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14 SOUND AND VIBRATION/JANUARY 2006
Modal Parameter Estimationfor Large, Complicated MIMO TestsPeter
Avitable, University of Massachusetts, Lowell, Massachusetts
Raj Singhal, Canadian Space Agency, Ottawa, Canada
Bart Peeters and Jan Leuridan, LMS International, Leuven,
Belgium,
Multiple-input, multiple-output (MIMO) experimentalmodal testing
is often used for large structures. The data col-lected are used in
multiple-reference reduction schemes tofind the best set of modal
parameters to describe the system.Often several or many of the
reference shakers do not ad-equately excite all of the modes from
each reference location.When this is the case, using all of the
reference data may pro-duce modes that are not optimum. A careful
selection of ref-erences for generating modal parameters is
critical for devel-oping a good modal database for design,
analysis, simulationand correlation efforts. While this is true of
earlier modal pa-rameter estimation algorithms, the latest PolyMAX
estimationalgorithm has significant advantages over historically
usedtechniques.
Experimental modal tests are often conducted using a
mul-tiple–input, multiple-output testing strategy. Depending on
thecomplexity of the structure to be tested, two or more shakersmay
be used for the excitation of the system. Many times it isvery
difficult, if not impossible, to have all the shakers exciteall the
modes of the system equally. This is especially truewhen the
structure exhibits directional global modes or whenthe structure
has an abundance of local modes due to append-age or subcomponent
modal energy. When this is the case,
Figure 1. RADARSAT1 under test. Figure 2. RADARSAT1 test
geometry.
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15SOUND AND VIBRATION/JANUARY 2006
multiple-shaker testing is necessary to adequately excite all
themodes of the system over the desired frequency range.
However, all of the shakers may not exhibit a high degree
ofmodal participation for each individual mode of the system.In
this case, the extraction of modal parameters may be affectedby the
inability to adequately excite every mode to a sufficientdegree. If
this is the case, the modal participation will reflectthis and the
resulting modal parameters are weighted by themodal participation
values. This is handled in the extractionphase of the modal
parameter estimation process. However,there is a serious concern
when modal participations are be-low 20% and especially if they are
below 10% of the total par-ticipation of the other shakers exciting
the system. Whenthese participations drop to such low values, the
modes of thesystem are not adequately excited, or excitation
directions andresulting measurements are generally not particularly
good.The coherence of these measurements that are not well
excitedis also affected and is generally not very good. The
measure-ments then are not considered optimum. The main problem
isthat the measurements contaminate the overall extraction ofmodal
parameters from good reference locations using tradi-tional
approaches.
To extract the best possible modal parameters, it may
benecessary to exclude certain measurements that are not
con-sidered particularly good from the global modal parameter
poleextraction. Using all of the measured data may not produce
thebest overall extracted modal parameters. A careful review ofall
the measurements and modal participation factors may helpto
determine the best set to use in extracting the best
modalparameters to describe the system.
However, more recent advances in modal parameter estima-tion
have yielded new processing algorithms that are not assensitive to
the requirements identified above. PolyMAX1,2 is
a newer algorithm that can overcome many of the
limitationsidentified above. Wide frequency bands with all
measurementdegrees of freedom can be effectively processed with
littlenumerical or user difficulty.
To illustrate some of the problems associated with using
acomplete set of multiple-input, multiple-output frequency
re-sponse measurements using more traditional modal
parameterestimation techniques, several modal parameter
estimationscenarios are explored. One extraction uses all of the
measureddata, and the other uses a selected set of frequency
responsemeasurements to show differences that exist in the
extractionprocess. Both of these utilize older modal parameter
estima-tion technologies common in almost all software
packagesavailable today.3 In addition, the same data set was also
pro-cessed using the new PolyMAX4 approach to illustrate the
dif-ferences and advantages of this new technique.
To illustrate the concerns in processing data, the
CanadianRADARSAT1 satellite experimental modal test shown in
Fig-ure 1 was used for demonstration purposes. This experimen-tal
modal test was conducted with several shaker excitationsapplied to
the structure. The modal test consisted of 250 re-sponse
accelerometers resulting from five separate shaker ex-citation
locations. Upon reduction of the data, the lower 25modes of the
system can be seen to be most directly excited byonly two of the
shaker excitation locations. Reduction of thedata was performed
using all of the shaker excitation locations.A more selective set
of excitation locations was used based onthe modal participation
factors for each of the modes of the sys-tem to illustrate the
degradation of the modal data when usingall data references
simultaneously. The data set was then finally
Figure 3. Summation function and mode indicator function (top)
us-ing all references along with stability diagram (bottom) over
entire 10-64 Hz band.
