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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing 25 (2011) 1028–1044

0888-32

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jnlabr/ymssp

Operational modal analysis by updating autoregressive model

V.H. Vu a, M. Thomas a,n, A.A. Lakis b, L. Marcouiller c

a Department of Mechanical Engineering, Ecole de Technologie Superieure, Montreal, Quebec, Canada H3C 1K3b Department of Mechanical Engineering, Ecole Polytechnique, Montreal, Quebec, Canada H3C 3A7c Hydro-Quebec’s Research Institute, Varennes, Quebec, Canada J3X 1S1

a r t i c l e i n f o

Article history:

Received 16 July 2008

Received in revised form

25 August 2010

Accepted 27 August 2010Available online 16 September 2010

Keywords:

Modal identification

Multivariate-autoregressive model

Model orders selection

QR factorization updating

Parameters estimation

Confidence intervals

70/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ymssp.2010.08.014

esponding author.

ail address: marc.thomas@etsmtl.ca (M. Thom

a b s t r a c t

This paper presents improvements of a multivariable autoregressive (AR) model for

applications in operational modal analysis considering simultaneously the temporal

response data of multi-channel measurements. The parameters are estimated by using

the least squares method via the implementation of the QR factorization. A new noise rate-

based factor called the Noise rate Order Factor (NOF) is introduced for use in the effective

selection of model order and noise rate estimation. For the selection of structural modes, an

orderwise criterion called the Order Modal Assurance Criterion (OMAC) is used, based on

the correlation of mode shapes computed from two successive orders. Specifically, the

algorithm is updated with respect to model order from a small value to produce a cost-

effective computation. Furthermore, the confidence intervals of each natural frequency,

damping ratio and mode shapes are also computed and evaluated with respect to model

order and noise rate. This method is thus very effective for identifying the modal parameters

in case of ambient vibrations dealing with modern output-only modal analysis. Simulations

and discussions on a steel plate structure are presented, and the experimental results show

good agreement with the finite element analysis.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Modal analysis identification techniques give useful information on modal parameters to understand the dynamicbehavior of a structure [1]. However, in many cases which present nonlinear behaviors, the modal parameters have to beestimated under operating conditions and very often the excitations cannot be measured [2]. We must thus consider theoperational modal analysis. Although the identification technique can be conducted both in the frequency [3] or time [4–6]domains, it is seen that the time domain is more suitable for operational modal analysis and can be classified into twogroups. The first group lies in the fitting of response correlation functions, including the Ibrahim Time Domain (ITD)method [7], the Least Squares Complex Exponential (LSCE) method [8], the Covariance-driven Stochastic SubspaceIdentification (SSI-COV) method [9], and several modified versions of these methods for more suitable applications,particularly under harmonic excitations [10–13]. Other methods, based on parametric models, involve choosing amathematical model to idealize the structural dynamic responses, including AutoRegressive Moving Average (ARMA) andAutoRegressive (AR) models [14–18]. For these autoregressive methods, a system identification algorithm is needed forestimating the model parameters. Among them, the Prediction Error Method (PEM) [19] is a common technique based oneither the least squares estimate or on the Gauss–Newton iterative search. When the mode identification is necessary,multiples sensors have to be simultaneously recorded, and several applications of the multivariate AR model can be foundusing the ordinary least squares in the form of normal equations [20–22] or the Levinson algorithm [23]. The PEM iterative

ll rights reserved.

as).

Nomenclature

Ai matrix of parameters relating the outputy(t� i) to y(t)

d vector dimension or number of sensorsD estimated covariance matrix of the determi-

nistic parte(t) the residual vector of all output channelsE estimated covariance matrix of the error partfi natural frequencyI unity matrixK data matrixK1,2 subdivided data matricesKn added data columns matrixL complex eigenvectors matrixN number of available data samplesp model orderpeff(com) efficient (computing) model orderQ orthogonal factor matrix of the QR factoriza-

tionQa Q factor of the added columns factorizationR upper-diagonal factor matrix of the QR factor-

izationRa R factor of the added columns factorization

Rij submatrices of RR�ð��Þ22 factorized matrices of R22 factor of the order

updatingt time indexTs sampling periodT1,2,3 R factor corresponding to added dataui discrete complex eigenvaluey(t� i) the output vector with time delay i� Ts

z(t) the regressor for the output vector y(t)zi damping ratioH real mode shapes matrixli continuous complex eigenvalueK model parameters matrixp Pi numberP state matrixW complex mode shapes matrix(p) parameter at order p

^ estimated valueT matrix/vector transpose

¯conjugated transpose

|| absolute valueIm(y) imaginary partRe(y) real partTrace(y) trace norm of a matrix

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1029

method, generally used to search for minimization, requires intensive computation and initial start-up values which arenormally calculated using the least squares method. Furthermore, in some cases, the local minimization problem poses abig challenge [19].

