aerodynamic system identification of the t-38a using sidpac
TRANSCRIPT
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
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: [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 <= 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