Estimation of Aircraft Unsteady Aerodynamic Parameters from Dynamic Wind Tunnel Testing

Document

AIAA-2001-4016 ESTIMATION OF AIRCRAFT UNSTEADY AERODYNAMIC PARAMETERS FROM DYNAMIC WIND TUNNEL TESTING Patrick C. Murphy* NASA Langley Research Center Hampton, Virginia USA 23681 2199 Vladislav Klein?

The George Washington University NASA Langley Research Center Hampton, Virginia USA 23681 2199 Abstract Nomenclature numerator transfer function coefficients A, B, C Improved aerodynamic mathematical models, for a, bl indicial function parameters use in aircraft simulation or flight control design, are required when representing nonlinear unsteady mean aerodynamic chord, m aerodynamics. A key limitation of conventional lift coefficient CL aerodynamic models is the inability to map frequency Cm pitching moment coefficient and amplitude dependent data into the equations of J cost function motion directly. In an effort to obtain a more general k non-dimensional frequency, k= ml/V formulation of the aerodynamic model, researchers l have been led to a parallel requirement for more characteristic length, I = _-/2 general testing methods. Testing for a more N number of frequencies comprehensive model can lead to a very time PSD power spectral density consuming number of tests especially if traditional q pitch rate, rad/sec single frequency harmonic testing is attempted. This S Laplace transform variable paper presents an alternative to traditional single- t time, sec frequency forced-oscillation testing by utilizing V Schroeder sweeps to efficiently obtain the frequency airspeed, m/sec response of the unsteady aerodynamic model.

measurement noise V Schroeder inputs provide signals with a flat power Z output measurement vector spectrum over a specified frequency band. For angle of attack, rid comparison, experimental results using the traditional unknown parameter vector single-frequency inputs are also considered. A method o-2 variance for data analysis to determine an adequate tmsteady aerodynamic model is presented. Discussion of T dummy integration variable associated issues that arise during this type of analysis co angular frequency, rad/sec and comparison of results using traditional single superscripts frequency analysis are provided.

estimate Semor Research Engineer, Senior Member AIAA subscripts ? Professor Emeritus, Associate Fellow AIAA L lift force Copyfight © 2001 by the American Institute of Aeronautics and pitching moment Astronautics, Inc. No copyright is asserted in the United States under Title 17, U.S. Code. The U.S. Government has a royalty fi'ee license to exercise all fights under the copyright claimed herein for Governmental proposes. All other fights ale msmved by the copyright owner.

American Institute of Aeronautics and Astronautics for a more comprehensive model can lead to a very Introduction large number of tests, especially if traditional single frequency harmonic testing is attempted. This paper Conventional aerodynamic math models for use in offers an alternative to traditional single-frequency aircraft simulation or flight control design have become testing by utilizing Schroeder 8 sweeps to obtain the increasingly deficient as aircraft maneuvering unsteady aerodynamic model. Schroeder inputs capabilities have extended the flight envelope beyond provide signals with a fiat power spectrum over a conventional boundaries. In these new flight regimes, specified frequency band with low peak-to-peak input especially cases where aircraft maneuver at high amplitudes. Amplitudes are controlled by properly angular rate or high angle of attack (c¢), nonlinear phasing each harmonic.

unsteady aerodynamic effects associated with rigid body responses have been well documented. The need The modeling of aircraft aerodynamics in planar to improve the aerodynamic mathematical model in one degree-of-freedom motion in pitch is identical to these cases is generally acknowledged 15. Although the that in Refs. [12] and [13]. For data analysis the least aerodynamic modeling problem is readily identified squares and maximum likelihood estimates were used.

with military fighter aircraft, it is also an issue for The methodology is tested on wind tunnel data from a civilian transport aircraft. Airliner loss-of-control 10% scale model of an F-16XL configuration. The accidents due to adverse weather or system failures results are compared with those obtained from place these aircraft in nonlinear flight regimes that are traditional static and single frequency data in Ref. [12] not well modeled.

and large amplitude oscillatory data in Ref. [13].

