A Quasi-Exact Dynamic Finite Element for Free Vibration Analysis of Sandwich Beams

Abstract

A Dynamic Finite Element (DFE) model for the vibration analysis of threelayered sandwich beams is presented. The governing differential equations of motion of the sandwich beam for the general case, when the properties of each layer are dissimilar, are exploited. Displacement fields are imposed such that the face layers follow the Rayleigh beam assumptions, while the core is governed by Timoshenko beam theory. The DFE model is then used to examine the free-vibration characteristics of an asymmetric soft-core sandwich beam with steel face layers and a rubber core. The natural frequency results for the first four modes, in this case, show the exact match between the DFE and 'exact' Dynamic Stiffness Matrix (DSM) formulations, using only a one-element mesh, justifying the use of Quasi-Exact (QE-DFE) title. Convergence-wise, the QE-DFE formulation also outperforms the conventional FEM, which makes it useful in benchmarking other studies or the examination of high frequency response where FEM requires the use of large number of elements in order to achieve better accuracy. The application of the DFE to a lead-core sandwich beam is also discussed.

New Frequency‐Dependent Trigonometric Interpolation Functions for the Dynamic Finite Element Analysis of Thin Rectangular Plates

“…Since its inception, the DFE method has been used in beam, beam-like, and blade vibration modelling and analysis. Hashemi and his coworkers (see, e.g., [18][19][20]) have extensively studied the free vibration of various beam configurations, such as isotropic, sandwich, composite, and thin-walled beams subjected to diverse loading configurations, using the Dynamic Finite Element (DFE) method. The results have consistently shown the DFE method to have a higher accuracy and rate of convergence compared to conventional FEM owing to the increased efficiency of the frequency-dependent, trigonometric shape functions based on the exact solutions to the governing equation which the DFE method employs.…”

“…The results have consistently shown the DFE method to have a higher accuracy and rate of convergence compared to conventional FEM owing to the increased efficiency of the frequency-dependent, trigonometric shape functions based on the exact solutions to the governing equation which the DFE method employs. In one study [19], application of the DFE formulation to the free vibration analysis of a sandwich beam resulted in a quasi-exact formulation. In a more recent study [20], investigating the vibration of bending-torsion coupled beams subjected to axial load and end moment, it was found that the DFE method required eight times less elements than conventional FEM to produce results with a percent difference of less than one percent.…”

A Framework for Extension of Dynamic Finite Element Formulation to Flexural Vibration Analysis of Thin Plates

“…The matrix thus developed is not case specific and can be assembled. The boundary and loading conditions are applied in a manner similar to FEM [11][12][13][14].…”

Free Vibration Analysis of Curved Sandwich Beams: A Dynamic Finite Element

“…Therefore, the approximation space is defined using frequency dependent trigonometric basis functions to obtain the appropriate interpolation functions with constant parameters over the length of the element. DFE theory was first developed by Hashemi (1998), and its application has ever since been extended by him and his coworkers to the vibration analysis of intact (Hashemi et al,1999, andHashemi &Richard, 2000a,b) and defective homogeneous , sandwich (Adique & Hashemi, 2007, 2010 and laminated composite beam configurations (Hashemi & Borneman, 2005, and Hashemi & Roach, 2008a exhibiting diverse geometric and material couplings. DFE follows a very similar procedure as FEM by first applying the weighted residual method to the differential equations of motion.…”

“…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 Hashemi, 2007, andHashemi &, 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). 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. Figure 1 below shows the notation and corresponding coordinate system used for a symmetrical curved three-layered sandwich beam with a length of S and radius R at the www.intechopen.com mid-plane of the beam.…”