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Research ArticleModal Identification Using OMA Techniques:Nonlinearity Effect
E. Zhang,1,2 R. Pintelon,2 and P. Guillaume3
1School of Mechanical Engineering, Zhengzhou University, Science Road 100, Zhengzhou 450000, China2Department of Fundamental Electricity and Instrumentation, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium3Department of Mechanical Engineering, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Correspondence should be addressed to E. Zhang; erliang.zhang@zzu.edu.cn
Received 24 March 2015; Accepted 14 June 2015
Academic Editor: Isabelle Sochet
Copyright Β© 2015 E. Zhang et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper is focused on an assessment of the state of the art of operational modal analysis (OMA) methodologies in estimatingmodal parameters from output responses of nonlinear structures. By means of the Volterra series, the nonlinear structure excitedby random excitation is modeled as best linear approximation plus a term representing nonlinear distortions. As the nonlineardistortions are of stochastic nature and thus indistinguishable from the measurement noise, a protocol based on the use of therandom phase multisine is proposed to reveal the accuracy and robustness of the linear OMA technique in the presence of thesystem nonlinearity. Several frequency- and time-domain based OMA techniques are examined for the modal identification ofsimulated and real nonlinearmechanical systems.Theoretical analyses are also provided to understand how the systemnonlinearitydegrades the performance of the OMA algorithms.
1. Introduction
Operational modal analysis (OMA) aims at identifying themodal properties of a structure based on response data of thestructure excited by ambient sources. Unlike experimentalmodal analysis (EMA) [1], the OMA is performed in opera-tional conditions of the vibrating structure, where the excita-tion is inaccessible to be measured or hard to be applied. Theresearch activity around the theoretical basis of the OMA hasbeen largely increased, and powerful OMA techniques havebeen developed in both the time- and frequency-domains,as collected in recent tutorial work [2, 3]. The time-domainmethodologies estimate the modal model based on a state-space representation of the system obtained from the time-domain data [4], while the frequency-domain approachesidentify the modal parameters from power spectral densityfunctions of output responses [5, 6]. System identificationmethods for the OMA are extensively reviewed in [7, 8],differentOMAand EMAmethods for the estimation of lineartime-invariant (LTI) dynamical models are compared in anextensive Monte Carlo simulation study [8]. The state of the
art of OMA methodologies is also assessed for estimatingmodal parameters of a helicopter structure in a laboratoryenvironment, where the helicopter structure behaves linearlyby controlling the excitation level [9].
The OMA techniques have been utilized to estimate themodal properties of important structures for the purpose ofhealth monitoring and maintenance, such as wind turbines[10], bridges [11], and historical aqueduct [12]. Generally,with the real-life vibration data, the detection of structuraldamage by estimating the modal properties is complicatedby the impact of changing environmental conditions andthe nonwhiteness property of the ambient excitation. Bothof them can induce significant changes of the monitoredalert feature. The former problem is treated using factoranalysis, and damage is detected using statistical processcontrol [13].The latter problem can be ideally solved throughthe concept of the transmissibility, as the input excitationis canceled in computing transmissibility functions betweenoutput responses. The transmissibility functions are fullyexploited for the OMA, as reported in [14, 15]. In additionto the above two factors, the output-only modal estimate
Hindawi Publishing CorporationShock and VibrationVolume 2015, Article ID 178696, 12 pageshttp://dx.doi.org/10.1155/2015/178696
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2 Shock and Vibration
is also perturbed by the system nonlinearity. Indeed, theOMA techniques are all derived based on LTI dynamicalmodels, while all the actual structures are nonlinear tosome extent. Moreover, the structures to which the OMAapproaches are applied are assemblies, often utilize compositematerials, and even have flexible components. Diverse typesof nonlinearities can be present, such as geometric nonlin-earity due to the large deformation experienced by a flexiblestructure, material nonlinearity owing to a nonlinear stress-strain constitutive law, contact nonlinearity resulting fromboundary conditions. Very often, the real-life structures tobe monitored are exposed to nonstationary and even severeambient excitation, for instance, caused by earthquake orhurricane. Within such context, it is therefore of paramountimportance to study the influence of the system nonlinearityon the output-only modal estimation for the purpose ofthe (fully automatic) structural damage detection. To theauthorsβ knowledge, this fundamental issue still remains to beaddressed. The way of dealing with the system nonlinearitycan be split into two groups.The first group considers explic-itly a nonlinearmodel such as nonlinear ARX (autoregressivewith exogenous inputs) for parameter estimation [16, 17].Thesecond group still considers linear models while boundingthe estimation variability due to the nonlinear distortions.The latter fits in the scope of this paper as the goal ofthe present work is to assess the performance of the LTImodel based OMA technique with respect to the systemnonlinearity.
The system nonlinearity strongly depends on the classof excitation signal. The ambient excitation is usually not ofGaussian type. However, approximate Gaussian distributioncan occur in many situations, as explained by the CentralLimit Theorem. As reported in literature, only the first- andsecond- order statistical properties of the ambient excitationare exploited in the OMA algorithms. So a normally dis-tributed ambient excitation is considered herein. The presentwork is confined to nonlinear Wiener systems that canbe approximated in the mean square sense by a Volterrasystem for Gaussian excitation. The major property of aWiener system is that the steady state response to a periodicinput is periodic with the same period as the input. Thenonlinear structure of the Wiener system can be modeled asa related linear system plus a part representing the nonlineardistortions under random excitation [18, 19]. The latter isproved to be asymptotically normally distributed, mixingof order infinity over frequency under Gaussian excitation[18]. Then, considering them in the weighting matrix of thecost function, applying LTI system identificationmethod willreturn best (in the least square sense) linear approximation(BLA) of the nonlinear system.
