Analysis the Brachistochronic Motion of A Mechanical System with Nonlinear Nonholonomic Constraint

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1 Analysis the Brachistochronic Motion of A Mechanical Syste with Nonlinear Nonholonoic Constraint Radoslav D. Radulović Teaching Assistant Dragoir N. Zeković Full Professor Mihailo P. Lazarević Full Professor Štefan Segľa Professor Technical University of Košice Košice Slovakia Bojan M. Jereić Teaching Assistant This paper analyzes the proble of brachistochronic planar otion of a echanical syste with nonlinear nonholonoic constraint. The nonholonoic syste is represented by two Chaplygin blades of negligible diensions which ipose nonlinear constraint in the for of perpendicularity of velocities. The brachistrochronic planar otion is considered with specified initial and terinal positions at unchanged value of echanical energy during otion. Differential equations of otion where the reactions of nonholonoic constraints and control forces figure are obtained on the basis of general theores of echanics. Here this is ore convenient to use than soe other ethods of analytical echanics applied to nonholonoic echanical systes where a subsequent physical interpretation of the ultipliers of constraints is required. The forulated brachistochrone proble with adequately chosen quantities of state is solved as siple a task of optial control as possible in this case by applying the Pontryagin axiu principle. The corresponding two-point boundary value proble of the syste of ordinary nonlinear differential equations is obtained which has to be nuerically solved. Nuerical procedure for solving the two-point boundary value proble is perfored by the ethod of shooting. On the basis of thus obtained brachistochronic otion the active control forces along with the reactions of nonholonoic constraints are defined. Using the Coulob friction laws a iniu required value of the coefficient of sliding friction is defined so that the considered syste is oving in accordance with nonholonoic bilateral constraints Keywords: Brachistochrone Nonlinear nonholonoic constraint Pontryagin s axiu principle Coulob friction Optial control. 1. INTRODUCTION This paper analyzes the proble of brachistochronic planar otion of a echanical syste with nonlinear nonholonoic constraint. The brachistrochronic planar otion is considered with specified initial and terinal positions at unchanged value of echanical energy during otion. Differential equations of otion where the reactions of nonholonoic constraints and control forces figure are obtained on the basis of general theores of echanics. The forulated brachistochrone proble with adequately chosen quantities of state is solved as siple a task of optial control as possible in this case by applying the Pontryagin axiu principle. The corresponding two-point boundary value proble of the syste of ordinary nonlinear differential equations is obtained which has to be nuerically solved. Nuerical procedure for solving the two-point boundary value proble is perfored by the ethod of Received: April 014 Accepted: October 014 Correspondence to: Radoslav Radulovic Kraljice Marije Belgrade 35 Serbia shooting. On the basis of thus obtained brachistochronic otion the active control forces along with the reactions of nonholonoic constraints are defined. Using the Coulob friction laws a iniu required value of the coefficient of sliding friction is defined so that the considered syste is oving in accordance with nonholonoic bilateral constraints.. DESCRIPTION OF THE DYNAMIC MODEL OF NONLINEAR NONHOLONOMIC SYSTEM In order to develop differential equations of otion of a dynaic odel of nonlinear nonholonoic echanical syste (henceforth referred to as the syste ) as well as for the needs of further considerations two Cartesian reference coordinate systes ust be first introduced. The stationary coordinate syste Oxyz whose coordinate plane Oxy coincides with the horizontal plane of otion and the non-stationary coordinate syste Aης attached rigidly to the considered syste so that the coordinate plane Aη coincides with the plane Oxy (see Fig. 1). The axis of the non-stationary coordinate syste A is defined by the direction AB that is B A whereas unit vectors of the non-stationary E-ail: rradulovic@as.bg.ac.rs doi: /fet140490r Belgrade. All rights reserved FME Transactions (014)