In this paper, the multivariate autoregressive model is expressed in a convenient fashion for computation. The QRfactorization gives an easy, fast and well-conditioned formulation for the least squares estimate of model parameters, andcan be effectively updated with respect to the model order. A new factor based on the separation and evolution of thesignal and noise is developed for the model order selection and noise rate estimation. The modal parameters are derivedusing the eigen-decomposition of the state matrix. An orderwise version of the correlation criterion called the Order ModalAssurance Criterion (OMAC) is defined for a user-friendly selection of modes. For interest on uncertainty in the parameterestimates, the confidence intervals for each modal parameter are computed. Finally, the method is applied both onsimulated and experimental data of a steel plate in comparison to finite element method.

2. Vector-autoregressive model and its order updating

In operational modal analysis, we assume that the excitation is unknown. Since the modal analysis is conducted byusing several d channels of measurements synchronized for data acquisition at sampling period Ts, a multivariateautoregressive model of pth order and dimension d can be utilized to fit the measured data [14,19].

yðtÞ ¼KzðtÞþeðtÞ ð1Þ

where K¼ ½�A1 �A2 � � � �Ap � of size d� dp is the parameters matrix, Ai of size d�d is the matrix of parameters

relating the output y(t� i) to y(t), z(t) of size dp�1 is the regressor for the output vector y(t), zðtÞT ¼

½ yðt�1ÞT yðt�2ÞT � � � yðt�pÞT �, y(t� i) of size d�1 (i=1:p) is the output vector with delayed time i� Ts, e(t) of size

d�1 is the residual vector of all output channels considered as the error of model.If N consecutive output vectors of the responses from y(t) to y(t+N�1) are taken into account, the model parameters

can be obviously estimated with the least squares method. The following section reports the solution of this least squaresproblem by using of the well-known QR factorization [24–26]. It is summarized for better understanding as follows:

The N� dp+d data matrix is first constructed from available data:

K¼

zðtÞT yðtÞT

zðtþ1ÞT yðtþ1ÞT

. . . . . .

zðtþN�1ÞT yðtþN�1ÞT

266664

377775 ð2Þ

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441030

The QR factorization of the data matrix K=Q�R gives an orthogonal matrix Q (size N�N) and R (size N� dp+d) anupper triangular matrix with the form

R¼

R11 R12

0 R22

0 0

264

375 ð3Þ

where R11, R12 and R22 are of size dp�dp, dp� d and d�d, respectively.The parameters matrix is finally derived, as well as the estimated covariance matrices of the deterministic part D and of

the error E (both of size d� d):

K¼ ðRT12R11ÞUðR

T11R11Þ

�1¼ ðR�1

11 R12ÞT

ð4Þ

D¼1

NRT

12R12 ð5Þ

E¼1

NRT

22R22 ð6Þ

The solution thus yields to the computation of R submatrices. The computation of those submatrices should be updatedwith respect to model order to avoid the problem of other algorithms which need a prior evaluation of model order andrequire a repetitive computation for a set of model orders. Although algorithms for the updating of QR factorization hadbeen developed in [25] by using the Givens rotations, these transformations need to be repetitively performed on half ofmatrix elements which can be large for multi-sensors modeling. In this paper, we present a computation for the updatingof the model when the order is increasing and it is shown that if only submatrices of R are required, the problem results inapplying the QR factorization or Givens rotations to only a submatrix of the previous solution.