Disturbed by the measurement noise in observation data,the stochastic nonlinear distortions are not recognizable,usually naively treated as independent noise. Therefore, theissues of separating them from noisy data and of generatingtheir stochastic realizations should first be addressed in orderto quantify the performance of the OMA algorithm. To thisend, the random phase multisine (RPM) is advocated asan excitation signal for the OMA. The RPM is normallydistributed by increasing the number of harmonic lines and
advantageous over the Gaussian noise due to its flexibledesign and its periodicity. The amplitudes of harmonic com-ponents in the RPM are set constant to meet the whitenessassumption of the excitation, as required by most OMAalgorithms. The random phases of the RPM then becomethe sole factors to determine the nonlinear distortions of thenonlinear system. Additionally, the periodic property of theRPM leaves Fourier transformation of input/output data freeof the leakage error.
Despite the existence of a large number of alternativeOMA techniques developed during the last decades, theyare, however, just based on few basic principles. Therefore,the present work is mainly focused on three representativemethods with completely distinct theoretical backgrounds:frequency-domain decomposition (FDD), stochastic sub-space identification (SSI), and transmissibility based opera-tional modal analysis (TOMA).Themain contribution of thepresent work is to provide a protocol to assess the modalestimation of nonlinear structures by the OMA techniques,which is based on the use of the RPM.Theprotocol comprisestwo parts: one part is to evaluate the accuracy of the LTImodel based OMA algorithm in comparison with the wellestablished input-output approach; the other is to examine itsrobustness to the nonlinear distortions. Further, the protocolis carried out for a series of excitation levels in order to revealextensively the performance of the OMA techniques in thepresence of the systemnonlinearity. Also, theoretical analysesare provided to understand how the system nonlinearityinfluences the accuracy of the OMA algorithms.
The rest of the paper is structured as follows. Section 2 isdevoted to building a nonlinear framework for system iden-tification. Section 3 recapitulates the theoretical backgroundof the considered OMA techniques. Section 4 describes theprotocol dedicated to investigating OMA techniques in thepresence of the structural nonlinearity. Section 5 illustratesthe output-only identification of simulated and real nonlinearstructures. Concluding remarks are given in Section 6.
2. Nonlinear System Modeling
2.1. Random Phase Multisine. The RPM plays a crucial rolein designing a protocol to assess the accuracy of estimatingnonlinear structures by OMA techniques. With a uniformamplitude π΄0, it takes the following form:
π (π‘) = π΄0
π/2β1β
π=βπ/2+1exp (π2ππ
π ππ‘ + π
π) , (1)
where ππ is the clock frequency of the arbitrary waveform
generator, π is the number of samples in one signal period,and the set of the phases {π
π} is a realization of an inde-
pendent distributed random process on [0, 2π) such thatE[exp(ππ
π)] = 0 and π
βπ= π
π. The amplitude of the RPM
is used to set the excitation level, and its random phasesdetermine the nonlinear distortions of the system.
2.2. Volterra Series of a Nonlinear System Output. A descrip-tion of a nonlinear system by means of the Volterra series
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Shock and Vibration 3
Nonlinear dynamic system Best linear approximationF0(k)G(k)
Y0(k) Y0(k)F0(k)
GBLA(k)
YLin(k)YS(k)
+
Figure 1: Modeling of a nonlinear system under a normally distributed excitation. πΉ0(π) and π0(π) are the input-output noise-free spectra,πΊBLA(π) is the best linear approximation of the nonlinear system, πLin(π) is the linear part of π0(π), and ππ(π) represents the nonlineardistortions in π0(π).
is formally introduced and the stochastic property of thesystem nonlinearity is presented. The case of single output isconsidered for the ease of illustration. By the Volterra series,the output of a nonlinear system is decomposed into
π¦ (π‘) =
+β
β
πΌ=1π¦(πΌ)(π‘) , (2)
where π¦(πΌ)(π‘) is the contribution of degree πΌ
π¦(πΌ)(π‘)
= β«
+β
ββ
β β β β«
+β
ββ
β(πΌ)(π1, . . . , ππΌ)
πΌ
β
π=1π (π‘ β π
π) dπ1,
. . . , dππΌ,
(3)
where π(π‘) denotes the excitation (driven by the RPM) andβ(πΌ)(π1, . . . , ππΌ) (πΌ > 1) is the generalization of the impulse
function β(1)(π) and is referred to as Volterra kernel, whosesymmetrized frequency-domain representation is written as
π»(πΌ)
π1 ,...,ππΌ= β«
+β
ββ
β β β β«
+β
ββ
β(πΌ)(π1, . . . , ππΌ)
β πβπ2ππ
π (π1π1+β β β +ππΌππΌ)dπ1, . . . , dππΌ.
(4)
Applying the discrete Fourier transform to (2) along with (4),it follows that at the πth frequency line π(π) = β+β
πΌ=1 π(πΌ)(π),
with
π(πΌ)(π) =
π/2β1β
π1 ,...,ππΌβ1=βπ/2+1π»
(πΌ)
π1 ,...,ππΌβ1 ,πΏππΉ (π1)
β πΉ (π2) β β β πΉ (ππΌβ1) πΉ (πΏπ) ,
(5)
and πΏπ= π β β
πΌβ1π=1 ππ.
