Document
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NASA T e chnical M e m o ran d u m 84635
NASA-TM-84635 19830018573
DETERMINATION OFSTABILITY ANDCONTROL PARAMETERS
OFA GENERAL AVIATION AIRPLA N E FROM FLIGHT DATA
_i " ' ' :.
IMRAN ABBASY
MARCH 1983
LI BRARV COP Y
' .iOtq "i4 _q83
N_A LANGLEY RESEARCH CENTER LIBRARY, NASA National Aeronau t ics and HAM P TONLVIRGINIA Space Administration Langley ResearchCenter Hampton, Virginia 23665 SUMMARY Values for the stability and control derivativesof a slngle-engine,low-wing, general avia t ion airplane have been de t erminedfrom flight d at a. La t eral and longi- t udinal transien t maneuvers were analyzed by t he equa t ion error and outpu t error methods. One quasl-s t eadymaneuver was also inves t igated. An angle of attack range between 4 and 26 degrees was covered and t here was a good agreement be t ween the par a metersextracted fr o m fligh t da t a and those predictedby wind tunnel.
INTRODUCTION There have been a number of previousattempts to determinestabilityand control parame t ers from flight data. Gerlach and How a rd in references1 and 2 have applied t he equation error method for the de t ermination of longitudinal and la t eral parame t ers, respec t ively. In references 3 and 4, parame t ers estimated from the outpu t error method were compared wi t h deriva t ivesobtained from t he wind t unnel and theore t icalpredictions. The parameter es t imation problem is reasonably routine in fligh t conditionsfor which t he aerodynamicsare linearly rela t ed t o the response and inpu t variables. However, problems persis t for t he m odeling of unsteady and high angles of a t tack aerodynamics. Mos t of t he recent research effor t has, therefore, been directed primarily towards these unresolved problems (refs. 5 t o II). Even though t he problem of parameter estimation at low angles of a tt ack has been well addressed in the pas t years, t here have not been many applicationsin high angle of at t ack regimes. Furthermore,few a t tempts have been made to correla t e the parameters ex t rac t ed from fligh t data t o the wind t unnel and t heore t icalpredic t ions.
This work was done under the General Aviation Stall / Spln Program at NASA Langley Research Center. The primary objectives of this program were to gain a better understanding of the aerodynamics in pre- and post-stall regimes and to design and validate airframe modifications to improve stall / spln characteristics. Part of the overall program was devoted to the measurement of the airplane transient maneuvers for the subsequent extraction of the stability and control derivatives. The test airplane which produced the data for this analysis was tested with several modifica- tions both in flight and in the wind tunnel. For the purpose of this report, data from only one of these modifications was used. In flight the airplane was excited from steady state flight at different airspeeds by conventional control surfaces, so that the entire operating angle of attack range was covered. The mathematical model was so postulated as to include nonlinear contributions to the equations of motion at high angles of a ttack.
The purpose of this report is to estimate the aerodynamicparametersfrom flight da t a covering angles of a tt ack between 4 and 26 degrees, and to compare t hese parame t ers wi t h those es t ima t ed from t he s t a t ic and oscilla t ory w lnd- t unnel t es t s.
Also t he validity of t he par a me t ersis checked by comparing t he simula t edda t a which is genera t edfrom t he estima t edmodel with t he ac t ual flight records. The estimation in this repor t was done by t he s t epwlse regressionmethod. Experiencehas shown that t h e s tepwise regr es sion an d maxim u m likeli h oo d me t h o ds g i ve s imilar r esu lt s .
Therefore, only a few runs were analyzed b y the maximum likeli h ood meth o d to chec k the regre s s i on e s t i mates.
S YMBOL S A defined in Appendix B aX longitudinal a c celeration, g units ay lateral a c celeration, g units a Z vert ic al ac c eleration, g units b span, m C A ax la l - f or cec oef f i c ient,-Fx / _S CD drag coefficient,D / qs CL lift coefficien t ,L / qs C£ rolling-momentcoefficien t , _ / _Sb Cm pltchlng-moment coefficient, _ l qS_ Cn yawlng-moment coefficient, _ / qSb C N norm al -f o r c e c oe f f ic ien t , - Fz / _S C T thrus t coeffi c ient,T / qS Cx longlt ud lnal-force c o efficient,Fx / _S Cy slde-forcecoefficien t ,Fy / qS CZ vertlcal-forcecoefficient,Fz / _S c mean aerodynamic chord, m D drag, N Fx,Fy,F z force along X, Y, and Z body axis, respectively, N G airplane response vec t or g acceleration due to gravity, m / sec 2 h distance of CG aft of leading edge of wing, percent of _,Iy,l Z moment of inertia about X, Y, and Z b ody axis, respe c t i vely kg-m 2 IXZ product of inertia, kg-m 2 J cost fun c tion k normalized frequency, _ / 2V or _ b / 2V L llft, N Mx,My, _ moment about X, Y, and Z body axis, respectively, N-m m mass, kg N number of data points p body axis roll rate, rad / sec q body axis pitch rate, rad / sec dynamic pressure, I / 2pV 2 R-1 weighting matrix r body a xis yaw rate, rad / sec S wing area, m2 T thrust, N u,v,w velocity along X, Y, and Z body axis, respectively, m / sec V airplane total velocity, m / sec Y aerodynamic force or moment coefficient angle of attack, rad or deg 6 angle of sideslip, rad defined in Appendix B 6 aileron deflection, red a elevator deflection, rad e rudder deflection, rad r € thrust vector angle with respect to longitudinal body axis, rad 0 stability and control derivative vector O pitch angle, rad P air density, kg / m 3 ¢ roll angle, red yaw angle, rad frequency, rad / sec Subscripts: o trim value M measured value WT wind tunnel Superscripts: T t ranspose • deriva t ivewith respect t o t ime ' deriva t ivesdefined in AppendicesA a nd B A es t ima t e Abbrevia t ions: CG center of gravity ML maximum likelihood SR stepwise regression Derivatives : CA = 8CA = _CA CD = 8CD q 8q_ / 2V CAR _ = 8e CD6 = a--_--_CD CL = _CL CL 6 _CL e e e 8e = 8--_-- e e CN = _CN _CN q _q_ 2_ CN_ = cx = 9Cx c x = 9 C x _ C x 9_ q 9qE / 2V CX_ = _--_-- e e 9Cz i _ C z c z = c z = C z_ 9e q 9q_ / 2V e e
