmonically excited and undamped circular plate. Meccanica, 45(4), pp. 567–575. Zaks, M.A., Park, E.-H., Rosenblum, M.G. and Kurths, J. (1999). Alternating locking ratios in imperfect phase synchronization. Phys. Rev. Lett., 82, p. 4228. Zhang, W. (2001). Global and Chaotic dynamics for a parametrically excited thin plate. J. Sound Vib., 239, pp. 1013–1036. Zhang, J.-H., Zhang, W., Yao, M.-H. and Guo, X.-Y. (2008). Multi-pulse Shilnikov chaotic dynamics for a non-autonomous buckled thin plate under parametric excitation. Int. J. Nonlinear Sci. Num. Sim., 9(4), pp. 381–394.

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Index Baker’s maps, 35 barrier, vi, 258–261, 263 location, 258, 263 BCs non-symmetric, 164 symmetric, 164 the first kind, 172 the second kind, 173 the third kind, 173 the fourth kind, 129, 174 beam, 47, 119, 122–124, 128–130 cantilever, 5, 297, 299 composite, 2, 4 curvilinear, 130, 207–211, 223, 224, 232, 233, 237 curvilinear planar, 129, 207 functionally graded, 5 geometrically nonlinear, 2, 265 multi-layer, 269, 276, 295, 325, 328 transversal, 122, 128, 191, 208, 272 transverse loading, 2 two-layer, 292, 293, 297, 303 beams v–ix, 1–4, 33, 46, 55, 125, 129, 130, 140, 156, 169, 172, 193, 209, 224, 231, 238, 241, 263, 269, 275, 314, 324, 330, 339, 425 curved, 2 curvilinear planar, 129, 207 planar, 129, 140, 169 beam middle line, 122, 141, 233 Belousov–Zhabotinsky reaction, 44 Benettin algorithm, 55, 130, 384

acceleration, 89, 100, 114, 143, 170, 179, 184, 186, 188, 209, 342, 351, 440, 462 aero-thermo-elastic, 11 airplane wings, 2 algorithm, 42, 48, 133, 134, 156, 164, 179, 184, 185, 278, 280, 281, 295, 340, 346, 348, 350, 351, 368–370, 382, 384, 385, 432, 435, 438, 460, 492, 499, 500 amplitude, 36, 60, 64, 68, 80, 82, 83, 95, 104, 105, 109, 111, 113, 115, 116, 130, 144, 147, 149 frequency, 4 anti-symmetric case, 6 approach correlation, 364 interpolating, 364 assumptions, vii, 2, 115, 275, 307, 345, 350, 351, 406 asymmetry, 148, 154, 155, 159 attractor, 222, 223, 263, 402, 403, 425, 429, 430, 450, 452, 467, 520 chaotic, 402, 404, 425, 449 “doubled scroll”, 99 screw-type, 402 spiral, 402 strange, 402, 423, 506 strange chaotic, 92 three-scroll, 86, 101 two-scroll, 86, 101, 102 attractors of “axiom A”, 38 axis horizontal, 20, 64 vertical, 20 551

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bifurcation period doubling, 263, 265, 269, 293, 400–404, 406, 414–422, 451 first, 424 second, 224, 389 ninefold, 421 quintuple, 421 septuple, 421 tripling, 421 biomechanics, v, 30 body elastic-plastic, 239 ideally elasto-plastic, 239 rigid, 2, 272 Boissinesque convection, 468 Bubnov–Galerkin approach, 357, 367, 412 method, 340 buckling, vi, viii, 1, 2, 11, 256, 259, 263, 392, 393, 412, 436, 479, 500, 503 jump, 503 Cantor dust, 14–17, space, 17 Ces´ aro fractal, vii, 14–17, 27, 30, 38, 83, 509 Cash–Karp method, 378, 381–383, 396 Cauchy problem, 146, 157, 159, 169, 209, 231, 246, 370 chains 1D, 1 2D, 1 chaos vii, ix deterministic, 5, 249, 253 labyrinth, 86 transient, 27 chaos–hyperchaos, 139, 148, 151, 153, 167, 197, 207, 340, 425, 426, 428, 430, 431, 438 chaotic

