On the use of the Sturm sequence to evaluate modal density

Abstract

The modal density of a structural-acoustic subsystem is usually obtained analytically by considering the dispersion of various propagating wave types. Closed form expressions are available for the modal densities of simple beams, plates and shells (with curvature in one or two directions). However, one is often interested in subsystems with complex geometry which may possess inhomogeneous material and physical properties. The classical asymptotic formulations are not appropriate for such subsystems and numerical methods are often adopted. The most straightforward approach is to perform a finite-element-based modal analysis and count the number of eigenvalues that fall within various frequency bands. However, the computational expense associated with solving the full eigenproblem is often prohibitive. Significant computational savings can be made by employing the Sturm sequence property to evaluate the modal density. This paper describes the approach in more detail and provides a numerical example.