Geometrical Nonlinear Aeroelastic Stability Analysis Of A .

Shock and Vibration 15 (2008) 325–333IOS Press325Geometrical nonlinear aeroelastic stabilityanalysis of a composite high-aspect-ratiowing1Chang Chuan Xie , Jia Zhen Leng and Chao YangAeroelasticity Branch, Aircraft Design Institute, Beijing University of Aero. & Astro., Beijing 100083, ChinaReceived 2007Revised 2007Abstract. A composite high-aspect-ratio wing of a high-altitude long-endurance (HALE) aircraft was modeled with FEM byMSC/NASTRAN, and the nonlinear static equilibrium state is calculated under design load with follower force effect, but withoutload redistribution. Assuming the little vibration amplitude of the wing around the static equilibrium state, the system is linearizedand the natural frequencies and mode shapes of the deformed structure are obtained. Planar doublet lattice method is used tocalculate unsteady aerodynamics in frequency domain ignoring the bending effect of the deflected wing. And then, the aeroelasticstability analysis of the system under a given load condition is successively carried out. Comparing with the linear results, thenonlinear displacement of the wing tip is higher. The results indicate that the critical nonlinear flutter is of the flap/chordwisebending type because of the chordwise bending having quite a large torsion component, with low critical speed and slowlygrowing damping, which dose not appear in the linear analysis. Furthermore, it is shown that the variation of the nonlinear flutterspeed depends on the scale of the load and on the chordwise bending frequency. The research work indicates that, for the veryflexible HALE aircraft, the nonlinear aeroelastic stability is very important, and should be considered in the design progress.Using present FEM software as the structure solver (e.g. MSC/NASTRAN), and the unsteady aerodynamic code, the nonlinearaeroelastic stability margin of a complex system other than a simple beam model can be determined.Keywords: Geometric nonlinearity, unsteady aerodynamics, aeroelasticity, flutter1. IntroductionIn recent decades, the High-Altitude Long-Endurance (HALE) aircraft has been given great interest, and theaeroelastic stability of high-aspect-ratio wings has been a key subject for many people [1–3]. Dowell et al. [4],pointed out clearly that very high aspect ratio wings with both structural and aerodynamic nonlinearities is one ofthe important research subjects. “A statically nonlinear, but dynamic linear, . . . Then the question of the dynamicstability of the statically nonlinear fluid-structural (aeroelastic) system may be addressed by a linear static dynamicperturbation analysis about this nonlinear static equilibrium.” For a complex structure, the Finite Element Method(FEM) gives out so many degrees of freedom that post instability motions, such as Limit Cycle Oscillation (LCO), arehard and inefficient to calculate. For this reason, the dynamic linearization is the practicable engineering approachto determine the critical aeroelastic instability conditions of a wing or a complete aircraft.1 This work was supported by the National Natural Science Foundation (10432040, 10502003) and Supported Program for New CenturyExcellent Talents in University (NCET-04-0170). Corresponding author. E-mail: xiechangc@163.com.ISSN 1070-9622/08/ 17.00 2008 – IOS Press and the authors. All rights reserved

