Developing Uncertainty Models for Robust 

Flutter Analysis Using Ground Vibration 

Test Data

Starr Potter and Rick Lind

NASA Dryden Flight Research Center

Edwards, California

April 2001
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April 2001

National Aeronautics and

Space Administration

Dryden Flight Research Center

Edwards, California 93523-0273

Developing Uncertainty Models for Robust 

Flutter Analysis Using Ground Vibration 

Test Data

Starr Potter and Rick Lind

NASA Dryden Flight Research Center

Edwards, California
NOTICE

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Space Administration.

Available from the following:

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(301) 621-0390 (703) 487-4650

ACKNOWLEDGMENTS

The authors would like to thank Cliff Sticht for his invaluable contributions to the Aerostructures Test Wing (ATW) project. He was instrumental in designing the system and developing an

associated finite-element model. Additionally, he provided the final updates that were used to

match the analytical model with the data. We would also like to acknowledge Tim Doyle and Roger

Truax for their support with finite element model updating and flutter analysis. We also thank Mike

Kehoe for his ideas towards nonlinearities in ground vibration testing. We greatly appreciate the

draft comments provided by Ralph Brillhardt.
1

American Institute of Aeronautics and Astronautics

*Aerospace engineer, starr.potter@dfrc.nasa.gov.

†Aerospace engineer, rick.lind@dfrc.nasa.gov, Member, AIAA.

Copyright 2001 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States under

Title 17, U.S. Code. The U.S. Government has a royalty-free license

to exercise all rights under the copyright claimed herein for Governmental purposes. All other rights are reserved by the copyright owner.

AIAA 2001-1585

DEVELOPING UNCERTAINTY MODELS FOR ROBUST FLUTTER 

ANALYSIS USING GROUND VIBRATION TEST DATA

Starr Potter* and Rick Lind†

NASA Dryden Flight Research Center

Edwards, California

Abstract

A ground vibration test can be used to obtain

information about structural dynamics that is important

for flutter analysis. Traditionally, this information—

such as natural frequencies of modes—is used to update

analytical models used to predict flutter speeds. The

ground vibration test can also be used to obtain

uncertainty models, such as natural frequencies and

their associated variations, that can update analytical

models for the purpose of predicting robust flutter

speeds. Analyzing test data using the -norm, rather

than the traditional 2-norm, is shown to lead to a

minimum-size uncertainty description and,

consequently, a least-conservative robust flutter speed.

This approach is demonstrated using ground vibration

test data for the Aerostructures Test Wing. Different

norms are used to formulate uncertainty models and

their associated robust flutter speeds to evaluate which

norm is least conservative.

Nomenclature

ATW Aerostructures Test Wing

FEM finite-element model

FRF frequency-response function

FTF flight test fixture

GVT ground vibration test

KEAS equivalent airspeed, knots

MAC modal assurance criterion

n number

POC pseudo-orthogonality check

v estimated mode shape

Vnom nominal flutter speed

Vrob robust flutter speed

scalar

uncertainty operator

Æ set of uncertainty operators

uncertainty parameter

analytical mode shape

set of frequency parameters

frequency

Introduction

Structural dynamics often are an important

consideration when evaluating system characteristics. A

concern related to structural dynamics is the analysis of

aeroservoelasticity, and particularly the phenomenon of

flutter, for a flight vehicle. The instability associated

with flutter can be quite sensitive to the structural

dynamics; therefore, analysis of robustness with respect

to uncertainty is becoming increasingly important for

the flight test community. Such an analysis is used to

increase flight safety and efficiency factors for envelope

expansion of experimental testing.

A ground vibration test (GVT) is commonly

performed as a preflight check that attempts to validate

the quality of an analytical model and its associated

flutter predictions.1 The basic concept is to excite the

structural dynamics and measure responses at locations
2

American Institute of Aeronautics and Astronautics

throughout the vehicle. The responses are then

analyzed to estimate a set of modal properties for a

linear model that corresponds to the observed data.

These modal properties, such as natural frequency and

mode shape, are compared with an analytical model to

ascertain the error and uncertainty in that model.

The modal estimates from a GVT are used to update

analytical models; however, updating is often not a

straightforward procedure. An accurate procedure for

model updating is not yet a solved problem. Several

approaches have been formulated to address particular

issues of the problem, but fundamental limitations exist

to all these approaches. The general difficulties

associated with model updating based on measurement

data are well-known to the flight test community.2

The concept of uncertainty recently has been

receiving attention with respect to model updating.

Ground vibration test data must be carefully analyzed

because many variations exist in the data as a result of

inherent properties of vibration testing.3 Such

variability naturally arises from both the process of

generating data and the process of analyzing data.4 Test

procedures can be formulated to reduce the variability,5

and orthogonality checks can be conducted during

testing to ensure data quality;6 however, some variations

in measured data will always exist. These variations

must be properly treated to ensure that any modal

parameters extracted from the data accurately reflect the

properties of the structure.

The most common approach to treat variations in

experimental data is to consider types of measurement

error. One method is to find parameter estimates, or

corrections to existing parameter estimates, by

minimizing the 2-norm of an error residual based on

data with assumed 2-norm bounded measurement

uncertainties.7 A similar method considers the effect of

measurement uncertainties that are modeled using fuzzy

set theory.8 Another approach is to consider the

clustering of pole estimates to minimize the effect of

measurement variation on resulting pole estimates.9

Methods for data analysis that use mode shape

information also exist. In particular, an approach has

been formulated that attempts to determine a mode

shape to within limits of assumed measurement error.10

This method perturbs a model with variations that are

anticipated to occur as a result of imprecise testing. A

model is considered to be accurate to within

measurement error if the modal assurance criterion

(MAC) of the experimental mode shapes lies within

range of the corresponding criterion for the perturbed

mode shapes.

This paper discusses an alternative method of

considering variations in data from GVTs. Essentially,

the principle here is that no single best answer exists;

instead, the objective is to determine an optimal range of

possible modal parameters as a result of variation of the

data. This approach assumes that determining an exact

set of modal parameters from GVT data is not possible,

or even mathematically valid. Because inherent

nonlinearities and complexities in the structure will

always introduce some level of variation of the data, a

linear model should always consider that variation.

The approach adopted in this paper is directly related

to a particular method of flutter analysis. Linear models

are formulated by including uncertain parameters in the

dynamics. These parameters are represented by a

best-guess nominal value and an associated set of

norm-bounded perturbations. A robust analysis is

performed on the model by computing worst-case flutter

speeds with respect to the uncertain parameters.11 The

objective of this paper is to determine the smallest

norm-bounded perturbations based on GVT data that

result in the least-conservative robust flutter speeds.

