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N A S A T E C H N I C A L N O T E QP^QP^ NASA TN D-6532
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LONGITUDINAL STABILITY AND CONTROL DERIVATIVES OF A JET FIGHTER AIRPLANE EXTRACTED FROM FLIGHT TEST DATA BY UTILIZING MAXIMUM LIKELIHOOD ESTIMATION
by George G. Steinmetz, Russell V. Parrish, and Roland L. Bowles
/ '*>'.'.' ^'''.. / '''... ^' . Langley Research Center \ '. ' '/'' / Hampton, Va. 23365 r/
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. MARCH 1972 f TECH LIBRAHY KAFB, NM
D1333b^ 1. Report No. 2. Government Accession No. 3. Recipient's Catalog No. NASA TN D-6532 4. Title and Subtitle 5. Report Date LONGITUDINAL STABILITY AND CONTROL DERIVATIVES March 1972______OF A JET FIGHTER AIRPLANE EXTRACTED FROM g performing Organization Code------FLIGHT TEST DATA BY UTILIZING MAXIMUM LIKELIHOOD ESTIMATION 7. Author(s) 8. Performing Organization Report No. George G. Steinmetz, Russell V. Parrish, L-7882 and Roland L. Bowles 10. Work Unit No. 9. Performing Organization Name and Address 136-62-01-03 .NASA Langley Research Center n. contract Grant No. Hampton, Va. 23365 13. Type of Report and Period Covered 12. Sponsoring Agency Name and Address Technical Note National Aeronautics and Space Administration 14. sponsoring Agency code Washington, D.C. 20546
15. Supplementary Notes
16. Abstract A method of parameter extraction for stability and control derivatives of aircraft from flight test data, implementing maximum likelihood estimation, has been developed and suc- cessfully applied to actual longitudinal flight test data from a modern sophisticated jet fighter. The results of this application establish the merits of the estimation technique and its com- puter implementation (allowing full analyst interaction with the program) as well as provide data for the validation of a portion of the Langley differential maneuvering simulator (DMS). The results are presented for all flight test runs in tabular form and as time history com- parisons between the estimated states and the actual flight test data. Comparisons between extracted and manufacturer's values for five major derivatives are presented and reveal good agreement for these principal derivatives with the exception of the static longitudinal stability derivative Cm This particular derivative is extensively investigated by utilizing the interactive capabilities of the computer program. The results of this investigation verify the numbers extracted by maximum likelihood estimation.
17. Key Words (Suggested by Author(s)) 18. Distribution Statement Parameter estimation Unclassified Unlimited Maximum likelihood Longitudinal stability and control characteristics
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price* Unclassified Unclassified 43 $3.00
For sale by the National Technical Information Service, Springfield, Virginia 22151
L LONGITUDINAL STABILITY AND CONTROL DERIVATIVES OF A JET FIGHTER AIRPLANE EXTRACTED FROM FLIGHT TEST DATA BY UTILIZING MAXIMUM LIKELIHOOD ESTIMATION
By George G. Steinmetz, Russell V. Parrish, and Roland L. Bowles Langley Research Center
SUMMARY
A method of parameter extraction for stability and control derivatives of aircraft from flight test data, implementing maximum likelihood estimation, has been developed and successfully applied to actual longitudinal flight test data from a modern sophisticated jet fighter. The results of this application establish the merits of the estimation tech- nique and its computer implementation (allowing full analyst interaction with the program) as well as provide data for the validation of a portion of the Langley differential maneu- vering simulator (DMS). The results are presented for all flight test runs in tabular form and as time history comparisons between the estimated states and the actual flight test data. Comparisons between extracted and manufacturer's values for five major deriva- tives are presented and reveal good agreement for these principal derivatives with the exception of the static longitudinal stability derivative Cm This particular derivative is extensively investigated by utilizing the interactive capabilities of the computer pro- gram. The results of this investigation verify the numbers extracted by maximum likeli- hood estimation.
INTRODUCTION
A method of parameter extraction for stability and control derivatives of aircraft from flight test data has been developed at the Langley Research Center (ref. 1). This method utilizes maximum likelihood estimation, an advanced mathematical theory, and has been fully implemented on the Langley Research Center real-time digital complex (ref. 2). The casting of the digital program in such a framework permits the analyst to interact with the program through a control console and cathode ray display. Applying the computer program to actual flight test data from a modern sophisticated jet fighter airplane (fig. 1) serves a dual purpose: first, to establish the merits of the computer program and, second, to provide additional means of validating the Langley differential maneuvering simulator (DMS) program (ref. 3). The successful extraction of stability derivatives from real-world flight test data goes a step beyond the use of analyti- cally generated data toward establishing the creditability and merits of any estimation technique. The documentation of the aerodynamic characteristics of the test airplane (ref. 4) is extensive and is currently being used as a prime source for the DMS program package. The DMS provides dual-cockpit fixed-base simulation in which the two piloted vehicles can be operated jointly or individually in either a cooperative mode or an uncoop- erative mode. The flight tests themselves and the parameters extracted serve as addi- tional means of validating the DMS for this particular airplane. The flight test runs utilized in this study are longitudinal responses generated by stabilator deflections in the neighborhood of +/-5. The changes in angle of attack and pitching velocity are typically +/-10 and +/-20 per second, respectively. Normal accelera- tion responses are between +4g and -Ig. The parameters extracted were the standard body-axis longitudinal stability and control derivatives.
SYMBOLS
Measurements and calculations were made in the U.S. Customary Units. They are presented herein in the International System of Units (SI) with the equivalent values in the U.S. Customary Units given parenthetically.
