Dynamics of a viscous ball rolling down on a rigid staircase

We theoretically investigate the motion of a ball rolling down on a periodical staircase. Our research is restricted in the case of completely inelastic collision when the ball falls down on the surface of the stairs. The ball is accelerated when it rolls cross the edge of the stair, while it is decelerated when it rolls on the horizontal surface due to the rolling friction. The competition between them causes two different regimes according to the parameters of the system. One is the steady moving regime in which the ball keeps moving forever, and the other is the still regime in which the ball finally stops after rolling on a finite number of stairs. The diagram of these two regimes is given in the reduced parameter space. The tendency that smaller scale of the staircase can keep moving on smaller inclined angle is found.