Chaotic Vibrations An Introduction for Applied Scientists and Engineers FRANCIS C. MOON Theoretical and Applied Mechanics Cornell University Ithaca, New York A WILEY-INTERSCIENCE PUBLICATION JOHN WILEY & SONS NEW YORK CmCHESTER BRISBANE TORONTO SINGAPORE
Contents Introduction: A New Age of Dynamics 1 1.1 What Is Chaotic Dynamics? 1 Why Should Engineers Study Chaotic Dynamics?, 6 Sources of Chaos in Continuum Physics, 7 Where Have Chaotic Vibrations Been Observed?, 9 1.2 Classical Nonlinear Vibration Theory: A Brief Review 10 Linear Vibration Theory, 10 Nonlinear Vibration Theory, 12 Local Geometric Theory of Dynamics, 19 Bifurcations, 21 1.3 Maps and Flows 24 Three Paradigms for Chaos, 25 Henon and Horseshoe Maps, 28 The Lorenz Attractor and Fluid Chaos, 30 Closing Comments, 35 2 How to Identify Chaotic Vibrations 37 Nonlinear System Elements, 39 Random Inputs, 41 Observation of Time History, 42 Phase Plane History, 43 Fourier Spectrum, 45 Poincare Maps, 47 xi
xii Contents Routes to Chaos, 56 Crisis, 62 Transient Chaos, 62 Conservative Chaos, 62 Lyapunov Exponents and Fractal Dimensions, 63 A Survey of Systems with Chaotic Vibrations 67 3.1 New Paradigms in Dynamics 67 3.2 Mathematical Models of Chaotic Physical Systems 68 Thermal Convection in Fluids, 68 Thermal Convection Model of Moore and Spiegel, 70 Supersonic Panel Flutter, 73 Impact Force Problems, 73 Double-Well Potential Problems, 77 Chaos in a Pendulum, 79 Spherical Pendulum, 80 The Kicked Rotor, 81 Circle Map, 84 Other Rigid Body Problems, 84 Aeroelastic Flutter, 84 Nonlinear Electrical Circuits, 85 Magnetomechanical Models, 88 Control System Chaos, 89 3.3 Physical Experiments in Chaotic Systems 91 Early Observations of Chaotic Vibrations, 91 Rigid-Body Systems, 92 Magnetic Compass Needle, 93 Magnetically Levitated Vehicles, 94 Chaos in Elastic Continua, 96 Three-Dimensional Elastica and Strings, 102 Impact Print Hammer, 102 Nonlinear Circuits, 104 Chaotic Dynamics in Fluid Systems, 111 Experimental Methods in Chaotic Vibrations 121 4.1 Introduction: Experimental Goals 121 4.2 Nonlinear Elements in Dynamical Systems 123 Material Nonlinearities, 124
Contents xiii Kinematic Nonlinearities, 124 Nonlinear Body Forces, 125 Geometric Nonlinearities, 126 4.3 Experimental Controls 127 Frequency Bandwidth, 128 4.4 Phase Space Measurements 128 Pseudo-Phase-Space Measurements, 129 4.5 Bifurcation Diagrams 130 4.6 Experimental Poincare Maps 130 Position Triggered Poincare Maps, 135 Construction of One-Dimensional Maps from Multidimensional Attractors, 139 Double Poincare Maps, 142 4.7 Quantitative Measures of Chaotic Vibrations 147 Frequency Spectra-FFT, 148 Fractal Dimension, 150 Lyapunov Exponents, 151 Probability or Invariant Distributions, 151 5 Criteria for Chaotic Vibrations 155 5.1 Introduction 155 5.2 Empirical Criteria for Chaos 156 Forced Oscillations of a Nonlinear Inductor: Duffing's Equation, 157 Forced Oscillations of a Particle in a Two-Well Potential: Duffing's Equation, 158 Rayleigh-Benard Convection: Lorenz Equations, 160 Forced Vibrations of a Two-Degree-of-Freedom Oscillator in a Two-Well Potential, 161 Forced Motions of a Rotating Dipole in Magnetic Fields: The Pendulum Equation, 162 Forced Oscillations of a Nonlinear RLC Circuit, 163 Harmonically Driven Surface Waves in Fluid Cylinder, 163 5.3 Theoretical Predictive Criteria 165 Period-Doubling Criterion, 166 Homoclinic Orbits and Horseshoe Maps, 172 Intermittent and Transient Chaos, 181 Chirikov's Overlap Criterion for Conservative Chaos, 183 Multiwell Potential Criteria, 187
xiv Contents 5.4 Lyapunov Exponents 191 Numerical Calculation of the Largest Lyapunov Exponent, 195 Lyapunov Spectrum, 200 6 Fractal Concepts in Nonlinear Dynamics 205 6.1 Introduction 205 Koch Curve, 206 Cantor Set, 207 The Devil's Staircase, 208 Fractal Dimension, 209 6.2 Measures of Fractal Dimension 214 Pointwise Dimension, 214 Correlation Dimension, 216 Information Dimension, 217 Relation Between Dimension Definitions and Lyapunov Exponents, 218 How Useful Is Fractal Dimension for Vibration Problems?, 222 6.3 Fractal Dimension of Strange Attractors 222 Discretization of Phase Space Variables, 223 Fractal Dimension of Poincare Maps, 226 Dimension Calculation from Single Time Series Measurement, 229 6.4 Optical Measurement of Fractal Dimension 235 An Optical Parallel Processor for the Correlation Function, 237 6.5 Fractal Basin Boundaries 242 Basins of Attraction, 242 Sensitivity to Initial Conditions: Transient Motion in a Two-Well Potential, 243 Fractal Basin Boundary: Forced Motion in a Two-Well Potential, 244 Homoclinic Orbits: A Criterion for Fractal Basin Boundaries, 246 Dimension of Basin Boundaries and Uncertainty, 250 Transient Decay Times: Sensitivity to Initial Conditions, 250 Other Applications, 253 Fractal Boundaries for Chaos in Parameter Space, 253 6.6 Complex Maps and the Mandelbrot Set 258
Contents xv Appendix A. Glossary of Terms in Chaotic and Nonlinear Vibrations 261 Appendix B. Appendix C. References Author Index Subject Index Numerical Experiments in Chaos B.I Logistic Equation Period Doubling B.2 Lorenz Equations B.3 Intermittency and the Lorenz Equation B.4 Henon Attractor B.5 Duffing's Equation: Ueda Attractor B.6 Two-Well Potential Duffing-Holmes Attractor B.7 Cubic Map (Holmes) B.8 Bouncing Ball Map (Standard Map) B.9 Circle Map: Mode Locking, Winding Numbers, and Fairey Trees B.10 Rossler Attractor: Chemical Reactions, Return Maps B.ll Fractal Basin Boundaries: Kaplan-Yorke Map B.12 Torus Maps Chaotic Toys C.I The Chaotic Elastica: A Desktop Chaotic Vibration Experiment C.2 The "Moon Beam" or Chaotic Buckling Experiment C.3 A Chaotic Double Pendulum or "Space Ball" C.4 Neon Bulb Chaotic Toy 269 269 270 271 271 272 272 273 273 274 275 276 277 279 280 281 284 285 287 301 305