c 9Cm c 9c c 9c
m = m = m_ = 86 _ a q _qc / 2V e e 9C Cy 8 9_ C£ = 9C£ Cn8 n = 9B B 9B = _B 9Cy C£ = 9C£ C 9Cn = p 9pb2_ _p 9pb / 2V p 9pb--72V n = Cy = 9Cy Cg = 9Cg C 9Cn r 9rb / 2V r _rb / 2V nr = 9rb / 2V 9C Cy 6 9Cy C£ _ 9C£ Cn6 n = 9_ = 96 = 96 a a a a a a 9Cy 9C_ C 9Cn CY6 = 96 C£6 = 9_ n6 = 96 r r r r r r 9Cy Cg = 9C£ C 9Cn Cy_ = _b / mV _ _b / mV n_ = _b / 2V 92Cx 1 92Cz C 9Cm CXa 2 1 CZ 2 = mR = -_ / 2V = 2 9a2 a 2 9 a 2 C£ a 2 = I 93C£ 8 2 9298 Flight Test and Data Reduction A slngle-englne, low-wlng, general aviation airplane with a fixed tricycle landing gear was used as the test vehicle. Since this airplane had been tested in spins, a spin recovery parachute system was installed. Furthermore, the configuration flight tested for parameter extraction had as an additional modification, an outboard leading edge droop, which enabled trimmed flight at high angles of attack. Figure 1 is a three view drawing of the test airplane and figure 2 is a drawing of the tested configuration. Table I lists the physical characteristics of the airplane. The flight data was recorded by an onboard instrumentation system, and was sampled at 40 cycles / sec. However, for the estimation purposes every other sample was used. For the longitudinal maneuvers, a simple elevator doublet or a combination of several doublets was chosen based upon reference 4. For lateral maneuvers the aileron and rudder inputs should have not only good harmonic content but also proper phase relationships. In accordance with the conclusions in reference 4, a rudder followed by aileron input or vice versa was chosen. Typical time histories for the transient longitudinal and lateral maneuvers are given in figures 3 and 4, respectively. In addition, one slow acceleration / deceleration run was also analyzed as quasi-steady maneuver, where airplane angle of attack is varied gradually by the elevator maintaining a constant pitch-rate.
The validity of the flight data was checked using the technique presented in reference 12, which gives a way of estimating the states via the aircraft kinematic equations. In case the integrated accelerations do not match with the flight data, appropriate biases are computed. Wherever needed such instrument biases and dynamics were taken into account. The measured angles of attack and sideslip were corrected for the upwash and sidewash. The measured velocity was corrected based upon the calibration derived from flying the test airplane over a measured ground trace.
Since the equations of motion used in the analysis are referred to the airplane center of gravity, the data were transformed accordingly.
Parameter Extraction Techniques There are several methods for the estimation of airplane stability and control parameters. Their basic differences are in the optimization criteria and assumptions regarding external disturbances and the presence of measurement noise in the data.
The well established methods for airplane parameter estimation are the equation error and maximum likelihood methods. The present report uses a stepwise regression method which is a version of the equation error method.
(a) Stepwlse regression(SR): This method is outlined in reference14. It is based on t he minimiza t ionof the cos t func t ion N J(@) = Yl= [YMi- Y(8)i] 2 (I) In the equation, YM is the aerodynamic force or moment coefficient based upon the measured states and y(O) is calculated from the estimated parameters (0). The difference between stepwise regression and ordinary linear regression is in the capa- bility of the first technique to select independent variables from a set of candidate parameters one at a time, until the regression is completed. The order of insertion and the adequacy of the model is determined by various statistical criteria. For a more detailed description the reader is referred to references 13, 14, and 15.
(b) Maximum likelihood (ML): This is a nonlinear estimation technique and therefore requires an iterative solution. The method minimizes the error between the measured and predicted outputs. For no process noise the simplified cost function that has to be minimized is N J(O) = i___ 1 [GMi- G(O)i]T R-I [GMi_ G(O)i] (2) where G is the airplane response vector, [V _ q 8]T or [8 p r €IT, and R-I is the weighting matrix.
F o r a d eta il e d discus s i o n the reade r i s referred to references 13, 16, and 1 7.
Results and Discussion There were 7 test flights for two CG locations, out of which a total of 62 longitudinal and 58 lateral maneuvers were available for analysis. In addit i on a total of 6 slow acceleration-dece l eratlon runs for the two configurat i ons were also made. The stepwls e regression method was con s idered more suita b le than the maximum likelihood method for the estimation due to its simplicity and its capability of selecting the appropriate model. A l so, the low noise leve l on the data guaranteed the efficiency of the technique (ref. 10).
Typical results for the longitudinal and lateral modes are shown in Table II.
Figure 5 i llustrates an example of a regression mode l and the autocorrelatlon coefficient of the residuals of the computed and measured coeff i cients. The features to note are the reasonable fit for the computed and measured coefficients, and the rapidly fal l lng autocorrelatlon of the residuals signifying a random sequence. The estimated parameters using two parameter estimation methods are shown in figures 6 through 13 for idle power conditions. The standard errors were under I 0 percent for most of the parameters. The model determined was linear until about I0 ° angle of attack, and at higher angles of attack nonlinear terms (Appendix A) were also i ncluded. The nonlinear terms improved the fit to the data, but did not significant l y change the values of the linear terms. Since d i fferent nonl i near terms were selected as being siginlflcant for each parameter extraction run, no attempt has been made to document them. The figures also show the repeatability of results from var i ous flights and their consistency with the ML estimates. Some maneuvers for different power settings were also ava i lable, but the data were insufficient to make any useful observations.
