1、NASA TECHNICAL NOTE6a = 5o3 6 ,6 x *T r,T a,T(Cj C; at a = a per radianV tr/ Lr TclK C7 at a = _, per radian4 rCn yawing -moment coefficient3Cn per radian8Cni- per radian),2Wacnstatic directional-stability derivative, , per radian9Cn5 = -ST“ per radiana 9oaacn per radiannV yawing-moment coefficient
2、at 0 = 0T, 6r = 6r T, 6aPT r,T a,T (cnp) cnp at a = Ti Per radianaCn at a = a- per radianQ!T aside-force coefficient, , . per radianflPb9l2v7per radianProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-9Cy - Per radian8/33C-yCv,- = - per radianYacv(CY)
3、K side -force coefficient at /3 = /3T, 6r = 6r _, 6a = 6a -HT r,T a,TT YP at * = *Tg acceleration due to gravity, meters/second (ft/sec)Ix aircraft moment of inertia about the body X-axis, kilogram -meters2(slug -ft2)product of inertia of aircraft referred to body X- and Z-axis,kilogram -meters2 (sl
4、ug -ft2)IY aircraft moment of inertia about the body Y-axis, kilogram -meters2(slug -ft2)r! aircraft moment of inertia about the body Z-axis, kilogram -meters(slug -ft2)2slope of linear variation of Cj with a, per radianroslope of linear variation of C with a, per radianKc slope of linear variation
5、of Cn with a, per radian2Kc slope of linear variation of Cn- with a, per radian26a aKg slope of linear variation of Cy with a , per radian2m mass of fueled airplane, kilograms (slugs)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-p rolling angular v
6、elocity, radians/secondq pitching angular velocity, radians/secondr yawing angular velocity, radians/secondS wing area, meters (ft2)u velocity along longitudinal body axis, meters/second (ft/sec)V true airspeed, meters/second (ft/sec)v velocity along lateral body axis, meters/second (ft/sec)w veloci
7、ty along vertical body axis, meters/second (ft/sec)Xy accelerometer offset coordinate from center of gravity along longitudinalbody axis, meters (ft)Yy . accelerometer offset coordinate from center of gravity along lateralbody axis, meters (ft)Zy accelerometer offset coordinate from center of gravit
8、y alongvertical body axis, meters (ft)a angle of attack, radiansoirp trim angle of attack, radians/3 sideslip angle, radians/3T trim sideslip angle, radians6a aileron deflection angle (positive when right aileron is deflected down),radians5a T aileron deflection angle at trim, radians6r rudder defle
9、ction angle (positive when trailing edge is deflectedto the right), radiansProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6r -T, rudder deflection angle at trim, radians1 “r fit errory,A arbitrary parameters6 pitch angle, radians0 roll angle, radian
10、sp mass density of air, kilograms/meter3 (slugs/ft3)A dot over a variable indicates the time derivative of that variable.FLIGHT TESTSThe flight test data were provided by the U.S. Naval Air Test Center at PatuxentRiver, Maryland. The flight tests were conducted by Navy test pilots as part of an inve
11、s-tigation with a McDonnell Douglas F-4 airplane. Five different lateral response runswere made: three during one flight test of the airplane and two during a second flighttest. The first three runs were made at an altitude of approximately 6096 m (20 000 ft)at Mach numbers of about 0.6, 0.7, and 0.
