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

michael.shepherd@edwards.af.mil

Abstract—The T-38A is the primary training aircraft at the

USAF Test Pilot School. The aircraft used was fully instru-

mented for all aerodynamic flight parameters including angu-

lar rates, accelerations, and control surface positions. Flight

test data were obtained over a series of sub-sonic and super-

sonic test points in the clean aircraft configuration. The flight

test data were reduced using the System Identification Pro-

grams for AirCraft (SIDPAC) toolbox for MATLAB result-

ing in an aerodynamic model of the T-38A. The investigation

identified several considerations when conducting a short-

term, limited scope model identification test. The lessons

learned from this application may be applied to further stud-

ies of aircraft dynamics.

TABLE OF CONTENTS

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 MODELING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

3 AIRCRAFT INSTRUMENTATION . . . . . . . . . . . . . . . . . . . 3

4 FLIGHT TEST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

5 PARAMETER IDENTIFICATION RESULTS AND

DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

6 CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . 7

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1. INTRODUCTION

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 reflectthe official policy or position of the United States Air Force, Department ofDefense, or the U.S. Government.

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 con-

siderations 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.

Figure 1. A T-38A Talon supersonic trainer over the Ed-

wards AFB runways.

The mission of the USAF Test Pilot School is to pro-

duce highly-adaptive critical-thinking flight test profession-

als 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. Un-

til recently a flight laboratory on system identification and

parameter estimation was not obtainable due to limitations

on aircraft hardware and software licensing. However, con-

version 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 de-

commute). This paper examines the issues of conducting a

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 spe-

cific 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.

2. MODELING

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.

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

u = rv − qw −qS

mCX − g sin θ +

T

M(1)

v = pw − ru −qS

mCY − g cos θ sinφ (2)

w = qu− pv −qS

mCZ − g cos θ cosφ (3)

p−Ixy

Ixr =

qSc

IxCl −

(Iz − Iy)

Ixqr +

Ixz

Ixqp (4)

q =qSc

IyCm −

(Ix − Iz)

Iypr −

Ixz

Iy(p2 − r2) (5)

r −Ixy

Izp =

qSc

IzCn −

(Iy − Ix)

Izpq −

Ixz

Izqr (6)

governed by the kinematic relationships

φ = p+ tan θ(q sinφ+ r cosφ) (7)

θ = q cosφ− r sinφ (8)

ψ =q sinφ+ r cosφ

cos θ. (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, Izand the sole non-zero product of inertia is Ixy . The normal as-

sumptions 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 dura-

tion maneuvers; aircraft is symmetric and thrust, T , is aligned

with body x-axis and rotational effects are ignored.

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

be formed and solved. Briefly we begin by judiciously choos-

ing to write the pitching moment equation for Cm as

Cm = Cmαα+ Cmα

α+c

2V0Cmq

q + Cmδeδe (10)

where Cmirepresent the dimensional aerodynamic stability

derivatives/parameters to be estimated and the recorded val-

ues for α, α, q, δe are the regressors. Note that the tradi-

tional scaling parameter for pitch damping has been applied

(0.5c/V0) where V0 represents the aircraft true airspeed.

Rewriting Eq 5 yields:

Cm =IyqSc

{q +(Ix − Iz)

Iypr +

Ixz

Iy(p2 − r2)} (11)

All of the terms on the RHS of the equation are known

values either recorded by the aircraft instrumentation sys-

tem or calculated directly from the inertia model pro-

vided in the appendix. The linear least-squares problem

uses the recorded regressors to estimate the stability pa-

rameters. Now define the estimated parameters as θ =[

CmαCmα

c2V0

CmqCmδe

]T.

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

C(i)m = X(i)

θ (12)

where the regressors are placed in X(i) =[

α(i) ˙α(i) q(i) δ(i)e

]

.

Defining z =[

C(1)m C

(2)m . . . C

(N)m

]T

we may now

solve for the estimated stability parameters:

θ = (XT X)−1XT 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 ap-

proach on all systems (especially unstable systems) may not

be advised. Often in unstable cases regressors such as ele-

vator deflection and pitch rate may look nearly identical, and

consequently result in large variances in the answers or in-

vertibility issues with the pseudo-inverse. These problems

may be overcome using flight test inputs where the inputs are

2

of higher frequency than resulting rate derivatives but require

careful attention to experiment design.

Once the non-dimensional stability derivatives were calcu-

lated the dimensional stability parameters may be calculated.

A complete derivation of these terms and their use in lin-

earized equations of motion may be found in the text by

Yechout [4]. Whereas the non-dimensional stability deriva-

tives 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:

Mα =qSc

Iyy

Cmαunits: sec-2 (14)

Mq =qSc2

2IyyV0Cmq

units: sec-1 (15)

Mδe=qSc

Iyy

Cmδeunits: sec-2 (16)

Similar parameters may be calculated in the x and z-

directions as well for the lateral directional degrees of free-

dom. For the computations in this paper the change in normal

force with respect to α was simplified to Zα = (qS/m)Czα.

