Modelling of Mechanical Systems: Discrete Systems

Note: Supplemental materials are not guaranteed with Rental or Used book purchases.

ISBN: 9781903996515 | 1903996511
Cover: Hardcover
Copyright: 11/6/2003

Buy Used

Usually Ships in 2-4 Business Days

$198.45
eBook

Available Instantly

Online: 1825 Days

Downloadable: Lifetime Access

$270.00

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; Hamiltons 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 * Presents the general methods that provide a unified framework to model mathematically mechanical systems of interest to the engineer, analyzing the response of these systems * Focuses on linear problems, but includes some aspects of non-linear configuration * Comprehensive coverage of mathematical techniques used to perform computer-based analytical studies and numerical simulations * Discusses the mathematical techniques used to perform analytical studies and numerical simulations on the computer

Foreword by Professor D J Ewins

Preface

xiii

Introduction

Chapter 1. Mechanical systems and equilibrium of forces

(38)

1.1. Modelling of mechanical systems

(18)

1.1.1. Geometry and distribution of masses

(1)

1.1.2. Motion relative to a given space

(1)

1.1.3. Coordinates

(5)

1.1.3.1. Degrees of freedom and generalized coordinates

(2)

1.1.3.2. Coordinate transformation

(2)

1.1.3.3. Changes of reference frame

(1)

1.1.4. Kinematical constraints

(8)

1.1.4.1. Holonomic constraints

(3)

1.1.4.2. Nonholonomic constraints

(1)

1.1.4.3. Example: a constrained rigid wheel

(5)

1.1.5. Forces formulated explicitly as material laws

(3)

1.1.6. Forces formulated as constraint conditions

(1)

1.2. Basic principles of Newtonian mechanics

(19)

1.2.1. Newton's laws

(4)

1.2.1.1. Law of inertia

(1)

1.2.1.2. Law of motion (basic principle of dynamics)

(1)

1.2.1.3. Law of action and reaction

(2)

1.2.2. D'Alembert's principle of dynamical equilibrium

(2)

1.2.3. Equations of motion In terms of moments

(9)

1.2.3.1. Moment of a force and angular momentum

(2)

1.2.3.2. Plane rotation of a particle

(1)

1.2.3.3. Centrifugal and Coriolis forces

(1)

1.2.3.4. Applications to a few basic systems

(5)

1.2.4. Inertia forces in an accelerated reference frame

(3)

1.2.5. Concluding comments

(1)

Chapter 2. Principle of virtual work and Lagrange's equations

(40)

2.1. Introduction

(1)

2.2. Mechanical energy and exchange of it

(18)

2.2.1. Work and generalized forces

(2)

2.2.1.1. Work performed by a force

(1)

2.2.1.2. Generalized displacements and forces

(1)

2.2.2. Work of inertial forces and kinetic energy

(10)

2.2.2.1. Linear motion (translation) in an inertial frame of reference

(1)

2.2.2.2. Rigid body rotating in an inertial frame of reference

(2)

2.2.2.3. Change of reference frame

(2)

2.2.2.4. Generalized inertial forces in a rotating frame

(2)

2.2.2.5. Properties of Hermitian matrices

(3)

2.2.3. Work performed by forces deriving from a potential

(3)

2.2.3.1. Potential energy

(1)

2.2.3.2. Generalized displacements and forces

(2)

2.2.4. Mechanical energy and the exchange of it with external systems

(3)

2.2.4.1. Conservative systems

(1)

2.2.4.2. Nonconservative systems

(2)

2.2.5. Work performed by constraint reactions and perfect constraints

(1)

2.3. Virtual work and Lagrange's equations

(20)

2.3.1. Principle of virtual work

(2)

2.3.2. Lagrange's equations

(3)

2.3.3. The Lagrange function (Lagrangian)

(1)

2.3.4. Special form of Lagrange's equations in the linear case

(2)

2.3.5. Lagrangian and Newtonian formulations

(1)

2.3.6. Application to a building resting on elastic foundations

(15)

2.3.6.1. Generalized displacements

(1)

2.3.6.2. Potential energy and stiffness

(1)

2.3.6.3. Generalized external loading and solution of the forced problem

(1)

2.3.6.4. Response to a distributed loading

(1)

2.3.6.5. Stiffness coefficients for distributed elastic foundations

(2)

2.3.6.6. Stiffness and mass matrices for any displacement field

(5)

Chapter 3. Hamilton's principle and Lagrange's equations of unconstrained systems

(32)

3.1. Introduction

(2)

3.2. The calculus of variations: first principles

(19)

3.2.1. Stationary and extremum values of a function

(3)

3.2.2. Static stability

(9)

3.2.2.1. Criterion for stability

(1)

3.2.2.2. Static stability of a pair of upside-down and coupled pendulums

(1)

3.2.2.3. Buckling of a system of two articulated rigid bars

(7)

3.2.3. Stationary value of a definite integral

(7)