The fundamental principles put forth in this paper

previously have been recognized as an inherent part of

analysis. For example, a concept of choosing methods

for data analysis by considering fundamental issues has

been discussed in the context of robust reliability for

decision making.12 Essentially, the analysis of

uncertainty must be linked with the need to use the

result. Similarly, the concept in this paper is to

formulate an uncertainty description by considering the

need to perform a flutter analysis with respect to that

description.

Norm-Bounded Uncertainty

The concept of uncertainty is an intuitively obvious

notion as an expression of error and variations in models

and data; however, expressing this notion as a

mathematical formulation is somewhat more difficult.

Several types of mathematical constructions have been

proposed to represent uncertainty. These constructions

include formulations that reflect properties such as

statistical distributions or deterministic measures. A

construction should be chosen that accurately reflects

the fundamental nature of the uncertainty to be

described and also is suitable for the associated

robustness analysis to be performed.
3

American Institute of Aeronautics and Astronautics

Analysis of GVT data to develop uncertainty

descriptions is actually performed for the eventual goal

of computing robust flutter speeds. These speeds are

computed using a mathematical framework based on

 theory; therefore, the uncertainty should be

represented as an element in the framework.11

Thus, the form for the uncertainty is chosen as a set of

stable perturbations formally described as an operator

set with finite –norm. This type of uncertainty is

sometimes noted as sector or interval uncertainty.

The general form used as an uncertainty description is

to formulate a set, Æ, that is composed of elements

denoted as such that . Each element is a

frequency-varying transfer function that is required to

be stable and rational. Furthermore, each element

describes a multiple-input and multiple-output system

with complex elements.

The range of dynamics admitted into models by the

general form of uncertainty is often more broad than is

justified by the nature of the errors and variations. For

example, physical parameters inherently are real

numbers, so allowing complex variations to these

parameters seems unrealistic. The analysis of

robustness with respect to uncertainty in the 

framework can be overly conservative when considering

the general form of uncertainty; therefore, specific

subsets of the generalized set of operators are used.

The uncertainty description associated with structural

dynamics for the analysis of robust flutter speeds is

restricted to affecting the modal parameters of natural

frequency and damping. These parameters inherently

are real scalars; therefore, the uncertainty is restricted to

being real scalars. Furthermore, the variations in each

modal parameter are assumed to be independent, so

different scalars will be associated with each parameter.

The basic description of the uncertainty associated with

a modal parameter is given as and depends on some

real scalar, , greater than 0 to provide a norm bound:

(1)

An obvious feature of this type of uncertainty is the

numerical distribution of the operators. Specifically, the

magnitudes of the operators are symmetrically

distributed around 0, which implies that no bias towards

either direction, positive or negative, is allowed in the

uncertainty. Another feature is the lack of an associated

probability distribution. Such distributions are included

with descriptions of uncertainty for the purpose of

computing statistical robustness.

Statistical robustness analysis typically considers an

uncertainty description that relates the probability of

operators occuring with all magnitudes. The -norm

approach is quite different from the statistical method in

that this approach considers only operators within a

limited range of magnitude. Furthermore, this approach

assumes any operator within this range is equally

probable and should be analyzed.

Remembering that the uncertainty description is being

developed foremost for robust flutter analysis is

important. Many types of uncertainty descriptions, such

as biased statistical distributions, are not suitable for this

application; therefore, this paper focuses on the

-norm approach.

Ground Vibration Testing

Variability is inherent in GVT data. Some causes for

variability of the data in the GVT hardware setup are

discussed, as well as its effect on modal parameter

estimation.

Variability in Generating Data

A GVT is used to estimate the properties of the

structural dynamics before flight operation. A GVT is

performed by exciting the aircraft through the

application of commanded energy and then measuring

the structural motion in response to that energy. For

example, a common method of testing aircraft is to

connect vibration shakers to the fuselage and wings and

measure the responses with accelerometers installed

across the vehicle. The data from this testing are

analyzed to estimate properties associated with modes

such as natural frequencies and the corresponding mode

shape. The estimation uses optimization and curve-fit

routines to analyze GVT data; therefore, the resulting

modal parameters can be overly influenced by

variability in the data. Methodical setup and operation

of test can help minimize, but never entirely eliminate,

such variability. 

Variability in data can manifest in several forms. For

example, aircraft testing often generates

frequency-response functions (FRF). The variability in

data associated with an FRF can be noted by frequencies

at which peaks occur. For example, different

accelerometers may indicate different frequencies at

which a bending mode occurs. This variation can be

exaggerated when data are collected at different times or

by using moving or roving transducers. Other types of

variability in FRF data can be noted as differences in

amplitudes and damping curves. Amplitude variations

can occur as the result of FRF processing techniques or

H

H

H

Æ

H

= { : R, }
4

American Institute of Aeronautics and Astronautics

simply because of transducer calibration variability.

Damping variation can be the result of signal

processing, including mathematical window application

to the data to prevent Fourier transform leakage.

One potentially significant source of variability in

GVT data is the process of exciting the structure.

Vibration exciters, also referred to as shakers, are

commonly used for aircraft testing, but the resulting

data are particularly susceptible to variability as a result

of poor setup of the shaker. The shaker armature, which

is the interface from the shaker to the test structure, can

misalign and cause a bending moment input to the force

transducer. The effect of this moment is an incorrect

measurement of excitation energy and, correspondingly,

variability in the amplitude and frequencies associated

with peaks in the FRF.

Excitation for vibration testing also can be provided

by the use of impact hammers. Different tips and

masses are used on impact hammers to control the input

frequency spectrum. Increasing the mass on the impact

hammer will increase the input force while decreasing

the frequency content. Changing the tip stiffness will

change the frequency range and peak force of

excitation. If the combination of mass and tip stiffness

are inadequate, causing a zero frequency component in

the frequency range of interest, any measured output

acceleration frequency spectrum will not clearly

indicate a resonance condition when one is actually

present.13

The impact test technique can also result in a poor

signal-to-noise ratio in the responses of the input and

the output. Following the impulse, background noise

may become as significant as the signal obtained from

the measurement transducers. If the noise level is high

enough, the input frequency spectra could result in a

frequency shift and peak variation. Impacting a lightly

damped structure can cause the output acceleration to

continue for much longer than the time sample window.