The deficiency in most aircraft simulations and Modeling and Identification models used in control design is the use of a traditional Results from wind tunnel forced-oscillation tests formulation for the aerodynamic model 6. This formulation assumes that the aerodynamic forces and for given amplitude and mean angle of attack show that moments can be represented by a differentiable the resulting combination of stability derivatives function and therefore expanded in a Taylor Series depends on the frequency of the oscillations. This expansion with only first order terms (stability and dependency contradicts the basic assumption that control derivatives) retained. Although this stability derivatives are time invariant. The effect of mathematical structure is adequate for benign portions frequency on the aerodynamic parameters was explained by Goman 9' 10 et al., at TsAGI, and by of the flight envelope, it is completely incapable of Klein11, 12 et al., at NASA LaRC, by formulating the modeling aircraft responses when nonlinear unsteady effects occur. Information about the aerodynamics can linear aerodynamic model equations with unsteady terms.

be obtained from static and dynamic wind tunnel testing.

As an example, lift is considered in the form A key limitation of the conventional model t structure is inability to map frequency and amplitude C a (t) = C a (0) + fCL_ (t - r)c_(r)dr dependent dynamic data into the equations of motion (1) directly. In addition, this formulation only t accommodates the measurement of combined stability derivatives during wind tunnel oscillatory tests preventing separate estimates of damping and unsteady terms. Because the conventional combined terms are where Cc_ (t) and Cc_ (t) are the indicial functions, frequency, amplitude, and angle of attack dependent, simulation engineers are forced to find ad hoc methods C L (0) is the initial value of CL, I is the characteristic to incorporate unsteady effects into the equations of length, and V is the airspeed. Two assumptions were motion 5.

adopted to allow simplification of the model used in the analysis of measured data: a) the effect of c_(t) on the In an effort to obtain a more general formulation of the aerodynamic model, researchers have been led to lift can be neglected and b) the indicial function a parallel requirement for more general testing CL_ (t) can be expressed as methods. Tobak 7 and others have suggested testing methods that allow measurement of the basic parameters of a general aerodynamic model. Testing American Institute of Aeronautics and Astronautics A ----gC (2) CL_ (t) = CL_ (_) -- ae -bl' .

-v L_(_) The simplified model, which takes into account B Z C L_ (_) -- a + b 1 -_ C L_ (_) (8) increments with respect to steady state conditions, can be expressed as C z blCL,_ (oo).

C L (t) = CLa (_)a(t)+ / CLq (_)q(t) Because Eq. (5) is nonlinear in the parameters, the estimation represents a nonlinear estimation problem.

, (3) The initial values of parameters for this technique were - a I e -bl (t-r)&(v)dv obtained from a linear regression using the cost o function where CL_ (_) and CL, (_) are the rates of change N J (O) = E CL (j)(bl + i_j ) + (A_ -C-iB_j )o_(j) 2 (9) with a and q evaluated in steady flow.

j=l Applying the Laplace transform to Eq. (3), the expression for the lift coefficient is obtained as Experiment As 2 +Bs+C A drawing of the 10% scale F-16XL model is C L (s) = a(s) (4) shown in Figure 1. Dynamic tests were conducted in s +b 1 the NASA Langley 12-Foot Low-Speed Wind Tunnel.

For these tests the model was mounted on a dynamic where s is the Laplace transform parameter.

test rig through a six-component strain-gauge balance.

The problem addressed in this paper is the The dynamic test rig is a computer controlled identification of the model given in (4). The parameters hydraulically actuated system, wlfich was sting- in (4) can be estimated by applying the maximum mounted on a C-strut support system. The mounting likelihood principle in frequency domain to the arrangement rotated the model about the reference dynamic wind tunnel data. The model used in the center of gravity location of 0.558 ?-. The tests were estimation procedure has the form conducted at a dynamic pressure of 192 Pa (4 psf) resulting in the flow velocity of 17.52 m/sec (58 fps) - A(o2 +C +iB(o CL (w) = a(w) (5) and a Reynolds number of 106 based on the mean b I +i(o aerodynamic chord.

z(j)=CL(j)+v(j), j=l,2 ..... N (6) Oscillatory data were created using Schroeder sweeps in c_as an input at 13 mean values of angle of where C L (w) and c_(w) are the Fourier transforms of attack, t_0, and an amplitude, (zA m 5 degrees. The range C L (t) and c_(t), v(j) is the measurement noise of c_-mean values was from t_0 = 0 to 65 degrees. Data assumed to be a Ganssian random complex sequence were sampled at 100 Hz with an analog filter at 10 Hz.

with zero mean and variance cy2, N is the number of Tests were repeated ten times at each angle of attack frequencies at which the transformed input output data and then an average signal was formed using the are known, and e0is the angular frequency.

ensemble data. The ensemble-averaged data was used for data analysis.

