FREQUENCY DOMAIN FLUTTER SOLUTION TECHNIQUE USING COMPLEX MU-ANALYSIS

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1 7 TH INTERNATIONAL CONGRESS O THE AERONAUTICAL SCIENCES REQUENCY DOMAIN LUTTER SOLUTION TECHNIQUE USING COMPLEX MU-ANALYSIS Yingsong G, Zhichn Yang Northwestern Polytechnical University, Xi an, P. R. China, 77 Keywords: ltter (Aerodynamics), reency domain, Complex m-analysis, Continity Abstract Appling pertrbation to dynamic pressre in the nominal aeroelastic eation of motion, a m-omega method is presented for fltter soltion. Utilizing the strctred singlar vale m as stability margin indicator, this method simply employs freency domain aerodynamics to predict the nominal fltter margin throgh freency domain m-analysis. The error of fltter dynamic pressre is selected as convergence criterion dring iterations. Discontinity of the real m with respect to prely real pertrbation is fond in a two dimensional wing model with steady aerodynamics when sing m-omega method. To prevent this problem, complex pertrbation to dynamic pressre is considered mathematically and the complex m-analysis is employed in this stdy. Nmerical examples demonstrated that the fltter reslts obtained by the m-omega method correlate well with those of the p-k method. General Introdction ltter is a typical dynamic instability phenomenon in flid strctral interaction systems sch as atmosphere flight vehicles with aerodynamic srfaces and the mathematical model can always redce to a nonlinear eigenvale problem. A historical review of several soltions of fltter eigenvale has been made by Heeg in [], in the classical k method and p-k method were discssed [,3]. or k method always assmes strctral damping g which lacks physical meaning, ths the p-k method are more widely sed, becase it is directly related to decay rate which has more physical meanings. However, as no p-domain aerodynamic is available (i.e. the nsteady aerodynamics is sally obtained in freency domain by panel methods); the decay rate compted by the p-k method is only valid near the critical fltter speed. Recent years a g- method is proposed by Chen [4], in which the nsteady aerodynamics is treated as analytic fnction in p-domain, and the freency domain aerodynamics is employed with a damping pertrbation approach. In the last decade, Lind and Brenner introdced strctred singlar vale (SSV) μ into the aeroelastic research field and developed a μ method [5]. In the μ method, aeroelastic system is parameterized at a nominal dynamic pressre with pertrbation to dynamic pressre, which makes it possible to solve nominal fltter problem by μ analysis. However, rational fnction approximation (RA) technie [6,7] of the freency domain aerodynamics is needed in the μ method, which sonds like a p method that employs a p-domain aerodynamics. Borglnd et al. [8-] demonstrated that the RA technie is not necessary in robst fltter analysis and proposed a μ-k method to make direct se of freency domain aerodynamics, bt the pertrbation to dynamic pressre is exclded to facilitate the robst fltter analysis. As a combined work of [5] and [7], the athors proposed a μ method to make a nominal fltter soltion in freency domain in the previos stdies, and the error of redced freency at the fltter speed is selected as the convergence criterion. The reslts coincide well with the p-k method for the test cases []. The μ- method simply solves the fltter margin throgh freency domain μ analysis of

2 YINGSONG GU, ZHICHUN YANG the system freency response matrix, ths it tilizes freency domain aerodynamics directly, and no assmption (i.e. small decaying rate) needs to be made other than the classical methods mentioned above. Using μ as fltter margin indicator, one increase the initial gess airspeed or dynamic pressre careflly and the μ- method predicts the fltter speed ntil the error of fltter redced freency converges to meet an acceptable accracy level. In this work, the discontinity of μ analysis with respect to prely real pertrbation to dynamics pressre is concerned and a complex pertrbation to dynamic pressre is introdced mathematically, the complex μ analysis is employed which can garantee the continity of μ and predict accrate fltter reslts. ormlation of the μ- Method. Eation of Aeroelastic Motion The eation of aeroelastic motion in freency domain reads [ M + jb + K Q( = () n n M, B, K R stand for the generalized mass, damping and stiffness matrices, respectively, Q (k) stands for the generalized aerodynamic inflence coefficient matrix which is a tablar fnction of redced freency k. n η C is generalized displacement and is dynamic pressre. Lind and Brenner proposed a real pertrbation to the dynamic pressre, which is expressed as ( + δ ) = () δ R and δ. Sbstitte E. () into E. () and transfer the pertrbed term to the right hand side of E. (), one can get [ M + jb + K Q( = [ Q( δ (3). Development of the μ ramework As note in [], the μ framework for E. (3) can be formlated by the following procedre. irst introdce two signals {w} and {z}, { w} = I { z} (4) δ n { z} = [ Q( (5) and define the fltter matrix at as = M + jb + K Q( ) (6) k Then E. (3) can be rewritten as follows. [ ]{ = { w} (7) Solving E. (7) for η, we get { = [ ] { w} (8) Now we Sbstitte E. (8) into E. (5), and the formlation is given by { z } = [ P]{ w} (9) Q( k) P = () The relationships between signals {w} and {z}, i.e. E. (4) and E. (9), essentially form a typical μ framework as depicted in ig.. z δ I n ig.. Diagram of the μ ramework of Nominal Aeroelastic System. Now we can se the freency domain μ- analysis to determine the critical fltter point with the freency response matrix P(j). Before that, let s first recall the definition of μ..3 Definition of μ and the μ- Method P or a plant model P formlated in the framework depicted in ig. with respect to a strctred ncertainty set w

