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Dynamics of Structures

I dedicate this book

to the memory of my mother Anne Franchette Beacuterangegravere Wolff

and

to my wife Solange and my daughters Geneviegraveve and Catherine


Patrick Paultre

First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from Dynamique des structures application aux ouvrages de geacutenie civil published 2005 in France by Hermes ScienceLavoisier copy LAVOISIER 2005

Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address

ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA

wwwistecouk wwwwileycom

copy ISTE Ltd 2011 The rights of Patrick Paultre to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988

Library of Congress Cataloging-in-Publication Data Paultre Patrick [Dynamique des structures application aux ouvrages de geacutenie civil English] Dynamics of structures Patrick Paultre p cm Includes bibliographical references and index ISBN 978-1-84821-063-9 1 Structural dynamics I Title TA654P37413 2010 624171--dc22

2010016491 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-063-9

Printed and bound in Great Britain by CPI Group (UK) Ltd Croydon Surrey CR0 4YY

Contents

Preface vii

Chapter 1 Introduction 1

11 Dynamic response 212 Dynamic loading 2

121 Periodic loadings 31211 Harmonic loadings 31212 Arbitrary periodic loadings 3

122 Non-periodic loadings 41221 Impulse loadings 41222 Arbitrary loadings 4

13 Additional considerations 414 Formulation of the equation of motion 5

141 System with one mass particle 61411 Newtonrsquos second law of motion 61412 DrsquoAlembertrsquos principle 61413 Virtual work principle 71414 Constraints 8

142 System with many mass particles 9143 System with deformable bodies 10

15 Dynamic degrees of freedom 1016 Modeling a dynamic problem 12

161 Mass concentration 12162 RayleighndashRitz method 13163 Finite element method 15

17 Dynamic analysis of structures 1818 Dynamic testing 1919 Measuring vibration levels 20110 Suggested reading 23

vi Dynamics of Structures

PART 1 SINGLE DEGREE OF FREEDOM SYSTEMS 25

Chapter 2 Equation of Motion 27

21 Response parameters 2722 Immobile support 2823 Effect of gravity forces 3024 Motion of the support 31

Chapter 3 Free Response 37

31 Characteristic equation 3732 Undamped free response 3833 Conservation of energy 4734 Damped free response 48

341 Subcritical damping 49342 Critical damping 52343 Overcritical damping 54

35 Dissipation of energy in a system with subcriticaldamping 55

36 Coulomb damping 5837 Logarithmic decrement 63

Chapter 4 Forced Response to Harmonic Loading 71

41 Forced response of conservative systems 72411 Forced response to cosine force 77

42 Beating 7843 Forced response of dissipative systems 7944 Steady-state response to cosine force 8645 Resonance 8746 Dynamic amplification factors 8947 Resonant angular frequency 9148 Power absorbed in steady-state forced vibration 9249 Complex frequency response 97410 Nyquist plot 101411 Vibration measurement instruments 105

4111 Displacement sensor or vibrometer 1064112 Velocity transducer or velometer 1084113 Acceleration transducer or accelerometer 108

412 Vibration isolation 1104121 Vertical oscillating force 1104122 Harmonic motion of the base 112

413 Mass eccentricity 116

Contents vii

Chapter 5 Measurement of Damping 123

51 Free-decay method 12352 Amplification method 12453 Half-power bandwidth method 12554 Nyquist plots 12955 Energy dissipated by damping 129

551 Viscous damping 129552 Internal material damping 132

Chapter 6 Forced Response to Periodic Loading 141

61 Representation of a periodic function as a Fourier series 142611 Trigonometric form of the Fourier series 142612 Complex or exponential form of the Fourier series 148

62 Fourier spectrum 14963 Response to periodic loading 151

631 Trigonometric Fourier series decomposition of the load function 151632 Exponential Fourier series decomposition of the load function 155

Chapter 7 Response to Arbitrary Loading in the Time Domain 161

71 Response to an impulse loading 16172 Dirac impulse or delta function 16373 Response to a Dirac impulse 16574 Duhamel integral 16575 Convolution integral 16776 Numerical evaluation of the Duhamel integral 170

761 Conservative system 170762 Dissipative system 171

77 Response to a step load 17678 Response to a linearly increasing force 17879 Response to a constant force applied slowly 179710 Response to impulse loads 181

7101 Sinusoidal impulse 1827102 Rectangular impulse 1847103 Triangular impulse 1877104 Symmetric triangular impulse 1897105 Shock response spectra 190

Chapter 8 Forced Response to Arbitrary Loading in Frequency Domain 195

81 Fourier transform 19582 Relationship between the frequency response function and the impulse

response function 19983 Discrete Fourier transform 200

viii Dynamics of Structures

84 Nyquist frequency 20385 Fast Fourier transform CooleyndashTukey algorithm 20586 Signal flow graph 21387 Calculation of double nodes 21588 Calculation of the inverse fast Fourier transform 216

