Non-holonomic dynamics of a ball moving inside a spherical cavity

Abstract

The ball type vibration absorbers are popular to be used at slender structures like masts or towers when a vertical space available and maintenance possibilities are limited. In general, the absorber consists of a cavity in which a heavy ball is rolling. In the paper, the response of a heavy ball rolling inside a semi-spherical cavity under horizontal kinematic excitation is investigated. The system with six degrees of freedom with three non-holonomic constraints is considered. The contact between the ball and the cavity surface is supposed to be perfect without any sliding. The system is strongly non-linear and, consequently, the full 3D mathematical model should be carried out and discussed. With respect to previous experiences the basic strategy declines the Lagrangian procedure including conventional constraints via Lagrangian multipliers. Instead of that the Gibbs-Appell formulation is used. The system has an auto-parametric character. The most important post-critical regimes are outlined and qualitatively evaluated in resonance domain. Numerical experiments have been performed when excitation frequency is slowly swept up and down. Some applications in civil engineering as a tuned mass damper used on slender structures is outlined and compared with a conventional pendulum damper. Strengths and weaknesses of both absorbers types are discussed.