Modeling, analysis and control of dynamical systems : with friction and impacts 9789813225282, 9813225289

Citation preview

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

10577hc_9789813225282_tp.indd 1

12/6/17 9:58 AM

WORLD  SCIENTIFIC  SERIES ON  NONLINEAR  SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A.

MONOGRAPHS  AND  TREATISES*

Volume 77:

Mathematical Mechanics: From Particle to Muscle E. D. Cooper

Volume 78:

Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations A. M. Samoilenko & O. Stanzhytskyi

Volume 79:

Robust Chaos and Its Applications Z. Elhadj & J. C. Sprott

Volume 80:

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science (Volume V) L. O. Chua

Volume 81:

Chaos in Nature C. Letellier

Volume 82:

Development of Memristor Based Circuits H. H.-C. Iu & A. L. Fitch

Volume 83:

Advances in Wave Turbulence V. Shrira & S. Nazarenko

Volume 84:

Topology and Dynamics of Chaos: In Celebration of Robert Gilmore’s 70th Birthday C. Letellier & R. Gilmore

Volume 85:

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science: (Volume VI) L. O. Chua

Volume 86:

Elements of Mathematical Theory of Evolutionary Equations in Banach Spaces A. M. Samoilenko & Y. V. Teplinsky

Volume 87:

Integral Dynamical Models: Singularities, Signals and Control D. Sidorov

Volume 88:

Wave Momentum and Quasi-Particles in Physical Acoustics G. A. Maugin & M. Rousseau

Volume 89:

Modeling Love Dynamics S. Rinaldi, F. D. Rossa, F. Dercole, A. Gragnani & P. Landi

Volume 90:

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

Volume 91:

Control of Imperfect Nonlinear Electromechanical Large Scale Systems: From Dynamics to Hardware Implementation L. Fortuna, A. Buscarino, M. Frasca & C. Famoso

Volume 92:

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts P. Olejnik, J. Awrejcewicz & M. Fečkan

*

To view the complete list of the published volumes in the series, please visit: http://www.worldscientific.com/series/wssnsa

Lakshmi - 10577 - Modeling, Analysis and Control of Dynamical Systems.indd 1

20-06-17 3:28:06 PM

NONLINEAR SCIENCE WORLD SCIENTIFIC SERIES ON

Series A

Vol. 92

Series Editor: Leon O. Chua

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts Paweł Olejnik Jan Awrejcewicz Lodz University of Technology, Poland

Michal Feckan ˘ Comenius University in Bratislava, Slovakia

World Scientific NEW JERSEY

•

LONDON

10577hc_9789813225282_tp.indd 2

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TAIPEI

•

CHENNAI

•

TOKYO

12/6/17 9:58 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

World Scientific Series on Nonlinear Science Series A — Vol. 92 MODELING,  A NALYSIS  A ND  CONTROL  OF  DYNAMICAL  SYSTEMS WITH  FRICTION  A ND  IMPACTS Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-3225-28-2

Printed in Singapore

Lakshmi - 10577 - Modeling, Analysis and Control of Dynamical Systems.indd 2

20-06-17 3:28:06 PM

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Preface

The monograph is directed primarily towards practicing engineers and applied mathematicians willing to strengthen their knowledge in the broad areas of piecewise-smooth mechanical systems extended from the essential mathematical and numerical modeling to the control of systems with friction and impacts. We have sectioned this monograph into fourteen chapters. In the first chapter is explained the significant role of various friction laws applied to mathematical modeling of a friction phenomenon in engineering sciences. Both advantages and disadvantages of the frictional processes are taken into deep consideration and discussed. It is shown how the static and dynamic friction laws and modern friction theories coexist in pure and applied mathematics. The important role of purely theoretical and experimental investigations is attenuated, wherein the appropriate friction models regarded in particular applications are outlined, putting special emphasis on the well-established approaches. The very popular belt-springblock model is investigated in Chapter 2. It has been designed to include variations of the normal load during the braking process. We show that due to the adiabatically slowing down velocity of the moving base, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. In particular, the creep-slip leading to a significant spectral widening of the dynamics is observed at the final phase of the process. Discrete modeling of discontinuous dynamical systems has to be performed with the use of highly accurate numerical procedures that will further guarantee the proper use of tools applied for their dynamical analysis. Therefore, Chapter 3 presents an application of H´enon’s method to obtain satisfactory numerical estimations of the stick-slip transitions existing in the Filippov-type discontinuous dynamical systems with dry friction. Subsequent sections focus on the problem definition, application of the method that was originally proposed for Poincar´e maps, its use in the estimation of phase trajectories, bifurcation diagrams of tangent points as well as in the estimations of Lyapunov exponents. The time moment of appearance of zero relative velocity during a relative displacement of contacting bodies in dynamical systems with dry friction can be precisely computed. Chapter 4 shows an approximation method that increases the quality of estimation of approximate v

page v

June 8, 2017 12:9

vi

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

discontinuous solutions observed in the stick-slip motion with friction. Results of numerical computations obtained by the method of approximation of solution of a one-degree-of-freedom dynamical system with friction confirm good accuracy of the determination of stick phases. Bifurcations from sliding homoclinic to bounded solutions for certain discontinuous planar systems under periodic perturbations are studied in Chapter 5. We make necessary assumptions and then analytically solve a system of piecewise linear differential equations with periodic perturbations. In Chapter 6, we follow a functional analytic approach to study the problem of chaotic behavior in time-perturbed impact systems whose unperturbed part has a piecewise continuous impact homoclinic solution that transversely enters the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has impact solutions that behave chaotically. A new class of relatively simple chaotic impact system consisting of non-flat billiards is introduced in Chapter 7. The particle of unitary mass moving on a Cartesian surface is subjected to the gravity and to a small amplitude periodic or almost periodic forcing and is reflected with respect to the normal axis when it hits the boundary of the billiard. We prove that the unperturbed problem has an impact on homoclinic orbit. Moreover, we derive a Melnikov type condition so that the perturbed problem exhibits chaotic behavior in the sense of Smale-like horseshoe. In Chapter 8, a planar frictional contact created by two rotating rigid bodies of a double torsion pendulum with kinematic forcing is subject to an analysis devoted to a parameter identification. Having a model time series of the system’s state variables, the disturbed parameters of the pendulum are estimated with the use of Nelder–Mead simplex direct search method. An asymmetric pendulum being a part of the twodegree-of-freedom mechanical system with friction that is taken into consideration in Chapter 1 is subjected to an identification in Chapter 9 to the observed influence of some resistance of its rotational motion in ball bearings. Complex damping of the motion can be considered as a nonlinear state-dependent damping. It is presumed that the effective nonlinear damping characteristics depend on a few effects such as the fluid friction-caused vibrations of the pendulum with two springs in the air as well as unknown kinds of a frictional resistance existing in ball bearings. Transient response oscillation of the pendulum is described by the explicitly state-dependent free decay. A free decay test of the pendulum with the state-dependent nonlinear parameters of damping and stiffness is described as well. In Chapter 10, a result for an existence of almost periodic sequences of ordinary differential equations with linear boundary value conditions is derived by means of the Banach fixed point theorem together with a method of majorant functions. We also propose an application to a damped pendulum with a jumping length and external force. The analysis of dynamical systems also requires some auxiliary methods in regards to the solution of algebraic equations. Chapter 11 brings a description of a gradient method for the solution of nonlinear algebraic equations in the analysis of stability of equilibria of a two-degree-of-freedom system with dry friction. General theoretical

page vi

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Preface

1st Reading

ws-book975x65

vii

considerations are introduced, and afterwards, the numerically estimated diagram of stability is subject to a qualitative assessment. The problem of control and dynamical modeling of a wheeled double inverted pendulum with rolling friction vibrating in a plane perpendicular to the direction of movement is taken into consideration in Chapter 12. The object of analysis consisting of a wheel and the double pendulum has been described by means of equations of motion derived using the Lagrange equation of second kind. The aim of control is to maintain the upper link at an unstable equilibrium point around the given angular position. Control moment of force is applied to the wheel in a numerical procedure utilizing the Kalman filtering approach. In some discontinuous dynamical systems with nonlinearities emitted by dry friction, a controller has to be designed to avoid the steady-state tracking errors or even some undesirable self-excited vibrations. In Chapter 13, we examine the influence of the dry friction-caused discontinuities on the controlled dynamics of a bearing rotor. Finally, Chapter 14 presents a simulation and control of a building structure subjected to stochastic excitation. The problem is reduced to a two-degreeof-freedom system with an approximated frictional discontinuity introduced by the Saint–Venant element. Encouraging the scientists to read this monograph, we would like to attenuate its considerable scope of the miscellaneous aspects of modeling of the dynamical systems with friction and impacts. The two studied significant phenomena of classical mechanics are supported by careful mathematical derivations proven by numerical computations confirming interesting results which are visualized on the plots of the time histories, phase planes and bifurcation diagrams. Moreover, this monograph examines related control problems seeking to improve properties of the considered piecewise-smooth dynamical systems. Mathematical models presented in this monograph will undoubtedly increase in importance in numerical experiments, experimental measurements and optimization problems found in applied mechanics. We believe that the proposed source of the theoretical and practical knowledge will strengthen the skills of physicists and mechanical engineers in modeling, analysis and control of discontinuous systems with friction and impacts. Pawel Olejnik Jan Awrejcewicz Michal Feˇckan

page vii

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Acknowledgments

The authors greatly acknowledge the assistance of Flaviano Battelli and Valery Pilipchuk who have helped in carrying out the research. Michal Feˇckan is partially supported by the Grant VEGA-MS 1/0071/14, and by the Slovak Research and Development Agency under the contract No. APVV14-0378.

ix

page ix

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Contents

Preface

v

Acknowledgments

ix

1.

Friction Laws in Modeling of Dynamical Systems 1.1 1.2

1.3 1.4

2.

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

Transient Friction-Induced Vibrations in a 2-DOF Braking System 2.1 2.2 2.3 2.4 2.5

3.

Introduction . . . . . . . . . . . . . . . . . . . Modeling of Dry Friction . . . . . . . . . . . . 1.2.1 Classic Friction Laws . . . . . . . . . 1.2.2 Modern Description of Dry Friction . Dry Friction and Dynamical Systems Theory Friction Advantages: A Brake Mechanism . . 1.4.1 Engineering Approach . . . . . . . . . 1.4.2 The Model . . . . . . . . . . . . . . . 1.4.3 Experimental Investigations . . . . . 1.4.4 Dynamical Analysis — Bifurcations .

1

Introduction . . . . . . . . . . . . . . . . . . . . . Mathematical Modeling of the Belt–Spring–Block Conditions for the Numerical Tests . . . . . . . . The Creep–Slip Response . . . . . . . . . . . . . Three-Dimensional Bifurcation Diagrams . . . . .

. . . . Model . . . . . . . . . . . .

. . . . .

. . . . .

1 3 3 4 30 40 41 44 45 47 49

. . . . .

. . . . .

. . . . .

. . . . .

49 51 53 54 59

Numerical Estimation of the Stick–Slip Transitions

61

3.1 3.2 3.3 3.4

61 63 66 68 70 72

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function of Boundary Transition Through a Discontinuity . . . . . Numerical Estimation of the Stick to Slip and Slip to Stick Transitions Bifurcations of Tangent Points . . . . . . . . . . . . . . . . . . . . 3.4.1 Tangent Points on the Oscillating Boundary of Discontinuity 3.4.2 A Two-Periodic Stick–Slip Numerical Solution . . . . . . . xi

page xi

June 8, 2017 12:9

ws-book961x669

xii

4.3 4.4

6.

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

77 81 82 83 84 87

5.1 5.2 5.3

89 90 91

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calculation of Homoclinic Solutions . . . . . . . . . . . . . . . . . . The Equation of Bifurcation . . . . . . . . . . . . . . . . . . . . . .

Occurrence of Chaos in Forced Impact Systems

95

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Problem Definition . . . . . . . . . . . . . . . . . . . . . Looking for an Impact Solution . . . . . . . . . . . . . . The Equation of Bifurcation . . . . . . . . . . . . . . . . Almost Periodic and Periodic Cases in Chaotic Behavior Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Impact Planar Systems . . . . . . . . . . . . . . 6.6.2 Impact Coupled Second-Order Systems . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Homoclinic Impact Solutions . . . . . . . . . . . . . . . Constructing the Melnikov Function . . . . . . . . . . . Chaotic Behavior . . . . . . . . . . . . . . . . . . . . . . Symmetry Conditions in Finding the Melnikov Function

Introduction . . . . . . . . . . . Mathematical Modeling . . . . Identification of Parameters . . Application of the Identification

. . . . . . . . . . . . . . . . . . . . . Procedure .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

95 99 101 104 112 117 118 122 127

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Parameter Identification of a Double Torsion Pendulum with Friction 8.1 8.2 8.3 8.4

9.

77

89

7.1 7.2 7.3 7.4 7.5 8.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Discontinuity in Models of Dry Friction . . . . . . . . 4.2.1 Zero Value in Numerical Integration . . . . . . 4.2.2 A Continuous Approximation of Step Function The Method of Smooth Approximation . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . .

73 73 74

Bifurcations in Planar Discontinuous Systems

6.1 6.2 6.3 6.4 6.5 6.6

7.

Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Computation of Lyapunov Exponents Using Time Series . 3.5.2 Lyapunov Exponents of Typical Trajectories . . . . . . . .

Smooth Approximation of Discontinuous Stick–Slip Solutions 4.1 4.2

5.

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

3.5

4.

BC: 10577 - Modeling, Analysis and Control of DS

. . . .

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

. . . .

. . . .

. . . .

127 130 140 146 151 159

. . . .

. . . .

159 162 164 165

167

page xii

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Contents

9.1 9.2

9.3

xiii

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Estimation of the Nonlinear Characteristics of Damping and Stiffness171 9.2.1 Estimation of the Polynomial Decay . . . . . . . . . . . . . 173 9.2.2 Estimation of the Variable Angular Frequency . . . . . . . 174 9.2.3 Estimation of the Time-Varying Stiffness and Damping . . 175 The Nonlinear Approximations — Verification Cases . . . . . . . . 176

10. Almost Periodic Solutions for Jumping Discontinuous Systems

181

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.2 Almost Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . 182 10.3 A Damped Pendulum With a Jumping Length and External Force 187 11. Solution of Nonlinear Algebraic Equations in Analysis of Stability 11.1 11.2 11.3 11.4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . The Gradient Method . . . . . . . . . . . . . . . . . . . . Analysis of Stability of Equilibrium States . . . . . . . . . Stable and Unstable Branches on the Diagram of Stability

. . . .

191 . . . .

. . . .

. . . .

. . . .

12. Control of a Wheeled Double Inverted Pendulum with Friction 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Overview of Exemplary Inverted Pendulums . . . . . . 12.3 Modeling of the 2-DOF Inverted Pendulum Driven by Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Numerical Modeling of System Dynamics . . . . . . . 12.5 Kalman Filter Based Control of the Pendulum . . . . 12.5.1 Simplification and Linearization of the Model . 12.5.2 Analysis of System Stability . . . . . . . . . .

. . a . . . . .

197 . . . . . . . . . . . . Rotating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13. Tracking Control of a Discontinuous System with Stick–Slip Friction 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 The Algorithm of Sliding Surface Control . . . . . . . . 13.2.1 Estimation of Linear and Nonlinear Parameters 13.2.2 The Low-Voltage Control of Rotational Velocity 13.3 Numerical Simulation . . . . . . . . . . . . . . . . . . .

191 192 193 194

. . . . .

. . . . .

. . . . .

197 199 201 204 204 206 206 209

. . . . .

. . . . .

. . . . .

14. Controlling Stochastically Excited Systems with an Approximate Discontinuity 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The Saint–Venant Element Modeling the Dry Friction Contact 14.2.1 The Nonlinear Parallel Spring Connection . . . . . . . . 14.2.2 The Viscous–Elastic Model . . . . . . . . . . . . . . . . 14.3 The 2-DOF Mechanical Model . . . . . . . . . . . . . . . . . . .

209 212 213 214 215

221 . . . . .

. . . . .

221 222 223 223 225

page xiii

June 8, 2017 12:9

ws-book961x669

xiv

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

14.3.1 Time Histories of State Variables . . . . . . . . . . . 14.3.2 The Amplitude–Frequency Diagrams . . . . . . . . 14.4 The Approximate System with Friction . . . . . . . . . . . . 14.4.1 Energetic Criterion . . . . . . . . . . . . . . . . . . 14.4.2 Normal Modes of Vibrations . . . . . . . . . . . . . 14.5 Control of the 2-DOF System Under a Stochastic Excitation 14.6 Numerical Simulation . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

226 226 228 228 229 231 233

Bibliography

235

Index

253

page xiv

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 1

Friction Laws in Modeling of Dynamical Systems

The first chapter is devoted to the significant role of various friction laws applied to mathematical modeling of a friction phenomenon in engineering sciences. Both advantages and disadvantages of the frictional process are taken into a deep consideration and discussed. It is shown how the static and dynamic friction laws and modern friction theories coexist in pure and applied mathematics. An important role of purely theoretical and experimental investigations is attenuated, wherein the appropriate friction models regarded to a particular applications are outlined, putting special emphasis on the well-established approaches like the models proposed by Bay–Wanheim, Dahl, Bliman–Sorine, Lund–Grenoble and many others. Friction treated as a complex process being in an interaction with wear, emission of heat and deformation is also partially considered. Then an influence of dry friction models on the theory of modeling of dynamical systems is reviewed. At the end of this chapter, including the experimental and numerical analyses, an application of the asymmetric friction force model used in investigation of a dynamical response of a mechanical system with two-degree-of-freedom is presented.

1.1

Introduction

The relative motion of solid bodies with regard to adhesion is an disequilibrial process, in which the kinetic energy is transformed into the energy of irregular microscopic movement. This phenomenon is related to dissipation of energy (as heat, for instance) and leads to friction and wear on surfaces of the contacting bodies. Widespread occurrence of friction in technology and in everyday life is the reason for extensive scientific research that would result in widened theoretical knowledge in the process of friction. The aim of such research is to provide a detailed comprehensive description of this phenomenon. In contrast to many advantageous uses of friction, for instance in metalworking, movement of vehicles, drive transmission with the use of frictional elements, even walking or vibration of strings in musical instruments, there are numerous negative aspects of friction in the form of noise, wear and unpredictable behavior of various 1

page 1

June 8, 2017 12:9

2

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

mechanisms. Special attention needs to be paid to harmful self-excited vibrations in engineering systems with friction that are associated with a periodical supply of energy from a constantly operating source that is controlled by the system’s motion by means of a feedback mechanism. Such a relation leads to a reciprocal reaction between the regulating device and the nonlinear vibrating system; thus the selfexcited system may control its own energy balance. As a result, despite inevitable losses, some non-disappearing periodic vibrations can be exhibited by the system. Self-excited vibrations in kinetic junctions are usually harmful and undesirable. Especially in extreme cases, they may result in destruction of a vibrating object. Among the phenomena, those in question are the vibrations caused by rotation of air around oscillating flexible connectors, strings or coatings (vibrations of electric wires or a plane’s wing “flatter”) [Ibrahim (1994b)]. Mechanical systems with coupled movements may generate self-excited vibrations at the expense of energy from the drive system. Such characteristics may be attributed to the “shimmy” vibrations, which cause snaking of a car’s front wheels. Among typical examples of the occurrence of self-excited vibrations in nature are sounds of speech and whistle of slender twigs in the wind. However, the same vibrations that cause such acoustic phenomena as a creak of old hinges or a loosened joint or a grid of a blot are associated with the negative characteristics of dry friction. The disturbance in the fluency of motion caused by self-excited vibrations resulting from an abrupt change of real sliding speed of frictional bodies is called a stick–slip effect. It can often be observed in measuring devices, precision tools and during machining processes. The basic property of friction conductive to stick–slip vibrations is the fact that the maximum friction force at the adhesion loss is larger than the force that occurs in the system during the slip phase at low relative speeds of sliding bodies. Mechanical devices with kinetic junctions characterized by friction of this type are exposed to additional mechanical vibrations, which in many cases cause disturbances in operation or damages. The occurrence of friction-excited vibrations in such systems as a knife and a support or a stock and a metal working machine results in worsening of the processed surface roughness. Scientific treatises concerned with research on self-excited vibrations, including stick–slip vibrations, aim at thorough exploration of the essence of this phenomenon. An occurrence of self-excited vibrations results from the lack of stability of the states of equilibrium or stabilized motion. The presumable presence of such vibrations in a kinetic slide junction should be detected, because the states of equilibrium may fall into undesirable vibrations leading to malfunction of the machine. Therefore, various scientific research works that determine the characteristics of friction, study on stability or identification of the parameters of friction with the dominating role of self-excited vibrations significantly contribute to general understanding and elimination of that phenomenon together with its usually unwelcome side-effects on operating mechanical devices.

page 2

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

1.2

ws-book975x65

3

Modeling of Dry Friction

All mechanical and mechatronic systems that effectively use friction require appropriate designing, which involves accurate understanding and detailed phenomenological description of the contact area. Excessive friction of the elements during vibration often causes decreasing precision of a mechanism’s work and generates limit cycles or loss of smoothness caused by stick–slip effect. Therefore, various compensation techniques have been developed in order to improve dynamics and to minimize the negative influence of friction. The procedures employed in such cases usually consist of accurate determination of conditions in a friction point, application of friction law and determination of all movement coefficients. A correct theoretical description supported by an experimental model embracing all of the elements operating together lay basis for compensation of friction effects in vibrating systems. Unfortunately, a long-term research on external friction of solid bodies, which is common in every-day life and in technology as well, has not developed a general theory that would have fully explained this phenomenon. Some hypotheses, formulated at the turn of centuries, were based on the process of dry friction that belongs to the general theory of dynamic friction of solid bodies. The modern description of friction presented in Sec. 1.2.2 is based on well-known models. Making a simplification, all the models delivered in Sec. 1.2.2 treat static and dynamic friction as two separate processes from the dynamic modeling point of view. It is worth noticing, that in general, between static and dynamic friction there exists a region during which the solid rubbing bodies get a very small relative shift due to surface layers’ elastic deformations before the friction surfaces slip. The phenomenon may have great influence on the stick–slip behavior on certain conditions especially when the slip phase is small enough.

1.2.1

Classic Friction Laws

It is a generally known fact that external dry friction occurs on non-lubricated surfaces of solid bodies that move relative to each other during adhesion. Research on friction conducted in the period between the 15th and the 19th century allowed Moore to formulate the classic friction laws presented below [Moore (1975)]. It was Leonardo da Vinci who concluded that the friction coefficient is not dependent on the nominal contact surface. He developed his theory from an observation of friction of a coiled rope stretched out by applying a force. Leonardo da Vinci’s mechanical theory was based on the observed phenomenon of the bodies varied capability to slip, which resulted in varied values of friction. The law formulated by Leonardo da Vinci is limited in certain aspects since it is true only for the materials of determined yield points. At the end of the 17th century Amontons formulated an observation-based dry

page 3

June 8, 2017 12:9

4

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

friction law which claimed that the friction force was proportional to the normal force perpendicular to the contact surface of frictional bodies (F = μN ) with a dimensionless coefficient of proportionality μ, independent of a load. In fact, the friction coefficient depends on the load as well as on the mechanical, geometrical and chemical properties of the frictional surfaces. Despite the fact that Amontons’ formula lacks accuracy, it has been widely applied in calculations. The next classic friction law derived from an observation that the static friction coefficient is larger than the dynamic friction coefficient, as a matter of fact cannot be applied either to elastic or viscoelastic materials. That leaves us with one more classic friction law that claims that the friction coefficient is independent of the sliding speed. It is obviously a false statement since it makes the law applicable only to metals at low and medium relative velocities. Mechanical theories of friction laid foundations for further study of this complex phenomenon. Amontons’ theory referring to the classic dry friction law was expanded by Coulomb in the 18th century by a dry friction law in the form of F = μN +C with correction parameter C that allowed for the dependence of the friction force on molecular reaction of the frictional surfaces [Coulomb (1809)]. Coulomb assumed that for flat surfaces correction C has a constant value independent of the normal loading and the contact surface. Mechanical theories of dry friction include also the theory of [Bowden and Tabor (1950)]. It is based on an assumption that it allows for plastic deformations of frictional solid bodies’ real contact surface. The basic feature of this theory is the view on creation and destruction of junctions appearing on the contact surface, and the frictional resistance is defined as the sum of the resistances of cutting the joined asperities and the resistance of pushing the deformed material. The knowledge on the phenomenon of dry friction was developed due to molecular theories of Tomilson and Deryagin, mechanical–molecular theories of Kragelskiy and energy theories of [Kragielskiy (1944)]. In [Epifanov (1975)], the author developed Bowden and Tabor’s theory, however his own friction theory was based on an assumption that a moving slide representing a single asperity of the surface cuts the build-up that is formed in front of it.

1.2.2

Modern Description of Dry Friction

The process of external dry friction that occurs in various junctions of machines and mechanisms has been investigated with utmost attention these days [Awrejcewicz and Pyryev (2002)]. The attempts to come up with a qualitative explanation of it by means of an appropriate mathematical notation face numerous difficulties related to the contact surface’s complex structure, heat emission and wear processes. These problems do not close the list of all processes that occur in the adhesion area. Therefore, an analytic and experimental explanation of the character and the mechanisms of their occurrence is a priority in the investigation of discontinuous dynamic sys-

page 4

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

5

tems. That is why, in order to develop a friction model, mathematical equations that describe the frictional contact are formulated and experiments on true real systems with friction are conducted. The methods of mathematical analyses, numerical simulations or empirical experiments are varied. The classic friction laws are applied when motion is only slightly affected by friction. Empirical experiments are conducted to verify the results of measurements with the use of analytic models. For the investigated frictional pairs and the conditions of motion their own friction models that take into account the specific conditions of the investigated junction are formed. Due to the researchers’ various approaches to the problem and their different methods of description, in what follows only the most important models of static and dynamic friction have been distinguished and described. The static friction models describe the dependence of the friction force on the relative sliding velocity, whereas the dynamic friction models take the form of differential equations, which also describe the friction in the stick phase of the frictional elements, i.e., when the measured relative sliding velocity is equal to zero. 1.2.2.1

Static Models

Vibrations excited by friction often cause problems in industrial mechanical devices such as junctions of turbine blades, junctions of robots’ parts, electrical engines’ drives, water-lubricated bearings used in ships and submarines, machine tools, brake and clutch systems, valve trains with cam-driven valves, and camless valve drives. Vibrations of this kind are undesirable, and their side effects often affect the mechanical system’s efficient operation in a negative way [Ibrahim (1994a); Popp et al. (1996); Feeny et al. (1998)].






0
k


0
k

0

vr -
k -
0

0

vr

-
k -
0

(a)

(b)

Figure 1.1: Friction characteristics: (a) Coulomb law, (b) the exponential law. The analysis of a system with one degree of freedom with friction in [Andreaus and Casini (2000)] is based on the assumption of a harmonic exciting force and a standard excitation in the form of a belt moving at constant velocity. The main

page 5

June 8, 2017 12:9

6

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

purpose of the work was to determine in what way the system response depends on the belt’s velocity and the friction model. The numerical investigation has been based on the friction coefficient’s discontinuous dependence on the relative sliding velocity vr (see Fig. 1.1) described by: (a) Coulomb model; (b) a model with the negative slope of friction characteristics, in which the static friction coefficient μ0 decreases exponentially with the value of the dynamic friction coefficient μk . In Fig. 1.1(b) the area marked by horizontal lines denotes low relative sliding velocities at which sticking of the mass occurs. The friction characteristics described with Coulomb’s law serve to show that the motion of the mass is stable during pure slips as well as during the subsequent stick and slip phases. The conditions in which it is possible to avoid the noise and stick– slip vibrations in the system have been presented. The influence of the belt’s velocity on the amplitude of transfer and the adhesion time has been investigated, and the results prove that beyond a certain value of the belt’s velocity, the oscillator’s response does not change and the stick–slip vibrations vanish. The transient motion has also been analyzed to determine the conditions that should be satisfied by the system’s initial state, in order to obtain a sequence of pure slips and avoid stick states at the same time. In the case of the friction characteristics presented in Fig. 1.1(b), the relative velocity at which one can observe only the pure slip phases increases along with the fall of the exponential branches. The work also emphasizes a significant dependence of the system response on the belt’s velocity. It shows the period-doubling bifurcations as well as n-periodic and chaotic solutions. An additional analysis of a one-degree-of-freedom system with harmonic excitation shows that the period’s length and the number of “sticks” per one cycle of motion may significantly increase. The problem of stick–slip vibrations that occur during friction described with Burridge–Knopoff earthquake model is presented in the works [Burridge and Knopoff (1967); Vieira (1999); Van de Vrande et al. (1999); Galvanetto (2001, 2002); Awrejcewicz and Olejnik (2003a)]. If the system presented in Fig. 1.2 includes more masses (in series elastically connected blocks vibrating on the belt), then modeling of stone blocks’ slips during earthquakes will be possible on the base of the theory of seismic vibrations. Vieira demonstrated with a three-block system the appearance of synchronized chaos. A consequence of this study was the conclusion that earthquake faults, which are generally coupled through the elastic media in the earth’s crust, could synchronize even when they have an irregular chaotic dynamics [Vieira (1999)]. Numerical analysis of Burridge–Knopoff model with the number of blocks reduced to two has been conducted considering the dependence on the dynamic friction force described by the following formula: ⎧ 1 ⎪ , if vd > v, ⎨ 1 − γ(v − vd ) (1.1) Fk (v − vd ) = −1 ⎪ ⎩ , if vd < v. 1 + γ(v − vd )

page 6

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

k1

m1

x1

k12

x2

m1g

T1

m2

7

k2

m2g

vd

T2

Figure 1.2: Approximate Burridge–Knopoff model. The kinetic friction force is usually a function of the relative velocity vr = v − vd between the mass and the belt. The coefficient γ ∈ [0.4, 3.4] has a positive value, as it is presented in Fig. 1.3, because the kinetic friction force decreases at small values of the relative velocity. Introduction of the friction model described by the dependence that takes the form of formula (1.1) enables observations of non-standard bifurcations and it proves the existence of the attractive sets of solutions. An analysis of the obtained results leads to a conclusion that such assumptions as the lack of back slip and the existence of symmetry in the system may cause negative consequences in the form of inaccurate solutions. Considering all the conclusions drawn from the observation, some new assumptions have been made for a geophysical interpretation of Burridge–Knopoff model.

1

-1

Figure 1.3: Friction characteristics for the Burridge–Knopoff model. A detailed numerical study of certain fundamental aspects of a one-dimensional homogeneous deterministic Burridge–Knopoff model has been presented by [Carlson et al. (1991)]. The work describes the model by a massive wave equation, in which the key nonlinearity is associated with the stick–slip friction characteristics shown in Fig. 1.4.

page 7

June 8, 2017 12:9

8

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts


(vr) 1 }

0

1


vr

  Figure 1.4: Stick–slip friction law defined in Eq. 1.2. The friction law with the stick–slip velocity-weakening friction force at the interface between tectonic plates takes the form ⎧ ⎨(−∞, 1], if vr = 0, (1.2) ϕ(vr ) = 1−σ ⎩ , if vr > 0. 1 + vr /(1 − σ) The function ϕ in (1.2) states the law dependent on the relative velocity vr of order 10−8 or smaller, and σ is a parameter determining the slope of the curve. Braking systems with anti-lock brake mechanisms (ABS) prevent the wheels from blocking during sudden applying of the brake, which improves the directional stability and shortens the braking time. A two-dimensional nonlinear dynamic system with switching control is one of the simplest mechanisms with anti-lock brake systems. The nonlinearity in the system is manifested in the relation between the slide and the dynamic friction coefficient. An analysis of the dynamics of an anti-lock brake system with various methods of control has resulted in obtaining limit cycles of various shapes. A physical model of an anti-lock braking system for a wheel, that includes nonlinear coefficients, is presented in [Fu et al. (2001)] (see also the theoretical foundations presented in [Fling and Fenton (1981); Choi and Lou (1991); Perko (1991); Yeh and Day (1992)]). A very popular mathematical model of an anti-lock brake system is a system of three first-order differential equations in the following form: mv˙ = −F, J ω˙ = −(Tb − rF ), T˙b = u,

(1.3)

where m and v denote the vehicle’s mass and velocity, respectively, F = μN is the kinetic friction force (described by means of Coulomb law) that occurs between the wheel and the road, N is the normal force, J denotes the wheel’s moment of inertia, ω is the angular velocity of the wheel with radius r (the distance between

page 8

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

9

the wheel’s axle and the road’s surface), Tb is the brake’s torque on the wheel, and u denotes the speed of the braking torque’s change. The conducted experiments confirmed the existence of a nonlinear relation between the kinetic friction coefficient and the slip s. The wheel’s torque may be presented as:   J(1 − s) , Te (s) = rμk N 1 + mr2 (1.4) rω . s=1− v The control in an anti-lock brake system consists of keeping the kinetic friction coefficient μk in the proximity of the boundary value μc that causes such braking that creates the greatest possible friction force F without blocking the wheel and slipping of the tire on the road’s surface. The authors of the work also present an analytically numerical approach to obtaining periodic solutions of a piecewise nonlinear autonomous anti-lock brake system with boundary control. Measurement of the static and kinetic friction coefficients is a frequent and well-known problem that appears during the investigations of systems with friction. [Brandl and Pfeiffer (1999)] contains a description of a tribometer (see also [Dweib and de Souza (1990)]), which serves for the determination of the dependence of the coefficient of friction on a relative velocity between sliding bodies. It also enables us to determine the static and dynamic friction coefficients for the majority of pairs of frictional materials. Coulomb law has been applied during the simulation, and the friction coefficient dependent on the relative velocity has been approximated by a function in the following form (see Fig. 1.5):   −δ|vr | + μk , (1.5) μ(vr ) = sgn (vr ) (μ0 − μk ) exp μ0 − μk where μk denotes the kinetic friction coefficient when the motion’s relative velocity vr goes to infinity, μ0 is the static friction coefficient, and δ denotes a constant. The results obtained from the measurements conducted in a true system and in a computer simulation have helped to prove that an appropriately adapted model of a tribometer and friction law depicted in Fig. 1.5 may serve as a perfect tool (as long as recurrence of results is obtained) for detecting and experimental visualizing of friction. Worthy of our attention are the still-developing techniques of measurement of frictional resistance leading to estimation of friction coefficients of various pairs of contacting bodies. A kind of micro-tribotester has been designed and fabricated by [Guo et al. (2007)] to evaluate the static and kinetic friction coefficients on lateral contact surfaces of a single-crystal silicon material-based MEMS device. The micro-tribotester was fabricated with the standard bulk-fabrication and bonding processes. Finally, dynamic and static frictional coefficients of the fabricated structure were measured with the micro-tribotester under microscope. The tested static

page 9

June 8, 2017 12:9

10

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts



0
k

0

vr -
k -
0

Figure 1.5: Exponential law of friction versus relative velocity. frictional coefficient was 0.9 and a kinetic frictional coefficient was in the range of [0.24, 0.35]. The properties of passive vibrations absorbers with dry friction are significantly different from those linear ones. An interesting phenomenon that can be observed in passive absorbers is their capability to damp all forms of excited vibrations. It is affected by a small area of the friction characteristics and a well-evaluated static friction threshold. [Hartung et al. (2001)] includes theoretical considerations and the results of experiments on the shape of the characteristics and its influence on the operation of a passive absorber. Passive vibrations absorber is a mass-elastic system connected with the housing in order to control its periodically forced vibrations. Figure 1.6 illustrates the positions of the masses and the connections of the absorber system.

x2

x1 k2 k1 Acos(
t)

m1

m2 dry friction

d

Figure 1.6: Mechanical model of a passive vibrations absorber. Mass m1 and spring k1 are used to model the housing, while the absorber system is made of mass m2 , an elastic connector characterized by rigidity k2 and viscous damping d. Mass m1 is forced by a harmonic motion with amplitude A and circular frequency ω. Dry friction that occurs between mass m2 and the guiding surface brings the problem down to an analysis of a strongly nonlinear mechanical system. Active structural vibration control concepts are the efficient means to reduce unwanted vibrations. In the contribution by [Gaul and Becker (2014)], two different semi-active control concepts for vibration reduction are proposed. They adapt to the

page 10

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Friction Laws in Modeling of Dynamical Systems

11

normal force of attached friction dampers. Simulation and experimental results of a test structure with passive and semi-active control of friction dampers have been compared for a stationary excitation of narrow band. For simulations of the control performance, transient simulations had to be employed to predict the achieved vibration damping. That transient simulation of systems with friction and normal contact requires excessive computational power due to the nonlinear constitutive laws and the high contact stiffness involved. Commercial finite-element codes do not allow simulating feedback control in a general way. Therefore, a special simulation framework has been proposed to allow efficient modeling of interfaces with friction and normal contact by appropriate constitutive laws which are implemented by contact elements in a finite-element model. A problem of active damping of vibrations in a 2-DOF mechanical system including Saint–Venant elements has been studied in Chapter 14. Classic study of the motion in mechanical systems with dry friction is in most works based on deterministic laws that have been defined as the product of the kinetic friction coefficient dependent on the relative velocity at the contact area and the normal loading dependent on time [Ibrahim (1994a,b)]. The author of these works has made an assumption that the normal force is constant during motion. By means of the characteristics described in Fig. 1.7(a)–(c) there has been also introduced a boundary value of the static friction force. F

F

F

Fs Fk

Fs

Fs

0

vr -Fs (a)

0

vr

0 -Fk -Fs

-Fs (b)

vr

(c)

Figure 1.7: Friction models: (a) Coulomb, (b) of a falling characteristics, (c) for Fs > Fk . The simplest way to notate the friction force is Coulomb law (see Fig. 1.7(a)), according to which static friction force Fs is equal to kinetic friction force Fk dependent only on the direction of motion. In the case of the falling characteristics presented in Fig. 1.7(b), Fs is still equal to Fk but there is a linear dependence of the kinetic friction force described by a decreasing function on the velocity of mass m2 (for instance, compare with Fig. 1.4). The case presented in Fig. 1.7(c) involves a constant kinetic friction force that is smaller than the static friction force Fs > Fk . An interesting consideration has been conducted whether various models of fric-

page 11

June 8, 2017 12:9

12

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

tion induce any significantly different phenomena and responses of the passive vibration absorber. The most optimal use of the vibration absorber has been estimated on basis of the frequency characteristics. Comparisons of the shapes of the frequency response curves for a Coulomb model with the falling curve show little qualitative changes. A change of the kinetic friction coefficient at the same assumptions as the previously mentioned for the model of friction in Fig. 1.7(c) moves the damping of the high vibration amplitude movements towards larger values of the static friction coefficient. If there is a small difference between these coefficients, the system behaves in such a way as if Coulomb law described the friction; see Fig. 1.7(a). Practical application of the models of dry friction described above and comparison of the obtained results prove that the system response is affected by a small area of the friction characteristics and the static friction coefficient influences the character of the transition between the stick and the slip phases. Another study has also been conducted in order to observe the stick phase and to identify a dry friction model. The friction model has been identified (the shapes of its characteristics have been obtained as a result) not only by the change of such mechanical parameters as the relative velocity or the normal force, but also by the change of the frictional junction’s material properties. In order to eliminate the additional influence of temperature and wear on friction, a special kind of brake pad friction material widely applied in engineering has been used [Schmieg and Vielsack (1998)]. The conducted experiments lead to a conclusion that the efficiency of a passive vibrations absorber with dry friction is conditioned by the existence of the stick phases. The displacement during the slip phase should be big enough to make the mechanical energy dissipate from the system. However, these requirements are satisfied for large boundary values of the static friction coefficient (the static friction force is applied from the outside). Moreover, the analyzed shapes of the kinetic friction characteristics in the slip area slightly influence the dampening that occurs in the passive absorber. In engineering, dry friction often causes undesirable effects. It may cause some self-excited vibrations that generate noise and affect the durability of a system’s elements. There are complex systems in which stick–slip vibrations may be observed [Arjun Patil and Teodoriu (2012)]. The most interesting among them are the structures of some drilling elements on the deep-sea drilling platforms [Baumgart (2000)]. A mathematical model for the drilling process comprises a mud (Moineau) motor which rotates the bit relative to the lower end of the pipe. The model accounts for buckling of the pipe due to excessive torque and longitudinal forces, as well as for the effect of hydraulic pressure on the deformed pipe. Weight on bit and torque on bit are computed from characteristic curves which are functions of the penetration of the bit into the rock and the angular velocity of the bit. Numerical simulations show self-excited oscillations of the drillstring, including bit take-off

page 12

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

13

from the bottom hole and large amplitudes in the bit’s angular velocity. The authors of [Van de Vrande et al. (1999)] take up an analysis of the torsional stick–slip vibrations that occur in rotational elements of the drills used on the deep-sea drilling platforms. Self-excited stick–slip vibrations caused by dry friction occur between a drill and the thin-walled pipes that lead the drill down into the seabed. They may cause damage to the working and fixing parts that is why control systems are necessary to secure the whole construction against such vibrations. In order to describe the friction that occurs in this system the following formula of the friction characteristics is applied  T± (vr ) = sgn(vr )μk (vr )N, vr = 0, T1 (vr ) = vr = 0, Ts ∈ [−μ0 N, μ0 N ], ⎧ ⎪ ⎪ ⎨ 1, vr > 0, (1.6) sgn(vr ) = −1, vr < 0, ⎪ ⎪ ⎩ 0, v = 0, r

μ0 , μk (vr ) = 1 + δ|vr | where μk (v) is the function of the relative velocity dependent coefficient of kinetic friction; μ0 is the coefficient of static friction; and δ is the parameter defining the rate of changes of the kinetic coefficient of dry friction with respect to changes in the relative velocity vr . The use of the dependence for μk (vr ) makes it possible to find such sets of parameters for which the stability of equilibrium points changes accordingly to super- and subcritical Hopf bifurcation. The occurrence of such a bifurcation of solutions influences formation of respectively stable and unstable periodic solutions. In many mechanical systems the frictional elements tend to stick and slip occasionally, which often results in undesirable malfunctioning and uncontrolled behavior. Computer simulations of mechanical systems with dry friction are difficult to run due to strongly nonlinear friction characteristics in a proximity of zero sliding velocity. [Tariku and Rogers (2000)] presents two improved friction models. One of them is based on the method of equilibrium of forces. The other uses the elastic and dampening properties of a system during the stick phase. The two models have been tested in many one- and two-dimensional systems with damping and elasticity and with excitation changeable in time and normal contact forces using the same methods of study. In [Karnopp (1985)], Karnopp has used the relation of the dynamic friction force during the slip formulated by Coulomb. However, the friction force during the stick phase has been determined on basis of the equilibrium of forces. A rigid body becomes physically stuck during friction if the relative velocity and the acceleration tangential to the friction surface equal zero. Since zero values of those variables cannot be obtained in numerical calculations, only small velocity “windows” are

page 13

June 8, 2017 12:9

14

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

defined (for example, when the relative velocity is within the range [−0.001, 0.001]), in which an assumption that the mass becomes stuck is made. Some piecewise continuous analytical solutions have been compared to the solutions of other friction models based on equilibrium of forces and the elasticdampening properties of the system during the stick phase. A method that enabled us to estimate the dimensions of a velocity “window” in friction Karnopp model has been developed [Karnopp (1985); Tan and Rogers (1996)]. The results obtained from the tests of the proposed method show that the new algorithm of the equilibrium of forces used for estimation of the velocity during the stick phase produces smaller errors in comparison with the original method. Moreover, the algorithm does not produce sharp “peaks” at the beginning of the stick phase. The method developed by Karnopp for the equation of motion m¨ x + cx˙ + kx = A cos (ωt) + F,

(1.7)

can be described in the following way: (i) if |x| ˙ > x˙ 1 , then a body with mass m remains in the slip phase and the ˙ kinetic friction force is described by the equation Fk = −μk N sgn x. (ii) if |x| ˙ ≤ x˙ 1 , then the “net” friction force affecting the mass can be calculated by means of the formula x + cx˙ + kx). Fnet = A cos (ω(t + t0 )) − (m¨ (iii) if |Fnet | > μ0 N , then a body slips and Fk is determined in the way as in point 1. (iv) if |Fnet | ≤ μ0 N , then a body becomes stuck and is affected by the force Fk = −Fnet . The other constants c, k, A and ω in Eq. (1.7) denote the damping coefficient, the coefficient of system rigidity, the amplitude and the frequency of the exciting force, respectively. Moreover, if the value of x˙ is very small and x ¨ = 0, then regardless of the equilibrium forces, the value of Fk may be accurately approximated by Fk ≈ −A cos (ω(t + t0 )) + kx. The criterion of stick detection in the described method consists of the observation of the change of the sliding velocity’s sign, rather than the attempts to define a velocity window. An alternative criterion may be an external net force less than the maximum friction force. When the system becomes stick, the sliding velocity falls to zero and changes its sign. Defining a low velocity window allows the algorithm to omit the stick phase. The use of the modified model results in lower constant stick velocities in comparison with Karnopp’s method. As soon as sticking is detected in Antunes model [Tariku and Rogers (2000)], a rigid spring and a damper are fitted along the motion’s direction. Thus the friction force during sticking can be notated as a sum of the elasticity and damping forces. The method of stick phase detection consists of simultaneous fulfilling the following conditions: (i) the sliding velocity changes its sign; (ii) the elasticity

page 14

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

15

and damping force during sticking is less or equal to the friction force during slipping. The stick phase is over as soon as the elasticity and damping force overpowers the dynamic Coulomb friction force. In the quoted publication, the original Antunes model, which includes the elasticity and damping force, is modified according to the following criteria: (i) The dynamic and static friction coefficient are taken into consideration; (ii) Due to a very small calculation time-step, the original algorithm of Antunes model applied in the work sometimes detects sticks during the sliding velocity’s change of direction even when the net friction force Fnet is greater than the maximum friction force. In order to avoid such a temporary sticking, the model has been modified by imposing a condition that the sticking may occur only when the net force is less than the maximum friction force. The modified friction model with elasticity and damping forces presented by the authors of the discussed publication assumes simultaneous fulfilling of the following criteria in order to detect sticking: (i) the change of sign of the velocity; (ii) the external net force must be less than the maximum friction force; (iii) the additional elasticity and damping force related to friction during sticking should be less than the maximum friction force. When sticking occurs, the algorithm activates the elasticity and damping force which makes the system return to the state a few time-steps before and carries the calculations on until the friction force during slipping matches up with the friction force from the additional virtual spring and damper. The algorithm converges again at the point of zero velocity with the still active elasticity and virtual damping. It is necessary to add that the additional elasticity and damping tangent to the direction of motion applied during sticking are by no means related to the real physical values. Numerous simulations confirm the fact that modified algorithms based on equilibrium of forces and application of an additional elasticity and damping force significantly improve the existing methods. As a result, the stick–slip problem in dry friction systems can be solved with greater accuracy. During sticking, an equilibriumof-forces model keeps constant low velocities, whereas a model with an additional elasticity and damping force gradually lowers the values of the stick velocity and thus more accurately approximates the zero velocity in the stick phase. There are certain difficulties in estimation of a “window” at determining the zero sliding velocity in Karnopp’s model with equilibrium of forces [Karnopp (1985); Tan and Rogers (1996)]. Therefore, the stick phase is often omitted in the calculations and too high sliding velocities are reached at high velocities of the base (frame). The improved method is not based on determining the velocity windows and it keeps low sliding velocities, which are usually much lower than in the previously mentioned Karnopp model, during the change of sign. Experiment A mechanical object consisting of a system of frictionally interacting masses that viscoelastically oscillate on a transmission belt is presented in [Bogacz et al. (1990);

page 15

June 8, 2017 12:9

16

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Bogacz and Ryczek (1997)]. The aim of the research was to formulate a model of dry friction that would describe the problem of stick–slip and quasi-harmonic vibrations. The research included the motion’s relative velocity, the acceleration, the adhesion time and the velocity of the friction force increase at the moment preceding the stick–slip transition. The empirical research aiming at determining the friction model’s parameters was conducted at the test stand used to investigate the vibrations of self-excited systems. The laboratory stand consisted of a mechanical system made of a system of masses connected by springs and an optical-electronic measuring system used for visualization, collecting and digital processing of data. The experiment brought coordinates of masses that allowed to determine the motion’s dynamic characteristics. The obtained dependencies of relocation, velocity and acceleration in time allowing to determine the static and dynamic friction force as a function of the investigated system parameters. The investigation of the dynamic friction force as a function of the relative velocity and the acceleration’s sign, as well as the investigation of the static friction force dependent on the sticking time and the velocity of the force change made it possible to formulate a dry friction model (see Fig. 1.8) for a steel-polyester frictional pair. F Fs

-vmax 0

vr -Fs

Figure 1.8: General form of a dry friction model (F — friction force, vr — relative velocity, Fs — friction force at the beginning of sliding, vmax — maximum relative velocity). The experimental research, modeling and computer simulation of the vibrations of a self-excited system in case of the assumed model of dry friction proved that the belt’s velocity significantly influences the system’s behavior. The analysis of the occurrence of stationary solutions proved that there are stick–slip relaxation vibrations in the system and that the system’s vibrations are strongly influenced by the static characteristics of friction. Along with an increase of the belt’s velocity greater than 0.25, one can observe a transition between the qualitatively new vibrations types of various frequencies — characteristic of supercritical Hopf bifurcation. On further increase of the belt’s velocity to greater than 0.4, the self-excited

page 16

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

17

vibrations disappear and the system approaches a stable slip, in which the relative velocity assumes a constant value equal to the belt’s velocity. In spite of the occurrence of such ranges of the belt’s velocities where the static or the dynamic friction force is more important, the system’s instantaneous motion depends on both characteristics of friction due to the model’s sensitivity to the system’s history. This phenomenon is characteristic of friction models with memory. An analysis of self-excited vibrations in mechanical systems with many degrees of freedom was conducted in [Awrejcewicz (1987)]. The experiments showed that a steel plate modeling a superficial contact of frictional bodies that was connected by elastic elements with a rigid body with three degrees of freedom vibrated almost harmonically with such velocity amplitude that a harmonic linearization of the sgn function occurred. It was assumed that the mass contacts of the frictional bodies in the feed motion systems of heavy machine tools might perform a similar role. Mathematical approach Considerations of the mathematical approach in the study of discontinuity, including the analysis of complex phenomena of friction, are presented in [Bothe (1999); Feˇckan (1997, 1999); Filippov (1988); Guckenheimer and Holmes (1983); Kunze and K¨ uper (1997); Kunze (2000)]. Assuming multivalent representations (for an sgn function, for instance) and using the equations of discontinuous oscillators as simple differential inclusions dependent on small parameters, the manifold of periodic solutions and sufficient conditions of their occurrence were investigated. Another example of the work that presents numerical solutions and mathematical considerations on the structure of the bifurcation parameters of nonsmooth oscillators with dry friction is [Kunze and K¨ uper (1997)]. The author investigates a periodically-excited dry friction oscillator with one degree of freedom, the exciting force of which is assumed in the harmonic form of A cos (ωt), where A, ω are the amplitude and the exciting force frequency, respectively. The elasticity proportional to relocation x(t) is modeled in accordance to Hook’s law, the dry friction force is described in accordance to Coulomb law, and the change of the relative velocity is described by function sgn (v(t) − 1).


0

vr

Figure 1.9: Friction characteristics for a periodically excited oscillator. With the use of the second Newton’s law, the following second-order differential

page 17

June 8, 2017 12:9

18

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

equation has been obtained x ¨ + x + μk [μ(1) + sgn(v − 1)μ(v − 1)] = A cos (ωt),

(1.8)

where μ(1) is the value of friction coefficient when the velocity is equal to 1. The dynamic friction coefficient (see Fig. 1.9) is described by the following relation μk (vr ) =

μ0 − μ1 + μ1 + λ1 |vr |2 , 1 + λ0 |vr |

(1.9)

where μ0 is the static friction coefficient and μ1 , λ0 , λ1 are constants. The discussed work presents a difference between continuous and discontinuous dynamic systems that is related to the existence of Lyapunov exponents. Interestingly enough, the assumed three-dimensional space degenerates due to the trajectory’s return to a discontinuous surface of a lower dimension. Therefore (as the calculations confirm), the first Lyapunov exponent converges to infinity. The second exponent is equal to zero by assumption (the system evidently depends on time) therefore the motion’s character is entirely described by third Lyapunov exponent. Such a situation is beneficial since it allows using Eq. (1.8) to reduce the investigation of the system to the investigation of one-dimensional Poincar´e map. Moreover, based on the obtained bifurcations of solutions, a conclusion was drawn that there are still many structures of that type that require thorough investigation and explanation. The block-on-belt model (see Fig. 1.18) designed to take into account variations of the normal load during the braking process was analyzed by [Pilipchuk et al. (2015)]. It has been shown that due to the adiabatically slowing down velocity of the belt, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. The model has been described by the friction force F depending upon variations of the vertical load as follows,



k3 x2 c2 M (1.10) λr − y˙ 1 . y1 + 1 − F = μmg 1 + m mg 2r mg The graphical representation of the friction law shown in Fig. 1.10, which is the dependence of friction coefficient on the relative velocity at the friction interface, Vrel = x˙ 1 − vb , is given

μ0 β (1.11) tanh αVrel , μ (Vrel ) = 1+ 1 + γ|Vrel | cosh αVrel where μ0 is a constant parameter controlling the spike on the friction coefficient’s characteristics, assuming that the range of relative velocities is narrow enough, the parameter γ is responsible for the decay of friction force as the modulus of relative velocity is increasing, α controls the curve’s sharpness near zero, and finally β controls the magnitude of spikes near zero. In other words, the rate of original drop of the friction coefficient just after the main mass quits the “creeping” area. In

page 18

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

1.320

μ [-]

0.575

ws-book975x65

19

μeff μ

0.000

-0.575 -1.424 -0.50

-0.25

0.00

Vrel [m/s]

0.25

0.50

Figure 1.10: Friction coefficient and friction interface (dashed line), and the effective friction coefficient of the model (solid line) obtained under the following parameter values: μ0 = 0.5, γ = 3.0, α = 200.0, β = 0.7. particular, the thin line illustrates the shape of friction law at μ0 = 0.5, γ = 3.0, α = 200.0, and β = 0.7. Such parameters are chosen below for numerical simulations. If γ = 0, the friction law (1.11) becomes similar to that introduced in [Pilipchuk and Tan (2004)] for modeling the so-called “creep–slip” dynamics of a typical 2-DOF mass-spring model on a moving belt within the class of smooth functions. Application of the above friction law has allowed one to observe the creep–slip leading to a significant widening the spectrum of the dynamics at the final phase of the process. A generalized Bay–Wanheim friction model that describes the mixtures of thin lubrication films was applied in industrial programming utilizing the finite element method [Dubois et al. (1996); Gu´erin et al. (1999)]. A three-dimensional axialsymmetric problem was presented and subsequently verified during the tests of circular compression by comparison of the obtained empirical and analytical results. The Bay–Wanheim model was also used to describe friction that occurs during the rotary and the slip motion of a processed sample with low and medium loading in the adhesion area. The methodology of friction coefficient identification was developed and the advantages as well as the disadvantages of the Bay–Wanheim model and its supremacy over Coulomb model were pointed out. The Bay–Wanheim friction model proposed in [Wanheim and Bay (1978); Bay (1987)] is characterized by lubrication processes that occur between the fixed base and the sample that rotates and slides on it, and it is described by the following relation vr P , = −μα (1.12) Pmax ||vr || where P emphasizes the reduction of the pressure force that appears during friction with respect to the coefficient of the real contact area α and an average friction coefficient μ, Pmax is a maximum pressure force during pure shearing, and vr denotes

page 19

June 8, 2017 12:9

20

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

the relative velocity. It is essential to add that the model utilizes the material’s physically non-measurable roughness. The coefficient α, defined as the ratio of the real and measured contact surfaces, increases along with the increase of the pressure adequate to the type of interactions on the rough surface and it depends on the assumed theory of elasticity. Identification of the average friction coefficient μ (in formula (1.12)) in the Bay– Wanheim model is conducted according to a reverse approach that depends on comparison and verification of empirical results with the use of the finite element method. The method has also been applied to compare the Bay–Wanheim model to the Coulomb model of friction. A procedure based on an assumption that the pressure force at the contact area is independent of the friction coefficient is applied to determine the friction coefficient for Coulomb law. Moreover, it is assumed that for a friction coefficient smaller than 0.15 the deflections of the effective plastic deformation are sufficiently small and may become independent of this coefficient. The use of the Bay–Wanheim model leads to a more accurate estimation of the average friction coefficient μ, and subsequently, to a better determination of the fields of plastic deformations in the contact area proximity. Combination of the sample’s rotation and slide with the generalized friction Bay–Wanheim law accurately represents the conditions in the frictional adhesion area of surfaces. [Panagiotopoulos et al. (1994); Mistakidis et al. (1998)] deal with the problems of fractal geometry and fractal behavior during unilateral contacts. The adhesion surfaces and the friction laws that apply to contact surfaces are modeled by means of fractal geometry. A fractal nature of the applied friction laws includes randomly located asperities on the contact surface that influence the friction force. The fractal law and the fractal contact surface in a fractal model were examined in the form of two different functions yielded as a result of a fractal interpolation. With regard to these assumptions, the fractal friction law was approximated with a sequence of nonmonotonic, usually multivalent C 0 -class curves. The effectiveness of the numerical analysis of all nonmonotonic problems was improved by the use of the advanced solution methods that approximate a non-monotonic problem with a sequence of monotonic problems. The work describes some numerical applications of the advanced solution methods for a static analysis of structures with cracks. The analysis employed fractal geometry and fractal friction laws formulated for friction contact surfaces with cracks. The authors were mainly focused on the laws’ fractal nature and their relation with the friction contact surface’s fractal geometry. The application of fractal geometry allowed them to define the contact surfaces with cracks and to assume the friction mechanism. A variational formulation of the fractal friction problem was based on a sequence of semi-variational problems, however the adhesion surface’s fractal approximation was made first. Unilateral contact conditions were assumed in a normal direction to the adhesion surface, whereas the fractal friction law (the characteristics of which

page 20

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

21

F Fs

0

vr -Fs

Figure 1.11: Fractal friction model for an unilateral contact. is presented in Fig. 1.11) was assumed to be tangential to it. The assumption of the adhesion surface’s fractal geometry and its fractal behavior in the problem of the unilateral contact led to yielding a correct model. The model accurately described the complicated geometry of a cracked surface with instantaneous loading and relative relocation leaps caused by partial damage to the adhesion surface. The velocity of the algorithm’s convergence to the solution of the friction problem in a system with a fractally-modeled structure strongly depends on the fractal friction law and the fractal dimension of the adhesion surface. The character of the generated loading additionally influences the velocity of convergence of the proposed algorithm. Machine and mechanism junctions The junctions in machines and mechanisms in which one can observe clearances and friction may be modeled as systems of non-deformable bodies connected by elasto-plastic systems (the so-called multibody discrete systems) with unilateral constraints. One of the most important characteristics of bolted joints is their capacity for damping as a result of friction between the parts of the joints. Extensive reviews of this subject in [Gaul and Nitsche (2001)] have been completed. The review article indicates approaches for describing the nonlinear transfer behavior of bolted joint connections. From a theoretical point of view, the carefully made overview of modeling issues [Stribeck (1902); Canudas de Wit et al. (1995)] is very useful and numerically applicable. The analyzed cases include classical and practical engineering models. Constitutive and phenomenological friction models describing the nonlinear transfer behavior of joints are discussed in [Majundar and Bhushan (1991)]. Hertzian contact [Johnson (1985)], the nonlinear relationships between friction and relative velocity in the friction interface, and a few solution techniques [Johnson (1985); Den Hartog (1956); Levitan (1960); Yeh (1966)] commonly applied to friction-damped systems have been presented. [Pfeiffer (1999); W¨osle and Pfeiffer (1999)] present how a three-degree-of-freedom mechanical system with friction and unilateral contacts was used to investigate the

page 21

June 8, 2017 12:9

22

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

conditions of the transient states’ occurrence. Discontinuous characteristics of friction and contact laws served to determine the transient states observed in machine and mechanism kinetic junctions. The discussed work studied the following configurations of unilateral constraints: separation, stick contact and slip contact. For a mechanical system with n unilateral constraints, the number of all possible combinations of all constraints’ states equals 3n . Thus, for the systems of a great number of unilateral constraints the number of combinations suddenly rises. Only one of them fulfills all of the well-defined kinetic and kinematic conditions. Formulation of the rigid body’s dynamics equations and the contact law, and finding solution to the contact problem were accompanied by the linearization of the static friction cone (with the use of Coulomb law). The characteristics of dry friction presented in Fig. 1.12 was applied to formulate the equations of the normally and tangentially directed constraints (using the notation of accelerations).
i
0i

0

vri

Figure 1.12: A decaying characteristics of dry friction for ith constraint. A preliminary theory (including extended method of Lagrange multipliers) for the problem of a three-dimensional contact was given, and subsequently an oscillator with dry friction made of one rigid body (placed on a slant surface), on which three unilateral constraints were placed, was analyzed. The rigid body with mass m1 placed on the slant surface was periodically forced by a rotating mass m2 with unbalance e. The contact surface’s inclination angle δ and the exciting mass’ angular velocity Ω (see Fig. 1.13) were assumed. An experiment gave evidence for occurrence of a stick–slip phenomenon during the motion. The oscillator’s locations were also measured and the results (obtained on basis of the previously applied theory for multibody systems with unilateral constraints) were compared with a computer simulation. Investigation of the stick–slip phenomena in rigid multibody systems with unilateral friction contacts allows us to find an explanation of certain difficulties that occur in three-dimensional systems. They are especially visible when the behavior of one junction affects the other junctions. The analysis of such a case based on the dynamics of constraints leads to compatibility of the stated problem. As it turns out, the use of a linear formulation of the problem (the linearization of the friction cone) does not result in finding only one combination of the states

page 22

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

23

Figure 1.13: Oscillator on an inclined plane. of all constraints, if the direction of the friction force that occurs in a constraint instantly after the transition from the stick phase to the slip phase is unknown. Such a situation leads to a nonlinear completion of the problem that may be solved by means of a few methods and algorithms roughly presented in this work. Additionally, if the conditions on the contacts are reciprocally dependent, then certain difficulties with a mathematical formulation of the stated problem appear. The methods discussed in the quoted work, including Lagrange’s extended method of multipliers, may be applied to multiple impacts with friction. Coulomb’s impact model with friction is extended to a three-dimensional multiple impact with friction to be subsequently solved with the use of a modified method of Lagrange multipliers. Atomic scale The friction force in the friction model presented below on the atomic scale [Zw¨orner et al. (1998)] is constant for small values of the motion’s velocity. A study conducted with the use of a microscope for various forms of carbon (diamond, graphite and amorphous carbon) was focused on investigating the influence of velocity on the point contact with friction. The results show that the friction force is constant in a vast range of sliding velocities of order nm/s or μm/s. The use of an FFM microscope — a more developed scanning force microscope (SFM) — allowed the authors to investigate the frictional properties of the point contact on the nanoscale. The behavior observed during the friction process was significantly different from the behavior characteristics for the macroscopic friction models. An especially interesting observation was that friction forces were proportional to the real contact surfaces, which in turn were not proportional to the forces exerted by the adhesion surface loading. If the second derivative of potential with a negative sign after relocation is larger than constant elasticity (between the mass and an atomic chain that is parallel to its motion and generates a potential’s field), then some unexpectedly significant changes of relocation appear in the form of discontinuous stick–slip movements (leaps from one potential’s minimum to another). At high velocities, the motion is dominated by viscous damping and the friction force at that time is proportional to the sliding velocity; see Fig. 1.14.

page 23

June 8, 2017 12:9

ws-book961x669

24

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

F [nN] 100

1

vr [nm/s] 10-2

vm

103

Figure 1.14: Friction force F as a function dependent on the sliding velocity vr . For low sliding velocities, friction force F is determined by dissipation of energy and thus it is constant in the range of such velocities. The more the sliding velocity increases, the more the friction force is described by a linear dependence (the straight line corresponding to viscous damping that crosses point vm on the vr -axis). Summing up, the work proves that macroscopic friction laws are not applicable on the microscopic scale. 1.2.2.2

Dynamic Models

Friction force in static models depends on sliding velocity. However, there is another approach according to which friction, as a phenomenon changeable in time, should be described with differential equations and viewed as a dynamic system. The description of friction with the use of differential equations is often applied to control oscillating systems with friction [Walrath (1984)]. Dahl suggests in [Dahl (1976)] a simple model of controlling systems with friction. The conclusions drawn from his experiments suggest that some microscopic irregularities located in the frictional surfaces’ contact area are the reason for the occurrence of dry friction (according to the classic mechanics of solids). The starting point for Dahl’s model is the curve representing the relation between the pressure force and the deformation [Ramberg and Osgood (1943)] presented in Fig. 1.15. According to [Bliman (1992); ˚ Astr¨om (1995)], Dahl suggests notating the curve in Fig. 1.15 as a differential equation of the following form,

β F dF (1.13) =k 1− sgn v , dx FC where x is the relocation, v denotes the velocity of motion, F describes the friction force, FC denotes friction Coulomb force, k is the coefficient of rigidity, and β (usually equals 1) denotes the shape parameter of the curve representing the relation between the pressure force and the deformation (stress–strain characteristics). Absolute value of the friction force will not be larger than FC , if it satisfies condition |F (0)| < FC . It is essential to notice that friction force in Dahl’s model is only a function of

page 24

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

25

F FC

v>0 v Fs , −Fs ≤ Fu − Fg > Fs ≤ Fs , Fu − Fg , Fs .

In the above, Fg is the gravity force, whereas k0 g(v) includes Stribeck effect [Choi and Lou (1991); Brandl and Pfeiffer (1999)], and k2 v describes the viscous friction that occurs at high sliding velocities. A sgn function in the Coulomb friction model serves as a switch between three different models for the negative, zero and positive velocities. The basic assumption is that friction force F depends on sliding velocity v and force Fu acting from the direction of the medium. The Lund–Grenoble model governed by equations: ⎧ ⎪ ⎨ F = k0 z + k1 z˙ + K2 v, , (1.20) |v| ⎪ z ⎩ z˙ = v − g(v) assumes that frictional contact occurs between two porous flat surfaces. Variable z denotes an additionally introduced state that models the average deflection of the porous surfaces. Porosity deflection of both surfaces generates the friction force ˙ where k0 and k1 are the coefficients of rigidity and damping, respectively. k0 z + k1 z, The two friction models presented above differ from each other in their complexity. The dynamic friction model is relatively simple to use but it turns out it cannot be differentiated during analyses and simulations. On the other hand, the Lund–Grenoble model is continuous and due to the possibility of using Lipschitz condition, it enables us to find solution for known initial conditions. Its primary fault is that it introduces an additional immeasurable variable z that ought to be taken into consideration during calculations. A singular perturbation model described in the discussion [Altpeter et al. (1998)] binds the models mentioned above. Thus they may be represented as a system of equations with a coefficient that functions as a transition from one model into another. The binding coefficient in the proposed notation of the singular perturbation model determines similarity between the Lund–Grenoble model and the dynamic friction model. According to the results of the analysis of cases Fc = Fs and Fc = Fs , the differences described by the coefficient of perturbation between the two friction models are significant (on basis of the relation between the friction force and the sliding velocity obtained from a numerical simulation) only for low sliding velocities. If the angular velocity is high (v > 1.5 rad/s) and constant, then the dynamic friction model appears to be more effective. However, when the sliding velocity is

page 27

June 8, 2017 12:9

28

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

low but changeable in time, then the two considered models describe friction with approximately the same effect. Interesting considerations that focus on the study of friction and wear of brake block friction linings are presented in [Ostermeyer (2001)]. There are certain characteristic structures on the contact surface in a disc-friction lining type of brake system. They are caused by material wear, which is closely related to the equilibrium of the stream of increase and destruction of firmly stuck “patches” on the contact surface. These patches — clusters of scraps detached from the surface of the block’s lining due to abrasion (presented on photographs taken with the use of a microscope) — modify the value of the adhesion surface friction coefficient. The overall aim of the research was to determine the basic principle of the brake blocks wear and to formulate a new type of second-order differential equation that would describe the changes of the dynamic friction coefficient. The differential equation proposed by the author describes a short-term stationary behavior during friction of the brake block lining against the brake disk, which seems to be the model’s serious limitation. In this case, engineers attempt to maximize the friction force and minimize the wear in order to achieve appropriately long endurance of the system’s frictional elements. Brake block linings usually consist of over twenty different chemical compounds. The chemical constitution of the brake systems’ contact surfaces affects the friction processes and thus influences the shaping of the structural coatings in the contact areas. Due to the fact that not all relations between the surface’s chemical constitution and the dynamic friction coating are known, extensive research has been made on the use of optical microscopes that are capable of characterizing tribological surfaces on the nanomechanical level [Eriksson and Jacobson (2000)]. Producers of brake blocks manufacture their own materials for their products. The trial-and-error method is often applied to optimize the chemical constitution of the brake block lining. As it turns out, friction in brake systems is not sufficiently explained as far as physics and dynamics are concerned. The value of the friction coefficient is usually within the range [0.1, 0.9] and it falls as the disc’s temperature and the friction force decrease. The value of the friction force in cars is usually 500 W/cm2 and the temperature in the contact area reaches 300 ◦ C. The effect of decreasing the friction coefficient along with an increase of the friction force is called “fading effect”. It is caused by a non-uniform increase of the thermal force. The results of an experimental analysis prove that rather periodical changes of thermal boundaries occur on brake discs. Another effect that appears in clutches and brake systems is a periodical change of the dynamic friction coefficient in time. The length of the vibration period changes from a few to a few hundred seconds. The fading and the periodical change of the dynamic friction coefficient have been explained in [Ostermeyer (2001)] by means of a second-order differential equation. Some theoretical hypotheses derived from the

page 28

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

29

analyses of this equation have been empirically verified.

friction force

friction surface wear

other forces

heat

kinetic energy change

motion resistance

Figure 1.17: Basic energy flow in systems with friction. An energetic analysis of friction has resulted in an observation that a friction force between a rigid body excited by an external force and a flat slip surface should be as big as possible to help maintain the non-zero sliding velocity. Streams of energy, the density of which should be independent of the sliding velocity according to Coulomb, correspond to the contact areas of the brake block lining and the brake disc. The energy’s density occurs physically in the thermal energy, in the changes that occur on the contact surfaces and in the material itself. In order to study the elementary friction processes, an analysis of basic energy streams has been made. Figure 1.17 contains a diagrammatic representation of such streams. The considerations presented so far serve as a basis for an assumption that due to the complex dynamics of friction, the friction coefficient μ ought to be determined with a system of differential equations that describe the energy changes. The considerations on the friction surface structure in the brake block lining adhesion area form foundations for an energetic friction model for brake systems represented by the following system of nonlinear differential equations:  μ˙ = −a(nvμ − η),   (1.21) η˙ = −c(η − η0 − αnv) − γ η 4 − η04 , where nv is a non-dimensional parameter dependent on time, which describes the normal force and the tangential force, η0 = b/aTef , Tef is an effective temperature on the surface on the adhesion area, and a, b, c, α, γ are the model parameters. Neglecting the small nonlinear terms and assuming n and v to be constants, the following stationary solution is obtained: η = η0 + αnv, η0 μ= + α. nv

(1.22)

Temperature is a linear function of velocity and normal force, whereas friction coefficient is described by the falling (due to the velocity and the normal force) characteristic. Ostermeyer also emphasizes the necessity of taking wear into consideration in modeling brake systems with dry friction.

page 29

June 8, 2017 12:9

30

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Empirical observations of fading prove that a leaping decrease of the friction coefficient’s value occurs during a leap of the brake disc’s rotational speed. The given friction model explains fading effect caused by a smaller increase of the speed of the contact area destruction in comparison to the speed of the development of the strongly fixed patches.

