Introduction to experimental nonlinear dynamics. A case study in mechanical vibration.

*(English)*Zbl 0966.70001
Cambridge: Cambridge University Press. xvi, 256 p. (2000).

The book is devoted to one of the most interesting areas of science – to the nonlinear dynamics of mechanical systems. The nonlinear systems contrast sharply with linear systems, which do not suffer from bifurcations or from sensitivity to initial conditions, and for which the distinction between local and global behaviour has no meaning. The main feature of this book is that the study of nonlinear dynamics is based on the archetypal Duffing differential equation. The specific energy form of Duffing equation – two wells separated by a hill top – allows to observe a great variety of phenomena such as chaos, fractal basin boundaries, unpredictability etc.

In the book, the author presents a very interesting and witty model which displays the full spectrum of nonlinear behaviour. The model consists of plexiglass sheets and simulates the hill top at the origin and two symmetric minima. This shape mimics the potential energy of Duffing differential equation in a uniform gravitational field. The point mass is represented by a short-wheel base cart with teflon wheels with small grooves to reduce the damping and antislip mechanism. The model has also the starting gate, the impact barrier, and the forcing mechanism. The author considers nonlinear electric circuits and continuous systems, and describes a strong correspondence between electrical elements (resistance, capacitance, and inductance) and mechanical elements (damping, stiffness, and mass).

The remainder of the book deals with continuous systems such as beams and plates which, being excited close to their first natural frequency, display behaviour dominated by the lowest mode. So, Galerkin method may be used to reduce the governing partial differential equations to a set of coupled nonlinear ordinary differential equations, which can then be analyzed using standard mathematical techniques of nonlinear mechanics.

This book can be useful to students and engineers to study the subject topics, and to teachers to popularize the wonderful world of nonlinear mechanics.

In the book, the author presents a very interesting and witty model which displays the full spectrum of nonlinear behaviour. The model consists of plexiglass sheets and simulates the hill top at the origin and two symmetric minima. This shape mimics the potential energy of Duffing differential equation in a uniform gravitational field. The point mass is represented by a short-wheel base cart with teflon wheels with small grooves to reduce the damping and antislip mechanism. The model has also the starting gate, the impact barrier, and the forcing mechanism. The author considers nonlinear electric circuits and continuous systems, and describes a strong correspondence between electrical elements (resistance, capacitance, and inductance) and mechanical elements (damping, stiffness, and mass).

The remainder of the book deals with continuous systems such as beams and plates which, being excited close to their first natural frequency, display behaviour dominated by the lowest mode. So, Galerkin method may be used to reduce the governing partial differential equations to a set of coupled nonlinear ordinary differential equations, which can then be analyzed using standard mathematical techniques of nonlinear mechanics.

This book can be useful to students and engineers to study the subject topics, and to teachers to popularize the wonderful world of nonlinear mechanics.

Reviewer: Yuri N.Sankin (Ul’yanovsk)

##### MSC:

70-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems |

70-05 | Experimental work for problems pertaining to mechanics of particles and systems |

70K40 | Forced motions for nonlinear problems in mechanics |

70K55 | Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics |

70K50 | Bifurcations and instability for nonlinear problems in mechanics |