Free vibration analysis of a laminated beam using dynamic stiffness matrix method considering delamination

Abstract

In this study, free vibration analysis of a laminated composite beam with a through-width delamination is investigated. The beam is modeled as a thin-walled cantilevered structure based on the Euler–Bernoulli equations. Also, composite material is modeled as a lamina fiber-reinforced. Governing equations are extracted by Hamilton’s principle with both geometric and material couplings. Using the variables separation method, general solution in the normalized form for bending and torsion deflections is first achieved. Then, expressions for the cross-sectional rotation, the bending moment, the shear force, and the torsional moment for the cantilevered beam are obtained. For delamination modeling, two models, namely the free and constrained models, are developed. In the free model, the delaminated beam is modeled as four interconnected intact beams; while in the constrained model three interconnected intact beams are used. Then, their results are compared with those of a cracked beam model, which is modeled as two interconnected intact beams. In fact, delamination effects on the free vibration analysis are reflected by the continuity equations at the two ends of delamination. By applying the boundary conditions (at the free and fixed ends) and continuity conditions (which are boundary conditions at the delamination locations) and using the Dynamic Stiffness Matrix (DSM), free vibration analysis of the defective beam is performed. Also, in order to represent the advantages of this method with respect to approximate methods, a numerical code is developed based on the finite element method (FEM) and the corresponding results are compared. Ultimately, the effects of various parameters such as fiber angle and delamination length on the natural frequencies and the mode shapes are studied. Comparison of the results of DSM method with those of FEM in the same cases indicates that DSM method is more accurate than FEM.