Suppose that we have the data matrix K(p) at the order p and its factored matrices Q(p), R(p) computed fromEqs. (2) and (3):

KðpÞ ¼

zðtÞT yðtÞT

zðtþ1ÞT yðtþ1ÞT

. . . . . .

zðtþN�1ÞT yðtþN�1ÞT

266664

377775¼ KðpÞ1 K2

h ið7Þ

RðpÞ ¼

RðpÞ11 RðpÞ12

0 RðpÞ22

0 0

264

375

At the order p+1, the data matrix can be subdivided as follows [25]:

Kðpþ1Þ¼ KðpÞ1 K� K2

h ið8Þ

where Kn of size N� d comprises the added d columns:

K� ¼

yðk�ðpþ1ÞÞT

yðkþ1�ðpþ1ÞÞT

. . .

yðkþN�1�ðpþ1ÞÞT

266664

377775 ð9Þ

Then we can compute the following matrix:

Q ðpÞTKðpþ1Þ¼ Q ðpÞTKðpÞ1 Q ðpÞTK� Q ðpÞTK2

h i¼

RðpÞ11 T1 RðpÞ12

0 T2 RðpÞ22

0 T3 0

2664

3775 ð10Þ

where T1 of size dp�d, T2 of size d� d and T3 of size N�dp�d�d are extracted from

Q ðpÞTK� ¼

T1

T2

T3

264

375

If we apply the QR factorization on the submatrixT2

" #, it yields to

T3

T2

T3

" #¼Q a

Ra

0

� �ð11Þ

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1031

where Ra of size d� d is an upper diagonal matrix and Qa of size N�dp�N�dp is the product of the Householdertransformations or Givens rotations.

Then Eq. (10) becomes

I 0

0 Q Ta

" #Q ðpÞTKðpþ1Þ

¼

RðpÞ11 T1 RðpÞ12

0

0

Ra

0

R�22

R��22

2664

3775 ð12Þ

where R�22 of size d�d and R��22 of size N�dp�d�d are obtained from multiplication

R�22

R��22

" #¼Q T

a

R22

0

� �

It can be noticed that the submatrix R��22 in the right hand side of Eq. (12) is not an upper diagonal matrix and must betriangularized by a small orthogonal transformation to yield the QR decomposition of the data matrix K(p +1). Thismodification evidently does not affect other components of Eq. (12). Hence the submatrices Rðpþ1Þ

11 , Rðpþ1Þ12 and Rðpþ1Þ

22 ofmodel at order p+1 are exactly updated:

Rðpþ1Þ11 ¼

RðpÞ11 T1

0 Ra

" #; Rðpþ1Þ

12 ¼RðpÞ12

R�22

" #; Rðpþ1Þ

22 ¼R��22 ð13Þ

Next, the model parameters matrix K(p + 1) and covariance matrix Dðpþ1Þ are updated, as shown in Eqs. (5) and (6):

Kðpþ1Þ¼ ½Rðpþ1Þ

11 ��1Rðpþ1Þ12

h iTð14Þ

Dðpþ1Þ¼Rðpþ1ÞT

12 Rðpþ1Þ12 ¼ RðpÞT12 RðpÞ12þR�22

TR ð�Þ22 ¼ DðpÞþR�22

TR ð�Þ22 ð15Þ

Since all transformations are orthogonal, covariance matrix Eðpþ1Þ

is also updated:

Eðpþ1Þ¼ Rðpþ1Þ

22TR ðpþ1Þ

22 ¼RðpÞ22RðpÞ22�R�22TR�22 ¼ E

ðpÞ�R�22

TR�22 ð16Þ

Once the model parameters are updated, the calculation of modal parameters is straightforward [14]. The state matrixof the system is first established from autoregressive parameters:

P¼

�A1 �A2 . . . �Ap�1 �Ap

I 0 . . . 0 0

0 I . . . 0 0

. . . . . . . . . . . . . . .

0 0 . . . I 0

26666664

37777775

ð17Þ

The eigen-decomposition of the state matrix can be obtained:

P¼ L

u1 0 0 0

0 u2 0 0

0 0 & ^

0 0 . . . udp

266664

377775L�1

ð18Þ

The continuous eigenvalues, natural frequencies, damping rates and complex modes of the system can be computed:

Eigenvalues : li ¼lnðuiÞ

Ts

Frequencies : fi ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRe2ðliÞþ Im2

ðliÞ

q2p

Damping rates : xi ¼�ReðliÞ

2pfi

Complex modes : W¼ W1 W2 . . . Wdp

h i¼ I 0 . . . 0� �

L ð19Þ

In modal analysis, the selection of modes may be realized by using either a modal signal-to-noise classification MSN[14] or the MAC (Modal Assurance Criterion [14,27]) which defines the correlation between the identified mode shapevector with a reference vector (extracted from numerical or experimental results). When conducting an operational modalanalysis under a noisy environment, the MSN index is still reliable but is difficult to apply because it needs a rigorousattention on the number of modes selected from the spurious ones and an apriori knowledge about frequency anddamping [14] which in fact is not available in output-only modal analysis. Since our algorithm is updated with respect to

