Harmonic vibrations of thick visco-elastic, infinitely long, circular cylinders subjected to boundary stresses are investigated. The cylinder is assumed to be homogeneous, isotropic, and linearly visco-elastic. The inelastic behavior of some metallic materials leads to dissipation of energy in the medium. The governing equation of motion is developed using modified theory of elastodynamics. The material damping is allowed using complex elastic moduli for the medium. Frequency responses for radial, tangential and axial displacements, as well as stresses at any location, are formulated using potential functions. The responses are computed for different circumferential and axial wave numbers for a given ratio of thickness to radius of the cylinder. The frequency range is extended to include five normalized resonant frequencies. The resonant frequency and loss factor for each mode are extracted using a circle-fit procedure in conjunction with finite difference analysis. The resonant frequency is deduced by estimating a maximum rate of sweep on the circle-fit and the loss factor is determined using selected data points on both sides of the resonance. Computation is performed to determine the effect of material loss factor on the overall damping effectiveness for the structure. Analysis was conducted by estimating resonant frequencies and modal loss factors for several circumferential and thickness to wavelength numbers.

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