Bg; normal acceleration (positive down), g units
C^ pitching-moment coefficient
^m o pitching-moment coefficient at cr 6 0
8Cm Cm Per radian 8[2V]^ QC^, QC^, + cm damping-in-pitch derivative, + ----, per radian q " ^ af^\2V/ af"l)\2V] QC C^, static longitudinal stability derivative, -m-, per radian "-a Qa
QCm C --'- per radian 01 8^[2V] 2 ^m)^ 5 pitching-moment coefficient at a crrp, 6 6rp
m C pitch-control-effectiveness derivative, per radian 6 6 Cx longitudinal-force coefficient aC-x- --Tper radian '1 ^ afa.)\2V;
Cy ^x per radian ^a Qa
(C-^\ longitudinal-force coefficient at a a-, 6 6rp "T' T
Cr, normal-force coefficient
dy normal-force coefficient at a 6 0
8Cy, ---per radian GZq \2V; ^ aCy; Cig slope of normal-force curve, --, per radian
(Cy} normal-force coefficient at a oirr, 6 6rr \ "/o'rp.Orp 1. 1
aC7 CZ5 "ag^11(R)1' radian c' mean aerodynamic chord, meters (ft) g acceleration due to gravity, meters/second^ (ft/sec2)
IY aircraft moment of inertia about the Y-axis, kilogram-meters2 (slug-ft2)
M Mach number
3 m mass of fueled airplane, kilograms (slugs) q pitching angular velocity, radians/second q pitching acceleration, radians/second2 q dynamic pressure, i-pV2, newtons/meter2 (Ib/ft2)
5 wing area, meters2 (ft2) u velocity along horizontal body axis, meters/second (ft/sec) u acceleration along horizontal body axis, meters/second2 (ft/sec2)
V true airspeed, meters/second (ft/sec) w velocity along vertical body axis, meters/second (ft/sec) w acceleration along vertical body axis, meters/second2 (ft/sec2) a angle of attack, radians a rate of change of angle of attack, radians/second
drr, trim angle of attack, radians
6 stabilator deflection angle, radians
5rp stabilator deflection angle at trim, radians
Q pitch angle, radians
0 rate of change of pitch angle, radians/second p mass density of air, kilograms/meter3 (slugs/ft^) w natural frequency of the pitching-moment oscillation for controls locked, radians/second
4 J r
FLIGHT TESTS
The flight test data were provided by the U.S. Naval Air Test Center at Patuxent River, Maryland. The flight tests were conducted by Navy test pilots as part of an inves- tigation with a McDonnell Douglas F-4 airplane. Five different longitudinal response runs were made: three during one flight test of the airplane and two during a second flight test. The first three runs were made at an altitude of approximately 1524 m (5000 ft) and at Mach numbers of about 0.6, 0.7, and 0.8, respectively. The other two runs were made at an altitude of approximately 6096 m (20 000 ft) and at Mach numbers of about 0.6 and 0.8, respectively. The stability augmentation system (SAS) was deactivated in order to pro- vide full response for all the test runs.
For each of the test runs the procedure was identical. The airplane was trimmed by the pilot at the desired altitude and Mach number and held for a short period, and then a disturbing input was placed on the stabilators. The pilot task during the maneuver was to null any lateral motions. Since thrust was held constant during the maneuver and since altitude and Mach number were approximately constant throughout the maneuver, the translational modes of the airframe were not sufficiently excited to allow extraction of the longitudinal-force coefficient. A time history of the stabilator position for a typical test run is shown in figure 2(a). Shown in figure 2(b) are typical time histories of hori- zontal velocity u, vertical velocity w, pitching angular velocity q, and normal accelera- tion az. True airspeed was first determined from figure 1 of reference 5 using Mach number, pressure altitude, and temperature from flight tests and then resolved through angle-of-attack measurements to yield horizontal and vertical velocities. Angle of attack was measured by means of a vane on a nose boom. The boom was assumed to be rigid. The readings were corrected for pitch by the Navy. Pitching veloc- ity was measured by rate gyros located slightly forward and at the foot level of the pilot. Normal acceleration was measured by an accelerometer located in the left wheel well. The location of the instrumentation is shown in figure 1. However, the center-of-gravity offset of the accelerometer was negligible for longitudinal responses. (No documentation was available from the Navy as to the accuracy of the instrumentation.) The data for each run were sampled at 0.1-sec intervals. Each of these points was used by the computer program but every other point is plotted in all time history figures to avoid crowding. No smoothing techniques were applied to the data provided. The data were supplied to Langley Research Center in tabular form but were transferred to mag- netic tape for use with the computer program.
5 COMPUTATIONAL METHOD