2.3. Best Linear Approximation and Nonlinear Distortions.The frequency response function of a nonlinear system is bydefinition computed as
πΊ (π) =π (π)
πΉ (π)=
+β
β
πΌ=1
π(πΌ)(π)
πΉ (π)= π»
(1)πβββββββ
πΊ0(π)
+
+β
β
πΌ=2
π/2β1β
π1 ,...,ππΌβ1=βπ/2+1π»
(πΌ)
π1 ,...,ππΌβ1,πΏπ
πΉ (π1) πΉ (π2) β β β πΉ (ππΌβ1) πΉ (πΏπ)
πΉ (π)βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββ
πΊ(πΌ)
(π)
,
(6)
where πΊ0(π) is the underling linear system and β+β
πΌ=2 πΊ(πΌ)(π)
represents the system nonlinearity.Under the excitation of the RPM, β+β
πΌ=2 πΊ(πΌ)(π) is further
decomposed into a systematic partπΊπ΅(π)which depends only
on the amplitude of the RPM and a stochastic part denotedby πΊ
π(π) which depends on both the amplitude and the
phase of the RPM. πΊπ(π) behaves as uncorrelated noise (over
frequency line), and further πΊπ(π) is an asymptotically zero-
mean circular complex normal variable with respect to therandom realization of the phases of the RPM [18].πΊ0(π) alongwithπΊ
π΅(π) constitutes the best linear approximation (BLA) of
the nonlinear system in the least square sense,
πΊ (π) = πΊ0 (π) + πΊπ΅ (π)βββββββββββββββββββββββββββπΊBLA(π)
+πΊπ (π) . (7)
It is stressed that, through the term πΊπ΅(π), πΊBLA(π) depends
on the loading condition (level or location of the excitation)applied to the distributed nonlinear structure. By (7), theoutput of a nonlinear system is accordingly split into twoparts: the first part that is related to the input and the secondpartπ
π(π) that is uncorrelatedwith the input over the random
phases of the RPM, as illustrated in Figure 1. The outputnonlinear distortions π
π(π) are related to πΊ
π(π) as
ππ (π) = πΊπ (π) πΉ0 (π) (8)
and have similar stochastic properties as πΊπ(π) (see [18] for
proof details). πΉ0(π) is the force applied by the exciter, whichis controlled with the RPM defined in (1).
3. Theoretical Background of OperationalModal Analysis Techniques
In this section, the theoretical background of the OMAtechniques considered in this paper is briefly recapitulated,Note that they have all been developed with the purpose ofextracting the underlying system πΊ0(π) in (6). More detailscan be found in the cited references.
3.1. Frequency-DomainDecomposition (FDD). The frequencyresponse matrix G(π) is expressed as a sum of the contribu-tions of system modes in a frequency band of interest,
G (π) =ππ
β
π=1
Rπ
ππ β ππ
+Rπ
ππ β ππ
, (9)
where Rπ= π
ππΎππwith π
πthe column vector of the πth mode
shape and πΎπthe column vector of the modal participation
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4 Shock and Vibration
factor and ππ= (βπ
π+πβ1 β π2
π)π0,π with ππ the damping ratio
and π0,π the angular frequency (rad/s).Theoretically, assuming that the ambient excitation has a
constant spectrum SπΉπΉ, the noiseless output power spectral
density matrix is written as
SYY (π) = G (π) SπΉπΉGπ»(π) . (10)
Substituting G(π) by its modal decomposition in (9) andusing the Heaviside partial fraction theorem for polynomialexpansions, SYY(π) can be written as
SYY (π) =ππ
β
π=1
Aπ
ππ β ππ
+A
π
ππ β ππ
+Aπ
π
βππ β ππ
+Aπ»
π
βππ β ππ
,
(11)
where
Aπ= R
πSπΉπΉ
ππ
β
π=1
Rπ»π
βππβ π
π
+Rππ
βππβ π
π
. (12)
In the vicinity of the πth eigenfrequency, π β sub(π0,π),it approximatively holds by exploiting the lightly dampedproperty that
AπβRπSπΉπΉRπ»π
2πππ0,π
= ππ
πΎππSπΉπΉπΎπ
2πππ0,πππ»
π= π
πππππ»
π. (13)
Then for π β sub(π0,π) it is derived that
SYY (π) βππππππ»π
ππ β ππ
+ππππππ»π
βππ β ππ
= 2 Re(ππ
ππ β ππ
)ππππ»
π,
(14)
where Re(π) is the real part of the complex numberπ.Assuming the orthogonality of the mode shapes, (14)
can be interpreted as a singular value decomposition (SVD)of SYY(π) where only one single component is dominantwhile π β sub(π0,π). This suggests a simple procedure toestimate the modal parameters: decomposing the spectralresponse into a set of single degree of freedom systems usingthe SVD, each corresponding to an individual mode. Thepoles are estimated by the enhanced FDD [20]. The unscaledmode shape is obtained by picking out the correspondingeigenvector of the estimated power spectral density matrix.
With the real-life data, the estimate of the output powerspectral density matrix can be (heavily) biased due to thepresence of the nonlinear distortions which are mutuallycorrelated over space in the frequency-domain.The use of theSVD can alleviate this situation by discarding lower singularvalueswhich correspond to less significant components in thenoisy data.
