Bifurcation and stability analysis of a rotating beam
Authors:
Peter Gross, Metin Gürgöze and Wolfhard Kliem
Journal:
Quart. Appl. Math. 51 (1993), 701-711
MSC:
Primary 73H10; Secondary 34A47, 73K05
DOI:
https://doi.org/10.1090/qam/1247435
MathSciNet review:
MR1247435
Full-text PDF Free Access
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Abstract: We discuss small oscillations of an elastic beam clamped radially to the interior of a rotating ring. It is well known that if the speed of rotation is sufficiently high, the trivial equilibrium of the beam may lose stability and the beam buckles.
N. Mostaghel and I. Tadjbakhsh, Buckling of rotating rods and plates, Internat. J. Mech. Sci. 15, 429–434 (1973)
F. G. Rammerstorfer, Comment on buckling of rotating rods and plates, Internat. J. Mech. Sci. 16, 515–517 (1974)
A. Nachman, The buckling of rotating rods, J. Appl. Mech. 42, 222–224 (1975)
J. T. S. Wang, On the buckling of rotating rods, Internat. J. Mech. Sci. 18, 407–411 (1976)
- W. D. Lakin and A. Nachman, Unstable vibrations and buckling of rotating flexible rods, Quart. Appl. Math. 35 (1977/78), no. 4, 479–493. MR 668740, DOI https://doi.org/10.1090/S0033-569X-1978-0668740-9
D. A. Peters and D. H. Hodges, In-plane vibration and buckling of a rotating beam clamped off the axis of rotation, J. Appl. Mech. 47, 398–402 (1980)
M. Gürgöze, On the dynamical behaviour of a rotating beam, J. Sound and Vibration 143, 356–363 (1990)
H. I. Weber, A note on the stability of a rod subjected to compression by centrifugal force, J. Sound and Vibration 46, 105–111 (1976)
A. G. Hernried and G. B. Gustafson, On the dynamic response of a single-degree-of-freedom structure attached to the interior of a rotating rigid ring, J. Appl. Mech. 55, 201–205 (1988)
D. A. Peters and D. H. Hodges, Discussion on the dynamic response of a single-degree-of-freedom structure attached to the interior of a rotating rigid ring, J. Appl. Mech. 55, 747–748 (1988)
W. Kliem, Good and bad mathematical modelling in mechanics, Teaching of Mathematical Modelling and Applications, M. Niss, W. Blum, and I. Huntley (Eds.), Ellis Horwood, Chichester, 1991, pp. 384–392
A. C. Hearn, Reduce, User’s Manual Version 3.3, The Rand Corporation, Santa Monica, CA, Rand Publication CP 78 (Rev. 7/87), 1987
G. Rayna, Reduce: Software for Algebraic Computation, Springer-Verlag, New York, 1987
- Lothar Collatz, Eigenwertaufgaben mit technischen Anwendungen, Mathematik und ihre Anwendungen in Physik und Technik, Reihe A, Band 19, Akademische Verlagsgesellschaft, Leipzig, 1949 (German). MR 0031337
- Gérard Iooss and Daniel D. Joseph, Elementary stability and bifurcation theory, Springer-Verlag, New York-Berlin, 1980. Undergraduate Texts in Mathematics. MR 636256
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
N. Mostaghel and I. Tadjbakhsh, Buckling of rotating rods and plates, Internat. J. Mech. Sci. 15, 429–434 (1973)
F. G. Rammerstorfer, Comment on buckling of rotating rods and plates, Internat. J. Mech. Sci. 16, 515–517 (1974)
A. Nachman, The buckling of rotating rods, J. Appl. Mech. 42, 222–224 (1975)
J. T. S. Wang, On the buckling of rotating rods, Internat. J. Mech. Sci. 18, 407–411 (1976)
W. D. Lakin and A. Nachman, Unstable vibrations and buckling of rotating flexible rods, Quart. Appl. Math. 35, 479–493 (1978)
D. A. Peters and D. H. Hodges, In-plane vibration and buckling of a rotating beam clamped off the axis of rotation, J. Appl. Mech. 47, 398–402 (1980)
M. Gürgöze, On the dynamical behaviour of a rotating beam, J. Sound and Vibration 143, 356–363 (1990)
H. I. Weber, A note on the stability of a rod subjected to compression by centrifugal force, J. Sound and Vibration 46, 105–111 (1976)
A. G. Hernried and G. B. Gustafson, On the dynamic response of a single-degree-of-freedom structure attached to the interior of a rotating rigid ring, J. Appl. Mech. 55, 201–205 (1988)
D. A. Peters and D. H. Hodges, Discussion on the dynamic response of a single-degree-of-freedom structure attached to the interior of a rotating rigid ring, J. Appl. Mech. 55, 747–748 (1988)
W. Kliem, Good and bad mathematical modelling in mechanics, Teaching of Mathematical Modelling and Applications, M. Niss, W. Blum, and I. Huntley (Eds.), Ellis Horwood, Chichester, 1991, pp. 384–392
A. C. Hearn, Reduce, User’s Manual Version 3.3, The Rand Corporation, Santa Monica, CA, Rand Publication CP 78 (Rev. 7/87), 1987
G. Rayna, Reduce: Software for Algebraic Computation, Springer-Verlag, New York, 1987
L. Collatz, Eigenwertaufgaben mit technischen Anwendungen, Akademische Verlagsgesellschaft, Leipzig, 1963
G. Iooss and D. D. Joseph, Elementary Stability and Bifurcation Theory, Springer-Verlag, New York, 1980
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955
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Article copyright:
© Copyright 1993
American Mathematical Society