Chapter 9 Direct Time Integration of Linear Systems 223

91 General 22492 Exact numerical integration for piecewise linear loading functions 22693 Central difference method 22994 Newmark method 236

941 Average acceleration method 238942 Linear acceleration method 241943 Generalization of the Newmarkrsquos methods 243

Chapter 10 Direct Time Integration of Nonlinear Systems 249

101 Incremental equation of dynamic equilibrium 249102 Newmarkrsquos methods 251103 Error reduction with Newton method 255

Chapter 11 Generalized Elementary Systems 271

111 Rigid-body assemblies 271112 Flexible system 277113 Elementary generalized system 281114 Rayleigh method 283

1141 Elementary system 2831142 Continuous system 2841143 Selection of a displacement function 2871144 Improved Rayleigh method 2941145 Discrete system 297

Chapter 12 Response to Earthquake Excitation 307

121 Earthquake response in the time domain 308122 Response spectrum 312123 Design spectrum 319124 Use of design spectra 320125 Earthquake intensity 323126 Fourier spectrum relative velocity spectrum and energy 324127 Response of a generalized SDOF system 328128 Nonlinear response 333129 Inelastic response spectrum 340

Contents ix

PART 2 MULTI-DEGREES OF FREEDOM SYSTEMS 347

Chapter 13 Equations of Motion 349

131 Simplified model of a building 350132 Equation of dynamic equilibrium 352133 Stiffness influence coefficients 354134 Static condensation 366135 Support motions 368

1351 Synchronous support motion of a planar system 3691352 Structure with multiple support motions 3741353 Additional mass method 379

Chapter 14 Finite Element Method 385

141 Finite element method overview 385142 Global formulation using the principle of virtual works 388143 Local formulation using the principle of virtual work 399144 Coordinate transformations 403145 Generalized displacements strains and stresses 406146 Two-node truss element 411147 Beam finite element 414148 Beam-column element 418149 Geometric stiffness matrix 422

1491 Two-node truss element 4231492 Two-node beam-column element 426

1410 Rules for assembling element matrices 4291411 Properties of the stiffness matrix 4331412 Numerical solution 4341413 Post-processing 4391414 Convergence and compatibility 4401415 Isoparametric elements 441

Chapter 15 Free Response of Conservative Systems 445

151 Physical significance of eigenvalues and eigenvectors 446152 Evaluation of vibration frequencies 448153 Evaluation of mode shapes 450154 Flexibility matrix formulation 454155 Influence of axial forces 456156 Orthogonality of mode shapes 457

1561 Normalization of eigenvectors 460157 Comparing prediction and measured data 461158 Influence of the mass matrix 464

x Dynamics of Structures

Chapter 16 Free Response of Non-conservative Systems 471

161 Proportional damping matrix 471162 Superposition of modal damping matrices 476163 Damping measurement by harmonic excitation 478164 Non-proportional damping matrix 481165 Construction of non-proportional damping matrices 483

Chapter 17 Response to Arbitrary Loading by Modal Superposition 489

171 Normal coordinates 490172 Uncoupled equations of motion 491173 Modal superposition method 493

1731 Calculation of the response 49317311 Direct numerical integration 49317312 Calculation of Duhamel integral 49317313 Fourier transform 494

1732 Initial conditions 4941733 Total response 4951734 Calculation of elastic forces 495

174 Error due to the use of a truncated eigenvector base 498175 Harmonic amplification 502176 Static correction 504177 Modal acceleration method 506178 Summary of the modal superposition method 507

Chapter 18 Modal Superposition Response to Earthquake Excitation 511

181 Modal superposition 511182 Effective modal mass 516183 Error due to the use of a truncated modal base 517184 Superposition of spectral responses 522185 Response of systems with multiple supports 528

Chapter 19 Properties of Eigenvalues and Eigenvectors 533

191 Standard symmetric eigenvalue problem 533192 Similarity transformations 536193 Some properties of the symmetric eigenvalue problem 537194 Generalized symmetric eigenvalue problem 539

1941 Fundamental properties 5401942 Multiplicity of eigenvalues 543

195 Standard eigenvalue problem for a real non-symmetric matrix 545196 Spectral shift 548197 Zero masses 550198 Transformation of generalized eigenvalue problems to standard form 551

Contents xi

199 Rayleigh quotient 5541991 Homogeneity property 5551992 Stationarity property 5551993 Bounding property 556

1910 Max-min and min-max characterization of the eigenvalues 5581911 Cauchyrsquos interlace theorem 5611912 Properties of characteristic polynomials 5631913 Sylvesterrsquos law of inertia 566

