This paper treats large‐amplitude radial oscillations of a perfectly elastic infinite cylinder made of an isotropic incompressible material. The oscillation follow a sudden setting of the cylinder into a uniform spinning motion about its longitudinal axis. By using Knowles's approach, the oscillations are analyzed for materials with essentially arbitrary constitutive equations. General conditions are found for the motions of the cylinder to be periodic. The results are specialized to three classes of rubberlike materials: Mooney‐Rivlin, Neo‐Hookean, and Ishiharaet al.‐Zahorski. Graphs are given (for some experimentally corroborated values of the elastic constants) for the frequency of oscillations and the maximum displacement of the inner radius of the tube versus the angular speed of the spin.