An Overview: Rolling of a Heavy Ball on Curvilinear Paths and Surfaces of the Basis of Nonlinear Dynamics of Radial and Spherical Ball Bearings in Machine Systems

Abstract

An overview of the derived nonlinear differential equations of the rolling of a heavy ball on curvilinear stationary and rotating plane curves, as well as on curved surfaces, is given. Non-linear phenomena are shown, such as bifurcation and a given series of phase portraits of nonlinear dynamics with a corresponding set of singular points and triggers of coupled singularities and a representation of the stratification of phase trajectories in the formation of homoclinic orbits. A special class of nonlinear dynamics of rolling a heavy ball on coordinate surfaces is represented by a system of nonlinear differential equations and corresponding cyclic and first integrals. New analytical expressions of the rolling ball collision theory are presented, as the basis of vibro-impact dynamics in vibro-impact systems containing rolling balls. A special new contribution of the lecturer will be shown by the results of research into the nonlinear dynamics of a heavy ball rolling: a* along a stationary circular path, while in contact with a moving circular path; b* on a stationary spherical surface, and at the same time it is in contact with a moving circular spherical surface. These theoretical results were applied to show the nonlinear dynamics of balls in radial ball bearings, as well as in spherical ball bearings, which are important in the study of nonlinear dynamics of complex machine systems.