Using energy methods to compare linear vibration models

In this paper, it is proved that the series representation of the solutions of the general linear vibration model with modal damping, is valid with respect to the energy norm. The partial sums of the series representation of solutions may therefore be used to compare different models. Parseval-type relationships are obtained for the eigenfunction expansions of functions. These relationships are used to derive expressions for the relative error in energy for partial sum approximations of functions. An example is included where a string model (wave equation model) and a modified Euler-Bernoulli beam model for the transverse vibration of a steel wire are compared.