NASA/TM-2010-216387

Application of Approximate Unsteady 

Aerodynamics for Flutter Analysis 

Chan-gi Pak and Wesley W. Li

NASA Dryden Flight Research Center

Edwards, California

July 2010
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NASA/TM-2010-216387

Application of Approximate Unsteady 

Aerodynamics for Flutter Analysis 

Chan-gi Pak and Wesley W. Li

NASA Dryden Flight Research Center

Edwards, California

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

July 2010
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(443) 757-5802
ABSTRACT

A technique for approximating the modal aerodynamic influence coefficient (AIC) matrices 

by using basis functions has been developed. A process for using the resulting approximated 

modal AIC matrix in aeroelastic analysis has also been developed. The method requires the 

unsteady aerodynamics in frequency domain, and this methodology can be applied to the 

unsteady subsonic, transonic, and supersonic aerodynamics. The flutter solution can be found by 

the classic methods, such as rational function approximation, k, p-k, p, root locus et cetera. The 

unsteady aeroelastic analysis using unsteady subsonic aerodynamic approximation is 

demonstrated herein. The technique presented is shown to offer consistent flutter speed prediction 

on an aerostructures test wing (ATW) 2 and a hybrid wing body (HWB) type of vehicle 

configuration with negligible loss in precision. This method computes AICs that are functions of 

the changing parameters being studied and are generated within minutes of CPU time instead of 

hours. These results may have practical application in parametric flutter analyses as well as more 

efficient multidisciplinary design and optimization studies.

NOMENCLATURE

A general AIC matrix

AIC aerodynamic influence coefficient

ATW aerostructures test wing

BFA basis function approximation

CEM central executive module

DFRC Dryden Flight Research Center

g damping

HWB hybrid wing body

k reduced frequency

m number of mode shapes

MDAO multidisciplinary design, analysis, and optimization

n number of basis functions

NASA National Aeronautics and Space Administration

Q modal AIC matrix

Qij i-th row and j-th column element of the modal AIC matrix Q

Q approximate modal AIC matrix

Qij i-th row and j-th column element of the approximate modal AIC matrix Q

Q basis AIC matrix

Q

sr s-th row and r-th column element of the basis AIC matrix Q

V velocity

βk

i Modal participation factors of the k-th basis function on the i-th mode shape

βs

i Modal participation factors of the s-th basis function on the i-th mode shape

βk

j Modal participation factors of the k-th basis function on the j-th mode shape
2

βr

j Modal Participation factors of the r-th basis function on the j-th mode shape

φ mode shape

φi i-th mode shape

Φ modal matrix

Ψ basis matrix

Ψk k-th basis function

Ψr r-th basis function

Ψs

T s-th transpose of the basis function

ω frequency

INTRODUCTION

Supporting the Aeronautics Research Mission Directorate guidelines, the National 

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

(Edwards, California) is developing an object-oriented multidisciplinary design, analysis, and 

optimization (MDAO) tool (ref. 1). This tool will leverage existing tools and practices, and allow 

the easy integration and adoption of new state-of-the-art software. At the heart of the objectoriented MDAO tool is the central executive module (CEM) as shown in figure 1. In this module, 

the user will choose an optimization methodology, and provide starting and side constraints for 

continuous as well as discrete design variables and external file names for interface variables, 

which communicate between the CEM and each analysis module. The structural analyses 

modules such as computations of the structural weight, stress, deflection, buckling, and flutter 

and divergence speeds have been developed and incorporated into the object-oriented MDAO 

framework.

In general, obtaining aerodynamic influence coefficients (AIC) by direct calculation for 

integration into preliminary design activities involving disciplines such as aeroelasticity, 

aeroservoelasticity, and optimization is, at present, a costly and impractical venture (ref. 1). With 

the increasing complexity of the configuration and the increasing fidelity of the aerodynamic 

equations, the computational costs increase rapidly. The unsteady aeroelastic analysis and design 

optimizations are a challenging task. The time required for unsteady computations whether in 

frequency domain or time domain will considerably slow down the whole design process. Also, 

these analyses are usually performed repeatedly to optimize the final design. Even though the 

computational costs may be reduced by the use of advanced algorithms and improved computer 

hardware processing speeds, these full aeroelasticity analyses cannot be incorporated effectively 

within a preliminary design and optimization environment. For example, using the ZAEROTM 

