7 - Modeling Free Vibration

Abstract

tions occur about an equilibrium point. The oscillations may be periodic such as the motion of a pendulum or random such as the movement of a tire on a gravel road. More often, vibration is undesirable, wasting energy and creating unwanted sound—noise. Careful designs usually minimize unwanted vibrations. This chapter illustrates the example of the classical pendulum as a mathematical model derived, grounded in, and verified against experimental results, and the process of obtaining qualitative information about its behavior. The chapter demonstrates the behavior of linear oscillators in several domains, and draws some distinctions between the behaviors exhibited by linear and nonlinear models. The concepts of linearity, dimensional consistency, scaling, and some basic ideas of second-order differential equations are used for this purpose. In terms of the behavior of the pendulum itself, the period of the linear model depends only on the pendulum's properties and not on its amplitude of vibration, as is the case for nonlinear models wherein the amplitude is large. For both, the two-mass pendulum and a predator-prey population system, the period of the vibrating system is sensitive to properties of that system—especially for the two-mass pendulum, for which instability occurs for certain combinations of masses.