NASA/TM—2011–215963

Unsteady Aerodynamic Model Tuning for 

Precise Flutter Prediction

Chan-gi Pak

Dryden Flight Research Center, Edwards, California

 Click here: Press F1 key (Windows) or Help key (Mac) for help 

April 2011
NASA STI Program ... in Profile

Since its founding, NASA has been dedicated 

to the advancement of aeronautics and space 

science. The NASA scientific and technical 

information (STI) program plays a key part in 

helping NASA maintain this important role.

The NASA STI program operates under the 

auspices of the Agency Chief Information 

Officer. It collects, organizes, provides for 

archiving, and disseminates NASA’s STI. The 

NASA STI program provides access to the 

NASA Aeronautics and Space Database and its 

public interface, the NASA Technical Report 

Server, thus providing one of the largest 

collections of aeronautical and space science 

STI in the world. Results are published in both 

non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types:

• TECHNICAL PUBLICATION. Reports of 

completed research or a major significant 

phase of research that present the results of 

NASA Programs and include extensive data 

or theoretical analysis. Includes compilations of significant scientific and technical 

data and information deemed to be of 

continuing reference value. NASA counterpart of peer-reviewed formal professional 

papers but has less stringent limitations on 

manuscript length and extent of graphic 

presentations.

• TECHNICAL MEMORANDUM. 

Scientific and technical findings that are 

preliminary or of specialized interest, 

e.g., quick release reports, working 

papers, and bibliographies that contain 

minimal annotation. Does not contain 

extensive analysis.

• CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees.

• CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, 

seminars, or other meetings sponsored 

or co-sponsored by NASA.

• SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest.

• TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission.

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing help desk and personal search 

support, and enabling data exchange services.

For more information about the NASA STI 

program, see the following:

• Access the NASA STI program home page 

at http://www.sti.nasa.gov

• E-mail your question via the Internet to 

help@sti.nasa.gov

• Fax your question to the NASA STI Help 

Desk at 443-757-5803

• Phone the NASA STI Help Desk at 

443-757-5802

• Write to:

NASA STI Help Desk

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

This page is required and contains approved text that cannot be changed.
NASA/TM—2011–215963

Unsteady Aerodynamic Model Tuning for 

Precise Flutter Prediction

Chan-gi Pak

Dryden Flight Research Center, Edwards, California

 Insert conference information, if applicable; otherwise delete 

 Click here: Press F1 key (Windows) or Help key (Mac) for help 

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, CA 93523-0273

April 2011
 Enter acknowledgments here, if applicable. 

 Click here: Press F1 key (Windows) or Help key (Mac) for help 

Available from:

 Click here: Press F1 key (Windows) or Help key (Mac) for help 

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076-1320

443-757-5802
1

Abstract

A simple method for an unsteady aerodynamic model tuning is proposed in this study. This method is 

based on the direct modification of the aerodynamic influence coefficient matrices. The aerostructures test 

wing 2 flight-test data is used to demonstrate the proposed model tuning method. The flutter speed margin 

computed using only the test validated structural dynamic model can be improved using the additional 

unsteady aerodynamic model tuning, and then the flutter speed margin requirement of 15 percent in 

military specifications can apply towards the test validated aeroelastic model. In this study, unsteady 

aerodynamic model tunings are performed at two time invariant flight conditions, at Mach numbers of 

0.390 and 0.456. When the Mach number for the unsteady aerodynamic model tuning approaches to the 

measured fluttering Mach number, 0.502, at the flight altitude of 9,837 ft, the estimated flutter speed is 

approached to the measured flutter speed at this altitude. The minimum flutter speed difference between 

the estimated and measured flutter speed is -0.14 percent.

Nomenclature

aij the i-th row and j-th column element of the real part of A

A aerodynamic influence coefficient matrix

AIC aerodynamic influence coefficient

ATW aerostructures test wing

bij the i-th row and j-th column element of the imaginary part of A

CG center of gravity

DFRC Dryden Flight Research Center

eij design variable for aij

f aeroelastic frequency, Hz

fij design variable for bij

FEM finite element model

FTF flight test fixture

g structural damping

GVT ground vibration test

G(x) constraints

m number of degrees of freedom

n number of modes

NASA National Aeronautics and Space Administration

O3 object-oriented optimization

V aircraft speed, KEAS

Vd dive speed, KEAS

VFEM test-validated-structural-dynamic finite element model

Introduction

The primary objective of this study is to reduce uncertainties in the unsteady aerodynamic model of 

an aircraft to increase the safety of flight. To this end, a new flutter analysis procedure using the validated 

aeroelastic model is proposed, and the block diagram of this new procedure is shown in figure 1.
2

Figure 1a. Previous procedure.

