On the brachistochronic motion of the Chaplygin sleigh

Šalinić, Slaviša; Obradović, Aleksandar; Mitrović, Zoran; Rusov, Srdjan

doi:10.1007/s00707-013-0878-2

Cited by 12 publications

(15 citation statements)

References 11 publications

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“…Differential equations of motion will be developed based on the general theorems of dynamics [6,8,10], that is, by applying the theorem of change in momentum of a mechanical system, as well as the theorem of change in moment of momentum of a mechanical system for the moving point A,…”

Section: Description Of the Dynamic Model Of Nonlinear Nonholonomic Smentioning

confidence: 99%

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

Radulović¹,

Zeković²,

Lazarević³

et al. 2014

FME Transaction

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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

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Section: Description Of the Dynamic Model Of Nonlinear Nonholonomic Smentioning

confidence: 99%

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

Radulović¹,

Zeković²,

Lazarević³

et al. 2014

FME Transaction

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show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Jeremić

Radulović

Zorić

et al. 2019

Filomat

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The paper considers realization of the brachistochronic motion of a nonholonomic mechanical system, composed of variable mass particles, by means of an ideal holonomic constraint with restricted reaction. It is assumed that the system performs planar motion in an arbitrary field of forces and that it has two degrees of freedom. In addition, the laws of the time-rate of mass variation of the particles, as well as relative velocities of the expelled and gained particles, respectively, are known. Restricted reaction of the holonomic constraint is taken for the scalar control. 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). Since the reaction of the constraint is restricted, different types of control structures are examined, from singular to totally nonsingular. The considerations are illustrated via an example.

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“…The configuration of the considered system relative to the system Oxyz is defined by a set of Lagrangian coordinates (q 1 , q 2 , q 3 , q 4 ) , where q 1 = x and q 2 = y are Cartesian coordinates of the point A, q 3 = ϕ is the angle between the axis Ox and the axis Aξ, whereas q 4 = ξ is the relative coordinate of the variable mass point B relative to the non-stationary coordinate system. In accordance with the restriction of motion of the points A and B of the system, homogeneous nonholonomic constraints can be written in the following form [3,4] (1.1)ẋ cos ϕ +ẏ sin ϕ = 0, −ẋ sin ϕ +ẏ cos ϕ + ξφ = 0.…”

Section: Introductionmentioning

confidence: 99%

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

Jeremić

Radulović

Obradović

2016

Theor appl mech (Belgr)

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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.

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On the brachistochronic motion of the Chaplygin sleigh

Cited by 12 publications

References 11 publications

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

Analysis the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

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

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

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