The hanging thin rod: A singularly perturbed eigenvalue problem

TL;DR

This paper analyzes the vibrational eigenmodes of a hanging thin flexible rod, employing asymptotic methods to address the singular perturbation caused by the small bending elasticity relative to tension.

Contribution

It introduces an asymptotic solution approach for a singularly perturbed eigenvalue problem modeling the vibrations of a hanging rod, highlighting key analytical techniques.

Findings

Asymptotic eigenvalue correction derived for the model

Demonstration of modeling and nondimensionalization techniques

Application of asymptotic series to singular perturbation problems

Abstract

We study the vibrations of a hanging thin flexible rod, in which the dominant restoring force in most of the domain is tension due to the weight of the rod, while bending elasticity plays a small but non-negligible role. We consider a linearized description, which we may reduce to an eigenvalue problem. We solve the resulting singularly perturbed problem asymptotically up to the first modification of the eigenvalue. On the way, we illustrate several important problem-solving techniques: modeling, nondimensionalization, scaling, and especially use of asymptotic series.