The signal truncation leads to filter leakage. This

mismatch in frequency between signal and analyzer

frequency components creates magnitude and slope

discontinuities in the sample function.13

A GVT of certain types of vehicles, such as

lightweight structures, requires careful attention to

ensure that high levels of variability in the data do not

result. One problematic issue is the effect of introducing

relatively significant amounts of mass by installing

sensors. This issue can be avoided by using a small set

of stationary sensors but applying excitation from an

impact hammer at various locations.

A susceptibility to nonlinearities potentially exists if

one accelerometer is used and the impact hammer

roams. Questions may exist as to stiffness of joints

between components when test points straddle these

joints. This problem is particularly evident with certain

types of structural elements. For example, consider a

bonded joint of a composite structure. This joint

probably is not purely linear. If one accelerometer is

placed at the tip of a composite test structure and the

hammer roams across several bonded joints, that input

pulse can be translated through each joint differently

and the accelerometer response will show a frequency

shift from one test point to the other.

Nonlinearities are an inherent part of the dynamics for

any structure. These nonlinearities should be

considered as a potential source for variability in the

data in any test. Sources of nonlinear behavior could

include springs, damping elements, boundary

conditions, fluid–structure interaction, mechanical

backlash in mating parts, and nonlinear electric and

magnetic forces. A wide range of variability in the data,

such as frequency and amplitude shifts in peaks, can

result from these nonlinearities.

Variability in Analyzing Data

The purpose of a GVT is to estimate a set of

properties, such as natural frequencies and mode

shapes, of a structure by analyzing GVT data. Checks

for coherence, reciprocity, force spectrum, and FRF

quality are evaluated from the data before parameter

estimation; however, these data always have some level

of variability from testing issues as discussed in the

previous section. Assuming the initial checks indicate

the data are reasonable, the FRF data are processed

using a variety of techniques that allow development of

a modal model containing the mode shapes and

associated frequencies. The modal parameter estimation

routines are mathematical optimizations that introduce

another level of variability that should be considered in

the results from a GVT.

“Modal parameter estimation” refers to applying

mathematical procedures for extracting eigenvalues and

eigenvectors from the measured structural response. If

FRFs are not consistent in the description of the poles

because of variability in the test setup, applying a modal

parameter estimation technique could magnify

frequency and damping shifts more than are seen in the

measured FRF. Many techniques are available for modal

parameter estimation. The modal parameter estimation

technique is selected based on the data quality and type

of structural response measured. Numerous options
5

American Institute of Aeronautics and Astronautics

exist in selecting which modal parameter estimation

technique is used to fit the measured data and whether

time- or frequency-domain techniques are used.

Sometimes multiple extraction techniques can be

applied to the same data to determine how much

variation occurs in the extraction.

Several techniques are conceptually similar and are

sometimes grouped into categories of local, global, and

polyreference approaches. The techniques considered

in this paper are based on analysis of frequency-domain

FRF data; however, many approaches are based on

analysis of time-domain data.

The concept of a local curve fit is to analyze an

individual FRF measurement and extract natural

frequencies of observed modes. Corresponding mode

shapes are then determined by noting the magnitude of

vibration at that modal frequency at different test points.

The concept of a global curve fit is to simultaneously

extract the natural frequency and mode shape associated

with modes by analyzing all or a subset of available

FRF measurements. These techniques compute the

modal properties in reference to a single FRF that

represents the excitation. The polyreference approaches

are somewhat similar in nature to the global approaches;

however, they analyze FRF measurements using

multiple references rather than a single reference. Each

approach has strengths and weaknesses, so analyzing

data using several approaches is common.

The modal parameters estimated from measured data

naturally will have some associated variability.

Analyzing an FRF using a particular technique will

result in a particular modal parameter; however,

analyzing that same FRF using a different technique

probably will result in a different modal parameter.

Issues such as optimization method, norm choices for

performance indices, and numerical conditioning and

stability associated with different methods will affect

the parameter estimate. Thus, variability of the data is

inherent to the analysis of an FRF measurement.

The issue of variability can become more pronounced

when considering multiple FRF measurements as

compared to a single measurement. Variability will

certainly exist when analyzing the same set of FRF

measurements using different techniques; however,

additional variability may result from having choices of

sets of FRF measurements. For example, analyzing

different subsets of measurements can result in different

parameter estimates even if the subsets are analyzed

using the same technique. Obviously, different subsets

will contain slightly different measurements because of

observability, noise, and structural nonlinearities, so the

analysis of these different measurements will estimate

different parameters.

The amount of variability in estimates of mode shape

can be easily quantified by computing a MAC.14 The

MAC is a good tool for checking similarity between

modal vectors. This check is performed by computing

the least square deviation with respect to the 2-norm of

the vectors. Consider mode shapes, v1 and v2, that are

estimated using different techniques. The MAC

associated with these shapes is computed as M:

(2)

The value of the MAC is a scalar that ranges between

0 and 1. The MAC will have a value of 1 if the mode

shapes are the same and a value of 0 if the shapes are

orthogonal. Similarly, a MAC matrix can be computed

to compare sets of mode shapes that are estimated from

different approaches. This matrix will have a value of 1

for a diagonal element if the methods compute an

identical mode shape, and a value less than 1 if variation

exists in the shape estimates.

Variability will always exist in modal parameter

estimates; however, the MAC is used to reduce

variability during a GVT by providing a quick check on

the data. For example, a value greater than 0.9 on the

diagonal and 0.05 on the off-diagonal elements

generally should be achieved for correlated modes.

Thus, if test data are analyzed and the MAC indicates

these desired values are not obtained, then the test can

be conducted again or the procedure and setup can be

altered for a new test. Conversely, a MAC matrix that

shows sufficient correlation between estimates indicates

the test can be considered successful because, although

the modal parameter estimates show variation, this

variation is as small as can be reasonably expected.

Developing Uncertainty Models

The final objective of the preflight testing considered

in this paper is to determine the aeroservoelastic and

flutter stability properties.11 The GVT helps achieve this

objective by generating information that describes the

structural dynamics and is used in predicting the

stability properties. Thus, the data from a GVT should

be analyzed with consideration for this final objective.

