On Brachistochronic Motion of a Multibody System with Two Degrees of Freedom with Real Constraints

The paper considers a case of brachistrochronic motion of the mechanical system in the field of conservative forces, subject to the action of constraints with Coulomb friction. In this case an analogy is made between the two approaches of solving this problem of the mechanical system with two degrees of freedom. The mathematical model used to compute the brachistohrone in this special case of the multibody system with two deegres of freedom is based on varational calculus. The complete analogy is made with a solution in relation to material point.


INTRODUCTION
The Bernoulli's case of brachistochronic motion of a particle (cf.[1]) was for the first time extended by Euler in [2] .Euler considered motion of a particle in resisting medium in which the force of resistance depends only on velocity of the particle.
The case of brachistochronic motion of a particle in the field of gravity subject to the action of Coulomb friction solved by Ashby in [3] presents a special case in which the power of friction forces is described as linear function in relation to generalized velocities and accelerations.The brachistochronic motion of a particle along a rough surface was treated by Čović and Vesković in [4] where the power of friction forces is given as the sum of two functions, one in relation to generalized coordinates and velocities and the other function which is linear in relation to accelerations and whose coefficients depend on generalized coordinates and velocities.The Bernoulli's brachistochrone problem was extended to the system of rigid bodies by Čović and Vesković in [5].
The brachistochronic motion of a mechanical system with two degrees of freedom subject to the action of constraints with Coulomb friction presented by Djuric in [6] use the form of the power of friction forces given in [4] but in case when the power of friction forces, potential and kinetic energy does not depend on generalized coordinates.In the paper [7] Čović and Vesković showed complete analogy between the brachistochronic motion of a mechanical system with two degrees of freedom and the brachistochronic motion of a particle under friction forces where the power of friction forces is expressed as function in relation to generalized velocities and accelerations, in a more general form than in [3] and [4] .
In the paper [8] Šalinić, Obradović, and Mitrović considered the case of brachistochronic motion of mechanical system with two degrees of freedom in which ideal bilateral constraints and one unilateral constrain with Coulomb friction are imposed on the system.
In this paper, the choice of the functions of generalized velocities in expression for power of Coulomb friction shows that in case of mechanical system with two degrees of freedom of motion the complete parameterization of differential equations of motions is similar to Ashby's frictional brachistochrone in relation to material particle.However introduced functions are different than in the papers mentioned above and this leads to new differential equations of brachistochronic motion which are solved.

FORMULATION OF THE PROBLEM
We consider, in this paper, the motion of a mechanical system in a stationary field of potential forces with potential ( ) q    , subject to the action of real constraints.The configuration of the system is defined by the set of Lagrangian coordinates , ) ,..., 2 , 1 ( n q q q q  to which correspond the generalized velocities ) ,..., 2 , 1 ( n q q q q      Lagrangian function of the system has the form ( [2]) where T is kinetic energy of the system Differential equations of motions of the mechanical system have the well known form (cf. [9]) where If we assume that the system moves from initial to final configuration along one definite trajectory for which Eqs. (4) has minimum value we will consider brachistochronic motion.If we now introduce Bernoulli's condition's (cf.[3]), i.e. the conditions which do not disturb the principle of work and energy subject to the action of control forces in virtue of , we formulate variational problem as constrained with constraint which represents the principle of work and energy where power of generalized forces of Coulomb friction has the form ,

GENERAL PART
Let us consider a case of brachistochronic motion presented in [7] in which the power of forces of Coulomb friction (cf.( 7)) has the form: and let us assume now that functions where the following Our aim is to obtain differential equations of brachistochrinic motion of the system.For that reason, we are able to minimize the integral (4) but in order to avoid second order functional we introduce the following constraints in terms of varational calculus and the integrand of functional (9) gets the form and where are further satisfied (cf.[6]), we shall apply transformation to coordinates ., , where   is Kroneker delta simbol.This transformation leads to a new integrand of functional ( 8) Euler's equations for (18) are given in [6].

MECHANICAL SYSTEM WITH TWO DEGREES OF FREEDOM
Let us consider a special case of motion of mechanical system with two degrees of freedom.Assuming that condition ( 16) is further satisfied and having in mind (17 Taking into account (21), kinetic energy of the system considered (cf.( 2)) can be written in the form where (cf.( 19), (21)) Power of generalized forces of Coulomb friction (cf.(10)) obtain the form (cf. (20)) we are able to eliminate the velocities 1 q  and 2 q  by the following relations The choice of functions 1 f i 2 f gives the different form of the principle of the work and energy (6).
Furthermore we introduce the functions with parameter wherefrom the principle of the work and energy (6) has the form and integrand (9) gets the following form The condition of transversality at the right end-point gets the form and Euler's equations (33) have the first integral .
Taking into consideration that V and z is not prescribed in the final position of the system, the endconditions have the following form Eliminating V  by (30), from Euler's equation (cf.(33)) in relation to z we are able to eliminate   from Euler's equation (cf.(36)) in relation to V , wherefrom we get (cf.(36)) In order to obtain the equations of motion of the system considered (cf.(32),(34) ) the differential equations ( 27) can be written in the form where

SOLUTION
Suppose that the mechanical system considered has a velocity at the start position then equation ( 39) Taking into account conditions (37) at the final position of the system considered, equation ( 40 Integration of differential equations (41) yields to general solutions where Taking into account that at final position the following relations holds where (cf.(48)) we get a value of the parameter z 1 at moment t 1 , and constant (52)

EXAMPLE
Let us consider the motion of the mechanical system which consists of three prismatic rigid bodies and moves in homogeneous field of gravity.The configuration of the system is defined by the set of coordinates ). 2 , 1 ( 1 q q q  System starts from position defined by coordinates 10 where it was at rest.Final position is set by: 11 The coefficient of Coulomb friction on the rough inclined side (at angle  to horizontal) of prism  Taking into account that the differential equations of motion of the system considered (fig. 1) have the form the relations (19) get the form The graph in (Fig. 2) is showing the brachistochone ) 1 ( 2 q f q  and graph in (Fig. 3) is showing (cf.(39)) of the system considered.

CONCLUSION
The paper considers a case of the brachistochronic motion of the mechanical system in the field of conservative forces, subject to the action of constrains with Coulomb friction.The constraint represents the modified form of the principle of work and energy (30) obtained in [6] and the new Euler's equations are formed.The complite analogy is made among solution obtained in the example considered, solution in [6] and the solution in relation to material point in [3].

Figure. 1
Figure.1 An example of the mechanical system

Figure. 2 Figure. 3
Figure.2 Brachistohrone of the mechanical system considered