Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

Abstract

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.

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

“…In this paper, the problem of brachistochronic motion of a variable-mass nonholonomic mechanical system has been solved. The paper represents the generalization of the results obtained in [8] to the general variable-mass systems with linear nonholonomic constraints and m degrees of freedom. Taking the first time-derivatives of quasi-velocities as control variables, the brachistochrone problem considered has been formulated as a singular optimal control task.…”

“…The variable-mass particles A and B , as well as the spring of stiffness c and free length l 0 , are interconnected by a lightweight mechanism of the “forks” type, which allows the distance AB true¯ = ξ ≠ const . to change. This example is a modified version of the example considered in [8]. Namely, modification involves the addition of the spring of stiffness c .…”

“…The available field-related literature can be separated into two groups. The first group includes [1][2][3][4][5], whereas the second covers [6][7][8]. Note that detailed literature overviews related to holonomic variable-mass mechanical systems is given in [9][10][11][12].…”

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

“…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.…”

“…To the best of the authors' knowledge, such problem has not been considered yet. The work is organized as follows: in Section 2 the brachistochrone problem of a variable mass nonholonomic mechanical system with two degrees of freedom is defined; in Section 3 the brachistochrone problem is formulated as the task of optimal control, which can be solved by scalar control, and numerical procedure for solving the obtained TPBVP based on the shooting method is presented; to show the obtained results, Section 4 gives an example of the mechanical system, which is a modification of the example from [9]; conclusion is contained in Section 5.…”