Flutter analysis of a flight vehicle is performed by

computing the stability properties of a linear model

assumed to represent the dynamics of that vehicle. A

GVT generates data, such as estimates of the natural

M(v1, v2 ) [v1 ]

T

( [v2 ])

2

[v1 ]

T

[v1 ][v2 ]

T

= ⁄( [v2 ])
6

American Institute of Aeronautics and Astronautics

frequencies of structural modes, that can be used to

update these models. The process of model updating,

even for low-order models, is challenging because of

difficulties in computing the modal parameters that

represent the dynamics. The data from a test, as

outlined in the previous section, have an inherent level

of variation that affects the complication.

Consider the problem of computing the optimal value

of natural frequency from a set of estimates with

variation. Assume that this optimal value for frequency,

 is to be formulated using a norm approach on the

data given by . Furthermore, assume that

the values are arranged in value of magnitude such that

 The most common approach is to

compute a value for , that is optimal in that it

minimizes the functional of .

(3)

The concept of optimal for the parameter estimation

is inherently tied to the concept of a norm for this type

of computation. Essentially, different values for will

be optimal with respect to different norms used in the

functional. The standard norms that are used are the

1-norm, 2-norm, and the -norm. Deriving the value

for that minimizes the functional for each type of

norm is straightforward (table 1).

Table 1. Optimal values of frequency with respect to 

different norms.

Traditional estimates from a GVT use the 2-norm

functional to choose the optimal value of natural

frequency. These estimates are reasonable choices to

updates models used for traditional flutter analysis;

however, the -norm functional can easily be shown to

be the best choice when considering robust flutter

analysis.

Consider the representation of natural frequency for a

model in the framework for robust flutter analysis. This

framework considers the natural frequency parameter

actually to be represented as a set of frequency

parameters, . The set can be described by a nominal

value, , and a set of perturbation operators, Æ, in an

additive relationship:

(4)

The uncertain model, unlike the traditional model,

requires two parameters, and , to be chosen from

the GVT data. This data analysis problem is actually

quite different from the traditional problem that was

solved by minimizing . Essentially, robust flutter

analysis is not predominately concerned with stability

of the modes as represented by ; instead, the analysis

considers the stability with respect to the modes as

represented by all values of . Thus, the data analysis

must address a functional that depends on the set of

natural frequencies in .

The fundamental nature of robust flutter analysis is to

compute a worst-case margin for stability with respect

to an uncertainty description. This margin can be highly

sensitive to the size of the uncertainty; therefore, the

uncertainty description should be as small as possible to

reduce conservatism in the robustness analysis. This

goal immediately presents the functional that should be

used for analysis of GVT data: namely, the data should

be analyzed to compute a natural frequency with the

smallest associated uncertainty description that

consequently results in the least-conservative robust

flutter speed.

The set, , is used to represent the natural frequency

and its uncertainty. Thus, the optimal value of is to

choose as the range of measurements in the GVT

data. The range of should match, but not

exceed, the range of variability noted in the data. Any

values of that lie outside the range of data

measurements would constitute extra uncertainty and

cause the flutter speed to be overly conservative.

Conversely, any data measurements that lie outside the

range of would not be considered in the robustness

analysis and cause the flutter speed to be suspect.

The main concept used to compute the parameters,

, and , of the optimal is to note the symmetric

nature of this set. The values of are determined by a

symmetric set of norm-bounded uncertainties that affect

a nominal value. Such an uncertainty description does

not have any bias in the sense that the set, Æ, contains

the same number of small perturbations and large

Norm

1 Median ( , ..., )

2 Mean ( , ..., )

1/2 ( + )

,

1

 ..., n{ , }

1

... n.

f ( )

f ( )

1

–

.

.

.

n

–

=

1 n

1 n

1 n

= { + : Æ}

f ( )
7

American Institute of Aeronautics and Astronautics

perturbations. This symmetric nature, coupled with the

desire to have match the range of data, indicates that

the optimal value of is in the middle of the data

measurements and is one-half the size of the data

variations. More specifically, can be computed using

the smallest and largest values, and , of the set of

measurements:

(5)

The range of clearly matches the range of the data

measurements. Also, the value of that results in this

optimal range of is the value that results from using

the -norm as noted in table 1. The -norm, rather than

the traditional 2-norm, is thus shown to be the optimal

choice to analyze GVT data for robust flutter analysis.

A small example serves to show that computing 

using an -norm approach results in the smallest range

for Assume that four data estimates are to be

considered as possibly representative of the true natural

frequency. These estimates are Table

2 shows the nominal value and uncertainty description,

along with the resulting range given by , for different

norm functionals.

Table 2. Analysis of example data.

The range of the uncertain parameter, , is smallest

when analyzing the data using the -norm functional.

Note that this result is in contrast to traditional methods

that always compute the optimal parameter using a

weighted 2-norm approach. The important

consideration here is that the data are not being analyzed

to compute the optimal flutter solution; rather, the data

are being analyzed to compute the least-conservative

robust flutter solution.

Experimental Test Bed

An experimental test bed has been used to examine

the flutter prediction methods of the flutterometer. The

following sections examine the construction of the test

bed, the GVT, modal analysis, data correlation, the

uncertainty model, and robust flutter analysis.

Description

A structure called the Aerostructures Test Wing

(ATW) is being used at NASA Dryden Flight Research

Center (Edwards, California) as a test bed for

investigating preflight and on-line methods of predicting

the onset of flutter. The ATW operates in a realistic

flight environment by using an F-15 aircraft and an

associated flight test fixture (FTF). The FTF is a host

carrier that rests under the center of the fuselage behind

the engine intakes (fig. 1).

Figure 1. F-15 aircraft with flight test fixture.

The ATW mounts horizontally to the side of this carrier

for flight tests. Figure 2 shows the ATW.

Figure 2. Aerostructures Test Wing.

The ATW has three main components: a body, an

aileron, and a boom. Because testing a wing that is

more realistic and complex than solid or hollow models

Norm

1 20.50 4.50 [16.00, 25.00]

2 21.25 3.75 [17.50, 25.00]

22.00 3.00 [19.00, 25.00]

1 n

:

n 1

( + ) 2

n 1

= { + = ⁄ , ( – )⁄2}

.

{19, 20, 21, 25}.

Photograph courtesy of Starr Potter.
8

American Institute of Aeronautics and Astronautics

is desired, the body is designed with an internal rib and

spar structure (fig. 3).

Figure 3. ATW internal assembly.

The aileron will allow active control of the system;

however, it has been glued in the faired position for the

purposes of the initial flutter testing. The boom is a

hollow tube with lumped masses that induces a flutter

instability at a desired flight condition.