The equations of motion used by the computer program for longitudinal motion are as follows: u -g sin 0 wq 4 ^[px)^^ - C^(o- o^) -<- C^ j^j ^ w g cos e + uq 4 ^j(Cz)^^ + Cz^(a o^) + Cz^+ C^(6
q I ^[(^T^T + cm^a' "T) + c^ a ^u o- tan-l ^ V ^/u2 + w2 Again, it should be noted that the form of input excitation (pitch doublet yielding the short-period motion) makes the extraction of longitudinal-force coefficients (Cy deriva- tives) infeasible, since u is negligible. However, the program extracts a set of C^ derivatives that provide a longitudinal response to match the flight data; thus, a proper representation of true airspeed is ensured. The method of parameter estimation incorporated into the computer program is based on maximum likelihood estimation. This method is inherently iterative and requires an initial estimate for each derivative in order to begin the extraction process. Refer- ence 1 describes in detail both the method of extraction and the computer program. It should be emphasized, however, that the procedure for extracting the derivatives is not straightforward. Engineering judgment tempered with estimator statistics, such as the variance of individual derivatives and intercorrelation coefficients, plays a significant role in the extraction procedure; thus, the capability of analyst program interaction is highly desirable. 6 RESULTS Figure 3 illustrates the response of the mathematical model to a typical set of starting values; the flight data are included for comparison. Usually 8 to 10 iterations are required before initial convergence is obtained. From this base, engineering judg- ment and statistical information are used to eliminate all possible inconsistencies between aerodynamic principles and statistical results. The conditions for the five flight tests are listed in table I. In figure 4 are presented the model responses generated by the final estimates of the stability derivatives for the five flight test runs and the respective flight test data. Again, it should be noted that the flight data displayed have not been filtered. The total set of extracted stability and control derivatives for each flight test run and the standard deviation of each derivative are listed in table n. The intercept deriva- tives Cz Q and Cm o are defined at zero angle of attack and zero stabilator deflection. From the set of extracted derivatives for the test airplane, the five principal deriv- atives Cz ^ZK' ^m ^'mn + ^m/yi SLn(^ ^m?, are plotted as a function of Mach number and compared with the manufacturer's data for an earlier version in figures 5 to 9. (It should be noted that the center-of-gravity variation between the flight test runs is less than 0.55% about 31.80% c".) Aerodynamic differences between the two airplanes are believed to be small and, hence, two different sets of estimates of the principal derivatives exist to predict airplane response. The five principal derivatives taken from the manufacturer's data were fixed as constants in the program and the minor derivatives f('Cx)o, 5 ^'Xru ^X/i? f^'z)(f 6 GZ (Cm\Q g were extracted from the flight data by the maximum likelihood algo- rithm. This procedure yielded the model responses presented in figure 10 for the five flight test runs. These results clearly indicate an error in both frequency and amplitude for normal acceleration a.z and an error in amplitude for vertical velocity w. The issue now is to uncover the manufacturer's estimates that give rise to the errors. Exam- ination of figures 5 to 9 reveals that the largest discrepancy occurs in the estimates of the static longitudinal stability derivative Cnn and, therefore, the error analysis was begun with this derivative. The principal derivatives taken from the extracted set were used in conjunction with the manufacturer's values of Cm-,, to generate the responses shown in figure 11; the minor derivatives were extracted in the usual manner. These results clearly indicate an error in frequency for all the variables. In a final attempt to use the manufacturer's value of C-m this derivative was fixed at the manufacturer's value and the algorithm was allowed to select the values of 7 11111 11111111111 all other derivatives. This procedure converged for all flight test runs with the exception of test run 4, which diverged. The resulting model responses with the respective flight test data are presented in figure 12 for test runs 1, 2, 3, and 5. Again, an error exists in both frequency and amplitude for normal acceleration a^ and an error exists in ampli- tude for vertical velocity w. On the basis of these results, it is concluded that the flight test data require a much lower value for the static longitudinal stability derivative Cm than that presented by the manufacturer. As further verification of this conclusion, values of the static longitu- dinal stability derivative were calculated