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

Lawrence N. Virgin's Introduction to Experimental Nonlinear Dynamics is a unique book in that it treats an extremely mathematical subject from an experimental point of view. Virgin integrates the theory and the experiments very well. Novices to the field of nonlinear dynamics and chaos theory will find the book's introduction of concepts both easy to understand and presented in a physically meaningful manner. The book will also be useful for specialists in chaos and nonlinearity in the comparison of experiments with theoretical models. Virgin was trained in theoretical and applied mechanics and uses beautiful mechanical experiments as a platform from which to investigate nonlinear behavior and mathematical models.

In chapters 1 and 2, Virgin reviews linear vibration theory, including forced oscillations and resonance. In chapter 3 he introduces the concepts of phase space, Poincaré sections, and bifurcations in the context of a simple oscillator. The mathematical model of a particle in a two-well potential and the mechanical analog of a buckled structure are presented in chapters 4 and 5. Virgin takes the reader through the world of chaotic vibrations, using his experimental model of a two-well oscillator, in chapters 6 through 10. He introduces the ideas of nonlinear free oscillations, subharmonic behavior, autocorrelation, and Lyapunov exponents, as well as beautiful experimental fractal Poincaré sections with comparisons with numerical solutions. He also discusses the idea of escape from a potential well.

In chapter 11 Virgin describes another mechanical experiment based on a hardening spring, and he discusses the Japanese attractor of Yoshisuke Ueda with a pure cubic nonlinearity. The impact oscillator under a sharp bilinear discontinuity is another experiment that is analyzed in chapter 12. Virgin ends the book with some global bifurcation issues related to quasiperiodic motions, fractal basin boundaries, Melnikov theory, and global transient motions.

Virgin does a reality check on the theory and illustrates the experimental robustness of nonlinear phenomena. At the end of his book he also describes an electrical-circuit experiment, for those who would like to explore chaos with voltages instead of mechanical motions.

Virgin's writing is clear and concise. The book contains some mathematical equations, but the general tone is toward physical explanation with visual and graphical presentations. This book is recommended for the libraries of both students and researchers in nonlinear science.