Chapter 20 Reduction of Coordinates 571

201 Kinematic constraints 572202 Static condensation 576203 Rayleigh analysis 581204 RayleighndashRitz analysis 583205 Load-dependent Ritz vectors 590206 GuyanndashIrons reduction method 602

Chapter 21 Numerical Methods for Eigenproblems 607

211 Iterative methods 6082111 Inverse iteration 6082112 Direct iteration 6152113 Inverse iteration with spectral shift 6202114 Inverse iteration with orthogonal deflation 622

212 Rotation and reflection 623213 Transformation methods 625

2131 Jacobi method 6262132 Generalized Jacobi method 6312133 QR iteration 639

214 HQRI iterations 651215 Subspace iterations 660

2151 Algorithm 6602152 Choice of the starting iteration vectors 663


221 Multi-step methods 6742211 Multi-step methods for first-order equations 6742212 Multi-step methods to solve second-order equations 675

222 The central difference method 676223 Houbolt method 681224 Newmark methods 683225 Wilson-θ method 687226 Collocation methods 689

xii Dynamics of Structures

227 HHT-α method 694228 Estimation of the highest eigenvalue 699229 Stability analysis 705

2291 Exact solutions 7062292 Discrete approximation 7072293 Central difference method 7072294 Houbolt method 7082295 Newmark method 7092296 Wilson-θ method 7102297 HHT-α method 711

2210 Stability conditions 71322101 Central difference method 71922102 Newmark methods 72122103 Wilson-θ method 72622104 HHT-α method 72622105 Comparison of the various methods 727

2211 Analysis of the consistency of a finite-difference scheme 7282212 Analysis of the accuracy 730

22121 Accuracy of the Newmark method 73122122 Measure of the accuracy of integration schemes 732

2213 Filtering of unwanted artificial modes and overestimation of theresponse 735

2214 Selection of a numerical direct integration method 740


231 Incremental equation of motion 743232 The central difference explicit method 744233 Implicit Newmark methods 747234 Error reduction with the Newton method 748235 Nonlinear analysis of a building under seismic loading 753

Appendix A Complex Numbers 759A1 Algebric representation 759A2 Operations 760A3 Geometric representation 760A4 Trigonometric form 761A5 Roots 764

Bibliography 767

Index 775

Preface

Structural dynamics is a subject that traditionally figures in the curriculum ofengineering schools An introductory course in structural dynamics is often availableas an elective in engineering programs followed by a more advanced course duringgraduate work at the masterrsquos or doctoral level The new standards and building codespromote the use of dynamic computation to determine the distribution of seismicforces when designing large or irregularly shaped buildings or in some cases asthe method of choice for determining the effects of seismic forces As a resultthe importance of an introductory course in structural dynamics should be obviousThis book is intended for engineering students and practising engineers dealing withproblems related to structural vibration and seismic design

This volume has two parts The first deals with single-DOF systems whichinclude complex systems that can be reduced to single-DOF systems The second partlooks at systems with multiple DOF that are solved using the finite-element methodThis division could be viewed as the separation between an introductory course onstructural dynamics for undergraduates and an advanced course for graduate studentsThat would not be a very profitable approach since it would not include modalanalysis which is discussed in the second part of this book The goal is to introducemodal analysis as part of an introductory course on structural dynamics analysisUnderstanding the bookrsquos contents requires no more knowledge of mathematics andstructural analysis than any engineering student would have The book breaks downas follows

Chapter 1 provides an introduction to structural dynamics The first part ofthe book deals with single-degree-of-freedom (SDOF) systems Chapter 2 providesthe equations of motion for single-DOF systems Chapter 3 develops conventionalsolutions for single-DOF systems ie under the initial conditions imposed withoutdynamic loading System response to harmonic loading is discussed in Chapter 4which leads to damping and its experimental measurement dealt with in Chapter 5The Fourier decomposition of periodic loading is considered in Chapter 6 which

xiv Dynamics of Structures

shows that the response is the superimposition of a set of harmonic loadings Chapter 7shows how to calculate the response of a single-DOF system subjected to any kind ofloading using Duhamelrsquos integral Chapter 8 introduces applying frequency-domainanalysis to dynamics problems and calculating the response to any kind of loadingusing the Fourier transform Chapter 9 provides an introduction to the direct numericalintegration of equations of motion The topics treated include an exact method forpiecewise linear loading functions the central difference method and conventionalNewmark methods Chapter 10 considers computation of the response of nonlinearsingle-DOF systems using direct numerical integration combined with Newtonrsquositerative method for error reduction Chapter 11 focuses on systems that can bereduced to a single DOF using Rayleighrsquos method The bookrsquos first part ends withan examination of single-DOF systems under earthquake action (Chapter 12)

