Nonlinear Dynamical Analysis of the “Power Ball”

Abstract

The gyroscopic exercise tool called the “Power Ball,” used to train the antebrachial muscle, is focused on. The basin of attraction of the synchronous rolling motion in the state space of initial condition is investigated. The reduced model governing the synchronous rolling motion is used and its averaged equation is deduced. The first integral for the dynamical behavior of the synchronous rolling motion occurring in the power ball is obtained. The separatrix, which identifies the basin of attraction of the synchronous rolling motion, is derived, and the ranges of initial precession angle and the initial spin angular velocity for realizing the synchronous rolling motion are clarified. These theoretically obtained results are then experimentally confirmed. Furthermore, the influences of parameters to the basin of attraction are also clarified.