The maximum likelihood estimator minimizes the negative logarithm of the likelihood function Example time histories of angle of attack, lift and pitching moment coefficients, c_, CL, and Cm, at t_0 =

O:mi {-1.L(ZN;O, o2)} (7)

36 degrees is given in Figure 2. These plots show O,t7 displacements relative to starting values at t_0. The where ZN = [Z(1), Z(2) ..... z(N)I is a vector of output harmonic content of the angle of attack is shown in measurements and O = [A, B, C, bl] is the vector of Figure 3 as a function of reduced frequency, k. Figure 3 unknown parameters. The parameters in (3) were indicates a flat spectrmn for a frequency range of 0.03 obtained by solving the equations Hz to 2.0 Hz which corresponds to a range of non- dimensional frequencies k = 0.004 to 0.268. For the American Institute of Aeronautics and Astronautics analysis, time histories of c_and the longitudinal Figure 5 presents estimates of unknown aerodynamic coefficients were transformed to the parameters in model Eq. (3). The estimation process frequency domain using a Discrete Fourier Transform failed for c_ < 25 degrees because of the lack of (DFT) algorithm 14. This algorithm allowed the unsteady information in the measurement data. This transform to be performed over the frequency range of identifiability issue is corroborated in Figure 4, where 0 to 2 Hz.

the variation of the in-phase and out-of-phase In addition, the traditional in-phase and out-of components with frequency is greatly reduced. Below c_ phase components of aerodynamic coefficients were = 25 degrees there is virtually no variation with computed from transformed data. Expressing frequency.

transformed angle of attack and lift coefficients as Estimates of CL_ (oo) show very good agreement O_(O)) = U1 + iv 2 and Cc (m) = u 2 + iv2, it can be shown with results obtained from static measurements. Static that the transfer function data estimates were obtained by performing a simple finite difference calculation on CL(C0. Estimates of CL (O)) -- U2 +iv2 =U +iV (10) c_(_o) u 1 + iv 1 CLq (oo), on the other hand, did not show good agreement with values in Ref. [12] and error bounds Then the in-phase and out-phase components are, were relatively large. This result is not too surprising respectively, since CL_ (oo) is usually a very small contributor to lift C--L= = CL= -k ZCL/I =U and and therefore difficult to estimate. Estimates of parameter a, the gain on the unsteady term, agreed very C-L_ = CL_ + CLa = 1-V. well with two other independent techniques in k References [12] and [13]. Again the error bounds are large where the information content of the measured Both Fourier components are obtained from measured data is substantially reduced. Estimates of bl generally data and plotted against angle of attack for three values follow the same variation with c_ as that found in of reduced frequency. These plots are presented in Reference [13]. The match with Reference [12] is only Figure 4.

approximate since the parameter was assumed to be Results and Discussion constant in that study to minimize identifiability issues.

The estimation method used in this study is In Figure 6, for c_ < 45 degrees, estimates different from that used in References [11-13] because of C,,_ (oo) show very good agreement with estimates it provides estimates of all unknown parameters at obtained from simple finite difference calculations on different values of cc The previous methods in the static data. Increasing error bounds for estimates in References [11] and [12] assumed that the time the high and low c_ranges occur as before due to low constant, l/b1, did not change with cc In Reference [13] information content. Poor signal-to-noise ratio in the it was assumed that initial estimates of CLq (o_) were pitching moment channel also contributed to poor available from small-amplitude oscillation data to identifiability, particularly in the high c_range. In initialize estimation of the unsteady term in Eq. (3).

Figure 2 an apparently more noisy response is seen in the pitching moment channel at c_ = 36 degrees. Signal- The parameter estimation method outlined in this to-noise ratio deteriorated further with increasing angle paper was applied to longitudinal (planar) low- of attack. This problem was aggravated by the limited amplitude forced-oscillation data. The method was only capability of the dynamic rig to follow low-amplitude applied to cases in the range of angle of attack between oscillation commands. The rig was originally intended c_= 20 degrees and c_ = 65 degrees. This range of angle for large amplitude and relatively fast oscillations. In of attack is where unsteady effects primarily occur for this low-amplitude experiment the Schroeder sweep this model. The estimated parameters in Eq. (3) and required a number of small oscillations as small as one their 2-sigma botmds are shown in Figures 5 and 6 for degree, particularly at the beginning of the sweep (see the lift and pitching moment equations, respectively.

