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

The free vibration analysis of a spinning metallic beam has been investigated by many authors, see for example Refs. [1,2], whereas that of a spinning composite beam made up of laminated fibrous composites is relatively a recent topic for research [3,4,5]. The difficulty arises because unlike metallic beams, composite beams exhibit (material) coupling between various modes of deformation as a result of ply orientation. In particular, the coupling between bending displacements and torsional rotations is of great significance in structural engineering design. The purpose of this paper is to develop the dynamic stiffness matrix of a spinning composite beam by including this coupling effect and then investigate the free vibration characteristics. Following the authors recent work on the dynamic stiffness formulation on free vibration analysis for a spinning metallic beam [2], it is highly pertinent to extend the theory to the important case of composites. This useful extension is of quite considerable complexity due to anisotropic nature of fibrous composites.

The investigation is carried out in following steps.

1. The governing differential equations of motion of a spinning composite beam in free vibration are derived using Hamilton's principle. The coupling effect between the bending and torsional displacements is fully taken into account in the derivation.
2. For harmonic oscillation, the governing differential equations are solved in closed analytical form for bending displacements, bending rotation and twist.
3. Expressions for shear force, bending moment and torque are also obtained in explicit analytical form from the solutions of the governing differential equations.
4. Next the shear forces, bending moments and torque are related to the bending displacements, bending rotations and twist by recasting the above expressions for loads and responses via the dynamic stiffness matrix.
5. The Wittrick-Williams algorithm [6] is used as a solution technique to compute the natural frequencies and mode shapes of an illustrative example.
6. Numerical results are given and discussed and this is followed by some concluding remarks.

1

Zu, J.W., 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

2

Banerjee, J.R., Su, H., "Dynamic Stiffness Formulation and Free Vibration Analysis of Spinning Beams", Proceedings of The Sixth International Conference on Computational Structures Technology, Prague, Czech Republic, 4-6 September 2002. doi:10.4203/ccp.75.32

3

Song, O., 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

4

Kim, W., Argento, A., Scott, R.A., "Forced Vibration and Dynamic Stability of a Rotatory Tapered Composite Timoshenko Shaft: Bending Motions in End-Milling Operations", Journal of Sound and Vibration, 246, 583-600, 2001. doi:10.1006/jsvi.2000.3521

5

Bert, C.W., Kim, C.D., "Whirling of Composite-material Drive Shaft including Bending-Twisting Coupling and Transverse Shear Deformation", Journal of Vibration and Acoustics, 117, 17-21, 1995. doi:10.1115/1.2873861

6

Wittrick, W.H., 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

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £135 +P&P)