Normal vibrations and friction at a hertzian contact under random excitation: Perturbation solution

Closed form solutions for the non-linear stochastic contact vibrations and friction of Hertzian contact systems are derived using a perturbation technique. The vibrations are excited either externally to the contact region by a white Guassian random normal load, or within the contact region by a rough surface input. Due to the non-linear Hertzian compliance, the mean normal contact compression under dynamic excitation is smaller than the static deflection in the absence of vibrations. One also finds a reduction in the average area of contact and, by implication, in the mean friction force. Expressions for the mean values of contact separation and friction force during steady sliding are presented directly in terms of the system and excitation parameters. The results are in good agreement with those previously calculated using the Fokker-Planck equation, which agreed well with instantaneous and average measured friction data.