Figure 2). The pitch channel was particularly sensitive The figures also show alternate estimates of the to this problem and for the c_= 46 degrees case parameters when possible.

convergence for the estimation algorithm was poor.

Further investigation is required to fully explain the American Institute of Aeronautics and Astronautics pitching moment response. Excluding the c_= 63 aerodynamic model for aircraft that includes nonlinear unsteady aerodynamics. The general structure proposed degrees case, estimates of C,,_ (_) match very well for the aerodynamic model has the form with values given in Ref. [13] and the 2-sigma botmds were relatively small. Estimates of parameter a, for the L = Lstatic + Lclynami c + Lunsteacly pitching moment equation, also agreed very well with the independent technique in Ref. [13] for c_ < 45 Lunsteacly = 1 pV 2 SC L (t) degrees. For c_> 45 degrees, estimates differed from Ref. [13] and large error bounds were obtained that where reflect the poor information content and low signal-to- °_ noise ratio. In addition, parameter a tended to be very c L =-b(a)c_ + a(a)a small in the low and high c_ranges making the estimation more difficult. Estimates of bl show a trend Similar equations can be written for drag and pitching moment equations. This structure allows easy toward zero or small values for c_above 45 degrees, interpretation of the model parameters by retaining thus indicating a trend toward neutral stability. With conventional stability and control derivatives for static parameter a reduced the tmsteady effect in Cm is limited and dynamic terms. Unsteady terms are obtained by for c_ greater than 45 degrees. As mentioned previously, the estimates of bl from Ref. [12] assumed no solving a first order differential equation with c_- dependent coefficients. This approach offers a dependence on cc Estimates of the same parameters in straightforward model for estimation and simulation.

Ref. [13] didn't reveal any dependencies on cc Below 45 degrees, estimates of bl were in the same range as Modeling unsteady aerodynamics demands estimates from both Refs. [12] and [13].

substantial testing in wind tunnels and in turn a parallel requirement for more general and efficient test Figures 7 and 8 show the frequency response of methods. In support of these goals this study the lift and pitching moment coefficients, respectively, demonstrated an alternative to single-frequency during forced oscillations at c_ = 36 degrees. These harmonic testing using Schroeder sweeps. A transfer figures show the relatively large measurement noise function model that includes unsteady aerodynamics (frequency domain) for this study. Amplitude ratio and was developed. A frequency domain method for data phase plots for the lift coefficient are relatively flat at analysis that can be used to estimate the unsteady the high frequency range (2 Hz). The pitching moment aircraft models was presented. Identifiability issues amplitude ratio is small and slightly increasing with associated with this methodology and with this frequency.

particular experiment were discussed.

An identifiability issue of concern for the transfer The methodology in this study compared well function model structure given in Eq. (10) is pole/zero with previous studies that used different techniques to cancellation. Figure 9 shows the pole and two zeros as a function of c_ for the lift transfer function. The 2na obtain the aerodynamic model for the F-16XL. Both present and past techniques separated the static, rotary, zero shown as a diamond in the upper plot is relatively and unsteady terms and both modeled the unsteady large and has a wide frequency separation from the term as an indicial function. However, the previous other transfer function roots in the lower angle of attack techniques made different assumptions about a priori range. The pole and 1 st zero are fairly close in information. The previous methods assumed that some magnitude and become almost equal at very low angle of the parameters were known. The present approach of attack. The lower plot highlights the pole and 1 st has the advantage that all unknown parameters in the zero to show how the two approach cancellation at c_ = model are estimated at once for various angle of attack.

20 degrees. At that point the values are within 5% of each other creating a more difficult estimation problem.

The previous techniques only used conventional As the pole and 1 st zero values approach equality the single-harmonic, constant-amplitude inputs requiring a model structure in Eq. (10) becomes inadequate and the substantially larger number of tests. This study estimation algorithm fails.

demonstrated the use of Schroeder sweeps that provided wide-band, flat-spectra inputs with low peak- Concluding Remarks to-peak input amplitudes. The maximum amplitude for This study is part of an ongoing effort at NASA the Schroeder sweep is constrained by the dynamic Langley to develop a more general formulation of the rig's capabilities and the bandwidth required for the American Institute of Aeronautics and Astronautics experiment. The Schroeder algorithm prescribes 2° Klein, V., and Noderer, K.D.: Modeling of smaller amplitude cycles within the sweep that improve Aircraft Unsteady Aerodynamic the input from an experiment design point of view, Characteristics", Part I - Postulated Models, especially for nonlinear or input-amplitude dependent NASA TM 109120, May, 1994; Part 2 - systems. In general, the Scbroeder sweep is an excellent Parameters Estimated From Wind Tunnel Data, frequency-rich input for forced oscillation testing NASA TM 110161, April 1995; Part 3 - however the dynamic rig must be able to handle the Parameters Estimated From Flight Data, NASA variety of input amplitudes and frequencies. When test TM 110259, May 1996.

