Oden and Lin [1] discovered that an elastic cylinder, when rotated at certain speeds, displays standing wave solutions that bifurcate from the axisymmetric solution. They noted a hierarchy of N-bump standing wave solutions, where the rotation speed decreases to a critical value as N goes to infinity. In the first part of my talk, I observe a more complex hierarchy of standing wave solutions using numerical eigenvalue software. If a viscoelastic model is added, the N-bump standing waves disappear and become decaying modes. In the second part of my talk, I observe that a commonly used viscoelastic model can lead to nonphysical blow-up of the cylinder at high speeds. The blow-up is shown not to occur for a fully nonlinear viscoelastic model based on the 2nd Law of Thermodynamics.