Keywords: sandwich beam, soft-core, free vibration, dynamic shape functions, dynamic finite element, dynamic stiffness matrix, finite element method.

Sandwich construction offers the structural designer many attractive features, such as high specific stiffness, good buckling resistance, formability into complex shapes, easy reparability, and so on. Thus the analysis of such structural systems has been investigated, more or less continuously, for well over half a century. This paper examines the dynamical behavior of asymmetric sandwich beams and exploits a model for three-layered sandwich beams with three distinct components, and with no restrictions on the geometry of each element [1]. The three elements all have, in general, a mean axial motion fully coupled with a common flexure. The end product of the investigation is the development, and application, of a super-convergent dynamic finite element (DFE) [2,3] of the three-layered beam. The DFE development is achieved by using symbolic computation, and the model used in this study assumes an isotropic material. Therefore, this model is not applicable to fiber-reinforced laminated composite beams, however, the given analysis can readily be extended to orthotropic materials.

A dynamic finite element theory of a three-layered sandwich beam is developed and subsequently used to investigate its free vibration characteristics. The top and bottom layers behave like Rayleigh beams, whilst the central core layer follows Timoshenko beam theory. For harmonic oscillations, the dynamic shape functions (DSF) [2,3] of approximation, satisfying the uncoupled governing differential equations of motion of the sandwich beam are developed. The application of weighted residual method, DSFs and the boundary conditions leads to the element dynamic stiffness matrix (DSM). The natural frequencies and mode shapes of illustrative examples are evaluated and compared with finite element results and those published in the literature. Further comparison is also made using the ANSYS®software. A single DFE element is shown to result in exact results, for the first four natural frequencies of an asymmetric sandwich beam with steel face layers and a soft (rubber) core. The proposed DFE could therefore be justifiably called a super-convergent formulation for dynamic analysis of soft-core sandwich beams. The corresponding predominantly flexural natural modes are also evaluated and agreed well with those reported in the literature. The DFE convergence, in this case, surpasses the finite element method, as it necessitates significantly fewer elements in the analysis of soft-core sandwich beams. However, for the vibration analysis of a lead-core sandwich beam, the DFE requires a higher number of elements to achieve the "exact" DSM results.

1

J.R. Banerjee, A.J. Sobey, "Dynamic stiffness formulation and free vibration analysis of a three-layered sandwich beam", International Journal of Solids and Structures 42, 2181-2197, 2005. doi:10.1016/j.ijsolstr.2004.09.013

2

S.M. Hashemi, "Free Vibrational Analysis of Rotating Beam-like Structures: A Dynamic Finite Element Approach", PhD. Dissertation, Department of Mechanical Engineering, Laval University, Quebec, Canada, 1998.

3

S.M. Hashemi, "The use of frequency dependent trigonometric shape functions in vibration analysis of beam structures-bridging gap between FEM and exact DSM formulations", Asian Journal of Civil Engineering, 3(3&4), 33-56, 2002.

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