Vibrations and Stability

Advanced Theory, Analysis, and Tools

By: Jon Juel Thomsen

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Hardcover | 11 September 2003 | Edition Number 2

At a Glance

Hardcover
428 Pages

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Revised

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24.13 x 16.51 x 3.18

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An ideal text for students that ties together classical and modern topics of advanced vibration analysis in an interesting and lucid manner. It provides students with a background in elementary vibrations with the tools necessary for understanding and analyzing more complex dynamical phenomena that can be encountered in engineering and scientific practice. It progresses steadily from linear vibration theory over various levels of nonlinearity to bifurcation analysis, global dynamics and chaotic vibrations. It trains the student to analyze simple models, recognize nonlinear phenomena and work with advanced tools such as perturbation analysis and bifurcation analysis. Explaining theory in terms of relevant examples from real systems, this book is user-friendly and meets the increasing interest in non-linear dynamics in mechanical/structural engineering and applied mathematics and physics. This edition includes a new chapter on the useful effects of fast vibrations and many new exercise problems.

Industry Reviews

From the reviews of the second edition:

"Vibrations and stability ... attracted a vast amount of attention of a multitude of researchers in the past and present and will remain highly topical in the future. ... The 2nd edition of the book, a thoroughly revised and expanded version of the 1st edition, is an essential by-product of this evolution. ... In the reviewer's opinion the author ... has written a highly recommendable book. I am very pleased to review the book. ... It presents a very readable and well-structured account ... ." (Dr C. B. Sharma, Contemporary Physics, Vol. 45 (6), 2004)

"The second edition of 'Vibrations and Stability' is an accomplished and valuable book, mainly devoted to vibrations in the non-linear regime. ... It is a pleasure to read this clearly written book, which achieves the aim of presenting important material on non-linear vibrations in a useful and quite understandable manner. ... relevant references are given for readers interested in more information. ... Engineers, researchers, and particularly students and teachers in mechanical and structural engineering will find this to be a very helpful book." (Pedro Ribeiro, Journal of Sound and Vibration, Vol. 274 (4-5), 2004)

"Every chapter is equipped with useful exercises. The reviewed book will be very useful in engineering and scientific practice." (Boris V. Loginov, Zentralblatt MATH, Vol. 1086, 2006)