The derivations for the other parameters were detailed in the

Yechout text.

The short period dynamics were of primary interest for air-

craft longitudinal flying qualities. Once the dimensional

derivatives were calculated an approximation for the short pe-

riod dynamic characteristics of natural frequency and damp-

ing 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 sta-

bility 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:

ωnSP≈

√

ZαMq

V0−Mα (17)

ζSP ≈−

(

Mq + Zα

V0+Mα

)

2ωnSP

(18)

After the dimensional derivatives have been determined all

the necessary components would be in place to compute ad-

ditional open loop flying quality terms n/α and CAP (Control

Anticipation Parameter) as required by military specification

or other design requirements.

3. AIRCRAFT INSTRUMENTATION

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 configura-

tion for all testing. The speed brakes, landing gear and flaps

were not extended. The aircraft was modified with an instru-

mentation system described below. Overall the aircraft was

considered production representative for flying qualities test-

ing. The relevant dimensions of the aircraft are provided in

Table 1.

Table 1. T-38A Dimensions

Name Nomenclature Value

b span 25.25 ft

S wing reference area 170.0 ft2

c mean aerodynamic chord 7.73 ft

The data system is detailed in internal TPS documenta-

tion [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 Modu-

lated (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 pro-

vides a precision time signal (IRIG-B) to the DAS for syn-

chronization of time-correlated information. This unit will

synchronize to GPS time after obtaining GPS lock, and pro-

vide IRIG-B to the DAS. A total air temperature probe is in-

stalled on the lower center fuselage.

Internal provisions have been made to allow a flight test nose-

boom to be mounted on the radome in place of the production

pitot-static probe as shown in Figure 2. The production sys-

tem 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 as-

semblies. 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 three-

axis rate gyro system was used to measure pitch, roll, and

3

yaw rates and was located in the left hand nose section. Note-

worthy was that heading angle is not measured or recorded.

Figure 2. The extended flight test noseboom on this T-38 is

the main external difference from production T-38s and pro-

vides angle of attack, angle of sideslip, pitot and static pres-

sures forward of the aircraft flow field.

Flight controls instrumentation includes potentiometer type

transducers which have been installed to monitor the posi-

tion 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 mea-

sure lateral stick force, longitudinal stick force, and rudder

pedal differential force. For the aircraft parameter identifica-

tion 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 per-

formance.

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 inter-

nal fuel to determine aircraft gross weight, center of gravity

and inertia parameters using the function provided in the Ap-

pendix. The function was based on formulas provided by a

mid-1990’s report written by a then-TPS student [3].

4. FLIGHT TEST

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 Fig-

ure 3. At each condition a series of longitudinal and lateral-

directional 1-g flying qualities test points were flown. All

points began from a wings-level, un-accelerated, constant al-

titude trim shot. Maneuvers included doublets, steps, raps

and frequency sweeps inputs from elevators, ailerons and rud-

der. 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

included rudder inputs not provided by the pilot. The air-

craft 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 spi-

ral modes were not targeted in the single flight test but would

be required to fully populate a complete six degree of free-

dom aerodynamic model for the aircraft.

Figure 3. Test points as a function of pressure altitude and

true airspeed with lines of constant dynamic pressure and

Mach.

5. PARAMETER IDENTIFICATION RESULTS

AND DISCUSSION

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 soft-

ware were generally successful for calculating estimates for

the longitudinal stability derivatives. The results for the pitch-

ing moment coefficients will be discussed in this section.

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.

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 pro-

vided adequate data for reduction. A typical frequency sweep

4

0 10 20 30 40 50 60 70−5

−4

−3

−2

−1

0

1

2

3

4

5

time (sec)

De

g/s

ec

alpha dot (derived)

pitch rate

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

flown by a test pilot is shown in Figure 5.

0 10 20 30 40 50 60 70−6

−4

−2

δe

0 10 20 30 40 50 60 702

4

6

α

0 10 20 30 40 50 60 70−5

0

5

q

time

Figure 5. Frequency sweep of elevator (deg) with angle of

attack (deg) and pitch rate (deg/sec) response at Pt 2 condi-

tions (M=0.7, q=200 lbsf).

The linear regression tool in SIDPAC was applied to the lon-

gitudinal frequency sweep test points. The test results for

aerodynamic pitching moment terms in Equation 10 are sum-

marized 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 ac-

tual 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.

After the aerodynamic stability derivatives were estimated the

original test maneuver may be simulated where the pitching

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 ob-

served pitching moment was relatively small indicating the

model was a valid representation at the condition tested.