The acce l eration / decelerat i on run was analyzed assuming quasi-steady flight (constant pitch-rate). Using the known elevator effectiveness, the effect of the elevator was removed to obtain the aerodynamic coefficients as a function of only angle of attack. Figure 14 shows a good agreement between the wind tunnel and flight determined CL and Cm.
Figures 15 through 22 give a comparison of the parameters est i mated from fl i ght using the SR method and the wind tunnel predictions.
I. Longitudinal Parameters: The X-force derivatives generally had higher standard errors (10 to 50 percent) and had a poor comparison with t he wlnd-tunnel results, as shown in figure 15. This is attributed to thrust effects (i.e., idle power isnot necessarily zero thrust) and the lack of excitation of this mode. A definite trend is appar e nt until about 12° angle of attack, and a scatter is observed f o r h i gher angles of attack.
The Z-force derivatives were well defined (standard error less than 5 percent) and the static derivatives showed good agreement with the wind-tunnel estimates (Figure 16). CZ , h o we v er, did not agree with the wind tunn e l, whi c h pr e di c ts a q positive trend in the parameter. A study of the aerodynamic characteristics of general aviation a i rplanes in the same class substantiated the results o bta i ned from flight (reference 18). The unusual contradiction between the flight and wind tunnel estimates has not been explained.
The pitching moment derivatives were generally of low standard error (less than 5 percent) and distinct trends with respect to angle of attack were observed, as shown in Figure 17. Though the configuration flight tested had different CG locations, the flight estimates failed to separate the respective C' s (figure 8) m An attempt to separate the C' s was made using the ML method, but there was no m significant change. Overall, the trends between the flight an d wind tunnel C' s m m agreed very well. C' was harder to estimate due to the high correlation with C' q m_ ' e as was shown b y the ML tec h nique. T h e values estimated were of a smaller magnitude then one would expect of an airplane of this class. The wind tunnel estimate on the m other hand predicted an unusual decrease in C' with increasing angle of attack, q which essentially implies an increase in short period damping. The comparison between flight and wind tunnel C' s was therefore poor The elevator effectiveness m q (C' ) was lower than what wind tunnel predicted. A study of the spin parachute m_ e recovery system as a possible explanation gave no clue as to the cause of this discrepancy.
2. Lateral Parameters: The 8-derivatlves were identifiable (standard errors less than 5 percent) and except for C£B agreed with the wind-tunnel results (figure 18). C£B was consis- tent until about 8 ° angle of attack and then there was an under-estlmation of the parameter magnitude from there onwards. This could be due to the flow separation over the wing planform, resulting in a sluggish dihedral effect.
The p-derivatives with an exception of Cyp were consistent with the wind-tunnel results for the entire angle of attack range (figure 19), the standard errors were under 7 percent. Cyp is a less significant parameter since the airplane motion is not very sensitive to it, therefore, the lack of consistency is not considered critical.
The r -d erivatives were generally consistent at low angles of attack, as shown in figure 20. Standard errors of the estimates were sometimes as high as 20 percent.
The disagreement at higher angles of attack could be due to the spin recovery parachute canister which protrudes about 1 / 2 meter aft of the elevator trailing edge.
The effect of the aileron was easily determinable and errors of estimation were very small (less than 5 percent). The aileron effectiveness (C£_) agreed with the a wind tunnel as shown in figure 21.
The effect of the rudder also agreed with the wind-tunnel results, standard errors of the estimates were less than 5 percent. A sharp drop in the magnitude of Cn_ is apparent around 12° angle of attack (figure 22). This could be due to the r effect of the wing wake on the vertical fin. At higher angles of attack the vertical fin is below the wake resulting in an improvement in the rudder effectiveness.
An high angles of attack the SR technique yielded numerous nonlinear terms (Appendix A), which have not been discussed. The discrepancies in flight and wind tunnel determined parameters in such regimes can be attributed to the effect that has been absorbed in such terms.
The ultimate test for the validity of the model was a comparison between the measured and simulated response of an airplane when subjected to the same input. The simulated data were generated uslng the parameters estimated from the SR algorithm.
Models estimated at different flight conditions were tested and in most cases the prediction capability was good. Typical comparisons for the longitudinal and lateral maneuvers are given in figures 23 and 24.
CONCLUDING REMARKS A complete set of stability and control parameters for a general aviation aircraft was obtained from flight data. The standard errors of the main derivatives varied between 1 and I0 percent, errors were higher for parameters of lesser importance. Most of these parameters agreed with the wind tunnel estimates.
Additional work is required to explain the discrepancies observed in the longitudinal rotary derivatives. Based on the estimates the airplane response was predicted well for the 4 to 20 degrees angle of attack range.
REFERENCES I. Gerlach, O. H.: Determination of Performance Stability and Control Characteris- tics from Measurements in Nonsteady Maneuvers. Stability and Control - Part 1, AGARD CP No. 17, September 1966, pp. 499-523.
2. Howard, J.: The Determination of Lateral Stability and Control Derivatives from Flight Data. Can. Aeron. and Space J., Vol. 13, No. 3, March 1967, pp. 127-134.
3. Suit, William T.: Aerodynamic Parameters of the Navlon Airplane Extracted from Flight Data. NASA TN D-6643, 1972.
4. Cannaday, Robert L.; and Suit, William T.: Effect of Control Inputs on the Estimation of Stability and Control Parameters of a Light Airplane. NASA TP-1043, 1977.
5. Park, G. D.: Determination of Tail-Off Aircraft Parameters Using Systems Identification. Proceeding of the Third AIAA Atmospheric Flight Mechanics Conference, June 1976, pp. 128-136.