attractor, viii, 31, 38, 68, 86, 94, 95, 99, 100, 402, 404, 425, 449, 506 phenomena, ix, 1, 5, 9, 459 vibrations, ix, 3, 5–8, 10, 11, 130, 149, 167, 169, 212, 213, 254, 256, 269, 281, 293, 307, 315, 331, 332, 329, 389, 394, 406, 428, 430, 459, 471, 475, 481, 490, 502, 504, 511, 520, 526 zones, 12, 40, 41, 197, 206, 314, 353, 389, 398, 411, 428, 432, 436, 469, 478, 516 Chebyshev algorithm, 183 mesh, 408, 410 clamped-clamped, 5 buckled, 3 Cohen–Daubechies–Feauveau wavelet, 63 coherent structures, 10 Coiflet wavelet, 63 compatibility, 325, 344, constraint, 8, 10, 74, 248, 258, 259, 263, 474 continuous object, 1, 173 system, ix, 1, 46, 47, 118, 144 control, viii, 24, 27, 42, 77, 100, 156, 169, 206, 207, 268, 315, 320, 322, 336, 437, 459, 477, 510, 513 chart, x, 157, 158, 164, 166–169, 190, 191, 197, 207, 211, 213, 291, 389, 396–398, 416–419, 422, 435, 436, 438, 443, 444, 458, 468, 469, 474, 475, 480, 483, 484, 510, 514, 515, 520, 523, 524, 526 cobweb, 14, 20, 24 diagram, 20 coefficient damping, 107, 108, 119, 122, 170, 208, 249, 277, 351, 462, 511 piezo-electric, 234 pyro-electric, 234

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Index

complex conjugated pair, 36 polynominals, 25 compressing, 20, 72, 81, 334, 346 stretching, 346 conditions asymptotic, 60 boundary, 352–355, 357, 359, 360, 367, 369, 391, 395, 397, 408, 412, 419, 421, 426, 432, 440, 461, 464–466, 472, 475, 481, 483, 490, 491, 493, 502, 504 initial, 21, 34, 35, 38, 50, 55, 57, 91, 93, 107, 108, 112, 117, 120, 121, 126, 146, 171, 182, 190, 209, 210, 231, 244, 246–248, 258, 284, 330, 353, 380, 406, 430, 442, 453, 459, 484, 492, 495 convergence, 20, 40, 68, 79, 94, 135, 146–148, 177–179, 182–184, 186–189, 211, 280, 330, 332, 340, 354, 386, 395, 396, 404–407, 410–412, 436, 437, 442, 469, 471, 472 Coulomb friction, 112, 113, 117 coupling bidirectional, 44, 45 electro-chemical, 236 unidirectional, 44 cycle neutral, 26 D’Alembert principle, 141, 170 damping critical, 109, 121, 127, 128, 280, 353 factor, 249 Kelvin’s structural, 11 supercritical, 128 undercritical, 127 viscous, 107, 119, 122, 124 Daubechies wavelet, 63, 68, 69, 79 deflection vi, vii, 5, 9, 11, 47, 109, 123, 129, 130, 141, 146, 149, 162,

