Closed-Form Representations of Friction

The current developed friction models are basically based on the assumption of the normal forces exerted between the contact surfaces known in advance, less work has been done for closed-form (i.e., analytic) modeling in complex mechanical systems where the normal forces vary greatly over time. In this paper, the closed-form representations of friction forces in mechanical systems are newly derived in a way of dynamics. The Udwadia-Kalaba equation is first used to calculate the normal force exerted by the contact surface in mechanical system. The friction force is then calculated via existing friction models. Such closed-form expressions of friction forces contain both the magnitude and direction at any instant of time, even as the normal force is nonconstant. The novel representations offer an effective way for analytically expressing friction force in dynamical systems, which makes it possible for accurate simulation and control design of dynamical systems with non-negligible friction forces.