Full text of DOI:10.2514/2.4934

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
–
Vol. 25, No. 4, July August 2002
Experimental Modal Analysis and Damping
Estimation for an In ated Thin-Film Torus
D. Todd GrifŽ th and John A. Main
University of Kentucky, Lexington, Kentucky 40506
Experimental modal testing of an in ated torus is examined. The principal focus of this investigation is an
evaluation of the use of conventional dynamic testing tools in modal testing of a thin-Ž lm in ated structure.
Because thin-Ž lm structures are inherently  exibleandlightweight,precautionsmust be takenin the test procedure.
Challenges in testing and identifyingnatural frequencies and modal dampingare detailed. The experimental study
shows that whereas localized  exible body dynamics is predominant at higher frequency, the structure behaves
in a fashion similar to that of a solid elastic structure for the low-order modes. In addition, natural frequencies
(
)
and modal damping are shown to be dependent on the level of internal pressurization. Furthermore, viscous air
(
)
damping and structural strain-rate damping coefŽ cients are estimated from the experimentally measured modal
parameters to provide a more complete assessment of the damping behavior of this structure.
Introduction
remains wet until deployed on orbit. Once in orbit, the structure
is deployed, and the resin material can be cured by, for example,
NFLATED structuresare receiving a great deal of attention due
to the need for low-cost space missions and the potential to im-
solar heating or humidiŽ cation. Replacing the enclosed gas with a
I
3;4
prove greatlycapabilitiesin a varietyof applications.In ated struc-
tures are a practical choice for deployment of very large aerospace
structures. A large in ated structure can be stowed in a compact
conŽ guration and deployed on a single launch. The result is reduc-
tion in stowage space and launch cost and an orbital structure with
unique capabilities.
One downfall of in ated structures is the loss of structural in-
tegritythatcan resultwhen thestructureis piercedby an impactwith
high-velocitymicrometeoritesandspacedebrison orbit.Toimprove
the survivability of in ated structures, various methods have been
attempted to stiffen the structure through rigidization. The concept
rigid foam has also been explored. With this approach, thin-Ž lm
in atables can be stiffened by the injection of a liqueous foam into
the structure once deployed. The liqueous foam expands to Ž ll the
structure and then hardens to form a rigid composite.
Typical in ated structure designs incorporatetwo structuralsup-
port components, a circular or elliptical torus and a set of support
strutsthatextendfrom the instrumentationhousingof the spacecraft
to the torus. Figure 1 shows a photographof a prototype solar con-
centrator with the torus/lens and support strut conŽ guration. The
dynamic behavior of the in ated torus is of practical interest to
spacecraft systems and structural design because the torus element
isa key componentof manyin atedstructuredesigns.5¡11 Inasmuch
as it servesas structuralsupportfor an opticalsystemsuch as a thin-
membrane re ector or Fresnel lens, the in ated torus substructure
is important to emerging space technologies that require solar en-
ergy acquisition for solar thermal propulsion and electrical power
generation in orbital vehicles and solar sails. A visualization of a
)
of rigidizationcan be approachedin two primary ways: 1 the shell
)
of the structure can be stiffened or 2 the enclosed gas can be re-
placed by a more rigid compressionelement. The stiffeningof shell
material can be accomplishedby constructingthe shell of the struc-
ture with a fabric/resin system.1;2 With the fabric/resin approach,
the key point in the design is the selection of a resin material that
Todd GrifŽ th received a B.S. in physics from Morehead State University and in mechanical engineering from
the University of Kentucky in 1998. He received a M.S. in mechanical engineering from the University of Kentucky
in 2000. His technical interests are in the general areas of dynamics and control of  exible structures and system
identiŽ cation. He is currently a Graduate Research Assistant at Texas A&M University. He is a Student Member
of AIAA.
John Main received a B.S. in physics and mathematics from Western Kentucky University in 1984 and M.S.
and Ph.D. degrees in mechanical engineering from Vanderbilt University in 1987 and 1993. He held a postdoctoral
research position at Vanderbilt University from 1993 to 1995 and was a faculty member at the University of Maine
from 1995 to 1997.In 1997 he joined the Department of Mechanical Engineering at the University of Kentucky. His
technical interests are in the general areas of smart structural design and control, design of extremely large space
telescope primary mirrors, shape control of thin-Ž lm mirrors, and in ated aerospace structures. He is a Member
of AIAA.
°c
Received 17 July 2001; revision received 30 September 2001; accepted for publication 21 November 2001. Copyright
2002 by D. Todd GrifŽ th and
John A. Main. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal
or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923;
include the code 0731-5090/02 $10.00 in correspondence with the CCC.
609

610
GRIFFITH AND MAIN
)
perimentalinvestigationthatprovides1 a modaltestapproachusing
)
conventional modal testing tools and 2 a database for a structural
modeling effort, as well as a description of the structuralcharacter-
istics that should be included in a structural model.
Description of Test Article
(
The test article fabricated by United Applied Technologies,
)
Huntsville, Alabama is a nonrigidizedthin-Ž lm in ated torus with
(
)
(
)
5:87 in.
