Harmonic Standing-Wave Excitations of Simply-Supported Thick-Walled Hollow Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

TL;DR

This paper derives exact 3D solutions for the steady-state vibration response of simply-supported thick-walled elastic cylinders under harmonic surface excitations, revealing resonance behaviors linked to natural frequencies.

Contribution

It introduces a semi-analytical method to solve the elastodynamic response of elastic cylinders under standing-wave excitations, applicable to various boundary conditions and excitation scenarios.

Findings

Resonance frequencies correspond to natural modes of the cylinder.

The method provides a direct approach for solving forced-vibration problems.

Solutions serve as benchmarks for structural mechanics and acoustics.

Abstract

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lam\'{e} equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.