2 coordinate syste axes are λ μ and ν respectively. The syste is coposed of two Chaplygin blades [3] of negligible asses and diensions which ipose constraint to the otion of particles A and B of equal asses in the for of perpendicularity of the velocities as shown in Fig. 1. Particles A and B are interconnected by a light forkslike structure which allows the distance AB const. to change. The configuration of the considered syste relative to the syste Oxyz is defined by a set of Lagrangian coordinates qq qq where q 1 xand q y 3 are Cartesian coordinates of the point A q φ is the angle between the axis Ox and axis A while q 4 is the relative coordinate of point B relative to the nonstationary coordinate syste. Figure 1. Nonlinear nonholonoic echanical syste Further analysis relates to the case when the otion of point A is constrained in the A axis direction while the otion of point B is constrained in the Aη axis direction that is lateral slipping of the points A and B of the syste is not perissible in the A and Aη axis directions respectively. Due to the iposed constraints to the otion there occur horizontal reactions of nonholonoic constraints at the contact points of the A and B points and horizontal plane of otion R A R A λ and R B R B μ respectively. In accordance with the constrained otion in the for of perpendicularity of the velocities of points A and B the nonholonoic nonlinear hoogeneous constraint has the for [3] [4] u 0 xx B yy B 0 (1) where and u are the velocities of points A and B respectively. Taking into account that the otion of point A is constrained in the A axis irection the second nonholonoic hoogeneous constraint can be represented in the for as follows xcos φ y sin φ = 0 () and as a result the velocity of point A has the Aη axis direction that is xsin φ y cos φ (3) where μ. The coordinates of point B relative to the coordinate syste Oxyz are xb xcos φ yb ysin φ zb 0. (4) Now based on nonholonoic constraints (1) and () taking into account the relations (3) and (4) the angular velocity of the syste is deterined in the for φ. (5) The velocity u of point B which has the A axis direction can be expressed based on relations () and (4) in the for as follows u (6) where u u λ. As it is well-known the realization of the brachistochronic otion of echanical systes can be left in general to the control forces whose total power equals zero during brachistochronic otion which can be represented in the for of active control forces then by the forces of reactions of the constraints or by their utual cobinations. For this case the realization of the brachistochronic otion is achieved by active control forces F1 F1 t μ and F F t λ acting at the points A and B respectively whose power equals zero during the brachistochronic otion G P F1 Fu 0 (7) that is F (8) 1 Fu 0. Differential equations of otion will be developed based on the general theores of dynaics [6810] that is by applying the theore of change in oentu of a echanical syste as well as the theore of change in oent of oentu of a echanical syste for the oving point A dk s F R dt (9) dl A s A K MA. dt The vector equations (9) have respectively the following corresponding scalar equations relative to the axes of defined non-stationary coordinate syste Aηζ φu F R η φ F1 RB ζ η 00 ζ u R. B A (10) FME Transactions OL. 4 No

3 Solving the syste of equations (8) and (10) the reactions of nonholonoic constraints as well as control forces are defined to realize the brachistochronic otion as a function of defined quantities of state and corresponding derivatives RA φ φ u u RB φu F φ φ (11) 1 F φ φ. u During the syste brachistochronic otion the law of the conservation of echanical energy holds Φ u u 0 (1) wheret 0 is the kinetic energy of the syste at initial tie oent t BRACHISTOCHRONIC MOTION AS THE PROBLEM OF OPTIMAL CONTROL This section involves the definition of the proble of brachistochronic otion of the syste as the proble of optial control. The equations of state that describe the otion of the considered syste in the state space can be defined in the for x sin φ y cos φ φ =u. (13) The coordinates of initial state x y φ and as well as the kinetic energy are defined at the initial position of the syste on anifolds: t0 0 x t0 0 y t0 0 φ t0 0 t0 a t0u t0 0 (14) as well as the coordinates of end state x y φ and at the terinal position of the syste on anifolds: t t x t a y t b φ t φ t a (15) where t 1 is the value unknown in advance of the final tie oent that corresponds to the end state of the syste on anifolds (15). The brachistochrone proble of the otion of the syste described by equations of state (13) consists in deterining the coordinates of optial control and u as well as their corresponding coordinates of state x y φ and so that the syste starting fro the initial state on anifolds (14) oves to the end state on anifolds (15) with unchanged value of echanical energy (1) in a iniu tie. This can be expressed in the for of condition so that the functional t1 I dt (16) t0 in the intervalt0 t1 has a iniu value. For the purpose of solving the proble of optial control defined by the Pontryagin axiu principle [1] the Pontryagin function is developed in the for as follows H λ0 λx sin φλy cosφλφ λu μφ u (17) where λ 0 const. 0 λx λy λ φ and λ are coordinates of the conjugate vector where it can be taken that λ0 1 while μ is a ultiplier corresponding to the relation (1). Taking into account the boundary conditions (14) and (15) as well as the fact that tie does not figure explicitly in equations of state (13) the defined proble of optial control can be solved by a straightforward application of Theore 3 that is Theore 1 [1]. Based on the Pontryagin function (17) the conjugate syste of differential equations has the for λ x 0 λ 0 y λ φ λxcos φ+ λysin φ λ (18) where fro it follows that λx const. and λy const.. Taking into account that the initial state (14) as well as the end state (15) is copletely deterined the transversality conditions are identically satisfied. If controls belong to an open set as in this case the conditions for deterining optial control can be expressed in the for H H 0 uu i j 0 i j=1. ui opt u iu u j u opt (19) When tie t 1 is not deterined in advance as in this case in solving the syste of equations (13) and (18) in the final for the condition should be added following a straightfoward application of Theore 1 [1] that the value of the Pontryagin function on the optial trajectory equals zero for tt0 t1 Ht 0 (0) that is in accordance with the relation (17) 1λx sinφλy cosφλφ λu μφ u 0. (1) 9 OL. 4 No FME Transactions