R D
r D
a 1.98-m 2
6:5 ft major diameter and a 15-cm 2
(
)
minor diameter Fig. 3 . The torus panels were thermally formed
from  at sheets of the polyimide Ž lm Kapton HN®. Six thermally
formed panels like the one in Fig. 4 make up the torus. Each ther-
mally formed panel accounts for one-half of the structure’s cross
sectionand 120 angulardegrees.Three thermallyformed panelsare
joined end to end with epoxy to complete one-half of the structure.
Two three-panelhalvesare joinedto completethe torus.The method
of fabricationresults in variationsin the thickness in the joining re-
gions.A list of the structure’s physicalpropertiesis given in Table1.
Table 1 Physical parameters of test article
(
Fig. 1 Prototype solar concentrator in test facility courtesy NASA
Marshall Space Flight Center .
)
Property
Value
( )
2.5 GPa 370,000 psi
Modulus of elasticity
Density
3
1.42 g/cm3 0.051 lb/in. /
(
( )
46 ¹m 1.8 mil
Shell thickness
Joining region width
Joining region thickness
( )
5.1 cm 2 in.
(
)
360/150 ¹m 14/6 mil
(
)
inner/outer
(
Fig. 2 Potential solar power application courtesy NASA Marshall
Space Flight Center .
)
potential application in solar power generation is shown in Fig. 2.
Figure 2 shows a series of in ated ring-type structures on the arm
of a large orbiting spacecraft.
A considerableamountofwork has been donein the areaofstruc-
tural design of in ated structures; however, the work performed in
the area of structural analysis is limited. Experimental investiga-
tions of the structuraldynamicsof in ated structureshave been per-
12¡15
formed in both ambient and vacuum conditions,
as well as in
Fig. 3 Test article and cross section.
zero-gravityconditions.12 Thesereportsdetailthechallengesexperi-
enced in testingthin-Ž lm in ated structures.In addition,some work
in analytical modeling of the natural frequencies12;14 and damping
behavior12 of in ated beams has been performed. Experimental in-
vestigations have also been performed on solid steel toroids16 and
thin-walledaluminumtoroidalshells.17 These reportsareinstructive
in that they describe some of the important dynamic characteristics
of the torus; however, the analysis was performed on stiffer, more
conventional structures than that studied in this work. The use of
noncontactingmethods for measuring structuralresponse has been
studied.13 Noncontactingmethods are advantageousfor highly ex-
ible, ultralightweightstructures in that accelerometermass loading
effectsand stiffnessof instrumentationwiringcan be avoided.How-
ever,in thispaperwe evaluatemodaltestingof in atedstructuresus-
ing conventionalcontactingmeasurementandexcitationtechniques.
Accurate knowledge of structural dynamics is very important
to structural design and structural control of any spacecraft struc-
ture. Further reŽ nements in in ated structuredesign require assess-
ment of in ated structure dynamics. This includes reŽ ning testing
methodologies and modeling approaches so that the dynamics of
spacecraft that incorporatein ated structureconcepts can be better
assessed.The objectiveof this paper is to presentresultsfrom an ex-
Fig.4 Thermallyformed panels used in torusconstructionandcloseup
of joining region.

GRIFFITH AND MAIN
611
Epoxy adds mass and stiffness to the structurein the bonded joints.
Figure 4 includes a closeup of the joining region, which describes
the variations in thickness in the bonded joints.
Modal Test Description
The objective of the experimental investigation was to evaluate
modal test procedures for lightweight in ated structures. The goal
was successfulidentiŽ cationof the naturalfrequenciesand damping
of the Ž rst three in-plane and out-of-planebending modes.
A number of issues related to the extreme  exibility and ultra-
lightweight nature of in ated structures create challenges in per-
forming modal tests. The key issues that must be addressed in the
)
test procedureare listed as follows: 1 Reduce the mass loading ef-
Fig. 6 Measurement locations and directions.
fect of accelerometers by measuring frequency response functions
FRFs with low-mass accelerometers. 2 Reduce the gravity ef-
(
)
)
fect by orienting accelerometers perpendicular to the gravity Ž eld.
3 Provide global excitation to structure using a modiŽ ed impact
hammer tip. In addition, internal pressurizationshould be varied to
determine the effect of the shell prestress on natural frequency and
damping. Each of these issues is addressed in more detail in the
following sections.
Ž eld minimizes any mass loading effects. Also, the stiffness of the
accelerometer lead was a concern because the structure was freely
suspended.Therefore, the leads were also suspended with monoŽ l-
ament to minimize their effect on the measurements.