The ATW was constructed to have a NACA-65A004

airfoil shape. The dimensions were determined by

balancing desires to make the test bed large enough to

be useful with consideration of safety-of-flight

restrictions associated with flight operations. Table 3

shows ATW dimensions.

Table 3. Parameters of the ATW.

Note the positive value of 3 wash-in at the tip of the

wing (table 3). The original design called for 3 of

washout at the tip; however, construction errors resulted

in a positive angle of twist. This error caused some

concern regarding aerodynamic and divergence issues,

but these concerns were alleviated by mounting the

entire ATW to the FTF at an angle of –4 . This angle

will minimize the steady static aerodynamic loads for

bending and torsion of the ATW measured at the wing

root and the spar centerline.

The materials used to construct the ATW were chosen

to satisfy safety-of-flight issues for the F-15 airplane

and the FTF. In particular, concern existed that a heavy

piece of metal could damage the aircraft system if the

ATW undergoes a flutter. This concern was addressed

by building the ATW out of lightweight materials with

no metal pieces. Composites and fiberglass were used

for many elements, and foam was used for the internal

core. The mass in the boom is provided by powdered

metal to minimize risk if the boom strikes the aircraft.

Table 4 shows the materials used in the ATW

construction.

Table 4. Materials used in the ATW.

The mass of the ATW is an issue that is closely related

to the materials. The onset of a flutter instability could

cause the ATW to break near the root and allow the

entire ATW to hit the F-15 airplane. The materials of

the ATW are designed to be frangible and light, so this

occurrence would have minimal damage. Table 5 shows

the weight of the ATW and associated mass used for

balancing.

Table 5. Mass properties of the ATW.

A measurement and excitation system has been

incorporated into the ATW. The measurement system

consists of 16 strain gages and 2 accelerometers placed

throughout the wing and boom assembly. The

excitation system consists of several patches of lead

Parameter Value

Airfoil NACA-65A004

Half span 18 in.

Root chord 13.2 in.

Tip chord 8.7 in.

Semispan area 197 in2

Aspect ratio 3.28

Span 20 percent ( )

Twist +3 wash-in at tip

Taper ratio 0.659

Sweepback at quarter chord 45

±10°

°

°

°

°

Elements Materials

Skin, spar web, aileron Fiberglass cloth and epoxy

Spar cap, boom Carbon fiber and epoxy

Wing core Rigid foam

Mass balance material Powdered tungsten carbide

Measure Weight, lb

Mass balance 0.65

Structure 2.00

Total 2.65

°
9

American Institute of Aeronautics and Astronautics

zirconate titanate (piezoelectric) materials that are used

to introduce sinusoidal sweeps of energy across a wide

range of frequencies.

A finite-element model has been formulated to

represent the ATW dynamics. The model is composed

of 446 elements. This initial model has been generated

from theoretical analysis of the wing design and does

not include any updates from the GVT. Table 6 shows

the primary modes of interest of the structural dynamics

and their associated frequencies for this initial model.

Table 6. Modes of the ATW model.

The flutter mechanism for the ATW is somewhat

complex. The first bending and torsion modes are

coupling to provide flutter; however, a contribution from

a third mode also exists, composed of a coupled bending

and torsion dynamic, that needs to be included in the

analysis. Computational approaches, such as p-k

analysis, predict flutter to occur at a flight condition of

433 knots equivalent airspeed (KEAS) and an altitude of

10,500 ft at Mach 0.8.

Ground Vibration Tests

For ground testing, the ATW was rigidly mounted by

clamping the wing root into the FTF plate, which was

then bolted to a steel ground test fixture. Three reference

accelerometers were used to measure the structural

response. Two accelerometers were permanently

mounted internally to the boom near the leading and

trailing edge of the ATW, and a third was mounted using

wax on the ATW surface centered on the nonfunctional

aileron. A calibrated impact hammer with a metal tip

and an added 0.065-lb mass was used to excite the

structure. Several types of tips were tested; however,

the metal tip excited the input frequency spectrum best

through the third mode of 77 Hz. A shaker was not used

for this test because the instrumentation to support a

shaker test increases the weight of the wing enough to

shift the modal frequency.

A total of 35 test points were excited with the impact

hammer and recorded at all three accelerometer

locations (fig. 4).

Figure 4. Locations of test points and sensors on the

ATW.

These test point locations were at the leading and

trailing edges, forward and aft of the spar, at the wing

root and wing-to-boom joint, at accelerometer locations,

and at each end of the boom near the loose mass. A

large number of test points were used to ensure the GVT

was able to correlate the mode shapes across the entire

structure.

The data acquisition system was set up for a 200-Hz

bandwidth window and 0.125-Hz resolution. No

windows were used on the excitation input or on the

response output because the structure is highly damped.

Three responses were averaged to reduce the effects of

any distortion in the recorded data.

A reciprocity test is commonly performed on a

structure to ensure the GVT is properly set up. This test

notes the transfer functions between two points on a

structure by alternating the locations of excitation and

response. Checking the reciprocity between the aft

boom accelerometer (test point 518) and the wing

accelerometer (test point 300) (fig. 4) showed

differences in magnitude for the second and third mode

and differences in damping for the third mode (fig. 5).

Mode Description Frequency, Hz

1 First bending 14.50

2 First torsion 23.34

3 Second bending or torsion 84.65
10

American Institute of Aeronautics and Astronautics

Figure 5. Transfer functions between test points 518 and

300 on the ATW.

The solid line represents the transfer function from the

acceleration at point 300 to an impact excitation at point

518. The dashed line represents the transfer function

from the acceleration at point 518 to an impact

excitation at point 300. The functions are similar near

the first mode; however, notable differences exist near

the second and particularly the third mode. These

differences are the first indication that an inconsistency

exists because of the wing-to-boom joint or the effect of

the loosely packed tungsten being located near the aft

accelerometer. 

Such examples of poor reciprocity are not entirely

uncommon for complicated structures when using

single-point to single-point analysis. The use of

multiple-point excitation with multiple-point sensors

can alleviate some of the effects of poor reciprocity.4

Thus, data analysis will consider both single- and

multiple-point approaches.

Modal Analysis

Figures 6–8 show the variance in the FRF at different

test locations. The appendix shows visually chosen

frequency peaks for the first, second, and third modes of

29 test points measured by each accelerometer. Figures

6–8 show 29 measured FRFs, referenced from the

accelerometer at point 518, to show scatter in the curves

as a function of structure location. These figures

graphically demonstrate the variability of the numerical

values of the handpicked frequency peaks. Every FRF

test point has nearly the same frequency for the first and

second modes, which draws the attention to the high

variation in the third mode. Figure 6 shows five test

points on the boom have a frequency variation from

78.94 to 85.00 Hz.