Part 2 is devoted to discrete systems with multiple DOF Chapter 13 establishesthe equations of motion for multiple-degree-of-freedom (MDOF) systems and definesmass damping and stiffness matrices based on a basic knowledge of structural matrixcomputations Chapter 14 provides an introduction to the finite-element method so thatthe mass and rigidity matrices can be established more formally The free responseof conservative multiple-DOF systems is seen in Chapter 15 which provides fordefining and computing the natural frequencies and associated mode shapes Chapter16 deals with the free vibration of discrete dissipative systems Chapter 17 showshow to use modal superposition to compute the response of discrete systems for anyload whereas Chapter 18 deals with seismic loading Chapter 19 looks at severalproperties of eigenvalues and eigenvectors required for a more in-depth study of theirnumerical determination Chapter 20 presents several coordinate reduction methodswhich are of prime importance in structural dynamics and introduces Ritz analysisChapter 21 presents several classic methods for computing eigenvalues and theassociated eigenvectors Direct numerical integration methods to solve equations ofmotion for discrete multiple-DOF systems receive in-depth treatment in Chapter 22including error and stability analysis of the different methods Application of directnumerical integration methods to solve nonlinear problems is seen in Chapter 23

The appendix provides some mathematical notions needed to understand the textThis book contains 88 examples illustrating application of the theories and methodsdiscussed herein as well as 181 problems

The contents can be used to develop a number of courses including

1) Introduction to Structural Dynamics an introductory course for engineeringstudents would cover Chapters 1 to 7 9 11 and part of 12 13 15 to 18

2) Advanced Structural Dynamics this course for graduate students who havetaken the introductory course in structural dynamics would comprise Chapters 1 812 to 18 and 20 to 23 in part

Preface xv

3) Computational Structural Dynamics this advanced course would be reservedfor graduate students who have already taken the advanced structural dynamics coursein which Chapters 1 8 and 14 to 23 would be seen

This text was used in delivering the structural dynamics course to senior students atthe University of Sherbrooke I take this opportunity to thank all my former studentswho through attending lectures and their enthusiasm for solving weekly problemswith CALWin LAS and MATLAB led me to write this book Professor Jacky Mazarsplayed an essential role in the process leading to this book first by inviting me to give acourse on structural dynamics at the Eacutecole Normale Supeacuterieure in Cachan to studentsat the DEA-MAISE and Laboratoire de Meacutecanique et Technologie for a number ofyears and then by inviting me to publish it in the civil-engineering collection atISTE and John Wiley amp Sons This text provided the foundation for an introductorycourse in structural dynamics given to Masterrsquos students in civil engineering andinfrastructure at the Grenoble IUP as well as for an advanced computational structuraldynamics course given to students at the doctoral school of Joseph Fourier Universityin Grenoble at the invitation of Professor Laurent Daudeville Lastly part of the bookwas presented in English to doctoral students attending ALERT sessions in AussoisFrance

I entered the text performed the layout and designed the artwork Professor NajibBouaanani read some of the chapters of the French version and made suggestionsthat without a doubt have improved the presentation and made the text clearerIn the final phase of writing the French version of the book Dr Benedikt Weberand Dr Thien-Phu Le read all the chapters Dr Benedikt Weber played an essentialrole suggesting clarifications and developing solutions for several problems usingMATLAB whereas Eacuteric Lapointe and myself developed all solutions with LASOlivier Gauron Research Assistant at the University of Sherbrooke proofread thetranslation of the French version of the book into English checked the solutions of theproblems and coordinated the production of the artwork in English His role was notlimited to these tasks as he made valuable suggestions that helped clarify part of thebook The author is grateful to Seacutebastien Mousseau Najib Bouaanani Ceacutedric AdagbeAdamou Saidou Sanda Danusa Tavares and Gustavo Siqueira for their dedication inmeticulously and expertly preparing the drawings

I am particularly grateful to my former Professor Reneacute Tinawi for initiallypiquing my interest in the subject For a number of years I taught a course onstructural dynamics in parallel with a course taught by Professor Pierre Leacuteger firstat McGill University and now at Eacutecole Polytechnique in Montreacuteal I have fondmemories of many discussions with Professor Leacuteger about teaching approaches andthe development of software for teaching structural dynamics The same is true forProfessor Jean Proulx who also wrote the first version of the CALWin programwhich is the ancestor of LAS I wish to thank Eacuteric Lapointe Masterrsquos student

xvi Dynamics of Structures

at the University of Sherbrooke for the development of the LAS1 program thatruns under Windows The LAS program is based on an earlier non-graphical basicversion developed by Dr Charles Carbonneau Eacuteric Lapointersquos enthusiasm technicalknowledge refined programming skills that allows him to put algorithms into codeat a record speed led to completion of a project that was dear to me for a number ofyears The program is available as freeware to anyone interested LAS is a powerfulprogram that can be used to quickly learn structural matrix computation methods thefinite-element method structural dynamics and matrix computations LAS softwarecan be downloaded from httpwwwcivilusherbrookecappaultre Lastly I wouldlike to thank Professor Jean Proulx who was a great help in the translation of the bookfrom French to English