(ZONA Technology Incorporated, Scottsdale, Arizona) code to generate the modal AIC matrix 

for the IKHANA (a modified Predator® B, General Atomics Aeronautical Systems, Incorporated, 

San Diego, California) aircraft with existing AIC data, one Mach number, and 16 reduced 

frequencies, takes an average of 30 min on an Intel® CoreTM2 2.80 GHz CPU computer (Intel 

Corporation, Santa Clara, California) (ref. 1). One of the examples presented in this paper, the 

hybrid wing body (HWB) aircraft, takes about 50 min to generate the modal AICs with existing 

AIC data.

As a result, there is considerable motivation to be able to perform aeroelastic calculations 

more quickly and inexpensively using basis function approximation (BFA) method. One of the 
3

primary goals behind the current development is to reduce the computation time for generating 

modal AIC matrices during the optimization procedure.

BASIS FUNCTION APPROXIMATION

The BFA method requires the unsteady aerodynamics to be represented in the frequency 

domain. In this study, the BFA for the subsonic and supersonic speeds are discussed. The flutter 

solution can be found by the classic methods, such as rational function approximation, k, p-k, p, 

root locus et cetera. A process that efficiently incorporates an approximated modal AIC matrix 

into the MDAO tool for design optimization at a reasonable computational cost has been 

developed and is outlined in the flowchart shown in figure 2.

In linear algebra, a basis is a set of vectors that in a linear combination can represent every 

vector in a given vector space or free module, and no element of the set can be represented as a 

linear combination of the others (ref. 2). In other words, a basis is a linearly independent spanning 

set.

This paper discusses an effective approach for approximating a modal AIC matrix for 

optimization with flutter speed constraints. Consider the steps of the approximation process 

depicted in the flowchart given in figure 2. Steps 1 and 2 are performed only once before starting 

the optimization process. Steps 3, 4, and 5 are done iteratively for aeroelasticity calculation.

In step 1, a set of representative basis functions Ψ is defined and intended to capture salient 

features of the modal responses the airplane is expected to encounter in the design space. 

Structural mode shapes obtained from different mass or various stiffness configurations of the 

airplane can be used as the basis functions. These basis functions are comparison functions

(ref. 3) since all the geometric and natural boundary conditions of the airplane are satisfied. Each 

mode shape of the airplane with the target configuration can be approximated as a linear 

combination of a set of the basis functions. The target configuration in this paper is simply an 

arbitrary design point within the design space and is used as a check case.

In step 2, the representative basis AIC matrices Q are computed corresponding to the 

representative basis functions defined in step 1 at any Mach number and reduced frequency. 

These basis AIC matrices are used as input for approximate modal AIC matrix calculation in step 

4.

In step 3, for a set of given design structural mode shapes φ , each mode shape is decomposed

in a linear combination of the basis functions. The i-th target mode shape vector φ (i=1, 2, ..., m, 

where m is the number of mode shapes) is approximated through the use of a least squares 

method of the representative basis functions as shown in equation (1),

φi ≈ βk

i

ψ k

k=1

n

∑ (1)
4

where a vector ψ k is the k-th basis function and a scalar coefficient βk

i is the modal participation 

factor of the k-th basis function on the i-th mode shape; n is the number of basis functions.

In step 4, an approximate modal AIC matrix Q is computed based on a basis AIC matrix Q

and modal participation factors. Let matrices Q, A, and Φ be a modal AIC matrix, a general AIC 

matrix, and a modal matrix, respectively. Then the modal AIC matrix Q can be defined as shown 

in equation (2),

Q = ΦT AΦ

where Φ = [φ1 φ2φm ] (2)

Therefore the i-th row and j-th column element of the modal AIC matrix Q can be written as 

shown in equation (3),

Qij = φi

T Aφ j (3)

Substituting equation (1) into equation (3) is given in equation (4),

Q ij = φi

T Aφ j ≈ βki

ψ k

T

k=1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

A βk

j

ψ k

k=1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ = Qij

Qij = βs

i

βr

j

ψ s

T Aψ r

s=1

n

∑ r=1

n

∑

(4)

The detailed derivations of equation (4) are listed in Appendix A. Let the basis AIC matrix Q be 

defined as shown in equation (5),

Q = ΨT AΨ (5)
5

then the s-th row and r-th column element of the Q can be written as shown in equation (6),

Q

sr =ψ s

T Aψ r (6)

Substituting equation (6) into equation (4) gives equation (7) for the approximate modal AIC 

matrix.

