Realizing brachistohronic motion of a variable mass body by centrodes

Abstract

The paper considers realization of the brachistochronic motion of a mechanical system, composed of free body and variable mass material points, by means of an ideal constraint in the form of the centrodes. It is assumed that the system performs planar motion in an arbitrary field of forces and that it has three degrees of freedom. In addition, the laws of the time-rate of mass variation of the material points, as well as relative velocities of the expelled particles, respectively, are known. Constraint reactions of the centrodes are expressed in the function of the generalized forces. Applying Pontryagin's maximum principle and singular optimal control theory, the problem of brachistochronic motion is solved as a two-point boundary value problem (TPBVP). The considerations are illustrated via an example, where it is examined how the change in the initial energy of the system affects the normal reaction of the connection and thus the coefficient of rolling friction.