Thermomechanical response of a viscoelastic rod driven by a sinusoidal displacement

The forced vibrations of a rod of thermoviscoelastic material are studied. The rod is considered to be laterally insulated but not constrained, such that a one-dimensional analysis may be employed. Temperature dependence of the material properties and the resulting thermomechanical coupling effects are included. The vibrations are forced by the imposition of a sinusiodal displacement of known amplitude and frequency at one end of the rod. This problem corresponds to a dissipative material bonded to the surface of a relatively rigid, vibrating structure.Initial transient behavior is not considered. A steady-state response is found by means of a finite difference formulation. Material properties of a Lockheed solid propellant are used.The presence of critical frequencies, characterized by high stresses and temperatures, is found for small amplitudes of vibration. Nonlinearities and instabilities lead to a lack of one-to-one correspondence between stress and displacement boundary conditions. No relationship is found between the critical frequencies of the driven rod and the natural frequencies of a rod with an equivalent temperature profile.