Abstract
The exact analytic solutions are obtained with the use of the eigenvalue approach for a free vibration problem of a thermoelastic hollow cylinder in the context of Green and Naghdi theory (GNIII). The dispersion relations for the existence of various types of possible modes of vibrations in the considered hollow cylinder are derived in a compact form and the validation of the roots for the dispersion relation is presented. To illustrate the analytic results, the numerical solution of various relations and equations has been carried out to compute the frequency, thermoelastic damping and frequency shift of vibrations in a hollow cylinder of copper material with MATHEMATICA and MATLAB software.