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FREQUENCY RESPONSE FUNCTION VERSUS OUTPUT-ONLY MODAL TESTING IDENTIFICATION

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FREQUENCY RESPONSE FUNCTION VERSUSOUTPUT-ONLY MODAL TESTING IDENTIFICATION
A. Sestieri
and
W. D’Ambrogio
Dipartimento di Meccanica e Aeronautica Dipartimento di EnergeticaUniversit`a di Roma “La Sapienza” Universit`a de L’AquilaVia Eudossiana 18, I 00184 Rome, Italy Localit`a Monteluco, I 67040 Roio Poggio (AQ), Italy
ABSTRACT
The identiﬁcation of modal parameters from frequency andtime domains have now reached a high degree of maturity.Several issues have been considered on their for and against,on the possibility of using ambient excitation for large (civil)structures, on the difﬁculty of identifying scaled modes fromoutput-only data, on the quality of identiﬁed parameters whenusing data derived from either domain. However, a strict com-parison between these two techniques was not completelyperformed, especially when one has the choice of using bothof them, as it is typical when testing lab structures. More-overtheidentiﬁcationofcoincidentmodesisataskthatcannotbe fulﬁlled by SIMO methods: actually only polyreference orMIMO methods are recognized of this possibility and it is im-portant to state whether only-output tests can provide someinformation on the subject. In this paper a comparison be-tween FRF and output-only techniques, when applied to a labstructure, is performed: scaled modes are provided for bothmethods and an estimate on the quality of results assessed.
1 INTRODUCTION
Maybe it appears superﬂuous and old fashioned to speak stillabout identiﬁcation of modal parameters, especially in thisconference where the subject was dealt so deeply and forso many years. Maybe, but this contribution intends to fo-cus on a subject that, yet widely discussed and assessed,still deserves some attention for a full comprehension: we arespeaking about the comparison between FRF methods andoutput-only methods for modal parameters estimation.It should be clear to everybody involved in modal analysis thatthe methods that identify the modal parameters from some el-ements of the FRF matrix require a simultaneous measure-ment of both excitation and response. From the FRF ma-trix the modal parameters can be identiﬁed using any of theavailable techniques in the frequency domain (single or multimodes estimation, circle ﬁt, line ﬁt, etc.), or, less usually, inthe time domain (CE: Complex Exponential and LSCE: Least-Square Complex Exponential, PRCE: Polyreference ComplexExponential [1]). It should also be quite clear that the chanceof applying and measuring the desired excitation becomesquite hard when the structure is very large, such as a build-ing or a bridge, and it is about 30 years that suitable tech-niques were proposed to deal with the problem of modal iden-tiﬁcation when the excitation (input) cannot be measured or isnot available. The ﬁrst basic and fundamental works in thisarea are due to S. Ibrahim (ITD: Ibrahim Time Domain [2,3])and to the application of ARMA models [4,5]. Quite soon theywere followed by the ERA: Eigensystem Realisation Algorithm[6]. All these methods are time-domain methods, where onlythe output is available, while the excitation is provided, butnot measured, by any natural, ambient excitation such as thewind on a bridge, trafﬁc, earthquakes etc. It is even possibleto stress that, in general, until the 90’s, time and frequency do-main identiﬁcation methods implied in turn the use of outputdata only or the use of a FRF data, determined from measure-ment of both input and output.Advantages and disadvantages of any of these methods wereoften discussed: they will not be speciﬁcally presented herebecause complete and useful references can be found in theIMAC Proceedings as well as in many books, e.g. [7]. Maybeit is just the case of recalling that, usually, time