Mechanical Vibrations: Types, Testing and Analysis : Types, Testing and Analysis [1 ed.] 9781616686444, 9781616682170

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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Mechanical Vibrations: Types, Testing and Analysis : Types, Testing and Analysis, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Mechanical Vibrations: Types, Testing and Analysis : Types, Testing and Analysis, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

MECHANICAL ENGINEERING THEORY AND APPLICATIONS

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MECHANICAL VIBRATIONS: TYPES, TESTING AND ANALYSIS

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MECHANICAL ENGINEERING THEORY AND APPLICATIONS

MECHANICAL VIBRATIONS: TYPES, TESTING AND ANALYSIS

AMY L. GALLOWAY

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

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Copyright © 2010 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Mechanical vibrations : types, testing, and analysis / editor, Amy L. Galloway. p. cm. Includes bibliographical references and index. ISBN:(eEook)

1. Machinery--Vibration. I. Galloway, Amy L. TJ177.M43 2010 620.1'1248--dc22 2010004372

Published by Nova Science Publishers, Inc. †New York

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CONTENTS

Preface Chapter 1

Chapter 2

Chapter 3

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Chapter 4

vii Methods for Research of States and Control of Vibration of Nonconservative Systems A.N. Kabelkov, V.A. Kabelkov and O.A. Burtseva Spectrum and Operational Modal Analysis with Vector Autoregressive Models V. H. Vu, M. Thomas, A. A. Lakis and L. Marcouiller

57

Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams Metin Aydogdu and Seckin Filiz

95

Parameter Estimation of Pre- Destruction State of the Steel Frame Constructions Using Finite Element and Vibrodiagnostic Methods V. Akopyan, A. Soloviev and A. Cherpakov

147

Chapter 5

Whole Body Vibration Training: Characterization and Analysis Antonio Fratini, Antonio La Gatta, Mario Cesarelli, Paolo Bifulco and Giulio Pasquariello

Chapter 6

Electromechanical Unstationary Thickness Vibrations of Piezoceramic Transformers at Electric Excitation M. O. Shul’ga and L. O. Grigoryeva

Chapter 7

Chapter 8

1

Corrections for Poisson Effect in Longitudinal Vibrations and Shearing Deformations in Transverse Vibrations Applied to a Prismatic Orthotropic Body Loïc Brancheriau Study of Vibratory Phenomena during Mechanical Grape Harvesting: Methodology of Signal Measurement and Analysis Claudio Caprara and Fabio Pezzi

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163

179

205

225

vi Chapter 9

Chapter 10

Chapter 11

Contents A Modified Fourier Series Method for the Dynamic Analysis of Structures Wen L. Li and Hongan Xu

239

Utility of Mechanical Vibration Method for Studying Physical Properties of Solid Materials Y. Hiki

293

On Mechanical Vibration Control Using Viscoelastic Surface Treatments Fernando Cortés, Manex Martinez and María Jesús Elejabarrieta

351

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Index

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405

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PREFACE Mechanical vibrations are the continuing motion, repetitive and often periodic, of a solid or liquid body within certain spatial limits. Vibration occurs frequently in a variety of natural phenomena such as the tidal motion of the oceans, in rotating and stationary machinery, in structures as varied in nature as buildings and ships, in vehicles, and in combinations of these various elements in larger systems. This new book examines the study of vibratory phenomena during mechanical grape harvesting, the utility of mechanical vibration methods for studying physical properties of solid materials, the vibration analysis of piecewise and continuously axially graded rods and beams and whole body vibration training, among others. Chapter 1- Mechanic equations of viscoelastic solid body are used for formulation of research problem of deformable constructions. These equations are considered together with mixed boundary conditions. Nonconservative loadings are determined at one part of body surface, and at the other part of the body surface displacements are determined. Partial differential equations are transformed with variation methods into systems of ordinary differential equations that may have variable coefficients. Research of stability is performed on the base of joint solving of the main state equations and spectrum problem for linearized equations of disturbed motion. Lyapunov-Shmidt method and method of equivalent linearization are used for research of periodical regimes that branch themselves off the main states. Chapter 2- A complete modal analysis method on the using of vector autoregressive models is presented with the introduction of some innovative aspects for applications in operational conditions of vibrating structures. The model is written in form of multivariate vectors and the model parameters are estimated via the computation of the QR factorization. A new index based on the global order-wise signal to noise ratio is built for the selection of model order from which a model at higher orders is derived by a model order updating algorithm. Modal parameters and structural modes are identified either from a stability diagram or from a reconstruction of the structural model via a classification of eigenmodes. Confidence intervals of each modal parameter are also derived and the spectrum of the structural model reveals distinguishingly physical peaks from the noisy spectrum floor. Furthermore, the method can be updated with respect to time for non stationary systems for the applications on structural monitoring with changing parameters. Automatic modal analysis is developed and applied to simulations and experimentally test cases in a widely noisy environment.

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Amy L. Galloway

Chapter 3- In this chapter, vibration of piecewise and continuously axially graded rods and beams is investigated. The equations of motion and boundary conditions are obtained using Newton’s method and Hamilton’s Principle. Initially a general formulation is given for the axially piecewise graded rods and beams. Then rods and beams with two constituents are studied. For the piecewise case governing equations are solved using classical separation of variable method. Legendre method is used for possible continuously grading cases. Since it is not possible to obtain closed form solution for all axially grading forms Ritz method is proposed for arbitrary variation of material properties in the axial direction. Results are obtained for different material composition, geometrical properties and different boundary conditions. Comparisons are made with previous uniform rods and beam results. Tables and graphs are used in order to represent parametrical results. Some specific mode shapes are also given to understand vibration behavior of axially graded rods and beams. It is obtained that the piecewise functionally graded rods and beams may give higher frequencies than their constituents. The piecewise axially grading is studied first time in the literature and the results of present study especially piecewise grading case may open a new research field for researchers studying the subject of vibration. Chapter 4- The known theoretical solutions for approximative estimation of the critical depth and crack location, which correspond to resonances eigenfrequencies of the oscillations for simple geometry (beams, bars) of element constructions have been present. However reliable Non-Destructive Testing (NDT) methods for determination of the crack critical size proceeded to destruction of the complicated element constructions are absent. These methods for technical diagnostics of the buildings with damage have important meaning. Methods for determination of remains health of these objects, for example sea platform frames for oil extraction ,frames antenna mast, have no been created. In the present article complex finite-element (FE) and vibrodiagnostic (VD) approaches to search and definition of the diagnostic parameters of the pre-destruction state (PDS) for steel triangular configuration elements of the frame constructions are discussed. Based on the numerical simulation, it is shown that some resonance frequencies of oscillations to being greater than others have sensitivity to the geometry (depth and angle incision) variation. We also find correlation between transformation of spatial form of the oscillations in the plane one and a critical angle of the incision opening. With aim of the reliability increasing of our results, we have been measured vibrodisplacements in the points near incision, which compared with a curve bending deflection for corresponding form of the oscillation in those points. In addition to this, the dependencies of resonance frequencies for different modes oscillations and the incision depth have been obtained. The peculiarities (points of the bending) at these dependencies have corresponded to start of the PDS process, which has been confirmed of our acoustic emission measurements of the process parameters. Finally, we proposed criteria of the PDS for frame constructions to being the basis for acoustic emission and resonance method of technical diagnostics of the frame constructions with damage. Chapter 5- Human reactions to vibration have been extensively investigated in the past. [9, 18, 19, 23]. Vibration, as well as whole-body vibration, was commonly considered as an occupational hazard and it has been highlighted for its detrimental effects on human condition and comfort.

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Preface

ix

It is normally associated with lower back disorders [8], muscle and nerve tissue damage [33], Raynaud’s Disease (vibration white finger) [25] and interference with cognitive processes, such as those required for short-term memory. [44] Although vibration may produce undesirable side-effects, different studies have shown the positive impacts of vibration upon the bone density of postmenopausal women and disabled children [42, 48, 52], back pain [37], stroke [47], multiple sclerosis [43] and muscle spasticity of cerebral palsy sufferers [1]. Physiologists and physiotherapists have also been reported to use vibration as a therapeutic intervention such as for clearing the lungs and improving joint mobility [23]. Vibratory stimulations were studied for their positive action in eliciting muscle activity. It is well known, in fact, that local tendon vibrations induce activity of the muscle spindle Iafibers, and that a reflex muscle contraction known as the Tonic Vibration Reflex (TVR) arises in response to such vibratory stimulus. The change in the length of the muscle produced by the applied vibratory stimulus is detected by muscle spindles and induces reflex and adaptive responses. [3] Training with vibration extended to the whole body, that is Whole Body Vibration (WBV), was initially proposed as a possible application of the Tonic vibration reflex occurrence to the entire body. [4, 6] Indeed, if a localized vibratory stimulus elicited muscle activity, then whole body vibration, if properly delivered, could obtain similar muscular activity in all the body muscles. The aim of WBV training (WBVT) is to mechanically activate muscles by eliciting similar stretch reflexes that occurred for localized application of vibration. Dr. Vladimir Nasarov, a chair for sports biomechanics at the State College in Minsk, was one of the first to apply vibratory stimulation to help athletes in physical training. Vibrations were applied to the distal muscles and then transmitted through the body chain to the proximal muscles. He used a special device to generate vibrations at a frequency of about 23Hz. Nasarov’s experiments highlighted the potential benefits of training with vibration to muscular development and peripheral circulation improvement. [32] Although literature has largely analyzed and documented the effects of WBV in electing neuromuscular, metabolic and hormonal responses, the exact mechanisms that are accountable for those effects are still unclear. WBV training is clearly different from localized vibratory stimulation; rarely have studies on WBVT monitored the local muscle stimulation (acceleration and displacement) and accounted for motion artifacts presence on EMG recordings. However, it is important to understand the neuro-physiological mechanisms involved in muscle activation under vibration stimulation in order to prescribe safe and effective WBVT programmes. [12] Chapter 6- In the article from the unified positions the systematic research and comparative analysis of thickness vibrations of piezoelectric transformers of flat, cylindrical and spherical form are conducted. The frequency spectrum of vibrations is determined analytically-numerically by reducing of the electroelasticity equations system to the Hamiltonian system on a radial co-ordinate and is used for the estimation of possible frequencies of electric excitation. Unified difference approximation on the spatial coordinate of the coupled system of equations of electroelastic vibrations in Cartesian, cylindrical and spherical coordinates is built. For integration on times, obvious and non-obvious difference schemes are used. On the

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developed algorithm the analysis of evolution of unstationary oscillating processes and their features in the transformers of different form is conducted. Chapter 7- This chapter deals with elastic wave propagation in a finite orthotropic bar. In the first part, the effect of lateral motion caused by the Poisson contraction phenomenon is discussed. This phenomenon induces wave dispersion and consequently vibration is considered as ―quasi-longitudinal‖, rather than pure longitudinal displacement of matter along the bar. Several authors have proposed analytic solutions to quasi-longitudinal wave propagation in rods. Solutions for isotropic bodies with a circular cross section have been discussed. The analytic solution presented here applies to an orthotropic rod with a rectangular cross-section. This solution is based on energy considerations. The material axes are collinear to those of the prismatic bar. The second part of this note is focused on the effects of rotary inertia and shearing deformations which occur during transverse vibrations of prismatic bars. The classical approach assumes that the cross-sectional dimensions of a bar are small in comparison with its length. To take into account these effects, corrections were made using a numerical factor (associated to the shearing force) to obtain a more complete differential equation of motion. A novel method is proposed to provide the exact solution of this differential equation for a rod of rectangular cross-section constituted by an orthotropic body in free-free conditions. This method uses a specific type of musical sound synthesis modeling coupled with classic considerations of elastic behavior of orthotropic materials. Chapter 8- The chapter presents a methodology for measuring vibrating stress during mechanical grape harvesting. The method involves the use of accelerometers positioned on the plants to measure the vibrating stress transmitted by the machine. The piezoelectric accelerometers (sensitivity: 0.316 pC/m s-2; frequency range: 0.1 Hz to 16.5 kHz; measurement range: ± 50000 m s-2 ) are connected to a load amplifier and digital recorder. Each sequence of measurements provides the graph of the accelerations as a function of time. The graphs provide the values of the single accelerations, and spectrum analysis (computed with the Fast Fourier Transform) can be used to demonstrate any effects of interference and overlapping of the vibrations during the machine transit. In order to demonstrate the validity of the methodology, analyses of the data measured using five different types of grape harvesters with horizontal shaking are reported. In some cases comparisons were made in different vineyards, in other cases in the same vineyard. The positioning of the accelerometers in different points of the curtain of vegetation was analyzed to demonstrate the importance of this element on the interpretation of the data. The results show that the construction differences of the machines influence the measured functional parameters. The importance of the vineyard also emerges, especially in the propagation speed of the wave phenomenon because of the different behaviour of the structures constituted by the cordon and shoots. Chapter 9- A modified Fourier series method is described for solving various structural dynamic problems. To better understand the essence of this new approach, a brief review is first given of the conventional Fourier series expansion and a few related mathematical theorems. An improved Fourier series representation is then presented which can be used to expand any function, over a solution domain including the boundary points, with a predetermined rate of convergence. Thus, for a given boundary value problem, an exact continuous solution can be systematically obtained by letting the series simultaneously satisfy both the governing differential equations and the boundary conditions on a point-wise basis.

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Preface

xi

This improved Fourier series method is first used to determine the vibrations of beams with general boundary conditions. It is subsequently extended to the vibrations of two arbitrarily coupled beams, multi-span beams under moving loads, plates with arbitrary boundary supports, and built-up structures composed of any number of beams and plates. The excellent accuracy and convergence of the analytical solutions have been repeatedly demonstrated by numerical examples with varying degrees of difficulties. The improved Fourier series method actually represents a general and powerful mathematical technique for solving a wide range of boundary value problems including the vibrations of various structural components and systems. Chapter 10- The method of observing the response of a solid to an applied mechanical vibration - mechanical vibration method - is useful for investigating various physical properties of materials. The present article is intended to show the utility of the method. First, the formalism of the elasticity and mechanical vibration is concisely described. Second, our previous research mainly concerning crystalline solids is presented. Topics are such as characterization of defects in crystals, nonlinear elasticity of crystals, electron conduction and superconductivity, and melting phenomenon. Internal friction and ultrasound propagation experiments are used. Third, our recent research concerning amorphous or glassy materials is presented. The materials are inorganic, organic, and metallic glasses of various kinds. The phenomena of the glass transition and crystallization are the main objects. Wide range of measurement frequencies is adopted in the experiments. Viscosity, internal friction, ultrasound propagation, and Brillouin scattering experiments are used. Chapter 11- This chapter deals with the structural vibration reduction making use of passive damping control techniques by means of surface treatments with viscoelastic materials. In fact, this kind of vibration control technique is nowadays largely utilised in several industrial applications, such as aeronautical and automotive products. In this context, this work involves a double objective. On the one hand, a classical and contemporarily relevant survey on different subjects concerning viscoelastic treatments is provided. Indeed, a revision of viscoelastic material models is presented, from the most classical models to the most modern ones such as the fractional derivative model, which is currently one of the most efficient ways to reproduce the behaviour of viscoelastic materials. Next, experimental techniques to obtain the material properties are described, as well as homogenised structural models for simple beams and plates with viscoelastic treatments. Finally, some numerical details regarding the analysis of complex systems by the finite element technique are given. On the other hand, two examples of application are shown: In the first one, a study carried out on an unconstrained layer damped beam is presented. First, a steel sample covered by a viscoelastic material is experimentally analysed aimed at characterising the damping material. Also, on the curve fitting of a fractional derivative model to experimental data is discussed. For the finite element analysis of the recovered beam, in frequency and time domains, the use of a fractional derivative model implies specific numerical techniques that nowadays have not achieved enough maturity for industrial practical applications. In this sense, some numerical examples are shown concerning the last investigations of the authors in this field. In the second example, an application of the constrained layer damping treatments to reduce the structural vibration of an automobile component is presented. More precisely, two tailgates are studied both experimentally and numerically. One of them is made exclusively of

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steel, and the other one is made of a sandwich conformed by two adhesively bonded steel sheets. The experimental results are obtained applying a random base motion, and from these results the vibration attenuation due to the damping material can be remarked. These results are used to validate the dynamic response computed by the finite element method. For the model creation, an experimental characterisation of the sandwich material in the frequency domain is required. A curve fitting method based on the equation proposed by Ross-KerwinUngar for symmetric sandwich materials is completed to identify the complex modulus in frequency domain.

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In: Mechanical Vibrations: Types, Testing and Analysis ISBN: 978-1-61668-217-0 Editor: Amy L. Galloway, pp. 1-55 © 2010 Nova Science Publishers, Inc.

Chapter 1

METHODS FOR RESEARCH OF STATES AND CONTROL OF VIBRATION OF NONCONSERVATIVE SYSTEMS A.N. Kabelkov, V.A. Kabelkov and O.A. Burtseva

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South-Russian State Technical University, Novocherkassk Polytechnical Institute, Russia

Mechanic equations of viscoelastic solid body are used for formulation of research problem of deformable constructions. These equations are considered together with mixed boundary conditions. Nonconservative loadings are determined at one part of body surface, and at the other part of the body surface displacements are determined. Partial differential equations are transformed with variation methods into systems of ordinary differential equations that may have variable coefficients. Research of stability is performed on the base of joint solving of the main state equations and spectrum problem for linearized equations of disturbed motion. Lyapunov-Shmidt method and method of equivalent linearization are used for research of periodical regimes that branch themselves off the main states. Problem of frictional oscillations in various mechanical systems and problem of oscillations of high-rise constructions under wind influence are solved. Systems of optimal control are constructed on the base of Lagrange-Euler formalism for damping of oscillation regimes or limitation their amplitudes. The following problems are solved: damping of oscillations in various frictional systems, control of parametrically exciting oscillations in high-rise constructions, optimal relocation of load by robot-manipulator from one place to other, control of fuel injection in diesel engines, problem of passive-active control methods of oscillations of high-rise and extended objects.