0 10 20 30 40 50 60 70−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

time (sec)C

m

data

model

Figure 6. Pitching moment coefficient test results versus

linear least squares model. Results are typical for the pitching

moment cases examined.

The calculated longitudinal dimensional stability parameters

were then algebraically calculated using the procedures in

Section 2 and are presented in Table 3. For comparison, pub-

lished values from Yechout’s text [4] for the F-104 were pre-

sented at two different test conditions. Although not repre-

sentative 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 nat-

ural frequency (converted to Hertz) and damping ratio were

computed using Equations 17 and 18. In general, the natu-

ral 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.

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 accel-

eration had gain as well as bias errors the normal acceleration

coefficients would be affected. In the data in Table 3 it would

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

PT1

Cmα-8.024e-1 8.633e-3 1.1 [ -0.820 , -0.785 ]

Cmq-1.306e1 4.400e-1 3.4 [ -13.941 , -12.181]

Cmδe-1.367e0 1.429e-2 1.0 [ -1.396 , -1.338]

PT2

Cmα-5.624e-1 3.209e-3 0.6 [ -0.569, -0.556 ]

Cmq-1.272e1 2.264e-1 1.8 [-13.170, -12.264]

Cmδe-1.285e0 5.539e-3 0.4 [-1.296 , -1.274]

PT3

Cmα-1.469e0 9.617e-3 0.7 [-1.488,-1.450]

Cmq-2.021e1 5.737e-1 2.8 [-21.361,-19.067]

Cmδe-1.500e0 1.656e-2 1.1 [ -1.533 , -1.467 ]

PT4

Cmα-1.154e0 1.276e-2 1.1 [ -1.179 ,-1.128]

Cmq-1.779e1 6.720e-1 3.8 [ -19.129 ,-16.441]

Cmδe-1.747e0 2.284e-2 1.3 [ -1.793 ,-1.701]

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

Parameter F-104 [4] F-104 [4] T-38 PT1 T-38 PT2 T-38 PT3 T-38 PT4

Altitude (ft) S/L 55,000 10,356 31,753 32,460 22,751

Mach 0.257 1.800 0.713 0.709 1.079 0.906

True Airspeed(fps) 287 1742 768 696.8 1061.8 929.3

q (lb/ft2) 97.8 434.5 512.5 202.5 458.0 498.0

S (ft2) 196 196 170 170 170 170

c (ft2) 21.9 21.9 7.79 7.79 7.79 7.79

α (deg) 10 2 1.7 4.0 2.7 1.1

Iyy (slug-ft2) 5.90e4 5.90e4 2.932e4 2.924e4 2.909e4 2.873e4

Longitudinal Derivatives

Zα (ft/s2) -140.2 -346.6 -1055 -502.2 -1076 -1704

Mα (1/s2) -2.009 -18.12 -18.4265 -5.117 -30.39 -26.27

Mq (1/s) -0.3046 -0.1844 -1.5101 -0.6419 -1.5219 -1.6849

Mδe(1/s2) -4.990 -18.15 -31.39 -11.69 -31.03 -39.78

Short Period Dynamics

ωn (Hz) 0.23 0.68 0.72 0.38 0.90 0.86

ζ 0.30 0.05 0.32 0.29 0.22 0.32

6

be the Zα dimensional parameter.

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 dif-

ficulties 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 condi-

tions 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 consider-

ations. 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 deriva-

tive computations needed to form α and β required for use as

regressors.

A design of experiments analysis was not conducted. Multi-

ple 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 tech-

nique to compute coefficients but was necessarily limited in

scope.

Finally the assumptions for linearizing the data should be un-

derstood. 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 re-

quired 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 re-

sults of obtained by Krause [3] using an output error tech-

nique. In general these observations were made.

The lateral directional testing showed the typical difficulty

when looking at a multiple input system. While it was ex-

pected that testing with the yaw damper off would have pro-

vided a better result since aircraft motion would not be at-

tenuated 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%.

6. CONCLUSIONS AND RECOMMENDATIONS

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 al-

gorithms. SIDPAC provides acceptable results for longitu-

dinal 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 algo-

rithms should be satisfactory but may lack the level of confi-

dence shown in the longitudinal cases.

APPENDIX

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

7

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: michael.shepherd@edwards.af.mil

% 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 <= 1000

ixxfr=1.429572953e-8*(-1479.23189+WFR).*(-1108.152712+WFR).*(-2.971601441+WFR);

else

ixxfr=5.738417234e-14*(2722.258505-WFR).*(-1350.000002+WFR).*...

(-1349.999998+WFR).*(1.307231912e6-2246.212161*WFR+WFR.ˆ2);

end

%Izz

izzfl=1.319074486e-10*(0.005072772213+WFL).*(337.372297+WFL).*...