by using the following equation: ctjn2IY r1""1C qsc (1/(1) These calculations were made from the flight test data after input excitation had essen- tially ceased. The results of these calculations are presented in figure 7. The calcula- tions yielded values of Cm which agreed more closely with the extracted values than with the manufacturer's data. At this point in the error analysis, at least one discrepancy has been shown to be significant. To determine discrepancies among the remaining four principal derivatives, the following procedure was used. The four principal derivatives taken from the manu- facturer's data were fixed as constants in the program, and the minor derivatives as well as Cma were extracted in the usual manner. This procedure yielded the model responses presented in figure 13 and the variation of Cmo, with Mach number presented in figure 14. Although the model responses for this procedure are quite good, the least- squares residual for fitting the flight data is inferior to that of the extracted set presented in table n. Therefore, it is believed that good agreement exists for all principal deriva- tives with the exception of the static longitudinal stability derivative Cm/y. Also, it should be noted that this procedure yielded values for Cm/y that agreed more closely with the extracted values than with the manufacturer's data. CONCLUDING REMARKS It is believed that the agreement between the extracted values and the manufacturer's data for four of the five principal derivatives (C^ C^; Cm + Cm-,, Cmg\ and the extensive investigation carried out on the fifth principal derivative Cm,-., establish not only the merits of maximum likelihood estimation in a real-world environment but also the merits of the interactive capabilities of the computer implementation utilized to carry out the investigation. The investigation of Cm,,, revealed that the flight test data require 8 a much lower value for the static longitudinal stability derivative than that presented by the manufacturer. The present study also provided some of the data to validate the Langley differential maneuvering simulator (DMS) program. Langley Research Center, National Aeronautics and Space Administration, Hampton, Va., January 19, 1972. REFERENCES 1. Grove, Randall D.; Bowles, Roland L.; and Mayhew, Stanley C.: A Procedure for Esti- mating Stability and Control Parameters From Flight Test Data by Using Maximum Likelihood Methods Employing a Real-Time Digital System. NASA TN D-6735, 1972. 2. White, Ellis: Eastern Simulation Council Meeting. Simulation, vol. 12, no. 2, Feb. 1969, pp. 53-56. 3. Copeland, J. L.; Kahlbaum, W. M., Jr.; Barker, L. E., Jr.; Steinmetz, G. G.; and Grove, R. D.: The Design of a Dual Wide Angle Visual Cue Simulator and the Analysis of Multiaxis Projection Equipment. AIAA Paper No. 70-360, Mar. 1970. 4. Bonine, W. J.; Niemann, C. R.; Sonntag, A. H.; and Weber, W. B.: Model F/RF-4B-C Aerodynamic Derivatives. Rep. 9842, McDonnell Aircraft Corp., Feb. 10, 1964. (Rev. Aug. 1, 1968.) 5. Aiken, William S., Jr.: Standard Nomenclature for Airspeeds With Tables and Charts for Use in Calculation of Airspeed. NACA Rep. 837, 1946. (Supersedes NACA TN 1120.) 9 TABLE I.- FLIGHT TEST CONDITIONS Test run ^itud^ ^r W Excitation C^of 1 1524 5 000 0.6 Pitch doublet Short-period motion 31.80% c" 2 1524 5 000 .7 Pitch doublet Short-period motion 31.47% c" 3 1524 5 000 .8 Pitch doublet Short-period motion 31.27% c" 4 6096 20 000 .6 Pitch doublet Short-period motion 32.22% c" 5 6096 20 000 .8 Pitch doublet Short-period motion 31.52% c" 10 TABLE n.- EXTRACTED STABILITY AND CONTROL DERIVATIVES Test run 1 Test run 2 Test run 3 Test run 4 Test run 5 (e.g. 31.80% c) (c.g. 31.47% c") (e.g. 31.27% c-) (e.g. 32.22% 5-) (e.g. 31.52% c-) Derivative Extracted Standard Extracted Standard Extracted Standard Extracted Standard Extracted Standard value deviation value deviation value deviation value deviation value deviation C^o 0.0193 0.00008 0.0156 0.00008 0.0148 0.0002 0.0217 0.0001 0.0177 0.0001 CmQi --106 .0017 -.126 .0026 -.167 .0037 -.109 .0009 -.163 .0013 Cmq + Cma -3.32 .06 -3.11 .08 -3.30 .11 -4.09 .04 -3.93 .05 Cm- .576 .009 .555 .015 .540 .020 .618 .010 .600 .010 Czft --080 .004 -.086 .004 -.106 .005 -.025 .004 -.049 .004 CZQ, -3.36 .07 -3.33 .10 -3.64 .10 -3.50 .04 -3.78 .06 Czg .346 .018 .334 .019 .325 .022 .372 .018 .361 .017 Cz Unidentifiable due to insensitivity i-i t-i iiiiiiiiiiiiriii ^--r ^d^^-~- ---J- -^^"^^~^' --;-- Accelerometer ^<^^ \. ^^y ' Basic instrumentation-^ .<^'^^--^^^^"^ ^r'"'-- ~\ Accelerometer ^n ^^^^-<^--^'"^ ^^^ ^^>'^. \ /1| '-Rate gyro i Figure 1.- Diagram of test airplane. .10 r^_---i---- T) -'___I ^ 0 ------,--l- LLJJ_.JJJ I- JJ J-l .-UJ -L JJ ^ -8 10 '0^ 2 4 6 Time, sec (a) Time history of stabilator deflection angle. Figure 2.