Study of pure transverse motion in free cylinders and plates in flexural vibration by Ritz’s method

In this work the flexural vibration of a free cylinder of any aspect ratio is analysed. A general solution by powers series of the coordinates is proposed here to represent the displacements, with restrictions on the powers of the radial coordinates which prevent potential energy and stress singularities at the axis of the cylinder. By means of an analytic method, it is concluded that certain points of the cylinder have no axial motion. As a result of the pure transverse movement and of the fact that the cylinder bends, it is inferred that the axis is extended. Furthermore, in the symmetric modes, the points situated at the centres of the bases are displaced in the same direction and sense, and hence the distance between them does not vary in time. Flexural natural frequencies are numerically calculated by Ritz’s method with the general solution series proposed. Since the series used are more adequate, convergence is better than with classic series. The results are verified by FEM. Some consequences are extended to a rectangular plate, whose points of the middle surface vibrate transversally in the double-symmetric mode. In order to verify the theoretical results, a set of experiments with a laser interferometer is carried out. The experimental frequencies agree with the theoretical values.