The material scientist is mainly interested in the mechanisms and an accurate measurement of the intrinsic material damping. The structural mechanician, on the other hand, is primarily interested in predicting the damping of a structure. The central objective of this paper is to establish a fundamental connection between material damping (which is an intrinsic or intensive property) and structural damping (which is an extrinsic or extensive property). As our starting point, we assume that the intrinsic material damping properties have been measured, and that an elasticity solution to the structural vibration problem has been found. By using the Correspondence Principle of linear viscoelasticity, we derive an exact solution for the viscoelastic structural damping.
In an earlier work, Kalyanasundaram, Allen, and Schapery obtained the viscoelastic structural damping of a Timoshenko beam with pinned ends from the complex frequency equation by separating its real and imaginary components. However, for more complex structures, this procedure is lengthy and we instead linearize the viscoelastic frequency equation in a complex Taylor series expansion. We show that the results of Kalyanasundaram et al. can be readily obtained. Then, by way of illustration, we solve the problem of a Timoshenko beam with inertial boundary conditions, that is, with a mass and a rotary inertia attached to both ends.