Analysis of minimum required sliding friction coefficient in the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

Abstract

This paper analyzes the problem of brachistochronic planar motion of a mechanical system with nonlinear nonholonomic constraint. The nonholonomic system is represented by two Chaplygin blades [3,4,5,6] of negligible dimensions, which impose nonlinear constraint in the form of perpendicularity of velocities. The brachistrochronic planar motion is considered, with specified initial and terminal positions, at unchanged value of mechanical energy during motion. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are obtained on the basis of general theorems of mechanics [8]. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems [9], where a subsequent physical interpretation of the multipliers of constraints is required. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved as simple a task of optimal control as possible in this case [6,7] by applying the Pontryagin maximum principle [1]. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved [2]. Numerical procedure for solving the two-point boundary value problem is performed by the method of shooting. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are defined. Using the Coulomb friction laws [8,9], a minimum required value of the coefficient of sliding friction is defined [10], so that the considered system is moving in accordance with nonholonomic bilateral constraints.