(7.20647684e6-5137.151321*WFL+WFL.ˆ2);

izzfr=1.090972066e-10*(2827.719311-WFR).*(0.005072711818+WFR).*...

(3.315606698e6-2680.168835.*WFR+WFR.ˆ2);

%Ixz

ixzfl=4.43948009e-8*(-4021.220951+WFL).*(-732.6733307+WFL).*(-10.50598976+WFL);

ixzfr= 3.149396586e-11.*(-1348.969404+WFR).*(0.005072705039+WFR).*...

(-3.644552245e6+2929.66091*WFR-WFR.ˆ2);

%inertia calculations

Ixx=1553.6+ixxfl+ixxfr;

Izz=29266+izzfl+izzfr;

Iyy=28333.72+(ixxfl+izzfl)+(ixxfr+izzfr);

Ixz=47.25+ixzfl+ixzfr;

%CG calculations (reference cg: FS354.39, WL100)

CG = (((OPM+wfllx+wfrlx)*Mom_Sim/GW)-LEMAC)*100/MAC;

CG_in = ((OPM+wfllx+wfrlx)*Mom_Sim/GW)-LEMAC;

DeltaXcg_in=(((OPM+wfllx+wfrlx)*Mom_Sim/GW)-354.39);

DeltaZcg_in=((-15852+wfllz+wfrlz)/GW);

9

ACKNOWLEDGMENTS

The authors wish to thank Dr. Eugene A. Morelli of NASA

Langley Research Center and Mr. Fred Webster of the Air

Force Flight Test Center for their invaluable guidance and

assistance and also the members of the public release team

led by Ms. Kandi Jones for their work over the holiday sea-

son and for helpful technical editing comments. The authors

would also like to thank the members of the USAF Test Pilot

School Technical Support staff for their energy and enthusi-

asm maintaining the aircraft data systems used to gather the

information which form the basis of this report.

REFERENCES

[1] Klein, V. and Morelli, E. A., Aircraft System Identifica-

tion: Theory and Practice, American Institute of Aero-

nautics and Astronautics, Inc., 1st ed., 2006.

[2] “USAF Test Pilot School Instrumentation Handbook,”

Tech. rep., USAF Test Pilot School Internal Documen-

tation, 2009.

[3] “A Limited Investigation of the Lateral-Directional Axes

of the F-15B and T-38A Near Wing Rock Angles of At-

tack Using a Personal Computer-Based Parameter Iden-

tification (PCID) Program,” Tech. Rep. AFFTC-TLR-94-

85, The Institute of Electrical and Electronics Engineers,

Inc., 1985.

[4] Yechout, T. R., Introduction to Aircraft Flight Mechan-

ics: Performance, Static Stability, Dynamic Stability, and

Classical Feedback Control, American Institute of Aero-

nautics and Astronautics, Inc., 1st ed., 2003.

BIOGRAPHY

Michael Shepherd is the Deputy Com-

mander, 412th Operations Group and

former Director, Technical Support Di-

vision at the United States Air Force Test

Pilot School where his division oversaw

aircraft instrumentation and telemetry

operations. Lieutenant Colonel Shep-

herd’s research interests include struc-

tural dynamics of aerospace structures and aircraft flight con-

trol. Lt Col Shepherd obtained a B.S. in engineering mechan-

ics from the USAF Academy in 1990, and an M.S. in aero-

nautical and astronautical engineering from the University of

Washington in 1991, and a Ph.D. from the Air Force Institute

of Technology in 2006. He is a master navigator with over

3000 flight hours in operational and developmental test air-

craft, a graduate of the USAF Test Pilot School and a member

of AIAA.

Timothy Jorrisis the Director, Cur-

riculum Support Division at the United

States Air Force Test Pilot School Lt

Col Tim Jorris received a B.S. and

M.S. in Aerospace Engineering from the

University of California, Los Angeles

(UCLA); a PhD in Astronautical Engi-

neering from the Air Force Institute of

Technology (AFIT). He’s worked on the F-15I, B-1, B-52,

and Global Hawk. A USAF Test Pilot School graduate, he

is currently an Instructor Flight Test Engineer at TPS and an

Adjunct Professor at AFIT.

William Gray is the Chief Test Pilot

for the USAF Test Pilot School at Ed-

wards AFB, CA. He earned his B.S. at

the US Air Force Academy (physics)

and his M.S. in mechanical engineering

at California State University, Fresno.

An associate fellow of the Society of Ex-

perimental Test Pilots and Member of

AIAA, Mr. Gray is currently conducting research on aircraft

handling qualities evaluation, developing a reconfigurable

variable stability flying qualities simulator and instructing

student test pilots in the T-38, F-16, and NF-16D VISTA air-

craft. A graduate of the USAF Test Pilot School, Mr. Gray

has accumulated over 4500 hours in 80 diverse aircraft types.

10