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Preface	p. VII
Contents	p. XIII
Notation	p. XXI
Vibration Basics	p. 1
Introduction	p. 1
Single Degree of Freedom Systems	p. 2
Undamped Free Vibrations	p. 2
Damped Free Vibrations	p. 2
Harmonic Forcing	p. 3
Arbitrary Forcing	p. 4
Multiple Degree of Freedom Systems	p. 4
Equations of Motion	p. 5
Undamped Free Vibrations	p. 6
Orthogonality of Modes	p. 7
Damped Free Vibrations	p. 7
Harmonically Forced Vibrations, No Damping	p. 7
Harmonically Forced Vibrations, Damping Included	p. 8
General Periodic Forcing	p. 9
Arbitrary Forcing, Transients	p. 9
Continuous Systems	p. 10
Equations of Motion	p. 10
Undamped Free Vibrations	p. 11
Orthogonality of Modes	p. 12
Normal Coordinates	p. 13
Forced Vibrations, No Damping	p. 13
Forced Vibrations, Damping Included	p. 14
Rayleigh's Method	p. 15
Ritz Method	p. 16
Energy Methods for Setting Up Equations Of Motion	p. 17
Lagrange's Equations	p. 17
Hamilton's Principle	p. 18
From PDEs to ODEs: Mode shape Expansion	p. 21
Bypassing PDEs: Using Lagrange's Equations with Continuous Systems	p. 23
Problems	p. 25
Eigenvalue Problems of Vibrations And Stability	p. 27
Introduction	p. 27
The Algebraic EVP	p. 27
Mathematical Form	p. 28
Properties of Eigenvalues and Eigenvectors	p. 28
Methods of Solution	p. 29
The Differential EVP	p. 29
Mathematical Form	p. 30
Stability-Related EVPs	p. 31
The Clamped-hinged Euler Column	p. 31
The Paradox of Follower-loading	p. 34
Buckling by Gravity	p. 35
Vibration-Related EVPs	p. 36
Axial Vibrations of Straight Rods	p. 36
Flexural Vibrations of Beams	p. 37
Concepts of Differential EVPs	p. 39
Multiplicity	p. 40
Classes of Boundary Conditions: Essential and Suppressible	p. 40
Function-classes: Eigen-, Test-, and Admissible functions	p. 41
Adjointness	p. 41
Definiteness	p. 42
Orthogonality	p. 43
Three Classes of EVPs	p. 43
The Rayleigh Quotient	p. 43
Properties of Eigenvalues and Eigenfunctions	p. 44
Real-valueness of Eigenvalues	p. 44
Sign of the Eigenvalues	p. 44
Orthogonality of Eigenfunctions	p. 45
Minimum Properties of the Eigenvalues	p. 46
The Comparison Theorem	p. 47
The Inclusion Theorem for One-term EVPs	p. 48
Methods of Solution	p. 49
Closed-form Solutions	p. 50
The Method of Eigenfunction Iteration	p. 50
The Rayleigh-Ritz Method	p. 51
The Finite Difference Method	p. 54
Collocation	p. 58
Composite EVPs: Dunkerley's and Southwell's Formulas	p. 59
Other Methods	p. 60
Problems	p. 60
Nonlinear Vibrations: Classical Local Theory	p. 65
Introduction	p. 65
Sources of Nonlinearity	p. 66
Geometrical Nonlinearities	p. 66
Material Nonlinearities	p. 68
Nonlinear Body Forces	p. 69
Physical Configuration Nonlinearities	p. 70
Main Example: Pendulum with an Oscillating Support	p. 71
Equation of Motion	p. 72
Qualitative Analysis of the Unforced Response	p. 73
Recasting the Equations into First-order Form	p. 73
The Phase Plane	p. 74
Singular Points	p. 76
Stability of Singular Points	p. 77
On the Behavior of Orbits near Singular Points	p. 79
Quantitative Analysis	p. 85
Perturbation Methods	p. 85
The Straightforward Expansion	p. 86
The Method of Multiple Scales	p. 88
The Method of Harmonic Balance	p. 92
The Method of Averaging	p. 94
The Forced Response--Multiple Scales Analysis	p. 96
Posing the Problem	p. 96
Perturbation Equations	p. 97
The Non-resonant Case	p. 99
The Near-resonant Case	p. 100
Stability of Stationary Solutions	p. 102
Discussing Stationary Responses	p. 105
Externally Excited Duffing Systems	p. 110
Two Physical Examples	p. 111
Primary Resonance, Weak Excitations	p. 116
Non-resonant Hard Excitations	p. 122
Obtaining Forced Responses by Averaging	p. 128
Concluding Remarks	p. 130
Problems	p. 130
Nonlinear Multiple-DOF Systems: Local Analysis	p. 137
Introduction	p. 137
The Autoparametric Vibration Absorber	p. 138
The System	p. 138
First-order Approximate Response	p. 140
