Vibration Mechanics
Vibration Mechanics Linear Discrete Systems by M. Del Pedro and P. Pahud Swiss Federal Institute oftechnology, Lausanne, Switzerland SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
ISBN 978-94-010-5554-3 ISBN 978-94-011-3514-6 (ebook) DOI 10.1007/978-94-011-3514-6 First edition Mecanique Vibratoire published in French by Presses Polytechniques Romandes, 1989. Translated from the French by RICHARD ELLIS MA DPhil (Oxon) Printed an acid-free paper AII Rights Reserved 1991 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1991 Softcover reprint ofthe hardcover lst edition 1991 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without wrinen permission from the copyright owner.
TABLE OF CONTENTS CHAPTER 1 INTRODUCTION 1.1 Brief history 1.2 Disruptive or useful vibrations 3 CHAPTER 2 THE LINEAR ELEMENTARY OSCILLATOR OF MF.CHANICS 2.1 Definitions and notation 2.2 Equation of motion and vibratory states 2.3 Modified forms of the equation of motion 5 6 CHAPTER 3 THE FREE STATE OF THE ELEMENTARY OSCILI,ATOR 3.1 Conservative free state Harmonic oscillat.or 3.2 Conservation of energy 3.3 Examples of conservative oscillators 3.3.1 Introduction 3.3.2 Mass at the end of a wire 3.3.3 Lateral vibrations of a shaft 3.3.4 Pendulum system 3.3.5 Helmholtz resonator.4 Dissipative free state 3.4.1 Super-critical damping 3.4.2 Critical damping 3.4.3 Sub-critical damping 3.5 Energy of the dissipative oscillator 3.6 Phase plane graph 9 11 13 13 13 18 20 22 26 26 29 31 35 38
-VJ- 3.7 Examples of dissipative oscillators 3.7.1 Suspension element for a vehicle 3.7.2 Damping of a polymer bar 3.7.3 Oscillator with dry friction 41 41 43 45 CHAPTER 4 HARMONIC STEADY STATE 4.1 Amplitude and phase as a function of frequency 4.2 Graph of rotating vectors 4.3 Use of complex numbers Frequ0ncy response 4.4 Power consumed in the steady state 4.5 Natural and resonant angular frequencies 4.6 The Nyquist graph 4.7 Examples of harmonic steady states 4.7.1 4.7.2 4.7.3 Vibrator for fatigue tests Meat;urement of damping Vibrations of a machine shaft 49 49 54 56 59 62 64 68 68 72 75 CHAPTER 5 PERIODIC STEADY STATE 5. Fourier serles Excitation and response spectra.2 Complex form of the Fourier series 5.3 Examples of periodic steady states 5.3.1 Steady state beats 5.3.2 Response to a periodic rectangular excitation 5.3.3 Time response to a periodic excitation 80 80 84 88 88 90 94 CHAPTER 6 FORCED STATE 98 6.1 Lnplace transform 98 6.2 General solution of the forced state 101 6.1 Response to an impulse and to a unit step force 105 6.3.1 Impulse response 105 6.3.2 Indicial response 107 6.3.3 Relation between the impulse and indicial responses 112
- vu- 6.4 Responses to an impulse and to a unit step elastic displacement 6.4.1 Introduction 6.4.2 Impulse response 6.4.3 Indicial response 6.5 Fourier transformation 6.6 Examples of forced states 6.6.1 Time response to a force F cos wt 6.6.2 Frequency response to a rectangular excitation 112 112 113 114 115 119 119 121 CHAPTER 7 ELECTRICAL ANALOGUES 7.1 G e n e r a l i t j ~ s 7.2 Force-current analogy 7.3 Extension to systems with several degrees of freedom. Circuits of forces 129 129 130 133 CHAPTER 8 SYSTEMS WITH TWO DEGREES OF FREEDOM 137 8.1 Generalities Concept of coupling 137 8.2 Free state and natural modes of the conservative system 139 8.3 Study of elastic coupling 146 8.4 Examples of oscillators with two degrees of freedom 149 8.4.1 Natural frequencies of a service lift 153 8.4.2 Beats in the free state 127 CHAPTER 9 THE FRAHM DAMPER 156 9.1 Definition and differential equations of the system 156 9.2 Harmonic steady state 157 9.3 Limiting cases of the damping 160 9.4 9.5 Optimization of the Frahm damper Examples of applications 162 166 9.6 The Lanchester damper 169
-VIII CHAPTER 10 THE CONCEPT OF THE GENERALIZED OSCILLATOR 10.1 Definition and energetic forms of the generalized oscillator 10.2 Differentiation of a symmetrical q u a d r form a t i ~ 174 Equations of Lagrange 10.3 Examination of particular cases 177 10.].1 Energetic forms of the oscillator with two 172 172 negrees of freedom 177 10.3.2 Potential energy of a linear elastic system 178 10.3.3 Kinetic energy of a system of point masses 180 CHAPTER 11 FREE STATE OF THE CONSERVATIVE GENERALIZED OSCILLATOR 184 11.1 Introduction 184 11.2 Solution of the system by linear combination of specific solutions 185 11.2.1 Search for specific solutions 185 11.2.2 General soilltion Natural modes 187 11.2.3 Other forms of the characteristic equation 188 11.2.4 Summary and comments Additional constraints 190 11.3 Solution of the system by change of cnnrninates 192 11.3.1 Decoupling of the equations Normal 11.3.2 11.3.3 11.3.4 11.3.5 coordinates Eigenvalue problem Energetic forms. Sign of the eigenvalues General form of the solution Linear independence and orthogonality of the modal vectors 192 194 196 198 199 11.3.6 Normalization of the natural mode shapes 201 11.4 Response to an initial excitation 202 11.5 Rayleigh quotient 204 11.6 Examples of conservative generalized oscillators 208 11.6.1 Symmetrical triple pendulum 208
