The differential equations governing Geometric-Materially coupled torsion-flexural vibrations of laminated composite wings are first reviewed. Based on the finite element methodology (FEM), Euler-Bernoulli and St-Venant beam theories, the Dynamic Trigonometric Shape Functions (DTSFs) for the beam’s uncoupled displacements are thereafter derived. Exploiting the Principle of Virtual Work (PVW) and interpolating the variables based on the DTSF, the Dynamic Finite Element (DFE) formulations for uniform beams’ coupled vibrations are first developed. The variable geometrical and mechanical parameters are then incorporated in the formulation. The applicability of the DFE method is then demonstrated by two illustrative examples where a Wittrick-Williams root counting technique is used to find the systems’ natural frequencies. The proposed DFE approach can also be advantageously extended to incorporate more complexities, further coupling as well as other geometrical and mechanical parameters in the formulation.

Damage to composite structures occurs from impact, fatigue, or over stress and can be critical in the safe operation of wings or any structural member. This paper presents a method for detection of multiple cracks present in laminated composite bending-torsion coupled cantilevered beams using natural frequency data, a type of Nondestructive testing (NDT). This methodology relies on both experimentally collected natural frequencies and frequencies calculated using a mathematical model. Precise natural frequencies are calculated using a new dynamic finite cracked element (DFCE) formulated within and based on dynamic trigonometric shape functions. An algorithm is devised based on the Adam–Cawley criterion and extended to laminated composites with multiple cracks. This method has shown exceptional convergence on the size and location of cracks using a number of modes of free vibration with and without error in measured frequencies.

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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.

Damage to composite structures occurs from impact, fatigue, or over stress and can be critical in the safe operation of wings or any structural member. This paper presents a method for detection of multiple cracks present in laminated composite bending-torsion coupled cantilevered beams using natural frequency data, a type of Nondestructive testing (NDT). This methodology relies on both experimentally collected natural frequencies and frequencies calculated using a mathematical model. Precise natural frequencies are calculated using a new dynamic finite cracked element (DFCE) formulated within and based on dynamic trigonometric shape functions. An algorithm is devised based on the Adam–Cawley criterion and extended to laminated composites with multiple cracks. This method has shown exceptional convergence on the size and location of cracks using a number of modes of free vibration with and without error in measured frequencies.