1.3

Dry Friction and Dynamical Systems Theory

The theory of modeling friction processes described in Sec. 1.2 embraces a wide range of problems investigated in order to discover physical and dynamic phenomena as well as to determine the mathematical relations that describe them. Many theories have been developed on the grounds of the nonlinear theory that enable us to make qualitative or quantitative analyses of the behavior of discontinuous dynamic systems described with the use of various friction models. Therefore, apart from the works that focus on investigation of typical systems with friction this work will also include the literature that describes such well-known problems in the nonlinear vibration theory as periodic and chaotic vibrations, Poincar´e portraits and sections, Lyapunov exponents and bifurcations of solutions. The phenomenon of self-excited vibrations caused by friction has been extensively described in technical literature. A simple oscillator of one degree of freedom [Stoker (1950)] was the first to show the occurrence of such vibrations. The necessary condition of the occurrence of the vibrations in that oscillator was a nonzero deflection of the dry friction characteristics given in the form of a relation of the friction force and the relative velocity between the moving belt and the rigid mass that oscillated on it. In [Hassard et al. (1981)] the authors also investigated an oscillator of that type, but in order to explain the phenomenon of self-excitation of vibrations, they chose an approach based on the use of the theory of discontinuous systems’ bifurcations. Interesting considerations on self-excited vibrations are presented in [Awrejcewicz and Lamarque (2003)], where a mechanical system of four degrees of freedom serves as a model of a singular surface contact between two masses. Friction force depends in this case on the frictional bodies’ relative velocity and on the change of the normal force. The solution of the system of nonlinear ordinary differential equations describing the system’s motion can be found with the use of the approximate analytical and numerical methods supported by experiments. The approximate analytical method taken into account in this work forms foundations for formulation of averaging nonlinear first-order differential equations. An example of self-excited vibrations caused by dry friction in a system of two degrees of freedom used to describe the procedures required in that case has been used. Applying the presented method allows us to analyze the stationary and non-stationary states that occur in the investigated system. Among various examples of the occurrence of self-excited vibrations there are

page 30

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

31

also high-voltage transmission lines and bridge suspension structures exposed to constant operation of winds, or pipe lines immersed at water reservoir outlets exposed to operation of water currents. Self-excited vibrations may damage or destroy the structure in such situations, and we mention Tacoma Narrows Bridge for instance. In certain conditions, self-excited systems disperse energy in the form of a harmful noise. Sometimes their presence in electronic devices, such as analog-todigital converters, may be desirable and intended, since they enable one to utilize the dynamics related with them to scan the frequencies that characterize the input state [Feely and Chua (1992)]. Deterministic models with static and dynamic dry friction are applied to describe the dynamics of faults and to explain earthquake tectonic processes [Carlson and Langer (1989)] (see also the Burridge–Knopoff model presented in Sec. 1.2.2.1). Self-excited vibrations cause premature wear of co-operating machine parts — as it is presented in [Knudsen et al. (1991)] that deals with the problem of dry friction occurring during motion of rail-vehicles (see also the results of numerical calculations presented in [Hinrichs et al. (1997)]). Another problematic aspect of mechanical vibrations excited by the friction of cutting tools and a processed object may be found in [Minis et al. (1990)]. The squeaking noise of machine cutting tools (a lathe tool, for example) during machining is an undesirable phenomenon. The accompanying disadvantageous effects, such as noise, increased roughness of the processed surface and the tool’s decreasing strength, add to the shortening of the tool’s useful time. The list of numerous coefficients that cause the squeaking noise during machining includes exciting, regenerative and self-excited vibrations. The self-excited and regenerative vibrations, which depend to a great extent on the relative motion between the tool and the processed object, are especially interesting. The squeaking noise caused by self-excited vibrations during turning may occur as a result of the changes of the friction force on the cutting tool’s blade, which in turn are caused by a dynamic change of the friction angle and the cutting angle. Work by [Stefa´ nski et al. (2000)] contains a description of a self-excited oscillator with friction designed and constructed for empirical analyses of dry friction effects. A mathematical model is also formulated and the influence of different types of classic friction characteristics on the behavior of a proposed oscillator is investigated numerically. The phenomenon of friction induced in many physical systems by self-excited and parametric vibrations is analyzed in [Cunningham (1958); Hayashi (1964)]. When those types of vibrations occur in a system simultaneously, then the problem becomes more complex as far as the mathematical notation and the physical interpretation of the system’s behavior are concerned. Such types of vibrations occur in mechanical devices, such as a combustion engine for instance. In certain conditions, self-excited vibrations of a piston and parametric vibrations of a crankshaft can be observed. Self-excited and parametric vibrations caused by dry friction in a three-degree-

page 31

June 8, 2017 12:9

32

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

of-freedom system are also described in [Awrejcewicz (1990)]. The work analyses a nonlinear parametric system consisting of a rectangular rotor placed on an oscillating rigid base. The areas of parametric instability are identified with the method of power distribution with extension due to two perturbation parameters related to the parametric excitation and the friction coefficient. A mathematical analysis serves to determine the influence of chosen parameters of the systems on the shape and the size of the instability areas of the first kind. The self-excited character of vibrations and the discontinuous form of differential equations enable one to observe stick–slip phenomena in real mechanical systems or in numerically modeled systems. As it turns out, the relation of friction force and relative velocity in case of two oscillating bodies also leads to the occurrence of non-decaying vibrations in a two-degree-of-freedom autonomous system. The transitions from the slip phase to the stick phase, which occur during the changes of the phase space’s dimension (within the range from 4 to 2), play a principal role in the system’s dynamic behavior. [Urabe (1967); Naranayanan and Jayaraman (1989)] are particularly interesting for their examples of vibrations in mechanical systems caused by friction, in which stick–slip phases are observed. Stick–slip movements that occur in the case of a pendulum with dry friction (also called Coulomb friction) belong to the less complex ones [Deimling (1992)]. [Den Hartog (1931)] describes the scientific research, the aim of which is to prevent stick phases from happening on the friction surface. It is necessary to add that although the study of such systems was initiated by den Hartog in 1930, it was not continued until the 1950s [Szablewski (1954)]. Den Hartog simplified the problem of vibration with friction and presented a periodic solution (that consisted only of the slip phase; see also [Hong and Liu (2001)]) of the oscillator’s response with harmonic excitation modeled with Coulomb law. Much later, a similar problem was analyzed with the use of the following techniques: numerical research in various domains of time, research in various phase spaces [Hundal (1979)], an incremental harmonic balance, an equivalent linearization method [Nayfeh and Mook (1979)] and others. Hartog’s work served as basis for the calculation of periodic solutions (occurring only during the pure slip phase) for the response of an oscillator with Coulomb friction affected by harmonic excitation described in [Hong and Liu (2001)]). The work presents a comparison of Hartog’s results [Den Hartog (1931); Stoker (1950)] with regard to an additional new calculation in relation to the maximum velocity and its delay time. The basic advantage of the new approach is that a simple relation has been derived to calculate the minimum of the amplitude of the exciting force necessary to provide the pure slip phase between the oscillating body and the adhesion surface. Compatibility of the assumptions and accuracy of the results have also been proved by means of a comparison with the exact solutions [Hong and Liu (2000)]. Finding solutions to the systems of differential equations that describe the dy-

page 32

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

33

namics of systems with friction often evokes numerical problems, such as impossibility to determine the stick phase correctly or too long calculation time, for instance. Therefore, [Awrejcewicz and Olejnik (2002b); Olejnik and Awrejcewicz (2013a)] apply H´enon method [H´enon (1982)] to solve a self-excited two-degree-offreedom system with dry friction. Significant shortening of the calculation time and improved accuracy of the phase trajectory with a clearly marked stick phase have been obtained on basis of conducted analyses. Work by [Awrejcewicz and Olejnik (2003a)] analyzes a self-excited two-degreeof-freedom system with Coulomb friction. The friction characteristics are approximated by an arctan function. The problem refers to a classic system of two masses located on a moving belt. The belt’s equations of motion are brought to a nondimensional form. A lot of interesting examples of nonlinear dynamics have been discovered regarding the stick–slip phenomenon. In [Carlson and Langer (1989); Carlson et al. (1991)] the authors have investigated the stick–slip effect in multi-dimensional systems and formulated a theory and conditioning of a mechanism responsible for earthquake-like events and noises. [Oden (1985); Persson (1998)] contain some interesting formulations and a theory concerning the stick–slip phenomenon. According to an observation, basic difficulties that appear during the description of this phenomenon result from the nature of friction law, which changes the direction of the friction force during the change of the relative velocity’s sign. Observations conducted by Oden have proved that an occurrence of the stick–slip motion in a model system does not require the assumption of the difference between the static and the dynamic friction coefficient. [Wikiel and Hill (2000)] utilizes that conclusion and assumes a singular value of the friction coefficient in relation to the relative velocity (in the form of an sgn function). In a mechanical system consisting of two masses connected by a spring and oscillating perpendicularly in a cylindrical pipe, non-smooth Coulomb friction characteristics has been applied. Undoubtedly, friction, wear, heat emission or deformations caused by a temperature increase are complex processes that require a special approach. In [Awrejcewicz and Pyryev (2002)], a nonlinear problem of a thermoelastic contact of a rotating shaft and a rigid bush fixed by springs onto the base have been investigated. In their research they have applied Laplace transform and the perturbation method, and they have assumed that the friction coefficient is a nonlinear function of the relative velocity. The problem has been reduced to a nonlinear system of differential and integral equations. The numerical analysis has resulted in the observation of self-excited vibrations during the stick–slip movements and the areas of stability of a stationary solution. The dynamics of an oscillator with two degrees of freedom with dry friction is investigated in [Guran et al. (1996)]. The system consists of a mass shaped as a channel bar oscillating on a base that moves at a constant velocity, and another mass (supported by a spring) located inside the channel bar and capable of per-

page 33

June 8, 2017 12:9

34

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

pendicular movements. A constant friction coefficient is assumed on the contact between the channel bar and the base’s surface. Solutions that lead to obtaining boundary cycles during subsequent stick–slip phases are described. The results of numerical analyses lead to a conclusion that one of the reasons for the occurrence of the stick–slip phase in the system is the channel bar shift coupling tangential to the friction plane with the oscillating mass’ perpendicular motion. The unstable nature of vibrations that is revealed during stick–slip motion has been observed in Van der Pol oscillators, in which the friction coefficient decreases along with an increase of the relative velocity [Ibrahim (1994a); Guran et al. (1996); Feeny et al. (1998)]. It also happens in the case of oscillators with an additional spring, which is fixed onto the oscillating mass, that causes a change of the pressure in the normal direction towards the friction surface [Oden (1985)]. Experiments justify considering the elasticity that influences the change of the perpendicular pressure onto the friction surface. Conducted observations confirm the assumptions that the mass’ relocations in a perpendicular direction towards the contact surface occur during the slip phase. Among many mechanical devices there are ones that are equipped with an additional system with a disc that serves to generate an intended effect of friction. The most popular are car brake blocks, computer hard discs or circular saw machines. The study of dynamic instability of such mechanisms described in [Mote (1970); Iwan and Moeller (1976)] refer to an analysis of stationary discs excited by a rotating loading and a disc affected by constant loading. In all the cases mentioned above, the investigated system’s transition into unstable states occurs at certain specific values of the parameters like mass, rigidity and damping. Friction modeling by description with regard to progressive (following) force is widely known in literature on disc vibrations. [Ono et al. (1991)] investigates friction with regard to following force, which appears in disc drives of personal computers. Interestingly enough, the results confirm the fact that the vibrations propagate in the direction that corresponds to the occurrence of instability observed on the disc surface. The occurrence of vibrations during motion of elevators belongs to the less thoroughly investigated problems. A simple mass-spring-damping system to model an elevator’s drive system with a drive wheel has been used in works [Miwa (1967); Sissala et al. (1985)]. Continuing their study in [Wee et al. (2001)], Wee and others investigated nonlinear, velocity-dependent, stick–slip vibrations that occur during a sliding-metal contact. The nonlinear behavior of that kind (observed during elevators’ motion) is characterized by the occurrence of discontinuity on the adhesion surface between the wheel and the guide rail. The described analysis aims at showing the effects of nonlinear dynamics on the contact of the studied elements, suggesting a mathematical model with a two-step evaluation of its parameters and suggesting a method of determining them on basis of computing techniques. The stick–slip motion is unstable due to the slip phase caused by a decrease of

page 34

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

35

the friction force value along with an increase of the relative velocity in a susceptible mechanical system. [Bowden and Tabor (1950); Halling (1957); Van de Velde and de Baets (1997)] explain that process thoroughly. Stick–slip vibrations are usually explained starting from the initial state in the stick phase and then the study of the entire phase stability is carried out, including verification and calculations for subsequent phases that follow within a long period of time [Banerjee (1968); Bo and Pavelescu (1982); Van de Velde and de Baets (1998a)]. It is also necessary to mention work by [Nussbaum and Ruina (1987)], which, as one of the few, includes the study of the consequences of starting stick–slip motion from the point in space that corresponds to the phase of a “weak” slide. That case has been developed into a problem of determining the time instant or the point in the phase space in which the stick–slip motion can be eliminated from a mechanical system through an increase of velocity beyond its critical value and then a decrease back to its initial value [Gao and Kuhlmann-Wilsdorf (1990)]. The problem of stabilization of nonsmooth systems is illustrated and discussed in [Lozano et al. (2000); Goeleven et al. (2003)]. An importance of the contact and impact phenomena in many mechanical systems are discussed in [Komanduri (1993); Ramachandran et al. (1994)] (grinding and deburing problems in manipulators performing tasks), [Shia et al. (1998)] (filamentary brushing tools for surface finishing), and [Studny et al. (1999)] (robotic systems). In general, collisions associated with friction and impact are considered as harmful behavior, but in some cases impacts are provoked intentionally in order to dissipate energy and contribute towards stabilization of the considered system [Brogliato et al. (1997, 2000)]. The accurate conditions for various types of stability properties of the closed-loop system involving a free motion phase, a permanently constraint phase and a transition phase are formulated in [Bourgeot and Brogliato (2005)]. In addition, the existence of a specific transition between permanent constraint phases and free-motion phases is rigorously proved. The analysis of the research described in [Van de Velde and de Baets (1996)] and further developed in [de Baets et al. (2000)] shows that friction that causes self-excitation of tangential vibrations in the stick–slip motion may be the reason for a sudden significant increase of the acceleration value and the vibration amplitude. According to observations, probability of an occurrence of a stick–slip phase during deceleration is significantly bigger because of lengthening of the phase that corresponds to the decrease of the acceleration value and to the tangential increase of the system’s rigidity. In [Brace and Byerlee (1996)] authors have initiated experimental research on the analogy between subsequent seismic events and stick–slip motion. Numerous critical analyses of the mechanisms for the seismic slide of blocks of rock appeared consequently. Some of them referred rather to the analysis of the slip phase (with the use of a discontinuous change of the sliding velocity step) before the occurrence of instability. Moreover, the physical aspect of normal relocation in the slip phase

page 35

June 8, 2017 12:9

36

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

was not fully explained. Experiments proved that vibrations perpendicular to the contact surface occur (for various materials) during the stick–slip motion. Direct measurements of metals in a test stand proved that a change in electric conductivity of the contact surface with stick–slip vibrations is caused by normal vibrations [Bowden and Tabor (1939)]. Similar vibrations were also observed during an investigation of rubber foam. A photography technology applied in that case helped to observe relocations of light-emitting diodes (LED) placed a few centimeters below and above the slide surface [Brune et al. (1993)]. During the experimental analysis with the use of LEDs described in work [Anooshehpoor and Brune (1994)], a peculiar behavior of two rubber foam masses was observed: within certain time intervals in the slip phase they were losing adhesion. The results of the presented investigation may lead to a conclusion that a decrease of a pressure component in the normal direction in the slip phase is caused not only by perpendicular vibrations but also by total loss of adhesion between contact surfaces. It means that the value of pressure in the normal direction tends to zero. The stick–slip phenomenon occurred only when pressure was very low, yet it could not be fully explained when the contact surfaces separate during heavy pressures in the adhesion area and when the slip surface roughness influences the system dynamics (see also the results of numerical analyses in [Tworzydlo and Hamzeh (1997)]). [Bouissou et al. (1998a)] includes the research results that describe the phase of adhesion loss in stick–slip motion within a vast range of pressure values perpendicular to the adhesion surfaces of investigated polymethacrylic samples. The analysis described in this work constitutes a part of extensive experimental research [Bouissou et al. (1998b,c)] focused on determining parameters and describing how normal pressure on the adhesion surface, loading velocity and roughness influences the stable slide in stick–slip motions. In order to comprehend the changes of friction force causing stick–slip motions, the relation between friction force and velocity in the slip phase has been determined through empirical measurements (see Sec. 1.2.2.1 on experimental static friction models, and also [Bell and Burdekin (1969); Bo and Pavelescu (1982); Lee et al. (1996); Marui and Endo (1996)]). The authors of the quoted works are preoccupied mostly on determining a loop that would not intersect on the surface in the friction force-relocation relative velocity coordinates. The results of the investigation do not bring any sufficient explanation for the mechanism of such a type of friction curve shape. [Van de Velde and de Baets (1998b)] shows that the shape of an intersecting or a not intersecting loop on the surface in the friction force-relative velocity coordinates may be obtained through changing the rigidity of a spring tangential to the friction surface of the investigated frictional pair. Moreover, the occurrence of the intersecting loop in the investigated system of co-ordinates is not produced by a real physical phenomenon, but it is a result of the vibrations’ “neglecting” some velocity changes which happens for most cases of rigidly connected and sliding

page 36

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

37

bodies. Investigation of the adhesion area during friction is a complex process, because it depends on many physical, mechanical and material parameters. It is also dependent on more or less complex relations that cause difficulties in formulating a mathematical description and understanding the essence of the problem (see also Sec. 1.2.2.1 on the mathematical approach to modeling friction, and [Tabor (1981); Zhuravlev (1998)]). In order to learn about the conditions on the adhesion surface during dry friction motion, which cause changes in the friction force, several theories based on various hypotheses have been developed. The basic conclusions include the following: (i) for a state in which the adhesion surface has a constant quantity and for which micro-relocations remain in a linear relation with a force tangential to that surface, the dynamic friction coefficient depends on the stationary adhesion (sticking) time [Brockley and Davis (1968)] — this phenomenon explains the connections made on the adhesion plane, which in time become stronger; (ii) during the slip-phase, the dynamic friction coefficient is assumed in many theories (see also Sec. 1.2.2.1 on dry friction static models) as a function of the relative sliding velocity [Rabinowicz (1958); Pavelescu and Tudor (1987)]. [Ferrero and Barrau (1996)] contains an analysis of dry friction occurring at small relocations (about 50 μm) on the adhesion surface and at almost zero sliding velocity (kept below 0.2 μm/s, see also Sec. 1.2.2.1, where a dry friction model on the atomic scale is described). Due to those conditions the quoted hypotheses and the standard approach theory could not have been applied in that case and the work describes the experiments conducted in order to explain the above mentioned aspect of dry friction. The results of the experiments show that the dynamic friction coefficient value increases after the conditions of the stationary adhesion on the investigated surface have been satisfied and it decreases along with an increase of the slide length during the micro–slip. The changes measured on the adhesion surface are continuous and depend on the relocation within asymptotic boundaries. An oscillating system, excited by a stream of air and made of a mass connected elastically with a stable base and a pendulum, is analyzed in [Tondl and Nabergoy (1994)]. Continuing the previously mentioned research, the authors additionally included in [Tondl and Nabergoy (1995)] the conditions of dry friction in the connection of the mass and the base, which significantly affected dynamics of the system. With the use of the mathematical analysis supported by numerical calculations, they showed that the trivial solution of the motion’s system of equations is stable, whereas the semi-trivial solution (in case of a motionless pendulum) is unstable within the entire range of the air stream velocity values. [Hinrichs et al. (1997)] presents an analysis of a linear dynamic system with damping connected with a nonlinear one-degree-of-freedom system, in which adhesion conditions and Lagrange multipliers have been applied. The dampers that utilize dry friction for functioning are assembled into a nonlinear elastic system in the form of a bar and also determine its fields of asymptotic stability.

page 37

June 8, 2017 12:9

38

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The mathematical approach to nonlinear physical phenomena involves many difficulties connected, among many other things, to the fact that they are described by differential equations with discontinuous and sometimes non-differentiable right sides. Monograph [Kunze (2000)] contains an extensive analysis of mathematical aspects of discontinuous dynamic systems with friction and impacts and it also describes mathematical methods applied by engineers during experiments. The author devotes a lot of attention to the problem of dry friction leading to differential inclusions, often called multivalued differential equations [Aubin and Celina (1984); Filippov (1988); Deimling (1992)]; see also Sec. 1.2.2.1. Furthermore, he investigates the problem of almost periodic solutions [Deimling et al. (1996)] and methods of calculating Lyapunov exponents for a pendulum with dry friction. The work also contains a formulation of the relation between the dynamic parameters of discontinuous systems with friction and impacts [Eckmann and Ruelle (1985)]. One of possible perspectives devoted to nonsmooth systems dynamics is addressed by [Georgiadis et al. (2005)], where shock isolation designs based on nonlinear energy pumping caused by non-smooth stiffness elements are studied. The term energy pumping is understood in a sense of the not reversible transfer of vibration energy from its point of generation into a predetermined spatial area (the nonlinear energy sink), where the vibration localizes and dissipates. In contrast to classical linear vibrations absorber it is shown that the nonlinear energy sinks are capable of efficiently absorbing energies caused by transient broadband disturbances. The rigorous approach to dry friction problems is presented by [Feˇckan (2005)], where the existence of a continuum of many chaotic solutions for certain differential inclusions, i.e., small non-autonomous multivalued perturbations of ordinary differential equations possessing homoclinic solutions to hyperbolic fixed points are shown. [Matrossov (1996, 2001)] focused on the uniqueness of solutions of the motion equations for a mechanical system with dry friction. The equations refer to a general case. The works quote several definitions applied to formulate (and subsequently utilize) the theorems on existence and uniqueness of solutions. Vibrations described with a mathematical model consist of a stick-phase with a relatively large static friction coefficient, and a slip-phase, in which the dynamic friction coefficient is considerably smaller. That is why the systems with dry friction are sometimes characterized by static behavior and more often by various dynamic behaviors represented by periodic, quasi-periodic and chaotic motions [Tolstoi (1967); Shaw (1986); Feeny and Moon (1994)]. The works by [Ueda (1979); Lorenz (2002)] gave beginning to the observations of chaotic movements in simple discrete nonlinear systems described by differential equations of at least the third order. The plane dynamics of a rigid block simply supported on a harmonically moving rigid ground exhibiting unilateral contacts, Coulomb friction and impacts has been studied by [Ageno and Sinopoli (2005)].

page 38

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

39

The results that enable one to observe the stick and slip phases in chaotic dynamics of simple dynamic systems with one degree of freedom and friction are presented in [Galvanetto et al. (1993)]. Its authors focus on numerical and analytical analyses of the search for chaos during the stick phase or the slip phase. Melnikov’s method applied to the investigations of discontinuous dynamic systems with dry friction is a main topic of [Awrejcewicz and Holicke (1999)]. The authors examined the problem of the search for chaos during a slightly forced stick–slip motion in a dynamic system. A stick–slip chaos analytical prediction has been confirmed analytically. Two-dimensional maps for second-order differential equations, for instance, can be made in a simple way using Poincar´e map. Therefore, such a simple mapping can be made for investigations of an extensive class of dynamic systems modeled by nonlinear oscillators [Awrejcewicz (1989, 1991); Maistrenko et al. (1994)]. Such a mapping may also serve as a useful tool for explaining sudden leaps of a phase trajectory. In real systems, trajectories are attracted by other attractors exactly after a leap, which consequently is the main reason for bringing an attractor to infinity [Filippov (1988); Szempli´ nska-Stupnicka (1990)]. Calculating Lyapunov exponents belongs to one of the most fundamental issues related to a quantitative analysis of dynamic systems. The theory developed by [Oseledec (1968)] and the numerical algorithms derived by [Benettin et al. (1980)] and [Wolf et al. (1985)] enable to estimate the spectrum of Lyapunov exponents for the systems described by continuous equations of motion. Moreover, if the equations of motion are unknown or are presented in a discontinuous form, then other methods are applied, such as the ones based on the reconstruction of an attractor from a time series, for instance. There is a growing tendency to consider real mechanical systems with friction or impacts as discontinuous dynamic systems. Numerous works on the theory of dynamic behavior of nonlinear systems with friction have appeared recently (see the works described above). Nevertheless, only a few of them, [Wolf et al. (1985); Hinrichs et al. (1997); M¨ ueller (1995)] for instance, contain a genuinely innovatory approach to calculation of Lyapunov exponents for that type of systems. Synchronization of chaos [Pecora and Carroll (1990)] has been applied in works [Stefa´ nski and Kapitaniak (2000); Stefa´ nski et al. (2000)] to present a method of calculating largest Lyapunov exponent for a system with friction and impacts, which is based on investigations conducted by [Fujisaka and Yamada (1983)]. They found a nonlinear relation between the value of a coupling coefficient (corresponding to the synchronization between two identically examined systems) and the value of the largest Lyapunov exponent. The synchronization condition was formulated only for negative symmetric back couplings between the analyzed systems. An exact solution of a discontinuous system of differential equations (e.g., describing a mechanism with dry friction) requires sometimes the use of sophisticated methods that enable one to determine the points of the real movement trajectory on the phase plane regarding all peculiarities (see also Sec. 1.2.2.1 and references

page 39

June 8, 2017 12:9

40

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

[Aubin and Celina (1984); Filippov (1988); Deimling (1992); Kunze (2000)]), such as the stick–slip transition, for instance. Therefore, the references like [Awrejcewicz and Olejnik (2002a); Olejnik and Awrejcewicz (2013a)] present solutions of the system of ODEs that describe the motion of a two-degree-of-freedom system with friction derived with the use of an “exact” H´enon method [H´enon (1982)]. In order to achieve constant time intervals between the trajectory’s points, the obtained trajectory is interpolated and Lyapunov exponents are calculated from a time series. At the end of this section, let us mention about another broad survey about friction modeling for dynamic system simulation performed by [Berger (2002)]. The author concludes that the system model and friction model are fundamentally coupled, and they cannot be chosen independently. Furthermore, the usefulness of friction model and the success of the system dynamic model rely strongly on each other. Across disciplines, it is clear that multi-scale effects can dominate performance of friction contacts, and as a result more research is needed into computational tools and approaches capable of resolving the diverse length scales present in many practical problems.

1.4

Friction Advantages: A Brake Mechanism

Nowadays, when computerization and technology are highly developed, the meaning of principal branches of science has been increasing significantly. Modern technology requires high-speed functioning or unerring precision of machines (for example, heavy machines such as cranes, traveling bridges, manipulators, robots, and others) in varied environments. Therefore, accurate modeling of a great number of dynamic phenomena that occur in machine systems has become a necessity. The present level of numerical methods’ development and the progress in highly advanced computerization enable one to choose adequate physical and mathematical models. Yet, the overall purpose is not to formulate a real and general description of a given phenomenon at any cost but to reflect its “nature” in a specific regime of the investigated system. Moreover, the well-known and traditionally described phenomena often require re-modeling based on new achievements of principal sciences and enhanced computing methods. Analyses of such a complex phenomenon as friction include all aspects of the above mentioned approaches. As confirmed above, friction has been a phenomenon of interest of many branches of science: mechanics [Renard (2001); Hartung et al. (2001)], tribology [Sokolov (2002)], mass and heat transfer [Ahn (2001)], the theory of elasticity and plasticity, materials science, fluid dynamics, intermolecular connections physics, or even physical-chemical processes (e.g., corrosion or frictional materials work in varied environments involving possibility of chemical reactions). In general, friction is accompanied by a number of other phenomena, such as stresses, material wear, heat emission, etc. In addition, friction along with impacts belongs to the group of discontinuous processes that require precise mathematical determining [Popp and Stelter

page 40

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Friction Laws in Modeling of Dynamical Systems

ws-book975x65

41

(1989); Awrejcewicz and Delfs (1990a,b); Monteiro-Marques (1994); Van de Vrande et al. (1999); Kunze (2000); Fu et al. (2001)]. It is extremely difficult (perhaps even impossible) to build a general friction model including all possible accompanying processes. Moreover, it seems to be pointless, since only some of the above-mentioned processes dominate in a specific object of study. The difficulties involved with explaining numerous effects of friction that appear during confrontations of mechanical (geometric), molecular (adhesive), mechanic–molecular and energy theories with experiments created the need to model friction with the use of simple dynamic systems for analysis of frictioninduced processes [Qiao and Ibrahim (1999)]. Diverse characteristics of friction can be observed in real dynamic systems. Many models do not require taking friction into account, although numerous systems are based on utilizing friction — hence omitting the friction-induced effects is impossible. Friction reveals its dominating nature in friction clutches, belt drive systems, as well as in the brake systems in which friction force between a wheel’s brake drum and brake blocks causes braking of a vehicle [Hulten (1997); Ostermeyer (2001)].

1.4.1

Engineering Approach

The most widely applied brakes are the ones, which are assembled in the wheels of vehicles with certain brake mechanisms. Let us examine a theoretical brake mechanism illustrated by the structural model in Fig. 1.18. It shows that the system is constructed in such a way so that the direction of friction force F is opposite to the force incoming from springs acting besides on mass m on the belt. Obviously, when the force in spring k1 is larger than friction force F , then a loss of adhesion occurs and mass m moves in direction −x. At that time, spring k1 expands and if its length exceeds its free length, then the perpendicular arm of the angle bracket is “pulled” in the direction −x and the angle bracket is turned in direction −ϕ. The horizontal arm of the angle bracket expands spring k2 , which decreases the pressure force exerted by the spring on mass m oscillating on the belt. The coupling repeats in the system throughout the entire friction process as long as the belt’s linear velocity vd is not equal to zero. The fundamental element that needs utmost attention while estimating the similarity of the described system (the model) to a brake mechanism (a real object) is the coupling of the mass m transfer on the belt induced by friction (see Fig. 1.18) with the normal pressure force affecting the belt. A model of a brake mechanism with intensified braking force is shown in Fig. 1.19. As is shown in Fig. 1.19, when the element that initiates braking starts to press the brake block against the disc, then friction force occurs between the block and the disc of a wheel. In effect, the brake block moves in direction x, the coupling element moves or turns and by an increase of the normal force intensifies braking. When the initial pressure decreases (the release of the brake lever), friction force between

page 41

June 8, 2017 12:9

42

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts




y1 b

M, J

k3

c.g.

a r

s

c1 y1

m

rotating angle body

c2

x1

k2

mg

rigid support

F vp

horizontally oscillating block

k1

Figure 1.18: 2-DOF structural model of a brake mechanism. A-A

Brake initiating element

Coupling element

Guide bar Brake block

x Spacer

Brake disc

Figure 1.19: Model of a brake mechanism with intensified braking force.

the brake block lining and the brake disc also decreases and the return spring k1 may pull the brake block back to its initial position. As it can be noticed, coupling of the friction force with the pressure force may function as power-assistance to the braking system. That is why the pressure force on the brake lever may be significantly lowered during braking. The described mechanical coupling can also be obtained with the use of a hydraulic oil system with a pump. The brake block lining has low susceptibility and it can be modeled with a belt when the experiment’s conditions are satisfied; see Fig. 1.18. Additionally, it is

page 42

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

43

possible to choose the materials of the brake disc and the block of mass m. According to the conducted investigation, there is a similarity between functioning of the dynamic systems presented in Fig. 1.18 and 1.19. The friction model, which is examined, assumed for the system in Fig. 1.18 approximately corresponds to the brake block lining’s friction against a car brake drum (or a disc). Assuming that the same materials are chosen, the model enables us to investigate the phenomena that result from friction and wear of sliding surfaces. The method of friction modeling that includes the relation between the friction force and the sliding bodies’ relative velocity, and between the friction force and the changes of the normal force may also be applied to the study of friction-induced dynamic phenomena in the braking system presented in Fig. 1.20. Brake block 1

Hydraulic drive

Return spring

Brake block 2 Coupling element

Figure 1.20: A scheme of the girling duo-servo brake mechanism [Awrejcewicz and Olejnik (2005a)]. Shown in Fig. 1.20, the structural model of the brake mechanism is assembled in a popular type of girling duo-servo brake. When a hydraulic servo initiates braking, the blocks are drawn aside and pressed against the inner surface of the drum. As a result, friction forces T1 and T2 are exerted between the blocks’ linings and the drum and the wheel stops. A careful analysis of the mechanism reveals a certain type of coupling [Olejnik (2002)]. Brake block 1 (also called a “backward” block) takes over a larger part of the friction force at the initial stage of braking, whereas brake block 2 (called a “concurrent” block) impedes with a weaker force. However, the coupling element (the angle bracket in the system in Fig. 1.18) combines the circumferential motion of brake block 1 with the motion of brake block 2 and the pressure force of the latter on the drum’s inner surface increases. The ratio of the braking forces exerted by brake blocks 1 and 2 is about 2:4. Brake blocks are connected by the return spring in such a way that it enables them to

page 43

June 8, 2017 12:9

44

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

return to their initial position as soon as the braking process is over. In practice, there are several types of braking mechanisms that function in a similar way. The purpose of the considerations is to show that a simple self-excited system in Fig. 1.18, with a changeable pressure force on the belt, may function as a starting point for analyses of friction in braking systems represented by drum brakes.

1.4.2

The Model

The study and prevention of self-excited vibrations of systems with friction is very important in industry and there is still a need for friction pair modeling that could correctly describe dynamic and static friction forces change between two movable surfaces. A further developed model can govern dynamics of the girling duo-servo brake mechanism. Therefore, the schematically illustrated in Fig. 1.18 dynamical system is analyzed numerically and investigated experimentally. The self-excited system is almost equivalent to a real test stand in which block mass m vibrates on the belt in the x1 direction, and where the angle bracket represented by the moment mass of inertia J is rotating about point s with respect to direction of angle ϕ. The analyzed system is constituted as follows: (i) two bodies are coupled by linear springs k2 and k3 ; (ii) the block on the belt is additionally coupled to a fixed base using the linear spring k1 ; (iii) angle bracket is excited only by spring forces; (iv) there are no extra mechanical actuators; (v) rotational motion of the angle bracket is damped by virtual actuators mainly characterizing air resistance and they are marked by constants c1 and c2 ; (vi) damping of the block is neglected; (vii) it is assumed that the angle of rotation ϕ of the angle bracket is small and it is within the interval [5, −5] degrees; (viii) a rotation is equivalent to the linear displacement y1 of arms of the angle bracket; (ix) the belt is moving with constant velocity vp and there is no deformation of the belt in a contact zone. Non-dimensional equations governing dynamics of our investigated system have the following form: ⎧ x˙ 1 = x2 , ⎪ ⎪ ⎨ x˙ 2 = −x1 − [η1 (x2 + y2 ) − y1 − T2 (vr )] /α1 , (1.23) ⎪ y˙ 1 = y2 , ⎪ ⎩ y˙ 2 = (−β3 y1 − η12 y2 − x1 − η2 x2 )/α2 , where x2 , y2 are velocities of the block and angle bracket, respectively; vr = x2 − vp is a relative velocity between bodies of the investigated system; α1 = ω 2 m/k2 , α2 = ω 2 J/(k2 r2 ), β1 = (k1 +k2 )/k2 , β2 = μ0 k3 /k2 , β3 = (k2 +k3 )/k2 , η12 = ω(c1 +c2 )/k2 , η2 = c2 ωμ0 /k2 , η1 = c1 ω/k2 , are the remaining parameters; ω is a natural frequency of vibrations of the mass m. Friction force is described in the following way  sgn (vr )T2± (vr ), if vr = 0, (1.24) T2 (vr ) = Ts ∈ [−Ts− , Ts+ ], if vr = 0. The T2+ (vr ) and T2− (vr ) friction force characteristics are described by a linear and an exponentially decaying function, respectively (see Fig. 1.22).

page 44

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

1.4.3

45

Experimental Investigations

In this section, the laboratory test stand [Olejnik (2002)] designed for observations and experimental research of frictional effects including friction force measurement is described. A photo of the test stand is presented in Fig. 1.21. The general view, component parts and some connectors like coil springs correspond (cf. Fig. 1.21) to those schematically indicated elements presented in Fig. 1.18.

7

5

6

1 - pendulum

2 k


ky

3

kx

4

Figure 1.21: The drum brake inspired block-on-belt system on the laboratory test stand: (1) the pendulum coupling linear motion of the block with a normal force acting in the contact surface on the belt; (2) an aluminium shaft rotating in ball bearings fixed in the frame; (3) the block creating a frictional contact with the belt; (4) incremental encoder; (5) a microcontroller for data acquisition; (6) a direct current motor with gear; (7) a direct current motor driver. Displacement of the block and angle of rotation are measured using a laser proximity switch and a Hall-effect device, which guarantee a non-sticking method of the measurement. Both of the sensors provide a linear dependency of the measured quantity versus analogue voltage output. Measurement instruments connected through PCI computer card to LabVIEW software allow one to perform a dynamic acquisition of the two measured signals. Disturbances of whole construction, noise in electrical circuits, and other additional maintenance have an observable influence on the accuracy of any measured signals. Therefore, some signals are filtered digitally and a real differentiation preventing formation of high peaks is applied.

page 45

June 8, 2017 12:9

46

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Appropriately transformed equations of motion (see Eq. (1.23)) can be used for friction force calculation after real-time measurement of state variables in the investigated system. Characteristics of friction force versus relative velocity between belt and block for positive and negative velocities of the belt are shown in Fig. 1.22. One may observe that zones occupied by functions of the friction model differ significantly. It is a regularity, since angle bracket causes reinforcement of friction force for negative vr . Owing to these considerations, friction forces are described by the linear T2+ (vr ) and the second-order exponential function T2− (vr ) as follows: ⎧ T2+ (vr ) Ts+ − T2+,min ⎪ ⎪ = 1 − |vr | , vr > 0, ⎪ ⎪ ⎪ Ts+ vr,max ⎨ Ts+ r,min r,min a1 − |vr |−v a2 − |vr |−v T2 (vr ) = T2− (vr ) b1 b2 = −1 + e + e , vr < 0, ⎪ ⎪ Ts− Ts− Ts− ⎪ ⎪ ⎪ ⎩ T ∈ [−T , T ] , v = 0. s s− s+

(1.25)

In Eq. (1.25), the following nomenclature is used: vr is the velocity of the relative displacement of contact surfaces; vr,min and vr,max are the minimal and maximal values of the relative velocity; T2± (vr ) stands for two cases of the function T2 (vr ) for the positive and the negative relative velocity vr ; Ts± is the maximum static friction force; a(1,2) , b(1,2) are the shaping coefficients of the experimental characteristics.

Linear approximation

Exponential approximation

Figure 1.22: Friction force characteristics T2 (vr ).

page 46

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Friction Laws in Modeling of Dynamical Systems

1.4.4

47

Dynamical Analysis — Bifurcations

The system of differential equations (1.23) and the friction force model given by Eq. (1.24) are transformed to the non-dimensional ones, and then, using bifurcation diagrams, a dynamical analysis of the 2-DOF mechanical system is carried out. The parameters of friction force characteristics obtained after measurements and identification are as follows: Ts+ = 3.63, T2+,min = 0.86 N, vr,max = 0.27 m/s (T2+ model); Ts− = 7 N, vr,min = −0.28 m/s, a1 = 3.2345, a2 = 2.8736 N, b1 = 0.0342, b2 = 0.3053 m/s (T2− model). Numerically performed dynamical analysis with implementation of introduced friction force dependency has resulted in the diagrams presented in Fig. 1.23.
1


1

4.2

3.5

4.0

3.4

3.8

3.3

3.6

3.2

3.4

3.1

3.2

3.0

3.0

3.9

2.8 0.17

0.43

0.68

(a)

0.93

x1

2.8 0.2

0.35

0.5

0.65

x1

(b)

Figure 1.23: Bifurcation diagrams for the system parameters: (a) α1 ∈ [2.8, 4.4] versus the displacement x1 , and moreover: α2 = 1.16, η1,2,12 = 0, β2 = 0.577, β3 = 1.825, γ1 = 0.2, γ2 = 0.8, vp = 0.6, μ0 = 0.7; (b) α1 ∈ [2.8, 3.6] versus the displacement x1 , and moreover: α2 = 1.093, η1,2,12 = 0, β2 = 1.73, β3 = 2.44, γ1 = 0.2, γ2 = 0.8, vp = 0.6, μ0 = 0.7. Zero initial conditions are assumed. A solution for each value of the bifurcation parameter has been observed after omitting any transient motion dynamics within the non-dimensional time τ = 5 · 104 . The bifurcation diagrams are constructed by changing the parameter α1 in an interval of changes with the step 0.001, where one gets hundreds of Poincar´e maps. Then one of the phase space axes is taken and all results are presented versus the bifurcation parameter. One may do it in two ways: (i) start each solution for a different bifurcation parameter with the same initial conditions; (ii) start from initial conditions at the beginning, and then, change the parameter during the solution after a prescribed period of time (the solution should not leave the attractor).

page 47

June 8, 2017 12:9

48

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

An example of the bifurcation diagram is shown in Fig. 1.23(a). Beginning from the smallest considered values of α1 , one observes a one-periodic motion, but for α1 ≈ 3.3, some period tripled bifurcation with an increase of α1 occurs. In the vicinity of the bifurcation point α1 ≈ 4.12, the period tripled bifurcation with a decrease of the bifurcation parameter is observed once again. It should be emphasized that for large interval of changes of the bifurcation parameter, only periodic motion is exhibited by the analyzed system. An interesting example of more complex bifurcations is showed in Fig. 1.23(b). One may trace how the successive period-doubling bifurcations (accompanying a decrease of α1 parameter) lead to a chaotic motions regime, which begins for α1 ≈ 3.11. Additionally, alternating n-period windows (approximately, for α1 = 3.25, 3.19, 3.11, 3.05) are reported. The classical and modern approaches of friction phenomena in various mechanical objects have been presented in this chapter. It was emphasized that foundations for present studies of difficulties connected with nonlinear dynamics are provided by the well-known mechanical theories. Friction laws described by a slip velocity-dependent coefficient were introduced to model the non-smooth stick–slip phenomenon. The friction mechanism yields self-sustained vibrations in mechanical systems with dry friction. The discontinuous dynamical aspect seems to be more important in the behavior of such systems. Some of the theoretically and experimentally-obtained friction force or friction coefficient characteristics tend to prove that their implementation depends especially on the type of engineering application. Owning to the complexity of multi-dimensional systems with nonlinearities associated with dry friction, the unilateral contacts can develop instabilities even if the basic Coulomb friction law is chosen. Therefore, the law with a slip velocity dependent coefficient is still used as a good approximation of stick–slip motion description. Engineering investigations and a numerical approach have allowed for estimation of the dry friction force model presented in this section. Two non-symmetric branches of the friction law confirm that some tribological effects in the contact zone are more complex. It is worth noticing that the experimental characteristics represented by the model T2− (vr ) is the first approach recommended to be used during analysis of friction effects occurring in systems, where the normal force acting between cooperated surfaces varies with time. From the point of view of practical application of the experimentally determined friction law for changes in normal force, especially, the nonlinear model T2− (vr ) can be applied to investigate tribological effects affected by friction, occurring between brake blocks and the brake disc. An idea for the friction pair modeling using both laboratory equipment and a numerical simulation is proposed, allowing for the observation and control of friction force.

page 48

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 2

Transient Friction-Induced Vibrations in a 2-DOF Braking System

The belt-spring-block model investigated in this chapter is designed to take into account variations of the normal load during the braking process. In this chapter we show that due to the adiabatically slowing down velocity of the belt, the system response experiences specific qualitative transitions that can be viewed as simple mechanical indicators for the onset of squeal phenomenon. In particular, the creep– slip leading to a significant spectral widening of the dynamics is observed at the final phase of the process.

2.1

Introduction

Friction-induced vibrations may occur in miscellaneous brake systems during deceleration. Such types of problems are of significant interest in the literature due to clear practical reasons, complexity of physical effects, and mathematical challenges of the modeling; see [Ibrahim (1994a,b); Akay (2002); Awrejcewicz and Olejnik (2005a)] for an overview of the problem’s status. The idea of considering the decelerating sliding is due to the fact that the brake squeal phenomenon is usually observed at the final stage of braking process. Practically, the deceleration is very slow as compared to the temporal scales of friction-induced vibrations associated with elastic modes of braking systems. This enables one to ignore the corresponding inertia forces when considering different transitional effects caused by the decay of relative speed at the friction interface. Modeling such transitional effects is important for understanding physical conditions of the onset of squeal phenomenon including possible mechanisms of excitation of acoustical modes. Although describing the acoustical squeal effects in terms of the reduced few degrees of freedom mechanical models seems unrealistic by many reasons, it is important to find simple analogues of squeal conditions by keeping the key physical features of real brake systems. In particular, based on the standard block-spring model on a moving belt, it was found earlier that stick–slip (or creep/slip) vibrations, captured at some low enough belt speed, are preserved during further deceleration by showing increasing temporal localization of slip phases [Pilipchuk and Tan (2004)]. This leads to 49

page 49

June 8, 2017 12:9

50

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

a spectral widening of the dynamics, which can provide the possibility of interaction with acoustical modes in real brake systems. Friction-induced vibrations in physical systems based on the mass-damperspring modeling have been widely considered in literature for many years. In particular, such models have been used extensively as deterministic simulators of earthquakes [Ryabov and Ito (1995)]. The earthquake generation mechanism as a chaotic phenomenon in a two-degree-of-freedom autonomous system with static and dynamic friction was illustrated in [Galvanetto and Bishop (1994)] by using the Poincar´e map diagrams showing, in many cases, a complicated chaotic behavior of the model. It was found that even perfectly symmetric systems possess complicated quasi-periodic orbits with significant out-of-phase components. Due to the specific temporal behavior of the stick–slip phenomena [Den Hartog (1956); Popp and Stelter (1990)] such components may lead to asymmetric broadband loads and therefore excite flexural vibration modes/waves in both interacting parts. Based on a numerical study of four-degree-of-freedom model, it was found in [Kinkaid et al. (2005)] that the motion of the system transverse to the direction of braking experiences a sharp change in excitation leading to a complicated vibration when the slip velocity in the braking direction is low. The authors interpret the disc brake squeal as a friction-induced phenomenon caused by the transient, dissipative nature of a braking process. Different spring–block friction models were used in order to examine the role of friction in the instability mechanism [Bengisu and Akay (1994); Knopoff et al. (1992); Leamy et al. (1998); Ostermeyer (2010)]. Based on a model of continuous and discrete elastic elements, the self-excited oscillation in two surfaces sliding against each other was investigated in [Adams (1996)]. Radiation of plane body waves caused by friction was considered in [Adams (2000)]. Some of the results support the interpretation that certain friction behavior is a consequence of the dynamics of the interacting systems rather than the interfacial properties. Contact modeling with emphasis on the contact forces and their relationships to the geometrical, material and mechanical properties of the contacting bodies was reviewed in [Adams and Nosovsky (2000)]. Applied aspects of friction-induced vibration and related stability problems are discussed in reviews [Ibrahim (1994a,b); Sinou et al. (2006); Vahid-Araghi and Golnaraghi (2011)]. In this chapter we analyze the transient dynamics of a two-degree-of-freedom system due to the deceleration of the belt carrying just one of the two inertial elements of the model. Note that the case of decelerating sliding is practically important because deceleration is often caused by friction. For instance, stick–slip instabilities in a decelerating two-degree-of-freedom mass–spring system with one mass subjected to friction were considered earlier in [Vielsack (2001)]. The research was motivated by the necessity to avoid stick–slip effects in engineering practice. Effects of four different discontinuous friction laws on the system response were investigated. Based on numerical solutions, it was concluded that the existence of stick–slip in decel-

page 50

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Transient Friction-Induced Vibrations in a 2-DOF Braking System

ws-book975x65

51

erated motions depends mainly on the properties of the mechanical systems rather than on the characteristics of the frictional force. In particular, it was noted that rapid and abrupt changes from stick to slip motions and vice verse induce a broadband excitation spectrum leading to squeal, rattling and possibly damage. It should be noted that rigid body models involving few degrees of freedom are generally not sufficient to adequately describe friction-induced phenomena at friction interface. However, numerous experimental results have clearly shown that dominant spikes of the “squealing spectrum” occur at a few frequencies or sometimes just at one frequency. Therefore only a few vibration modes may eventually be involved in the interaction dynamics.

2.2

Mathematical Modeling of the Belt–Spring–Block Model

A general view on the laboratory test stand and schematic diagram for mathematical modeling are given in Fig. 1.18 and 1.21. The designed laboratory test stand approximates the real braking system through the feedback reinforcement of the friction force acting on the vibrated block. The feedback reinforcement of the friction force is provided by the increase of vertical load on the block through the angle bracket with the set of connecting springs and dampers. When the belt moves the block to the right, the body rotates counterclockwise causing compression of the vertical spring and inducing the vertical damping force. Such mechanism associates with the functionality of drum brakes in real braking systems [Olejnik (2002)]. A schematic diagram of the model is shown in Fig. 1.18. The main (horizontal) block is driven longitudinally by the moving belt by friction forces. The load transfer between the main block of mass m and the angle bracket is provided by the spring k2 and damper c1 while the normal load from the rotating body on the main block is transferred by the vertical spring k3 and damper c2 . The moment of inertia of the angle bracket with respect to its center of rotation s is J and the total mass is M . Additionally, the main block is also connected to the base by the spring of stiffness k1 . Using generalized Euler–Lagrange equations, the equations of motion of the two-degree-of-freedom system are derived in the form [Pilipchuk et al. (2015)]:

x1 x21 m¨ x1 + c1 z˙1 + (k1 + k2 )x1 + k2 y1 + k3 y1 + = −F, r 2r

x21 Q J y ¨ + c z ˙ + c y ˙ + k z + k + y + M gλr = − , 1 1 1 2 1 2 1 3 1 2 r 2r r

(2.1a) (2.1b)

where z1 = x1 + y1 , z˙1 = x˙ 1 + y˙ 1 , F is the friction force between the main block and the moving belt given by Eq. (1.10), and Q, at the suspension point s, is the resistance torque which can be caused by dry friction, viscous damping, other elastic

page 51

June 8, 2017 12:9

ws-book961x669

52

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

forces, and moreover

√

2l λr = 2 r



y2 y3 y1 − 12 − 13 1+ r 2r 6r

,

√ l = (3 2/4)r.

Extending a representation of the friction law (1.11), the curve peaks can be found analytically as we explain below. Differentiating Eq. (1.11) with respect to the relative velocity Vrel gives   μ (Vrel ) = αμ0 sech (Vrel α) 2βsech 2 (Vrel α) + sech (Vrel α) − β = 0. Therefore, peak locations are determined from the following quadratic equation with respect to sech (Vrel α) 2βsech 2 (Vrel α) + sech (Vrel α) − β = 0. This eventually gives four roots for Vrel of which only two are real

 β0 − 1 1 Vrel = ±V ∗ , V ∗ = arcsech , β0 = 1 + 8β 2 . α 4β Now, substituting Eq. (2.2) in (1.11) gives the peak magnitudes  β0 − 3β μ 0 (3 + β0 )(β0 − 1 + 4β). μ∗ = ± 16β 1−β

(2.2)

(2.3)

The factor 1/(1 + γ|Vrel |) in (1.11) is added in order to better capture the decaying intervals outside the narrow area between positive and negative spikes of the curve. Such decays were observed in the previous tests conducted with the same test stand [Awrejcewicz and Olejnik (2005a,b)]. Note that there are very many friction laws suggested in the literature of Chapter 1, possibly due to the fact that the behavior of friction force depends upon individual geometrical and physical properties of real interacting bodies as well as environmental conditions. Let us assume the steady-state conditions at constant belt’s speed and obtain an effective friction coefficient, which takes into account the influence of the angle bracket as shown in Fig. 1.21. Ignoring the inertia and viscosity forces and solving linearized (2.1) and Eq. (1.10) with respect to F , x1 and y1 provides the effective friction coefficient (1 + κ1 )μ F (2.4) = , μeff = mg 1 + κ2 μ where κ1 =

3M r [k1 k2 + 2(k1 + k2 )k3 ] 4rk2 k3 − 3M gk2 , κ2 = , mκ3 κ3 κ3 = 3M g(k1 + k2 ) + 4r [k2 k3 + k1 (k2 + k3 )] .

The curve of effective friction coefficient (2.4) is shifted downward as illustrated in Fig. 1.10 and has as many as twice larger amplitudes, which is caused by both

page 52

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Transient Friction-Induced Vibrations in a 2-DOF Braking System

ws-book975x65

53

structural and physical specifics of the model. It is shown in the next section that the asymmetry of the effective friction coefficient affects the system dynamics. Quick variations of the friction coefficient in the neighborhood of zero relative speed constitutes the most common physical phenomenon, which is responsible for important qualitative features of the dry friction dynamics, in particular, stick–slip vibrations. The idea of modeling the dry friction coefficient by simple discontinuous functions enables one to approximate the strongly nonlinear characteristics by combining exact analytical expressions for physically different regimes of the friction process [Shaw (1986)]. However, this non-analytic relation of the friction force versus the relative speed requires different differential equations to be used for describing the dynamics in different regions of the phase space. The corresponding formulations appear to be mathematically challenging [Filippov (1988)] and require suitable conditioning for the numerical integration since a single form of the differential equation of motion cannot be derived in this case. Alternatively, exponential functions can be employed to approximate the abrupt changes of the friction coefficient [Bristow (1947); Tolstoi (1967); Bengisu and Akay (1994); Feeny and Moon (1994); Ibrahim (1994a,b); Pilipchuk et al. (2002)]. An obvious advantage of these analytic expressions is the direct application of nonlinear vibration techniques such as expansion and perturbation methods. From a physical viewpoint, the static friction is considered as a viscous force with a very large coefficient of viscosity. Note that the strongly viscous region |Vrel | ≤ V ∗ represents a relatively narrow domain of creeping, i.e., a quasi-sticking motion. According to [Ryabov and Ito (1995)] together with references therein, such an area is presumably a necessary component of any realistic friction law. The creeping area was represented by a linear viscosity assumption such that the friction law had a “kink” in transiting from the stick to the slip regime, and remained as a non-smooth function of the relative speed. 2.3

Conditions for the Numerical Tests

In order to numerically illustrate the effect under consideration, let us chose the experimentally estimated model parameters as follows: M = 0.14027, m = 0.1052 kg, J = 2.4423 · 10−4 kg·m2 , r = 0.078 m, k1 = 367.0, k2 = 76.0, k3 = 69.0 N·/m, c1 = 0.0156, c2 = 0 N·s/m. In this case, the corresponding linearized conservative system has the following eigen-frequencies: ω1 = 9.87007 and ω2 = 14.1971 Hz. Therefore, in contrast to [Pilipchuk and Tan (2004)], no low-order internal resonances are assumed here. In order to provide “adiabatic” loading conditions, let us consider the case of slowly decelerating belt whose speed is decaying by the linear law

t vb (t) = V0 1 − , (2.5) Tmax

page 53

June 8, 2017 12:9

54

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

where V0 = 0.17 m/s is the initial speed and Tmax = 45 s is the duration of process. Based on such parameters, the speed expressed by Eq. (2.5) can be considered as slowly varying in the temporal scale of eigen-vibrations of the model. The above listed values of parameters are close to those obtained by measurements for the test stand, which is shown in Fig. 1.21. Note that some of the important parameters are hardly possible to be exactly identified due to both geometrical and physical issues. For instance, it is known that the friction-induced dynamics is quite sensitive to parameters of friction laws and shapes of the corresponding curves. Also, the real block interacting with the moving belt is a three-dimensional body creating a non-homogeneous distribution of the normal pressure on the belt across the bottom surface of the block. In addition, the belt speed is difficult to control exactly to match the target expression (2.5). The main purpose of the present study is to find the major transitional effects in decelerating sliding.

2.4

The Creep–Slip Response

Illustration of the numerical results is given in Fig. 2.1. Figures 2.1(a)–(d) illustrate the velocity of main block x2 = x˙ 1 obtained from the numerical solution of the system of Eq. (2.1). The time histories are shown in time windows of 2 s covering the developed and final phases of the dynamics. The simulation shows the typical spike-wise behaviors of the velocity, which is usually observed during creep–slip vibrations. Due to the deceleration of the moving belt, the intervals between spikes are increased by the end of the process as it is seen from comparison of Fig. 2.1(a), (c) and (e). Increasing irregularity of the spike amplitudes is also seen near the end of the process. There are visible factors leading to the widening of spectrum of the dynamic states which will be illustrated in detail below. Note that the friction force, which is obtained from the numerical modeling, is shifted downward into the negative area; see Fig. 2.1(e) and (f). When the spikes become widely spaced, the friction force develops a small high-frequency component in creeping phases as seen from the fragments (d) and (f) of Fig. 2.1. Such component is caused by the variation of vertical load on the main block due to the rotational vibrations of the angle bracket. These vibrations are excited by sudden slips of the main block through the horizontal and somewhat vertical springs as follows from the form of equations (2.1). In order to characterize transient effects in spectral properties of the dynamics, the Fourier spectrogram tool is applied to the coordinate and velocity of the main block. This sub-component is directly excited by the friction force and therefore represents the major source of dynamic excitation to the rest of the system. The spectrogram representation is convenient in the present case of temporal scales, although other methods, such as wavelet or Hilbert transforms can be ap-

page 54

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

12

5

1

-8

-10

[cm/s]

18

-21

-20 -31

-34 -47

15.0

15.4

15.8

t

[s]

16.2

16.6

-42

17.0

36.5

36.9

37.3

t

[s]

37.7

38.1

38.5

40.2

40.6

41.0

40.2

40.6

41.0

(b)

3

-2

-6

-6

[cm/s]

2

-10

x2

-15

x2

[cm/s]

(a)

11

-14

-24 -33

55

x2

x2

[cm/s]

Transient Friction-Induced Vibrations in a 2-DOF Braking System

15.0

15.4

15.8

t

[s]

16.2

16.6

-18

17.0

39.0

39.4

(c)

39.8

t

[s]

(d)

0.4

-0.5

-0.2

-0.9

F [N]

-0.2

F [N]

1.0

-1.3

-0.8 -1.3

-1.7

-1.9

-2.0

-2.5

-2.4 15.0

15.4

15.8

t [s]

(e)

16.2

16.6

17.0

39.0

39.4

39.8

t [s]

(f)

Figure 2.1: Time histories in short-term windows of two seconds: (a), (b) velocity of the main block — experiment; (c), (d) velocity of the main block — numerical modeling; (e), (f) friction force — numerical modeling.

plied as well for the purpose of processing such types of signals. Also, spectral characterization seems to be most reasonable in case of mechanical systems, since it gives a direct answer regarding possible resonance conditions. Figure 2.2 represents a spectrogram of the block coordinate x1 obtained from the numerical solution of Eq. (2.1). Major spectral lines of the diagram in Fig. 2.2 reflect the two principal frequencies of the linearized conservative system; see the beginning of Sec. 2.3. The natural frequencies of vibrations of the system are found: ω1 = 7, ω2 = 14 Hz.

page 55

June 8, 2017 12:9

56

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

2

zoom

1 0

Figure 2.2: Time history of the main body coordinate with the corresponding spectrogram obtained from numerical simulation for the theoretical model.

Figure 2.3: Time histories of the main body’s velocity with the corresponding spectrogram obtained from numerical simulation for the theoretical model; see comments to Fig. 2.2.

page 56

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Transient Friction-Induced Vibrations in a 2-DOF Braking System

ws-book975x65

57

The bottom spectral line near zero corresponds to a constant component in the oscillating coordinate. Note a gradual decrease of the frequencies as the belt speed decelerates with time. Keeping in mind the discussion of Fig. 2.1, note also a significant spectral widening by the final phase of the process, which begins quite abruptly at about t = 35 s; see also Fig. 2.1(b) and (d) for a detailed characterization of temporal histories in this interval. In addition to the two major frequencies clearly seen in the interval 0 < t < 35 s, there are several sub-harmonics however of much lower power; in spectrograms, a “cooler” color represents lower amplitudes of the power spectral density in the logarithmic scale, 10 log10 P . Similar illustrations are used for the block’s speed in Fig. 2.3. Sub-harmonics are seen more developed in this case since high frequency components produce a stronger contribution to the speed. The upper edge of the time history diagrams associated with the stick phase of vibrations actually reflects the belt speed; see Fig. 2.1(a)–(d) for details. There are some irregular upward spikes in the record for the block’s velocity x2 = x˙ 1 ; see Fig. 2.1(a) and (b). Such “overshoots” mean that practically the block sticks sometimes in a reverse mode. In practice, such events are quite irregular and rather happen due to natural imperfections in the belt surface, fluctuations of the belt speed caused by its elasticity, as well as many other factors. Variations of the normal pressure on the block could also be a contributing factor; however, the model did not capture such effect, although it takes into account the normal pressure variations. Figures 2.4(a), (c), (e) and (b), (d), (f) shows trajectories on the phase plane of main block and configuration plane of the system, respectively, in three different time windows of the same length 5 s obtained from the numerical experiment. Typical stick–slip vibrations can be identified from the phase-plane diagrams in Fig. 2.4(a), (c) and (e). The horizontal part of every loop corresponds to the stick phase and therefore its level indicates the belt speed, which is gradually decreasing with time. This is also the reason why the loops have different sizes so that the diagrams look like a bunch of curves rather than an attractor. Similar effect is seen in configuration planes shown in Fig. 2.4(b), (d) and (f), although configuration trajectories look more irregular due to contribution of the angle bracket. Nevertheless, “fingerprints” of some attractors, corresponding to different frequency ratios, can be seen in fragments (b) and (d) of Fig. 2.4. Further, the irregularity is increasing by the end of the process as predicted already by the Fourier spectrogram and time histories. The elevation of some loops on the right of the phase plane diagrams correspond to the spikes as described in the discussion of Fig. 2.1. An extension of the presented dynamical analysis of the numerical model and also an experimental 2-DOF belt-spring-block system have been done in [Pilipchuk et al. (2015)]. Usually typical curves of friction laws, for instance, expressed by Eq. (2.3), are used for pure theoretical studies, however, in the extended study a comparison of the modeling and experimental results is performed. In real braking systems with much high and denser eigen-frequencies, the densi-

page 57

June 8, 2017 12:9

ws-book961x669

-0.55

1.3

-0.68

-5.8

-0.81

[cm]

8.4

-12.8

y1

-1.06

-27.0 -0.04

0.13

0.30

x1

0.47 [cm]

0.64

0.82

a) 23 < t < 28

4.5

-1.19 -0.04

-0.69

-6.1

-0.84

0.47 [cm]

0.64

0.82

0.64

0.79

0.63

0.75

y1

-1.16

-16.7 0.07

0.21

0.35

x1

0.50 [cm]

0.64

0.79

-1.32

c) 34 < t < 39

2.6

0.07

-6.0

-0.85

0.36

x1

0.50 [cm]

[cm]

-0.73

0.21

d) 34 < t < 39

-0.61

-1.7

-0.96

x2

y1

-10.4

-1.08

-14.7 -19.0

x1

-1.00

-11.4 -22.0

0.30

[cm]

-0.8

0.13

b) 23 < t < 28

-0.53

x2

[cm/s]

ws-book975x65

-0.94

-19.9

[cm/s]

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

x2

[cm/s]

58

BC: 10577 - Modeling, Analysis and Control of DS

0.14

0.26

0.38

x1

0.50 [cm]

e) 38 < t < 43

0.63

0.75

-1.20

0.14

0.26

0.38

x1

0.50 [cm]

f) 38 < t < 43

Figure 2.4: Numerical solutions for the analyzed numerical model shown in different time windows for (a), (c), (e) phase plane of the main block; and (b), (d), (f) configuration plane of the model.

fying spectral bend becomes more likely to cover some of the acoustic modes with generation of squeal. Numerical simulations with a very gradual belt deceleration reveal the hierarchy of modal transitions associated with different quasi steadystate multiple frequency vibrations separated by relatively narrow time intervals of irregular dynamics.

page 58

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Transient Friction-Induced Vibrations in a 2-DOF Braking System

2.5

59

Three-Dimensional Bifurcation Diagrams

Bifurcation diagrams is one of the most important tools of dynamical analysis (see introduction to Chapter 3) and bring us an interesting view on the system behavior accordingly to some intended changes in values of its parameters. Such approach reflects the uncertainty of real dynamical systems, the parameters of which often vary in time due to interaction of the system with an environment and changes of natural properties of the system like, for instance, wear or fatigue processes.

(a) (x1 , x2 , k3 ) hyper-surface k3 ∈ [501; 309], k1 = k2 = 200, γ = 3

(b) (y1 , y2 , k3 ) hyper-surface k3 ∈ [501; 309], k1 = k2 = 200, γ = 3

(c) (x1 , x2 , k2 ) hyper-surface k2 ∈ [201; 9], k1 = 200, k3 = 500, γ = 2

(d) (y1 , y2 , k2 ) hyper-surface k2 ∈ [201; 9], k1 = 200, k3 = 500, γ = 2

Figure 2.5: Three-dimensional bifurcation diagrams of the dynamical system (2.1) for constant velocity of the belt vb = −0.2 and the set of constant system parameters assumed in Sec. 2.3. Intervals of the bifurcation parameters k3 (a, b), k2 (c, d) and k1 (e, f) as well as some varying parameters are provided in appropriate subtitles. At certain conditions, a point of Poincar´e map is placed on it for the particular value of the bifurcation parameter k bif ; in our case it is either k1 , k2 or k3 . (C1) If x2 (i) < 0 and x2 (i + 1) > 0, then a point (y1 (jx ), y2 (jx ))(P ) is placed on a hyper-surface of the pendulum for the jx th point of the Poincar´e map

page 59

June 8, 2017 12:9

60

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

(e) (x1 , x2 , k1 ) hyper-surface k1 ∈ [201; 9], k2 = 200, k3 = 500, γ = 2

(f) (y1 , y2 , k1 ) hyper-surface k1 ∈ [201; 9], k2 = 200, k3 = 500, γ = 2

(Fig. 2.5 continued) of the block in the coordinates x1 and x2 . It means, that the bifurcation diagram of the block is superimposed on the (y1 , y2 , k bif ) hyper-surface, where, in our case, k bif states for k1 , k2 or k3 . (C2) If y2 (i) < 0 and y2 (i + 1) > 0, then a point (x1 (jy ), x2 (jy ))(P ) is placed on a hyper-surface of the block for the jy th point of Poincar´e map of the pendulum in the coordinates y1 and y2 . Analogously, the bifurcation diagram of the pendulum is superimposed on the (x1 , x2 , k bif ) hyper-surface. Having defined the Poincar´e maps for the case (C1) and (C2), we can analyze the system dynamics based on the bifurcation diagrams of both system bodies. Generally, in a wide range of stiffness (a bifurcation parameter) of springs of the system, only periodic motions have been observed. Numbered circle marks in Fig. 2.5 indicate switching of system dynamics between one-, two- or even threeperiodic vibrations. Beginning from Fig. 2.5(a), point 1 indicates the appearance of two-periodic motion of the pendulum so a 1-period to 2-period bifurcation of motion occurs; see blue points of Poincar´e map superimposed on the block’s hyper-surface. Similar bifurcation appears in point 2 and 4 (see Fig. 2.5(b) and (c)), but in point 3 and 5 a 2-period to 3-period bifurcation of motion is observed. Point 6 corresponds to the end of a three-periodic motion, hence a 3-period to 2-period bifurcation of the system orbit is found. Point 7 corresponds to a sudden jump on the branch manifesting a one-periodic motion of the block. Finally, the block exhibits oneperiodic stick–slip motions that is acknowledged by a single red path (a branch) on the pendulum’s hyper-surface shown in Fig. 2.5(b), (d) and (f). Real physical objects can behave chaotically, since our analysis confirms that the dynamical system, being an identified counterpart of a real block-on-belt system, behaves regularly in a wide range of changes of stiffness of its springs.

page 60

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 3

Numerical Estimation of the Stick–Slip Transitions

Discrete modeling of discontinuous dynamical systems have to be performed with the use of highly accurate numerical procedures that will further guarantee the proper use of the tools applied for their dynamical analysis. No less important in this matter is the numerical estimation of the phase trajectories, bifurcation diagrams and Lyapunov exponents. This chapter presents an application of H´enon’s method to obtain satisfactory numerical estimations of the stick–slip transitions existing in the Filippov-type discontinuous dynamical systems with dry friction. Subsequent sections are focused on the problem definition, application of the method that was originally proposed for Poincar´e maps, its use in estimation of phase trajectories, bifurcation diagrams of tangent points as well as on estimation of Lyapunov exponents.