domain meth-ods, yet particularly appropriate to identify close modes, werenot considered particularly efﬁcient in estimating the modaldamping, they do not provide directly scaled modes, and, par-ticularly, they were felt as more mathematical and less intuitivethan the frequency methods. Therefore, whenever possible,those involved with lab or mechanical structures continue touse frequency domain techniques while only those involvedwith large structures become real fans of time domain tech-niques.Recently this scenario begun to change or will change in thefuture because of efﬁcient techniques developed with output-only data in the frequency domain: we refer speciﬁcally to theFDD technique: Frequency Domain Decomposition [8] and
EFDD: Enhanced Frequency Domain Decomposition [9]. Ap-parently there are not particular reasons to think that output-only data are necessary or convenient to identify the modalparameters of lab and/or mechanical structures. Howevertherearesituationswherethisisthe caseor, simply, theuseofoutput-only data can be more convenient. A simple example isthe case of the identiﬁcation of the modal parameters of tyres,that, when not loaded, present coincident modes: althoughMIMO or polyreference techniques can be appropriately usedto recognize the existence of coincident modes, output-onlymethods seem to have this chance also, providing this infor-mation simpler and faster: however this possibility has not yetbeen assessed clearly. Another and deﬁnite reason to useoutput-only data would be the simplicity of conducting a modaltest, provided that the minor efforts devoted to it does not af-fect the quality of the identiﬁed parameters. These are condi-tions that we met in tests appropriately performed. Although itdoes not mean that in the future this will be the technique thatwe will use more, for the same reason it will not drive us in thedirection of avoiding output-only tests.The basic ideas of these techniques will be recalled hereafter,and the application to a lab structure (a simple plate) will beshown, by comparing results obtained by circle and line ﬁtswith those available from output-only data. Other tests, underdevelopment, are performed on a tyre, trying to reach someuseful information on the argument of coincident modes.The possibility of obtaining scaled modes is also discussedand a comparison of scaled modes obtained from standardFRF measurements and output-only data is presented.
2 FREQUENCY DOMAIN DECOMPOSITION: FDD
The vibration response of a linear system can be obtained asthe superposition of its normal modes as:
{
w
(
t
)
}
= [
Φ
]
{
q
(
t
)
}
(1)
where
[
Φ
]
is the matrix of the system eigenvectors and
{
q
(
t
)
}
the vector of modal coordinates.For a random stationary process, the covariance matrix of re-sponses is:
[
R
w
(
τ
)] =
E
{
w
(
t
)
}{
w
(
t
+
τ
)
}
T
=
E
[
Φ
]
{
q
(
t
)
}{
q
(
t
+
τ
)
}
T
[
Φ
]
T
Since modeshapes are independent of the random process,one can also write
[
R
w
(
τ
)] = [
Φ
]
E
{
w
(
t
)
}{
w
(
t
+
τ
)
}
T
[
Φ
]
T
= [
Φ
][
R
q
(
τ
)][
Φ
]
T
(2)
where
[
R
q
(
τ
)]
is the covariance matrix of the modal coordi-nates. The Fourier transform of covariance is the power spec-tral density, so that, by Fourier transforming, one obtains:
[
S
w
(
f
)] = [
Φ
][
S
q
(
f
)][
Φ
]
T
(3)
If one now assumes the the excitation is uncorrelated, as inthe case of an ambient random excitation or, as typical inStatistical Energy Analysis, in the case of rain-on-the-roof ex-citation, the covariance matrix of modal responses is diago-nal, and, consequently, also the power spectral density of themodal responses
[
S
q
(
f
)]
is diagonal. Moreover, since the di-agonal terms of the spectral density matrix are the auto spec-tral density terms, they are intrinsically real and positive.If one now performs the singular value decomposition of ageneral matrix
[
A
]
, one has:
[
A
] = [
U
][
`
Σ
`
][
V
]
T
where
[
`
Σ
