Approximation of the integral of the asperity height distribution for the Greenwood—Tripp asperity contact model

The load carried by asperity contacts is a very important quantity when performing elastohydrodynamic analyses in the mixed-lubrication regime. The Greenwood—Tripp model for the contact of two nominally flat rough surfaces is traditionally used for the evaluation of these loads. In this model it is assumed that the asperity heights follow a Gaussian distribution, thus, the load carried by the asperities can be evaluated by the integration of a non-linear function that relates the surface separation with the asperity height distribution. In order to avoid the computational burden of integrating this function numerically, several approximations have been proposed in literature. In the current technical note, the authors examine the quality of two of these approximations, a power law approximation and a sixth-order polynomial approximation, proposed in research efforts for the lubrication analysis of piston rings. The lack of fit for these two approximations is identified and in turn a new exponential approximation is proposed with the coefficients derived via the method of least squares. This new approximation exhibits a better fit over the entire range of the tabulated values for the asperity height distribution integral provided by Greenwood and Tripp. The computational cost of this approximation is also found to be acceptable. Researchers can use this approximation with confidence in mixed-lubrication analyses.