Trajectories of a ball moving inside a spherical cavity using first integrals of the governing nonlinear system

Analytical study of ball vibration absorber behavior is presented in the paper. The dynamics of trajectories of a heavy ball moving without slipping inside a spherical cavity are analyzed. Following our previous work, where a similar system was investigated through various numerical simulations, research of the dynamic properties of a sphere moving in a spherical cavity was carried out by methods of analytical dynamics. The strategy of analytical investigation enabled definition of a set of special and limit cases which designate individual domains of regular trajectories. In order to avoid any mutual interaction between the domains along a particular trajectory movement, energy dissipation at the contact of the ball and the cavity has been ignored, as has any kinematic excitation due to cavity movement. A governing system was derived using the Lagrangian formalism and complemented by appropriate non-holonomic constraints of the Pfaff type. The three first integrals are defined, enabling the evaluation of trajectory types with respect to system parameters, the initial amount of total energy, the angular momentum of the ball and its initial spin velocity. The neighborhoods of the limit trajectories and their dynamic stability are assessed. Limit and transition special cases are investigated along with their individual elements. The analytical means of investigation enabled the performance of broad parametric studies. Good agreement was found when comparing the results achieved by the analytical procedures in this paper with those obtained by means of numerical simulations, as they followed from the Lagrangian approach and the Appell-Gibbs function presented in previous papers.

Automated summary

The paper presents an analytical study on the behavior of a ball vibration absorber, analyzing the dynamics of trajectories of a heavy ball moving inside a spherical cavity. By neglecting energy dissipation and kinematic excitation, the study derived a governing system and defined three first integrals to evaluate trajectory types with respect to system parameters. The analytical investigation enabled broad parametric studies and showed good agreement with numerical simulations based on the Lagrangian approach and Appell-Gibbs function.