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Modelling of Mechanical Systems Volume 1 : Modelling of Mechanical Systems: Discrete Systems Discrete Systems v.1 - Francois Axisa

Modelling of Mechanical Systems Volume 1

Modelling of Mechanical Systems: Discrete Systems Discrete Systems v.1


Published: 1st January 2004
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This publication is the first in a series of four volumes, written by an eminent authority in the field. The series presents the general methods that provide a unified framework to model mathematically mechanical systems of interest to the engineer, analysing the response of these systems. Whilst focusing on linear problems, some aspects of non-linear configuration are also included.

A particular feature is the clear, concise and accessible presentation, emphasising essential aspects of the subject of most use in mechanical engineering. An important feature is the treatment of mathematical techniques that are used to perform analytical studies and numerical simulations on the computer.

This first volume is concerned with discrete systems - the study of which constitutes the cornerstone of all mechanical systems, linear or non-linear. It covers the formulation of equations of motion and the systematic study of free and forced vibrations. The book goes into detail about subjects such as generalized coordinates and kinematical conditions; Hamilton's principle and Lagrange equations; linear algebra in N-dimensional linear spaces and the orthogonal basis of natural modes of vibration of conservative systems. Also included are the Laplace transform and forced responses of linear dynamical systems, the Fourier transform and spectral analysis of excitation and response deterministic signals.

Forthcoming volumes in this series:
Vol II: Structural Elements; to be published in June 2005
Vol III: Fluid-structure Interactions; to be published in August 2006
Vol IV: Flow-induced Vibrations; to be published in August 2007

Written by an authority in the field
Comprehensive coverage of mathematical techniques used to perform computer-based analytical studies and numerical simulations
Concise, accessible style

"For this is not just a translation; much has been added, refined and polished, to make this book an excellent addition to anyone's bookshelf, whose interests lie in Dynamics, Vibrations, or Fluid Structure Interactions." - Michael Paidoussis, Emeritus Professor in the Department of Mechanical Engineering, McGill University, Canada, in the Journal of Fluids and Structures, 2004 "I feel sure that this work by Francois Axisa will reward those who study it with new and unusual insights into the fascinating and notoriously difficult-to-master subject of predicting and controlling the vibration properties of the complex practical structures encountered across a wide range of engineering sectors." - D.J. Ewins, Professor of Vibration Engineering in the Department of Mechanical Engineering, Imperial College London, UK