3.1

Introduction

Collisions of phase–space trajectories with discontinuity zones often result in occurrence of non-smooth bifurcations [Budd and Lamba (1994)]. Bifurcation theory is a strongly growing branch of dynamical analysis while providing a broad field for analytic considerations and numerical schemes [Kuznetsov (1998); Leine et al. (2000)]. The most basic problem of the theory focuses on the study of qualitative changes in a behavior at slightly changing system parameters around the critical values. The fundamental diagram, or a variation of the dynamical process is a transition of solution from an orbit with a period T into an orbit with a period 2T . For example, a two-degree-of-freedom plane disk, performing one-dimensional transitional and rotational motion, placed on the moving belt has been mathematically modeled and numerically analyzed in [Kudra and Awrejcewicz (2012)]. The friction model for sliding mode is developed assuming the classical Coulomb friction law, which is valid for any infinitesimal element of circular contact area. As a result the integral expressions for the friction force and torque is obtained. The exact integral model is then approximated by the use of different functions, like Pad´e approximates or their modifications. Some generalizations of the approximate func61

page 61

June 8, 2017 12:9

62

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

tions used by other authors are proposed. The special event-driven model of the investigated system together with a numerical simulation algorithm is developed, and in particular, the transition conditions between the stick and slip modes are defined. Some examples of numerical simulation and analysis are also presented by means of Poincar´e maps and bifurcation diagrams. It has been shown, that for certain sets of parameters, the investigated system exhibits very rich multi-periodic stick–slip oscillations. In order to illustrate the beneficial and harmful effects of the existence of bifurcations in dynamical systems, a variety of interesting techniques for analysis and observation are used. Among them, the stroboscopic view often used during stability tests of switching (discrete) electronic systems and determining piecewise continuous mappings (e.g., in systems with dry friction) is worth noting. A detailed analysis as well as many numerical experiments of miscellaneous grazing bifurcations have been conducted in [Kuznetsov and Rinaldi (2003)]. The electronics are not the only examples of systems in which the effects generated during normal operation are the cause of the occurrence of unexpected and irregular behavior in the dynamic changes resulting from the bifurcation of parameters. A completely different nature of switching can be identified in other discrete mechanical systems. They are formed on the basis of a sequence of consecutive slides (slips) and sticks which are connected to each other and to vibrating solid bodies. An example of such dynamic analysis of a discontinuous system with two degrees of freedom and friction is found in [Galvanetto (2004)]. The test system consisted from elastically connected vibrating solid bodies placed on the belt moving with a constant velocity. Contact friction described by a model of friction existing between the ideal surfaces of cooperating bodies is a source of excited stick–slip vibrations. Occurrence of such type of vibrations is the cause of exposing by the system a variety of bifurcations between the stick and slip modes. Any conventional model cannot describe the stick mode of friction phenomenon [Awrejcewicz and Olejnik (2005a)]. In the case of conventional friction force model, the body will be sliding even though the friction state will be stick. It holds because the relative velocity must have a non-zero value in order to generate the friction force. To solve this kind of problem, a stick–slip friction force model including a spring-like force has been proposed in [Cha et al. (2011)]. In the case of the stick– slip friction force model, the body can be stuck on the inclined surface because the friction force will be a non-zero value, even though the relative velocity approaches zero. Therefore, a relative displacement variable called the stick deformation is defined. Moreover, the stick–slip friction model was proposed and applied in the contact algorithm of the MBD system. Two friction models are then compared with some numerical examples, which relate to more realistic results. Several effective numerical methods for solving the elastic contact problems with friction are presented in [Xucheng et al. (1990)]. A direct substitution method is employed to impose the contact constraint conditions on condensed finite element

page 62

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Numerical Estimation of the Stick–Slip Transitions

ws-book975x65

63

equations, thus resulting in a reduction by half in the dimension of final governing equations. As an extension, an algorithm composed of contact condition probes is utilized to solve the governing equation, which distinguishes two kinds of nonlinearity making the solution unique. In addition, Positive–Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytic integration is introduced to determine the elastic state after each iteration, hence the computational efficiency is enhanced to a great extent. A model of a forced pendulum with viscous damping and Coulomb friction was considered in [Lamarque and Bastien (2000)]. Existence of a unique local solution of the mathematically well-posed problem has been proved. An adapted numerical scheme has also been built. Attention was devoted to the study of the nonlinear behavior of a pendulum via a numerical scheme with small constant time steps. Global behavior of the free and forced oscillations of the pendulum due to friction was described. Authors proved that chaotic behavior occurs when friction force is not too large. Lyapunov exponents were computed and a Melnikov relation obtained as a limit of regularized Coulomb friction. [Dimova and Georgiev (1992)] deals with the techniques for solving ordinary differential equations with essential nonlinearity arising from the representation of frictional force by the sign function of the relative velocity. This problem is connected with dynamic behavior studies of base-isolated structures with dry friction effects. Mathematical evidence is presented concerning the inaccuracies of the solution which are connected with the representation of the frictional force by the sign function. An approximation of this sign model is made on the basis of the appropriate linearization of a small part of the frictional force function. A numerical algorithm for the dynamic analysis of base-isolated structures with dry friction has been proposed. In [Feˇckan and Kelemen (2013)], the authors define a discrete Poincar´e map around a periodic solution. In the idea, one starts from the transverse cross-section of the periodic solution, then a numerical one-step method is applied until the iteration returns back near to the cross-section. Then the last step’s size is changed so that the final step iteration returns back to the cross-section. This reminds us of the H´enon’s method, but it is different. Moreover, error bounds are computed between the discrete Poincar´e map and the continuous Poincar´e map of the periodic solution. These error bounds are derived also for derivatives of both Poincar´e maps.

3.2

Function of Boundary Transition Through a Discontinuity

A one-degree-of-freedom dynamical system with dry friction subject to our investigation is shown in Fig. 4.4. A solid body of mass m is attached to a fixed support by means of a nonlinear elastic element characterized by the parameters k1 and k2 . The solid body oscillates under the influence of dry friction force T existing in the contact zone created by the body’s surface and the outer belt’s surface that

page 63

June 8, 2017 12:9

64

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

runs with constant velocity vp . One assumes a non-deformable moving belt being under action of normal force N = mg and other tangent forces to its surface in the contact zone (the state is secured by the schematically-drawn rigid support attached to the inner side of the belt). Forcing response of the nonlinear elastic element on change of displacement y of mass m is k1 y − k2 y 3 as well as the force of dry friction existing in the contact zone of the analyzed frictional pair is denoted by T . The cubic term is used to model a more realistic nonlinear response of the spring as well as to show influence of additional terms on the form of the transition function σ(x). Another work that studies the stick–slip oscillations of discrete systems interacting with translating energy source through a nonlinear smooth friction curve is found in [Kang et al. (2009)]. The second-order differential equation modeling dynamics of the discontinuous one-dimensional mechanical system (see in Fig. 4.4) takes the form m¨ y − k1 y + k2 y 3 + T = 0,

(3.1)

where the friction force T and the coefficient of kinetic friction μk at the velocity v = y˙ − vp of relative displacement of contacting surfaces are given, respectively: T (v) = sgn(v)μk (v), μ0 . μk (v) = 1 + δ|v|

(3.2)

There are other applications of friction models applied to worm-like locomotion systems, in which the friction force T depends also on control forces [Steigenberger and Behn (2012)]. For a particular dynamical system we consider some values of model parameters: m = 1 kg, k1 = k2 = 1 N/m. Equation (3.1) assumes the following: μ0 = 0. y¨ − y + y 3 + sgn(y˙ − vp ) (3.3) 1 + δ|y˙ − vp | Equation (3.3) describes dynamics of the excited vibrations that occur during the dry friction between contacting surfaces of mass m and the moving belt. Changing the state variables x1 = y and x˙ 1 = x2 in Eq. (3.3), the following system of two first-order differential equations is obtained:  x˙ 1 = x2 , (3.4) x˙ 2 = x1 − x31 − T (v). In the next step, Eq. (3.4) can be written in the form  ⎧ μ0 3 ⎪ ⎪ x , −x + x + 1 for 1 ⎨ 2 vp − x2 + 1  x˙ = f (x) =  μ0 ⎪ 3 ⎪ ⎩ x2 , −x1 + x1 + for vp − x2 − 1 T  where x = [x1 , x2 ]T and f = f (1) , f (2) .

x2 − vp < 0, (3.5) x2 − vp > 0,

page 64

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Numerical Estimation of the Stick–Slip Transitions

65

The discontinuity zone Σ that splits two adjacent regions S1 and S2 is defined as follows: Σ = {x ∈ R2 : x2 − vp = 0},

(3.6)

where S1 and S2 take these forms, respectively: S1 = {x ∈ R2 : x2 − vp < 0}, S2 = {x ∈ R2 : x2 − vp > 0}. On the background of Filippov’s theory the transition function σ(x) is defined as (1)

(2)

(3.7)

σ(x) = f2 (x)f2 (x). (1)

(2)

The functions f2 and f2 are the second components of vector fields f (1) and f , respectively. If we assume that xv = [x1 , x2 = vp ]T , then on the boundary zone the function (3.7) yields, (2)

σ(xv ) = (x1 (x21 − 1) − μ0 )(x1 (x21 − 1) + μ0 )

(3.8)

= x21 (x21 − 1)2 − μ20 .

Moreover, if the derivative Dσ of function σ(xv ) exists in the direction x1 , then it is possible to find some critical points of the function at subsequent conditions: Dσ,0 =

∂σ(xv,0 ) = 0, ∂x1

(3.9)

σ(xv,0 ) = 0.

Using Eq. (3.9), the characteristic equation Dσ,0 = 2x1,0 (x21,0 − 1)(3x21,0 − 1) = 0, at conditions (3.9), allows us to find the set of roots (tangent points) as follows:   1 (k) , for k = 1, . . . , 5. (3.10) x1,0 = 0, ±1, ± √ 3 By substituting values from the set (3.10) to the Eq. (3.8) and then solving each of the obtained equations with respect to μ0 , the set of boundary values of the shape parameter of friction characteristics reads   2 (k) , for k = 1, . . . , 3. (3.11) μ0 = 0, ± √ 3 3 (1)

Inspecting the problem carefully, the stick zone Σu vanishes at μ0 = μ0 but in (1,2,3) the discontinuity zone Σ, there remains only three tangent points, i.e., x1,0 . One (2,3)

the stick zone Σu is continucould say that this zone is degenerated. At μ0 = μ0 √ (6,7) ous and exists between the boundary tangent points x1,0 = ±2/ 3. The tangent (4,5)

points x1,0 zone.

lying between that boundary tangent points are also included in this

page 65

June 8, 2017 12:9

66

3.3

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Numerical Estimation of the Stick to Slip and Slip to Stick Transitions

H´enon’s algorithm derived in [H´enon (1982)] can be used after determination of the points of Poincar´e maps, since after appropriate modification and taking into account Filippov’s theory, it even could be useful in computation of the numerical solution of various discontinuous systems with dry friction [Awrejcewicz and Olejnik (2002b)]. The considerations should be taken depending on the broader analysis of the discontinuity, visible and invisible tangent points and a function defining some transition between zones of sliding (standard) solutions. The numerical procedure is based on a class of general implicit Runge–Kutta schemes, which covers Euler schemes for differential inclusions (if sgn is seen as a maximal monotone graph, and if an approach by Moreau sweeping process is considered [Awrejcewicz and Lamarque (2003)]). The exact solutions presented in this chapter could serve for a comparison with the proposed approximation method described in Sec. 4.3, and with the illustrative example given in Sec. 4.3. The procedures compute accurately the event or discontinuity points, therefore, they belong to the class of event-driven methods, that are useful if there are finitely many event points. If this requirement is not satisfied, then some numerical scheme convergence problems could arise. The friction force and associated energy could be not well represented, because the friction force should belong to an interval expressing zero relative velocity; for instance, see Eq. (1.6). The simulation begins with the adoption of some dynamic system parameters (0) (0) and initial conditions x1 = x1 (t0 ), x2 = x2 (t0 ). Then one checks in each iteration (k) whether the solution is contained in: 1) one of the stick zones Σu , for k = 1, 2, . . . ; 2) one of the regions S1 or S2 , in which the differential equation (3.5) describing the continuous system are solved numerically by means of the selected standard integration procedure of higher order, i.e., fourth-order Runge–Kutta integration algorithm. During a slip mode in current step of integration, if the difference x2 − vp is neither positive or negative, then respectively to j = 1, 2, one repeats the step by integration of the differential equation (3.5) transferred to the form   x2 1 , X˙ 1 = F1 (X1 ) = (j) (j) , f2 f2 where X1 = [x1 , τ ]T , τ is the time going in the temporary dynamical state, and x2 is the independent variable. Respectively, at x2 − vp > 0 or x2 − vp < 0 in the (i) (i) current iteration of integration, the step h1 = vp − x2 or is equal to x2 − vp , (i+1) = vp . In this way, the transition between but the velocity of mass m equals x2 the slip and stick mode of motion is estimated with high accuracy. Afterwards, the solution to Eq. (3.5) valid in the stick zone can be continued. If in the current step of integration in a stick mode, the position reaches some (i+1) (k) (k) > x1,0 , where x1,0 is the kth tangent point, then one of the tangent points x1

page 66

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Numerical Estimation of the Stick–Slip Transitions

67

needs to repeat the step by integration of the differential equation (3.5) transferred to the form   1 ˙ ,0 , X2 = F2 (X2 ) = (i) f2 where X2 = [τ, x2 ]T , τ is the time going in the temporary dynamical state, but (k) (i+1) x1 is the independent variable. In this iteration, the difference h2 = x1,0 − x1 at kth visible tangent point makes the temporary step of integration. In this way, the transition between the stick and slip mode of motion is estimated with high accuracy, and it allows us to continue the solution to Eq. (3.5) valid in the slip zone. 2

2

x

x

6 5

1

3

4

2 1

Σ

6 5

1

2 (0)

x

Σ Σ

0

09

−0 6

08

Σ

u

u

.

.

.

−1 2 −1 7 .

0 (a)

1 (0)

.

x

1

07 −1 7 .

1

x

2

.

0 (b)

1 (0)

x

1

1

x

2

x

x

Σ

Σ

u

Σ

3

4

1

Σ

u

2 1

1 Σ

u

09 .

09

08

08

Σ

u

Σ

u

.

.

.

07 −1 7 .

.

−1

1 (0)

x

0 (c)

1

07 −1 7 .

1

x

.

1

0 (d)

1 (0)

x

1

x

Figure 3.1: Numerical solutions of the dynamical system (3.5) beginning from different initial conditions marked at intersections of dashed lines. Visible (1, 3, 5) and invisible (2, 4, 6) tangent points in the stick zones are marked by grey rectangles. The enumerated schemes of transition from slip to stick and from stick to slip determine appearance of the mentioned critical points, i.e., they determine points crossing the intersection zone Σc as well as boundary points beginning and ending

page 67

June 8, 2017 12:9

68

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

the system trajectory running in Σu . If a particular point of solution that lies on the intersection of regions S1 and S2 of the sliding solution is included in Σc ⊂ Σ, then deflections of the periodic trajectory on the phase plane (x1 , x2 ) presented in Fig. 3.1(a) are observed in the neighborhood of the third tangent point. The deflections are characteristic for such types of systems with friction. The reason is that the trajectory of solution (after crossing the intersection zone Σc ) is influenced by a different vector field of the investigated dynamical system. The described methodology is based on detection of the critical points that are not present in continuous systems. Exemplary periodic solutions have been shown in Fig. 3.1. In the considered system with dry friction, any stick mode does not occur, and a discontinuity in the form of the described deflection of the transient trajectory is seen in its transition through the zone of intersection Σc ⊂ Σ. Various initial conditions are selected to observe the transient trajectory running in the neighborhood of tangent points. In √ addition, the coefficient of static friction μ0 = 1/(3 3) and velocity vp = 1 of (k) the belt are assumed to investigate some particular case. The tangent points x1,0 determined for the selected system parameters by finding roots of the transition function σ(x) given by the formula (3.7) are as follows: 2  (k) x1,0 = √ · cos (c), cos (2c), sin (0.5c), − sin (0.5c), 3  π − cos (2c), − cos (c) , c = , for k = 1 . . . 6. 9 3.4

Bifurcations of Tangent Points

In this part of the study, some bifurcations of stick modes and their corresponding tangent points of a typical discontinuous dynamical system with one degree of freedom and a harmonic forcing are analyzed. horizontally moving base

horizontally oscillating block

x1

rigid support

T

m

k(z)

y=Asin(
t)

mg

vp

Figure 3.2: Structural model of the investigated mechanical system with friction and harmonic forcing. A one-dimensional system consists of a mass m that is placed on the belt moving at constant velocity vp . The oscillating body is connected to the harmonically forced spring described by a nonlinear characteristic k(z). The distance z = x1 − y results

page 68

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Numerical Estimation of the Stick–Slip Transitions

ws-book975x65

69

from the difference between position x1 of mass m and y(t) = A sin (ωt), describing the time-dependent displacement of the second end of the nonlinear spring. The harmonic function has an amplitude A and frequency ω. Adjacent surfaces of the frictional pair created by the body’s mass m and the movable belt constitute the dry frictional contact. The contact is mathematically described by the friction force T which depends on the shape parameter μ0 (the coefficient of static friction) and the relative velocity of motion x2 − vp measured in the distinguished frictional connection. Model of the investigated mechanical system with friction and harmonic excitation is shown in Fig. 3.2. Dynamics of the modeled mechanical system is mathematically described as follows: x˙ 1 = x2 , (3.12) 1 x˙ 2 = (−k1 (x1 − y) + k2 (x1 − y)3 + T (v)), m where T is the velocity-dependent friction force modeled by Eq. (3.2); see Fig. 3.3.

0

Figure 3.3: The friction force model T (v) with monotonically decreasing branches (bold line). Putting in Eq. (3.12) the dependencies on T and y, the following system of first order differential⎧equations reads

  μ0 ⎪ x , 1 k z3 − k z + ⎪ for x2 − vp < 0, ⎨ 2 m 2 1 vp − x2 + 1  x˙ = f (x) =  μ0 ⎪ 1 ⎪ ⎩ x2 , m k2 z 3 − k1 z + for x2 − vp > 0, vp − x2 − 1 y = A sin (ωt),

T  where x = [x1 , x2 ]T , f = f (1) , f (2) and z = x1 − y. The discontinuity zone Σ splitting two adjacent regions S1 and S2 and the transition function σ(x) are given by Eqs. (3.6) and (3.7).

page 69

June 8, 2017 12:9

70

3.4.1

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Tangent Points on the Oscillating Boundary of Discontinuity

If we assume that zv = z at x2 = vp , then the transition function (3.7) in Σ reads  2 1  (3.13) σ(zv ) = 2 k2 zv3 − k1 zv − μ20 . m If there exists a derivative of function σ(zv ), then it is possible to find critical points of the function by means of these conditions: D σ0 =

∂σ(zv0 ) = 0, ∂zv

σ(zv0 ) = 0.

In the next step, one calculates 2zv0 Dσ0 = (3k2 zv20 − k1 )(k2 zv20 − k1 ) = 0, m2 where the set of roots at conditions (3.14) follows     k1 k1 (i) zv0 = 0, ± ,± , for i = 1, . . . , 5. k2 3k2

(3.14)

(3.15)

(k)

Putting zv0 given by Eq. (3.15) in (3.13) and then evaluating the obtained equation with respect to μ0 , the following set of boundary values of shape parameter of the friction characteristics T follows !  k1 2k1 (i) , for i = 1, . . . , 3. (3.16) μ0 = 0, ± 3 3k2 The set of values (3.16) places the transition function of the stick zone as a tangential (with an approach from below and from above too) to the abscissa in the graph of the dependency σ(zv , μ0 ) shown in Fig. 3.4(a). The resulting definitions in the form of generalized equations (3.10) and (3.11) contain parameters k1 , k2 and μ0 . It confirms that the location of a stick zone depends on parameters of the particular dynamical system. Small bifurcations of their values cause the displacement of stick zones that determine the dynamic state of the considered discontinuous system with friction. In Fig. 3.4(a), the top graph marked by a dashed line confirms that in the contact zone and at some boundary value μ0 , any stick mode not exists. It holds because the function σ(zv , μ0 ) ≥ 0. A degeneracy of the stick zone Σu to three tangent points is observed. It means, that some deflections will appear only at such points of the solution that crosses the intersection zone Σc from region S1 to S2 or vice versa. Quite opposite interpretation is regarded to the graph marked by a solid line (see Fig. 3.4(a)), which is tangent from below to the abscissa. In this case, the stick zone is wide and extended from the left to the right branch of the graph (both tend to +∞ in the direction +σ). The solution will be strongly attracted by both vector fields which are faced opposite to each other in the discontinuity zone Σ splitting regions S1 and S2 . As a result, a generation of stick–slip cycles with a possibility of (2,3) will occur. their bifurcation for the control parameter μ0 < μ0

page 70

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Numerical Estimation of the Stick–Slip Transitions

tangent points

ws-book975x65

71

stick zones

(a)

(b)

0.7

0.35

0

-0.1 -1.5 c)

0

1.5

(c)

Figure 3.4: (a) Tangent points (circles) on the intersection of function σ(zv , μ0 ) (1) (2,3) (solid line). (b) The with the axis zv for μ0 = μ0 (dashed line) and μ0 = μ0 time-dependent oscillations y(t) of the stick of tangent points  zone.√(c) Bifurcations  (1) at changes of the control parameter μ0 ∈ μ0 , 4 3/9 for the system parameters: m = 1, k1 = k2 = 1, v = 1.

The transition function σ(x) introduced in Eq. (3.7) is defined as multiplication (1) (2) of components f2 and f2 . They could be used to estimate the tangent points lying in the discontinuity zone at intersection of the vector fields f (1) and f (2) , acting in the regions S1 and S2 , respectively. Bifurcations of tangent points in a range of changes of the bifurcation parameter μ0 have been presented in Fig. 3.4(c). The described method allows us to estimate the tangent points on the basis of known function σ(zv , μ0 ). Assuming for the√ analyzed dynamical system √ the set π , of parameters: m = 0.1, vp = 0.4, μ0 = 93 , k1 = k2 = 1, A = 4 3 3 sin 18

page 71

June 8, 2017 12:9

72

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

ω = 3π/2.4728, there has been estimated an initial set of tangent points: (k) x1,0

√   π 2π π 2 3 · ± cos , ± cos , ± sin = , 3 9 9 18

for k = 1, . . . , 6.

(3.17)

Values in the set (3.17) are very useful. For instance, they allow us to exactly compute the length and duration time of the stick modes.

3.4.2

A Two-Periodic Stick–Slip Numerical Solution

A two-periodic stick–slip trajectory of the numerical " (1,2) " solution to the system (3.12) is shown in Fig. 3.5. It has two stick modes "Σu " of lengths 0.4697 and 0.1663.

7

6 3

4 1 5

-0.77

-1.9 -0.84

2

-0.32

0.2

Figure 3.5: A two-periodic stick–slip trajectory for the initial conditions x ¯0 = (3) (x1,0 , vp ). Corresponding to the mentioned stick modes, the oscillating body of mass m sticks to the moving belt within 1.1746 and 0.4161 s, and travels at its constant velocity vp until the next tangent point is met. Loss of stick and the associated beginning of the slip mode occurs at visible tangent points. The slip mode’s trajectories projected on the (x1 , x2 ) plane have the shape of arcs alternately appearing in regions S1 (the upper arcs) and S2 (the lower arcs). In the left part of the two-periodic closed trajectory, there are visible two deflection points indicating the intersection zone Σc . With regard to constant harmonic excitation, the stick zone Σu oscillates along ±x2 in the discontinuity zone Σ. As a result, initial locations of tangent points given by the set (3.17) oscillate in time accordingly to (k) (k) x ˜1,0 (t) = x1,0 + A sin (ωt), for k = 1, . . . , 6. The two-periodic trajectory of the exemplary dynamical system containing the oscillating stick zone (see Fig. 3.5) was found using the numerical procedure published in [Olejnik and Awrejcewicz (2013a)].

page 72

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Numerical Estimation of the Stick–Slip Transitions

3.5

ws-book975x65

73

Lyapunov Exponents

Lyapunov characteristic exponents were introduced by Lyapunov [Sansone and Conti (1964)] in the context of non-stationary solutions of ordinary differential equations. They provide a way to characterize the asymptotic behavior of nonlinear dynamical systems by giving a measure of the exponential growth (or shrinkage) of perturbations about a nominal trajectory. Since they measure the sensitivity of solutions of dynamical systems to small perturbations, they are often used to indicate chaotic motions when the dynamics occurs on an invariant set; see [Oseledec (1968); Eckmann and Ruelle (1985)]. Their positive values establish chaos, but all negative a non-chaotic system. Computation of Lyapunov exponents is one of the most important numerical problems considered in the analysis of nonlinear dynamical systems. Oseledec’s theory [Oseledec (1968)] and Benettin’s numerical algorithms described in [Benettin et al. (1980); Kim and Choe (2010)] refer to methods of computational estimation of Lyapunov exponents’ spectra for continuous systems. If equations of motion are not known explicitly, then some methods based on attractor’s reconstruction (for any continuous or discontinuous system) from a time series are developed [Sano and Sawada (1985); Wolf et al. (1985)].

3.5.1

Computation of Lyapunov Exponents Using Time Series

The dynamics of real physical systems can be sometimes assessed visually or by measuring signals stored in a computer memory in the form of a sequence of measured values. Measurement of the actual state of a dynamical system (displacement, temperature, voltage, etc.) is done with a time sampling leading to a continuous signals of physical quantities. The numerical estimate of the nature of the n-dimensional discrete trajectories requires the calculation of n Lyapunov exponents. The mathematical basis for calculation of the spectrum of Lyapunov exponents based on a discrete time trajectories of n-dimensional chaotic dynamical system has been developed in [Sano and Sawada (1985)]. The essence of the calculation algorithm is as follows. Let x ¯ ∈ Rni be the vector state of some ni -dimensional discrete dynamical system and x ¯j (j = 1, . . . , nj ) denote the vectors of subsequently chosen points of the investigated time history. If to assume a small ball of radius ε, centered in the point matched successively by the vector x ¯j , then nk points included in the ball is found. Radius of the ball must be large enough, i.e., nk > ni . Distance from nk points of that set to center of the ball is y¯k = x ¯k − x ¯j at k = 1, . . . , nk . After the time τ = dΔt (d — a small natural number, Δt — the sampling time in which the ¯j+d , but the neighboring trajectory points have been saved), the vector x ¯j shifts to x ¯k+d − x ¯j+d . If points to x ¯k+d . This also results in changes of the distances y¯k+d = x radius ε is small enough, then with good approximation the vectors y¯k+d and y¯k

page 73

June 8, 2017 12:9

74

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

are tangent to each other and they create a tangent space. The linear dependency between these vectors is y¯k+d = Aj y¯k , since the smallest quadratic estimation of the matrix Aj ∈ Rni ×ni is given by %# n %−1 #n k k $ $ T T Aj = y¯k+m y¯k y¯k y¯k , j = 1, . . . , nj . k=1

k=1

Spectrum of Lyapunov exponents is computed by means of the formula nj 1 $ λi = lim ln ||Aj e¯ji ||, nj →∞ nj τ j=1

i = 1, . . . , ni .

(3.18)

After the assumed number of iterations, there must be carried out an orthogonalization of directions and normalization of lengths of vectors connecting the neighborhood points of the trajectory. In Eq. (3.18) the symbol e¯ji denotes the set of basis vectors acting at points x ¯j included in the tangent space. In the first iteration of the algorithm of estimation of Lyapunov exponents the initial set of basis vectors is arbitrary. Gram–Schmidt orthogonalization of directions of the basis vectors e¯ji as well as normalization of lengths Aj e¯ji are done before all components of the sum (3.18) are computed [Wyk and Steeb (1997)]. In the next part of this chapter, we use the high accuracy method of numerical integration (see Sec. 3.3) to generate the time series trajectory of an exemplary discontinuous system with friction. Estimations of Lyapunov exponents for three kinds of solutions, i.e., a periodic, quasi-periodic and chaotic one are finally computed. 3.5.2

Lyapunov Exponents of Typical Trajectories

We consider the time series of solution of a discontinuous dynamical system with dry friction shown in Fig. 1.18. The system of four first-order ordinary differential equations takes the form: x˙ 1 = x2 , x˙ 2 = −x1 − ξ1 (ξ2 (x2 + x4 ) − x3 − T (v)) , x˙ 3 = x4 ,

(3.19)

x˙ 4 = ξ3 (−x1 − ξ2 x2 − ξ4 x3 − ξ5 x4 ). In Eq. (3.19) the state vector x ¯ = [x1 ,x2 ,x3 ,x4 ] ∈ R4 , relative velocity of contacting surfaces v = x2 − vp , and the constant parameters ξ1 = k2 /(ω 2 m), /(ω 2 J), ξ4 = (k2 + k3 )/k2 , ξ5 = ω(c1 + c2 )/k2 , ξ6 = μ0 k3 /k2 , ξ2 = c1 ω/k2 , ξ3 = k2 r2 ξ7 = ωc2 μ0 /k2 , ω = (k1 + k2 )/m, J = m(a2 + b2 )/3 are assumed. The term y = 1 − ξ6 x3 − ξ7 x4 includes also the additional normal force attached to the mass m from the displacement of the bracket’s end, and one has to take it into account in the friction force model: |T (v)| ≤ yμ0 , T (v) = sgn (v)yμk (v),

for v = 0, for v = 0,

page 74

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Numerical Estimation of the Stick–Slip Transitions

75

where μk is expressed by Eq. (3.2). The self-excited system presented in Fig. 1.18 is equivalent to the real test stand investigated in [Awrejcewicz and Olejnik (2005b)], where a block of mass m moves on the belt in the direction x1 , and the bracket characterized by the mass moment of inertia J rotates about the point s with respect to the angle ϕ. Two bodies are coupled by linear springs of the stiffness k2 , k3 and dampers of the damping coefficients c1 and c2 . The block on the moving belt is additionally attached to a fixed wall by means of a linear spring of stiffness k1 . One assumes that maximum angles of rotation of the second body (the bracket) are small, therefore, the simplification sin (ϕ) ≈ ϕ ≈ y/r is assumed.

ξ1...7 , δ, vp 0.237 0.000 0.613 1.888 0.000 1.300 1.115 0.330 0.930

-0.24

-0.76

-1.76 -2.1

-0.7

λpi −0.0004 −0.0049 −0.1339 −0.3207

0.7

Figure 3.6: Two-periodic trajectory for x ¯0 = (0, −0.1, 0, −0.1).

ξ1...7 , δ, vp 0.413 0.000 1.069 1.980 0.000 0.707 0.693 0.101 0.707

-0.24

-0.4

-1 -0.7

0.13

λqua i −0.0004 −0.0000 −0.0116 −0.3645

0.96

Figure 3.7: Quasi-periodic trajectory for x ¯0 = (0, 0, 0, −0.1). Spectra λi of Lyapunov exponents of the periodic, quasi-periodic and chaotic

page 75

June 8, 2017 12:9

76

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

ξ1...7 , δ, vp 0.222 0.000 0.575 3.333 0.000 2.900 6.766 0.976 0.689

0.16

-0.57

-1.3 -1.1

-0.17

λch i +0.0025 −0.0019 −0.0705 −0.7542

0.76

Figure 3.8: Chaotic trajectory for x ¯0 = (0, −0.1, 0, −0.1). trajectories (saved in time (3 − 12) · 103 s, in the form of 45001 × 4 elements of vector x ¯), that have been obtained by solving the system (3.19) by means of the numerical procedure described in Sec. 3.3 are presented in Fig. 3.6–3.8. ch The Lyapunov exponents’ spectra λpi , λqua i , λi confirm shapes of the observed phase–plane trajectories. As it was expected, each of them contains at least one characteristic exponent placed near zero. In the case of the quasi-periodic solution, cha > 0. λqua {1,2} ≈ 0, but in the case of the chaotic one λ1 The Filippov’s theory applied to the considered discontinuous system has made it possible to carry out interesting numerical analysis of stick–slip bifurcations observed in piecewise continuous dynamical systems. An oscillating boundary of discontinuity has been described by the velocity-dependent function of motion of the belt. Additionally, it has been made dependent on the function of external harmonic excitation. Tested dynamical systems were divided into two subsystems whose continuous solutions have been found in corresponding two adjacent regions of the analyzed state space of the system. Finally, trajectories of some dynamic behaviors have been confirmed by spectra of characteristic Lyapunov exponents.

page 76

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 4

Smooth Approximation of Discontinuous Stick–Slip Solutions

The time moment of appearance of zero relative velocity during a relative displacement of contacting bodies in dynamical systems with dry friction can be precisely computed. This chapter describes an approximation method that significantly increases quality of estimation of approximate discontinuous solutions observed in the stick–slip motion with friction. Results of numerical computations obtained by means of the method of approximation of solution of a one-degree-of-freedom dynamical system with friction confirm good accuracy of the determination of stick phases. Desired smoothness of the approximate stick–slip transitions is achievable by decreasing a fitting parameter of the approximation.

4.1

Introduction

The multivalued mathematical functions are parts of the differential equations modeling discontinuous effects caused by dry friction. Models of dry friction are usually given as a function of friction coefficient or a function of friction force with respect to the velocity of relative displacement between surfaces of bodies being in a frictional contact. Among many forms described in Chapter 1, one distinguishes a theoretical model of friction given by Eq. (1.6) as well as an experimental one, given by Eq. (1.25). Numerical integration methods of any differential equations modeling dynamics of discontinuous systems including the multivalued functions T1 (vr ) and T2 (vr ) defined in Eq. (1.6) and (1.25) are more difficult in estimation of accurate numerical solutions. Inspecting these functions at vr = 0, while a stick phase occurs in the temporary state of existence of zero relative velocity, the force of static friction Ts± is directed opposite to the force W , acting in a surface being parallel to the contact surface of the cooperating frictional couple. The value of W dependent on sign causes appearance of two different scenarios of slipping (after the end of the preceding it stick phase) that is modeled neither by functions T+ or T− , respectively to the positive or negative relative motion. A schematic view of distribution of forces is presented in Fig. 4.1. 77

page 77

June 8, 2017 12:9

78

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Figure 4.1: Distribution of forces acting on mass m during (a), (b) the static and (c), (d) kinetic friction phenomena (y — displacement of the moving body, vp — velocity of the base). The stick and slip phases are drawn using vertical and horizontal lines, respectively. On the basis of the problem explained above, the desired accuracy of numerical solution of any system of first order differential equations modeling motion of the dry friction-induced dynamical systems will require one to apply some particular approximations or some special methods. One such method is proposed in this chapter, but before, let us focus our attention on some attempts oriented on a precise modeling of such kinds of systems. Friction at sliding joints is considered neither as functions of position- or velocitydependent friction coefficients and of position-, velocity- or even accelerationdependent reaction forces. Friction models of such kinds determine forms of nonlinear differential and algebraic equations. Therefore, they require us to use miscellaneous numerical and iterative methods [Klepp (1994)]. The governing equations for a one-degree-of-freedom system have been established in this work with the Lagrangian formalism and with conditions of dynamic equilibrium for the system. Properties of responses are determined by the numerical simulation of representative spring-loaded three-body system with two sliders and two hinges under a harmonic excitation. Dependence of solutions on perturbations of friction coefficient characteristics on the basis of the bilateral quasi-static contact of a viscous-elastic body with a rigid obstacle has been studied in [Amassad et al. (1999)]. Contact surface was modeled with a modified Coulomb law of dry friction. The coefficient of friction was assumed to be dependent either on the total or current slip. Classical variational formulations

page 78

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Smooth Approximation of Discontinuous Stick–Slip Solutions

ws-book975x65

79

of the investigated problems were used to establish the existence and uniqueness of a weak solution to each of these two cases for a sufficiently small coefficient of friction. The problem of motion of two interconnected mass points in a resistant medium under a periodic change of distance between them has been considered in [Zimmermann et al. (2013)]. It was shown that the necessary condition for the displacement of mass is constituted by the nonlinear law of dry or viscous friction. The friction is assumed to be small and the investigations are based on the method of averaging. By means of this method an analytical dependence of the velocity of motion is obtained. Based on a few examples mentioned above, and also in Chapter 1, many approaches to proper formulation of piecewise continuous analytical solutions to miscellaneous problems of contact mechanics have been elaborated. Numerical methods have to be developed simultaneously. They produce some approximate time series of numerical solutions at a desired accuracy. Therefore, the approximate methods of numerical solution of discontinuous dynamical systems are also required to get a deeper knowledge about the interesting and still explored problems of mathematics and mechanical engineering. Approximate methods or simplification rules applied to numerical solutions of discontinuous dynamical systems are strictly related to this chapter. They are different and depend on the dimension, complexity and accuracy of modeling of the system. One of such methods is based on numerical computation of Poincar´e maps from a system of first-order differential equations [H´enon (1982)]. An exact solution of any discontinuous model can be found by the brute-force approach, i.e., a numerical integration until a stable steady state is reached to locate the stick to slip or slip to stick transitions. The H´enon method cannot be used in combination with any of shooting and path following algorithms. In [Van de Vrande et al. (1999)] a set of discontinuous ODEs for a dynamical system with dry friction has been particularly studied. The proposed numerical algorithm to search for both stable and unstable periodic stick–slip oscillations was used to investigate the finite-element models with local nonlinearity such as dry friction, hence, these kinds of nonlinearities had to be approximated by means of the smooth function given in Eq. (4.3). In contrary to the smoothing procedure, there is tested by [Parker and Chua (1989)] a simple shooting method with a stiffODE solver in combination with a path following method to compute periodic solutions for a time-varying design variable of some autonomous one- and twodegree-of-freedom systems. Numerical continuation of Hamiltonian relative periodic orbits has been considered in [Wulff and Schebesh (2008)]. The paper presents a method and numerical computation of relative periodic orbits persisting from orbits in a symmetry breaking bifurcation. In a generalization of the approach, a path following the algorithm based on a multiple shooting algorithm for the tangential continuation method with

page 79

June 8, 2017 12:9

80

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

implicit re-parametrization has been derived. Another efficient numerical path following method beyond critical points is described in [Deufhard et al. (1987)]. A single inclined oscillator system with a dynamical model of dry friction has been considered by [Chatterjee and Saha (2007)]. The authors assumed that a displacement x forms a slip phase of a slider. The relative motion occurs between two consecutive stick phases and velocity reversal does not take place during the particular slip phase. Resetting the time origin τ ∗ at the time of beginning of a slip phase requires an unknown phase to be introduced in the excitation function for both forward and backward slips. Duration of each slip phase could be obtained by solving a first-order differential equations on x at τ ∗ , but the total slip movement per excitation period should be an algebraic summation of the steps x(Tslip ), for all phases making the single period. Therefore, each slip phase could be terminated by sticking without any velocity reversal. If velocity reverses before sticking, then one has to integrate the slip phase in two parts based on direction of the friction force. Stable stick–slip motion exists only if the absolute value of the extended force is less than the limiting friction force at the end of the last slip phase. Otherwise, the system moves in the continuous slipping mode. If we observe that the method changes itself for the absence of sticking, then the proposed approach seems to be quite difficult to be generalized on a wider class of discontinuous systems. [Teixeira and da Silva (2012)] presents some applications to the constrained differential equations. It concerns some aspects of the qualitative-geometric theory of non-smooth systems. A particular survey of the state of the art on the connection between the regularization process of non-smooth vector fields and the singular perturbation problems has been done. Peculiarities of systems with dry friction and systems with collisions (impacts) make them very difficult for the conventional asymptotic analysis developed for continuous systems, i.e., requiring from the right-hand sides of the equations of motions a definite level of continuity. [Fidlin (2001)] brings an analysis of transformation of discontinuous systems to a special forms being natural for the corresponding problem. Averaging theorems were formulated and proved for the considered forms of dynamical systems. A vibration-induced displacement served for illustration of the proposed approach. A numerical technique for overcoming the transitions between stick and slip phases, or the slip phase accompanied by high-frequency oscillations of the relative velocity difference has been developed by [Dimova et al. (1995)]. Computationally efficient procedure increased accuracy of solution of differential equations with Coulomb damping. In order to validate the proposed technique, the dynamic response of a four-story braced frame with friction devices was presented. A class of contact problems with friction in elastostatics was considered in [Angelov and Liolios (2004)]. Finite element approximation, existence and uniqueness results as well as mostly used iterative methods have been briefly summarized in the reference. On the background of the nonlinear alternating direction Kellog’s

page 80

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Smooth Approximation of Discontinuous Stick–Slip Solutions

81

method, an exemplary problem with different contact boundary conditions has been proposed, analyzed and solved. The short review of existing techniques confirms, that any attempts of a precise integration of non-smooth systems produce interesting mathematical derivations as well as some computational difficulties at the discontinuities splitting the regions of continuous solutions.

4.2

Discontinuity in Models of Dry Friction

The form of discontinuities defined in Sec. 4.1 can be shown more clearly after drawing characteristics T1 (v) and T2 (v) given by Eq. (1.6)–(1.25) and a chosen set of parameters. For a deeper investigation the following simplified forms of friction characteristics are proposed: ⎧ ⎪ ⎨T± (v) = sgn(v) 1 , v = 0, 1 + c|v| (4.1) T1 (v) = ⎪ ⎩Ts ∈ [−1, 1], v = 0, ⎧ 1 ⎪ T+ (v) = (1 − cv), ⎪ ⎪ ⎪ 2 ⎪ ⎨  c  −c(11v+3) e T (v) = −1 + + e−cv−1 , − T2 (v) = 10 ⎪   ⎪ ⎪ ⎪ c 1 ⎪ ⎩Ts ∈ −1 + e−3c + e−1 , , 10 2

v > 0, v < 0,

(4.2)

v = 0.

Friction models (4.1) and (4.2) represent the particular forms of Eqs. (1.6) and (1.25) as functions of the parameter c and the velocity v of the relative displacement of contacting surfaces. a)

1

b)

T1 (v)

0.5

T2 (v)

T+

0.6

0.2 v

0.3

0.2 0.3

0.1

v

0.1

0. 1

0. 1

0.3

0.2

0.3

0.2

0.6




0.6

T

1

0.9

Figure 4.2: (a) The theoretical and (b) the experimental model of dry friction (c = 3).

page 81

June 8, 2017 12:9

82

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

On the basis of friction force T1 (v) given by Eq. (4.1), value of the static friction force Ts is included in the interval [−1, 1]. If the normal force N is constant, then the interval can be reduced to the set of boundary values, i.e, Ts± ∈ {−1, 1}. Amplitude of an analogous signal (continuous in time domain) can take any value from the continuous interval that is limited from both sides by its extremes. Choosing a different way, the amplitude is calculated at a time instant with the use of a mathematical formula (if known), describing the considered signal. Quite different situation holds in the numerical procedure, when a computer processes this formula for a series of discrete input values. Similarly, acquisition of an analogous voltage signal by a computer measurement card leads to the discrete signal represented by a series of experimental data. The series is known only approximately (in values and time of occurrence) at some moments of time dependently on the time step between each read of data from the voltage output ports being located on the measurement card. For instance, if a set of input values for an evaluation according to the function (4.1) would be vi = {−0.2, −0.07, 0.03, 0.1}, then for c = 3, the series of i = 1, . . . , 4 values of the function T1 (vi ) becomes equal to {−0.62, −0.82, 0.92, 0.77}, with accuracy 0.01. The time of transition from slip to stick phase (state) is improperly estimated, not known. This simple but very important observation has to be taken into consideration and carefully inspected after investigations of discontinuous systems with a nonlinearity introduced, i.e., by friction, impacts or clearances. As a consequence, self-sustaining inaccuracies will be accumulated in each step of integration leading to the improper numerical image of the investigated real system. Subsequent slip–stick and stick–slip transitions that are observed in the selfexcited systems with dry friction follow at large accelerations. It results from fast changes of velocity of the relative displacement (from zero to a maximum and vice versa) between surfaces making the dry frictional contacts, for instance, in braking mechanisms, where at some conditions the squeal phenomennon appears [Schlagner and von Wagner (2009)]. Various friction models with extended review of nowadays approaches have been included in [Berger (2002); Awrejcewicz and Olejnik (2005a)]. Any numerical scenario of error accumulation devoted to the estimation of zero relative velocity generates incorrectly computed boundaries or placements (duration) of stick phases. Therefore, a proper modeling of systems with dry friction requires one to keep some high accuracy of computations reflected in an adequately small time step of numerical integration. We can find better numerical solutions applying a proper representation of differential equations. This is the essential content of the following sections.

4.2.1

Zero Value in Numerical Integration

Usually, any research performed in the field of numerical modeling of nonlinear dynamical systems requires us to define some particular (critical) values. One of such values could be set at v(t) = 0 (at some instance of time), appearing in modeling of

page 82

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Smooth Approximation of Discontinuous Stick–Slip Solutions

ws-book975x65

83

dry friction, i.e., with the use of the function T2 (v) given by Eq. (4.2). Numerical solution of that kind of systems is given by a series of real numbers. Therefore, v0 determining a state of transition neither from or to the stick phase of contacting surfaces would be difficult to achieve. An approach used to approximate these particular values in numerical computations is based on a definition of an interval I(δ) (a symmetric “window”), determining boundaries of appearance of any particular value like v0 . A method that enabled us to estimate the dimensions of a velocity window in the frictional Karnopp’s model has been developed in [Karnopp (1985)]. As an example, let us choose the function T2 (v). If during a numerical integration the time history v(t) of the relative displacement of contacting surfaces reaches the interval I(δ) := [v0 −δ, v0 +δ], for some small δ and in some step of integration, then the particular (crossing) value v0 is achieved and one could switch at τ ∗ between functions T+ and T− (a crossing through zero of the relative velocity appears) or between any of these functions and T2 (v) = Ts , and then, a stick phase of surfaces of the cooperating couple occurs. Correct estimation of the particular value v0 ∈ I(δ) is quite difficult, because boundaries of the interval I depend either on requirements of accuracy or a time that would be acceptably long for computations. They depend also on the step of integration and are related to the differences between consecutive values of velocity v(t). Any incorrect estimation of boundaries of the interval I often reflects significant inaccuracies. Because of any too large window for v0 , the time moment of transition into the state of sticking increases and extends boundaries of the corresponding stick phase. In contrary to the scenario, assumption of any too small window for v0 could disregard the existing (in reality or in any analytical solution) stick phase of frictional surfaces. The second scenario provides too frequent crossings through the surface splitting positive and negative relative velocities. This kind of discontinuous (switching) behavior is mathematically described in Filippov’s theory [Kuznetsov and Rinaldi (2003)] as well as solved numerically by special methods [H´enon (1982); Awrejcewicz and Olejnik (2002b)]. 4.2.2

A Continuous Approximation of Step Function

Derivation of a mathematical model of dry friction requires to use the function sgn defined in (1.6). For the purpose of analytical solutions the stepping character of the function can be approximated sgn(v) ≈ sgn(v) ˜ =

2 atan(av), π

(4.3)

where a is a large number (about 104 in macro-scale). A few approximations corresponding to Eq. (4.3) for several values of the parameter a are illustrated in Fig. 4.3. There is assumed in Fig. 4.3, a = 103 to show another view of the approximating function. One could check that at a = 105 the roundness disappears in

page 83

June 8, 2017 12:9

ws-book961x669

84

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

a

=101

a

=102

a

=104

1.0

˜ ( )

sgn v

a

=103

1.0 -

0.5

0

˜ ( )

sgn v

0.5 v

1.0

0.5

0. 5

1.0

v

1.0

0. 5

1.0

0.5 1.0

a)

-

1.0

b)

Figure 4.3: Approximation of the function sgn(v) for several values of a and a series of vi = ih, for i = 1, . . . , n: a) h = 0.04 and n = 51, b) h = 0.0005 and n = 4001. the assumed scale. It is sufficient for the macro-scale applications that could be taken into consideration, for instance, in a worm-like locomotion systems [Behn and Steigenberger (2012)]. Another more flexible approximation of the sgn function is given by Eq. (1.11) in Chapter 1. It is visible in Fig. 4.3(b) that the properly chosen parameter a and the step of integration h allow for some intended distribution of the desired amount of points (for the relative velocities about zero value) on the interval [−1, +1] of values of function sgn. ˜ As a consequence, it allows us to approximate the force of static friction characterizing the states of sticking. Application of the smoothing function is simple, but due to the stiffness of the approximation around v = 0, it consumes more computational time while computing values of atan or tanh functions. More multiplications have to be done in numerical algorithms at each integration step. Following the H´enon’s method for Poincar´e maps [H´enon (1982)] or the Filippov’s theory related to discontinuous planar systems [Kuznetsov and Rinaldi (2003)], a method of approximation of the discontinuous planar systems having a switching nature can be proposed.

4.3

The Method of Smooth Approximation

Let us investigate a one-degree-of-freedom dynamical system with dry friction shown in Fig. 4.4. A solid body of mass m is attached to a fixed support by means of a nonlinear elastic element characterized by parameters k1 and k2 . The body oscillates under the influence of dry friction force T1 existing in the contact zone, that is created by the body’s surface and the outer surface of the base moving with constant velocity vp . One assumes a moving rigid base being under action of the normal force mg and other forces tangent to its surface in the contact zone. A force of reaction of the nonlinear elastic element on a change of the displacement y of the body m

page 84

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Smooth Approximation of Discontinuous Stick–Slip Solutions

ws-book975x65

85

is denoted by W , whereas the dry friction force existing in the contact zone of the analyzed couple of surfaces is denoted by T1 .

m

y

mg

T

k1, k2 W

vp

Figure 4.4: Structural model of the investigated one-degree-of-freedom mechanical system. The equilibrium of forces acting on the body of mass m is expressed by the following equation m¨ y + W (y) = T1 (v).

(4.4)

Having in Eq. (4.4) the friction force dependency (4.1), the relative velocity of displacement v = y˙ − vp and the force W (y) = −k1 y + k2 y 3 , the second-order discontinuous differential equation takes the form μ0 = 0. m¨ y − k1 y + k2 y 3 − sgn(y˙ − vp ) (4.5) 1 + c|y˙ − vp | Equation (4.5) describes dynamics of a self-sustained relative vibrations with dry friction occurring at the contact surface of the body of mass m and the moving base. An “exact” numerical solution to Eq. (4.5) has been described in [Olejnik and Awrejcewicz (2013a)], but in this chapter we continue the research discussing some particular approximate solution of the analyzed system. There are considered model parameters: m = k1 = k2 = c1 = vp = 1, μ0 = 0.5. Equation (4.5) takes on that background the following non-parametric form, ˙ y¨ − y + y 3 + sgn(1 − y)

0.5 = 0. 1 + |y˙ − 1|

(4.6)

Substitution y˙ = 1−w in Eq. (4.6) produces two first-order differential equations: y˙ = 1 − w, w˙ = y 3 − y − sgn(w)

0.5 . 1 + |w|

(4.7)

Basic approximation of the proposed method, the mathematical background of which is derived in [Awrejcewicz et al. (2005); Olejnik and Awrejcewicz (2014)] is done in this step: y˙ = 1 − w, w˙ = y 3 − y − fε (w).

(4.8)

page 85

June 8, 2017 12:9

86

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

For small ε > 0, the definition of function fε : R → R yields ⎧ 0.5 ⎪ , ⎨sgn(w) 1 + |w| fε (w) := 0.5w ⎪ ⎩ , (1 + ε)ε

|w| > ε, |w| ≤ ε.

(4.9)

Function fε states some approximation of multivalued mapping  sgn(w), w = 0, Sgn := [−1, 1] , w = 0. In a result, on the basis of general theory [Deimling (1992)], the system (4.8) approximates the discontinuous differential inclusion y˙ = 1 − w, 3

w˙ − y + y ∈ −Sgn(w)

0.5 . 1 + |w|

Regarding to the definition (4.9), one needs to analyze dynamics of changes of the state variable w. As it has been mentioned in Sec. 4.2.1, zero at w0 is now the particular value of w(t) = 0 at some t = t0 . With respect to that, investigations should be carried out in the two intervals, i.e., for w ≥ ε and |w| ≤ ε. In the first one, equation (4.8) is rewritten in the form y˙ = 1 − w, w˙ = y 3 − y −

0.5 , 1+w

(4.10)

which for w > 0 is in agreement with Eq. (4.7). In the second interval, for |w| ≤ ε, one substitutes w = εz for the time-dependent variable z, but satisfying |z| ≤ 1. Then the system (4.8) takes the following form y˙ = 1 − εz, εz˙ = y 3 − y −

0.5 . 1 + εz

(4.11)

The approximate method expressed by Eqs. (4.10) and (4.11) can be applied to the periodic sliding solution p(t) of planar discontinuous systems. The advantage of this representation is visible even in the detection of crossing through w(t) = 0. We do not have to take any interval I(δ) centered at ε, but we only need to check if w(t) is greater than ε, then we integrate Eq. (4.10); but otherwise, Eq. (4.11). Of course, the smaller is the step of the numerical integration, the more exactly the boundaries of the stick phase are estimated. Also the solution to Eq. (4.11) can be expanded in the Taylor series with respect to ε (see [Vasileva and Butzov (1973)]) and Taylor terms can be found by solving linear variational equations of (4.11). So this should be another approximation approach but we do not follow it in this chapter.

page 86

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Smooth Approximation of Discontinuous Stick–Slip Solutions

4.4

87

Numerical Simulation

The results of numerical computations presented in Fig. 4.5 confirm good accuracy of approximation of the stick phase after application of the proposed method. 3

w(t)

2

0.01

y(0)

0 w(0)

ˆy0 0.01

1

=1
4  =1e
3 h

y(t) w(t)

a)

2 0

2

4

6

t0

t1

b)

t

e

6

t0

7. 5

t1

t

Figure 4.5: Time histories of the numerical solutions (y(t), w(t)) of Eq. (4.10) and (4.11) for h = 0.0001 and ε = 0.001: (a) t ∈ [0, 10]; (b) zoom of a part for t ∈ [t0 , t1 ]. Taking into consideration Eqs. (4.10) and (4.11), the procedure introduced in Sec. 4.3 is proposed.

w(t)

w(t)

0.01

1e

0

0.01

0

1e


4  =1e
4 h

a)

3

6

=1e

3


4  =1e
4 h

t0

7.5

t1

t

b)

6

=1e

t0

7.5

t1

t

Figure 4.6: Influence of decreasing of the parameter ε on accuracy of estimation of the stick phase on the time histories of the numerical solution w(t) of Eqs. (4.10) and (4.11), for h = ε = 0.0001: (a) for w(t) ∈ [−0.02, 0.02]; (b) zoom of a part for w(t) ∈ [−0.002, 0.002]. Qualitatively better results have been obtained for the approximate solution shown in Figs. 4.5(b) and 4.6(b). The stick phase computed on the basis of Eqs. (4.10) and (4.11) is smooth and its ends are more precisely estimated. Ac-

page 87

June 8, 2017 12:9

88

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

curacy of the solution could be increased by a decrease of ε, since h should be also decreased to ensure that h ≤ ε. The parameter ε may be called as a fitting parameter of the approximate solution to the real shape of the stick phase. Maintaining the same time step of integration and taking ε = 0.0001, one gets a satisfactorily exact estimation of boundaries of the stick phase; see Fig. 4.6. There is drawn in Figs. 4.5 and 4.6 every 100th point of the time series wi , for i = 1, . . . , 105 . By using the Tichonov theorem for singularly perturbed differential equations, a relationship between dynamics of the discontinuous differential equations and their continuous approximation along periodic solution has been derived. The elaborated approximation method expressed in Eqs. (4.10) and (4.11) is relatively simple, adjustable, can be applied to any periodic sliding solution of a planar discontinuous systems and could be also extended on higher order systems. On the basis of a close to exact numerical solution, the tools of dynamical analysis like bifurcation diagrams, phase portraits or Lyapunov exponents can be now more efficiently used. Drawing a final conclusion, decreasing the fitting parameter ε, a smoother and better approximated numerical solution can be achieved.

page 88

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 5

Bifurcations in Planar Discontinuous Systems

Bifurcations from sliding homoclinic to bounded solutions on R for certain discontinuous planar systems under periodic perturbations are studied in this chapter. We make necessary assumptions and then analytically solve a system of piecewise linear differential equations with periodic perturbations.

5.1

Introduction

Sliding periodic solutions of discontinuous differential equations are investigated in [Makarenkov and Lam (2012); Novaes et al. (2014); Llibre et al. (2015); Feˇckan and Posp´ıˇsil (2016)] with both analytical and numerical methods. Qualitative properties of discontinuous systems are studied in [Leine et al. (2000); Han et al. (2012)]. Bifurcations for planar discontinuous ordinary differential systems with small periodic perturbations from homoclinic solutions transversely intersecting levels of discontinuity have been considered in [Shi et al. (2013)] to generalize the well-known Melnikov method for a smooth case [Battelli and Lazarri (1990)] to a discontinuous one. We note in this chapter that bifurcations from sliding homoclinic solutions are different to those studied in [Battelli and Lazarri (1990); Shi et al. (2013)]. In Chapter 3 devoted to an application of H´enon’s method, an integration of a properly prepared discontinuous differential equations has been performed. Such methodology was taken into consideration in [Awrejcewicz et al. (2006)]. Based on an example of a system of first-order differential equations describing dynamics of an autonomous two-degree-of-freedom discontinuous mechanical system, a scheme of bifurcation from sliding homoclinic to a bounded solution on R is described below. Let a discontinuous planar dynamical system be characterized not by one, but by two discontinuities on the lines (x, 12 ) and (x, 1) (a special case of multiple discontinuous levels). It will serve us for the derivation of an exemplary bifurcation of a homoclinic solution to a boundary solution of three systems of two differential equations of first-order (a planar case with two defined lines of discontinuity). These equations define three separate regions of continuous solutions and include small perturbation ε of vector fields of the analyzed dynamical system. This approach can be 89

page 89

June 8, 2017 12:9

ws-book961x669

90

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

generalized to the cases when a homoclinic orbit transversely crosses another curve of discontinuity. For simplicity, we suppose that such a discontinuity occurs at the level y = 1/2 and y = 1, that is, we deal with the system z˙ = f+ (z) + εg(z, t, ε), z˙ = f− (z) + εg(z, t, ε), z˙ = F (z) + εg(z, t, ε),

y > 1, 1 0, f±2 (x, 1) < 0 for x &− (0) ≥ x ≥ x &+ (0), f−2 (& x+ (0), 1) = 0, x ≥ x &+ (0); f−2 (x, 1) > 0 for x and a directional derivative in point (& x+ (0), 1) of second component of the x+ (0), 1) < 0. Furthermore, vector field in the region 12 < y < 1 is ∂x f−2 (& f−2 (η− (a− )) > 0 and f−2 (η+ (a+ )) < 0. &+ (t) of equation z˙ = F (z), y ≤ 1/2 defined (Z3) There are two solutions γ &− (t), γ &± (t) = 0 and on (−∞, 0] and [0, +∞), respectively, such that: lims→±∞ γ γ− (0)) > 0 and γ± (0) = η± (a± ). Moreover, F (z) = (F1 (z), F2 (z)) with F2 (& γ+ (0) < 0. F2 (& (Z4) In point z = 0 the function F (z) = 0, since dF (z)/dz in this point possesses eigenvalues on the real axis of complex plane. Assumptions (Z1)–(Z4) imply that for ε = 0 the dynamical system described by Eqs. (5.1)–(5.3) has a sliding homoclinic solution γ &, created by η± and y&± , to a hyperbolic equilibrium 0. We study in this section a bifurcation of γ & in the system (5.1)–(5.3) for ε = 0 small.

5.2

Calculation of Homoclinic Solutions

The system of equations (5.1)–(5.3) is given in the explicit form   y, y, z˙ = f− (z) = z˙ = f+ (z) = x − 3y, −x + 2y, ⎧

⎨−2ay, 0 g(x, t, ε) = cos t . z˙ = F (z) = ⎩− 1 x, 1 2a

(5.4)

page 90

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Bifurcations in Planar Discontinuous Systems

ws-book975x65

91

Accordingly to the first assumption (Z1) the conditions x ˜+ (0) = 2 and y˜+ (0) = 1, the solution η+ of Eq. (5.4) follows ˜+ (0)tet + x ˜+ (0)et = tet − 2tet + 2et , x ˜+ (t) = y˜+ (0)tet − x

(5.5)

˜+ (0)tet + y˜+ (0)et = tet − 2tet + et . y˜+ (t) = y˜+ (0)tet − x

(5.6)

In Eqs. (5.5) and (5.6) one substitutes t = s, yielding  x ˜+ (s) = es (2 − s), η+ (s) = y˜+ (s) = es (1 − s).

(5.7)

Subsequently, at the initial conditions x ˜− (0) = −a and y˜− (0) = 12 , components of the solution η− can be written: 1 x ˜− (t) = y˜− (0)tet − x ˜− (0)tet + x ˜− (0)et = tet + atet − aet , 2 1 1 y˜− (t) = y˜− (0)tet − x ˜− (0)tet + y˜− (0)et = tet + atet + et , 2 2 and then, applying the substitution t = s − a− , one gets ⎧

1 ⎪ ⎪ ˜− (s) = −aes−a− + + a es−a− (s − a− ), ⎨x 2

η− (s) = 1 1 ⎪ s−a − ⎪ + a es−a− (s − a− ). + ⎩y˜− (s) = e 2 2 Calculation of the solution γ & can be done in a similar way. At the initial x− (0), y˜− (0)) = (−a, 12 ), the solutions conditions (˜ x+ (0), y˜+ (0)) = (a, 12 ) and (˜ &− ) of equations (5.4) are as follows: γ & = (& γ+ , γ ⎧ ⎧ ⎨x ⎨x ˜+ (s) = ae−s , ˜− (s) = −aes , s −s γ &+ (s) = γ & (5.8) (s) = − ⎩y˜− (s) = e , ⎩y˜+ (s) = e , 2 2 where after integration of Eq. (5.4) with respect to t the substitution t = s has been done.

5.3

The Equation of Bifurcation

Bifurcation of a homoclinic solution can appear in a specific dynamical system that is characterized by proper parameters. For the purpose of estimation of an approximate of a− , one needs to assume a value of the parameter a of Eq. (5.4). Let a+ be a positive solution to the equation η+2 (s) = y˜− (0), in which the substitution s = a+ allows to write 1 (5.9) ea+ (1 − a+ ) = . 2 Then one assumes a = η+1 (a+ ), yielding to the estimation a = ea+ (2 − a+ ) ≈ 2.65554,

(5.10)

page 91

June 8, 2017 12:9

92

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

where a+ ≈ 0.768039 is the positive solution to Eq. (5.9). This value estimated on the basis of Eq. (5.10) at the assumption, that a− is the negative solution to the algebraic equation η−2 |s=0 = y˜+ (0), permits for the calculation

1 1 s−a− e + a es−a− (s − a− ) = 1, + 2 2

1 1 −a− e + a e−a− a− = 1, − 2 2

1 1 a− + a a− = , (5.11) e + 2 2 where on the basis of Eq. (5.11), a− ≈ −0.122043 and

1 η+ (a+ ) = a, . 2 Bifurcation of the homoclinic solution γ & bounded in R to the solution of the system (5.1)–(5.3) where ε = 0 holds for some values of the bifurcation parameter α. To compute these values we use the following derivations obtained in [Awrejcewicz et al. (2006)]: a) the linear variational initial problem df− (η+ (s)) w + g(η+ (s), s + α, 0), w˙ =

ds g2 (η+ (0), α, 0) ,0 , w(0) = − ∂x f−2 (η+ (0)) b) an equation of bifurcation  +∞ & B(α, 0) = (g(γ+ (s), a+ + s + α, 0), μ &(s))ds

(5.12)

0

+γ &˙ +2 (0)ψ&+ (α, 0) = 0,

(5.13)

for a scalar product (·, ·), and also ( ' &˙ +1 (s) , μ &(s) = γ &˙ +2 (s), −γ ψ&+ (α, 0) = −f−1 (η+ (a+ ))

w2 (a+ ) + w1 (a+ ). f−2 (η+ (a+ ))

On the basis of Eqs. (5.4) and (5.7), Eq. (5.12) can be transformed to the form



0 0 1 w˙ = w + cos (s + α) , w(0) = ( cos α, 0). (5.14) −1 2 1 Solution w = (w1 , w2 ) of the system (5.14) composed of two differential equations of the first-order is expressed by the formulas: 1 (5.15) w1 (s, α) = ((2 − s) es cos α + (1 − s) es sin α − sin (α + s)), 2 1 w2 (s, α) = ((1 − s) es cos α − es s sin α − cos (α + s)). (5.16) 2

page 92

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Bifurcations in Planar Discontinuous Systems

Second component of Eq. (5.13) is calculated below

1 1 w2 (a+ , α) + w1 (a+ , α) γ &˙ +2 (0)ψ&+ (α, 0) = − − 2 2 1−a

w2 (a+ , α) 1 = − w1 (a+ , α) , 4(1 − a) 2

93

(5.17)

where to estimate functions w(a+ , α) dependent on the bifurcation parameter α, Eqs. (5.15) and (5.16) are applied. The definite integral in Eq. (5.13) is calculated as follows:  ∞ (g(γ+ (s), a+ + s + α, 0), μ &(s))ds 0  ∞ cos (a+ + s + α)γ &˙ +1 (s)ds =− 0  ∞ cos (a+ + s + α) e−s ds =a 0 "s=∞ 1 −s " = a e (− cos (a+ + α + s) + sin (a+ + α + s))" 2 s=0 a = ( cos (a+ + α) − sin (a+ + α)). (5.18) 2 Next, we substitute values of the parameters a and a+ to Eqs. (5.17) and (5.18). Then basing on Eq. (5.13) the equation of bifurcation is obtained as ' ( 2 (a+ ,α) & B(α) = a2 ( cos (a+ + α) − sin (a+ + α)) + w4(1−a) − 12 w1 (a+ , α) ≈ 0.032595 cos α − 1.877467 sin α + 0.072545 sin α + 0.016556 cos α −0.490202 cos α + 0.054819 sin α ≈ −0.441052 cos α − 1.7501 sin α. & Numerical computation of zeros of the function B(α) yields: α1 = 2.8947188 and α2 = 6.03631115 in α ∈ [0, 2π]. Summarizing this chapter, it has been shown that the equation of bifurcation (5.13) has two positive roots. Therefore, the homoclinic solution γ & given by the equations (5.8) bifurcates to the bounded in R discontinuous solution of the dynamical system expressed by equations (5.4).

page 93

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 6

Occurrence of Chaos in Forced Impact Systems

In this chapter we follow a functional analytic approach to study the problem of chaotic behavior in time-perturbed impact systems whose unperturbed part has a piecewise continuous impact homoclinic solution that transversely enters the discontinuity manifold. We show that if a certain Melnikov function has a simple zero at some point, then the system has impact solutions that behave chaotically. Applications of this result to quasi-periodic systems are also given.

