Analysis 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 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. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems, 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 by applying the Pontryagin maximum principle. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved. 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, a minimum required value of the coefficient of sliding friction is defined, so that the considered system is moving in accordance with nonholonomic bilateral constraints

The Brachistochronic Movement of a Material Point in the Horizontal Vector Field of a Mobile Fluid

“…Thus, comparing the obtained results, we can conclude that function ( ) y x found with the plus sign in formula (18) gives the maximum value to functional (7), and the function with the minus sign is the minimum value of all motion paths that correspond to the formulation of the presented variational problem.…”

“…(the start and finish points lie on axis ). OX 1) First, in the formula (18), we consider the case with the plus sign.…”

“…We use formula (7) to calculate it with the selected data of the studied problem. In other words, function ( ) f x is selected by formula (15), function ( ) y x is chosen by formula (18)…”

“…As a result of calculating integral (7), we obtain 2.46 T (units of time). 2) Now consider the case with the minus sign in formula (18).…”

“…Therefore, in this case, only five members of the Taylor series should be taken into account without losing the accuracy of the approximation of the desired function ( ) y x . Now we calculate the time of the boat moving along this trajectory, obtained with the minus sign in front of the integral in formula (18). After calculations we get a smaller result:…”