Forewordp. xi
Prefacep. xiii
Introductionp. xv
Mechanical systems and equilibrium of forcesp. 1
Modelling of mechanical systemsp. 2
Geometry and distribution of massesp. 2
Motion relative to a given spacep. 3
Coordinatesp. 3
Degrees of freedom and generalized coordinatesp. 3
Coordinate transformationp. 5
Changes of reference framep. 7
Kinematical constraintsp. 8
Holonomic constraintsp. 8
Nonholonomic constraintsp. 11
Example: a constrained rigid wheelp. 11
Forces formulated explicitly as material lawsp. 16
Forces formulated as constraint conditionsp. 19
Basic principles of Newtonian mechanicsp. 20
Newton's lawsp. 20
Law of inertiap. 20
Law of motion (basic principle of dynamics)p. 21
Law of action and reactionp. 22
D'Alembert's principle of dynamical equilibriump. 24
Equations of motion in terms of momentsp. 26
Moment of a force and angular momentump. 26
Plane rotation of a particlep. 28
Centrifugal and Coriolis forcesp. 29
Applications to a few basic systemsp. 30
Inertia forces in an accelerated reference framep. 35
Concluding commentsp. 38
Principle of virtual work and Lagrange's equationsp. 39
Introductionp. 40
Mechanical energy and exchange of itp. 41
Work and generalized forcesp. 41
Work performed by a forcep. 41
Generalized displacements and forcesp. 42
Work of intertial forces and kinetic energyp. 43
Linear motion (translation) in an intertial frame of referencep. 43
Rigid body rotating in an inertial frame of referencep. 44
Change of reference framep. 46
Generalized inertial forces in a rotating framep. 48
Properties of Hermitian matricesp. 50
Work performed by forces deriving from a potentialp. 53
Potential energyp. 53
Generalized displacements and forcesp. 54
Mechanical energy and the exchange of it with external systemsp. 56
Conservative systemsp. 56
Nonconservative systemsp. 57
Work performed by constraint reactions and perfect constraintsp. 59
Virtual work and Lagrange's equationsp. 59
Principle of virtual workp. 59
Lagrange's equationsp. 61
The Lagrange function (Lagrangian)p. 64
Special form of Lagrange's equations in the linear casep. 65
Lagrangian and Newtonian formulationsp. 67
Application to a building resting on elastic foundationsp. 67
Generalized displacementsp. 68
Potential energy and stiffnessp. 69
Generalized external loading and solution of the forced problemp. 70
Response to a distributed loadingp. 71
Stiffness coefficients for distributed elastic foundationsp. 72
Stiffness and mass matrices for any displacement fieldp. 74
Hamilton's principle and Lagrange's equations of unconstrained systemsp. 79
Introductionp. 80
The calculus of variations: first principlesp. 82
Stationary and extremum values of a functionp. 82
Static stabilityp. 85
Criterion for stabilityp. 85
Static stability of a pair of upside-down and coupled pendulumsp. 86
Buckling of a system of two articulated rigid barsp. 87
Stationary value of a definite integralp. 94
Variational formulation of Lagrange's equationsp. 101
Principle of virtual work and Hamilton's principlep. 101
General form of the Lagrange's equationsp. 102
Free motions of conservative systemsp. 103
Forced motions of conservative systemsp. 106
Nonconservative systemsp. 109
Constrained systems and Lagrange's undetermined multipliersp. 111
Introductionp. 112
Constraints and Lagrange multipliersp. 112
Stationary value of a constrained functionp. 112
Nonholonomic differential constraintsp. 116
Lagrange's equations of a constrained systemp. 116
Prescribed motions and transformation of reference framesp. 127
Prescribed displacements treated as rheonomic constraintsp. 127
Prescribed motions and transformations of reference framep. 130
Autonomous oscillatorsp. 139
Linear oscillatorsp. 140
Mechanical oscillatorsp. 140
Free vibration of conservative oscillatorsp. 142
Time-histories of displacementp. 142
Phase portraitp. 144
Modal analysisp. 146
Free vibration of nonconservative linear oscillatorsp. 148
Time-histories of displacementp. 148
Phase portraitp. 151
Modal analysisp. 152
Static instability (divergence or buckling)p. 153
Nonlinear oscillatorsp. 154
Conservative oscillatorsp. 154
Damped oscillatorsp. 164
Self-sustaining oscillatorsp. 167
Numerical integration of the equation of motionp. 169
Explicit scheme of central differences of second orderp. 170
Recursive processp. 170
Initialisation of the algorithmp. 171
Critical value of the time-step for stabilityp. 172
Accuracy of the algorithmp. 174
Application to a parametrically excited linear oscillatorp. 176
Application to an oscillator impacting against an elastic stopp. 179
Impact force modelp. 179
Constrained modelp. 185
Newmark's implicit algorithmp. 188
Natural modes of vibration of multi degree of freedom systemsp. 191
Introductionp. 192
Vibratory equations of conservative systemsp. 193
Linearization of the equations of motionp. 193
Solution of forced problems in staticsp. 194
Modal analysis of linear and conservative systemsp. 196