Figure 6. FRFs for five impacted boom test points

recorded by the accelerometer at point 518.

Figure 7 shows nine test points on the nonfunctional

aileron have a frequency variation from 77.69 to 80.44

Hz for the third mode.

Figure 7. FRFs for nine impacted aileron test points

recorded by the accelerometer at point 518.

Figure 8 shows a representative subset of test points on

the ATW body, less the nonfunctional aileron and boom,

that has a frequency of 77.125 Hz for the third mode.
11

American Institute of Aeronautics and Astronautics

Figure 8. FRFs for a representative subset of impacted

wing test points (minus the boom and glued aileron)

recorded by the accelerometer at point 518.

The main issue to note from figures 6–8 is the

variability in the third mode. In particular, three

frequency ranges exist for the third mode, depending on

the structural location of the response. Groups of

accelerometers that correspond to major structural

elements have been noted in several other structures to

demonstrate this behavior.15 Several possible

explanations for this behavior exist, such as the

nonlinearity of certain elements.

Consider the nonlinearities of the ATW

mass-balancing material. Located in the tips of the

boom and wing leading edge are small cavities holding

loosely packed, powdered tungsten (fig. 4). Speculation

has been made that the loose tungsten caused a moving

mass condition or that it had a frequency of its own that

coupled with the third mode. In addition, the connection

from the wing body to the boom (fig. 9) could have a

nonlinear stiffness as a result of the adhesive bonding

between the structural elements.

Figure 9. Wing-to-boom interface for the ATW.

Test data were analyzed for frequency and damping

using local, global, and polyreference curve-fitting

methods. Table 7 shows variances in frequency for

each mode shape as a function of the curve-fitting

technique compared with the finite-element model

(FEM) and a randomly picked FRF from the appendix.

Each of the curve-fitting methods used the same set of

FRF data found in the appendix under point 518. The

local curve-fitting method (table 7) was a

single-degree-of-freedom technique that could only

curve-fit the first two modes because of scatter in the

third mode. The global curve-fitting method was able to

fit all three modes using measured responses from the

accelerometer at point 518. The polyreference

curve-fitting method also fit all three modes using

measured responses from the accelerometers at points

518 and 300. An orthogonal polyreference was used to

provide an additional data point to describe the third

mode. No trends have been found, but the polyreference

curve-fitting technique presented the best results for

frequency comparison with the analytical model.

Data Correlation

Table 7 shows a set of parameters that represent

estimates of the natural frequencies. These estimates

Table 7. Estimates of natural frequencies.

Mode FEM, Hz

Measured 

FRF peaks, 

Hz

Local 

curvefitting, Hz

Global 

curvefitting, HzPolyreference

curve-fitting, Hz

Orthogonal

Polyreference

curve-fitting, Hz

1 14.50 14.43 14.34 14.23 14.51 --

2 23.34 21.06 21.03 21.30 22.28 --

3 84.65 76.81 -- 77.93 77.66 78.03
12

American Institute of Aeronautics and Astronautics

were computed from modal analysis of GVT data using

different techniques. Some of these estimates possibly

are computed with less confidence than others;

therefore, performing some correlation analysis on the

data to determine its validity is important. The

correlation analysis includes comparisons of test data

with test data, and test data with an analytical model.

Poor modal confidence factors resulting from the

correlation analysis would indicate some estimates can

be ignored, whereas all estimates with high modal

confidence factors should be retained and used for

uncertainty modeling. 

Table 8 shows three modal assurance criteria to

compare mode shapes as computed using different

curve-fit methods. In the first method, the MAC for

global curve-fitting as a function of polyreference

curve-fitting was computed. In the second method, the

MAC for global curve-fitting as a function of

single-degree-of-freedom curve-fitting of the first two

modes was computed because the local curve-fitting

method could only curve-fit the first two modes. In the

third method, the MAC for polyreference curve-fitting

as a function of single-degree-of-freedom curve-fitting

was computed, and again could only compare the first

two modes. The global and polyreference approaches

have the closest match to one another; the

single-degree-of-freedom method becomes less

effective by the second mode and can be eliminated.

Table 8. Values of MAC to compare mode shapes 

generated by different analysis methods.

Comparing the global and polyreference frequencies

in table 7, deciding which of the two curve-fitting

techniques best compares mode shape to the FEM is

difficult. The “Modal Analysis” section noted that

polyreference curve-fitting shows the closest match with

the FEM (comparing frequency only). Table 7 shows

the polyreference technique more closely matching the

FEM frequencies for the first and second modes, but the

global curve fit slightly better matches the third mode of

the FEM. The final choice is made when comparing the

global and polyreference curve fits with the FEM using

the pseudo-orthogonality check (POC), which is

sometimes referred to as self-orthogonality or

cross-orthogonality.

The POC is calculated after the MAC matrix indicates

valid test data and the FEM has been updated with the

GVT data. This check assures that applying the

analytical mass matrix to the test mode shape for

correlation with the FEM will give correct results. To

compare test mode shapes with analytical mode shapes,

the full-degree-of-freedom analytical model was

reduced to the measurement degree of freedom from test

using a reduced mass matrix. The reduced mass matrix

is also used for orthogonality calculations and a back

expansion to fill in unmeasured degrees of freedom for

better mode shape visualization.

Orthogonality is a vector-based technique that uses

the analytical mass matrix that has been reduced to the

set degree of freedom from test. Therefore, the

accuracy of the technique depends on the accuracy of

the analytical mass matrix and an appropriate number of

degrees of freedom being included in the reduction

process (and in the test). The POC calculation for

predicted and measured eigenvectors is given as:

(6)

If the measured modes are indeed mass-normalized

and orthogonal with respect to the mass matrix, ;

the test eigenvectors, , and the analysis

eigenvectors, , represent the same mode shapes

and the orthogonality criterion, , will be close to

the identity matrix. The POC criterion places

constraints on that the values of the diagonal

should be in excess of 0.90 and off-diagonal terms

smaller than 0.05. Correlating test mode shapes to test

mode shapes using the same measured data shows an

unanticipated match as higher frequency modes are

introduced. This match means the same test data

curve-fitted slightly different than the predicted mode

shape will have two different orthogonality matrices.