9in x 6in

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553

169, 191, 208, 223, 233, 247–249, 255, 256, 258, 259, 261, 263, 265, 266, 270, 280, 281, 288, 314, 331, 342, 344, 345, 353, 357, 391, 395, 406, 416, 417, 419, 431, 447, 458, 461, 466, 472, 481, 490, 499, 500, 502, 520 maximal, 389, 390, 426, 428 deformation elastic-plastic, 130, 248, 259, 340, 341, 346, plastic, 2, 237, 256, 258, 342, 343, 346, 348, 349, 417–419 residual, 346 shear, v, 4, 5, 308, 327 tangential, 276, 308, 343 derivatives, 100, 114, 115, 133, 134, 175, 179, 247, 274, 279, 359–361, 364, 366, 380, 408, 410, 432, 433, 437, 441, 460 second, 409 Devaney’s definition, 34 diagonal, 20, 180, 360, 430 Dirac delta, 175 Dirichlet conditions, 410 -Neumann problem, 408 problem, 179, 494 discontinuous, 3, 16 discretization, 3, 72, 365 displacement, 2, 10, 87, 118, 119, 141, 169, 170, 208, 213, 233, 269–271, 276, 308, 312, 313, 325, 329, 342, 344, 349, 417, 439, 440 divergence, 10, 12, 23, 35, 50, 51, 90, 132, 154, 410, 444 averaged exponential, 51 domain post-critical, 10 spatial-temporal, 198 Donnell–Mushtari–Vlasov theory, 9 Duffing equation, 3, 56, 60, 62, 102, 357, 391 dyadic monoid, 16

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dynamics beam, 2, 3, 4, 213, 249, 253, 269, 289, chaotic, viii, 1, 3, 5, 7, 9–11, 13, 14, 20, 31, 38, 44–46, 100, 104, 105, 129, 166, 167, 206, 207, 237, 293, 307, 340, 389, 424, 425, 432, 437, 459, 504 master–slave, 44 nonlinear, ix, 7–10, 14, 24, 87, 94, 156–158, 190, 393, 421, 438, 449, 460, 503 periodic, 10, 21, 206, 396 population, 40 quasi-periodic, 10, 396 regular, 55, 319 effects inertial, vii thermal, viii, 129, 130 eigenfunctions, 120, 121, 124, 125 eigenvalues, 36, 42, 51, 52, 91, 120, 186, 385, 386, 499 eigenvector, 52, 53, 96 electric field, 3, 130, 233, 234 electricity, 233 electrostatics, 234 energetic characteristics, 82, 83 energy balance, 109 entropy, 50, 384, 444 KS, 138, 147, 148, 150–152, 154, 155 equation characteristic, 120, 124, 386, difference, 20, 33, 48, 111, 178–180, 365 fundamental, 109, 114, 311 heat transfer, 169 integral-partial differential, 8 “jerk”, 86, 100, 101 linearized, 36, 51, 54, 90, 164, 385, 391 non-homogeneous, 126 non-homogeneous differential, 102

nonlinear ordinary differential, 5 ordinary, 5 ordinary differential (ODEs), 5, 33, 35, 50, 80 128, 370 partial differential, 2, 3, 8, 32, 33, 367, 432 recurrence, 27 equilibrium position, 117, 246, 249, 255, 261, 263, 265, 283 Euler–Bernoulli, beam, viii, 2, 3, 4, 129, 130, 143, 148, 158, 167, 189, 193, 207, 238, 278 hypothesis, 238, 270, 273, 277, 313 theory, 3 Euler’s approach, 32 modified method, 372, 374, 381–383 excitation external, 8, 95, 104, 394, 398, 399 harmonic, 6, 293, 297, 314, 332, 380, 459, 467, 509, 511, 513 heat, 191 in-plane, 8 excitations parametric, 9, 104 experiment, x, 7–10, 35, 39–42, 44–47, 50, 93, 97, 116, 179, 182, 183, 271, 339, 346–348, 350, 354, 382, 387, 391–393, 396, 397, 416, 430, 436, 438, 450, 481, 525 external force, 32, 238, 312, 392, 406, 418, 436, 439 factor dissipative, 504 damping, 249 fatigue, vi Fehlberg method, 379, 381, 396 Feigenbaum constant, 14, 23, 41, 49, 265, 268, 400, 414, 421, 506, 510, 525, 526 filter, 364–366

