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CivilComp Proceedings
ISSN 17593433 CCP: 81
PROCEEDINGS OF THE TENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING Edited by: B.H.V. Topping
Paper 180
Dynamic Stiffness Formulation and Free Vibration Analysis of a Composite Beam J.R. Banerjee and H. Su
School of Engineering and Mathematical Sciences, City University, London, United Kingdom J.R. Banerjee, H. Su, "Dynamic Stiffness Formulation and Free Vibration Analysis of a Composite Beam", in B.H.V. Topping, (Editor), "Proceedings of the Tenth International Conference on Civil, Structural and Environmental Engineering Computing", CivilComp Press, Stirlingshire, UK, Paper 180, 2005. doi:10.4203/ccp.81.180
Keywords: dynamic stiffness method, free vibration, composite beams, geometrical and material coupling, WittrickWilliams algorithm.
Summary
The free vibration analysis of composite beams [1,2,3] is an important and intense
area of research activity, particularly because of its applications in aeronautical
design [4,5]. Such an analysis is also used as a prerequisite to carry out aeroelastic
or response analysis [6,7,8]. It is well recognised that the free vibration and response
characteristics of composite beams can be very different from their metallic counter
parts. This is because coupling between various modes of deformation that is not
generally found in isotropic metals can occur in composites as a result of their
anisotropic properties. From an aeroelastic point of view, particularly when
designing composite wings this is significant. The particular coupling between the
bending and torsional motion in a high aspect ratio aircraft wing, which can cause
instability such as flutter, is a research topic of considerable importance [5,6,7,8].
The current paper is concerned with the dynamic stiffness formulation and free vibration analysis of composite beams which exhibit bendingtorsion coupling. Such coupling arises from two principal sources. One of them stems from noncoincident shear centre and centroid of the crosssection [9]. This type of coupling is referred to as geometric coupling as it involves only the geometry of the crosssection. The nature of this bendingtorsion coupling is inertial and as a consequence, the bending and torsional motions are generally uncoupled under static loading condition. The other type of coupling is confined to composite beams and it occurs due to anisotropic material properties (generally caused by fibre orientations). Clearly such type of coupling is possible under both static and dynamic loads. This particular form of coupling is referred to as material coupling because the coupling is dependent on the material properties only. Even for a doubly symmetric crosssection (for example, a solid rectangular or a box section), bendingtorsion coupling can still occur in a composite beam due to ply orientations in the laminate [10]. As this particular coupling is due entirely to material properties, the geometry of the crosssection is not directly relevant. It is important to note that by splitting the coupling terms into geometric and material components provides a better understanding of and insight into the problem. Within the above context the dynamic stiffness matrix of a bendingtorsion coupled composite beam with effects of both geometric and material coupling is developed in order to investigate its free vibration characteristics. The governing differential equations of motion of the composite beam are derived using Hamilton's principle and the equations are solved analytically. The solution is used to obtain the dynamic stiffness matrix, which relates harmonically varying forces with harmonically varying displacements of the composite beam at its ends. Finally the resulting dynamic stiffness matrix is applied with particular reference to the WittrickWilliams algorithm [11] to compute the natural frequencies and mode shapes of an illustrative example. The results are discussed and some conclusions are drawn. References
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