Frequency and Force Responses	p. 144
Concluding Remarks on the Vibration Absorber	p. 147
Nonlinear Mode-Coupling of Non-Shallow Arches	p. 147
The Model	p. 148
Linear Response and Stability	p. 149
Nonlinear Response and Stability	p. 150
Other Systems Possessing Internal Resonance	p. 154
The Follower-loaded Double Pendulum	p. 155
The Model	p. 156
The Zero Solution and its Stability	p. 158
Periodic Solutions	p. 160
Non-periodic and Non-zero Static Solutions	p. 164
Summing Up	p. 164
Pendulum with a Sliding Disk	p. 165
Introduction	p. 165
The System	p. 165
Equations of Motion	p. 166
Inspecting the Equations of Motion	p. 167
Seeking Quasi-statical Equilibriums by Averaging	p. 167
String with a Sliding Pointmass	p. 168
Model System and Equations of Motion	p. 169
Illustration of System Behavior	p. 172
Response to Near-resonant Base Excitation	p. 174
Response to Slow Frequency-sweeps	p. 177
Response to Near-Resonant Axial Excitation	p. 179
Non-trivial Effects of Rotary Inertia	p. 182
Summing Up	p. 182
Vibration-Induced Fluid Flow in Pipes	p. 183
Problems	p. 185
Bifurcations	p. 189
Introduction	p. 189
Systems, Bifurcations, and Bifurcation Conditions	p. 190
Systems	p. 190
Bifurcations	p. 190
Bifurcation Conditions: Structural Instability	p. 191
Codimension One Bifurcations of Equilibriums	p. 192
The Pitchfork Bifurcation	p. 192
The Saddle-node Bifurcation	p. 194
The Transcritical Bifurcation	p. 195
The Hopf Bifurcation	p. 196
Codimension One Bifurcations for N-Dimensional Systems	p. 197
Saddle-Node Conditions	p. 198
Transcritical and Pitchfork Conditions	p. 199
Hopf Conditions	p. 200
Center Manifold Reduction	p. 201
The Center Manifold Theorem	p. 202
Implications of the Theorem	p. 203
Computing the Center Manifold Reduction	p. 204
An Example	p. 206
Summing up on Center Manifold Reduction	p. 207
Normal Form Reduction	p. 208
Bifurcating Periodic Solutions	p. 209
Grouping Bifurcations According to their Effect	p. 210
On the Stability of Bifurcations to Perturbations	p. 211
Stability of a Saddle-node Bifurcation	p. 211
Stability of a Supercritical Pitchfork Bifurcation	p. 212
Summing up on Different Notions of Stability	p. 213
Examples	p. 215
Midplane Stretching (Duffing's Equation)	p. 215
Pendulum with a Moving Support (Parametric Excitation)	p. 217
The Autoparametric Vibration Absorber	p. 219
The Partially Follower-loaded Double Pendulum	p. 221
Problems	p. 222
Chaotic Vibrations	p. 225
Introduction	p. 225
A First Example	p. 227
Tools for Detecting Chaotic Vibrations	p. 229
Phase Planes	p. 229
Frequency Specters	p. 231
Poincare Maps	p. 231
Lyapunov Exponents	p. 234
Horizons of Predictability	p. 238
Attractor Dimensions	p. 240
Basins of Attraction	p. 241
Summary on Detecting Tools	p. 242
Universal Routes to Chaos	p. 242
The Period-doubling Route	p. 243
The Quasiperiodic Route	p. 245
The Transient Route	p. 246
The Intermittency Route	p. 248
Summary on the Routes to Chaos	p. 249
Tools for Predicting the Onset of Chaos	p. 249
Criteria Related to the Universal Routes of Chaos	p. 249
Searching for Homoclinic Tangles and Smale Horseshoes	p. 251
The Melnikov Criterion	p. 255
Criteria Based on Local Perturbation Analysis	p. 259
Criteria for Conservative Chaos	p. 263
Mechanical Systems and Chaos	p. 263
The Lorenz System (D = 3)	p. 263
Duffing-type Systems (D = 3)	p. 264
Pendulum-type Systems (D = 3)	p. 265
Piecewise Linear Systems (D [greater than or equal] 3)	p. 267
Coupled Autonomous Systems (D [greater than or equal] 4)	p. 268
Autoparametric Systems (D [greater than or equal] 5)	p. 273
High-order Systems (D [greater than sign] 5)	p. 277
Other Systems	p. 277
Elastostatical Chaos	p. 278
Spatial and Spatiotemporal Chaos	p. 281
Controlling Chaos	p. 282
Closing Comments	p. 283
Problems	p. 284
Special Effects of High-Frequency Excitation	p. 287
Introduction	p. 287