Qij = βs

i

βr

j

Q

sr

s=1

n

∑ r=1

n

∑ (7)

The basis AIC matrix Q in equation (5) can be computed and saved before starting 

optimization as mentioned in step 2. During optimization, the mode shapes, φi (i=1, 2, ..., m), are 

fitted using basis functions ψ k and modal participation factors βk

i as shown in equation (1), and 

the approximate modal AIC matrix Q can be computed using equation (7). This approximate 

modal AIC matrix will be used for flutter analysis in step 5. The resulting flutter speed and 

frequency could be used for optimization. In step 5, the flutter solution can be found by the 

classic methods, such as rational function approximation, k, p-k, p, root locus et cetera.

APPLICATIONS

In order to validate the proposed BFA technique in subsonic flight regimes, the flutter results 

using this approximate method are compared with the direct flutter results. First a simple model 

from the aerostructures test wing (ATW) 2 program was chosen. Subsequently, a complex 

practical problem HWB model was analyzed. In this study, structural mode shapes obtained from 

different mass configurations of the airplane are used as basis functions.

A Modified Aerostructures Test Wing 2

The proposed technique has been applied and validated using a modified ATW2. The original 

ATW2 (ref. 4) test article was actually designed, built, and flight-tested at NASA DFRC as 

shown in figure 3. This wing was cantilevered from a center station pylon and flown on the 

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

In this study, calculated structural mode shapes obtained from different mass configurations 

of the wing are used as basis functions. To create each different mass configuration, the mass at 

each mass point is represented using an additional concentrated mass element attached through 

rigid body elements to the finite element model as shown in figure 4. These alterations 

unintentionally increased the effective wing stiffness and the first natural frequency over that of 

the actual ATW2 configuration (ref. 5) in spite of the mass increase because of the rigid elements.

The total weight and center of gravity of the various fictitious and target ATW2 mass 

configurations are listed in table 1. The direct flutter analysis of the target configuration (mass = 

0.0 units) is performed at Mach 0.82 with 16 reduced frequencies using MSC/NASTRAN (MSC 
6

Software Corporation, Santa Ana, California) and ZAEROTM codes (refs. 6-7). In order to have 

better flutter solution convergence for flutter speed, the first 10 structural modes are included in 

the calculation. While preparing for the approximate flutter solution, three different mass 

configurations, 0.3, 0.6, and 0.9 units were selected for generating the basis functions.

In order to be able to capture more of the target mode shapes, larger basis functions were 

generated. Fifteen structural mode shapes for each mass configuration were generated using 

MSC/NASTRAN. The natural frequencies of the various fictitious and target ATW2 mass 

configurations are listed in table 2. According to Pak and Lung (ref. 5), the primary natural modes 

for the first flutter mode are mode numbers 1, 2, and 3; and all of the higher modes are the 

secondary modes. The first three natural frequencies and mode shapes are the first bending, the 

first torsion, and the second bending modes, respectively and are shown in figure 5. A total of 45 

basis functions were generated. Then the 45 by 45 basis AIC matrix corresponding to those 45 

basis functions was computed for each of 16 reduced frequencies at Mach 0.82 using ZAEROTM. 

The mode shapes of the target configuration are fitted using the basis functions together with a 

least squares method; and then the approximate modal AIC matrices were computed, and the 

flutter analysis was performed.

For the present test case, ATW2 mode shapes, generated from direct method and BFA 

method, are plotted in figures 5a, 5b, and 5c. The approximated mode shapes are very well 

matched to the target mode shapes. The difference of the first three mode shapes between these 

two methods is plotted in figure 6.

The results of the matched flutter analysis using the direct method and the BFA method are 

summarized in table 3. Three percent damping was used for the flutter speed and frequency 

computation. Both methods predicted the same flutter speed of 543.59 KEAS at a flutter 

frequency of 49.01 Hz. The speed versus damping, V-g, and speed versus frequency, V-ω curves 

from the direct and BFA methods at Mach 0.82 are given in figures 7 and 8.