The POC matrix with the diagonal terms closest to a

value of 1.0 and the off-diagonal terms closest to a value

of 0 will be used to update the FEM.

The POC in table 9 compares an analytical mode

shape, , as a function of an estimated mode shape, v,

as computed by a single reference (point 518

accelerometer) global curve fit. These values indicate a

reasonable degree of orthogonality is achieved;

however, v3 had a relatively low comparison to the

analytical model because of inadequate data collection

at that mode.

Mode First 

method

Second 

method

Third 

method

1 0.992 0.979 0.992

2 0.758 0.734 0.666

3 0.556 -- --

X

[ ]

T MA

[ ] A[ ] X= [ ]

MA[ ]

X

[ ]

A

[ ]

X

[ ]

X

[ ]
13

American Institute of Aeronautics and Astronautics

Table 9. Orthogonality measures to compare

analytical and global estimated mode shapes.

The POC in table 10 compares the analytical mode

shape as a function of the two reference (points 518 and

300) polyreference curve fits. The results in table 10

show that the test mode shape has a slightly better

comparison to the FEM when information from two

accelerometers are used to curve-fit. This

cross-orthogonality check shows the two mode shapes

similarly match enough to consider the model a

reasonably accurate representation of the test data.

Table 10. Orthogonality measures to compare analytical 

and polyreference estimated mode shapes.

The high correlation shown in tables 8–10 indicates

that the GVT produced relatively consistent modal

estimates. Less confidence exists in the estimates

associated with the third mode than in the first and

second modes; however, the data in table 7 is believed to

represent a set of valid GVT data. Thus, all the

estimates in table 7 will be used for uncertainty

modeling and, consequently, robust flutter analysis.

Uncertainty Model

The GVT data were analyzed to derive uncertain

models of the natural frequencies of the ATW. These

models include a nominal value of the natural frequency

and an associated uncertainty description. The nominal

value and the uncertainty are formulated by analyzing

the data using the functional approach described in this

paper for different norms. Table 11 shows the uncertain

model of the natural frequency for the first mode.

Table 11. Modal parameters for the first mode.

The data from the GVT show variations in estimates

of approximately 0.28 Hz, so the uncertain parameter,

 should contain at least, but hopefully not much more

than, 0.28 Hz of variation. Table 11 shows that the range

of values covered by has a variation of 0.41 Hz when

using the 1-norm, 0.32 Hz when using the 2-norm, and

only 0.28 Hz when using the -norm.

An interesting feature of the results (table 11) is the

difference in the nominal value of the optimal estimate

of natural frequency, The data are biased towards the

large-value extremum of the observed range, and so the

optimal estimate obtained using the 1-norm and 2-norm

functionals is higher than that obtained using the

-norm. The ability of the 2-norm to account for this

bias accounts for its use in traditional analysis. The

-norm appears to derive a skewed estimate for that

does not account for the bias in the data; however, the

resulting uncertainty is smaller. Thus, the -norm

approach is clearly the least-conservative approach

when analyzing data for the purpose of robust flutter

analysis.

The estimates of natural frequency for the second

mode were analyzed, and table 12 shows the results.

This mode, similar to the first mode, shows that the

-norm approach generates an with the least amount

of range needed to match the range of experimental

estimates.

Table 12. Modal parameters for the second mode.

Table 13 gives the analysis results for the third mode.

This analysis presents the same conclusion as that of the

other modes; namely, the -norm approach is

preferable for modeling uncertain parameters to be used

in robust flutter analysis.

Estimated 

mode shape

Analytical mode shape

v1 0.983 0.040 0.064

v2 0.249 0.966 0.015

v3 0.322 0.420 0.676

Estimated 

mode shape

Analytical mode shape

v1 0.992 0.015 0.018

v2 0.216 0.949 0.113

v3 0.084 0.331 0.873

1 2 3

1 2 3

Norm Hz , Hz

1 14.43 0.20 [14.23, 14.64]

2 14.39 0.16 [14.23, 14.55]

14.37 0.14 [14.23, 14.51]

Norm Hz , Hz

1 21.30 1.56 [19.74, 22.86]

2 21.71 1.15 [20.56, 22.86]

21.95 0.92 [21.03, 22.86]

,

,

.

,
14

American Institute of Aeronautics and Astronautics

Table 13. Modal parameters for the third mode.

Robust Flutter Analysis

Robust flutter speeds were computed for the

analytical models that contain the uncertain natural

frequencies shown in tables 11–13. Table 14 shows

these speeds.

Table 14. Nominal and robust flutter speeds.

The main feature of interest in table 14 is the variation

in value of the robust flutter speed,11 Vrob. Specifically,

the model based on -norm analysis has a robust flutter

speed that is 42 KEAS higher than the corresponding

speed computed using the 1-norm analysis. This result

clearly shows that the -norm approach results in the

least-conservative answer.

The reason for the decrease in conservatism can be

easily discerned by examining tables 11–12. Basically,

the value of for the first mode is allowed to be as

high as 14.63 Hz when using the 1-norm analysis but

only 14.51 Hz when using the -norm analysis.

Conversely, the value of for the second mode is

allowed to be as low as 19.74 Hz when using the 1-norm

analysis but only 21.03 Hz when using the -norm

analysis. The smaller uncertainty associated with the

-norm analysis limits the size of possible frequency

variations and consequently restricts the modal

frequencies to being farther apart. Thus, more airspeed

is needed for the modes to couple, and the robust flutter

speed is higher.

Another feature to note is that the nominal flutter

speed, Vnom, is higher for the -norm analysis than for

the other norms. This feature results from the nominal

values of the natural frequencies for the first two modes

being farthest apart for the -norm analysis. This result

is a mere coincidence of the bias in the data. Remember

that the application was actually optimizing the values

to reduce the conservatism in the robust flutter speed

without consideration of the nominal flutter speed. For

example, the difference between the nominal flutter

speeds is 21 KEAS, whereas the difference between the

robust flutter speeds is 42 KEAS; so clearly the biggest

advantage is seen for the robust analysis.

Conclusions

This paper considered approaches to developing

uncertainty models from analysis of ground vibration

test data. The objective was to formulate an uncertain

value of natural frequency that represents the amount of

variation observed in the test data. This paper showed

that analyzing the data using an -norm approach

generates a model with less uncertainty than

corresponding 1-norm or 2-norm approaches. The

smaller uncertainty is useful for the flight test

community because the associated robust flutter speeds

can be computed with less conservatism. The

Aerostructures Test Wing (ATW) was used to

demonstrate this procedure. A ground vibration test was

performed on the ATW and used to generate a range of

natural frequency estimates. These estimates were

analyzed using the different norm approaches to

formulate uncertainty models that cover the entire range

of observed variations. The approach using the -norm

produces the smallest uncertainty and the

least-conservative robust flutter speed.

