Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



Unstable vibrations and buckling of rotating flexible rods

Authors: W. D. Lakin and A. Nachman
Journal: Quart. Appl. Math. 35 (1978), 479-493
MSC: Primary 73.35
DOI: https://doi.org/10.1090/qam/668740
MathSciNet review: 668740
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Abstract: We consider a group of fourth-order boundary-value problems associated with the small vibrations or buckling of a uniform flexible rod which is clamped at one end and rotates in a plane perpendicular to the axis of rotation. The vibrations may be in any plane relative to the plane of rotation and the rod is off-clamped, i.e. the axis of rotation does not pass through the rod's clamped end. The governing equation for the vibrations involves a small parameter for rapid rotation and must be treated by singular perturbation methods. Further, a turning point of the equation always coincides with a boundary point. Both free and unstable vibrations are examined, and a stability boundary is obtained. Results for the unstable vibrations predict the unexpected existence of a time-independent buckled state in non-transverse planes when the rod is wholly under tension. The general buckling problem in the transverse plane is also considered.

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DOI: https://doi.org/10.1090/qam/668740
Article copyright: © Copyright 1978 American Mathematical Society

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