The motion of a cylinder on a moving plane with friction

The motion of a cylinder on a moving plane with sliding friction and rolling friction is considered. In the case of vertical motion of the plane, it is shown that after a finite time for arbitrary initial conditions one of the following motion modes is established: rest or rolling downwards with or without sliding, accelerated or uniform, depending on the values of the system parameters. In the case of a plane-parallel motion of a horizontal plane, parametric conditions are found for the existence of two periodic rolling modes. It is shown that one of these modes sets up after a finite time for arbitrary initial conditions. An example demonstrating the influence of the torque of rolling friction is considered.