The Method of Direct Partition of Motions (MDPM)	p. 288
Outline of the MDPM	p. 288
The Concept of Vibrational Force	p. 291
The MDPM Compared to Other Perturbation Approaches	p. 291
Simple Examples	p. 292
Pendulum on a Vibrating Support (Stiffening and Biasing)	p. 292
Mass on a Vibrating Plane (Smoothening and Biasing)	p. 297
Brumberg's Pipe (Smoothening and Biasing)	p. 300
A Slight but Useful Generalization	p. 302
A Fairly General Class of Discrete Systems	p. 303
The System	p. 303
Example Functions	p. 304
The Averaged System Governing the 'Slow' Motions	p. 304
Interpretation of Averaged Forcing Terms	p. 307
The Effects	p. 308
Stiffening	p. 309
Biasing	p. 312
Smoothening	p. 315
A General Class of Linear Continuous Systems	p. 318
The Generalized No-Resonance Prediction (GNRP)	p. 320
The Generalized Analytical Resonance Prediction (GARP)	p. 322
Example 1: Clamped String With HF Base Excitation	p. 324
Example 2: Square Membrane with In-Plane HF Excitation	p. 325
Specific Systems and Results--Some Examples	p. 327
Using HF Excitation to Quench Friction-Induced Vibrations	p. 327
Displacement due to HF Excitation and Asymmetric Friction	p. 329
Chelomei's Pendulum--Resolving a Paradox	p. 329
Stiffening of a Flexible String	p. 332
Concluding Remarks	p. 333
Problems	p. 334
Performing Numerical Simulations	p. 339
Solving Differential Equations	p. 339
Computing Chaos-Related Quantities	p. 340
Interfacing with the ODE-Solver	p. 340
Locating Software on the Internet	p. 343
Major Exercises	p. 347
Tension Control of Rotating Shafts	p. 347
Mathematical Model	p. 348
Eigenvalue Problem, Natural Frequencies and Mode Shapes	p. 348
Discretisations, Choice of Control Law	p. 349
Local Bifurcation Analysis for a Balanced Shaft (p = me = 0)	p. 350
Quantitative Analysis of the Controlled System	p. 351
Using a Dither Signal for Open-Loop Control	p. 352
Numerical Analysis of the Controlled System	p. 352
Conclusions	p. 353
Vibrations of a Spring-Tensioned Beam	p. 353
Mathematical Model	p. 353
Eigenvalue Problem, Natural Frequencies and Mode Shapes	p. 355
Discrete Models	p. 355
Local Bifurcation Analysis for the Unloaded System	p. 356
Quantitative Analysis of the Loaded System	p. 356
Numerical Analysis	p. 357
Conclusions	p. 358
Dynamics of a Microbeam	p. 358
System Description	p. 359
Mathematical Model	p. 360
Eigenvalue Problem, Natural Frequencies and Mode Shapes	p. 361
Discrete Models, Mode Shape Expansion	p. 361
Local Bifurcation Analysis for the Statically Loaded System	p. 362
Quantitative Analysis of the Loaded System	p. 362
Numerical Analysis	p. 363
Conclusions	p. 364
Mathematical Formulas	p. 365
Formulas Typically Used in Perturbation analysis	p. 365
Complex Numbers	p. 365
Powers of Two-Term Sums	p. 365
Dirac's Delta Function ([Delta])	p. 366
Averaging Integrals	p. 366
Fourier Series of a Periodic Function	p. 366
Formulas for Stability Analysis	p. 366
The Routh-Hurwitz Criterion	p. 366
Mathieu's Equation: Stability of the Zero-Solution	p. 367
Vibration Modes and Frequencies for Structural Elements	p. 371
Rods	p. 372
Longitudinal Vibrations	p. 372
Torsional Vibrations	p. 372
Beams	p. 372
Bernoulli-Euler Theory	p. 372
Timoshenko Theory	p. 373
Rings	p. 375
In-Plane Bending	p. 375
Out-of-Plane Bending	p. 376
Extension	p. 376
Membranes	p. 376
Rectangular Membrane	p. 376
Circular Membrane	p. 377
Plates	p. 377
Rectangular Plate	p. 377
Circular Plate	p. 378
Other Structures	p. 378
Properties of Engineering Materials	p. 379
Friction and Thermal Expansion Coefficients	p. 379
Density and Elasticity Constants	p. 380
References	p. 381
Index	p. 393
Table of Contents provided by Rittenhouse. All Rights Reserved.

Vibrations and Stability

Advanced Theory, Analysis, and Tools

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Hardcover

Industry Reviews

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