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Index

flexible satellites, 2 fluid, 13, 32, 33, 45, 87, 88, 173, 174, 509 flutter instability, 10 formula explicit, 81, 496 Fourier coefficients, 64, 66, 80 series, 64, 66, 70, 104, 365, 498 spectra, 8, 10, 142, 317, 320, 334, 335 transformation, 47, 168, 317, 365 wavelet, 76 fractal dimensions, 7, 15, 18, 38, 50, 97, 444 Fredholm equation, 270 frequency amplitude, 4 excitation, 4, 515 fundamental, 223, 295, 335, 391, 394, 395, 400, 402, 405, 406, 412–414, 416, 425, 506 “saw”, 266 switching, 399 synchronization, 46 vibrations, 281, 289, 334, 389, 412 frequency–time localization, 70, 81 window, 71, 72, 81 friction hysteretic, 116, 117 internal, 239, 249 linear, 106, 353 nonlinear, 109, 117, 352 function autocorrelation, 47, 213, 223 integrable, 65, 66 mesh, 368, 497, 498 quadratic, 21, 134 functionally graded beams, 5 graded material (FGM), 7 fuselages, 2 gap, 46, 130, 263, 279, 280, 292

b2304-index

555

Gabor function, 75, 76 wavelet, 76 Galerkin procedure, 4, 9 Gauss wavelet, 321 geometric imperfection, 8, 10 nonlinearity, 4, 7, 12, 246, 289, 304, 355, 416–418, 437, 439 geometry, 13, 29, 208, 271, 472, 480, 481, 484 Ginzburg–Landau equation, 46 Gramm–Schmidt method, 55, 136 orthonormalization, 129 Green function, 175, 270, 272 relation, 273 Haar wavelet, 63, 67, 76, 149, 150, 152–154 Hamilton’s principle, 8 Hamiltonian, 50, 60, 61 Hertz, 64 H´enon maps, 136, 137 Hilbert space, 461 Hopf, bifurcation, 36, 38, 42, 206, 335, 400, 402, 414, 446, 447, 475, 479, 487, 510 Hook’s law, 116, 170 principle, 105 hyperchaos, ix, 137, 139, 140, 148, 151, 224, 340, 425–427 hypotheses, vii, 2, 141, 169, 207, 223, 241, 307, 314, 324, 339, 460, 504 imperfections, 8, 460 image encryption, 23 infinity, 4, 17, 25, 32, 253 in-plane excitation, 8 instability flutter, 10 pull-in, 3, 4

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Deterministic Chaos in One-Dimensional Continuous Systems

interaction acoustic fluid-structure, 11 intermittency in–out chaos–chaos, 44 one-off chaos–chaos, 44 interval, vi, 15, 16, 22–24, 47, 50, 55, 56, 71, 80, 84, 91, 93, 108, 130–133, 135, 146, 147, 159, 160, 190, 191, 197, 211, 213, 222, 247, 253, 256, 258, 259, 274, 275, 284, 289–291, 293, 301, 305, 314, 317, 319, 320, 322, 323, 327, 334–336, 339, 356, 358, 366, 367, 384, 393, 395, 398, 399, 406, 408, 409, 413, 417, 423, 424, 427, 428, 439, 441, 442, 446, 451, 453, 487, 496, 520, 526 invariance, 81, 82 investigation experimental, 7 iteration, 17–21, 25, 26, 40, 137, 179, 181–187, 345, 346, 351, 353, 384, 451, 452 Julia set, 25–27, 29, 30 jumps, 92, 224, 256, 259, 331, 390, 428 Kaplan–Yorke dimension, 101, 138, 147, 148, 150–153, 155 Karhunen–Loeve method, 8 kinematic viscosity, 32, 89 Kirchhoff–Love hypothesis, 439, 460 Koch cube, 19 curve, 18, 19 island, 17 snowflake, 17, 18 star, 17 Krenck–Nicholson method, 359 Kronecker delta, 176, Lagrange equation, 106, 107 functional, 327 laminar phase, 42 Laplace equation, 175, 179, 205, 209,