12、8, respectively. Control inputs for these runswere rudder only, rudder and aileron, and rudder only, respectively. The other two runswere made at an altitude of approximately 11 277.6 m (37 000 ft) at Mach numbers ofabout 0.9 and 0.8, respectively. Control inputs for these runs were rudder only and
13、rud-der and aileron, respectively. The stability augmentation system (SAS) was deactivatedin order to provide full response for all the test runs.For each of the test runs, the airplane was trimmed by the pilot at the desired alti-tude and Mach number and held for a short period. Then the control in
14、put or inputs wereapplied. No attempt was made to null any longitudinal motions. Roll and pitch angles aswell as Mach number, pressure altitude, rudder deflection, aileron deflections, and cali-brated airspeed were recorded every tenth of a second. True airspeed was determinedfrom figure 1 of refere
15、nce 3 using Mach number, pressure altitude, and temperaturefrom flight tests and resolved through angle-of-sideslip measurements to yield lateralvelocity.Lateral displacement of the control stick in the F-4 airplane produces a combinationof aileron and spoiler deflections. The aileron deflection is
16、limited from 0 to 30 down-ward and from 0 to 1 upward. The spoiler being located on the upper surface of thewing has no downward deflection and is limited to upward deflections between 0 and 43.In the flight records only the aileron deflections were recorded. Aileron-deflection datawere used in the
17、following manner to yield a single control input, which reflects a spoilerProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-effect. The assumption was made that a negative reading for either the right or leftaileron was the indication of an aileron inp
18、ut. It was further assumed that the spoilereffect on the opposite side of the negative aileron deflection was equivalent to a positiveaileron deflection of the same magnitude. Hence, by doubling the magnitude of the nega-tive aileron deflection and applying the sign convention of a right aileron to
19、the magnitude,a single right aileron input, which is effectively the total aileron input, could be used inthe equations. It should be noted that the aileron coefficients (Cyfi , Cj_ , and Cn5 extracted by this program reflect the effect of both aileron and spoiler. Since these con-trol surfaces are
20、physically linked, it is impossible to uniquely determine the coefficient of each aileron and spoiler without additional information.Instrumentation consisted of rate gyros located slightly forward and at foot level ofthe pilot for measuring pitching, rolling, and yawing velocities; accelerometers l
21、ocatedin the left wheel well for measuring lateral and normal accelerations; and vanes on a noseboom for measuring angle of attack and angle of sideslip. (See fig. 1.) No documentationwas available from the Navy as to the accuracy of the instrumentation, although the methodof parameter extraction (r
22、ef. 1) typically yielded the following signal-to-noise amplituderatios (the noise amplitude was the 2-sigma level):Lateral velocity 18 decibelsRolling velocity 24 decibelsYawing velocity 20 decibelsRoll angle 22 decibelsLateral acceleration 8 decibelsAIRCRAFT MATHEMATICAL MODELThe equations of motio
23、n used by the computer program (ref. 1) were modified con-tinually during the analysis. However, three basic models evolved. The first modelconsisted of mainly lateral motion, the second model contained longitudinal coupling, andthe third model contained longitudinal coupling and nonlinear lateral d
24、erivatives. Thenonlinear derivatives KQ , KQ, , KQ , KC , and KQ permit variations withYp t/3 ir np naangle of attack in those particular derivatives that exhibit such dependence in wind-tunnelresults (ref. 4). The three models can be obtained from the following equations:V . g COSProvided by IHSNot
25、 for ResaleNo reproduction or networking permitted without license from IHS-,-,-p -KIz12VK (a-T)(5a“6a,T)+KCn6a( Tirbs p tan COScos 6 sinProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The first model, mainly lateral motion, can be obtained from the
26、basic equationsby requiring all longitudinal variables (u, w, 9, a, and q) to be constants and the non-linear derivatives KCV , KQ , KQ , KQ , and K to be zero. The secondP /3 r nP amodel, containing longitudinal coupling, can be obtained by using the longitudinal flightdata as inputs to the equatio
27、ns in the same manner as rudder and aileron deflections areused. The third model, containing both coupling and nonlinear derivatives, is obtainedfrom the second model by not restricting the nonlinear derivatives to zero.The nonlinear derivatives K , K(-. , KQ , KQ , and KQ , as well asthe longitudin
28、al coupling terms, were discovered to be necessary in order to fit the flighttest data, as is demonstrated in the next section. Also demonstrated in the next sectionis the problem of uniqueness mentioned in references 1 and 2.RESULTSThe conditions for the five flight test runs are listed in table I.
