Keywords: composite beam, coupled torsion-extension vibrations, mesh reduction methods, FEM, DSM, DFE formulation.

Structural members such as composite circular tubes composed of laminated lay-ups of fibre-reinforced plastic materials, springs, cables and wire ropes exhibit coupled extension-twist behaviour when subjected to static or dynamic loads [1,2,3,4]. The vibration analysis of a coupled extension-torsion structural member, in general, and composite circular tubes, in particular, is of great significance. This is because of the practical nature of the engineering applications of such elements particularly for future needs in the context of the ever-increasing application of thin-walled composite beams in the aerospace industry. Interest shown by some investigators concentrates on both free and forced vibration characteristics of such individual members by solving its differential equations [1,2], whereas others rely on exact [4] or approximate numerical methods [3]. The Galerkin-based Finite Element Methods (FEM) where the structural element matrices are evaluated from assumed fixed shape functions (like polynomials) are commonly used [3]. A generalized linear eigenvalue problem then results and one can evaluate the natural frequencies and modes of vibration of beams and beam structures.

Alternatively, the Dynamic Stiffness Matrix (DSM) method [4] can be used to evaluate the natural modes of vibration of the beam structure. Obviously the method gives more accurate results because it exploits the exact member theory. A generalized nonlinear eigenvalue problem then results. But it implies, in many cases, mathematical procedures that are difficult to deal with, and are often limited to special cases.

This work is partly motivated by earlier works (see, for example, [5]) and presents the derivation of a mesh-reduction Dynamic Finite Element (DFE) technique for coupled extension-twist vibration analysis of a uniform structural member, starting from basic governing differential equations. Exploiting the uncoupled member theory, the Dynamic Trigonometric Shape Functions (DTSFs), also satisfying the uncoupled governing differential equations of member's vibration, are derived. The application of the Weighted Residual Method (i.e., similar to FEM formulation), in conjunction with these dynamic shape functions, results in the element Dynamic Stiffness matrices. Assembling the element matrices in the conventional way generates the global stiffness matrix leading to the eigenvalue problem of the system. Solving the resulting non-linear eigenproblem, one can then evaluate the natural frequencies and modes of vibration of the structure. Based on the proposed DTSFs, the weighted residual method and the Principle of Virtual Work (PVW), it would be possible to extend the DFE method to more complex problems including more geometric and/or material coupling effects. The DFE approach can also be extended to incorporate variable material, mechanical and geometrical parameters which distinguishes this method from the DSM method.

In this paper, the application of the DFE approach to the vibration analysis of composite tubes, with particular reference to an established algorithm [4], is discussed. Numerical DFE results for coupled torsion-extensional vibration of a circular composite tube composed of laminated lay-ups of fibre-reinforced plastic materials are presented. The comparison is made between natural frequencies of the system obtained from DFE, FEM, 'exact' DSM and other published results. Comparison between the results obtained from the first two methods revealed that the DFE approach leads to much faster convergence of results, in general, and for higher frequencies, in particular.

1

M.W. Nixon, "Extension-twist coupling of composite circular tubes with application to tilt rotor blade design", Proc. 28th AIAA/ASME/ASCE/AHS Structures, Structural dynamics and Materials Conference, 295-303, 1987.

2

E.C. Smith and I. Chopra, "Formulation and evaluation of an analytical model for composite box-beams", Journal of American Helicopter Society, 36, 23-35, 1991. doi:10.4050/JAHS.36.23

3

J.E. Mottershead, "Finite elements for dynamical analysis of helical rods", International Journal of Mechanical Science, 22, 267-283, 1980. doi:10.1016/0020-7403(80)90028-4

4

J.R. Banerjee and F.W. Williams, "An exact dynamic stiffness matrix for coupled extensional-torsional vibration of structural members", Computers & Structures 50(2), 161-166, 1994. doi:10.1016/0045-7949(94)90292-5

5

S.M Hashemi, M.J. Richard, G. Dhatt, "A new dynamic finite element (DFE) formulation for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometric shape functions", Journal of Sound and Vibration, 220(4), 601-624, 1999. doi:10.1006/jsvi.1998.1922

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