Control of the rolling of a dynamically symmetrical sphere on an inclined rotating plane

The work investigates the rolling dynamics of a dynamically symmetrical heavy sphere (or a heavy spherical shell) along an inclined rough plane (platform) rotating with constant or periodic angular velocity around an axis, which is perpendicular to the plane and passing through some fixed point of this plane. Nonholonomic and holonomic constraints are imposed at the point of contact of the sphere with the reference plane. The equations of motion of the sphere are constructed. In the case of the constant angular velocity of the plane at any slope and in the case of the periodic angular velocity of the plane located horizontally the boundedness of the velocities of the geometric center of the sphere is proved. Moreover, in the case of the constant angular velocity of the plane, solutions are found analytically. Based on numerical integration, it is shown that for the periodic angular velocity of the plane and for the nonzero slope the square of the velocity vector of the geometric center of the sphere increases indefinitely. Two controls for the slope of the plane proportional to the projections of the velocity vector of the sphere on the coordinate axes lying in the reference plane are introduced. In the case of the constant angular velocity of the plane, a qualitative analysis of the equations of motion has been carried out, the control parameters at which the square of the velocity vector of the geometric center of the sphere will be bounded and at which it will be unbounded have been analytically found. The results of this control are presented for the case of periodic angular velocity of the plane. It is shown that by controlling the slope of the plane, it is possible to achieve the boundedness of the square of the velocity vector of the geometric center of the sphere. The obtained results are illustrated, the trajectories of the contact point and graphs of the desired mechanical parameters are constructed.

dynamics, control, sphere, rotating plane, slope, nonholonomic system