6.1

Introduction

Bifurcation theory is a powerful method for investigating parametrically dependent nonlinear phenomena (see for example [Chicone (2006); Guckenheimer and Holmes (1983); Appell et al. (1994); Appell and Zabrejko (1993); Chow and Hale (1982); Krasnosel’skii and Zabrejko (1984); Zabrejko and Povolockij (1970)]). This chapter is a contribution to this field. More concretely, we continue our study on the existence of chaos for time-perturbed piecewise-smooth NDS (PSNDS). By PSNDS we mean a differential equation of the form x˙ = F (t, x) where F (t, x) is a function continuous together with its first and second derivatives (C 2 in short, C r -functions are similarly defined) on R × (Ω \ S), Ω is an open subset of Rn and S is a submanifold of Ω that we call discontinuity manifold. In [Battelli and Feˇckan (2011, 2008)] we started from an unperturbed PSNDS possessing a homoclinic solution to a hyperbolic equilibrium and we developed Melnikov methods for PSNDS when the homoclinic solution crosses the discontinuity levels in several different ways. First we studied the case when the homoclinic solution crosses transversally the discontinuity manifold S. Later we considered the case when a part of the homoclinic solution slides on S [Battelli and Feˇckan (2010b,a)]. Related results have been also presented in [Awrejcewicz et al. (2005, 2006); Feˇckan (2008)]. On the other hand there are several interesting mechanical systems with impact conditions. Many stimulating examples of impact oscillators are given in [di Bernardo et al. (2008); Brogliato (1996); Fidlin (2006); Kunze (2000); Leine and Nijmeijer (2004)] where different numerical and analytical methods are described to study their dynamics. Methods 95

page 95

June 8, 2017 12:9

96

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

of singularity theory are used in [Chillingworth (2002)] to classify the local geometry of the discontinuity set with associated local dynamics for a class of oscillators with one degree of freedom impacting against a fixed obstacle. Homoclinic bifurcation methods related to this chapter are used in [Du and Zhang (2005)] for inverted pendulum impacting on rigid walls under external periodic excitation, and in [Xu et al. (2009)] for Duffing vibro-impact oscillator. Here we study much more complicated dynamics for impact systems in arbitrary finite dimensional space. Other interesting impact models emerge from understanding the dynamics of rigid blocks [Lenci and Rega (2005); Kovaleva (2010)]. A sophisticated analysis by means of scaling and averaging methods is used in [Makarenkov and Verhulst (2009)] to combine the dynamics of an impact oscillator with the dynamics of a smooth family of singularly perturbed oscillators. x(t)

t

Figure 6.1: The solution x(t) of Eq. (6.1) with x(0) = 1, x(0 ˙ + ) = −1.01. So our next natural step is to develop the Melnikov method for showing chaos in impact NDS (INDS) and we carry out it in this chapter [Battelli and Feˇckan (2013)]. As a matter of fact, there is numerical evidence of chaotic behavior in impact systems. For example, consider the simplest impact linear system modeling inverted pendulum with impacts under a periodic force given by x ¨ − x = 0.01 sin t, for x < 1, + − ) = −1.01x(t ˙ ), for x(t) = 1, x(t ˙

(6.1)

x˙ 1 = x2 x˙ 2 = x1 + 0.01 sin t, for x1 < 1, x2 (t+ ) = −1.01x2 (t− ), for x1 (t) = 1,

(6.2)

which is rewritten as

page 96

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

97

and clearly S = {(1, x2 ) | x2 ∈ R}. Note the unperturbed system (6.2) given by x˙ 1 = x2 , for x1 < 1, x˙ 2 = x1 , x2 (t+ ) = −x2 (t− ), for x1 (t) = 1, has a hyperbolic equilibrium (0, 0) with stable and unstable manifolds ± et (1, 1) and ± e−t (1, −1), respectively, transversely crossing the impact level x1 = 1. Next, there is an impact homoclinic solution  t t for t < 0, (e , e ) γ(t) = ( e−t , − e−t ) for t ≥ 0 . x(t) t

0

Figure 6.2: The solution x(t) of Eq. (6.1) with x(0) = 1, x(0 ˙ + ) = −1.02. Now we present some numerical results. In Fig. 6.1, the solution of Eq. (6.1) is drawn with x(0) = 1, x(0 ˙ + ) = −1.01. This solution tends asymptotically to a hyperbolic periodic solution located near (0, 0). In Fig. 6.2, we see the solution of Eq. (6.1) with x(0) = 1, x(0 ˙ + ) = −1.02. This solution tends to infinity. x(t) 1.0 0.8 0.6 0.4 0

t1 1

2

3

4

5

t2 6

7

8

9

10

t

11

12

13

14

15

16

17

Figure 6.3: The solution x(t) of Eq. (6.1) with x(0) = 1, x(0 ˙ + ) = −1. Figure 6.3 regards the solution of Eq. (6.1) with x(0) = 1, x(0 ˙ + ) = −1. This solution consists of three parts: the first part is an impact branch on the time interval [0, 5.285], so at t1 = 5.285 there occurs the first impact. The middle part concerns the solution in the time interval [5.285, 10.459], and has again an impact

page 97

June 8, 2017 12:9

98

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

with the threshold x = 1 at t2 = 10.459. These impact branches are located near the impact homoclinic solution γ(t). The third part is an escaping branch for t > 10.459. Summarizing, we see that there is a sensitive dependence of solutions of Eq. (6.1) on initial value conditions. Our aim is to analytically construct impact bounded solutions on R of perturbed impact systems like (6.1) consisting either from infinitely many parts of impact branches (see the two left parts of Fig. 6.3) or from a finitely many impact branches and ending with asymptotic ones; see Fig. 6.1. We get in this way a continuum of bounded impact solutions with chaotic dynamics. Now we explain this construction in more details. In Sec. 6.2 we set our studied impact problem in arbitrary finite dimension and define what we mean by impact solutions (in particular impact homoclinic orbits). In short an impact solution of a differential equation z˙ = F (t, z), with impact manifold S, is a solution of it in R \ T , T being a discrete subset of R, such that t ∈ T if and only if z(t− ) ∈ S and in this case z(t+ ) = R(z(t− )), R being a (smooth) map from S to S. Then in Sec. 6.3 we start to construct these impact bounded solutions for a perturbed system, i.e., with F (t, z) = f (z) + εg(t, z, ε). As the starting step, we show the existence of a discrete subset D ⊂ R and bounded piecewise smooth functions t → z(t), possibly discontinuous at t ∈ D, with the following properties: (i) z(t) is C 1 in R \ D; (ii) if t ∈ D, then z(t± ) := lim z(s) belong to a small neighborhood either of s→t±

S or of the equilibrium; (iii) for t ∈ R \ D, z(t) satisfies an associated perturbed differential equation; (iv) z(t) is shadowed by the impact homoclinic orbit of the unperturbed system. Here shadowed means that z(t) belongs to a small neighborhood of the impact homoclinic orbit (see Eq. (6.29) in Theorem 6.1). Note that z(t) is an approximate impact solution consisting either of impact branches or of solutions asymptotic to periodic solutions as it is mentioned in the example (6.2) above. Then in Sec. 6.4 we define the Melnikov function and state our main result, based on an argument in [Battelli and Feˇckan (2010a)], saying essentially that if the Melnikov function has a simple zero at some point then, for any sufficiently small ε, among these solutions there are infinitely many impact solutions of the perturbed system. So the approximate solution z(t) constructed in the first step is shadowed by true impact solution. From this construction it will follow, in fact, that the system possesses a continuum of bounded impact solutions and it behaves chaotically in the sense that the perturbed INDS has a Smale-like horseshoe. This result is applied in Sec. 6.5 to show chaotic behavior of impact systems with periodic or almost periodic perturbations. In Sec. 6.5, we give a geometric interpretation of some of our basic assumptions. In Sec. 6.6, we present two applications. Firstly, in Sec. 6.6.1, we explicitly construct the Melnikov function for impact planar differential systems by giving two examples of forced second-order equations with impacts. Then in Sec. 6.6.2, we consider two coupled forced second-order equations with impacts.

page 98

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Occurrence of Chaos in Forced Impact Systems

6.2

ws-book975x65

99

Problem Definition

Let ε0 ∈ (0, 1) be a positive real number, Ω ⊂ Rn be a bounded open set in Rn and ¯ with r ≥ 2. We set G(z) be a C r -function on Ω, Ω− = {z ∈ Ω | G(z) < 0},

Ω0 := {z ∈ Ω | G(z) = 0} .

¯ − ) and g ∈ C r (R × Ω ¯ − × R) be C r Let R ∈ Cbr ((−ε0 , ε0 ) × Rn ), f ∈ Cbr (Ω b n ¯ functions, defined on (−ε0 , ε0 ) × R , a neighborhood U of Ω− and on R × U × R respectively, bounded together with their derivatives up to the order r. Moreover, we assume that R(ε, z) ∈ Ω0 , for (ε, z) ∈ (−ε0 , ε0 ) × Ω0 , and that the rth order derivatives of f , R and g are uniformly continuous, uniformly with respect to t. Throughout this chapter ε will denote a real parameter such that |ε| ≤ ε0 . In particular ε is bounded. We consider the equation  ¯− , z˙ = f (z) + εg(t, z, ε), z ∈ Ω (6.3) + − − z(t ) = R(ε, z(t )), if z(t ) ∈ Ω0 .

Figure 6.4: The impact homoclinic cycle γ(t) of z˙ = f (z) to equilibrium z = z0 . We assume (cf. Fig. 6.4): (H1) equation z˙ = f (z),

¯ −, z∈Ω

(6.4)

has the hyperbolic equilibrium z = z0 ∈ Ω− and a solution γ(t), which is homoclinic to z = z0 and consists of two branches  γ− (t) ∈ Ω− , if t < 0, γ(t) = γ+ (t) ∈ Ω− , if t ≥ 0, where γ− ∈ C r+1 ((−∞, 0], Rn ), γ+ ∈ C r+1 ([0, ∞), Rn ), γ+ (0), γ− (0) ∈ Ω0 and γ+ (0) = R(0, γ− (0));

page 99

June 8, 2017 12:9

ws-book961x669

100

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

(H2) it results as G (γ(0− ))f (γ(0− )) > 0

and

G (γ(0+ ))f (γ(0+ )) < 0.

Remark 6.1. (i) For technical purposes, we Cbr -smoothly extend f on Rn , g on Rn+2 and γ± on R in such a way that ¯ −} , sup{|f (z)| | z ∈ Rn } ≤ 2 sup{|f (z)| | z ∈ Ω ¯ − , |ε| ≤ ε0 } . sup{|g(t, z, ε)| | (t, z, ε) ∈ Rn+2 } ≤ 2 sup{|g(t, z, ε)| | t ∈ R, z ∈ Ω We also assume that up to the rth order all the derivatives of R and of the extended f and g are bounded, uniformly continuous, uniformly with respect to t, and continue to keep the same notations for extended mappings and functions. It should be emphasized, however, that these conditions are not relevant since only the values of f , g for z in a compact neighborhood of γ(t) (and of R(ε, ·) in a neighborhood of γ− (0) in Ω0 ) are needed. (ii) Suppose that (6.4) is time-reversible, i.e., there is a linear mapping (involution) S : Rn → Rn such that S 2 = I and Sf (z) = −f (Sz), for any z ∈ Rn . Then we can take an inner product ·, · on Rn that Sz1 , Sz2 = z1 , z2 , for any z1 , z2 ∈ Rn . Assume z0 = 0 ∈ Ω− is a hyperbolic equilibrium of (6.4) possessing a solution γ− (t) ∈ Ω− , for t < 0 and γ− (0− ) ∈ Ω0 , where G(z) = a, z − c, for some a ∈ Fix S := {x ∈ Rn | Sx = x}, the set of fixed points of S, and 0 = c ∈ R. Moreover, we suppose a, γ− (0− ) > 0. Then we take γ+ (t) = Sγ− (−t), for any t > 0 and R = S. This situation may occur in second-order differential equations; see Sec. 6.6.1. Note now Ω0 = G−1 (0) is a hyper-plane and since G(Sz) = a, Sz − c = Sa, Sz − c = G(z), we see that Ω− and Ω0 are S-invariant. (iii) From the definition of Ω− , Ω0 and assumption (H1) it follows easily that G (γ(0− ))f (γ(0− )) ≥ 0 and G (γ(0+ ))f (γ(0+ )) ≤ 0. So condition (H2), is a transversality condition of γ± (t) with Ω0 at γ± (0), respectively. Changing G(z) with −G(z) we see that our results remain valid if γ± (t) ∈ Ω+ := {x ∈ Ω | G(x) > 0}. So we can assume, more generally, that either γ± (t) ∈ Ω− or γ± (t) ∈ Ω+ , for t = 0, γ± (0) ∈ Ω0 , γ+ (0) = R(0, γ− (0)) and [G (γ(0− ))f (γ(0− ))] · [G (γ(0+ ))f (γ(0+ ))] < 0. The geometric meaning of the above inequality is that the homoclinic solution γ(t) transversely hits and leaves the continuity manifold S in γ(0− ) and γ(0+ ), respectively. We are interested in the chaotic dynamics of Eq. (6.3) near γ(t), for ε = 0 small. Definition 6.1. By an impact solution z(t) of (6.3) we mean a C 1 -function z : R \ T → Rn where T is a discrete subset of R for which the following holds (see Fig. 6.4):

page 100

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

101

– right and left derivatives of z(t) at the points T& ∈ T exist; – if t ∈ / T , then z(t) ∈ Ω− and satisfies the equation z˙ = f (z) + εg(t, z, ε); – if T& ∈ T , then z(T&− ), z(T&+ ) ∈ Ω0 and z(T&+ ) = R(ε, z(T&− )). We will use the notation T = {T&m }, where T&m is an increasing sequence, defined either for m ∈ Z or for m ≤ m0 or for m ≥ m1 or for m0 ≤ m ≤ m1 (where m0 , m1 ∈ Z). 6.3

Looking for an Impact Solution

To allow more generality we take an increasing sequence {Tm }m∈Z with Tm+1 −Tm ≥ T  1 (that will be fixed later). We look for solutions z± (t) of Eq. (6.3) such that z± (t) ∈ Ω− and z± (t) is orbitally close to γ± (t). Since z0 is a hyperbolic fixed point of z˙ = f (z), the linear system x˙ = f  (γ+ (t))x has an exponential dichotomy on [−1, ∞) with (stable) projection P+ , constant k and exponent δ > 0, that is, if X+ (t) denotes the fundamental matrix of the linear equation x˙ = f  (γ+ (t))x, t ≥ −1 with X+ (0) = I we have (cf. [Appell et al. (1993); Meyer and Sell (1989); Palmer (1984)]): −1 (s) ≤ k e−δ(t−s) , X+ (t)P+ X+ −1 (s) ≤ k eδ(t−s) , X+ (t)(I − P+ )X+

for −1 ≤ s ≤ t, for −1 ≤ t ≤ s.

Here  ·  is a fixed matrix norm, for example the usual matrix norm A = max x =1 Ax) and, of course, the constants k and δ depend on this matrix norm. To simplify the matter and to avoid misunderstandings, we use the notation  ·  for any norm on a Banach space X instead of the notation like  · X . As a consequence, the linear system x˙ = f  (γ+ (t − T2m ))x has an exponential dichotomy on [T2m −1, ∞) with (stable) projection P+ , constant k and exponent δ > 0 independent of m ∈ Z. Similarly, the linear system x˙ = f  (γ− (t))x has an exponential dichotomy on (−∞, 1] with projection P− (unstable), constant k and exponent δ > 0, that is, if X− (t) denotes the fundamental matrix of the linear equation x˙ = f  (γ− (t))x, t ≤ 1 with X− (0) = I we have: −1 X− (t)P− X− (s) ≤ k e−δ(t−s) , −1 (s) ≤ k eδ(t−s) , X− (t)(I − P− )X+

for s ≤ t ≤ 1, for t ≤ s ≤ 1.

Without loss of generality we may, and will, assume that the constant k and the exponent δ of the dichotomies on (−∞, 1] and (−1, ∞] are equal. As a consequence, the equation x˙ = f  (γ− (t − T2m ))x

page 101

June 8, 2017 12:9

102

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

has an exponential dichotomy on (−∞, T2m + 1] with projection P− (unstable), the same constant and exponent. We set: −1 −1 (−t), P+ (t) = X+ (t)P+ X+ (t), P− (t) = X− (−t)P− X− −1 P+,m = X+ (T2m+1 − T2m + 1)P+ X+ (T2m+1 − T2m + 1), −1 (T2m−1 − T2m − 1). P−,m = X− (T2m−1 − T2m − 1)P− X−

Note that P−,m = P− (T2m − T2m−1 + 1)

P+,m = P+ (T2m+1 − T2m + 1).

Next, for a given linear map L : Rn → Rm , RL and N L will denote the range and the kernel of L respectively, i.e.: RL = {y ∈ Rm | ∃ x ∈ Rn such that Lx = y}, N L = {x ∈ Rn | Lx = 0}. According to a result obtained by [Palmer (1984)], it follows that lim P± (t) − P0  = 0,

t→∞

P0 being the stable projection of the dichotomy of the linear system x˙ = f  (z0 )x. As a consequence, T > 1 exists such that for any t , t ≥ T one writes N P+ (t ) ⊕ RP− (t ) = Rn . From now on we take such a T . Let ρ > 0 be a positive number such that, for any t ≥ 0 the closed balls B(γ+ (t), ρ), B(γ− (−t), ρ) are subset of Ω. For any x with |x| < ρ we set h± m (t, x, α, ε) = f (x + γ± (t − T2m )) − f (γ± (t − T2m )) −f  (γ± (t − T2m ))x + εg(t + α, x + γ± (t − T2m ), ε) . Finally, we put

) * N = sup "|g(t, z, ε)| |"(t, z, ε) ∈ Rn+2 , " " n+2 N  = sup " ∂g , ∂z (t, z, ε)" | (t, z, ε) ∈ R Δ− (ρ) := sup|x|≤ρ supt≤0 |f  (x + γ− (t)) − f  (γ− (t))|, Δ+ (ρ) := sup|x|≤ρ supt≥0 |f  (x + γ+ (t)) − f  (γ+ (t))|.

We are looking for an impact solution z(t) of Eq. (6.3) such that for any t ∈ R, z(t) belongs to a (small) neighborhood of Γ := {γ(t) | t ∈ R}, i.e., shadowed by γ(t). The following two results have been proven in [Battelli and Feˇckan (2011)] (cf. also [Lin (1990); Vanderbauwhede and Fiedler (1992); Sandstede (1993); Knobloch (2000)]) and help us to find branches of these solutions, i.e., solutions shadowed by the impact homoclinic orbits that are defined in bounded intervals whose union is the whole real line. For the convenience of the reader we recall them here. Proposition 6.1. Assume that (H1) holds and let (ξ− , ϕ− , α, ε) ∈ N P− ×RP−,m × R2 and ρ > 0 be such that   2k |ξ− | + |ϕ− | + 2δ −1 N |ε| ≤ ρ, 4kδ −1 [Δ− (ρ) + N  |ε|] < 1,

page 102

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

103

k and δ being the constant and the exponent of the dichotomy, respectively. Then, for t ∈ [T2m−1 + α − 1, T2m + α], the equation z˙ = f (z) + εg(t, z, ε) has a unique − − (t) = zm (t, ξ− , ϕ− , α, ε), which is C r -smooth in the parameters bounded solution zm (ξ− , ϕ− , α, ε) and satisfies   − |zm (t + T2m + α, ξ− , ϕ− , α, ε) − γ− (t)| ≤ 2k |ξ− | + |ϕ− | + 2δ −1 N |ε| ≤ ρ, (6.5) for any t ∈ [T2m−1 − T2m − 1, 0] along with: − (T2m + α, ξ− , ϕ− , α, ε) = ξ− , P− zm − (T2m−1 + α − 1, ξ− , ϕ− , α, ε) = ϕ− . P−,m zm − (t + α, ξ− , ϕ− , α, ε) and its derivatives with respect to (ξ− , ϕ− , α, ε) Moreover, zm are also uniformly bounded in [T2m−1 − 1, T2m ] with respect to (ξ− , ϕ− , α, ε) and m ∈ Z, uniformly continuous in (ξ− , ϕ− , α, ε) with respect to t ∈ [T2m−1 − 1, T2m ], m ∈ Z, and satisfy: − ∂zm (t + α, 0, 0, α, 0) = X− (t − T2m )(I − P− ), ∂ξ− − ∂zm −1 (t + α, 0, 0, α, 0) = X− (t − T2m )P− X− (T2m−1 − T2m − 1), ∂ϕ− − ∂zm (t + α, 0, 0, α, 0) ∂ε t

= T2m−1 −1  T2m

−

t

(6.6)

−1 X− (t − T2m )P− X− (s − T2m )g(s + α, γ− (s − T2m ), 0)ds

−1 X− (t − T2m ) (I − P− ) X− (s − T2m )g(s + α, γ− (s − T2m ), 0)ds.

In the next step, we make a similar assumtion (see again [Lin (1990); Vanderbauwhede and Fiedler (1992); Sandstede (1993); Knobloch (2000); Battelli and Feˇckan (2011)]). Proposition 6.2. Assume that (H1) holds and let (ξ+ , ϕ+ , α, ε) ∈ RP+ ×N P+,m × R2 and ρ > 0 be such that   2k |ξ+ | + |ϕ+ | + 2δ −1 N |ε| ≤ ρ, 4kδ −1 [Δ+ (ρ) + N  |ε|] < 1 . Then, for t ∈ [T2m + α, T2m+1 + α + 1], the equation z˙ = f (z) + εg(t, z, ε) has + + (t) = zm (t, ξ+ , ϕ+ , α, ε), which is C r -smooth in the a unique bounded solution zm parameters (ξ+ , ϕ+ , α, ε) and satisfies   + |zm (t + T2m + α, ξ+ , ϕ+ , α, ε) − γ+ (t)| ≤ 2k |ξ+ | + |ϕ+ | + 2δ −1 N |ε| ≤ ρ, (6.7) for any t ∈ [0, T2m+1 − T2m + 1] along with: + (T2m + α, ξ+ , ϕ+ , α, ε) = ξ+ , P+ zm + (T2m+1 + 1 + α, ξ+ , ϕ+ , α, ε) = ϕ+ . P+,m zm

page 103

June 8, 2017 12:9

ws-book961x669

104

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

+ Moreover, zm (t + α, ξ+ , ϕ+ , α, ε) and its derivatives with respect to (ξ+ , ϕ+ , α, ε) are also uniformly bounded in [T2m , T2m+1 + 1] with respect to (ξ+ , ϕ+ , α, ε) and m ∈ Z, uniformly continuous in (ξ+ , ϕ+ , α, ε) with respect to t ∈ [T2m , T2m+1 + 1], m ∈ Z, and satisfy: + ∂zm (t + α, 0, 0, α, 0) = X+ (t − T2m )P+ , ∂ξ+ + ∂zm −1 (t + α, 0, 0, α, 0) = X+ (t − T2m )(I − P+ )X+ (T2m+1 − T2m + 1), ∂ϕ+ + ∂zm (t + α, 0, 0, α, 0) ∂ε t

−1 X+ (t − T2m )P+ X+ (s − T2m )g(s + α, γ+ (s − T2m ), 0)ds

=

T2m  T2m+1 +1

−

t

−1 X+ (t − T2m )(I − P+ )X+ (s − T2m )g(s + α, γ+ (s − T2m ), 0)ds.

(6.8)

6.4

The Equation of Bifurcation

± In this section we look for conditions that will allow us to join the branches zm (t) (cf. Prop. 6.1–6.2) at the suitable points in order to obtain an impact solution of system (6.3). We will see that this is possible provided a certain function BT (α, ε), that we call bifurcation function and will be defined later in this section, vanishes for some values of (α, ε). This function is obtained by applying a kind of Lyapunov– ± (t) at Schmidt reduction to the infinite set of equations obtained by equating zm (approximately) the end points of their intervals of definition. Then we will look for the solution of the bifurcation equation BT (α, ε) = 0. It is at this point where the Melnikov condition arises. To start with, we take the Banach space  ) * + − + ∞ 4n := θ := (ϕ− ∞ n m , ϕm , ξm , ξm ) m∈Z ∈  (R ) :  + − + (ϕ− m , ϕm , ξm , ξm ) ∈ RP−,m × N P+,m × N P− × RP+ ∀m ∈ Z

with the norm +) + * ) − * + + + − + + − + θ = + (ϕ− m , ϕm , ξm , ξm ) m∈Z + = sup max |ϕm + ϕm |, |ξm |, |ξm | . m∈Z

Let ρ0 > 0 be the largest positive number satisfying (cf. Prop. 6.1–6.2)   N δ −1 ρ0 ≤ 1. Δ± (ρ0 ) + 4kδ 4N k We consider   ∞ ∞ 1 (R) := α ∈  (R) : sup |αm − αm−1 | < 1 . m∈Z

page 104

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Occurrence of Chaos in Forced Impact Systems

ws-book975x65

105

ρδ Next, let 0 < ρ < ρ0 and ερ := 8N k . For any ε ∈ (−ερ , ερ ) we set  ) − + − + * ∞ ∞ ρ,ε := θ := (ϕm , ϕm , ξm , ξm ) m∈Z ∈ n :   ±  −1 2k |ξm | + |ϕ± N |ε| < ρ, ∀m ∈ Z , m | + 2δ

and ∞ ρ =



 ∞ (θ, α, ε) ∈ ∞ ×  (R) × (−ε , ε ) . ρ ρ ρ,ε 1

∞ ∞ Note that, because of the choice of ρ and ερ , ∞ ρ,ε , ρ and 1 (R) are open ∞ ∞ ∞ ∞ nonempty subsets of n , n ×  (R) × R and  (R), respectively. Let us take an increasing sequence T := {Tm }m∈Z with Tm+1 − Tm > T + 1 where T is defined and fixed in Sec. 6.3. We note that for any (θ, α, ε) ∈ ∞ ρ assumptions of Prop. 6.1–6.2 are satisfied. Indeed, we have:

4kδ −1 [Δ± (ρ) + N  |ε|] < 4kδ −1 [Δ± (ρ) + N  ερ ] ,   N δ ρ0 ≤ 1. ≤ 4kδ −1 Δ± (ρ0 ) + 8N k − − − (t) = zm (t, ξm , ϕ− Hence, by Prop. 6.1–6.2 we obtain solutions zm m , αm , ε), + + + = zm (t, ξm , ϕm , αm , ε), ε) of equation z˙ = f (z) + εg(t, z, ε) that are defined on [T2m−1 + αm − 1, T2m + αm ], [T2m + αm , T2m+1 + αm + 1], respectively. Since |αm+1 − αm | < 1 and Tm+1 − Tm > T + 1 > 1, for any m ∈ Z, we obtain easily:

+ (t) zm

T2m+1 + αm+1 − 1 < T2m+1 + αm < T2m+2 + αm+1 . − + Hence both zm (t) and zm+1 (t) are defined in t = T2m+1 + αm . Thus we can consider the following infinite set of equations: − − (T2m + αm , ξm , ϕ− G(zm m , αm , ε)) = 0, + + − − − zm (T2m + αm , ξm , ϕ+ m , αm , ε) = R (ε, zm (T2m + αm , ξm , ϕm , αm , ε)) , − − + + + zm (T2m+1 + αm , ξm , ϕm , αm , ε) = zm+1 (T2m+1 + αm , ξm+1 , ϕ− m+1 , αm+1 , ε), (6.9) + − + ∞ ∞ and to look for θ = {(ϕ− m , ϕm , ξm , ξm )}m∈Z ∈ ρ,ε and α ∈ 1 (R) in terms of ε in such a way that the equations (6.9) are satisfied. Note that if we are able to solve Eq. (6.9), then in the sense of Def. 6.1, we can obtain an impact solution of Eq. (6.3) near γ(t) with:  − − , ϕ− zm (t, ξm m , αm , ε), for T2m−1 + αm−1 ≤ t < T2m + αm , z(t) := + + zm (t, ξm , ϕ+ m , αm , ε), for T2m + αm ≤ t < T2m+1 + αm ,

and vice versa, if we have an impact solution of Eq. (6.3) in a sufficiently small neighborhood of γ(t), then because of the uniqueness part of the Prop. 6.1–6.2, the system (6.9) can be solved for some parameter values in terms of ε.

page 105

June 8, 2017 12:9

106

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

∞ 2n+1 So we consider the equation GT (θ, α, ε) = 0 where GT : ∞ ) is ρ →  (R defined as follows: ⎧⎛ ⎞⎫ + + zm (T2m+1 + αm , ξm , ϕ+ ⎪ ⎪ m , αm , ε) ⎪ ⎪ ⎪ − − ⎪ ⎜ −z − (T ⎪ ⎟⎪ ⎪ + α , ξ , ϕ , α , ε) 2m+1 m m+1 ⎪ ⎜ m+1 ⎪ ⎟⎪ m+1 m+1 ⎪ ⎪ ⎪ ⎜ ⎪ ⎟ ⎪ ⎬ ⎨⎜ ⎟⎪ ⎜ ⎟ − − GT (θ, α, ε) := ⎜ G(zm (T2m + αm , ξm , ϕ− ⎟ m , αm , ε)) ⎜ ⎟⎪ ⎪ ⎪ ⎪ ⎟⎪ ⎪⎜ ⎪ ⎜ ⎪ ⎟⎪ ⎪ ⎪ ⎪⎝ + + + ⎪ ⎪ ⎠ z (T + α , ξ , ϕ , α , ε) ⎪ ⎪ 2m m m m m m ⎪ ⎪ ⎭ ⎩ − − − −R (ε, zm (T2m + αm , ξm , ϕm , αm , ε)) m∈Z + − + ∞ ∞ for T = {Tm }m∈Z , θ = {(ϕ− m , ϕm , ξm , ξm )}m∈Z ∈ ρ,ε and α = {αm }m∈Z ∈ 1 (R). Next, from (6.5), (6.6) we see that: − (T2m+1 + αm , 0, 0, αm+1 , 0) = γ− (T2m+1 − T2m+2 + αm − αm+1 ), zm+1 − (T2m + αm , 0, 0, αm , 0) = γ− (0), zm − ∂zm+1 (T2m+1 + αm , 0, 0, αm+1 , 0) ∂ξ− = X− (T2m+1 − T2m+2 + αm − αm+1 )(I − P− ), − ∂zm (T2m + αm , 0, 0, αm , 0) = I − P− , ∂ξ− − ∂zm+1 (T2m+1 + αm , 0, 0, αm+1 , 0) ∂ϕ− −1 = X− (T2m+1 − T2m+2 + αm − αm+1 )P− X− (T2m+1 − T2m+2 − 1), − ∂zm −1 (T2m + αm , 0, 0, αm , 0) = P− X− (T2m−1 − T2m − 1) ∂ϕ−

and from (6.7) and (6.8) we see that: + (T2m+1 + αm , 0, 0, αm , 0) zm + zm (T2m + αm , 0, 0, αm , ε), 0) + ∂zm (T2m+1 + αm , 0, 0, αm , 0) ∂ξ+ + ∂zm (T2m + αm , 0, 0, αm , 0) ∂ξ+ + ∂zm (T2m+1 + αm , 0, 0, αm , 0) ∂ϕ+ + ∂zm (T2m + αm , 0, 0, αm , 0) ∂ϕ+

= γ+ (T2m+1 − T2m ), = γ+ (0), = X+ (T2m+1 − T2m )P+ , = P+ , −1 = X+ (T2m+1 − T2m )(I − P+ )X+ (T2m+1 − T2m + 1), −1 = (I − P+ )X+ (T2m+1 − T2m + 1).

We obtain G (0, α, 0) = ⎧T⎛ ⎞⎫ ⎨ γ+ (T2m+1 − T2m ) − γ− (T2m+1 − T2m+2 + αm − αm+1 ) ⎬ ⎝ ⎠ 0 ⎩ ⎭ 0 m∈Z

(6.10)

page 106

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

107

and D1 GT (0, α, 0)θ ⎧⎛ ⎞⎫ − + + Lα (ϕ− ⎪ ⎪ m+1 , ϕm , ξm+1 , ξm ) ⎪ ⎪ ⎪⎜ ⎪ ⎪ ⎟⎪ ⎬ ⎨⎜ ⎟   −1  − − ⎟ ⎜ = ⎜ G (γ− (0)) ξm + P− X− (T2m−1 − T2m − 1)ϕm ⎟ ⎪ ⎪ ⎪ ⎝ ⎠⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ α + + L1 (ϕm , ξm , ξm ) m∈Z

(6.11)

where − + + Lα (ϕ− m+1 , ϕm , ξm+1 , ξm ) − + − X− (T2m+1 − T2m+2 + αm − αm+1 )ξm+1 = X+ (T2m+1 − T2m )ξm −1 + +X+ (T2m+1 − T2m )(I − P+ )X+ (T2m+1 − T2m + 1)ϕm −1 −X− (T2m+1 − T2m+2 + αm − αm+1 )P− X− (T2m+1 − T2m+2 − 1)ϕ− m+1 (6.12)

and −1 + + + + Lα 1 (ϕm , ξm , ξm ) = (I − P+ )X+ (T2m+1 − T2m + 1)ϕm + ξm   − −1 −D2 R(0, γ− (0)) ξm + P− X − (T2m−1 − T2m − 1)ϕ− m .

(6.13)

Since Tm+1 − Tm > T + 1 and |αm+1 − αm | ≤ 1 as in [Battelli and Feˇckan (2011)], we obtain: + + + ∂ + −δT + ce , + GT (0, α, 0)+ c e−δT , (6.14) GT (0, α, 0) ≤ & +≤& ∂α for a positive constant & c independent of α and m. α ∞ 3n Let Lm : RP−,m × N P+,m → Rn and Hα : ∞ n →  (R ) be the linear maps defined as below −1 − + + Lα m : (ϕ , ϕ ) → X+ (T2m+1 − T2m )(I − P+ )X+ (T2m+1 − T2m + 1)ϕ −1 − −X− (T2m+1 − T2m+2 + αm − αm+1 )P− X− (T2m+1 − T2m+2 − 1)ϕ

and

⎧⎛ ⎪ ⎪ ⎪ ⎪ ⎨⎜ ⎜ α H (θ) = ⎜ ⎜ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

− + Lα m (ϕm+1 , ϕm ) − G (γ− (0))ξm

⎞⎫ ⎪ ⎪ ⎪ ⎟⎪ ⎟⎬ ⎟ ⎟⎪ ⎠⎪ ⎪ ⎪ ⎭ −

+ ξm − D2 R (0, γ− (0)) ξm

.

(6.15)

(6.16)

m∈Z

As in [Battelli and Feˇckan (2011)], the following properties can be shown: Lα m is an isomorphism and a positive constant & c, independent of α, m, exists such that + + + α+ −1 + + ∂(Lα + ∂Lm + m) α α −1 + + +≤& + (6.17) ≤& c, + c c, (Lm )  ≤ & c, + Lm  ≤ & ∂α + ∂α + and D1 GT (0, α, 0) − Hα  ≤ & c e−δT .

(6.18)

page 107

June 8, 2017 12:9

ws-book961x669

108

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts α

As a consequence, we obtain that both Hα and ∂H ∂α are bounded linear maps ∞ 3n into  (R ) with bound independent of T and α ∈ ∞ from ∞ n 1 (R). Let S  = N P− ∩ N G (γ− (0)), (6.19) S  = RP+ ∩ N G (γ+ (0)), S  = D2 R (0, γ− (0)) S  . Then by [Battelli and Feˇckan (2011)] dim S  = dim N P− − 1,

dim S  = dim RP+ − 1 .

Hence dim (RP+ + S  ) = dim RP+ + dim S  − dim RP+ ∩ S  ≤ dim RP+ + dim S  ≤ dim RP+ + dim N P− − 1 = n − 1 . (6.20) We suppose (H3) RP+ + S  has a codimension 1 in Rn . Let us list few consequences of assumption (H3). By (H3) and Eq. (6.20) it follows dim RP+ ∩ S  = 0 and D2 R (0, γ− (0)) : S  → S  is an isomorphism. Note S  ⊂ N G (γ− (0)), S  ⊂ N G (γ+ (0)) and by (H2), D2 R (0, γ− (0)) : N G (γ− (0)) → N G (γ+ (0)). Next dim S  = dim S  and dim S  ∩ S  = 0. Thus S  ⊕ S  ⊂ N G (γ+ (0)) and dim S  ⊕ S  = n − 2. As a consequence, ψ& ∈ N G (γ+ (0)) exists such that & = N G (γ+ (0)) . S  ⊕ S  ⊕ [ψ] & ⊂ N G (γ+ (0)) so S  = RP+ which Note ψ& ∈ / RP+ , otherwise RP+ = S  ⊕ [ψ]  contradicts dim S = dim RP+ − 1. Thus & = Rn , RP+ ⊕ S  ⊕ [ψ] but for simplicity we take a unitary ψ ∈ [RP+ ⊕ S  ]⊥ and denote with Π : Rn → RP+ ⊕ S  the orthogonal projection onto RP+ ⊕ S  along ψ. The next result easily follows; see also [Battelli and Feˇckan (2011)]. Lemma 6.1. Assume that (H3) holds and let ψ ∈ Rn be a unitary vector such that ψ ∈ (RP+ ⊕ S  )

⊥

(6.21)

holds. Then the equation

⎧⎛ ⎞⎫ ⎨ am ⎬ ∈ ∞ (Rn × Rn × Rn ) Hα (θ) = ζ := ⎝ bm ⎠ ⎭ ⎩ cm m∈Z

has a unique solution θα (ζ) if and only if   bm ∗ ψ cm +  D2 R(0, γ− (0))γ˙ − (0) = 0, G (γ− (0))γ˙ − (0)

for any m ∈ Z .

(6.22)

page 108

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

109

Moreover, + − + θα (ζ) = {(ϕ− m (am−1 ), ϕm (am ), ξm (bm , cm ), ξm (bm , cm ), ξm (bm , cm ))}m∈Z

with + α −1 (am ) (ϕ− m+1 , ϕm ) = (Lm ) bm − − ˙ − (0) ξm = ηm + G (γ− (0)) γ˙ − (0) γ − + and ηm ∈ S  , ξm are the unique solutions of bm + − D2 R(0, γ− (0))γ˙ − (0). ξm − D2 R(0, γ− (0))ηm = cm +  G (γ− (0))γ˙ − (0) & exists such that Furthermore, a positive constant C

& θα (ζ) ≤ Cζ. To state our main results, we define a function ψ(t) by ⎧ −1 ∗ ⎨ X− (t)∗ P−∗ R− D2 R(0, γ− (0))∗ ψ, for t ≤ 0, ψ(t) = ⎩ −1 ∗ X+ (t) (I − P+∗ )ψ, for t > 0, where R− is the projection onto N G (γ− (0)) along γ˙ − (0), i.e., G (γ− (0))w R− w = w −  γ˙ − (0). G (γ− (0))γ˙ − (0) Now we are ready to prove the following theorem.

(6.23)

(6.24)

(6.25)

(6.26)

(6.27)

Theorem 6.1. Assume f (z) and g(t, z, ε) are C r -functions with bounded derivatives and that their rth order derivatives are uniformly continuous. Assume, moreover, that conditions (H1)-(H3) hold. Set  ∞ ψ ∗ (t)g(t + α, γ(t), 0)dt + ψ ∗ D1 R(0, γ− (0)). M(α) := −∞

Then for given c0 > 0, there exist constants ρ0 > 0 and c1 > 0 such that for any 0 < ρ < ρ0 , there is ε¯ρ > 0 such that for any ε, 0 < |ε| < ε¯ρ , for any increasing sequence T = {Tm }m∈Z ⊂ R with Tm − Tm−1 > 1 − 2δ −1 ln |ε| and such that     0 0 = 0 ∀m ∈ Z and inf |M T2m + αm | > c0 , M T2m + αm (6.28) m∈Z

0 for some α0 = {αm }m∈Z ∈ ∞ 1 (R), there exists a unique sequence

{αm }m∈Z = {αm (T , ε)}m∈Z ∈ ∞ 1 (R) 0 | < c1 |ε|, for any m ∈ Z, and a unique impact solution z(t) = with |αm (T , ε) − αm z(t, T , ε) of system (6.3) such that

z(t) ∈ Ω− ∀t ∈ [T2m−1 + αm−1 , T2m+1 + αm ] \ {T2m + αm }, z(t± ) ∈ Ω0 , z(t+ ) = R(ε, z(t− )) ∀t ∈ {T2m + αm }, and supt∈[T2m−1 +αm−1 ,T2m +αm ) |z(t) − γ− (t − T2m − αm )| < ρ, supt∈[T2m +αm ,T2m+1 +αm ] |z(t) − γ+ (t − T2m − αm )| < ρ, for any m ∈ Z.

(6.29)

page 109

June 8, 2017 12:9

ws-book961x669

110

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Proof. We follow [Battelli and Feˇckan (2011, 2010a)]. Let Πψ : ∞ (R3n ) → ∞ (R3n ) be the projection given by ⎧⎛ ⎞⎫ ⎨ am ⎬ (I − Πψ ) ⎝ bm ⎠ := ⎭ ⎩ cm m∈Z ⎧⎛ ⎞⎫ (6.30) 0 ⎪ ⎪ ⎨ ⎬ ⎜ ⎟ 0 . ⎝   ⎠⎪ ⎪ b ⎩ ψ∗ c +  ⎭ m D R(0, γ (0)) γ ˙ (0) ψ m − − G (γ− (0))γ˙ − (0) 2 m∈Z

Then Eq. (6.9) has the form θ + (Hα )−1 Πψ FT (θ, α, ε) = 0,

(6.31)

(I − Πψ ) FT (θ, α, ε) = 0 ,

(6.32)

and

where FT (θ, α, ε) = GT (θ, α, ε) − Hα (θ). Let μ = e−δT . By using Lemma 6.1 as in [Battelli and Feˇckan (2010a)] if ρ¯0 > 0, 0 < μ0 < 1 and 0 < ε¯0 ≤ ερ are sufficiently small and 0 < μ < μ0 , |ε| < ε¯0 , from the Implicit Function Theorem the existence follows the unique solution θ = θT (α, ε) of ¯0 , 0 < μ ≤ μ0 and T = {Tm }m∈Z (6.31), which is defined for any α ∈ ∞ 1 (R), |ε| < ε such that Tm+1 − Tm > T + 1 where T = −δ −1 ln μ. Moreover, θT (α, ε) satisfies

along with

θT (α, ε) ≤ & c(μ + |ε|) < ρ¯0

(6.33)

+ + + ∂θT (α, ε) + +≤& + + c[μ + |ε|] + ∂α

(6.34)

for a positive constant & c independent of μ, ε and T . Next, as in [Battelli and Feˇckan (2010a)], we take μ = ε2 , and then we obtain from (6.32) the bifurcation function 1 (I − Πψ )FT (θT (α, ε), α, ε), ε whose zeroes, for ε = 0, correspond to solutions of the equation GT (θ, α, ε) = 0 and hence impact solutions of Eq. (6.3) as we have observed earlier. Moreover, following the procedure by [Battelli and Feˇckan (2010a)], we obtain: BT (α, ε) :=

BT (α, ε) = (I − Πψ ) D3 GT (0, α, 0) + O(ε) and D1 BT (α, ε) = (I − Πψ ) D2 D3 GT (0, α, 0) + o(1)

page 110

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

111

uniformly with respect to (T , α). Thus we are led to look at: D G (0, α, 0) = ⎧3⎛ T + ⎞⎫ − ∂zm+1 ∂zm ⎪ ⎪ (T + α , 0, 0, α , 0) − (T + α , 0, 0, α , 0) 2m+1 m m 2m+1 m m+1 ⎪ ⎪ ∂ε ⎪ ⎜ ∂ε ⎟⎪ ⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎪ ⎪ ⎪ ⎪ ⎟ ⎪ ⎪⎜ − ⎬ ⎨ ∂zm  ⎜ ⎟ G (γ (0)) (T + α , 0, 0, α , 0) − 2m m m ⎜ ⎟ ∂ε . ⎜ ⎟⎪ ⎪ ⎪⎜ ⎟⎪ ⎪ ⎪ + ⎜ ⎪ ⎟⎪ ∂zm ⎪ ⎪⎝ ⎪ ⎠⎪ ⎪ ⎪ ∂ε (T2m + αm , 0, 0, αm , 0) ⎪ ⎪ − ⎩ ⎭ ∂zm −D1 R(0, γ− (0)) − D2 R(0, γ− (0)) ∂ε (T2m + αm , 0, 0, αm , 0) m∈Z Considering only the third component of (I − Πψ )D3 GT (θ0 , α, 0), since the first two equal zero (see Eq. (6.30)), we derive  ∂z + BT (α, ε) = ψ ∗ m (T2m + αm , 0, 0, αm , 0) ∂ε ∂z − ∗ ∗ − ψ D1 R(0, γ− (0)) − ψ D2 R(0, γ− (0)) m (T2m + αm , 0, 0, αm , 0) ∂ε − ∂zm   G (γ− (0)) ∂ε (T2m + αm , 0, 0, αm , 0) ∗ ψ D R(0, γ (0)) γ ˙ (0) . + 2 − − G (γ− (0))γ˙ − (0) m∈Z Inserting (6.6) and (6.8) into the above formula, we obtain:  T2m+1 −T2m +1 −1 BT (α, ε) = − ψ ∗ (I − P+ )X+ (t)g(t + T2m + αm , γ+ (t), 0)dt 0

 −

0 T2m−1 −T2m −1



−ψ ∗ D1 R(0, γ− (0)) −1 ψ ∗ D2 R(0, γ− (0))P− X− (t)g(t + T2m + αm , γ− (t), 0)dt

0

+ T2m−1 −T2m −1

−1 G (γ− (0))P− X− (t)g(t + T2m + αm , γ− (t), 0) dt G (γ− (0))γ˙ − (0)

×ψ ∗ D2 R(0, γ− (0))γ˙ − (0) + O(ε) that is BT (α, ε) = −ψ ∗ D1 R(0, γ− (0)) −



∞ −∞

ψ ∗ (t)g(t + T2m + αm , γ(t), 0)dt + O(ε) (6.35)

where ψ(t) is defined by Eq. (6.26). Note that, from the previous part we also get:  ∞ ∂BT (α, ε) = − ψ ∗ (t)gt (t + T2m + αm , γ(t), 0)dt + o(1). ∂α −∞

(6.36)

As a consequence: lim BT (α0 , ε) = 0,

ε→0

0 lim D1 BT (α0 , ε) = {M (αm + T2m )}m∈Z

ε→0

uniformly with respect to T . That is D1 BT (α0 , ε) > c0 /2 provided |ε| is sufficiently small. From the Implicit Function Theorem we deduce the existence of

page 111

June 8, 2017 12:9

ws-book961x669

112

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

0 < ε¯ρ < ε0 such that for any 0 = ε ∈ (−¯ ερ , ε¯ρ ) and any sequence T = {Tm }m∈Z that satisfy the assumption of Theorem 6.1 there exists a unique sequence α(T , ε) = {αm (T , ε)}m∈Z ∈ ∞ 1 (R) such that α(T , 0) = α0 and BT (α(T , ε), ε) = 0. Taking θT (ε) = θT (α(T , ε), ε), recalling Eq. (6.9) and the constructions above it, we get the desired z(t) = z(T , ε)(t)(= z(t, T , ε)) of system (6.3) satisfying the conclusion (6.29) of Theorem 6.1. Remark 6.2. (i) Let R0 : R × Ω0 → Ω0 be the restriction of R to R × Ω0 . Since RR− = N G (γ− (0)) = Tγ− (0) Ω0 we see that D2 R0 (0, γ− (0))R− = ∗ ∗ D2 R(0, γ− (0))∗ = R− R0 (0, γ− (0))∗ . So to D2 R(0, γ− (0))R− , which implies R− compute ψ(t) (see Eq. (6.26)) we only need to know the restriction of R(0, z) to Ω0 . In other words, although in the proof of Theorem 6.1 we need to know the values of R(ε, z) in a full neighborhood Oγ− (0) of Ω0 (see, for example, Eq. (6.22)), the Melnikov function M(α) is determined as long as we know R(ε, z), for z ∈ Ω0 ∩ Oγ− (0) , Oγ− (0) being a small neighborhood of γ− (0), and R(ε, z) ∈ Ω0 , for any z ∈ Oγ− (0) ∩ Ω0 . (ii) Since codim (RP+ ⊕ S  ) = 1, any non-zero vector in (RP+ ⊕ S  )⊥ is a nonzero multiple of ψ. Hence the conditions on M(α) in Theorem 6.1 do not depend on the non-zero vector ψ ∈ (RP+ ⊕ S  )⊥ we choose. 6.5

Almost Periodic and Periodic Cases in Chaotic Behavior

In this section, we extend the results of deterministic chaos presented in [Stoffer (1988); Meyer and Sell (1989); Palmer and Stoffer (1989)] to the impact almost periodic and periodic system (6.3). Let E := {e : Z → {0, 1}} be the set of doubly infinite sequences of 0 and 1. We write an element e ∈ E as e = {em }m∈Z . It is well-known that E becomes a totally disconnected compact metric space with the distance d(e , e ) =

$ |e − e | m m . |m|+1 2 m∈Z

The Bernoulli shift σ : E → E is defined as σ(e) := {em+1 }m∈Z . We assume that M(α) has a simple zero α0 and that g(t, z, ε) is almost periodically uniform in (z, ε) and the following holds: (H4) For any ν > 0, there exists Lν > 0 such that in any interval of a length greater than Lν there exists Tν , which is almost a period for ν that satisfies |g(t + Tν , z, ε) − g(t, z, ε)| < ν, for any (t, z, ε) ∈ Rn+2 .

page 112

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

113

Then as in [Battelli and Feˇckan (2011)] we see, that for any ε = 0 sufficiently ε ε }m∈Z and αε = {αm }m∈Z ∈ ∞ (R) such that small, there are sequences T ε = {Tm ε ε 0 −1 ε 0 ε ε ) = 0, + αm Tm+1 − Tm > 1 + 4|α | − 2δ ln |ε| and α  ≤ 2|α |, satisfying M(T2m  ε ε for any m ∈ Z and inf m∈Z |M (T2m + αm )| > c0 , for some c0 > 0. Then taking ε ε + αm and α0 = 0, the following property is satisfied: Tm = Tm (H5) for any ε = 0 sufficiently small there is a sequence T = {Tm }m∈Z (that depends on ε) such that Tm+1 − Tm > 1 − 2δ −1 ln |ε| and M(T2m ) = 0, for any m ∈ Z and inf m∈Z |M (T2m )| > c0 , for some c0 > 0. Let T = {Tm }m∈Z be as in (H5). For any given e ∈ E, we take a (fixed) increasing, in general doubly-infinite, sequence of integers {nem }m such that ek = 1 e } ⊂ R as if and only if k = nem and define an increasing T e = {Tm  T2nek if m = 2k, e = Tm T2nek −1 if m = 2k − 1. As in [Battelli and Feˇckan (2011)] we have the following theorem. Theorem 6.2. Let f (z) and g(t, z, ε) be C 2 -functions with bounded derivatives and such that their second-order derivatives are uniformly continuous. Assume that conditions (H1)–(H4) hold and that M(α) has a simple zero at α = α0 . Then taking T = {Tm }m∈Z as in (H5), for any e ∈ E there exists a unique sequence e e }m = {αm (ε)}m and a unique impact solution z(t, T , e, ε) of Eq. (6.3), depend{αm e | < c1 |ε| ing only on T and e ∈ E (and not on the choice of {nem }m ) such that |αm and the following holds: e e supt∈[T2m−1 e e +αe ) |z(t, T , e, ε) − γ− (t − T2m − αm )| < ρ, +αem−1 ,T2m m e e |z(t, T , e, ε) − γ+ (t − T2m − αm )| < ρ, supt∈[T2m e +αe ,T e e m 2m+1 +αm ]

(6.37)

if nem , nem+1 ∈ Z, or supt∈(−∞,T2em¯

−

supt∈[T2em¯

−

e +αem ¯ − ,T2m ¯

+αem ¯ −)

− +1

+αem ¯ −]

e |z(t, T , e, ε) − γ− (t − T2em ¯ − )| < ρ, ¯ − − αm e |z(t, T , e, ε) − γ+ (t − T2em ¯ − )| < ρ, ¯ − − αm

(6.38)

¯ − , or if ej = 0, for any j < m supt∈[T2em¯

+ −1

+αem ¯

+ −1

,T2em +αem ¯ +) ¯ +

supt∈[T2em¯

+

+αem ¯ + ,∞)

e |z(t, T , e, ε) − γ− (t − T2em ¯ + )| < ρ, ¯ + − αm e |z(t, T , e, ε) − γ+ (t − T2em ¯ + )| < ρ, ¯ + − αm

if ej = 0, for any j > m ¯ + . If, instead, ej = 0, for any j ∈ Z, then sup |z(t) − z0 | < ρ. t∈R

Moreover, setting T

k

:= {Tm+2k }m∈Z , we have z(t, T k+1 , σ(e), ε) = z(t, T k , e, ε),

for any t ∈ R and e ∈ E.

(6.39)

page 113

June 8, 2017 12:9

114

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

We can restate Theorem 6.2 as follows. Let Fk : Rn → Rn be defined so that Fk (ξ) is the value at time T2(k+1) of the impact solution z(t) of Eq. (6.3) such that z(T2k ) = ξ, i.e., Fk (ξ) = z(T2(k+1) ), and let Φk (e) := z(T2k , T (k) , e). Recall z(T2k ) = + + ) and z(T2k , T (k) , e) = z(T2k , T (k) , e), since we consider impact solutions which z(T2k are right continuous at any point t. Although Fk may not be defined in the whole Rn , for sure it is defined in the set  Sk = z(T2k , T (k) , e) | e ∈ E ,

k ∈ Z.

Then we have: Φk+1 ◦ σ(e) = z(T2(k+1) , T (k+1) , σ(e)) = z(T2(k+1) , T (k) , e)

(6.40) = Fk (z(T2k , T (k) , e)) = Fk ◦ Φk (e). According to [Palmer (1984)], it is standard to prove that Sk is compact in Rn and Φk : E → Sk is one-to-one and a homeomorphism, for any k ∈ Z. Summarizing, we get the next result.

Theorem 6.3. Assume the conditions of Theorem 6.2 hold. Then for any ε = 0 sufficiently small, the following diagrams commute: σ /E E Φk

 Sk



Fk

Φk+1

/ Sk+1

for all k ∈ Z. Moreover, all Φk are homeomorphisms. The solution g(t, z, ε) is quasi periodic in t if the following holds: g(t, z, ε) = q(ω1 t, . . . , ωm t, z, ε), for ω1 , . . . , ωm ∈ R with q ∈ Cbr (Rm+n+1 , Rn ) and q(θ1 , . . . , θm , z, ε) is 1-periodic in each θi , i = 1, 2, . . . , m. Moreover, ωi , i = 5m 1, 2, . . . , m are linearly independent over Z, i.e., if i=1 li ωi = 0, li ∈ Z, i = 1, 2, . . . , m, then li = 0, i = 1, 2, . . . , m so g(t, z, ε) satisfies assumption (H4) [Levitan and Zhikov (1983); Meyer and Sell (1989)], and hence, the conclusion of Theorem 6.2 holds. Finally, if g(t, z, ε) is periodic in t with period p we can apply the result for the quasi periodic case with T = mp a sufficiently large multiple of the period p. Arguing as in [Battelli and Feˇckan (2011, 2010a)], we obtain the following theorem. Theorem 6.4. Besides (H1)–(H4), assume g(t + p, z, ε) = g(t, z, ε) that is g(t, z, ε) is p-periodic. If ε = 0 is sufficiently small and there is a simple zero α0 of M(α), then the following diagram commutes: σ /E E Φ

 S

F

 /S

Φ

Here F = ϕm ε = ϕε ◦ . . . ◦ ϕε (m times) is the mth iterate of the p-periodic map ϕε of (6.3) and S = Sk , that in the periodic case is independent of k.

page 114

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Occurrence of Chaos in Forced Impact Systems

ws-book975x65

115

Let us concern our analysis on the condition (H3) and the function ψ(t). First, we give a geometric meaning of condition (H3). For any ξ ∈ Ω0 let ϕ(t, ξ) be the solution of z˙ = f (z) such that ϕ(0, ξ) = ξ. Next, for any ξ ∈ Ω0 near γ− (0), we construct an impact solution φ(t, ξ) of z˙ = f (z) as follows:  ϕ(t, ξ), for t < 0 , φ(t, ξ) = ϕ(t, R(0, ξ)), for t ≥ 0 . & := ∂φ (t, γ− (0))η of φ(t, ξ) at γ− (0) along η ∈ N G (γ− (0)) The linearization φ(t) ∂ξ is given by  X− (t)η, for t < 0 , & φ(t) = X+ (t)D2 R(0, γ− (0))η, for t ≥ 0 . & is a bounded impact solution of the variational equation of Eq. (6.3) Hence φ(t) with ε = 0 given by ⎧ for t < 0, v˙ = Df (γ− (t))v, ⎪ ⎪ ⎨ v(0+ ) = D2 R(0, γ− (0))v(0− ), (6.41) ⎪ R− v(0− ) = v(0− ), ⎪ ⎩ for t ≥ 0, v˙ = Df (γ+ (t))v, if and only if v(0− ) ∈ N P− ∩ N G (γ− (0)) = S  and D2 R(0, ξ)v(0− ) ∈ RP+ . In & with other words, equation (6.41) has as many independent bounded solutions φ(t) −   & φ(0 ) ∈ N G (γ− (0)) as dim[S ∩RP+ ] (cf. Eq. (6.19)). Now, as in (6.20) we derive 0 ≤ dim S  ∩ RP+ ≤ n − 1 − dim [S  + RP+ ] .

(6.42)

So using (6.42), we see that (H3) is equivalent to dim S  ∩ RP+ = 0. Summarizing, we derive the following result (cf. [Battelli and Feˇckan (2011, 2010a)]). & is bounded on R Proposition 6.3. Condition (H3) is equivalent to saying that φ(t) if and only if it is equal to zero. This corresponds to some nondegeneracy condition on an impact homoclinic solution γ(t) with respect to z˙ = f (z). Now we look at the function ψ(t). More generally, we consider any vector w ∈ ⊥ (RP+ ⊕ S  ) so we relax assumption (H3). Such a w satisfies P+∗ w, z = 0, for n any z ∈ R and D2 R(0, γ− (0))∗ w, R− z = 0, for any z ∈ Rn such that P− R− z = 0. Note that P− R− = P− R− P− and then: P− R− P− R− = P− R− R− = P− R− so that P− R− is a projection. Hence: w ∈ N P+∗ ∩ N [D2 R(0, γ− (0))R− (I − P− R− )]∗ . So we can write: P+∗ w = 0, w D2 R(0, γ− (0))R− [I − P− R− ] = 0. ∗

(6.43)

page 115

June 8, 2017 12:9

116

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Then for any w ∈ (RP+ ⊕ S  )⊥ consider the function ⎧ −1 ∗ ⎨ X− (t)∗ P−∗ R− D2 R(0, γ− (0))∗ w, w(t) = ⎩ −1 ∗ X+ (t) w,

for t ≤ 0, (6.44) for t > 0.

∗ ∗ ∗ ∗ P− = P−∗ R− P− . Next we know from Since P− R− = P− R− P− we get R− ∗ ∗ ∗ ∗ ∗ ∗ D2 R (0, γ− (0)) w. Eq. (6.43) that R− D2 R (0, γ− (0)) w = R− P− uw , uw = R− Therefore: ∗ ∗ ∗ ∗ D2 R(0, γ− (0))∗ w(0+ ) = R− D2 R(0, γ− (0))∗ w = R− P− uw R− ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ = R− P− R− P− uw = R− P− R− D2 R(0, γ− (0))∗ w = R− w(0− ).

As a consequence, for any w ∈ (RP+ ⊕S  )⊥ , the function w(t) defined in Eq. (6.44) is a bounded solution of the adjoint equation ⎧ w˙ = −f  (γ− (t))∗ w, for t ≤ 0, ⎨ ∗ ∗ + ∗ (6.45) w(0− ), R D R(0, γ− (0)) w(0 ) = R− ⎩ − 2  w˙ = −f (γ+ (t))∗ w, for t > 0. On the other hand, according to the above arguments the dimension of bounded solutions of Eq. (6.45) is  dim w(0+ ) ∈ Rn | w(0+ ) ∈ R(I − P+∗ ),  ∗ ∗ R− D2 R(0, γ− (0))∗ w(0+ ) = R− w(0− ), w(0− ) ∈ RP−∗   ∗ ∗ ∗ D2 R(0, γ− (0))∗ w(0+ ) ∈ RR− P− = dim w(0+ ) ∈ Rn | w(0+ ) ∈ R(I − P+∗ ), R−   = dim w(0+ ) ∈ Rn | w(0+ ) ∈ (RP+ ⊕ S  )⊥ = dim[RP+ ⊕ S  ]⊥ . Proposition 6.4. The space of bounded solutions of the adjoint system (6.45) is spanned by the functions w(t) defined by (6.44) with w ∈ (RP+ ⊕ S  )⊥ . Theorem 6.5. The nonhomogeneous linear equation ⎧ v˙ = f  (γ− (t))v + h− (t), for t < 0, ⎪ ⎪ ⎨ + − v(0 ) = D2 R(0, γ− (0))v(0 ), ⎪ R− v(0− ) = v(0− ), ⎪ ⎩ for t ≥ 0, v˙ = f  (γ+ (t))v + h+ (t),

(6.46)

has a bounded solution for continuous and bounded h± if and only if 0

∗

∞

w(t) h− (t)dt + −∞

w(t)∗ h+ (t)dt = 0,

(6.47)

0

for any bounded solution w of the adjoint equation (6.45). Remark 6.3. It is because of Theorem 6.5 that the equation (6.45) is called the adjoint equation to (6.46).

page 116

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

117

Proof. First we note that the middle condition of Eq. (6.45) is equivalent to D2 R(0, γ− (0))∗ w(0+ ) − w(0− ) ∈ span {∇G(γ− (0))}.

(6.48)

Next, a bounded solution of the first equation of the system (6.46) is given by 6t −1 (s)h− (s)ds v(t) = X− (t)η + −∞ X− (t)P− X− 60 −1 − t X− (t) (I − P− ) X− (s)h− (s)ds, for η ∈ N P− , and of the third one 6t −1 (s)h+ (s)ds v(t) = X+6 (t)ξ + 0 X+ (t)P+ X+ ∞ −1 − t X+ (t) (I − P+ ) X+ (s)h+ (s)ds, for ξ ∈ RP+ . Using the second condition of (6.46), we derive

 0 −1 (I − R− ) η + P− X− (s)h− (s)ds = 0, D2 R(0, γ− (0)) η +



−∞

0 −∞

−1 P− X − (s)h− (s)ds



 =ξ−

∞ 0

−1 (I − P+ )X+ (s)h+ (s)ds.

(6.49) Now we split η ∈ N P− as η = η1 + η2 with η1 ∈ S  , η2 ∈ span {γ˙ − (0)}. Then the first condition in Eq. (6.49) reads  η2 = (R− − I)

0 −∞

−1 P− X − (s)h− (s)ds,

from which we see that the second condition in Eq. (6.49) reads: 60 −1 (s)h− (s)ds ξ − D2 R(0, γ− (0))η1 = D2 R(0, γ− (0))R− −∞ P− X− 6∞ −1 + 0 (I − P+ ) X+ (s)h+ (s)ds. So the necessary and sufficient condition for the solvability of (6.46) is

 ∞  0 −1 −1 w∗ D2 R(0, γ− (0))R− P− X − (s)h− (s)ds + (I − P+ )X+ (s)h+ (s)ds = 0 −∞

0

for any vector w ∈ [RP+ ⊕ S  ]⊥ . Using Eq. (6.44), this fact is equivalent to (6.47). The proof is finished.

6.6

Examples

This section is devoted to applications of Theorems 6.3 and 6.4 to certain concrete forced impact systems of ordinary differential equations.

page 117

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

118

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

6.6.1

Impact Planar Systems

Let us consider a two-dimensional system (z = (x, y) ∈ R2 ) given by:  x˙ = P (x, y), y˙ = Q(x, y).

(6.50)

Note the system (6.50) has the form of Eq. (6.3) with ε = 0 and f (z) = (P (z), Q(z)), for z = (x, y). We write the homoclinic orbit as

u± (t) ¯ −. ∈Ω γ± (t) = v± (t) Then N P− = span {(u˙ − (0), v˙ − (0))} and RP+ = span {(u˙ + (0), v˙ + (0))}. Clearly, N G (γ± (0)) = span {(Gy (γ± (0)), −Gx (γ± (0)))} = Tγ± (0) G−1 (0). So (H2) implies S  = S  = {0}, and thus S  = {0}. Consequently, (H3) is satisfied and we take ψ = (v˙ + (0), −u˙ + (0))∗ . Next let a± (t) = Px (u± (t), v± (t)) + Qy (u± (t), v± (t)) be the trace of the Jacobian matrix of the linearization of (6.50) along (u± (t), v± (t)), and  a− (t) if t < 0, a(t) := a+ (t) if t ≥ 0. ' (  v˙ (t) − 0t a± (τ )dτ satisfy the adjoint system: Then −u± e ˙ ± (t)  x˙ = −Px (γ± (t))x − Qx (γ± (t))y, y˙ = −Py (γ± (t))x − Qy (γ± (t))y. As a consequence:

v˙ ± (t) −u˙ ± (t)



e−

t 0

a± (τ )dτ

∗ = X± (t)−1



v˙ ± (0) . −u˙ ± (0)

Then RP−∗ = span {(v˙ − (0), −u˙ − (0))∗ } and N P+∗ = span {(v˙ + (0), −u˙ + (0))∗ } = ∗ D2 R(0, γ− (0))∗ ψ = kψ (v˙ − (0), −u˙ − (0))∗ , for some kψ ∈ span {ψ}. Hence P−∗ R− R. Thus

 0 v˙ − (t) ∗ −1 ∗ ∗ ∗ e t a− (τ )dτ , X− (t) P− R− D2 R(0, γ− (0)) ψ = kψ −u˙ − (t) for any t ≤ 0. Note P− R− = P− . Similarly, for t > 0, we have

 0 v˙ + (t) ∗ −1 ∗ X+ (t) (I − P+ )ψ = e t a+ (τ )dτ . −u˙ + (t) Putting everything together, we obtain

∗  0 t v˙ − (t) ∗ g(t + α, γ(t), 0) e− 0 a(τ )dτ dt M(α) =ψ D1 R(0, γ− (0)) + kψ ˙ − (t) −∞ −u

∗  ∞ t v˙ + (t) + g(t + α, γ(t), 0) e− 0 a(τ )dτ dt −u˙ + (t) 0

page 118

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Occurrence of Chaos in Forced Impact Systems

that can be written as follows:



0

f (γ(t)) ∧ g(t + α, γ(t), 0) e− M(α) =γ˙ + (0) ∧ D1 R(0, γ− (0)) + kψ −∞  ∞ t + f (γ(t)) ∧ g(t + α, γ(t), 0) e− 0 a(τ )dτ dt.

119

t 0

a(τ )dτ

dt

0

(6.51) As an example, consider a second-order PSNDS of the form x ¨ + Q(x) = εφ(t),

for x < c,

(6.52)

where c > 0 is a constant and φ(t) is a C 2 -smooth almost periodic function. We have G(x, y) = x − c. We suppose Q(0) = 0, Q (0) < 0 and there is a solution τ (t) of x ¨ + Q(x) = 0 such that τ (0) = c, τ  (0) < 0 and limt→∞ τ (t) = 0. Then according to [Vanderbauwhede and Fiedler (1992)] we know that τ (t) < c, ∀t < 0. Then we clearly have u± (t) = τ (±t) and v± (t) = ±τ˙ (±t). Next we take R(ε, x, y) = (x, −(1 + εr)y), for a r ∈ R. Next, by (6.45), the adjoint variational equation is given by ⎧  ⎨ v˙ 1 = Q (u± (t))v2 , (6.53) v˙ = −v , ⎩ 2 + 1 ∗ = span {(1, 0)∗ }. (v1 (0 ), −v2 (0+ )) − (v1 (0− ), v2 (0− )) ∈ N R− We know that the dimension of bounded solutions of (6.53) is 1 (see Prop. 6.4) and the function  (−¨ τ (−t), −τ˙ (−t)) , for t ≤ 0, ψ(t) = (−¨ τ (t), τ˙ (t)) , for t > 0, is such a bounded solution. Summarizing, the Melnikov function of (6.62) has the form 60 6∞ φ(t + α)τ˙ (t)dt M(α) = rτ˙ (0)2 − −∞'φ(t + α)τ˙ (−t)dt + ( 0 6∞ (6.54) 2 ˙ ˙ = rτ˙ (0) − 0 φ(α − t) + φ(α + t) τ (t)dt. Here we note that sometimes it is better to use the adjoint variational equation (6.45) than to compute kψ . To be more concrete, let us consider the simplest case of Eq. (6.52) with Q(x) = −x, c = 1 and φ(t) = sin t, i.e., the equation x ¨ − x = εA sin t, for x < 1, ˙ − ), for x(t) = 1, x(t ˙ + ) = −(1 + εr)x(t with an amplitude constant A > 0. Then  t t ( e , e ), γ(t) = ( e−t , − e−t ),

(6.55)

for t < 0, for t ≥ 0.

So τ (t) = e−t for this case. By inserting the above into Eq. (6.54), we obtain M(α) = r −A cos α. Consequently, if |r| < A, then there is a simple zero of M. This

page 119

June 8, 2017 12:9

120

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

implies that, for sufficiently small ε = 0, (6.55) has a Smale horseshoe-type chaos orbitally close to γ(t). Condition |r| < A means that the forcing in Eq. (6.55) has to be sufficiently large with respect to the impact impulse in order to have chaos. Note that numerical results concerning Eq. (6.55) are presented in Sec. 6.1. We conclude this section with another more sophisticated example. We consider a pendulum equation (6.56)

ϕ(t) ¨ + sin ϕ(t) = 0 with impact conditions

 −  ˙ ) − εg(t, ε) ϕ(t ˙ + ) = − ϕ(t

if ϕ(t) = a + εh(t, ε),

(6.57)

where h(t, ε) and g(t, ε) are C 3 -smooth and almost periodic in t, uniformly with respect to ε (see (H4)), and −π < a < π is given. Introducing x = ϕ − εh(t, ε),

y = ϕ˙ − εg(t, ε)/2,

into Eqs. (6.56) and (6.57), we derive: ' ( ⎧ 1 ˙ ⎪ x ˙ = y + ε g(t, ε) − h(t, ε) , ⎪ 2 ⎪ ⎪ ⎨ ˙ y˙ = − sin (x + εh(t, ε)) − ε g(t,ε) 2 , ⎪ ⎪ ⎪ ⎪ ⎩ + y(t ) = −y(t− ) if x(t) = a.

(6.58)

The unperturbed part of Eq. (6.58) is just Eq. (6.56) with the upper heteroclinic solution ϑ(t) = 2 arcsin(tanh t). Note ϑ(t) is odd and increasing from −π to π so a unique ta ∈ R exists such that ϑ(ta ) = a. Also R(ε, x, y) = (x, −y). So now we have τ (t) = ϑ(t + ta ), for t < 0, and then  ˙ + ta )), (ϑ(t + ta ), ϑ(t for t < 0, γ(t) = ˙ − ta )), for t ≥ 0, (−ϑ(t − ta ), −ϑ(t and

⎧' ( ˙ + ta ) , ¨ + ta ), ϑ(t ⎨ −ϑ(t ( ψ(t) = ' ˙ − ta ) , ¨ − ta ), −ϑ(t ⎩ ϑ(t

for

t < 0,

for

t ≥ 0.