This book was typeset with LATEX2ε Donald Knuth cannot be thanked enough forTEX

Patrick Paultre Sherbrooke 2010

ldquoLet no one say that I have said nothing new the arrangement of the subject isnewrdquo

Blaise Pascal Penseacutees 22-696

1 LAS is an acronym for Language for the Analysis of Structures which in French is Languagepour lrsquoAnalyse des Structures

Chapter 1

Introduction

The aim of this book is to study vibrations of structures caused by dynamicloadings that vary over time as opposed to static loadings These dynamic loadingsgive rise to displacements internal forces reactions and stresses that are timedependent Hence a unique solution does not exist as for a static problem In adynamic problem it is necessary to calculate the displacements in time ndash collectivelycalled dynamic response ndash before determining maximum values of forces reactionsand stresses that are necessary for design purposes It is however easy to concludethat time is the only difference between the dynamic and static analysis of a structureThis is obviously not true because on the one hand a load is never applied staticallyand on the other hand the effects of a static load do vary in time due to the viscoelasticproperties of the materials (creep shrinkage relaxation etc) forming the structuresThe distinctive nature of a dynamic problem comes from the presence of inertiaforces fI(t) which oppose the motion generated by the applied dynamic loadingp(t) The dynamic character of the problem is dominant if the inertia forces arelarge compared to the total applied forces The problem can be treated as static if themotion generated by the applied load is so small that the inertia forces are negligibleFigure 11 illustrates the effects on the bending moment of a concentrated forceapplied dynamically and statically to the tip of a column

A dynamic load has intensity direction and point of application that can vary intime If it is a known function of time the loading is said to be prescribed dynamicloading The analysis of a structure under a prescribed dynamic loading is considereddeterministic If the variation in time of the loading function is unknown and can onlybe described in statistical terms it is said to be random dynamic loading Randomvibration analyses study the response of a structure under random dynamic loadings

2 Dynamics of Structures

Figure 11 Difference between static and dynamic loading (a) static loading andcorresponding bending moment diagram (b) dynamic loading and corresponding bending

moment diagram (not at scale)

11 Dynamic response

The end result of the deterministic analysis of a structure excited by a givendynamic loading is the dynamic response expressing the displacements of the structurewith time which is also called the displacement time history The strains stressesinternal forces and reactions are determined once the displacement time history isknown (Figure 12) We recall that there is no uncertainty in expressing the loadingfunction in a deterministic analysis

Figure 12 Response time history displacements stresses or forces

Dynamic response varies with time However for design or verification all that isrequired is the maximum dynamic response which for a linear system can be addedto the maximum static response to yield the maximum total response For a nonlinearsystem the static effects need to be calculated first and added to the dynamic effectsto determine the total nonlinear response

12 Dynamic loading

Dynamic loadings can be divided into periodic loadings and non-periodicloadings Table 51 summarizes the different types of dynamic loadings that are

Introduction 3

encountered in civil engineering Permanent and live loads that are applied slowlycompared to the period of vibration of structures are generally considered staticloadings as are dead loads

Sources of dynamic loadings

Periodic Non-periodic

Simple harmonic Arbitrary periodic Arbitrary Impulsive

Rotating machine Reciprocating machine Construction Construction

Walking jogging Wind Impact

Wind Waves Explosion

Earthquakes Loss of support

Traffic Rupture of an element

Table 11 Types and sources of dynamic loadings

121 Periodic loadings

A periodic loading repeats itself after a regular time interval T called the periodPeriodic loadings can be divided into simple harmonic loadings and arbitrary periodicloadings

1211 Harmonic loadings

Figure 13 Harmonic loading applied by a rotating machine

The simplest periodic loading varies as a sinusoid and is called simple harmonicloading (Figure 13) This type of loading is generated by rotating machines andexciters with unbalanced masses and it gives rise to the resonance phenomenon whenthe excitation period matches the structurersquos natural period of vibration

1212 Arbitrary periodic loadings

Arbitrary periodic loadings repeat themselves at regular interval of time This typeof loading is generated by reciprocating machines by walking or jogging by one ormany persons crossing a pedestrian bridge (Figure 14) by rhythmic jumping anddancing by one or many persons on a floor by hydrodynamic forces generated by thepropeller of a boat by waves etc


Figure 14 Periodic loading caused by the steps of a person crossing a pedestrian bridge

122 Non-periodic loadings

Non-periodic loadings vary arbitrarily in time without periodicity Non-periodicloadings can be divided into impulsive short-duration loadings and arbitrary long-duration transient loadings

1221 Impulse loadings

Impulse loads have a very short duration with respect to the vibration period ofthe structures and are caused by explosions (Figure 15) shock failure of structuralelements support failure etc

