On the problem of a heavy homogeneous ball rolling without slipping over a fixed surface of revolution

Abstract

The paper considers a heavy homogeneous ball rolling without slipping on the outside of a real rough fixed surface of revolution, which is generated by rotating a plane curve around a vertical axis. By applying the Coulomb sliding friction law, the position is established at which slipping occurs during rolling down the surface, and thereafter this mechanical system ceases to be holonomic. Dynamic differential equations of motion are obtained using general theorems of mechanics. The paper presents the procedure of determining the tangential and normal component of the reaction of constraint depending on the height of the contact point between the ball and the surface. On the basis of the initial total mechanical energy of the system and the value of Coulomb friction coefficient, the condition is determined to prevent the ball from slipping, as well as the height interval in which the considered system behaves as a nonholonomic system. The procedure is illustrated by examples of surfaces generated by rotating segments of the circular arc, line and parabola. In the last example there is not a closed-form solution, so that numerical integration of a corresponding Cauchy problem is performed.