Keywords: dynamic stiffness method, free vibration, spinning beam, Wittrick-Williams algorithm.

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 Bernoulli-Euler beam using the classical approach of differential equations. Chen and Liao [2] employed an assumed-modes method for solving the free vibration problem of pre-twisted 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:

1. 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.
2. 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.
3. 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 cross-section of the beam are also obtained in terms of eight separate, but interrelated sets of constants.
4. 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.
5. The Wittrick-Williams 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.

1

Bauer, H.F., "Vibration of a Rotating Uniform Beam, Part I: Orientation in the Axis of Rotation", Journal of Sound and Vibration, 72, 177-189, 1980. doi:10.1016/0022-460X(80)90651-3

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, 497-513, 1991. doi:10.1016/0022-460X(91)90497-8

3

Yu, S.D., and Cleghorn, W.L., "Free Vibration of a Spinning Stepped Timoshenko Beam", Journal of Applied Mechanics-Transactions of the ASME, 67, 839-841, 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, 475-482, 1987.

5

Song, O., and Librescu, L., "Anisotropy and Structural Coupling on Vibration and Instability of Spinning Thin-walled Beams", Journal of Sound and Vibration", 204, 477-494, 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 Mechanics-Transactions of the ASME, 65, 770-772, 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 Thin-walled Beam", Journal of The Acoustical Society of America, 109, 972-981, 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, 161-171, 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, 263-284, 1971. doi:10.1093/qjmam/24.3.263

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