Abstract
The purpose of this study is to show the influence of bending-torsion coupling on natural frequencies and mode shapes of aircraft wings by using two finite element beam formulations. The bending-torsion coupling parameters are the geometric parameter (distance between the mass axis and elastic axis of the cross-section of the beam) and the material coupling due to laminated composites. Cubic and high-order Hermite finite element interpolations are presented in this study, in order to show the influence of geometric and material coupling on natural frequencies and mode shapes. Starting by the governing partial differential equations of motion for the coupled bending-torsion beam with the bending and torsion equations, the Galerkin’s method is used with high-order finite element interpolation to obtain the high-order Hermitian shape functions. The mass and stiffness matrices are obtained using the kinetic energy and potential energy, respectively. The beam finite element has two nodes, the cubic element has three degrees of freedom at each end (transvers displacement, slope and torsion), where the high order element has five degrees of freedom at each end (transvers displacement, slope, curvature, gradient of curvature and torsion). The mass matrix contains geometric coupling terms and the stiffness matrix contains terms of material bending-torsion coupling. The obtained results using cubic and high-order finite element Euler-Bernoulli beam formulations are compared for a free vibration analysis of Goland metallic wing (geometric coupling) and validated with Dynamic Stiffness Method for composite wings.
Highlights
Many analytical and numerical methods (Finite Element Method, Dynamic Stiffness method dynamic stiffness approximation method (DSM), etc.) are adopted to determine approximate natural frequencies and mode shapes of uniform beams [1]
This paper presents the effect of both geometric and material coupling in free vibration analysis of coupled bending-torsional beams by using a high-order finite element formulation
The free vibration analysis of composite wing is investigated using a cubic and high-order finite element Euler-Bernoulli beam formulations in order to show the effect of geometric coupling shown in Fig. 1 on vibration frequencies and mode shapes for metallic wings and generalized to composite wings where the anisotropy of materials has significant effects on natural frequencies and mode shapes
Summary
Many analytical and numerical methods (Finite Element Method, Dynamic Stiffness method DSM, etc.) are adopted to determine approximate natural frequencies and mode shapes of uniform beams [1]. Vibrations measurements of a wing are investigated experimentally by using the method of time averaged projection moiré analyzed by Maskeliūnas et al [3]. The measurement of plane vibrations of a two elastic structure are analyzed by Maskeliūnas et al [5]. A dynamic stiffness element for free vibration analysis of composite beams and its application to aircraft wings is developed by Banerjee et al [1], where a dynamic stiffness matrix of a composite beam is used to investigate it’s free vibration characteristics. A finite element parametric modeling technique of aircraft wing structures is given by Tang and Xi. The composite beam models are used to study the dynamic response and aeroelasticity for high-aspect-ratio of composite wings [7,8,9,10]
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