Coupling and uncoupling of the degrees of freedomp. 196
Natural modes of vibrationp. 199
Basic principle of the modal analysisp. 199
Basic properties of the natural modes of vibrationp. 199
Modal analysis of 2-DOF systemsp. 203
Natural modes of vibration as standing wavesp. 209
A few extensions of the modal conceptp. 215
Natural modes of vibration of constrained systemsp. 215
Free modes of rigid bodyp. 218
Prestressed systems and buckling modesp. 219
Rotating systems and whirling modes of vibrationp. 227
Particle tied to a rotating wheel through springsp. 228
Fly-wheel on flexible supportsp. 234
Forced vibrations: response to transient excitationsp. 239
Introductionp. 240
Deterministic transient excitation signalsp. 241
Locally integrable functions and regular distributionsp. 241
Signals suited to describe transient excitationsp. 242
Impulsive excitations: Dirac delta distributionp. 244
Excitations of infinite duration and finite energyp. 247
Forced response and Laplace transformationp. 247
Laplace and inverse Laplace transformationsp. 248
Transfer functions of the harmonic oscillatorp. 249
External loads equivalent to nonzero initial conditionsp. 250
Initial velocity and impulsive loadingp. 250
Initial displacement and relaxation of a step loadp. 251
Time-history of the response to a transient excitationp. 252
Response to a rectangular pulsep. 253
Response to a trapezoidal transientp. 258
Response to a truncated sine functionp. 260
Impulsive response and Green's functionp. 267
Green's function of a harmonic oscillatorp. 267
Green's function and forced response to any transientp. 268
Response of MDOF linear systemsp. 269
Transfer function matrix of a conservative systemp. 269
Uncoupling by projection on the modal basisp. 271
Principle of the methodp. 271
Modal expansion of the transfer and Green's functionsp. 272
Viscous dampingp. 280
Model of viscous and proportional dampingp. 281
Non proportional viscous dampingp. 283
Implicit Newmark algorithmp. 285
Spectral analysis of deterministic time signalsp. 295
Introductionp. 296
Basic principles of spectral analysisp. 298
Fourier seriesp. 298
Hilbert space of the functional vectors of period Tp. 300
Application: propagation of nondispersive 1-D wavesp. 309
Fourier transformationp. 313
Definitionsp. 313
Properties of Fourier transformsp. 313
Plancherel-Parseval theorem (product theorem)p. 314
Fourier transform in the sense of distributions and Fourier seriesp. 314
Spectral content of time signalsp. 315
Spectral density of energy of a transient signalp. 315
Power spectral density of periodical functionsp. 318
Mutual or cross-spectrap. 321
Spectra and correlation functionsp. 322
Coefficients of correlationp. 323
Correlation of periodic signalsp. 326
Functions approximated by truncated Fourier seriesp. 326
Digital signal processingp. 328
Sampling of a time signalp. 328
The Shannon sampling theoremp. 329
Fourier transforms of the original and of the truncated signalsp. 333
Discretization of the Fourier transformp. 335
Discrete finite Fourier transform and Fourier seriesp. 335
Definition and properties of the discrete Fourier transformp. 335
Illustrative examplep. 336
Spectral analysis of forced vibrationsp. 341
Introductionp. 342
Linear (harmonic) oscillatorp. 342
Spectra of excitation and responsep. 342
Spectral properties of transfer functionsp. 343
General features of the displacement/force transfer functionp. 343
Spectral ranges of the oscillator responsep. 346
MDOF linear systemsp. 352
Excitation and response spectrap. 352
Interesting features of the transfer functionsp. 352
Basic principles of the measurement of transfer functionsp. 358
Response spectra resulting from an MDOF excitationp. 359
Vibration absorber using antiresonant couplingp. 360
Shock absorber of a car suspensionp. 365
Forced vibrations of Duffing's oscillatorp. 368
Periodic solutions and nonlinear resonancesp. 368
Ritz Galerkin methodp. 368
Relationship between pulsation and amplitude of the oscillatory responsep. 370
Nonlinear resonance peakp. 374
Hysteresis effectp. 374
Numerical simulations and chaotic vibrationsp. 376
Periodic motionsp. 376
Chaotic motionsp. 379
Appendicesp. 383
Vector spacesp. 383
Vector and multiple products of vectorsp. 389
Euler's angles and kinetic energy of rotating bodiesp. 390
Hermitian and symmetrical matricesp. 394
Crout's and Choleski's decomposition of a matrixp. 398
Some basic notions about distributionsp. 402
Laplace transformationp. 409
Modal computation by an inverse iteration methodp. 414
Bibliographyp. 419
Indexp. 425
Series synopsis: modelling of mechanical systemsp. 435
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781903996515
ISBN-10: 1903996511
Series: Modelling of Mechanical Systems : Book 1
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 300
Published: 1st January 2004
Publisher: Elsevier Science & Technology
Country of Publication: GB
Dimensions (cm): 23.5 x 15.2  x 3.84
Weight (kg): 0.82