Introduction to Experimental Nonlinear Dynamics A Case Study in Mechanical Vibration

Introduction

In this chapter we go back and reexamine the transition between the periodic behavior described in Chapter 8 and the thoroughly nonlinear chaotic behavior described in Chapter 9. This is an interesting aspect of the double-well oscillator (or indeed any system with a hilltop): Trajectories, initially contained within a single potential energy well, might escape over a nearby local maximum (hilltop). In the doublewell case, this means that the motion may traverse through to the adjacent well. Clearly, this is a situation completely alien to the confined, parabolic-well, linear oscillator. Dynamical systems characterized by this possibility of escape from a local potential energy well occur in many physical problems including a rigid-arm pendulum passing over its inverted equilibrium position (Baker and Gollub, 1996), snap-through buckling in arch and shell structures (Bazant and Cedolin, 1991), capsizing of ships (Virgin, 1987), and the toppling of rigid blocks (Virgin, Fielder, and Plaut, 1996).

The escape of trajectories from a local minimum of an underlying potential energy function is essentially a transient phenomenon. Given a single-degree-of-freedom system at rest in a position of stable equilibrium, it is often desirable to find the range of harmonic excitation that causes the subsequent motion to overcome an adjacent barrier defined by the limit of the catchment region (basin of attraction) surrounding the minimum. Escape occurs as the motion within the potential well grows “large enough.” This is clearly more likely to occur when the forcing magnitude is “large” in relation to some system characteristic. However, even in linear dynamics, the response of a sinusoidally forced system will be magnified close to resonance as we have seen. In nonlinear systems, the size of the basins of attraction surrounding an attractor depend crucially on certain system parameters. Nonstationary changes are incorporated in this chapter to simulate quasi-steady escape.

First, the unforced system is used as an introduction to escape based on initial conditions. Second, a slowly evolving harmonic excitation is applied to the system. The evolution is achieved by changing the forcing amplitude or frequency very slowly, either in an increasing or decreasing manner. In this way transients are minimized such that the evolving trajectory remains “close” to the underlying steady-state solution.