3.2. Stochastic Subspace Identification (SSI). The SSI methodestimates the modal parameters based on the stochasticdiscrete-time state-space model of a mechanical structure,
xπ+1 = Axπ +wπ,
yπ= Cx
π+ k
π,
(15)
where the subscript π denotes the time instant, xπis a vector
of the system state, yπis an output vector, A is the discrete
state matrix, C is the selection matrix, wπis a vector of noise
due to the (white) random excitation, and kπis a vector of
noise representing the sum of the random excitation and themeasurement noise.
Two versions of the SSI are typically used [4, 21]: datadriven SSI and covariance driven SSI (SSI-COV). The latteris taken for algorithmic explanation in what follows. TheSSI-COV algorithm starts with the covariance matrix of thestructural response with π
πreference outputs yref
πof them,
Ξrefπ= E [y
π+π(yref
π)π
] , (16)
where the expectation operation (E) is performed with thepurpose of correlating out the unwanted dynamics to obtainthe system description. However, the nonlinear distortionsin the output noise term k are mixing of infinite order overtime (self-correlated), whose components are also mutuallycorrelated over space.This leads to a biased covariancematrixΞref
π, and eventually decreases the estimation accuracy.A block Toeplitz matrix is constructed by assembling the
output covariance matrices
Lref1|ππ
=
[[[[[[[
[
Ξrefππ
Ξrefππβ1 β β β Ξ
ref1
Ξrefππ+1 Ξ
refππ
β β β Ξref2
.
.
.... β β β
.
.
.
Ξref2ππβ1 Ξ
ref2ππβ2 β β β Ξ
refππ
]]]]]]]
]
. (17)
UsingΞrefπ= CAπβ1Gref withGref the so-called state-reference
output covariance matrix, Lref1|ππ
is decomposed into
Lref1|ππ
=
[[[[[[[
[
CCA...
CAππβ1
]]]]]]]
]βββββββββββββββββββ
Oππ
[Aππβ1Gref Aππβ2Gref β β β AGref Gref]βββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββββCrefππ
,(18)
where Oππ
is called the extended observability matrix andCref
ππ
is the reference-based stochastic controllability matrix.Theoretically, from the observability matrix O
ππ
, the statematrix A and the output selection matrix C can be obtained.The natural frequencies, damping ratios, and unscaled modeshapes are finally derived from the estimated matrices A andC.
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Shock and Vibration 5
3.3. Transmissibility Based Operational Modal Analysis(TOMA). The case of the scalar transmissibility functionis considered for the algorithmic illustration. The scalartransmissibility function is defined in the Laplace domain asthe ratio of the output spectrum at the πth degree of freedom(dof) and the one at the πth dof under the excitation of anunknown force at the πth dof,
π(π)
ππ(π ) =
π(π)
π(π )
π(π)
π(π )
=πΊ
ππ (π ) πΉπ (π )
πΊππ (π ) πΉπ (π )
=πΊ
ππ (π )
πΊππ (π )
. (19)
The kernel idea of the TOMA approach is that the scalartransmissibility functions of a linear structure, estimatedwiththe response data from different loading conditions, crosseach other at the poles of the system.Using (9), the limit valueof the transmissibility function when π β π
π
limπ βπ
π
π(π)
ππ(π ) = lim
π βππ
(π β ππ) πΊ
ππ (π )
(π β ππ) πΊ
ππ (π )
= limπ βπ
π
βππ
π=1 [(π β ππ) ππππΎππ/ (π β ππ) + (π β ππ) ππππΎππ/ (π β ππ)]
βππ
π=1 [(π β ππ) ππππΎππ/ (π β ππ) + (π β ππ) ππππΎππ/ (π β ππ)]
=ππππΎππ
ππππΎππ
=πππ
πππ
,
(20)
which is independent of the input. Combining the transmis-sibility functions of two distinct loading locations (π) and (π),it follows that
limπ βπ
π
Ξπ(ππ)
ππ(π ) = lim
π βππ
[π(π)
ππ(π ) β π
(π)
ππ(π )]
=πππ
πππ
βπππ
πππ
= 0.(21)
Different ways are reported in the literature to extract themodal parameters based on (21); see, for instance, [14, 15].The approach employed here is based on the parametricallyestimated scalar transmissibility functions. The transmissi-bility function admits a rational form, as indicated by (19).Choosing the πth output as reference, a common denomina-tor rationalmodel is used to parameterize the transmissibilityfunctions, whose parameters are estimated using the samplemaximum likelihood method [18].
Then, using the transmissibility functions estimated fromthe loading conditions (π) and (π), respectively, the functionused for modal identification is defined as
Ξβ1π(ππ)(π , οΏ½ΜοΏ½) =
1β
πβD [π(π)
ππ(π , οΏ½ΜοΏ½) β π(π)
ππ(π , οΏ½ΜοΏ½)]
=π΅ (π , π
π)
π΄ (π , ππ),
(22)
where D denotes the set of the dofs of the outputs exceptthe πth one. π
πcan be easily derived from οΏ½ΜοΏ½ by means of
[1,1](t)
[2,1](t)
[Nπ ,1](t)
[1,2](t)
[2,2](t)
[Nπ ,2](t)
[1,Nπ](t)
[2,Nπ](t)
[Nπ ,Nπ](t)
Β· Β· Β·
Β· Β· Β·
Β· Β· Β·
...
