The analysis and validation of natural frequencies and mode shapes of 3D plates in the framework of the generalized thermoelastic theory

Abstract

This article considers thermal effects that induce thermoelastic damping and, which, consequently, because of the absorbed thermal energy, cause thermal stresses. The problem is stated in the framework of the generalized thermoelastic theory. The natural frequency and mode shape analyzes of 3D plates are presented under uniform and nonuniform heating. From the modeling point of view, this results in the quadratic eigenvalue problem where complex frequencies and modes are obtained. Moreover, the quality factor related to the thermoelastic damping is also calculated in terms of the imaginary and the real parts of the frequencies for both the uniform and nonuniform thermal distributions. The variation of natural frequencies with temperature is investigated and compared with the experimental results for the plates with free boundary conditions. The first two mode shapes are also examined for all clamped edges for the uniform and nonuniform cases. Complex frequencies and complex modes are observed due to the thermal effects. For the uniform case, the real part of the modes decreases and the complex part increases, while the mode’s norm remains the same as in the classical modes as temperature increases. For the nonuniform case, the mode shape analysis is performed under the assumption that the elasticity modulus, thermal expansion, and specific heat parameters are functions of temperature. The peak point of the out-of-plane displacement is shown to be shifted toward the warm zone. The obtained outcomes may help design thermal structures and allow for future extensions.