Figure 1b. Proposed new procedure.

Figure 1. Flutter analysis procedure at NASA Dryden Flight Research Center.
3

Significant efforts (refs. 1–3) have been made in developing corrections to linear aerodynamic models 

to improve correlation with steady-state wind tunnel and flight test data. There has been a limited amount 

of effort in the correction of unsteady aerodynamics for aeroservoelastic applications, which has been 

relatively sparse and ad-hoc when compared to the steady-state work that has been performed.

The aerostructures test wing (ATW) 2 test article, shown in figure 2, was developed and flown at the 

National Aeronautics and Space Administration (NASA) Dryden Flight Research Center (DFRC) on the 

F-15B (McDonnell Douglas, now The Boeing Company, Chicago, Illinois) test bed aircraft on 

December 15, 2009. To support the envelope expansion, a test-validated-structural-dynamic finite 

element model (VFEM) was used for the flutter analysis of the ATW2. Flutter boundaries of the ATW2, 

before and after the structural dynamic model tuning (ref. 4), are compared with the flight envelopes as 

shown in figure 3 (ref. 5). In this figure, the solid line bounds the ATW2 test envelope that is planned for 

flight, and the dashed line is the 15 percent margin of the ATW2 test envelope. This 15 percent margin 

line was designed to match the numerical flutter boundaries computed using the VFEM with the 3 percent 

structural damping, the solid line with circular marker. The solid line with the diamond marker represents 

flutter boundaries using the VFEM with the measured structural damping (ref. 5). The measured flutter 

point of the ATW2 is also shown in figure 3, using the x marker.

Figure 2. Aerostructures test wing 2 mounted on the F15-B pylon for flight-testing.
4

Figure 3. Flutter boundaries before and after structural dynamic model tuning.

Required flutter margins for the safety of flight were computed and summarized in table 1 and

figure 3. It should be noted that the series of ground vibration tests (GVT) and structural dynamic model 

tuning has been performed (refs. 5–7) resulting in the computed flutter boundaries, based on the ATW2 

configuration in figure 2 and the corresponding VFEM, shown in figure 3. The GVT for the final ATW2 

configuration was performed while the ATW2 was mounted to the flight test fixture (FTF) (ref. 8) in the 

FTF ground handling cart as shown in figure 4. The FTF was sufficiently massive when compared to the 

ATW2 so that cantilevered boundary conditions were used. It may be concluded from table 1 and figure 3 

that when only the structural dynamic model is validated with respect to GVT data, the flutter margin 

required for the flutter certification of the ATW2 should be approximately 40 percent. In addition to the 

historically stand-alone structural dynamic model, the unsteady aerodynamic model should also be 

validated with the test data to use the 15 percent flutter margin in the military specification (ref. 9).

Table 1. Required flutter margins for different types of structural dynamic models.

Flutter boundaries Flutter margins

Measured/1.15 (= Vd) 0 %

Measured (= 1.15 Vd) 15 %

Test validated FEM; after model tuning with measured damping 32 %

Test validated FEM; after model tuning with 3% structural damping 41 %
5

Figure 4. Flight test fixture and the ATW2 on a ground-handling cart.

Unsteady Aerodynamic Model Tuning Procedure

A simple technique has been proposed and developed to update unsteady aerodynamic models. The 

technique is based on matching the measured and numerical aeroelastic frequencies of an aircraft 

structure. In defining the optimization problem to match the measured aeroelastic frequencies, the 

variation of the unsteady aerodynamic force was selected as the design parameter. The unsteady 

aerodynamic force is a function of Mach number, reduced frequency, and dynamic pressure; which can be 

obtained based on any aerodynamic model. ZAERO (ZONA Technology Incorporated, Scottsdale, 

Arizona) (ref. 10) code is used in this study. If the Mach number is constant, the reduced frequency and 

dynamic pressure become variables for changing the unsteady aerodynamic force.

