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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] William E. Boyce, VIBRATIONS OF ROTATING BEAMS, ProQuest LLC, Ann Arbor, MI, 1955. Thesis (Ph.D.)–Carnegie Mellon University. MR 2938649
  • [2] D. J. Gorman, Free vibration analysis of beams and shafts, Wiley, New York, 1975
  • [3] E. Hille, Some problems concerning spherical harmonics, Ark. Mat., Astron., Fys. 13, 1-76 (1918)
  • [4] E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
  • [5] W. D. Lakin, On the differential equation of a rapidly rotating slender rod, Quart. Appl. Math. 32, 11-27 (1974)
  • [6] W. D. Lakin, Vibrations of a rotating flexible rod clamped off the axis of rotation, J. Engrg. Math. 10 (1976), no. 4, 313–321. MR 0416186, https://doi.org/10.1007/BF01535567
  • [7] W. D. Lakin and B. S. Ng, A fourth-order eigenvalue problem with a turning point at the boundary, Quart. J. Mech. Appl. Math. 28 (1975), 107–121. MR 0361332, https://doi.org/10.1093/qjmam/28.1.107
  • [8] R. A. Mathon, private communication
  • [9] 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 (1950), 225–235. MR 0035901, https://doi.org/10.1093/qjmam/3.2.225
  • [10] N. Mostaghel and I. Tadjbachsh, Buckling of rotating rods and plates, Int. J. Mech. Sci. 15, 429-434 (1973)
  • [11] A. Nachman, The buckling of rotating rods, J. Appl. Mech. 42, 222-224 (1975)
  • [12] Jahnke-Emde, Tables of higher functions, McGraw-Hill, New York, 1960
  • [13] W. H. Reid, Composite approximations to the solutions of the Orr-Sommerfeld equation, Studies in Appl. Math. 51 (1972), 341–368. MR 0347225
  • [14] W. H. Reid, Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory, Studies in Appl. Math. 53 (1974), 217–224. MR 0363150
  • [15] Yudell L. Luke, Integrals of Bessel functions, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1962. MR 0141801
  • [16] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. MR 0435697

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73.35

Retrieve articles in all journals with MSC: 73.35

Additional Information

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

American Mathematical Society