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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W. D. Lakin, Vibrations of a rotating flexible rod clamped off the axis of rotation, J. Eng. Math. 10, 313–321 (1976)
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R. A. Mathon, private communication
J. C. P. Miller. On the choice of standard solutions for a homogeneous linear differential equation of the second order, Quart. J. Mech. Appl. Math. 3, 225–235 (1950)
N. Mostaghel and I. Tadjbachsh, Buckling of rotating rods and plates, Int. J. Mech. Sci. 15, 429–434 (1973)
A. Nachman, The buckling of rotating rods, J. Appl. Mech. 42, 222–224 (1975)
Jahnke-Emde, Tables of higher functions, McGraw-Hill, New York, 1960
W. H. Reid, Composite approximations to the solutions of the Orr-Sommerfeld equation, Studies Appl. Math. 51, 341–368 (1972)
W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation, part 2: the general theory, Studies Appl. Math. 53, 217–224 (1974)
Y. Luke, Integrals of Bessel functions, McGraw-Hill, New York, 1962
F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974
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Article copyright:
© Copyright 1978
American Mathematical Society