Figure 15 Impulse loading caused by an explosion

1222 Arbitrary loadings

Arbitrary loads are of long duration and are caused by earthquakes wind wavesetc Figure 16 shows the time variation of the acceleration that occurs at the base ofa structure during an earthquake giving rise to time-varying inertia forces over thestructurersquos height

13 Additional considerations

Additional considerations are needed for dynamic loads These considerations aremostly related to the cyclic nature of the loading ndash which can lead to fatigue-relatedfailure ndash and to the properties of specific materials whose behavior changes with theloading rate

Introduction 5

Figure 16 Long-duration arbitrary load caused by an earthquake

When the external loads are lower than the structurersquos elastic limit fatigue-relatedrupture is caused by stress concentrations near defects where fatigue microcracks canbegin to propagate One such crack will dominate and propagate by cyclically openingand closing to a critical size that will lead to instability of the structural memberThis failure depends on the difference between the maximum and minimum stressand on the number of cycles during which this difference remains above a specificlevel

The rate of loading also influences the stiffness and resistance characteristics ofcertain materials The stiffness and resistance of such materials increase with the rateof loading For example the compressive strength of concrete can increase by close to30 for strain rates of 005s which is typical of the rates induced in a structure byearthquake loading

14 Formulation of the equation of motion

In order to determine the dynamic response of a structural system we need towrite the equations of motion governing the dynamic displacement of the systemThe solution of these equations provide the systemrsquos response as a function oftime Three methods will be used in this book to write the dynamic equations ofmotions ie Newtonrsquos second law of motion drsquoAlembertrsquos principle and the principleof virtual work particularly the principle of virtual displacements A variationalapproach using the notion of work and energy and leading to Hamiltonrsquos principlecan also be used Although very powerful and often leading to a more profoundunderstanding of the dynamic phenomena this formulation will not be used inthis book


141 System with one mass particle

1411 Newtonrsquos second law of motion

Newtonrsquos1 second law of motion states that the rate of change of momentum of amass particle m is equal to the sum of forces acting onto it that is

p(t) =ddt

(m

dudt

)[11]

where p(t) is the sum or resultant of all forces acting on the mass particle m u is itsposition vector and m(dudt) its momentum Assuming the mass does not vary withtime as is usually the case equation [11] can be written as

p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]

where the dots represent differentiation with time Equation [13] can be written interms of the components of the vectors that is

pi(t) = mui(t) i = 1 2 3 [14]

1412 DrsquoAlembertrsquos principle

Transposing the right-hand side of equation [12] to the left we obtain

p(t) minusmu(t) = 0 [15]

or in component form

pi(t) minusmui(t) = 0 i = 1 2 3 [16]

These equations are an expression of drsquoAlembertrsquos2 principle which states that the sumof all applied force vectors and vector minusmu for a dynamic system is equal to zeroThe vector mu whose magnitude is mu and direction is opposite to the accelerationis called inertia force vector In other words this powerful principle states that anaccelerating mass particle is equivalent to a static system in equilibrium when the

1 Isaac Newton physicist mathematician and natural philosopher born in WoolsthorpeLincolnshire England on December 25 1642 died in London England on March 20 17272 Jean Le Rond drsquoAlembert lawyer mathematician physicist and philosopher born onNovember 17 1717 in Paris France died on October 29 1783 in Paris France

Introduction 7

inertial force is added The mass particle is said to be in dynamic equilibrium Note thatthe inertial force must act through the center of mass and in the case of a rotating massan inertial moment acting anywhere must also be considered The sum of all appliedforces includes all forces resulting from kinematic constraints opposing displacementall viscous forces opposing velocities and all external applied forces The applicationof drsquoAlembertrsquos principle is in general the simplest way of writing the equations ofmotion of a dynamic system and will be used quite extensively in this book

1413 Virtual work principle

Figure 17 Mass particle and virtual displacement

Let us assume that the mass particle follows a path u from a given position u(t1)at time t1 to a final position at u(t2) at time t2 (Figure 17) Let us assume an arbitraryvirtual path uprime that has same position as u at time t1 and t2 ie uprimei(t1) = ui(t1)and uprimei(t2) = ui(t2) We define the components of a virtual displacement δui of thesystem at time t1 lt t lt t2 as

δui = uprimei minus ui i = 1 2 3 [17]

where ui and uprimei are respectively the components of u and uprime in direction 1 2 and 3The virtual displacement is arbitrary except for the following conditions

δui(t1) = δui(t2) i = 1 2 3 [18]

From equation [15] it follows that

ddt

(δui) =ddt

(uprimei minus ui) = uprimei minus ui = δui [19]

where it is seen that the symbol δ commutes with the first differential operatord In fact the symbol δ is more than an indicator of a virtual quantity butbehaves like a variational operator obeying the rule of operation similar to the firstdifferential operator d If we multiply the dynamic equilibrium equations [16] by thecorresponding virtual displacement and we take the sum of the components we obtain