6. Queijo, M. J.; Wells, W. R.; and Keskar, D. A.: Approximate Indlclal Lift Function for Tapered, Swept Wings in Incompressible Flow. NASA TP-1241 August 1978.
7. Queljo, M. J.; Wells, W. R.; and Keskar, D. A.: Inclusion of Unsteady Aerodyna- mics in Longitudinal Parameter Estimation from Flight Data. NASA TP-1536, December 19 7 9.
8. Klein, V.: Maximum Likelihood Method for Estimating Airplane Stability and Control Parameters from Flight Data in Frequency Domain. NASA TP-1237, May 1980.
9. Wells, W. R.; Bonda, S. S.; and Quam, D. L.: Aircraft Lateral Parameter Estima- tion from Flight Data w lth Unsteady Aerodynamic Modeling. AIAA Paper 81-0221, 19th Aerospace Sciences Meeting, January 1981.
10. Klein, V.: Determination of Stability and Control Parameters of a Light Airplane from Flight Data Using Two Estimation Methods. NASA TP-1306, 1979.
11. Klein, V.; and Batterson, J. G.: Determination of Airplane Aerodynamic Parameters from Flight Data at High Angles of Attack. ICAS paper 82-6.3.3, August 1982.
12. Klein, V.; and Schless, J. R.: Compatibility Check of Measured Aircraft Responses Using Kinema t ic Equations and Extended Kalman Fil t er. NASA TN-8 5 14, 1977.
13. Klein, V.: Identification Evaluation Methods. Parameter Identification, AGARD-LS-104, 1979; pp. 2-1 through 2-21.
14. Klein, V.; Batterson, J. G.; and Murphy, P. C.: Determination of Airplane Model Structure from Flight Data by Using Modified Stepwise Regression. NASA TP-1916, October 1981.
15. Draper, N. R.; and Smith, H.: Applied Regression Analysis. John Wiley and Sons, Inc., c. 1966.
16. Taylor, L. W.; lliff, K. W.: Systems Identification Using a Modified Newton-Raphson Method - A FORTRAN Program. NASA TN D-6734, 1972.
17. Grove, Randall D.; Bowles, Roland L.; and Mayhew, Stanley C.: A Procedure for Estimating Stability and Control Parameters from Flight Test Data by Using Maximum Likelihood Methods Employing a Real-Time Digital System. NASA TN D-6735, 1972.
18. Batterson, J. G.: Estimation of Airplane Stability and Control Derivatives from Large Amplitude Longitudinal Maneuvers. NASA TM-83185, October 1981.
19. Newson, W. A.; Satran, D. R.; and Johnson, J. L.: Effect of Wing-Leading Edge Modification on a Full-Scale, Low-Wing General Aviation Airplane - Wind Tunnel Investigation of High-Angle-of-Attack Aerodynamic Characteristics. NASA TP-2011, June 1982.
20. Cline, A. K.: Smoothing by Splines Under Tension. Department of Computer Sciences and Center for Numerical Analysis. University of Texas at Austin.
CNA-168, 1981.
APPENDIX A Equations of Motion The airplane equations of motion are referred to the body axes (see figure 25).
These equations are based on the following assumptions: I. the airplane is a rigid body, 2. the effect of spinning rotors are negligible, 3. the airplane has a plane of symmetry XZ, 4. the airplane motion from initial reference conditions consists of small perturbations.
Based on the above assumptions, the equations of motion are expressed as: • P V2 S u = -qw + rv - gsin8 + 2--_ _ (AI) • p V2S v = -ru + pw + gcosBsin# + 2_ Cy (A2) • p V 2 S w-- -pv + qu + gcosBcos# + 2---m-- CZ (A3) • (Iy-Iz_ Ixz p V2Sb q = pr \ my + I-_ (r2 - p2) + 2my Cm (A5) IX-IY ) IXZ pV2Sb r = pq_ _Z + --_Z (_ - qr) + 2i Z Cn (A6) = qcos_ - rsin_ (A7) = p + (qsin_ + rcos_) tan8 (A 8 ) where CX = CTC°SC + CLsin_- CDC°S_ (A9) CZ = CTsin_ + CLC°S_- CDsin= (A10) For the equation error method, the aerodynamic coefficients were calculated from the measured quantities as follows: CX = mg aX (All) i0 CY = _SS ay (AI2) CZ = _SS az (A13) - I X (pq + r (A14) - Iy (AI5) The aerodynamic coefficients were postulat e d as functions of the state and input variables and their combinations: (a) The longitudinal coefficients CX, CZ, and Cm as functions of a , q, a 2 62 _5 = 7 8 6e ' _q, e 6e, , a62, a3, a 4, , a6, , a.
(b) The lateral coefficients Cy, C%, and Cn as functions of _, p, r, 8a, 8r a8, ap, a r, aSa, a Sr, a 2B' e 2p' a 2r' a28a, = 28r, B2, 63, B4, BS, f13 a 2, fl3a, a, a2 , a 3 .
All the above variables and their combinations are the increments with respect to their trim values. Typical linear models for the longitudinal and lateral aerodynamic coefficients are as follows: Cz = Cz + Cz ( a - a o) + Cz q_ + CZ8 ( 8 -8 ) (AI7) o a q 2V e e° e = + C (B-B o) + C p__b r b Cn Cno nfl np 2V + Cnr --2V
(AIS)
+ Cn8 (8a-8 a ) + Cn8 (8r-8 r ) o O a r To avoid identification problems, the linear pitching moment coefficient is defined as: C ' = C ' + C ' ( _- C ' qc C' m m m a o) + + (8 - 8 ) (AI9) o a m 2V m 8 e e q o e where C'm -- Cm + m_--_m CZ + cos (A20)
c % ) o o o 2V2
ii o SE C 'm = C m a + 4-'-'_ Cm_ CZ a ( A 21) m m m ° 4 ---m CZ (A22 )
c'=c + c _+ Ps _ 1
q q a q
C' = C + pS _
m 6 m 6 4 - " -m Cm _ C Z6 (A2 3 ) e e e APPENDI X B Wind Tunnel Data A st a tic fo rce inve s tig a tion was conducted on a full-scale model of the test airplane (reference 19). Various wing and tall modifications were tested for determining their effect on high angles of attack aerodynamics. The investigation covered angles of attack ranging from -9 to 41 degrees at a Reynolds number of 2.5 x 106 , based on the mean aerodynamic chord.
