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PROCEEDINGS OF THE ELEVENTH INTERNATIONAL CONFERENCE ON CIVIL, STRUCTURAL AND ENVIRONMENTAL ENGINEERING COMPUTING
Edited by: B.H.V. Topping
Free Vibration of Sandwich Beams using the Dynamic Finite Element Method
E. Adique and S.M. Hashemi
Department of Aerospace Engineering, Ryerson University, Toronto, Ontario, Canada
E. Adique, S.M. Hashemi, "Free Vibration of Sandwich Beams using the Dynamic Finite Element Method", in B.H.V. Topping, (Editor), "Proceedings of the Eleventh International Conference on Civil, Structural and Environmental Engineering Computing", Civil-Comp Press, Stirlingshire, UK, Paper 118, 2007. doi:10.4203/ccp.86.118
Keywords: sandwich beam, free vibration, FEM, DSM, DFE.
Sandwich beams have widespread applications in engineering as load carrying structural members with high strength to weight ratios. The research interest in this area has led to many publications. The focus of the present study is the free vibration analysis of sandwich beams, also carried out by a number of investigators [1,2,3]. A relatively simple (but practically realistic) three-layered symmetric sandwich beam model assumes that the top and the bottom faces deform according to the Bernoulli-Euler beam theory whereas the core deforms only in shear. This model has been used by many and has provided an important basis for further development on the subject. The classical approaches and the finite element method (FEM) have been used to solve the governing differential equations of motion of the problem, leading to the natural frequencies and modes of the system. Banerjee  carried out the free vibration analysis of the above model, using an exact dynamic stiffness matrix (DSM) method that includes all the frequency dependent terms of higher orders to solve the free vibration problem of sandwich beams.
The main purpose of the work presented in this paper is to develop the dynamic finite element (DFE) of a three-layered symmetric sandwich beam and then to use it to investigate its free vibration characteristics. The superiority of the DFE method over FEM and DSM methods has been well established . The DFE exploits general Galerkin-type FEM, together with the closed form solutions of the uncoupled governing equations as the basis functions of approximation space. Considering harmonic oscillations, and the weighted residual formulation, the integral form of the two coupled governing differential equations is first developed. Integration by parts are then applied to reduce the inter-element continuity requirement of solution functions. The resulting weak integral form of the governing equations is then discretized using two-node beam elements. Exploiting the dynamic shape functions leads to the frequency-dependent dynamic stiffness matrix of the sandwich beam element, representing both stiffness and mass properties. Assembly of the element matrices and application of the boundary conditions leads to a nonlinear eigenvalue problem. A determinant search algorithm is used to compute the natural frequencies of an example sandwich beam. The DFE results are in excellent agreement with both analytical and FEM solutions. The proposed DFE formulation can also be extended to cover more complex configurations, involving variable geometric and material properties. This feature distinguishes the DFE from the DSM method.
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