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The paper deals with the general equations for the vibration of thin cylinders and a theoretical and experimental investigation is made of the type of vibration usually associated with bells. The cylinders are supported in such a manner that the ends remain circular without directional restraint being imposed. It is found that the complexity of the mode of vibration bears little relation to the natural frequency; for example, cylinders of very small thicknessdiameter ratio, with length about equal to or less than the diameter, may have many of their higher frequencies associated with the simpler modes of vibration. The frequency equation which is derived by the energy method is based on strain relations given by Timoshenko. In this approach, displacement equations are evolved which are comparable to those of Love and Flugge, though differences are evident due to the strain expressions used by each author. Results are given for cylinders of various lengths, each with the same thickness-diameter ratio, and also for a very thin cylinder in which the simpler modes of vibration occur in the higher frequency range. It is shown that there are three possible natural frequencies for a particular nodal pattern, two of these normally occurring beyond the aural range.

Flexural vibrations of the walls of thin cylindrical shells having freely supported ends