Mechanics Colloquim
PROGRAM
PROGRAM ZA JANUAR 2009.
Pozivamo Vas da učestvujete u radu sednica Odeljenja i to:
PETAK, 30. jan. 2009. u 14 sati:
Slaviša Salinić, Mašinski fakultet Kraljevo
CONTRIBUTION TO THE BRACHISTOCHRONE PROBLEM WITH COULOMB FRICTION
Abstract: This paper formulates and solves in closed form (expressed by elementary functions) the brachistochrone
problem with Coulomb friction of a particle which moves down a rough curve in a uniform gravitational field
assuming that the initial velocity of the particle is different from zero. The problem is solved by the
application of variational calculus. Two variants are considered: first, the initial position and the
final position of the particle are given; second, the initial position is given, and the final position
lies on a given vertical straight line. The new approach in treating this problem by variational
calculus lies in the fact that the projection sign of the normal reaction force of the rough
curve onto the normal to the curve is introduced as the additional constraint in the form of
an inequality. This inequality is transformed into an equality by introducing a new state variable.
Although this is fundamentally a constrained variational problem, by further introducing a new
functional with an expanded set of unknown functions, it is transformed into an unconstrained problem
where broken extremals appear. Brachistochrone equations in parametric form are obtained for both
variants which are examined, with the slope angle of the tangent to the brachistochrone being taken as the parameter.
These equations contain a certain number of unknown constants which are determined from the corresponding
systems of nonlinear algebraic equations. They are solved by an alternative approach which is based
on the application of differential evolution. The obtained brachistochrones are generally two-segment
curves with the initial line segment representing a free-fall parabola in nonresistant medium.
It is shown that regarding the special values of the parameters the results of the paper
coincide with the known results from literature.
Sednice se održavaju u zgradi SANU, Knez Mihailova 35, u sali 2 na prvom
spratu.