Aerodynamic System Identification of the T-38A Using SIDPAC

Limited Aerodynamic System Identification of the T-38A using SIDPAC Software. Michael J. Shepherd Timothy R. Jorris Will

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Limited Aerodynamic System Identification of the T-38A using SIDPAC Software. Michael J. Shepherd Timothy R. Jorris William R. Gray, III USAF Test Pilot School 1220 South Wolfe Ave Edwards AFB, California, 93524, USA [email protected]

Eugene A. Morelli of the NASA Langley Research Center. The objective of the flight was to demonstrate the adequacy of the current USAF Test Pilot School (TPS) data acquisition system (DAS) installed on the T-38A and to highlight considerations should USAF TPS decide to include parameter estimation as part of the core curriculum in the future. The T-38A Talon aircraft is shown in Figure 1.

Abstract—The T-38A is the primary training aircraft at the USAF Test Pilot School. The aircraft used was fully instrumented for all aerodynamic flight parameters including angular rates, accelerations, and control surface positions. Flight test data were obtained over a series of sub-sonic and supersonic test points in the clean aircraft configuration. The flight test data were reduced using the System Identification Programs for AirCraft (SIDPAC) toolbox for MATLAB resulting in an aerodynamic model of the T-38A. The investigation identified several considerations when conducting a shortterm, limited scope model identification test. The lessons learned from this application may be applied to further studies of aircraft dynamics.

TABLE

OF

C ONTENTS

TABLE OF C ONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 I NTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 M ODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 A IRCRAFT I NSTRUMENTATION . . . . . . . . . . . . . . . . . . . 3 4 F LIGHT T EST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 PARAMETER I DENTIFICATION R ESULTS AND D ISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 6 C ONCLUSIONS AND R ECOMMENDATIONS . . . . . . . . 7 A PPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 R EFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 B IOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Figure 1. A T-38A Talon supersonic trainer over the Edwards AFB runways. The mission of the USAF Test Pilot School is to produce highly-adaptive critical-thinking flight test professionals to lead and conduct full-spectrum test and evaluation of aerospace weapon systems. Each semester, approximately 24 U.S. and Allied forces pilots, navigators and flight test engineers begin a 48-week curriculum focused on aircraft performance, flying qualities, systems and test management. As part of the flying qualities curriculum courses are taught which cover aircraft dynamics and parameter estimation. Until recently a flight laboratory on system identification and parameter estimation was not obtainable due to limitations on aircraft hardware and software licensing. However, conversion to solid state recording devices now means flight data is available to students immediately upon flight completion (older tape recording systems took time and dollars to decommute). This paper examines the issues of conducting a

1. I NTRODUCTION This paper outlines the results of a parameter identification study conducted to obtain aerodynamic stability derivatives at three test conditions flown by a T-38A on a single flight on 20 July 2009. The primary tool used to reduce the data was the AirCraft (SIDPAC) toolbox for MATLAB written by Dr. IEEEAC Paper #1214, Version 5 Updated 25 Jan 10. U.S. Government work not protected by U.S. copyright. Approved for public release; distribution is unlimited. AFFTC-PA-09520. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.

1

parameter estimation exercise using available equipment and may help guide future instrumentation of T-38Cs and other aircraft as they are introduced into the curriculum. The specific objectives were to evaluate the data system for adequacy for parameter identification on the T-38A and evaluate the utility of the SIDPAC Matlab toolbox.

be formed and solved. Briefly we begin by judiciously choosing to write the pitching moment equation for Cm as

Cm = Cmα α + Cmα˙ α˙ +

c¯ Cmq q + Cmδe δe 2V0

(10)

2. M ODELING where Cmi represent the dimensional aerodynamic stability derivatives/parameters to be estimated and the recorded values for α, α, ˙ q, δe are the regressors. Note that the traditional scaling parameter for pitch damping has been applied (0.5¯ c/V0 ) where V0 represents the aircraft true airspeed.

This aircraft system identification exercise, using SIDPAC software, was based on the theory presented in the text by Klein and Morelli[1]. A brief overview of the modeling and theory is presented here. For more in-depth discussion the reader is referred to the source text.