rig capabilities are limited, an alternate choice may be a traditional frequency sweep with limited amplitude 3° Greenwell, D.J.: Difficulties in the Application variation.

of Stability Derivatives to the Maneuvering Aerodynamics of Combat Aircraft, ICAS 98- Identifiability issues for the tmsteady model, 3.9.2, September 1998, Melbourne, Australia.

discussed in tiffs paper, were associated with either a lack of information content or an inadequate model 4° Goman, M.G. (TsAGI), and Khrabrov, structure. Degradation of information content in system A.N.(TsAGI), State-Space Representation of response was found in fllree cases. First, low levels of Aerodynamic Characteristics of an Aircraft at frequency dependence were found for in-phase and out- High Angles of Attack, Journal of Aircraft, of-phase coefficients in the low and high c_ranges.

Vol.31, No.5, September-October, 1994, pp.

Second, low signal-to-noise ratios were found in 1109-1115.

pitching moment responses, especially in the high c_ range or low amplitude cases. Third, when parameter 5° Ogburn, M.E., Nguyen, L.T., and Hoffier, K.D.: values were very small causing little contribution to the Modeling of Large-Amplitude High Angle-of- overall response. Inadequate model structure was Attack Maneuvers, AIAA 88-4357, Atmospheric highlighted in this study primarily for the problem of Flight Mechanics Conference, August 1988.

pole-zero cancellation at very low angles of attack.

° Etkin, B.: Dynamics of Atmospheric Flight, John Other contributors to a mismatch between model Wiley & Sons, Inc., New York, 1972.

structure and system response are suspected, i.e., the dynamic rig, designed for large amplitude motion, may 7° Tobak, Murray and Schiff, Lewis B.: On the have introduced responses not associated with the Formulation of the Aerodynamic Characteristics aerodynamics. This was especially a problem in the in Aircraft Dynamics. NASA TR-R-456, 1976.

pitching moment channel at mid to high angles of attack and for small amplitude motion. In this study 8° Schroeder, M.R.: Synthesis of Low-Peak-Factor responses in the pitch channel did not completely Signals and Binary Sequences with Low follow the expected harmonic responses. Only minor Autocorrelation, IEEE Transactions on distortion was expected since even at 5 degrees Information Theory, January 1970, pp. 85-89.

amplitude some nonlinear behavior can occur in the 9.

Goman, M. G.; Stolyarov, G. I.; Tartyslmikov, S.

pitch channel.

L.; Usolcev, S. P. and Khrabrov, A. N.: Acknowledgments Mathematical Description of Longitudinal Aerodynamic Characteristics at High Angles of Wind tunnel tests were carried out with the Attack Accounting for Dynamic Effects of assistance of NASA researchers Jay Brandon, Dr.

Separated Flow. TsAGI Preprint No. 9, 1990 (in Sungwan Kim, and George Washington University Russian).

student, Leslie Gould.

10.

Goman, M. and Khrabrov, A.: State-Space References Representation of Aerodynamic Characteristics ° Brandon, J.M., and Foster, J.V.: Recent at High Angles of Attack. Journal of Aircraft, Dynamic Measurements and Considerations for Vol. 31, No. 5, Sept.-Oct. 1994, pp. 1109-1115.

Aerodynamic Modeling of Fighter Airplane 11.

Klein, Vladislav and Noderer, Keith D.: Configurations, AIAA 98-4447, August 1998 Modeling of Aircraft Unsteady Aerodynamic Characteristics. Part 2-Parameters Estimated American Institute of Aeronautics and Astronautics from Wind Tunnel Data. NASA TM 110161, 1995.

30 .... :.... :--- _-- -_---: .... :---;---: .... :--- 12.

Klein, Vladislav; Murphy, Patrick C.; Curry, Timothy J. and Brandon, Jay M.: Analysis of Wind Tunnel Longitudinal Static and Oscillatory _ 10 Data of the F-16XL Aircraft. NASA]TM-97- ¢ 0 206276, 1997.

13.