Supporting the Aeronautics Research Mission Directorate guidelines, NASA DFRC has developed an 

object-oriented optimization (O3) tool (ref. 11), which leverages existing tools and practices, and allows 

the easy integration and adoption of new state-of-the-art software. Unsteady aerodynamic model tuning 

used in this study is based on the minimization of the discrepancies in numerical and measured aeroelastic 

frequencies. A computer code for unsteady aerodynamic model tuning has been developed using the O3 

tool together with the pre-processor, ZAERO, and post-processor codes shown in figure 5.
6

Figure 5. Unsteady aerodynamic model tuning using the object-oriented optimization tool.

Pre–processor code

This code reads in design variables generated by the O3 tool, and then reads modal aerodynamic 

influence coefficient (AIC) matrices that were computed and saved using ZAERO code. Modified modal 

AIC matrices are then created as shown in figure 6.

Figure 6. Flow chart of the pre-processor.
7

Design variables in this unsteady aerodynamic model tuning are scaling factors for each element in the 

AIC matrices. The AIC matrix at a reduced frequency can be written as:

A =

a11 a12 ... a1n

a21 a22 ... a2n

   

am1 am2 ... amn

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

+ i

b

11 b12 ... b1n

b21 b22 ... b2n

   

bm1 bm2 ... bmn

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

Design variables, e11, e12, ..., e21, e22, ..., emn, f11, f12, ..., f21, f22, ..., fmn are defined as:

e

11a11 e12a12 ... e1na1n

e21a21 e22a22 ... e2na2n

   

em1am1 em2am2 ... emnamn

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

+ i

f

11b11 f12b12 ... f1nb1n

f

21b21 f22b22 ... f2nb2n

   

f

m1bm1 fm2bm2 ... fmnbmn

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⎥

The following design variable linking options are available for the unsteady aerodynamic model tuning.

Option 1: single design variable

d=e11=e12=...=emn=f11=f12=...=fmn

Option 2: two design variables

d1=e11=e12=...=emn; real part

d2=f11=f12=...=fmn ; imaginary part

Option 3: columnwise the same design variables (total n design variables)

d1=e11=e21=...=em1= f11=f21=...=fm1

d2=e12=e22=...=em2= f12=f22=...=fm2

...

dn=e1n=e2n=...=emn=f1n=f2n=...=fmn; 

Option 4: columnwise the same design variables (total 2n design variables)

d1=e11=e21=...=em1

d2=e12=e22=...=em2

...

dn=e1n=e2n=...=emn; real parts

dn+1=f11=f21=...=fm1

dn+2=f12=f22=...=fm2

...
8

d2n=f1n=f2n=...=fmn; imaginary parts

Option 5: No design variable linking; total 2mn design variables.

ZAERO Flutter Analysis

Flutter analyses in this study are based on ZAERO code. This code acquires the modified modal AIC 

matrices and performs the matched flutter analysis as shown in figure 7. This computer simulation 

requires the natural frequencies and mode shapes of the aircraft, and in this study these modal data are 

computed using MSC/NASTRAN (MSC Software Corporation, Santa Ana, California) code (ref. 12). 

The V-g and V-f data are computed and saved for the next post-processing step.

Figure 7. Flow chart of the flutter analysis procedure.

Post–processor code

This program reads in the V-g and V-f data and the target altitude where the flight test was 

performed. Based on the velocity information V in the V-g and V-f data, corresponding altitudes at fixed 

Mach numbers are computed. Numerical aeroelastic frequencies are computed from target and computed 

altitudes using the cubic splining procedure. 

This program also computes the frequency difference between the numerical and measured 

aeroelastic frequencies. Frequency difference will be an objective function, which will be minimized 

through the use of the O3 tool. The proposed tuning technique is an unconstrained optimization problem 

that can be solved using a gradient based optimizer (ref. 13), a genetic algorithm (ref. 14), or a 

big-bang-big-crunch algorithm (refs. 15 – 17). A flow-chart of this post-processor code is shown in 

figure 8.
9

Figure 8. Flow chart of the post-processor.

