VPOD 3DOF Vibratory Device

Nondimensional 3DOF Bouncing Ball Simulator

The Nondimensional 3DOF Bouncing Ball Simulator is a simple Matlab program meant to mimic the behavior of a bouncing ball on a vibratory device capable of sinusoidal motion in three degrees, such as the VPOD. The simulator uses a numerical method in which the equations of motion describing the ball's flight are determined from the state variables of the previous impact. The program uses a binary search to locate the time at which the ball's position in the x,z plane is equal to that of the oscillating bar. In short, it calculates the intersection of a parabola with a sinusoid, uses a simple impact model to compute the new state variables, and repeats this computation as many times as desired. The equations of motion for the oscillating bar are as follows:

$x(t)=A_{x}sin(\omega _{x}t+\phi _{x})\,$

$z(t)=A_{z}sin(\omega _{z}t+\phi _{z})\,$

$\theta (t)=A_{\theta }sin(\omega _{\theta }t+\phi _{\theta })\,$

where A_x, A_z, A_\theta represent the maximum x, z, and angular displacements of the bar's motion. The key input variables are the three position amplitudes, three frequencies, and three phase angles, as well as coefficients of restitution in the normal and tangential directions.

• It should be noted that for this particular version of the simulator, bar motion in the x direction will not have any effect on the behavior of the bar. Since the impact model assumes that the ball is bouncing on a frictionless surface, horizontal vibration of the bar will not produce changes in the ball's velocity. The option to move the bar horizontally is a remnant of a previous version, but will do no harm.

Download the simulator.