3.3. Variational formulation of Lagrange's equations

101

(10)

3.3.1. Principle of virtual work and Hamilton's principle

101

(1)

3.3.2. General form of the Lagrange's equations

102

(1)

3.3.3. Free motions of conservative systems

103

(3)

3.3.4. Forced motions of conservative systems

106

(3)

3.3.5. Nonconservative systems

109

(2)

Chapter 4. Constrained systems and Lagrange's undetermined multipliers

111

(28)

4.1. Introduction

112

(1)

4.2. Constraints and Lagrange multipliers

112

(15)

4.2.1. Stationary value of a constrained function

112

(4)

4.2.2. Nonholonomic differential constraints

116

(1)

4.2.3. Lagrange's equations of a constrained system

116

(11)

4.3. Prescribed motions and transformation of reference frames

127

(12)

4.3.1. Prescribed displacements treated as rheonomic constraints

127

(3)

4.3.2. Prescribed motions and transformations of reference frame

130

(9)

Chapter 5. Autonomous oscillators

139

(52)

5.1. Linear oscillators

140

(14)

5.1.1. Mechanical oscillators

140

(2)

5.1.2. Free vibration of conservative oscillators

142

(6)

5.1.2.1. Time-histories of displacement

142

(2)

5.1.2.2. Phase portrait

144

(2)

5.1.2.3. Modal analysis

146

(2)

5.1.3. Free vibration of nonconservative linear oscillators

148

(5)

5.1.3.1. Time-histories of displacement

148

(3)

5.1.3.2. Phase portrait

151

(1)

5.1.3.3. Modal analysis

152

(1)

5.1.4. Static instability (divergence or buckling)

153

(1)

5.2. Nonlinear oscillators

154

(15)

5.2.1. Conservative oscillators

154

(10)

5.2.2. Damped oscillators

164

(3)

5.2.3. Self-sustaining oscillators

167

(2)

5.3. Numerical integration of the equation of motion

169

(22)

5.3.1. Explicit scheme of central differences of second order

170

(6)

5.3.1.1. Recursive process

170

(1)

5.3.1.2. Initialisation of the algorithm

171

(1)

5.3.1.3. Critical value of the time-step for stability

172

(2)

5.3.1.4. Accuracy of the algorithm

174

(2)

5.3.2. Application to a parametrically excited linear oscillator

176

(3)

5.3.3. Application to an oscillator impacting against an elastic stop

179

(9)

5.3.3.1. Impact force model

179

(6)

5.3.3.2. Constrained model

185

(3)

5.3.4. Newmark's implicit algorithm

188

(3)

Chapter 6. Natural modes of vibration of multi degree of freedom systems

191

(48)

6.1. Introduction

192

(1)

6.2. Vibratory equations of conservative systems

193

(3)

6.2.1. Linearization of the equations of motion

193

(1)

6.2.2. Solution of forced problems In statics

194

(2)

6.3. Modal analysis of linear and conservative systems

196

(19)

6.3.1. Coupling and uncoupling of the degrees of freedom

196

(3)

6.3.2. Natural modes of vibration

199

(16)

6.3.2.1. Basic principle of the modal analysis

199

(1)

6.3.2.2. Basic properties of the natural modes of vibration

199

(4)

6.3.2.3. Modal analysis of2-DOF systems

203

(6)

6.3.2.4. Natural modes of vibration as standing waves

209

(6)

6.4. A few extensions of the modal concept

215

(24)

6.4.1. Natural modes of vibration of constrained systems

215

(3)

6.4.2. Free modes of rigid body

218

(1)

6.4.3. Prestressed systems and buckling modes

219

(8)

6.4.4. Rotating systems and whirling modes of vibration

227

(14)

6.4.4.1. Particle tied to a rotating wheel through springs

228

(6)

6.4.4.2. Fly-wheel on flexible supports

234

(5)

Chapter 7. Forced vibrations: response to transient excitations

239

(56)

7.1. Introduction

240

(1)

7.2. Deterministic transient excitation signals

241

(6)

7.2.1. Locally integrable functions and regular distributions

241

(1)

7.2.2. Signals suited to describe transient excitations

242

(2)

7.2.3. Impulsive excitations: Dirac delta distribution

244

(3)

7.2.4. Excitations of infinite duration and finite energy

247

(1)

7.3. Forced response and Laplace transformation

247

(20)

7.3.1. Laplace and inverse Laplace transformations

248

(1)

7.3.2. Transfer functions of the harmonic oscillator

249

(1)

7.3.3. External loads equivalent to nonzero initial conditions

250

(2)

7.3.3.1. initial velocity and impulsive loading

250

(1)

7.3.3.2. Initial displacement and relaxation of a step load

251

(1)

7.3.4. Time-history of the response to a transient excitation

252

(15)

7.3.4.1. Response to a rectangular pulse

253

(5)

7.3.4.2. Response to a trapezoidal transient

258

(2)