Application

During the flight test of the ATW2, a classical bending and torsion type of flutter, as shown in 

figure 9, was observed near a Mach number of 0.502 and a flight altitude of 9,837 ft. Measured Mach 

number, flight altitude, and acceleration at the middle and leading edge of the wing tip boom after 

take-off, near flutter, and during flutter are shown in figures 10 and 11, respectively. Measured aeroelastic 

frequencies during the flight test as well as natural frequencies computed using the VFEM and measured 

during GVT are shown in table 2.

Table 2. Numerical and measured frequencies (Hz) of the ATW2 during the flight test.

Natural frequencies Measured aeroelastic frequencies Mode

VFEM GVT After take off* Near flutter** During flutter***

1 17.45 17.45 (0.623)†

2 43.48 43.72 (0.610) 40.45 38.99 37.69

3 82.98 83.66 (0.778)

4 133.60 N/A

5 153.80 142.30 (0.674)

* Time steps 83,742 to 83,745 (Mach = 0.390; altitude = 9,934 ft; time invariant)

** Time steps 84,668 to 84,672 (Mach = 0.456; altitude = 9,858 ft; time invariant)

*** Time steps 84,683 to 84,684 (Mach = 0.502; altitude = 9,837 ft; time varying)

( )† Measured damping (%)
10

Figure 9. Classical bending and torsion flutter during flight test.

Figure 10a. Mach number.

Figure 10. Measured Mach number, flight altitude, and acceleration at the middle and the leading edge of 

the boom after take off (time steps 83,720 s to 83,750 s).
11

Figure 10b. Flight altitude.

Figure 10c. Acceleration at middle of boom.

Figure 10. Continued.
12

Figure 10d. Acceleration at leading edge of boom.

Figure 10. Concluded.

Figure 11a. Mach number.

Figure 11. Measured Mach number, flight altitude, and acceleration at the middle and the leading edge of 

the boom just before and during flutter (time steps 84,660 s to 84,690 s).
13

Figure 11b. Flight altitude.

Figure 11c. Acceleration at middle of boom.

Figure 11. Continued.
14

Figure 11d. Acceleration at leading edge of boom.

Figure 11. Concluded.

The first model tuning is performed using flight data after take-off, from 83,742 s to 83,745 s as 

shown in figure 10. The ATW2 in this time period is a time invariant system. The average flight Mach 

number and altitude during this time period were 0.39 and 9,934 ft, respectively. Unfortunately, measured 

acceleration data in figure 10 was noisy, and it was quite difficult to estimate the first and third aeroelastic 

frequencies because of high aerodynamic damping. The second measured aeroelastic frequency is 

40.45 Hz as shown in table 2. An initial aeroelastic frequency of 41.12 Hz is computed using ZAERO 

code as shown in table 3.

Table 3. The second aeroelastic frequency before and after unsteady aerodynamic model tuning 

and corresponding scaling factors.

Mach Measured 

(Hz)

Altitude 

(ft)

Before tuning 

(Hz)

Scaling factor 

(design variable)

After tuning 

(Hz)

0.390 40.45 9934 41.12 1.2579 40.45

0.456 38.99 9858 40.10 1.2719 38.99

In this model tuning procedure, the aeroelastic frequency difference in the second mode is minimized 

using the design variable linking option 1. The number of target aeroelastic frequencies to be matched is 

one, and therefore the simplest option is selected. In other words, there is an unconstrained optimization 

problem with a single design variable. After model tuning, the second aeroelastic frequency of 41.12 Hz 

becomes 40.45 Hz, and the corresponding scaling factor (single design variable) is 1.2579 as shown in 

table 3.
15

The saved AIC matrices at Mach = 0.502 is updated using the scaling factor of 1.2579, and the 

updated flutter boundary at this Mach number is summarized in table 4 and figure 12. The tuned flutter 

speed, corresponding altitude, and flutter frequency are 277.3 KEAS, 9,670 ft, and 37.69 Hz, respectively. 

It should be noted in table 4 that flutter speed and frequency difference after the unsteady aerodynamic 

model tuning are 0.33 percent and 0.00 percent, respectively.

Table 4. Measured and computed flutter boundaries at Mach = 0.502.

Flutter speed Flutter frequency 

Comment

Scaling 

factor 

(design 

variable)

KEAS Difference 

%

Altitude, 

ft Hz Difference 

%

Measured N/A 276.4 0.00 9836.9 37.69 0.00

Before tuning 1.0000 311.3 13.00 3561.5 37.67 -0.05

Use M = 0.390 aero 1.2579 277.3 0.33 9670.0 37.69 0.00

Use M = 0.456 aero 1.2719 276.0 -0.14 9912.5 37.68 -0.03

Figure 12. Flutter boundaries at Mach = 0.502 before and after unsteady aerodynamic model tuning.

