Integration of Constraint Equations in Problems of a Disc and a Ball Rolling on a Horizontal Plane

arXiv:1107.3963v1 [nlin.SI] 20 Jul 2011 Integration of Constraint Equations in Problems of a Disc and a Ball Rolling on a Horizontal Plane Eugeny A. Mityushov Ural Federal University Department of Theoretical Mechanics Prospect Mira 19 620 002 Ekaterinburg Russian Federation e-mail: mityushov-e@mail.ru Abstract The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural equation, i.e. when the dependence of the curvature of the trajectory is explicitly expressed in terms of the distance passed by the point. 1 1 The problem of a rolling disc The rolling motion of a disc and a ball on a static plane was described many times, see e.g. [1, 2], and generalized in later publications [3]-[8]. These are the classical examples of motion of mechanical systems with non-holonomic constraints. Let a disc with radius R be tangent to a plane π with the system of coordinates Oxy, let P be the point of contact between the disc and the plane. The position of a disk is determined by five independent coordinates. For example, it can be fixed by coordinates xP and yP, by the angle of rotation ϕ, by the angle of precession ψ, and by the angle of nutation ϑ, see Figure 1. φ J y x y z O p P Figure 1: Generalized coordinates of a rolling disk. Evolution of the coordinates always satisfies the two non-holonomic con- straints which can be written in the following form: dxP = R cos ψdϕ , dyP = R sin ψdϕ . (1) The problem above is, therefore, to integrate the given dynamical system under applied arbitrary external forces. The problem is interesting because the methods of the classical mechanics it requires to use are rather involved. The solution can be essentially simplified if the rolling trajectory is known. Such a formulation is possible, for example, under modeling of the rolling with the help of a computer animation. 2 In fact, the kinematics of the rolling motion of the disk is determined by the three functions ϕ = ϕ(t) , ϑ = ϑ(t) , ψ = ψ(t) . (2) Under the rolling motion without slipping the arc coordinate s of the point of contact of the disc and the plane is related to the angle of rotation ϕ as ϕ = s/R . (3) Therefore, ˙ψ ˙ϕ = Rk(s) . (4) Here k(s) = dψ ds is a curvature of the point of contact trajectory for the disc. It is a well-known that given any function k = k(s) one can find a curve ⃗r = ⃗r (s) with the curvature equal k(s). The curve is unique up to a congru- ence. Equation k = k(s) is known as a natural equation of the curve. The parametric equations of the trajectory of the point of contact, xP = Z s 0 cos( Z τ 0 k(s)ds)dτ, yP = Z s 0 sin( Z τ 0 k(s)ds)dτ , (5) see [9], allow one to find the location of the disk on the plane at any moment of the motion. Equations (2) enable one to describe kinematics of the disk with equations of the point of contact in form (5). The found solution satisfies non-holonomic constraint equations (1). Let us give an example of how rolling of the disc can be described by three equations of motion (2) by using the natural equation for the trajectory of the point of contact with a horizontal plane. Example 1: Let rolling of the disk be determined by following equations: ϕ = ωt, ψ = εt2 2 , ϑ = f(t) . (6) The curvature of the trajectory of the point of contact is k(s) = ˙ψ ˙s = εt Rω = εs (Rω)2 , (7) 3 where ε and ω are some parameters, f(t) is a function. This trajectory is the clothoid whose asymptotic point has coordinates x = y = (Rω q π/ε)/2. The equations of motion of the point of contact in this case are xP = Z s 0 cos εs2 2(Rω)2ds , yP = Z s 0 sin εs2 2(Rω)2ds ; (8) or xP = Rω Z t 0 cos εt2 2 dt , yP = Rω Z t 0 sin εt2 2 dt . (9) A direct substitution of these functions into the equations (2) shows that obtained solution satisfies the constraints. Figure 2: Phases of motion of the disk along a clothoid (ω = π, ε = π 16, t∗= 16) The phases of motion are shown on Fig. 2. They are obtained by using equation (6) with f(t) = π t∗2(t −t∗)2. 4 2 The problem of a rolling ball Let us now discuss a similar problem for a ball. We assume that the ball rolls on a plane and spins simultaneously. Like in case of the disc, the position of a ball (see Fig. 3) can be determined by three functions. To introduce these functions we decompose the vector of angular velocity of the ball ⃗Ωinto two parts, as shown on Fig. 3, ⃗Ω= ⃗ωs + ⃗ωr . (10) The vector of angular velocity related to the spinning, ⃗ωs, is a orthogonal to the plane. The vector of the angular velocity associated to the rolling, ⃗ωr, is parallel to the plane. y x y z O p P r wr s wr Figure 3: The rolling and spinning angular velocity vectors of the ball. Evolution of the velocity vectors at the center of the ball is determined by the same angle ψ. By taking this into account one can find the complete set of functions which describe the motion of the ball: ϕ = Z t 0 ωr(t)dt ,

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