An analytical, numerical, and experimental study of the axisymmetric vibrations of a short cylinder

This paper presents a study of standing axisymmetric waves in homogeneous, isotropic, linear, and elastic cylinders whose length and diameter are of the same order of magnitude. It is analytically demonstrated that only materials with a Poisson's ratio equal to zero can vibrate in pure radial and longitudinal axisymmetric modes and, hence, the universal frequencies, independent of Poisson's ratio, and the corresponding universal aspect ratios can be accurately calculated. The Ritz method is applied to numerically determine the natural frequencies and the mode shapes of finite cylinders. Vibration of a cylinder in the neighbourhood of the universal point is analysed. Both free and forced vibrations are experimentally detected by using a laser speckle interferometer that allows the determination of the natural frequencies and the mode shapes. From the numerical calculations, it is confirmed that, for materials with Poisson's ratio not equal to zero, no crossings occur between the curves of non-dimensional frequency versus aspect ratio, neither for antisymmetric modes themselves nor for symmetric modes. There is a good agreement between the analytical solutions, the numerical results, and the experimental measurements.