Vibrations of Transversely Isotropic Finite Circular Cylinders

Abstract

This paper investigates the exact frequency equations for all the possible natural vibrations in a transversely isotropic cylinder of finite length. 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 shear tractions and zero axial displacement on the end surfaces and 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 the elastic properties of the material. The vibration frequencies for sapphire cylinders are studied as an example. Two limiting cases are also considered: the long bar limit equals the frequency equation for the longitudinal vibration of bars obtained by Morse (1954) and by Lord Rayleigh (1945); and the frequency equation for thin disks (small length/radius ratio) is also obtained. The frequency for the first axisymmetric mode agrees with the experimental observation by Lusher and Hardy (1988) to within one percent. Natural frequencies for the first three longitudinal and circumferential modes are plotted for all cylinder geometries. The lowest frequency always corresponds to the first nonsymmetric mode regardless of the dimension of the cylinder. For axisymmetric vibration modes, numerical plots show that double roots exist in the frequency equations; such doublets were observed experimentally by Booker and Sagar (1971).