)
The structure was tested at two in ation pressures, 5.52 kPa
(
)
(
)
0.8 psig and 6.89 kPa 1.0 psig , to examine the in uence of shell
prestresson the modal parameters.Air was supplied to the structure
using a small aquarium pump, and pressure was maintained with
the use of a controllercapableof maintainingthe pressure to within
0.34 kPa 0.05 psi of the desired value. The gas supply system re-
mained on during testing to compensate for leaks and maintain the
desiredconstantpressure.Any effectof thepumponmeasuredFRFs
FRF Measurement
The structure was tested in ambient laboratory conditions in two
test conŽ gurations to determine the modal parameters for the in-
plane and out-of-plane bending modes. In-plane testing was per-
formed with the test article freely suspended parallel to the  oor,
whereas out-of-plane testing was performed with the test article
freely suspended in a vertical position Fig. 5 . MonoŽ lament line
was used to hang the test article. Three monoŽ lament strands were
used for in-plane testing, whereas a single strand was utilized for
out-of-planetesting.Thestrandswerekeptaslongas possible 2m
for the in-planeconŽ guration to minimizethe frequencyof the pen-
dulumrigid-bodymode 0.3 Hz . For the out-of-planetest conŽ g-
uration, the strands used to hang the structure could not be greatly
lengthenedconsideringthe space available;however, the pendulum
mode for this conŽ guration was below 1 Hz.
(
)
(
)
was expectedto show up at 60 Hz the pump diaphragmfrequency .
No such result was found.
(
)
Inadditiontopumpnoise,the ow ofairintothestructurecouldbe
consideredan additional,nonquantiŽed input.Leak ratesweresmall
enough that resupply was only needed every 30 s to maintain the
desired pressure. Because the time period of one measurement was
only 0.5 s, it was possibleto takemeasurementswhen the pump was
not in operation.Replacementair ow into the structureand leakage
fromthestructuredidnotnoticeablyeffectthemeasurementofFRFs
in these tests.
(
»
)
(
»
)
For in-planetesting,the in ated torus was excitedwith an impact
hammer at a location on the outer surface of the structure near the
joining region. The impact was directed in the plane of the struc-
ture in the radial direction. Acceleration was measured by roving
an accelerometer, taking measurements at 16 evenly spaced points
along the inner surface of the structure in the radial direction. For
out-of-plane testing, the in ated torus was impacted in a direction
perpendicularto its plane with 16 evenlyspacedrovingacceleration
measurements taken in the same direction. Figure 6 shows details
of the input and output measurement locations and directions.Two
or three measurement locations would have been sufŽ cient to char-
acterizefrequencyand damping;however, 16 measurementsresults
in adequate description of the mode shapes for the Ž rst three bend-
ing modes in both conŽ gurations, in plane and out of plane. Mea-
Excitation Method
Because impact hammer testing is a common form of excitation
in modal testing, it was decided to evaluate its utility in excitation
of in ated structures. Attempts were made initially to excite the
structurewith commonimpacthammer tipssuchas small aluminum
and rubber tips. It was found that these tips were too stiff and did
not provide sufŽ cient global excitation to the structure. The small
hammer tips resultedin local deformation,which only excited local
shell modes of the thin-Ž lm structure.
Further attempts were made with a hammer conŽ guration in-
corporating a rectangular piece of aluminum measuring 3.81 cm
(
)
(
)
1.5 in. in width and 10.2 cm 4 in. in length. This modiŽ cation
was made to distribute the impact energy to a larger area of the
structure’s surface. Figure 7 shows the modiŽ ed hammer tip used
in the in ated structure tests. This hammer conŽ guration provided
sufŽ cient energy distribution and minimized any type of local ex-
citation, while exciting the global bending modes of the structure.
Improvementin excitationwith the modiŽ ed hammer conŽ guration
was more profound in the case of out-of-plane testing. Here, the
excitation is applied to the most  exible portion of the structure,
the outer curved surface. For in-plane testing the impact is near the
joining region, where the structure is stiffer; however, distributing
the impact energy in this fashion also improved the measurement
for the in-plane tests.
(
)
surements were taken with low-mass 1.1-g accelerometers with
(
)
»
excitation repeated at a single point. Mounting coins 1.5 g were
used to orient the accelerometerson the curved surfaceof the torus.
FRFs were recorded with a laptop computer and a Zonic Medal-
(
)
lion Mobile Fast Fourier Transform Analyzer Model 2300 eight
channel data acquisition system.
The mass of an accelerometeris a concern because the structure
is extremely  exible. However, using low-mass accelerometersand
the taking measurements in a direction perpendicularto the gravity
For any modal test, it is highly desirable that there be a strong
correlation between the quantiŽ ed input and the measured output.
The coherencefunction,or simplycoherence,givesa measureof the
–
input output relationship and can be simply deŽ ned as the square
of the correlation coefŽcient between the input and output signals
as a function of frequency.18 Coherence was used primarily to de-
termine the effectiveness of each impact excitation in these tests.
The coherencewas viewed after each successiveimpact. Any sharp
decrease in coherence broadband indicated poor excitation, and so
the measurement was removed and repeated.
)
)
Fig. 5 Photographsof test orientations: a in plane and b out of plane.

612
GRIFFITH AND MAIN
Fig. 8 Consistency diagram for in-plane conŽ guration at 5.52 kPa
using H1 estimate.
Fig. 7 ModiŽ ed impact hammer conŽ guration.
