Approximate modelling of resulting dry friction forces and rolling resistance for elliptic contact shape

There are many examples of mechanical systems with non-point friction contacts (billiard ball, Thompson top, wobblestone, electric polishing machine, the wobblestone, the Celtic stone), where the assumption of one-dimensional dry friction model do not necessarily lead to satisfactory accuracy of the numerical simulation. Moreover the rolling resistance often plays an important role in such systems. The paper is devoted to the problem of developing an approximate coupled model of resulting dry friction force and moment as well as rolling resistance, suitable for fast numerical simulation of rigid bodies with friction contacts, i.e. allowing to avoid the space discretization. An integral model of dry friction components is built under assumption of classical Coulomb friction law and fully developed sliding on the contact area of general shape and arbitrary contact pressure distribution. Then the special model of stress distribution over the elliptic contact area is developed, being a kind of generalization of Hertzian normal stress distribution, with addition of special distortion related to the rolling resistance. Finally some original approximate models of friction force and moment are proposed, based on Padé approximants and their generalizations as well as in the form of piecewise polynomial functions.