Vibration of Cracked Composite Beams: a Dynamic Finite Element

S. M. Hashemi(1*), S. R. Borneman(2), H. Alighanbari(3)

(1) Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto (ON), Canada
(2) Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto (ON), Canada
(3) Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto (ON), Canada
(*) Corresponding author

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A Dynamic Finite Element (DFE) is developed to analyze the vibration characteristics of cracked composite beams. Stress intensity factors, corrected for geometry and material anisotropy, have been used to develop the local flexibility of a through-thickness cracked uniform laminated unidirectional unbalanced beam. By exploiting the principle of virtual work and the Dynamic Trigonometric Shape Functions (DTSF’s), developed from the exact solutions to the equations governing uncoupled flexural and torsional vibrations of the system, the element Dynamic Stiffness Matrix (DSM) is developed. By implementing the local flexibility of crack, the element matrices exhibiting both mass and stiffness properties are then assembled and the boundary conditions are applied to form the eigenproblem of the overall system. The natural frequencies and modes are then extracted using the well-known Wittrick–William (W-W) root counting algorithm. Numerical tests are conducted for a flat, solid rectangular cross-section, uniform, cantilever, laminated composite beam.  Both intact and damaged scenarios (for cracks located at 20% and 50% of beam length), with various crack ratios, , and ply angles, , are investigated. Numerical results on natural frequencies and convergence tests demonstrate the higher accuracy and faster convergence of the proposed DFE and its superiority over the classical Finite Element Methods (FEM).
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Bending-Torsion Couplings; Cracked Composite Beam; Dynamic Finite Element (DFE); Dynamic Stiffness Matrix; FEM; Materially Coupled Vibrations

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