Dynamics of impacts with a table moving with piecewise constant velocity

Abstract

Dynamics of a ball moving in gravitational field and colliding with a moving table is considered. The motion of the limiter is assumed as periodic with piecewise constant velocity. It is assumed that the table moves up with a constant velocity and then goes down with another constant velocity. The Poincaré map describing evolution from an impact to the next impact is derived. Several classes of solutions are computed in analytical form.

Simple Model of Bouncing Ball Dynamics

“…We treat the ball as a material point and assume that the limiter's mass is so large that its motion is not affected at impacts. Dynamics of the ball from an impact to the next impact can be described by the following Poincaré map in nondimensional form [13] (see also Ref. [14] where analogous map was derived earlier and Ref.…”

“…Accordingly, we have decided to choose the limiter's periodic motion in a polynomial form to make analytical investigations of the dynamics possible. In our previous papers we have assumed displacement of the table as piecewise linear periodic function of time [9,10] as well as quadratic [11]. In this work we study dynamics for a cubic function of time Y c (T ): In Fig.…”

“…We approached this problem assuming a special motion of the table. Recently, we have considered several models of motion of a material point in a gravitational field colliding with a limiter moving periodically with piecewise constant velocity [9,10] and velocity depending linearly on time [11]. In the present work we study the model in which periodic displacement of the table is a cubic function of time, carrying out our project to approximate the sinusoidal motion of the table as exactly as possible but preserving possibility of analytical computations [12].…”

Simple model of bouncing ball dynamics: displacement of the table assumed as quadratic function of time

“…In our previous papers, we have assumed displacement of the table as piecewise linear periodic function of time [12][13][14][15]. In our recent work, preliminary results for function Y (T ) assumed as quadratic, Y q , and two cubic functions of time, Y c 1 and Y c 2 , have been obtained [16].…”

“…Recently, we have considered several models of motion of a material point in a gravitational field colliding with a limiter moving with piecewise constant velocity [12][13][14][15]. Moreover, we have proposed more realistic yet still simple models approximating sinusoidal motion of the table as exactly as possible but still preserving possibility of analytical computations [16].…”