Theoretical and experimental investigations of the rolling of a ball on a rotating plane (turntable)

Abstract

In this work we investigate the motion of a homogeneous ball rolling without slipping on uniformly rotating horizontal and inclined planes under the action of a constant external force supplemented with the moment of rolling friction, which depends linearly on the angular velocity of the ball. We systematise well-known results and supplement them with the stability analysis of partial solutions of the system. We also perform an experimental investigation whose results support the adequacy of the rolling friction model used. Comparison of numerical and experimental results has shown a good qualitative agreement.

The Rolling of a Homogeneous Ball with Slipping on a Horizontal Rotating Plane

“…The theoretical results obtained supplement the models of motion of a ball without slipping on a rotating plane [2]. The problem under consideration is of great importance for applications.…”

“…The motion described above is similar to that of the ball on the plane if one incorporates the moment of resistance to rolling which is proportional to the angular velocity of the ball [2]. But in this case the untwisting of the spiral occurs much faster (see also [11]).…”

“…In the case where there is no slipping, the motion of the ball is subject to an inhomogeneous nonholonomic constraint, which corresponds to the condition that the velocities of the contacting points on the surface of the ball and the rotating plane be the same. In this case, the system is linear and admits explicit integration (see, e.g., [1][2][3] and references therein). The trajectories of the center of mass of the ball relative to a fixed reference system are circles, and the ball moves with constant angular velocity, which depends only on the angular velocity of the surface on which the ball rolls, and on the radius of inertia of the ball.…”

Circular orbits of a ball on a rotating conical turntable

“…Another open problem is to examine the rolling of the ball with rolling resistance. In the recent paper [1], in the case of motion of a homogeneous ball on a plane, a good agreement was shown between the experimental trajectory and the theoretical trajectory obtained by adding the moment of rolling friction which is proportional to the angular velocity of the ball. Using this friction model, it was shown that all trajectories asymptotically tend to an untwisting spiral.…”

On rolling friction

“…Заметим, что эти выводы сделаны для сильно деформируемых материалов (например, резины). При анализе экспериментов по трению качения стальных шариков в [12,13] показано, что коэффициент трения пропорционален угловой скорости. Для неоднородного шара (Чаплыгина) в [14] показано, что учёт только трения скольжения (сухого или вязкого) не даёт удовлетворительного согласия с экспериментом, более существенное влияние оказывает трение качения.…”