Table 4 lists the computational cost of computing the modal AIC with a given structural 

mode shape at one Mach number using the direct method and the BFA method. The comparison 

was done on an Intel® CoreTM2 2.80 GHz CPU computer. The computational cost of generating 

the basis functions before the parametric study is not included in the comparison.

Hybrid Wing Body Vehicle

The modified ATW2 is a simple cantilevered wing model and a straight forward application. 

A hybrid wing body (HWB) aircraft (ref. 8) was selected for a second and more challenging 

demonstration of the BFA method.

In this HWB application, the basis functions are created based on three different total fuel 

conditions of the vehicle, configuration number 1, 2, and 3. The target model weight is bounded 

by the range of the weight of configurations 2 and 3. In the finite element model, the various fuel 

weights are modeled using a concentrated mass element attached directly to each of the node 

points in the fuel tank area as shown in figure 9 (ref. 9). The total weight and center of gravity 

location of the different fuel weight and target configurations are listed in table 5.

With target configuration, the direct flutter analysis is performed at Mach 0.50, with 40 

structural modes and 14 reduced frequencies using MSC/NASTRAN and ZAEROTM codes. A 

classical wing bending torsion type of flutter was predicted in this HWB example. The first 

fourteen flexible natural frequencies and the first four flexible mode shapes are listed in table 6 
7

and figure 10, respectively. The first four modes are the first symmetric and anti-symmetric 

bending, and the first symmetric and anti-symmetric torsion.

For the approximate flutter solution, a total of 150 basis functions are generated from 3 

different fuel configurations and 50 structural mode shapes for each fuel configuration using 

MSC/NASTRAN. Then the 150 by 150 basis AIC matrix corresponding to those 150 basis 

functions is computed for each of 14 reduced frequencies at Mach 0.50 using ZAEROTM.

The flutter boundary of the HWB at Mach 0.50 using BFA method is compared to that from 

the direct method. The flutter boundaries are summarized in table 7. The flutter speed is 

normalized with respect to direct method flutter speed. With 3-percent structural damping, the 

normalized flutter speed using the BFA method is 0.991, and the flutter frequency is 2.671 Hz. 

The percentage errors of the flutter speed and frequency are 0.86 percent and 0.15 percent. The 

speed versus damping, V-g, and the speed versus frequency, V-ω, curves of the HWB at Mach 

0.50 obtained from the direct and BFA methods are given in figures. 11 and 12. The BFA offers 

consistent flutter speed and frequency predictions on the HWB target configuration. See table 4 

for the computational cost of the HWB example. The computational cost of generating the basis 

functions and AIC is about 250 min in this HWB example.

CONCLUSION

A technique for approximating the modal AIC matrix by using basis functions has been 

developed and validated, and a process for using the resulting AIC matrix in aeroelastic analysis 

and design optimization has been proposed. The approximation method has been applied to the 

aeroelastic analyses, and the results are essentially identical to those using direct solution. The 

technique presented has been shown to offer consistent flutter speed prediction on an ATW2 

configuration and a HWB type vehicle with a negligibly small loss in precision. These results 

may have practical significance in the analysis of aircraft aeroelastic calculation and could lead to 

a more efficient design optimization cycle. The basis function approximation approach yields 

significant improvements in computational efficiency as compared to the original approach, 

thereby meeting the objective of this study.
8

TABLES

Table 1. Summary of total weight and center of gravity location for the ATW2 with different 

fictitious mass configurations.

Mass configuration 1 2 3 Target

Total weight, lb 4.046 5.250 6.450 2.850

X center of gravity, in 14.510 14.950 15.220 13.710

Y center of gravity, in -8.760 -9.010 -9.170 -8.290

Z center of gravity, in 0.000 0.000 0.000 0.000

Number of modes 15 15 15 10

Table 2. Summary of natural frequencies (Hz) of the ATW2 with different fictitious mass 

configurations.

