# Harmonic Standing-Wave Excitations of Simply-Supported Isotropic Solid Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

## TL;DR

This paper presents an exact 3D elastodynamic solution for the steady-state forced vibration of a simply-supported elastic cylinder under harmonic surface excitations, revealing resonance phenomena and providing a benchmark for future studies.

## Contribution

It introduces a novel exact analytical method for solving the 3D forced-vibration problem of elastic cylinders, expanding the understanding of their dynamic response.

## Key findings

- Resonance frequencies correspond to natural modes of the cylinder.
- Two analytical solutions are constructed for different harmonic excitations.
- Numerical evaluation shows series of resonances in the frequency response.

## Abstract

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 previously-obtained particular solutions to the Navier-Lam\'{e} equation of motion 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. Two complete analytical solutions are in fact constructed corresponding to two different but closely-related families of harmonic standing-wave excitations. The second of these analytical solutions is evaluated numerically in order to study the steady-state frequency response in some example excitation cases. In each case, the solution generates a series of resonances that are in correspondence with a subset of the natural frequencies of the simply-supported cylinder. The considered 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.