Single degree of freedom model for thermoelastic damping

Finding the thermoelastic damping in a vibrating body, for the most general case, involves the simultaneous solving of the three equations for displacements and one equation for temperature (called the heat equation). Since these are a set of coupled nonlinear partial differential equations there is considerable difficulty in solving them, especially for finite geometries. This paper presents a single degree of freedom (SDOF) model that explores the possibility of estimating thermoelastic damping in a body, vibrating in a particular mode, using only its geometry and material properties, without solving the heat equation. In doing so, the model incorporates the notion of "modal temperatures," akin to modal displacements and modal frequencies. The procedure for deriving the equations that determine the thermoelastic damping for an arbitrary system, based on the model, is presented. The procedure is implemented for the specific case of a rectangular cantilever beam vibrating in its first mode and the resulting equations solved to obtain the damping behavior. The damping characteristics obtained for the rectangular cantilever beam, using the model, is compared with results previously published in the literature. The results show good qualitative agreement with Zener's well known approximation. The good qualitative agreement between the predictions of the model and Zener's approximation suggests that the model captures the essence of thermoelastic damping in vibrating bodies. The ability of this model to provide a good qualitative picture of thermoelastic damping suggests that other forms of dissipation might also be amenable for description using such simple models.