Two-dimensional analysis of natural frequencies and mode shapes of a wide beam: Application to the determination of mechanical characteristics of viscoelastic materials

The theory of plane elastodynamics is used to provide a simple method for calculating the natural frequencies and the normal mode shapes of a wide rectangular beam. The boundary conditions at both ends are prescribed in a mean-value sense. It is shown that the elementary (vibrating string) beam theory turns out to be an approximation of the theoretical model; the results obtained agree with those given by the Love theory. The method enables one to predict all frequency branches in terms of the width-to-length ratio, by comparatively simple calculations, in contrast to the situation when sufficiently elaborate one-dimensional theories are used. A second application is the determination of the complex Young's modulus and Poisson's ratio for viscoelastic materials.