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2

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

1. GENERAL FORMULATION OF THE PROBLEM AND THE METHOD OF STUDYING AUTO-OSCILLATORY MODES OF THE SYSTEMS BEING DEFORMED 1.1. The General Equations of Motion of the Viscoelastic Bodies For setting and solving the problem of investigating the auto-oscillations in the systems being deformed, we use the equation of the mechanics of viscoelastic solid body [1]. Let the body occupy the area V of three-dimensional Euclidean space, limited by the surface S. We assign the location (configuration) of the deformed body in the space w by radius-vectors of its dots at any moment of time. Three configurations of the body are considered as follows: reading V, for which we select the configuration of the underformed body, basic , corresponding to the process of deformations; agitated . We designate radius-vectors of dots in each of them respectively as ,

,

, We call mappings (1.1) of configuration V in

, moreover

, or

(1.1)

as deformations. The Jacobeans of

these mappings at every moment of time are considered to be not degenerated:

Displacement field it is described by vector

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respect to configuration

. Gradient of deformation with

is ,

where it is marked that

- is a single tensor,

(1.2)

- the Hamiltonian operator, т - the sign

of transportation. We select Cauchy- Green tensor [2] as the measure of ultimate deformation. .

(1.3)

We assume the volume forces, which act on the body, to be stationary: .

(1.4)

We consider that on the S1 part of S surface from the V configuration the displacements is assigned; the S2 part – has nonconservative forces. ,

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(1.5)

Methods of Research of States and Control of Vibration …

3

where the matrix is operator , D - the derivative of time,

- the constants, which have the dimensionality of time,  - vector

of the load parameters. The connection between the stresses and the deformations should be presented in the following form: ,

(1.6)

where the tensor of generalized stresses with respect to the configuration V is ; T - is the tensor of actual stresses in of

(1.7)

configuration .

In (1.6) and (1.7) formulas, designations « » and «-1» correspond to operations

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convolution and the inversion of tensors. We determine tensors of the fourth and the sixth

Here

and

1

rank by the expressions

are tensors of the elastic characteristics of the material;

and

are

tensor nuclei of relaxation. The motion of body in the reading V configuration is described by the nonlinear equations: (1.8) . Equations (1.8) together with the boundary conditions (1.9) compose the nonlinear mixed boundary problem of moving the viscoelastic body in relation to the reading configuration V.

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1.2. Formulation of the Problem about the Auto-Oscillations of Viscoelastic Bodies Let us formulate the problem of the body basic motion, caused by the development of viscous deformations. Let - be the field of displacement and the field of the generalized stresses in the basic motion. Aggregate of

and

describes the process of the slowly

developing deformations of body. In this case inertia terms can be disregarded in the equations of dynamics. Taking into account relationships (1.2-1.6), (1.9) we write the equations of boundary problem for the basic motion as follows:

The equation of motion, disturbance in relation to the basic state, takes the form accordingly [1]

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(1.10) The boundary conditions correspond to the equation (1.10)

(1.11) In equations (1.10) and (1.11) the following designations are introduced:

,

,

,

,

.

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Methods of Research of States and Control of Vibration … and vector function of

5

– are nonlinear parts of the expansions in the series of

and p in the environment

;

In the problems about the auto-oscillations of the friction and frictional-retarded systems of stress and deformation being deformed, the states are small. In connection to this, disregarding the minors of the second order

and

, we obtain boundary problem with the

equations (1.12) which are linear in the region V and nonlinear (due to the nonconservative forces) on the bounding surface S2:

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(1.13) Nonlinear task (1.12), (1.13) is basic for analyzing the conditions of appearing and characteristic calculations of auto-oscillatory motions. From the position of stability, the fact that auto-oscillations appear, is identical to the loss of stability of the body’s basic state of equilibrium, and auto-oscillations themselves considered as sub- or supercritical motions. In connection to this the conditions under which auto-oscillations appear are determined [1] by the arrangement of the linearized problem spectrum (1.14)

, (1.15) and nonlinear task (1.14), (1.15) is used for calculating the auto-oscillations.

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

1.3. Deriving the Problem of Body’s Agitated Motion to the Operator Equation Boundary problem (1.12), (1.13) is equivalent to the following operator identity:

(1.16) where h - is the vector, which belongs to the Hilbert space H1. The latter is introduced as a subset of smooth vectors, which are inverted into zero on the part of the S1 surface, on a norm generated by the scalar product:

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We present functionals in (1.16) in the form of scalar products in H1:

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Methods of Research of States and Control of Vibration …

7

As a result we come to the operator equation (1.17) Here is the following

are respectively the operators of inertial, dissipative, elastic, viscous and external nonconservative nonlinear forces.

1.4. Determination of the Critical Parameters of Viscoelastic Systems The basic state of system is obtained during the solution of boundary problem (1.10), (1.11). For the analysis of stability of this state we examine the linearized equation of the disturbance motion .

(1.18)

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Assuming [1, 1.3]

where,

we come to the spectral task .

(1.19)

Here «marginal operator of viscosity» is determined by the following identity

where the tensor is (1.20)

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8

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva Let us note that

where

is the tensor of the given maximum characteristics of hardness [1]. Let us name the spectrum of stability an aggregate of the values

, with which the task

(1.19) is nontrivially solved. The presented determination procedure of the critical parameters can be interpreted otherwise. With large t, linearized problem (1.18) of integrodifferential operator by time can be replaced with the differential equation ,

(1.21)

in which the complex operator is introduced

We investigate differential equation (1.21) by a standard method. Assuming that

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we obtain the spectral task (1.22) The pair of roots corresponds to the critical assembly of the system parameters

;

the rest have For finding the critical parameters, problems (1.10), (1.11) and (1.22) should be solved, for example, with (1.23) where m and n - are vectors with the assigned coefficients. A certain straight line in the multidimensional parameter spaces corresponds to the equation (1.23). The values of кр , determine distances from the dot with the coordinates mj to the boundary surfaces, which divide the stability and instability areas. Gradually increasing the parameter of , from equations (1.10), (1.11) we find the appropriate displacement of and the roots

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Methods of Research of States and Control of Vibration …

9

of spectral task (1.22). Using a method of sequential approximations, we calculate кр with the necessary accuracy. The presented method of finding critical parameters is the propagation of the D partition method [4] into the connected problems for the viscoelastic systems, when operators and H depend on the basic state of .

1.5. Calculation of Self-Oscillatory Regimes by Lyapunov - Schmidt Method We write nonlinear operator equation (1.17) with large t in the form:

To find amplitudes and frequencies of the auto-oscillations we produce the replacement of variable in previous operator equation (1.20). Here - is the searched frequency of auto-oscillations. As a result we obtain (1.24) where the operators to кр. Derivatives on

and vector

depend on the displacement of

, corresponding

are designated with dots.

We report increases to the parameters

, assuming that

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. We search the solution of operator equation in the form of series on the degrees of the low parameter : .

(1.25)

We assume respectively

(1.26)

Substituting operator series (1.26) into the equation (1.24), taking into account (1.25) we obtain the sequence of operator equations:

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10

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva (1.27)

where

- are the nonlinear vector functions, which depend on the previous solutions. The

first equation (k=1), is uniform and corresponds to the problem (1.22) about the eigenvalues. We assume that (1.28) where

are the particular solutions, moreover

Substituting expression (1.28) into the equation (1.27) and equalizing expressions with the identical degrees

Here

, we obtain the sequence of the equations:

are

the

vector

coefficients

of

vector

functions

expansions in series according to the degrees With j≠1 there is an inverse operator of function

.

and we find the vector

we find usual way. With j=1 and all k=1, 2,… we have the degenerated

case. In this case with k=1

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where

is its eigenvector function of equation (1.22) with

and

If

is a simple root of spectral task, their eigenvector functions are known with an accuracy to a constant coefficient A1: (1.29) Since the left side of equation (1.27) does not depend on k, we have for k>1

taking into account expression (1.29)

Condition for existence

- the periodic solution of the second equation (1.27): (1.30)

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Methods of Research of States and Control of Vibration …

11

Here

- is the solution of the operator equations system

combined with ,

that corresponds to equation (1.27) with k=1. Examining the third equation (1.27), we compose the expression

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where we find the unknowns

and

from the condition

1.6. Analysis of the Auto-Oscillations Stability For the analysis of auto-oscillations stability varied near the state

where the derivatives

we examine operator equation (1.20),

:

are found in the meaning of Frechet [5]. Assuming that

,

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12

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

we obtain:

(1.31) We search the solution of the equation (1.31) in the form (1.25), assuming that

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(1.32) where

- are the unknown numbers.

Substituting series (1.25) and (1.32) into operator equation (1.31) and equalizing terms with the identical degrees

, we obtain the sequence of the equations:

, where

k

are the vectors, which depend on the form of the operators

Using conditions of solvability of operator equations (1.30)

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Methods of Research of States and Control of Vibration …

we consequently find values are unstable, with of sign

If

13

then the auto-oscillations of system

- they are stable. In the case of

we pass to a study

and the like.

1.7. Approximation of Viscoelastic Systems Distributed by the Parameter Systems with Finite Number of Degrees of Freedom Viscoelastic body can be approximated by a system with n degrees of freedom while we describe the displacement of by using the expression

where

is

a

matrix

with

the

assigned

components

–

functions

r;

is the finite-dimensional vector of generalized coordinates. In this case we represent the equation of system motion, obtained on the basis of the principle of additional work stationarity in the matrix form

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Here is the inertia matrix

R –is the vector of the generalized dissipative forces;

are the vectors of system

parameters and load; H is the secant stiffness matrix; vector of the generalized nonconservative stationary efforts . We determine the basic (equilibrium) state of system from the equation

The equation of disturbance motion system takes the form

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14

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva (1.33)

where the matrix

By relationship

we introduce the «tangential» stiffness matrix. The algorithm of calculation and stability analysis of auto-oscillations in the systems with n degrees of freedom is analogous to the one, examined in p. 1.5, 1.6. Let us prove the periodic solution uniqueness of the type (1.25) and the equation (1.33) when . For this purpose let us derive equation (1.33) to the form (1.34) where the matrix is .

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We represent different from zero component of the vector

by the

series , in which we preserve terms not higher than the third order of smallness. Together with the equation (1.34) we examine the auxiliary equation ,

(1.35)

where it is marked as: ,

Here

.

are the continuous functions, which satisfy conditions;

,

is the matrix of (2n-2) (2n-2) order which has the same eigenvalues as

,

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Methods of Research of States and Control of Vibration … while

, excluding

components

; X - is a nonlinear vector function of parameter

and

.

Since the matrices

and

nonspecific matrix В, where substituting

15

are similar, there is [6] such . Consequently, equation (1.34) by

can be transformed to the form (1.35).

For the equations of type (1.35), theorem [7] concerning the originality of periodic type solution (1.25), which is both stable or unstable limiting cycle is true.

2. RESEARCH OF AUTO-OSCILLATIONS BY MECHANIC SYSTEMS 2.1. Research of Auto-Oscillations of Mechanical System «Cutter-Support» The problem of auto-oscillation at processing the details having a high hardness is solved taking into consideration:   

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 

nonlinear relations between contact forces and displacements, thickness and relative velocity of cutting; retardations of cutting forces relative to displacements and each other; effect of bending deformations of a cutter and yielding of support on the thickness of cuttings; mass distribution along the length of the cutter; dissipative forces of viscous friction.

The design diagram of the mechanical system of the cutter-support is represented in Figure 2.1. The cutting forces and shall be determined by the expressions: (2.1)

Figure 2.1.

Рис. 2.1

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16

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva The signs of the correlations are: – retardation times that are determined by the formula

where

and and

are constant «ways» of retardation; - are dimensionless coefficients considering the influence of oscillations of

the system on cutting speed and the chip flow. At

the retardation

и

are

constant; at the retardation depend nonlinearly on the speeds of the relative motion of the cutter and the piece; - chip shrinkage coefficient [8]; – cutter specific pressure (kg/mm²) at cuttings removal of width mm and thickness

mm; – conditional friction coefficient depending on relative velocity of turning; ,

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where and are coefficients determined on the experiments. The cutter having the mass m is approximated by a bar of a constant cross-section with the bending rigidity EJ and per unit length mass ρ; the support having the mass m1 shall be considered elastically fixed by a spring having a constant rigidity c1 and affected at motion by the forces of viscous friction with the dissipation factor h. The differential equations of the motion of the mechanical system are given as follows: , (2.2) At the following boundary conditions:

.

(2.3).

The following factors aren’t taken into account in the equations (2.2), (2.3): the stretching and compressive deformations; the energy dissipation caused by bending vibrations of the cutter; rotations inertia effect and displacements of the cross-sections effect on the bending deformation of the cutter. Taking into account the displacements of the system the cuttings thickness shall be determined by the formulae

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Methods of Research of States and Control of Vibration … .

17 (2.4)

Here Δ – is a time-independent quantity of the support ―moving up‖ Let’s introduce now the boundary conditions into the equation (2.1), (2.3), (2.4): dimensionless time, coordinate, displacements, forces:

we mark here

Here: .

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As a result we have a system of equations:

and boundary conditions

. Hereinafter the derivatives

and

are supposed to be of small quantities, at

that .

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18

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Let’s present the auto-oscillations research of the system having the following parameters: m=20 kg; kg/m; k=1.67103 МPа; =0.75; E=2105 MPа; I=6.75108

m4; =10-4 m; f0=0.6;

;

, where

and is a velocity

of cutting at which the minimum of the conditional coefficient of friction f is attained. The value of L, , b,  is varied at the computation process. Results of research of stability of balanced conditions are presented in a form of diagrams of the boundary curves dividing areas of stability and instability. The top and bottom borders of areas of balanced conditions stability depending on width of cutting at various dissipation factors h (Figure 2.2) length L of a cutter (Figure 2.3) and support rigidity (Figure 2.4) are given in Figure 2.2-2.4. Wavy arrows specify «vicinities» of values

the steady auto-oscillations are correspondent to, direct arrows – unstable

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(sub- and overcritical) auto-oscillations are correspondent to. At increase the values L, , b the areas of balanced conditions are narrowed, at increase the parameter h they are expanded. The auto-oscillations frequency 0 determined at first approximation (Figure 2.5), is close to free frequencies of a cutter.

Figure. 2.2 Figure 2.3

Figure 2.4 Figure 2.5

Figures 2.2–2.5

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Methods of Research of States and Control of Vibration …

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.6.-2.11.

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19

20

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

The results of stability research of cyclic conditions branching from the balanced conditions are presented in a form of diagrams of dimensionless amplitude square curve depending on the parameters L, , b и h, see Figure 2.6-2.8. Dotted lines correspond to auto-oscillations «in a vicinity» of bottom borders of areas of stability, solid lines – of the top borders of areas of stability. The graphs Figure 2.6-2.8 show that amplitudes of insensitive auto-oscillations:  

decrease at increasing the parameters L, ; can reach extrema at some values of cutting width. The increase of a coefficient of damping h in some cases have an destabilizing impact on oscillation amplitude.

The similar correction graphs for frequencies of auto-oscillations are presented in Figure 2.9-2.11.

Conclusions At analysis of influence of various parameters of the system on areas of balanced conditions and areas of stability of auto-oscillations it was determined that in mechanical systems like support-cutter: (a) can arise both stable and unstable auto-oscillation modes, at that the balanced (stationary) modes of cutting become unstable; (b) in every case there is a critical value of cutting speed 1 and below this value the equilibrium conditions are unstable. The value 1 depends considerably on the parameters of the system; (c) all sub critical auto-oscillations are stable, and the Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

supercritical ones at

can be stable and unstable. The above given

information shows that the modes at ,

are inadmissible and the modes at

are undesirable since the equilibrium conditions are

unstable. (d) at increase of damping of the area of equilibrium conditions the systems expand and the sub critical oscillations don’t appear; (e) at increase of cutting width the areas of stable balanced conditions restrict and at large values disappear; (f) auto-oscillations frequency practically doesn’t depend on dissipation factor, it’s close to the frequency of cutter’s free oscillations and decreases at its length increase.

2.2. Research of High-Rise Buildings Auto-Oscillations Flowed Round by Air Current As a result of interaction of the high-rise buildings modelled in a form of a bar, with an air current flowing round it, the aerodynamic forces (frontal, tangential, elevating) and the moment proportional to speed of an oncoming current appear. At detached flow of round

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Methods of Research of States and Control of Vibration …

21

section the periodical (Karman force) force appears, that is orthogonal to a vector of relative speed with the frequency depending on Strouhal number. Projections of frontal force to the axes of fixed coordinate systems are determined by the following correlations [9,10]: Figure 2.12









q n1  q n 0 1  2 0 u1 cos  u 3 sin    02 u12  u 32











2

3 1 3





2 2 2 q n3  q n 0 1  2 0 u1 cos  u 3 sin    0 u1  u 3



1 2

     u uu ;



 cos  u1 u1 cos  u 3 sin    0  u1 1  u1







1 2

(2.5)



     u u u  ,



 sin   u 3 u1 cos  u 3 sin    0  u 3 1  u 3

2

1 1 3

where  is an angle formed by the flow speed V0 with an axis Ox1 . Projections of the Karman force to axes of the fixed coordinate system are the following:









 sin   0u3 sin 0;









 cos  0u3 sin 0.

qk1  qk 0 1  20 u1 cos  u3 sin   02 u12  u32 qk 3  qk 0 1  20 u1 cos  u3 sin   02 u12  u32

1 2

1 2

The equations of bending oscillations are as follows:

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 4u1

 u1   2u1 u1 u3 2       1    q  1  cos  q  sin  cos   n0 0 n0 0  4      2   u u  qk 00 sin  cos sin 0  1  qk 00 1  sin 2  sin 0  3     qn 0 cos  qk 0 sin  sin 0  N1 ;







 4u3  4



 u   2u3 u u  1    3    qn 00 1  sin 2  3  qn 00 sin  cos 1  2          u u  qk 00 sin  cos sin 0  3  qk 00 1  cos2  sin 0  1     qn 0 sin   qk 0 cos sin 0  N 2 .











