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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Additional Information

DOI:
https://doi.org/10.1090/qam/668740

Article copyright:
© Copyright 1978
American Mathematical Society