Period
Exp. #1
Exp. #2
z
z
z
z z
zz
z z Exp. #NR
Figure 2: Scheme of multiple experiments, z(π‘) = [π(π‘), yπ(π‘)]π,y(π‘) = [π¦1(π‘), π¦2(π‘), . . . , π¦π
π¦
(π‘)]π with π
π¦the number of outputs and
the superscriptπ the transpose operator.ππandπ
π are the number
of periods and experiments, respectively.
symbolic computation. The poles are then obtained from ππ
and the unscaledmode shapes are estimated by evaluating theidentified transmissibility functions at the estimated poles.
Note that the TOMAmethod is superior to other output-only identification techniques in dealing with colored ambi-ent excitation; however, the need of combining different load-ing conditionsmakes it vulnerable to the system nonlinearity.
4. Assessment Protocol
The output-only modal identification of nonlinear structuresis conducted for a series of excitation levels. At each excitationlevel, a protocol is applied to assess the algorithm perfor-mance with respect to the nonlinearity, which is establishedin the following sections.
4.1. Uncertainty Bound Based on Multiple Experiments. Withthe RPM defined in Section 2.1 as an excitation signal,the measurement noise accounts for the difference of theobserved data over the periods while the nonlinear distor-tions are deterministic once the phases of the RPM are fixedin an experiment.Therefore, themeasurement noise is kickedout by averaging vibration data over period and the denoisedoutput data are mainly corrupted by (a realization of) thenonlinear distortions. The efficient way for examining therobustness of the algorithm is to provide an uncertaintybound induced by the system nonlinearity in the form offormula. However, unlike the independent disturbing noise,the nonlinear distortions are dependent on input signals.This will create higher order moments in the variancecalculations formodal estimates, which cannot be captured ina linear system identification framework. As a result, a MonteCarlo analysis based on multiple experiments with differentrandom realizations of the RPM is needed.
The strategy of multiple experiments aims at generatinga set of realizations of the nonlinear distortions, over whichthe robustness of the OMA algorithm is completely assessedfor a specified excitation level, as depicted in Figure 2.Each experiment is conducted with an independent phaserealization of the RPM, and a number of consecutive periodsof the steady state response are measured.
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6 Shock and Vibration
m
c
k
m
c
k
m
c
k
m
c
k
x1 x2 x3 x4
kNL
cNL
(a)
20
1 53 7 9
20
2.6 3 3.4 3.8
11
β20
β60
Mag
nitu
de (d
B)
Frequency (Hz)
(b)
Figure 3: (a) 4DOF model with local nonlinearity (π = 1 kg, π = 103 Nβ mβ1, π = 0.5 kgβ sβ1, πNL = π + 50π₯31(π‘), and πNL = π + 5 Γ 10
β3οΏ½ΜοΏ½24(π‘)).
(b) Evolution of the BLAs associated with the 3rd mass when the input is located at the 4th mass with three levels of excitation and perturbedlines displayed below represent the standard deviations of the nonlinear distortions (black: strong, dark gray: medium, and light gray: weak).
The dataset, collecting output data from all the experi-ments, is used to investigate the uncertainties of the modalestimates in light of the following procedure:
(i) Considerβπ = 1, . . . , ππ .
(a) Feed the shaker with the RPM of the πth phaserealization and acquire the noisy output signals{y[π,π](π‘)}ππ
π=1.(b) Average the output data over the periods, leav-
ing them mainly corrupted by the nonlineardistortions,
yΜ[π] (π‘) = 1π
π
ππ
β
π=1y[π,π] (π‘) , (23)
where themeasurement noise in yΜ[π](π‘) vanishesat the rate of 1/βπ
π.
(c) Apply each OMA algorithm to yΜ[π](π‘) to extracteigenfrequencies, damping ratios, and unscaledmode shapes.
(ii) Repeat steps (a)β(c) for all the ππ experiments, a set
of modal estimates being delivered, over which anyorder statistics can be empirically computed.
4.2. Identification of Best Linear Approximation. The OMAapproaches fail to extract the underlying linear system dueto the presence of the bias term πΊ
π΅in (7) induced by the
nonlinear behavior of the system.Therefore, their accuracy isherein evaluated in comparison with the input-output EMAapproach. In the presence of the nonlinear distortions, theEMA approach extracts the modal parameters based on theBLAs of the nonlinear structure. The identification of theBLA can be done, for instance, using the sample maximumlikelihood method which is optimal in the sense that thevariance of the estimate is close to the CrameΜr-Rao lowerbound [18].
5. Applications
The OMA techniques are applied to identify nonlinearsystems excited by a series of excitation levels. The excitationlevel is expressed in the form of signal-to-noise ratio definedas follows:
SNRdB = 10log10
{{{
{{{
{
βππ¦
π=1 βππΊ
(π)
BLA (ππ)
2
βππ¦
π=1 βποΏ½ΜοΏ½2πΊ(π)
BLA(π
π)
2
}}}
}}}
}
, (24)
where πΊBLA and ππΊBLA are the identified BLA and quantifiedstandard deviation, respectively.
5.1. Simulated Case
5.1.1. Nonlinear System. The simulated example is a 4DOFmass-spring-damper model with local nonlinearity and thespring stiffness and damping at the ends are state-dependent,as described in Figure 3(a).