3sumi=1

(pi(t) minusmui(t)) δui = 0 [110]


which is the principle of virtual displacements ndash a particular case of the principle ofvirtual work ndash that can be stated as below

THEOREMndash The work done by the effective forces acting on a mass particle during avirtual displacement δui is equal to zero

1414 Constraints

The position of a mass particle that is restricted to move in a plane can be describedby two coordinates x and y or xi i = 1 2 The system is said to have two DOF(DOFs)

Figure 18 Pendulum restricted to move on a plane (a) simple pendulum (b) doublependulum

If the mass at coordinates x1 and x2 is attached to a frictionless hinge at position(0 0) (Figure 18a) by a rigid massless bar with length L ndash this system is called apendulum ndash a constraint is introduced which can be expressed by

x21 + x2

2 = L2 [111]

which is a constraint equation The introduction of a constraint in this case reducethe number of DOFs by one Either x1 or x2 or more often the angle θ between thependulum and the vertical axis can be chosen as DOF The constraint equation can bewritten as

f(x1 x2 x3 t) = const [112]

Systems for which the constraint equation is a function of the coordinates and time arecalled holonomic system and the constraint equation is called a holonomic constraintA holonomic system is further subdivided into rheonomic if time appears in theconstraint equation or scleronomic otherwise If the constraint equation is also afunction of the derivatives of the coordinates with time such that

f(x1 x2 x3 x1 x2 x3 t) = const [113]

the system is called non-holonomic We are concerned in this book with onlyholonomic systems

Introduction 9

142 System with many mass particles

If we have N mass particles we will have 3N equations of dynamic equilibrium

pik(t) minusmuik = 0 i = 1 2 3 k = 1 2 N [114]

where pik are the components of all applied forces In this case the system is said tohave 3N DOFs

Let us define the virtual displacements which satisfy the kinematic conditions ofthe system such as

δuik = uprimeik minus uik i = 1 2 3 k = 1 2 N [115]

with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]

Equation [110] becomes

Nsumk=1

3sumi=1

(pik(t) minusmuik(t)) δuik = 0 [117]

which can be stated as below

THEOREMndash A system of particles is in equilibrium if the total virtual work done forevery virtual displacement is equal to zero

The position of a mass particle is described by three coordinates xi i = 1 2 3 in3D space and has three DOFs A system of N particles in space has 3N DOFs Thenumber of DOFs is reduced by one for every kinematic constraints that are introducedbetween the mass particles Hence the number of DOFs in 3D is given by

n = 3N minus nc [118]

where n is the number of DOFs and nc is the number of constraints A doublependulum consisting of two masses m1 and m2 connected by massless rigid bars oflength L1 and L2 and restricted to move in a plane has n = 4 minus 2 = 2 DOFs (Figure18b) The two constraint equations are

x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]

For many mass particles systems with constraints the principle of virtualdisplacement can therefore be restated as below


THEOREMndash A mechanical system is in equilibrium if the total virtual work done forevery virtual displacement consistent with the constraints is equal to zero

Mechanical systems in the previous theorem include rigid bodies with their massand mass moment of inertia concentrated at their center of mass

143 System with deformable bodies

No proof will be given but the principle of virtual displacements can be stated asbelow

THEOREMndash A system is in equilibrium if the virtual work of external forces is equalto the virtual work of internal forces when it is subjected to a virtual displacementfield that is consistent with the constraints

15 Dynamic degrees of freedom

From the preceding discussion it can be stated that the number of degrees offreedom (DOFs) of a structural system is the number of independent displacementcoordinates or generalized coordinates that is necessary to completely and uniquelydescribe the displaced or deformed shape of a structure Generalized coordinates areCartesian coordinates but can also be rotations or even amplitude of deflected shapesand Fourier series expansion as we shall observe A simply supported beam has aninfinite number of DOFs Let us assume that two bending moments are applied tothe ends of a simply supported beam If we dispose of an analytical function relatingthe deflexion of the beam at any point along its length to the rotation at the endsof the beam we need only two DOFs namely these two rotations to define thedeformed shape of the beam This definition applies to a static problem and needssome specialization for a dynamic problem The generalized coordinates that mustbe considered in order to represent the effects of every important inertia forces ona structural system are called dynamic DOFs and their number is the total numberof DOFs in the system In the case of a dynamic problem the nodal displacementsthat control inertia forces are generally not significantly affected by local deformationvariations As a result fewer DOFs are required for a dynamic model than for a staticmodel Let us illustrate the difference between a static and a dynamic problem with asimple example Only basic knowledge of matrix structural analysis is required (seeChapter 13)