In addition, coherence was noted to be sensitive to input force
level. The measured coherence was improved for lower force lev-
els. In these tests, the acceptable input force level range was deter-
mined by softening the impact until the measured coherence was
optimized. The requirement for a suitable impact was a force input
–
–
(
)
range of 2 5 N 0.45 1.12 lbf for the out-of-plane excitation and
–
–
(
)
3 12 N 0.67 2.69 lbf for the in-plane excitations. Higher force
levels for the in-plane excitation are acceptable because the struc-
ture was impacted near the seam, a stiffer portion of the structure.
Additionalinformationon coherencemeasurementfor these tests is
detailed later in this report.
Discussion of Modal Test Results
(
X-Modal version 2, developed by the Structural Dynamics
)
Research Laboratory, University of Cincinnati was used to per-
form the system identiŽ cation. Essentially, X-Modal builds a linear
system model of the structuraldynamics. The software was used to
determine a consistency diagram from the FRF data, which were
used to identify the natural frequencies and damping values and
to assess the variation of the modal parameters with system model
order. The polyreferencetime domain algorithmwas chosen to per-
form the system identiŽ cation. In this approach, the desired model
orders for the system identiŽ cation are user speciŽ ed. In addition,
the highest model order chosen should be sufŽ cient to capture the
order of the system. In addition, frequency and damping estimates
Fig. 9 Consistency diagram for in-plane conŽ guration at 5.52 kPa
using H2 estimate.
H
H
were obtainedusingboth 1 and 2 FRFs to obtaina betterestimate
of the modal parameters.
IdentiŽ cation of Natural Frequencies
Natural frequencies were identiŽ ed using the consistency dia-
grams, FRFs, and mode shape plots. Certainly, this is a common
and elementary procedure for determining frequency and damping
information; however, for a thin-Ž lm in ated structure, it is neces-
sary to consider all available information in the identiŽ cation pro-
cess. Therefore, we go into greater detail than is usually warranted
for this type of analysis to present more completely the dynamics
of this nontraditionalstructure.
Fig. 10 Drive point FRF for in-plane conŽ guration.
A stable column of poles on the consistency diagram indicates
that there is a high-energy mode at that particular frequency. The
plots in Figs. 8 and 9 are consistencydiagrams for the in-plane test
activity.Figures 10 and 11 show typicaldrive point FRFs for the in-
planeand out-of-planeconŽ gurations,respectively.In a strictsense,
theseare not truedrivepointmeasurementsbecausethe acceleration
measurements were taken at points 180 deg about the cross-section
from the input location, therefore, some degree of  exibility exists
between the input and output. Figure 10 is a drive point FRF for
(
)
H
1
H
estimates,
2
at a pressurizationof 5.52 kPa 0.8 psig for
and
respectively. These diagrams show three strong peaks near 15, 40,
(
and 60 Hz with correspondingcolumns of complex poles denoted
)
byaninvertedtriangleontheconsistencydiagram ata near-constant
frequency for increasing system model order.
( )
The use of FRFs to identify natural frequencies is inadequate
due to noisy measurements that result from localized shell mode
the in-plane conŽ guration at 6.89 kPa 1.0 psig . Figure 10 shows
(
distinct resonance peaks for the Ž rst two natural frequencies near

GRIFFITH AND MAIN
613
circular ring model given in Ref. 19 show good agreement with
those in Figs. 12 and 13; therefore,we can conŽ rm that these modes
correspond to the desired bending modes.
In addition to structural response, the pneumatic response of the
17
enclosed gas should also be considered. Pneumatic resonance
can be predicted using the famous relation for speed of sound,
(
frequency, and wavelength speed of sound is equal to frequency
)
timeswavelength .±The speedofsoundinairatatmosphericpressure
±
(
)
(
)
101.3 kPa and 21 C 70 F is 344 m/s. The wavelengthassociated
with the fundamental harmonic pneumatic resonance is the mean
(
)
circumference of the in ated torus 6.2 m . With these values for
speed of sound and wavelength, the fundamental harmonic pneu-
matic resonancefrequencyis estimatedto be 55.3Hz. An inspection
of the consistency diagrams of Figs. 8 and 9 shows that there is a
small peak near this frequencyto the left of the third dominant col-
umn of poles in each Ž gure. We cannot say with certainty that this
is a pneumatic responseof the enclosedgas. However, note that the
fundamental pneumatic response and higher-order harmonics are
likely to be found in the frequencyband of interest. The precise fre-
quency at which pneumatic resonancewill occur will depend on, as
shown here, the speed of sound and the wavelength of the mode of
interest. A test to identify pneumatic resonancecould includeusing
different in ation gases with distinctly different speeds of sound,
or, if possible, testing structures with different length/diameter. In
this report, we made no attempt to identify pneumatic resonance.
Several modal tests were performed in each conŽ guration at two
Fig. 11 Drive point FRF for out-of-plane conŽ guration.
First
Second
Fig. 12 Experimental mode shapes for out-of plane bending modes.