Rewriting Eq 5 yields:

The equations of motion in the body axes may be written as

T q¯S CX − g sin θ + m M q¯S CY − g cos θ sin φ v˙ = pw − ru − m q¯S CZ − g cos θ cos φ w˙ = qu − pv − m q¯S¯ (Iz − Iy ) Ixy c Ixz r˙ = Cl − qr + qp p˙ − Ix Ix Ix Ix (Ix − Iz ) q¯S¯ c Ixz 2 Cm − pr − (p − r2 ) q˙ = Iy Iy Iy q¯S¯ (Iy − Ix ) c Ixz Ixy p˙ = Cn − pq − qr r˙ − Iz Iz Iz Iz u˙ = rv − qw −

Cm =

(1)

(Ix − Iz ) Ixz 2 Iy pr + (p − r2 )} {q˙ + q¯S¯ Iy Iy c

(11)

(2) All of the terms on the RHS of the equation are known values either recorded by the aircraft instrumentation system or calculated directly from the inertia model provided in the appendix. The linear least-squares problem uses the recorded regressors to estimate the stability parameters. Now define the estimated parameters as θˆ = T  Cmα Cmα˙ 2Vc¯0 Cmq Cmδe .

(3) (4) (5) (6)

Eq 11 may be re-written for each time interval of data i as

governed by the kinematic relationships (i) Cm = X(i) θˆ

φ˙ = p + tan θ(q sin φ + r cos φ) θ˙ = q cos φ − r sin φ q sin φ + r cos φ . ψ˙ = cos θ

(7)

h where the regressors are placed in X(i) = α(i)

(8)

(12)

˙ α(i)

q (i)

h iT (1) (2) (N ) Defining z = Cm we may now Cm . . . Cm solve for the estimated stability parameters:

(9)

In the above equations the translational velocities in the x, y, z directions are represented by u, v, w and the rotational pitch, roll, and angles are represented by θ, φ, ψ. Dynamic pressure, planform area, span, chord and mass are represented by q¯, S, b, c¯, m. Principal moments of inertia are Ix , Iy , Iz and the sole non-zero product of inertia is Ixy . The normal assumptions for a rigid body aircraft are applied. That is fixed, non-moving atmosphere; flat non-rotating inertial earth with constant gravitational field, g; mass is constant for short duration maneuvers; aircraft is symmetric and thrust, T , is aligned with body x-axis and rotational effects are ignored.

θˆ = (XT X)−1 XT z.

(13)

A similar approach may be used to estimate parameters for the remaining five force and moment equations. It should be noted that although results for demonstration of principles on the T-38 were adequate with this method, using this approach on all systems (especially unstable systems) may not be advised. Often in unstable cases regressors such as elevator deflection and pitch rate may look nearly identical, and consequently result in large variances in the answers or invertibility issues with the pseudo-inverse. These problems may be overcome using flight test inputs where the inputs are

The problem is to determine the coefficients Cx , Cy , Cz and Cl , Cm , Cn which represent the aerodynamic forcing and moments upon the aircraft. A linear regression problem may 2

(i)

δe

i

.

3. A IRCRAFT I NSTRUMENTATION

of higher frequency than resulting rate derivatives but require careful attention to experiment design.

The T-38A aircraft is a tandem 2-seat supersonic trainer. The tail number used for testing was S/N 68-205. All flight control surfaces were irreversible and hydraulically actuated. The rudder and elevator (or slab) were all moving surfaces. There was a rudder-yaw damper which could be disabled in the cockpit. The aircraft was flown in the cruise configuration for all testing. The speed brakes, landing gear and flaps were not extended. The aircraft was modified with an instrumentation system described below. Overall the aircraft was considered production representative for flying qualities testing. The relevant dimensions of the aircraft are provided in Table 1.

Once the non-dimensional stability derivatives were calculated the dimensional stability parameters may be calculated. A complete derivation of these terms and their use in linearized equations of motion may be found in the text by Yechout [4]. Whereas the non-dimensional stability derivatives are useful for comparing different aircraft throughout entire flight envelopes, the dimensional parameters simplify the book-keeping when writing linear equations of motion. For the pitching moment coefficients, the non-dimensional stability derivatives are related to the dimensional parameters through the following relationships:

Table 1. T-38A Dimensions

Mα =

q¯S¯ c Cmα Iyy

q¯S¯ c2 Cmq 2Iyy V0 q¯S¯ c = Cmδe Iyy

Mq = Mδe

units: sec-2

(14)

