By Farzad Ebrahimi

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1 Cantilever end conditions The first test case investigates the natural frequencies of the beam described above, with cantilever end conditions. The DFE and FEM results (Table 1) are presented and compared 45 Free Vibration Analysis of Curved Sandwich Beams: A Dynamic Finite Element with those reported by Ahmed (1971), obtained from a 10-element FEM model of 2-noded 8DOFs beam elements. The model developed by Ahmed employs polynomial cubic Hermite shape functions for the approximation space of the field variables v, v’, w and w’.

Simply supported at y = 0 and y = S; ∂v/∂y = w = ∂2w/∂y2 = 0. • • cantilever configuration; at y = 0: v=w=∂w/∂y=0; and at y=S: ∂v/∂y=∂2w/∂y2=0 and a resultant force term of [2γ 2 ∂ 3 w /∂y 3 + 2 β 2 h(2 v + h∂w /∂y )] = 0 , … For harmonic oscillation, the weak form of the governing equations (1) and (2) are obtained by applying a Galerkin-type integral formulation, based on the weighted-residual method. The method involves the use of integration by parts on different elements of the governing differential equations and then the discretization of the beam length into a number of twonode beam elements (Figure 2).

The authors have previously developed DFE models for two straight, 3-layered, sandwich beam configurations; a symmetric sandwich beam, where the face layers are assumed to follow Euler-Bernoulli theory and core is allowed to deform in shear only (Adique & Hashemi, 2007, and Hashemi & Adique, 2009), and a more general nonsymmetric model, where the core layer of the beam behaves according to Timoshenko theory while the faces adhere to Rayleigh beam theory (Adique & Hashemi, 2008, 2009). The latter model not only can analyze sandwich beams, where all three layers possess widely different material and geometric properties, but also it has shown to be a quasi-exact formulation (Hashemi & Adique, 2010) when the core is made of a soft material.

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Advances in Vibration Analysis Research by Farzad Ebrahimi
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