Introduction

  Consider the motion of a ball bouncing on a periodically vibrating table. The bouncing ball system  is illustrated in Figure 1.1 and arises quite naturally as a model problem in several engineering applications. Examples include the generation and control of noise in machinery such as jackhammers, the transportation and separation of granular solids such as rice, and the transportation of components in automatic assembly devices, which commonly employ oscillating tracks. These vibrating tracks are used to transport parts much like a conveyor belt [1].

Assume that the ball's motion is confined to the vertical direction and that, between impacts, the ball's height is determined by Newton's laws for the motion of a particle in a constant gravitational field. A nonlinear force is applied to the ball when it hits the table. At impact, the ball's velocity suddenly reverses from the downward to the upward direction (Fig. 1.1).

  
Figure 1.1: Ball bouncing on an oscillating table.

The bouncing ball system is easy to study experimentally [2]. One experimental realization of the system consists of little more than a ball bearing and a periodically driven loudspeaker with a concave optical lens attached to its surface. The ball bearing will rattle on top of this lens when the speaker's vibration amplitude is large enough. The curvature of the lens is chosen so as to help focus the ball's motion in the vertical direction.

Impacts between the ball and lens can be detected by listening to the rhythmic clicking patterns produced when the ball hits the lens. A piezoelectric film, which generates a small current every time a stress is applied, is fastened to the lens and acts as an impact detector. The piezoelectric film  generates a voltage spike at each impact. This spike is monitored on an oscilloscope, thus providing a visual representation of the ball's motion. A schematic of the bouncing ball machine is shown in Figure 1.2. More details about its construction are provided in reference [3].

  
Figure 1.2: Schematic for a bouncing ball machine.

The ball's motion can be described in several equivalent ways. The simplest representation is to plot the ball's height and the table's height, measured from the ground, as a function of time. Between impacts, the graph of the ball's vertical displacement follows a parabolic trajectory as illustrated in Figure 1.3(a).

  
Figure 1.3: Simple periodic orbit of a bouncing ball: (a) height vs. time, (b) phase space (height vs. velocity), (c) impact map (velocity and forcing phase at impact). (Generated by the Bouncing Ball program.)

The table's vertical displacement varies sinusoidally. If the ball's height is recorded at discrete time steps,

 

then we have a time series  of the ball's height where is the height of the ball at time .

Another view of the ball's motion is obtained by plotting the ball's height on the vertical axis, and the ball's velocity on the horizontal axis. The plot shown in Figure 1.3(b) is essentially a phase space  representation of the ball's motion. Since the ball's height is bounded, so is the ball's velocity. Thus the phase space picture gives us a description of the ball's motion that is more compact than that given by a plot of the time series. Additionally, the sudden reversal in the ball's velocity at impact (from positive to negative) is easy to see at the bottom of Figure 1.3(b). Between impacts, the graph again follows a parabolic trajectory.

Yet another representation of the ball's motion is a plot of the ball's velocity and the table's forcing phase at each impact. This is the so-called impact map  and is shown in Figure 1.3(c). The impact map goes to a single point for the simple periodic trajectory shown in Figure 1.3. The vertical coordinate of this point is the ball's velocity at impact and the horizontal coordinate is the table's forcing phase. This phase, , is defined as the product of the table's angular frequency, , and the time, t:

 

where T is the forcing period. Since the table's motion is -periodic in the phase variable , we usually consider the phase ,  which means we divide by and take the remainder:

 

A time series, phase space, and impact map plot are presented together in Figure 1.4 for a complex motion in the bouncing ball system. This particular motion is an example of a nonperiodic orbit known as a

  
Figure 1.4: ``Strange'' orbit of a bouncing ball: (a) height vs. time, (b) phase space, (c) impact map. (Generated by the Bouncing Ball program.)

strange attractor . The impact map, Figure 1.4(c), is a compact and abstract representation of the motion. In this particular example we see that the ball never settles down to a periodic motion, in which it would impact at only a few points, but rather explores a wide range of phases and velocities. We will say much more about these strange trajectories throughout this book, but right now we turn to the details of modeling the dynamics of a bouncing ball.