So the Melnikov function has the form

1 ˙ + α, 0) ϑ(t ¨ + ta )dt g(t + α, 0) − h(t 2 −∞

 0 g(t ˙ + α, 0) ˙ cos ϑ(t + ta )h(t + α, 0) + − ϑ(t + ta )dt 2 −∞

 ∞ 1 ˙ + α, 0) ϑ(t ¨ − ta )dt + g(t + α, 0) − h(t 2 0

 ∞ g(t ˙ + α, 0) ˙ cos ϑ(t − ta )h(t + α, 0) + + ϑ(t − ta )dt. 2 0 

M(α) = −

0



page 120

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Occurrence of Chaos in Forced Impact Systems

ws-book975x65

121

Since 

0 −∞



˙ + ta )dt cos ϑ(t + ta )h(t + α, 0)ϑ(t 0

=

d ( sin ϑ(t + ta )) h(t + α, 0)dt dt

−∞  0

=−

−∞

... ϑ (t + ta )h(t + α, 0)dt

¨ a )h(α, 0) + = −ϑ(t



0 −∞

˙ + α, 0)dt ¨ + ta )h(t ϑ(t

(6.59)

and similarly 

∞ 0

 ˙ − ta )dt = −ϑ(t ¨ a )h(α, 0) + cos ϑ(t − ta )h(t + α, 0)ϑ(t

∞ 0

˙ + α, 0)dt, ¨ − ta )h(t ϑ(t

we obtain   1 0 1 0 ¨ ˙ + ta )dt g(t + α, 0)ϑ(t + ta )dt − g(t ˙ + α, 0)ϑ(t M(α) = − 2 −∞ 2 −∞   1 ∞ 1 ∞ ¨ ˙ − ta )dt + g(t + α, 0)ϑ(t − ta )dt + g(t + α, 0)ϑ(t 2 0 2 0   0 d 1 ∞ d ˙ − ta )]dt − 1 ˙ + ta )]dt [g(t + α, 0)ϑ(t [g(t + α, 0)ϑ(t = 2 0 dt 2 −∞ dt ˙ a )g(α, 0). = − ϑ(t (6.60) Thus, if g(α, 0) has a simple root, then so does M, and consequently, equation (6.56) with (6.57) is chaotic, for any ε = 0 small. For example, if the pendulum oscillates in the (z1 , z2 ) plane, ϕ is the angle it makes with the z2 -axis and the impact line moves according to the law z1 = A + εH(t), with |A| < 1, then we get a + εh(t, ε) = arcsin(A + εH(t)) with a = arcsin A, ˙ ˙ − ) plus the projection of εH(t) onto the tangent vector to and ϕ(t ˙ + ) equals −ϕ(t ˙ ε) cos 2 (a + εh(t, ε)) the trajectory of the pendulum. This means that g(t,√ ε) = h(t, ˙ ˙ ˙ 0)(1 − A2 ) = H(α) 1 − A2 . and g(α, 0) = h(α, 0) cos 2 a = h(α, On the other hand, we may also consider the case where the impact line rotates around the same point as the pendulum according to the rule: ϕ(t) = a + εh(t) ˙ ˙ then g(t, ε) = h(t) and so g(α, 0) = h(α). 6T Note that when H(t) and h(t) are T -periodic in t, then 0 g(α, 0)dα = 0 in both impact cases, and hence, g(α, 0) must change sign in [0, T ]. Then using a topological degree arguments provided in [Battelli and Feˇckan (2002)], one may show that such a kind of impacts always chaotically excites the pendulum.

page 121

June 8, 2017 12:9

ws-book961x669

122

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Figure 6.5: On the left the impact line is z1 = A(t) := A + εH(t). On the right the impact line oscillates around the same point of the pendulum according to ϕ(t) = a(t) := a + εh(t). 6.6.2

Impact Coupled Second-Order Systems

In this section, we apply Theorem 6.2 to discuss chaotic behavior of a coupled second-order system of ODEs such as:  x ¨1 = x1 g1 (x1 , x2 ) + εh1 (t), (6.61) x ¨2 = x2 g1 (x1 , x2 ) + εh2 (t), or as a first-order system: ⎧ x˙ 1 = y1 , ⎪ ⎪ ⎨ x˙ 2 = y2 , ⎪ y˙ = x1 g1 (x1 , x2 ) + εh1 (t), ⎪ ⎩ 1 y˙ 2 = x2 g2 (x1 , x2 ) + εh2 (t)

(6.62)

with the impact condition y2 (t+ ) = −(1 + εr)y2 (t− )

if x2 (t) = a,

(6.63)

for r ∈ R. So x1 (t), x2 (t), x˙ 1 (t) = y1 (t) are continuous and x2 (t) is reflected, i.e., x˙ 2 (t+ ) = −(1 + εr)x˙ 2 (t− ), when x2 (t) = a. Hence we take G(x1 , x2 , y1 , y2 ) = x2 − a. Note also that f (x) = (y1 , y2 , x1 g1 (x1 , x2 ), x2 g2 (x1 , x2 )). We assume the following: (i) h1,2 (t) and g1,2 (x1 , x2 ) are C 2 -functions bounded with their derivatives and h1,2 (t) are almost periodic; (ii) g1,2 (0, 0) > 0;

page 122

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Occurrence of Chaos in Forced Impact Systems

ws-book975x65

123

(iii) x ¨ = xg2 (0, x) has an even homoclinic solution ϑ, ϑ(t) = ϑ(−t), to the hyperbolic equilibrium x = 0; (iv) 0 < a < ϑ(0); ∂g2 (0, θ(t)) = 0. (v) ∂x 1 We note that the hypothesis ϑ(t) = ϑ(−t) can be assumed without loss of ˙ generality by shifting time so that ϑ(0) = 0. Now, according to [Vanderbauwhede ˙ ˙ = 0, for and Fiedler (1992)], we recall the following known fact that ϑ(0) = 0 ⇒ ϑ(t) ˙ any t = 0. In fact ϑ(t0 ) = 0, t0 = 0, implies ϑ(t + t0 ) = ϑ(t0 − t). Then ϑ(2t0 ) = ϑ(0) ˙ ˙ 0 ) = 0 = ϑ(0). As a consequence, ϑ(t + 2t0 ) = ϑ(t) contradicting ϑ(t) → 0 and ϑ(2t as |t| → ∞. Hence ϑ(t) > 0 and it is decreasing on (0, ∞) and increasing on (−∞, 0). Condition (iv) concerns the value of ϑ(t) at the initial time t = 0. Let us take ta > 0 so that ϑ(ta ) = a. Note such a ta > 0 exists and it is unique. As impact homoclinic solution of Eq. (6.62) with ε = 0, we take:  ˙ − ta )) for t < 0, γ− (t) := (0, ϑ(t − ta ), 0, ϑ(t γ(t) = ˙ γ+ (t) := (0, ϑ(t + ta ), 0, ϑ(t + ta )) for t ≥ 0. Finally, we take R(ε, x1 , x2 , y1 , y2 ) = (x1 , x2 , y1 , −(1 + εr)y2 ) and note R(0, γ− (0)) = γ+ (0), because ϑ(t) is even. ˙ a ) < 0, We already known that the transversality condition (H2), that is ϑ(t holds. The linearization of Eq. (6.62) with ε = 0 at γ is given by



v v˙ 0 I (6.64) , t ∈ R± , = w w˙ D± (t) 0 where v = (v1 , v2 ), w = (w1 , w2 ), ϑ± (t) = ϑ(t ± ta ) and # % 0 g1 (0, ϑ± (t)) D± (t) = ∂g2 0 g2 (0, ϑ± (t)) + ϑ± (t) ∂x (0, ϑ± (t)) 2 or, more explicitly:

⎧ v˙ 1 = w1 , ⎪ ⎪ ⎪ ⎨ w˙ 1 = g1 (0, ϑ± (t))v1 , v˙ 2 = w'2 , ⎪ ( ⎪ ⎪ ⎩ w˙ = g (0, ϑ (t)) + γ ∂g2 (0, ϑ (t)) v . 2 2 ± ± 2 ∂x2

(6.65)

We remark that in (6.65) we take ϑ+ (t) when t ≥ 0 and ϑ− (t) when t < 0 and that in the system (6.65) the variables (v1 , w1 ) and (v2 , w2 ) are decoupled. Clearly N G (x) = {(0, 1, 0, 0)}⊥ and D2 R(0, x) = diag {1, 1, 1, −1}. Next, when t → ±∞ the system v˙ 1 = w1 , w˙ 1 = g1 (0, ϑ± (t))v1 tends to v˙ 1 = w1 , w˙ 1 = g1 (0, 0)v1 , which has stable and unstable spaces both of dimension 1. As a consequence, the space of solutions of system (6.65) that are bounded on (−∞, 0] is span {γ˙ − (t), (v1 (t), 0, w1 (t), 0)}

page 123

June 8, 2017 12:9

ws-book961x669

124

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

with (v1 (t), w1 (t)) being a non-zero solution of the system:  v˙ 1 = w1 , , w˙ 1 = g1 (0, ϑ− (t))v1 , which is bounded on (−∞, 0]. Note that (v1 (t), w1 (t)) being non-zero we get (v1 (0), w1 (0)) = (0, 0). Hence ˙ a ), 0, ϑ(t ¨ a )), (v 0 , 0, w0 , 0)}, N P− = span {(0, −ϑ(t 1 1 where (v10 , w10 ) = (v1 (0), w1 (0)). Since ϑ+ (t) = ϑ(t + ta ) = ϑ(−t − ta ) = ϑ− (−t) we see that (v1 (−t), −w1 (−t)) is a non-zero solution of the system:  v˙ 1 = w1 , w˙ 1 = g1 (0, ϑ+ (t))v1 , bounded on [0, ∞). Thus the space of solutions of the system (6.65) that are bounded on [0, ∞) is span {γ˙ + (t); (v1 (−t), 0, −w1 (−t), 0)}, and ˙ a ), 0, ϑ(t ¨ a )), (v 0 , 0, −w0 , 0)}. RP+ = span {(0, ϑ(t 1 1 As a consequence, S  = span {(v10 , 0, w10 , 0)}, S  = S  and RP+ ∩ S  = span {(v10 , 0, w10 , 0)} ∩ span {(v10 , 0, −w10 , 0)}. So the non-degeneracy condition (H3) is satisfied if and only if the following holds (vi) the equation v¨ = g1 (0, ϑ± (t))v, for t ∈ R± , has no nontrivial C 1 -bounded solutions. Assuming (vi) we get codim RP+ ⊕ S  = 1. We look at the adjoint variational equation that, on account of Eq. (6.45), is given by ⎧ ∗ ⎨ v˙ = −D± (t) w, (6.66) w˙ = −v, ⎩ ∗ R− [D2 R(0, γ− (0))∗ (v(0+ ), w(0+ )) − (v(0− ), w(0− ))] = 0 or ⎧ ⎪ v˙ 1 = −g'1 (0, ϑ± (t))w1 , ( ⎪ ⎪ ∂g2 ⎪ ⎪ v˙ 2 = − g2 (0, ϑ± (t)) + ϑ± (t) ∂x (0, ϑ (t)) w2 , ± ⎪ 2 ⎪ ⎨ w˙ 1 = −v1 , (6.67) ⎪ w ˙ 2 = −v2 , ⎪ ⎪ ⎪ ⎪ ⎪ (v (0+ ), v (0+ ), w1 (0+ ), −w2 (0+ )) − (v1 (0− ), v2 (0− ), w1 (0− ), w2 (0− )) , ⎪ ⎩ 1 ∗ 2 ∈ N R− = span {(0, 1, 0, 0)∗ }. We know that the dimension of bounded solutions of (6.67) is 1 (see Prop. 6.4) and the function ⎧' ( ¨ − ta ), 0, ϑ(t ˙ − ta ) , for t < 0, ⎨ 0, −ϑ(t ( ψ(t) = ' ¨ + ta ), 0, ϑ(t ˙ + ta ) , for t ≥ 0, ⎩ 0, −ϑ(t

page 124

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Occurrence of Chaos in Forced Impact Systems

125

is such a bounded solution. Summarizing, the Melnikov function of (6.62) has the form 6∞ ˙ − ta )dt + h2 (t + α)ϑ(t ˙ + ta )dt h (t + α) ϑ(t 2 −∞ 0 ' ( 6 ˙ a )2 − ∞ h˙ 2 (α + ta − t) + h˙ 2 (α − ta + t) ϑ(t)dt. = rϑ(t ta

˙ a )2 + M(α) = rϑ(t

60

To find a condition ensuring the assumption (vi), we write ¯ ± (t), v¨ − g1 (0, ϑ± (t))v = v¨ − g1 (0, 0)v + h ¯ ± (t) = [g1 (0, ϑ± (t)) − g1 (0, 0)]v, which implies with h v∞ ≤

¯ ∞ Ma h ≤ v∞ , g1 (0, 0) g1 (0, 0)

¯ − (−t) = h ¯ + (t). So if for Ma := max |g1 (0, ϑ(t)) − g1 (0, 0)|. Note h t≥ta

(6.68)

Ma < g1 (0, 0), then v = 0. Another simple condition is ma := inf g1 (0, ϑ(t)) ≥ 0. t≥ta

(6.69)

Indeed, if v is a bounded C 1 -solution of v¨ − g1 (0, ϑ± (t))v = 0, then we get  ∞  ∞ − v(t) ˙ 2 dt = v¨(t)v(t)dt −∞



−∞ 0

= −∞



2

g1 (0, ϑ− (t))v(t) dt +

∞ 0

g1 (0, ϑ+ (t))v(t)2 dt ≥ 0,

so v = 0. Let us consider a concrete example given by two equations: √  2t, x ¨1 = x1 (2  + x2 )2 + ε cos  x ¨2 = x2 1 − 2x2 + x21 + ε cos t.

(6.70)

Then ϑ(t) = sech t so 0 < a < 1 and ta = arcsech a. Subsequently, Ma = a < 2 = g1 (0, 0) so the relation (6.68) holds. Note ma ≥ 2 so the definition (6.69) holds too. The Melnikov function is now 6∞ M(α) = ra2 (1 − a2 ) + ta [ sin (α + ta − t) + sin (α − ta + t)]sech t dt 6∞ = ra2 (1 − a2 ) + 2 sin α ta cos (t − ta )sech t dt, and it has a simple root provided 0 < a < 1 and r ∈ R are such that (vii)

6∞ 0

cos tsech (t + ta ) dt = 0 and |r|
0, is not a bounded solution of the variational equation (see Eq. (6.45) in Chapter 6), since it does not satisfy the impact condition (see the second equation (6.45)) w(t) :=

∗ [R (γ− (0), γ˙ − (0))∗ w(0+ ) − w(0− )] = 0. R−

Indeed:

∗ [R (γ− (0), γ˙ − (0))∗ w(0+ ) − w(0− )] R−

(7.32)

∗

= (0, 0, 2 sin β(a cos β + b sin β), 2 sin β(b cos β − a sin β)) = 0. ∗ On the other hand, it is true that R− [R (γ− (0), γ˙ − (0))∗ ψ(0+ ) − ψ(0− )] = 0 as we already know. Thus to apply Theorem 7.1 the knowledge of a first integral it is not enough to obtain a solution w(t) of the adjoined variational equation as given in the first and third equation (6.45) in Chapter 6, but one also has to check whether w(t) satisfies the impact condition (7.32) presented by the second equation (6.45).

page 145

June 8, 2017 12:9

146

7.4

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Chaotic Behavior

In this section, we perturb Eq. (7.3), or equivalently Eq. (7.21), and construct the corresponding Melnikov function associated with the chaotic behavior of the perturbed system. We construct such a perturbation by allowing the boundary of Ω to oscillate around the equilibrium. However, to fit into the framework of this chapter, we need that the perturbation does not act when the particle runs from the first hitting line (y = 0) to the second (y = x tan β). We may obtain such a situation endowing the line y = 0 with a switcher interrupting the boundary movement when the particle hits it and another switch restoring the boundary movement immediately before the time when the particle hits the line y = x tan β. Another kind of perturbation fitting into this framework may be obtained by taking a steel particle and letting the gravity acceleration g vary slowly by the effect of an electromagnetic field. Again, we need two switchers near the impact boundary: the first near the line y = 0, stopping the electromagnetic field and the second one, restoring it immediately before the particle hits the line y = x tan β. If these conditions are satisfied we may neglect the part of the trajectory from y = 0 to y = x tan β and assume that the impact manifold varies with time as follows: I = {(y1 − εp(t))(y1 − x1 tan β − εp(t)) = 0}, where p(t) is a periodic or almost periodic C 4 -function.

Figure 7.4: The billiard with a moving boundary. Remark 7.1. Suppose a solution hits the line y = 0 at a point x0 near the homoclinic solution with the velocity (x˙ 0 , y˙ 0 ). Then it is reflected to the solution starting ¨ = y¨ = 0, we from (x0 , 0) with the velocity (x˙ 0 , −y˙ 0 ). Since this solution satisfies x have: x(t) = x0 + tx˙ 0 ,

y(t) = −ty˙ 0 .

So the reflected solution hits the line y = x tan β at the time t such that x0 −ty˙ 0 = (x0 + tx˙ 0 ) tan β ⇔ t = − . y˙ 0 x˙ 0 + tan β

page 146

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

147

In our framework, we essentially assume that the perturbation or boundary of the billiard stops for a time duration given by − x0y˙ 0 before starting again, where x˙ 0 + tan β

x0 is the point of the x-axis hit by the ball, (x˙ 0 , y˙ 0 ) is the velocity of the ball at the hitting time. ˙ Instead of Eq. (7.21), changing y1 (t) and y2 (t) with y1 − εp(t) and y2 − εp(t), respectively, we obtain the system: ⎧ x˙ 1 = x2 , ⎪ ⎪ ⎨ y˙ 1 = y2 , (7.33) ⎪ ˙ x˙ = λ(x1 , y1 + εp(t), x2 , y2 + εp(t))f x (x1 , y1 + εp(t)), ⎪ ⎩ 2 ˙ p(t), y˙ 2 = λ(x1 , y1 + εp(t), x2 , y2 + εp(t))f y (x1 , y1 + εp(t)) − ε¨ that we rewrite below: ⎧ x˙ 1 = x2 , ⎪ ⎪ ⎨ y˙ 1 = y2 , ⎪ x ˙ = λ(x1 , y1 , x2 , y2 )fx (x1 , y1 ) + εh1 (t, x1 , y1 , x2 , y2 , ε), ⎪ ⎩ 2 p(t). y˙ 2 = λ(x1 , y1 , x2 , y2 )fy (x1 , y1 ) + εh2 (t, x1 , y1 , x2 , y2 , ε) − ε¨

(7.34)

This change of variables has the effect that the domain Ω does not change and it is as in the previous section: Ω = {(x1 , x2 , y1 , y2 ) | y1 (y1 − x1 tan β) > 0}. So the perturbed impact equation is (7.34) together with the same impact conditions as the unperturbed system y1 (t∗− ) = 0 ⇒ (x1 (t∗+ ), y1 (t∗+ ), x2 (t∗+ ), y2 (t∗+ )) = R(x1 (t∗− ), 0, x2 (t∗− ), y2 (t∗− )). ˙ and Let z = (x1 , y1 , x2 , y2 ), ζ(t) = (0, p(t), 0, p(t)) T  F (z) = x2 , y2 , λ(z)fx (x1 , y1 ), λ(z)fy (x1 , y1 ) . ˙ we obtain Taking z + εζ(t) = (x1 , y1 + εp(t), x2 , y2 + εp(t)), ⎛ ⎛ ⎞ ⎞ 0 x2 ⎜p(t) ⎜ ⎟ ⎟ y2 + εp(t) ˙ ˙ =⎜ ⎜˙ ⎟ ⎟ F (z + εζ(t)) − εζ(t) ⎝λ(z + εζ(t))fx (x1 , y1 + εp(t))⎠ − ε ⎝ 0 ⎠ . p¨(t) λ(z + εζ(t))fy (x1 , y1 + εp(t)) So that our perturbed system reads ˙ z˙ = F (z + εζ(t)) − εζ(t), that we write

Note

T  z˙ = F (z) + ε 0, 0, h1 (t, z, ε), h2 (t, z, ε) . ⎞ 0 ⎟ ⎜ 0  ˙ ⎟ ⎜ ⎝h1 (t, γ± (t), γ˙ ± (t), 0)⎠ = F (γ± (t), γ˙ ± (t))ζ(t) − ζ(t). h2 (t, γ± (t), γ˙ ± (t), 0) ⎛

(7.35)

page 147

June 8, 2017 12:9

148

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The Melnikov function associated to the perturbed problem (7.35) is then (see Eq. (7.30)): ' ( 6∞ ∗ ˙ + α) dt (t) F  (γ+ (t), γ˙ + (t))ζ(t + α) − ζ(t M(α) := 0 ψ+ ' ( 60 ∗ ˙ + α) dt + −∞ ψ− (t) F  (γ− (t), γ˙ − (t))ζ(t + α) − ζ(t 6∞ ∗ 6∞ ∗ ˙ + α)dt = 0 ψ+ (t)F  (γ+ (t), γ˙ + (t))ζ(t + α)dt − 0 ψ+ (t)ζ(t 60 60 ∗  ∗ ˙ + α)dt + −∞ ψ− (t)F (γ− (t), γ˙ − (t))ζ(t + α)dt − −∞ ψ− (t)ζ(t 6∞  6∞ ∗ ∗ ˙ = 0 (F (γ+ (t), γ˙ + (t))ψ+ (t)) ζ(t + α)dt − 0 ψ+ (t)ζ(t + α)dt 60 60 ∗ ∗ ˙ + α)dt + −∞ (F  (γ− (t), γ˙ − (t))ψ− (t)) ζ(t + α)dt − −∞ ψ− (t)ζ(t 6∞ ∗ 6∞ ∗ ˙ ˙ = − 0 ψ+ (t)ζ(t + α)dt − 0 ψ+ (t)ζ(t + α)dt 60 60 ∗ ∗ ˙ + α)dt (t)ζ(t + α)dt − −∞ ψ− (t)ζ(t − −∞ ψ˙ −   6∞ d  ∗ 60 d  ∗ = − 0 dt ψ+ (t)ζ(t + α) dt − −∞ dt ψ− (t)ζ(t + α) dt ∗ β p(α) ˙ = (ψ+ (0) − ψ− (0)) ζ(α) = (v+ − v− )p(α) ˙ = 2a√cos . a2 +b2 As a consequence, M(α) has a simple zero at some α if and only if p(α) has a non-degenerate maximum or minimum. So using Theorem 6.1 in Chapter 6, we conclude with the following theorem. Theorem 7.2. Assume that F (ρ) is a C 5 -function whose support is contained in ◦

the interval [0, r02 ] such that B((a, b), r0 ) ⊂Ω and such that F  (0) < 0 and p(t) is an almost periodic C 4 -function with a nondegenerate max or min. Then there exists ε0 > 0 such that for |ε| < ε0 the equation (7.33) exhibits chaos in a suitable neighborhood of the impact homoclinic orbit (γ(t), γ(t)). ˙ As a second example we consider the case of the periodically perturbed gravity g. So we assume g is changed with g + εp(t) and p(t) is a T -periodic C 2 function. Then emphasizing the dependence on g, the perturbed system is: ⎧ x˙ 1 = x2 , ⎪ ⎪ ⎨ y˙ 1 = y2 , (7.36) ⎪ x˙ 2 = λ(x1 , y1 , x2 , y2 , g + εp(t))fx (x1 , y1 ), ⎪ ⎩ y˙ 2 = λ(x1 , y1 , x2 , y2 , g + εp(t))fy (x1 , y1 ). So the perturbation is given by:

⎛

⎞ 0 ⎜ ⎟ 0 ⎟ ε−1 [λ(x1 , y1 , x2 , y2 , g + εp(t + α)) − λ(x1 , y1 , x2 , y2 , g)] ⎜ ⎝fx (x1 , y1 )⎠ , fy (x1 , y1 )

and its limit as ε → 0 evaluated at (γ(t), γ(t)) ˙ is: ⎛ ⎞ 0

⎜ ⎟ ∂λ 1 0 0 ⎟=− (γ± (t), γ˙ ± (t), g)p(t+α) ⎜ p(t+α). ⎝fx (γ± (t))⎠ ∂g 1 + ∇f (γ± (t))2 ∇f (γ± (t)) fy (γ± (t))

page 148

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

Now:

So:

7

149



8 d −¨ γ± (t) 0 , = γ˙ ± (t), ∇f (γ± (t)) = f (γ± (t)). γ˙ ± (t) ∇f (γ± (t)) dt



−1 0 ψ (t) p(t + α)dt 1 + ∇f (γ+ (t))2 ∇f (γ+ (t)) 0 6∞ d p(t + α) dt f (γ+ (t))dt = √ −1 2gF (0) 0

 ∞ 1 p(α)[f (γ+ (0)) − f (a, b)] + = [f (γ+ (t)) − f (a, b)]p(t ˙ + α)dt . 2gF (0) 0 

∞

∗

and similarly:

=

1 2gF (0)



0 p(t + α)dt ∇f (γ− (t)) −∞

 0 −p(α)[f (γ− (0)) − f (a, b)] + [f (γ− (t)) − f (a, b)]p(t ˙ + α)dt . 

0

ψ ∗ (t)

−1 1 + ∇f (γ− (t))2

−∞

Note f (a, b) = F (0). Since the support of f (x, y) is contained in the ball (x − a)2 + (y − b)2 ≤ r02 and γ± (0) do not belong to this ball, we get: 6∞ M(α) = √ 1 (f (γ(t)) − f (a, b))p(t ˙ + α)dt 2gF (0) −∞ 6 kT 1 limN k→∞ −kT f (γ(t))p(t ˙ + α)dt. =√ 2gF (0)

Then we see that



T −T

M(α)dα = 0

so that if M(α) ≡ 0 the equation M(α) = 0 must have a solution that generically is simple. As a consequence, using again Theorem 6.1 in Chapter 6, we get the following result. Theorem 7.3. Assume that F (ρ) is a C 5 -function whose support is contained in the ◦

interval [0, r02 ] such that B((a, b), r0 ) ⊂Ω and such that F  (0) < 0. Then given any p(t) in an open dense subset of the space of T −periodic C 2 -functions, there exists ε0 > 0 such that for |ε| < ε0 Eq. (7.36) exhibits chaos in a suitable neighborhood of the impact homoclinic orbit (γ(t), γ(t)). ˙ We continue with Example 7.1. To allow more generality we take: a = cos θ, b = ) sin θ and β *= 2θ with θ ∈ (0, π4 ). Then the condition (7.17) reads r0 < min 2 sin 2 θ, sin θ so that we take r0 = sin 2 θ. Moreover, we derive t± a,b = ta,b = r0 −tan θ √ < 0 and thus ρ+ (t) = ρ− (−t). Recall that 2F (0)g

ρ+ (t) = ρ0 (t+ (a,b) + t)

and

ρ− (t) = ρ0 (t− (a,b) − t).

page 149

June 8, 2017 12:9

ws-book961x669

150

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Hence we arrive at M(α) = 

1



0

(F (ρ− (t)) − F (0))p(t ˙ + α)dt 2gF (0) −∞

 ∞ + (F (ρ+ (t)) − F (0))p(t ˙ + α)dt 0  ∞   1 = F (ρ+ (t)2 ) − F (0) (p(t ˙ + α) + p(−t ˙ + α)) dt. 2gF (0) 0

So if p(t) is even, then M(0) = 0, and √  ∞   2 F (ρ+ (t)2 ) − F (0) p¨(t)dt M (0) =  gF (0) 0 √  ∞ (   2 ' F (ρ+ (t)2 ) − F (0) p¨(t)dt = − F (0)p(−t ˙ a,b ) + gF (0) −ta,b √  ∞ ( '   2 F (ρ0 (t)2 ) − F (0) p¨(t − ta,b )dt . = − F (0)p(−t ˙ a,b ) + gF (0) 0 Note ρ0 (t) is determined by: ρ˙ 0 = −G(ρ20 )ρ0 , with G(r) :=

ρ0 (0) = r0 ,

⎧9 (r)) ⎨ 2g(F (0)−F  2 ,

for r > 0,

⎩

for r = 0.

r(1+4rF (r) )

−2gF  (0),

and 0 < ρ0 (t) < r0 , for t > 0. Set: ¯ := max G(r), G 2 r∈[0,r0 ]

¯ := sup F (0) − F (r) > 0, K r r∈(0,r02 ]

g¯ := min2 G(r), r∈[0,r0 ]

k¯ :=

F (0) − F (r) >0 r r∈(0,r0 ] inf 2

¯

(recall that F  (0) < 0). Then clearly r0 e−Gt ≤ ρ0 (t) ≤ r0 e−¯gt , for all t ≥ 0. Now assume (7.37)

p¨(−ta,b ) > 0,

and take the smallest t0 > 0 such that p¨(t − ta,b ) > 0, for any t ∈ [0, t0 ) and 6T p¨(t0 − ta,b ) = 0. Note such t0 > 0 exists since 0 p¨(t − ta,b )dt = 0. Next, since ¯ ≤ F (0) − F (r) ≤ Kr, ¯ 0 (t)2 ≤ F (0) − F (ρ0 (t)2 ) ≤ Kρ ¯ 0 (t)2 , and then: ¯ we get kρ kr  ∞ (F (0) − F (ρ0 (t)2 ))¨ p(t − ta,b )dt 0  ∞  t0 ¯ ρ0 (t)2 p¨(t − ta,b )dt − K ρ0 (t)2 |¨ p(t − ta,b )|dt ≥ k¯ 0 t 0  ∞  t0 ¯ ¯ e−2Gt p¨(t − ta,b )dt − r02 K e−2¯gt |¨ p(t − ta,b )|dt. ≥ r02 k¯ 0

t0

page 150

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

Then it holds

√

'

2 gF (0) 6 ¯ ∞ −r02 K t0

M (0) ≤ − √

ws-book975x65

151

6 ¯ 2 ¯ t0 −2Gt p¨(t − ta,b )dt+ F (0)p(−t ˙ a,b ) + r0 k 0 e ( e−2¯gt |¨ p(t − ta,b )|dt .

Hence, in addition, if it holds   t0 ¯ 2¯ −2Gt 2 ¯ e p¨(t − ta,b )dt − r0 K F (0)p(−t ˙ a,b ) + r0 k 0

∞

e−2¯gt |¨ p(t − ta,b )|dt > 0,

t0

(7.38) then we can apply Theorem 7.3 and conclude that Eq. (7.36) exhibits chaos. For the Example 7.1, we compute: . . ¯= ¯ = 96. g¯ = 4.2335, G 21.5527, k¯ = 16, K . ˙ = sin t and Recall ta,b = −0.0739158. Finally, we take p(t) = − cos t. Then p(t) . . p¨(t) = cos t. Next, p¨(−ta,b ) = cos 0.0739158 = 0.997269 > 0 so (7.37) holds. Clearly, . t0 = π2 + ta,b = 1.49688. Then we can check that  ∞  t0 ¯ . 2¯ ¯ e−2Gt p¨(t − ta,b )dt − r02 K e−2¯gt |¨ p(t − ta,b )|dt = 0.0969 F (0)p(−t ˙ a,b ) + r0 k 0

t0

so assumption (7.38) is verified as well.

7.5

Symmetry Conditions in Finding the Melnikov Function

In this section, we suggest a direct way to obtain the function ψ(t) which is not based on the existence of first integrals. This approach is based on a careful analysis of the variational system of the first order ODEs associated with Eq. (7.3) along the solution (γ± (t), γ˙ ± (t)). It is proved that this variational equation satisfies symmetry conditions that allow us to reduce its order from the fourth to the second. Obviously, this method applies any times the fourth-order variational equation has the form (7.39) and the coefficient matrix satisfies suitable symmetry conditions (see Eq. (7.40) and (7.41)). From Eq. (7.7) we know that      g + 2F  ρ2 ρ2 θ2 + η 2 + 4ρ2 η 2 F  ρ2 λ(x1 , y1 , x2 , y2 ) = λ(ρ2 , η 2 , θ2 ) := − 2 1 + 4ρ2 F  (ρ2 ) with ρ2 = (x1 − a)2 + (y1 − b)2 , η 2 = x22 + y22 and ρ2 θ = y2 (x1 − a) − x2 (y1 − b). Then x1 −a

y2

0

−x2 ρ∇ρ = y10−b , η∇η = x02 , 2θρ∇ρ + ρ2 ∇θ = −(y1 −b) . 0

x1 −a

y2

Note that the derivatives are taken with respect to all variables (x1 , y1 , x2 , y2 ), and for simplicity, we omitted the argument (x1 , y1 , x2 , y2 ) so y2

x1 −a

−x2 ρ2 ∇θ = −(y1 −b) − 2θ y10−b . x1 −a

0

page 151

June 8, 2017 12:9

ws-book961x669

152

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Now ∇λ = 2D1 λ(ρ2 , η 2 , θ2 ) · ρ∇ρ + 2D2 λ(ρ2 , η 2 , θ2 ) · η∇η + 2D3 λ(ρ2 , η 2 , θ2 ) · θ∇θ, but on the homoclinic orbit (x1 , y1 , x2 , y2 ) = (γ+ (t), γ˙ + (t)), we have θ = ϕ˙ = 0 and x1 − a y1 − b x2 y2 = u+ , = v+ , = u+ , = v+ + + + + ρ0 (ta,b + t) ρ0 (ta,b + t) ρ˙ 0 (ta,b + t) ρ˙ 0 (ta,b + t) so ∇ρ =

u+

v+ 0 0

,

η∇η = ρ˙ 0 (t+ a,b + t)



0 0 u+ v+

,

and then

u+

' ( v+ + 2 + + t), ρ ˙ (t + t), 0 ρ (t + t) ∇λ(γ+ (t), γ˙ + (t), 0) = 2D1 λ ρ20 (t+ 0 0 a,b a,b a,b 0 0 0

' ( 0 + 2 + +2D2 λ ρ20 (t+ . u+ a,b + t), ρ˙ 0 (ta,b + t), 0 ρ˙ 0 (ta,b + t) v+

Similarly on (γ− (t), γ˙ − (t)), we have θ = ϕ˙ = 0 and x1 − a y1 − b x2 y2 = u− , = v− , = −u− , = −v− , − − − ρ0 (t− − t) ρ (t − t) ρ ˙ (t − t) ρ ˙ (t 0 a,b 0 a,b 0 a,b − t) a,b and then

u−

' ( v− − 2 − − t), ρ ˙ (t − t), 0 ρ (t − t) λ (γ− (t), γ˙ − (t), 0)∗ = 2D1 λ ρ20 (t− 0 a,b 0 a,b a,b 0 0 0

' ( 0 − 2 − . − 2D2 λ ρ20 (t− u− a,b − t), ρ˙ 0 (ta,b − t), 0 ρ˙ 0 (ta,b − t) v−

We set, for simplicity,

  ˆ = λ ρ2 (t), ρ˙ 2 (t), 0 , λ(t) 0 0   ˆ 1 (t) = D1 λ ρ2 (t), ρ˙ 2 (t), 0 , λ 0 0   ˆ 2 (t) = D2 λ ρ2 (t), ρ˙ 2 (t), 0 , λ 0

0

and ˆ ± (t) = λ(t ˆ ± ± t), λ a,b ˆ 1 (t± ± t), ˆ ± (t) = λ λ 1

a,b

ˆ 2 (t± ± t). ˆ ± (t) = λ λ 2 a,b The linear variational system along (γ± (t), γ˙ ± (t)) follows (+ is for t ≥ 0 and −, for t < 0) ⎛ ⎞ ⎛ ⎞ x1 x˙ 1

⎟ ⎜ ⎜ y˙ 1 ⎟ 0 I ⎜ y1 ⎟ , ⎜ ⎟= (7.39) ⎝ ⎝x˙ 2 ⎠ C± (t) D± (t) x2 ⎠ y˙ 2 y2

page 152

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

where

153



ˆ 2 (t± ± t)ρ˙ 0 (t± ± t) fx (γ± (t))u± fx (γ± (t))v± D± (t) = ±2λ a,b a,b fy (γ± (t))u± fy (γ± (t))v± ˆ 2 (t± ± t)ρ˙ 0 (t± ± t)f  (γ± (t))∗ (u± , v± ), = ±2λ a,b a,b

and ˆ 1 (t± ± t)ρ0 (t± ± t) C± (t) = 2λ a,b a,b



fx (γ± (t))u± fx (γ± (t))v± fy (γ± (t))u± fy (γ± (t))v±



ˆ ± ± t)Hf (γ± (t)) + λ(t a,b

ˆ 1 (t± ± t)ρ0 (t± ± t)f  (γ± (t))∗ (u± , v± ) + λ(t ˆ ± ± t)Hf (γ± (t)). = 2λ a,b a,b a,b From Eq. (7.6) we know that  2 ± f  (γ± (t))∗ = 2ρ0 (t± a,b ± t)F (ρ0 (ta,b ± t))

 u±  v±

,

and hence

2

ˆ 2 (t± ± t)ρ0 (t± ± t))ρ˙ 0 (t± ± t))F  (ρ2 (t± ± t)) u± , u± v± D± (t) = ±4λ 0 a,b 2 a,b a,b a,b u± v ± v ± 2

ˆ 2 (t± ± t) d [F (ρ2 (t± ± t))] u± , u± v± . = 2λ 0 a,b 2 a,b dt u± v ± v ±

As a consequence D± (t) = D± (t)∗

and similarly

C± (t) = C± (t)∗ .

(7.40)

Following the above, the adjoined system to (7.39) takes the form ⎛ ⎞ ⎛ ⎞ x˙ 1 x1

⎜ y˙ 1 ⎟ ⎜ y1 ⎟ 0 C (t) ± ⎜ ⎟=− ⎜ ⎟ ⎝x˙ 2 ⎠ I D± (t) ⎝x2 ⎠ y˙ 2

y2

with the impact condition (7.32) and w = (x1 , y1 , x2 , y2 )∗ . For sake of simplicity we set 2

u± u± v ± M± = , 2 u± v ± v ± ˆ 2 (t) d [F (ρ2 (t))], d(t) = λ 0 dt 2 ˆ c(t) = 4λ1 (t)ρ (t)F  (ρ2 (t)), d± (t) = c± (t) =

0 ± ±2d(ta,b ± c(t± a,b ± t),

0

t),

then: D± (t) = ±2d(t± a,b ± t)M± = d± (t)M± , ˆ ± ˆ± C± (t) = c(t± a,b ± t)M± + λ(ta,b ± t)Hf (γ± (t)) = c± (t)M± + λ (t)Hf (γ± (t)).

page 153

June 8, 2017 12:9

ws-book961x669

154

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Let us write



and subsequently

ψ1± (t) ψ2± (t)

= ψ(t) =

−1 (t)∗ X±



0 0 ±u± ±v±



⎧ ⎪ ψ˙ ± (t) = −C± (t)ψ2± (t), ⎪ ⎨ 1 ψ˙ 2± (t) = −d± (t)M± ψ2± (t) − ψ1± (t), ⎪ ⎪ ⎩ ψ ± (0) = 0, ψ ± (0) = ±  u±  . v± 1 2

Suppose, for the moment, that C± (t) and M± commute, that is (7.41)

C± (t)M± = M± C± (t), then we see that



M± ψ1± (t) M± ψ2± (t)



satisfies the same equation, and hence ψ1± (t) = M± ψ1± (t), ψ2± (t) = M± ψ2± (t), or else 

ψ1± (t),

v± −u±



= ψ2± (t),



v± −u±



= 0.

This means that to determine ψ(t) we do not need to study any four-dimensional equation but a two-dimensional equation that is the equation satisfied by the functions  u±   u±  ξ1± (t) := ψ1± (t), v± , ξ2± (t) := ψ2± (t), v±

. From the expression of C± (t) we easily see that C± (t)M± = M± C± (t) is equivalent to Hf (γ± (t))M± = M± Hf (γ± (t)). But from Eq. (7.6), we derive 

2



2

2

Hf (γ± (t)) = 2F (ρ )I + 4F (ρ )ρ



u2± u± v± 2 u± v ± v ±



= 2F  (ρ2 )I + 4F  (ρ2 )ρ2 M±

with ρ = ρ0 (t± a,b ± t), and then, Hf (γ± (t))M± = M± Hf (γ± (t)) easily follows. Note that, being  v  ± (t), −u±± = 0, ψ1,2 and, because of the orthonormality of the set {(u± , v± ), (v± , −u± )}, we get  u±  ± ± ψ1,2 . (t) = ξ1,2 (t) v±

page 154

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

155

 u±   u ±  ∗ = v± , we see that (ξ1 (t), ξ2 (t)) satisfies So using also M± = M± and M± v± the system in R2 : ' (   ⎧ ± ˙± (t) = − C± (t) u± , uv± ⎪ ξ

ξ2± (t), ⎪ 1 v± ⎨ (7.42) ξ˙2± (t) = −ξ1± (t) − d± (t)ξ2± (t), ⎪ ⎪ ⎩ ± ± ξ1 (0) = 0, ξ2 (0) = ±1. Since C± (t)

 u±   u ±   u±   u±  ˆ v± , v± = c± (t) + λ± (t) Hf (γ± (t)) v± , v±

we can rewrite equations (7.42) as below:  ⎧  ±   u±   ± ± ˆ ± (t) Hf (γ± (t)) uv± ⎪ , v± ξ2 (t), ⎨ ξ˙1 (t) = − c± (t) + λ ± ± ± (7.43) ξ˙2 (t) = −ξ1 (t) − d± (t)ξ2 (t), ⎪ ⎩ ± ξ1 (0) = 0, ξ2± (0) = ±1, or: ⎧ ˙± ξ1 (t) = − {c± (t)  ⎪  ⎪ ⎪ ⎨ ˆ ± (t) F  (ρ2 (t± ± t)) + 2F  (ρ2 (t± ± t))ρ2 (t± ± t) ξ ± (t), +2λ 0 a,b 0 a,b 0 a,b 2 ± ± ± ⎪ ˙ ⎪ ξ2 (t) = −ξ1 (t) − d± (t)ξ2 (t), ⎪ ⎩ ± ξ1 (0) = 0, ξ2± (0) = ±1. The equation adjoined to the system (7.43) (without initial conditions) is  ± η˙ 1 (t) =η2 (t),  ±   u±   ± ˆ ± (t) Hf (γ± (t)) uv± , v± η (t) + d± (t)η ± (t). η˙ 2 (t) = c± (t) + λ 1

(7.44)

2

Since (γ˙ ± (t), 㨱 (t)) is a solution of Eq. (7.39) satisfying the initial condition:  (x1 (0), y1 (0), x2 (0), y2 (0)) = (γ˙ ± (0), 㨱 (0)) = ∓ 2gF (0)(u± , v± , 0, 0),   u±   u±  

, ¨ γ± (t), v±

satisfies: we see that γ˙ ± (t), v±  η˙ 1 (t) = η2 (t),  ± ˆ ± (t) Hf (γ± (t))γ˙ ± (t), uv±

+ d± (t)η2 (t), η˙ 2 (t) = c± (t)η1 (t) + λ but since γ˙ ± (t) = γ˙ ± (t),

 u±   u±   v   v  ˙ ± (t), −u±± −u±± v± v± + γ

and

 v   u±  Hf (γ± (t)) −u±± , v±

= 0,   u±   u±   we see that γ˙ ± (t), v± , ¨ γ± (t), v± satisfies (7.44) with the initial conditions:  η1± (0) = ∓ 2gF (0), η2± (0) = 0. As a consequence, the function 1  2gF (0)



 u±   t

¨ γ± (t), v± e− 0 d± (s)ds u±  − γ˙ ± (t), v±

page 155

June 8, 2017 12:9

ws-book961x669

156

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

satisfies Eq. (7.43) (initial condition included). So  u± 

±

 1

¨ γ± (t), v± ξ1 (t) − 0t d± (s)ds e , = u±  (t), − γ ˙ ξ2± (t) v± ± 2gF (0) and then

 u± 

 u±   u± 

 1

v± ξ1± (t)  v± ¨ γ± (t), v± − 0t d± (s)ds   e = u± u±   u±  . (t), ξ2± (t) v± − γ ˙ v v± ± 2gF (0) ±



ψ(t) =

From Eq. (7.24) we see that: γ˙ ± (t) = ±ρ˙ 0 (t ± t± a,b )

 u±  v±

㨱 (t) = ρ¨0 (t ± t± a,b )

,

and hence 1

ψ(t) =  e 2gF (0)

−

t 0

d± (s)ds



 u±  v±

,

㨱 (t) . −γ˙ ± (t)

Next we have  2 2  2 ˆ 2 (t) = −2 F (ρ0 (t) ) + 2ρ0 (t) F (ρ0 (t) ) , λ 1 + 4ρ0 (t)2 F  (ρ0 (t)2 )2

and

     2 2 2  2 4ρ (t)ρ (t)F (ρ (t) ) F (ρ (t) ) + 2ρ (t) F (ρ (t) ) d 0 0 0 0 0 0 2 ˆ 2 (t) F (ρ0 (t) ) = − d(t) = λ . dt 1 + 4ρ0 (t)2 F  (ρ0 (t)2 )2 Hence 8ρ± (t)ρ± (t)F  (ρ± (t)2 )[F  (ρ± (t)2 )+2ρ± (t)2 F  (ρ± (t)2 )] 1+4ρ± (t)2 F  (ρ± (t)2 )2 2  2 2 4ρ± (t) F (ρ± (t) ) ],

d± (t) = ±2d(t± a,b ± t) = − d = − dt ln[1 +

for ρ± (t) = ρ0 (t± (a,b) ± t). This gives e−

t 0

d± (s)ds

= 1 + 4ρ± (t)2 F  (ρ± (t)2 )2 ,

and hence, using also the equality 1 + 4ρ± (t)2 F  (ρ± (t)2 )2 = 1 + ∇f (γ± (t))2 ,

1 + f  (γ± (t))2 㨱 (t)  ψ(t) = , −γ˙ ± (t) 2gF (0) that coincides with Eq. (7.29). Summarizing, the main purpose of this chapter was to introduce a new class of relatively simple chaotic impact system consisting of non-flat billiards. Thus we have studied the behavior of a particle of unitary mass moving on a Cartesian sur¯ with piecewise face z = f (x, y) in R3 and (x, y) belonging to a convex domain Ω smooth boundary. The particle was subjected to the gravity field and was reflected with respect to the normal axis when it hit the smooth part of the boundary that, in turn, is subjected to a small amplitude periodic or, more generally, almost periodic force. Due to the complexity of the problem we have considered the radially symmetric functions with compact support in the interior of Ω. In such conditions

page 156

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Impacts in Chaotic Motion of a Particle on a Non-Flat Billiard

ws-book975x65

157

we have proved the existence of a piecewise smooth homoclinic orbit for the unperturbed problem (when the boundary of Ω is frozen) consisting of three smooth parts. Since the time spent by the solutions near the middle part of the homoclinic orbit is almost the same, we have replaced the equation with an impact equation, assuming that when a solution hits ∂Ω at a certain velocity, then it is immediately sent to another point of ∂Ω with another velocity. This reflection law has been explicitly computed by studying the flow near the middle part of the homoclinic orbit. Then we used a result in Chapter 6 concerning chaotic behavior of an impact system to construct the Melnikov function for such an impact dynamical system. To clarify the result we have applied it to two concrete situations, the first, when we have a moving boundary to show the wider applicability of the result, and the second, when the gravity varies periodically. We have seen that chaotic behavior of the dynamical system appears generically in such situations and we have also studied an example with a concrete function f (x, y).

page 157

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 8

Parameter Identification of a Double Torsion Pendulum with Friction

This chapter is devoted to the parameter identification of a double torsion pendulum with kinematic forcing. A planar frictional contact is created by two oscillating bodies, i.e., the kinematically excited column and a free body placed on it — a disc sliding on its head. Angular displacement of the upper disc is caused only by the frictional forces, appearing in the contact between the disc and the pendulum column’s head that is subjected to regular excitation by means of a kinematically forced spiral spring. Having a model time series of the system’s state variables, the disturbed parameters of the pendulum are estimated with the use of Nelder–Mead simplex method.

8.1

Introduction

The sliding friction exists in the form of a resistance in a relative motion of two contacting bodies. Two kinds of the friction can be distinguished: (i) a static friction between two mutually not moving bodies; (ii) a kinetic friction between the two mutually moving bodies. It appears while a non-zero relative velocity of motion of two surface contacted bodies occurs. The static friction may have even a structural form that is related to a dissipation of energy released in the contact surface of mutually fixed bodies. The friction may be also internal being observed in solids and fluids in the form of loss of mechanical energy. The first full friction model was presented by Charles Coulomb, who showed that the static friction is not constant and pointed to the variability of kinetic friction. It is often used in engineering practice; see Sec. 1.2.1. The concept of friction can incorporate the frictional effects such as: (i) a viscous friction increasing linearly while the speed of motion increases; (ii) the Stribeck effect observed when a friction force decreases at a low speed of relative motion (called the Stribeck speed); (iii) the friction dependent on angular position in a rotational relative motion of contact surfaces; see Sec. 13.1. Steady states of a nonlinear discrete three-degree-of-freedom system containing a torsional damper are investigated in [Skup (2002)]. The system under consider159

page 159

June 8, 2017 12:9

160

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

ation is harmonically excited. The analysis takes into account both the structural and linear viscous friction of a ring floating in a plunger filled with a high density silicon oil. The influence of external loading amplitude, unit pressures, linear viscous damping, geometric parameters and amplitude-frequency characteristics is analyzed. The equations of motion of the examined power transmission system are solved by a slowly-varying parameter method and a numerical simulation. The system is related to the model introduced in this chapter in two key points, i.e.: (i) it regards to the influence of friction in the two-degree-of-freedom system; (ii) the torsion pendulum’s forced oscillations caused by a specific harmonic excitation are investigated. In [Bassan et al. (2013)], an analysis of dynamics of a simple torsion pendulum is presented. The usual basic dynamical model is investigated in the context of some unexpected features found in experimental data. Comparison with observed values yields estimates for the misalignment angles and other parameters of the model. The authors developed a more flexible model for the torsion pendulum. The basic feature of that is to consider a rigid body suspended to the fiber at an arbitrary point, and therefore, not necessarily associated to any particular symmetry of the body. Despite maximum experimental accuracy, some misalignment can occur when the fiber is fastened to the test mass. A detailed mechanical model of the torsion pendulum with geometrical imperfections can explain two unexpected features, i.e., the modulation of the torsion signal at the natural frequency of the swinging motion and the splitting of the swinging resonance. Analysis of torsion pendulums is also conducted in some experimental works. For instance, the dynamic process from a period-doubling bifurcations to chaos is observed in [Miao et al. (2014)] by changing the driving period of the modified Pohl’s torsion pendulum that formally exhibits periodic behavior. As a result of application of special data acquisition system, abundant chaotic sequence diagrams and phase diagrams can be clearly seen in real time. The system is related to the model considered in this chapter in two key points: (i) it is also kinematically forced; (ii) applying periodic forcing and the free body being in a frictional contact with the base, our model is also capable to exhibit irregular behavior, including various ways of parameter dependent bifurcation diagrams and chaotic motion. An eight-degree-of-freedom Lagrangian model that provides a suitable account for the motion of the double torsion pendulum is described in [De Marchi et al. (2013)]. The mathematical model fully governs its free dynamics, a response to external disturbances, can accurately predict the torsion swinging pendulum and bouncing resonances. The number and location of resonance peaks are correctly predicted and are used for the first validation of the model with a preliminary data. Using the Rayleigh dissipation function to account for frictional forces, the Lagrangian formalism is extended. By means of the fluctuation–dissipation theorem, dissipate effects are taken into account in order to predict the fundamental limits of sensitivity of the system. The system is related to our model in two key points: (i)

page 160

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Parameter Identification of a Double Torsion Pendulum with Friction

ws-book975x65

161

a parameter identification is performed during validation of the investigated model; (ii) estimation of the frictional forces that act in the contact interface is performed; see Sec. 8.2. In [Coullet et al. (2009)], a nonlinear oscillator that oscillates in the proximity of its supercritical bifurcation with a period inversely proportional to its angular amplitude is presented. The authors performed experiments with a Holweck–Lejay-like pendulum, which was used to measure the gravity field during the twentieth century. Main conclusions prove, that the spiral spring, the parabolic heavy body and Holweck–Lejay oscillators belong to the same class of universality as the Ising ferromagnet [Acharyya (2015)] close to their symmetry-breaking bifurcation. The Larmor’s law was confirmed experimentally with a good accuracy. The system is related to our model considered in this chapter by kinematic forcing of spiral spring. Thanks to that, the dynamics may be investigated using similar methods taken into consideration in [Coullet et al. (2009)]. The real application of torsional dynamics can be found in [Liu et al. (2014)], showing modeling and analysis of a drilling system discretized into several components with lumped inertia properties and with the inertia elements interconnected with axial and torsional springs. The work extends the model of one coupling of two mutually rotating bodies onto a set of multiple couplings modeling the dynamics of a drill string. In other fields of science like in biology or even textile metrology, the double torsion pendulum is found as the basic model for the derivation of principles governing description of dynamics of complex hybrid systems. The nonlinear dynamics of DNA relevant to the transcription process in terms of a chain of coupled pendulums was described in [Cadoni et al. (2013)]. The authors provided a simple model for a nonlinear double chain, showing some features which are quite interesting both in the frame of nonlinear dynamics for discrete systems and for applications, in particular, to DNA torsional dynamics. In [Michalak and Kruci´ nska (2004)], the influence of chemical treatment on bending and torsional rigidity of flax and hemp fibers is studied. The double torsion pendulum’s mathematical model is useful for the determination of fiber bending and torsional rigidity in investigation of a bast fiber rigidity. Concluding the above overeview, the mechanism of torsional kinematic forcing with a frictional interaction of contacting bodies is deeply explored. It is caused by a presence of interesting dynamical behaviors and a practical adequacy of such models to real rotational connections. Identification of parameters of such systems including many sources of discontinuities is still the demanding task. Therefore, the next sections take into consideration the aforementioned aspects of the investigated dynamical system including its identification. A laboratory test stand being a practical realization of the model introduced below has been investigated in [Czerwi´ nski et al. (2015)].

page 161

June 8, 2017 12:9

162

8.2

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Mathematical Modeling

The investigated double torsion pendulum with friction is an extension of a single kinematically forced torsion pendulum. Comparing to the general models of torsion pendulums, the construction has been inverted so that a free body was placed on the pendulum column’s head. Free motion dynamics of the body (a disc) will depend on the column’s dynamics and the friction forces acting tangentially to the contact surface of the two interacting bodies. This implies that the resulting frictional coupling with the two movable bodies establishes the two-degree-of-freedom mechanical system with dry friction. In Fig. 8.1, we have introduced two variables that constitute the vector of generalized coordinates q¯ = [ϕ1 , ϕ2 ]T , where the angular displacement of lower disc is denoted by ϕ1 [rad], and relatively, angular displacement of the upper disc (a free body) is denoted by ϕ2 [rad]. Zero position of the lower disc is determined by the neutral position of the kinematically forced end of the spiral spring, which is attached to the disc. Kinetic energy in rotational motion of the double torsion pendulum 1 1 (8.1) T = B1 ϕ˙ 21 + B2 (ϕ˙ 1 + ϕ˙ 2 )2 , 2 2 where the mass moment of inertia of the lower disc is B1 , and the upper, B2 [kg·m2 ]. upper disc (free body) lower disc (forced base)

m2, B2

m1, B1


2
1

contact surface

kinematic forcing

spiral spring

Figure 8.1: Physical model of the double torsion pendulum. The potential energy 1 (8.2) k(fe (t) − ϕ1 )2 , 2 where k [N·m/rad] is the spiral spring’s stiffness. The kinematic sinusoidal forcing fe (t) (external excitation) of free end of the spiral spring V =

fe (t) = A sin (ωt), where ω [rad/s] is the angular frequency, A [rad] — amplitude of the forcing.

(8.3)

page 162

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Parameter Identification of a Double Torsion Pendulum with Friction

¯ of generalized forces can be written as follows: The vector Q     τ1 −ML + MT ¯ Q= = , τ2 −MT

ws-book975x65

163

(8.4)

where ML [N·m] is the frictional resistance torque of the sliding bearing in which the lower disc is placed, MT [N·m] — the frictional resistance torque between both pendulum bodies of inertia B1 and B2 . We assume that the frictional resistance torque of the sliding bearing depends on unknown viscous friction and the Coulomb friction expressed by the maximum nska static friction torque M1 , acting in the contact zone [Michalak and Kruci´ (2004)]. Moreover, the Coulomb friction model of discontinuity in a form of the dry friction is smoothed by the function arctan, approximating the function sgn (ϕ˙ 1 ) of sign of the angular velocity ϕ˙ 1 of the lower disc. We have 2 (8.5) ML = c1 ϕ˙ 1 + M1 arctan (ε1 ϕ˙ 1 ), π where c1 [N·m·s/rad] is the coefficient of viscous friction, M1 [N·m] — maximum torque of the static friction on the contact surface, ε1 [s] — a parameter determining accuracy of smoothing of the static friction torque (a Coulomb term), acting on the sliding bearing’s contact surface. The larger the value of ε1 is the closer the arctan function approximates the sgn function (see Sec. 4.2.2), thus, the frictional nonlinearity caused by the discontinuity of Coulomb friction. For a further use in the text, the term “contact surface” will denote a contact zone in the frictional coupling between the pendulum’s bodies of inertia B1 and B2 . One takes into account more frictional effects existing on the contact surface between both interacting bodies. The frictional resistance torque between both pendulum bodies depends on the viscous friction Tv ϕ˙ 2 , smoothed dependency for the Coulomb’s dry friction Ts1 2/π arctan(ε2 ϕ˙ 2 ), the free body’s angular position dependent frictional torque Ts2 (1 − sgn |ϕ˙ 2 |) and the Stribeck effect that is characterized by a Stribeck curve of an additional frictional torque that may be observed on the contact surface, i.e., Tst (1 − exp(−T0 |ϕ˙ 2 |)) sgn (ϕ˙ 2 ). Therefore, concatenating the possible forms of frictional discontinuities, one obtains MT = Tv ϕ˙ 2 + Ts1 π2 arctan(ε2 ϕ˙ 2 ) + Ts2 (1 − sgn |ϕ˙ 2 |) + Tst (1 − e−T0 |ϕ˙ 2 | ) sgn (ϕ˙ 2 ),

(8.6)

where Tv [N·m·s/rad] is the viscous friction coefficient on contact surface, Ts1 , Ts2 [N·m] — the maximum static friction torques, Tst [N·m] — the static friction torque associated with the exponential curve of Stribeck effect, T0 [s] — a parameter of exponential curve, ε2 [s] — a parameter of static torque. Expressing the kinetic and potential energies in the system of generalized coordinates, the Lagrange function L is defined as the difference of the kinetic energy T and the potential energy V . We have L = T − V,

(8.7)

page 163

June 8, 2017 12:9

164

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

which after substituting Eq. (8.1) and (8.2), yields  1 (8.8) B1 ϕ˙ 21 + B2 (ϕ˙ 1 + ϕ˙ 2 )2 − k(fe (t) − ϕ1 )2 . L= 2 Then for the function L, the Lagrange equation follows

d ∂L ∂L = Qi , (8.9) − dt ∂ q˙i ∂qi ¯ is the vector of generalized forces, and qi is the ith generalized coordinate. where Q The Lagrange equation for the coordinate ϕ1 is given by

d ∂L ∂L = τ1 . (8.10) − dt ∂ ϕ˙ 1 ∂ϕ1 Considering only the left-hand sides of Eq. (8.10) and (8.8), one obtains

∂L d ∂L = B1 ϕ¨1 + B2 (ϕ¨1 + ϕ¨2 ) − k(fe (t) − ϕ1 ). − dt ∂ ϕ˙ 1 ∂ϕ1 The Lagrange equation for the coordinate ϕ2 is given as well

d ∂L ∂L = τ2 . − dt ∂ ϕ˙ 2 ∂ϕ2 Combining the left-hand sides of Eq. (8.12) and (8.8), we write

d ∂L ∂L = B2 (ϕ¨1 + ϕ¨2 ). − dt ∂ ϕ˙ 2 ∂ϕ2

(8.11)

(8.12)

(8.13)

Substituting the generalized forces τ1 and τ2 , defined in Eq. (8.4), into Eq. (8.10) and (8.12), and taking into account (8.11) and (8.13), the two ordinary differential equations of second-order describing the dynamics of the double torsion pendulum with kinematic forcing are found:  B1 ϕ¨1 + B2 (ϕ¨1 + ϕ¨2 ) − k(fe (t) − ϕ1 ) = −Ml + MT , (8.14) B2 (ϕ¨1 + ϕ¨2 ) = −MT . Finally, the double torsion pendulum with a frictional coupling is represented by a two-degree-of-freedom discontinuous dynamical system mathematically modeled by the two second order ODEs:  B1 ϕ¨1 − k(fe (t) − ϕ1 ) = −Ml + 2MT , (8.15) B2 (ϕ¨1 + ϕ¨2 ) = −MT . 8.3

Identification of Parameters

Prior to the process of identification, the objective function was assumed as the arithmetic average of squares of the differences between the “measured” (actual) and estimated angles of rotation of both bodies. One takes this function #N % N $ $ 2 2 (8.16) y= (ϕ1 − ϕˆ1 ) + (ϕ2 − ϕˆ2 ) /2N, i=1

i=1

page 164

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Parameter Identification of a Double Torsion Pendulum with Friction

ws-book975x65

165

where ϕ1 and ϕ2 are the actual angles of rotation, ϕˆ1 and ϕˆ2 are the estimates, N is the number of samples in the time series of a numerical solution. To identify the unknown parameters of the pendulum the Nelder–Mead simplex method for finding the local minimum of a function of several variables is used. It allows to determine the local minimum without the necessity of using the derivatives so it can be applied to any function not differentiable at a point. Exemplary references extending the problem of finding unknown parameters with the use of the Nelder–Mead method can be found in [Luersen and Le Riche (2004)]. In the identification process the known parameters are assumed: ω, ε1 , ε2 , while the unknowns to be declared with initial values are as follows: k, A, B1 , B2 , c1 , M1 , Tv , Ts1 , Ts2 , Tst , T0 . 8.4

Application of the Identification Procedure

An exemplary simulation allowing us to test the procedure of a parameter identification of the system can be carried out after assuming an ideal solution obtained from numerical integration of the model (8.15). a)

b)

Figure 8.2: Time histories of the vector of angular displacements ϕ, ¯ velocities ϕ¯˙ and accelerations ϕ¨ ¯ of (a) the kinematically forced lower disc and (b) the free body. The identification procedure is preceded by an assumption of the ideal parameters of the model: ω = 0.5, ε1 = ε2 = 1000, B1 = 20, B2 = 1, k = 100, A = π/6, c1 = 10, M1 = 3, Tv = 1, Ts1 = Ts2 = 2, Tst = 0.5, T0 = 10. After computations the trajectories of state variables shown in Fig. 8.2 were obtained. In the next step, the set of model parameters will be slightly changed in relation to the original ones assumed above, i.e., we take the disturbed set of parameters of the model: B1 = 22, B2 = 11, k = 10, A = π/4, c1 = 5, M1 = 2, Tv = 0.1, Ts1 = 3, Ts2 = 1, Tst = 0.2, T0 = 1000.

page 165

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

166

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

a)

b)

c)

d)

Figure 8.3: Time histories of actual angular displacement ϕ¯ and their estimate ϕˆ¯ of forced lower disc (a, c) and the free body (b, d), after a few steps (a, b) of the identification procedure. Overlapping of red and blue trajectories at the end of the process (c, d) ensures good estimation of the identified model parameters. After a few steps of identification of the pendulum parameters, the time trajectories shown in Fig. 8.3(a) and (b) are significantly divergent. Although the discontinuous effects caused by friction strongly affect the process, after several thousand iterations a set of system parameters guaranteeing the desired coincidence of the trajectories shown in Fig. 8.3(c) and (d) was obtained. As it is shown in Fig. 8.3(c) and (d), the used procedure guarantees good identification of the test model with a frictional discontinuity and kinematic forcing. The introduced mathematical model of the double torsion pendulum with a spiral spring and complex frictional discontinuity modeled by Eq. (8.6) has allowed one to simulate the dynamics of the mechanical system as well as permitted one to identify its parameters with the use of selected method of parameter identification.

page 166

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 9

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

An asymmetric pendulum being a part of the two-degree-of-freedom mechanical system with friction introduced in Sec. 1.4.1 is subject to identification with respect to the observed influence of some resistance of its rotational motion in ball bearings. It is damped in a complex manner what could be considered as a nonlinear damping. There is supposed between others, that the effective nonlinear damping characteristics depends on a few effects such as fluid friction caused by vibrations of the pendulum with two springs in the air as well as unknown kinds of a frictional resistance existing in ball bearings. The model under investigation finds its real realization on a laboratory test stand designed for experimental investigations of viscous and structural frictional effects. A transient response oscillations of the pendulum are described by the explicitly state-dependent free decay. A free decay test of the pendulum with the state-dependent nonlinear parameters of damping and stiffness is described in this chapter. It provides interesting observations that lead to elaboration of an identification method of the overall damping and stiffness coefficients. Effects of application of the proposed semi-empirical method of identification of the coefficients are illustrated and discussed.