In formulas (2.5), (2.6) dimensionless values are introduced:

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(2.6)

22

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva 1

 l 4  2 gl  ;   p t; s  l ; u1  u1 l ; u3  u3 l ;   ; p0   0  EJ  EJ   pl C dV02l 3 C dV02l 3 0   p0 ; 0  0 ; qn0  n ; qk 0  k , V0 2 EJ 2 EJ 2

where  - is a specific density of a bar; EJ – bending rigidity; l – lenths of a bar; d – bar diameter; g - gravitational constant;  - specific air density; Cn and Сk - dimensionless constants determined by experiments. In the formulas (2.6) the summands N1 and N2 containing nonlinear terms are introduced. Assuming that

u j (, ) 

1 f ()(1  cos ), j : 1, 3 2 j

and using the Bubnov-Galyerkin method we reduce homogeneous equations in partial differentiations (2.6) to system nonlinear of ordinary second-order differential equations which are transformed below into a system of first order differential equations

x  A, x  N, x, t ,

 

where – x  R 2n is the state vector; A ,  – is a periodic matrix 2n  2n , depending on a vector of parameters   C , C ,  ,  ,  т ; N,x, t  – is a nonlinear relatively to x vectorCopyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.



n

k

1



1

function and – 2 - a periodic relatively to . On the basis of a linear system of equations 0

x2 V0

u1

u3

r i2 r O i3 u1

r i1 u3

x1

x3 Figure 2.12.

Рис. 2.12.

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

23



x  A0  A1ei0   A1ei0  x

(2.7)

the problem of detection of exitation conditions of parametric vibrations is solved. It is designated here:





т x  f1 , f 3 , f1 , f3 ;

0 1 0  0  0 0 1  0  2   4  2  2 )  C  sin  cos   ,     A 0     0  C  ( 1  cos  1 3 2 1 n 1 n 1 4    2 )  0    C  sin  cos   C  ( 1  sin   1 n 1 n 1 ;

0 0 A1  0.5iCk 1  0  0 0 0 A1  0.5iCk 1  0  0

0 0 0  0 0 0  2 ; 0 sin  cos  1  sin    2 0 1  cos  sin  cos   0 0 0  0 0 0  2 . 0 sin  cos  1  sin    2 0 1  cos  sin  cos  

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The solution of the equation (2.7) is searched in the form of:

x



  x k e 2

1 i

0 k

k 0 

 xk e

 12 i 0 k 

 .

(2.8) 1 i

As a result of substitution (2.8) into (2.7) and equating the members at identical e 2 for the sequences k = 0, 2, 4... and k = 1, 3, 5.... the systems of algebraic equations result:

  i 0   A 0  2 E   A1 A0   

   A1    x  1  0 ; i 0 x  E   1  2    

0 k



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(2.9)

24

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

    A 0  i0 E A1  A1 A0  0 A1    

 0 A1 A 0  i 0 E 

      x 2     2x 0   0 ,   x 2     

(2.10)

Setting equal to zero the equations determinant (2.9) and (2.10) we receive formulas for finding the critical parameter  and oscillation frequency  .

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0

Figure.2.13. Dependence of arrangement of instability area on parameters values

, Ck , Cn :

  5, Ck  0.3, Cn  0.005 (diagram 1),   5, Ck  0.3, Cn  0.05 (diagram 2). On an axis of abscisses -  , on an axis of ordinates 1

0 .

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Methods of Research of States and Control of Vibration …

25

Figure 2.14. Dependence of arrangement of instability area on parameters values , C , C : k n

  2, Ck  1.5, Cn  0.005 (diagram 1),   2, Ck  0.55, Cn  0.005 (diagram 2). On an axis of

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abscisses -  , on an axis of ordinates 1

0 .

Conclusions We consider the parameter

1

to be changing in a given range and

0

to be variable

(relative to the latter the equation is solved). As a result of numerical experiments the areas of instability depending on the parameters Cn , Ck , 1, , 1,  are drawn up. Characteristic boundary curves dividing the areas of stability and instability are presented in Figure 2.132.15. The computation results give us the possibility to make the following conclusions: 

at small parameters

Cn , Ck

the proportional dependence of the frequency

parametric oscillations on critical values of the parameter 

0

of

1 that characterizes the eddy

currents velocity is observed that is coordinated with the known results; at increase C , C the instability areas expand and the proportional dependence n

k

0 1  is disturbed; 

at increase of the bar rigidity the increase of the parametric oscilation frequency is observed. (blue diagram).

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26

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Figure 2.15. Dependence of arrangement of instability area on parameters values , C , C : k n

  5, Ck  0.3, Cn  0.005 (diagram 1),   5, Ck  0.3, Cn  0.05 (diagram 2). On an axis of

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abscisses -  , on an axis of ordinates 1

0 .

3. OPTIMAL CONTROL OF NONCONSERVATIVE SYSTEM VIBRATION It is known [11, 12] that optimal Kalman controls of the observed system ;

(3.1) (3.2)

with quadratic quality functional

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(3.3) (3.4)

Methods of Research of States and Control of Vibration … Symmetric matrix

27

is found from the nonlinear equation (3.5)

vector

– from equation

.

(3.6)

Boundary conditions have the following form

General number of unknown functions, entering equations (3.5) and (3.6) make up In the relations (3.1) – (3.6) is designated as: vectors and control effects;

and

is an output signal of measurement (observation) tools;

– are given and generally variable matrixes; definite;

are state variable

- positive - definite symmetric matrixes;

are nonnegative - function-vector of the

- error vector; т- matrix transportation operation

generated motion program. index; Based on functional (3.3) being with

, we obtain [12]

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(3.7) where symmetric matrix

comply with Riccati equation relatively to

of its

unknown components: (3.8) At introduction of quadratic criterion of generalized operation the quality functional is used in the form (3.9) where

– unknown time function-vector and vector

dependent on

is expected to be clearly

coordinates.

The equation relative to matrix V becomes linear: (3.10)

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Let’s consider the definition of the problem (3.1) – (3.3) within the scope of classical variational theory: definition of the system of functions и delivering minimum to functional (3.3) for a controlled and observed system is required. (3.11)

Here

is the given matrix

Introducing vector

;

- vector of disturbing effects:

.

of indeterminate Lagrange multipliers we convert

functional (3.3) to the form of (3.12) where (3.13) Euler equations relevant to the condition

for the functional (3.12), look like (3.14)

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(3.15) The problem’s boundary conditions will be considered separately. From the equation (3.14) it follows that (3.16) compare it with the first formula (3.4). Differentiating the expression (3.13) we obtain

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(3.17)

Methods of Research of States and Control of Vibration …

29

By substituting the obtained correlations to the expression (3.15), we obtain (3.18) Joint equation system (3.11) and (3.18) with regard for dependence (3.16) wee represent in a matrix form (3.19)

Similar equations have been derived in the paper [13], where controls are assumed by proportional errors:

where

is an undetermined matrix

. In addition, conditions (3.14) are substituted

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by

Note that equations (3.19) can be obtained on the basis of Pontryagin principle [12]. The problem of determining optimal equations is reduced to he solution of 2n linear equations (3.19) with corresponding two-point border conditions. Let’s show that equations (3.19) (with ) are equal to nonlinear equations (3.5), (3.6). Let’s assume (3.20) (see formulas (3.4)and (3.16). Substituting the expression (3.20) by (3.19) we obtain

(3.21) As far as equation (3.21) must be satisfied at any (3.6). If in the functional (3.12)

it falls into equations (3.5) and

is introduced, equation (3.19) is transformed into (3.22)

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

where under conditions that

there follows a Rikkarti nonlinear

equation (3.8). Let’s consider a generalized quality functional like (3.9):

From the condition (3.14) we still obtain the correlation (3.16). The condition (3.16) with regard to correlation (3.20) leads to the equation, containing undetermined vectors X, L and matrix V: (3.23)

Nevertheless this expression with regard to formula (3.16) also is transformed into linear equations. (3.24)

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At

in the equations (3.23), (3.24) and (3.16) it should be adopted that:

As a result of prime transformations we obtain the Krasovsky equation(3.10). Let’s turn to composition of equations with regards to which linear equations such as 3.19), (3.24). must be integrated. In general the minimum condition of the functional(3.12) is the zero equality of its variation.

(3.25) Here the following designation is introduced

With regard to equations (3.14) и (3.15) we have

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Methods of Research of States and Control of Vibration … ,

where the derivative

31 (3.26)

is defined by the expression (3.17).

Consider some particular cases. A. Values are given and the prime state of the system is

As far as at

being fixed,

from the equation (3.26) there follows a boundary condition that takes the following

form for the equations (3.19), (3.22):

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Equations (3.24) and (3.25) must satisfy one-point boundary conditions

B. Values

are given and the finite system states

In terms of

and respectively to equation (3.26) we get

With regards to (3.20) we have (3.27) For the problems (3.24) и (3.25) boundary conditions take the following form:

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32

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva C. Values

, are given but the initial and finite system states are unknown.

From the correlation (3.26) we get

where down to the independence of variations

and

we derive the boundary

conditions for the problems (3.19) и (3.22):

D. In moving boundary problems (at

being undetermined) we introduce additional

conditions (3.28) where

are given vector –functions.

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Considering the relation

on the bases of equation (3.26) we derive: (3.29)

(3.30) The expressions (3.28) – (3.30) form a system of boundary conditions, sufficient for defining integration constants 2n and values . For a particular problem with moving ends, the control duration being given

, we expect that

, where from

expressions (3.26) и (3.29), (3.30) there follows

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Methods of Research of States and Control of Vibration … E. In the moving boundary problems consider the finite state . are varied. With regard to

and constant control time

to be given. In this case the initial state a и

33 we and the time

from the equation (3.26) we get (3.31)

The first of relations (3.31) leads to the boundary condition (3.27) for the problems (3.19) and (3.22). The second correlation leads to a scalar equation relatively to the unknown time value

. Due to its awkwardness this equation is not presented here.

Conclusions For the problems concerning optimal controls and presenting minimum for quadratic quality criteria of types (3.3) и (3.9), the application of Euler-Lagrange variation theory is more preferable which provides: 



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

linearization of equations like (3.19) and significant reduction of the unknown ( instead of for the nonlinear Kalman equations (3.24) or for equations (3.25); wide scope for boundary condition variation- from Cauchi problem to two-points connected problems with moving boundaries; possibility for estimating outer effects and state variables of the controlled system. Really, let of output signals of measurement tools, estimating state variables system and perturbations. Suppose that

where

is a certain approximate effect estimation . The correction

vector

and values

where control values

for the

are defined from equations (3.19), assuming that

are considered to be given. These equations at the noted

designations take the following form

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

4. EXAMPLES OF THE MECHANICAL SYSTEMS CONTROL 4.1. Robot-Manipulator Based on Pantograph Mechanism Equation of robot-manipulator motions based on pantograph mechanism (figure 4.1) has view [15]:

At   Bt   Ct   Dt Q  O3.

(4.1)

Here At , Bt , Ct , Dt  - matrix; their coefficients are nonlinear, recurrence time function; ,  ,  - vector of small deviation, its first and second time derivatives; Q vector of small compensated control action;

O3

- vector not higher than 3-order infinitesimal

relatively deviation and their derivatives. In the first stage of research we have the task to analyze of stability of program motion of manipulator in a first approximation, i.e. to set a limit on study of system behaviour as:

At   Bt   Ct   0.

(4.2)

neglecting influence of non-linear part O3 because of lower order infinitesimal and setting

Q  0 , because stability of program motion is regarded without compensation.

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We carry system of equations of the second-order (4.2) to normal form of Koshi. Result is the system of equations of the first-order

dX  Lt X, dt



где X  1 2 1 2

1   1 , а 2   2 ;



т

(4.3)

- vector of phases coordinates of system, moreover



dX  1  2   1  2  dt



т

- vector of derivatives from

 - modular matrix, in it 0 E phase time coordinates; Lt    0 – zero  1 1          A t C t  A t B t   matrix of the second-order, Е – identity matrix of the second-order. Aim of our research is to set the limits of system stability in the planes of reseaching parameters. As varied parameters we chose contour speed V and coefficients of viscous resistance 1 , 2 for fixed time moments. Rigidity of a springs here c1  c2  4700 Н м .

For obviousness the results of calculating of stability limits are represented as family of

 

curves   2 1

t const V const

.

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Methods of Research of States and Control of Vibration …

35

Figure 4.1. Accounted scheme of robot-manipulator

Calculating we used next numerical parameters values: 1. Length of manipulator links: L  2 m, L  0,5 m, L  1,5 m, L  2 m ; 1 2 3 4 2. Mass of links: m1  12 kg, m2  3 kg, m3  9 kg, m4  12 kg ; 3. Mass of crosshead and tools: m  m  10 kg, m  15 kg ; п1 п2 0 4. Parameters of tool’s trajectory: b  3 m - tunnel’s width; R  1,5 m - radius of

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tunnel’s arch; y  0,5 m - initial height of tool above the tunnel’s floor. 0

Figure 4.2. Section of vertical grade of tool

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

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Figure 4.3. Section of motion on the arch of circle

Figure 4.4. Section of vertical grade of tool

The families of stability limits for 3 volues of contour speed were calculated: V  0,01 m s ,V  0,02 m s ,V  0,03 m s . All results are in the figure 4.2, 4.3. Stability limits are built on points and approximated by cube spline. Coefficients of viscous resistance 1 , 2 were changed in the range from 0 till 10 N s m . If we regard some curve, domain of stability will be above it, but unstable motion will be under it. Valid part of one couple of complex conjugate roots goes to zero on

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Methods of Research of States and Control of Vibration …

37

the curve in the pointed moment after beginning of motion in the considered section of trajectory, speed of the tool is pointed.

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Figure.4.5. Section of motion on the arch of circle

Stability limits are portion of a straight line in the first section of motion. Gradually zone of non-stable motion is decreasing. Beginning with definite moment motion become stable with any values of coefficients of viscous resistance 1 , 2 . We have another picture in the second section. In the initial time moments stable motion is possible with all 1 , 2 , and then zone of unstable motion appears; it’s widen during some time. Besides it, designe of stability limits were carried out in the space of parameters V , c1 , c2 , t . Same families of curves were built in the plane c1 ,c2 . Besides The values of









coefficients of viscous resistance 1   2  1 N  s m (рис. 4.4, 4.5). The curves were built in the rage of rigidity from 0 to 5000 N m . But regidity of spring of horizontal drive is limited by the values not more than 2000 N m in the left. It’s connected with next thing: Stability is possible only with smalle values с with very large 1

values of rigidity

с2 , more than 5000

N m.

Stability limit shifts down in the first section during some time. Besides it, we see, that rigidity с1 in the regarded range influences the system stability poorly . Stability limit shifts down in the second section during some time too. Beginning with definite moment motion becomes stable with any values of coefficients of rigidity in the regarded range [16]. Oscillation modes were researched. They appeared with hitting of parameters 1 , 2





and c1 ,c2 to the limit of section of stability, which were built in the plane of these

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



A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

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38

Figure.4.6. Typical amplitude-frequency characteristic of oscillation modes, which are brunched from the main motion on the limits of stable and unstable sections of programmed motion.

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Methods of Research of States and Control of Vibration …

39

parameters. Besides it, we regarded the fact, that branch of modes of oscillation is impossible on the boundaries, where statical stability losses take place (some number has zeroth really and imaginary section). Designing of amplitude-frequency characteristic was carried out for every admissible limit, received in [17]. Received dependence (Figure 4.6) allow to estimate the character of oscillation modes, which are brunched from the main motion on the limits of stable and unstable sections of programmed motion. Analysis of received amplitude-frequency characteristic led to the next conclusion: it’s necessary to build the system of stabilizing control by robot-manipulator for its containing in admissible space of programmed trajectory.Perturbed motion of manipulator is described by the system of equations in the state space.

dX  Dt X  GU , dt

(4.1)

 



where X - vector n 1 of program motion deviation; D t , G - matrix, relatively

n n and n r ; U - vector r 1 of supplementary control action for the compensation

of deviation. The aim of the task is to find control U(t ) , which will compensate programmed motion deviation and satisfy the optimization: t2





  X т V3 X   X т V1X  U т V2 U dt  min t1

,

(4.2)

where V , V - nonnegative definite matrix; V - positive definite symmetrical matrix. 1 3 2

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We should to solve the system of equations 1 т d  X   Dt   GV2 G  X      dt  L    V1  Dт t   L   

(4.3)

with boundary conditions,



Xt  t  X t1 , 1





L t2  V3X t2

,

(4.4)



i.e. at given initial state of system X t1 , and given time moments of beginning t1 , and control end t (formalism of Lagrange-Euler). Control, which sutisfies the functional (4.2), is 2 described as: Ut   V2 G Lt . 1

т

Set notation:

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(4.5)

40

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva  Dt   GV 1G т  2  Pt    т V     D t  1 

Suggest, that matrix Pt   P constant. Fundamental matrix of system solving is describe as:  t  t Yt    С1  e 1 С2n  e 2 n ,  

where  i  i -th proper number of matrix P; С i - proper vector of matrix P which is corresponds to this proper number. We must eject components, which correspond to proper numbers with positive real sections from Y t , because in this conditions it’s impossible to receive acceptable technical solving of boundary problem for the system (4.3). Generally the solution equals:



Xt  Lt 

т

~ т  Yt  ,

~ matrix where Y 2n n has structure Y~ t   C k  e  k t  , k  N 2 - set of proper t 







numbers of matrix P, which have negative real sections ;  - column vector n 1 of ~ is represented as modular matrix: constant integration. Мatrix Y t  т





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т ~ ~ ~ Yt   Y1 t  Y2 t  ,

then we receive equations:

~ т Xt   Y1 t   ,

(4.6)

~ т Lt   Y2 t   .

(4.7)

Substitute in (4.6) left boundary condition (4.4), we receive:

~ 1 т   Y1 t Xt , For vector of Lagrangian indefinite multipliers we receive from it:

~ ~ Lт t   Y2 t Y11 t Xt .

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41

Method of synthesis of optimal control with using of Lagrangian formalism (look at p.3) is based on feedback (depends on state system (4.1)). It’s connection between vector Lt  and vector Xt 

Lt   Vt  Xt ,



(4.8)



where V t  - matrix n n . We present (4.8) in (4.7) and calculating from the equation (4.6), multiplied in the left V t  , we have:

Y~ t  V(t)Y~ t  2

т

1

 0.

(4.9)

So far as  т  0 , (4.9) gives:

~ ~ Vt   Y2 t Y11 t .

