Vibrations of transversely isotropic finite circular cylinders

The exact frequency equations for all the possible natural vibrations in a transversely isotropic cylinder of finite length are investigated. Two wave potentials are used to uncouple the equations of motion; the resulting hyperbolic equations are solved analytically for the vibration frequencies of a finite cylinder with zero tractions on the curved surfaces. In general, the mode shapes and the frequency equations of vibrations depend on both the range of the frequency and elastic properties of the material. The vibration frequencies for sapphire cylinders are studied as an example. The lowest natural frequency for longitudinal and circumferential modes always corresponds to the first nonsymmetric mode regardless of the cylinder dimension.