`
]
is the diagonal matrix of singular values, while
[
U
]
and
[
V
]
are orthogonal matrices whose columns are calledleft and right singular vectors. They are the eigenvectors of
[
A
][
A
]
T
and
[
A
]
T
[
A
]
, respectively. If
[
A
]
is square and sym-metric, it is of course
[
U
] = [
V
]
, so that we have:
[
A
] = [
U
][
`
Σ
`
][
U
]
T
(4)
By comparing the expression of the singular value decom-position (4) with equation (3) and considering that
[
S
q
(
f
)]
isdiagonal, one could stress that, if
[
Φ
]
would be orthogonal,the SVD of the measured response spectral matrix
[
S
w
]
ata frequency line
f
would permit to identify the modal matrix
[
Φ
]
as the matrix of singular vectors
[
U
]
, so that it would bepossible to determine the whole modal matrix from a singlespectral line. Unfortunately,
[
Φ
]
is not generally orthogonal(
[
Φ
]
T
[
Φ
]
= [
`
I
`
]
). However, in this case, by considering thespectral line where a single mode is dominant, a good esti-mate of a modeshape can be determined by the singular vec-tor corresponding, for the analyzed frequency, to the highestsingular value (
σ
1
). In fact, one can write explicitly the SVD byisolating the largest singular value as:
[
A
] =
{
U
1
}
σ
1
{
U
1
}
T
+
m
i
=2
{
U
i
}
σ
i
{
U
i
}
T
(5)
while, by isolating the
r
-th mode, from equation (3) one ob-tains:
[
S
w
(
f
)] =
{
Φ
r
}
S
q
r
(
f
r
)
{
Φ
r
}
T
+
m
i
=1
,i
=
r
{
Φ
i
}
S
q
i
(
f
i
)
{
Φ
i
}
T
(6)
A comparison between these last two relationships showsthat, when neglecting the sums, the singular vector corre-sponding to the ﬁrst singular value (the largest) is certainlya good approximation for the
r
-th mode in that the effect ofthe adjacent modes is minimum.Of course, by this procedure the identiﬁed modes are not cor-rectly scaled - the mass matrix of the system is unknownand the scaling factors are dependent on the unknown (un-measured) excitation - so that an appropriate scaling pro-cedure must be developed to obtain correctly scaled mode-shapes and, more speciﬁcally, modeshapes scaled to unitmodal mass matrix. A speciﬁc procedure to obtain this goal isdescribed in section 4.
3 THE ENHANCED FREQUENCY DOMAINDECOMPOSITION (EFDD)
The FDD is generally a good approach to estimate naturalfrequencies and modeshapes but cannot be used to identifythe modal damping. The EFDD provides this information anda better estimate of the previous modal parameters. It per-forms a detailed analysis in the neighborhood of a natural fre-quency identiﬁed by the FDD, by means of a suitable use oftheMACthatidentiﬁes, foreachmode, asingledegreeoffree-dom spectral density of the response and puts to zero the restof the response. In this way the spectral response of a singledegree of freedom is obtained so that it can be transformedback to the time domain and the modal parameters of suchsingle DOF system determined by simple operations.A suitable frequency interval for the analysis of each singlemode, identiﬁed by the FDD, is determined by using an appro-priate MAC threshold value. In the neighborhood of a pickedfrequency corresponding to an FDD identiﬁed modeshape, aMAC is computed between the reference vector and the sin-gular vectors in the selected range. If the largest MAC valueof this vector is above the threshold MAC value, the corre-sponding singular value is considered in the response of thatmodal coordinate, while the others are put to zero. To im-prove the modes estimate, the singular vectors are weightedby the MAC values determined above. At this point dampingand natural frequency of that mode are estimated by inversetransforming the single DoF spectral bell into the time domain,which provides a single DoF correlation function, by a sim-ple regression analysis. This correlation function is an almostdamped harmonic response, so that the natural frequency canbe estimated by the harmonic period and the damping by thelog decrement of this damped signal.