(4.10)



If matrix P does not depend on time, calculating V t using formula (4.10) gives matrix with constant coefficients Vt   V . Control equals:

ˆ Xt  , U т t   V2 1G т L т t   V2 1G т VXt   K

(4.11)

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ˆ  V 1G т V - regulator matrix. Thus, according to (4.11), optimal control of where K 2 linear independent system (criterion(4.2)) does not depend on its initial state, but it’s defined only by parameters of all these systems and its running state. Perturbed motion of manipulator is described by the dependent system of equations (4.1). ˆ K ˆ t , because eigenvectors and eigenvalues of matrix In addition regulator matrix K

Pt depend on time. We use the method of coefficients ―freezing‖, which is wide-spread in practice of control system design by technical tools. We defined control in each moment of time as:

ˆ t Xt , Uт  K ~ ~ . Results of semulation of perturbed motion of manipulator ˆ t   V 1G т Y where K t Y1t  2 2

are shown in the Figure 4.7 and 4.8.

CONCLUSIONS Analysis of received graphs is shown that synthesized stabilizated control meets all technical requirements for manipulator operating. Control allows to provide the compensation of manipulator deviations from motion program.

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Figure.4.7. Deviations of generalized coordinates from the motion program using application of stabilizated control.

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Methods of Research of States and Control of Vibration …

Figure.4.8. Supplementary control effects, which compensate deviations of robot-manipulator from the motion program. Mechanical Vibrations: Types, Testing and Analysis : Types, Testing and Analysis, Nova Science Publishers, Incorporated, 2010. ProQuest Ebook

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

4.2. Industrial Robot-Manipulator

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An industrial robot-manipulator with numerical control (fig.4.9), model М20П.40.01 is designed for automation of loading-unloading operations with work pieces and changing instruments on the metal-cutting machine tools with numerical control. This manipulator can provide service for one or two machine tools thus maintaining a complex system (a robotmachine tool) which is the basis for making machine tool modules designed for continuous work without operator’s participation.

Figure.4.9. The schematic image of the manipulator.

Robot control is done from a separately installed numerical control system. Each link pair is connected to an other by viscoelastic elements with rigidity viscosity

and

. Control actions are applied onto the kinematic pairs, (couples) tending to move

a load from one point in space to an other [18,19]. It is required to find corrections to the control actions in order to dampen and limit the brunching oscillation modes. Mathematical model of the robot is made up on the basis of Lagrange equations of the II-d type

d  T  dt  q j

 T П     Qj ,    q  q  q j j j 

j  1,...,4;

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Methods of Research of States and Control of Vibration … where T, П - kinetic and potential energy system respectively; – generalized forces. In terms of great stiffness of a link

45

 - Dissipative function;

1 2 x  q4 ( x, t )  1  cos  q4 (t ) 2 l2  While analyzing stationary state stability it has been determined that at clearance of the link 2 being given, stationary states are stable and therefore we can consider that the link is absolutely rigid (stiff) and thus reduce the number of coordinates to 3. A system of matrix equations is obtained

Mq  Фq  Hq  G  N

(4.12) (4.12)

Where

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1   J1  J 2  4 m2l2  M 0  0  

   l1m2  l1m 0 , 0 l2 m2  l2 m    0

,

0

l12 c1 0  H   0 l1c2  0 0 

0   0  l2 c3 

are inertial, dissipative matrixes and stiffness matrix respectively; is a vector of generalized coordinates; - a nonlinear vector function; G - controlling effect vector. In the equation dimensionless coordinates

,

have been introduced. Make up the equation of a system motion and perturbed motion relative to the state q : q  q by introducing dimensionless time t : t (4.13) where matrixes

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva are defined at relocating values of q and manipulator's characteristics as .

Taking into consideration state vector the equation (4.13) to the following form:

we transform the linear part of

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where 0 - zero matrix, E - unit matrix.

Figure 4.10. Stability areas

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Figure 4.11. Stability areas

Let’s consider the movement of the manipulator’s gripping device from one point in the space to another for the calculations' simplification let’s consider that  1. Movement with constant velocity along of trajectory is stable. 2. With speed-up movement the instability occurs at the starting phase. In Fig. 4.10 a, b, the movement takes place in time s moving from point m,

m, m,

m,

  1 s, in fig. 4.10 c, d, in   2 m, to the point with coordinates

m.

In fig. 14.11 a, b the movement is executed in

  1 s time, on the fig. 4.11 c, d in .

  2 s time during the movement from point .

m,

point with coordinates

m.

m,

m,

m,

m to the

From the results obtained we notice that viscosity being increased, the stability area expands; the oscillation frequency grows with stiffness and mass increasing, which appears to be a manifestation of nonlinear effects.

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Let's compare the equation of perturbed movement relatively to the q state (4.14) Let's transform the linear part of the equation (4.14) to the following form: ,

(4.15)

where

Optimal system control (4.15) at quadratic quality functional [19,20] (4.16) are defined by the expression

, where symmetric matrix its unknown components:

satisfies the equation of Rikkarti kind relatively to

of

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, where is a not negative definite symmetric matrix,. - a definitely positive symmetric matrix. Introducing vector of undetermined Lagrange multipliers, convert the functional (4.16) into the following form: (4.17)

where (4.18) Euler equations, corresponding the condition following form

for the functional (4.17) have the

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(4.19)

Methods of Research of States and Control of Vibration …

49

(4.20) From the equation (4.19) there follows that . Differentiating the expression (4.18) and substituting the obtained correlations into the expression (4.20) we get .

(4.21)

The joint equation system (4.15) with consideration of dependence (4.21) is presented in the matrix form

.

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The obtained optimal additional control effects G stabilize the system observed (to look Figure 4.12-4.14). In the fig. 4.12-4.14: 1 ,2,3, the movement from point . m, m, m, to the point with m, m, m coordinates. On the fig. 4.12 =4 N s/m, m=5 kg, c=0.9 N/m, in the fig. 4.13. m=10 kg., c=0.95 N/m, in the fig. 4.14. m=20 kg., c=1.05 N/m.

Figure 4.12.

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

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Figure 4.13.

Figure 4.14.

4.3. Passive-Active Control System for High-Rise Structures High-rise structures operation experience proves the demand of implicating different oscillation control systems caused by seismic, wind and technological loads depending on constructive realization and construction height. A system of passive-active oscillation control for high-rise structures by introducing auxiliary connections and devices has been observed. The oscillation control system for high-rise structures is formed on the basis of requirements to the systems of extinguishing oscillations of civil engineering structures [21] Passive elements add to the active system increasing the reliability of the total construction control system. In the active system the oscillation control is executed by electric hydraulic actuators. The active system comprises a subsystem of measurement and estimation of state variables and of identification of disturbing effects at incomplete information. Control subsystem providing optimal the control law.

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51

Fig. 4.15. Design of a high-rise structure Figure 4.15. Design of a high-rise structure equipped with passive-active connections located on all cated on sectional views all sectional views

Let's consider a high-rise structure, equipped with additional connections (Figure 4.15). Stress-deformation state of the structure is described by means of the differential equation system in partial derivatives. By means of variation methods (eg. the finite element method)) the problem of researching the stress-deformation state is reduced to the system of ordinary

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52

A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

differential equations. For a general disturbing effect case the structure’s movement is described through the following equation [22]:

 

(t )  K (t )q (t )  H(t )q(t )  B F F(t )  B R R(t ), q t0  q 0 , Mq

(4.22)

where M, K, H - are the inertial, dissipative and stiffness matrixes of the object; q(t ) vector of generalized coordinates of the structure; . F(t ), R(t ) - determinate vector of disturbing and controlling effects respectively; B , B - are distribution matrixes of F

R

disturbing and controlling efforts in the construction respectively. A mathematical model of the electric-hydraulic actuator complex is described by the differential equation [22]  (t )  GR  (t )  DR(t )  NU (t ) , ER

(4.23)

where E, G, D, N are diagonal coefficient matrixes; U(t ) a stress vector, given to the actuators inputs. Combination of equations (4.22) and (4.23) allows to obtain a close system of differential equations of the controlled structure

 (t )  AX(t )  B F F(t )  BU U(t ) . X x

(4.24)

x

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Vector of optimal control stresses U(t ) is defined from quadratic functional minimum

where

; is a not negative definite symmetric matrix,. symmetric matrix. The solution is obtained in the form

- a definitely positive

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53

, using boundary conditions for the A case (to look p. 3). Numerical study of the active-passive control system was executed by the example of a high-rise structure - a tower head-gear (head-frame) of Obukhovskaya mine. The tower headgear made of monolith reinforced concrete and erected in slips forms with 120 m height and plan dimensions 21 m x 21 m was subjected to a seismic effect with a magnitude of 7 with 0

directing cosines cos x  35 , cos y  550 , cos z  45 relatively to the global coordinate system. Effect frequency is resonant with the frequency of the main oscillation tone of the structure. Additional links are established on the marks of 12 m, 24 m, 36 m providing its controllability. Relative transfer (permutation) of the tower head-gear equipped with an active-passive connection system is shown in the Figure 4.16. Control voltages are shown in the Figure 4.17

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0

Figure 4.16. The comparative analysis of passive and active damping of one period of fluctuations: level of admissible fluctuations; construction with system of rigid communications; сonstruction with system of the rigid communications equipped «a hydraulic spring»; сonstruction with system of the communications equipped hydraulic damper with a throttle;Construction with system of is passiveactive communications

Conclusions Analysis of numerical results makes it possible to conclude that using a passive system of oscillation dampening changes frequency characteristics of a structure, lowering the oscillation amplitude preventing the oscillation amplitude of the tower head-gear from entering admissible limits. Using a combined system decreases the oscillation amplitude of a construction to the admissible level and saves the integrity of a structure and connection work capacity.

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A. N. Kabelkov, V. A. Kabelkov and O. A. Burtseva

Figure 4.17. Control voltages: --- 1(3) drive; +++ 2(4) drive; --- --- borders of admissible values

REFERENCES [1]

[2] [3]

[4]

Gromov V.G. The Lyapunov's first method in dynamic stability of flexible thermoviscoelastic bodies //Reports of the USSR Academy of Sciences, 1976. - V. 223. - Num. 4. - P. 819-822. Lourye A.I. Theory of elasticity. - í: The Science, 1970. - 940 p. Gromov V.G. Dynamic criterion of stability and over critical behaviour of flexible viscoelastic bodies under thermopower loading //Reports of the USSR Academy of Sciences, 1975. - V. 220. - Num. 4. - P. 805-808. Neumark J.I. Structure of D - splittings of polynoms space and Vyshegradsky and Nyquist diagrammes // Reports of the USSR Academy of Sciences, 1948. - V.59. Num. 5. - P. 859-856.

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Methods of Research of States and Control of Vibration … [5] [6] [7]

[8] [9] [10] [11] [12] [13]

[14] [15] [16]

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[17]

[18]

[19]

[20] [21]

[22]

55

Kolmogorov A.N., Fomin S.V. Elements of functions theory and functional analysis. í: Science, 1976. - 544 p. Mishina A.P., Proskurjakov I.V. Higher algebra. - í: Science, 1965. - 300 p. Brushlinskaya N.N. Qualitative integration of one differential equations system in area containing special point and limit cycle // Reports of the USSR Academy of Sciences, 1961. - V. 139. - Num. 1. - P. 9-12. Eljasberg M. E. About stability of cutting process // Reports of the USSR Academy of Sciences, 1958. - Num. 9. - P.12-15. Svetlitsky V.A. Mechanika of bars: Textbook for higher technical education. In two parts. Part 1. Statics. - í: Higher school, 1987. - 320 p. Svetlitsky V.A. Mehanika of bars: Textbook for higher technical education. In two parts. Part 2. Dynamics. - í: Higher school, 1987. - 304 p. Kalman R. E. Mathematical Description of Linear Dynamical Systems. J. SOC Indust. Appl. Math. Series A. On Control, 1963. V. 1. - P. 152-192. Roytenberg J. N. Automatic control. - M.: Science, 1978. - 552 p. Vorontsov G. V, Kabelkov A. N. Optimal controls in automatic regulation systems for tensed-deformed condition of constructions // Building mechanic and constructions calculation, 1990. - Num. 2. - P. 70-75. Krasovsky A. á, Bukov V.N., Shendrik V. S. Universal algorithms of optimal control. í: Science, 1977. - 272 p. Patent of Russian Federation 2101434 ó1 registered 10.01.1998 /M.D. Bondarenko, V.T. Zagorodnjuk, D.M. Krapivin; Novocherkassk State Technical University. Pritykin D.E., Kabelkov A.N. Solution of first problem of dynamics of robotmanipulator based on pantograph mechanism // News of higher educational institutions. North-Caucasian region. Engineering science, 2004. - Num. 2. Pritykin D.E., Kabelkov A.N. Research of program movement stability of robotmanipulator based on pantograph mechanism in first approach // News of higher educational institutions. North-Caucasian region. Engineering science, 2005. - Num. 2. Nefedov V.V. Stability of loading-unloading manipulator // Works of VI International conference "Modern problems of continuous environment mechanics", Rostov-on-Don, June, 12-14th, 2000, Rostov-on-Don, Publishing house: North-Caucasian Science Centre of Higher School, 2001. - V.2. - P.118 - 119. Nefedov V.V., Kabelkov A.N. Calculation of control impacts at specified movement program of loading-unloading robot gripper. // News of higher educational institutions. North-Caucasian region. Engineering science, 2002. - Num 1. - P.15-17. Rojtenberg Y.N. automatic transmission. -M.: Nauka, 1978. – 522 pp.. Cousina (Burtseva) O.A. Analysis of devices and systems for vibration damping of high-rise tower type constructions. - South - Russian State Technical University Novocherkassk, 2002. - 65 p. Deposited in International council for Scientific and Technical Information (ICSTI) 23.05.02. - Annotated in the Bulletin of Account ICSTI"Deposited Scientific Works". - 2002. -Num. 7. - b/Ï num. 60. Kabelkov A.N., Cousina O.A. Mathematical model of passive-active system vibration damping of multidimensional constructions- South - Russian State Technical University - Novocherkassk, 2002. - 69 p.

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In: Mechanical Vibrations: Types, Testing and Analysis ISBN: 978-1-61668-217-0 Editor: Amy L. Galloway, pp. 57-93 © 2010 Nova Science Publishers, Inc.

Chapter 2

SPECTRUM AND OPERATIONAL MODAL ANALYSIS WITH VECTOR AUTOREGRESSIVE MODELS V. H. Vu1, M. Thomas1, A. A. Lakis2 and L. Marcouiller3 1

Department of Mechanical Engineering, École de Technologie Supérieure, 1100 Notre Dame West, Montreal, Qc, H3C 1K3, Canada 2 Department of Mechanical Engineering, École Polytechnique, Montreal, Qc, H3C 3A7, Canada 3 Hydro-Québec’s Research Institute, Varennes, Qc, J3X 1S1, Canada

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ABSTRACT A complete modal analysis method on the using of vector autoregressive models is presented with the introduction of some innovative aspects for applications in operational conditions of vibrating structures. The model is written in form of multivariate vectors and the model parameters are estimated via the computation of the QR factorization. A new index based on the global order-wise signal to noise ratio is built for the selection of model order from which a model at higher orders is derived by a model order updating algorithm. Modal parameters and structural modes are identified either from a stability diagram or from a reconstruction of the structural model via a classification of eigenmodes. Confidence intervals of each modal parameter are also derived and the spectrum of the structural model reveals distinguishingly physical peaks from the noisy spectrum floor. Furthermore, the method can be updated with respect to time for non stationary systems for the applications on structural monitoring with changing parameters. Automatic modal analysis is developed and applied to simulations and experimentally test cases in a widely noisy environment.

1. INTRODUCTION Experimental modal analysis is the procedure to extract the modal characteristics of a dynamic system. It has been widely spread in aerospace, mechanical, acoustic and civil

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engineering for applications in such vibration trouble shooting, structural dynamics modifications, model updating, optimal dynamic design, vibration control as well as vibration-based structural health and condition monitoring. From the last two decades, the industry was asking for the need to conduct modal analysis in-situ on machines operating in working and severe conditions at a low cost. The technique was called operational modal analysis or output only modal analysis since it deals only with the output responses of the structure or machine. It is seen that the core of the modal analysis technique lies on the modal parameter which always attract innovative researches. Unlike the conventional modal analysis with the ability of controllable excitation, identification methods used in operational modal analysis are of MIMO type (Multi-Input Multi-Output) and don’t require the input measurement. Although the experimental modal analysis can deal both with time and frequency domain, operational modal analysis advantageously deals with time measurement and hence time domain methods are utilized and developed in three different groups. In the first methods groups called natural excitation technique (NExT), a time response function is firstly estimated from the output responses via several different ways then modal parameters are identified by their modal decomposition. That time response function can be the impulsive response function [1] as well as random decrement [2]. The second class involves the stochastic subspace identification methods (SSI) which includes the covariance-driven SSI and the data-driven SSI. The common of those methods is the modeling of response data in a discrete state space model. The covariance-driven SSI uses a classical identification method (normally covariance method) to give the system matrices. The Kalman filter is used to give the Kalman states [3], while the data-driven SSI employs first an orthogonal or oblique projection of the space and the SVD to give the Kalman states then a least squares technique to produce the system matrices [4]. The third family is the parametric model based methods where a time series ARMA-type (AutoRegressive Moving Average) model is engaged for the modeling of the output responses. Model parameters are estimated by one of the optimization techniques and modal parameters are identified from the state matrix constructed from the model parameters. It is seen that the general ARMA model can be utilized for both conventional modal analysis with known input and operational modal analysis with unknown input but the later requires nonlinear parameters estimation. Since only the model parameters of the AR part are of interest to give the modal parameters, the AR models are preferred for the output only modal analysis where only a linear estimation is needed. It is seen that the model based modal analysis becomes cumbersome in multi-channel modeling and the selection of the appropriately optimal model order is kind of importance. In this chapter, we present a complete application of the AR model for the operational modal analysis discussions on various aspects. The model is rewritten in a vector version which is convenient for multichannel measurement and model parameters are estimated by least squares via the computation of the QR factorization which is fast and well conditioned. Introduction of a new index called Noise rate Order Factor (NOF) is useful for the selection of model orders. The structural modes are then found at highest modal signal to noise ratio indexes. Modal parameters with corresponding uncertainty are identified with respect to model order and include the Order-wise Modal Assurance Criterion (OMAC) which can confirm if one mode comes from structural behavior or not. Furthermore, all natural frequencies can be exhibited

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Spectrum and Operational Modal Analysis with Vector Autoregressive Models

59

in a balanced spectrum via an amplified modal factor. The procedure can be implemented in a short time manner for the modal based health monitoring.