lattice, 35 layer boundary, 10, 410 pre-stressed graded, 7 turbulent boundary, 10 limit-cycle, 21 linear isotropic, 8 Littlewood–Paley (LP) wavelet, 64 loads aerodynamic, 10, 11 aero-thermal, 11 harmonic, 13, 209, 248, 280, 288, 314, 325, 330, 340, 350, 394, 446, 449, 450, 453, 467, 490, 511, 513 localization exact, 322 frequency, 72, 73, 526, 528 frequency–time, 70, 73, 81 logarithmic decrement, 108, 113 long-span bridges, 2 Lorenz equation, 90, 91, 95, strange attractor, 88 Lyapunov characteristic exponents, 50 dimension 10, 340 Mach numbers, 10 Mall procedure, 79 Mandelbrot set, 27–30, 452, 453 manifold Smale horseshoe, 5 unstable, 57, 58 manipulator, 2 maps horseshoe, 35 iterated, 20, 42 one-dimensional, 42, 96 Poincar`e, 10, 11, 13, 41, 42, 47–49, 52, 53, 139, 140, 149, 197, 198, 213, 222, 249, 253–255, 333–335, 414, 422, 430, 438, 447, 450, 453, 458,

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Index

481, 506, 509, 511, 513, 520, 526 Poincar`e pseudo, 50 matrix anti-symmetric, 59 Jacobi, 51, 58, 385, 386 Toeplitz, 365 mechanics v, 5, 30, 44, 45, 79, 99, 102, 103, 405, 421, 437, 438, 529 solid, 2 Melnikov approach, 104 distance, 58, 60 method, 5, 6, 31, 57, 62 technique, 62 mesh function, 368, 497, 498 method explicit, 182, 183, 360, 372, 406, 408, 410, 432, 433, 438 finite difference, 10, 129, 143, 168, 210, 314, 340 finite elements, 143 global perturbation, 6 implicit, 179, 406, 408, 432 integration, 396, 405, 410 Peaceman–Rachford Alternating Direction Implicit (ADI), 184 pseudo-spectral, 408–410, 432, 433, 437 triangle, 182 micro-electro-mechanical systems (MEMS), 3 microbeams, 3–5 middle line, 122, 141, 233, 277, 308, 309, 325 model elastic-plastic, 248 modulus shear, 4, 119, 126, 238, 277, 309, 310, 346, 348 Young’s, 119, 277, 342, 350, 351 moment bending, 2, 310, 313, 328, 343 zero, 69, 77, 79, 529

9in x 6in

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557

movable, 8, 345, 466, 467, 481, 484, 490, 491, 493, 502, 511 multi-fractal signals, 83 multiple scales, 3, 6, 9 multiplier, 67, 82, 181, 183, 344 nanotubes, 3 Navier–Stokes equation, 32 neural network, ix, 129, 132–134, 140, 147, 149, 156, 167 Newton’s, law, 45 method, 353 secants, 27 non-dimensional quantities, 171, 172, 236, 237, 240, 244, 277, 440, 462 non-homogeneity, 248, 391 nonlinear PDEs, ix, x, 1 quadratic, 86, 100 responses, 3 subsonic, 11 vibrations, v, 5, 8, 47, 304, 332, 341, 350, 353, 486 nonlinearity cubic, 3 geometric, 4, 7, 12, 246, 289, 304, 355, 416–418, 437, 439 physical, vi, 239, 253, 265, 281, 292, 304, 341, 415–417, 419 numerical methods, 86, 340, 369, 432, 438 Nyquist frequency, 366 ODEs first-order, 1, 100, 102, 247, 359, 371, 442, 463 first-order nonlinear, 1 non-autonomous, 102 operator, 35, 59, 135, 146, 186, 190, 208, 269, 270, 330, 360, 364–366, 408, 441, 462, 495, 499

