Coulomb law in generalized differential form in problems of dynamics of rigid bodies with combined kinematics

Abstract

We suggest an exact integral model of sliding and spinning friction constructed under the assumption that the Coulomb law in generalized differential form holds for the surface element in the interior of the contact spot.

We show that, just as in the case of the classical Coulomb law in differential form, the friction force is directed oppositely to the relative sliding velocity but, compared with the results obtained in [3], the expressions for the friction force and torque contain additional polynomial terms.

To avoid using integral representations in equations of motion, we construct models of friction based on the first- and second-order Padé expansions of the corresponding exact integral model. These models can be viewed as generalized rheological two-dimensional first- and second-order models of friction, because, when solving actual problems, their coefficients can be determined experimentally. Moreover, compared with the results of [3], the second-order model is determined by the same number of coefficients as the first-order model. This fact simplifies the use of the second-ordermodel when solving dynamic problems.

As an example of use of the suggested models, we study the dynamics of torsional vibrations of an elastically fixed cylindrical rod one of whose ends rests on an infinite band moving at a constant velocity.