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441032

model order with availability of stabilized diagrams, we propose a new version of correlation criterion, which we havecalled OMAC (Order Modal Assurance Criterion). In this index, the MAC [14] of the ith mode is replaced by the correlationof the identified mode shapes given by two successive model orders p and p�1, formulated as follows:

OMACðpÞi ¼

WðpÞi

h iTWðp�1Þ

i

��������ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

WðpÞi

h iTWðpÞi

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiWðp�1Þ

i

h iT

Wðp�1Þi

r ð20Þ

where WðpÞi and Wðp�1Þi are the ith identified complex mode shape vectors at orders p and p�1, respectively, and

WðpÞ

i signifies the conjugated transpose of WðpÞi .

This OMAC parameter can be thus considered as a complementary information of the usual MAC (which compares thecomputed mode with a reference), by analysing the stability of the mode with the order of computation, since it allows toeliminate perturbation modes, before to compute the MAC.

It is evident that the measurement with unobserved perturbations produces an uncertainty in the parameter estimationand hence on the modal parameters. Therefore, an analysis of confidence intervals of modal parameters should be takeninto account. Uncertainty in structural dynamics has been generally introduced and bibliographically reviewed in [31]where all experimental modal analysis contributed on non-parametric methods. A recent derivation on variances of modalparameters has been presented in [32] which supposed an availability of the transfer function in the frequency domain. Itis seen that the uncertainty of autoregressive model parameters can be transferred from the estimation of the least squareestimate by differentiation [33]. In this paper, the derivatives of modal parameters are derived from Eq. (19) and theirconfidence intervals are straightforward transferred as described in [33]:

Derivative of frequencies : _f i ¼ReðliÞReð _liÞþ ImðliÞImð

_liÞ

4p2fi

Derivative of damping rates : _z i ¼ ziReð _liÞ

ReðliÞþ_f i

fi

!

Fig. 1. NOF evolution and efficient order selection.

Fig. 2. Configuration of testing plate.

Fig. 3. NOF on simulated data.

Fig. 4. Frequency stabilization diagram on simulated data.

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1033

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441034

Derivative of Hl,i, the lth component of the ith real mode shape:

ðHl,iÞ2¼ ðWl,iÞ

2¼ Re2

ðLl,iÞþ Im2ðLl,iÞ, l¼ 1,2,. . .,d

9Wl,i9 _H l,i ¼ 9ReðLl,iÞ9Reð _L l,iÞþ9ImðLl,iÞ9Imð _L l,iÞ ð21Þ

Several observations should now be pronounced:

�

TabRes

M

SSIdImIdImIdRIdR

With only a small factorization of R submatrices, the model parameters and covariance matrices can be exactly updatedto a higher order. This technique is much preferred than the repetitive QR factorization for each order value.

� From the update of covariance matrices, it is noticed that, as the model order increases, the estimated signal part D

increases and the estimated noise part E decreases monotonically. The changing amount is significant at low orders andnegligible at high orders. This feature is thus an idea for a new orders selection criterion as discussed later.

� The introduction of the OMAC is very convenient with the order updating algorithm since the modes shapes are

successively constructed. It becomes easy to check if a stabilized frequency comes from the structural properties whenits corresponding OMAC is closed to unity within all considered model orders.

� The updating of the solution with respect to model order allows an evaluation of uncertainty of the modal parameters

with respect to model order and to the noise rate, as seen in the experiment section.

3. A new index for model order selection and noise estimation

The selection of the optimal model orders is the first step in the model based identification process. Among the betterknown methods are the criteria based on the statistical properties of prediction errors eðtÞ and a penalty function, such asthe Final Prediction Error (FPE) and the Minimum Description Length (MDL) [34]. However some remarks should betackled:

�

These criteria are primarily based on the evolution of the error covariance which monotonically decreases with respectto the model order.

le 1ults on simulations.

ode 1 2 3 4 5 6 7

imulated frequencies (Hz) 40.8 77.5 112.9 172.3 222.9 290.4 294.4

imulated damping rate (%) 1 1 1 1 1 1 1

entified frequencies (Hz) 40.78 77.5 112.8 171.6 221.5 287.4 291.3

pact force, order 5entified damping rate (%) 1 1 1 1 1.1 1.1 1.1

pact force, order 5entified frequencies (Hz) 39.7 75.7 112.3 169.7 218.8 278.1 286.5

andom force, order 5entified damping rate (%) 1.9 0.7 0.6 0.6 0.6 1 0.2

andom force, order 5

Fig. 5. Real testing data.