2. VECTOR AUTOREGRESSIVE MODEL AND PARAMETERS ESTIMATION By using d sensors, a pth order multivariate autoregressive model can be expressed as follows:

 y(t )   a1  y(t 1)  a2  y(t  2)  ...  a p   y(t  p)  e(t )

(0.1)

 y(t ) d 1   A d dp  (t ) dp1  e(t ) d 1

(0.2)

or

where: 

 A ddp   a1 

  a2  ...   ai  ...  a p   is the model parameter

matrix, 

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

  

 ai  d d is the matrix of autoregressive parameters relating the output  y(t  i) to  y(t ) ,  (t ) dp1   y(t 1);  y(t  2); ...;  y(t  p) is the regressor for the output vector  y (t ) ,  y(t  i) d1 (i=1:p) is the output vector with delay time i  T , T is the sampling period (s), and e(t ) d 1 is the residual vector of all output channels, and is considered as the error of the model.

Model parameters need to be estimated from the available data and the most important technique is the prediction error methods (PEM). There are several different estimates to find the solution of the PEM methods, such as the least squares (LSE), maximum likelihood (MLE), instrument variable (IV) and Newton-Raphson iterative search. While the LSE minimizes the sum of squares of the model error [5], the MLE introduces a noisy likelihood function from the power density function and estimates the parameters by maximizing its logarithm in such that the noise is likely as input [6, 7], the IV was introduced for unbiasing the ordinary LSE in case of correlated noises by employing the variable instrument matrices which are well correlated with the responses but are independent of noise [8, 9].

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It is seen that the least squares estimation is the basic method to give the initiation to other methods and therefore if only the output data were available, the LSE is a must and is preferred for the linearity and speed. Obviously, the ordinary LSE is biased because of the contaminated correlated noises [10]. For improving the accuracy and consistency, the LSE can be implemented in recursive manner [11] or in generalized iterative version [12]. In fact, most real applications of operational modal analysis deal with random excitations which can be assumed to be a Gaussian white noise, and therefore the LSE may be applied. Furthermore, the underlying model is rewritten in this chapter in a multivariable vector with a high number of model parameters which can reduce the bias if present [13] and the LSE solution was found via the computation of the QR factorization which is stable and well-conditioned. If N successive output vectors of the responses from y (k ) to y(k  N  1) are considered, the model parameters can be obviously estimated with the least squares method by minimizing the norm of error sequences. ˆ  arg min V ()  arg min( 1  N N

k  N 1



e(t ) )  arg min( 2

t k

1 N

k  N 1



y(t )   (t )

2

(0.3)

t k

The term ―arg min‖ means ―the minimizing argument of the function‖ and VN ( ) is a well-defined scalar valued function of the model parameters. In this paper, the assumption on the Gaussian white noise is kept and hence the least squares estimate is assured to be unbiased. If the trace norm of the error covariance matrix is used, the function VN ( ) can be transformed as follows: T  1 k  N 1   y (t )   (t ) y (t )   (t )  t  k N 

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VN (  )  Tr 

T     1 k  N 1   [ (t ) T (t )]1[ (t ) yT (t )]  (t ) T (t )   [ (t ) T (t )]1[ (t ) yT (t )]        =Tr    N t k  y (t ) yT (t )  [ y (t ) T (t )][ (t ) T (t )]1 (t ) yT (t ) 

(0.4)

It is readily to see that the norm is minimized when: k  N 1 T  k  N 1 T  ˆ d dp      y (t ). (t )    (t ). (t )  t k   t k 

at the value

1

 1 k  N 1  T ˆ  (t ) yT (t ) VˆN ( )  Tr  y (t ) y (t )       N t k 

(0.5)

(0.6)

The data of these N consecutive values of the responses are used to construct the moment matrices:

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Spectrum and Operational Modal Analysis with Vector Autoregressive Models

U

V

W



dp dp

d d



d  dp



61

k  N 1

1 N

  (t ).

T

(t )

t k

k  N 1

1 N



y (t ). yT (t )

(0.7)

t k

k  N 1

1 N



y (t ). T (t )

t k

The estimated parameters matrix is therefore simply derived and can be written as follows:

ˆ  W .U 1 

(0.8)

Following up is the estimated covariance matrix of the deterministic part: Dˆ

d d



1 N

k  N 1

  y(t )  eˆ(t ). y(t )  eˆ(t )

T



t k

1 N

k  N 1

 t k

T

1 T ˆ   ˆ    (t )  .  (t )   W .U .W

(0.9)

and the estimated covariance matrix of the error:

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Eˆ

d d



1 N

k  N 1

 eˆ(t ).eˆ

T

(t )  (V  W .U 1.W T )

(0.10)

t k

ˆ . (t ) . with the estimated residue vector eˆ(t )  y(t )   Referring to equation (0.8), it can be noticed that the matrix U

dp dp

possesses very high

dimensions and may be ill-conditioned. This technique is hence avoided in case of multivariable models and therefore the singular value decomposition (SVD) or the triangular decomposition (QR) is more suitable. In practice, the QR decomposition works well and requires less computation than the SVD [14]. The following section describes an efficient way using the QR-factorization [15]. The moment matrices can be expressed as follows:

M

( dp  d )( dp  d )

U W T  1   W V  N

k  N 1

 t k

 (t )  T  y(t )   (t )  

yT (t )  

1 T K .K N

where K is the data matrix

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(0.11)

62

V. H. Vu, M. Thomas, A. A. Lakis et al.

K

N ( dp  d )

  T (k )  T  (k  1)   ...  T  ( k  N  1) 

  y (k  1)   ...  T y (k  N  1)  yT (k )

T

(0.12)

Introducing now the QR factorization of the data matrix K  Q.R where Q orthogonal matrix ( Q.QT  I ) and R

N ( dp  d )

 R11 dpdp  R 0  0 

N N

is an

is an upper triangular matrix:

  R22 d d  0  

R12

dp  d

(0.13)

This decomposition gives the matrix M in a new form of the Cholesky decomposition

M

T 1 1 1 R R (Q.R)T .(Q.R)  RT .R   11T 11 N N N  R12 R11

  T R12T R12  R22 R22  R11T R12

(0.14)

From (0.8) with attention to the uniformity of (0.11) and (0.14), the parameters matrix is finally derived:

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ˆ  ( RT R ).( RT R )1  ( R1R )T  12 11 11 11 11 12

(0.15)

also the estimated sample covariance matrices of the deterministic part and of the error in (0.9) and (0.10) become:

Eˆ

1 T R12 R12 N

Dˆ

d d

d d

 (V  W .U 1.W T ) 

 W .U 1.W T 

1 T R22 R22 N

(0.16)

(0.17)

It is advantageously found that with the QR factorization: 



The model parameters are estimated in a fast and well conditioned fashion since the condition number of matrix R11 is the square root and is much lesser than matrix U .

ˆ are computed Estimated covariance matrices of error Eˆ and of deterministic part D separately from the model parameters and hence are referred to the signal-noise partition.

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Furthermore, triangularity of matrix R11 and Cholesky form of these matrices are convenient and cost-effective for problem solving and updating, as given in next section.

3. MODEL UPDATING IN TIME The QR factorization of a model order p should be recursively updated when a new set of samples data is available along with measuring time. That means that we want to calculate, from the matrices Q

(k )

and R

(k )

of data matrix K ( k ) at time t  k , the Q

(k s)

and R

(k s)

of data matrix K ( k  s ) which is found by deleting the first s rows and appending more s rows to matrix K ( k ) .

K (k )

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K (k s)

N ( dp  d )

N ( dp  d )

  T (k )  T  (k  1)   ...  T  (k  N  1)

  T (k  s)  T  (k  s  1)   ...  T  (k  s  N  1)

  y (k  1)   ...  yT (k  N  1)  yT (k )

T

(0.18)

  yT (k  s  1)   ...  yT (k  s  N  1)  yT (k  s)

(0.19)

The relationship can first be established as follows:   K (k )  T  T  (k  1  N  1) y (k  1  N  1)    ... ...  T  T  (k  s  N  1) y (k  s  N  1) 

( N  s )( dp  d )

  T (k ) yT (k )    ... ...   T  (k  s  1) yT (k  s  1)    K (k s)  

(0.20) ( N  s )( dp  d )

That gives in terms of the QR decomposition of the data matrix:   Q(k ) R(k )  T  T  (k  1  N  1) y (k  1  N  1)    ... ...  T  T  (k  s  N  1) y (k  s  N  1) 

( N  s )( dp  d )

  T (k ) yT (k )    ... ...   T  (k  s  1) yT (k  s  1)    Q(k s ) R(k s )  

(0.21) ( N  s )( dp  d )

and in an innovative form:

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Q   0

(k )

where I s

s s

  R(k )  T  T 0   (k  1  N  1) y (k  1  N  1)   I s     0 Is  ... ...  T  T  (k  s  N  1) y (k  s  N  1) 

  T (k ) yT (k )    (0.22) 0  ... ...  (k s)  T T Q   (k  s  1) y (k  s  1)    R(k s )  

is the unity matrix.

In this algorithm, one may wants an update of the submatrices R11 , R12 and R22 of matrix R as defined in equation (0.13). Matrices R ( k ) and R( k  s ) should therefore be partitioned in a well conditioned form:

Q   0

(k )

 R1( k )  T 0   (k  1  N  1)  Is   ...  T  (k  s  N  1) (k )

where the new submatrices R1

  y (k  1  N  1)   I s    0 ...  yT (k  s  N  1)  R2( k )

T

and R2

(k )

  T (k )  0  ... (k s)  T  Q  ( k  s  1)   (k s)  R1

  (0.23)  T y (k  s  1)   R2( k  s )  yT (k ) ...

(k )

are related to the submatrices R11

, R12

(k )

and

R22( k ) as follows:

(k ) 1 N dp

R

 R11( k ) dpdp   and R2( k )   0 ( N dp )dp   

Nxdp

 R12 ( k ) dpxd     R22 ( k ) ( N dp ) xd   

(0.24)

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If the first dp columns of equation (0.23) are extracted, we obtain:

Q ( k )   0

  R1( k )  T  0   (k  1  N  1)   I s     0 Is   ...  T   (k  s  N  1) 

  T (k )    0  ...  Q ( k  s )   T (k  s  1)    (k s)  R1 

(0.25)

It can be seen that equation (0.25) is a sub-problem of equation (0.23) for the first dp columns. The right hand side can then be transformed from the left one by using two sets of orthogonal Givens rotations.   R1( k )  T  The first set G1 applies on the matrix  (k  1  N  1)  to annihilate all dp  s lower   ...  T   (k  s  N  1)  elements and to obtain an upper triangular matrix.

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Spectrum and Operational Modal Analysis with Vector Autoregressive Models     R1( k ) R1( k )  T   T   (k  1  N  1)   (k  1  N  1)  G1   [( J N 1,dp ...J N  s ,dp )...( J N 1,1...J N  s ,1 )]      ... ...  T   T   (k  s  N  1)   (k  s  N  1) 

where J i , j

( N  s )( N  s )

65

(0.26)

is the ( i, j th ) Givens matrix zeroing the (i, j )th element of the matrix

  R1( k )  T   (k  1  N  1)  . in the previous step J ...J  i 1, j N  s , j ...( J N 1,1 ...J N  s ,1 )   ...  T   (k  s  N  1)  These Givens matrices are easily constructed in common form:

Ji, j

( N  s )( N  s )

1   0   0   0 

...

0

...

c

...  s ...

0 j

... 0 ... 0    ... s ... 0  j   ... c ... 0  i   ... 0 ... 1  i 

(0.27)

 c s   Givens(element ( j , j ), element (i, j )) of the matrix of the previous s c 

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where 

step. The left side term of (0.25) can be rewritten as:

Q   0

(k )

  R1( k )  T  0   (k  1  N  1)  Q ( k )     0 Is  ...  T   (k  s  N  1) 

    R1( k ) R1( k )  T   T  0  T  (k  1  N  1)   (k  1  N  1)  (0.28)  Q ( k )G1   G1 G1     Is  ... ...  T   T   (k  s  N  1)   (k  s  N  1) 

The second set of Givens rotations G2 is used to set unitary the first s rows and columns of the augmented matrix Q ( k )  Q  0

(k )

Consider vector zr ( k ) Qr 1( k ) at the (r  1)

th

0 T.  G1 Is  T

( N  s )1

 qr ( k )  where qr

(k )

is r

th

row of the augmented matrix

computational step (r  1: s) , since

zr ( k ) is orthonormal

T

(  zr ( k )  z ( k )  1 ), we can have

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G2 r zr ( k )  J r 1 J r 2 ...J N  s zr ( k )  0 ... 1 ... 0 ... 0   r  where the Givens matrix J i zeroing the i th element of zr

Ji

( N  s )( N  s )

1   0  0   0

(k )

T

is in the form:

0 ... 0    ... c s ... 0  (i  1)  ...  s c ... 0  (i )   ... 0 0 ... 1 

...

(0.29)

0

(0.30)

( i 1) ( i )

Computing the matrix G2 hence shows it to be equal to the multiplication of s components 1

G2   G2r

(0.31)

r s

The left side term of (0.25) or (0.28) is thus further rewritten:

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Q ( k )   0

  R1( k )  T  0   (k  1  N  1)  Q ( k )     0 Is   ...  T   (k  s  N  1) 

  R1( k )  T  0 T T  (k  1  N  1)    G1 G2 G2G1   Is  ...  T   (k  s  N  1) 

(0.32)

Since the second Givens rotation set is established on the first s rows of the augmented matrix, two interesting consequences are released: 

Its right transpose multiplication will unitary the first s rows and first s columns of augmented matrix Q



(k )

Q ( k )   0

0 T  G1 ; 1

Its left multiplication will nonzero the first s elements of each row of the upper

 R1( k )  triangular matrix G1  T  , makes each one an upper Hessenberg matrix.  (k  N )  That explains:

Q ( k )   0

0  T T Is  G1 G2   Is  0

0 *( k  s )

Q

  

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(0.33)

67

Spectrum and Operational Modal Analysis with Vector Autoregressive Models

    T (k )  R1( k )   T   ...  (k  1  N  1)     G2G1     T (k  s  1)  ...   T   *( k  s )   (k  s  N  1)   R1

(0.34)

It can be seen that two Givens rotations sets are built only on the first dp columns of the data matrix. The derived matrix Q*( k  s ) therefore coincides to the exact matrix Q ( k  s ) on the T

first dp columns and the orthonormal condition Q*( k  s )  Q*( k  s )   I is assured. Matrix

R1*( k  s ) which is only nonzero on the first dp rows, is the actual desired matrix R1( k  s ) *( k  s )

hence R1

 R1( k  s ) . Then the factorized matrices R11( k  s ) at the sample index (k  s)

are therefore exactly updated at this stage:

R11( k  s )  R1*( k  s ) (1: dp,:) *( k  s )

with the derived matrix Q

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Q ( k )   0

, the last d columns of equations (0.24) can be rewritten as:

  R2( k )  T  0   y (k  1  N  1)   I s     0 Is   ...  T   y (k  s  N  1) 

*( k  s )

and the matrix R2

(0.35)

 yT (k )    0  ...  Q*( k  s )   yT (k  s  1)    *( k  s )  R2 

(0.36)

is directly extracted from the equality:

 yT (k )    Is ...    yT (k  s  1)   0    *( k  s ) R  2 

  R2 ( k )  T  0   y (k  1  N  1)    Is   ...  T   y (k  s  N  1) 

 Q ( k ) T Q*( k  s )    0  0

*( k  s )

As discussed earlier, the first dp rows of matrix R2 means the submatrix R12

(k s)

(0.37)

are also exactly derived, which

and covariance matrix of the deterministic part were exactly

updated. We can now write:

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R2

Since Q

(k s)

*( k  s )

and Q

*( k  s )

 R12 ( k  s ) dpd    * (k s)  R22  ( N  dp )d  

(0.38)

are both orthogonal, we can readily see that the submatrix

* (k s) satisfies the equations: R22

T

T

* (k s) * (k s )  R22   R22    R22( k  s )   R22( k  s ) 

(0.39)

and the error covariance matrix is therefore updated.

4. MODEL ORDER UPDATING 4.1. Updating with Ascending Model Order It is derived from the previous section that the solution yields to the computation of submatrices of the R factor. Up to date, all existing algorithms require a repetitive computation for a set of model orders which needs itself a prior evaluation. In this section, we present a computation for the update of the model when the order is increasing. Suppose that

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we have the data matrix K ( p ) at order p and its factored matrices Q

K ( p)

N ( dp  d )

  T (k )  T  (k  1)   ...  T  (k  N  1)

R( p)

  y (k  1)    K1( p )  ...  T y (k  N  1) 

N ( dp  d )

( p)

, R

( p)

:

yT (k )

T

 R11( p )   0  0 

N  dp

K2

N d

 (0.40) 

R12( p )  ( p)  R22  0 

At order p  1 , data matrix takes the form:

K ( p 1) where K

' N d

N ( d ( p 1)  d )

  K1( p )

N dp

K'

N d

K2

N d

 

is the added d columns

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(0.41)

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Spectrum and Operational Modal Analysis with Vector Autoregressive Models

K'

N d

 yT (k  ( p  1))   T  y (k  1  ( p  1))      ...  T   y (k  N  1  ( p  1)) 

(0.42)

Then one can compute the following matrix:

Q

( p )T

K

( p 1)

 Q

( p )T

( p) 1 N  dp

K

Q

( p )T

K

'

Q

N d

( p )T

 R11( p ) T1  K 2 N d    0 T2  0 T3 

R12( p )  ( p)  R22  0 

(0.43)

where

 T1 dpd    Q ( p )T K '   T2 d d .   T3 ( N ( dp  d ))d  If one takes the QR factorization of the submatrix T2

T3  , it yields to: T

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T2   RT  T   QT  0     3 where RT

d d

is an upper diagonal matrix and QT

( N  dp )( N  dp )

(0.44)

is an orthogonal matrix.

