Keywords: free vibration, metallic beam, composite beam, bending-torsion coupling.

Many investigators have studied the important effect of bending-torsion coupling on natural frequencies, mode shapes and response of metallic and composite beams [1,2,3,4,5,6,7,8,9,10,11,12]. However, all of these studies have been carried out independently, either for metallic, or for composite beams. The current investigation combines the studies of both metallic and composite beams using exact analytical approaches developed in References [1,2]. The main purpose of this paper is to compare the effect of two different types of coupling between bending and torsional deformations in beams made of metallic or composite constructions. In metallic beams, the mode coupling occurs as a result of non-coincident mass and shear centres. This coupling is inertial and is called geometric coupling. On the other hand, the coupling in composite beams arises as a result of stacking sequence and ply orientation and is called material coupling. Numerical results in this investigation are obtained using the methods described in References [1] and [2] for metallic and composite beams respectively. Two aircraft wings, namely the Goland wing [3] and the Loring wing [4] are used as examples for metallic beams, for which the effect of the geometric coupling is studied in details. These results are compared with published results [7] and excellent agreement was found. There are also two types of composite beams analysed in this paper, of which one is a flat beam with solid rectangular cross-section and the other is a thin-walled box beam. The former, namely the MIT beam, has been studied extensively in Reference [5] whereas the latter is referred to as the GIT beam [6]. As the names imply the work on these two beams was carried out by researchers at the Massachusetts Institute of Technology (MIT) and Georgia Institute of Technology (GIT) in the USA respectively. The effect of the material coupling on the free vibration characteristics is studied for both composite beams. Some of the results are compared with those obtained using the finite element method and experimental measurement [5]. Finally, numerical results for both metallic and composite beams are compared and discussed. One of the intriguing features of the current study is that the geometric and the material coupling can both cause modal interchanges between bending and torsional modes. The results presented in this paper highlight some of these interesting features which may have profound influence in aeroelastic studies.

1

Banerjee, J.R., "Explicit frequency equation and mode shapes of a cantilever beam coupled in bending and torsion", Journal of Sound and Vibration, 224(2), 267-281, 1999. doi:10.1006/jsvi.1999.2194

2

Banerjee, J.R., "Explicit analytical expression for frequency equation and mode shapes of composite Beams", International Journal of Solid and Structure, 38, 2415-2426, 2001. doi:10.1016/S0020-7683(00)00100-1

3

Goland, M., "Flutter of a uniform cantilever wing", Journal of Applied Mechanics, 12, A197-A208, 1945.

4

Loring, S.J., "Use of generalised coordinates in flutter analysis", SAE Journal, 52, 113-132, 1944.

5

Jesen, D.W., Crawley, E.F., and Dugundji, J., "Vibration of cantilevered graphite/epoxy plates with bending-trosion coupling", Journal of Reinforced Plastics and Composite, 1, 254-269, 1982. doi:10.1177/073168448200100305

6

Cesnik, C.E.S., Hodges, D.H., and Patil, M.J., "Aeroelastic analysis of composite wings", Proceedings of the 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1113-1123, 1996.

7

Eslimy-Isfahany, S.H.R., and Banerjee, J.R., "Use of generalized mass in the interpretation of dynamic response of bending-torsion coupled beams", Journal of Sound and Vibration, 238(2), 295-308, 2000. doi:10.1006/jsvi.2000.3160

8

Armanios, E.A., and Badir, A.M., "Free vibration analysis of anisotropic thin-walled closed-section beams", AIAA Journal, 33, 1905-1910, 1995. doi:10.2514/3.12744

9

Njuguna, J., Su, H., Cheung, C.W. & Banerjee J.R., "The influence of ply orientation on the free vibration of composite beams", Proceedings of the ACUN-4 Conference on Composite Systems-Macrocomposites, Microcomposites, Nanocomposites, 307-314, University of New South Wales, Australia, 2002.

10

Bishop, R.E.D., Cannon, S.M., and Miao, S., "On coupled bending and torsional vibration of uniform beams", Journal of Sound and Vibration, 131, 457-464, 1989. doi:10.1016/0022-460X(89)91005-5

11

Song, O., and Librescu, L., "Free vibration and aeroelastic divergence of aircrft wings modelled as composite thin-walled beams", Proceedings of the AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics, and Materials Conference, Baltimore, MD, Paper No. AIAA-91-1187, 2128-2136, 1991.

12

Weisshaar, T.A., and Foist, B.L., "Vibration tailoring of advanced composite lifting surfaces", Journal of Aircraft, 22, 141-147, 1985. doi:10.2514/3.45098

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