4 Now and based on (17) (19) and (1) the value of a ultiplier μ is deterined as well as the control functions and u in the for as follows 1 1 μ = λxsinφλy cos φ+ λφ 4T0 u λ. () Based on condition (0) defined at the initial tie oent as well as (14) (17) and () the conjugate vector coordinate λ φ is deterined at the initial tie oent λ t t λ λ t φ 0 1/ 0 y 0 (3) where fro a global estiate for the λ t can be given coordinate 0 λ t. T (4) 0 0 T0 Now based on (13) (18) and () the basic and conjugate syste of differential equations can be created in the for 1 x = λxsin φ λy cos φ+ λφ sin φ 1 y = λxsinφ λy cosφ+ λφ cos φ 1 1 φ λxsin φ λy cos φ+ λφ λ λ x 0 λ y 0 1 λ φ λxsin φ λy cos φ+ λφ λx cos φ+ λ ysin φ 1 λ 1 λxsin φλy cos φ+ λφ λφ. (5) Nuerical procedure for solving the corresponding two-point boundary value proble of the syste of ordinary nonlinear differential equations of the first kind (5) based on the known initial state (14) and (3) as well as end state (15) is grounded on the ethod of shooting []. The four-paraeter shooting consists of deterining unknown coordinates of the conjugate vector λ x λy and λ t 0 as well as a iniu required tie t 1. The procedure of nuerical deterination of the unknown paraeters consists of shooting the end-state coordinates (15) at the known initial state (14) and (3). The application of shooting ethod requires an estiate for the interval of paraeters values being deterined [10]. On the basis of estiates for the values of the conjugate vector coordinates λ t 0 given in (4) it can be claied that all solutions to the corresponding two-point boundary value proble are certainly located within the interval given thereby the global iniu tie in the brachistochronic otion of the syste. For a concrete case the estiate for the values of coordinates λ x and λ y cannot be given so all solutions satisfying the axiu principle should be found. For the case of ultiple solutions to the axiu principle global iniu is the one corresponding to the iniu tie. The two-point boundary value proble is solved for the values of the paraeters as follows kg T0 kg a 1 φ 1 π / rad.. (6) s In accordance with (16) the tie of the brachistochronic otion of the syste as well as the conjugate vector coordinates for the given values of paraeters (6) are t s λ x s/ λ s/ and λ t s/. Figure. Trajectories of points A and B. y FME Transactions OL. 4 No

5 Figure 3. Graphs of angle φ and relative coordinate. Figure 5. Graphs of reactions of nonholonoic constraints R A and R B. Figure 4. Graphs of control functions and u. Figure 6. Graphs of control forces F 1 and F. 4. DYNAMIC CONDITIONS FOR REALIZING THE BRACHISTOCHRONIC MOTION Differential equations of otion of the syste (10) that is the reactions of nonholonoic constraints and control forces as well (11) are obtained in accordance with restrictions (1) and (). 94 OL. 4 No FME Transactions

6 Taking this into account the necessary dynaic conditions for realizing the otion of the syste in accordance with restrictions (1) and () [910] based on the Coulob friction laws are that intensities of the interaction forces between points A and B of the syste and the horizontal plane of otion should not exceed the corresponding Coulob forces of sliding friction. The interaction force between point A of the syste and the horizontal plane of otion is defined by the vector su of the control force F 1 and the reaction force of nonholonoic constraint R A so that in accordance with (6) the necessary dynaic condition for realizing the defined otion of point A of the syste can be written in the for of inequality fr A 1 A A A A F F R F μ N (7) fr where FA and μ A are the friction force and the sliding friction coefficient between point A of the syste and the horizontal plane of otion whereas NA is noral reaction of the plane of otion of point A of the syste. Analogously to what has been above entioned for point A of the syste the necessary dynaic condition for point B of the syste can be written in the for as follows fr B B B B B. F F R F μ N (8) Noral reactions of the horizontal plane of otion of points A and B of the syste are NA NB 19.61N. The diagras below based on previous considerations show the laws of change of iniu required values of the sliding friction coefficients μa and μ B. Miniu required value of the sliding friction coefficient between point A of the syste and the horizontal plane of otion based on (7) * is μa μat 1.153s 0.8 whereas iniu required value for point B of the syste based on (8) * is μb μbt 0.913s Cobining the necessary dynaic conditions (7) and (8) to realize the defined brachistochronic otion in accordance with restrictions (1) and () it is clearly deduced that iniu required value of the sliding friction coefficient is μ * μ * B CONCLUSIONS This paper considers the brachistochronic planar otion of the echanical syste with nonlinear nonholonoic constraint with specified initial and terinal positions at unchanged value of echanical energy. The procedure for developing differential equations of otion of the syste based on the general theores of echanics is presented. The forulated brachistochrone proble with adequately chosen quantities of state is solved as the siplest task of optial control by applying the Pontryagin axiu principle. The conducted nuerical procedure for solving the two-point boundary value proble is grounded on the shooting ethod. Afterwards the reactions of nonholonoic constraints as well control forces are defined to realize the brachistochronic otion. Using the Coulob friction laws a iniu required value of the sliding friction coefficient between the horizontal plane of otion and the considered syste is defined. The authors consider that the results obtained in this work can be extended to the case when the sliding friction coefficients are below iniu required values. In that case when the control forces are constrained the task of optial control becoes considerably ore coplex which is the subject of future investigations. ACKNOWLEDGMENT Authors gratefully acknowledge the support of Ministry of Education Science and Technological Developent of the Republic of Serbia under the project ON and TR REFERENCES Figure 7. Graphs of iniu required sliding friction coefficients μ A and μ B. [1] Pontryagin L.S. Boltyanskii.G. Gakrelidze R.. and Mishchenko E.F.: The Matheatical Theory of Optial Processes Wiley New Jersey 196. [] Stoer J. and Bulirsch J.: Introduction to Nuerical Analysis Second ed. Springer New York and London [3] Zeković D.: Dynaics of echanical systes with nonlinear nonholonoic constraints I The history of solving the proble of a aterial realization of a nonlinear nonholonoic constraint ZAMM Z. Angew. Math. Mech. 91 No FME Transactions OL. 4 No

