International Journal of Vehicle Structures and Systems

Seyed M. Hashemi ¹, Ernest J. Adique ¹

Affiliations
1 Ryerson University, CA

Abstract

The free vibration analysis of three-layered symmetric sandwich beams is studied. The governing differential equations of motion in free vibration are coupled both in axial and bending deformations. The weighted residual method is used and the resulting weak integral form of the governing equations is discretized using the Finite Element Method (FEM) formulation. The dynamic, frequency-dependent, trigonometric shape functions, derived from the solution of the uncoupled equations, are then used to develop the element matrices. The assembly of resulting Dynamic Finite Element (DFE) matrices leads to a nonlinear eigenvalue problem. A determinant search method is then used to compute the coupled natural frequencies and modes of a symmetric, thin face-layer sandwich beam. The DFE numerical results for the first four bending-axial coupled modes, obtained from a 20-element mesh, show good agreement with FEM, exact Dynamic Stiffness Matrix (DSM) method, and other results available in the literature. All the four modes are found to be dominated by bending deformation. The discussion of results is followed by some concluding remarks.

Keywords

Sandwich beam; Free vibration; Finite element method; Dynamic stiffness matrix; Dynamic finite element

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Stephen R. Borneman ¹, Seyed M. Hashemi ¹, Hekmat Alighanbari ¹

Affiliations
1 Department of Aerospace Engineering, Ryerson University, CA

Abstract

The free vibration of piecewise uniform defective laminated composite beams is investigated. The governing differential equations of motion are coupled both in torsional and bending deformations. A Dynamic Finite Cracked Element (DFCE) is developed and is applied to slender beams, characterized by an offset between inertial and bending axes. The hybrid DFCE is a combination of the conventional Finite Element Method (FEM) formulation and frequencydependent interpolation functions stemmed from the exact Dynamic Stiffness Matrix (DSM) method. The defect, a through-thickness edge crack, is then represented by a set of stiffness terms evaluated from the beam compliance matrix at the crack location. A number of stepped beam configurations are investigated by reducing the base, thickness, or both. The natural frequencies and modes of free vibration of the beams are examined for single through-thickness edge crack configurations.

Keywords

Laminated composites; Cracked beam; Free vibrations; Coupled bending-torsion; Dynamic stiffness matrix; Dynamic finite element; Compliance matrix

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