An Alternative Representation of the Elastic-Viscoelastic Correspondence Principle for Harmonic Oscillations

An alternative representation of the elastic-viscoelastic correspondence principle is derived for solids with identical damping characteristics in bulk and shear undergoing steady-state harmonic motion. This form is particularly useful when the elastic solution of the mechanical system is not available in closed form but is known only numerically, say as a tabular list. The analyticity property of the frequency response function is utilized to formulate a Dirichlet problem in the lower half of the complex plane with the elastic solution on the real line supplying the boundary data. The expression for the viscoelastic frequency response function is then obtained as an infinite integral in which the elastic frequency response and the viscoelastic parameters constitute the integrand. This integral may be evaluated numerically by quadrature to any desired degree of accuracy by suitably increasing the range of integration and employing a finer mesh. Any isolated singularity in the elastic response, like poles at the resonant frequencies, can be very accurately handled by using exact complex integration in the sense of Cauchy principal value. A simple example is presented to illustrate an application of this alternative formulation.