3 REQUENCY DOMAIN LUTTER SOLUTION TECHNIQUE USING COMPLEX μ-analysis Δ { Δ : Δ = δ I, δ R, δ } () = n strctred singlar vale μ is defined as μ ( P) = / min{ σ ( Δ) : Δ Δ,det( I PΔ) = } () Δ If there is no Δ Δ that makes I PΔ singlar, then μ Δ( P) =. The peak vale of μ, β is sed to predict the fltter margin. β = sp μ ( P( j)) (3) Δ = Q / μ is non-dimensional dynamic pressre. Let Δ = Iλ in E. (3) and solve for λ one can obtain c λ = a det( I PΔ) = (7) + jd ( ac + bd ) + j( ad bc ) = + jb a + b δ / (4) β f ( / β ) = (5) + E. (4) stands for the maximm pertrbation in E. () to ensre the robst stability of ncertain aeroelastic model, E. (3). In the μ- method, one increases the initial gess airspeed or dynamic pressre careflly ntil the predicted fltter dynamic pressre f converges to an acceptable accracy level. According to the definition of E. (), if there exists a real λ, μ = / λ, else μ =. Now the existence of real λ is eal to 4 a d bc = ( a + b + c) = (8) 3 Analysis and Main Reslts 3. Analytical μ Analysis of D Wing System or a two dimensional (D) wing aeroelastic system, when steady aerodynamics is considered, the system eation of motion and the generalized matrices can be formlated as follows [3]. h&& + x && + c h& h + h = (6) x h&& + r && + c & + r = ( + a) x ch M =, B = x r h K =, η =, r Q = + a c, Assming a >. 5(the elastic axis locates aft /4 chord stream wisely), it is conclded that E. (8) has three non-negative real roots at most, i.e.,, when strctral damping is omitted. This means the real μ has at most three non-zero vale along the positive freency axis, i.e. the real μ is discontinos. 3. Nmerical μ Analysis of D Wing System With the parameters employed in [3] (see Table ), real part and image part of λ is shown in ig. with respect to a given dynamic pressre Q = 4., and nmerical vale of real and complex μ is shown in ig. 3. igre depicts that the image part of λ crosses the horizontal axis three times. It is illstrate that real μ only has non-zero vales at three discrete freency points in ig. 3, while the complex μ is a continos fnction of freency. 3

4 YINGSONG GU, ZHICHUN YANG The predicted fltter reslts by analytical method are shown in Table compared with the reslts by real and complex μ. Table Non-dimensional parameters for D wing parameters vale μ. a -. x.5.5. c h. c. r Nondimensional freency / ig.. Diagrams of μ v.s. / real μ complex μ Based on this fact, the complex μ- method can be described as: A nominal fltter soltion by increasing the initial gess airspeed or dynamic pressre careflly every time, tilize a freency domain μ analysis to search a minimm W that destabilizes the aeroelastic system with respect to prely complex dynamic pressre pertrbation. In this stdy, we jst se the predicted fltter dynamic pressre in each step as initial gess of dynamic pressre in the next step. With a convergence level of Q / Q % < % (9) the fltter reslts predicted with complex μ is shown in Table 3. It is demonstrated that the algorithm converges well and the final reslts coincide well with the analytical reslts shown in Table. An interesting graph of the predicted fltter dynamic pressre is illstrated in ig. 4, in the center point of each disc stands for initial gess dynamic pressre Q in the first colmn of Table 3, and the radis of each disc stands for the predicted fltter margin. This graph shows a good convergence property of the complex μ- method with rapidly decreasing radis within 4 iterations. Nondimensional freency / ig. 3. Diagrams of μ v.s. / Table ltter reslts predicted for Q = 4. Method β μ Q / Analytic Real μ Complex μ The Complex μ- Method and Nmerical Application It is implied that when the compted dynamic pressre is close to the fltter point, complex μ can predict a fltter reslt with certain accracy. Image part Table 3 ltter reslts predicted with complex μ Predicted ltter Q Margin Q Real part of Q ig. 4. Predicted ltter Dynamic Pressre on Complex Srface /

