Introduction to Experimental Nonlinear Dynamics A Case Study in Mechanical Vibration

This chapter will set out the motivation and define the context for the material contained in this book. The recurrent themes to be described should help to establish what this book is about (and what it isn't).

The Physical Context

• Mechanical Vibration

This book uses an aspect of traditional mechanics to provide a focus and solid context for the material development. The field of mechanical vibration provides a rich source of material at the interface between applied mathematics and practical design. Classical vibration is very well developed, especially for linear, low-order systems. Progress is being made in stochastics, control, and nonlinear effects. It is this latter aspect of vibration theory (and its experimental verification) that is pursued in this book.

• Modeling

In this book we are primarily interested in low-order, time-varying, deterministic, nonlinear, experimental, mechanical systems! During the modeling process we make a number of trade-offs between the desire for mathematical simplicity and practical usefulness (and phenomenological interest). Although the experimental paradigm developed is a discrete mechanical system, the extension to continuous systems becomes clear. Classical methods (e.g., Lagrange's equation) are used to derive governing equations, with the major modeling challenge presented by energy dissipation.

• Duffing's Equation

Throughout we will use Duffing's equation. Why focus so much on one type of system? The development of nonlinear dynamics is framed by intense scrutiny of certain archetypal systems, typically named after the researchers who first studied them: van der Pol, Duffing, Lorenz, Hénon, and Rossler. The nonlinear ordinary differential equation that has come to be known as Duffing's equation has particular importance in engineering, and indeed it was first considered by the German experimentalist Georg Duffing to study the hardening spring effect observed in many mechanical systems.

The specific double-well form of Duffing's equation, which provides the backbone of this book, offers compelling pedagogical insights. This is partly based on the fact that it is symmetric (about the origin) in a global sense, a feature nominally present in a variety of practical situations. But since the underlying equilibria are offset, it also subsumes a variety of types of asymmetric behavior and is globally bounded.