10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

Radulović, Radoslav; Obradović, Aleksandar; Jeremić, Bojan

doi:10.5937/fmet1403199r

Cited by 5 publications

(10 citation statements)

References 10 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%

“…Solving the system of equations (8) and (10), the reactions of nonholonomic constraints, as well as control forces, are defined to realize the brachistochronic motion, as a function of defined quantities of state and corresponding derivatives…”

Section: Description Of the Dynamic Model Of Nonlinear Nonholonomic Smentioning

confidence: 99%

“…The procedure of numerical determination of the unknown parameters consists of 'shooting' the end-state coordinates (15), at the known initial state (14) and (23). The application of shooting method requires an estimate for the interval of parameters' values being determined [10]. On the basis of estimates for the values of the conjugate vector coordinates   0 ξ λ t , given in (24) , it can be claimed that all solutions to the corresponding two-point boundary value problem are certainly located within the interval given, thereby the global minimum time in the brachistochronic motion of the system.…”

Section: Brachistochronic Motion As the Problem Of Optimal Controlmentioning

confidence: 99%

“…Differential equations of motion of the system (10), that is, the reactions of nonholonomic constraints and control forces as well (11), are obtained in accordance with restrictions (1) and (2).…”

Section: Dynamic Conditions For Realizing the Brachistochronic Motionmentioning

confidence: 99%

“…Taking this into account, the necessary dynamic conditions for realizing the motion of the system in accordance with restrictions (1) and (2) [9,10], based on the Coulomb friction laws, are that intensities of the interaction forces between points A and B of the system and the horizontal plane of motion should not exceed the corresponding Coulomb forces of sliding friction.…”

Section: Dynamic Conditions For Realizing the Brachistochronic Motionmentioning

confidence: 99%

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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%

Section: Description Of the Dynamic Model Of Nonlinear Nonholonomic Smentioning

confidence: 99%

Section: Brachistochronic Motion As the Problem Of Optimal Controlmentioning

confidence: 99%

Section: Dynamic Conditions For Realizing the Brachistochronic Motionmentioning

confidence: 99%

Section: Dynamic Conditions For Realizing the Brachistochronic Motionmentioning

confidence: 99%

See 3 more Smart Citations

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

Radulović¹,

Zeković²,

Lazarević³

et al. 2014

FME Transaction

Self Cite

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A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

Radulović

Šalinić

Obradović

et al. 2016

Mathematics and Mechanics of Solids

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A new approach for the determination of the global minimum time for the case of the brachistochronic motion of the Chaplygin sleigh is presented. The new approach is based on the use of the shooting method in solving the corresponding two-point boundary-value problem and defining either the crossing points of surfaces or the crossing points space of curves in a three-dimensional space of two costate variables and the time of the brachistochronic motion of the sleigh. A number of examples for multiple extremals of the Chaplygin sleigh brachistochrone problem are provided. In these examples, the global minimum is the solution to which the minimum time of motion corresponds.

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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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10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

Cited by 5 publications

References 10 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

A new approach for the determination of the global minimum time for the Chaplygin sleigh brachistochrone problem

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

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