9.1

Introduction

Nonlinear dynamical models are commonly applied to assess the properties and to predict the performance of miscellaneous engineering objects. Dynamic modeling of a mechanical system with nonlinear strain frequency dependent damping is carried out in [Yuan (2013)]. Information about nonlinear damping of a damping alloy specimen from the free decay signal by means of the moving autoregressive model method has been extracted. The viscoelastic theory is introduced to describe the strain-frequency characteristics of damping more accurately, and a viscoelastic three parameter structural damping constitution model is developed whose parameters are identified from the test data by means of an optimization algorithm. The finite element dynamic equations for strain-frequency-dependent damping are derived through the established three parameters constitution. The 167

page 167

June 8, 2017 12:9

168

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

established element dynamic equations are assembled into the system dynamic equations of an elastic linkage mechanism by means of the kineto-elastodynamic theory, and a closed-form numerical algorithm is constructed in order to solve the high-order differential equations with time-varying coefficients. In the design process, it is essential to use accurate numerical simulation tools to predict the complex aero-hydro-servo-elastic response of a floating wind turbine [Koh et al. (2013)]. Cited paper focuses on the use of the open-water test data of the SWAY prototype wind turbine to calibrate a floating offshore wind turbine numerical model for future validation efforts. After turbine deployment and installation of the NREL instrumentation, a few free decay tests were conducted on the SWAY prototype by displacing the system and allowing it to return to an equilibrium. The inability to model frictional damping in the universal joints of the system contributes to discrepancies between measured and simulated results. The inability to model frictional damping in the universal joints in the tension rod became significant in affecting the overall motion of the system. In offshore engineering and naval architecture it is common practice to determine damping coefficients, both linear and nonlinear, from free decay tests [Faltinsen (2010)]. For example, in ocean engineering, it is common practice to obtain damping coefficients of floating structures from free decay tests [Asmuth et al. (2015)]. The paper presents interesting work on the determination of nonlinear damping coefficients for flap-type oscillating wave surge converters from free decay tests. Simulations of the free decay tests in the computational fluid dynamics are presented as well as their validation against experimental results is performed. Analysis of the obtained data reveals that linear damping, as commonly used in time domain models, is not able to accurately model the occurring damping over the whole regime of rotation amplitudes. In a conclusion, a hyperbolic function is most suitable to express the instantaneous damping ratio over the rotation amplitude. A nonlinear inverse method of nonparametric identification is proposed in [Jang (2015)] for the following: (i) recovering the full nonlinear damping; (ii) restoring functions in a harmonic forced nonlinear oscillator. For that, a proper inverse problem and its mathematical formalism are developed by introducing the intersection and zero-crossing times with respect to motion response, based on the acceleration response measurements. The identification does not depend on the usual regularization, which is generally essential to the usual (ill-posed) inverse problems arising in mathematical science and engineering. As a model equation, a highly nonlinear system is examined for the workability of the inverse method proposed through numerical experiments. For vibration serviceability calculations, the damping value of the structural systems is a critical parameter. The amplitude-dependent damping behavior of the laboratory footbridge investigated in [Avci (2016)] is subjected to different amplitudes of sinusoidal excitation. The amplitude-dependent damping ratio values obtained from effective mass calculations proved to be correct with the FE model

page 168

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

169

acceleration predictions. The FE model predictions successfully matched the test results with the nonlinear characteristics introduced for modal damping. An equivalent linearization method is adopted in [Gao et al. (2013)] to model the added damping and stiffness effect via a free decay response in zero wind speed condition. It is found that the nonlinear characteristics of added damping effect are rather obvious, while that of added stiffness effects is considerably weak. These examples and many other problems reported in [Butterworth et al. (2004); Eret and Meskell (2008); Cruciat and Ghindea (2012); Duda et al. (2011)] after measurement of stiffness or damping, confirm the necessity of deeper investigations on the techniques related to identification of parameters of vibrating bodies. It becomes especially important for the investigation of a vibrating physical systems, where among others, viscous, Coulomb, Rayleigh or modal damping is necessary to determine [Butterworth et al. (2004); Meskell (2006, 2011); Mottershead and Stanway (1986); Nakamura (2016)]. Various functions of nonlinear damping are incorported in more or less advanced mathematical models of dynamical systems. Among these applications, one distinguishes: (i) a nonlinear damped wave equation with Dirichlet boundary condition [Meng et al. (2016)]; (ii) a one-degree-of-freedom (1-DOF) passive vibrations isolation system with geometrically nonlinear damping [Carranza et al. (2015)]; (iii) dynamical model of bearing supported with a nonlinear damping force and a linear elastic restoring force [Yan et al. (2014)]; (iv) clamped beam with tip mass coupled to nonlinear spring and damper [Mileti´c et al. (2014)]; (v) an extension of Caughey’s linear damping models to a nonlinear class [Le Gorrec et al. (2015)]; (vi) generalized Rayleigh dissipation function, including isotropic, nonlinear damping in the Euler–Lagrange equations [Joubert et al. (2013)]; (vii) a novel causal damping model capable of expanding the constant frequency area [Nakamura (2016)]. This chapter is focused mainly on the use of experimental measurement data in identification of a nonlinear damping coefficient characteristics during free vibrations of a single parametric pendulum with friction [Awrejcewicz and Lamarque (2003)]. The angle bracket — a single pendulum, a part of the 2-DOF mechanical system with friction, shown in Fig. 1.21 (see Sec. 1.4.3), is damped in a complex manner. It could be considered as a time-varying damping of some nonlinear characteristics, that by an assumption, depends on angular displacement and velocity of the pendulum rotating about the pivot point s; see Fig. 1.18. It was observed during many experimental tests performed in [Olejnik (2002); Pilipchuk et al. (2015)], that the unknown characteristics of damping of the parametric pendulum does noticeably influence the dynamics of the entire mechatronic system. One assumes in the presented study an existence of an effective nonlinear damping characteristics which can be dependent on the system states affected by the phenomena:

page 169

June 8, 2017 12:9

ws-book961x669

170

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

(P1) Buoyancy and viscous resistance of the air wherein vibrations of the pendulum of an irregular shape 1 with the two linear springs attached to it occur; see Fig. 1.21. (P2) Other unknown resistances introduced by ball bearings (viscous damping, Coulomb damping, etc.) in which both ends of the aluminum shaft 2 are mounted to allow the pendulum to rotate about the pivot point s. The phenomena (P1) and (P2) are the most important, since most of the energy loss in a free swinging pendulum is due to air friction, and respectively, due to the commonly interacting rolling and sliding friction in the bearings. In a free decay test, an existence of a nonlinear explicitly state-dependent characteristics of damping and stiffness of the parametric pendulum will be proven. To get knowledge about an approximate characteristics of that parameters, the block 3 has been fixed onto the base and a free decay experimental response (see Fig. 9.1) of the pendulum 1, shown in Fig. 1.21 (see Sec. 1.4.3), was analyzed using a semiempirical identification method presented in the subsequent sections. Originally, self-excited vibrations of the block 3, sliding on the moving base, cause some irregularly forced response of the pendulum [Olejnik and Awrejcewicz (2013b)]. The pendulum 1 is coupled with the block by means of two springs, and therefore, it changes the normal and tangent contact forces on the frictional creep– slip contact surface of the block sliding on the moving base. From this point of view, a precise mathematical modeling, tending to describe the dynamical behavior of the pendulum 1, is very important in the context of identification of the static and kinetic friction forces acting on the contact surface in the block-on-belt model [Olejnik and Awrejcewicz (2014)]. 0.4 0.3

ym (t) [cm]

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4 0.0

0.2

0.4

0.6

t [s]

0.8

1.0

1.2

1.4

Figure 9.1: Time history of a transient response of the pendulum in a free decay test. The time series ym (t) has been acquired from measurements on the laboratory test stand; see Fig. 1.21.

page 170

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

9.2

171

Estimation of the Nonlinear Characteristics of Damping and Stiffness

It would be interesting to check if the damping and stiffness coefficients of the investigated free decay vibrations are constant. At small enough angles of rotation, the starting equation in our research follows m¨ y (t) + cy(t) ˙ + ky(t) = 0,

for y(t) ≈ rϕ(t) and max{ϕ(t)} < π/36,

(9.1)

where y [m] is the linear displacement of the pendulum, ϕ [rad] — angle of rotation, m = J/r2 [kg] — a virtual mass, J = 2.4423·10−4 kg·m2 — mass moment of inertia, r = 0.078 m — length of arm, c → c˜ϕ (t) [N·s/m] — unknown overall damping, ˜ k → k(t) = k˜ϕ (t) + k1 + k2 = k˜ϕ (t) + 145.82 N/m — unknown overall stiffness, k1 , k2 [N/m] — constant stiffness coefficients estimated from static characteristics of both elastic elements (linear springs), k˜ϕ (t) [N/m] — unknown implicitly statedependent function of stiffness of the rotational connection (in Fig. 1.18, see the pivot point s which is created by the ball bearings and the pendulum mounted in the bearings), and c˜ϕ (t) — unknown implicitly state-dependent function of overall coefficient of damping in both symmetrically situated rotational joints. Note that in Fig. 1.18 the block on the base has been stopped and a free response of the asymmetric pendulum is analyzed. Equation (9.1) can be represented in a classic form of a non-forced and damped harmonic oscillator ω ˙ + ω 2 y(t) = 0, y¨(t) + y(t) (9.2) Q where the damping component at y˙ is written in a canonical form by means of the quality factor Q — a dimensionless parameter of strength of viscous friction in motion of the pendulum [Asmuth et al. (2015)]. Our oscillator has small mass and is fairly small damped so the factor Q is defined by 2πE/|ΔE|, where E is the energy of oscillation, ΔE is the energy loss per cycle of the oscillation because of dissipation [Peters (2007)]. The dissipation is expressed either in terms of the dimensionless quality factor Q or by the damping ratio δ which has the dimension of frequency. Then by a definition ω ˙ 2δ y(t) ˙ = y(t). (9.3) Q Applying in Eq. (9.3) the definition of the period T = 2π/ω between time instances of every two adjacent turning point amplitudes A(i) and A(i + 1), one finds, that the constant logarithmic decrease of damping, called the damping ratio δ, is given by A(i) π 1 = . δ = ln T A(i + 1) Q If to check the successive periods’ duration of the free decay oscillations shown in Fig. 9.1, then all points are irregularly distributed as shown in Fig. 9.4. Therefore,

page 171

June 8, 2017 12:9

ws-book961x669

172

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

some usage of any constant damping ratio of the real free response of our single pendulum is not well justified. As a consequence, constant damping ratio δ is to be ˜ which will exhibit a nonlinear characteristics dependent on the replaced with δ(t) time going in the free decay test. 700 600

|Ω|

500 400 300 200 100 0 0

10.9

25

50

f [Hz]

75

100

Figure 9.2: A discrete FFT of the time series ym (t) of the pendulum’s linear displacement acquired from the measurement of the angle ϕ on the test stand. One dominant frequency is found as marked by the dashed line. Analyzing in Fig. 9.2, the frequency spectrum of vibrations from the experimental time series ym (t) of our single parametric pendulum, one observes, that the body vibrates at the dominant frequency fF ≈ 10.9 Hz, corresponding to ωF ≈ 68.45 rad/s. Let us now select two adjacent turning point amplitudes separated by one period T of motion, i.e., A(1) = 0.421, A(2) = 0.371 cm. Then let us calculate the angular frequency ω with respect to the period T related to the time elapsed between two successive measurements of peak amplitudes A1 and A2 . We get the constant angular frequency ω, the period of vibrations T and a damping ratio δ as shown below: A1 1 2π = 67.630 rad/s, T = tA2 − tA1 = 0.093 s, δ = ln = 1.481 1/s. ω= T T A2 Comparable values of both angular frequencies, i.e., ω ≈ ωF are confirmed so after that simple calculus, the exact transient response ye (t) [cm] of the oscillator with an exponential decay δe (t) = A1 exp (−δt),

(9.4)

and the constant angular frequency ω takes the analytical form ye (t) = δe (t) cos (ωt) = A1 exp (−δt) cos (ωt) = 0.421 exp (−1.481t) cos (67.63t). The solution ye (t), given in Eq. (9.5), is drawn in Fig. 9.3 using a red line.

(9.5)

page 172

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

0.4 Tm

ym

δ˜f

ye

173

δe

0.3 0.2

y(t) [cm]

0.1 0.0 −0.1 −0.2 −0.3 −0.4 0

0.2

0.4

0.6

0.8

1.0

1.2

t [s]

Figure 9.3: A comparison of two time histories confirming discrepancy between the experimental trajectory ym (t), and correspondingly, the analytical solution ye (t). . We see in Fig. 9.3 that the obtained estimate solution ye (t) bounded by the line of the exponential decay (9.4), passing through a turning point amplitudes of the solution, does not coincide with the experimental trajectory ym (t) (black line) representing the measurement. Drawing a conclusion, the estimated parameters of the investigated transient response, given by Eq. (9.5), are not valid at each successive constant period T . Therefore, the highest inaccuracy in the coincidence of both compared time trajectories is visible at the end of the pendulum’s free decay oscillations, i.e., at the final time tk = 1.4 s, when ye (t) should much closely tend to zero. The procedure of searching for the new approximating function δ˜f (t) — a statedependent polynomial decay which replaces the standard exponential decay with a constant damping ratio as well as for an approximation of ω ˜ (t), resulting from T˜(t), is proposed below. Tilde over the symbols make them distinguishable from constants ω, T and the function δe (t). 9.2.1

Estimation of the Polynomial Decay

The polynomial decay is an important function in a context of the initiated dynamical analysis and the frictional phenomena observed on a stick–slip contact surface of the block-on-belt model investigated on the test stand. Moreover, it has a significant influence on evaluation of damping properties of the pendulum, vibrating at significant velocity variations as it was observed during experiments [Awrejcewicz and Olejnik (2005c, 2007b); Pilipchuk et al. (2015)].

page 173

June 8, 2017 12:9

ws-book961x669

174

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The quality factor introduced in Eq. (9.2) is expressed by [Peters (2007)] π # %. Q(t) = a 3 (9.6) T˜(t) a1 δ˜f (t) + a2 + δ˜f (t) We obtain the first sought approximation in a form of the polynomial decay (see the black doted line in Figs. 9.3 and 9.7) a2 (p(t) − 1) + r(p(t) + 1) , δ˜f (t) = (9.7) 2a1 (1 − p(t))  1 A(1)+a2 −r where p(t) = 2a a22 − 4a1 a3 , A(1) = 0.421, the numeri2a1 A(1)+a2 +r exp(−rt), r = cally estimated a1 = −3.45, a2 = 2.52, a3 = 0.05, and δ˜f (t) is the polynomial decay of oscillations. 9.2.2

Estimation of the Variable Angular Frequency

Now, we search for an estimation of the second system parameter that is directly connected with the unknown variable stiffness coefficient, i.e., ω ˜ (t) — the implicitly state-dependent angular frequency of motion. As it is shown in Fig. 9.4, the polynomial function T˜(t) of period instances can be obtained from a continuous approximation of the measurement points Tm (i) = tA(i) − tA(i+1) , for i = 0, . . . , 16, with the use of the following polynomial of third degree: T˜(t) = γ3 t3 + γ2 t2 + γ1 t + γ0

for γ¯(3...0) = [−2, 28, −147, 9385] × 10−5 .

(9.8)

We have observed that fitting of the trajectory ym (t) at its final stage, for t ∈ [1.2, 1.4] s, regarded to the low-amplitude vibrations of the pendulum could be more precise. Therefore, the free response’s approximation was checked for a substitution of the polynomial approximation ω ˜ (t) for a logarithmic term ω ˜ l (t) (see Fig. 9.5), that reads ˜ (t) + b2 log ω ˜ (t) + b3 ω ˜ l (t) = b1 ω

for b(1...3) = [2.57, 0.13, 93] × 10−3 .

Figure 9.5 illustrates a time-dependent function ω ˜ (t) that results from the poly˜ nomial approximation T (t), given by Eq. (9.8), with its logarithmic fit ω ˜ l (t) for the purpose of improvement of the stage of low-amplitude vibrations of the pendulum. After checking the resulting effectiveness of approximations ω ˜ (t) and ω ˜ l (t), the first one has been selected. If δ˜f (t), given by Eq. (9.7), states for the desired approximation of turning point amplitudes of free decay vibrations of the analyzed pendulum, then using the obtained polynomial approximation (9.8), an implicitly state-dependent angular frequency reads 2π , for t ∈ [0, tk ]. ω ˜ (t) = (9.9) T˜(t)

page 174

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

175

0.095

Tm (i), T˜(i) [s]

0.090 0.085 0.080 0.075 0.070 0.065

T˜

Tm

0.060 0

2

4

6

8

10

12

14

i

0.20

0.25

0.30

0.35

0.40

(a)

T˜(δf ), T˜f (δf )

0.0940 0.0935

T˜ T˜f

0.0930 0.0925 0.0920 0.00

0.05

0.10

0.15

δf (t)

(b)

Figure 9.4: (a) Non-smooth distribution of periods Tm (i) (green circles) calculated between the time tA(i) and tA(i+1) of appearance of the successive peak amplitudes A(i) and A(i + 1) of oscillations in the experimentally acquired series ym (t) versus ith cycle number; (b) the expected similarity of the dependencies T˜f (δf ) and T˜(δf ). Third degree polynomial approximation T˜(t) of the distribution is matched by solid line. 9.2.3

Estimation of the Time-Varying Stiffness and Damping

This section takes into account the obtained estimates (9.6) and (9.9) to provide some definitions of parameters of the investigated dynamical system. Comparison of terms in Eq. (9.1), at the state variables y and y˙ divided by m, and in Eq. (9.2), yields: ˜ ω ˜ (t) k(t) c˜ϕ (t) = and =ω ˜ 2 (t), m Q(t) m and after rearrangement: m˜ ω (t) ˜ = m˜ and k(t) ω 2 (t). c˜ϕ (t) = (9.10) Q(t)

page 175

June 8, 2017 12:9

ws-book961x669

176

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

68.2 68.0

ω ˜ ω ˜l

ω ˜ (t) [rad/s]

67.8 67.6 67.4 67.2 67.0 66.8 0.0

0.2

0.4

0.6

t [s]

0.8

1.0

1.2

1.4

Figure 9.5: A polynomial approximation of the angular frequency ω ˜ (t) (black line) and its logarithmic fit ω ˜ l (t) (red line). The nonlinear functions of system parameters, i.e., the variable damping ˜ have been drawn in coefficient c˜ϕ (t) and the variable stiffness coefficient k(t) Fig. 9.6. These parameters are denoted as functions of time, since it is valid only for the time going in the free decay test. Therefore, the parameters will be implicitly state-dependent when one will need to apply them in the simulation of dynamics of full mechanical system shown in Fig. 1.21, being composed of the block-on-belt subsystem and the identified pendulum forced by motion of the block. ˜ It is worth reminding that by obtaining in Eq. (9.10), the stiffness coefficient k(t), we have identified the overall stiffness of the pendulum (see Sec. 9.2), that includes constant components k1 and k2 of springs’ stiffness. The unknown at the beginning the state-dependent stiffness k˜ϕ (t) of the rotational connection of the pendulum at ˜ − k1 − k 2 . the suspension point s (see Fig. 1.18) is found at k˜ϕ (t) = k(t) 9.3

The Nonlinear Approximations — Verification Cases

The first case. Verification of accurateness of the analytical formula (see blue line in Fig. 9.7) ˜ (t)t), yf (t) = δ˜f (t) cos (w1 ω where δ˜f (t) is defined by the formula (9.7), ω ˜ (t) is defined by Eq. (9.9), and w1 = 1.02 is a non-dimensional fitting parameter at the term of angular frequency. The second case. The identified nonlinear approximations (9.7) and (9.9), re˜ (t), are put into the numerical model of the analyzed spectively, for δ˜f (t) and ω single parametric pendulum to check accurateness of the numerical solution ys (t) in comparison to the measurement series ym (t). A state-space representation of the single pendulum dynamics described by one

page 176

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

c˜ϕ (t) [N s/m]

2.0

187

c˜ϕ k˜

186 185 184

1.5

183 1.0

182

˜ [N/m] k(t)

2.5

177

181

0.5

180

0.0 0

0.2

0.4

0.6

0.8

1.0

1.2

179 t [s]

Figure 9.6: Nonlinear functions of system parameters: (a) the variable damping ˜ (solid line). coefficient c˜ϕ (t) (dashed line); (b) the variable stiffness coefficient k(t) second order ODE (9.2) follows:  y˙ 1 (t) = y2 (t), ω ˜ (t) y˙ 2 (t) = − Q(t) y2 (t) − w22 ω ˜ 2 (t)y1 (t).

(9.11)

Solving numerically the system (9.11) we obtain a numerical solution ys (t) = y1 (t) — a linear displacement of the pendulum of which time history shown in Fig. 9.7 is drawn by red line. Also here, a non-dimensional fitting parameter of angular frequency, i.e., w2 = 1.029 is applied in the numerical model to obtain the computed time history ys (t) better fit to the experimental counterpart ym (t). The third case. Average displacement of the pendulum in ith period of oscillations of the time trajectories y(t) shown in Fig. 9.7 can be taken into account in the qualitative assessment of the obtained approximations. The average displacements y¯(i) (bar over the symbol y denotes an average value of the time series), in the ith cycle period T˜(i), have been numerically computed for the three time series ym (t), ys (t) and yf (t) by means of the formula: y¯(i) =

1 ˜ T (i)



T˜ (i)

y(t)dt 0

for i = 1, . . . , 16 and y(t) = {ym (t), ys (t), yf (t)}.

If we take a look at the averaged series y¯s and y¯f drawn in Fig. 9.8, then, interchangeably, the average y¯s or y¯f is closer to y¯m . As it is seen, the real measurement series ym (t) is irregular in the time duration of each period of its 16 oscillations. One would find the best result of our identification, if the averaged series y¯s or y¯f could as much as possible coincide with the averaged measurement y¯m . Hereby, a qualitative method of assessment of the presented identification of a single real pendulum’s parameters has been proposed. Due to a time-dependency of parameters of the pendulum it has been verified to be the parametric one.

page 177

June 8, 2017 12:9

ws-book961x669

178

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

0.4 z1

0.3

yf

ym

z1 (t), y(t) [cm]

0.2 0.1 0.0 −0.1 −0.2 −0.3 −0.4

0

0.2

0.4

0.6

0.8

1.0

t [s]

(a) t ∈ [0, tk ]

0.08 ym ys yf δ˜f

0.06

y(t) [cm]

0.04 0.02 0.00 −0.02 −0.04 −0.06 0.8

0.9

1.0

1.1

1.2

1.3

t [s]

(b) t ∈ [0.8, tk ]

Figure 9.7: Time histories of a transient response of the pendulum in the free decay test: ym (t) — measurement (thick grey line), ys (t) — a numerical solution (red line), yf (t) — an analytical solution (blue line) that takes into account the identified parameters of the pendulum.

The high degree of coincidence of trajectories ym (t), yf (t) and ys (t), presented in Fig. 9.7, proves that the two system parameters such as damping c and stiffness k, introduced at the beginning in Eq. (9.1), have to be dependent on the pendulum’s state variables. For that requirement, the two implicitly state-dependent parameters ˜ are identified. c˜ϕ (t) and k(t) The nonlinear function c˜ϕ (t) drawn in Fig. 9.6 has an important property. Up to about 1 s of the free decay response, the pendulum’s damping depends almost linearly on time. In rough approximation, it can be assumed as constant. In Fig. 9.3, the moment of time points to the amplitude of about 0.05 cm. One can use the linear piece of function c˜ϕ (t) while the oscillator’s angular velocity is high enough. When the vibration body exhibits some irregular dynamics by reaching

page 178

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Identification of Time-Varying Damping of a Parametric Pendulum with Friction

179

0.006 0.004 0.002 y¯(i) [cm]

0.000 −0.002 −0.004 −0.006

y¯m y¯s y¯f

−0.008 −0.010 −0.012 0

0.2

0.4

0.6

0.8

1.0

1.2

t [s]

Figure 9.8: Average displacements of the pendulum in i-th cycle period of oscillations for the measurement ym (t) in relation to the approximates ys (t) and yf (t). low velocity regimes of motion, then the whole nonlinear characteristics shown in Fig. 9.6 has to be used. The identified parameters of the irregularly damped single pendulum’s motion have significant influence on dynamics of the entire dynamical system which additionally includes the block-on-belt model. For the practical use of the identified functions of both system parameters, it is necessary to use a transition from the phase space of the free decay to the phase space of full system dynamics, including a self-excited vibrations of the block on the moving base.

page 179

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 10

Almost Periodic Solutions for Jumping Discontinuous Systems

In this chapter an existence result for almost periodic sequences of ordinary differential equations with linear boundary value conditions is derived by means of the Banach fixed point theorem together with a method of majorant functions. An application is given to a damped pendulum with a jumping length and external force.

10.1

Introduction

Let us consider the motion of the damped mathematical pendulum with changing length l = l(t) [Hatv´ ani and Stach´ o (1998)] and the external force e = e(t) given by l(t)φ¨ + cφ˙ + sin φ = e(t).

(10.1)

We suppose that l(t), e(t) are step functions. The stability of linear ordinary differential equations with step function coefficients was studied in [Elbert (1997b,a); Graef and Karsai (1997); Hatv´ani (1998); Hatv´ ani and Stach´o (1998)]. In this chapter, we assume that l(t), e(t) have certain almost periodicity in the following sense that there are sequences: {tn }n∈Z ⊂ R,

{lk }k∈Z ⊂ C,

such that: tn = nT +

$

Tk eıωk n

{ek }k∈Z ⊂ C, ∀n ∈ Z,

{Tk }k∈Z ⊂ C,

T > 0,

k∈Z

|Tk | < T /2,

k∈Z

and, for any tn < t < tn+1 , we have $ ek eıωk n , e(t) =

l(t) =

k∈Z

where:

$

{ωk }k∈Z ⊂ R

$

$

lk eıωk n ,

k∈Z

|ek | < ∞,

k∈Z

$

|lk | < ∞.

k∈Z

Consequently, we suppose that l(t), e(t) are step functions with almost periodic jumping. We are interested in finding conditions on l(t), e(t), c that Eq. (10.1) 181

page 181

June 8, 2017 12:9

ws-book961x669

182

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

has a bounded solution on R with the same almost periodic properties as l(t), e(t). To handle this problem, in Sec. 10.2, we study a sequence of ordinary differential equations with linear boundary value conditions. We rewrite this sequence into an ordinary differential equation on a certain Banach space with a linear boundary value condition. We solve this boundary value problem by using the Banach fixed point theorem together with a method of majorant functions. We are motivated in applying this approach by the books [Stoker (1950); Strizhak (1984)], where continuous almost periodic ordinary differential equations are widely studied. In Sec. 10.3, we apply results of Sec. 10.2 to Eq. (10.1). We consider, for simplicity, a concrete form of Eq. (10.1). Finally, we note that our approach can be directly modified for investigation of the existence of bounded solutions to almost periodic difference equations. For instance, let us consider the difference equation √ (10.2) xn+2 + axn+1 + xn = bx3n + d1 cos n 2 + d2 sin 3n, n ∈ Z, where a ∈ R, |a| > 2 and b, d1 , d2 ∈ R. It can be shown that if  2  3 27|b| |d1 | + |d2 | < 4 |a| − 2 , then Eq. (10.2) has a solution of the form √ $ $ 3 |d1 | + |d2 | . xn = zkp eı(k 2+3p)n , |zkp | ≤ 2 |a| − 2 k,p∈Z

k,p∈Z

We refer to [Dibl´ık et al. (2013)] for more detail about this subject.

10.2

Almost Periodic Solutions

Let {ωk }k∈Z be a sequence of real numbers such that pωk1 + qωk2 ∈ {ωk }k∈Z

∀p, q, k1 , k2 ∈ Z , ωk = ωl

whenever

k = l.

Let us consider a sequence of ordinary differential equations: x˙ n = A(n)xn + f (n, xn , t) + h(n, t), xn+1 (0) = E(n)xn (1), t ∈ [0, 1], n ∈ Z, where: $ $ Ck eıωk n , h(n, t) = Dk (t) eıωk n , A(n) = f (n, x, t) =

$$

k∈Z

k∈Z

Bjk (t) e

ıωk n j

x ,

j≥1 k∈Z

Dk ∈ C([0, 1], Cm ), E(n) =

$

D=

$

Ck ∈ L(C ),

k∈Z

Bjk ∈ C([0, 1], Cm ),

|Dk | < ∞,

Ek ∈ L(C ),

$'$ j≥1

k∈Z

$

||C || < ∞,

∈Z

|Dk | = max |Dk (t)|,

k∈Z

Ek eıωk n ,

C=

(10.3)

E=

( |Bjk | Rj < ∞,

$

[0,1]

||E || < ∞,

∈Z

|Bjk | = max |Bjk (t)|, [0,1]

page 182

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Almost Periodic Solutions for Jumping Discontinuous Systems

for some constant R > 0. We put Ω(r) =

# $ $ j≥1

ws-book975x65

183

% |Bjk | rj ,

k∈Z

for 0 ≤ r < R. We are interested in almost periodic solutions of Eq. (10.3) of the form $ ak (t) eıωk n , xn (t) = (10.4) k∈Z

such that: max [0,1]

Let

$

|ak (t)| < ∞,

max [0,1]

k∈Z

 l1 = {ak }k∈Z

" " "

a k ∈ Cm ,

$

|a˙ k (t)| < ∞.

k∈Z

|a| =

$

 |ak | < ∞ .

k∈Z

We intend to rewrite Eq. (10.3) as an ordinary differential equation on l1 with a linear boundary value condition. We have: '$ ($ $ $ $ A(n) ak eıωk n = Cj eıωj n ak eıωk n = Cj ak eıωl n . j∈Z

k∈Z

Moreover, we get: $ "" " l∈Z

$

" $ " Cj ak " ≤

ωj +ωk =ωl

l∈Z ωj +ωk =ωl

k∈Z

$

||Cj || · |ak | ≤ C|a|.

l∈Z ωj +ωk =ωl

Consequently, {A(n)}n∈Z generates a bounded linear mapping L : l1 → l1 given by  $  Cj ak . L : {ak }k∈Z → ωj +ωk =ωl

l∈Z

Note that ||L|| ≤ C. Similarly, we can check that the sequences {f (n, x, t)}n∈Z and {h(n, t)}n∈Z generate mappings: F : BR × [0, 1] → l1 ,

H : [0, 1] → l1 ,

respectively, where BR = {a ∈ l1 | |a| < R}. Moreover, F is analytic in a ∈ BR , F and H are continuous. Indeed, we calculate '$ (j $ $ $$ Bjk (t) eıωk n ap eıωp n = Bjk (t)ap1 · · · apj eıωl n . j≥1 k∈Z

p∈Z

l∈Z ωp1 +···+ωpj +ωk =ωl

Moreover, for a ∈ l1 , |a| < R, we get $ $ "" " l∈Z

ωp1 +ωp2 +···+ωpj +ωk =ωl

" " Bjk (t)ap1 ap2 · · · apj " ≤ Ω(|a|).

page 183

June 8, 2017 12:9

ws-book961x669

184

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Consequently, we arrive at the formulas: $ F (a, t)l =

Bjk (t)ap1 ap2 · · · apj ,

∀l ∈ Z,

ωp1 +ωp2 +···+ωpj +ωk =ωl

∀l ∈ Z.

H(t)l = Dl (t),

The above computations also give: |F (a, t)| ≤ Ω(|a|),

|H(t)| ≤ D.

Furthermore, for v = {vk }k∈Z ∈ l1 , we have ' 5 (Da F (a, t)v)l = ωp +···+ωp +ωk =ωl Bjk (t) vp1 ap2 · · · apj

1 j ( + ap1 vp2 ap3 · · · apj + · · · + ap1 · · · apj−1 vpj .

Hence

" $ " "Da F (a, t)v " ≤

' |Bjk | |vp1 | · |ap2 | · · · |apj |

$

l∈Z ωp1 +···+ωpj +ωk =ωl

( +|ap1 | · |vp2 | · |ap3 | · · · |apj | + · · · + |ap1 | · · · |apj−1 | · |vpj | ≤ Ω (|a|)|v|. The boundary value condition of Eq. (10.3) gives: '$ ( $ $ Ek eıωk n ap (1) eıωp n = al (0) eıωl eıωl n , p∈Z

k∈Z

e

$

−ıωl

l∈Z

Ek ap (1) = al (0)

∀l ∈ Z.

ωk +ωp =ωl

Next, a linear continuous mapping M : l1 → l1 is introduced $ (M a)k = e−ıωk Ep a q . ωp +ωq =ωk

We have ||M || ≤ E. Clearly, the boundary value condition of Eq. (10.3) is expressed by a(0) = M a(1). Summarizing, we arrive at the following result. Theorem 10.1. Problem (10.3) generates a boundary value problem on l1 given by: a˙ = La + F (a, t) + H(t),

a(0) = M a(1),

where L, F , H, M are defined above. Let us solve Eq. (10.5). The variation of constant formula for (10.5) gives t   Lt eL(t−s) F (a(s), s) + H(s) ds. a(t) = e a(0) + 0

The boundary value condition of Eq. (10.5) implies 1 L

M e a(0) + M 0

  eL(1−s) F (a(s), s) + H(s) ds = a(0).

(10.5)

page 184

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Almost Periodic Solutions for Jumping Discontinuous Systems

185

By assuming that I − M eL : l1 → l1 ,

(10.6)

is continuously invertible, we get 1 a(0) = KM

  eL(1−s) F (a(s), s) + H(s) ds,

K = (I − eL )−1 .

0

Hence, Eq. (10.5) is rewritten in the form 1 Lt

a(t) = e KM

  eL(1−s) F (a(s), s) + H(s) ds

0

t +

e

(10.7)

 L(t−s)



F (a(s), s) + H(s) ds.

0

We can prove the main result of this section. Theorem 10.2. Assume the mapping (10.6). Let us put ' ( eC − 1 Φ(r) = eC ||K||E + 1 (Ω(r) + D) C for r, 0 ≤ r < R. If there is a r0 , for 0 < r0 < R, such that Φ(r0 ) ≤ r0 and Φ (r0 ) < 1, then Eq. (10.3) has an almost periodic solution which can be found by an iterative method. Proof. One solves Eq. (10.7) on X = C([0, 1], l1 ) by using the Banach fixed point theorem to the mapping G : X → X given by G(x)(t) = eLt KM +

6t

61

  eL(1−s) F (a(s), s) + H(s) ds

0

e

L(t−s)



 F (a(s), s) + H(s) ds

0

for x ∈ X, |x| < R. The above computations imply ( C ' |G(x)| ≤ eC ||K||E + 1 e C−1 (Ω(|x|) + D), ' ( C ||DG(x)|| ≤ eC ||K||E + 1 e C−1 Ω (|x|). Hence, we arrive at: |G(x)| ≤ Φ(|x|),

||DG(x)|| ≤ Φ (|x|).

On the ball Br0 = {x ∈ X | |x| ≤ r0 }, we have: |G(x)| ≤ Φ(r0 ) ≤ r0 ,

||DG(x)|| ≤ Φ (r0 ) < 1.

The Banach fixed point theorem gives a unique solution of x = G(x) on Br0 . The proof is finished.

page 185

June 8, 2017 12:9

ws-book961x669

186

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The main difficulty in applying Theorem 10.1 is the verification of mapping  L (10.6), which is equivalent to 1 ∈ / σ M e . Here σ denotes the spectrum. Lemma 10.1. Assume: If e

ıωk



∈ / σ Se

A



A(n) = A,

E(n) = S

∀n ∈ Z.

∀k ∈ Z and

 −1 || < ∞, m0 = sup || S eA − eıωk I

(10.8)

k∈Z

then the mapping (10.6) is satisfied along with ||K|| ≤ m0 . Proof. We have If eıωk

    I − M eL a k = I − e−ıωk S eA ak .   −1  ∈ / σ S eA , then I − e−ıωk S eA exists and we can put −1  −ıωk S eA ak ∀k ∈ Z. (Ka)k = I − e 

Hence |Ku| ≤

$  −1 || eıωk I − S eA || · |ak | ≤ m0 |a|. k∈Z

The proof is finished. Lemma 10.2. Assume A(n) = A + A1 (n), where A1 (n) =

$

Ck1 eıωk n ,

E(n) = S + E1 (n)

Ck1 ∈ L(C ),

C∞ =

E1 (n) =

$

||C ∞ || < ∞,

∈Z

k∈Z

$

∀n ∈ Z,

Ek1 e

ıωk n

,



Ek1 ∈ L(C ),

E∞ =

$

||E ∞ || < ∞.

∈Z

k∈Z

If Eq. (10.8) holds along with   ||S||C1 + E1 e||A||+C1 m0 < 1, then Eq. (10.6) holds with ||K|| ≤ m0

:'

(10.9)

(   1 − ||S||C1 + E1 e||A||+C1 m0 .

(10.10)

Proof. Now we have M = M0 +M1 , L = L0 +L1 , where like above M0 , M1 , L0 , L1 correspond to S, E1 (·), A, A1 (·), respectively. Moreover, we get ||M0 || ≤ ||S||,

||M1 || ≤ E1 ,

||L0 || ≤ ||A||,

||L1 || ≤ C1 .

We compute

  I − M eL = I − M0 eL0 + M0 eL0 − eL0 +L1 − M1 eL0 +L1 .

Since

  ||M0 eL0 − eL0 +L1 || ≤ ||S|| e||A||+C1 C1 ,

||M1 eL0 +L1 || ≤ E1 e||A||+C1 ,

the Neumann theorem implies the assertion of Lemma 10.2.

page 186

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Almost Periodic Solutions for Jumping Discontinuous Systems

10.3

187

A Damped Pendulum With a Jumping Length and External Force

Let us consider a pendulum given by ln φ¨ + cφ˙ + sin φ = ξ + γ sin ω3 n, tn = n + α sin ω1 n < t < n + 1 + α sin ω1 (n + 1) = tn+1 , ln = l + β sin ω2 n, ˙ = ln lim φ(t) ˙ lim φ(t) = lim φ(t), ln−1 lim φ(t) ∀n ∈ Z, t→tn −

t→tn +

t→tn −

t→tn +

(10.11) where c > 0, l > 0, ω1 , ω2 , ω3 ∈ R \ {Zπ}, α, β, γ, ξ ∈ R, and " " "α sin ω1 " < 1/2, |β| < l. 2 Here φ is the angle between the axis directed vertically downward and the thread. Equation (10.11) describes the motion of the pendulum with a sudden almost periodic change of the length together with an external force at any t = tn , n ∈ Z. For any t, tn ≤ t ≤ tn+1 , we put: t = stn+1 + (1 − s)tn ,

s ∈ [0, 1],

φn (s) = φ(t).

Then Eq. (10.11) becomes φ˙ n = −δψn − β0 φn , dn 1 cn β0 hn sin φn − (φn − sin φn ) + φn − , ψ˙ n = δφn − α0 ψn + cn ψn + δ lδ δ δ (10.12) β0 φn+1 (0) = φn (1), ψn+1 (0) = ψn (1) + gn ψn (1) + gn φn (1), δ where  1 1 c2  δ 2 = max , − 2 , 4l √l+1 l c − c2 + 4δ 2 l2 − 4l , α0 = 2l √ c + c2 + 4δ 2 l2 − 4l , β0 = 2l ω1 2n + 1 sin β sin ω2 n − 2αl cos ω1 2 2 , cn = c l(l + β sin ω2 n) (10.13) 2n + 1 2n + 1 ω1 ω1 sin + 4α2 l cos 2 ω1 sin 2 − β sin ω2 n 4αl cos ω1 2 2 2 2 , dn = l(l + β sin ω2 n) ω1 2n + 1 sin )2 (ξ + γ sin ω3 n) (1 + 2α cos ω1 2 2 hn = , l + β sin ω2 n 2n + 3 ω1 cos ω1 )(l + β sin ω2 n) (1 + 2α sin 2 2 gn = − 1. 2n + 1 ω1 cos ω1 )(l + β sin ω2 (n + 1)) (1 + 2α sin 2 2

page 187

June 8, 2017 12:9

ws-book961x669

188

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

We assume that ω1 , ω2 , ω3 are incommensurable, i.e., it holds (10.14) pω1 + qω2 + rω3 = 0 ∀(p, q, r) ∈ Z3 \ {(0, 0, 0)}. * ) We take {ωk }k∈Z = pω1 + qω2 + rω3 (p,q,r)∈Z3 . Formula (10.12) has the form (10.3) with

−β0 −δ A(n) = A = , h(n, t) = (0, −hn /δ) , δ −α0 ' ( dn 1 c n β0 f (n, φ, ψ, t) = 0, cn ψ + φ+ sin φ − (φ − sin φ) , δ δ lδ



10 0 0 S= . , E(n) = S + E1 (n), E1 (n) = 01 β0 gn /δ gn  Considering on C2 the norm |(φ, ψ)| = |φ|2 + |ψ|2 , yields: 1 ıωn 1 e − e−ıωn , 2ı 2ı 2n + j 1 1 cos ω = eıωj/2 eıωn + e−ıωj/2 e−ıωn , 2 2 2 sin ωn =

j = 1, 3.

Hence ω1 | 2 =Γ , |cn | ≤ c 1 l(l − |β|) ω1 ω1 + |β| 4l|α sin | + 4α2 l sin 2 2 2 = Γ2 , |dn | ≤ l(l − |β|) ω1 2 (1 + 2|α sin |) 2 = Γ3 , |hn | ≤ (|ξ| + |γ|) l − |β| ω1 ω1 4l|α sin | + 4|αβ sin | + 2|β| 2 2 = Γ4 . |gn | ≤ ω1 (1 − 2|α sin |)(l − |β|) 2 In this case, as a majorant function for Ω(r) from Sec. 10.2, we take ' 1 1 β0 ( θ(r) = 1 + Γ1 r + Γ2 sinh r + (sinh r − r). δ δ lδ In context of Lemma 10.2, it is not hard to see that: ( 'β 0 + 1 Γ4 = Γ5 , ||S|| = 1, E1 ≤ 9 δ |β| + 2l|α sin

C1 = 0,

||A|| ≤

α02 + β02 + 2δ 2 = Γ6 .

Furthermore, since || eA || ≤ e−2α0 for the relation (10.8), we get ( ;' m0 ≤ 1 1 − e−2α0 . Condition (10.9) is satisfied if the following holds Γ5 eΓ6 < 1 − e−2α0 .

(10.15)

page 188

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Almost Periodic Solutions for Jumping Discontinuous Systems

189

According to the relation (10.10) of Lemma 10.2, we get ( ;' ||K|| ≤ 1 1 − e−2α0 − Γ5 eΓ6 . We see, that in Theorem 10.2, the function Φ can be of the form ' ( e Γ6 − 1 ' 1 Γ3 ( Φ(r) = eΓ6 (1 + Γ ) + 1 θ(r) + 5 1 − e−2α0 − Γ5 eΓ6 Γ6 δ = τ1 r + τ2 sinh r + τ3 (sinh r − r) + τ4 , where:

' β0 ( τ1 = τ 1 + Γ1 , τ2 = τ Γ2 /δ, τ3 = τ /(lδ), τ4 = τ Γ3 /δ, δ ' ( e Γ6 − 1 1 τ = e Γ6 (1 + Γ ) + 1 . 5 1 − e−2α0 − Γ5 eΓ6 Γ6

If τ1 + τ2 < 1,

(10.16)

then there is a unique r0 > 0 such that τ1 + τ2 cosh r0 + τ3 (cosh r0 − 1) = 1,

(10.17)

and for any τ4 satisfying τ4 < r0 − τ1 r0 − τ2 sinh r0 − τ3 (sinh r0 − r0 ),

(10.18)

a unique r1 , 0 < r1 < r0 exists, such that Φ(r1 ) = r1 and Φ (r1 ) < 1. We note that such r1 is determined by the equation τ1 r1 + τ2 sinh r1 + τ3 (sinh r1 − r1 ) + τ4 = r1 .

(10.19)

Consequently, for such r1 , the conditions of Theorem 10.2 are satisfied. Summarizing our derivations, we arrive at the following result. Theorem 10.3. Let Eqs. (10.14), (10.15) and (10.16) be satisfied and r0 > 0 be given by Eq. (10.17). If (10.18) holds, then Eq. (10.11) possesses an almost periodic solution x of the form (10.4) such that supR |x(·)| ≤ r1 , where r1 , 0 < r1 < r0 , is given by Eq. (10.19). We note that τ1 = τ2 = 0 and Γ5 = 0 if α = β = 0, and then, Eqs. (10.15) and (10.16) are trivially satisfied.

page 189

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 11

Solution of Nonlinear Algebraic Equations in Analysis of Stability

In this chapter a gradient method for solution of nonlinear algebraic equations in the analysis of stability of equilibria of a two-degree-of-freedom system with dry friction is applied. General theoretical considerations are introduced, and afterwards, the numerically estimated diagram of stability is subject to a qualitative assessment.

11.1

Introduction

Let us consider a system of n differential equations x ¯˙ = f¯(¯ x),

(11.1)

¯(t) of where x ∈ Rn and f¯ is a smooth vector field. We can say that the trajectory x the system (11.1) is in an equilibrium state if it does not depend on time. It follows that the coordinates of the equilibrium state can be found as the solution of the system of algebraic equations f¯(¯ x0 ) = 0.

(11.2)

In the general case, the system (11.1) has only a finite number of equilibrium ¯ states in any bounded region. Moreover, if the Jacobian matrix ∂∂ fx¯ is non-singular at a point x ¯0 , then the equilibrium state is isolated [Li and Zhi (2014)]. From the point of view of numerical investigations the determination of all solutions of the system (11.2) states the relatively simple task for small n and for simple linear right-hand sides of Eq. (11.1). However, the number of equilibrium states of a strongly nonlinear system of a higher dimension may be very large, and searching for all of them becomes problematic [Shilnikov et al. (1998)]. Therefore, useful gradient method for solving the system of nonlinear equations is presented and then applied to solve a system of two algebraic equations corresponding to a twodegree-of-freedom mechanical system with dry friction. 191

page 191

June 8, 2017 12:9

ws-book961x669

192

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

11.2

The Gradient Method

Let a system of n nonlinear algebraic equations has the form x) = 0, fk (¯

(11.3)

k = 1, . . . , n,

¯ = [x1 , . . . , xk , . . . , xn ]. where each fk is a real scalar function of the vector x Let the following function be defined G(¯ x) =

n $

fk2 (¯ x).

(11.4)

k=1

All solutions to the system (11.3) are zero values of the function G(¯ x) and vice versa. If to assume that all solutions of the system are isolated [Li and Zhi (2014)], then the problem leads to calculation of minimum of the real function G(¯ x) with the variables xk , for k = 1, . . . , n. To indicate minimum of function (11.4), a fastest decrease (gradient) method can be applied. In this method, in the successive (i + 1) iteration of the solution, for i = 1, . . . , N , the fastest decrease correction P¯i is used: x ¯i+1 = x ¯i + P¯i , where



(11.5) 

∂G(¯ xi ) (11.6) . ∂x ¯ When in each ith iteration ∇G(¯ xi ) = 0 and the parameter αi satisfy the condition xi ) P¯i = −αi ∇G(¯

and

∇G(¯ xi ) =

∂G(¯ xi ) ∂x1 ,

...,

∂G(¯ xi ) ∂xn

=

min {αi : G(¯ xi+1 ) < G(¯ xi )} ,

(11.7)

then the algorithm of gradient approximation of successive solutions takes the form ¯i − αi ∇G(¯ xi ). x ¯i+1 = x

(11.8)

The directional minimization with respect to the parameter αi is obtained using the scalar function n $ fk2 (¯ xi+1 ). G(¯ xi+1 ) = (11.9) k=1

Basing on Eq. (11.7) the parameter αi must minimize the introduced scalar function, i.e., d (11.10) G(¯ xi+1 ) = 0. dα To find the dependency for αi Eq. (11.10) is solved by means of the linear approximation:   ∂fk (¯ xi ) ∇G(¯ xi ), ¯i + P¯i ≈ fk (¯ xi+1 ) = fk x xi ) − α i fk (¯ ∂x ¯

(11.11)

page 192

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Solution of Nonlinear Algebraic Equations in Analysis of Stability

ws-book975x65

193

  ∂fk ∂fk k where the directional vector ∂f ∂x ¯ = ∂x1 , . . . , ∂xn . In a result, taking into assumption Eq. (11.4), the equation for the unknown parameter αi yields n $   −2 fk (¯ xi ) − αi Sjn Sjn = 0, (11.12) k=1

where: Sjn = ∂G(¯ xi ) ∂ = ∂xj ∂xj

#

n $ ∂fk (¯ xi ) ∂G(¯ xi ) j=1

n $ k=1

∂xj %

fk2 (¯ xi )

∂xj =2

n $

,

fk (¯ xi )

k=1

xi ) ∂fk (¯ . ∂xj

Solution of Eq. (11.12) gives the ith iteration for αi as follows: 5n xi )Sjn k=1 fk (¯ αi = 5 n n 2 . k=1 (Sj )

(11.13)

The described method will be used in next section to assess stability of equilibrium states of a two-degree-of-freedom system with dry friction.

11.3

Analysis of Stability of Equilibrium States

The analysis will be performed for the discontinuous dynamical system with dry friction shown in Fig. 1.18; see Sec. 1.4.1. There are applied some modifications of the physical model: (i) elastic element of the stiffness k1 and the dashpot of damping c2 are removed; (ii) the elastic element of the stiffness k2 becomes a nonlinear one of the stiffness k1 x1 + k3 x31 ; (iii) the vertical elastic element of stiffness k3 is now denoted by k2 . Non-dimensional equations of motion of the analyzed 2-DOF model are as follows: ⎧ ⎪ ⎪ x˙ 1 = x2 , ⎪ ⎨ x˙ = 1 x − x + e x3 − x3  + e (x − x ) + T  , 2 3 1 3 6 4 2 1 3 e1 (11.14) ⎪ x˙ 3 = x4 , ⎪    ⎪ ⎩ x˙ = 1 x − (e + 1)x + e x3 − x3 + e (x − x ) , 4 1 4 3 3 6 2 4 3 1 e3 where the dry friction model of decaying characteristics is used ⎧ ⎨− sgn (vr ) 1 + e2 x3 , vr = 0, 1 + γ|vr | T (vr ) = ⎩ vr = 0, 1 + e 2 x3 ,

(11.15)

and vr = x2 − vp is the relative velocity, vp is the velocity of motion of the base, as well as other nondimensional parameters are defined:    1  δmgμ0 mω 2 k2 (a − b)μ0 Jω 2 k3 (mgμ0 )2 c1 ω , J = M a 2 + b2 . , γ= √ e¯ = 3 k1 ak1 k1 a2 k13 k1 k1 m

page 193

June 8, 2017 12:9

194

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The positive parameter δ measures the rate at which the kinetic coefficient of friction decreases with an increase in |vr | [m/s]. The remaining notations are used: m [kg] — mass of the block, M [kg] — mass of the angle body, ω [rad/s] — natural frequency of oscillations of the block, k1 , k2 [N/m] — coefficients of stiffness at linear components, k3 [N/m] – a coefficient of stiffness at nonlinear component, c1 [Ns/m] — damping coefficient, μ0 [Ns/m] — static friction coefficient, a, b [m] — dimensions of the angle body. In next step, the equilibrium states will be found numerically using the aforementioned gradient method. According to the general form (11.2), satisfying the requirement of stability of vanishing of all first-order derivatives of state variables, the system of differential equations (11.14) is transformed to the system of algebraic equations: ⎧ ⎪ x02 = 0, ⎪ ⎪ ⎪  3  x 1 + e ⎪ 2 03 ⎨ x03 − x01 + e5 x − x3 − = 0, 01 03 1 + γvr (11.16) ⎪ x04 = 0, ⎪ ⎪ ⎪   ⎪ ⎩ x01 − x03 − e4 x03 + e5 x303 − x301 = 0. Observe that we have to solve only two equations dependent on the variables x01 and x03 . Starting from various initial conditions, the gradient method will converge to solutions of the system (11.16).

11.4

Stable and Unstable Branches on the Diagram of Stability

The procedure of examining stability of the system (11.1) near an equilibrium state is based on the standard linearization procedure. The stability of the equilibrium states is determined by the Jacobian matrix ∂ f¯(¯ x0 ) (11.17) , J= ∂x ¯ of which eigenvalues are the characteristic exponents of the characteristic equation det |J − λI| = 0.

(11.18)

The equilibrium state x ¯0 is stable if all its characteristic exponents have negative real parts. Solving numerically the system of two non-trivial algebraic equations 2 and 4 in the system (11.16), and afterwards, checking the characteristic exponents of the obtained equilibrium states with respect to changes in vr (velocity of motion of the base), the stable and unstable branches on the x01 (vr ) diagram of stability of the analyzed dynamical system have been found and shown in Fig. 11.1. It can be seen in Fig. 11.1 that branches of equilibria are horizontally symmetric along the axis x01 = 0, but are not symmetric in the vertical direction along the axis of zero relative velocity. It is caused by the mutually asymmetric configuration of the two bodies of the 2-DOF mechanical system (cf. Chapter 9).

page 194

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Solution of Nonlinear Algebraic Equations in Analysis of Stability

3u

2u

5 1s

-0.7

-2 2s

195

1u

-0.36

-3

ws-book975x65

-1 1

-0.07

-5

1

2 4u

-15

Figure 11.1: Coordinate x01 of the estimated equilibrium points versus the relative velocity vr (1s–2s, 1u–4u denote branches of stable and unstable equiblirium points, respectively). Due to non-dimensionality of the system the plot variables are also non-dimensional. There are two branches of stable equilibria: 1s, 2s; and four branches of the unstable branches 1u–4u. Almost always three equilibrium states for the positive as well as negative relative velocity of motion of the two contacting bodies exist. Near vr = −0.36 only two unstable equilibrium states are confirmed. It means, that by maintaining this relative velocity of motion, the block will oscillate on the base between two unstable equilibria. There is also seen that these stable branches 1s and 2s are horizontally symmetric to the branches 1u and 2u, respectively. The next conclusion is that for a relative velocity being not greater than about −0.7 (in the regime of vr < −0.7) and not less than −0.07 (in the regime of vr > −0.07) the block of mass m can oscillate in the neighborhood of a stable point, since at a continuously varying velocity and acting external forces it can even approach some permissible regions of two unstable equilibrium states on the base. Summarizing our study, the gradient method for solving the nonlinear algebraic equations (11.16) has been successfully applied, and then, stability branches of the estimated system equilibria have been subject to basic examination. In practice, the gradient method as an associated method for any dynamical analysis of mechanical systems is easy to apply. The equilibrium points have been found with a satisfactorily high accuracy, since the bifurcation branches 1u–1s and 2u–2s showing both stable and unstable equilibrium states have been determined. There is shown in this chapter, that due to the asymmetric configuration of the dynamical self-excited system with dry friction, the vertical axis of symmetry of the diagram of stability is translated to the left from zero relative velocity. Another well-known method based on finding a Lyapunov function in the analysis of stability of a naturally unstable 2-DOF system is carried out in Sec. 12.5.2.

page 195

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 12

Control of a Wheeled Double Inverted Pendulum with Friction

The problem of control and dynamical modeling of a wheeled double inverted pendulum with rolling friction vibrating in a plane perpendicular to the direction of movement is taken into consideration in this chapter. The object of analysis consists of two basic parts, i.e., the wheel and the double pendulum. The equations of motion have been derived using the Lagrange equation of second kind. The kinematic excitation has been applied to the upper link. The aim of control is to maintain the upper link in an unstable equilibrium point around given angular position. Control moment of force has been applied to the wheel in a numerical procedure utilizing the Kalman filtering approach.

12.1

Introduction

The study is devoted to a problem of modeling and control of a dynamical system consisting of a single wheel vehicle (a single-track unicycle) and a cyclist’s body. The unicycle is a specific simplification of bicycle, because it consists only of one wheel and a seat on which a driver operates to keep balance and to drive forward or backward. The unicycle is a child of an original bicycle having the driving big wheel and a small wheel that was only the rolling one helping the driver in keeping the vehicle’s direction of movement. Basic feature of the monocycle’s construction is that it looks like a bicycle wheel with a hub designed so that the axle is a fixed part of the hub. Therefore, the rotation of the cranks directly controls the rotation of the wheel. The cranks are attached to the ends of the axle so pedals always rotate during riding the unicycle. The direct connection between the axle and the crack is not a rule and some ratio between their rotations may exist. Riding the unicycle is not easy. It is caused by single tracking of the vehicle that requires to keep balance of the system simultaneously in two planes. Moreover, to ride comfortably the distance between the saddle and the lowest pedal position has to be smaller than length of the cyclist’s leg. As a consequence, the center of gravity of the cyclist’s body lies a slightly higher than his normal upright position. 197

page 197

June 8, 2017 12:9

198

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

To keep balance in plane that is parallel to the direction of forward or backward movement, the cyclist has to speed up or slow down the driving wheel to perpendicularly maintain center of gravity of his body above the axle of rotation of the wheel. To keep balance in the plane that is transverse to the direction of riding, the cyclist has to balance his body from the left to the right side with the use of his loins. Construction and mechanics of an unicycle allows one to name it as the double inverted pendulum. Physical model of the system is shown in Fig. 12.1. The first link is created by the cyclist’s body, and the second link by the fork frame stiffly joined with the seat post (a link between the frame and saddle). The wheel states the third link so a kind of triple pendulum could be also assumed. The system shown in Fig. 12.1 has three degrees of freedom and to control it one would need to apply a control moment of force applied to the driving wheel or in the joint created by the rotational connection between the second and third link.

Figure 12.1: Physical model of an inverted pendulum with rolling friction. When a cyclist’s body loses balance while riding the unicycle, then his membranous labyrinth senses it and an error of regulation of the unstable equilibrium appears. Brain functions here as a regulator that receives the error signal and accordingly to ability of its learned neural network, it produces the appropriate control signals that through the nervous system cause desired reaction of muscles (actua-

page 198

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Control of a Wheeled Double Inverted Pendulum with Friction

199

tors). Reaction of muscles enables the cyclist to correct his body’s position. Following this principle, a cyclist driving an unicycle could be approximately analyzed as a follow-up control system shown in Fig. 12.2. disturbance input signals

object of control

controller

output signals

sensors

Figure 12.2: A follow-up control system of a cyclist driving an unicycle.

12.2

Overview of Exemplary Inverted Pendulums

The above analysis of various pendulum dynamics usually attains stabilization so later attempts have produced many problems regarding the field of control and optimization of linear and nonlinear discontinuous (with impact and friction) and continuous realizations of inverted pendulums as multibody systems. Balancing an inverted pendulum of any kind is a classic control problem of recent 40 years. A new fuzzy controller for stabilizing series-type double inverted pendulum systems is proposed in [Yi et al. (2001)] based on the SIRMs (Single Input Rule Modules) dynamically connected fuzzy inference model. The proposed controller deals with six input items. Each input item is provided with a SIRM and a dynamic importance degree (DID). The SIRM and the DID are set up such that the angular control of the upper pendulum takes the highest priority order over the angular control of the lower pendulum and the position control of the cart, when the relative angle of the upper pendulum is big enough. By using the SIRMs and the DIDs, the control priority orders are automatically adjusted according to control events. Simulation results show that the controller stabilizes series-type double inverted pendulum systems of different parameter values in about 10 seconds for a wide range of the initial angles. In [Li and Xu (2009)] an adaptive fuzzy logic control of dynamic balance and motion is investigated for wheeled inverted pendulums with parametric and functional uncertainties. The proposed adaptive fuzzy logic control based on physical properties of wheeled inverted pendulums makes use of a fuzzy logic engine and a systematic online adaptation mechanism to approximate the unknown dynamics. Based on Lyapunov synthesis, the fuzzy control ensures that the system outputs track the given bounded reference signals to within a small neighborhood of zero, guaranteeing a semi-global uniform boundedness of all closed-loop signals. The

page 199

June 8, 2017 12:9

200

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

effectiveness of the proposed control is verified through extensive simulations. [Siuka and Sch¨ oberl (2009)] deals with the application of energy based control methods for a model of inverted pendulum on a cart. A swing-up controller as well as a nonlinear balancing controller with the focus on the implementation on a laboratory model is presented. The well-known control concept has been adapted such that they work on a concrete experiment with all the undesirable effects like friction and quantization. In [Lin et al. (1996)] a linear state feedback design technique for balancing an inverted pendulum is provided. The pivot of the investigated pendulum is mounted on a carriage that has limited horizontal travel. For any given (arbitrarily small) allowable travel of the carriage, a linear state feedback controller that balances the pendulum with an infinite amount of gain margin has been adopted in the sense that, if the feedback gain is perturbed by any multiplying factor greater than one, the controller balances the pendulum without requiring greater traveling distance than the maximum allowable. A mathematical model of a planar double inverted pendulum was established in [Kumar and Jerome (2013)] by means of an analytic dynamics method. Based on the linear quadratic optimal theory, a LQR self-adjusting controller was derived. Further, the output of LQR controller was refined, optimizing the factor which was the function of the states of planar pendulum, and on account of that, control action exerted on the pendulum was improved. Simulation results together with pilot scale experiment verify the efficiency of the suggested scheme. [Woodham and Su (2002)] uses the symbolic manipulation toolbox available in Matlab to algebraically investigate a pole-zero cancellation of the uncontrollable double inverted pendulum, following exploratory numerical computation. The ability of the software to factorize complicated multi-variable polynomials is exploited to identify, in an algebraic form, the anticipated pole-zero term canceling throughout the transfer functions of the uncontrollable pendulum system. The investigated system has been considered with respect to the force on the trolley, for which it is a conditionally uncontrollable problem, and with respect to each of the torques on the arms which are unconditionally uncontrollable problems. A methodology of Lyapunov stability control is presented in [Wu et al. (1998)] to achieve the upright balance of a base-excited inverted pendulum with two degrees of rotational freedom. The inclusion of the base point movement led to the dynamical system of such a pendulum, which is a non-autonomous one and is under persistent disturbance. An idealized piecewise continuous control strategy was designed, and for the obtained controller, the solution trajectories to be arbitrarily close to the upright position have been guaranteed. The continuous control law guarantees that the solution trajectories are kept in a controlled region around the upright position. The stability has been traded off with a weaker stability to prevent chattering. The robustness of the controllers with respect to certain class of uncertainties was also examined.

page 200

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Control of a Wheeled Double Inverted Pendulum with Friction

201

A passive fault tolerant control scheme has been suggested by [Niemann and Stoustrup (2005)]. A nominal controller is augmented with an additional block which guarantees stability and performance after the occurrence of a fault. The method is based on parameterization which requires the nominal controller to be implemented in observer based form. The proposed method is applied to a double inverted pendulum system, for which the H∞ controller has been designed and verified in laboratory tests. The literature overview shows that the problem is still valid, states a good field for practicing in control of multibody systems as well as opens new perspectives for application of interesting structures of controllers. One of such simple structures that is based on the standard LQG control has been studied in this work. Application of the Kalman filtering problem to control such a kind of continuous systems is, in general, not examined in literature. An exemplary contribution [Lee and Jung (2012)] that uses a Kalman filter to help the estimation of a gyro angle presents some study devoted to balancing and navigation of a MIPS robot. It is a mobile inverted pendulum system whose structure is a combination of a wheeled mobile robot and an inverted pendulum configuration. Low cost gyro and tilt sensors are used and fused to detect balancing angle. Digital filters are selectively designed for sensors to measure an inclined angle accurately with respect to different frequencies. Performances of balancing and navigation of the MIPS are tested by experimental studies through remote control.

12.3

Modeling of the 2-DOF Inverted Pendulum Driven by a Rotating Wheel

Let a simplified model of an unicycle-cyclist system shown in Fig. 12.1 be of our concern. Assuming that the most upper link is the cyclist’s body and the remaining two are the unicycle’s links, it creates a physical model placed in Cartesian coordinates. Physical model of the unicycle-cyclist system consists of three solid bodies of which masses are focused in points of their centers of gravity: (i) the driving wheel; (ii) fork frame with the seat post; (iii) the cyclist’s body. Equations of motion have been derived by means of Lagrange equations of second kind [Awrejcewicz (2007)]:

∂V ∂E d ∂E + = Qi , i = 1, 2, . . . , N, (12.1) − dt ∂ q˙i ∂qi ∂qi where N is the number of generalized coordinates, qi — the ith coordinate. In the analysis we will assume that ϕi are the generalized coordinates. Taking into consideration that a cyclist riding the unicycle can incline his body forward and backward with the frequency ω in the direction of movement, one writes: ϕ3 = ϕ2 − Ψ(t),

Ψ(t) = a sin (ωt),

where Ψ(t) is the angle of inclination of the cyclist’s body sitting in the saddle.

page 201

June 8, 2017 12:9

202

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Having the above, one reduces the model by rewriting Eq. (12.1) as follows:

d ∂E ∂V ∂E + = Qi , i = 1, 2. − dt ∂ ϕ˙ i ∂ϕi ∂ϕi Moments of inertia Ii of ith link with respect to the axis perpendicular to the center of gravity of ith mass are given by the formula Ii = mi ρ2i ,

i = 1, 2, 3,

where corresponding radii of inertia are as follows: √ √ e2 3 e3 3 ρ1 = r, ρ2 = , ρ3 = . 3 3 Distances between centers of gravity and the corresponding axis of rotation ei are as follows: ei = 0.5li ,

i = 2, 3.

Kinetic energy E of the unicycle-cyclist three-degree-of-freedom system is a sum of kinetic energies of the linear and angular displacement of each mass E=

3 $   mi  2 x˙ i + y˙ i2 + (ρi ϕ˙ i )2 . 2 i=1

(12.2)

Gravitational forces are conservative so the total potential energy of each mass of the system reads V =g

3 $

(12.3)

mi yi .

i=1

Assuming that the direct driving wheel is stiff and rolls without slips, the integrable geometrical and kinematic constraints are superposed: x1 = rϕ1 ,

y1 = r.

In accordance to the aforementioned assumptions, the following relations between Cartesian and generalized coordinates are found: ⎧ x2 = x1 + e2 sin ϕ2 , ⎪ ⎪ ⎨ y2 = y1 + e2 cos ϕ2 , ⎪ = x1 + e3 sin (ϕ2 − a sin (ωt)) + l2 sin ϕ2 , x ⎪ ⎩ 3 y3 = y1 + e3 cos (ϕ2 − a sin (ωt)) + l2 cos ϕ2 . For estimation of generalized forces we need to assume some nonconservative forces, i.e., the normal force Nc and the resistances in joints. Generalized coordinates describe absolute angular displacements, therefore, moments of forces acting on appropriate links are taken as the generalized moments: Q1 = MN − M01 + M21 ,

Q2 = M12 .

page 202

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Control of a Wheeled Double Inverted Pendulum with Friction

203

We will examine the dynamics of the open-loop control system moving with a constant velocity. With this condition, the moments acting on the wheel with respect to z1 -axis have to balance themselves. If links 2 and 3 move forward with constant velocity, keeping their upright positions in the directions x2 and x3 , respectively, then the driving moment MN is equal to the moment of rolling resistance MN = f (m1 + m2 + m3 )g. Additionally, the moment of rolling resistance is given by M01 = f sgn (ϕN ˙ c) , where the normal force Nc reads Nc = g

3 $ i=1

mi +

3 $

mi y¨i .

i=1

A relative moment generated by viscous damping of the unicycle’s frame yields M12 = −c2 (ϕ˙ 2 − ϕ˙ 1 ) = −M21 . Generalized forces are finally found in the form:   1 Q1 =MN − f sgn (ϕ˙ 1 ) m3 l3 sin (a sin (ωt) − ϕ2 ) + aω 2 sin (ωt) 2 2

− l3 cos (a sin (ωt) − ϕ2 ) (ϕ˙ 2 − aω cos (ωt)) − 2l2 ϕ˙ 22 cos ϕ2   2  1 − 2l2 ϕ¨2 sin ϕ2 + − l2 m2 ϕ˙ 2 + ϕ¨2 sin ϕ2 + g(m1 + m2 + m3 ) 2 + c2 (ϕ˙ 2 − ϕ˙ 1 ) , Q2 = − c2 (ϕ˙ 2 − ϕ˙ 1 ) . By substitution of Eqs. (12.2), (12.3) and (12.2) in Eq. (12.1) we found the two second-order ODEs describing the reduced dynamical model of the two-degree-offreedom system driven by a rotating wheel: 1 (l3 m3 (3s1 (rϕ˙ 1 + g) + 2l3 (ϕ¨2 + aω 2 sin (ωt)) 6 + 3l2 (m3 (l3 (aω(−2ϕ˙ 2 cos (ωt)s1 + aω cos 2 (ωt) sin (a sin (ωt)) + ω sin (ωt) cos (a sin (ωt))) + 2ϕ¨2 cos (a sin (ωt))) + 2rϕ¨1 cos ϕ2 − 2g sin ϕ2 ) + m2 (rϕ¨1 cos ϕ2 − g sin ϕ2 )) + 2l22 (m2 + 3m3 )ϕ¨2 ) + c2 (ϕ˙ 2 − ϕ˙ 1 ) = 0, 1 r(l3 m3 (ϕ˙ 2 s1 (ϕ˙ 2 − 2aω cos (ωt)) + ϕ¨2 c1 2 + aω 2 (a cos 2 (ωt)s1 + sin (ωt)s2 )) − l2 (m2 + 2m3 )(ϕ˙ 22 sin ϕ2 − ϕ¨2 cos ϕ2 ) 1 m3 (l3 s1 + 2(2m1 + m2 + m3 )rϕ¨1 ) − c2 (ϕ˙ 2 − ϕ˙ 1 ) + sgn (ϕ˙ 1 2 + aω 2 sin (ωt)) − l3 c1 (ϕ˙ 2 − aω cos (ωt))2 − 2l2 ϕ˙ 22 cos ϕ2 − 2l2 ϕ¨2 sin ϕ2

1 − l2 m2 (ϕ˙ 22 cos ϕ2 + ϕ¨2 sin ϕ2 ) + g(m1 + m2 + m3 ) = MN , 2 where s1 = sin (a sin (ωt) − ϕ2 ), c1 = cos (a sin (ωt) − ϕ2 ).

(12.4)

page 203

June 8, 2017 12:9

ws-book961x669

204

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

12.4

Numerical Modeling of System Dynamics

Numerical solution of the unicycle-cyclist system has been preceded by an experimental estimation of some parameters. At normal use of the unicycle that rides on a concrete road, the mean linear velocity of driving vm = 3 m/s, and the corresponding mean angular velocity of the wheel ϕ˙ 1 = 10 rad/s. Other parameters of a test unicycle read: m1 = 5, m2 = 30 kg, r = 0.3, l2 = 1, l3 = 0.8 m, e2 = 0.5, e3 = 0.4 m, c2 = 0.01 N·m/s, a = 0.25, f = 0.02 m, ω = 4.7 rad/s. 11


0


2,3 [rad]


1 [rad]

0 0


2
3

t [s]

10

5/6


0

(a)

1.5

xi [cm]

y2,3 [cm]

0

x1 x2 x3 t [s]

10

(b)

20

0

t [s]

10


1.5

y2 y3

0

(c)

t [s]

10

(d)

Figure 12.3: Time histories of state variables of the system (12.4). Figure 12.4 presents subsequent frames of the numerical solution of the differential equations (12.4) without any control input torque and at the initial conditions: ϕ˙ 1 (0) = 10 rad/s, ϕi (0) = 0, for i = 1, 2, 3. Comparison of time histories of the triple inverted pendulum with one rolling link are shown in Fig. 12.3.

12.5

Kalman Filter Based Control of the Pendulum

The aim of control is to force the double link wheeled pendulum by means of its third rotating link to maintain its upright always unstable equilibrium position. It has been numerically simulated using a control moment of force applied to the driving wheel (first link). The procedure of control is prepared accordingly to instructions provided by the Matlab’s Control System Toolbox supporting gain selection from root locus, pole

page 204

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Control of a Wheeled Double Inverted Pendulum with Friction

205

Figure 12.4: A stroboscopic view on motion of the unicycle-cyclist system. placement, Linear–Quadratic–Gaussian (LQG) control, and other. Let us provide the model 12.4 in the matrix form M(ϕ)ϕ¨ + N(ϕ)ϕ˙ 2 + O(ϕ)ϕ˙ + Cϕ + P(ϕ) = F¯ , which is prepared for a numerical integration as follows   ϕ¨ = (M(ϕ))−1 F − N(ϕ)ϕ˙ 2 − O(ϕ)ϕ˙ − Cϕ − P(ϕ) ,

(12.5) (12.6)

page 205

June 8, 2017 12:9

ws-book961x669

206

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

where elements of the matrices M(ϕ), N(ϕ), O(ϕ), P(ϕ) stay at appropriate components of state variables in equations (12.4), and additionally:           ϕ˙ 1 ϕ¨1 ϕ1 MN c2 −c2 , ϕ˙ = , ϕ¨ = , F¯ = , ϕ= , C= ϕ2 ϕ˙ 2 ϕ¨2 0 −c2 c2 ϕ1 (0) = ϕ˙ 1 (0) = ϕ2 (0) = ϕ˙ 2 (0) = 0. 12.5.1

Simplification and Linearization of the Model

The dynamical system under consideration is strongly nonlinear, since to control it closely around the desired zero angle upright position of second link, one needs to assume the following simplifications: sin ϕ2 ≈ ϕ2 , cos ϕ2 ≈ 1, cos (a sin (ωt)) ≈ c1 ≈ 1, cos (ωt) sin (a sin (ωt)) ≈ cos 2 (ωt) sin (a sin (ωt) ≈ sin (a sin (ωt)) ≈ 0,

(12.7)

2

cos (ωt) sin (a sin (ωt) − ϕ2 ) ≈ 0, s1 ≈ −ϕ2 . Applying the simplifications (12.7), the Laplace transfer function between the input MN (s) and the output Φ2 (s) is obtained: G(s) =

−0.424 · 105 s + 0.0003 · 105 Φ2 (s) = 3 . MN (s) s + 14.95 · 105 s2 − 1.626 · 105 s − 91.81 · 105

(12.8)

Figures 12.3(a) and (c) confirm unstability of the system, hence it should be confirmed analytically as it follows in next section. 12.5.2

Analysis of System Stability

State-space representation of the single input (the control moment of force) to single output (angle of rotation of the second link) control system derived from Eq. (12.8) is introduced:  ¯˙ = Aζ(t) ¯ + B¯ ¯ 0 ) = ζ¯0 , ζ(t) u(t), for ζ(t (12.9) ¯ + D¯ ν¯(t) = Cζ(t) u(t), where ζ¯ is the state vector, ν¯ is the output vector, and: ⎡ ⎤ ⎡ ⎤ −14.95 · 105 1.626 · 105 91.81 · 105 1 ⎦, B = ⎣0⎦, A=⎣ 1 0 0 0 1 0 0   5 5 C = 0 −0.424 · 10 0.0003 · 10 , D = [0].

(12.10)

¯ called One can analyze stability of the system by finding some function V (ζ), the Lyapunov function, which for the time invariant system satisfies: ¯ > 0, V (0) = 0, V˙ (ζ) ¯ = ∂V dζ ≤ 0. V (ζ) (12.11) ∂ ζ¯ dt For the linear time invariant systems the procedure for finding the Lyapunov function converges to the problem of solving a Lyapunov algebraic equation. The

page 206

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Control of a Wheeled Double Inverted Pendulum with Friction

207

¯ such linear system (12.9) is stable if one is able to find a scalar function V (ζ) that the conditions (12.11) associated with the system are satisfied [Gaji´c and Leli´c ¯ for ¯ = ζ¯T W1 ζ, (1996)]. The Lyapunov function can be chosen to be quadratic V (ζ) T W1 = W1 > 0, which in view of Eq. (12.9) leads to the following ¯ = ζ¯T (AT W1 + W1 A)ζ. ¯ V˙ (ζ)

(12.12)

The single-input single-output system (12.9) with the matrices (12.10) is asymptotically stable, if for any positive definite matrix W2 = W2T > 0 there exists W1 = W1T > 0 such that AT W1 + W1 A = −W2 .