Name

Nomenclature

units: sec-1

(15)

units: sec-2

(16)

b S c¯

span wing reference area mean aerodynamic chord

The short period dynamics were of primary interest for aircraft longitudinal flying qualities. Once the dimensional derivatives were calculated an approximation for the short period dynamic characteristics of natural frequency and damping ratio may be obtained. Note that when performing these calculations for this analysis it was assumed the body-fixed axis (aligned with the aircraft x-axis) and the body-fixed stability axis (where the x-axis is aligned with the free stream relative wind prior to maneuver execution) were the same. This assumption was only valid for very low angles of attack. The approximations for natural frequency and damping ratio were:

ζSP ≈

r

Zα M q − Mα V0  − Mq + ZVα0 + Mα˙ 2ωnSP

25.25 ft 170.0 ft2 7.73 ft

The data system is detailed in internal TPS documentation [2]. The following description is an excerpt from these documents. The data system used a SCI Mini-ATIS Data Acquisition System (DAS) and a TTC solid-state recorder. DAS information is received from Special Instrumentation (SI) transducers that have been installed on the aircraft. This data is then processed and combined through the Mini-ATIS DAS. The data is output in a Chapter 4 Pulse Code Modulated (PCM) stream at 250 Kbytes/sec composed of 16 bit word length, four frames deep, and 72 words per frame. This data stream can then be transmitted to a TPS ground station or recorded on a PCMCIA card. A time code generator/GPS receiver is located in the right avionics bay. This unit provides a precision time signal (IRIG-B) to the DAS for synchronization of time-correlated information. This unit will synchronize to GPS time after obtaining GPS lock, and provide IRIG-B to the DAS. A total air temperature probe is installed on the lower center fuselage.

Similar parameters may be calculated in the x and zdirections as well for the lateral directional degrees of freedom. For the computations in this paper the change in normal force with respect to α was simplified to Zα = (¯ q S/m)Czα . The derivations for the other parameters were detailed in the Yechout text.

ωnSP ≈

Value

Internal provisions have been made to allow a flight test noseboom to be mounted on the radome in place of the production pitot-static probe as shown in Figure 2. The production system has a single angle of attack fuselage mounted vane and no angle of sideslip vane. The T-38 flight test noseboom is used to measure aircraft total and static pressure, angle of attack (AOA) and angle of sideslip (AOSS). It consists of three data sensors: a pitot-static air data probe and two air data vane assemblies. The two vane assemblies on the boom acquire AOA and AOSS. The shafts are connected to dual potentiometers installed in the noseboom body. A three-axis accelerometer system was installed in the center fuselage. The attitude gyro was used to measure angle of pitch and roll, and the threeaxis rate gyro system was used to measure pitch, roll, and

(17) (18)

After the dimensional derivatives have been determined all the necessary components would be in place to compute additional open loop flying quality terms n/α and CAP (Control Anticipation Parameter) as required by military specification or other design requirements. 3

yaw rates and was located in the left hand nose section. Noteworthy was that heading angle is not measured or recorded.

included rudder inputs not provided by the pilot. The aircraft was not equipped with neither longitudinal (stabilator) nor wing-leveling (aileron) stability augmentation systems. The test points flown were determined to characterize the fast dynamics of the T-38, specifically the longitudinal short period mode and the lateral-directional roll and Dutch roll modes. Longer period motions such as the phugoid and spiral modes were not targeted in the single flight test but would be required to fully populate a complete six degree of freedom aerodynamic model for the aircraft.

Figure 2. The extended flight test noseboom on this T-38 is the main external difference from production T-38s and provides angle of attack, angle of sideslip, pitot and static pressures forward of the aircraft flow field. Flight controls instrumentation includes potentiometer type transducers which have been installed to monitor the position of the rudder pedals, lateral stick, longitudinal stick, left and right aileron, rudder, and stabilator positions. The front cockpit has strain gauge type transducers installed to measure lateral stick force, longitudinal stick force, and rudder pedal differential force. For the aircraft parameter identification problem the flight control positions and input forces were not required. The fuel flow systems of both engines have been modified by the addition of fuel flow meters and temperature sensors in the main fuel lines. Fuel flow lines for both afterburners have been modified by the installation of fuel flow meters. These modifications do not cause any change in engine performance.

Figure 3. Test points as a function of pressure altitude and true airspeed with lines of constant dynamic pressure and Mach.

