The vibration of a solid elastic cylinder is one of the classical applied problems of elastodynamics. Many fundamental forced-vibration problems involving solid elastic cylinders have not yet been studied or solved using the full three-dimensional (3D) theory of linear elasticity. One such problem is the steady-state forced-vibration response of a simply-supported isotropic solid elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surface. In this paper, we exploit certain recently-obtained particular solutions to the Navier–Lamé equation and exact matrix algebra to construct an exact closed-form 3D elastodynamic solution to the problem. The method of solution is direct and demonstrates a general approach that can be applied to solve other similar forced-vibration problems involving elastic cylinders. The obtained analytical solution is then applied to a specific numerical example, where it is used to determine the frequency response of the displacement field in some low wave number excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a subset of the natural frequencies of the simply-supported cylinder. The analytical solution is also used to compute the resonant mode shapes in some selected asymmetric excitation cases. The studied problem is of general interest both as an exactly-solvable 3D elastodynamics problem and as a benchmark forced-vibration problem involving a solid elastic cylinder.
Effective properties of an isotropic solid weakened by micro-cracks located at inter-granular boundaries