5.1.2. Parameter Setting. The sampling frequency is 64Hz,π
π= 4, π
π = 100. In order to illustrate the algorithmic
behavior very clearly, we take the extreme point of view thatno measurement noise is present. The simulated data areonly corrupted by the nonlinear distortions after discardingthe periods in response corrupted by the transient effect.Three levels of excitation are considered, the correspondingSNRdB, 89.8, 46.7, and 37.6, are classified as weak, medium,and strong, respectively.The evolution of the BLAs and of theassociated nonlinear distortions of the 3rd mass is shown inFigure 3(b) when the input is located at the 4th mass.
The modal assurance criterion (MAC) index is set equalto 0.85 for the FDD method to choose the frequency vicinityaround a pole. Consider π
π= 15 in (17) for the SSI-COV
algorithm. The number of measured outputs (ππ¦) directly
influences the performance of the FDD, SSI, and TOMAthrough (10), (16), and (22), respectively. The more reliableresults would be obtained with more output data. However,
-
Shock and Vibration 7
1 1.02 1.040
200
400
Mod
e 1
SSI-COV
1 1.02 1.040
200
400FDD
1 1.02 1.040
200
400TOMA
1 1.02 1.04 1.06 1.080
200
400
Mod
e 2
1 1.02 1.04 1.06 1.080
200
400
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
1 1.02 1.04 1.06 1.080
200
400
1 1.02 1.040
200
400
Mod
e 3
1 1.02 1.040
200
400
1 1.02 1.040
200
400
1 1.02 1.040
200
400
Mod
e 4
1 1.02 1.040
200
400
Normalized eigenfrequencies Normalized eigenfrequencies1 1.02 1.04
0
200
400
Normalized eigenfrequencies
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequencies
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequencies
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequencies
Figure 4: The rows show four modes and the columns represent the probability distributions of eigenfrequencies obtained by the SSI-COV,FDD, and TOMA for three excitation levels (light gray: weak, dark gray: medium, and black: strong). Dashed lines: EMA estimates with theforce applied at π₯4.
ππ¦is limited by the data acquisition system. Here, π
π¦is set as
4 for the output-only and input-output modal identification.Two loading conditions are created for the TOMA approachby shifting the excitation location from the 1st mass to the 4thone.
5.1.3. Results and Discussion. Themultiple simulated data aregenerated (as shown in Figure 1), based on which the previ-ously described OMA and EMA approaches are applied toobtain the modal estimates. The estimated eigenfrequenciesand damping ratios are normalized with respect to thoseidentified using the input-output data at the lowest level ofexcitation.Themode shape estimated by theOMA is normal-izedwith respect to the one obtained by the EMA through theuse of the MAC.These normalized modal values can providea clear view on the accuracy of the OMA algorithms, whosedispersion vividly demonstrates the algorithmic robustnesswith respect to the nonlinear distortions.
As expected, Figures 4β6 depict that in the case of thelowest excitation level the considered OMA techniques allidentify well the system which behaves linearly. The modalestimates are shown to become more dispersed and evenbiased but still situated around those of the BLAs when
severe nonlinear distortions are stimulated by increasingthe excitation level. The performance of the FDD methodis remarkable except for the mode shape estimates as theyare obtained by picking the discrete values determined bythe frequency resolution of the spectrum. As commentedin Section 3.3, the eigenfrequency estimates by the TOMAapproach generally deviate more from the EMA estimatesthan the other methods. However, it is worth pointing outthat the hardening behavior of the nonlinear system is mostcaptured by all the OMA techniques, as reflected by theestimated eigenfrequencies.
5.2. Real Case
5.2.1. Nonlinear System. The nonlinear system under inves-tigation is a mechanical structure consisting of a beam,frame, and support attachment, as displayed in Figure 7(a).The main part of the system is formed by fastening twobeams together, whose left side is clamped in a rigid supportand right side is adhered to a steel frame put freely onthe ground. The use of various assemblies and joins in thissetup creates diverse underlying sources of nonlinearity, suchas beam clamping which introduces nonlinear stiffness of
-
8 Shock and Vibration
0 1 2 30
5
Mod
e 1
SSI-COV
0 1 2 30
5
FDD
0 1 2 30
5
TOMA
0 1 2 30
5
Mod
e 2
0 1 2 30
5
0 1 2 30
5
0 1 2 30
5
Mod
e 3
0 1 2 30
5
0 1 2 30
5
0 1 2 30
5
Mod
e 4
0 1 2 30
5
Normalized damping ratios Normalized damping ratios Normalized damping ratios
Normalized damping ratios Normalized damping ratios Normalized damping ratios
Normalized damping ratios Normalized damping ratios Normalized damping ratios
Normalized damping ratios Normalized damping ratios Normalized damping ratios
0 1 2 30
5
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Figure 5: The rows show four modes and the columns represent the probability distributions of damping ratios obtained by the SSI-COV,FDD, and TOMA for three excitation levels (light gray: weak, dark gray: medium, and black: strong). Dashed lines: EMA estimates with theforce applied at π₯4.
(possibly high-order) polynomial form, frictional slips atloosened interfaces which introduce additional flexibilityand hysteretic damping to the overall structural dynamics.Also, it commonly occurs that nonlinearity is unintentionallyintroduced in the measurement chain, for example, preload-ing the beam because of insufficient human checks, whichgenerally creates a nonlinear stiffness of cubic form. All kindsof nonlinearity are treated in a unified way based on theestablished protocol, whose effects on structural dynamicsare determined by identifying the BLA and bounding thestochastic nonlinear distortions.
