Keywords: coupled vibrations, wing, flutter, frequency coalescence, DFE method, DSM formulation, numerical aeroelasticity.

The vibration and flutter behavior of wings for which the elastic or flexural axis is not coincident with the inertial axis are characterized by a combination of bending translation and torsional rotation. Naturally, it is very important to take into account the coupling effects in vibration and response calculations of structures made up of these beams. This is particularly so for aeroelastic problems associated with aircraft wing (with cantilever end conditions) and this problem has been investigated by many researchers and various approaches have been exploited.

In the so-called normal mode method the coupled bending-torsional modes are required for flutter analysis of an aircraft wing [1,2]. For approximate solutions one may discretize the system by either the lumped-mass method [2] or one of the methods based on the assumed deformation shapes. The latter category includes the Rayleigh-Ritz method [3], the Galerkin method [1], and the Finite Element Method (FEM) [4] where wing structure element matrices are evaluated from assumed fixed shape functions (like polynomials). A generalized linear eigenvalue problem then results and one can evaluate the natural frequencies and modes of vibration of the structure which are ultimately used to write down the flutter equation of the problem.

Alternatively, the Dynamic Stiffness Matrix (DSM) method can be used to evaluate the natural modes of vibration of the wing structure [5,6,7,8]. Obviously the method gives more accurate results because it exploits the exact member theory. A generalized nonlinear eigenvalue problem then results. But it implies, sometimes, mathematical procedures which are difficult to deal with, and/or are often limited to special cases.

The Dynamic Finite Element (DFE) approach in vibration analysis of beam assemblies is well established [9,10]. The DFE formulation proved to produce the accurate solutions for coupled flexural-torsional vibration of beams, and for structures composed of such beam elements. This method, presented as an intermediate approach, combines the generality of the well known "weighted residual method" procedure, as used in the FEM, and the high precision provided by DSM method. The weighting functions and shape functions are evaluated referring to the appropriate exact DSM formulation. The eigenvalue problem resulting from this method, is also a nonlinear one. The DFE approach can be extended to more complex cases which distinguishes this method from the DSM method [10].

The aim of this investigation is to extend the DFE methodology to the vibration analysis of wings in the presence of aerodynamic forces. Some effects of unsteady aerodynamic forces are then taken into account. To simplify the problem, all the structural and aerodynamic damping terms are neglected. The remaining terms, in this case, represent the added stiffness and inertia of the system. It is shown that by increasing the air speed the first two frequencies of the system change, and near the flutter condition they come together. This scenario is very well in agreement with the coalescence flutter. Based on the proposed dynamic shape functions, the principle of virtual work (PVW) and the weighted residual method, it would be possible to extend the DFE method to more complex cases including damping effects and variable structural parameters.

1

Fung, Y. C., "An introduction to the theory of aeroelasticity", Dover Publication, New York, 1969.

2

Banerjee, J. R., "Coupled bending-torsional dynamic stiffness matrix for beam elements", International journal for numerical methods in engineering, 28, 1283-1298, 1989. doi:10.1002/nme.1620280605

3

Bisplinghoff, R. L., Ashley, H., Halfman, R. L., "Aeroelasticity", Addison-Wesley, Massachusetts, 1955.

4

Mei, C., "Coupled vibrations of thin-walled beams of open-section using the finite element method", International Journal of Mechanical Science, 12, 883-891, 1970. doi:10.1016/0020-7403(70)90025-1

5

Banerjee, J. R., "Flutter characteristics of high aspect ratio tailless aircraft", Journal of Aircraft, 21, 733-736, 1984. doi:10.2514/3.45022

6

Banerjee, J. R., "Flutter modes of high aspect ratio tailless aircraft", Journal of Aircraft, 25(5), 473-476, 1988. doi:10.2514/3.45607

7

Butler R., Banerjee J. R., "Optimum design ofg bending-torsion coupled beams with frequency or aeroelastic constraints", Computers & Structures, 60(5), 715-724, 1996. doi:10.1016/0045-7949(95)00451-3

8

Georghiades G. A., Banerjee J. R., "Role of modal interchange on the flutter of laminated composite wings", Journal of Aircraft, 35(1), 157-161, 1997. doi:10.2514/2.2276

9

Hashemi, S. M., Richard, M. J., Dhatt, G., "A Bernoulli-Euler stiffness approach for vibrational analysis of spinning linearly tapered beams", ASME Paper No. 97-GT-500, in "42nd ASME Gas Turbine and Aeroengine Congress", 1998.

10

Hashemi, S. M., Richard, M. J., Dhatt, G., "A new dynamic finite elemnent (DFE) formulation for lateral free vibrations of Euler-Bernoulli spinning beams using trigonometyric shape functions", Jornal of Sound and Vibration, 220(4), 601-624, 1999.

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