Klein, V. and Murphy, P.C.: Estimation of -10 -20 .... :.... :----:---;---: .... :---;---: .... :--- Aircraft Nonlinear Unsteady Parameters from I:D Wind Tunnel Data, NASA/TM-1998-208969, -30 .................................. :- - - December 1998.

-40 ....................................... :- - - 14. Morelli, Eugene A.: High Accuracy Evaluation -50 .....................................

of the Finite Fourier Transform Using Sampled -60 Data, NASA TM 110340, June 1997.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 i frequency, Hz Figure 3. Harmonic content of transformed _ input for fro = 36 degrees.

_'1 I0 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii_+_ -1 10 20 30 40 50 60 70 Figure 1. Three-view drawing of 10% F-16XL model.

,:,-15 IO_1o _0 .

-5 10 20 30 40 50 60 70 0 5 10 15 20 25 30 35 (deg) 0.2 Figure 4. Variation of in-phase and out-of-phase components of lift coefficient with angle of atack for three different reduced frequencies.

02 t , , , , , , 04_ .0 0 5 10 15 20 25 30 35 0.05

oo:t

0 5 10 15 20 25 30 35 time (sec) Figure 2. Time histories of angle of attack, lift, and pitch-moment coefficients for _o=36 degrees.

American Institute of Aeronautics and Astronautics estimate 0 estimate -x - static data - static data

t

\ _0 o 0 0 -1 -1 \ / -2 -2 i i i i i i --3 i i -3 20 25 30 35 40 45 50 55 60 65 20 25 30 35 40 45 50 55 60 65 r r r r r 0 r estim _--'_atell 2.5 0.5 I-- r_j.]

8_ 1.5 "-_-0.5 E o 1 o -1 0.5

i { {'"t

-1.5 / / -0.5 -2 20 25 30 35 40 45 50 55 60 65 20 25 30 35 40 45 50 55 60 65 0 estimate -- - ref [13] -0.5 /_ .

\ -1 _ 2 _ 5 | • _. J - -2 ref [12] -3 _ 3 _ --- ref [131

-if I

-4 20 25 30 35 40 45 50 55 60 65 20 25 30 35 40 45 50 55 60 65 I 0 estimateEH /-- ref[12] n 1- " ref[13] Jl

t

.............. ° -'- ; -'- -i

ref [1 3] -1 -5 20 25 30 35 40 45 50 55 60 65 20 25 30 35 40 45 50 55 60 65 ot (deg) ot (deg) Figure 5. Estimated parameters and their 2_ Figure 6. Estimated parameters and their 2_ confidence bounds for lift coefficient.

confidence bounds for pitch-moment coefficient.

American Institute of Aeronautics and Astronautics

2[

o ++ .4-_-1_

50 O lstzero l: ..... ' .... ' .... '..... '..... ' .... / "_7_ 1.5 t i "_-_ "_-T'I-_"I--M-T"_'I' 0 2ndzero l: : : :-- : 0 : / 0_..-e--,_--:#---:#-:#--:-_--:-_-:-- _4

i+. >>_+ +

t

::; ii o ....

0.5 I'+ + 20 25 30 35 40 45 50 55 60 65 0 2 4 6 8 10 12 14

l_llx'po,e' ': ': ': ;O ; ': /

O 1st zero 1:O :O :U .................. ] 150r+_... 1_ estimat-_ed ": ..... _! ..... !x i .... i .... !..... !....

_°°[_v_,._,_ ........

_: :...£..!x .... !?' .... !.... ix...!.x..!.x.!..o] 200L_ [ + measured]l

-11_ ........................... i...... o ...... /

, , 20 25 30 35 40 45 50 55 60 65 o 2 4 6 8 10 12 14 frequency (rad/sec) (deg) Figure 7. Comparison of measured and computed Figure 9. Pole and zero variation with angle of attack frequency response, CL/C_. for lift coefficient transfer function.

0.35 [

6 °3I_ + +

0.25t _+ ++ ++++%+_

02t_+. __ _+.T

_015_.--V---_* _ +

o.1P-4- + .....

0 2 4 6 8 10 12 14 [ + measured 2001150_N,, .... , - '-- ]1 I.d_- I -- estimated _100I ' _+,--__1_.. -- -- 5o I . ,_,._, o_ 0 _

__ol "_'__

0 2 4 6 8 10 12 14 frequency (rad/sec) Figure 8. Comparison of measured and computed frequency response, Cm/_.