Introduction to Experimental Nonlinear Dynamics A Case Study in Mechanical Vibration

Introduction

We have seen that in order to design a close mechanical analogue of the equation of motion (developed in the previous chapter) a number of objectives have to be achieved. An initial approach might consist of allowing a small ball to roll on a grooved guide (Marion and Thornton, 1988). However, it is then difficult to monitor the motion of the ball (at least in an accurate manner) and significant slippage can occur. Suppose we wish to build a small cart or roller coaster. In this case it is relatively easy to measure the position of the cart based on the output of a rotational potentiometer attached to an axle. The rotary inertia of the cart can be minimized by keeping the cart small, and the a term can also be adjusted through the track geometry. To avoid slippage between the cart wheels and the track, a chain-sprocket system can be used. This does have the drawback of complicating the damping modeling but it will be seen that the overall damping in the model is quite small. Another advantage of the chain-sprocket guide is that it minimizes the possibility of the cart actually leaving the track during fast motions (Gottwald, Virgin, and Dowell, 1992).

We will also see that to replicate Duffmg's equation it is desirable to have a relatively shallow curve so as to minimize those nonlinear terms in the accurate equation of motion (5.38) that do not appear in the standard Duffing's equation (4.1). We do not wish to stray too far from the familiar form of Duffing's equation, although we realize that typical nonlinear features are by no means restricted to a narrow class of ordinary differential equations. The features to be described are actually quite generic and robust. The theoretical development in the previous chapter showed that these effects can be grouped together by consideration of a single nondimensional parameter a in the experiment. Equation (5.37) showed that a can be made smaller by reducing the vertical distance between the unstable equilibrium (hilltop) and the symmetrically positioned stable equilibria to reduce the vertical component of the acceleration.