(
)
different pressures. The natural frequencies bending modes and
damping identiŽ ed from modal parameter estimation for each of
these tests are listed in Tables 2 and 3. A description of how the
modal damping values in Tables 2 and 3 were determined is given
in the next section.At higherpressurization,the increasein stiffness
is larger than the increase in mass associated with the higher inter-
nal pressurization;therefore, the naturalfrequenciesincreasedwith
increasing pressure. This result is found for both the in-plane and
(
out-of-planebending modes. For a 25% increase in pressure from
)
5.52 to 6.89 kPa , the out-of-plane natural frequencies increased
by 1.7, 9.1, and 2.3%, whereas the in-plane natural frequenciesin-
creased by 12.9, 13.3, and 27.6%, respectively, for the Ž rst three
)
)
b Second
a First
H
bending modes in each conŽ guration based on the
estimation.
1
Furthermore, the in-plane bending frequencies are higher than the
out-of-planebendingfrequencies.This effectis due to the additional
in-plane stiffness due to the joining regions. In addition,it is shown
that modal damping follows the opposite trend with lower modal
damping ratios for a higher pressurizationlevel with the exception
of the second out-of-planemode at 6.89 kPa 1.0 psig . The depar-
ture of the modal damping of this mode from the general trend was
evident in a subsequent modal test, and, as well, the latter value
was consistentwith the reportedvalue. This indicatesthat either the
damping for this mode is simply unusually high, or that possibly
some other mechanism, that is, excitation of other modes such as
in-planebendingby out-of-planeimpact,contributesto the response
of the second out-of-planemode at 6.89 kPa. There is no evidence
thatthein-planeand out-of-planemodeswere coupledin thesetests;
therefore, it is most likely that the damping for this mode simply
deviates from the trend.
Fig. 13 Experimental mode shape plots for in-plane bending modes.
)
18 and 45 Hz . The resonancepeak near 80 Hz is the third in-plane
bendingmode, which is lesseasilyidentiŽ ed becausethereis a large
amount of shell mode activity at this frequency.
Figure 11 is a drive point FRF for the out-of-planeconŽ guration
(
)
(
)
at 6.89 kPa 1.0 psig . Figure 11 only shows one distinctpeak at the
Ž rst bending mode near 13 Hz and shows that higher-orderbending
modes are not easily discerniblewhen a large amount of shell mode
activityis presentat a higherfrequency.As expected,the abundance
of shell mode activity is larger for the out-of-planetest because the
acceleration measurements were taken on the most  exible outer
curved surface of the structure. A comparison of Figs. 10 and 11
showsthatFRFs aregenerallymore easilydiscerniblewhen exciting
andrecordingmeasurementsat a more stifflocationon thestructure.
With a large amount of localizedshell mode activity,it is difŽ cultto
determine higher-order bending modes from FRFs alone. For this
reason, the use of mode shapes is of great importance to the identi-
Ž cation of global bending modes. The FRFs indicate the frequency
at which shell modes begin to become important. For these tests,
shell modes dominate the structural dynamics above 50 Hz for the
in-plane test and above 30 Hz for the out-of-planetest.
Damping Estimation
Identifying the natural frequencies of the Ž rst three bending
modes is a straightforward task; however, determination of the
damping values for each mode requires more attention. For each
conŽ guration and pressurizationlevel, the modal analysis was per-
H1
H2
Shell mode activity in the frequency band of global bending
modes tends to obscure the identiŽ cation of natural frequencies.
Therefore,mode shape plots are a key piece of informationin iden-
tifying the bending modes. Analytical circular ring models19 offer
a good approximation of the mode shapes for the in ated torus for
the purposeof comparisonwith experimentalmode shapes.The Ž rst
two experimental mode shapes for each conŽ guration are given in
Figs. 12 and 13. Figure 12 shows the Ž rst two out-of-planebending
modes, and Figure 13 shows the Ž rst two in-plane bending modes.