(12.13)

Matrix A has the following eigenvalues: 6 λA 1 = −1.495 · 10 ,

λA 2 = 2.533,

λA 3 = −2.424,

and hence, this system is not asymptotically stable. In order to apply Lyapunov method, an initial positive definite matrix W2 = I3 is built. Using Matlab’s function lyap(AT , W2 ), the matrix W1 can be calculated ⎤ ⎡ 33.44 · 10−8 0.16 · 10−4 −5.45 · 10−8 ⎦. (12.14) W1 = ⎣ 0.16 · 10−4 24.33 −3.15 −8 −3.15 −149.74 −5.45 · 10 Function lyap(AT , W2 ) solves the equation that represents the transpose of the algebraic Lyapunov equation (12.13), satisfying W2 = W2T > 0. If some of eigenvalues of the matrix W1 are in open left half plane, then the system (12.9) is unstable. Computing eigenvalues of the matrix W1 , we get: −8 λW , 1 = 33.45 · 10

λW 2 = 24.39,

λW 3 = −149.79.

As it could be expected, the investigated system is unstable. Therefore, in accordance to the configuration shown in Fig. 12.5, the LQG control is applied. disturbance d(t)

F

B

Kalman estimator

linear model of pendulm G(s)

output signals

noise n(t)

Figure 12.5: Kalman filter configuration in the analyzed control system.

page 207

June 8, 2017 12:9

208

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The LQG design tools of Matlab used in the numerical experiment (see [Olejnik et al. (2013)]) include functions lqry(), kalman() and lqgreg() that have been applied to compute the LQ-optimal state-feedback gain, hence to design the Kalman filter [Kwakernaak and Sivan (1972)]. Since the system measurements are corrupted by white noise, any exact values of state variables are always not available in reality. The aim is to find a dynamical system, that produces the estimates, for which the variance of the estimation error ¯ − ζ¯e (t) is minimized. The result of application of the described LQG (t) = ζ(t) Design Tools is presented in Fig. 12.6.
2 [rad] 4
103

0

4
103 0

2.5

5

7.5

t [s]

Figure 12.6: Time history of angular displacement of the second link of the analyzed two-degree-of-freedom inverted pendulum operating in the closed-loop control system with a LQG regulator. A kinematic excitation, modeling natural behavior of a cyclist riding a monocycle, has been applied to the third link allowing for cancellation of one degree of freedom of the double inverted pendulum driven by a rotating wheel. This motion has been treated as a disturbance signal, since upright position of the second link (the unicycle’s frame) has been subjected to control by driving the rotating wheel. The Linear–Quadratic–Gaussian control has produced a satisfactorily good time response and positioning of the object of control.

page 208

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 13

Tracking Control of a Discontinuous System with Stick–Slip Friction

We have mentioned in the first chapter that the natural resistance to the relative motion between non-lubricated surfaces of two contacting bodies is caused by dry friction. In some discontinuous dynamical systems with non-linearities emitted by dry friction, a controller has to be designed to avoid the steady-state tracking errors or even some undesirable self-excited vibrations. In this chapter, we examine an influence of the dry friction-caused discontinuities on the controlled dynamics of a bearing rotor.

13.1

Introduction

An adaptive friction compensation to improve performance of tracking errors without the Stribeck effect has been proposed in [Huang et al. (2000)]. A new control strategy for compensation of frictional phenomena including the Stribeck effect, hysteresis, stick–slip limit cycling, pre-sliding displacement and rising static friction has been particularly described and examined. The proposed compensator could be useful for handling significant nonlinearities in motor controls. Similarly, the Lund–Grenoble model of dynamical friction has been used in [Hirschorn and Miller (1999)] to control nonlinear effects that the model captures: the Stribeck and Dahl effects, viscous and Coulomb friction [Awrejcewicz and Olejnik (2005a)]. A new Lyapunov-based continuous dynamic controller has been delivered for a more general class of nonlinear systems. It produced better control of a high-speed precision linear tracking table than some tuned PID controllers without direct nonlinear effects compensation. As it has been shown also in [Song et al. (1998)], conventional feedback control methods cannot ensure good results in the presence of dry friction (stick–slip) even in the one-degree-of-freedom DC motor system. Because of steady state errors, a traditional PD controller would not achieve satisfactory performance. These errors could be reduced by increasing the P gain, but significant instabilities would be reported while driving the motor with some angular velocities or along the desired rapidly changing time history of its angular position. Very good positioning accuracy have been obtained with the use of a new sliding-mode-based 209

page 209

June 8, 2017 12:9

210

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

smooth adaptive robust controller designed for dry friction compensation. A study of control of a mechanism under the influence of low velocity friction has been conducted in [Adams and Payandeh (1996)]. The theoretical and experimental comparative study of linear (PD, PID) and nonlinear (smooth continuous and piecewise linear discontinuous) compensation algorithms have been proposed. In the case of modeling a two-degree-of-freedom controlled planar manipulator, the nonlinear controllers have proved to have superior performance regarding some P and D gains, compared to any PD controller. Moreover, their tracking performance was also superior to the PID controller, but it provides an oscillatory time dependency of torque. Stability of smooth controllers was much simpler to demonstrate. Simple active control of the belt-driven oscillator with stick–slip friction in a control system with a feedback loop created by a transducer, frequency filter, phase shifter, amplifier and a shaker attached to the oscillating body has been studied in [Heckl and Abrahams (1996)]. The feedback system allowed for suppression of unstable vibrations at high effectiveness insensitive to errors in phase shift and amplification. Similarly, in [Li et al. (2009)], some type of friction-driven oscillator controlled by Lyapunov redesign based on delayed state feedback has been numerically investigated. Authors redesigned a continuous controller on the basis of a delayed state feedback to ensure that the nonsmooth friction-driven system is ultimately bounded. Moreover, by constructing a Lyapunov–Krasovskii function, the sufficient condition of stability for the investigated system was obtained. Neural networks have the capability of approximating nonlinear functions, therefore they are also demanding when it comes to estimation of frictional behaviors. Much work has been done in this subject [Kim and Lewis (2000); Otten et al. (1997)]. [Otten et al. (1997)], for instance, investigates control of linear motion of motors by means of the learning forward controller that is designed in the discrete state-space. The authors [Driessen and Sadegh (2004)] have also solved the problem of discrete-time iterative learning control for position trajectory tracking of multiple-input, multiple-output systems, including Coulomb friction, bounds on the inputs, static and sliding friction coefficients. On the background of a two-link revolute-joint planar robot arm, some satisfactorily learned angular position time histories (at a decrease in position-tracking error) have been shown. In accordance to linear servo motor control, a novel, very interesting approach for designing a wavelet basis function network learning controller for a linear motor control system was considered in [Lin (2002)]. The proposed wavelet network-based controller dealt with viscous friction and force ripples that occur in motion control of linear synchronous motors. The considerations presented in the exemplary articles were motivated by examination of control approaches, but the investigation of stability should be also considered. An interesting reference [Chenafa et al. (2005)] presents analysis of global stability of linearizing control of induction motors with a new robust nonlinear observer-based approach. Authors used traditional Park’s induction model

page 210

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Tracking Control of a Discontinuous System with Stick–Slip Friction

ws-book975x65

211

in a stator fixed reference frame related to the stator given by [Mansouri et al. (2004)]. They designed a control algorithm based on feedback linearization [Marino et al. (1993)]. After assumption of parameters of the induction motor, a detailed scheme of the nonlinear control with an observer has been done in Simulink. The new robust observer based on a nonlinear control scheme offered advantage of only one tuning parameter. The global stability of the whole system consisting of the motor, the controller and the observer was established by means of the precise Lyapunov function that kept observer’s dynamics free. More on the initial strategy on input–output linearization can be found in [Chiason (1997)], and on the global stability of the process-observer-controller system in [Lubineau et al. (2000)]. Deeper survey through the cited literature provides many references to theoretical derivations and practical implementations confirming permanent interest in control of non-smooth systems. Basically, control strategies depend on the aim, the friction law, the system at hand and its field of application. This study concerns a numerical simulation of compensation of frictional effects present in a real system designed for observations and experimental estimation of friction force characteristics, see [Awrejcewicz and Olejnik (2003b)]. The system consists of a DC motor driving a wide transmission belt on which a rigid body, being in frictional contact with the surface of the belt, vibrates. For instance, to find bifurcations of sliding solutions [Awrejcewicz and Olejnik (2005c)], after a relative motion was observed between contacting surfaces of the investigated coupling, it is required to precisely implement some desired function of changes of angular velocity of the DC motor that drives the belt. Therefore, rotational velocity of the driving motor should vary in a periodic cycle, exactly tracking the desired time-dependent characteristics (triangular, sinusoidal, etc.). However, such a situation does not occur with regard to the existing nonlinear friction characteristics. From the point of view of the control theory, it states a problem of providing a robust tracking control of rotational velocity of the DC motor. Therefore, some close to ideal generation of regular input signal (excitation of the belt) would be possible after application of some tracking control technique that has been already implemented, for instance, to control robot manipulators [Lewis et al. (1993)]. The following control errors could be used as input to the method: e(t) = ϕ(t) − ϕd (t), ε(t) = ϕ˙ d (t) − λe(t), where: ϕ and ϕ˙ are, respectively, the angular position and velocity of motor’s rotor driving the base by means of a non-stretchable transmission toothed belt; index d denotes desired values of corresponding variables along with the desired phase trajectory. Particular investigation will be focused on examination of influence of frictional contacts existing during rotation of a bearing rotor. One can distinguish a phenomenon of stick–slip friction that mostly affects accuracy of positioning. Friction of such type was investigated by authors of the monograph in [Awrejcewicz and Olejnik (2005a)], and may result from the following: Coulomb friction that repre˙ at a slip phase and Tsm (1 − sgn |ϕ(t)|) ˙ sents maximum static friction Tsm sgn ϕ(t) at a stick phase, when an input torque generated by a system driven by mo-

page 211

June 8, 2017 12:9

ws-book961x669

212

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

tor’s rotor could by applied, exponential friction described by the Stribeck curve ˙ sgn ϕ(t)), ˙ viscous friction Tvm ϕ(t), ˙ and position-dependent TStm (1 − exp(−T0 |ϕ(t)|) ˙ as proposed in [Slotine and Li (1987)], where: friction T1m sin (T2 ϕ(t)+T3 ) sgn |ϕ(t)|, sgn ϕ˙ denotes the sign of the value of angular velocity, ϕ is an angular displacement, Tsm is the maximum static friction torque, TStm and T0 > 0 are the parameters of Stribeck curve, Tvm is the coefficient of viscous friction, T1m , T2 and T3 are constants. The mechanical part of the reduced dynamical system of differential equations used for modeling the dynamics of rotational motion of a DC motor holds:

' ( cb cm ˙ sgn ϕ(t) ˙ Jm ϕ(t) ¨ + + Tvm ϕ(t) ˙ − TStm 1 − e−T0 |ϕ(t)| Ra (13.1) + T1m sin (T2 ϕ(t) + T3 ) sgn |ϕ(t)| ˙ + Tsm (1 − sgn |ϕ(t)| ˙ + sgn ϕ(t)) ˙ = cm ψm (t), and the remaining unknown model parameters are as follows: Ra and ψm denote the armature resistance and the armature current, respectively, Jm is the moment of inertia of the rotor, cb is a constant of the back electromotive force, and cm is the motor torque constant. One rewrites Eq. (13.1) in a form scaled with respect to cm as follows J ϕ(t)+B ¨ ϕ(t) ˙ + τ (t) = ψ(t), τ (t) = Tv ϕ(t) ˙ − TSt (1 − exp(−T0 |ϕ(t)|)) ˙ sgn ϕ(t) ˙

(13.2)

˙ + Ts (1 − sgn |ϕ(t)| ˙ + sgn ϕ(t)) ˙ , + T1 sin (T2 ϕ(t) + T3 ) sgn |ϕ(t)| where the function τ (t) states for the scaled friction force, and J, B, Tv , Ts , T1 , TSt m equal Jcm , Rcba , Tcvm , Tcsm , Tc1m , TcStm , respectively. m m m m It is not possible to exactly describe friction and to correctly assume all values of parameters. A tracking control, which is a point of the study, should also correct any inaccuracies caused by an imprecise system modeling. 13.2

The Algorithm of Sliding Surface Control

The objective of control is to design an adaptive controller that would allow us to change the angular velocity of rotation of motor’s rotor according to some desired function ϕd (t). Let us begin from the so-called sliding surface method [Slotine and Li (1987)]. On its background, the control error e(t) = ϕ(t) − ϕd (t), an auxiliary variable ε(t) = ϕ˙ d (t) − λe(t) and the definition of the sliding surface r(t) = ϕ(t) ˙ − ε(t) = 0, where variables with index d denote corresponding desired values, λ > 0 is for more general multidimensional case a positive definite main diagonal matrix. Having all variables of the sliding surface method introduced, let us propose a control law derived from Eq. (13.2) in the form: ' ( ˆ ˙ ˆ sgn ϕ(t) ˙ + Tˆs sgn ϕ(t) ˙ ψ(t) = Jˆε(t) ˙ + Dε(t) − TˆSt 1 − e−T0 |ϕ(t)| (13.3) ˙ us (t) − ub (t), + Tˆs (1 − sgn |ϕ(t)|)

page 212

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Tracking Control of a Discontinuous System with Stick–Slip Friction

ws-book975x65

213

where ub (t) is a condition of bounding function, us (t) = 1 − sgn |r(t)| at term describing sticking phase is a function introduced with respect to the definition of ˆ =B ˆ + Tˆv , and the circumflexˆabove symbols marks the sliding surface r(t) = 0, D estimates of corresponding parameters. Equation (13.3) will be used for adaptation of unknown estimates in a scheme in which ψ(t) is put to Eq. (13.1) to compensate for linear and nonlinear forces included in it. Such an adaptive feed-forward control loop is good to compensate linear friction forces like Coulomb and viscous ones [Song et al. (1998)]. Nonlinear friction forces, like the Stribeck effect and the angular position-dependent friction force, cannot be controlled in the loop, but some adaptation law based on a robust compensator to learn an upper bounding function has to be used [Lewis et al. (1993)]. The following bounding function is assumed ub (t) = kD r(t) + ρˆkT tanh(r(t)(a + bt)),

(13.4)

where a, b and kD are positive constants, and kT > 1. If the parameter ρˆ is an estimate of the upper bound of the nonlinear residual terms, then ub (t)|r(t)=λe behaves as a proportional gain robustly compensating nonlinear friction forces. It inputs to the control law (13.3) a torque greater than the maximum static friction allowing for compensation. 13.2.1

Estimation of Linear and Nonlinear Parameters

In the sliding surface method, the adaptive law validating all unknowns at each step of integration is based on a simple first-order differential equation. Therefore, the following adaptive law [Slotine and Li (1987)] validating estimates of the system parameters at linear terms takes the form: ˆ˙ ˆ˙ = −δ1 ε(t)r(t), ˙ D(t) = −δ2 ε(t)r(t), J(t)  − δ3 sgn (ϕ(t))r(t), ˙ in a slip, ˙ Tˆs (t) = ˙ |r(t), in a stick, δ3 (1 − sgn |ϕ(t)|) ˙ − ε(t). where δ1,...,3 are positive constants, r(t) = ϕ(t) Putting ψ(t) from Eq. (13.3) to Eq. (13.2), including ε(t) = ϕ(t) ˙ − r(t), ε˙ = ϕ(t) ¨ − r(t) ˙ with T˜St = TˆSt − TSt and T˜0 = Tˆ0 − T0 measuring differences between estimates and their corresponding real values, one gets ˆ ˙ + ω(t) − ub (t), J r(t) ˙ + Dr(t) = Jˆε(t) ˙ + Dε(t) + Tˆs sgn ϕ(t)

(13.5)

where

( ' ˆ ˙ ˆ ˙ ˜ ˙ − TSt e−T0 |ϕ(t)| e−T0 |ϕ(t)| − T˜St − Tp sgn ϕ(t). ˙ ω(t) = TˆSt e−T0 |ϕ(t)|

(13.6)

2 ˜ in a Taylor series about ˙ = 1+ T˜0 |ϕ(t)|+ ˙ T˜0 |ϕ(t)| ˙ /2+ R Expanding exp (T˜0 |ϕ(t)|) |ϕ(t)| ˙ = 0 and using only the first three terms of the expansion with a reminder

page 213

June 8, 2017 12:9

ws-book961x669

214

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

3 3 ˜ ≤ exp (T˜0 |ϕ(t)|)T R ˙ ˙ /6: 0 |ϕ(t)|  2 |ϕ(t)| ˙ −Tˆ0 |ϕ(t)| ˙ −Tˆ0 |ϕ(t)| ˙ ˆ − TSt e ˙ + T˜0 1 + T˜0 |ϕ(t)| ω(t) = TSt e 2

 3 |ϕ(t)| ˙ ˜ ˙ eT0 |ϕ(t)| − T˜St − Tp sgn ϕ(t) ˙ = ρ sgn ϕ(t), ˙ + T˜03 6 2 ˙ − γ3 |ϕ(t)| ˙ exp (−Tˆ0 |ϕ(t)|) ˙ − γ4 |ϕ(t)| ˙ where variable ρ = −γ1 + γ2 exp (−Tˆ0 |ϕ(t)|) 3 3 ˆ ˜ ˜ exp (−T0 |ϕ(t)|), ˙ γ1 = max|ϕ(t)|∈[0,∞] {|ϕ(t)| ˙ exp (−T0 |ϕ(t)|) ˙ T0 TSt /6} + TSt + Tp , ˙ γ2 = T˜St , γ3 = T˜0 TSt , γ4 = T˜02 TSt /2. Constants γ1,...,4 depend on estimates or on their difference from real values. At this point, let us come back to Eq. (13.4) containing an unknown estimate ρˆ. In Eq. (13.6), to get cancellation of reminders not dependent on r(t), r(t), ˙ ε(t) and ε(t), ˙ ω(t) − ρˆ(t)ωr (t) → ∞, where ωr (t) = kT tanh (r(t)(a + bt)) as introduced in [Cai and Lamba (1994)]. Therefore, ρ sgn ϕ(t) ˙ → ρˆkT tanh (r(t) (a + bt)), and if ub (t) is the upper bounding function of ω(t), then ˆ

ˆ

ˆ

˙ −T0 |ϕ(t)| ˙ 2 −T0 |ϕ(t)| ˙ ρˆ(t) = −ˆ γ1 + γˆ2 e−T0 |ϕ(t)| − γˆ3 |ϕ(t)|e ˙ − γˆ4 |ϕ(t)| ˙ e

states the estimate of ρ variable. Similarly to construction of Eq. (13.5), we can calculate the remaining estimate ρˆ by solving the following system of equations: ⎧ γˆ˙ 1 (t) = δ4 |r(t)|, ⎪ ⎪ ⎪ ⎨ γˆ˙ (t) = −δ |r(t)|e−Tˆ0 |ϕ(t)| ˙ , 2 5 ˆ0 |ϕ(t)| − T ˙ ˙ ⎪ ˙ , γˆ3 (t) = −δ6 |r(t)||ϕ(t)|e ⎪ ⎪ ⎩˙ 2 −Tˆ0 |ϕ(t)| ˙ ˙ e , γˆ4 (t) = −δ7 |r(t)||ϕ(t)| where γ1,...,4 are positive constants. 13.2.2

The Low-Voltage Control of Rotational Velocity

In a real application, one would need to source the motor not with the electric current but with voltage of known function. In this situation, the following full electromechanical system has to be introduced into the analysis: J ϕ¨f (t) + B ϕ˙ f (t) + τf (t) = ψf (t), ˙ La ψf (t) + Ra ψf (t) + cb ϕ˙ f (t) = vf (t),

(13.7) (13.8)

where index f is used to denote a full three-dimensional dynamical system, La is the armature inductance, vf (t) is a time-dependent function of voltage required to realize the desired task of control. One assumes that Eqs. (13.1) and (13.2) mathematically describe the dynamics of the motor, the electrical and mechanical parameters of which will be taken according to the direct current commutation motor suitable for use in cross-feed drives of numerically controlled machines [Olejnik and Awrejcewicz (2013b)]. In a full electromechanical system, voltage control requires to regard to Eq. (13.7). If the current-input control of the DC motor works correctly, then the

page 214

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Tracking Control of a Discontinuous System with Stick–Slip Friction

ws-book975x65

215

best solution is to maximally reduce the influence of the second equation. In the full system, it provides these unwanted disturbances influencing the optimal current input. The most obvious would be to apply to Eq. (13.8) the voltage input vf (t) calculated on the basis of ψ(t) which is estimated after solution of only the reduced mechanical system (13.2) given by control law (13.3). Therefore, voltage input necessary to cancel the dynamical disturbances of the complete three-dimensional electromechanical system (13.7)-(13.8) is expected in the form ˙ + Ra ψ(t) + cb ϕ(t) (13.9) ˙ + d(t) vf (t) = La ψ(t) with a limitation that ψ(t) ensures proper tracking current-input control of the reduced model in Eq. (13.2) and ϕ˙ states for angular velocity resulting from that control. Function d(t) is a compensator of dynamical differences between state variables of Eqs. (13.8) and (13.9). After substitution of vf given by Eq. (13.9) to Eq. (13.8), all dynamical terms in Eq. (13.7) have their counterparts canceling them, but some occurring differences are expected to be compensated by d(t) which, if disregarded, makes the substitution incorrect, and some significant oscillations about zero value are observed. To increase effectiveness of the control strategy, it is proposed to apply a two-dimensional proportional control with a feedback from the object of control described by a full dynamical system of the modeled motor. Therefore, applying d(t) = k1 (ϕd (t) − ϕf (t)) + k2 (ϕ˙ d (t) − ϕ˙ f (t)) to Eq. (13.9) to be used in Eq. (13.7), the following equation of dynamical equilibrium is found: ( ' ˙ La ψ˙ f (t) − ψ(t) + Ra (ψf (t) − ψ(t)) + cb (ϕ˙ f (t) − ϕ(t)) ˙ = k1 (ϕd (t) − ϕf (t)) + k2 (ϕ˙ d (t) − ϕ˙ f (t)) , (13.10) where to get the demanding cancellation of Eq. (13.8), tuning factors k1 and k2 should ensure equality of both sides of Eq. (13.10), but at each time instant, solution ψf (t) have to be updated in Eq. (13.7), where ϕd and ϕ˙ d are the desired coordinates of the phase trajectory of the rotor motion. Having this condition met, solution ϕf (t) to Eq. (13.7) should track the optimal solution ϕ(t) of Eq. (13.2). In tracking control, the time history of vf (t) can be saved and used as the input to drive the complete electromechanical dynamical system of the DC motor along with either the desired phase trajectory, angular velocity or angular position of its rotor.

13.3

Numerical Simulation

Efficiency of the two-stage control method is checked with numerical simulations performed for a model of the DC motor PZTK 60-46 J with stick–slip friction occurring in contact zones located between rotor’s shaft and bearings. Rotational velocity of the DC motor is required to follow the desired trajectory ϕd (t) drawn with a dashed line in Fig. 13.1.

page 215

June 8, 2017 12:9

216

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

ϕ˙ [rad/s]

0.2

ϕ˙ d ϕ˙ f

0

−0.2 0

1

2

(a) k1 = 2.7 ·

ϕ˙ [rad/s]

0.2

3

103 ,

k2 = 0.1 ·

4

5

t [s]

4

5

t [s]

4

5

t [s]

103

ϕ˙ d ϕ˙ f

0

−0.2 0

1

2

3

(b) k1 = 1.1 · 103 , k2 = 0.3 · 103

ϕ˙ [rad/s]

0.2

ϕ˙ d ϕ˙ f

0

−0.2 0

1

2

3

(c) k1 = 2.7 · 103 , k2 = 0.3 · 103

Figure 13.1: Desired time history of angular velocity ϕ˙ d (t) (dashed line) and the corresponding response ϕ(t) ˙ (solid line) of the analyzed voltage-controlled simulation model of the DC motor defined by the assumed set of model parameters in Tab. 13.1, PD controller’s constants: k1 and k2 , and initial values of state variables.

page 216

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Tracking Control of a Discontinuous System with Stick–Slip Friction

217

Table 13.1: System and tuning parameters for the numerical simulation.

Motor torque constant Constant of the back electromotive force Armature resistance Moment of inertia of the rotor Armature inductance Coefficient of viscous friction Max. static friction torque on the Stribeck curve Max. static friction torque Position-dependent friction torque Constant of the Stribeck curve Constants of position-dependent friction Constants of adaptation laws Tuning factors: P gain Tuning factors: D gain

Notation

Value

Unit

cm cb R Jm La Tv TStm Tsm T1m T0 [T2 , T3 ] δi=1,...,7 k1 k2

0.5 0.011 1.1 2 10−3 8 0.5 1.5 0.35 10 [1, 0.5] 9·i 0.21 · 103 1.40 · 103

N·m/A V/rpm Ω kg·m2 H N·m·s/rad N·m N·m N·m – – – – –

Time history of the desired velocity is formed in the scheme: (i) it increases from 0 to 0.2 rad/s in 0.2 s, (ii) it is held at this value for 0.6 s, (iii) it is decreased to 0 in 0.2 s, (iv) without a delay it changes its value (in the second half of the period) to negative, achieving symmetrically the same thresholds and times of presence as for positive values. After 2 s the cycle is repeated; see the dashed line in Fig. 13.1. ˆ (0) = 1, Tˆs(0) = Initial values of parameter estimates at linear terms: Jˆ(0) = 1, D (0) (0) 0.2, TˆSt = 1, Tˆ0 = 1. The initial value of the parameter estimate at nonlinear (0) terms ρˆ(0) = 0, and initial values of state variables ϕ(0) = ϕ˙ (0) = 0, ψf = 0. 5 ρˆ ˆ B ˆ TSt

4

ˆ TˆSt ρˆ, B,

3 2 1 0 −1

0

1

2

3

4

t

Figure 13.2: Convergence of selected estimates.

5

t [s]

page 217

June 8, 2017 12:9

218

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

It is important to observe that at the beginning of simulation, exact values of some parameters are not known, but are given by initial values of their counterpart estimates (see convergence in Fig. 13.2). Besides the uncertainty of parameters, there exists some influence of discontinuous terms of frictional torques described earlier. Looking at Fig. 13.1, it can be noticed that the system response is inaccurate at first occurrence of the threshold of constant angular velocity (0.2 [rad/s]). Such transient behavior results from the model and tuning parameters that are not correctly estimated at the corresponding time. The response changes over time to produce an acceptable overlapping of both trajectories at the beginning of the second period (at 2 [s]). At subsequent ±0.2 [rad/s] thresholds, the system step response is well damped, smoothly fitting edges of the desired shape. Figures 13.1 bring a comparison of three solutions: oscillatory, over-dumped and the most accurate, which could be also subjected to some small improvement to get faster convergence to the steady-state velocity.

ϕ˙ f [rad/s]

0.2

0

−0.2 −0.35

−0.3

−0.25

ϕf [rad]

Figure 13.3: Projection of phase trajectory of the controlled system on the plane ϕ(ϕ). ˙ The phase trajectory presented in Fig. 13.3 gives another view on the desired trajectory. It should take a shape of a closed curve bounded between ϕ˙ = ±0.2. To achieve the demanding effect of control, voltage input should be applied accordingly to the time history shown in Fig. 13.4. Amplitude of the demanding voltage control input changes impulsively after crossing ϕ˙ = 0, for t = 1, 2, . . . , n [s]. The proposed strategy of voltage tracking control ensures robust adaptation, works correctly, and can be applied to solve other control objectives related to shaping of time histories of responses of some group belonging to discontinuous dynamical systems. After a trial-and-error approach of tuning, parameters k1 and k2 of the second stage of conducted control have been estimated. They significantly affect local step response (appearing while going on the thresholds of constant an-

page 218

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Tracking Control of a Discontinuous System with Stick–Slip Friction

ws-book975x65

219

20

vf [V]

10 0 −10 −20 0

1

2

3

4

5

t [s]

Figure 13.4: Voltage input vf (t) applied to the optimally controlled DC motor. gular velocity). Moreover, on the basis of sliding surface based smooth adaptive robust controller for compensation of frictional effects, there was a useful and easily applicable extension of this method for numerical tracking control of DC motors by means of voltage input proposed. A kind of drawback or an inconvenience in application of the elaborated control strategy is the requirement of estimation of the upper bounding function for the nonlinear stick–slip friction in order to guarantee the closed-loop stability.

page 219

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Chapter 14

Controlling Stochastically Excited Systems with an Approximate Discontinuity Among many kinds of dynamical loading that can affect various constructions, these seismic ones belong to the most crucial. They are observed in the ground after a sudden propagation of earthquakes. This chapter presents a simulation and control of a building structure subjected to a stochastic excitation. The problem is reduced to a two-degree-of-freedom system with an approximated frictional discontinuity introduced by the Saint–Venant element.

14.1

Introduction

The forces acting on a buidling construction apear formally as a result of inertial responses that have been uncovered during the kinematic forcing of foundation of the construction [Copra (2000)]. Simply, a seismic excitation can be realized by means of the deterministic function A cos (ωt) exp(−0.5λt), in which A determines the intensity of earthquakes, ω is the frequency of excitation, and the parameter λ is responsible for the rate of the excitation damping. One can exhibit a very exact form of earthquakes with the use of a random process. One of the most popularized models is the stationary stochastic process (see [Rofooei et al. (2001)]), that is characterized by a wide spectrum given by the formula S(ω) =

˜2 1 + 4ξg2 ω S0 , (1 − ω ˜ 2 ) + 4ξg2 ω ˜2

for ω ˜=

ω , ωg

(14.1)

where ξg and ωg are, respectively, the site dominant damping coefficient and frequency, ω states the frequency of excitation, and S0 is the constant power spectral intensity of excitation. In practice, these parameters need to be estimated from the local earthquake records and geological properties. The relation seen in Eq. (14.1) is called the Kanai–Tajimi power spectral density function and might be interpreted as the corresponding to an ideal white noise excitation that is filtered at the bedrock level through the overlaying soil deposits at a site. A modified version of the above 221

page 221

June 8, 2017 12:9

222

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

has been presented in [Ruiz and Penzien (1969)]. It follows ωg + 4ξg2 ωg2 ω 2 S0 , S(ω) =  2 2 ωg2 − ω 2 + 4ξg2 ωg2 ω 2 (ω12 − ω 2 ) + 4ξ12 ω12 ω 2 with an assumption that ω1 = 1.636 and ξ1 = 0.619. The problem of seismic engineering devoted to the building-ground dynamics has been explained by [Newmark and Rosenblueth (1975)] and [Wolf (1994)]. A point of useful consideration done under reduction of the influence of loading on a building structure was presented in [Inman (1989)]. There has been estimated a control law allowing for minimization of amplitude vibration of the construction. The equation in a matrix form of the n-degree-of-freedom system, the motion of which is caused by the foundation’s vibration with the vector of frequency ω ¯ (t), becomes the general representation ¨¯ (t) + B¯ Mq¨ ¯ + Cq¯˙ + K¯ q = −Mω u(t), where q¯ is the vector of the displacement (system state) in relation to a stationary reference system; u ¯ — the vector of control inputs; ω ¯ — vector of the frequencies of system excitation for each component of the relative displacement vector; B — a matrix determining the point of attachment of control force in the inner space of the construction; M, C, K — mass, damping and stiffness matrices of construction, respectively.

14.2

The Saint–Venant Element Modeling the Dry Friction Contact

Dry friction is well-known in the theory of earthquakes and their control; see Chapter 1. The Saint–Venant element (also known as SV-element) states one of the attempts of the approximate simulation of damping effects observed in ground structure materials and even in building structures. This element incorporates a discontinuity into the work regime [Monteiro-Marques (1994)].




k m

x

Figure 14.1: Structural model of one-degree-of-freedom system with one SV-element denoted by α. Figure 14.1 shows a simple one-degree-of-freedom system with the Saint–Venant damping (also known as SV damping), since the model can be successfully extended to a four-degree-of-freedom system of oscillators coupled in parallel.

page 222

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Controlling Stochastically Excited Systems with an Approximate Discontinuity

14.2.1

ws-book975x65

223

The Nonlinear Parallel Spring Connection

Let us assume a point-focused mass m subjected to an external force F , and even in parallel, connected to a spring of the stiffness k and a discontinuous SV-element determined by the the maximal force α [N] and the displacement x [m] of mass m. One writes the following inclusion m¨ x + kx + ασ(x) ˙  F, ˙ = x˙ 0 . The multivalued function σ is with the initial conditions: x(0) = x0 , x(0) defined: ⎧ ⎪ x˙ < 0, ⎪ ⎨−1, (14.2) σ(x) ˙ = 1, x˙ > 0, ⎪ ⎪ ⎩[−1, 1], x˙ = 0. The multiplication ασ(x) ˙ denotes the friction force propagated in the SVelement of which α determines its boundary (limit) force. Some physical properties of the element allow one to assume, that for a relative velocity of motion x˙ = 0, the dry friction keeps the motion of mass m within the limits, maintaining the sate of zero slip velocity (the stick or creep phase). Friction force takes some values from the interval (−α, α), while after crossing the boundary values of this interval the mass begins sliding and the friction force is unequivocally determined. 14.2.2

The Viscous–Elastic Model

A point mass m has been subjected to an external force F and also connected in serial to several connectors, i.e., a spring of the stiffness k, the SV-element bounded by α and a dashpot of the damping c attached to a ground fixed base.

k

u A' A

v u


c

wv u

x m

B''B' B

F

C''' C''C' C

Figure 14.2: A scheme of the viscous–elastic model. In relation to Fig. 14.2, the following state independent displacement variables are assumed: u [m] — displacement of end of the spring k [N/m] with regard to the initial position at point A; v [m] — a difference between segments AB and A’B”, but the difference w [m] is found between segments BC and B”C”’; x [m] — the final displacement of mass m x = u + v + w.

(14.3)

Additionally, let {k, c, α} =  0 and the function: f = cw˙ = ku = ασ(v) ˙

(14.4)

page 223

June 8, 2017 12:9

224

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

in any section of the analyzed coupled system. The equation of motion modeling dynamics of the system from Fig. 14.2 is found as m¨ x = −cw˙ + F. Using Eq. (14.4) m¨ x = −ku + F.

(14.5)

If the SV-element is taken into the account, the following inclusion holds ku ∈ ασ(v), ˙ where v˙ reflects differentiation of Eq. (14.3) with respect to time v˙ = x˙ − u˙ − w. ˙ On the basis of Eq. (14.4), one writes w˙ =

ku c

so v˙ = x˙ − u˙ − ku/c, and finally,



ku ku ∈ ασ x˙ − u˙ − . c

(14.6)

Inspecting Eq. (14.6) as well as the function in (14.2), the following conclusion appears: ku ∈ [α, α].


-1 0

1

x

Figure 14.3: The discontinuous shape of β(x) function. Now, a new representation is necessary to introduce. Therefore, let the β(x) function (see Fig. 14.3) be defined: ⎧ ⎪ 0, x ∈ [−∞, −1] ∪ [1, +∞], ⎪ ⎪ ⎪ ⎨{0}, x ∈ [−1, 1], (14.7) β(x) = ⎪ R− , x = −1, ⎪ ⎪ ⎪ ⎩ + R , x = 1. Rearrangement of Eq. (14.6) and substitution of Eq. (14.7) provides

ku ku . u˙ + β  x˙ − c c

(14.8)

page 224

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Controlling Stochastically Excited Systems with an Approximate Discontinuity

ws-book975x65

225

The pair of equations (14.5) and (14.8) with these initial conditions: x(0) = x0 [m],

x(0) ˙ = x˙ 0 [m/s],

u(0) = u0 [m]

(14.9)

will describe our model shown in Fig. 14.2. Taking the substitutions η = α/k [m], y = x˙ [m/s] and the system of equations (14.5) and (14.8) with initial conditions, Eq. (14.9) is equivalent to:

u F − ku ku , u˙ + β x˙ = y, y˙ =  y− (14.10) m η c with the complementary initial conditions: x(0) = x0 ,

x(0) ˙ = x˙ 0 ,

u(0) = u0 ∈ [−η, η].

(14.11)

The form of the full system (14.10)–(14.11) is said to be the boundary one if the damping c tends to infinity. It does denote physically a very large viscosity of the damper approximating some behavior of the ground. The system in Fig. 14.2 is reduced to a simpler counterpart structural form composed of a spring and the discontinuous SV-element. The model refers therefore to the Prandtl’s model described by the mathematical representation:

u F − ku , u˙ + β  y. x˙ = y, y˙ = m η 14.3

The 2-DOF Mechanical Model

For the aim of modeling the dynamics of building constructions we use a system of an undeformable mass connected to a spring of the stiffness k and a dashpot characterized by the constant c.

k1

c1

k0


F1

m1 c2

F2

x1 k2

m2

x2

Figure 14.4: Structural model of the 2-DOF building-ground system without excitation from the ground.

page 225

June 8, 2017 12:9

ws-book961x669

226

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

The structural model shown in Fig. 14.4 is mathematically described by the system of equations: ⎧ ⎪ x˙ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 2 y˙ 1 ⎪ ⎪ y ˙2 ⎪ ⎪ ⎪ ⎪ ⎩ u˙ 1

= y1 , = y2 , = m11 (F1 − k0 x1 − k1 u1 − c1 y1 + k2 (x2 − x1 ) + c2 (y2 − y1 )) , = m1'2 (F(2 − k2 (x2 − x1 ) − c2 (y2 − y1 )) , +β

u1 η1

(14.12)

 y1 ,

and initial conditions: xi (0) = xi,0 ,

14.3.1

x˙ i (0) = x˙ i,0 ,

u1 (0) = u1,0 ∈ [−η1 , η1 ].

Time Histories of State Variables

The two-degree-of-freedom system is now under the periodic forcing from two external forces A1 cos (wt) and A2 cos (wt) and is attached to the first (from the left side) and the second mass, respectively. Time histories and phase-planes are usually the first graphical representations of a dynamical behavior. Let the system parameters be assumed: A1 = 0.06, A2 = 0.01 N, m1 = m2 = 1 kg, k0 = 2, k1 = k2 = 1 N/m, α = 0.02 N, c1 = c2 = 0.05 N·s/m, ω = 0.81 rad/s, and the initial conditions as well: x1,0 = x2,0 = 0 m, y1,0 = y2,0 = 0 m/s, u1,0 = 0 m. Free and periodically excited vibrations with a frequency of excitation ω are observed in the considered system. Because of existing damping the free vibrations decay during the transitory stage (see Fig. 14.5), while each time trajectory xi (t) of the only harmonic vibrations is given in the form xi (t) = ai cos (ωt + ϕi ),

(14.13)

where ai is the maximal amplitude, ϕi is the phase angle’s shift between the excitation and the system response, and ω represents the excitation frequency.

14.3.2

The Amplitude–Frequency Diagrams

The square norm of an xi process given by the formula (14.13) is

2

||xi || =



2π/ω 0

a2i cos 2 (ωτ + ϕi )dτ.

(14.14)

page 226

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Controlling Stochastically Excited Systems with an Approximate Discontinuity

227

Let the solution of Eq. (14.14) be expected in the form:  2π/ω 1 + cos (2(ωτ + ϕi )) a2i dτ 2 0  2π/ω  2π/ω 1 cos (2(ωτ + ϕi )) 2 2 = ai dτ + ai dτ 2 2 0 0 2 "2π/ω

"2π/ω " ai t "" 2 sin (2(ωt + ϕi )) " + a = i " " 2 2ω 0 0



 a2i 2π a2i 2π −0 + ω + ϕi = − sin (2(0 · ω + ϕi )) sin 2 2 ω 2ω ω a2 π = a2i + i ( sin (4π + 2ϕi ) − sin (2ϕi )) ω 2ω a2 π = a2i + i ( sin (4π) cos (2ϕi ) + cos (4π) sin (2ϕi ) − sin (2ϕi )) ω 2ω 2π = ai = ||xi ||2 , ω 0.3

1

x1

x2

-0.4 1945

t

-1 1945

1990

t

(a) 0.2

0.6

y1

y2

-0.2 1945

t

1990

-0.8 1945

t

(c)

1990

(d)

0.2

0.6

y1

y2

-0.2 -0.4

1990

(b)

-0.8

x1

0.3

x2

-1

(e)

1

(f) 0.02

u1

-0.02 1945

t

1990

(g)

Figure 14.5: Time histories of the displacements xi (t) and u1 (t) (a, b, g), velocities yi (t) (c, d) and phase-planes yi (xi ) (e, f).

page 227

June 8, 2017 12:9

ws-book961x669

228

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

hence

 |ai | =

ω π



2π/ω 0

x2i (τ )dτ .

Using the numerically estimated time histories xi (t) of the system (14.12) at each time step, it enables us to calculate the angular frequency dependent amplitude of the process, by an assumption, a stochastic one. Analogously to the traditionally prepared resonance diagrams of 1-DOF systems (as the normalized dependency ai /xstatic (ω/α), where xstatic = Astatic /k is caused by the force of static loading of the spring of a stiffness k), and the |ai (ω)| characteristics describes the amplitude versus frequency of the system’s free vibrations. 0.5

1.4

k=1 k=3 k = 10

a1

0 0.5

k=1 k=3 k = 10

a2




3.5

0 0.5

(a)




3.5

(b)

Figure 14.6: Amplitude-frequency diagrams for k = 1, 3, 10 N/m: (a) |a1 (ω)|; (b) |a2 (ω)|. The graphical representation shown in Fig. 14.6 permits for the determination of free vibrations. Regions of a resonance occurrence are placed at the frequency values at which the excitation frequency ω approaches the frequency of free vibrations — at these points the amplitude of wave is the maximal one.

14.4

The Approximate System with Friction

For the aim of creation of the counterpart — an approximate system to the discontinuous one, we intend to find for each subsystem the equivalent coefficients of damping. These subsystems reduce the initially introduced 2-DOF dynamical system to two equivalent 1-DOF systems. 14.4.1

Energetic Criterion

The source of inner resistance appearing in building constructions is often connected with non-elastic micro-deformations of material. An absorption of a part of mechanical energy, which is successively dissipated into heat, characterizes any element of building construction. One can estimate the dissipating energy in an experimental

page 228

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

Controlling Stochastically Excited Systems with an Approximate Discontinuity

ws-book975x65

229

way by calculating the surface of the so-called “spring-hysteresis”. The energy dissipated in one cycle by a force of viscous damping of one degree of freedom is given by the formula:  T /4  T /4  A dx dx dτ = 4 cxdx ˙ =4 c c(x) ˙ 2 dτ, E=4 dτ dτ 0 0 0 where cx˙ is the damping force acting during a displacement from 0 to A. If x = A sin (ωt), then in the cycle T , the energy dissipation equals:  π/(2ω) cA2 ω 2 cos 2 (ωt)dτ, E=4 0

= 2cωA2 ( cos (ωt)sin(ωt) + ωt)|0

= πcωA2 . 6A Analogously, the SV-element’s dissipation of energy equals 0 Fˆ dx, which is the work done by the friction force Fˆ on the distance A. In any moment of time the π/(2ω)

force acts in opposite direction to the ongoing motion so  A πcsv ωA2 = Fˆ dx, 0

and after rearrangement, the equivalent coefficient of damping of the SV-element csv is 6A Fˆ dx . csv = 0 πωA2 Each SV-element of the two subsystems will have the following representation of equivalent damping coefficients: 6A (Fi (t) − mi x ¨)dx csv,i = 0 , for i = 1, 2. 2 πωA 14.4.2

Normal Modes of Vibrations

Let us inspect the normal modes of vibrations of the two-degree-of-freedom linear system shown in Fig. 14.7. Matrix form of the analyzed system’s differential equations is given ¨¯ + Cx Mx ¯˙ + K¯ x = F¯ ,

(14.15)

where x ¯ is the vector of displacement, M, C, K — mass, damping and stiffness matrices, respectively, F¯ — the vector of external forces. By an assumption that C = 0 and F¯ = 0, one writes ¨¯ + K¯ Mx x = 0.

(14.16)

Let a solution of Eq. (14.16) be given in the form of a harmonic function of known frequency ω: x ¯(t) = x ˜ sin (ωt), where x ˜ is the vector of free oscillation amplitudes, representing the displacement of the system’s mass elements in a direction

page 229

June 8, 2017 12:9

ws-book961x669

230

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

c1

F1

k1

m1 c2

F2

x1 k2

m2

x2

Figure 14.7: Structural model of the investigated 2-DOF system. of generalized coordinates. Putting the above in Eq. (14.16) and assuming that the new solution has to be met in each moment of time, the following system of linear algebraic equations in a matrix form holds:   2 ˜ = 0, −ω M + K x and it is solvable if det(−ω 2 M + K) = 0. Expansion of the determinant provides the nth degree polynomial with respect to ω 2 . Afterwards, the roots ω1 , ω2 can be ˜=x ˆi , therefore found. Each frequency ωi corresponds to the solution x  2  ˆ¯i = 0, −ω M + K x ˆ where x ¯i is the ith eigenvector or the ith mode of normal oscillations. Two eigenˆ¯i , i.e.: vectors exist in our case. Let P¯ be defined as a vector of solutions x P¯ = [ˆ x1 , x ˆ2 ],

and

x ¯ = P¯ q¯ = P¯ [q1 , q2 ]T .

Substituting x ¯ in Eq. (14.15) and multiplying it by P¯ −1 , one obtains P¯ −1 MP¯ q¨ ¯ + P¯ −1 CP¯ q¯˙ + P¯ −1 KP¯ q¯ = P¯ −1 F¯ (t). One assumes that

  ˜ sv = c˜st,1 0 , P¯ −1 CP¯ = C 0 c˜st,2

(14.17)

where parameters c˜st,i are the equivalent coefficients of damping for the first and second subsystem, respectively. The two-degree-of-freedom system is then described by:  q¨1 + c˜st,1 q˙1 + ω12 q1 = F˜1 (t), q¨2 + c˜st,2 q˙2 + ω12 q1 = F˜2 (t), where

6A c˜st,i =

0

(F˜i − q¨i )dqi μi dqi = . πωqi2 πωqi2

page 230

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

ws-book975x65

1st Reading

Controlling Stochastically Excited Systems with an Approximate Discontinuity

231

Taking into the analysis the effect of dry friction, one requires to define μi as follows: ¨1 ) /p11 + (F2 (t) − m2 x ¨2 ) /p12 , μ1 = (F1 (t) − m1 x ¨1 ) /p21 + (F2 (t) − m2 x ¨2 ) /p22 , μ2 = (F1 (t) − m1 x q1 = x1 /p11 + x2 /p12 , q2 = x1 /p21 + x2 /p22 . ˜ The matrix Csv can be defined if μi and qi are known. Having it found, firstly, let us do the right-hand side multiplication of Eq. (14.17) by P¯ P¯ −1 . Hence we get ˜ sv P¯ −1 . (14.18) P¯ P¯ −1 CP¯ P¯ −1 = P¯ C In the second step, simplifying Eq. (14.18), a definition of the matrix C yields:   ˜ = P¯ C ˜ sv P¯ −1 = c˜11 c˜12 , c˜12 = c˜21 . C=C c˜21 c˜22 Finally, the matrix representation of equations of motion of the vicarious system takes the form ˜x ¨¯ + C Mx ¯˙ + K¯ x = F¯ .

14.5

Control of the 2-DOF System Under a Stochastic Excitation

Current knowledge about analysis of this particular dynamical system allows us to describe this kind of seismic excitation (an exemplary form) by the filtrated white noise [Lin and Yong (1987); Beards (1996)]. 2.5

xs 0

-2.5 0

2.4

4.8

t

7.2

9.6

12

Figure 14.8: Time history of the signal xs (t) defined by Eq. (14.19) of noisy displacement of ground: ωd = 18.87 rad/s, ξd = 0.65, t1 = 3 s, t2 = 10 s and c = 0.76 1/s. A stochastic loading generated in Fig. 14.8 of our system in Fig. 14.9 comes from the ground and is estimated by means of a linear filtering of white noise. The used irregular time-dependent displacement xs is cast in the following form: xs (t) = Gs (t)za (t),

(14.19)

page 231

June 8, 2017 12:9

232

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

where za is the n-dimensional vector, representing the state of the seismic excitation model z¨a (t) = 2ξd ωd z˙a (t) + ωd2 z(t) + (t).

m1 c1

x1 k1

m2 c2

(14.20)

x2 k2

xs Figure 14.9: Structural model of the 2-DOF system loaded from the ground. Parameters ωd and ξd represent any local ground conditions. Equation (14.20) possesses the first-order differential equation’s form z˙a (t) = Az za (t) + Bz (t),

(14.21)

where matrices Az and Bz have the following representations:     0 1 0 , Bz = Az = , −ωd2 −2ξd ωd 1 and  denotes a stationary white noise defined as a boundary of Ornstein– Uhlenbeck, whereas Gs (t) is an envelope of za accordingly defined: ⎧ ⎪ 0, t < 0, ⎪ ⎨ ' (2 t Gs (t) = , 0 ≤ t ≤ t1 , t1 ⎪ ⎪ ⎩ exp(−c(t − t2 )), t > t2 , where c is a constant value, and t1 and t2 are the initial and final time of duration of the excitation signal, respectively. General form of the system under external loading is x ¯˙ (t) = A¯ x(t) + B(t)¯ u(t) + D(t)¯ z (t),

(14.22)

where A is the (n × n) matrix of structure parameters, B — (n × p) matrix of executing (regulatory) elements (inputs), x ¯ — the n-dimensional state vector of the system, p — number of control inputs, z¯ — the s-dimensional vector of loading. The (n × s) matrix D(t) indicates points of application of external loading

page 232

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Controlling Stochastically Excited Systems with an Approximate Discontinuity

233

forces (even stochastic). Merged Eq. (14.21) and (14.22) create a simplified twodegree-of-freedom system, describing the investigated model          x ¯(t) x ¯˙ (t) A DGs (t) B 0 (t). = + u ¯(t) + 0 Az za (t) za (t) 0 Bz Knowledge of xs and its derivative permits us to propose the system of equations:  ¨1 (t) = −u(t) + k1 (xs (t) − x1 (t)) + c1 (x˙ s (t) − x˙ 1 (t)), m1 x ¨2 (t) = u(t). m2 x The control law u(t) is proposed (see in [Awrejcewicz and Olejnik (2007a)]) ¯(t) + Kz Gs (t)za (t)) , u ¯(t) = −R−1 BT (Kx x where the Riccati and Lyapunov solution matrices need to be numerically computed:  ˙ x = −Kx A − AT Kx − Q + Kx BR−1 BT Kx , K with Kx (tf ) = θ(tf ), ˙ z = −Kx D − Kz Az + AT Kz − S + Kx BR−1 BT Kz , with Kz (tf ) = 0, K where tf is the final time of integration and the matrices are as follows: ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ q1 + q2 −q2 0 0 0 0 0 1 0 ⎢ −q2 q2 0 0 ⎥ ⎢ 0 ⎥ ⎢ 0 0 0 1⎥ ⎢ ⎢ ⎥ ⎥ ⎥, A=⎢ ⎣ −k1 /m1 0 −c1 /m1 0 ⎦ , B = ⎣ −1/m1 ⎦ , Q = ⎣ 0 0 0 0⎦ 1/m2 0 0 0 0 0 0 00 ⎤ ⎡ 0 0     ⎢ 0 0 ⎥ ⎥ , ST = −q1 0 0 0 , Az = 0 1 , θ = 0, R = r, D=⎢ ⎣ k1 /m1 c1 /m1 ⎦ z1 z2 0 000 0 0 and q1 , q2 , r are some weighting coefficients of quality and reaction matrices. 14.6

Numerical Simulation

A comparison of the exact (with the SV-element) and the approximated counterpart (with the viscous damping) of the analyzed 2-DOF system studied in Sec. 14.4 is performed; see Fig. 14.10. The numerical simulation takes the system parameters: A1 = 6, A2 = 0.01 N, m1 = m2 = 1 kg, k0 = 2, k1 = k2 = 1 N/m, α = 0.04 N, c1 = 0.025, c2 = 0.25 N·s/m, and the initial conditions: x1,0 = x2,0 = 0 m, y1,0 = y2,0 = 0 m/s, u1,0 = 0 m. Next to the dependence on the frequency of external excitation, the numerical results of first part of this section confirm a stiff dependence of damping coefficients also on both A1 and A2 — the amplitudes of forces F1 and F2 . The comparisons shown in Fig. 14.10 confirm an acceptable coincidence of the |ai (ω)| curves. Now, in the second part, let us solve our control problem from Sec. 14.5 by incorporating into the simulation such a set of system parameters: m1 = 0.15 · 104 ,

page 233

June 8, 2017 12:9

ws-book961x669

234

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

12

30

original aproximate

a1

original aproximate

a2

0 0.5




0 0.5

3




(a)

3

(b)

Figure 14.10: Amplitude-frequency diagrams for the original system and its approximated counterpart: (a) |a1 (ω)|; (b) |a2 (ω)|. m2 = 1.1 · 104 kg, k1 = 9.75 · 105 , k2 = 5.3 · 104 N/m, c1 = 1.08 · 104 , c2 = 1.5 · 104 N·s/m, z1 = −ωd2 , z2 = −2ξd ωd , q1 = 10, q2 = 1, r = 2 · 10−10 . Figure 14.11 shows, that for a properly chosen weighting coefficients, the displacement of the top mass m2 subjected to the active control is better damped. 7.5

passive active

xs 0

-7.5 0

2.4

4.8

t

7.2

9.6

12

Figure 14.11: Time history of the passive and active control of the system. A few numerical experiments devoted to analysis and control of a two-degreeof-freedom mass-spring system are provided. The transformation of the analyzed system with friction to the approximated counterpart is acceptable and can be verified also during application of further control schemes. The presented approach can be used to model a more advanced multibody system that is externally excited by a stochastic type irregular forces. In the theory of earthquakes, such irregularities are commonly explored in a variety of forms, since the theory of vibrations of machine constructions finds even more solutions for damping of undesirable vibrations.

page 234

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Bibliography

Acharyya, M. (2015). Patterns, dynamics and phase transitions in Ising ferromagnet driven by propagating magnetic field wave, Journal of Physics: Conference Series 638, 1, p. 012008. Adams, G. G. (1996). Self-excited oscillations in sliding with a constant friction coefficient — a simple model, J. Tribol.-T. ASME 118, 4, pp. 819–823. Adams, G. G. (2000). Radiation of body waves induced by the sliding of an elastic halfspace against a rigid surface, J. Appl. Mech.: T. ASME 67, 1, pp. 1–5. Adams, G. G. and Nosovsky, M. (2000). Contact modeling — forces, Tribol. Int. 33, 5-6, pp. 431–442. Adams, J. and Payandeh, S. (1996). Methods for low-velocity friction compensation: Theory and experimental study, J. Robotic. Syst. 13, 6, pp. 391–404. Ageno, A. and Sinopoli, A. (2005). Lyapunov’s exponents for non-smooth dynamics with impacts: Stability analysis of the rocking block, Int J Bifurcation Chaos 15, 6, pp. 2015–2039. Ahn, S. W. (2001). The effects of roughness types on friction coefficients and heat transfer in roughened rectangular duct, Int. Commun. Heat Mass Transfer 28, 7, pp. 933– 942. Akay, A. (2002). Acoustics of friction, J. Acoust. Soc. Am. 111, 4, pp. 1525–1548. Altpeter, F., Ghorbel, F., and Longchamp, R. (1998). Relationship between two friction models: A singular perturbation approach, in Proceedings of 37th IEEE Conference on Decision and Control (Florida, USA), pp. 1572–1574. Amassad, A., Shillor, M., and Sofonea, M. (1999). A quasistatic contact problem with slip-dependent coefficient of friction, Math. Method. Appl. Sci. 2, pp. 267–284. Andreaus, V. and Casini, P. (2000). Dynamics of friction oscillators excited by a moving base and/or driving force, J. Sound Vib. 245, 4, pp. 685–699. Angelov, T. A. and Liolios, A. A. (2004). An iterative solution procedure for Winkler-type contact problems with friction, Z. Angew. Math. Mech. 84, pp. 136–143. Anooshehpoor, R. and Brune, J. N. (1994). Frictional heat generation and seismic radiation in a foam rubber model of earthquakes, Pageoph 142, pp. 735–747. Appell, J., Lakshmikantham, V., Van Minh, N., and Zabrejko, P. P. (1993). A general model of evolutionary processes. Exponential dichotomy — I, II, Nonlinear Analysis TMA 21, pp. 207–218, 219–225. Appell, J., Moroz, V. B., Vignoli, A., and Zabrejko, P. P. (1994). On the application of Kielh¨ ofer’s bifurcation theorem to Hammerstein equations with potential nonlinearity, Boll. Unione Math. Ital. 8-B, pp. 833–850. Appell, J. and Zabrejko, P. P. (1993). Bifurcation points for integral equations of Ham-

235

page 235

June 8, 2017 12:9

236

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

merstein type, J. Integral Equations Operator Theory 16, pp. 15–37. Arjun Patil, P. and Teodoriu, C. (2012). Model development of torsional drillstring and investigating parametrically the stick–slips influencing factors, J. Energy Resour. Technol. 135, 1, pp. 013103–013103, doi:10.1115/1.4007915. Arnold, V. I. (2010). Mathematical Methods of Classical Mechanics, 2nd edn. (Springer). Asmuth, H., Smitt, P., Elsaesser, B., and Henry, A. (2015). Determination of non-linear damping coefficients of bottom-hinged oscillating wave surge converters using numerical free decay tests, Tech. rep., Queen’s University Belfast – Research Portal. ˚ Astr¨ om, K. J. (1995). Control of systems with friction, Tech. Rep. 759, Swedish Research Council for Eng. Science. Aubin, J. P. and Celina, A. (1984). Differential Inclusions (Springer-Verlag, Berlin– Heidelberg–New York). Avci, O. (2016). Amplitude-dependent damping in vibration serviceability: Case of a laboratory footbridge, J. Archit. Eng. 04016005, doi:10.1061/(ASCE)AE.1943-5568. 0000211. Awrejcewicz, J. (1987). Analysis of self-excited vibration in mechanical system with four degrees of freedom, in Scientific Bulletin, Vol. 72 (Lodz University of Technology), pp. 5–27. Awrejcewicz, J. (1989). Bifurcation and Chaos in Simple Dynamical Systems (World Scientific, Singapore). Awrejcewicz, J. (1990). Parametric and self-excited vibrations induced by friction in a system with three degrees of freedom, J. Mech. Sci. Technol. 4, 2, pp. 156–166. Awrejcewicz, J. (1991). Bifurcation and Chaos in Coupled Oscillators (World Scientific, Singapore). Awrejcewicz, J. (2007). Technical Mechanics (WNT, Warsaw). Awrejcewicz, J. and Delfs, J. (1990a). Dynamics of a self-exicted stick–slip oscillator with two degrees of freedom. Part I: investigation of equilibria, Eur. J. Mech. A-Solid 9, 4, pp. 269–282. Awrejcewicz, J. and Delfs, J. (1990b). Dynamics of a self-exicted stick–slip oscillator with two degrees of freedom. Part II: slip–stick, slip–slip, stick–slip transitions, periodic and chaotic orbits, Eur. J. Mech. A-Solid 9, 5, pp. 397–418. Awrejcewicz, J., Feˇckan, M., and Olejnik, P. (2005). On continuous approximation of discontinuous systems, Nonlinear Anal.: Theor. 62, pp. 1317–1331. Awrejcewicz, J., Feˇckan, M., and Olejnik, P. (2006). Bifurcations of planar sliding homoclinics, Math. Probl. Eng. 2006, pp. 1–13. Awrejcewicz, J. and Holicke, M. M. (1999). Melnikov’s method and stick–slip chaotic oscillations in very weekly forced mechanical systems, Int. J. Bifurcation Chaos 9, 3, pp. 505–518. Awrejcewicz, J. and Lamarque, C. H. (2003). Bifurcation and Chaos in Nonsmooth Mechanical Systems, Nonlinear Science (World Scientific, Singapore). Awrejcewicz, J. and Olejnik, P. (2002a). Calculating Lyapunov exponents from an interpolated time series, in XX Symposium — Vibrations in Physical Systems (Pozna´ nBla˙zejewko), pp. 94–95. Awrejcewicz, J. and Olejnik, P. (2002b). Numerical analysis of self-excited by friction chaotic oscillations in two-degrees-of-freedom system using exact H´enon method, Machine Dynamics Problems 26, 4, pp. 9–20. Awrejcewicz, J. and Olejnik, P. (2003a). Regular and chaotic stick–slip dynamics in a selfexcited two-degrees-of freedom system with friction, Int. J. Bifurcation Chaos 4, 13, pp. 843–861. Awrejcewicz, J. and Olejnik, P. (2003b). Stick–slip dynamics of a two-degree-of-freedom

page 236

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

237

system, Int. J. Bifurcation Chaos 13, 4, pp. 843–861. Awrejcewicz, J. and Olejnik, P. (2005a). Analysis of dynamic systems with various friction laws, Appl. Mech. Rev. 58, 6, pp. 389–411. Awrejcewicz, J. and Olejnik, P. (2005b). Friction pair modeling by 2-DOF system: Numerical and experimental investigations, Int. J. Bifurcation Chaos 15, 6, pp. 1931–1944. Awrejcewicz, J. and Olejnik, P. (2005c). Sliding solutions of a simple two degrees-offreedom dynamical system with friction, in Proceedings of 5-th EUROMECH, Nonlinear Dynamics Conference, ENOC (Eindhoven, The Netherlands), pp. 277–282. Awrejcewicz, J. and Olejnik, P. (2007a). Active control of two degrees-of-freedom buildingground system, Archives of Control Sciences 17, 4, pp. 357–372. Awrejcewicz, J. and Olejnik, P. (2007b). Occurrence of stick–slip phenomenon, J. Theoret. Appl. Mech. 45, 1, pp. 33–40. Awrejcewicz, J. and Pyryev, Y. (2002). Thermoelastic contact of a rotating shaft with a rigid bush in conditions of bush and stick–slip movements, Internat. J. Engrg. Sci. 40, pp. 1113–1130. Banerjee, K. (1968). Influence of kinetic friction on the critical velocity of stick–slip motion, Wear 12, pp. 107–116. Bassan, M., de Marchi, F., Marconi, L., Pucacco, G., Stanga, R., and Visco, M. (2013). Torsion pendulum revisited, Phys. Lett. A 377, 25–27, pp. 1555–1562. Battelli, F. and Feˇckan, M. (2002). Chaos arising near a topologically transversal homoclinic set, Topol. Methods Nonlinear Anal. 20, pp. 195–215. Battelli, F. and Feˇckan, M. (2008). Homoclinic trajectories in discontinuous systems, J. Dynam. Differential Equations 20, pp. 337–376. Battelli, F. and Feˇckan, M. (2010a). Bifurcation and chaos near sliding homoclinics, J. Differential Equations 248, pp. 2227–2262. Battelli, F. and Feˇckan, M. (2010b). An example of chaotic behaviour in presence of a sliding homoclinic orbit, Ann. Mat. Pura Appl. (4) 189, pp. 615–642. Battelli, F. and Feˇckan, M. (2011). On the chaotic behaviour of discontinuous systems, J. Dynam. Differential Equations 23, pp. 495–540. Battelli, F. and Feˇckan, M. (2013). Chaos in forced impact systems, Disc. Cont. Dyn. Sys.-S 6, pp. 861–890. Battelli, F. and Lazarri, C. (1990). Exponential dichotomies, heteroclinic orbits, and Melnikov functions, J. Differential Equations 86, 2, pp. 342–366. Baumgart, A. (2000). Stick–slip and bit-bounce of deep-hole drillstrings, J. Energy Res. Technol. 122, 2, pp. 78–82, doi:10.1115/1.483168. Bay, N. (1987). Friction stress and normal stress in bulk forming processes, J. Mech. Work. Technol. 14, pp. 203–223. Beards, C. F. (1996). Engineering Vibration Analysis with Application to Control Systems (Edward Arnold, London). Behn, C. and Steigenberger, J. (2012). Worm-Like Locomotion Systems: An Intermediate Theoretical Approach (Oldenbourg Wissensch Vlg.). Bell, R. and Burdekin, M. (1969). A study of the stick–slip motion of machine tool feed drives, Proc. Inst. Mech. Eng. 184, 1, pp. 543–560. Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J. M. (1980). Lyapunov exponents for smooth dynamical systems and Hamiltonian systems: a method for computing all of them. Part 1: Theory, Meccanica 15, pp. 9–30. Bengisu, M. T. and Akay, A. (1994). Stability of friction-induced vibrations in multidegree-of-freedom systems, J. Sound Vibration 171, 4, pp. 557–570. Berger, E. J. (2002). Friction modeling for dynamic system simulation, Appl. Mech. Rev. 55, 6, pp. 535–577.

page 237

June 8, 2017 12:9

238

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Bliman, P. A. (1992). Mathematical study of the Dahl’s friction model, Eur. J. Mech. A-Solid 11, 6, pp. 835–848. Bliman, P. A. and Sorine, M. (1993). A system theoretic approach of systems with hysteresis. Application to friction modelling and compensation, in Proceedings of 2nd European Control Conference (Groningen, The Netherlands), pp. 1844–1849. Bliman, P. A. and Sorine, M. (1995). Easy-to-use realistic dry friction models for automatic control, in Proceedings of 3rd European Control Conference (Rome, Italy), pp. 3788– 3794. Bo, L. C. and Pavelescu, D. (1982). The friction-speed relation and its influence on the critical velocity of stick–slip motion, Wear 82, pp. 277–289. Bogacz, R., Irretier, H., and Sikora, J. (1990). On discrete modelling of contact problems with friction, Z. Angew. Math. Mech. 70, 4, pp. T31–T32. Bogacz, R. and Ryczek, B. (1997). Dry friction self-excited vibrations. Analysis and experiment, Engineering Transactions 45, 3–4, pp. 487–504. Bothe, D. (1999). Periodic solutions of non-smooth friction oscillators, Z. Angew. Math. Phys. 50, pp. 779–808. Bouissou, S., Petit, J. P., and Barquins, M. (1998a). Experimental evidence of contact loss during stick–slip: possible implications for seismic behavior, Tectonophysics 295, pp. 341–350. Bouissou, S., Petit, J. P., and Barquins, M. (1998b). Normal load, slip rate and roughness influence on the PMMA dynamics of sliding. Part 1: Stable sliding to stick–slip transition, Wear 214, pp. 156–164. Bouissou, S., Petit, J. P., and Barquins, M. (1998c). Normal load, slip rate and roughness influence on the PMMA dynamics of sliding. Part 2: Characterisation of the stick– slip phenomenon, Wear 215, pp. 137–145. Bourgeot, J. M. and Brogliato, B. (2005). Tracking control of complementarity Lagrangian systems, Int. J. Bifurcation Chaos 15, 6, pp. 1839–1866. Bowden, F. P. and Tabor, D. (1939). The area of contact between stationary and betwen moving surfaces, Proc. R. Soc. London, Ser. A , pp. 391–413. Bowden, F. P. and Tabor, D. (1950). The Friction and Lubrication of Solids (Clarendon Press). Brace, W. F. and Byerlee, J. D. (1996). Stick–slip as a mechanism for earthquakes, Science 153, pp. 990–992. Brandl, M. and Pfeiffer, F. (1999). Tribometer for dry friction measurement, in T. ASME, DETC and Computers and Information in Engineering Conferences, pp. DETC99/VIB–8353. Bristow, J. R. (1947). Kinetic boundary friction, Proc. R. Soc. London, Ser. A 189, pp. 88–102. Brockley, C. A. and Davis, H. R. (1968). The time dependence of static friction, J. Lubr. Technol. 90, pp. 35–41. Brogliato, B. (1996). Nonsmooth Impact Mechanics, Lecture Notes in Control and Information Sciences (Springer, Berlin). Brogliato, B., Niculescu, S. I., and Monteiro-Marques, M. D. P. (2000). On tracking control of a class of complementary-slackness mechanical systems, System & Control Letters 39, 4, pp. 255–266. Brogliato, B., Niculescu, S. I., and Orhant, P. (1997). On the control of finite dimensional mechanical systems with unilateral constraints, IEEE T. Automat. Contr. 42, 2, pp. 200–215. Brune, J. N., Brown, S., and Johnson, P. A. (1993). Rupture mechanism and interface separation in foam rubber models of earthquakes: a possible solution to the heat

page 238

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

239

flow paradox and the paradox of large overthrusts, Tectonophysics 218, pp. 59–67. Budd, C. and Lamba, H. (1994). Scaling of Lyapunov exponents at nonsmooth bifurcations, Phys. Rev. E 50, pp. 84–90. Burridge, R. and Knopoff, L. (1967). Model and theoretical seismicity, Bull. Seismol. Soc. Am. 57, pp. 341–371. Butterworth, J., Lee, J. H., and Davidson, B. (2004). Experimental determination of modal damping from full scale testing, in 13th World Conference on Earthquake Engineering, 310 (Vancouver, B.C., Canada), pp. 1–15. Cadoni, M., de Leo, R., and Gaeta, G. (2013). Solitons in a double pendulums chain model, and DNA roto-torsional dynamics, J. Nonlinear Math. Phys. 14, 1, pp. 128–146. Cai, L. and Lamba, H. (1994). Joint stick–slip friction compensation of robot manipulators by using smooth robust controllers, J. Robotic. Syst. 11, 6, pp. 451–470. Canudas de Wit, C., Olson, H., ˚ Astr¨ om, K. J., and Lischinsky, P. (1995). A new model for control systems with friction, IEEE T. Automat. Contr. 40, 3, pp. 419–425. Carlson, J. M. and Langer, J. S. (1989). Properties of earthquakes generated by fault dynamics, Phys. Rev. Lett. 62, 22, pp. 2632–2635. Carlson, J. M., Langer, J. S., and Shaw, B. E. (1991). Intrinsic properties of a BurridgeKnopoff model of an earthquake fault, Phys. Rev. A 44, 2, pp. 884–897. Carranza, J. C., Brennan, M. J., and Tang, B. (2015). Sources and propagation of nonlinearity in a vibration isolator with geometrically nonlinear damping, J. Vib. Acoust. 138, 2, pp. 024501–024501, doi:10.1115/1.4031997. Cha, H.-Y., Choi, J., Ryu, H., and Choi, J. (2011). Stick–slip algorithm in a tangential contact force model for multi-body system dynamics, J. Mech. Sci. Technol. 25, pp. 1687–1694, doi:10.1007/s12206-011-0504-y. Chatterjee, S. and Saha, A. (2007). On the theoretical basis of vibro-frictional actuation in microsystems, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 221, pp. 119–133. Chenafa, M., Mansouri, A., Bouhenna, A., Etien, E., Belaidi, A., and Denai, M. A. (2005). Global stability of linearizing control with a new robust nonlinear observer of the induction motor, Int. J. Appl. Math. Comput. Sci. 15, 2, pp. 235–243. Chernov, N. and Markarian, R. (2006). Chaotic Billiards, Mathematical Surveys and Monographs, Vol. 127 (Amer. Math. Soc.). Chiason, J. (1997). A new approach to dynamic feedback linearization control of an induction motor, IEEE T. Automat. Contr. 43, 3, pp. 391–397. Chicone, C. (2006). Ordinary Differential Equations with Applications (Springer, New York). Chillingworth, D. R. J. (2002). Discontinuous geometry for an impact oscillator, Dynamical Systems 17, pp. 389–420. Choi, H. S. and Lou, J. Y. K. (1991). Nonlinear behavior and chaotic motions of an SDOF system with piecewise-nonlinear stiffness, Internat. J. Non-Linear Mech. 26, pp. 461–473. Chow, S. N. and Hale, J. K. (1982). Methods of Bifurcation Theory (Springer, New York). Copra, A. K. (2000). Theory and Applications to Earthquake Engineering (Prentice Hall, New York). Coullet, P., Gilli, J. M., and Rousseaux, G. (2009). On the critical equilibrium of the spiral spring pendulum, P. Roy. Soc. A: Math. Phy. 466, pp. 407–421. Coulomb, C. A. (1809). Theorie des Machines Simples-en Ayant en Regard au Frottement de Leures Parties et a la Roideur des Cordages (Bachelier, Libraire, Quai des Augustins, Paris). Courant, R. and John, F. (1989). Introduction to Calculus and Analysis, Vol. II (Springer, New York).

