Shaping the trajectory of the billiard ball with approximations of the resultant contact forces

Abstract

This paper presents mathematical models of the contact pressure distribution on a circular contact area and the corresponding rolling resistance. Hertzian pressure distribution is distorted in a special way in order to move rolling centre outside the geometrical centre of the contact area. With the assumption of fully developed sliding and classical Coulomb friction law on each element of the contact, integral models of the total friction force and moment reduced to the contact centre are given. In order to improve the convenience of use of the contact models in numerical simulations of rigid body dynamics and decrease their computational cost, special approximations of the integral models of friction force and moment are proposed. Moreover, special modifications of the corresponding expressions for friction forces and rolling resistance are proposed, which allows avoiding their singularities for vanishing relative motion of the contacting bodies. The application of the proposed contact models in mathematical modelling of a rigid ball rolling and sliding over a deformable table is presented. Furthermore, possibilities of use of the developed simulation models in shaping the billiard ball's trajectory are presented.