Analytical and numerical solutions for thick beams with thermoelastic damping

Abstract

In this work, an analytical solution for thermoelastic damping (TED) quality factor in beams based on Timoshenko beam theory has been proposed along the lines of a previous analytical solution obtained by Lifshitz and Roukes. Heat transfer in the axial direction of the beam was neglected while deriving the analytical solution. A numerical approach using the spectral element method (SEM) was implemented for obtaining the TED quality factor in a Timoshenko beam. Heat transfer in both the axial and thickness directions was considered for obtaining the numerical solution. The two-dimensional heat conduction problem was transformed into an one-dimensional problem by using a weighted residual technique.Quality factors were obtained numerically using eigenfrequency analysis and energy approach. The results from both analytical method and SEM were compared with the analytical models and three-dimensional finite element solutions. The analytical solution based on Timoshenko beam model gave a better result when compared to the analytical model based on the Euler–Bernoulli model. Numerical solutions are in good agreement with both analytical results and three-dimensional finite element results when the aspect ratio (L/h) is high. The numerical results were closer to the three-dimensional solutions as thickness increased. It has been shown that heat transfer in the axial direction cannot be ignored while computing quality factor attributable to TED in thick beams.