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

U ovom radu analizira se problem brahistohronog ravnog kretanja mehaničkog sistema sa nelinearnom neholonomnom vezom. Neholonomni mehanički sistem je predstavljen sa dva Čapljiginova sečiva, zanemarljivih dimenzija, koja nameću nelinearno ograničenje u vidu upravnosti brzina. Razmatra se brahisthrono ravno kretanje pri zadatom početnom i krajnjem položaju uz neizmenjenu vrednost mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačkih sila, dobijene su na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne mehaničke sisteme u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza. Formulisan brahistohroni problem, uz odgovarajući izbor veličina stanja je rešen kao, najjednostavniji u ovom slučaju, zadatak optimalnog upravljanja primenom Pontryagin-ovog principa maksimuma. Dobijen je odgovarajući dvotačkasti granični problem sistema običnih nelinearnih diferencijalnih jednačina koji je neophodno numerički rešiti. Numerički postupak za rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije neholonomnih veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, tako da se razmatrani sistem kreće u skladu sa neholonomnim zadržavajućim vezama.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

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

U ovom radu analizira se problem brahistohronog ravnog kretanja mehaničkog sistema sa nelinearnom neholonomnom vezom. Neholonomni mehanički sistem je predstavljen sa dva Čapljiginova sečiva, zanemarljivih dimenzija, koja nameću nelinearno ograničenje u vidu upravnosti brzina. Razmatra se brahisthrono ravno kretanje pri zadatom početnom i krajnjem položaju uz neizmenjenu vrednost mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačkih sila, dobijene su na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne mehaničke sisteme u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza. Formulisan brahistohroni problem, uz odgovarajući izbor veličina stanja je rešen kao, najjednostavniji u ovom slučaju, zadatak optimalnog upravljanja primenom Pontryagin-ovog principa maksimuma. Dobijen je odgovarajući dvotačkasti granični problem sistema običnih nelinearnih diferencijalnih jednačina koji je neophodno numerički rešiti. Numerički postupak za rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije neholonomnih veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, tako da se razmatrani sistem kreće u skladu sa neholonomnim zadržavajućim vezama.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

Analysis of minimum required sliding friction coefficient in the brachistochronic motion of a mechanical system with nonlinear nonholonomic constraint

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 [3,4,5,6] 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 [8]. Here, this is more convenient to use than some other methods of analytical mechanics applied to nonholonomic mechanical systems [9], 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 [6,7] by applying the Pontryagin maximum principle [1]. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which has to be numerically solved [2]. 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 [8,9], a minimum required value of the coefficient of sliding friction is defined [10], so that the considered system is moving in accordance with nonholonomic bilateral constraints

BRACHISTOCHRONIC MOTION OF A NONLINEAR NONHOLONOMIC MECHANICAL SYSTEM MODEL

The paper analyzes the brachistochronic motion of a nonlinear nonholonomic mechanical system [1] in a horizontal plane, between two specified positions. Differential equations of motion are created based on the general theorems of dynamics. The formulated brachistochrone problem is solved applying the variational calculus. The two-point boundary value problem (TPBVP) of the system of nonlinear differential equations is solved by the shooting method. Special attention is directed to the realization of thus obtained brachistochronic motion

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

U ovom radu analizira se problem brahistohronog ravnog kretanja mehaničkog sistema sa nelinearnom neholonomnom vezom. Neholonomni mehanički sistem je predstavljen sa dva Čapljiginova sečiva, zanemarljivih dimenzija, koja nameću nelinearno ograničenje u vidu upravnosti brzina. Razmatra se brahisthrono ravno kretanje pri zadatom početnom i krajnjem položaju uz neizmenjenu vrednost mehaničke energije u toku kretanja. Diferencijalne jednačine kretanja, u kojima figurišu reakcije neholonomnih veza i upravljačkih sila, dobijene su na osnovu opštih teorema dinamike. Ovde je to podesnije umesto nekih drugih metoda analitičke mehanike primenjenih na neholonomne mehaničke sisteme u kojima je neophodno dati naknadno fizičko tumačenje množitelja veza. Formulisan brahistohroni problem, uz odgovarajući izbor veličina stanja je rešen kao, najjednostavniji u ovom slučaju, zadatak optimalnog upravljanja primenom Pontryagin-ovog principa maksimuma. Dobijen je odgovarajući dvotačkasti granični problem sistema običnih nelinearnih diferencijalnih jednačina koji je neophodno numerički rešiti. Numerički postupak za rešavanje dvotačkastog graničnog problema vrši se metodom šutinga. Na osnovu tako dobijenog brahistohronog kretanja određuju se aktivne upravljačke sile, a ujedno i reakcije neholonomnih veza. Koristeći Kulonove zakone trenja klizanja, određuje se minimalno potrebna vrednost koeficijenta trenja klizanja, tako da se razmatrani sistem kreće u skladu sa neholonomnim zadržavajućim vezama.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