Major simplifications in a current linear model for the motion of a thermoelastic plate

A dynamic model for a thin thermoelastic plate proposed by Lagnese and Lion s in 1988 [1] has been used recently by several authors (e.g., [2]-[5]) to study existence and stability of solutions to initial/boundary-value proble ms. Simple, systematic order-of-magnitude arguments show that it is consist ent to neglect several terms appearing in the governing differential equati ons that couple a temperature moment to the average vertical displacement. Further, because the time scale on which the temperature adjusts itself to the strain rate contribution to the energy equation is quite small compared with the longest (isothermal) period of free vibration of the plate, the e nergy equation can be solved for the temperature in terms of derivatives of the vertical displacement and hence the system reduced to a single equatio n, only slightly more complicated than the classical (Kirchhoff) equation o f motion. Among other things, it is shown that the temperature has a cubic rather than a linear variation through the thickness. Finally, another orde r-of-magnitude estimate for a clamped aluminum plate of one meter radius an d 1mm thickness shows that thermal damping acting alone takes on the order of 200 cycles of vibration to halve the initial amplitude.