Nonlinear dynamic analysis of a rattleback subjected to harmonic excitation

Abstract

A rattleback is a rigid, semi-elliptical toy that exhibits intriguing behavior. When spun in one direction on a flat surface, it rapidly tilts and reverses its spin direction. The perplexing behavior of the rattleback originates from an asymmetry in its mass distribution, leading to a misalignment between the principal axes of inertia and the axis of curvature. This study introduces a novel approach to determine the dynamic response of a rattleback on a harmonically excited platform by employing Kane’s model. Kinematic equations are derived under the assumption of rolling without slipping using Kane’s model, while dynamical equations are obtained using the Newtonian approach. Furthermore, the explicit Runge-Kutta fourth-order (RK4) method is applied to numerically solve the system of nonlinear equations under various initial conditions. Owing to the misalignment between the principal axes of inertia and curvature, spin motion is coupled with rolling and pitching. The vibration of the platform induces the rattleback’s pitching motion, which, through the coupling between pitch and spin, results in the rotation of the rattleback in the desired direction. Additionally, the increased energy dissipation reduces the reversal frequency, with odd-numbered reversals being more robust during the initial clockwise spins of 5 rad/s. For a given set of rattleback parameters, the initial anti-clockwise spin of 0.5 rad/s exhibits no reversals, with the spin rate steadily increasing with minimal oscillation in the angle between the vertical axes of the ellipsoidal surface and the flat surface. This phenomenon can be used for energy harvesting, since the platform vibrations are coupled with spin motion.