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orbits homoclinic, 6 periodic, 22–24, 29, 52, 451, 453, 511 Shilnikov-type homoclinic, 6 orthogonal decomposition, 10 wavelets, 70 orthogonality, 68 oscillator continous mechanical, 45 Duffing–Van der Pol, 104 microbeam, 4 non-autonomous, 1 single-well, 86, 103 two-well, 103 van der Pol, 104 panel curved skin, 11 fluttering, 10 panels, v, vi, viii, ix, 1, 9–11, 340, 341, 346, 391, 436–438 parameters control, x, 24, 41, 56, 156, 157, 164, 197, 211, 268, 315, 320, 336, 387, 395, 428–430, 443, 453, 468, 474, 484, 510, 514, 515, 520, 523–525 non-dimensional, 33, 143, 209, 277, 330, 352 super-critical, 139 Parseval’s formula, 66, 83 theorem, 82 pattern, 116, 119, 360, 364, 367, 380, 381, 404, 406–409, 450 Paul wavelet, 79, 529 PDEs, vi, vii, viii, 5, 7–9, 11, 46, 94, 95, 129, 144, 146, 189, 209, 244, 307, 340, 353, 359, 367, 369, 437, 440, 441, 459, 460, 467 nonlinear, ix, x, 1, 208 Pelekh–Sheremetev beams, ix

periodicity, 20, 41, 47, 48, 104, 150, 156, 458, 459 quasi-, 45, 47, 48, 156, 320, 322 phase difference, 282, 288, 291, 293, 301, 304 locking, 38, 44, 45, 301, 302 portraits, 47, 152, 198, 222, 223, 247, 248, 258, 259, 266, 317, 322, 335, 403, 404, 414, 421, 430, 438, 443, 453, 481, 490, 503, 511, 526 space, 44, 49, 51, 55, 57, 90, 130, 143, 155, 255, 384, 420, 429, 430, 458 space compression, 138, 155 synchronization, 44, 45, 281, 289–291, 295, 302, 304, 305 phenomena chaotic, ix, 1, 5, 9, 459 doubling, 49 hysteresis, vi, intermittency, 254, 339 locking, 38 piezoelectric actuator, 7 plate axially moving, 9 buckled thin, 6 circular, 6, 7 point central beam, 266, 394 critical, 23, 26, 36, 37, 90, 91, 95, 400 elliptic, 57 fixed, 20, 21, 41, 42, 44, 92, 451, 452 heteroclinic, 5 hyperbolic, 57, 58, 61 Poisson’s coefficient, 238, 241, 276, 277, 310, 342, 391, 440, 462 polynomial, 25, 27, 75, 347, 360, 364, 408 approximation, 48, 81 pre-critical states, 32

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Index

pressure, 7, 9, 32, 33, 130, 272, 276, 281 contact, 270–273, 275, 278, 282, 289–291, 293, 306 Prince–Dormand method, 380, 381 problem contact, 270, 271, 274, 276 initial-boundary, 353 procedure averaging, 6, 115 process quasi-periodic, 35 transitional, 40, 87, 249, 256, 320, 334 protecting sensitivity, 39 Ramberg–Osgood formula, 347 randomness, 8 Rayleigh approximation, 89 beam theory, 4 number, 89, 91, 468 -Ritz procedure, 9 reduction, vi, vii, x, 12, 189, 270, 340, 353 regime chaotic–hyperchaotic, 153 harmonic, 223, 394, 515 regularity, 32, 34, 48, 82, 83, 206, 266 local, 82, 83 relation finite-difference, 279, 492 recursion, 20 relaxation, vi, 97, 179, 180, 185, 187, 188, 280, 459, 494, 497, 499 reliability, 129, 190, 210, 314, 330, 340, 368, 384, 387, 391, 393, 412, 432, 433, 436–438, 441, 459, 467, 503 resonances internal, 3, 6, 12 ultra-sub-harmonic, 10 results, vi, viii–x, 5, 7, 9, 33, 40–42, 46, 56, 68, 80, 87, 93, 129, 136–139, 147, 149, 156, 158, 160, 162, 164,