7 [4] Zeković D.: Exaples of nonlinear nonholonoic constraints in Classical echanics estnik Mosk. Un-ta Ser. 1 Mat.-Meh [5] Chaplygin S. A.: On the theory of otion of nonholonoic systes 008. The reducingultiplier theore. Matheatical Collection. 8(1) English Translation by A.. Getling. Regular and Chaotic Dynaics. 13(4) [6] Šalinić S. Obradović A. Mitrović Z. and Rusov S.: On the brachistochronic otion of the Chaplygin sleigh. Acta Mechanica ACME-D R [7] Obradović A. Čović. esković M. and Dražić M.: Brachistochronic otion of a nonholonoic rheonoic echanical syste Acta Mech. 14 (3 4) pp [8] Pars L.A.: Treatise on analytical dynaics Heineann London [9] Soltakhanov Sh. Kh. Yushkov M.P. and Zegzhda S.A.: Mechanics of non-holonoic systes Berlin: Springer-erlag 009. [10] Radulović R. Obradović A. Jereić B.: Analysis of the iniu required coefficient of sliding friction at brachistochronic otion of a nonholonoic echanical syste FME Transactions ol. 4 No 3 pp [11] Čović. and esković M.: Brachistochronic otion of a ultibody syste with Coulob friction Eur. J. Mech. A Solids 8(9) pp [1] Šalinić S. Obradović A. and Mitrović Z.: On the brachistochronic otion of echanical syste with unilateral constraints Mechanics Research Counications 45 pp АНАЛИЗА БРАХИСТОХРОНОГ КРЕТАЊА МЕХАНИЧКОГ СИСТЕМА СА НЕЛИНЕАРНОМ НЕХОЛОНОМНОМ ВЕЗОМ Радослав Радуловић Драгомир Зековић Михаило Лазаревић Штефан Сегл а Бојан Јеремић У овом раду анализира се проблем брахистохроног равног кретања механичког система са нелинеарном нехолономном везом. Нехолономни механички систем је представљен са два Чапљигинова сечива занемарљивих димензија која намећу нелинеарно ограничење у виду управности брзина. Разматра се брахистхроно равно кретање при задатом почетном и крајњем положају уз неизмењену вредност механичке енергије у току кретања. Диференцијалне једначине кретања у којима фигуришу реакције нехолономних веза и управљачких сила добијене су на основу општих теорема динамике. Овде је то подесније уместо неких других метода аналитичке механике примењених на нехолономне механичке системе у којима је неопходно дати накнадно физичко тумачење множитеља веза. Формулисан брахистохрони проблем уз одговарајући избор величина стања је решен као најједноставнији у овом случају задатак оптималног управљања применом Понтрyагин-овог принципа максимума. Добијен је одговарајући двотачкасти гранични проблем система обичних нелинеарних диференцијалних једначина који је неопходно нумерички решити. Нумерички поступак за решавање двотачкастог граничног проблема врши се методом шутинга. На основу тако добијеног брахистохроног кретања одређују се активне управљачке силе а уједно и реакције нехолономних веза. Користећи Кулонове законе трења клизања одређује се минимално потребна вредност коефицијента трења клизања тако да се разматрани систем креће у складу са нехолономним задржавајућим везама. 96 OL. 4 No FME Transactions