formed using two FRF estimation approaches, the
and
tends minimize the error of the output and results
in underestimationof the amplitude at resonanceand overestimates
estimates.18
H1
H
2
the damping values, whereas
tends to minimize the error of the
input and results in overestimation of the amplitude at resonance
and underestimates the damping values.20 Therefore, it should be
expected that a damping range can be determined using both FRF
estimates. Because the damping values vary signiŽ cantly with sys-
tem model order, a range of model orders in which the damping
values became most consistent was chosen to estimate the percent
damping. For each mode, the damping values, as well as the natural
(
Note that in Fig. 13b that the mode shape plot for the secondbend-
)
ing mode is shown in an isometric view. The mode shapes for a

614
GRIFFITH AND MAIN
Table 2 Natural frequencies and damping for in-plane bending
H1 estimate
H2 estimate
Mode
1
Model orders
Frequency, Hz Damping, % Frequency, Hz Damping, %
In plane: 5.52 kPa
–
50 64
Average
Standard deviation
Range
16.00
0.02
——
——
1.50
0.50
0.94 2.12
15.92
0.02
——
——
1.55
0.76
0.47 2.68
–
–
–
–
Range for all model orders
2
0.23 2.54
0.38 10.14
–
50 64
Average
Standard deviation
Range
40.94
0.04
——
——
2.54
0.69
40.91
0.09
——
——
2.13
0.69
–
–
1.85 3.34
1.35 3.20
–
–
Range for all model orders
0.14 7.16
0.49 7.29
–
3
50 64
Average
Standard deviation
Range
63.20
0.32
——
——
4.79
4.06
62.54
0.45
——
——
2.96
1.96
–
–
1.79 15.28
0.35 5.26
–
–
Range for all model orders
1.46 15.28
0.35 5.91
In plane: 6.89 kPa
–
1
50 64
Average
Standard deviation
Range
18.07
0.02
——
——
1.11
0.18
18.05
0.01
——
——
1.13
0.22
–
–
0.94 1.37
0.90 1.55
–
–
Range for all model orders
2
0.38 3.36
0.57 2.83
–
50 64
Average
Standard deviation
Range
46.40
0.10
——
——
1.68
0.45
46.55
0.11
——
——
1.81
0.49
–
–
1.23 2.33
1.32 2.72
–
–
Range for all model orders
0.41 5.18
1.32 2.72
–
3
65 86
Average
Standard deviation
Range
80.65
1.33
——
——
2.13
2.72
80.73
0.84
——
——
2.60
2.72
–
–
0.10 8.71
0.08 7.39
–
–
Range for all model orders
0.10 8.71
0.08 7.39
Table 3 Natural frequencies and damping for out-of-plane bending
H1 estimate H2 estimate
Mode
1
Model orders
Frequency, Hz Damping, % Frequency, Hz Damping, %
Out of plane: 5.52 kPa
–
60 74
Average
Standard deviation
Range
13.30
0.04
——
——
3.39
0.66
13.31
0.05
——
——
4.38
0.97
–
–
2.66 4.15
3.37 5.43
–
–
Range for all model orders
2
1.32 8.32
0.16 12.09
–
80 94
Average
Standard deviation
Range
31.48
0.19
——
——
2.85
1.00
31.33
0.07
——
——
3.75
1.28
–
–
2.05 4.72
2.15 5.81
–
–
Range for all model orders
2.05 4.72
0.48 7.16
–
3
75 90
Average
Standard deviation
Range
64.71
0.21
——
——
4.05
0.86
64.78
0.24
——
——
3.94
0.65
–
–
3.18 5.39
3.22 4.92
–
–
Range for all model orders
1.84 5.97
1.10 6.44
Out of plane: 6.89 kPa
–
1
60 74
Average
Standard deviation
Range
13.53
0.01
——
——
1.39
0.25
13.71
0.01
——
——
0.84
0.14
–
–
1.07 1.76
0.65 1.02
–
–
Range for all model orders
2
0.31 9.68
0.31 7.60
–
80 94
Average
Standard deviation
Range
34.34
0.31
——
——
7.07
2.35
34.02
0.14
——
——
6.80
2.76
–
–
4.64 11.02
3.53 11.15
–
–
Range for all model orders
4.64 11.02
3.53 11.15
–
3
70 90
Average
Standard deviation
Range
66.21
0.47
——
——
3.55
2.63
64.57
0.34
——
——
3.67
2.63
–
–
0.54 6.43
0.22 6.81
–
–
Range for all model orders
0.54 6.43
0.22 8.19

GRIFFITH AND MAIN
615
2
3
2
3
7
4
frequencies, were averaged for a range of about 15 model orders.
I
I
1
¯
2³1!1¹
1
H
The same range of modelorderswas maintainedfor boththe 1 and
6
7
µ ¶
6
6
6
4
4
6
7
2³2!2¹
1
¯
:
:
C
H
2 estimatesto draw a comparisonbetween the two FRF estimation
7
7
5
2
D BT W B ¡1 BT W
B D
6
7
7
5
.
/
;
where
:
:
6
techniques. Again, the natural frequencies and damping values are
listedin Tables2 and 3 for eachtest conŽ gurationand pressurization
level.
Cs
:
4
:
4
n
2³n !n¹
I
1
¯
Given the high damping of this structure, a more conservative
( )
2
H2
damping ratio estimate should be achieved using an
estimate.
However, an interestingresult is found in this modal analysisin that
For the current problem, we only have three modes available for
this regression.This is a limitation; however, the modal parameters
Ž t well to this damping model with the exception of out-of-plane
H
2
the
estimation techniquedoes not always give a lower damping
H
value estimate than the 1 technique.An examinationof the damp-
ing valuesshownin Tables2 and 3 showsthatthe dampingestimates
for each technique are not signiŽ cantly different in magnitude. In
addition, the damping range for each technique is not signiŽ cantly
differentstatisticallyas shownby the standarddeviationof the range
of values.
(
)
regressionfor a pressurizationof 6.89 kPa 1.0 psig . The estimated
viscous damping and structural damping coefŽ cients are given in
R2
Table 5 along with the
values for each Ž t. The regression plots
are given in Figs. 14 and 15.