All longitudinal forces and moments were presented in the wind-axes system, whereas all the lateral-directional forces and moments were in the body-axes system.
Therefore, all the longitudinal forces were transformed into the body axes as follows: = -CD C O S= + _sina ( B1 ) Cz = -CDs ina - _c o ss ( B2 ) The de ri v ati ves with re s pe ct to th e angl e of a ttac k w ere de t er mi ned by pu tti n g a se con d o rde r sp li ne t hr o u g h t he da ta a nd c a l cu lat in g t he s l ope a t ea ch p o in t (re f e r ence 2 0 ).
Ba sed o n ( B1 ) an d ( B2 ), t he der i v ati ves in t he b od y ax is s y s t em w e r e c a lc u l a t ed from the expressions, CX = ( C L - C D )c o ss+ ( C D + CL )sins (B3) s S S CZs = -(CLs + % )coss + (CL - CDs)sins (B4) The derivatives with respect to the control deflection were determined based on the coefficient changes over the entire angle of attack range.
AS - ( B S)
f 1
where A is _, _, CZ, C£, Cm, or Cn ; and _ is 6e, 6a , o r _r" Transformation equations for the control derivatives for % and CZ follow from (BI) and (B2) a s follows coss+ sins (B6) CX6 = - CD6 C L_ e e e = -(CD6 sins coss) (B7) C Z _ + CL_ e e e f C m was transformed to the airplane Center of Gravity by S
Cms s
= ( C m)WT - % s(hw T - h) (B8) 1 3 T h e va l u e s f o r Cy , C AB ' and C nB were read di r ectly from reference 1 9., The dyna mic d e rivative s w e re d e r ive d from t he unpubl i s h ed w ln d't u nn el tests tha t wer e run on a one-thlrd s cal e mode l . A po w er sp ec tr al dens i ty a n al ys i s f o r both longitudinal a nd la t e r al tr a n sie nt m a neuv e rs wa s r e qu i r e d to d e t e rm i n e th e domin a ting fre q uen c ie s .
F or t he lo ng it u dinal c a se k wa s de t e r m in e d to r an ge betwe e n 0.0 28 and 0.04 f or angl e s of atta c k ranging betw e en 4 and 24 d e gr ee s. A s imilar an a lysi s on lat e ral transient maneuv e rs betw ee n angles of atta c k 6 and 26 degr e e s y ie l ded a k value ranging betw ee n 0.15 and 0.24, the wind tunnel data how e ver unlik e the longitudinal c a se wa s in se n s iti ve t o t h is va r iati o n.
T he following d e rivative s given in an unpublished wlnd-tunnel test report are: ffi + _ s in a (B9) C ' = C + C n sin_ n n (BIO)
p p
C_ = C£ + CA sl n a
p p _ (B11 )
C' = C + C m m m. ( B I2) q q a C_ = -(C A + CA_ ) (B13) q q CZq' = -(CNq + CN_) (B14)
co. (B 15 )
C' = C - C c o s a n n (BI6) r r n_ C_r = CAr - CA_ c os a (BIT) TABLE I.- PHYSICAL CHARACTERISTICS OF TEST AIRPLANE Wing (Modified NACA 642-415 ) Sp a n, m . . . . . . . . • • • . • 2 ................. 7.45 Area, m ............................. 9.21 Mean aerodynamic chord, m ..................... 1.23 Aspect ratio ........................... 6.10 Dihedral deg Wing incldence_ ........................... 5.00 deg ........................ 3.50 Aileron (each) Span , m . . . . . . . . . • • • • 2 ............... 1.16 Area, m ............................ 0.24 chord, m ............................ 0.21 Flap (each) Span, m .............
2 ............... 1 . 1 5 Area, m ............................ 0.25 Chord, m ....... ..................... 0.21 Horizontal tail (NACA 651-012 ) Span, m . . . . . . ..... • .
2 ................. 2.34 Area, m .............................. 1.96 Mean aerodynamic chord, m ..................... 0.84 Incidence, deg .......................... -3.00 Elevator Span, m . . . . . . . . . . . . . .
2 .............. 2.34 Area, m ............................ 0.77 Root chord, m ......................... 0.34 Tip chord, m .......................... 0.21 Vertical tail (NACA 651-012 ) Span, m .............
2 ................. 1.25 Area, m .............................. 1.20 Root chord, m ........................... 1.09 Tip chord, m ........................... 0.51 Rudder Span, m . . . . . . . . • • • • 2 ................ I .25 . A rea, m ............................ 0.33 Root chord, m .......................... 0.34 Tip chord, m ......................... 0.21 Propeller diameter, m ......................... 1.80 Propeller pitch, m ........................... 1.17 T ABL E II . - E ST IM AT E A N D STA N DARD E RR O RS OF PARA M E TE RS Lo ng i t u d inal Pa r am e t e r V alu e S tan d ard E rr o r CX - 0 .0806 o Cx a 0.396 0.0076 C X 1.19 0.263 q C X6 - 0.115 0.0 1 09 e CX a 2 1. 7 4 0. 11 6 IP a r a m e t e r V a lu e St a nd a rd Err o r Cg -0.0554 CZ -3.43 0.01 1 7 C z -6.07 0.406 q C Z6 -0.214 0.0169 e Cz 2 2 .4 0. 1 78 P ara meter Va lue Stand a rd Error C' 0.00279 m o C ' - 0.22 3 0.00 3 67 m C ' -6.21 0.127 m q C ' -0.60 7 0.00526 m_ e TABLE II .- ESTI M ATE AND STAN D ARD ERRORS O F PARAMETERS (c o nc l'd ) L a t e r a l.
