The hanging thin rod: A singularly perturbed eigenvalue problem

2010/08/11 by Yossi Farjoun, Farjoun, Yossi, David G. Schaeffer +1
Computer Science · Engineering · Mathematics · #Adhesion, Friction, and Surface Interactions #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #math.CA

paper · pdf · doi:10.48550/arxiv.1008.1912

28 pages, 6 figures, 2 tables

arxiv created 2010/08/11 · openalex publication_date 2010/08/11 · arxiv updated 2010/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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