Dynamic behavior and stability of a ball rolling inside a spherical surface under external excitation

Abstract eng:
The area of tuned mass dampers is a wide field of inspiration for theoretical studies in nonlinear dynamics and dynamic stability. They attempt to estimate their behavior and reliability of their function. 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 mathematical model using the Appel-Gibbs function of acceleration energy is developed and discussed. Comparison with conventional way via Lagrangian procedures is given. The system has an auto-parametric character and therefore semi-trivial solutions and their dynamic stability is analyzed. 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. Results obtained by means of semi-analytical investigation and numerical simulation have been evaluated and physically interpreted. 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.

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