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PROCEEDINGS OF THE EIGHTH INTERNATIONAL CONFERENCE ON COMPUTATIONAL STRUCTURES TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Free Vibration of a Composite Timoshenko Beam Using the Dynamic Stiffness Method
School of Engineering and Mathematical Sciences, City University, London, United Kingdom
J.R. Banerjee, "Free Vibration of a Composite Timoshenko Beam Using the Dynamic Stiffness Method", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Eighth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 28, 2006. doi:10.4203/ccp.83.28
Keywords: dynamic stiffness method, free vibration, composite Timoshenko beams, geometrical and material coupling, Wittrick-Williams algorithm.
In a recently published paper the free vibration behaviour of a composite beam was investigated using the dynamic stiffness method . An important feature of this work was the inclusion of the bending-torsion coupling effects arising from both the cross-sectional geometry and the material properties of the composite beam. The published theory has several applications, but in the paper it was applied to a composite wing as an example to illustrate the free vibration performance. For aeroelastic and, or response analyses of composite wings, the paper is of considerable importance, because prior to carrying out such analyses, the free vibration behaviour is an essential prerequisite when the normal mode method is used. The main contribution made in this earlier publication is the development of an exact dynamic stiffness matrix of a geometrically and materially coupled composite beam and its subsequent use in free vibration analysis. This work has principally motivated the current research, which is aimed at developing a more refined dynamic stiffness theory for composite beams. In essence the present paper focuses on the development of an exact dynamic stiffness matrix of a geometrically and materially coupled composite beam with the effects of shear deformation and rotatory inertia taken into account so as to represent the structural member as a composite Timoshenko beam. Finally the resulting dynamic stiffness matrix is applied to investigate the free vibration characteristics. It is well known that the effects of shear deformation and rotatory inertia on the free vibration behaviour of a beam (metallic or composite) can be significant when the cross-sectional dimensions of the beam are large  and, or when higher natural frequencies and mode shapes are required . For composite beams in particular, these effects are generally much more pronounced because fibre reinforced composites have low shear moduli that result in low shear rigidities. This makes a strong case for research into the free vibration analysis of composite beams using the Timoshenko theory as opposed to the simpler Bernoulli-Euler theory. It will be shown that this new development is far from trivial and indeed, is a substantial task, requiring considerable ingenuity and extensive expenditure of analytical and computational effort.
There are of course, many papers in the literature on the free vibration analysis of composite beams using the Timoshenko theory [4,5]. However, most of these papers are based on the finite element or other approximate methods, and are rather restrictive in that they deal with the free vibration behaviour of composite Timoshenko beams by taking into account only the effect of material coupling. This is unsatisfactory, particularly when analysing composite wings for which the additional effect of geometric coupling arising from the geometry of the wing cross-section can play a very important part. Thus the present paper accounts for the combined effects of geometric as well as material coupling when developing the dynamic stiffness theory of composite Timoshenko beams. The theory is subsequently applied to investigate the free vibration behaviour of a composite wing. One of the significant differences between this paper and earlier papers is in the quality of the composite beam model. It is expected to be much superior in the present paper because exact member theory resulting from the exact solution of the governing differential equations has been used. A brief description of the proposed theory is given below.
The governing differential equations of motion of a composite Timoshenko beam are derived using Hamilton's principle. These equations are solved explicitly in closed analytical form. The solution is then used to develop the dynamic stiffness matrix of the composite Timoshenko beam, which relates harmonically varying loads with harmonically varying responses at its ends. Finally the resulting dynamic stiffness matrix is applied with particular reference to the Wittrick-Williams algorithm  to compute the natural frequencies and mode shapes of an aircraft wing. The results are discussed and some conclusions are drawn.
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