- Typical flight data. 12 699 ZJ-'-""n.) 3 o6 :: + Flight data Computer estimate 679 206.96 E^ --1----- I ! --+ 4- 1J' 659 -+++..+++^.+++------^+-^-L;^- 200.86 ^6 +.^+..++++ 3 3 Z.^ 639 ^-- 194.77 619 lUl. 1.1 .U..IJ I.I LI. .I UJ ll.U__U.LI-J+/-J l.l+/-l loo 67 100 P------I 30.48 z. +4" + 50 u r-' +.. 15.24 (B + + + -^ + + ++ +f -^+ S! ^ ", ..+ -'.,.- o ^ ^++'^E + ---^- -50 ^ r- -15.24 -ioo mJ- iJ-L___- JiU-J -U_ ^llj __L^J_ -30.48 z. + s = + ..+++ i -,. -.- ^+--.,: ^-^,,^-^-. z ++ + + "r+ w +. _.5o ^Li-L 1.1^1 -UJJ- LL.L JJ__ L1.LI.__L I..I U.___JJ__ 4J ++- ^ \ :----- :,^ ^-^. - +"~"^++;+. ^ -2 ^^~--- "^ -4 ^+/-U. U JJJLJ.LI J+/-LL -U.I Jl-U .1.1 J_LU-[+/- 0 2 4 8 10 Time, sec (b) Time histories of horizontal velocity, vertical velocity, pitching velocity, and normal acceleration. Figure 2.- Concluded. 13 664 r:----I-----1------' / 202.39 + Data g44 ^------Computer 196.29 g 1 estimate ^^ S! ^o j^^+/-^^^^^ ^ ^.,^. 604 ::------^------184.10 ~_ ^ f H. I.I.I .-U-LL ^ 584 -ti 1- .1 1.1- LI Ll.1.1 1-L I-.LLLL i78.oo 150 r:----j-----r------1-----,----- 45.72 = ^. .4 75 ^------__ -^ _..-^..-,+.^_ 22.86 S-'-'--- + + + +4" ^~r--t-++^+- Sij _\ 4 !______-LLU _____^U__-U___ ,UU-J____ -150 ^ -.45.72 j o i- 7. ^-. ++-+--^- -,-^-. ^ "r_y \ T-<-.^^+ ^^ J- 0 17^:^"^' 7--^" +^++-.----:+4++:^++- _4 __LL^J-U- J U .-___ JJJ.I- ^l^ L ___ il I- 8 10 0 ^ 2 4 6 Time, sec Figure 3.- Responses to typical starting values. 14 699 b . ^------I 213-06 679 Computer 206.96 619 t-LLJL L .U 11 HH ^8.67 100 --!----1-----I-----I-----I 30.48 = ^\ 50 ^------I \ 15.24 S = / \ /^^ /^<^ s , 0 ^.^ ^ /-----^^^^^ 0 > s -_ \^4 ^ -50 _^- -15.24 -100 "LLU -30.48 .50 _- ---1-----1-----j-----1-----I-----l-----1 \ =^ T \ v'^\, -/--\--^----^, -.?^+/-*:^ ^^^=F;y:^^ ^n 1^ \ / "*++;'r ^ t? ~-_^^ ^>f._ _.so ^111-1 .1 LI 5.0 S 0 ^=----g-:-\------;:^-.------,.^^------.^-^ \ y \ y '^-v-'' \ -t ^ ^ . -2.5 ^------N^ ^-^^^^--^ +/-LL). U J-LL1. LLU Jill+/- J_LLL -5.0 0 2 4 6 8 10 Time, (a) Test run 1. Figure 4.- Final model responses generated by extracted derivative estimates. 15 807 245.97 + Data 787 Computer 239.88 ^ estimate y 767 I. ^4r- Ji"*"^ + < I- -l^M- -<-yH- -i-y>-- =1 r 233.78 |. ^ J =pry^ ^-F^F 3 3 747 ^------227.68 100 ::------30.48 50 Z------_fc._------___- 15.24 | , ^W [_ \ '\ S ^'^-^ ^ 0 s ^/t' ^;-' -50 ^^ 3 ^----^^----^^-----^^-=------^ -15.24 ^ -LLLL J-U.1. LI..LL U_U_ _^p ^ -30.48 25 :=------A- f \\- = / \ /^^- ^-^ ^, : ^Cv^'/ V\ '^Z^ '~^"-;-'. ^ -t JJLU_J-LJ_L_LLLL .J-UJ-J_1+/-L.JJJ1 J.L11 ,l.UJ JLLH 5.0 Z------ 2.5 o Z____^?__,e.^^___^__ ^ -2.5 =------^^------5.0 ^I N 0 2 4 6 8 10 Time, (b) Test run 2. Figure 4.- Continued. 16 916 ----|--- -]-----1------| 279.20 + Data 896 _- Computer 273.10 estimate S 0) g. 876 .~i=?-r -+/--^-+- --+-<-<- ^^^-r. 267.00 3 ^ ^ ^ 856 _-^ 260.91 g36 ?.LLL 1.1. JJ.LL J_LajJJ+/- J_l_l_L_I_l_l_L+/-LLL -Ll_LL 254.81 100 30.48 50 15.24 ~------^/^\ / \ ^-^t 3-s .^-^- i .^4/- -V\ -/-^,--^^^/ ^_y^- T-^+ ^s \ .j/' f -50 -15.24 -ioo JJ-.U- -U-LL-.Li L J IJ L J 1.1 -30.48 .50 .25 ------A- 1 = /V ^ j o ^-^--/-i-y-^ -^+/-:^:^ ^ Z -\y+ \ .' \^^ -1-^ t? ^___" ^ ^^ 50 Il-U. JJJ_L -IJU+/- -11 U- J--LLI. l.LLI JJ.J+/- JJJJL _LU_1_ 5.0 2.5 +J ^.2.5 =_.. V /+ \-_^____'^^____ -5.0 ^l l-l 0 1.6 3.2 4.8 6.4 8.0 Time", sec (c) Test run 3. Figure 4.- Continued. 17 664 202.39 ^--- + Data 644 Computer 196.29 estimate u ^ y. 584 3-LUL -U U-LL J_1_U_ J-LU- 1LLL 178.00 120 ^------7^~^~ 36.58 = / \ 60 ------.:------\------^:+/-4^;.^.------IB. 29 FE+++., / \i. A '*~^. g -^-^M- S = \_v._'___/ \\<. ." 0 -^- -^^-^ \> ^3 -60 ^- -18.29 1U^JJJ+/-J^1LJJJ+/-JJJJ_11JJ.JJJJ. _^0 JJ1J_JJ^J_LU_ -36.58 , .. /__'S.\ J^ 0 ^-^.--1-=^----^------\- ----^------^-^^^:^=FP^ ^^ ^ = w 's-'-^-+^ ..so lLJ-L+/-[JU_JJJ_L-UJ_i.JJJ_L ll_LL.LLLLJ+/-l+/-JlLL 11LL 5.0 2.5 +J -----.--, z---~~^^----m -----~^ ^^^--- "^ -2.5 v^S--^"______^^-^ ^------^ ^1 0 2 4 6 8 10 Time, sec (d) Test run 4. Figure 4.- Continued. 18 866 263-^ h n-i-- -!---+ ---iData 846 Computer 257.86 estimate u a> ~---- 826 ^ ^r^^^^.^^^.^^^^^^^^^ ^, 3 806 .."^----- 245.67 ^ lLLL_LN_L +/-N+/-JJJJ.iJJ+/-J_U+/-JJJ+/- JJJ+/-_LlJ+/-JJ^ ^ 239.5, 120 [------l--/t------I-----|-----1-----|A 36-58 60 = ------______f.----\______--.'y\.______-1-0.291Q u 7 } /' H- a -f \ ^" S; _!-- S S o ^\^L^_^^_V^^ o i i E V -60 ^ -18.29 -T U N _^Q ^ ^ / \ /"^k ^ i V 7 \ ^-^s ^2 o ^ 7--\- -f-^ --3-.-^-^^ w ^+" '< / ^-^>-7' 2 5 -.___. ____:? .;/_._ 50 :-U-LL -LLLL J 1111. JJJJ. J.UJ. 1U.L 5.0 G 3 01 0 --?^^^ 7/1q^ -2.5 Vtw"JL -LL11. 1U ,1.1 LL _1 1J_|_ J J _^ Q ^ 0 2.2 4.4 6.6 8.8 11.0 Time, sec (e) Test run 5. Figure 4.- Concluded. 19 A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data (untrimmed) c.g.=31%c; 1524m (5000 ft) Manufacturer's data (untrimmed) c.g.=31%C; 6096m (20000 ft) -4.0 , / -3.8 Q / ^ -3.6 ' / 0 ./ C7, -3.4 ^>- ^ A -- -3.2 ^^^^ -3.0 ______I______. i_ -l .5 .6 .7 .8 .9 M Figure 5.