Analysis of flexural vibrations of a circular disk

The classical theory of flexural vibrations of a circular plate by Poisson and Kirchhoff is not accurate enough to design thick disks vibrating in axisymmetric or nonaxisymmetric modes, which are often used in a mechanical filter. The present analysis is based on the approximate method previously applied to vibrations of a rectangular plate.Itl The equation governing the relation between wave number and frequency of cylindrical waves in a plate is just the same as dispersion equations for plane waves in a plate. The dispersion equations have an infinite number of imaginary and complex branches as well as the usual real branches. Hence, an infinite number of independent cylindrical waves are obtained which satisfy both the differential equation of motion and the boundary condition on the major surfaces of a disk. A linear combination of such waves is employed to approximate the boundary condition on the lateral surface. Frequency spectra of modes of circumferential order up to three are calculated and compared with those predicted by the classical theory and those experimentally obtained from steel disks. Two kinds of approximations are tried. In the first approximation, the first two branches, real and purely imaginary, of flexural waves are taken into account. Whereas, in the second approximation, the first purely imaginary branch of torsional waves is further taken into account. The agreement between the observed and calculated values was excellent even for a disk with a very small diameter-to-thickness ratio, for which the classical theory predicts too high resonant frequencies. It is also found that significant contributions of torsional waves are seen only at the lowest vibration of the circumferential order of two and three.