Effect of symmetrical boundary conditions on the vibration of thick hollow cylinders

A finite element for axisymmetric elasticity is formulated directly in cylindrical coordinates. The element is used to study the vibration of hollow, isotropic finite length cylinders. The ratio of the inside radius to the outside radius is assumed as 0.2, 0.5 and 0.9 for a given cylinder length. Frequencies are computed for free-free end boundary conditions that can be compared with the existing literature. Additionally, fixed-fixed boundary conditions are studied. Five different configurations for simple supported boundary conditions are analyzed and compared. Numerical results are given in tabular form and some fundamental mode shapes are compared graphically.