326C.C. Xie et al. / Geometrical nonlinear aeroelastic stability analysis of a composite high-aspect-ratio wingIn the last ten years, the beamlike wing – as the simplest model of the high-aspect-ratio wing – has been used tostudy the mechanism of its aeroelastic behavior. Patil and Hodges [5] studied the geometrical nonlinearity effects onthe static and dynamic aeroelastic behavior using the dynamic linearization method. In their model, the chordwisebending frequency was higher than the torsion frequency, the geometric nonlinearities induced the coupling ofthese two modes where torsion frequency was decreasing, and the corresponding flutter speed decreased as the loadmagnitude was growing. And the non-planar effects of aerodynamics were negligible even for large deformation.Tang and Dowell [6] tested a beamlike wing in the wind tunnel, and compared with the theoretical results, in whichthe dynamic linearized calculation gave out a quite precise prediction of the flutter speed. Recently, the dynamiclinearization method started to be applied to the complex structure in engineering. Strong et al. [7], contractedlystudied a complex pre-stressed wing to demonstrate the differences comparing with linear analysis.The composite material can be beneficial to the structure, but the stiffness characteristics of a composite structureare different from the metallic ones and the geometrical nonlinear effects are different too. For the traditionalmetallic high-aspect-ratio wing, the bending and torsion are elastically uncoupled, and the stiffness of chordwisebending is inherently quite large. So the chordwise bending mode has a little contribution to the linear flutter. Asfor the deformed wing, the geometrical nonlinear effect is mainly showed in the change of the torsion mode leadingto the changing of the flutter speed. For composite structures, the bending and torsion stiffness coupling is usuallyintroduced to satisfy the linear aeroelastic constrains and light weight, and then the chordwise bending stiffness wouldsometimes be much smaller than that of torsion. However, for the high-aspect-ratio wing with large deformationduring flight, the chordwise bending and torsion coupling is strengthened due to the geometrical nonlinearities.The low frequency chordwise bending mode with large torsion component can lead to severe nonlinear aeroelasticproblems.In the present work, the aeroelastic stability of a real imbedded composite high-aspect-ratio wing with largesymmetrical deformation is studied. The static nonlinear equilibrium is calculated firstly, and then the system isdynamic linearized to determine the aeroelastic unstable condition. The flutter of the wing is caused by flap bendingand chordwise bending which has a large torsion component due to geometrical nonlinearities.2. Structural and aeroelastic theoryFor the high-aspect-ratio wing of HALE aircraft, the structure has quite large deformation in the climbingmaneuver process, even during the straight flight. The static and dynamic analysis of the structure should considerthe deformed configuration of the aircraft with the specific flight conditions, where the flight load and geometricnonlinearity are important. Consequently, the aeroelastic stability will be changed dramatically comparing with thelinear analysis. The complete nonlinear aeroelastic problem is quite complex, but for the engineering calculationof subsonic aeroelastic stability, many factors can be trimmed off. The assumptions can be made as follow, 1) thestructure vibrates slightly about the state of nonlinear equilibrium state; 2) the aerodynamic nonlinearity due tolarge attack angle can be ignored for the design limitation of flight load and stall, 3) the non-planar effect of theaerodynamics can be ignored. Under these assumptions, it is adequate that the frequency domain method of thetraditional aeroelastic stability analysis can be used with little modification. After calculation of nonlinear staticequilibrium, the normal modes can be carried out, and the unsteady aerodynamics in the frequency domain can becalculated by the doublet lattice method. Thus the p-k method [8] can be used to give out the critical unstable speedof the system.2.1. Geometrical nonlinear elasticityFlexible long slender wing standing large aerodynamic forces has finite bending and torsion deflection, so theinfinitesimal deformation condition is disobeyed, while the material is thought to be not beyond the elastic limitationfor the little strain. This means that the nonlinear geometric equation should include the quadric term of thedisplacement differential, and that the nonlinear force equilibrium equation should be defined with reference tothe deformed state of the structure. The incremental finite element method is commonly used to solve structural

C.C. Xie et al. / Geometrical nonlinear aeroelastic stability analysis of a composite high-aspect-ratio wing327geometric nonlinear problems, which has two formulas called Total Lagrange Formulation (TLF) and UpdatedLagrange Formulation (ULF) [9]. The ULF is presented and used in the current work.The relationship between nonlinear Lagrange/Green strain and displacement istεij 12 (t ui,j t uj,i t utk,i uk,j )(1)where, t ui,j means the partial derivative of displacement component u i to the coordinate x j at time t. The conjugateKirchhoff stress tensor S ji at time t satisfies,tt tSjinj ds t xti,j dTj(2)twhere, nj is direction cosine of small area element ds at time t, dT j is the corresponding surface force in which thefollower force effect is considered. The linear elastic constitutive relation is as follows,ttSij t Dijklεkl(3)Dijkl is the elastic tensor, which has different form for isotropic or anisotropic material.2.2. Finite element methodFEM based on energy principles is an effective approach to solve structural problems. For the geometric nonlinearproblems with follower forces, the incremental FEM is used. The strain ε ij can be decomposed into linear part e ijand nonlinear part η ij ,tεij t eij t ηij(4)ttThe stress is decomposed by increments, where Sij represents the equilibrium stress at time t, and τij representsthe incremental stress to be calculated at each time step.t tSij t Sij t τij(5)The integral equation is established by linearization in each incremental step, t tttttttt tDijkl ekl δ eij d V τij δ ηij d V Q τij δ t eij dt VtVtV(6)tVwhere, t t Q is the incremental outer force including the aerodynamic force, engine thrust and gravity, etc., atthe new time step. Considering a number of shape functions, the relationship between strain and deformation ispresented astttttη t BNL u e BL u,(7)Substituting them into Eq. (6), the element governing equation results in(t KL t KN L )u t t Q̄ t F(8)The corresponding dynamic equation is,tM t t ü (t KL t KN L )u t t Q̄ t F(9)2.3. Dynamic linearizationThe assumption of small amplitude around the static equilibrium state is suitable in many dynamic problems,including dynamic stability. That isu ū x(10)where, ū is the large deflect equilibrium deformation from Eq. (8). x is a small vibration deformation. Accordingto equation (9) and static equilibrium condition, the vibration equation of the system under steady forces reduces toMT ẍ KT x 0(11)where MT is the inertial matrix of the structure at static equilibrium configuration and K T is the correspondingstiffness matrix. From Eq. (11), the mode shapes and frequencies are deduced.