Introduction to Experimental Nonlinear Dynamics A Case Study in Mechanical Vibration

Introduction

So far, we have seen the appearance of essentially linear behavior for relatively small-amplitude forced oscillations together with the occurrence of (local) instability phenomena and subharmonic oscillations. The main features were a growth of amplitude with forcing magnitude and proximity to resonance and a response frequency the same as the input (forcing) frequency - features not dissimilar to the purely linear oscillator. The appearance of hysteresis signaled the enhanced role played by nonlinearity and initial conditions. However, considered locally from a topological viewpoint, all these responses can still be classified as periodic attractors, and in many important ways they are not substantially different from their point attractor counterparts in dissipative, gradient systems. Predicting the future behavior in these cases is relatively easy. But for nonlinear dynamical systems (flows) that exist in a phase space of three or more dimensions thoroughly more complicated and less predictable behavior becomes possible (Lorenz, 1963).

Chaos

It is the fascinating (and universal) nature of chaos that will be the main focus of attention in this chapter. The discussion will be somewhat constrained to the types of behavior exhibited directly by the experimental system, with a focus on invariant measures. A number of excellent books on chaos are available. A sample includes those covering theoretical (numerical) approaches (Guckenheimer and Holmes, 1983; Thompson and Stewart, 1986; Wiggins, 1990; Marek and Schreider, 1991; Ott, 1993), experimental aspects (Moon, 1992; Tufillaro Abbot, andReilly, 1992), and general treatments (Jackson, 1989;Mullin, 1993). This subject has reached a sufficient level of maturity that there are even books using pedagogical approaches (Abraham and Shaw, 1982; Strogatz, 1994; Baker and Gollub, 1996) and more general expositions for the general public (Gleick, 1987; Stewart, 1989).

In keeping with the progression of the previous chapters we introduce further strengthening of the external driving, thus encouraging significant nonlinear effects. It will be shown later that a progression toward chaotic behavior follows some very generic routes, but since many of the subtle interactions are of a global nature, they will be left until later (Dowell and Pezeshki, 1986).