29、 The analysis ofthese runs involved two major problems: uniqueness and longitudinal effects. Theresults are presented in a manner to illustrate how each problem was encountered andthen resolved.Uniqueness ProblemBefore presenting the uniqueness problem as encountered in this study, it would bewell t
30、o describe the problem and the means of detecting its presence. The problemitself can be best described as follows: Given a set of parameters that minimize the fiterror between measured and computed variables, does another set of parameters existthat will yield the same fit error? If the answer is y
31、es, a uniqueness problem exists.Detection of the problem is facilitated by the use of the covariance matrix provided bythe maximum likelihood estimation technique. Minor manipulation of this matrix, asdescribed in reference 1, yields pairwise parameter correlation coefficients which esti-mate the de
32、gree of linear dependence between two parameters. Figure 2 illustrates theexistence of a uniqueness problem due to linear correlation between arbitrary parametersX and y. Values of X and y that lie on the line of dependence yield the same fiterror. However, it should be emphasized that two parameter
33、s may exhibit high correla-tion without indicating a uniqueness problem. Thus, it is necessary for the analyst totest any parameters with significant correlation coefficients to determine whether auniqueness problem is present. The test is simply to determine whether the fit errorchanges as the para
34、meters vary along the line of dependence. The procedure for carry-ing out the test is to assign to one of the correlated parameters several values in therange of interest and then extract the other parameters values; this determines the lineProvided by IHSNot for ResaleNo reproduction or networking
35、permitted without license from IHS-,-,-of dependence. In figure 2, the fit error r does not change as A. and y vary alongthe line of dependence. Thus, a uniqueness problem is present. If the fit error didchange, both parameters would be identified by the estimation technique at the point ofminimum f
36、it error and no uniqueness problem would exist, although the parameterswould still be correlated. In this hypothetical illustration, the correlation between Xand y is perfectly linear and will cause divergence of the estimation technique when anattempt is made to extract both parameters. However, in
37、 the use of real data, the pres-ence of noise usually prevents perfect linear correlation, and thus divergence.Figure 3 presents the model responses generated by the estimates of the stabilityderivatives of test run 1 and the respective flight test data, using the first model with alllongitudinal va
38、riables fixed as constants (average values obtained from the flight data foreach variable). (Note that symbols in figure 3 and subsequent machine plots presentingmodel responses and respective flight test data are not the standard symbols defined inthe Symbols section.) Table II presents the estimat
39、es of the derivatives obtained, andtable in presents a form of the covariance matrix for these estimates. Diagonal ele-ments of this matrix are the standard deviations of the estimates, and the off-diagonalterms are correlation coefficients. As denoted by the asterisks of table in, Cyo, Ynand CYr; C
40、, CZp, and C; Cn/3, Cnp, and Cnr; and C and Cn/3 all have sig-nificant correlation. Investigation of these parameters revealed the existence of a unique-ness problem.A major cause of uniqueness problems is generally admitted to be insufficientexcitation of the aircraft (for example, ref. 5). Test ru
41、n 1 had rudder deflections only.Test run 2 contained both rudder and aileron deflections, and the model responses gener-ated by the derivative estimates for this test run and the respective flight test data areshown in figure 4. Again the longitudinal variables were fixed as constants during theextr
42、action process. Table IV presents the estimates of the derivatives obtained, andtable V presents the modified covariance matrix. As pointed out in section 7.8.3 of ref-erence 5, the likelihood of obtaining a unique set of derivatives is increased when both arudder input and an aileron input are used
43、 to excite the airframe, as is evidenced by thelack of correlation exhibited in table V.Longitudinal Coupling EffectsExamination of figure 4 (test run 2 responses) reveals poor fits for all the lateralvariables; these poor fits indicate a possibly incomplete model. The longitudinal datafor test run
44、2 are presented in figure 5 and indicate a substantial amount of longitudinalmotion. Use of the longitudinal data as input, together with the modeling of angle ofattack dependence of some of the derivatives, resulted in the extraction of a new set ofderivatives for test run 2. Figure 6 presents the
45、model responses generated by this set10Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-of derivatives and table VI contains the derivatives and their standard deviations. Nosignificant correlation was present and, thus, a unique set of derivatives ha
46、s beenextracted. It should be noted that a lack of confidence exists for all the Cy deriva-tives with the exception of Cyo, due to the large standard deviations of the estimates,as is the case with some of the nonlinear derivatives.Solution of the Uniqueness ProblemThe uniqueness problem of test run
47、 1 was resolved by fixing the values of the non-linear derivatives and (Cy A C , and Cnr at the values obtained in test run 2 / Cc rri “(the wind-tunnel results presented in ref. 4 show these derivatives to be fairly insensi-tive to Mach number variations in this flight regime) and extracting the re
48、maining deriv-atives. This same procedure was used to solve the uniqueness problem of test run 3,which also had a rudder-only input. The model responses generated by the final esti-mates of the derivatives for test run 1 and test run 3 are shown with the respective flightdata in figures 7 and 8, res
49、pectively. The values of the derivatives and their standarddeviations are presented in table VII for test run 1 and table VIII for test run 3.Test run 4 had essentially a rudder-only input, whereas test run 5 had both rudderand aileron inputs. Again, the results of test run 5 were used to solve the unique