Figure 4. Stability diagram for three different bands using all
references:a) 12.6-20.6 Hz; b) 36.7-40.6 Hz; and c) 44.1-48.1
Hz.
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16 SOUND AND VIBRATION/JANUARY 2006
processed using the PolyMAX technique. First, the data
werereduced using traditional approaches to show the amount
ofeffort and manipulation required to extract useful modal
pa-rameters. Then, the data were processed using the newly
de-veloped PolyMAX technique.
TheoryThe extraction of modal parameters involves several
basic
equations related to modal analysis theory. These equations
arebriefly summarized to show the effects of different
referencelocations on the extraction of modal parameters.
Frequency Response Measurement Formulation. The fre-quency
response function can be expressed in terms of thesummation of the
modes of the system. One form of this equa-tion represents the
modal characteristics of poles and residuesas:
For a particular mode k the frequency response can also beshown
to be expressed as the singular valued decompositionof the system
matrix as:
In this formulation, the residue matrix is therefore related
tothe mode shapes in the classical representation as:
Upon expanding some of the terms of this expression,
the relationship of the residue to the mode shapes can beclearly
seen. When a particular mode is evaluated, every oneof the rows and
columns of the frequency response matrix canbe used to extract that
particular mode of the system (provid-ing that the reference is not
at the node of a mode). For instance,using the first column of the
residue matrix, the mode shapefor a particular reference (assuming
unit modal mass scaling)can be found from:
While any row or column can be used, it is very obvious
thatcertain rows or columns (certain references) are better
refer-ences to select for the generation of good frequency
responsefunctions. When directional modes exist in the system,
certainreferences may not be very good for some modes but
excellentreferences of other modes of the system. The modal
participa-tion factors help to identify the amount of participation
eachexciter location provides to the overall excitation of the
modesof the system. The frequency response equation can be
rewrit-ten in another popular form that identifies the modal
partici-pation part of its formulation as:
From these relationships, the modal participation L of each
Figure 5. Typical synthesized FRFs using all references, 10-64
Hz.
Figure 6. MAC of modes using all references.
(1)
(2)
(3)
(4)
(5)
(6)
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j p
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kk
m k
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( )ÈÎ ˘̊ = ( )ÈÎ ˘̊ =[ ]
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kk
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k( )ÈÎ ˘̊ = { } - { }=
A s q u uk k k k
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a a a
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k k kk
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m( )
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17SOUND AND VIBRATION/JANUARY 2006
mode can be clearly seen. While theoretically all the modes
canbe obtained from any reference location, certain references
aremuch better than other reference locations. Certain
referencesthat do not excite the modes well enough will result in
mea-sured frequency response functions that may be susceptible
tonoise and poor dynamic range. These measurements may notbe
optimum, and use of these measurements for extractingmodal
parameters is questionable to say the least.
Case StudiesThe test geometry of the RADARSAT1 experimental
modal
test configuration is shown in Figure 2. This structure
wastested with five separate shaker excitation locations and
250measurement points. The main modes of interest for this
struc-ture exist in the 10- to 64-Hz frequency band. Several
differ-ent modal parameter extraction scenarios were performed
toshow the degradation of the extracted modes when all of
themeasured degrees of frequency (DOF) are used as opposed to amore
selective set of DOF for generating poles and extractingresidues.
The dataset is then processed using the PolyMAXtechnique to show
the ease with which this difficult data setcan be efficiently
processed.
Use of All Measured Frequency Response Functions. Thefrequency
response measurements were evaluated over eightdifferent bands
between 10 and 64 Hz. Poles were extractedusing a time-domain,
complex, exponential, curve-fitting tech-nique. Typical mode
indicator tools were used for identifyingmodes of the system. The
summation function, multivariatemode indicator function and the
complex mode indicator func-tion were all used for identifying
modes and are shown in Fig-ure 3 for the entire bandwidth. Figure 3
also shows the firststability diagram using the entire bandwidth
for evaluation.
Clearly, the stability diagram is very difficult to interpret
whenusing the entire bandwidth for all of the references.