9in x 6in

b2304-index

559

167–169, 172, 177, 178, 187, 188, 190, 193, 197, 198, 211, 224, 232, 256, 263, 275, 280, 281, 291, 293, 295, 305, 314, 315, 317, 320, 321, 323, 330, 331, 339, 340, 348, 354, 364, 368, 380, 382–384, 387, 389, 393, 395–397, 400, 406, 410, 412, 414, 421, 422, 428–430, 433, 436–438, 442, 443, 467, 470, 474, 479, 481, 486, 494, 500, 502, 503, 514, 520, 523, 524, 526 rheological properties, vi rib, 341, 342, 344, 345 Rikitake chaotic attractor, 86 rings, 147, 151, Ritz approach, 468 robust, 187, 222–224, 228, 230 rotary inertia, v, 4, 5, 7 Rayleigh theory beam, 4 Runge principle, 314, 412, 442 Runge–Kutta fourth-order method, 12, 146, 157, 169, 190, 210, 246, 248, 381, 396, 463 sixth order method, 158, 159, 169 schemes difference, 367, 406, 499 explicit, 406 implicite, 359, 365, 437 Seidel’s method, 181, 182, 187 self-similarity, 16, 27, 75, 76, 83, 528 series time, 42, 47, 48, 131, 146, 148, 295, 332 trigonometric, 272 wavelet, 66, 70 shafts, 2 Shannon wavelet, 64 Sharkovsky’s bifurcation, 340 exponents (SE), 453 periodicity, 420, 451, 511

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Deterministic Chaos in One-Dimensional Continuous Systems

series, ix theorem, 23–25, 424, 451, 453 shear deformation, v, 4, 5, 308, 327 modulus, 4, 119, 126, 238, 277, 309, 310, 346, 348, 453 shell conical, ix, 459, 464, 467, 471, 472, 475, 477, 480, 481, 484, 486 shallow, v, 12, 459, 468, 474, 480, 494, 525, 526, 529 spherical, 12, 207, 466, 474, 475, 477–479, 484, 490, 504, 506, 509 shell-panel cylindrical, 9, 10 double-curved, 10 Sheremetev–Pelekh beam, 307, 324, 332, 339 model, 324 Sierpi´ nski carpet, 17 sigmoid, 132, 133 simply supported, 3, 4, 6–12, 148, 164, 171, 197, 212, 213, 244–246, 248, 258, 280, 299, 330, 359, 413, 419, 436, 437, 481, 506, 509, 524, 525 singularity, 83, 175 slope, 83, 416 snap-through, 10 solution asymmetric periodic, 11 fundamental, 51, 175, 177 sound, 9, 10, 87 space energetic, 461 functional, 66–68, 79 metric, 34 phase, 44, 49, 51, 55, 57, 90, 130, 143, 155, 160, 255, 384, 420, 429, 430, 458 S-shaped, 182

spectra frequency, 11, 129, 206, 267 power, 47, 146, 160, 197, 249, 304, 323, 331, 332, 380, 381, 387, 389, 394, 402, 406, 411, 430, 438, 443, 453, 470, 473, 479, 511, 526 spectrum frequency-temporal wavelet, 288, 317, 335 global energy, 84 spring, 104 stabilities dynamic, 11 static, 449 stability, vi, viii, 1, 11, 24, 35, 36, 38, 39, 90, 91, 143, 147, 164, 212, 233, 283, 292, 293, 303, 304, 314, 331, 340, 353, 367, 389, 390, 406, 407, 409, 425, 427, 431, 446, 449, 451, 452, 459, 477, 486, 490, 500, 502, 503, 509, 510, 520 stochasticity, 32, 50, 425, 444, 478 strain hardening, 347 stress internal, 310, 328 normal, vii, 141, 169, 241, 273, 307, 325 stress-strain intensity, 349 stretching–compressing, 346 strings, vi, 125 strip beam-, 341, 342, 344–346, 350, 351, 380, 383, 385, 386, 391, 392 plate-, ix, 340, 342, 343 structural damping, 11, members, v, vi, viii–x, 1, 4, 32, 33, 46, 125, 384, 392, 467, 529 Str¨ omberg wavelet, 64 subharmonic, 265, 266, 398, 399, 406, 430, 436 superposition, 66, 68

