The dynamics of a bouncing superball with spin

Abstract

When a superball is thrown forwards but with backspin, it is observed to reverse both direction and spin for a few bounces before settling to bouncing motion in one direction. The bouncing ball is modelled by a two-dimensional iterated map in terms of the horizontal velocity and spin immediately after each bounce. The asymptotic motion of this system is easily determined. However, of more interest is the transient behaviour. The two-dimensional linear map is reduced to a one-dimensional nonlinear map and this map is used to determine the number of direction and spin reversals that can occur for given initial conditions. The model is then generalised to describe a ball bouncing up and down a single step or successively down a staircase.

The bouncing motion of a superball between a horizontal floor and a vertical wall

“…In the direction tangential to the wall it is assumed, following Cross [3] and Aston and Shail [1], that the tangential velocity of the ball at the point of contact P w with the wall is reversed and reduced in magnitude by a factor α w . This condition gives…”

“…It is recognised that the model introduced in [3] which was subsequently used in [1] and the present paper is an oversimplification of the physics of superball impact. Thus, the tangential coefficient of restitution α is known not to be constant, but to depend on factors such as the speeds, the angles of incidence and the angular velocities of the bodies in collision (see, for example, Cross [3], [4], Labous et al [5], Dong and Moys [6], Sondergaard et al [7], although we note that the latter two studies were concerned with steel balls, not superballs).…”

“…In a previous paper [1] two of the present authors considered in detail the mechanics of a superball bouncing back and forth on a rough horizontal plane. Reversals in direction in the horizontal motion of the ball result from the application of a tangential law of restitution at the point of impact of the ball and the plane.…”

“…Garwin's model was modified by Cross [3] who employed a tangential coefficient of restitution α satisfying 0 < α < 1, with the horizontal velocity of the point of impact of the ball being reversed and reduced in magnitude by a factor of α in the impact. Further details of the physics of this model are given in [1], together with references to other theoretical and experimental work.…”

“…Stronge et al [9] model the collision by again considering the elastic impact region, which they represent by a deformable particle, the remainder of the system being treated as a rigid body. A very different approach to collision dynamics is that of Bibó et al [10], who construct a mechanical model of a ball which can exhibit the back and forward bouncing studied in [1]. Basically they consider the ball to have a rigid core attached by torsion springs to an outer casing, each component being capable of rotation about a common axis.…”