Conditions for pure rolling of a heavy cylinder along a brachistochrone

Abstract

Algebraic equations for the line of steepest descent of a cylinder are derived in parametric form. Conditions for rolling without slipping and separation of the cylinder along a brachistochrone are established based on the equations of motion with constraint reaction. The important conclusion is drawn that the center of mass of a cylinder moving along a brachistochrone describes a cycloid Introduction. Brief Review of Relevant Studies. In 1696, Bernoulli posed the following brachistochrone problem: find the shape of the curve down which a bead slipping from rest and accelerated by gravity will slip from one point to another in the least time. It is assumed that the bead (material point) moves in a vertical plane and in a uniform gravity field.Newton, Leibniz, l'Hopital, and two Bernoullis showed that the solution of this problem is a cycloid [14]. Analytic solutions of the brachistochrone problem obtained using geometrical optics and the calculus of variations can be found in [15,16]. Brachistochrone with Coulomb friction was studied in detail by Ashby et al. [10], Hayen [18], Heijden and Diepstraten [25]. The problem of a brachistochrone on a cylindrical surface with Coulomb friction was generalized by Covic and Veskovic in [12]. The problem of a brachistochrone in nonconservative force fields was addressed in [27]. The problem of a brachistochrone on a cylinder in uniform force fields was solved in [28]. Brachistochrones on cylinders and spheres were considered by Palmieri [20]. The same problem was generalized to nonuniform force fields by Aravind [9], Denman [13], and Venezian [26]. Venezian solved the problem of brachistochronein linear radial force fields. Denman [13], Parnovsky [21], and Tee [24] solved the same problem for radial force fields inversely proportional to the squared distance between the interacting points. The generalizations of the brachistochrone problem involving relativistic effects were outlined by Goldstein and Bender [17], Scarpello and Ritelli [23].The next natural step in the generalization of the brachistochrone problem is to derive the brachistochrone equation for rolling finite-size bodies (balls, cylinders, disks, wheels, etc.). Various formulations of the brachistochrone problem for arbitrary rolling bodies were given in [22,19].The brahistohrone problem for a heavy cylinder is of theoretical importance due to the following applied problem.There is a variety of antivibration devices for which the minimum time they take to reduce the amplitude of forced vibrations to a standard level is the performance criterion [6,7,11]. This criterion is applied to rolling antivibration devices [2,3]. In this connection, it is necessary to find the shape of the directrix of a cylindrical surface over which a heavy homogeneous cylinder rolls without slipping (pure rolling) in the least time. The requirement of no slipping is standard for vibration-protection problems.The present study continues the research reported in [4,5] where the following differential equation of brachistochrone along whi...

The brachistochronic motion of a vertical disk rolling on a horizontal plane without slip

“…On the other hand, in the literature available, the papers discussing the problem of brachistochronic motion of the disk consider the problem of seeking the curve lying in the vertical plane along which the disk, rolling without slip, is descending from a specified initial position to a specified terminal position for a minimum time (see [17][18][19]). Identical problem but for a circular cylinder is studied [20,21]. Also, in these papers conditions of realizability of rolling without slip and separations in presence of the Coulomb dry-friction forces are considered.…”

“…Such case has not been considered in the literature available to date. Like in [17][18][19][20][21], the analysis was performed of the realizability condition of the brachistochronic motion of the disk in the presence of the Coulomb dry-friction forces. Numerical examples are provided.…”

On the brachistochronic motion of a variable-mass mechanical system in general force fields

“…Research is inspired not only by the expansion of existing fundamental knowledge in this area, but also by various engineering applications (see e.g. [1][2][3][4][5][6][7]). Thus in [8][9][10][11][12][13][14][15][16] the brachistochronic motion of a particle in the presence of resistance forces (forces of dry friction, viscous friction) is analyzed, while in [6,7,17,18] the brachistochronic motion of a particle on a surface is considered.…”

“…Note that in [6] the problem of optimization of a bobsled traveling on a path was solved as the problem of the brachistochronic motion of a particle on a surface, whereas [7] considers the brachistochrone problem for a steerable particle moving on a 1D curved surface with an application to ski racing. The next important group of references comprises the papers that consider the problem of brachistochronic motion of a rigid body [1–3, 5, 19] and of a system of rigid bodies [20–23]. Furthermore, in [24–27], the brachistochronic motion of mechanical systems with nonholonomic constraints is analyzed.…”