Vibration analysis of thick-walled spheres and cylinders

A wave-propagation approach is employed to determine the fundamental frequency of hollow spheres and long hollow cylinders (plane strain state). The study was initiated in order to study the periodic response of such specimens as a function of the number of wave reflections off the inner and outer surfaces as well as the thickness-to-inside radius ratios, (h/Ri). It was hoped to achieve a relationship similar to that obtained in the classical longitudinal step loading bar problem. In the bar case, the material response is periodic for either two or four complete reflections, depending on whether the bar has a free or a fixed end. In the hollow sphere and cylinder problem, however, the analysis is not as simple. The motion is approximately periodic. It was found that the number of wave reflections for one cycle was some irrational number, the number being a function of (h/Ri). This unusual observation led to the development of useful design curves and equations for determining the fundamental frequencies of thick-walled spheres and cylinders. Comparisons of results from the vibration and wave propagation approaches are used to establish regions of validity of the approaches as a function of (h/Ri).