5.2.2. Parameter Setting. The sampling frequency is 8192Hz,π
π= 256, π
π = 40. Following the same lines of the
simulation, three levels of excitation are set by the SNRdB:60.3 (weak), 48.7 (medium), and 33.5 (strong). Applying theshaker at the position (1), Figure 7(b) shows the apparentshift-down of eigenfrequencies of the considered system byraising the excitation level, which is in large part due to theuse of various joints in the setup. In addition to the softeningeffect, two close but distinct poles are present around 300Hzfor low levels of excitation.These poles are transformed into asingle pole when a larger amount of power is injected into the
system. The presence of the system nonlinearity is classicallyindicated in terms of the coherence function in Figure 8(a),and Figure 8(b) displays several realizations of the nonlineardistortions.
The MAC index is set equal to 0.9 for the FDD method.Consider π
π= 25 for the SSI-COV algorithm. The shaker
is applied at locations (1) and (2) on the beam, respectively,creating distinct loading conditions for the TOMA.
5.2.3. Results and Discussion. When an electrodynamicshaker is used to excite the structure in modal testing, itis not trivial to investigate the considered OMA techniquesin a fair way. In fact, the force applied on the structure isthe reaction force between the shaker and the beam whenthe armature mass and spider stiffness of the shaker arenot negligible, whose magnitude and phase depend uponthe characteristics of the structure and of the exciter. Thusthe shaker driving force does not meet the fundamentalassumption of white noise. Consequently, the FDD and SSI-COV algorithms actually identify the whole system includingthe nonlinear structure and the shaker part. Superior toboth of OMA techniques, the TOMA approach can identifyonly the nonlinear structure of interest as the shaker part
-
Shock and Vibration 9
0.998 1 1.002 1.0040
5000
Mod
e 1
SSI-COV
0.998 0.999 1 1.001 1.0020
5000FDD
0.99 0.995 1 1.0050
5000TOMA
0.995 1 1.0050
2000
4000
Mod
e 2
0.995 1 1.0050
2000
4000
0.995 1 1.0050
2000
4000
0.995 1 1.0050
2000
4000
Mod
e 3
0.995 1 1.0050
2000
4000
0.995 1 1.0050
2000
4000
0.98 1 1.020
100
200
Mod
e 4
0.98 1 1.020
100
200
Modal assurance criterion (MAC) indicesModal assurance criterion (MAC) indicesModal assurance criterion (MAC) indices
Modal assurance criterion (MAC) indicesModal assurance criterion (MAC) indicesModal assurance criterion (MAC) indices
Modal assurance criterion (MAC) indicesModal assurance criterion (MAC) indicesModal assurance criterion (MAC) indices
Modal assurance criterion (MAC) indicesModal assurance criterion (MAC) indicesModal assurance criterion (MAC) indices
0.98 1 1.020
100
200
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Figure 6:The rows show fourmodes and the columns represent the probability distributions ofMAC indices obtained by the SSI-COV, FDD,and TOMA for three excitation levels (light gray: weak, dark gray: medium, and black: strong).
(1) (2)
(a)
0
200 400300
260 280 300 320
30
20
500 600 700
20
40
β20
β40
Mag
nitu
de (d
B)
Frequency (Hz)
(b)
Figure 7: (a) Experiment setup of the real structure, (b) best linear approximations under three excitation levels (black: strong, dark gray:medium, and light gray: weak).
is simultaneously present in different outputs and furthercanceled by computing transmissibility functions.
The output data are processed in line with the proposedprotocol (see Section 4.1). As shown in Figures 9 and 10,the eigenfrequencies estimated by the SSI-COV and FDDalgorithms agree well with those extracted from the BLAs forthe three excitation levels. It is also found by comparing with
EMA estimates that the eigenfrequency estimates are moreaccurate than those of damping ratios for the consideredOMA techniques. As analyzed in Sections 3.1 and 3.2, the esti-mated SYY(π) used for the FDD andΞrefπ for the COV-SSI arebiased in the presence of the correlated nonlinear distortionsover space and time. The nonlinear perturbations in thesecovariance matrices are propagated to the identification of
-
10 Shock and Vibration
200
0.6
1
0.2400300 500 600 700
Frequency (Hz)
Coh
eren
ce fu
nctio
n
(a)
5
0
43
21
43
β50
200
400
600
800
Freque
ncy (H
z)
Realization
Non
linea
r di
stort
ions
(dB)
(b)
Figure 8: (a) Coherence functions under three excitation levels (black: strong, dark gray: medium, and light gray: weak), (b) samples of thenonlinear distortions separated from output data.
0.9 1 1.10
100
200
300
SSI-
COV
0.95 1 1.050
500
1000
1500
FDD
0.98 0.99 1 1.010
100
200
300
0.98 0.99 1 1.010
500
1000
1500
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequenciesNormalized eigenfrequencies
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequenciesNormalized eigenfrequencies0.98 0.99 1 1.010
100
200
300
0.98 0.99 1 1.010
500
1000
1500
0.99 1 1.010
100
200
300
0.99 1 1.010
500
1000
1500
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Mode 296.04Hz Mode 408.59Hz Mode 530.71Hz Mode 675.71Hz
Figure 9: The columns show four modes and the rows represent the probability distributions of eigenfrequencies obtained by the SSI-COVand FDD with the shaker at location (1) for three excitation levels (light gray: weak, dark gray: medium, and black: strong). Dashed lines: theEMA estimates.
damping ratios, especially for the SSI-COV without applyingany filter, as verified in Figure 10.