Then equation (0.43) becomes:

Q ( p )T K ( p 1)

'

where R22

I   dpdp  0

 R11( p ) 0  0 QT    0 

d d

RT 0

 R( p ) 0   11 0 QT    0 

T

''

and R22

   I dpdp ( p)  T  R22   QT    0  0  R12( p )

T1

( N  dp  d ) d

T1 RT 0

R12( p )  '  (0.45) R22  ''  R22 

( p)  R22  .  0 

are submatrices of the product QT  ''

It can be noticed that the submatrix R22 in the right hand side of equation (0.45) is not an upper diagonal matrix and can be trangularized by a small orthogonal transformation to yield the true QR decomposition of the data matrix K

( p 1)

. This modification does not evidently

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affect other submatrices. Hence the submatrices R11

( p 1)

and R12

of model order p  1 are

exactly updated:

 R( p ) R11( p 1)   11  0

 R12( p )  T1  ( p 1) and R    '  12 RT   R22 

(0.46)

ˆ ( p1) and covariance matrix Dˆ ( p 1) are updated as Followed, the model parameters  shown in equation (0.15)-(0.16).

ˆ ( p 1)  ( R( p 1)  1 R( p 1) )T   11  12

(0.47)

T T T T ' ' ' '  R22  R22 (0.48) Dˆ ( p 1)   R12( p 1)  R12( p 1)   R12( p )  R12( p )   R22  Dˆ ( p )   R22

Since all transformations are orthogonal, covariance matrix Eˆ ( p 1) is also updated T T ( p 1) T ( p 1) '' T '' ( p) T ( p) ' ' ' '  R22   R22    R22  R22   R22   Eˆ ( p )   R22   R22  (0.49) Eˆ ( p 1)   R22   R22   R22

Several observations should now be made: 

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

With only a small factorization of a submatrix of the R factor matrix, the model parameters and covariance matrices can be exactly updated to a higher order. This technique is much preferred than the repetitive QR factorization for each order value. From the update of covariance matrices, it is noticed that as the model order increases, the deterministic part increases and the stochastic part decreases monotonically. The changing amount is equal and is significant at a low order and negligible at high orders. This feature is thus an idea for a new orders selection criterion as discussed later.

4.2. Updating with Decreasing Model Order The order downdating of QR factorization is taken from the order p in the same data set. Consider at the available data, the data matrix K the data matrix K

( p 1)

( p)

of model order p can be partitioned to

by removing its d last columns K

' N d

of the regressors term.

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K ( p)

N ( dp  d )

  T (k )  T  (k  1)   ...  T  (k  N  1)

  yT (k  1)    K ( p)   1 ...  yT (k  N  1) 

  K1( p 1)

K'

K ( p 1)

yT (k )

N d ( p 1)

N ( dp ( p 1)  d )

K2

N d

  K1( p 1)

N d

N dp

K2

N d

 

(0.50)

 

K2

N d ( p 1)

N d

 

(0.51)

Since the sample number N is always larger than the data dimension d, the QR factorization applied on the matrix K ( p ) can have the form, see (0.13).

K

( p)

Q R ( p)

( p)

Q

( p)

 R11 0   0

R12  R22  0 

(0.52)

It is shown from the time updating in the previous section, that the matrices Q *( p )

are derived instead of the exact factored Q can be rewritten as:

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R

K ( p )  Q*( p ) R*( p )

R  Q*( p )  11 0

( p)

and R

( p)

*( p )

and

, then the data matrix K

 R ' R '' 11  11 R12  ''' *( p ) 0 R  Q  11 *  R 22    0 0

( p)

  R12''  R12  R*22 

(0.53)

R12'  *( p ) #( p 1) Q R R22# 

(0.54)

R12' R11

Then it is readily to see that:

K ( p 1)

 R11'   Q*( p )  0 0 

 R11' R12'    R12''   Q*( p )  0 0 *  R22  

R12'  '  *( p )  R R12''   Q  11  0 * R22 #  R22 

And by consequence of the exact QR decomposition:

Q*( p ) R#( p 1)  Q( p 1) R( p 1) It can be seen that matrix R

#( p 1)

*( p )

first subcolumns of matrix R

(0.55)

can be found by removing the last d columns from the

and according to (0.53), it is not neither an upper triangular

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matrix. This makes the formulation (0.54) not a true QR factorization of the data matrix

K ( p 1) . Fortunately, since the two matrices K ( p ) and K ( p 1) have the same first d ( p  1) columns, their R factors are thus identical in first d ( p  1) rows and d ( p  1) columns. It means the submatrices R11 and R12 in matrix R #( p 1) are exactly as found in the matrix '

'

R( p 1) and make the eq. (0.15) and (0.16) accurate for the updated model order ( p  1) .

ˆ ( p 1)  ( R' 1R' )T  11 12

(0.56)

Dˆ ( p 1)

(0.57)

The only different component between R *( p 1)

*

is not upper triangular (like R22 in R

 R12' T R12'

d d

#( p 1)

and R

( p 1)

#

lies on the matrix R22 which

). It is noted that the energy of matrix K

( p 1)

is

constant, one can have: T

T

Q*( p ) R#( p 1)  Q*( p ) R#( p 1)   Q( p 1) R( p 1)  Q( p 1) R( p 1)  Q*( p )T Q*( p )  Q( p 1)T Q( p 1)  I ,

Since #( p 1) T

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 R   R updated:

#( p 1)

Eˆ ( p 1)

   R

d d

( p 1) T

  R

R

' T 22

( p 1)

R R ' 22

it

can

be

found

(0.58) that

 and finally the covariance matrix of error can be exactly

#T 22

R   R # 22

'' T 12

 R12''  *T * R   *   R12'' T R12''  R22 R22 (0.59)  R22  *T 22

It means the QR factorization is accurately updated from model order p to the model order p  1 . Several advantages can be raised: 

The updated solution can be found without any transformation and thus there is no attention to the Q factor matrix.



Though the R matrix given from time updating is not the true R factor, updated solution is accurately achieved by a simple matrix extraction.

According to theorem 1.4.2 of Bjork on the interlacing property [14], the smallest singular value of R will not decrease, the technique is hence a stable procedure.

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5. SELECTION OF MODEL ORDER The selection of the model orders is the first step in the identification process. Several authors have discussed on the question and have proposed different criteria [16]. The model order would be selected by several criterions of the following common fashion:

pˆ  arg min eˆ(t ) f (d , p, N )

(0.60)

p

where f (d , p, N ) is a linear penalty function that increases with respect to p and decreases with N , and eˆ(t ) is a norm of the estimated model error. Among the better known ones are the criteria based on the statistical properties of prediction errors written herein after for multivariate model: 

The Final Prediction Error (FPE) which is based on the covariance of the prediction error, which was proposed by Akaike [17] and its variants (e.g. AIC):

FPE ( p)  

d

N  d. p N ( N  d . p)

(0.61)

The Minimum Description Length (MDL) which was developed by Schwartz and Rissanen [18] and its variants (e.g. SBC, BIC, MAP) :

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eˆ(t )

eˆ(t ) d

(1 

2d . p log N ) N

(0.62)

Relating to criteria for the prediction ability of a multivariate AR model, several discussions should be made as follows: 





These criteria are aimed to prediction purpose and are primarily based on the evolution of the error covariance which monotonically decreased with respect to model order; FPE and its variants asymptotically choose the correct order model if the underlying multiple time series has high dimensions d [19], but they have tendency to overestimate the model order as the data length increases [20]; MDL and its variants outperform with long recorded data and are strongly consistent when the data length tends to infinite [21].

However, the application of these criteria requires first the selection of a possible interval for the model order to be used and then an evaluation of parameters. The selection of the order is made on the basis of minimal variance. It is necessary, therefore, to have a prior evaluation with different orders, which results in a significant computation time. Other authors [22, 23] have proposed new approaches derived from the MDL criterion and based on

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the minimum eigenvalue of the covariance matrix without prior evaluation. To determine the model order, it is necessary to determine a limiting upper value in order to establish the order of the matrix. However, it is difficult to experimentally determine that value when significant noise is present. In modal analysis, not only the error sequence but also the deterministic part of the signal has to be taken into account since this latter contains the modal parameter information of system. Based on these facts and applied to mechanical systems, this research proposes an innovative factor for selecting the model orders of the mechanical system based on the analysis of the noise-to-signal rate (NSR).

5.1. Definition of the New Noise-Rate Order Factor While the MDL criterion considers only the prediction error, the estimated noise-tosignal rate (NSR) can be defined from the above signal-noise separation with the presence of the deterministic norm: 

NSR 

 eˆ(t ) Tr ( Eˆ ) Tr ( Eˆ ) (%) or NSR  10log10 (dB)  yˆ (t ) Tr ( Dˆ ) Tr ( Dˆ )

(0.63)

Then a Noise-rate Order Factor (NOF) is defined as being the variation of the NSR between two consecutive orders: 



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NOF ( p)  NSR( p)  NSR( p  1)

(0.64)

It is seen from eq. (0.48) and (0.49) that as the model order increases, the deterministic part increases and the error part decreases. That is why the NSR increases monotonically with respect to model order and contains properties of both stochastic (on numerator) and deterministic (on denominator) norms. The NOF is therefore always positive. It falls quickly at low orders, converges at high orders and hence is a criterion of model performance. The NOF is a representative factor for the convergence of the NSR which changes significantly at low orders and also converges at high orders. Since this factor is positive and close to zero, selection of optimal model orders is thus easier conducted on this single curve evolution (Figure 1).

5.2. Determination of Minimum Required Model Order The minimum required order is referred to the structural system order which depends on the degrees of freedom of system and is independent of noise. It is chosen as corresponding to the convergent point after the significant change of the NOF curve. Since both deterministic and stochastic norms converged at this minimum value, any higher order could be used to fit

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75

the data. This minimum order is hence easy and obvious to determine on the NOF evolution as referred to pmin in Figure 1.

5.3. Selection of Computational Model Order and Noise Rate Estimation Since all orders higher than the minimum value could be used to fit the data with insignificant discrepancy, computational order should rather be selected above the minimum order. It is seen that the algorithm is updated with respect to model order. The computational model order pcom for identifying modal parameters in modal analysis application should be selected above the minimum order satisfying following requirements:   

Not too high to avoid spurious poles on the frequency diagram and the time consuming, Enough high above the minimum value in order to distinctly construct the stabilized diagram of frequencies, Should reveal an easy decision on the stabilization of damping rates and of the mode shapes (OMAC and/or MAC).

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Once the computational order is selected, the updating procedure can be stopped to produce the estimated NSR (0.63), the modal parameters and their corresponding confidence intervals, as presented below.

Figure 1. NOF evolution and orders selection

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6. MODAL PARAMETERS IDENTIFICATION AND A NEW OMAC FORMULATION Once the model parameters are estimated, the state matrix of the system can be established in form of autoregressive parameters:

A

dp  dp

 a1  I   0   ...  0

a2 0 I ... 0

a3 ... a p  0 ... 0  0 ... 0   ... ... ...  0 I 0 

(0.65)

where the poles of model are also the roots of the characteristic polynomial of the state matrix. Therefore the eigendecomposition of the state matrix can be obtained as follows:

[S

dp dp

,u

1dp

]  eig ( A)

(0.66)

The eigenvalues, frequencies, damping rates and mode shapes of the system can be computed as follows: Eigenvalues:

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k 

ln(uk ) Ts

Frequencies:

k  Re2 (k )  Im2 (k )

(0.67)

Damping rates:

k   Mode shapes:

 d dp   I

Re(k )

k

0 ... 0d dp abs  Sdpdp 

If an analytical modelling is available, the accuracy of mode shapes can be classically evaluated by the examination of the MAC (Modal Assurance Criterion) factor [24] which

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defines the correlation between the identified mode shape and its corresponding analyzed vector:

MACik 

Cov(:i , :k )

(0.68)

Var (:i ) Var (:k )

where :i and :k are the numerical and identified real mode shape vectors respectively. When conducting an operational modal analysis while the analytical results are not available, we propose a new approach of this correlation, that we have called OMAC (Order Modal Assurance Criterion). In this new method, modal analysis is available from the model updating with respect to model order, the MAC is replaced by the correlation of the identified mode shapes given by the model order p and its previous value p  1 . We propose the new formulation as follows:

OMACi  p 1

where :i and :i p

Cov(:pi , :pi 1 )

(0.69)

Var (:pi ) Var (:pi 1 )

are the identified real mode shape vector at orders p and p  1

respectively.

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7. UNCERTAINTY OF MODAL PARAMETERS It is evident that the measurement with unobserved perturbations rises to an uncertainty in the parameters estimation and hence on the modal parameters. Therefore an analysis of confidence intervals of modal parameters should be taken into account. In order to facilitate this work, the model parameters matrix 

d  dp

should be transformed in a vector

stacking each column on the others ( x:,i denotes i

th



d 2 p 1

by

element or column of vector or

matrix x ):



d 2 p1

 :,1; :,2 ; ...; :,dp 

(0.70)

Asymptotically with large sample sizes and with the Gaussian white noise assumption, estimate error of all elements of parameters matrix has thus an asymptotic standard normal distribution with zero mean. The covariance matrix of the least squares estimator thus depends on the noise covariance matrix and the moment matrix U

dp dp

in equation (0.7) as

follows [16]:

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Cov(ˆ)

d 2 pd 2 p

 U 1

dpdp

 Eˆ

(0.71)

d d

where  denotes the Kronecker matrix multiplication operator. From equations (0.14) and (0.17), Cov(ˆ) is easily computed and is proved to be a positive semi-definite matrix:

Cov(ˆ)  ( R1T R1 )1  ( R3T R3 )

(0.72)

Consider now a real-valued function on the model parameters f  f (d 2 p1 ) , the estimated function is taken as function of the estimated variable fˆ  f (ˆ) . It can be seen that, if ˆ is consistently estimated, it is asymptotically normal distributed and so is the function fˆ . This function may be the frequency, damping rate or a component of a mode. It estimated value can be approximately developed as Taylor series truncated at the first derivative: T

 f ( )  ˆ f (ˆ)  f (0 )      0    0 

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where  0 is the true parameters vector and





(0.73)

 is a negligible term taking into account the

high order terms. With this linearization, the covariance of estimated function can be derived and be guaranteed to be positive (semi-) definite. Cov( f (ˆ))  [ f (ˆ)  f ( 0 )][ f (ˆ)  f ( 0 )]T  ( (

f ( ) T f ( ) ) Cov(ˆ)( ) 0 0

f ( ) T ˆ f ( ) ) (   0 )(ˆ   0 )T ( ) 0 0 (0.74)

The confidence interval of function f can be constructed from the distribution of t-ratio

t

f where the estimated error is ˆ f

f   fˆ  f

and the estimated variance is

ˆ 2f  Cov( f (ˆ)) . If the function f is linear, the relation (0.74) holds exactly and the distribution of the t-ratio follows the student t-distribution with N  d p degrees of freedom [16]. If the function is nonlinear, this assumption is not consistent, but we still assume this distribution to the t-ratio for a heuristic reason of construction of confidence intervals. It means that for the quantity f ( ) , its 100  % confidence has the error margin: 2

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fˆ  t ( N  d 2 p,(1   ) / 2)ˆ f

79 (0.75)

With the t-distribution assumption, the construction of confidence intervals of any function comes down from the computation of its derivative f ( ) with respect to  0 parameters vector. If the function is explicitly constructed on the model parameters such as the transfer function, this derivative is straightforward. But this derivative becomes quite involved for the derivative of modal parameters since these latter are implicitly depending on the model parameters. In order to construct the derivative of natural frequencies and damping ratios, the derivative of the eigenvalues of the state matrix A on the model parameters are first built. The eigenvalue relation of the state matrix in equation (0.66) leads to the equation

AS:k  uk S:k where uk is the k the k

th

th

k  1: dp

(0.76)

discrete eigenvalue and S:k is the corresponding eigenvector which is

column of the matrix S .

This matrix should be normalized by each column so that S:k

2

 1 [25]. If the real and

imaginary parts of the matrix S are distinguished by X  Re S and Y  Im S , each column can be rewritten S:k  X :k  iY:k and the normalization becomes:

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 X :Tk X :k  Y:kT Y:k  1  X :Tk Y:k  0 

(0.77)

Taking the derivative of equations (0.76) and (0.77) with respect to each model parameter leads to:

AS:k  AS:k  uk S:k  uk S:k

(0.78)

2( X :Tk X :k  Y:kT Y:k )  0  T T  X :k Y:k  Y:k X :k  0

(0.79)

If the eigen-decomposition of the state matrix A  S S matrix 

dp dp

1

with discrete eigenvalues

is substituted in equation (0.78), we have:

S S 1S:k  SS 1 AS:k  SS 1uk S:k  SS 1uk S:k

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(0.80)

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Or

(  uk I )S 1S:k  I:k uk  S 1 AS:k

(0.81)

If only the k th component of this equation is considered, the first term on the left side disappeared and thus the derivative of the k th eigenvalue reveals explicitly:

uk  (S 1 AS )kk

(0.82)

The equation (0.82) is also found in [25] which stays at the derivative of the discrete eigenvalue uk . The derivative of continuous eigenvalue can be derived from equation (0.67):

k 

uk ( S 1 AS )kk  uk Ts ukTs

(0.83)

The natural frequencies, damping ratios and their derivatives are straightforward calculated:

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k Re2 k  Im2 k Re k Re k  Im k Im k ; fk  fk   4 2 f k 2 2

k  

Re k

k

;

 Re k

k k 

 Re k



fk   fk 

(0.84)

(0.85)

For the computation of derivatives of mode shapes, it is practical to define for each model parameter a matrix Z whose each column has the form:

Z:k  S 1S:k

(0.86)

This definition leads to the transformation of the equation (0.81) whose each component is derived:

Z jk 

( S 1 AS ) jk u j  uk

( j  k)

If one develops:

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(0.87)

81

Spectrum and Operational Modal Analysis with Vector Autoregressive Models X :Tk X :k  X :Tk Re( S:k )  X :Tk Re( S:k )  X :Tk Re( SZ:k )  X :Tk (Re S Re Z:k  Im S Im Z:k )  X :Tk X Re Z:k  X :Tk Y Im Z:k )

(0.88)

and similar for the others terms of equation (0.79), the normalization condition becomes:

( X :Tk X  Y:kT Y ) Re Z:k  (Y:kT X  X :Tk Y ) Im Z:k  0  T T T T ( X :k Y  Y:k X ) Re Z:k  ( X :k X  Y:k Y ) Im Z:k  0

(0.89)

and the k th component of the vector Z:k should be solved to complete the derivative of the eigenvector with equation (0.87):  Re Z k ,k   ( X T Y  Y T X ) k ,l Im Z l ,k  ( X T X  Y T Y ) k ,l Re Z l ,k     l k  (Y T Y  X T X ) k ,l Im Z l ,k  ( X T Y  Y T X ) k ,l Re Z l ,k    Im Z  l  k k ,k  ( X T X  Y T Y )k ,k 

(0.90)

Once matrix Z is found, the complex partial derivative S is straightforward determined from equation (0.86). The real mode shapes are taken from the amplitude of complex eigenvectors as shown in equation (0.67) and thus the partial derivative of a component of mode shapes is finally obtained as following:

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 2l ,k  Re2 (Sl ,k )  Im2 (Sl ,k )

l  1: d

l ,k l ,k  Re(Sl ,k ) Re(Sl ,k )  Im(Sl ,k ) Im(Sl ,k )

(0.91) (0.92)

Once the derivative of a mode shape vector to the model parameter vector is computed, its confidence intervals is straightforward derived as shown in equations (0.74) and (0.75).

