Frictionless elastic contact of crowned roller: Approximate analytical calculation of compliance and contact area

Abstract

Formulas for Hertzian frictionless elastic contact are well-known and have been used for a long time. However, they are not directly applicable for a crowned roller, a cylindrical roller with symmetrically rounded corners, deflecting a half-space. For this punch shape, the approximate analytical method of Barber and Billings (Int J Mech Sci 1990; 32(12): 991–997) was adapted to compute the load–deflection curve and the contact boundary avoiding calculation of pressure distribution. Comparisons were performed with finite element method (FEM) simulations, an adaptive Hertzian quadratic punch surface approximation (AHA) and other published results. Good agreement of analytical load–deflection curves with FEM results was noted but the AHA results deviated significantly from the former two. The FEM contact area resembled a dog-bone shape, the analytical a bow-tie shape and the AHA, by design, an ellipse. Though the AHA contact area was closer to the FEM result numerically, the contact boundary aspect ratio of the analytical model was closer to the FEM. The analytical results simulating conditions from other publications agreed quantitatively for the load–deflection curves and qualitatively for the contact boundary. The non-linear spring characteristic of the crowned roller was quantitatively evaluated for a diverse range of geometry.