Shifting the shaker between the two locations induces aremarkable evolution of the BLA, as clearly demonstrated inFigure 11. The modal parameters are estimated by combiningthe output dataset from these two loading configurations, asseen in Figure 11.The eigenfrequency estimates still approachquite well those of the BLAs with a relative error up to only2.5%. They also can reflect the shift-down property of thesystem, while the damping estimates are less meaningful. Infact, (21) does not hold anymore in the presence of the systemnonlinearity such that lightly damping properties are hardlyextracted with reliable accuracy.
Although the approaches in time- and frequency-domains behave similarly, it is found from the simulated andreal case studies that the FDD approach is characterized bythe highest accuracy for the eigenfrequency estimates.
6. Conclusions
TheOMA techniques have been applied to identify simulatedand real nonlinear structures; a full view on the algorithmicperformance is delivered by using a protocol that establishesthe estimation accuracy and robustness with respect to thesystem nonlinearity. A theoretical motivation for the pro-posed protocol is also provided. The following conclusionsand remarks can be drawn from our study.
(i) The output-only identification techniques are able toextract most linear dynamics of a nonlinear structure,but the estimates obtained by them are more biasedand dispersed in the presence of increasing nonlinear-ity induced by the excitation change. The estimatedeigenfrequency is a robust indicator of the systemstate induced by the nonlinearity.
-
Shock and Vibration 11
0 2 4 60
0.2
0.4
SSI-
COV
1 2 30
5
10
15
FDD
0 2 4 60
0.2
0.4
0 1 20
5
10
15
Normalized damping ratios Normalized damping ratios Normalized damping ratiosNormalized damping ratios
Normalized damping ratios Normalized damping ratios Normalized damping ratiosNormalized damping ratios0 2 4 60
0.2
0.4
0.5 1 1.5 20
5
10
15
0 2 4 60
0.2
0.4
0.5 1 1.5 20
5
10
15
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
ns
Prob
abili
ty d
ensit
y fu
nctio
nsMode 296.04Hz Mode 408.59Hz Mode 530.71Hz Mode 675.70Hz
Figure 10: The columns show four modes and the rows represent the probability distributions of damping ratios obtained by the SSI-COVand FDD with the shaker at location (1) for three excitation levels (light gray: weak, dark gray: medium, and black: strong). Dashed lines: theEMA estimates.
0.96 0.98 1 1.02 1.040
1000
2000
3000
Eige
nfre
quen
cy
0.99 1 1.010
1000
2000
3000
0.98 1 1.02 1.04 1.060
1000
2000
3000
0.99 0.995 1 1.0050
1000
2000
3000
0 1 2 30
4
8
12
Dam
ping
ratio
0 1 2 30
4
8
12
Normalized damping ratios Normalized damping ratios Normalized damping ratios Normalized damping ratios
Normalized eigenfrequencies Normalized eigenfrequenciesNormalized eigenfrequenciesNormalized eigenfrequencies
0 1 2 30
4
8
12
0.5 1 1.5 20
4
8
12
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Prob
abili
ty d
ensit
y fu
nctio
nsPr
obab
ility
den
sity
func
tions
Mode 297.64Hz Mode 405.00Hz Mode 537.51Hz Mode 685.01Hz
Figure 11: The columns show four modes and the rows show the probability distributions of eigenfrequencies and damping ratios by theTOMA for three excitation levels (light gray: weak, dark gray: medium, and black: strong). Solid and dashed lines: EMA estimates from twoloading conditions.
(ii) Contrary to the SSI-COV and FDD, the TOMAis insensitive to the coloring of the unobservedexcitation. However, it is more sensitive to systemnonlinearities than them.
(iii) Two (or more) loading conditions can be present inone real-life measurement record, which can confusethe interpretation of the modal estimates obtainedfrom it.Thus, developing an assistant tool of detectingloading conditions comes to be very demandingwhen
the OMA is performed for the purpose of structuralhealth monitoring and system modeling.
Appendix
The modal assurance criterion (MAC) is used to determinea suitable frequency band for FDD estimation, which isperformed based on the following steps.
-
12 Shock and Vibration
(1) Estimate roughly the πth eigenfrequency (e.g., bythe pick-up method) and obtain the eigenvector ofSYY(οΏ½ΜοΏ½0,π) at the estimated eigenfrequency οΏ½ΜοΏ½0,π, whichis seen as the modal vector π
ππ.
(2) Select a frequency line in the vicinity of the eigen-frequency οΏ½ΜοΏ½0,π, at which the eigenvector of SYY(π) isdecomposed and denoted by the modal vector π
ππ.
(3) Compute the MAC index
MAC =ππ»
πππππ
2
{ππ»
πππππ} {ππ»
πππππ}, (A.1)
where the superscriptπ» denotes theHermitian trans-pose.
(4) Collect all the frequency lines at which the MAC val-ues are above the predefined threshold, constitutingthe frequency band for the FDD estimation.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work was jointly supported by the Fund for ScientificResearch-Flanders (G024612N), the Belgian Federal Gov-ernment (Interuniversity Attraction Poles Programme VII,Dynamical Systems, Control, and Optimization), and theEducation Bureau of Henan Province, China (14A460019).
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