Brahistohrono kretanje mehaničkih sistema sa realnim vezama i primene na tehničke objekte

Abstract

Analizirano je brahistohrono kretanje materijalne tačke i sistema krutih tela u prisustvu Kulonove sile trenja primenom varijacionog računa. U slučaju brahistohronog kretanja materijalne tačke razmatran je slučaj kada se materijalna tačka kreće u vertikalnoj ravni u homogenom gravitacionom polju duž veze u obliku hrapave linije sa Kulonovim trenjem, pri čemu je početna brzina tačke različita od nule. Analiza brahistohronog kretanja tačke je sprovedena za slučaj kada se veza tretira kao zadržavajuća i za slučaj kada se veza tretira kao nezadržavajuća. Izvršena je generalizacija rezultata iz literature, koji su tretirali ovu problematiku primenim varijacionog računa u slučaju kada je početna brzina tačke jednaka nuli, uvođenjem pretpostavke o znaku normalne reakcije veze kao dodatnog ograničenja u varijacionoj formulaciji problema.

The brachistochrone motion of the particle and the rigid multibody systems in the presence of Coulomb friction was analyzed by the application of variational calculus. In case of the brachistochrone motion of the particle, the case when the particle moves in the vertical plane in the homogeneous gravitational field along the coustraint in the form of a rough curve with Coulomb friction was considered, where the initial velocity of the particle was diff'erent fiom zero. The analysis of the brachistocluone ntotion of the particle was performed for the case when the constraint is treated as bilateral and for the case when the constraint is treated as unilateral. By introducing the assumption regarding the sign of the normal reaction of the constraint as an additional constraint in the variational formula of the problem, the results from the references treating these problems by variational calculus with the assumption that the initial velocity of the parlicle is equal to zero were generahzed. The equations of the brachistochroue were obtained in their parameter form, where the slope angle of the tangent on the brachistochrone curve was taken as the parameter. It was shown that the brachistochrone in the general case is a two-segment curve w'ith the initial line segment representing a free-fall parabola in nonresistant medium. The application of obtained results in the problems of optimization in a plant for transportation of granular nraterial was presented. In this technical object, the problem of minimization of losses of rnechanical energy due to the action of Coulomb friction during transporlation of granular material was analytically solved. In the brachistochrone motion of a constrained system of rigid bodies, the case when a certain number of unilateral constraints treated as real constraints with Coulomb friction are fbLrnd among the constraints imposed on the system. A general approach to solution by the application of the ntethodology used in the problem of brachistochrone motion of the particle was given. Within this chapter of the dissertation, a special type of the mechanical system with two degrees of freedom on which the analogy of solving the brachistocluone problem of this system with the brachistochrone problem of the particle considered in the first two chapters of the dissertation was analyzed. The results were obtained in the form which is suitable for illustration on specific technical objects and further numerical analysis. The application of differential evolution, as the optimization method, in solving systems of nonlinear algebraic equations was mentioned.