Mode Configuration 1 Configuration 2 Configuration 3 Target

1 18.10 15.98 14.45 21.32

2 62.81 61.91 60.60 63.55

3 85.87 77.10 71.84 103.18

4 126.60 112.29 101.86 147.79

5 191.68 173.89 160.47 208.55

6 230.10 220.25 215.18 265.53

7 357.83 317.80 290.89 408.75

8 403.33 383.13 367.52 434.86

9 450.24 444.21 426.69 472.85

10 532.04 477.94 448.37 589.94

11 628.72 616.37 600.44

12 633.07 630.08 628.51

13 708.57 707.15 684.02

14 729.18 726.37 705.74

15 782.46 728.68 725.76

Table 3. Summary of the ATW2 flutter results comparison for BFA and direct method.

BFA method Direct method Error, %

Flutter speed, KEAS 543.59 543.59 0.0

Flutter frequency, Hz 49.01 49.01 0.0
9

Table 4. Computational cost comparison of BFA and direct method.

Test case ATW2 HWB

Number of nodes 269 ~24000

Number of structural modes 10 40

Number of reduced 

frequencies

16 14

Direct method elapsed time 43 s 50 min

BFA method elapsed time 8 s 9 min

Table 5. Different fuel weight configurations versus target fuel weight configurations of the 

HWB.

Mass configuration 1 2 3 Target

Total weight, lb 403781 478941 591681 535311

X center of gravity, in 1337.60 1334.51 1331.34 1332.76

Y center of gravity, in 0.75 0.46 0.16 0.29

Z center of gravity, in 20.89 21.35 21.82 21.61

Number of modes 50 50 50 40

Table 6. Natural frequencies (Hz) of the HWB with various fuel weight and target fuel weight 

configurations.

Mode Configuration 1 Configuration 2 Configuration 3 Target

1 1.60 1.54 1.47 1.50

2 1.74 1.64 1.53 1.58

3 3.52 3.35 3.13 3.24

4 3.64 3.45 3.22 3.33

5 5.24 5.00 4.54 4.83

6 5.26 5.04 4.54 4.89

7 6.42 5.42 4.68 4.92

8 6.42 5.43 4.76 4.92

9 6.92 6.66 5.81 6.31

10 7.12 6.68 5.82 6.31

11 7.33 6.95 6.11 6.39

12 7.65 6.95 6.32 6.48

13 8.22 7.03 6.69 6.85

14 8.23 7.40 7.06 7.22

Table 7. Summary of the HWB flutter results comparison for BFA and direct method.

BFA method Direct method Error, %

Flutter speed, normalized 0.991 1.000 0.86

Flutter frequency, Hz 2.671 2.667 0.15
10

FIGURES

Figure 1. Object-oriented multidisciplinary design, analysis, and optimization tool.

Figure 2. Flowchart of the flutter analysis module in object-oriented MDAO tool.
11

Figure 3. Aerostructures test wing 2.

Figure 4. Aerostructures test wing 2 finite element model with fictitious mass point locations.
12

Figure 5a. Mode 1 (First bending), 21.32 Hz.

Figure 5b. Mode 2 (Second bending and first torsion), 63.55 Hz.

Figure 5. Structural finite element model mode shapes and frequencies of the ATW2 with target 

mass configuration (direct method).
13

Figure 5c. Mode 3 (Second bending and first torsion), 103.18 Hz.

Figure 5. Concluded.

Figure 6. Structural finite element model mode shapes error of the ATW2 with target mass 

configuration.
14

Figure 7a. V-g plot.

Figure 7b. V-ω plot.

Figure 7. V-g and V-ω plots for the ATW2 at Mach 0.82 using direct modal AIC matrices.
15

Figure 8a. V-g plot.

Figure 8b. V-ω plot.

Figure 8. V-g and V-ω plots for the ATW2 at Mach 0.82 using approximate modal AIC matrices.
16

Figure 9. Hybrid wing body aircraft finite element model.

Figure 10. Hybrid wing body mode shapes.
17

Figure 11a. V-g plot.

Figure 11b. V-ω plot.

Figure 11. V-g and V-ω plots for the HWB at Mach 0.50 using direct modal AIC matrices.
18

Figure 12a. V-g plot.

Figure 12b. V-ω plot.

