Numerical analysis of ball motion across a two-zone turntable with periodic angular velocities

Abstract

A two-zone turntable is a turntable with two independently rotating concentric parts or zones, so that different zones can have different angular velocities. This study analyzed numerically the movement of a homogeneous ball across a two-zone horizontally flat turntable with sinusoidal angular velocities per zone. The ball is assumed to roll without slipping throughout the entire turntable, except for very short instants when the ball moves from one zone to another. The ball's equations of motion across the turntable were first obtained analytically. Then, numerical analysis was done on the obtained set of differential equations. In general, the resulting ball paths depend on the zone's angular velocity and on the ball's initial conditions. For the case where turntable angular velocities are of the form Ω₁(t) = ϵ sin(ωt) + k and Ω₂(t) = ϵ cos(ωt) + k, resulting ball paths generally do not exhibit periodic patterns except when k = 0 or ϵ = 0.