5. PARAMETER I DENTIFICATION R ESULTS AND D ISCUSSION

ILIAD software was used for data reduction into a .csv file. The data output rate was 25 Hz. Higher output rates should be possible, but the highest rate data were corrupted due to an unknown cause. A Matlab script was used to read and convert the .csv file to an FDATA file compatible with SIDPAC. In the Matlab script the fuel burned was subtracted from full internal fuel to determine aircraft gross weight, center of gravity and inertia parameters using the function provided in the Appendix. The function was based on formulas provided by a mid-1990’s report written by a then-TPS student [3].

The results of the testing were grouped into longitudinal and lateral directional test results and discussion. Longitudinal Results The parameter identification methods using the SIDPAC software were generally successful for calculating estimates for the longitudinal stability derivatives. The results for the pitching moment coefficients will be discussed in this section.

4. F LIGHT T EST

Before conducting the parameter identification it was decided to remove the α˙ regressor as the derived parameter for the change in alpha with respect to time was noisy and would likely interact with the pitch rate regressor. This is not an uncommon practice in flight test. A plot of the α˙ and pitch rate q is presented in Figure 4.

The flight test was conducted on a single flight on 20 Jul 09 at Edwards AFB in California. A series of test points were flown at the four different flight conditions shown in Figure 3. At each condition a series of longitudinal and lateraldirectional 1-g flying qualities test points were flown. All points began from a wings-level, un-accelerated, constant altitude trim shot. Maneuvers included doublets, steps, raps and frequency sweeps inputs from elevators, ailerons and rudder. Lateral-directional points also included aileron-rudder doublets. All points were flown in the clean configuration. The majority of lateral-directional points were flown with the yaw-damper on and consequently lateral-directional points

For the longitudinal cases it was determined a frequency sweep provided sufficient data for data reduction. Although a programmed test input would normally be used for systems so-equipped, the T-38 required manually flown frequency sweeps. In all cases the manually flown frequency sweep provided adequate data for reduction. A typical frequency sweep 4

moment was calculated from the actual regressors acting on the stability derivative estimates (Eqn 10) and then compared against the actual pitching moment observed (Eqn 11). A typical result is shown in Figure 6. It may be seen the error between the model’s pitching moment compared to the observed pitching moment was relatively small indicating the model was a valid representation at the condition tested.

5 alpha dot (derived) pitch rate

4 3

Deg/sec

2 1 0 −1

0.04 data model

−2 0.03

−3 −4 −5

0.02 0

10

20

30

40

50

60

70

Cm

time (sec) Figure 4. Pitch rate compared to time rate of change in alpha.

0.01

0

−0.01

flown by a test pilot is shown in Figure 5. −0.02

−0.03

δ

e

−2

0

10

20

30

40

50

60

α

4

q

30

40

50

60

70

0

10

20

30

40

50

60

70

0

10

20

30

40

50

60

70

The calculated longitudinal dimensional stability parameters were then algebraically calculated using the procedures in Section 2 and are presented in Table 3. For comparison, published values from Yechout’s text [4] for the F-104 were presented at two different test conditions. Although not representative of the T-38 dynamics or of the conditions tested, the values from the F-104 were provided as a check on the rough order of magnitude of the test results. The short period natural frequency (converted to Hertz) and damping ratio were computed using Equations 17 and 18. In general, the natural frequency at the low dynamic pressure test point was a lower value than the three test points at the higher dynamic pressures. The natural frequency was highest for the high dynamic pressure, supersonic test point (Test Point 4) which was consistent with expected results. This condition also was the most lightly damped condition tested.

5 0 −5

20

Figure 6. Pitching moment coefficient test results versus linear least squares model. Results are typical for the pitching moment cases examined.

70

6

2

10

time (sec)

−4 −6

0

time Figure 5. Frequency sweep of elevator (deg) with angle of attack (deg) and pitch rate (deg/sec) response at Pt 2 conditions (M=0.7, q=200 lbsf). The linear regression tool in SIDPAC was applied to the longitudinal frequency sweep test points. The test results for aerodynamic pitching moment terms in Equation 10 are summarized in Table 2. The results were characterized by tight confidence bounds on the resulting coefficients for all three coefficients at all four test conditions. For each test point only a single frequency sweep was conducted and analyzed and confidence bounds are for the fit of the data. For an actual flight test program a careful design of experiments may require multiple test points at each flight condition. The pitch static stability showed the largest change. Consistent with theory the value increased in magnitude in the super-sonic region as the aerodynamic center moved aft.

