The heterogeneous helicoseir

TL;DR

This paper investigates the equilibrium shapes and eigenmodes of a rotating, heavy, non-uniform string (helicoseir) under gravity, generalizing previous results for uniform strings and revealing new properties of solutions.

Contribution

It extends the analysis of rotating strings to a class of non-uniform densities, proving orthogonality and node properties of solutions, and linking nonlinear eigenmodes to linear spectrum.

Findings

Eigenmodes correspond to equilibrium configurations.

Solutions are orthogonal and have hierarchical node properties.

Generalization to a class of density functions including the uniform case.

Abstract

We study the rotations of a heavy string (helicoseir) about a vertical axis with one free endpoint and with arbitrary density, under the action of the gravitational force. We show that the problem can be transformed into a nonlinear eigenvalue equation, as in the uniform case. The eigenmodes of this equation represent equilibrium configurations of the rotating string in which the shape of the string doesn't change with time. As previously proved by Kolodner for the homogenous case, the occurrence of new modes of the nonlinear equation is tied to the spectrum of the corresponding linear equation. We have been able to generalize this result to a class of densities $\rho(s) = \gamma (1-s)^{\gamma-1}$, which includes the homogenous string as a special case ($\gamma=1$). We also show that the solutions to the nonlinear eigenvalue equation (NLE) for an arbitrary density are orthogonal and that a solution of this equation with a given number of nodes contains solutions of a different helicoseir, with a smaller number of nodes. Both properties hold also for the homogeneous case and had not been established before.