Brachistochronic motion of a nonholonomic variable-mass mechanical system in general force fields

Abstract

In this paper, the brachistochronic motion of a mechanical system composed of variable-mass particles is analysed. Workless (ideal) holonomic and linear nonholonomic constraints are imposed on the system. It is assumed that the system moves in an arbitrary field of known potential and nonpotential forces with prescribed both laws of the time-rate of mass variation of the particles and relative velocities of the expelled (or gained) masses. The first time-derivatives of quasi-velocities are taken as control variables. Using Pontryagin’s maximum principle and singular optimal control theory, the problem of brachistochronic motion of the nonholonomic variable-mass mechanical system is solved as a two-point boundary value problem. In addition, a discussion about the realization of control forces is given. The results are illustrated via an example.

Realizing brachistochronic planar motion of a variable mass nonholonomic mechanical system by an ideal holonomic constraint with restricted reaction

“…The present work has solved the problem of realizing brachistochronic planar motion of a nonholonomic variable mass mechanical system by means of an ideal holonomic constraint with restricted reaction. Considerations presented in this work rely on the work [8] and thus are a kind of continuation of mentioned study. The considered system has two degrees of freedom so that the motion can be realized by means of a single ideal holonomic constraint.…”

“…where δπ α are variations of independent quasi-coordinates, whereπ α = V α holds. Due to variations' independence, that is δπ α 0, and taking into account (7), (12), (13) and (14), and using the contravariant coordinates of metric tensor G αβ , after rearrangement the differential equations of motion for the considered system are obtained [8]:…”

“…Throughout literature it is possible to encounter works related to the brachistochronic motion of a material point, of both constant and variable mass [2][3][4][5][6][7], as well as works related to the brachistochronic motion of mechanical systems [8][9][10][11][12][13][14][15]. Regarding variable mass nonholonomic mechanical systems, there is not a lot od works on that subject, whether it is the application of other types of equations in that field, such as Kane's [16] or Hamilton's equations [17], or the control of such systems [8][9][10][11][12][13][14]18]. Pontryagin's maximum principle [19][20][21][22], as well as the optimal control theory [19][20][21] can be applied in solving the brachistochrone problem.…”

“…Pontryagin's maximum principle [19][20][21][22], as well as the optimal control theory [19][20][21] can be applied in solving the brachistochrone problem. Considerations in this work rely on [8]. The aim of the work is to realize the brachistochronic motion of a variable mass nonholonomic mechanical system with the help of an ideal holonomic constraint with restricted reaction, and therefore it represents a kind of continuation of work [8].…”

“…Considerations in this work rely on [8]. The aim of the work is to realize the brachistochronic motion of a variable mass nonholonomic mechanical system with the help of an ideal holonomic constraint with restricted reaction, and therefore it represents a kind of continuation of work [8]. To the best of the authors' knowledge, such problem has not been considered yet.…”