7.3.4.3. Response to a truncated sine function

260

(7)

7.4. Impulsive response and Green's function

267

(2)

7.4.1. Green's function of a harmonic oscillator

267

(1)

7.4.2. Green's function and forced response to any transient

268

(1)

7.5. Response of MDOF linear systems

269

(26)

7.5.1. Transfer function matrix of a conservative system

269

(2)

7.5.2. Uncoupling by projection on the modal basis

271

(9)

7.5.2.1. Principle of the method

271

(1)

7.5.2.2. Modal expansion of the transfer and Green's functions

272

(8)

7.5.3. Viscous damping

280

(18)

7.5.3.1. Model of viscous and proportional damping

281

(2)

7.5.3.2. Non proportional viscous damping

283

(2)

7.5.3.3. Implicit Newmark algorithm

285

(10)

Chapter 8. Spectral analysis of deterministic time signals

295

(46)

8.1. Introduction

296

(2)

8.2. Basic principles of spectral analysis

298

(30)

8.2.1. Fourier series

298

(2)

8.2.2. Hilbert space of the functional vectors of period T

300

(9)

8.2.3. Application: propagation of nondispersive 1-D waves

309

(4)

8.2.4. Fourier transformation

313

(2)

8.2.4.1. Definitions

313

(1)

8.2.4.2. Properties of Fourier transforms

313

(1)

8.2.4.3. Plancherel-Parseval theorem (product theorem)

314

(1)

8.2.4.4. Fourier transform in the sense of distributions and Fourier series

314

(1)

8.2.5. Spectral content of time signals

315

(13)

8.2.5.1. Spectral density of energy of a transient signal

315

(3)

8.2.5.2. Power spectral density of periodical functions

318

(3)

8.2.5.3. Mutual or cross-spectra

321

(1)

8.2.5.4. Spectra and correlation functions

322

(1)

8.2.5.5. Coefficients of correlation

323

(3)

8.2.5.6. Correlation of periodic signals

326

(1)

8.2.5.7. Functions approximated by truncated Fourier series

326

(2)

8.3. Digital signal processing

328

(13)

8.3.1. Sampling of a time signal

328

(1)

8.3.2. The Shannon sampling theorem

329

(4)

8.3.3. Fourier transforms of the original and of the truncated signals

333

(2)

8.3.4. Discretization of the Fourier transform

335

(7)

8.3.4.1. Discrete finite Fourier transform and Fourier series

335

(1)

8.3.4.2. Definition and properties of the discrete Fourier transform

335

(1)

8.3.4.3. Illustrative example

336

(5)

Chapter 9. Spectral analysis of forced vibrations

341

(42)

9.1. Introduction

342

(1)

9.2. Linear (harmonic) oscillator

342

(10)

9.2.1. Spectra of excitation and response

342

(1)

9.2.2. Spectral properties of transfer functions

343

(9)

9.2.2.1. General features of the displacement/force transfer function

343

(3)

9.2.2.2. Spectral ranges of the oscillator response

346

(6)

9.3. MDOF linear systems

352

(16)

9.3.1. Excitation and response spectra

352

(1)

9.3.2. Interesting features of the transfer functions

352

(6)

9.3.3. Basic principles of the measurement of transfer functions

358

(1)

9.3.4. Response spectra resulting from an MDOF excitation

359

(1)

9.3.5. Vibration absorber using antiresonant coupling

360

(5)

9.3.6. Shock absorber of a car suspension

365

(3)

9.4. Forced vibrations of Duffing's oscillator

368

(15)

9.4.1. Periodic solutions and nonlinear resonances

368

(8)

9.4.1.1. Ritz Galerkin method

368

(2)

9.4.1.2. Relationship between pulsation and amplitude of the oscillatory response

370

(4)

9.4.1.3. Nonlinear resonance peak

374

(1)

9.4.1.4. Hysteresis effect

374

(2)

9.4.2. Numerical simulations and chaotic vibrations

376

(7)

9.4.2.1. Periodic motions

376

(3)

9.4.2.2. Chaotic motions

379

(4)

Appendices

383

(36)

Appendix 1: Vector spaces

383

(6)

Appendix 2: Vector and multiple products of vectors

389

(1)

Appendix 3: Suler's angles and kinetic energy of rotating bodies

390

(4)

Appendix 4: Hermitian and symmetrical matrices

394

(4)

Appendix 5: Grout's and Choleski's decomposition of a matrix

398

(4)

Appendix 6: Some basic notions about distributions

402

(7)

Appendix 7: Laplace transformation

409

(5)

Appendix 8: Modal computation by an inverse iteration method

414

(5)

Bibliography

419

(6)

Index

425

(10)

Series synopsis: modelling of mechanical systems

435

Please wait while the item is added to your bag...

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Modelling of Mechanical Systems: Discrete Systems

Summary

Author Biography

Table of Contents

Supplemental Materials