The paper considers the brachistochronic motion of a variable mass
nonholonomic mechanical system [3] in a horizontal plane, between two
specified positions. Variable mass particles are interconnected by a
lightweight mechanism of the ?pitchfork? type. The law of the time-rate of
mass variation of the particles, as well as relative velocities of the
expelled particles, as a function of time, are known. Differential equations
of motion, where the reactions of nonholonomic constraints and control forces
figure, are created based on the general theorems of dynamics of a variable
mass mechanical system [5]. The formulated brachistochrone problem, with
adequately chosen quantities of state, is solved, in this case, as the
simplest task of optimal control by applying Pontryagin?s maximum principle
[1]. A corresponding two-point boundary value problem (TPBVP) of the system
of ordinary nonlinear differential equations is obtained, which, in a general
case, has to be numerically solved [2]. On the basis of thus obtained
brachistochronic motion, the active control forces, along with the reactions
of nonholonomic constraints, are determined. The analysis of the
brachistochronic motion for different values of the initial position of a
variable mass particle B is presented. Also, the interval of values of the
initial position of a variable mass particle B, for which there are the TPBVP
solutions, is determined.
Nonlinear Dynamic Analysis and Optimization of Closed-Form Planetary Gear System