Figure 4 shows three separate stability diagrams over
threeseparate bandwidths, where all the references and all
measuredDOFs are used for the extraction process. The stability
diagramwas also used for identifying the poles of the system. For
thiscase, the use of all the references and all the measured
DOFswere used to extract modal parameter estimates. The
modeindicator tools produced adequate identification of the modesof
the system, but the stability diagram produces only
marginalidentification of the poles of the system. Selecting poles
fromthese plots was fairly difficult due to the variance of the
esti-mated pole parameters. Due to the large number of
measure-ments that were obtained from references that did not
ad-equately excite the modes, the stability diagram results are
notparticularly good. The pole selection adequacy is very
ques-tionable.
Once mode shapes were extracted, frequency response func-tions
(FRFs) were synthesized and compared to measured data.Two different
synthesized functions are shown in Figure 5.These are not
particularly good synthesized comparisons. Thisis due to the poor
extraction of modal parameters from themodal extraction process.
These two plots are typical of thesynthesized functions for other
measurement locations on thestructure.
In addition, the modal assurance criteria (MAC) were usedfor
assessing the modes extracted. The matrix plot of the MACvalues is
shown in Figure 6. The majority of the off diagonalterms are
reasonably low, and the extracted data from this per-spective
appears acceptable even though the synthesized FRFsare not very
good. The first 25 modes extracted from the mea-sured data are
shown in Figure 7. Many of these modes are
Figure 7. First 25 structural modes of the RADARSAT1
satellite.
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18 SOUND AND VIBRATION/JANUARY 2006
Figure 10. Stability diagram for three different bands using
selectivereferences: a) 12.6-20.6 Hz; b) 36.7-40.6 Hz; and c)
44.1-48.1 Hz.
Figure 8. Modal participation matrix of RADARSAT1 satellite.
Figure 9. Summation function and mode indicator function using
se-lective references, 10-64 Hz.
primarily local modes of the main radar and solar panels of
thesatellite. While the modes appear reasonable from mode
shapeplots and from the MAC, the synthesized FRFs clearly showthat
the extracted parameters need further scrutiny.
To further evaluate these data, the modal participation fac-tors
are plotted in matrix form in Figure 7. The participationsseen in
Figure 8 clearly show that the first 25 modes are pri-marily
excited by the X-shaker reference location and the y-shaker
reference location. The higher frequency modes areactivated more
significantly from the Z-shaker reference loca-tion. To show the
detrimental effects of using all the referencesand all the measured
DOFs, a selective set of references andmeasurement locations was
used to determine the modal pa-rameters of the system in the next
case study.
Use of Selected Sets of Measured FRFs. For this evaluation,only
the X-shaker excitation location and the y-shaker excita-tion
location were used – the Z-shaker excitation locations werenot used
as references in the evaluation. Again, the frequencyresponse
measurements were evaluated over eight differentbands between 10
and 64 Hz. Poles were extracted using a time-domain, complex,
exponential, curve-fitting technique. Typi-cal mode indicator tools
were used for identifying modes of thesystem. The summation
function, multivariate mode indicatorfunction and the complex mode
indicator function were allused for identifying modes and are shown
in Figure 9 for theentire bandwidth. Comparing Figure 9 with Figure
3 shows thatthe indicator tools are much easier to interpret.
Figure 10 shows three separate stability diagrams over
threeseparate bandwidths where a selective set of references and
aselective set of measurements were used for the
evaluation.Comparing Figure 10 to Figure 4 clearly shows the
improvedsituation for selecting poles from the stability diagram.
Thecareful selection of references and measurements for the de-
termination of poles clearly has an impact on the
extractionprocess and improves the selection of poles for the
system. Theselection of poles is definitely improved through the
selectiveselection of reference location.
Once the mode shapes were extracted, frequency responsefunctions
were synthesized and compared to measured data.Two different
synthesized functions are shown in Figure 11.Both of these show
very good correlation with the actual mea-
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19SOUND AND VIBRATION/JANUARY 2006
Figure 11. Typical synthesized FRFs using selective references,
10-64Hz.
sured data and are improved when comparing them to the re-sults
shown in Figure 5. These two plots are typical of the syn-thesized
functions for other measurement locations on thestructure. Clearly,
careful selection of references and measure-ment locations for
extracting modal parameters has a signifi-cant effect on the
extracted modal parameters.
In addition, the Modal Assurance Criteria (MAC) was usedfor
assessing the modes extracted. The matrix plot of the MACvalues is
shown in Figure 12. The majority of the off-diagonalterms is
reasonably low. Some of the off-diagonal terms mayindicate spatial
aliasing; additional measurements would mini-mize this. The MAC is
not a particularly good tool for the de-tailed evaluation of the
extracted results. The MAC heavilyweights the largest values of the
shape and is not a particularlygood tool for detailed overall
assessment of the extracted pa-rameters. It is shown mainly for
reference.
