Frequency spectra of extensional vibration in isotropic shortcolumns and thick circular disks

A simple analytical method based on the equivalent elastic theory is presented for predicting the natural frequencies of a short column or a thick circular disk of axially symmetrical coupled vibration in which the longitudinal, the radial and the thickness extensional vibrations are considered. A critical value of thickness/diameter ratio is given that defines a boundary between a short column and a thick disk. Theoretical analyses show that the longitudinal and radial vibrations in infinite length rods, the radial and thickness extensional vibrations in thin plate are the special vibrating modes of a short column or a thick disk. The calculated frequencies for an isotropic column or a disk are in good agreement with the measured results