The second model tuning is performed using another time invariant system, between time steps of 

84,660 s and 84,690 s. Relatively flat time histories of flight Mach numbers and altitude, between time 

steps of 84,668 s and 84,672 s, are observed as shown in figure 11. In this time period, the average flight 

Mach number and altitude were 0.456 and 9,858 ft, respectively.

The second measured aeroelastic frequency in these time steps is 38.99 Hz as shown in table 2. The 

corresponding aeroelastic frequency computed from the ZAERO simulation with Mach 0.456 

aerodynamics is 40.10 Hz as shown in table 3. Unsteady aerodynamic model tuning is performed using 
16

these two numerical and measured frequencies, and a scaling factor of 1.2719 based on the design 

variable linking option 1, which is also given in table 3.

Flutter analysis at Mach 0.502 is performed using this new scaling factor, and the flutter speed and 

frequency are summarized in table 4 and figure 12. Flutter speed difference of 13 percent before the 

unsteady aerodynamic model tuning becomes -0.14 percent after tuning. 

In case of the ATW2, computation time required for completing an unsteady aerodynamic model 

tuning based on option 1 was less than 7 min. Once the scaling factor (design variable) is computed, an 

additional 1 or 2 min is needed for the Fast Fourier Transformation, one more flutter analysis at a higher 

Mach number to compute the updated V-g and V-f data, and automatic computations of updated flutter 

speed and frequency. Therefore, less than 9 min are enough to predict more accurate flutter speed based 

on the current flight test data.

Conclusion

A simple unsteady aerodynamic model tuning based on the direct AIC modification is proposed in 

this study. The value of the unsteady aerodynamic model tuning procedure has been shown with the 

application to the ATW2 flight test data.

Flutter boundaries and the ATW2 flight test envelope were computed using the VFEM (ref. 5, 6). The 

flutter margins required for the safety of flight were approximately 40 percent when only the structural 

dynamic model was validated. Excellent flutter speed matching is accomplished when the simple 

unsteady aerodynamic model tuning is applied resulting in flutter speed differences of 0.33 

and -0.14 percent. The flutter margin requirement of 15 percent in the military specification can now be 

used with the test validated aeroelastic model, that is test validated structural dynamic and unsteady 

aerodynamic models. The modeling uncertainties associated with the unsteady aerodynamics can be 

easily minimized through the use of the simple model tuning procedure proposed in this study.

Unsteady aerodynamic model tunings are performed at two time invariant flight conditions, at Mach 

numbers of 0.390 and 0.456. When the Mach number for the unsteady aerodynamic model tuning 

approaches to the measured fluttering Mach number, 0.502, at the flight altitude of 9,837 ft, the estimated 

flutter speed is approached to the measured flutter speed at this altitude. Therefore, we may conclude that 

the Mach number selected for the unsteady aerodynamic model tuning is closer to the measured fluttering 

Mach number at the same flight altitude, and we may get a more accurate scaling factor for the precise 

flutter prediction.
17

References

1. Baker, Myles L., Kuo-An Yuan, and Patrick J. Goggin, “Calculation of Corrections to Linear 

Aerodynamic Methods for Static and Dynamic Analysis and Design,” AIAA–98–2072, 

Proceedings of the 39th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and 

Material Conference and Exhibit, April 20–23, 1998.

2. Giesing, Joseph P., Terez P. Kalman, and William P. Rodden, “Correction Factor Techniques for 

Improving Aerodynamic Prediction Methods,” NASA CR–144967, May 1976.

3. Klein, Vladislav, and Patrick C. Murphy, “Estimation of Aircraft Nonlinear Unsteady Parameters 

from Wind Tunnel Data,” NASA/TM–1998–208969, December 1998.

4. Pak, Chan-gi, “Finite Element Model Tuning Using Measured Mass Properties and Ground 

Vibration Test Data,” ASME Journal of Vibration and Acoustics, Vol. 131, No. 1, February 2009.