There were several sources of error which may not be evident by analyzing the tables. This was despite the very good fit to the data. The error sources were inherent to the analysis assumptions and the instrumentation. The data instrumentation system as installed on the T-38 was a possible source of error. Normal and axial accelerations were suspect for the entire test. The normal acceleration showed a bias of 0.2 g in 1-g level flight which had to be compensated for prior to data reduction. If the normal acceleration had gain as well as bias errors the normal acceleration coefficients would be affected. In the data in Table 3 it would

After the aerodynamic stability derivatives were estimated the original test maneuver may be simulated where the pitching 5

Table 2. Test Results for Aerodynamic Pitching Moment Non-Dimensional Stability Derivative. Angular measurements are in radians.

Parameter

Estimate

Std Error

% Error

95% Conf Interval

-8.024e-1 -1.306e1 -1.367e0

8.633e-3 4.400e-1 1.429e-2

1.1 3.4 1.0

[ -0.820 , -0.785 ] [ -13.941 , -12.181] [ -1.396 , -1.338]

-5.624e-1 -1.272e1 -1.285e0

3.209e-3 2.264e-1 5.539e-3

0.6 1.8 0.4

[ -0.569, -0.556 ] [-13.170, -12.264] [-1.296 , -1.274]

-1.469e0 -2.021e1 -1.500e0

9.617e-3 5.737e-1 1.656e-2

0.7 2.8 1.1

[-1.488,-1.450] [-21.361,-19.067] [ -1.533 , -1.467 ]

-1.154e0 -1.779e1 -1.747e0

1.276e-2 6.720e-1 2.284e-2

1.1 3.8 1.3

[ -1.179 ,-1.128] [ -19.129 ,-16.441] [ -1.793 ,-1.701]

PT1 Cmα Cmq Cmδe PT2 Cmα Cmq Cmδe PT3 Cmα Cmq Cmδe PT4 Cmα Cmq Cmδe

Table 3. Dimensional Stability Parameters: Comparison of published F-104 data and T-38 test data

Parameter Altitude (ft) Mach True Airspeed(fps) q¯ (lb/ft2 ) S (ft2 ) c¯ (ft2 ) α (deg) Iyy (slug-ft2 )

F-104 [4]

F-104 [4]

T-38 PT1

T-38 PT2

T-38 PT3

T-38 PT4

S/L 0.257 287 97.8 196 21.9 10 5.90e4

55,000 1.800 1742 434.5 196 21.9 2 5.90e4

10,356 0.713 768 512.5 170 7.79 1.7 2.932e4

31,753 0.709 696.8 202.5 170 7.79 4.0 2.924e4

32,460 1.079 1061.8 458.0 170 7.79 2.7 2.909e4

22,751 0.906 929.3 498.0 170 7.79 1.1 2.873e4

-140.2 -2.009 -0.3046 -4.990

-346.6 -18.12 -0.1844 -18.15

-1055 -18.4265 -1.5101 -31.39

-502.2 -5.117 -0.6419 -11.69

-1076 -30.39 -1.5219 -31.03

-1704 -26.27 -1.6849 -39.78

0.23 0.30

0.68 0.05

0.72 0.32

0.38 0.29

0.90 0.22

0.86 0.32

Longitudinal Derivatives Zα (ft/s2 ) Mα (1/s2 ) Mq (1/s) Mδe (1/s2 ) Short Period Dynamics ωn (Hz) ζ

6

6. C ONCLUSIONS

be the Zα dimensional parameter.

AND

R ECOMMENDATIONS

The methods shown in this paper show the instrumentation system of the T-38A is capable of capturing the required data for the SIDPAC software running linear regression algorithms. SIDPAC provides acceptable results for longitudinal cases. Scaling errors not fully understood by the test team limit the application for the lateral directional case at the current time. Once the scaling issues are resolved the lateral directional applications of the linear regression algorithms should be satisfactory but may lack the level of confidence shown in the longitudinal cases.

