Normal vibrations and friction at a Hertzian contact under random excitation: Theory and experiments

Non-linear stochastic contact vibrations at a Hertzian contact are studied using the Fokker-Planck equation. The vibrations are excited either externally to the contact region by a white Gaussian random normal load, or within the contact region by a rough surface input. The statistics of the stationary response are obtained for each case. The parametric dependence of the normal motions and contact area on various contact parameters is investigated. 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. It is found that for a 5% probability of contact loss, a reduction in the mean friction force of approximately 9% is expected. Average friction measurements taken during continuous sliding are in agreement with the analysis. Changes in the probability density function of the friction force with sliding, as predicted by the analysis, are also shown to be in good agreement with measured friction data. The results are consistent with the adhesion theory of friction in both an instantaneous and an average sense; even in the presence of substantial fluctuations in the normal contact force.