P a r a m ete r V a lu e St a nd a rd Error
Cy 0.0 2 4
o
C y8 -0.328 0.0042
Cy 0.519 0.0204 P Cy -0.374 0.0505 r Cy6 0.181 0.00442 a Cy_ 0.0593 0.0039 r Parameter Value Standard Error C£ -0.00113 o
C£8 -0.0827 0.00174
C£ -0.382 0.00844 P
C£ 0.214 0.0209
r C £_ -0 •108 0.00183 a C£6 0.0152 0.00162 r C£ 14.5 2.03 _2_ Parameter Value Standard Error ¢ - 0 . 0 00559 n O C 0 . 0506 0. 000596 n B C -0.0 7 69 0.00 28 9 n
p
C -0.0878 0.00716 n r C -0.000272 0.000676 n 6 a C -0.05 0 5 0.0 0 0553 ng r
v _ 1 . 22
directi o n 0
and velocity
s en sor
2 . 05
Spin recovery
parachute syste m
5 .7 6
Figure i.- Test airplane. Dimensions are in meters.
1 8
).57 b / 2 _ _ 0.38 b / 2
b / 2
_ -Leadin g - edge d r oop /-- Basic ai rf oil
\ J ( NA CA 64 2 -4 15)
.
Se ct ion A-A
(enl a rged)
Figure 2. - C o nfiguration with droop o utb o ard le a ding edge.
.3 .2 d 2 _ rad .1 q , rad / sec -.5 - 1.0 .2-- rad -.6 g units -.1 - .2 a Z , g units -,1 -.2 Z . 2-- / - .2- 5 e , rad -.6 -1 . 0 ** H l'*'l'l'J*''l*l'''l'Jl*ilJJ'll'J''ll'l'*'l''l'J'''*',,l'**,',',,i**,lJl*,ll 0 2 4 6 8 10 12 14 16 t , sec F i gure 3 .- Lon git ud i n a l t r an s i en t m a neuver.
2O r_
ra d / se c
-,1 -, 2 -. 2 ¢ ,
r ad
-.6 _----
- 1 O : I I I I 1 I I I I I I I I I I I I I I ' I ' I ' I I I I I I I I ' I I I I ' I I I I I I I I I ' I
• 4 _ 8 12 16 20
t, sea
Figure 4.- Lateral transient maneuver.
2 1
t, sec
Figure 4.- Cohcluded.
__ k n 0 -- , 1, - 4 _ ) -- - + - 0 - + _. ) -- + 0 .
"_ O_ +++++++++ + + ++ + ++++++ +++++ 4 - +
-I llIIIIlll'JJIII J111J WJlJ, ,IIIIIl,,rjl ,,,fil,,l
0 I0 20 3 0 40 50 lag -I0 0 4 8 12 16 20 t_ sec Fi g ur e 5.- Mod e l charac t eris t ics from s t epwisere g ression t echnique.
2 3
Op e n s y mb o ls -SR es t imates
Close d sym bo l s -M L es ti mates
_Z Fl t 222 a Fl t 2 34
<> Fit 2 2 3 [ 3 Flt 23 5
1 - '% Flt2 2 4 O Fl t 236
oo
o O v
0 0
- 00_
C x _ o <> 0
A DO
o
-1 I I I I I I I
10-
0_ , A
O O o
0 D ° D
CXq 0 -
Ooo_0 o
VV
- 10 I I I I I I I
.5 -
A
0 •
<>
C x 0 - 0 • oDO
6e 0 0 01-11-100
-. 5 I I I I I I I ....
0 4 8 1 2 1 6 2 0 24 28
a , deg
Figure 6.- Longitudinal-force derivatives estimated from f l ight data.
Open s ym b ols-SR e s timate s
Clo s ed symbols-ML e s t i mat es
V F lt 222 O F lt 234
F lt 2 2 3 r'! F it 2 35
A Flt 224 O Flt 236
0 --
o • oI
-2_ o 0 0 zxQ>°°_
Cza o
ov
-4- o
-6 I I I I I I I
10-
o 0
-_0 - v •_ 0 0
CZq O [] 0 o
0 % o m
- 3 0 - A 0 o
0 0 _:0 °
-5 0 I I I I I I I
i o o_ Oo v° _ cpo% o,
°000 • o
CZSe - I -
-2 I I I I I I I
0 4 8 12 16 20 24 28
a, deg
Figure 7.- Vertical-force derivatives estimated from flight data.
Op e n symbols-S R e sti m at e s
Closed symbols-ML estim a tes
v rlt 222 o mt 2s4
0 - O Fl t 223 O Fl t 235
O __O O V O V _ A Fit 224 O Fit 236
V
<>
C m a - I -
, . o nA ___no . .nn i_p_ ' -
-2 I I I I I I I
m
OV o O @ no
%
o_ 0 v m
, 0 [3
Cmq -10 -
- 20 _ I I I I I I
_
o °
!
C rn - I -
6e
-2 I I I I I I I
0 4 8 1 2 16 2 0 2 4 2 8
a, d e g
Figure 8. - Pitching - moment derivatives estimated fr o m flight data.
Open symbols -S R e s t imates
C losed symbols-ML estimates
2 - V Flt222 _ Flt 234
O Flt 223 El Fit 235
A Flt 224 O Fit 2 36
C y¢ 0 -
a _ _>o_ o o _ n%<>ov q,_
[]
-2 I I I I I I I
m
n _o _ vn o
x_
c_ -.2- O O O AO_ov °o° o
oo
-.4 I I I I I I I
.2-
O ne3 0 - n I:_onv H a O0_ A_ V
O0 (_a
-.2 I I I I I I I
0 4 8 12 1 6 2 0 24 28
a, deg
Figu r e 9.- Sideslip derivatives estimated from flight data.