4 SCALING OF OPERATIONAL MODESHAPES
Few proposals have been made recently to scale appropri-ately the determined modeshapes identiﬁed from output-onlydata. The most simple and advantageous seem to be thosebased on the sensitivity of the modal parameters, particularlyeigenvalues, to lumped mass changes on the tested struc-ture. This approach requires an additional test but it is ap-parent that it permits to have good results, provided that theapplied mass(es) are exactly known, by measuring simply thenatural frequency shifts between the srcinal structure and themass-modiﬁed one. The approach followed in this work refersexplicitly to a recent proposal by Parloo et al. [10]. Althoughthe method used is exactly the method proposed in that paper,the derivation of the scaling formula is slightly different. In factan undamped (or very lightly damped) system with real modesis considered here. However the approach can be quite easilygeneralized to provide the same formulas presented in [10].For an undamped discrete system, the eigenvalue problem isgiven by:
([
K
]
−
λ
[
M
])
{
Ψ
}
= 0
.The sensitivity of the system eigenvalues to a parameter vari-ation (speciﬁcally, here, a lumped mass variation located atDoF
k
, that must be a DoF where the response was srcinallymeasured) can be determined from the previous equation as:
∂
[
K
]
∂m
k
−
λ∂
[
M
]
∂m
k
−
∂λ∂m
k
[
M
]
{
Ψ
}
+([
K
]
−
λ
[
M
])
∂
{
Ψ
}
∂m
k
= 0
The derivative of the stiffness matrix is obviously zero. Pre-multiplying by
{
Ψ
}
T
, the second term on the right hand sideis identically zero for the symmetry of the mass and stiffnessmatrices, so that one has:
−
λ
{
Ψ
}
T
∂
[
M
]
∂m
k
{
Ψ
}−
∂λ∂m
k
(
{
Ψ
}
T
[
M
]
{
Ψ
}
= 0
(7)
By assuming now that only a single mass is varied at DoF
k
and that the normalization condition
{
Ψ
}
T
[
M
]
{
Ψ
}
=
b
i
(
b
i
=1
for unit modal mass) is used, one obtains:
∂λ
i
∂m
k
=
−
λ
i
Ψ
2
i,k
b
i
(8)
where
λ
i
=
ω
2
i
for undamped systems (real modes).Usually in output-only modal analysis the identiﬁed modes
{
Φ
}
are not scaled, so that we can relate the unscaled modesto those (
{
Ψ
}
) scaled to
b
i
by the following relationship:
{
Ψ
}
i
=
α
i
{
Φ
}
i
(9)
so that, by substituting it into equation (8) and passing fromthe derivatives to the ﬁnite differences, one ﬁnally obtains(compare [10]):
α
i
=
−
b
i
∆
λ
i
λ
i
Ψ
2
i,k
∆
m
k
(10)
This is a ﬁrst order approximation in that differential quantitiesare substituted by ﬁnite differences, that could be affected by(more or less signiﬁcant) changes in the modeshapes.If instead of a unique mass variation at a single DoF, the massis varied at a set of DoFs
N
, the previous equation simplycorrects into:
α
i
=
−
a
i
∆
λ
i
λ
i
N k
=1
Ψ
2
i,k
∆
m
k
(11)
Finally if the modes are to be normalized to unit modal mass(
b
i
= 1
) the unknown scaling factor is obtained as:
α
i
=
−
(∆
ω
i
)
2
ω
2
i
N k
=1
Ψ
2
i,k
∆
m
k
−
2∆
ω
i
ω
i
N k
=1
Ψ
2
i,k
∆
m
k
(12)
being
∂ω
2
i
/∂m
k
= 2
ω
i
∂ω
i
/∂m
k
.