- Variation of C^ with Mach number. A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data, c.g.=31%5; 1524m (5000 ft) Manufacturer's data, c.g.=31%5; 6096m (20000 ft) A A .3 ------I------t^------.5 .6 .7 .8 .9 M Figure 6.- Variation of C^; with Mach number. 20 A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data; 1524m (5000 ft) Manufacturer's data; 6096m (20000 ft) 0 Equation (1) 1524 m (5000 ft) 0 Equation (1) 6096 m (20000 ft) -.35 , ^ 30 "' ' -25 :------0"^ B cm" -.,o ^ Q Q 8 1 A -.10 .5 .6 .7 .8 .9 M Figure 7.- Variation of with Mach number. Crn Q! All data corrected to e.g. 31% c". > 21 A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data (extrapolated) c.g.=31%c; 1524m (5000 ft) Manufacturer's data (extrapolated) c.g.=31%c; 6096m (20000 ft) -4.5 p 0 Q + A -3.0 C m. a -2.5 I------.------.------.5 .6 .7 .8 .9 M Figure 8.- Variation of C^ + C^^ with Mach number. y; > 22 A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data (extrapolated) c.g.=31%c; 1524m (5000 ft) Manufacturer's data (extrapolated) c.g.=31%5; 6096m (20000 ft) .7 p 0 .6 Q A. .4 ------I------t 5 -6 -7 .8 .9 M Figure 9.- Variation of Cm,, with Mach number. 23 699 i-----,-----I-----I-----,-----,-----r------213.06 ^ + Data 679 Computer 206.96 u estimate 619 ^1 H M L L J I. ias.67 100 ~Z_------' 30.48 ~L ++ 50 Z------:_J_---- 15.24 s o -. :- t^^. 0 ;. -50 =------_------r-^--^^--^--^ ^ ^^^^^ .15.24 -100 -ti t| -30.48 .50 ^------Z + -25 = "n \ -,/''l''ls!s ^ f +/-^- ^"g ^.w C? ^^--^'i-::^'-^- ^^ 50 ELLLL -U-LL _1_U_L J_IJ_L -LIJ+/- JJ J.JL +/-LLL 5.0 S 2.5 =------C 3 01 o E_As_ _/- ^+/---,--, _[ ^ ' v / ^-^---\^^y^^--^ '^ -5.0--\rti iVl 0 2 4 6 8 10 Time, sec (a) Test run 1. Figure 10.- Final model responses generated by using manufacturer's derivative estimates. 24 307 245.97 + Data ^g^ Computer 239.88 estimate m u !+/-~w" 767 '+A'1L-T~ ^^f A*<-+: -t^-l^--y!-(- -i-W^ :++++: 233.78 it? ^-!-i-!- T-FFT' e ^3 227-68 747 E------^M L LLLL ^ 221.59 100 _------30.48 50 ^___------+/-:^------15.24 I o 1-^^^^^^^^ o | E -50 -15.24 :MJ.L U _^0 .30.48 .50 ------.------,----,-----] -25 ^------/^------+ Is. / \ ^^\ ^+. ^ ^-^/_\^/^.^,^^,,.^. ^ EUJJ-- JJJJ- .U LL J \) L -LLU. LLL _LU_L ^ ^ 5.0 P g 2.5 ^^------tr ,r-\ '- ^.^^^'^i^'^-^-^---^--\ /, \^ /-+ \^^^-+^^^ -2.5 ^------V^y +--- ^'1------5.0 t n n m n n m 0 2 4 6 8 10 Time, sec (b) Test run 2. Figure 10.- Continued. 25 916 ^--- 279.20 + Data 896 =--- -_ computer 273.10 estimate u 'Z, 876 = w:r ^+ a ^g- 856 =------260.91 ^ 836 lLajJJ+/- JJJ+/- J-U+/- lliL .LLlJ-J_LlLlllJ_JLllLJ+/-^ 254.81 100 ^------j='----, 30.48 50 ^------f--- 15.24 . ^^-^^^__-^- . I I ^^\ / +"'i^'^ + = ^+ ^^ '+ 3 -50 ^-^ -15.24 -100 ^I N U_U_ .11 -30.48 .50 -_------]------l-----l------I----- .25 =------^------j / -----_____-w-_____W-v-^s^^^^-- - -.25 -.50 IUJ--LJU-LJ-LU_-LU_L J-LLL ..LLLL J_LJ_1_ ^LLL-LLLL 5. 0 ^--- ^ 2.5 ^--- ^.: feE^Z^^^V -T HI n m n n ijj -5.0 ~- n 0 1.6 3.2 4.8 6.4 8.0 Time, sec (c) Test run 3. Figure 10.- Continued. 26 j ^ 664 ------1----- 202.39 'Z. + Data W Computer 196.29 Si -~Z^ estimate i" ^W -t ^. 190-20 624 ^---->^rr -^I-^-^ ^-p-R^- S -^W^"^ ^Wi- ^ 604 ^ 184.10 3 584 iLaj+/-iJ-JJJJ.JJJJ_ JJJJ._LlLLJJJJ.illJ_JUJ^JJJL^ ^s.oo 120 ------r--r+ --^------,-----, 36_5g z + 60 -__ ___+/-+++-l..____ ^g_29 g Jl ^_l_ /^^ -^^-S^------^^_^.^^----- ^. ++++- 4J 0 +^-' +'^----^' ^_-- --+ --r-^ 0 ^ z + >e s --^++ -60 =_ -18.29 +/-Ll^JJ^ _i2o JJJJ_+/-UJ__LUJ-J_Ll+/-JJJJLJ_UJ-JJJ_LiJJ_L _36.s8 .25 aa E------^^V \ = \ ro yf '^^^^~. f0 0 = -~--/-7 ^~^~----\ ^^L^^.^F^ha*'.^ t? ^ ^^"v s--.++^^' _.5o 3-l-LL-U-LL-LJ_LLJJJJ__LUJ_+/-LLL^LLl_ i LI 5.0 2.5 0 ^TT< .++^1 / +++T ^+: +~ --^. :TTT+= = \~^~++ /,.++ \^+j;^?+~^ ^^ 0 2 4 6 8 10 Time, sec (d) Test run 4. Figure 10.- Continued. 27 263-96 866 r:----,-----|-----|-----,-----j------j------+ Data 846 Computer 257.86 estimate '-^^"y^ '*-1-______\t-,__ ^- 82b --kA-^a'i-' 251- 76 ^ ^- ^..^ -q,^ SSS^ ^^ -W-, W=FF > ^? H ^^ 9 g06 ^------^ ^ 245.67 786 36-58 120 r:----1-----1--+-]-----|-----|-----T----I-----I--- = + + 60 -_--- +^\,.----- _^------18.29 g tt) / \ ^/^'^v>' j"t~ + + ^ 1~{~4+' ^'l-H-Hk, t / \. ~^<^, ^ft, ^, 0 ^^---^'--^?-~____ ^K ^ :_____ '.l____; 4--- ^\___ '{+T ^ -60 -18.29 N _i2o N -LL^L _3g 5g .50 ^ ^----I-----j-----l----v-- .. l- --\----+ / \ ./ -\ 4 \ \ >+-^^, ^+>-W.r____A---V-___/7 V____^-----^^----- +^;++/-'tl:A+-te ^2 0 r^v w ^-^^ 0' -______\^______^^ _.50 ^l .LLI L-UI-L .1 1.1,1. -Ul-L L1J <\ 4J \ | 0 ~-----/->.^----/-^+----- /^. "-^ 'a" -2.5 i"'"^^^^\^ ^ Z--- y+/+ ^ ^l W ^^^^l. JJJJ-Ji.l I- JJ-U-JJ-LL 11.0 0 2.2 4.4 6.6 8.8 Time, sec (e) Test run 5. Figure 10.- Concluded. 28 699 1_ ------1-----I-----i-----I 213.06 ~Z. + Data 679 Computer 206.96 ^ estimate S 659 ~^' '-4.-ffc _] --'-'"" A+++" '> -"-_ -^ -,-yr -t-te- --r,: +TI-T :T+-+1"-*'"- 200.86 s ^. 