Solution of dynamic problems for perforated structures by the method of averaging

Abstract

We consider transverse oscillations of a membrane with a square network of perforations whose contour is clamped and whose aperture edges are free. We apply the method of averaging in conjunction with variational methods for solving “cell problems.” The eigenfunctions and characteristic frequencies of oscillation are represented as asymptotic series in powers of a small parameter that characterizes the period of the lattice. The problem is solved in stages: first the “cell problem,” then the principal parts of the eigenfunctions and characteristic frequencies are determined from the averaged equation. To determine the first correction to the frequency we use the averaged equation for the next approximation and the self-adjoint property of the original operator.