-\x- 11.6.2 Masses concentrated along a cord 214 11.6.3 Masses concentrated along a beam 218 11.6.4 Study of the behaviour of a milling table 224 CHAPTER 12 FREE STATE OF THE DISSIPATIVE GENERALIZED OSCILLATOR 12.1 Limits of classical modal analysis 12.2 Dissipative free state with real modes 12.3 Response to an initial excitation in the case of real modes 12.4 General case 12.5 Hamiltonian equations for the system 12.6 Solution of the differential system 12.6.1 Change of coordinates Phase space 12.6.2 Eigenvalue problem 12.6.3 General solution 12.6.4 Orthogonality of the modal vectors Normalization 12.7 Response to an initial excitation in the general case 12.8 Direct search for specific solutions 12.9 Another form of the characteristic equation 233 233 236 239 240 242 248 248 250 252 255 257 261 262 CHAPTF'R 13 EXAMPLE OF VISUALIZATION OF COMPLEX NATURAL MODES 13.1 Description of the system 13.2 Energetic form Differential equation 13.3 Isolation of a mode 13.3. 1 General case 13.3.2 Principal axes of the trajectory 13.3.3 Conservative system 13.4 Numerical examples 13.4.1 Equations of motion 13.4.2 Isolation of the first mode 13.4.3 Isolation uf the second mode 13.4.4 Conservative system 13.5 Summary and comments 264 264 265 268 268 270 272 273 273 274 276 277 279
-x- CHAPTER 14 FORCED STATE OF THE GENERALIZED OSCILLATOR 14.1 Introdurtion 14.2 Dissipative systems with real modes 14.3 Dissipative systems in the general case 14.4 Introduction to experimental modal analysis 281 281 281 284 288 BIBLIOGRAPHY 299 INDEX 302 SYMBOL LIST 321
-XI- FOREWORD Objectives This book is used to teach vibratory mechanics to undergraduate engineers at the Swiss Federal Institute of Technology of Lausanne. It is a basic course, at the level of the first university degree, necessary for the proper comprehension of the following disciplines. Vibrations of continuous linear systems (beams, plates) random vibration of linear systems vibrations of non-linear systems dynamics of structures experimental methods, rheological models, etc. Effective teaching methods have been given the highest priority. Thus the book covers basic theories of vibratory mechanics in an appropriately rigorous and complete way, and is illustrated by numerous applied examples. In addition to university students, it is suitable for industrial engineers who want to strengthen or complete their training. It has been written so that someone working alone should find it easy to read. pescription The subject of the book is the vibrations of linear mechanical systems having only a finite number of degrees of freedom (ie discrete linear systems). These can be divided into the following two categories :
- XIIsystems of solids which are considered to be rigid, and which are acted upon by elastic forces and by linear resist.ive forces (viscous damping forces). deformable continuous systems which have been made discrete. In other words, systems which are replaced (approximately) by systems having only a limited number of degrees of freedom, using digital or experimental methods. The behaviour of an elementary oscillator, which has one degree of freedom only, is first studied in a detailed way. In effect, a deep knowledge of this behaviour is indispensible to t.he proper understanding of complex systems. The free, steady and forced states are established by taking into consideration sub-critical, critical and super-cr i tical damping for each case. The power consumed by an oscillator is analyzed with care, and an original energetic interpretation of the indicial (ie unit step) and unit impulse responses is proposed. Two concepts are introduced : that of frequency response, which is most frequently used in vibratory mechanics, and that of admittance, or transfer function, which is usually used in control theory. The electrical analogue of force-current and the concept of circuits of force are briefly considered. The oscillator with two degrees of freedom is then considered, emphasis being put on the conservative free state and the analysis of elastic coupling. Furthermore, the Frahm damper, a classical example of the steady state, is the subject of an optimization study. The concept of a generalized linear discrete oscillator, having any finite number of degrees of freedom n, is introduced on the basis of quadratic forms of the kinetic energy, of the potential energy and of the total power consumed. The methods of Lagrange and of Hamilton are used to establish the second order linear differential system, having n variables. The solutions are established and analyzed in a systematic way, by order of increasing complexity.
- XIII- That is to say, the null, proportional (ie the hypothesis of Caughey), and other dampings are considered in turn. The concepts of modal basis, and of real and complex modes, which are analysed in a rigorous manner, are made as natural as possible on the physical level. An original example of the visualization of complex modes is dealt with completely. Finally, the book finishes with an exposition of the principles of experimental modal analysis. AcknQWledgepents The authors extend thanks to Christine Benoit et Claudine Candaux for typing the manuscript, to Jose Dias Rodrigues for his contribution to the last chapter and to Jean-Francois Casteu for the preparation of the figures. They particularly thank Martin Schmidt who reviewed the text and corrected the proofs. Lausanne, march 1988 Pierre PAHUD and Michel DEL PEDRO