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12.—Spectral Theory of Rotating Chains

Published online by Cambridge University Press:  14 February 2012

C. A. Stuart
C. A. Stuart
Affiliation:
Battelle Institute, 7 Route de Drize, Carouge, Geneva
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Synopsis

Two eigenvalue problems associated with steady rotations of a chain are considered. To compare the spectra of these two problems, let σK(n) denote the set of all angular velocities with which a chain of unit length with one end fixed and the other free can rotate in a vertical plane so as to have exactly n nodes on thevertical axis (including the fixed end). In a linearised theory σK(n) is a single point, i.e.

In the full non-linear theory σK(n) is an infinite interval lying to the right of ⍵n. Indeed,

This is established in [1].

Next, let σM(β, n) denote the set of all angular velocities with which a chain, having ends fixed at unit distance apart on the vertical axis, can rotate in a vertical plane so as to have exactly n+l nodes on the vertical axis (including the ends) and so that the tension takes the value β at the lower end. This problem, in which the length of the chain is not prescribed is a model for a spinning process in which the ‘chain’ is continuously created in a rotating configuration. For β>0, we again have in the linearised theory that σM(β, n) is a singleton, i.e.

In the full non-linear theory σM(β, n) lies to the left of λn(β). Although unable to determine exactly σM(β, n) for β>0, we have

where ⍵n λn(β), and are all characterised as the nth zeros of known combinations of Bessel functions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 73 , 1975 , pp. 199 - 214
Copyright
Copyright © Royal Society of Edinburgh 1975

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