Vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach

According to the integral type constitutive relation of linear coupled thermoviscoelasticity, a mathematical model of thin plates is set up by the introduction of ‘‘structural functions” and ‘‘thermal functions” in the sense of the Kirchhoff’s hypothesis. The corresponding integral type variational formulations are presented by means of modern convolution bilinear forms as well as classical Cartesian bilinear forms. The Ritz method in the spatial domain and the differentiating method in the temporal domain are used to approximate the mathematical model in a system of rectangular Cartesian coordinates. By properties of inequality and parabola, the structure of dynamic solution to vibration of a thermoviscoelastic thin plate under a harmonic thermal load is studied in the space splayed by material parameter and loading parameter. The influences of thermal excitation frequency, mechanical relaxation time and thermal relaxation time on amplitude and phase difference of steady-state vibration of a square plate are investigated by amplitude-frequency analysis and phase-frequency analysis. Double-peak resonance vibration of thermoviscoelastic plates exists for given parameters.