8. IDENTIFICATION OF STRUCTURAL MODES It is seen from the modal decomposition that the number of eigenvalues and modeshapes is normally large and contains modal features of system with noise and excitation frequencies if present. Conventional model based modal analysis distinguishes the structural modes from the spurious ones by observing the stability of the identified modal parameters with respect to increasing model order. This method seems the most effective on the selection of physical modes but it requires a time consuming by repetitive calculus at each model order. There exist several indexes for characterizing the eigenmodes which can be used for the classification and isolation of physical modes. The early Modal Confidence Factor (MCF) [26] compares the two modal vectors identified from the same data when the origin is shifted. Limitations of the MCF were found in noisy data when it may give unnecessary low values from the true modes

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and an artificially value close to 1 for the modes of noise. Pandit [27] has developed the average modal amplitude (AM) and the Modal Signal-to-Noise ratio (MSN) which could be used together with prior knowledge on frequencies and damping to classify and identify the modes based on its participation to the following modal decomposition into deterministic and stochastic parts.

y(t ) 

t p   d 1  j  p 1  t l s   l   i  i i  i    L (i, k ) e(t  j )k  i  i 1  j 0  k 1   dp

(0.93)

Since the deterministic participations deal with continuous functions, in this research, instead of using the average modal amplitude, we construct the continuous average modal power (AP) which is the power of each eigenmode in the deterministic signal (first term of (0.93)) and is the area under the envelope li 

H

eigenvalue version from the relationship i  e

li  s(i)

( i  j ) 

e2 it when we use the continuous

.

li  li  s(i) H

APi 

2

2

(0.94)

2

where the scale factor s (i ) is derived from the initial regressor as follows:

sdp1   L Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved.

1 dpdp

 ( p  1)dp1

(0.95)

The modal variance which characterizes the participation of modes in the stochastic part, is better written in discrete form over the sampled data:

MVi 

k  N 1

 t k

li  MVi t 

H

2 2N H  i (1  i )  1 1 ˆ       l L ( i ,1: d ) E L ( i ,1: d ) N     i  2    1  i  

1  i

2

(0.96)

Finally, the modal signal to noise ratio (MSN) is built for each eigenmode:

MSNi 

APi MVi

(0.97)

It is seen that a very low participating factor can reveal a physical mode by giving a smaller AP index in comparison with a computational mode but the MSN index is an effective criterion since higher the MSN a mode has, more evident it belongs to a structural feature. Furthermore, the presence of damping behaviour on the denominator of the AP index penalizes the very high damped modes which normally belong to computational modes.

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Another discrete version of the particular modal power of a mode which can be computed from the discrete modal decomposition (0.93) may be expressed as follows:

1 MPi  N

k  N 1

 l  l  s(i) t k

H

i

i

2

i

2( t  k )

1 li   N

H

li  s(i)

2

1    2N

i

1  i

2

(0.98)

It is seen that unlike the average power AP which is the square of average modal magnitude, the modal power MP is the energy of the decomposed signal for each mode and according to Parseval theorem [28], it is equal to the area under the modal power spectral density computed for each eigenvalue. Hence if the data is assumed to be pre-processed for having zero mean at each channel, the classification of the modal power MP with respect to descending MSN exhibits the strength of each mode in the data.

9. CONSTRUCTION OF THE NOISE FREE SPECTRUM It is evident that the construction of the noise free spectrum is another interest in modal analysis. Suppose that 2n modes in conjugate pairs (eigenvalues and eigenvectors) are selected to be from the deterministic part, the spectral matrix function can be computed directly from the eigendecomposition as follows:

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T  2n ci    Eˆ   2n ci    P( )  s   jTs       jTs  2  i 1 (1  i e ) )  i 1 (1  i e





Ts 2

H

H  2n  Ts   2n  jT 1  jT 1    (1  i e s )  ci   Eˆ   ck  (1  i e s )   2   k 1    i 1 

2n  (1  i e jTs )   H  2n   jTs 1  jTs 1 ˆ   (1   e ) (1   e ) c E c        i i i   k  jTs H   (1   e ) k 1  i 1   k  

H   2n  ci   Eˆ   ck   Ts  2 n jTs  jTs jTs 1  jTs 1   )(1  i e ) (1  i e )   (1  i e )(1  i e  2  i 1  k 1 (1  i e jTs )(1  k e jTs ) H      



H   2n ci   Eˆ  ck  Ts  2 n   jTs jTs jTs 1  jTs 1 2  (1   e   e   )(1   e ) (1   e )    i i i i i  jTs  jTs H 2  i 1    ) 1  (k e )   k 1 (1  i e   



Ts  2 n   di  (1  i e jTs  i e jTs  i2 )(1  i e jTs )1 (1  i e jTs )1    2  i 1 

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(0.99)

84

V. H. Vu, M. Thomas, A. A. Lakis et al.

where

ci dd  Lstr (1: d , i)   Lstr 1 (i,1: d ) / i

ci   Eˆ  ck 

2n

 di d d   k 1

(0.100)

H

(1  i e jTs ) 1  (k e jTs ) H 

(0.101)

It is clear that the spectrum is decomposed into the sum of individual frequency spectrum which exhibits only the free-of-noise peaks at corresponding natural frequencies and excitation frequencies if present. We can see that this decomposition yields a Hermitian spectral matrix and is the generalization of Pandit [27] where further calculations on the multi spectral matrix such as channels coherence function and phase can be found of interest. A difficulty appears when a low amplitude peak is found near a higher amplitude peak and cannot be displayed. In order to exhibit all frequency peaks for representation purpose, we propose the introduction of a scale factor for each frequency by dividing its participating spectrum by the real norm of the complex matrix d i . Since a peak is much less influenced at its frequency by the other ones, the presence of an amplified factor affects only a scale of the particular spectrum in order to exhibit all the peaks.

 P( ) 

Ts 2

dpmin   1   di  (1  i e jTs  i e jTs  i2 )(1  i e jTs )1 (1  i e jTs )1  (0.102)   trace(  di  ) i 1 

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10. EXPERIMENTAL APPLICATIONS The method presented above is applied to the dynamic testing of a clamped-freeclamped-free rectangular steel plate with dimensions of 500 x 200 x 1.9 mm (Figure 2).

Sensor 1

100

Sensor 2

Sensor 5

Sensor 3

150

150

Sensor 4

100

Sensor 6

Figure 2. Configuration of plate testing

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Figure 3. Time response data of the plate

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Material properties are: elastic modulus E = 200GPa, Poisson coefficient ν = 0.29 and density ρ = 7872 kg/m3. Six accelerometers are mounted on the plate to simultaneously capture the responses at the measurement locations. Before doing the test, a finite element method was used to give the numerical results. The plate is excited by an impact hammer but the excitation force and location are not considered. Data are simultaneously acquired and sampled at a frequency of 1280 Hz (Figure 3). For the assessment of noise effect, several additive random white noises are artificially added to each measurement channel. Figure 4 shows the evolution of the NOF factor at various noise rates. It is found that the minimum model order is independently set at 6 regardless the noise level. It means that the model at least requires an order of 6 in order to identify all the modal parameters.

Figure 4. NOF evolution

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Figure 5. MSN index of modes at order 10.

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Figure 6. Frequency stabilized diagram.

By observing the distribution of the MSN index (Figure 5) at a higher model orders, it can be estimated that there are 13 physical modes available in the data. Since the deterministic part is seen to be converged from that minimum order, all those modes start to appear on the stabilized diagrams as shown on Figure 6 at a noise rate of 100% when the solution was updated from order 2 to 30. Figure 7 shows individually the stabilization diagrams in frequency and damping of the first 5 candidate modes with their corresponding uncertainties (95% confidence interval). It is seen that higher the model order is, smaller uncertainty of modal parameters can be obtained. It was found also that the 4th candidate mode at the frequency of 120 Hz is not a natural mode since its OMAC is not well stabled around unity (Figure 8). Since the number of structural modes is estimated, one can construct the noise-free multi spectrum given in Eq. (0.102) for only the identified frequencies at different orders above the minimum value identified (Figure 9). The plate was then immerged into water and then emerged to the air while subjected to a random excitation given by a turbulent flow (Figure 10). The fluid-solid interaction causes added mass and damping effects on the plate and changes the modal parameters of the structure with respect to the depth [29], as shown in Table 1 for the first five modes.

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a) Frequency candidate 1

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b) Frequency candidate 2

c) Frequency candidate 3 Figure 7. Continued

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d) Frequency candidate 4

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e) Frequency candidate 5 Figure 7. Frequency and damping identification at NSR =100% with confidence intervals

Figure 8. OMAC diagram at NSR=100%

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Figure 9. Noise-free spectrum.

Figure 10. Plate temporal response

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Table 1. Modal identification of the plate

Mode 1st 2nd 3rd 4th 5th

Frequency (Hz) in submerged conditions (Depth/plate length ratio) 0.6(totally 0.4 0.2 0.1 0 (totally in air) submerged) 11.9 12.0 12.2 12.7 39.4 34.1 34.1 34.2 35.0 75.0 77.7 77.9 78.1 79.5 108.6 135.3 135.4 135.6 137.5 164.0 151.3 151.3 151.4 152.6 210.0

In order to survey this evolution, the underlying model described above was implemented in a short time manner. Simulations have shown, as seen in Figure 11 that once the order is higher than the minimum value, the length of windowed data must be set at least 4 times of the highest period in order to track the change of all modes. With this updating of the block length, the natural frequencies and their changes can be tracked in time (Figure 12). It is seen that the change of natural frequencies is most significant at low depth when the plate is coming to the air surface which is in good agreement with the numerical results shown in Table 1.

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a) Frequencies

b) Damping rates

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Figure 11. Modal parameter identification with block size of a 3 d.o.f. system

Figure 12. Monitoring of frequencies

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11. CONCLUSION An overall description on the application of the mathematical autoregressive model for operational modal analysis is presented with a complete computation and some novel aspects. The model is rewritten in a multivariate version and the parameters are estimated by least squares via the QR factorization which is fast and well conditioned. It is seen that the solution can be updated with respect to both time and model order by an economic manipulation on the R factor matrix. A new global index called Noise rate Order Factor (NOF) was introduced and its convergence deal with a minimum required model order from which the modal properties start to be stabilized regardless the noise level. The modal parameters and their corresponding uncertainties are identified from the state matrix including the Order Modal Assurance Criterion (OMAC) which confirms which mode is a structural mode. The number of physical modes can be estimated from a Modal Signal to Noise index (MSN) at its significant decrease. Those physical modes can then be exhibited on a balanced and smooth spectrum by introducing of a modal amplified factor. Furthermore, the whole computation can be implemented in a short time manner to apply for the monitoring of changes in the modal parameters. It is seen that the window length should be at least 4 times of the period of the smallest natural frequency in order to track changes on frequencies.

ACKNOWLEDGMENT

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The support of NSERC (Natural Sciences and Engineering Research Council of Canada) through Cooperative Research grants is gratefully acknowledged. The authors would like to thank Hydro-Quebec’s Research Institute for its collaboration.

REFERENCES [1] [2] [3] [4] [5] [6] [7]

James, GH; et al. Modal Testing Using Natural Excitation. in Proceedings of the 10-th IMAC. 1992, San Diego, CA, USA.. Ibrahim, SR. Random Decrement Technique for Modal Identification of Structures. Journal of Spacecraft and Rockets, 1977, 14(11), 696-700. Benveniste, A; FJJ. Single sample modal identification of a nonstationary stochastic process. IEEE Transactions on Automatic Control, 1985, 30(1), 66-74. Van Overschee, P; DMB. Subspace Identi cation for Linear Systems: Theory, Implementation, Applications, 1996, Dordrecht: Kluwer Academic Publishers. He Xia, DRG. System identification of mechanical structures by a high-order multivariate autoregressive model. Computers & Structures, 1997, 64(1-4), 341-351. Capecchi Danilo, Difference models for identification of mechanical linear systems in dynamics. Mechanical Systems and Signal Processing, 1989, 3(2), 157-172. Larbi, N; LJ., Experimental modal analysis of a structure excited by a random force. Mechanical Systems and Signal Processing, 2000, 14, 181-192.

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92 [8]

[9] [10] [11] [12] [13] [14] [15] [16] [17]

[18] [19]

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[20] [21] [22]

[23] [24] [25] [26] [27]

V. H. Vu, M. Thomas, A. A. Lakis et al. Kadakal, U; Yüzügüllü, Ö. A comparative study on the identification methods for the autoregressive modelling from the ambient vibration records. Soil Dynamics and Earthquake Engineering, 1996, 15(1), 45-49. Stoica, P; S. T., Optimal instrumental variable estimation and approximate implementations. IEEE Transactions on Automatic Control, 1983, 28(7), 757-772. Sinha, NK; KB. Modeling and identification of dynamic systems. 1983, New York, N.Y.: Van Nostrand Reinhold. xi, 334. Gontier, C; Smail, M; GPE. A time domain method for the identification of dynamic parameters of structures. Mechanical Systems and Signal Processing, 1993, 7(1), 45-56. Hsia, T. On least squares algorithms for system parameter identification. Automatic Control, IEEE Transactions on, 1976, 21(1), 104-108. Ljung, L. System Identification - Theory for the User. 2nd ed. 1999, Upper Saddle River, N J: Prentice-Hall. 609. Bjorck, A. Numerical Methods for Least Squares Problems. 1996, Philadelphia, PA: Society for Industrial and Applied Mathematics, 408. Golub, G; VLC., Matrix computations. Third ed. 1996, London: The Johns Hopkins University Press. 694. Lutkepohl, H. Introduction to Multiple Time Series Analysis. Second ed. 1993, Berlin: Springer-Verlag. 454. Kenneth, P; Burnham, DR. Anderson. Multimodel Inference: Understanding AIC and BIC in Model Selection. in Proceedings of Amsterdam Workshop on Model Selection. 2004. Amsterdam. Rissanen, J. Modeling By Shortest Data Description. Automatica, 1978, 14, 465-471. Paulsen, J; TD. On the estimation of residual variance and order in autoregressive time series. J. Roy. Statist. Soc., 1985, B 47, 216-228. Kashyap, RL. Inconsistency of the AIC Rule for estimating the order of autoregressive Models. IEEE Transactions on Automatic Control, 1980, AC-25, 996-998. Hannan, EJ. The estimation of the order of an ARMA process. The Annals of Statistics, 1980, 8(5), 1071-1081. Gang Liang, Wilkes, DM; CJA. ARMA Model Order Estimation Based on the Eigenvalues of Covariance Matrix. IEEE Transactions on Signal Processing, 1993, 41(10), 3003-3009. Smail, M; Thomas, M; Lakis, A. Assessment of optimal ARMA model orders for modal analysis. Mechanical Systems and Signal Processing, 1999, 13, 803-819. Allemang, RJ; BDL. A correlation coefficient for modal vector analysis. in Proceedings of the First International Modal Analysis Conference. 1982. Orlando. 110-116. Arnold Neumaier, T. Schneider, Estimation of parameters and eigenmodes of multivariate autoregressive models. ACM Trans. Math. Softw., 2001, 27(1), 27-57. Ibrahim, SR. Modal confidence factor in vibration testing. Journal of spacecraft and rockets, 1978, 15(5), 313-316. Pandit, SM. Modal and spectrum analysis: data dependent systems in state space. 1991, New York, N.Y.: J. Wiley and Sons. 415.

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[28] Alan, V. Oppenheim and Schafer, RW. Discrete-Time Signal Processing. 1999, Upper Saddle River, NJ: Prentice Hall. 870. [29] Vu, VH; Thomas, M; Lakis, A; Marcouiller, L. Identification of added mass on submerged vibrated plates. in 25th Seminar on machinery vibration. 2007. St John, Canada, Canadian Machinery Vibration Association, 40.1-40.15.

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In: Mechanical Vibrations: Types, Testing and Analysis ISBN: 978-1-61668-217-0 Editor: Amy L. Galloway, pp. 95-145 © 2010 Nova Science Publishers, Inc.

