Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration

by Lawrence N. Virgin

• ISBN13: 9780521779319
• ISBN10: 0521779316
• Format: Paperback
• Copyright: 2000-03-28
• Publisher: Cambridge University Press
• More Book Details

Summary

Nonlinear behavior can be found in such highly disparate areas as population biology and aircraft wing flutter. For this reason, nonlinear dynamics and chaos have become very active fields of research. This work uses an extended case study--an experiment in mechanical vibration--to introduce and explore the subject of nonlinear behavior and chaos from an engineering perspective. After a review of basic principles, the text then describes a cart-on-a-track oscillator and shows what happens when it is gradually subjected to greater excitation, thereby encountering the full spectrum of nonlinear behavior, from simple free decay to chaos. Advanced undergraduate and graduate students, as well as practicing engineers, will find this book a lively, accessible introduction to the subject.

Table of Contents

Preface	xi
Brief Introductory Remarks	1	(5)
The Physical Context	1	(2)
Experiments	3	(1)
Stability	3	(1)
Predictability	4	(2)
Background: Linear Behavior	6	(17)
Introduction	6	(1)
Simple Harmonic Motion	6	(4)
Damped Oscillations	10	(3)
Viscous Damping	10	(2)
Coulomb Damping	12	(1)
Forced Oscillations	13	(5)
Measurement of Damping	18	(3)
Estimating Viscous Damping from a Free Decay	18	(1)
Forced Vibration -- Half-Power Method	19	(1)
Estimation of Coulomb Damping	19	(2)
Superposition	21	(2)
Some Useful Concepts	23	(23)
Introduction	23	(1)
Phase Space	23	(4)
The Poincare Section	27	(5)
Spectral Analysis	32	(3)
Stability	35	(7)
Stability of Equilibrium	36	(3)
Stability of Cycles	39	(3)
Numerical Issues	42	(2)
Extensions to Higher-Order Systems	44	(2)
The Paradigm	46	(12)
Introduction	46	(1)
Some Basic Forms of Nonlinearity	47	(2)
Nonlinear Structural Dynamics	49	(4)
Lumped Parameters	53	(2)
A Contrived Mechanical Paradigm	55	(3)
Mathematical Description	58	(15)
Introduction	58	(4)
Damping	62	(5)
Viscous Damping	63	(1)
Coulomb Damping	64	(3)
Nondimensionalization	67	(2)
Alternative Analytic Approaches	69	(4)
The Arc-Length Coordinate	69	(2)
Isodynamical Systems	71	(2)
The Experimental Model	73	(13)
Introduction	73	(1)
The Model	74	(2)
The Starting Gate	76	(1)
The Impact Barrier	76	(1)
The Forcing Mechanism	76	(2)
Data Acquisition	78	(1)
Time-Lag Embedding	79	(2)
Measurement of Damping	81	(2)
Poincare Sampling	83	(1)
Other Experimental Issues	84	(2)
Signal Conditioning	84	(1)
System Identification	85	(1)
Free Oscillations	86	(15)
Introduction	86	(2)
An Approximate Treatment	88	(3)
A Perturbation Approach	91	(3)
Phase Trajectories	94	(2)
Experimental Free Oscillations	96	(5)
Forced Response: Periodic	101	(16)
Introduction	101	(1)
Mildly Nonlinear Behavior	102	(5)
Simple Periodic Motion -- Experiments	107	(1)
Subharmonic Behavior	108	(3)
Nonstationarity	111	(4)
Cross-Well Periodic Behavior	115	(2)
Forced Response: Nonperiodic	117	(18)
Introduction	117	(1)
Chaos	117	(3)
Poincare Sections	120	(2)
Autocorrelation	122	(3)
Dimension	125	(2)
Lyapunov Exponents	127	(3)
Unstable Periodic Orbits	130	(3)
Transient Chaos	133	(2)
Escape from a Potential Energy Well	135	(17)
Introduction	135	(1)
An Asymmetric Potential Energy Shape	136	(5)
The Unforced Case	138	(1)
The Forced Response	139	(2)
Quasi-Steady Escape	141	(4)
The Period-Doubling Cascade	144	(1)
Transient Escape	145	(7)
A Hardening Spring Oscillator	152	(16)
Introduction	152	(1)
Mathematical Modeling	153	(1)
Amplitude Frequency Response	154	(2)
Stability Analysis	156	(5)
Experimental and Numerical Approach	156	(4)
Analytic Approach	160	(1)
Zero Linear Stiffness -- Period-Doubling Revisited	161	(7)
The Effect of a Stiffness Discontinuity	168	(12)
Introduction	168	(1)
Free Response	169	(3)
Forced Response	172	(8)
Two-Frequency Excitation	180	(14)
Subharmonic Forcing	180	(3)
Quasi-Periodic Forcing	183	(5)
Single-Well Motion	184	(2)
Double-Well Motion	186	(2)
Characterization of Chaos	188	(4)
Additional Features	192	(2)
Global Issues	194	(21)
Introduction	194	(1)
Dependence on Initial Conditions	195	(11)
Stochastic Interrogation	198	(4)
Transient Lengths	202	(4)
Manifold Interaction	206	(1)
Melnikov Theory	207	(2)
Bifurcations in Control Space	209	(3)
Indeterminate Outcomes	212	(3)
A Nonlinear Electric Circuit	215	(9)
A.1 Introduction	215	(2)
A.2 Typical Responses	217	(3)
A.3 Bottlenecking	220	(4)
B A Continuous System	224	(13)
B.1 Introduction	224	(1)
B.2 Underlying Statics	225	(2)
B.3 Experimental Description	227	(3)
B.3.1 Free Response	227	(3)
B.4 Typical Forced Responses	230	(3)
B.5 Snap-Through Behavior	233	(4)
Bibliography	237	(14)
Index	251