The modal damping for the second out-of-plane bending mode
Also of interest is estimation of the individual damping mecha-
nismsthatcontributeto theexperimentallydeterminedmodaldamp-
(
)
at 6.89 kPa 1.0 psig was considerably higher than that of the Ž rst
and third modes. This raises questions about the accuracy of the
damping values for this data set; however, the uncertainty in this
measurementwas accountedforby usingthe weightedleast-squares
approach. This allows some general trends to be observed from the
estimated damping coefŽ cients in Table 5. Structural damping was
(
)
ing. These damping mechanisms can include viscous air damp-
(
)
ing, structural strain-rate damping, spatial hysteresis, and time
hysteresis.21 However, we will only consider the time-independent
viscous and structuraldamping. Estimation of viscous damping in-
dicates something importantregardingthe in uence of air damping
for tests performed in the ambient environment. In addition, it pro-
vides another parameter to be included in a structural model. The
ability to estimate the effect of structural damping would be very
useful for an in ated aerospace structure because external viscous
damping will not be present on-orbit. Cudney and Inman22 have
(
generally higher for higher pressurization levels higher Ž lm pre-
)
stress . On the other hand, viscous damping follows the opposite
trend with higher viscous damping at lower pressurization.In addi-
(
)
tion, modal damping for the out-of-planebending modes Table 3
was generally higher than those for the in-plane bending modes.
This is not surprising because viscous damping is directly propor-
tional to the amount of external  uid mass displaced. More  uid is
–
demonstrated that, for an Euler Bernoulli beam, determining the
individual effects of viscous and structural damping can be accom-
plishedusing the measurednaturalfrequenciesand modaldamping.
In essence, a linear least-squaressolution is utilized to Ž t the mea-
sured modal parameters to an appropriatedamping model based on
the equation of motion for the structure.
Table 4 Theoretical eigenvalues
¯
for the in- and out-of-plane
circular ring models
n
In the interestof brevity,we will not presentthe full development,
but only summarize the most important results. Assuming propor-
tionaldamping,thefollowingequationisfoundupona modalexpan-
Mode
In plane
Out of plane
1
2
3
1.638
2.755
3.815
1.611
2.732
3.796
( )
sion solution of the equation of motion. Equation 1 relates the ex-
perimentallydeterminedmodal parameters to the to-be-determined
damping coefŽ cients:
Table 5 Damping regression results
4
n
( )
1
D C C C I
2³n !n ¹
¯
H1
H2
s
Pressure, C,
kPa
Cs ,
C,
Cs ,
ConŽ guration
Ns/m2 Ns/m5
R2 Ns/m2 Ns/m5
R2
( )
Equation 1 is written here in general form and can be applied for
both the in-plane and out-of-planedata sets. Here, !n and ³n are the
experimentallydeterminednaturalfrequencyand damping ratio for
In plane
In plane
5.52
6.89
5.52
6.89
0.216 68,500 0.991 0.419 38,500 0.988
0.201 46,400 0.989 0.194 52,100 0.995
0.529 324,000 0.999 0.821 288,000 0.998
0.168 392,000 0.666 0.046 386,000 0.704
Out of plane
Out of plane
n
the th mode, ¹ is the mass per unit length about the torus circum-
I
ference, is the moment of inertia for the torus cross section, and
n
C
¯n is the circularring theoreticaleigenvaluefor the th mode. and
Cs
are the viscousand structuraldampingcoefŽcients, respectively.
Note that, due to the joining region, the moments of inertia for the
in-plane and out-of-planecases are different. Therefore, depending
of whether the in-plane or out-of-plane modes are being studied,
¡7
4
( )
(
3:2 10 m / or
I
I
£
I
the value¡o8f in Eq. 1 becomes either
y
x
.4:9 10 m4/, respectively. The theoretical eigenvalues ¯n are
£
available from frequency expressionsfor a circular ring.19
form with representedby 2³n!n¹ and
C
by ¯n . In this convenientform, the damping coefŽ cients and
s
–
x y
( )
Equation 1 is in
y
x
4
I
C
are found from a least-squaresŽ t of the set of experimentallydeter-
mined natural frequencies, modal damping ratios, and theoretical
eigenvalues.The regression yields the viscous damping coefŽ cient
C
y
C
as the interceptand the structuraldamping coefŽ cient as the
s
slope of the regression.
The theoretical eigenvalues for in-plane and out-of-plane bend-
ing are given in Table 4. The Ž tting problem is approached us-
ing a weighted least-squaresapproach to account for measurement
23
W
uncertainty. Therefore, the weighting matrix is chosen to have
2
2
(
n
the form diag 1=¾n /, where ¾ is the variance of the th modal
damping ratio. In this way, thendamping values that are known the
best are given more importance in the regression. The problem is
formulated as
Fig. 14 Damping regressions in plane.

616
GRIFFITH AND MAIN
Fig. 17 Coherence at drive point for in-plane testing at 5.52 kPa.
is less stiff and more apt to wrinkle. The drop in coherence broad-
band indicates either that the structure behaves nonlinearly for the
entire frequency range of interest or that in some way the suspen-
sion is providing input to the structure. Additional input from the
suspensionis not likely because a broadbanddrop in coherencewas
not found at the higher pressurizationlevel. Therefore, the drop in
coherencebroadbandat the lower pressurizationlevel is most likely
due to nonlinear behavior because more nonlinearity is expected
at lower pressure levels, where the structure is more apt to wrin-
kle. These results indicate that the in ated torus begins to exhibit
Fig. 15 Damping regressions out of plane.