5. Pak, Chan-gi, and Shun-fat Lung, “Flutter Analysis of the Aerostructures Test Wing with Test 

Validated Structural Dynamic Model,” Journal of Aircraft, “to be published.”

6. Pak, Chan-gi, and Shun-fat Lung, “Reduced Uncertainties in the Flutter Analysis of the 

Aerostructures Test Wing,” NASA TM–2011–216421, January 2011.

7. Lung, Shun-fat, and Chan-gi Pak, “Updating the Finite Element Model of the Aerostructures Test 

Wing Using Ground Vibration Test Data,” NASA TM–2009–214646, April 2009.

8. Richwine, David M., “F–15B/Flight Test Fixture II: A Test Bed for Flight Research,” 

NASA–TM–4782, December 1996.

9. Military Specification, “Airplane Strength and Rigidity Vibration, Flutter, and Divergence,” 

MIL–A–8870C, March 25, 1993.

10. ZAERO User’s Manual version 8.2, ZONA Technology, Inc., March 2008.

11. Pak, Chan-gi, “Preliminary Development of an Object–Oriented Optimization Tool,” 

NASA–TM–2011–216419, January 2011.

12. MSC/NASTRAN Quick Reference Guide version 69, The MacNeal Schwendler Corporation, 

1996.

13. DOT Design Optimization Tools User’s Manual Version 5.0, Vanderplaats Research & 

Development, Inc., Colorado Springs, 2001.

14. Charbonneau, Paul, and Barry Knapp, A User’s Guide to PIKAIA 1.0, National Center for 

Atmospheric Research, Boulder, Colorado, 1995.

15. Erol, Osman K., and Ibrahim Eksin, “A New Optimization Method: Big Bang–Big Crunch,” 

Advances in Engineering Software, Vol. 37, No. 2, pp. 106–111, 2006.
18

16. Camp, Charles V., “Design of Space Trusses Using Big Bang–Big Crunch Optimization,” 

Journal of Structural Engineering, ASCE, Vol. 133, No. 7, pp. 999–1008, July 2007.

17. Kaveh, A., and S. Talatahari, “Size Optimization of Space Trusses Using Big Bang–Big Crunch 

Algorithm,” Computers and Structures, Vol. 87, pp. 1129–1140, 2009.
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

1. REPORT DATE (DD-MM-YYYY)

01-04-2011

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Unsteady Aerodynamic Model Tuning for Precise Flutter Prediction

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Pak, Chan-gi

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, CA 93523-0273

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

H-3088

10. SPONSORING/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

Pak, NASA Dryden Flight Research Center

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 1 Availability: NASA CASI (443) 757-5802 Distribution: Standard

19b. NAME OF RESPONSIBLE PERSON

STU Help Desk at e-mail: help@sti.nasa.gov

14. ABSTRACT

A simple method for an unsteady aerodynamic model tuning is proposed in this study. This method is based on the direct 

modification of the aerodynamic influence coefficient matrices. The aerostructures test wing 2 flight-test data is used to demonstrate 

the proposed model tuning method. The flutter speed margin computed using only the test validated structural dynamic model can 

be improved using the additional unsteady aerodynamic model tuning, and then the flutter speed margin requirement of 15 percent

in military specifications can apply towards the test validated aeroelastic model. In this study, unsteady aerodynamic model tunings 

are performed at two time invariant flight conditions, at Mach numbers of 0.390 and 0.456. When the Mach number for the unsteady 

aeroynamic model tuning approaches to the measured fluttering Mach number, 0.502, at the flight altitude of 9,837 ft, the estimated 

flutter speed is approached to the measured flutter speed at this altitude. The minimum flutter speed difference between the 

estimated and measured flutter speed is -0.14 percent.

15. SUBJECT TERMS

Aerodynamic influence coefficient matrices, Aerostructures Test Wing, Flight test data, Test validated aeroelastic model, Unsteady 

aerodynamic model tuning

18. NUMBER

 OF 

 PAGES

23

19b. TELEPHONE NUMBER (Include area code)

(443) 747-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39-18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

11. SPONSORING/MONITORING

 REPORT NUMBER

NASA/TM-2011-215963

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing 

data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or 

any other aspect of this collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, 

Directorate for Information Operations and Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware 

that notwithstanding any other provision of law, no person shall be subject to any penalty for failing to comply with a collection of information if it does not display a 

currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.