Chapter 3

VIBRATION ANALYSIS OF PIECEWISE AND CONTINUOUSLY AXIALLY GRADED RODS AND BEAMS Metin Aydogdu* and Seckin Filiz** *

Department of Mechanical Engineering, Trakya University, 22180, Edirne, Turkey ** Natural Science Institute, Trakya University, 22180, Edirne, Turkey

ABSTRACT

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In this chapter, vibration of piecewise and continuously axially graded rods and beams is investigated. The equations of motion and boundary conditions are obtained using Newton’s method and Hamilton’s Principle. Initially a general formulation is given for the axially piecewise graded rods and beams. Then rods and beams with two constituents are studied. For the piecewise case governing equations are solved using classical separation of variable method. Legendre method is used for possible continuously grading cases. Since it is not possible to obtain closed form solution for all axially grading forms Ritz method is proposed for arbitrary variation of material properties in the axial direction. Results are obtained for different material composition, geometrical properties and different boundary conditions. Comparisons are made with previous uniform rods and beam results. Tables and graphs are used in order to represent parametrical results. Some specific mode shapes are also given to understand vibration behavior of axially graded rods and beams. It is obtained that the piecewise functionally graded rods and beams may give higher frequencies than their constituents. The piecewise axially grading is studied first time in the literature and the results of present study especially piecewise grading case may open a new research field for researchers studying the subject of vibration.

*

Corresponding author: Email: [email protected].

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1. INTRODUCTION Vibration is one of the most important design problems in many engineering applications such as mechanical, civil and aeronautical engineering applications. A clear understanding of this phenomenon of structural member reduces unwanted failures. Non uniformity of structural members such as bars, beams and plates may result from geometry or/and material. Geometric non-uniformity is analyzed in order to weight, buckling load and vibration frequency optimizations in the previous studies. A wide variety of studies are available on this subject but only a few will be mentioned here due to space limitations. Conway et al. studied canonical beam in terms of Bessel functions. Wang investigated longitudinal vibration of rods with exponentially varying stiffness and mass. Eisenberger obtained exact natural frequencies of a variable cross-section rod with polynomial variation in the cross section and mass distribution. Bapat found exact solutions for the longitudinal vibration of exponential and catenoidal rods. Lau and Abrate derived closed form solutions for vibrating rods with polynomial area variation. Kumar and Sujith studied free vibration of rods with polynomial and trigonometric area variation. Li investigated free longitudinal vibration of stepped non-uniform rods. Cranch and Adler determined the closed-form solutions (in terms of Bessel functions and power series) for the natural frequencies and mode shapes of the unconstrained non-uniform beams and the truncated-cone beams with four kinds of rectangular cross section. Heidebrecht obtained the approximate natural frequencies and mode shapes of a non-uniform simply supported beam from the frequency equation using a Fourier sine series. Naguleswaran determined the approximate natural frequencies of single tapered beams and double tapered beams with direct solution of the mode shape equation based on the Frobenius method. Laura et al. used approximate numerical approaches to determine the natural frequencies of Bernoulli beams with constant width and bilinear varying thickness. Caruntu examined the nonlinear vibrations of beams with rectangular cross section and parabolic thickness variation. Vibration of beams for a family of cross-section geometries with exponentially varying width is studied by Ece et al. Different from geometrical non-uniformity second possibility is changing material properties in the preferred directions. These types of materials are called functionally graded materials (FGM). Functionally graded materials are first proposed by a group of Japan scientists (Koizumi, Suresh and Mortensen). Two or more constituent materials are used in powder metallurgy methods in order to manufacture FGM. Material properties are continuously changed in preferred directions. This continuous change provides smooth variation of stress and strain which is the main drawback of layered composite materials. FGM are initially used as thermal barrier, wear and corrosion resistant coatings. Other application of FGM includes gears, balls, roller bearings or thin walled members like beams, plates and shells, which are used in reactors, turbine blades or machine elements. During these applications vibration is one of the most important problems which should be considered by design engineers. In the most of the previous studies FGM are graded in the thickness direction. Since it is not possible to cite all of the work related with FGM and their vibrations some of them briefly reviewed in the following paragraphs.

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Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams

97

Vel and Batra have studied three dimensional vibrations of functionally graded rectangular plates with simply supported edges. Temperature dependent vibration of functionally graded rectangular plates investigated by Kim. He used third order shear deformation theory and Ritz method in the analysis. Batra and Jin used first order shear deformation theory to study free vibration of functionally graded anisotropic rectangular plate with the objective of maximizing one of its first natural frequencies. The nonlinear vibration of a shear deformable functionally graded plate is presented by Chen. Nonlinear transient thermo elastic analysis of functionally graded ceramic-metal plates is investigated by Praveen and Reddy. Loy et al. studied vibration of functionally graded cylindrical shells. Reddy and Cheng obtained frequency correspondence between membranes and functionally graded spherical shallow shells of polygonal planform. Uymaz and Aydogdu studied three dimensional vibration analyses of functionally graded plates under various boundary conditions. They used Ritz method in their analysis. Free vibration analysis of functionally graded beams with simply supported edges is investigated by Aydogdu and Taskin. A new beam finite element for the analysis of functionally graded materials is developed by Chakraborty et al. Recently some studies have been carried out on axially graded rods, beams and plates. Qian and Batra found that material properties should be graded in the axial direction to maximize first frequency of one edge clamped FG plate. Wu et al. used semi-inverse method to find the solutions to dynamics equation of inhomogeneous, functionally graded simply supported beams. Analytical polynomial solutions for vibrating axially graded beams are given by Elishakoff and Guede. Vibration of inhomogeneous beam with a tip mass is investigated by Elishakoff and Johnson. Calio and Elishakoff investigated closed form solutions for axially graded beam columns. Recently, buckling of axially graded columns studied by Maalawi, Singh and Li. For isotropic uniform material rods and beams working conditions may also effect on material properties. For example temperature has important effects on Young modulus of materials. Blade temperature in gas turbines is studied by Kane and Yavuzkurt. They obtained that temperature is changing along the pressure and suction surface. This change leads to variation of Young modulus along the blade. Although blade made from a homogeneous material due to working conditions it behaves like a FGM. Considering previously cited studies axially graded rods and beams are not widely studied. Combining two dissimilar materials in the form of rods or beam can also be used in the engineering applications. These two materials can be joined with friction welding. These types of structural members may be called as piecewise continuous functionally graded rods, beams and plates. This topic also is not considered in the previous studies. By these motivations, vibration of axially graded (piecewise or continuously) rods and beams is studied in this chapter. Organization of the chapter is as follows: In Section 2 vibration of inhomogeneous rods are analyzed. After finding governing equations, piecewise axially graded rods are studied using separation of variables method. Then Ritz method and Legendre methods are used for vibration of axially graded inhomogeneous rods. Results are given in figures and tables. In section 3 similar steps are repeated for inhomogeneous beam vibrations. Finally, conclusion is indicated in Section 4.

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Metin Aydogdu and Seckin Filiz dx x

dx 

u dx x

u

P

P

P dx x

Figure 2.1. The axially vibrating rod

2. IN-HOMOGENEOUS RODS The equations of motion of a vibrating elastic solid can be derived using two different methods: Newton second law and Hamilton principle. In this part of the chapter these methods will be used to derive the equations of motion of FG rods and in Section 3 similar derivations will be given for the FG beams.

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2.1. Newton Method The governing equation of the free vibrating uniform-non-uniform and homogeneousinhomogeneous rods can be found in many previous studies (Meirovich and Tse et al.). In order to completeness of this chapter derivation of the governing equations of the inhomogeneous rods are briefly explained in the following parts. Now consider a non-uniform and inhomogeneous rod of the cross sectional area A(x), the material density m(x), and the modulus elasticity E(x) under an axial force P, as shown in Figure 2.1. Now take a differential strip of length dx and mass m( x) A( x)dx For one dimensional case, Hooke’s Law can be written as

  E(x)

(2.1)

where  and  are the component of the stress and the strain tensor respectively. Considering equilibrium of the element given in Figure 2.1, Eq. (2.1) takes the following form p u  E ( x) A( x) x

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(2.2)

Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams

99

where P is the force at x and A(x) the cross sectional area. Applying Newton’s Second Law gives (P 

dP  2u )  P  dm 2 dx t

(2.3)

Substituting Eq. (2.3) into (2.2) and simplifying gives the equation of motion of an inhomogeneous rod   u ( x, t )   2u ( x, t )  A( x) E ( x)   m( x) A( x) x  x  t 2

(2.4)

If A is constant, we obtain,

E ( x)

 2u ( x, t ) dE ( x) u ( x, t )  2u ( x, t )   m( x) 2 x dx x t 2

(2.5)

This equation is the governing equation of in-homogeneous rods. Additionally if we take E and m as constant Eq. (2.5) reduces to well known wave equation in the following form:

E

 2u ( x, t )  2u ( x, t )  m x 2 t 2

(2.6)

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2.2. Virtuel Displacement Principles The Newton method used in the previous section is suitable for simple systems when obtaining the governing equations of a continuum. For complicated system it is better to use variational principles to find required equations. The variational principles also give boundary conditions which are necessary when solving related governing equations. In this section of this chapter the governing equations will be obtained using Hamilton principle. This principle can be written as (Meirovich): t2

  (T  V  W )dt  0,

0 xL

t1

(2.7)

where T is the kinetic energy 1 L  u  T   m( x)   dx 0 2  t  2

V is the strain energy

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(2.8)

100

Metin Aydogdu and Seckin Filiz 1 L  u  E ( x) A  dx  0 2  x  2

V

(2.9)

and W is the virtual work of the axial force f W

1 L f ( x, t )udx 2 0

(2.10)

Inserting Eq. (2.8)-(2.10) into Eq.(2.7) and performing partial integrations leads to again Eq.(2.3) and the following boundary conditions. ( EA

u )  0 or x u  0

(2.11)

Following classical boundary conditions are considered in this chapter: Clamped-Clamped (CC): u(0,t)=u(L,t)=0; Clamped-free (CF): u (0, t ) 

u ( L, t ) 0 x

(2.12)

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u (0, t ) u ( L, t )  0 x Free-Free (FF): x

2.3. Piecewise Axially Graded Rods As it is mentioned in the introduction part of this chapter, functionally grading is generally performed in the continuous manner. It is also possible to join different homogeneous materials to form a rod or beam. In the production of this type of structures friction welding methods can be used. Strength and other mechanical properties of friction welded dissimilar parts are studied by some researchers (Yılmaz et al.). It is shown that under optimized friction welding conditions %70-80 of stiffness and mechanical properties of weak component of welding are obtained for friction welded dissimilar material parts (Sahin). Some of the examples of friction welded dissimilar parts: Drill bits, pump shafts drive shafts, piston rods, hydraulic cylinders, high torque drive shafts and similar applications (www.thompson-friction –welding.co.uk). Vibration problem is generally exists in working conditions of those and similar parts. Also axially grading may provide new design opportunities in different engineering applications. A similar procedure is applied in order to investigate column buckling problem (Singh and Li, Maalawi).

2.3.1. Governing equations for piecewise axially graded rods Now consider, A piecewise axially graded rod (PAGR) consisting from n parts (Figure 2.2).

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Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams 101

1

2

i

n-1

n

L Figure 2.2. The piecewise axially graded rod

The equation of motion of the piecewise axially graded rod can be written in the following form for the each segment with harmonic motion assumption (u=F(x)sint) :

Ei

d 2 Fi  mi 2 Fi  0, dx 2

i  1,2,..., n.

(2.13)

where n denotes number of the elements in the rod. In the solution of Eq.(2.13) for given boundary conditions (one of the conditions given in Eq.(2.12)) following matching conditions are also satisfied (continuity of displacement and axial force at the junction of the segments). F j  F j 1 and

Ej

dF j dx

 E j 1

dF j 1 dx

j  1,2,..., n  1.

(2.14)

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The solution of Eq.(2.13) for the given boundary conditions (Eq.(2.12) with the matching conditions (Eq.(2.14) determine the free vibration frequency of the PAGR. For the two steps PAGR Eq.(2.13) takes the following form (Figure 2.3): d 2 F1   2 F1  0, dx 2 d 2 F2   2  2 F2  0, 2 dx

(2.15)

where sub index 1 and 2 denote the displacement of first and second portion of the bar respectively and the non-dimensional terms are defined in the following manner: 2 

m1 2 L2 m E ,    m E ,  m  2 and  E  1 . E1 m1 E2

(2.16)

x η L L

Figure 2.3. Two steps piecewise axially graded rod

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102

Metin Aydogdu and Seckin Filiz

In Figure 2.3,  is the length of the first segment of the rod. Therefore length of the second portion is 1-. General solution of Eq.(2.15) can be written as:

u1 ( x)  C1 sin(x)  C 2 cos(x),

(2.17)

u 2 ( x)  C3 sin(x)  C 4 cos(x). where Ci (i=1,2,3 and 4) are the undetermined coefficients. In this section of this chapter, Eq.(2.17) is applied for the vibration of the PAGR with three different boundary conditions. Namely: Clamped-Clamped (CC), Clamped-Free (CF) and Free-Free (FF). Using boundary conditions (Eq.2.12) and matching conditions (Eq.2.14) four homogeneous equations with four unknown constants Ci (i=1,2,3 and 4) are obtained. A nontrivial solution of this set of equations is possible only if the characteristic determinant of the coefficients vanishes. Characteristic equations are given for each boundary condition as follows: C-C boundary condition: 0

sin()

cos()

sin( )

 sin( )

 cos( )

E1 cos( )

 E2 cos( )

E2 sin( )

0

(2.18)

 sin( )  cos( )  0 E2 sin( )

(2.19)

C-F boundary condition:

 cos( ) sin( )  sin( ) E1 cos( )  E2 cos( )

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0

F-F boundary condition:

 cos() cos( )  sin( )  E1 sin( )  E2 cos( ) 0

  sin()  cos( )

0

(2.20)

E2 sin( )

It should be noted that C1=0 for the FF and C2=0 for the CC and the CF boundary conditions.

2.3.2. Numerical results for pagr In this section the non-dimensional frequency parameters for the axially graded rods are given for different E1/E2 and m1/m2 ratios and for different boundary conditions. Materials are grouped according to following rule: Material I : E1/E2>1 and m1/m2 >1 (Ex: Steel (High stiffness and high density)Aluminum (Low stiffness and low density) )

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Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams 103 Material II : E1/E2>1 and m1/m2 E2 and m1 >m2 case. According to these figures, the non-dimensional frequency changes in a wave like manner with the parameter . It is interesting to note that the maximum non dimensional frequencies are obtained between =0 and 1 values. For different modes different  value gives the maximum frequency parameters. For example the maximum fundamental frequencies are obtained when =0.3. Two peaks are observed for the second mode at =0.2 and 0.6. Similar pattern is observed for the third frequency with three peaks at =0.1, 0.4 and 0.75. This oscillating behavior of the frequency parameter can be used for optimization purposes. The dimensionless frequencies of the E1 >E2 and m1 E2 and m1 >m2 case. According to these figures, similar to the C-C boundary case the non-dimensional frequency changes in a wave like manner with the parameter . For this case the frequency parameters are symmetric with respect to =0.5. It is interesting to note that the maximum non dimensional frequencies are obtained between =0 and 1. For different modes different  value gives the maximum frequency parameters. For example the maximum fundamental frequencies are obtained =0.5. Two peaks are observed for the second mode at =0.2 and 0.8. Similar pattern is observed for the third frequency with three peaks at =0.1, 0.4 and 0.75. Comparing with the C-C case the maximum frequencies are obtained at different . The dimensionless frequencies of the E1 >E2 and m1 E2 and m1 1 and m1/m2 >1 (Ex: Steel (High stiffness and heavy)-Aluminum (Low stiffness and light)) Material II: E1/E2>1 and m1/m2 E2 and m1 >m2 case. According to these figures, non-dimensional frequency changes in a wave like manner with parameter . Maximum non-dimensional fundamental frequencies are obtained between =0 and 1. It should be noted that this behavior is different from PAGR. For higher modes maximum frequencies are obtained for different  values. Frequency parameter results are symmetric with respect to =0.5. This oscillating behavior of frequency parameter can be used for optimization purposes. 10.00

Mode 3

S-S



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8.00

Mode 2 6.00

4.00

Mode 1

2.00 0.00

0.20

0.40



0.60

0.80

1.00

Figure 3.5.Variation of first three non dimensional frequencies of axially functionally graded S-S beam as a function of  (E1/E2=2; m1/m2=2).

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Vibration Analysis of Piecewise and Continuously Axially Graded Rods and Beams 133 10.00

Mode 3

8.00



S-S

Mode 2 6.00

4.00

Mode 1

2.00 0.00

0.20

0.40



0.60

0.80

1.00

Figure 3.6.Variation of first three non dimensional frequencies of axially functionally graded S-S beam as a function of  (E1/E2=5; m1/m2=5). 10.00

8.00

S-S



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Mode 3

Mode 2

6.00

4.00

Mode 1

2.00 0.00

0.20

0.40

0.60

0.80

1.00



Figure 3.7. Variation of first three non dimensional frequencies of axially functionally graded S-S beam as a function of  (E1=200GPa, E2=70 GPa; m1=7860 kg/m3, m2 =2710 kg/m3 [Steel-Copper]).

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134

Metin Aydogdu and Seckin Filiz

Dimensionless frequencies of E1 >E2 and m1 E2 and m1 >m2 case. According to these figures, similar to S-S boundary conditions, non-dimensional frequency changes in a wave like manner with parameter . Maximum non dimensional fundamental frequencies are obtained between =0 and 1. For higher modes maximum frequencies are obtained for different  values. Dimensionless frequencies of E1 >E2 and m1 E2 and m1 >m2 case. According to these figures, similar to S-S and C-C boundary conditions, nondimensional frequency changes in a wave like manner with parameter . Maximum nondimensional fundamental frequencies are obtained between =0 and 1. For higher modes maximum frequencies are obtained for different  values. Frequency parameter results are symmetric with respect to =0.5 as in the C-C boundary case. Dimensionless frequencies of E1>E2 and m10.7)

3

4

5

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6

8

Figure 3. Different shape of the modes oscillation.

Some preliminary conclusions follow from analysis of the dependencies ε(γ). 1. The value of the resonance frequencies for majority oscillation modes decrease with growth of the incision depth, but in different degree. From physical point of view this fact can be explained by that a stiffness of the damaged element degrades.

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Parameter Estimation of Pre-Destruction State of the Steel Frame Constructions … 157 2. For two investigated models, there are oscillations frequencies, which could be named sensitive to the incision depth in the whole range their change. 3. The plots of the resonances curves ε(γ) for different oscillation modes have doubledealing character. Some of them decrease monotonously with growth of the depth incision. Other plots have the bend points, which is displaced at the depth incision into range 0.67< ε(γ)