Contribution to the brachistochrone problem with Coulomb friction

Analytical solution for the problem of maximum exit velocity under Coulomb friction in gravity flow discharge chutes

“…4.1, it can be shown that the conditions (21), (22), (36), and (37) yield a system of 14 equations in 14 unknowns. Numerically solving this equation system by means of the differential evolution [12], it is obtained that ϕ 0 > ϕ 1 = 0 which does not make physical sense because it is not in accord with the fact that the angle ϕ increases on the line segment SG1. Hence, the variant SG1-SG3-SG2-SG3 is not possible.…”

“…In this paper, the problem of maximization of exit velocity is solved analytically using variational calculus and the methodology from the references [3,4,7,12]. In contrary to the references [1,2,9,15], in this paper the fast flow condition [10] is included as an additional constraint in variational calculus formulation of the problem, which caused obtaining significantly different profiles of chutes than those determined in [9] for certain combinations of boundary conditions.…”

“…Without connection with the problem of optimization of discharge chutes, in [14], by applying the Pontryagin maximum principle, the problem of minimization of mechanical energy losses due to Coulomb friction for a particle moving from the rest position along a rough curve was solved. According to the used particle model, the results from [12] can be used as an analytical solution of the problem of finding gravity flow discharge chute profiles to achieve the minimum transit time.…”

10.5937/fmet1403199r = Analysis of the minimum required coefficient of sliding friction at brachistochronic motion of a nonholonomic mechanical system

“…During motion, the vehicle is acted on by the control force   L k θ   , and the resistance moment 3 L , proportionate to the relative angle of the front axle rotation around the vertical axis, where 3 3 L k θ  . Now, the differential equations of vehicle motion, based on the theory of change in momentum and moment of momentum for a movable axis that passes through point B and is perpendicular to the plane of motion [9], have the form…”

“…Much later, the generalization of the classical brachistochrone problem was carried out within the calculus of variations [1] and [2]. A detailed review of literature related to the problems of brachistochronic motion can be found in [3] and [7]. The problems considered in the present paper involve a review of references on the Bernoulli's case of the classical brachistochrone extended to the system of rigid bodies.…”