O pe n s y mbo l s - S R es ti ma t es
Cl o se d sym bol s- M L es ti ma t es
V F i t 222 a F lt 234 -
2- _ Fl t 2 2 3 [3 Fl t 2 3 5
Z l F l t 2 24 O Fl t 23 6
v
-2 I I I I I I .... !
m
C_p 0 -
/
- 1 i I I I I I I
.4 -
Cnp 0 -
a w°aa_ va O _ v _> z _o_ov _ a
- .4 I I I I I I I
0 4 8 12 1 6 20 2 4 2 8 . ,
, d e g
Figure i0. - Roll - rate d erivatives estimated from flight data.
Open symbols-SR e stimat e s
Clos e dsymbols-ML e stim a t e s
V F lt 222 _ F it 2 3 4
3-
Fit22 3 n Fit2 35
2k Fl t 224 O Fit 23 6
CY r 0 - _O _7 V _A_ ) Z _( 9 0A O_
- 3 I I I I I I I
m
o o _vV Va A_"O_ A o oa
C _ r 0 -- VO
- 1 I I I I I I I
.5-
V
O
Cn r 0 - a _Oa_ 7 _ 7 , a _ A O_ O
- .5 I I I I I I I
0 4 8 1 2 1 6 20 24 2 8
g, deg Fig u re i i.- Yaw-rate derivatives estimated from f l ight data.
Op e n s y mb ol s -S R es t ima t es
Clo sed sy m bo ls- ML es ti ma t es
V Flt 222 _ Fl t 23 4
.5- <> F lt 223 D Fl t 23 5
Z i Fi t 2 2 4 O Fl t 236
5a 0 - _ 0 OQ O
-.5 ! I I I I I i
. 2 --
CiS a 0 --
o O oO
-.2 I I I I I I I
.05 -
0,
°o
% o - a vaB <>z_ o o,
- .05 i I I I I I I -_
0 4 8 1 2 16 20 24 28
o r, d eg
Figure 1 2 . - Aileron deriv a tiv e s estimated from flight data.
3 O
Op e n sy mbols -SR es ti mates
Cl os e d s ym b o ls -M L es ti ma t es
V Fit222 G F l t2 34
.5-- <> Flt2 23 [3 Fi t2 35
A Fi t 224 O Fl t 236
c_ 0- ° ._vo _> o_a 4 _v %0o
r
-. 5 I I I [ I I I
.I-
<>
C_6r O - _ 3 W ' _V _ Z G _ > _<> /k < > <_D _ 7 00
-. 1 I I I I I I I
O mm
c c , vmsz_ v_
Cn5 - .I -
r
-.2 I I I I I I I
0 4 8 12 1 6 2 0 2 4 28
a , deg
F i gure 13 . - Rudder der i vat i ves estimate d from f l ight data.
3 1 1.5 D
00 0000
1.0 -
CL
.5 Wi nd tunne l
O Flight
5e = 00
C.G. = 0.346
o I I I I I
. 5 -
Cm 0 _ _0 0
-. 5 I I I I I . ,
0 5 10 15 2 0 2 5
_, deg
-/ Figure 1 4.- Lift and pltchlng-m o ment c o efficients d etermined fr o m flight and wind tunnel data.
1 - 0 0 Q O_ 0 Flight
u ; O C_ Wind tunnel
o ' O
Cx a 0
O
O O
o
-1 I I I 0 I I I I
25-
O oO OO
k = 0 . 019
Cxq o o %0 c
k = 0.0 5
-25 [ [ i ] I [
0.5 Cx 0 n 0 0 0 , 5 e 0 _ 0 0
_b o o
00000 0 00
-0.5 I I I I I I I
0 4 8 12 16 2 0 24 28
a , deg
Figure 15.- Comparison of longltudinal-force derivatives determined from flight and wind tunnel.
0 Flight
-2
C Z a -4 '- 00 _
- 6 i I I I I I I
7 5 -
k = 0 . 019
C Z q 0 k = 0 . 05
- 75 I i I I I I i
0 0
0 _00 0 0
CZSe - I
-2 I I I I l I ........ I
0 4 8 1 2 16 2 0 2 4 2 8
a, deg
F i gure 16. - C om paris o n o f vertical-force derivati v es d eter m inedfrom fligh t and wind tunne l .
O0 0 ,,, W ind tunn e l
. , 0 _0 0 Fl ig h t
, 0 0 0 0 _0 , ._0 0 .346
C m a - 1 - __ 0.274_
-2 I I I I I I I
!
0 , / f'____ O0 0
I
Cmq -10
= 0. 0 5
- 20 I I I I I I I k = 0 . 0 1 9
m
O°oO
o oBoooo 6 _o o o o o 6_ oo
C m
5e -I
r_
- 2 I I I I I I I
0 4 8 12 1 6 20 2 4 2 8
a, deg
Figure 17.- Comparisonof pitching-momentderivativesdetermined from flight and wind tunnel.
m
0 Fig h t
W i nd tun n el
¢¥ 0 o -
o d_d_a__ -
-2 I I I I I I 1
0 ___._0 0[- 0 0 O0 0
c _ -0.2
-.4 I
o 2 -- C n _ 0 - 0 Q30_ . .
oo
- .2 I I I I I I I "
0 4 8 1 2 1 6 20 2 4 28
a , deg
Figure 1 8 .- C om paris o n o f sideslip derivativ esd eterminedfr o m flight a nd wind tunnel.
O Fl i g h t
2 - W ind tunne l
%p 0 o o°oooo 00 @8 0 oq_p
- 2 I I I I I I I
I m
C t p 0 -
_ oood_O co_....
- 1 I I I I I I I
. 4 -
C np 0 -
O Oo° cO oo o Oo°UO° °_°__ _g__
-. 4 I I I I I I I
0 4 8 12 1 6..... 2 0 24 2 8
a, deg
Figure 19.- Comparison of roll - rate derivatives determined from flight and wind tunnel.
