Rolling, tilting and spinning spherical wheels: Analytical results using the brush theory

This paper investigates the rolling dynamics of spherical wheels using the theoretical framework provided by the brush models. The analysis is mainly conducted under the assumption of vanishing sliding inside the contact patch. Different types of kinematics are considered: simply rolling wheels, rolling and tilting, and purely spinning. For the first two cases, a complete solution is derived concerning both the steady-state and transient behaviours. Some qualitative trends for the forces and moments generated inside the contact patch are then provided when accounting for limited friction. For the case of a purely spinning spherical wheel, it is shown that steady-state conditions are never possible owing to the assumption of vanishing sliding. Moreover, it is demonstrated that the shear stresses acting inside the contact patch grow unbounded if the additional contribution relating to the deflection of the bristle is not taken into account when calculating the total sliding velocity. In this case, a stationary solution may be eventually recovered as an asymptotic distribution only by assuming limited friction inside the contact patch.