The aircraft data system did not have an air data computer for airspeed computations. This use of raw pitot and static pressure not compensated for position errors introduced difficulties in the data reduction. In the end the recorded Mach number was determined to be the most reliable instrument. Barometric altitude and Mach number at standard day conditions were used to calculate dynamic pressure. The ILIAD data reduction software did not support data collection at rates greater than 25 Hz although data were recorded at rates up to 108 Hz. Furthermore data that were returned were decimated without any anti-aliasing considerations. Although aliasing of signals did not appear to be an issue better data reduction processes should be developed in the future. A faster data rate may allow for smoother derivative computations needed to form α˙ and β˙ required for use as regressors.

A PPENDIX An inertia model for the T-38 is provided for future reference.

A design of experiments analysis was not conducted. Multiple points throughout an envelope would normally be flown to produce the greatest statistical confidence in the final model. The data analysis presented showed the validity of the technique to compute coefficients but was necessarily limited in scope. Finally the assumptions for linearizing the data should be understood. None of the assumptions were violated during the testing but would induce small inaccuracies. These errors were not thought to be significant. The flight test and data reduction concentrated on computing the derivatives most required for calculating short period dynamics. More testing of longer duration flight test maneuvers would be required to capture the remaining derivatives which make up the low frequency phugoid motions. Lateral Directional Test Results The lateral directional test results could not be verified at the time of this writing with an apparent scaling issues attributed to system instrumentation when compared to the earlier results of obtained by Krause [3] using an output error technique. In general these observations were made. The lateral directional testing showed the typical difficulty when looking at a multiple input system. While it was expected that testing with the yaw damper off would have provided a better result since aircraft motion would not be attenuated by the damper it appeared having the yaw damper engaged was beneficial since rudder excitation was evident over the course of the entire maneuver. Flight test techniques such as aileron-rudder doublets were good, but similar results were generally obtainable using aileron inputs only with the yaw damper on. Whereas confidence bounds of less than 10% were typical with the longitudinal cases confidence bounds for the lateral directional cases often exceeded 25%.

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function [Ixx, Iyy, Izz, Ixz, CG, CG_in, mass, DeltaXcg_in, DeltaZcg_in] = ... T38_mass_properties(WFL, WFR); %************************************************************************** % function [Ixx, Iyy, Izz, Ixz, mass, CG, DeltaXcg, DeltaZcg] = % T38_mass_properties(WFL, WFR); % % Author: Lt Col Michael J. Shepherd USAF TPS % Email: [email protected] % Date: 20 July 09 % % Source: AFFTC-TLR-94-85 % Krause, Paul A. "A limited investigation of the lateral-directinal axes % of the F-15B and T-38A near wing rock angles of attack using a personal % computer- based parameter identification (PCPID) program" % % Inputs % WFL:Fuel Weight in LEFT system (pounds) % WFR:Fuel Weight in RIGHT system (pounds) % % Outputs % Ixx, Iyy, Izz,: Mass Moment of Inertia % Ixz: Product of Inertia (note Iyz = Ixy = 0) % CG: Center of Gravity % CG_in: Center of Gravity in inches from reference CG % mass: mass (Slugs) % DeltaXcg_in: Change in Center of Gravity from Reference CG (FS 354.39) % DeltaZcg_in: % % Assumes empty weight of 8750 lbs. % % Aircraft Mass Properties: % Xcg = FS 354.39 in % Ycg = BL 0.0 in % Zcg = WL 100 in % %************************************************************************** %Form F variables OPW = 8750; %Operating Weight OPM = 3039.6; %Operating Moment Mom_Sim = 1000; %Moment simplifier (constant to keep moments simple to use) MAC = 92.76; LEMAC = 331.20; %Computer Gross Weight GW = OPW+WFL+WFR; mass = GW/32.174; % %X-moments wfllx =1.40309e-5*WFL.ˆ2+2.856289E-1*WFL+4.008761; wfrlx=-7.705063e-6*WFR.ˆ2+3.971317e-1*WFR + 5.; %Z-moments wfllz=0.01126431823*(729.8419938-WFR).*(26.56136596+WFL); wfrlz=2.917364207e-9*(1357.678895-WFR).*(0.005072708037+WFR).*... (3.133863746e6-2003.023886*WFR+WFR.ˆ2);

8

%Ixx ixxfl=2.43186279e-8*(55.74415428+WFL).*(450464.1502-1336.852141*WFL+WFL.ˆ2); if WFR