O Flig ht
Wi n d tun n el
C Y r 0 O 00 0 , 0 000
- 3 I I I I I I
o_ooo°°°°° o_ o o oo
C _ r 0 - O O
-1 I I I I I I I
. 5 -
O
000 000
Cn r 0 - 0 0 _ 0000 0000
0 0 o CO o 0 _,0 _°
-.5 I I I I I I •
0 4 8 12 16 2 0 24 2 8
_ , d e g
Figure 20.- Comparison of yaw-rate derivatives determined from flight and wind tunnel.
3 8
. 5 - O F lig ht
W i nd tunne l
o 8_ °o@o ° o ° OoO ooo,. ,o o
_ u
Cy O0
5a 0- C)_
- .5 I I I I I I I
.2-
C _6 a 0 -
___n 0 (_000 0_000_
0 _ , _ _ , 0
-.2 I I I I I I I
.05 -
c _OOcpo ° o
0- o o0%°8
- .05 I I I I I I I
0 4 8 12 16 2 0 2 4 2 8
a , d e g
Figure 21.- Comparison of aileron derivatives determined from flight and wind tunnel.
O Flight
• 5 m %)
Wind tunnel
o ooOO_OO e_ o _o _o
CY5r 0 -
-.5 I I I I I I I
O
L;O_ O -
C _ Or 0- 0 0 0 0 0
-.1 I I I I I I I
cbo
O0
Cnsr -. 1
- .2 I I I I I I I
0 4 8 12 16 20 24 2 8
e , de g
Figure 22.- Comparison of rudder deriv a tives d etermined from flight and wind tunnel.
40 ,
. 5
+ m ea su re d
. 3 - c o m p u t ed
O t_
rad
. I
q,
ra d / sec
- .5
1_
a Z,
g units -1
.2_--
-.2- :" ""
5 e ,
r a d !
-.6 -1 . 0 I 'l''ll'[',,,,,I,lillllll,llll,lll,,,,l,,l=,,tl,l,,l=ll,,,ii,,,I,,,,I,,,,,,,,,I
0 2 4 6 8 10 12 14 16
r
t, s ea
Figure 23.- Measured longitudinal flight data time histories and those computed by using parameters obtained by stepwise regression method.
4 1 .4 _ + measured _ , .2 -- - c o mputed
•
_o 2"_ 111111 I111 II IIIlllllll II IIlllllllllll II,lll,lllll 9 " ra d / s ec-.5 _l.O_l , , , ,,,,I,,,, , , ,,l l, , ,,, ,i ,, lll,,,,,i,l,,,,,,, ,, I .1--- ra d / s ec -. I r , -.2 -7,,,,,,,,I,,,,,% g, ,I,,,,,,,,,I, , ,,,, , ,,1,,,,, ,,,, I .5_ _ _ i L I 5a, _ rad -.5 _ .2 - - - 8 r , : . .
Fad 0 -_ :
_ i
-.I ;,,,,,,,*l, , ,,,,,* , l, ,i ,,,,, , l,, ,, ,,,,,l,,,,li,, , l 0 4 8 12 16 2 0 t, s e c Fi g u re 24 .- M e a s ur e d lat e ral f lig h t d ata tim e h i s t o r± es and t hose computed by using para m eters obtained by stepwise regr e ssion method• r e a
q
u
Lat e r a l
w V P v L ongitu d inal
Ve r t i cal
Figure 25.- Body system of axes. Arrows indicate positive direction of quantities.
1 . Repo r t No. 2 . Go v ernment Accession No. 3. Rec i p i ent' s Catalog N o.
NAS AT M - 84635 4. Ti t l e and Subt i t l e 5. Report Date Determinati o n o f Stabilityand ControlParameters March 1983 of a GeneralAviationAirplanefrom FlightData.
6 . Perfor m ing O rganiz a t ion Co d e 505-34-0 3 -06 7. Author(s) 8 . P e rforming O r g an i z a t i on Re po rt No .
*ImranAbbasy .... I 10 . Work Un i t No . _ 9. Perfor m ing Org a niz a tion Name and Ad dres s N ASA LangleyResearchCenter - Hampt o n, V A 23665 11 Co ntract or G r an t N o .
13. Type of Re po rt a n d Period Co vered 12. Sp o ns o r ing A g e n c y Nam e a nd Ad dr es s TechnicalMem o randum N ationalAeronauticsand SpaceA d ministration 1 4 Sp o ns o ring A g e ncy Co de Washington,DC 20546 " ' 1 5 . S _ Jpple m en t a r y Notes ' *The GeorgeWashingtonUniversity, Joint " Institute for Advancementof Flight Sciences,NASA LangleyResearchCenter,Hampton,Virginia.
16 . A bs tract Val ue s f o r t he st ab i l ity and con tr ol pa r ame t e rs f o r a gene r a l av i a ti on a ir p l ane have been de t e mi ned fr o m f l i gh t data. La t e r al and long it ud i na l tr an si en t maneuve rs we r e anal y zed b y t he e q ua ti on e rr o r a nd ou t pu t e rr o r methods. The r e wa s a good ag r ee m en t be t ween t he pa r ame t e r s ex tr ac t ed fr o m f l ig h t da t a and t ho s e p r ed i c t ed b y w i nd t unnel , 1 7. K ey W or d s ( Sugg e ste d b y A u t h o r (s)) 18 . D i strib ution S ta t e m ent ' G ene r a l av iati on Stepwiseregression Maximumlikelihood Unclassified - Unlimited System identification STAR Category - 08 Subject Category08 19. Se c urity Cl as s if. (of t h is re po rt) 20 . Sec u rity Cl as s if. (of this page) 21, No . of Pages 22 . Price* Uncl assi fied U n cl assi fied 44 A03 i N- S OS For sale b y t he Na t ional T echnical In f o rm a t i on Se r vice,Spri n gfield.Virginia 22 161