
CivilComp Proceedings ISSN 17593433
CCP: 75 PROCEEDINGS OF THE SIXTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping and Z. Bittnar
Paper 32 Dynamic Stiffness Formulation and Free Vibration Analysis of Spinning Beams
J.R. Banerjee and H. Su School of Engineering, City University, London, United Kingdom
Full Bibliographic Reference for this paper
J.R. Banerjee, H. Su, "Dynamic Stiffness Formulation and Free Vibration Analysis of Spinning Beams", in B.H.V. Topping, Z. Bittnar, (Editors), "Proceedings of the Sixth International Conference on Computational Structures Technology", CivilComp Press, Stirlingshire, UK, Paper 32, 2002. doi:10.4203/ccp.75.32
Keywords: dynamic stiffness method, free vibration, spinning beam, WittrickWilliams algorithm.
Summary
There has been a growing interest in the investigation of free vibration
characteristics of spinning beams because the topic plays an important role in the
design of shafts, turbine blades, propellers and many other spinning
structures [ 1, 2, 3, 4, 5, 6, 7, 8].
Bauer [ 1] was one of the first few investigators who presented an analytical method
for the free vibration analysis of a spinning BernoulliEuler beam using the classical
approach of differential equations. Chen and Liao [ 2] employed an assumedmodes
method for solving the free vibration problem of pretwisted spinning beams. There
are other related studies [ 3, 4] in which finite element methods have been used to
extend the general application of the problem to complex structural systems. For
example, Yu and Cleghorn [ 3] investigated the free vibration problems of a spinning
stepped beam by applying the Timoshenko theory to idealise the stepped beam as a
series of uniform elements.
The current investigation is, however, focused on an alternative approach, namely
the dynamic stiffness method, thus allowing an exact free vibration analysis of
spinning beams. This appears to be the first study that uses the dynamic stiffness
method to solve the free vibration problem of spinning beams.
The investigation is carried out in following steps:
 The governing differential equations of a spinning beam undergoing free
vibration are derived using Hamilton's principle. This leads to two fourth
order partial differential equations which couple the vibratory motion of the
beam in two principal planes.
 For harmonic oscillation, the above two equations are combined into one
eighth order ordinary differential equation by eliminating the time dependence
of the displacement functions.
 The complete analytical solution of the above equation is then obtained in
terms of eight arbitrary constants. Exact expressions for bending rotations,
bending moments and shear forces at any crosssection of the beam are also
obtained in terms of eight separate, but interrelated sets of constants.
 The dynamic stiffness matrix is developed by applying the boundary
conditions for displacements and forces at the ends of the beam, and by
eliminating the eight arbitrary constants.
 The WittrickWilliams algorithm [9] is applied to the resulting dynamic
stiffness matrix to yield natural frequencies and mode shapes of spinning
beams.
Numerical results for natural frequencies and mode shapes of spinning beams are
obtained using the present theory, which show excellent agreement with published
results. The method is exact and can be used as an aid to validate finite element and
other approximate methods.
References
 1
 Bauer, H.F., "Vibration of a Rotating Uniform Beam, Part I: Orientation in the Axis of Rotation", Journal of Sound and Vibration, 72, 177189, 1980. doi:10.1016/0022460X(80)906513
 2
 Chen, M.L., and Liao, Y.S., "Vibrations of Pretwisted Spinning Beams under Axial Compressive Loads with Elastic Constraints", Journal of Sound and Vibration, 147, 497513, 1991. doi:10.1016/0022460X(91)904978
 3
 Yu, S.D., and Cleghorn, W.L., "Free Vibration of a Spinning Stepped Timoshenko Beam", Journal of Applied MechanicsTransactions of the ASME, 67, 839841, 2000. doi:10.1115/1.1331282
 4
 Filipich, C.P., Maurizi, M.J. and Rosales, M.B., "Free Vibrations of a Spinning Uniform Beam with Ends Elastically Restrained against Rotation", Journal of Sound and Vibration, 116, 475482, 1987.
 5
 Song, O., and Librescu, L., "Anisotropy and Structural Coupling on Vibration and Instability of Spinning Thinwalled Beams", Journal of Sound and Vibration", 204, 477494, 1997. doi:10.1006/jsvi.1996.0947
 6
 Zu, J.W., and Melanson, J., "Natural Frequencies and Normal Modes for Externally Damped Spinning Timoshenko Beams with General Boundary Conditions", Journal of Applied MechanicsTransactions of the ASME, 65, 770772, 1998. doi:10.1115/1.2789123
 7
 Song, O., Jeong, N.H., and Librescu, L., "Implication of Conservative and Gyroscopic Forces on Vibration and Stability of an Elastically Tailored Rotating Shaft Modeled as a Composite Thinwalled Beam", Journal of The Acoustical Society of America, 109, 972981, 2001. doi:10.1121/1.1348301
 8
 Lee, H. P., "Dynamic Stability of Spinning Beams of Unsymmetrical Cross section with Distinct End Conditions", Journal of Sound and Vibration, 189, 161171, 1996. doi:10.1006/jsvi.1996.0013
 9
 Wittrick, W.H. and Williams. F.W., "A General Algorithm for Computing Natural Frequencies of Elastic Structures", Quarterly Journal Mechanics and Applied Mathematics, 24, 263284, 1971. doi:10.1093/qjmam/24.3.263
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