page 239

June 8, 2017 12:9

240

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Cruciat, R. and Ghindea, C. (2012). Experimental determination of damping characteristics of structures, Mathematical Modelling in Civil Engineering 4, pp. 51–59. Cunningham, W. J. (1958). Introduction to Nonlinear Analysis (McGraw Hill Company, New York). Czerwi´ nski, E., Olejnik, P., and Awrejcewicz, J. (2015). Modeling and parameter identification of vibrations of a double torsion pendulum with friction, Acta Mechanica et Automatica 9, 4, pp. 209–217. Dahl, P. R. (1976). Solid friction damping of mechanical vibrations, AIAA Journal 14, 12, pp. 1675–1682. de Baets, P., Degrieck, J., and van de Velde, F. (2000). Experimental verification of the mechanisms causing stick–slip motion originating from relative deceleration, Wear 243, pp. 48–59. De Marchi, F., Pucacco, G., Bassan, M., De Rosa, R., Di Fiore, L., Garufi, F., Grado, A., Marconi, L., Stanga, R., Stolzi, F., and Visco, M. (2013). A quasi-complete mechanical model for a double torsion pendulum, Phys. Rev. D Part. Fields 87, 122006, pp. 1–15. Deimling, K. (1992). Multivalued Differential Equations (Walter de Gruyter). Deimling, K., Hetzer, G., and Shen, W. X. (1996). Almost periodicity enforced by Coulomb friction, Adv. Differential Equations 1, 2, pp. 265–281. Den Hartog, J. P. (1931). Forced vibrations with combined viscous and Coulomb damping, Phil. Mag. 7, 9, pp. 801–817. Den Hartog, J. P. (1956). Mechanical Vibrations (McGraw-Hill). Deufhard, P., Fiedler, B., and Kunkel, P. (1987). Efficient numerical pathfollowing beyond critical points, SIAM J. Numer. Anal. 18, pp. 949–987. di Bernardo, M., Budd, C. J., Champneys, A. R., and Kowalczyk, P. (2008). PiecewiseSmooth Dynamical Systems: Theory and Applications (Springer, London). Dibl´ık, J., Feˇckan, M., and Posp´ıˇsil, M. (2013). Forced Fermi-Pasta-Ulam lattice maps, Miskolc Mathematical Notes 14, pp. 63–78. Dimova, S. and Georgiev, V. (1992). Numerical algorithm for the dynamic analysis of base-isolated structures with dry friction, Nat. Hazards 6, pp. 71–86, doi:10.1007/ BF00162100. Dimova, S., Meskouris, K., and Kr¨ atzig, W. B. (1995). Numerical technique for dynamic analysis of structures with friction devices, Earthquake Eng. Struct. Dyn. 24, pp. 881–898. Driessen, B. J. and Sadegh, N. (2004). Convergence theory for multi-input discrete-time iterative learning control with Coulomb friction, continuous outputs, and input bounds, Internat. J. Adapt. Control Signal Process. 18, pp. 457–471. Du, Z. and Zhang, W. (2005). Melnikov method for homoclinic bifurcation in nonlinear impact oscillators, Computers and Mathematics with Applications 50, pp. 445–458. Dubois, A., Oudin, J., and Picart, P. (1996). Elastoplastic finite element analysis of an upsitting-sliding test for the determination of friction at medium and high contact pressure, Tribol. Int. 29, 7, pp. 603–613. Duda, K., Magalas, M. B., Majewski, M., and Zieli´ nski, T. P. (2011). DFT-based estimation of damped oscillation parameters in low-frequency mechanical spectroscopy, IEEE Trans. Instrum. Meas. 60, 11, pp. 3608–3618. Dweib, A. H. and de Souza, A. F. (1990). Self-excited vibrations induced by dry friction. Part I: Experimental study, J. Sound Vibration 137, 2, pp. 163–175. Eckmann, J. P. and Ruelle, D. (1985). Ergodic theory of chaos and strange attractors, Rev. Modern Phys. 57, pp. 617–656. ´ (1997a). On asymptotic stability of some Sturm–Liouville differential equations, Elbert, A.

page 240

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

241

General Seminar of Mathematics, Univ. Patras 22-23, pp. 57–66. ´ (1997b). Stability of some difference equations, in Advances in Difference EquaElbert, A. tions, Proc. Second Int. Conf. Difference Eqns., Veszpr´em, Hungary, August 7-11, 1995 (Gordon & Breach Science Publ., London), pp. 165–187. Epifanov, G. I. (1975). On the Two-Parts Friction (AN SSR, Moscow). Eret, P. and Meskell, C. (2008). A practical approach to parameter identification for a lightly damped, weakly nonlinear system, J. Sound Vibration 310, 2008, pp. 829– 844. Eriksson, M. and Jacobson, S. (2000). Tribological surfaces of organic brake pads, Tribol. Int. 33, pp. 817–827. Faltinsen, O. (2010). Hydrodynamics of High-Speed Marine Vehicles (Cambridge University Press). Feˇckan, M. (1997). Bifurcation of periodic solutions in differential inclusions, App. MathChech. 42, 5, pp. 369–393. Feˇckan, M. (1999). Chaotic solutions in differential inclusions: Chaos in dry friction problems, T. Am. Math. Soc. 351, 7, pp. 2861–2873. Feˇckan, M. (2005). Chaos in non-autonomous differential inclusions, Int. J. Bifurcation Chaos 15, 6, pp. 1919–1930. Feˇckan, M. (2008). Topological Degree Approach to Bifurcation Problems (Springer). Feˇckan, M. and Kelemen, S. (2013). Discretization of Poincar´e map, Electron. J. Qual. Theo. 60, pp. 1–33. Feˇckan, M. and Posp´ıˇsil, M. (2016). Poincar´e–Andronov–Melnikov Analysis for NonSmooth Systems, 1st edn. (Academic Press). Feely, O. and Chua, L. O. (1992). Nonlinear dynamics of a class of analog-to-digital converters, Int. J. Bifurcation Chaos 2, 2, pp. 325–340. Feeny, B., Guran, A., Hinrichs, N., and Popp, K. (1998). A historical review on dry friction and stick–slip phenomena, Appl. Mech. Rev. 51, pp. 321–341. Feeny, B. and Moon, F. C. (1994). Chaos in a forced dry friction oscillator: Experiment and numerical modelling, J. Sound Vibration 170, 3, pp. 303–323. Ferrero, J. F. and Barrau, J. J. (1996). Study of dry friction under small displacements and near-zero sliding velocity, Wear 209, pp. 322–327. Fidlin, A. (2001). On the asymptotic analysis of discontinuous systems, Z. Angew. Math. Mech. 82, 2, pp. 75–88. Fidlin, A. (2006). Nonlinear Oscillations in Mechanical Engineering (Springer, Berlin). Filippov, A. F. (1988). Differential Equations with Discontinuous Right-Hand Sides, Mathematics and Its Applications (Kluwer Academic Publishers, Dordrecht). Fling, R. T. and Fenton, R. E. (1981). A describing-function approach to antiskid design, IEEE Trans. Veh. Technol. 30, 3, pp. 134–144. Fu, W. P., Fang, Z. D., and Zhao, Z. G. (2001). Periodic solutions and harmonic analysis of an anti-lock brake system with piecewise-nonlinearity, J. Sound Vibration 246, 3, pp. 543–550. Fujisaka, H. and Yamada, T. (1983). Stability theory of synchronized motion in coupledoscillator systems, Progr. Theoret. Phys. Suppl. 69, 1, pp. 32–47. Gaji´c, A. and Leli´c, M. (1996). Modern Control Systems Engineering (Prentice Hall, Europe). Galvanetto, U. (2001). Some discontinuous bifurcations in a two-block stick–slip system, J. Sound Vibration 248, 4, pp. 653–669. Galvanetto, U. (2002). Some remarks on the two-block symmetric Burridge–Knopoff model, Phys. Lett. A 293, pp. 251–259. Galvanetto, U. (2004). Sliding bifurcations in the dynamics of mechanical systems with

page 241

June 8, 2017 12:9

242

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

dry friction — remarks for engineers and applied scientists, J. Sound Vibration 276, pp. 121–139. Galvanetto, U. and Bishop, S. R. (1994). Stick–slip vibrations of a two-degrees-of-freedom geophysical fault model, Int. J. Mech. Sci. 36, 8, pp. 683–698. Galvanetto, U., Bishop, S. R., and Briseghella, L. (1993). Some remarks on the stick–slip vibrations of a two-degree-of-freedom mechanical model, Mech. Res. Comm. 20, 6, pp. 459–466. Gao, C. and Kuhlmann-Wilsdorf, D. (1990). On stick–slip and the velocity dependence of friction at low speeds, J. Tribol. 112, pp. 354–360. Gao, G.-Z., Zhu, L.-D., and Ding, Q.-S. (2013). Identification of nonlinear damping and stiffness of spring-suspended sectional model, in Proceedings of The Eighth AsiaPacific Conference on Wind Engineering (Chennai, India), pp. 263–272. Gaul, L. and Becker, J. (2014). Reduction of structural vibrations by passive and semiactively controlled friction dampers, Shock and Vibration 2014, p. 7. Gaul, L. and Nitsche, R. (2001). The role of friction in mechanical joints, Appl. Mech. Rev. 54, 2, pp. 93–106. Georgiadis, F., Vakakis, A. F., McFarland, D. M., and Bergman, L. (2005). Shock isolation through passive energy pumping caused by non-smooth nonlinearities, Int. J. Bifurcation Chaos 15, 6, pp. 1989–2001. Goeleven, D., Motreanu, D., and Motreanu, V. V. (2003). On the stability of stationary solutions of first order parabolic variational inequalities, Adv. Nonlinear Var. Inequal. 6, pp. 1–30. Graef, J. R. and Karsai, J. (1997). On irregular growth and impulses in oscillator equations, in Advances in Difference Equations, Proc. Second Int. Conf. Difference Eqns., Veszpr´em, Hungary, August 7-11, 1995 (Gordon & Breach Science Publ., London), pp. 253–262. Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer-Verlag, New-York). Gu´erin, J. D., Bartys, H., Dubois, A., and Oudin, J. (1999). Finite element implementation of a generalized friction model: application to an upsetting-sliding test, Finite Elem. Anal. Des. 31, pp. 193–207. Guo, Z., Meng, Y., Wu, H., Su, C., and Wen, S. (2007). Measurement of static and dynamic friction coefficients of sidewalls of bulk-microfabricated MEMS devices with an onchip micro-tribotester, Sensors and Actuators A 135, 2, pp. 863–869. Guran, A., Pfeiffer, F., and Popp, K. (eds.) (1996). Dynamics with Friction — Modeling, Analysis and Experiment — Part I: Series on Stability, Vibration and Control of Systems (World Scientific, Singapore). Gutkin, E. (2003). Billiard dynamics: a survey with the emphasis on open problems, Reg. Chaot. Dyn. 8, pp. 1–13. Halling, J. (1957). Principles of Tribology (Macmillan Press, London). Han, Y. W., Cao, Q. J., Chen, Y. S., and Wiercigroch, M. (2012). A novel smooth and discontinuous oscillator with strong irrational nonlinearities, Science China Physics, Mechanics & Astronomy 55, 10, 1832, doi:https://doi.org/10.1007/ s11433-012-4880-9. Hartung, A., Schmieg, H., and Vielsack, P. (2001). Passive vibration absorber with dry friction, Arch. Appl. Mech. 71, 6-7, pp. 463–472. Hassard, B. D., Kazarinoff, N. D., and Wan, Y. H. (1981). Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge). Hatv´ ani, L. (1998). On the existence of a small solution to linear second order differential equations with step function coefficients, Dyn. Contin. Discrete Impuls. Syst. 4, pp.

page 242

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

243

321–330. Hatv´ ani, L. and Stach´ o, L. (1998). On small solutions of second order differential equations with random coefficients, Arch. Math. (Brno) 34, 1, pp. 119–126. Hayashi, C. (1964). Nonlinear Vibrations in Physical Systems (McGraw Hill Company, New York). Heckl, M. A. and Abrahams, I. D. (1996). Active control of friction-driven oscillations, J. Sound Vibration 193, 1, pp. 417–426. H´enon, M. (1982). On the numerical computation of Poincar´e maps, Physica D 5, pp. 412–413. Hinrichs, N., Oestreich, M., and Popp, K. (1997). Dynamics of oscillators with impact and friction, Chaos, Solitons & Fractals 8, 4, pp. 535–558. Hirschorn, R. M. and Miller, G. (1999). Control of nonlinear systems with friction, IEEE Trans. Contr. Syst. Technol. 7, 5, pp. 588–595. Hong, H. K. and Liu, C. S. (2000). Coulomb friction oscillator: Modelling and responses to harmonic loads and base excitations, J. Sound Vibration 229, pp. 1171–1192. Hong, H. K. and Liu, C. S. (2001). Non-sticking vibration formulae for Coulomb friction under harmonic loading, J. Sound Vibration 244, 5, pp. 883–898. Huang, S. N., Tan, K. K., and Lee, T. H. (2000). Adaptive friction compensation using neural network approximations, IEEE T. Syst. Man. Cyb. 30, 4, pp. 551–557. Hulten, J. O. (1997). Friction phenomena related to drum brake squeal instabilities, in Proceedings of ASME Design Eng. Tech. Conference, 16th Biennial Conference on Mechanical Vibrations and Noise (Sacramento), pp. 1–10. Hundal, M. S. (1979). Response of a base excited system with Coulomb and viscous friction, J. Sound Vibration 64, pp. 371–378. Ibrahim, R. A. (1994a). Friction-induced vibration, chatter, squeal, and chaos. Part I: Mechanics of contact and friction, Appl. Mech. Rev. 47, 7, pp. 209–226. Ibrahim, R. A. (1994b). Friction-induced vibration, chatter, squeal, and chaos. Part II: Dynamics and modeling, Appl. Mech. Rev. 47, 7, pp. 227–253. Inman, D. J. (1989). Vibrations: with Control, Measurement Stability (Prentice Hall, Englewood Cliffs, New Jersey). Iwan, W. D. and Moeller, T. L. (1976). The stability of a spinning disc with a transverse load system, J. Appl. Mech.: T. ASME 43, pp. 485–490. Jang, T. S. (2015). A new mathematical procedure for simultaneous identification of the nonlinear damping and restoring characteristics based on acceleration measurements, Ships and Offshore Structures 10, 4, pp. 426–435, doi:10.1080/17445302.2014. 942084. Johnson, K. L. (1985). Contact Mechanics (Cambridge University Press, Cambridge). Joubert, S. V., Shatalov, M. Y., and Manzhirov, A. V. (2013). Bryan’s effect and isotropic nonlinear damping, J. Sound Vibration 332, 23, pp. 6169–6176. Kang, J., Krousgrill, C. M., and Sadeghi, F. (2009). Oscillation pattern of stick–slip vibrations, Internat. J. Non-Linear Mech. 44, 7, pp. 820–828. Karnopp, D. (1985). Computer simulation of stick–slip friction in mechanical dynamic systems, J. Dyn. Syst. Meas. Control 107, pp. 100–103. Kim, B. J. and Choe, G. H. (2010). High precision numerical estimation of the largest Lyapunov exponent, Commun. Nonlinear Sci. Numer. Simul. 15, pp. 1378–1384. Kim, Y. H. and Lewis, F. L. (2000). Optimal design of CMAC neural network controller for robot manipulators, IEEE T. Syst. Man. Cyb. 30, 1, pp. 22–31. Kinkaid, N. M., O’Reilly, O. M., and Papadopoulos, P. (2005). On the transient dynamics of a multi-degree-of-freedom friction oscillator: A new mechanism for disc brake noise, J. Sound Vibration 287, 4–5, pp. 901–917.

page 243

June 8, 2017 12:9

244

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Klepp, H. J. (1994). Non-linear iterative mechanics for the investigation of multi-body systems with friction, Z. Angew. Math. Mech. 74, pp. 397–407. Knobloch, J. (2000). Lin’s method for discrete dynamical systems, J. Differ. Equations Appl. 6, pp. 577–623. Knopoff, L., Landoni, J. A., and Abinante, M. S. (1992). Dynamical model of an earthquake fault with localization, Phys. Rev. A 46, 12, pp. 7445–7449. Knudsen, C., Feldberg, R., and Jaschinski, A. (1991). Nonlinear dynamic phenomena in the behavior of a railway wheelset model, Nonlinear Dynam. 2, pp. 389–404. Koh, J. H., Robertson, A., Jonkman, J., Driscoll, F., and Ng, E. Y. K. (2013). Building and calibration of a fast model of the sway prototype floating wind turbine, in International Conference on Renewable Energy Research and Applications (National Renewable Energy Laboratory, Madrid, Spain), pp. 1–7. Komanduri, R. (1993). Machining and grinding — a historical rewiev of classical papers, Appl. Mech. Rev. 46, 3, pp. 80–132. Kovaleva, A. (2010). The Melnikov criterion of instability for random rocking dynamics of a rigid block with an attached secondary structure, Nonlinear Anal. 11, pp. 472–479. Kragielskiy, I. V. (1944). On the Influence of Pressure and Surfaces Dimension on Amount of a Friction Force (Machine Construction, Moscow). Krasnosel’skii, M. A. and Zabrejko, P. P. (1984). Geometrical Methods of Nonlinear Analysis (Springer-Verlag, Berlin). Krasnoselskiy, M. A. and Pokrovskiy, A. V. (1980). Systems with Hysteresis (SpringerVerlag, New York). Kudra, G. and Awrejcewicz, J. (2012). Bifurcational dynamics of a two-dimensional stick– slip system, Differential Equations Dynam. Systems 20, pp. 301–322, doi:10.1007/ s12591-012-0104-z. Kumar, E. V. and Jerome, J. (2013). Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum, Procedia Engineering 64, pp. 169 –178, doi:http://dx.doi.org/10.1016/j.proeng.2013.09.088. Kunze, M. (2000). Non-Smooth Dynamical Systems (Springer-Verlag, Berlin-Heidelberg). Kunze, M. and K¨ uper, T. (1997). Qualitative bifurcation analysis of a non-smooth frictionoscillator model, Z. Angew. Math. Mech. 47, pp. 87–101. Kuznetsov, Y. A. (1998). Elements of Applied Bifurcation Theory (Springer). Kuznetsov, Y. A. and Rinaldi, A., S Gragnani (2003). One-parameter bifurcations in planar Filippov systems, Int. J. Bifurcation Chaos 13, 8, pp. 2157–2188. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems (Wiley, New York). Lamarque, C. H. and Bastien, J. (2000). Numerical study of a forced pendulum with friction, Nonlinear Dynam. 23, pp. 335–352, doi:10.1023/A:1008328000605. Le Gorrec, Y., H´elie, T., and Matignon, D. (2015). Nonlinear damping models for linear conservative mechanical systems with preserved eigenspaces: A port-Hamiltonian formulation, IFAC-PapersOnLine 48, 13, pp. 200–205. Leamy, M. J., Barber, J. R., and Perkins, N. C. (1998). Distortion of a harmonic elastic wave reflected from a dry friction support, J. Appl. Mech.: T. ASME 65, 4, pp. 851–857. Lee, H. and Jung, S. (2012). Balancing and navigation control of a mobile inverted pendulum robot using sensor fusion of low cost sensors, Mechatronics 22, 1, pp. 95–105, doi:http://dx.doi.org/10.1016/j.mechatronics.2011.11.011. Lee, R. T., Yang, C. R., and Chiou, Y. C. (1996). A procedure for evaluating the positioning accuracy of reciprocating friction drive systems, Tribol. Int. 29, 5, pp. 394–404. Leine, R. I. and Nijmeijer, H. (2004). Dynamics and Bifurcations of Non-smooth Mechanical Systems (Springer-Verlag, Berlin).

page 244

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

245

Leine, R. I., Van Campen, D. H., and Van de Vrande, B. L. (2000). Bifurcations in nonlinear discontinuous systems, Nonlinear Dynam. 23, 2, pp. 105–164. Lenci, S. and Rega, G. (2005). Heteroclinic bifurcations and optimal control in the nonlinear rocking dynamics of generic and slender rigid blocks, Int. J. Bifurcation Chaos 15, pp. 1901–1918. Levitan, B. M. and Zhikov, V. V. (1983). Almost Periodic Functions and Differential Equations (Cambridge University Press, New York). Levitan, E. S. (1960). Forced vibrations of a spring-mass system having combined Coulomb and viscous damping, J. Acoust. Soc. Am. 32, 10, pp. 1265–1269. Lewis, F. L., Abdallah, C. T., and Dawson, D. M. (1993). Control of Robot Manipulators (Macmillan Publishing Company, New York). Li, N. and Zhi, L. (2014). Verified error bounds for isolated singular solutions of polynomial systems, SIAM J. Numer. Anal. 52, 4, pp. 1623–1640, doi:10.1137/120902914, eprint http://dx.doi.org/10.1137/120902914. Li, Z., Wang, Q., and Gao, H. (2009). Friction driven oscillator control by Lyapunov redesign based on delayed state feedback, Acta Mech. Sinica 25, pp. 257–264. Li, Z. and Xu, C. (2009). Adaptive fuzzy logic control of dynamic balance and motion for wheeled inverted pendulums, Fuzzy Sets and Systems 160, pp. 1787–1803. Lin, H. T., C L Huang (2002). Linear servo motor control using adaptive neural networks, P. I. Mech. Eng. I-J. Sys. 216, pp. 407–427. Lin, K. Y. and Yong, Y. (1987). Evolutionary Kanai–Tajimi earthquake models, J. Eng. Mech. 113, 8, pp. 1119–1137. Lin, X. B. (1990). Using Melnikov’s method to solve Silnikov’s problems, Proc. Roy. Soc. Edinburgh 116A, pp. 295–325. Lin, Z., Saberi, A., Gutmann, M., and Shamash, Y. A. (1996). Linear controller for an inverted pendulum having restricted travel: A high-and-low gain approach, Automatica 32, 6, pp. 933–937. Liu, X., Vlajic, N., Long, X., Meng, G., and Balachandran, B. (2014). State-dependent delay influenced drill-string oscillations and stability analysis, J. Vib. Acoust. 136, 5, pp. 1–9. Llibre, J., da Silva, P. R., and Teixeira, M. A. (2015). Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds, Nonlinearity 28, pp. 493–507. Lorenz, E. N. (2002). Deterministic nonperiodic flow, J. Atmos. Sci. 10, pp. 130–138. Lozano, R., Brogliato, B., Egeland, O., and Maschke, B. (2000). Dissipative Systems Analysis and Control (Springer CCES, London). Lubineau, D., Dion, J. M., Dugard, L., and Roye, D. (2000). Design of an advanced non linear controller for induction motors and experimental validation on an industrial benchmark, Eur. Phys. J. Appl. Phys. 9, pp. 165–175. Luersen, M. A. and Le Riche, R. (2004). Globalized Nelder-Mead method for engineering optimization, Comput. Struct. 82, 23-26, pp. 2251–2260. Maistrenko, V., Maistrenko, Y., and Sushko, I. (1994). Noninvertible two-dimensional maps arising in radiophysics, Int. J. Bifurcation Chaos 4, 2, pp. 383–400. Majundar, A. and Bhushan, B. (1991). Fractal model of elastic-plastic contact between rough surfaces, J. Tribol.: T. ASME 113, pp. 1–11. Makarenkov, O. and Lam, J. (2012). Dynamics and bifurcations of nonsmooth systems: A survey. hysica D: Nonlinear Phenomena 241, 22, pp. 826–1844. Makarenkov, O. and Verhulst, F. (2009). Bifurcation of asymptotically stable periodic solutions in nearly impact oscillators, arXiv:0909.4354 . Mansouri, A., Chenafa, M., Bouhenna, A., and Etien, E. (2004). Powerful nonlinear ob-

page 245

June 8, 2017 12:9

246

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

server associated with the field-oriented control of the induction motor, Int. J. Appl. Math. Comput. Sci. 14, 2, pp. 209–220. Marino, R., Peresada, S., and Valigi, P. (1993). Adaptive input-output linearizing control of induction motors, IEEE T. Automat. Contr. 38, 2, pp. 208–221. Marui, E. and Endo, H. (1996). Some considerations of slideway friction characteristics by observing stick–slip vibration, Tribol. Int. 29, 3, pp. 251–262. Matrossov, I. V. (1996). On existence of the right side solutions of the differential equations of mechanical systems with dry friction, Differential Equations 32, 5, pp. 606–614. Matrossov, I. V. (2001). On existence and uniqueness of solutions for equations of motion of mechanical system with dry friction, Nonlinear Anal. 47, pp. 5391–5402. Meng, F., Yang, M., and Zhong, C. (2016). Attractors for wave equations with nonlinear damping on time-dependent space, Discrete and Continuous Dynamical Systems Series B 21, 1, pp. 205–225, doi:10.3934/dcdsb.2016.21.205. Meskell, C. (2006). A decrement method for quantifying nonlinear and linear damping parameters, J. Sound Vibration 296, 2006, pp. 643–649. Meskell, C. (2011). A decrement method for quantifying nonlinear and linear damping in multidegree of freedom systems, ISRN Mechanical Engineering 2011, 659484, pp. 1–7, doi:10.5402/2011/659484. Meyer, K. R. and Sell, G. R. (1989). Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, T. Am. Math. Soc. 314, pp. 63–105. Miao, C., Luo, W., Ma, Y., Liu, W., and Xiao, J. (2014). A simple method to improve a torsion pendulum for studying chaos, European J. Phys. 35, 055012, pp. 1–9. Michalak, M. and Kruci´ nska, I. (2004). Studies of the effects of chemical treatment on bending and torsional rigidity of blast fibres, Mater. Sci. 10, 2, pp. 182–185. Mileti´c, M., St¨ urzer, D., and Arnold, A. (2014). An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip, Discrete and Continuous Dynamical Systems — Series B 20, 9, pp. 1–25. Minis, I. E., Magrab, E. B., and Pandelidis, I. O. (1990). Improved methods for the prediction of chatter in turning, Part 2: Determination of cutting process parameters, J. Eng. Ind. 112, pp. 21–32. Mistakidis, E. S., Panagouli, O. K., and Panagiotopoulos, P. D. (1998). Unilateral contact problems with fractal geometry and fractal friction laws: methods of calculation, Comput. Mech. 21, pp. 353–362. Miwa, T. (1967). Evaluation methods for vibration effect, Ind. Health 5, pp. 183–205. Monteiro-Marques, M. D. P. (1994). An existence, uniqueness and regularity study of the dynamics of systems with one-dimensional friction, European Journal of Mechanics - A/Solids 13, 2, pp. 277–306. Moore, D. F. (1975). Principles and Applications of Tribology (Pergamon Press, Oxford). Mote, C. D. (1970). Stability of circular plate subjected to moving loads, Journal of Franklin Institute 290, pp. 329–344. Mottershead, J. and Stanway, R. (1986). Identification of nth-power velocity damping, J. Sound Vibration 105, 1986, pp. 309–319. M¨ ueller, P. (1995). Calculation of Lyapunov exponents for dynamical systems with discontinuities, Chaos, Solitons & Fractals 5, 9, pp. 1671–1681. Nakamura, N. (2016). Extended rayleigh damping model, Frontiers in Built Environment 2, 14, pp. 1–13, doi:10.3389/fbuil.2016.00014. Naranayanan, S. and Jayaraman, K. (1989). Chaotic motion in nonlinear system with Coulomb damping, in Nonlinear Dynamics in Engineering Systems, IUTAM Symposium (Stuttgart), pp. 217–224. Nayfeh, A. H. and Mook, D. T. (1979). Nonlinear Vibrations (Wiley, New York).

page 246

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

247

Newmark, N. M. and Rosenblueth, E. (1975). Fundamentals of Earthquake Engineering (Prentice Hall, New York). Niemann, H. H. and Stoustrup, J. (2005). Passive fault tolerant control of a double inverted pendulum — a case study, Control. Eng. Pract. 13, 8, pp. 1047–1059. Novaes, D. D., Jeffrey, M. R., and Teixeira, M. A. (2014). On sliding periodic solutions for piecewise continuous systems defined on the 2-cylinder, Proceedings of the Edinburgh Mathematical Society . Nussbaum, J. and Ruina, A. (1987). A two degree-of-freedom earthquake model with static/dynamic friction, Pageoph 125, 4, pp. 629–656. Oden, J. T. (1985). Models and computational methods for dynamic friction phenomena, Comput. Methods Appl. Mech. Engrg. 52, pp. 527–634. Olejnik, P. (2002). Numerical and Experimental Analysis of Self-Excited Regular and Chaotic Vibrations in a Two-Degrees-of-Freedom System with Friction, Ph.D. thesis, Lodz University of Technology, Lodz, in Polish. Olejnik, P. and Awrejcewicz, J. (2013a). Application of H´enon method in numerical estimation of the stick–slip transitions existing in Filippov-type discontinuous dynamical systems with dry friction, Nonlinear Dynam. 73, 1-2, pp. 723–736. Olejnik, P. and Awrejcewicz, J. (2013b). Low-speed voltage-input tracking control of a DC-motor numerically modelled by a dynamical system with stick–slip friction, Differential Equations Dynam. Systems 21, 1-2, pp. 3–13. Olejnik, P. and Awrejcewicz, J. (2014). An approximation method for the numerical solution of planar discontinuous dynamical systems with stick–slip friction, Appl. Math. Sci. 8, 145, pp. 7213–7238. Olejnik, P., Awrejcewicz, J., and Nielaczny, M. (2013). Solution of the kalman filtering problem in control and modeling of a double inverted pendulum with rolling friction, Pomiary, Automatyka, Robotyka 17, 1, pp. 63–70. Ono, K., chen, J. S., and Bogy, D. B. (1991). Stability analysis for the head–disc interface in a flexible disc drive, J. Appl. Mech.: T. ASME 58, pp. 1005–1014. Oseledec, V. I. (1968). A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19, pp. 197–231. Ostermeyer, G. P. (2001). Friction and wear of brake systems, Forsch. Ingenieurwes. 66, 6, pp. 267–272. Ostermeyer, G. P. (2010). Dynamic friction laws and their impact on friction induced vibrations, Tech. Rep. 2010-01-1717, SAE Technical Paper, doi:10.4271/2010-01-1717. Otten, G., Perkins, T. J. A., van Amerongen, J., Rankers, A. M., and Gaal, E. W. (1997). Linear motor motion control using a learning feedforward controller, IEEE-ASME T. Mech. 2, 3, pp. 179–187. Palmer, K. J. (1984). Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55, pp. 225–256. Palmer, K. J. and Stoffer, D. (1989). Chaos in almost periodic systems, Z. Angew. Math. Phys. 40, pp. 592–602. Panagiotopoulos, P. D., Panagouli, O. K., and Mistakidis, E. S. (1994). On the consideration of the geometric and physical fractality in solid mechanics. Part I: Theoretical results, Z. Angew. Math. Mech. 74, pp. 167–176. Parker, T. S. and Chua, L. O. (1989). Practical Numerical Algorithms for Chaotic Systems (Springer-Verlag, Berlin). Pavelescu, D. and Tudor, A. (1987). The sliding coefficient: Its evolution and usefulness, Wear 120, pp. 321–336. Pecora, L. M. and Carroll, T. L. (1990). Synchronization of chaos, Phys. Rev. Lett. 64, 8, pp. 821–824.

page 247

June 8, 2017 12:9

248

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Perko, L. (1991). Differential Equations and Dynamical Systems (Springer, New York). Persson, B. N. (1998). Sliding Friction. Physical Principles and Applications (SpringerVerlag). Peters, R. D. (2007). Damping Theory, chap. 2, Vibration Damping, Control, and Design (Taylor & Francis Group, London), pp. 1–65. Pfeiffer, F. (1999). Unilateral problems of dynamics, Arch. Appl. Mech. 69, pp. 503–527. Pilipchuk, V. and Tan, C. (2004). Stick–slip capture and source of squeal at decelerating sliding, Nonlinear Dynam. 35, pp. 259–285. Pilipchuk, V. N., Ibrahim, R. A., and Blaschke, P. G. (2002). Disc brake ring-element modeling involving friction-induced vibration, J. Vib. Control 8, 8, pp. 1085–1104. Pilipchuk, V. N., Olejnik, P., and Awrejcewicz, J. (2015). Transient friction-induced vibrations in a 2-DOF model of brakes, J. Sound Vibration 344, pp. 297–312. Popp, K., Hinrichs, N., and Oestreich, M. (1996). Analysis of a Self-Excited Friction Oscillator with External Excitation (World Scientific Publishing, London). Popp, K. and Stelter, P. (1989). Nonlinear Vibrations of Structures Induced by Dry Friction (Springer-Verlag, Berlin). Popp, K. and Stelter, P. (1990). Stick–slip vibrations and chaos, Philos. T. R. Soc. S., A A332, pp. 89–105. Qiao, S. L. and Ibrahim, R. A. (1999). Stochastic dynamics of systems with friction-induced vibration, J. Sound Vibration 223, 1, pp. 115–140. Rabinowicz, E. (1951). The nature of static and kinetic coefficients of friction, J. Appl. Phys. 22, 11, pp. 1373–1379. Rabinowicz, E. (1958). The intrinsic variables affecting the stick–slip process, Proc. Phys. Soc. London 71, pp. 668–675. Ramachandran, R., Pande, S. S., and Ramakrishnan, N. (1994). The role of deburring in manufacturing: a state-of-the-art survey, J. Mater. Process. Technol. 44, 1-2, pp. 1–13. Ramberg, R. and Osgood, W. R. (1943). Description of stress-strain curves by three parameters, techreport 902, National Advisory Committee for Aeronautics. Renard, Y. (2001). Numerical analysis of a one-dimensional elastodynamic model of dry friction and unilateral contact, Comput. Methods Appl. Mech. Engrg. 190, 15-17, pp. 2031–2050. Rofooei, F. R., Mobarake, A., and Ahmadi, G. (2001). Kanai–Tajimi model, Eng. Struct. 23, pp. 827–837. Ruiz, P. and Penzien, J. (1969). Probabilistic study of the behaviour of structures during earthquakes, Tech. Rep. EERC 6903, Earth Eng. Res. Center, University of California, Berkely. Ryabov, V. B. and Ito, H. M. (1995). Multistability and chaos in a spring-block model, Phys. Rev. E 52, 6, pp. 6101–6112. Sandstede, B. (1993). Verzweigungstheorie homokliner Verdopplungen, Ph.D. thesis, University of Stuttgart. Sano, M. and Sawada, Y. (1985). Measurement of the Lyapunov spectrum from a chaotic time series, Phys. Rev. Lett. 55, 10, pp. 1082–1085. Sansone, G. and Conti, R. (1964). Non-Linear Differential Equations (Pergamon Press, Oxford-New York). Schlagner, S. and von Wagner, U. (2009). Characterisation of disk brake noise behavior via measurement of friction forces, Proc. Appl. Math. Mech. 9, pp. 59–62. Schmieg, H. and Vielsack, P. (1998). Modellbildung und experimentelle Untersuchungen zum Bremsenquietschen, Z. Angew. Math. Mech. 78, pp. 709–710. Shaw, S. W. (1986). On the dynamic response of a system with dry friction, J. Sound

page 248

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

249

Vibration 108, 2, pp. 305–325. Shi, L. S., Zou, Y. K., and K¨ upper, T. (2013). Melnikov method and detection of chaos for non-smooth systems, Acta Mathematicae Applicatae Sinica, English Series 29, 4, pp. 881–896, doi:10.1007/s10255-013-0265-8. Shia, C. Y., Stango, R. J., and Heinrich, S. M. (1998). Analysis of contact mechanics for a circular filamentary brush/workpart system, J. Manuf. Sci. Eng. 120, 4, pp. 715–721. Shilnikov, L. P., Shilnikov, A. L., Turaev, D. V., and O, C. L. (1998). Methods of Qualitative Theory in Non-Linear Dynamics (World Scientific, Singapore). Sinou, J. J., Dereure, O., Mazet, G. B., Thouverez, F., and Jezequel, L. (2006). Frictioninduced vibration for an aircraft brake system — Part 1: Experimental approach and stability analysis, Int. J. Mech. Sci. 48, 5, pp. 536–554. Sissala, M., Heimola, T., and Otala, M. (1985). Optimization of lift car vibrational behavior by modal analysis, Elevator World 6, pp. 27–31. Siuka, A. and Sch¨ oberl, M. (2009). Applications of energy based control methods for the inverted pendulum on a cart, Robot. Auton. Syst. 57, 10, pp. 1012–1017, doi: 10.1016/j.robot.2009.07.016. Skup, Z. (2002). Structural friction and viscous damping in a frictional torsion damper, J. Theoret. Appl. Mech. 2, 40, pp. 497–511. Slotine, J. J. E. and Li, W. (1987). On the adaptive control of robot manipulators, Int. J. Rob. Res. 6, 3, pp. 49–59. Sokolov, I. Y. (2002). Anomalous increase of friction in the vicinity of nano-size defects, Tribol. Lett. 12, 2, pp. 131–134. Song, G., Cai, L., Wang, Y., and Longman, R. W. (1998). A sliding-mode based smooth adaptive robust controller for friction compensation, Internat. J. Robust Nonlinear Control 8, pp. 725–739. Stefa´ nski, A. and Kapitaniak, T. (2000). Using chaos synchronization to estimate the largest Lyapunov exponent of nonsmooth systems, Discrete Dynamics in Na 4, pp. 207–215. Stefa´ nski, A., Wojewoda, J., and Furmanik, K. (2000). Experimental and numerical analysis of self-excited friction oscillator, Chaos, Solitons & Fractals 12, pp. 1691–1704. Steigenberger, J. and Behn, C. (2012). Worm-Like Locomotion Systems: an Intermediate Theoretical Approach (Oldenbourg, Munich). Stoffer, D. (1988). Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, II, Z. Angew. Math. Phys. 39, pp. 518–549, 783–812. Stoker, J. J. (1950). Nonlinear Vibrations in Mechanical and Electrical Systems (Interscience Publ., New York). Stribeck, R. (1902). Die wesentlichen Eigenschaften der Gleit- und Rollenlager — the key qualities of sliding and roller bearings, Z. Ver. Dtsch. Ing. 46, 38-39, pp. 1342–48, 1432–37. Strizhak, T. G. (1984). Asymptotic Method of Normalization (Vyscha Shkola, Kiev). Studny, D., Rittel, D., and Zussman, E. (1999). Impact fracture of screws for disassembly, J. Manuf. Sci. Eng. 121, 1, pp. 118–126. Szablewski, W. (1954). Einfluss der Coulombschen Reibung auf Schwingungsvorg¨ ange, Math. Nachrichten 12, pp. 183–208. Szempli´ nska-Stupnicka, W. (1990). The Behavior of Nonlinear Vibrating Systems (Kluwer Academic Publishers, Dordrecht). Tabor, D. (1981). Friction, Journal of Lubrication Technology 103, pp. 169–179. Tan, X. and Rogers, R. J. (1996). Dynamic friction modelling in heat exchanger tube simulations, ASME Pressure Vessels and Piping Division 328, pp. 347–358.

page 249

June 8, 2017 12:9

250

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Tariku, F. A. and Rogers, R. J. (2000). Improved dynamic friction models for simulation of one-dimensional and two-dimensional stick–slip motion, J. Tribol. 123, 4, pp. 661–669. Teixeira, M. A. and da Silva, P. R. (2012). Regularization and singular perturbation techniques for non-smooth systems, Phys. D 241, pp. 1948–1955. Tolstoi, D. M. (1967). Significance of the normal degree of freedom and natural normal vibrations in contact friction, Wear 10, pp. 199–213. Tondl, A. and Nabergoy, R. (1994). Non-periodic and chaotic vibrations in a flow induced system, Chaos, Solitons & Fractals 4, 3, pp. 2193–2202. Tondl, A. and Nabergoy, R. (1995). A flow induced system with friction, Chaos, Solitons & Fractals 7, 3, pp. 353–363. Tworzydlo, W. W. and Hamzeh, C. N. (1997). On the importance of normal vibrations in modelling of stick–slip in rock sliding, J. Geophys. Res. 102, pp. 15091–15103. Ueda, Y. (1979). Randomly transitional phenomena in the system governed by Duffing equation, J. Statist. Phys. 20, pp. 181–186. Urabe, M. (1967). Nonlinear Autonomous Vibrations (Academic Press, New York). Vahid-Araghi, O. and Golnaraghi, F. (2011). Friction-Induced Vibration in Lead Screw Drives (Springer). Van de Velde, F. and de Baets, P. (1996). Mathematical approach of the influencing coefficients on stick–slip induced by decelerative motion, Wear 201, pp. 80–93. Van de Velde, F. and de Baets, P. (1997). Comparison of two stick–slip testers and recommendations for repeatable and significant stick–slip testing, Tribotest 3, 3-4, pp. 361–392. Van de Velde, F. and de Baets, P. (1998a). A new approach of stick–slip based on quasiharmonic tangential vibrations, Wear 216, pp. 12–26. Van de Velde, F. and de Baets, P. (1998b). The relation between friction force and relative speed during the slip-phase of a stick–slip cycle, Wear 219, pp. 220–226. Van de Vrande, B. L., van Campen, D. H., and de Kraker, A. (1999). An approximate analysis of dry-friction-induced stick–slip vibrations by a smoothing procedure, Nonlinear Dynam. 19, pp. 157–169. Vanderbauwhede, A. and Fiedler, B. (1992). Homoclinic period blow-up in reversible and conservative systems, Z. Angew. Math. Phys. 43, pp. 292–318. Vasileva, A. B. and Butzov, V. F. (1973). Asymptotic Expansion of Solutions of Singularly Perturbed Equations (Nauka). Vieira, M. D. (1999). Chaos and synchronized chaos in an earthquake model, Phys. Rev. Lett. 82, pp. 201–204. Vielsack, P. (2001). Stick–slip instability of decelerative sliding, Internat. J. Non-Linear Mech. 36, pp. 237–247. Walrath, C. (1984). Adaptive bearing friction compensation based on recent knowledge of dynamic friction, Automatica 20, 6, pp. 717–727. Wanheim, T. and Bay, N. (1978). A model for friction in metal forming processes, Ann. CIRA P. 27, pp. 189–194. Wee, H., Kim, Y. Y., Jung, H., and Lee, G. N. (2001). Nonlinear rate-dependent stick–slip phenomena: modeling and parameter estimation, Internat. J. Solids Structures 38, pp. 1415–1431. Wikiel, B. and Hill, J. M. (2000). Stick–slip motion for two coupled masses with side friction, Internat. J. Non-Linear Mech. 35, pp. 953–962. Wolf, A., Swift, J. B., Swinney, H. L., and Vastano, J. A. (1985). Determining Lyapunov exponents from a time series, Phys. D 16, pp. 285–317. Wolf, J. P. (1994). Foundation Vibration Analysis Using Simple Physical Models (Prentice

page 250

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Bibliography

1st Reading

ws-book975x65

251

Hall, Englewood Cliffs, New Jersey). Woodham, C. A. and Su, H. (2002). A computational investigation of pole-zero cancellation for a double inverted pendulum, J. Comput. Appl. Math. 140, 1-2, pp. 823–836, doi: http://dx.doi.org/10.1016/S0377-0427(01)00477-0, Int. Congress on Computational and Applied Mathematics 2000. W¨ osle, M. and Pfeiffer, F. (1999). Dynamics of spatial structure-varying rigid multibody systems, Arch. Appl. Mech. 69, pp. 265–285. Wu, Q., Thornton-Trump, A. B., and Sepehri, N. (1998). Lyapunov stability control of inverted pendulums with general base point motion, Internat. J. Non-Linear Mech. 33, 5, pp. 801– 818, doi:http://dx.doi.org/10.1016/S0020-7462(97)00052-8. Wulff, C. and Schebesh, A. (2008). Numerical continuation of hamiltonian relative periodic orbits, J. Nonlinear Sci. 18, pp. 343–390. Wyk, M. A. and Steeb, W. H. (1997). Chaos in Electronics (Kluwer Academic Publishers, Dordrecht, The Nethelands). Xu, W., Feng, J., and Rong, H. (2009). Melnikov’s method for a general nonlinear vibroimpact oscillator, Nonlinear Anal. 71, pp. 418–426. Xucheng, W., Liangming, C., and Zhangzhi, C. (1990). Effective numerical methods for elasto-plastic contact problems with friction, Acta Mech. Sinica (English Ed.) 6, pp. 349–356, doi:10.1007/BF02486894. Yan, S., Dowell, E. H., and Lin, B. (2014). Effects of nonlinear damping suspension on nonperiodic motions of a flexible rotor in journal bearings, Nonlinear Dynam. 78, 2, pp. 1435–1450. Yeh, E. C. and Day, G. C. (1992). Parametric study of anti-skid brake systems using Poincar´e map concept, Int. J. Veh. Des. 13, pp. 210–232. Yeh, G. C. K. (1966). Forced vibrations of a two-degree-of-freedom system with combined Coulomb and viscous damping, J. Acoust. Soc. Am. 39, pp. 14–24. Yi, J., Yubazaki, N., and Hirota, K. (2001). Stabilization control of series-type double inverted pendulum systems using the SIRMs dynamically connected fuzzy inference model, Artificial Intellignce in Engineering 15, pp. 297–308. Yuan, D. N. (2013). Dynamic modeling and analysis of an elastic mechanism with a nonlinear damping model, J. Vib. Control 19, 4, pp. 508–516. Zabrejko, P. P. and Povolockij, A. I. (1970). Bifurcation points of Hammerstein’s equations, Soviet Mathematics: Doklady 11, pp. 1220–1223. Zhuravlev, V. G. (1998). The model of dry friction in the problem of the rolling of rigid bodies, J. Appl. Math. Mech. 62, 5, pp. 705–710. Zimmermann, K., Zeidis, I., and Pivovarov, M. (2013). Dynamics of two interconnected mass points in a resistive medium, Differential Equations Dynam. Systems 21, 1, pp. 21–28. Zw¨ orner, O., H¨ olscher, H., Schwarz, U. D., and Wiesendanger, R. (1998). The velocity dependence of frictional forces in point-contact friction, Appl. Phys. A 6, pp. S263– S267.

page 251

This page intentionally left blank

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Index

active control, 234 adhesion area, 36 adiabatic loading conditions, 53 adjacent regions, 65 adjoint equation, 116 variational, 119 air resistance, 44 almost periodic forcing, 128 function, 119, 122, 146 jumping, 181 ODE, 182 perturbation, 98, 128 solution, 38, 112, 183, 189 system, 112 periodicity, 181 Amontons’ law, 3 amplitude -dependent damping, 168 -frequency diagram, 228, 234 angle bracket, 52, 169 angular displacement, 165, 202, 208 frequency, 162, 172 velocity, 163, 216 anti-lock brake mechanism, 8 Antunes’ model, 14 approximate solution, 98 arctan function, 33, 163 armature, 217 asymmetric pendulum, 167, 171 asymptotic solution, 98 stability, 37, 207 atan function, 83

atomic scale, 23 attractor, 47, 57 average displacement, 177 friction coefficient, 20 ball bearings, 170 Banach fixed point theorem, 182, 185 space, 101, 182 norm, 104 Bay–Wanheim model, 19 bearing, 45, 169 belt, 45, 57 -spring-block model, 49, 57 Bernoulli shift, 112 bifurcation diagram, 47, 59 equation, 92, 104 function, 104, 110 homoclinic, 96 parameter, 93 supercritical, 161 symmetry-breaking, 161 billiard, 127 Bliman–Sorine model, 25 block-on-belt model, 6, 18, 45, 170, 176 boundary, 128 condition, 184 Dirichlet, 169 linear, 182 Ornstein–Uhlenbeck, 232 solution, 89 bounded linear mapping, 183 solution, 116, 145 253

page 253

June 8, 2017 12:9

254

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

dimension, 119 impact, 98, 115 unique, 103 brake, 5, 12, 28, 41 braking, 49, 50 force, 26 system, 42, 57 brute-force approach, 79 buckling, 12 building-ground system, 225 buoyancy, 170 Burridge–Knopoff model, 6, 7 canonical form, 171 Cartesian coordinates, 201 surface, 156 Caughey’s damping, 169 chain of coupled pendulums, 161 chaos, 112, 120, 121, 126, 129, 140, 144, 156, 160 chaotic behavior, 95, 96 dynamics, 6, 100 solution, 38 characteristic equation, 194 clamped beam, 169 clutch, 5 coefficient of friction, 52 dynamic, 6 kinetic, 9, 64 static, 9 coil spring, 45 collisions, 35, 80 compact metric space, 112 conditions adiabatic loading, 53 impact, 120 necessary and sufficient, 117 nondegeneracy, 115 steady-state, 52 symmetry, 151 transversality, 100 constraints, 22 contact frictional, 160 stiffness, 11 continuity manifold, 100 control follow-up, 199

law, 222, 233 moment, 198 system, 207 convex subset, 130 Coulomb damping, 80 force, 24, 213 friction, 33, 38, 163, 211 impact model, 23 law, 4, 5, 8, 11, 17, 22, 26, 32 coupled pendulums, 161 crack, 20 creep–slip, 19, 223 contact surface, 170 vibrations, 54 creeping area, 18, 53 cutting tool, 31 D’Alembert principle, 128 Dahl model, 24 damped mathematical pendulum, 181 damper, 11, 159 damping, 226 Caughey’s, 169 coefficient, 75, 229 dominant, 221 matrix, 222 measurement, 169 modal, 169 nonlinear, 167, 169 ratio, 168, 171, 172 Rayleigh, 169 virtual, 15 viscous, 203, 229 DC motor, 45 decaying characteristics, 22, 193 decelerating sliding, 54 deceleration, 50 deterministic chaos, 112 diagram of stability, 191, 194 dichotomy, 101 projection, 141 difference equation, 182 differential inclusion, 17, 223 digital filter, 201 Dirichlet boundary condition, 169 disc, 34, 41, 159, 162 discontinuity manifold, 95 zone, 65, 72

page 254

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Index

discontinuous differential equations, 39 dynamical system, 164 function, 224 solution, 93 discrete FFT, 172 disequilibrial process, 1 dissipation, 171, 228 dominant frequency, 172 double pendulum, 197 torsion, 159, 162, 164 doubly-infinite sequence, 113 drilling platform, 12 system, 161 drum brake, 43, 45, 51 dry friction, 3, 10, 12, 38, 51, 53, 63, 80, 162, 191, 222, 231 characteristics, 22 Coulomb, 163 model, 16, 81 Duffing vibro-impact oscillator, 96, 127 dynamic friction coefficient, 6, 28, 37 force, 6 model, 24 earthquakes, 6, 221 effective friction coefficient, 19, 52 eigen -frequencies, 53 -vibrations, 54 eigenvalues, 90, 137, 194, 207 eigenvector, 230 eight-degree-of-freedom Lagrangian model, 160 elasticity force, 15 electromagnetic field, 146 energetic analysis, 29 energy dissipation, 1, 24, 35, 228 kinetic, 162, 202 loss, 170 potential, 162 pumping, 38 equation adjoint, 116 bifurcation, 92, 104 characteristic, 194 Lyapunov, 207

255

non-parametric, 85 nonlinear damped wave, 169 equations of motion, 51, 201 equilibrium, 99, 146, 191 hyperbolic, 90, 137 state, 194 estimate, 165 estimation error, 208 Euler–Lagrange equations, 51, 143, 169 exponential curve, 163 decay, 44, 172 dichotomy, 101 law, 5, 10 external excitation, 96, 226, 229 fading effect, 28 FFT, 172 fibre, 161 Filippov’s theory, 65, 83 finite-element model, 11 first integral, 142, 145 fitting parameter, 88, 176 fixed point, 100 hyperbolic, 101 flexural vibration modes, 50 fluctuation–dissipation theorem, 160 follow-up control, 199 forced vibrations, 10 four-degree-of-freedom system, 50 Fourier spectrogram, 54, 57 fractal friction law, 20 free decay test, 168–171, 174, 178 vibrations, 228 frequency spectrum, 172 friction -induced vibrations, 49 coefficient, 53 Coulomb, 163, 211 damper, 11 exponential, 212 force, 44, 212 characteristics, 46 kinetic, 159 law, 52 model, 50 position-dependent, 212 rolling, 198 sliding, 170

page 255

June 8, 2017 12:9

256

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

static, 159, 213 stick–slip, 215 viscous, 159, 212, 217 fundamental matrix, 101 fuzzy controller, 199 inference model, 199 gear, 45 generalized coordinates, 162, 201 forces, 163, 164, 203 moments, 202 geometrical constraints, 202 imperfections, 160 geometrical construction, 133 girling duo-servo brake, 43 gradient, 128 method, 191 gravity field, 161 gyroscope, 201 H´enon method, 40, 79 Hall-effect, 45 harmonic excitation, 5, 160 linearization, 17 vibrations, 226 Hessian, 128, 142 heteroclinic solution, 120 Hilbert transform, 54 Holweck–Lejay-like pendulum, 161 homeomorphism, 114 homoclinic bifurcation, 96 cycle, 99 orbit, 102, 118 solution, 38, 89, 90, 95, 98, 123 impact, 131 Hook’s law, 17 Hopf bifurcation, 13, 16 hyper -plane, 100 -surface, 60 hyperbolic equilibrium, 90, 97, 99, 123 fixed point, 38, 101 function, 168

identification, 159, 161, 164, 167, 168 impact, 35, 38, 120 boundary, 146 branch, 98 conditions, 120, 122, 145 equation, 147 homoclinic cycle, 99 orbit, 127, 137, 148 model, 23 oscillator, 95 solution, 100, 102, 104, 105 homoclinic, 115, 123, 131 unique, 109, 113 system, 95 Implicit Function Theorem, 110, 111 incremental harmonic balance, 32 inductance, 217 initial conditions, 47, 67, 72, 91, 133, 142, 155, 194, 204, 225, 226, 233 parameters, 165, 217 tangent points, 72 inner product, 100 instability, 32, 34, 50 inverse problem, 168 inverted pendulum, 96, 127, 197, 198, 200, 201, 204 base-excited, 200 involution, 100 irregular forcing, 170 Ising ferromagnet, 161 isomorphism, 107 isotropic damping, 169 iterative method, 185 Jacobian matrix, 118, 139, 191 Kalman filter, 197, 201, 204 Kanai–Tajimi power spectral density, 221 Karnopp model, 14, 83 kernel, 102, 141 kinematic constraints, 202 excitation, 208 forcing, 164, 221 sinusoidal, 162 kinetic energy, 143, 162, 202 friction, 159

page 256

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Index

coefficient, 9, 64 force, 7, 11, 46, 170 junction, 2, 22 kink, 53 Lagrange equations, 201 function, 163 multipliers, 22 Lagrangian, 143 Larmor’s law, 161 lateral contact surface, 9 law Amontons’, 3 control, 233 Coulomb, 4, 5, 8, 11, 17, 22, 26, 32 exponential, 10 Hook’s, 17 Larmor’s, 161 Leonardo da Vinci, 3 linear approximation, 192 displacement, 171 map, 107 mapping, 100 nonhomogeneous equation, 116 spring, 170 state feedback controller, 200 system, 96, 101 variational, 141, 152 variational initial problem, 92 linearization, 22, 52, 115, 123, 137, 169, 194 harmonic, 17 linearized conservative system, 55 lining, 42 logarithmic decrease, 171 fit, 174 scale, 57 LQG control, 201 lubrication, 19 Lund–Grenoble model, 26, 27, 209 Lyapunov –Schmidt reduction, 104 exponents, 18, 30, 39 function, 195, 206 solution matrix, 233 stability control, 200

257

magnetic field, 128 majorant functions method, 182 manifold, 95, 134 impact, 146 Riemannian, 127 unstable, 97 map, 134 mapping, 183 mass -spring -damping system, 34 model, 19 matrix, 222 moment of inertia, 162, 171 mathematical pendulum, 181 matrix norm, 101 measurement, 169, 172 Melnikov condition, 104, 127 function, 95, 112, 119, 120, 125, 141, 144, 146, 148 method, 39, 89, 96 membranous labyrinth, 198 method gradient, 192 H´enon, 40, 79 majorant functions, 182 Melnikov, 39, 89 path following, 79 trial-and-error, 28 micro -deformations, 228 -relocation, 37 -tribotester, 9 microscopic movement, 1 minimization, 192 modal damping, 169 model Antunes, 14 Bay–Wanheim, 19 belt-spring-block, 49 Bliman–Sorine, 25 block-on-belt, 6, 18, 45, 170 Burridge–Knopoff, 6, 7 Coulomb, 6, 11, 27 Dahl, 24 deterministic, 31 dry friction, 16, 81, 193 dynamic friction, 5, 24, 26 FE, 11, 168

page 257

June 8, 2017 12:9

258

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

fractal, 20 fuzzy inference, 199 Karnopp, 14, 83 Lagrangian, 160 Lund–Grenoble, 26, 27, 209 mass-spring, 19 mathematical, 83, 200 mechanical, 160 moving autoregressive, 167 numerical, 168, 176 parameters, 85, 165, 216 Park’s, 210 passive vibrations absorber, 10 physical, 162, 193, 198, 201 Prandtl’s, 225 reduced, 203, 215 seismic excitation, 232 simulation, 216 singular perturbation, 27 static friction, 5, 36 structural, 41, 42, 68, 85, 222, 225, 230, 232 damping, 167 test, 166 theoretical, 77 tribometer, 9 unicycle-cyclist, 201 viscous–elastic, 223 Moineau motor, 12 moment force, 197 rotor’s inertia, 217 monocycle, 197 motor torque constant, 217 moving autoregressive model method, 167 multibody discrete system, 21 multiple discontinuous levels, 89 shooting algorithm, 79 multivalued differential equations, 38 function, 223 natural frequency, 44, 55 imperfections, 57 negative slope, 6 Nelder–Mead simplex method, 159, 165 Neumann theorem, 186 neural network, 198

Newton’s laws, 17 Noether theorem, 145 noise, 12, 31 noisy displacement, 231 non -degeneracy condition, 124 -dimensional equations, 44, 193 form, 33 time, 47 -elastic micro-deformations, 228 -flat billiards, 127, 156 -homogeneous pressure distribution, 54 -parametric equation, 85 -singular matrix, 191 nonconservative force, 202 nondegeneracy condition, 115 nonhomogeneous linear equation, 116 nonlinear approximation, 176 billiards, 129 damping, 167, 169, 172 coefficient, 168, 176, 178 elastic element, 63 energy sink, 38 oscillator, 39 spring, 169 stiffness, 193 nontrivial bounded solution, 124 normal force, 36, 170, 202 mode, 229 vector, 128 numerical model, 168, 176 simulation, 56, 168, 217, 233 solution, 54, 58, 67, 87, 165, 204 “exact”, 85 obstacle, 96 one -degree-of-freedom system, 6, 37, 63, 77, 84, 85, 169, 209, 222 -periodic motion, 48, 60 open-loop control system, 203 orbit, 118 Hamiltonian, 79 homoclinic, 127 impact, 133, 136, 148 Ornstein–Uhlenbeck boundary, 232

page 258

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Index

orthogonal matrix, 142 projection, 108 orthonormality, 154 oscillator, 161 damped, 171 forced, 17 harmonically, 168 impact, 95 out-of-phase component, 50 over-dumped oscillations, 218 overshoot, 57 parameter identification, 159, 165 parameterization, 201 parametric pendulum, 169 representation, 128 Park’s induction model, 210 particle, 128, 143, 146 passive fault tolerant control, 201 vibrations absorber, 10, 169 patches, 28 path following method, 79 PD controller, 210 pendulum, 37, 121, 187 asymmetric, 167 equation, 120 freely swinging, 170 Holweck–Lejay-like, 161 inverted, 96, 127, 198 parametric, 169 torsion, 159, 160, 162, 164 period-doubling bifurcation, 6, 48, 160 periodic excitation, 96 force, 96 forcing, 22, 128, 160 oscillator, 17 motion, 48 orbit, 79 perturbations, 89 solution, 32, 63, 68, 88, 183 sliding, 86, 89 perturbation, 140, 148 perturbed system, 98, 147, 148 phase -plane, 218, 227 trajectory, 218

259

piecewise continuous solution, 14 linear differential equation, vi, 89 smooth function, 98 planar discontinuous system, 86 frictional contact, 159 pendulum, 200 Pohl’s torsion pendulum, 160 Poincar´e map, 18, 30, 39, 47, 59, 63 polar coordinates, 130 pole-zero cancellation, 200 polynomial, 230 approximation, 175, 176 decay, 173 porous surface, 27 positive definite matrix, 207 potential energy, 143, 162, 202 power spectral intensity, 221 Prandtl’s model, 225 projection, 115, 141 pure slip, 6 quadratic equation, 52 quality factor, 171 matrix, 233 quasi -harmonic vibrations, 16 -periodic motion, 38 orbit, 50 solution, 76 system, 95 -sticking motion, 53 radius of inertia, 202 random process, 221 Rayleigh damping, 169 dissipation function, 160, 169 reaction matrix, 233 reflection, 134 regularization, 168 relative displacement, 46 resistance, 217 torque, 163 resonance, 228 Riccati matrix, 233 Riemannian manifold, 127

page 259

June 8, 2017 12:9

260

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

rigid obstacle, 78 rigidity, 161 robustness, 200 rolling friction, 197, 198, 203 rotational connection, 171 rubbing body, 3 Saint–Venant element, 223 seismic event, 35 excitation model, 221, 232 self-excited system, 33, 44 vibrations, 2, 12, 17, 30, 170 semi-empirical identification, 170 sgn function, 27 simple zero, 119, 148 simplex method, 159 singular perturbation model, 27 sliding bearing, 163 friction, 159, 210 solution, 66, 211 homoclinic, 90 periodic, 89 surface, 212, 213 method, 212, 213 slowly-varying parameter method, 160 Smale-like horseshoe, 98, 127, 141 small relocation, 37 solution asymptotic, 98 boundary, 89 bounded, 103, 116, 145 nontrivial, 124 chaotic, 38, 74 discontinuous, 93 approximate, 77 estimate, 173 generically simple, 149 heteroclinic, 120 homoclinic, 89, 91, 93, 95, 100, 123 sliding, 90 impact, 98, 100, 102, 104, 105, 114 bounded, 115 homoclinic, 98, 115, 123, 131 unique, 109, 113 isolated, 192 numerical, 54, 55, 67, 77, 83, 87 periodic, 32, 68

almost, 38, 112, 189 hyperbolic, 97 sliding, 86, 89 unstable, 13 piecewise continuous, 79 quasi-periodic, 74 sliding, 66 stationary, 16, 29 unstable, 132 weak, 79 spectral line, 55 widening, 50, 57 spectrogram, 54, 57 spiral spring, 162 spring-hysteresis, 229 squeal, 18, 49, 51, 58 squealing spectrum, 51 stability, 181, 194 asymptotic, 37 stable branch of equilibria, 195 manifold, 137 projection, 102 solution, 132 steady state, 79 state -feedback gain, 208 -space representation, 206 static friction, 159 force, 82 coefficient, 9, 38, 194 force, 11 model, 5 torque, 163 torque, 163 stationary solution, 16, 29 white noise, 232 steady-state conditions, 52 velocity, 218 step function, 181 stick, 13 –slip, 6, 50 bifurcations, 76 cycle, 70 dynamics, 22, 36 effect, 2

page 260

June 8, 2017 12:9

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

Index

friction law, 8 motion, 26, 60, 80 oscillations, 64 trajectory, 72 vibrations, 12, 53 phase, 57 stiff-ODE solver, 79 stiffness coefficient, 174 matrix, 222 stochastic loading, 231 strain-frequency characteristics, 167 Stribeck curve, 212, 217 effect, 25, 163, 209 stroboscopic view, 205 structural coatings, 28 damping, 167 model, 41, 42, 68, 85, 222, 225, 230, 232 vibration control, 10 viscous friction, 160 structure parameters, 232 sub-harmonics, 57 supercritical bifurcation, 161 SV-element, 222 swinging resonance, 160 symmetry -breaking bifurcation, 161 conditions, 151 synchronization, 6, 39 system algebraic equations, 192 belt-spring-block, 57 block-on-belt, 176 braking, 42 anti-lock, 8 building-ground, 225 chaotic, 127 drilling, 161 follow-up control, 199 impact, 95 linear, 101 variational, 141, 152 linearized, 55 mass-spring-damping, 34 one-degree-of-freedom, 37, 63, 222 open-loop control, 203 perturbed, 98 self-excited, 33

1st Reading

ws-book975x65

261

SISO, 206, 207 three-degree-of-freedom, 159, 198, 202 two-degree-of-freedom, 40, 44, 50, 203, 226, 233 linear, 229 two-dimensional, 118 unicycle-cyclist, 201 unstable, 206 tangent contact force, 170 point, 72 tangential continuation method, 79 temporal scale, 54 test data, 167 model, 166 stand, 45, 75, 161, 167, 168, 170, 172, 173, 200, 201 theorem, 109, 113, 114, 116, 142, 148, 184, 189 Banach fixed point, 185 fluctuation-dissipation, 160 Neumann, 186 Noether, 145 Tichonov, 88 theory Bowden and Tabor, 4 earthquakes, 234 elasticity, 20 Filippov, 65, 83 multibody systems, 22 Oseledec, 39 seismic vibrations, 6 singularity, 96 three -degree-of-freedom system, 21, 32, 159, 198, 202 -periodic motion, 60 Tichonov theorem, 88 time -perturbed impact system, 95 -reversible equation, 100 -varying stiffness, 175 history, 56, 87, 165, 170, 173, 178, 208, 216, 227, 231, 234 invariant system, 206 series, 39, 74, 88, 165, 170 topological degree arguments, 121 torsion pendulum, 159, 162

page 261

June 8, 2017 12:9

262

ws-book961x669

BC: 10577 - Modeling, Analysis and Control of DS

1st Reading

ws-book975x65

Modeling, Analysis and Control of Dynamical Systems with Friction and Impacts

Pohl’s, 160 torsional damper, 159 DNA dynamics, 161 spring, 161 vibrations, 13 tracking control, 218 transient disturbances, 38 dynamics, 47, 50, 54 friction-induced, 49 motion, 6 response, 167 simulation, 11 state, 22, 218 trajectory, 68 trial-and-error method, 28, 218 tribometer, 9 trigonometric simplification, 206 tuning parameters, 218 turning point amplitude, 171, 172 two -degree-of-freedom system, 19, 33, 40, 44, 162, 167, 191, 203, 226, 229, 233 autonomous, 32, 50, 79 discontinuous, 164 -dimensional system, 118 -periodic trajectory, 75 stick–slip, 72 unicycle-cyclist system, 201 uniformly continuous derivative, 100, 103, 109 unilateral constraint, 22 contact, 20, 21, 38 unique impact solution, 109, 113 unstable branch of equilibria, 195 equilibrium, 197, 204 manifold, 137 solution, 13, 132 space, 123 Van der Pol oscillator, 34 variational equation, 115, 119, 124, 145 adjoint, 119 formulation, 20

initial problem, 92 system, 152 vector field, 90, 191 velocity -weakening friction force, 8 window, 13 vertical load, 18 vibrations absorber, 38 creep–slip, 54 free decay, 171, 174 harmonic, 226 normal mode, 229 self-excited, 170 vibro-impact oscillator, 96 virtual damping, 15 viscous –elastic model, 223 damping, 51, 160, 203, 229 friction, 27, 159, 163, 171 coefficient, 217 resistance, 170 visible tangent point, 72 voltage control, 219 wave equation, 169 wavelet, 54 weak slide, 35 solution, 79 weighting coefficients, 233 wheeled mobile robot, 201 pendulum, 204 white noise, 221 wind turbine, 168 work, 229

page 262