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Index

support clamping–clamping, 352 simple–clamping, 352 simple–simple, 352 surface middle, 241, 266, 275, 276, 307, 312, 313, 325, 328, 343, 344, 439, 440 middle beam-strip, 342 space–time, 146 spatial-temporal, 197, 206 Symlets wavelet, 64 synchronization identical, 45 imperfect phase, 45 phase, 45, 281, 289–291, 295, 302, 304, 305 system continuous, ix, 1, 46, 47, 118, 144 continuous 1D, 118 dissipative, viii, 50, 57, 105, 428, 429 two-degree-of freedom, 8 tangent hyperbolic, 134, 135 vector, 50, 54, 55 Tchebishev’s secants, 27 temperature, 7, 13, 88–90, 129, 130, 172–174, 190, 232, 242, 295, 495 fields, 129, 169, 170, 172, 174, 180, 190–193, 197, 198, 206, 207, 211, 224, 231, 233, 237 -dependent, 7, 11 -variable, 7 theory beam refined, 5 perturbation, 58 thermal environment, 7, 8 thermoelastic, 7 Timoshenko beam, 307, 314, 339 hypothesis, 313 model, 4, 5, 312, 325, 327, 328

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561

trajectories non-periodic, 52 periodic, 51, 430 transformation inverse, 68–70 transition flat, 365, 406 smooth, 1 universal, 45 transitive, 34 triangles, 17, 18 turbulent, 10, 32, 33, 87, 94, 431, 432 boundary layer, 10 two-dimensional elasticity, 4 universality, 22, 402 validity, 177, 190, 210, 242, 259, 314, 368, 389, 392, 437, 503 value characteristic, 36, 346, 348 critical, 35, 36, 41, 89, 91, 259, 265, 268, 392, 468 vector excitation, 130 normal, 59 tangent, 50, 54, 55 velocity, 33, 35, 50, 52–54, 89, 105, 109, 110, 112, 114, 117, 179, 236, 248, 259, 352, 444 average, 53, 54 vibration chaotic, ix, 3, 5–8, 10, 11, 130, 149, 167, 169, 212, 213, 254, 256, 269, 281, 293, 307, 315, 331, 332, 339, 389, 394, 406, 428, 430, 459, 471, 475, 481, 490, 502, 504, 511, 520, 526 free, 5, 108, 119, 121, 391, 392, 436 lateral forced, 3 multi-frequency, 334

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Deterministic Chaos in One-Dimensional Continuous Systems

nonlinear, v, 5, 8, 47, 304, 332, 341, 350, 353, 486 periodic, 149, 167, 193, 197, 213, 224, 250, 263, 269, 314, 317, 319, 332, 333, 387, 389, 390, 398, 399, 412, 423, 426–428, 430, 436, 446, 471, 473, 474, 479, 481, 490, 491, 502, 503, 509, 511, 516, 517, 520, 523 quasi-periodic, 249, 253–256 rod, 119 steady-state periodic, 250 string transversal, 125 transverse, 3, two-frequency, 281, 289 voltage, 4, von K´ arm´ an form, 141, 169, 308, 325 PDEs, 7 plate, 8, 9 strains, 3 wave sinusoidal, 65, 66, 68, 71, 77 turbulence, 7 well-localized soliton-type small, 66 wavelet analysis, 31, 47, 63, 65, 77, 280, 293, 315, 322, 339 beta, 64

hat French, 76 Mexican, 64 orthogonal, 70 “small wave”, 63, 66, 68 spectrum, 80, 84, 281, 282, 288, 293, 315, 317–320, 323, 333–336, 339, 526, 527 valued complex, 64 real, 64 wavelet transforms continuous, 63, 64, 68, 82 discrete, 63, 68, 70 harmonic, 63 multiresolution-based, 63 Winkler’s model, 275 relation, 276 Wolf algorithm, 129–131, 162, 164, 168, 193, 211 method, 156, 167

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