Analytical approach to dynamic and vibration analysis of a spherical ball under contact stress

This paper presents a nonlinear model to illustrate the effect of contact stress in the vibration behavior of mechanical components, especially rotating systems. The problem is considered in the case of the vertical vibration of a sphere on a plate. The Hertzian contact theory is used to obtain the relationship between contact force and the deflection of the mass center of the sphere. Modeling the system by a mass and a nonlinear spring, the vibration equation of the mass center of the sphere is derived. The method of Lindstedt–Poincaré is implemented to solve the equation of motion, and obtain vibration characteristics under a compressive preload. The dependency of frequency on several parameters, such as initial applied force, initial amplitude of oscillation and the diameter of the sphere, is distinguished. Results show that increasing the initial applied force or the diameter of the ball raises the frequency, while increasing oscillation amplitude has an inverse effect. Finally, the accuracy and convergence of the solution are illustrated by comparison between different orders of approximation. Also, results are in good agreement with those extracted from numerical modeling.