# Free Vibration Analysis of Tapered Composite Wings

Introduction

The materially coupled composite, uniform and piece-wise uniform stepped wing beams were previously analysed. The tapered wing configurations were then presented and discussed in (http://www.aeroway.ca/Taperedwing.htm) . In this research, the wing model is extended to more complex configurations exhibiting not only the material but also geometrical couplings. Using a wing-box model for the wing cross-section and a circumferentially asymmetric stiffness (CAS) configuration for the composite ply lay-up, a more realistic composite wing model is generated. In the previous chapters (http://www.aeroway.ca/Taperedwing.htm), only material coupling was considered which arises from an unbalanced ply lay-up or symmetric stacking sequence. An additional geometric coupling arises from the cross-sectional geometry of the wing.

The present wing model, is modeled as a symmetric configuration where the materially coupled behaviour is characterized by bending-torsion coupled stiffness K. The added geometric coupling is a consequence of an offset of the mass centre axis, Gs, from the geometrical elastic axis, Es, denoted by x?. Any structural component located in front of the leading spar or behind the rear spar is considered not to contribute to the rigidity of the wing (Lillico, Butler, Guo and Banerjee, 1997). The omitted components do however contribute to the mass and inertia of the wing such that the mass centre, initially located at the geometric centre of the box, shifts slightly towards the rear of the wing-box (refer to Figure 6?2(b))(refer again to http://www.aeroway.ca/Taperedwing.htm).

Model, Hypotheses and Simplifying Assumptions

The proposed wing model is constructed as a wing-box, where L is the span-wise length and c is the wing chord. The lateral bending and twist displacements are governed by Euler-Bernoulli and St. Venant beam theories, respectively. Shear deformation, rotary inertia, commonly associated with Timoshenko beam theory, as well as warping effects are neglected.

Different stacking sequence and/or thickness of the thin-walled box-beam result in different coupling behaviours. For a circumferentially asymmetric stiffness (CAS) configuration the axial stiffness, A, must remain constant in all walls of the cross-section. The coupling stiffness, B, in opposite members is of the opposite sign as stated by Armanios and Badir (1995) and Berdichevsky et al (1992). As a result of axial stiffness, A, remaining constant, the corresponding thickness must also remain constant. Chandra et al. (1990) consider a symmetric configuration for a box-beam which consists of opposite walls having the same stacking sequence, although the stacking sequences between the horizontal and vertical members need not be the same. The CAS and symmetric configurations both lead to a bending-torsion coupled response for thin-walled beams.

The second configuration considered by Armanios and Badir (1995) and Berdichevsky et al (1992) was a circumferentially uniform stiffness configuration (CUS) where A, B, C, axial, coupling and shear stiffness, respectively, are constant throughout the circumference of the cross-section. Chandra et al. (1990) built-up similar configurations where the stacking sequence of opposite walls is of oppositely stacked, what they call anti-symmetric configuration. Anti-symmetric or CUS configurations are beyond the scope of this research and will not be discussed further. The CAS or symmetric configuration leads a bending-torsion coupled wing which will be used to model the wing-box composite plies.

The Circumferentially Asymmetric Stiffness Configuration for this particular wing is described fully in http://www.aeroway.ca/Laminatetheory.html .

The free vibration analysis of thin-walled composite wing-boxes with quadratic and cubic tapers is fully described in the above references. By implementing the CAS configuration and non-coincident mass and shear axes, the wing exhibits dually coupled vibration. The natural frequencies and modes of deformation have been extracted using the three methods, conventional FEM, DFE, and the refined DFE (DFE with deviators). These deviators take into account the variable geometric and/or material parameters of the wing model over each DFE. The convergence of the refined DFE (RDFE) is validated in comparison with the FEM method for multiple tapered geometries and ply orientations. The RDFE method provides a much higher convergence rate than classical finite elements. The corresponding natural modes of vibration were also evaluated and plotted using the advanced plotting features in MATLAB®.