5 REQUENCY DOMAIN LUTTER SOLUTION TECHNIQUE USING COMPLEX μ-analysis Conclsions The discontinity problem of real μ analysis in fltter soltion is noted in this report and an alternative way that sing complex μ analysis to get a continos μ is explored. It is fond that, when sing real μ analysis in fltter soltion, the discontinity problem of real μ that reslts from prely real pertrbation to dynamic pressre shold be careflly examined. The complex pertrbation to dynamic pressre is mathematically introdced to make a continos μ. The algorithm developed for fltter soltion with complex μ analysis can converge to the fltter reslt with satisfied accracy, which is already applied to several examples sccessflly in another work [4]. Acknowledgments This work is fnded by the Project, the National Natral Science ondation of China (Grant No: 6735) and the Program for New Centry Excellent Talents in University of China (Grant No: NCET-4-965). References [] Heeg J. Dynamic Investigation of Static Divergence: Analysis and Testing. NASA/TP--3. [] Theodorsen T. General Theory of Aerodynamic Instability and the Mechanism of ltter. NACA Report 496, 935. [3] Hassig H J. An Approximate Tre Damping Soltion of the ltter Eation by Determinant Iteration. Jornal of Aircraft, Vol. 8, No., pp , 97. [4] Chen P C. Damping pertrbation method for fltter soltion: the g-method. AIAA Jornal, Vol. 38, No. 9, pp 59-54,. [5] Lind R and Brenner M. Robst Aeroservoelastic Stability Analysis. Springer-Verlag, London, April 999. [6] Roger K L. Airplane Math Modeling and Active Aeroelastic Control Design. AGARD-CP-8, pp , 977. [7] Karpel M. Design for Active ltter Sppression and Gst Alleviation sing State-space Aeroelastic Modeling. Jornal of Aircraft, Vol. 9, No. 3, pp -7, 98. [8] Borglnd D. Robst Aeroelastic Stability Analysis Considering reency-domain Aerodynamic Uncertainty. Jornal of Aircraft, Vol. 4, No., pp89-93, 3. [9] Borglnd D. The μ-k Method for Robst ltter Soltions. Jornal of Aircraft, Vol. 4, No. 5, pp 9-6, 4. [] Borglnd D. Upper-Bond ltter Speed Estimation Using the μ-k Method. Jornal of Aircraft, Vol. 4, No., pp , 5. [] Borglnd D and Ringertz U. Efficient Comptation of Robst ltter Bondaries Using the μ-k Method. Jornal of Aircraft, Vol. 43, No. 6, pp , 6. [] G Y S, Yang Z C, Wang W, and Xia W. Dynamic Pressre Pertrbation Method for ltter Soltion: the μ- Method. AIAA 9-3. [3] Zhao L C and Yang Z C. Chaotic Motions of an Airfoil with Non-linear Stiffness in Incompressible low. Jornal of Sond and Vibration, Vol. 38, No., pp 45-54, 99. [4] G Y S and Yang Z C. Generalized M-Omega Method with Complex Pertrbation to Dynamic Pressre, AIAA Contact Athor Address gyingsong@nwp.ed.cn Copyright Statement The athors confirm that they, and/or their company or organization, hold copyright on all of the original material inclded in this paper. The athors also confirm that they have obtained permission, from the copyright holder of any third party material inclded in this paper, to pblish it as part of their paper. The athors confirm that they give permission, or have obtained permission from the copyright holder of this paper, for the pblication and distribtion of this paper as part of the ICAS proceedings or as individal off-prints from the proceedings. 5