Consider the frame illustrated in Figure 19a which consists of a beam supportedby two columns fixed at their base The beam and columns are modeled with linearbeam elements and meet at points called nodes Consider the case of static forcesapplied only at the nodes In an elastic system that undergoes small displacements thetransverse displacement of the elements are uniquely related to the node displacements

Introduction 11

by cubic polynomials If the six displacements u1 to u6 are known the transversedisplacements of any given point on an element can be determined This structuretherefore has six static DOFs as shown in Figure 19b If other forces are present orif the displacement of other points is sought additional nodes must be added whichincreases the number of static DOFs (three per additional node) A beam often consistsof a web and a slab that is very rigid in the longitudinal axis In this case it can beconsidered as rigid in the longitudinal direction with respect to the columnrsquos flexuralstiffness which removes one DOF Moreover for low-rise structures as is the casehere column longitudinal deformations can be neglected with very little impact onaccuracy Thus there remain three static DOFs the horizontal displacement u1 andthe rotations u2 and u3 (see Figure 19c) In a static problem the stresses depend onthe derivatives of the displacement A more refined model improves the deformationgradient thereby improving stress predictions

Figure 19 Static and dynamic DOFs


In the case of a dynamic load as illustrated in Figure 19d the effects of rotationalinertia can be shown to be negligible The system reduces to four translation DOFsu1 to u4 (see Figure 19e) The effects of longitudinal deformation are also negligibleas is the case in the static problem We can also assume that the column masses arenegligible with respect to the total structural mass which is concentrated at the rooflevel DOFs u2 to u4 can therefore be eliminated from the preceding model Thestructure is reduced to a single-degree-of-freedom (SDOF) system in the horizontaldirection as shown in Figure 19f

16 Modeling a dynamic problem

We have explained that the inertia forces characterize a dynamic problem Theseforces must therefore be well defined in any model For continuous systems such as abeam the mass is distributed along its entire length which means that accelerationsand displacements should be defined for each point on the beam Analysis of a beamfor example leads to simultaneous partial differential equations that are a functionof the position x along the beam and time t It is almost impossible to solve thesedifferential equations analytically except with very simple structures and load casesDiscretization techniques are generally used to formulate and solve equations fordynamic problems These techniques can be simple mass concentrations or moresophisticated coordinate-reduction methods such as Rayleigh3 and Ritz4 methods orthe widely used finite element method In structural dynamics the finite elementmethod is very often used for the spatial discretization of structures combined with thefinite difference method for time discretization These methods are briefly describedbelow

161 Mass concentration

Important simplifications can be achieved by concentrating the masses on a givennumber of points The inertia forces can only be developed at these points and theresponse parameters are only defined at these locations Figure 110 represents athree-span bridge with variable inertia The bridge is modeled as a discrete systemin which the mass is concentrated (or lumped) at seven specific points Neglectingthe longitudinal deformations and rotational inertia results in a model with sevendynamic DOFs This type of modeling generally leads to an n DOFs system Theproblem is determining n in order to represent the inertia forces as accurately as

3 John William Strutt Lord Rayleigh mathematician and physicist born on November 12 1842in Langford Grove Essex UK died on June 30 1919 in Terling Place Essex UK4 Walter Ritz physicist born on February 22 1878 in Sion Switzerland died on July 7 1909in Goumlttingen Germany




and



Patrick Paultre









Contents

Preface vii


























Contents vii




























Contents ix


























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11











Patrick Paultre









Contents

Preface vii


























Contents vii




























Contents ix


























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11


















Contents

Preface vii


























Contents vii




























Contents ix


























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11

























Contents vii




























Contents ix


























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11






















Contents ix


























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11
























Contents xi

























Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11





















Bibliography 767

Index 775

Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11










Preface











Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11

















Preface xv












Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11

















Chapter 1

Introduction






11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11













11 Dynamic response




12 Dynamic loading


Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11










Introduction 3





























Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11





















Introduction 5










p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11














p(t) =ddt

(m

dudt

)[11]


p(t) = md2udt2

[12]

which we will write

p(t) = mu(t) [13]


pi(t) = mui(t) i = 1 2 3 [14]





pi(t) minusmui(t) = 0 i = 1 2 3 [16]



Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11










Introduction 7







δui(t1) = δui(t2) i = 1 2 3 [18]


ddt

(δui) =ddt



3sumi=1





1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11













1414 Constraints




x21 + x2

2 = L2 [111]


f(x1 x2 x3 t) = const [112]


f(x1 x2 x3 x1 x2 x3 t) = const [113]


Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11










Introduction 9







with the conditions

δuik(t1) = δuik(t2) i = 1 2 3 k = 1 2 N [116]


Nsumk=1

3sumi=1







x211 + x2

12 = L21 and x2

21 + x222 = L2

2 [119]











Introduction 11



















Introduction 11










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