'l-^.t.^'^^- -f-T ++ ++++ ^^^i"^TT IH 3 639 3 ^- 194.77 1_L _U+/-L 619 ^LLL J IIL 1J i88.67 100 30.48 ^'v 50 = --f \ 15.24 8 ^- ^p^'-*-.,. __+ L./ \-1-\ + '+ "T"^" ++ , , ^" M^+ \ /--^e^-^'^ -50 = ^ ^- ^ -15.24 -100 -ti.l U N IJ -30.48 .50 s = n+ /-^ -!". o \ / + \ i- ^-r^' ^++., ^ ?''IS'S' .Jf~ V /+- "VS? ^"- ^'+/-: mw^- '^"^ -.50 ^J 1X 5.0 2. 5 ^~---- -J o =--,^____.'^^+___-^.^^---- ; y^- ^-^ 'N -2.5 ~____V A______K ^^ ^+ ^^^^^ 0 2 4 6 8 10 Time, sec (a) Test run 1. Figure 11.- Final model responses generated by using manufacturer's Cm and other principal derivatives from extracted estimates. 29 807 C---I------i-----I-----I-----I------245.97 + Data 737 Computer 239.88 S ~Z. estimate 767 ^-^>i^M^A^^-i^^-U^ ^^-+A^-^^ 233.78 ^3 ^3 = 747 1------227.68 727 ^l 1 1L 221.59 30.48 ioo r -p--r 50 ~Z______fti'.------.- 15.24 / i+ a) \ /~^ <" ". ^ U \+ /^ ^^ ^^'<++ -'-:. -50 ^^^-^=______-. ^^^-^^ -15.24 ^ ^H L H IJ 1.1.1 J 1.1 1.11- -30.48 .50 .25 =----yV- 0) fr V-_ -^ / <^\+ g 0 =^_i^._(-^_J__1..__-/^:^ /i. \_-- ^A^;^.--'-^'.,.-.:.. Ai+/-i-. f \y ^- -.25 ~______^ "N H 1-1-1 i-LI 1. 1 li 50 5 0 +^ i 0 ^^Y^ ^ '-L.______-a.--- -, -2.5 ------\F<-t------aS ^^^^^- -5.0 iLLLJJJ I. 1- .1 1JJ-1 .1 U-L-LLLJ- I. JJLLL 0 2 4 6 8 10 Time, (b) Test run 2. Figure 11.- Continued. 30 t 916 279.20 ~i + Data 896 Computer 2/-S.-LUin ~_~--- estimate +i" 3 e 856 ~--.. 260.91 ^ 836 3-ULL J_LLL -L1_L-LJ_LLLJ_LLLJ-1.1 IJ 254.81 100 _--.----- ..---^___^-----, 30.^18 g 50 ^~------^--- 15.24 -ioo 3-JLl+/-JJ+/-LJJJJ_JJJJ_JJJJ.JJJJ_J.|Jj_jjjj.jii_Lj_u_L -30.48 f'\ i = /v /^ 0 -4-^--^---^----f+/----S^_---.y^_^ -^>t^ --yf. -^^ =.^t-+- =___^__- ^ -.50 1-1-L1- JJ LL-1JJ_LJ_1JJ_J_I.LLJ.IJ^ 1. -1-1JJ- _LU-1- LI.L 11.J.L. 5.0 --I------I------]------ 2.5 ;---- 4-1 9 o + /" i'+ &' ~-- V y^'^4" ---t-t';++ -2.5 ^^ -5.0 -T_1_LL J 11 1J 1. 8.0 0 ^ 1.6^^ 3.2 4.8 6.4 Time, sec (c) Test run 3. Figure 11.- Continued. 31 1. 664 202-39 F-I--1--I------"=-+ Data I--l 644 computer 196.29 u ^--- estimate (U S 624 ^-M, ^+/-+/-^__----^^;^^^ 1_____^------^: 190.20 ^ = v+ ,^:^^y^^^++++ ++++ e s-+++" 3 3 Z. ^ 604 ^------184.10 584 t _U1 L U L 11 U_ 178.00 120 ^------T^"^-- 36.58 = /+ \ \ 60 =---_____ ^__ \ '*'. ^+^ 18.29 3-^ ^-^ L \ + + ^.---- g Si f" \ + / ^^ ++++ ^. " =---V^---^ --\^^ ^^---^ o ^ = + ++ ^f s -60 =--- _18.29 _^o FT J-LLL 1J I. .LIJJ. L J.LLI JL1JJ. -36.58 .50 = .25 o ----^-SL------\+ -.50 3-LLL JJJJ_ _UJ_L -IJJ L 5.0 rr--i---1---|------I--i 2.5 +J ^------:c V- / ^.+ +/-^^^++ ia vt-.-- ''i^'*' -2.5 ------^-e^------^^'^ --- t -5.0 ^^^ LU 1. 0 2 4 6 8 10 Time, sec (d) Test run 4. Figure 11.- Continued. 32 t 866 I----I-----I-----i-----1------I-----'-----'-----I-----I 263.96 + Data 846 Computer 257.86 ^--- estimate J 826 251-76 i ^^ 3 806 ^^'^^^^^^^^_--- 245.67 ^ 239.57 36-58 120 p-r -;f\-?------i-i-i-i-i 60 ~-^------^-l-.-----/'%------, ,------18.29 <1) \ + \\ ! /^Tis + + S 2-i-H-tk. \^ / 4 <^ <<:P^'--++d:t 3! :- ^-^^-^-:^^-^^ 0 -60 -18.29 XLLL IJJ -36.58 ^ _^ .50 .25 ^------j-\ = / \ A^ ^-fct++ o ^^,^--^-^-^ -^'_^ ^ = V \+ /+ Y^ ^^^^ ^ -.25 =---_____--llA 50 =U-LL -UJJ. JJ-LL .1 I, jLLLL .-UJ.L -U-tJ- 5.0 2.5 ~_ i: -"=- -.->.\ /^^^^^^^<^_ ^ -2.5 ^--- V+ lUL J _5 o .l llJ +/-1 n 0 2.2 4.4 6.6 8.8 11.0 Time, sec (e) Test run 5. Figure 11.- Concluded. 33 fc 699 ^------._. 213.06 + Data 679 :------computer 206.96 u ^ estimate gj j 659 ^^^^^^ww^-^ 200-S6 639 194.77 ^ 619 liJ+/-JJJ+/-JJJ+/-JJ-LlJJJLLJ-LLL+/-U+/-+/-LLL-UJ+/-JJJJL loo.b/ 100 ^------30,48 1 ++ 50 + ^------15.24 /^Y + +, .4.+ / \ ^---<<+ .j-'t 4-, 0 3=5S^^A___s.___ Z^ ^ -100 U-LL H -LLLLJLLlL-LLl.LJ-U-L+/-LLL +/-iU -30.43 I -25 -----^\------^: -^\-/-^^^^-^--~- \ \ y^ -.25 Z______^^___ -.50 ^l _LLL1. _LU.L N L LLI L 10 5 -_ ~Z. 4-1 "rf 0 ^?-'r- ---/- -(^+ -,--s^ ^,... ^ y.,? ^ \ <<: ,,^-.,-,~ --^.^. \-:/^ ^ ^^ -5 -_------\L------____-^ nS -10 -n m H n n n +/-1 ij 0 2 4 6 8 10 Time, (a) Test run 1. Figure 12.- Final model responses generated by using manufacturer's C^ and other derivatives selected by algorithm. 34 v 807 --, 245.97 + Data -,0-j Computer 239.88 ~^~-- estimate ^-M-l- iti 767 +/-H-(- .-+<^, ^.y^. A<--++, ,.;..p^ .^o(., .+^+4, ,4-y^. ^^.(.^ 233.78 -s 3 ? 747 Z~ 227.68 727 :U^-U^^'^^-L-U^^U^+/-1J-L.LJL1-L+/-LLJ_+/-LLL 221.59 100 ^------. -----.-----, 30.48 Si 50 ^------+"^------15.24 I ^^^\^^^^ 0 J' Z + s -50 l------__------15.24 -100 LlJJJL J-LJJL-LU-LJJ_U_JJJ-LJJJ+/-J_t-U-JJJLl.JJJ+/- JJ_U_ -30.48 .50 ^--- ~~~}-----I------I----]-----,-- -25 E----^- / \ ig o = . =^^.M^-'v-7y--^--\../ L--^-^LL^.^^ ,^=t_.A^ -.50 5-U.^-U-LLJJ-U-J_UJ_J^L1-JJJ-.LJL1-U--LLL1.JJJJ.JJ^ +J "' \^ -5 = ^ ^------10 ti LL J 1JLL LLL n 0 2 4 6 8 10 Time, sec (b) Test run 2. Figure 12.- Continued. 35 916 [-----i-----1-----r 279.20 Z. + Data 273.10 ggg Z------Computer estimate 1U+/-J_1_LL J_1_1_L+/-1JJ-.