Estimation of fundamental frequency of non-circular plates with free, circular cutouts

A method of obtaining approximate fundamental frequencies of slightly non-circular plates with free circular cutouts is developed. An approximate expression for the radius of each bounding curve is written as a truncated Fourier series. The support conditions, which are written in terms of a perturbation series of the modes of a circular annulus, are satisfied on the approximate boundaries. The approximate characteristic equation (either first or second-order approximations) is obtained from satisfaction of the support conditions and the fundamental frequency determined as the first root to this characteristic equation. Approximate frequencies of the fundamental mode of a clamped elliptical plate, square plate with a circular cutout and an eccentric annulus are determined to demonstrate the second-order approximation. In addition, the first-order approximation to the fundamental mode of an eccentric annulus is obtained.