Use of PolyMAX. With the very recent advancement in
modalparameter estimation using the PolyMAX approach, widebands of
frequency response measurements can be effectivelyprocessed with
little restriction on bandwidth and little needto sift the large
set of measurements to produce good pole es-timates. The PolyMAX
approach to modal parameter estima-
tion has revolutionized the modal parameter estimation pro-cess.
The same data sets described here were reprocessed us-ing all
measured DOF for all references. The mode indicatorfunction and
complex mode indicator function are shown inFigure 13 for
reference. The stability diagram is presented inFigure 14 and is
very easy to interpret. Clearly the poles ex-tracted appear to be
very well identified over this wide fre-quency range. Figure 15
compares some synthesized frequencyresponse functions for selective
measurements. These synthe-sized measurements show very good
correlation to the actualmeasurements acquired. In addition, the
Modal AssuranceCriteria was used for assessing the modes extracted.
The ma-trix plot of the MAC values is shown in Figure 16. The
major-ity of the off-diagonal terms are reasonably low. In
compari-son to modes extracted using other techniques, the
MACoff-diagonal terms are comparable or lower than that of
previ-ously extracted mode shapes.
Comparison of PolyMAX and Traditional TechniquesThe estimation
of parameters from historically used ap-
proaches is plagued by noise and mode participation
consid-erations in the estimation process. A significant amount
ofwork is required to sort the data sets into selective bands
thatare reasonably well excited by the various reference
shakerlocations to extract acceptable modal parameters. This
involvessignificant time and effort. The newer PolyMAX technique
sim-plifies this process and extracts equivalent parameters
usingwide bandwidths and without having to selectively sift
throughthe data sets for the best references to excite all
modes.PolyMAX is a significant tool for shortening the effort of
thereduction of frequency response functions for modal param-eter
extraction.
ConclusionsUsing traditional approaches, extracting modal
parameters
from multiple-referenced data may need careful selection
ofmeasurements used in the process. In general, the use of
allmeasured DOF along with all references may not
necessarilyproduce the best-extracted modal parameters with the
tech-niques historically used. A more selective selection of
refer-ences and measured DOF for extracting modal parameters
isgenerally required to produce acceptable results for
extractedmodal parameters with techniques commonly used.
Figure 12. MAC of modes using selective references.
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20 SOUND AND VIBRATION/JANUARY 2006
Figure 13. Multivariate mode indicator function (top) and
complexmode indicator function using all references (bottom).
Figure 14. Stability diagram using PolyMAX and all references,
10-64Hz.
Figure 15. Selective frequency response functions.
1.00
10.00 64.00Hz-180
180
Phas
e°
10.0–6
FRF pms:217:+X/pms:217:-XSynthesized FRF
pms:217:+X/pms:217:-X
((m
/s2 )
/N)
Log
100–9
100–3
-180
180
Pha
se°
FRF bus:101:+X/bus:101:-XSynthesized FRF
bus:101:+X/bus:101:-X
((m
/s2 )
/N)
Log
64.0010.00 Hz
Figure 16. MAC of modes using PolyMAX.
The authors may be contacted at: [email protected].
The PolyMAX technique has revolutionized the process
ofextracting modal parameters. It is shown to be robust and hasthe
ability to extract equal or better modal parameters
withsignificantly less time and user interaction with the pole
se-lection portion of the overall process.
References1. P. Guillaume, P. Verboven, S. Vanlanduit, H. Van
Der Auweraer, and
B. Peeters, “A Poly-Reference Implementation of the
Least-SquaresComplex Frequency-Domain Estimator,” Proceedings of
IMAC 21, theInternational Modal Analysis Conference, Kissimmee, FL,
February2003.
2. B. Peeters, P. Guillaume, H. Van der Auweraer, B. Cauberghe,
P.Verboven, and J. Leuridan, “Automotive and Aerospace
Applicationsof a New Fast-Stabilizing Polyreference
Frequency-Domain Param-eter Estimation Method,” Proceedings of IMAC
22, Dearborn, MI,January, 2004
3. LMS Cada-X Modal Analysis System, Leuven Measurements
Systems,Leuven, Belgium
4. LMS International, LMS Test.Lab – Structural Testing Rev 4B,
Leuven,Belgium, www.lmsintl.com, 2003.