+/-LLL JJJ+/-JJ-LL JJ -1 L 254.81 oJ6 100 r:---1-----r"--' 30.48 g 50 =------:-.,------__---_.__-__-^ 15.24 E __J'"\ 0 . r^.^r ~\~r ^ ^^ ^ ^ -+ - ^? _50 ^^^^ ^^ -15.24 -loo 5-LLL LLI.L-LLLLJ.JJLLJJ-L1--LU-L JLU-L-LLLL -LU-L JJJJ- -30.48 50 ^--- ___^- ^^-W- ^ 0 ^-,^^___^__^---^^^^^ ELL.I-.L -JJLLi- LLI+/- -J_IJ+/- -LLLL +/-Ll+/- 1_LU_ _LI-LL J-l- IJ- J -LLL -.50 4-1 C -10 ILLL-LU-L-l-U-L-LLl.L llLL.LLLr.LLL-L -Ll -1-1 6.4 8.0 0 1.6 3.2 4.8 Time, sec (c) Test run 3. Figure 12.- Continued. 36 t 86e l_' -----1-----I-----I------] 263.96 + Data 846 computer 257.86 y ^--- estimate d) ". 826 ^-rypK yt-M- 5t-___ 251.76 ^ ^ \^ {-f-8-^ ^<^ i-M^"1- ta-^ t-M-fT- -t^""^ Fw^ -". " 245.67 3 gOg ^-- iLlJ-JJJ.L.LJJ+/-+/-l+/-LJJJ+/-JLrLL JJJ+/-+/-U+/-J-LLL-LLLL ^ 239.57 36-58 120 (_------I--,--]-----I------1------|-----\-----1 = + + 60 ++------18.29 S -Z------/^.-! / \ J^+ .J-"*^ 0) ^ S ^^~ '^'i- ..,.+, ^ 0 w^-_____{/^^^^^l^ 0 ^ -60 ^ ^ ^ -18.29 J+/-L JJLU--LU_LJ-LLL _^ ^LLU- .I 1.1 -36.58 .50 _____" \____ .257 ------.-^------= \ /^ ^ i \ T \ ^ = ^--vyA^. ^-^^^-^^ ^ -.25 --^______.___\-jL-- 50 ^-LLL JJ-L1- HJ+/- JLLLL -LLLL .UJ+/- JJ-.LL 10 _-- 5 ^- ^\ T'j~^_ ^ -f^^^t^- ,^<-="?-": ;c -f-w-t^= \--^/^/ ^ \--+N-^-1/ ^ ^^^++., {f^~ -^-^d n-ryF -. -5 -^--- is ^- ^l l.l -10 0 2.2 4.4 6.6 8.8 11.0 Time, sec (d) Test run 5. Figure 12.- Concluded. 37 699 I-----1------1-----1-- r 213.06 -_ + Data Computer 206.96 ~------^ estimate J l---^^^^^.---^ 2086 j p -___ 194.77 a 639- Z--- 619 30-48 100 F-|--i------|--]--|--|--| ^, = T 15.24 50 ^------. --\^------^ = / \ /" -^iA' ^e ___^- \+ ^[-__^^--.--E" 0 >^ 0n -S-S'S--S-EL,^/-r -----^----/fe---_.L .S^-.^^------_---^y.F-'-'^ " ^'M z \4 \ /" ^ "-A^ ^3 s -50 E______--r___- -15.24 -ioo IU-L-LLLL-LLLL U.L.I -l..l-U.J.Ji.]. -L-U-L-lJ-LLJJJLL-UJ-L -30.43 .25 ^------/-^ 2. 5 V g = .+ ,^*+ ', 0 ~^\~ ^--": V^^------'*' 'f""F:!<^. AkA-k- "i / Z__ f "-^^--- -2.5, 'p^-- ^ JJJ-LJJJJ--LU-L-l-l-Ll- _s o ^^H J-LLLJLlJLLJJJLJ-JJJ+/- 8 10 '0 ^ 2 4 6 Time, (a) Test run 1. Figure 13.- Final model responses generated by using algorithm-selected Cm and manufacturer's principal derivatives. 38 I- + Data ~_--.. 737 Computer 239.88 estimate a) > 767 ^-1-M- ^-t-W A*^fc ..p,-i-+- ^-(-<-+. .^^->-(-^. ^-^^ ^y-i--t-!-M- 233.78 ^, d Z g~ 747 _- 227.68 727 t -I 1 1.1 1. .1.1 1.1 221.59 100 30.48 50 -.+-' 15.24 ~--- / "^.. A V V ^ ^i-Ktt.-l. s f^ ^/ X^ '- 0 -50- -15.24 ^-- '^^ ^ 4.1 1.1. LUJ -1.) U -30.48 _^ 1.1 .25 :r--- A- ./ V | ^-^^^/ T ^^^-^ :r.LIJ--LI-lJL.IJ-IJ. ^^ 1J LL -.50 5.0 2.5 4-1 ^^ 1-1 _^ ^ 0 2 4 6 8 10 Time, sec (b) Test run 2. Figure 13.- Continued. 39 k 916 279.20 ^--- + Data 896 computer 273.10 a) ^------estimate =+/-^ ^^i-. 267.00 ^ 876 J--^ .^-^- -t-=*=fe>--*--- ^-4.+ -t-h+.^fc^F-AfeF 9 E 3 260.91 ^ 856 ^------ 836 ti JJJ.LJLLLI 254.81 100 1:------30.48 50 ------^------15.24 -C \ '+-,, 3i o 3-*^___L \--___f \- ^+/-^-+/-___ -+---g^= y^-/--\--/- --\--y ^=---^r .^y o ^e ^ti = ^4 \ /, <--^ -^ ++ ; S -50 -15.24 ,11 LJ_LLLJJ+/-L -30.48 _^Q I.IJ. .U .50 .25 a) =---__^_------__j/ \ E / ./^-s. + ~^ w ^\_'_ \^_ _^_ -.50 ~r\ .LUJ. .LLLL-LU-L _l_LLl LLLL.LLLL 5.0 ^------ 2.5 : ----^---/^ ----:;H,, \ / \ y - -2.S ^------^/-----^-^-----^ ^^^ T 1- ^^.LLLLJ11.L 11 LI- 1-LU 11. _5.o 0 1.6 3.2 4.8 6.4 8.0 Time, (c) Test run 3. Figure 13.- Continued. 40 664 3^39 + Data 644 computer 196.29 ^ ^- estimate ^ 624 ^^^^^,^^ 190.20 i 184.10 604 ^^^^:--- 5A ^I 178.00 120 -^-p,^------i-----,-----i 36.58 = / \ 60 1___. ____;_____V-__ '++^;.. 18.29 " s-'-^^, I \ ^<^. s s \ / \ .;'" >*<-^+^ e > 0 =____-\-/______--s^-___y^-----^ 0 Z V ^ 3 ^ -60 ^^^ -18.29 _^0 ^l U N -36.58 -g E \ 3 0 .----^. -----.^ ^---Z-^^^^^p ^^^= \4 ^^ -.50 iuj- n n n n n n 5.0 +J J 1JLLL 111J -LLLI -5.0 ^ 10 0 2 4 6 8 Time, sec (d) Test run 4. Figure 13.- Continued. 41 866,----j-----,-----,-----[-----r-----r--- --1 263.96 + Data 846 Z------Computer 257.86 a) estimate :- ^ i^4^ ^ 251-76 1 -___ 245.67 806 ::------^^^^^^^ I------'--"------786 36.58 120 =: f\A 60 ^______7__u_____ -..i^. 18.29 S ^r^.. ; =^n. /" \ /S^A____^-iA't-L____. e > o =__-^i-/'___-\______-^.- 0 :- = "V ^ -60 ^ ^ -18.29 ^H -36.58 -120 ,, |_ __4_____ i o ^. ^^y^^^--^^ C? \ ./ -.25 ^------"------ _.50 ^N 5 0 2.5 ^------ I ~~^.-----f^------^^, ^^ ^ -2.S ------y---- T U N .LLLl.-m LJLLLL ll.U '0 2.2 4.4 6.6 8.8 11.0 Time, sec (e) Test run 5. Figure 13.- Concluded. 42 I A Extracted from flight; 1524m (5000 ft) 0 Extracted from flight; 6096m (20000 ft) Manufacturer's data; 1524 m (5000 ft) Manufacturer's data; 6096m (20000 ft) 0 Algorithm extracted using fixed manufacturer's principal derivatives; 1524m (5000 ft) 0 Algorithm extracted using fixed manufacturer's principal derivatives; 6096m (20000 ft) -.35 ^ / ^ -.30 ^ / -.25 :______.------^^"^ C m 15 A Q a .5 ^.6 .7 .8 .9 M Figure 14.- Variation of C^ with Mach number. 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