Appendix

This appendix provides handpicked estimates of

natural frequencies that arose from different

accelerometer measurements. These measurements

correspond to the fixed-location accelerometers, noted

as references in table A-1, in response to hammer

excitations at locations described as node numbers.

Note that there are only 33 points listed, whereas figure

4 indicates 35 test points. This discrepancy results from

not including data from points 512 and 239, which did

not provide additional information beyond these

33 points. Also note that certain entries are labeled as

“NODE” to indicate that this excitation location lay

along the node line for that particular mode.

Norm Hz , Hz

1 77.93 3.89 [74.04, 81.82]

2 78.45 3.37 [75.08, 81.82]

79.32 2.50 [76.82, 81.82]

Norm , KEAS , KEAS

1 371 317

2 384 343

392 359

,

Vnom Vrob
15

American Institute of Aeronautics and Astronautics

Table A-1. Estimates of natural frequencies.

Reference point 511 (Forward boom 

accelerometer) Reference point 518 (Aft boom accelerometer) Reference point 300 (Aileron accelerometer)

Point Mode 1 Mode 2 Mode 3 Point Mode 1 Mode 2 Mode 3 Point Mode 1 Mode 2 Mode 3

500 14.438 21.063 80.125 500 14.438 21.063 79.563 500 14.438 21.063 79.56

511 14.438 NODE 79.25 511 14.438 21.063 79.25 511 14.438 21.375 79.25

518 14.438 21.063 79.25 518 14.438 21.063 78 518 14.438 21.063 79.68

529 14.438 22.313 78.938 529 14.438 21.063 78.938 529 14.438 21.063 78.93

1 14.438 NODE 76.5 1 14.438 NODE NODE 1 14.438 21.375 77.125

11 14.438 21.063 79.25 11 14.438 21.063 79.25 11 14.438 21.063 77.44

23 14.438 21.375 77.125 23 14.438 21.063 76.81 23 14.438 21.063 77.125

33 14.438 21.063 78.313 33 14.438 21.063 76.81 33 14.438 21.063 77.125

152 14.438 NODE 77.125 152 14.438 21.063 76.81 152 14.438 21.063 77.125

150 14.438 NODE 77.125 150 14.438 21.063 76.81 150 14.438 21.063 77.125

146 14.438 21.375 76.813 146 14.438 21.063 77.125 146 14.438 21.063 77.125

308 14.438 21.063 77.125 308 14.438 21.063 77.125 308 14.438 21.063 77.125

312 14.438 21.063 77.125 312 14.438 21.063 77.44 312 14.438 21.063 77.44

300 14.438 21.375 77.125 300 14.438 21.063 77.125 300 14.438 21.063 79.56

307 14.438 21.063 77.125 307 14.438 21.063 77.44 307 14.438 21.063 77.44

311 14.438 21.063 77.125 311 14.438 21.063 77.125 311 14.438 21.063 77.125

55 14.438 21.375 77.688 55 14.438 21.063 77.44 55 14.438 21.063 77.44

168 14.438 21.375 77.125 168 14.438 21.063 77.125 168 14.438 21.063 77.125

172 14.438 NODE 76.813 172 14.438 21.063 77.125 172 14.438 21.063 77.125

174 14.438 21.375 77.125 174 14.438 21.063 76.81 174 14.438 21.063 77.125

45 14.438 21.375 77.125 45 14.438 21.063 77.125 45 14.438 21.063 77.125

67 14.438 21.375 77.125 67 14.438 21.063 77.125 67 14.438 21.063 77.125

196 14.438 21.375 76.81 196 14.438 21.063 77.125 196 14.438 21.063 77.125

194 14.438 21.375 77.125 194 14.438 21.063 77.125 194 14.438 21.063 77.125

190 14.438 21.375 77.125 190 14.438 21.063 76.813 190 14.438 21.063 77.125

77 14.438 21.375 77.125 77 14.438 21.063 77.125 77 14.438 21.063 77.125

89 14.438 21.375 77.125 89 14.438 21.063 77.125 89 14.438 21.063 77.125

218 14.438 21.375 76.81 218 14.438 21.063 76.81 218 14.438 21.33 77.125

216 14.438 21.375 76.81 216 14.438 21.063 76.81 216 14.438 21.33 77.125

212 14.438 NODE 76.81 212 14.438 21.063 76.81 212 14.438 21.063 77.125

99 14.438 21.063 76.81 99 14.438 21.063 76.81 99 14.438 21.063 77.125

111 14.438 21.063 77.125 111 14.438 21.063 77.125 111 14.438 21.063 77.125

121 14.438 21.063 76.813 121 14.438 21.063 76.813 121 14.438 21.063 76.81
16

American Institute of Aeronautics and Astronautics

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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Developing Uncertainty Models for Robust Flutter Analysis Using

Ground Vibration Test Data

WU 274-00-00-E8-RR-00-DDF

Starr Potter and Rick Lind

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

H-2452

National Aeronautics and Space Administration

Washington, DC 20546-0001 NASA/TM-2001-210392

A ground vibration test can be used to obtain information about structural dynamics that is important for flutter analysis.

Traditionally, this information—such as natural frequencies of modes—is used to update analytical models used to predict

flutter speeds. The ground vibration test can also be used to obtain uncertainty models, such as natural frequencies and

their associated variations, that can update analytical models for the purpose of predicting robust flutter speeds. Analyzing

test data using the -norm, rather than the traditional 2-norm, is shown to lead to a minimum-size uncertainty description

and, consequently, a least-conservative robust flutter speed. This approach is demonstrated using ground vibration test

data for the Aerostructures Test Wing. Different norms are used to formulate uncertainty models and their associated

robust flutter speeds to evaluate which norm is least conservative.

Flutter, Ground vibration test, Uncertainty 21

Unclassified Unclassified Unclassified Unlimited

April 2001 Technical Memorandum

Presented at 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

and Exhibit, Seattle, Washington, April 16–19, 2001, AIAA-2001-1585.

Unclassified—Unlimited

Subject Category Subject Category 05

This report is available at http://www.dfrc.nasa.gov/DTRS/