By the previous expressions it is possible to obtain any wishedscaling of the modeshapes identiﬁed from the output-only ex-perimentbyperformingasecondtestwheresomemass(es)isadded on the measured DoFs. The mass variation (either addor removal) creates a shift on the natural frequencies of thesystem. The estimate of such shift, together with the knowl-edge of the srcinal modal parameters is sufﬁcient to yield thescale factor
α
i
. Of course the mass variation must be smallenough to permit a ﬁrst order approximation while it shouldbe appropriately located to permit a signiﬁcant measurableshift of the natural frequencies. This is actually a critical point,possibly the real weakness of the method, because small dif-ferences in the shift estimation can lead to signiﬁcant errorsin the determination of the scaled factor. This considerationallows to state that mass variations at different DoFs wouldallow better estimates than a single variation, provided thatthey are still sufﬁciently small to stress that the modeshapesare not signiﬁcantly changed by these new mass variations(see also [11]). Therefore the method must be appropriatelyoptimized, by performing a careful analysis on the mass dis-tribution and location(s).
5 RESULTS AND COMPARISONS
To test the features and accuracy of output-only methods, twosets of tests are performed: the ﬁrst one on a simple free-supported plate, the second on an automotive tyre. The ﬁrsttest on the plate is aimed to perform a complete comparisonbetween the modal parameters identiﬁed from traditional FRFdata - a SIMO method - where a single excitation (hammer im-pulse) and a set of responses (accelerations) were measured,and those determined from output-only data when applyingan uncorrelated random excitation along the plate (type-rain-on-the-roof force applied for about 15 seconds). The modalparameters extracted from the FRF measurements were de-termined by using the ICATS code (Imperial College) while,for the output-only data, the ART&MIS (SVS) code (FDD andEFDD) was used. The comparison was aimed to establishboth the accuracy of the identiﬁed parameters as well as thesimplicity and rapidity in performing the tests. The plate waschosen because it is a very simple structure, and the tests arewell controllable and easy to perform. For the same reason,some conclusions can go lightly astray.The second test on the tyre, yet under development, is aimedto analyze the features of the output-only method on a morecomplex mechanical structure, trying to enlightening the pos-sibility of determining double modes. This analysis is not yetconcluded and results will be presented soon in the future.The plate considered is a rectangular aluminum plate, size
0
.
3
×
0
.
34
×
0
.
003
m, whose total weight is
0
.
8
kg. The edgedimensions of the plate were chosen similar to have closemodes. For both the tests the response was measured ona grid of 20 points, uniformly distributed (ﬁgure 1). Since only5 equal accelerometers were used together in performing thetest, to avoid resonance shifts during each set of measure-ments, the plate was loaded by discrete masses, equal to theaccelerometers’ mass (7 gr) lumped at the 20 measurementpositions. (With this added masses the weight of the plate be-comes
0
.
94
kg). When performing each set of measurements,the accelerometers were appropriately located and the corre-sponding masses removed. To obtain free support conditionsthe plate was suspended through a wire, leading to rigid bodymodes within the ﬁrst 3 Hz, while the ﬁrst natural frequency ofthe free plate ranges around 80 Hz.
1
2
3
4
5
6
7
8
9
10 11 12
13 14 15 16 17 18
19
20
Figure 1: Flexural tested plate
For the SIMO test, an impact excitation was provided by ahammer with a rubber tip and the drive point was appropri-ately chosen to avoid nodal points (point 16 of the structureshown in ﬁgure 1). With this excitation a signiﬁcant FRF wasdetermined until 600 Hz, in which range 7 modes are ob-served. The modal parameters were extracted by using asimple circle-ﬁt code, providing the modal parameters shownin table 1.
TABLE 1: Modal parameters identiﬁed from FRFs
Mode n. Nat. freq.
f
n
[Hz] Damping
η
n
%
1 81.0 2.412 118.4 1.713 163.9 1.444 201.5 0.995 218.3 0.866 346.3 0.577 423.8 0.65For the output-only test, a reference node at location 21 (ﬁg-ure 1) was used for all the measurement sets - 4 accelerationssimultaneously measured and moved for each of the 5 sets,plus the reference accelerometer, kept ﬁxed at node 21. Theexcitation used was made by an uncorrelated series of im-pulses applied across the surface of the plate for 15 seconds.

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