Figure 12. V-g and V-ω plots for the HWB at Mach 0.50 using approximate modal AIC matrices.
19

APPENDIX A

BASIS FUNCTION APPROXIMATION

= βk

i

ψ k

T

k=1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

A β1

j

ψ1 + β2

j

ψ 2 ++ βn

j ( ψ n )

= βk

i

ψ k

T

k=1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ Aβ1

j

ψ1 + Aβ2

j

ψ 2 ++ Aβn

j ( ψ n )

= βk

i

ψ k

T

k=1

n

∑

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟ β1

j

Aψ1 + β2

j

Aψ 2 ++ βn

j ( Aψ n )

= β1

i

ψ1

T + β2i

ψ 2

T ++ βni

ψ n

T ( ) β1

j

Aψ1 + β2

j

Aψ 2 ++ βn

j ( Aψ n )

(A1)
20

REFERENCES

1. Pak, Chan-gi, and Wesley Li, “Multidisciplinary Design, Analysis and Optimization Tool 

Development Using a Genetic Algorithm,” Proceedings of the 26th Congress of 

International Council of the Aeronautical Sciences, Anchorage, Alaska, 2008.

2. Kahaner, David, Cleve Moler, and Stephen Nash, Numerical Methods and Software, 

Prentice-Hall, Englewood Cliffs, New Jersey, 1989.

3. Meirovitch, Leonard, Analytical Methods in Vibrations, Macmillan Series in Applied 

Mechanics, The Macmillan Company, New York, 1967.

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

Test Wing Using Ground Vibration Test Data,” AIAA-2009-2528, Proceedings of the 50th 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 

Palm Springs, California, 2009.

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

the Aerostructures Test Wing,” Proceedings of the 27th Congress of International Council 

of the Aeronautical Sciences, Nice, France, 2010 (to be published).

6. ZAERO Theoretical Manual, ZONA Technology, Incorporated, 2007.

7. MSC/NASTRAN 2004 Quick Reference Guide, MSC.Software Corporation, Santa Ana, 

California, 2004.

8. Felder, James L., Hyun Dae Kim, and Gerald V. Brown, “Turboelectric Distributed 

Propulsion Engine Cycle Analysis for Hybrid-Wing-Body Aircraft,” AIAA-2009-1132, 

Proceedings of the 47th AIAA Aerospace Sciences Meeting, Orlando, Florida, 2009.

9. Velicki, Alex, “Damage Arresting Composites for Shaped Vehicles - Phase 1 Final Report,” 

NASA/CR-2009-215932, 2009.
REPORT DOCUMENTATION PAGE Form Approved

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1. REPORT DATE (DD-MM-YYYY)

01-07-2010

2. REPORT TYPE 

Technical Memorandum

 4. TITLE AND SUBTITLE

Application of Approximate Unsteady Aerodynamics for Flutter Analysis

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Chan-gi Pak and Wesley W. Li

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

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 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-3058

10. SPONSORING/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

Pak and Li, NASA Dryden Flight Research Center. Also presented as AIAA-2010-3085 at the 51st AIAA/ASME/ASCE/AHS/ASC 

Structures, Structural Dynamics, and Materials Conference, Orlando, Florida, April 12-15, 2010.

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 02 Availability: NASA CASI (443) 757-5802 Distribution: Standard

19b. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

A technique for approximating the modal aerodynamic influence coefficient (AIC) matrices by using basis functions has been 

developed. A process for using the resulting approximated modal AIC matrix in aeroelastic analysis has also been developed. The

method requires the unsteady aerodynamics in frequency domain, and this methodology can be applied to the unsteady subsonic, 

transonic, and supersonic aerodynamics. The flutter solution can be found by the classic methods, such as rational function 

approximation, k, p-k, p, root locus et cetera. The unsteady aeroelastic analysis using unsteady subsonic aerodynamic approximation 

is demonstrated herein. The technique presented is shown to offer consistent flutter speed prediction on an aerostructures test wing 

(ATW) 2 and a hybrid wing body (HWB) type of vehicle configuration with negligible loss in precision. This method computes 

AICs that are functions of the changing parameters being studied and are generated within minutes of CPU time instead of hours. 

These results may have practical application in parametric flutter analyses as well as more efficient multidisciplinary design and 

optimization studies.

15. SUBJECT TERMS

Aerodynamics influence coefficient matrix, Basis function approximation, Flutter analysis, Least squares method, Multidisciplinary 

design optimization

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