(
)
some nonlinearitybelow 6.89kPa 1.0 psig . At higherpressure,the
structure behaves more linearly, as indicated by Fig. 16.
Conclusions
In ated structures have several interesting characteristics that
contributestructuralnonlinearities.AsigniŽ cantmaterialnonlinear-
ity is presentbecause it has been observedin an experimentalstudy
thatthe dynamicmodulusof prestressedŽ lms varieswith frequency.
Also, the method of gluing the panels that comprise the structure
signiŽ cantly impacts structural stiffness and structural damping.
Another important structural characteristic of in ated structures is
wrinkling, which is present in the loaded as well as unloaded state,
due to imprecise joining of the thermally formed panels. At in-
creasing pressurization, a considerable portion of the wrinkling is
removed; however, when bending moments reach sufŽ cient magni-
tude, the Ž lm will wrinkle.
This paper demonstrates that at least certain types of in ated
aerospace structures can be successfully tested in ambient condi-
tions. Conventional modal testing of in ated aerospace structures
requires making efforts to reduce mass loading effects from the in-
strumentation, to reduce the gravity effect by orienting accelerom-
eters perpendicularto the gravityŽ eld, and to avoid local excitation
by distributing the impact over a larger contact area. The result of
testing in ambient conditions is that air damping is present and will
impact the structuraldynamics. However, it has been demonstrated
that even in the presence of air damping measured modal param-
eters can be used to estimate the individual viscous and structural
damping effects. The ability to identify a larger number of modes
would improve the accuracy of damping estimation. In addition, a
study of time-dependentdamping mechanisms may reveal some of
the key structuraldynamicsof in atedaerospacestructures.Estima-
tion of these damping mechanisms is important with regard to the
development of accurate structural models. In addition, an attempt
Fig. 16 Coherence at drive point for in-plane testing at 6.89 kPa.
displaced in out-of-plane bending because the joining regions will
have a greater contribution to viscous damping than is the case for
in-plane bending. Note that, if we had used a non-weighted least-
squaresapproachto performthe dampingestimation,we wouldŽ nd
the estimated viscous damping coefŽ cients to be higher, and in turn
the structural damping coefŽ cients would be lower than those re-
ported in Table 5. This result was found for each conŽ guration and
H
1
H
and 2 estimates.
pressurizationlevel for both
No attempt was made to determine the effect of temperature on
the natural frequencies and damping in these tests with the tem-
±
±
)
(
perature maintained at 21 C 70 F . However, in general, we can
expect that temperature will have an effect on the properties of the
shell material and the enclosedgas. For example, at lower tempera-
ture, we should expect the Ž lm material to stiffen, and the speed of
sound of the enclosedgas will decrease. As a result, the natural fre-
quenciesshouldincreasewith higherstructuraldamping and higher
viscousdamping,and the frequencyof pneumaticresonanceshould
decrease.
Coherence Measurement
Figure16 is a drivepointcoherenceplotfor the test characterizing
(
)
in-plane bending modes at a pressurizationof 6.89 kPa 1.0 psig .
The coherence is close to one broadband with the exception of the
value at resonant frequencies. This could be the result of nonlinear
behaviornear resonance,but is more likely the result of a large out-
put measurement correspondingto a small input. Because the input
and output locations are not collocated, some  exibility exists be-
tween thesepointsinasmuchas they are locatedat 180 deg aboutthe
structure’s cross section. The large amplitude output measurement
is likely due to the excitation of modes related to distortion of the
cross section.
H
H
techniques.
2
was made to estimate modal parameters by
and
1
In theory, this approach would result in an upper and lower bound
on the estimates for natural frequency and damping. However, for
this study no such range could be established.
The full assessment of the effect of the enclosedgas on the struc-
tural dynamics of an in ated structureis still an open question.The
results of this study have shown that natural frequenciesand modal
damping are dependent on the level of pressurization. For higher
internal pressure, natural frequenciesincrease, whereas, in general,
modal damping follows the opposite trend. In the orbital environ-
ment, external viscous damping will not be present. Therefore, an
Figure 17 shows a similar plot of coherence at a lower pressur-
(
)
ization level of 5.52 kPa 0.8 psig . At lower pressure, the structure

GRIFFITH AND MAIN
617
important question that remains is what role does the enclosed gas
play in the damping behaviorof an in ated structure.The resultsof
the presentstudyfortwo pressurizationlevelsindicatethatstructural
damping is greater for higher internal pressurization. A more rig-
orous test protocol should be designed to ascertain the effects of
pressurizationlevel and internal and external viscous air damping.
It is suggestedthatan in atedaerospacestructurebe tested at a min-
imum of three pressurizationlevels. In addition, testing a structure
in both the ambient environment and in vacuum conditions would
provide an important characterization of the damping behavior of
these structures.
9Ashley, S., “Rockets Powered by the Sun,” Mechanical Engineering,
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