Mechanical Nonlinear Oscillations Using a Hertzian-Type Restoring Force

This paper examines the generic case of nonlinear mechanical oscillation under the influence of Hertzian-type restoring forces, a model relevant to phenomena involving elastic contact. The study addresses the complexity of strongly nonlinear systems by focusing on the differential equation governing the oscillation of a rigid sphere interacting with an elastic half-space, which includes a full series expansion to account for large deformations. Since no closed-form solution exists for the amplitude-dependent oscillation period, a new approximate analytical approach is introduced. This method preserves the system’s dominant Hertzian scaling while incorporating higher-order corrections through an averaged factor. For amplitudes where the deformation is less than or equal to the sphere’s radius, this approximation is nearly identical to the numerical solution. For larger amplitudes, the accuracy is further enhanced by introducing a semi-empirical linear adjustment to the relative error. This framework provides a reliable analytical description of the system’s behavior, offering a useful tool for theoretical studies and comparison with numerical results.