Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



On the differential equation of a rapidly rotating slender rod

Author: W. D. Lakin
Journal: Quart. Appl. Math. 32 (1974), 11-27
DOI: https://doi.org/10.1090/qam/99690
MathSciNet review: QAM99690
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Abstract | References | Additional Information

Abstract: In this work we consider a boundary-value problem arising from the transverse vibrations of a slender, finite, uniform rod which rotates with constant angular velocity about an axis through the rod's fixed end. The relevant dimensionless parameter is assumed to lie in a range corresponding to rapid rotation. The differential equation in this problem is fourth-order, linear, and takes its distinctive character from the simple turning point where the coefficient of the second derivative term vanishes. A significant feature is that the turning point is also a boundary point and hence outer expansions alone are not adequate for formation of a characteristic equation. Approximations valid at and away from the turning point are obtained and related through the method of matched asymptotic expansions. Outer expansions are required to be ``complete'' in the sense of Olver, and approximations are found for the Stokes multipliers which describe the analytic continuations of these expansions across Stokes lines in the complex plane. A consistent approximation to the characteristic equation is obtained, and the limiting behavior of the spectrum is derived analytically.

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Additional Information

DOI: https://doi.org/10.1090/qam/99690
Article copyright: © Copyright 1974 American Mathematical Society

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