A rigid cylinder on a viscoelastic plane

The paper considers two two-dimensional dynamical problems for an absolutely rigid cylinder interacting with a deformable flat base (the motion of an absolutely rigid disk on a base which in non-deformed condition is a straight line). The base is a sufficiently stiff viscoelastic medium that creates a normal pressure $p(x)=kY(x)+\nu\dot Y(x)$, where $x$ is a coordinate on the straight line, $Y(x)$ is a normal displacement of the point $x$, and $k$ and $v$ are elasticity and viscosity coefficients (the Kelvin–Voigt medium). We are also of the opinion that during deformation the base generates friction forces, which are subject to Coulomb's law. We consider the phenomenon of impact that arises during an arbitrary fall of the disk onto the straight line and investigate the disk's motion “along the straight line” including the stages of sliding and rolling.