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1035

�

FPE and its variants asymptotically choose the correct order model if the underlying multiple time series has highdimensions d [35], but tends to overestimate the model order as the data length increases [36]. � MDL and its variants outperform with long recorded data and are strongly consistent when the data length tends to

infinite [28]. However, the application of these criteria first requires the selection of a possible interval for the modelorder to be used, and then an evaluation of model parameters. The selection of the order is made on the basis of

Fig. 6. NOF evolution at different noise rates.

Fig. 7. Comparison NOF and MDL.

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441036

minimum variances. It is necessary, therefore, to have a prior evaluation with different orders, which results in asignificant computation time.

Other authors [29,30] have proposed new approaches derived from the MDL criterion, and based on the minimumeigenvalue of the covariance matrix without prior evaluation. To determine the model order, it is necessary to determine alimiting upper value in order to establish the order of the matrix. However, it is difficult to experimentally determine thatvalue when significant noise is present.

In modal analysis, we must consider not only the error sequence, but also the deterministic part of the signal, since thislatter contains the modal information of the system. Based on these facts and to compare to the well-known optimal modelcriterion MDL, this paper proposes an innovative factor for the selection of an efficient model order peff based on ananalysis of the noise-to-signal ratio (NSR). This order is defined as the smallest order value which can be used to fit the datawith a negligible discrepancy and hence can be effectively used for modal analysis.

While the MDL criterion considers only the prediction error, the estimated noise-to-signal ratio (NSR) can be definedfrom the trace norm part of the estimated deterministic and error covariance matrices E and D, as previously derived inEqs. (15) and (16):

NSR¼TraceðEÞ

TraceðDÞð%Þ or NSR¼ 10log10

TraceðEÞ

TraceðDÞðdBÞ ð22Þ

Then a Noise-ratio Order Factor (NOF) is defined as being the variation of the NSR between two successive orders:

NOFðpÞ ¼NSRðpÞ�NSRðpþ1Þð23Þ

It is seen that the NSR decreases monotonically with respect to model order, and contains properties of both stochastic(on numerator) and deterministic (on denominator) norms. The NOF is therefore always positive and is a representativefactor for the convergence of the NSR, which changes significantly at low orders and converges at high orders. Since thisfactor is positive and close to zero, the convergent to zero property can be used to set the selection of efficient model orderwhich is thus easier to do on the curve evolution (Fig. 1).

4. Application on a plate structure and discussions

The method presented above is applied to a clamped-free–clamped-free rectangular steel plate [37] with dimensions of500�200�1.9 mm3 (Fig. 2). Material properties are as follows: elastic modulus E=200 GPa, the Poisson coefficient n=0.29

Table 2Selection of orders and noise rate estimation.

Simulated noise rate (NSR) 0% 1% 10% 100% 200% 400%

Estimated noise rate (%) 0.03 1.20 10.52 102.1 201.2 386.5

Efficient order peff from NOF 6 6 6 6 6 6

Optimal order from MDL 21 11 6 6 6 4

Computational order pcom 9 12 12 12 14 14

Fig. 8. Frequency stabilization diagram at NSR=100%.

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1037

and density r=7872 kg/m3. Six accelerometers are mounted on the plates to simultaneously record the responses at themeasurement locations.

4.1. Numerical simulations

The plate has been numerically excited by assumed impact and random forces. The responses have been computed forthe first seven modes till 350 Hz by the modal superposition method and by considering a 1% damping ratios for all modes.

Fig. 9. Modal parameters identification with confidence intervals (NSR=100%).

Fig. 9. (Continued)

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441038

A sampling frequency of 1280 Hz has been applied and no noise has been added on the responses. It can be noticed that thesixth and seventh frequencies are close. Fig. 3 shows the NOF factor under the two excitations and it is seen that in bothcases, the efficient model order can be clearly identified as 5 under an impact force (Fig. 3a), while it is less clear under arandom force (Fig. 3b).

Fig. 9. (Continued)

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1039

The stabilization diagrams in Fig. 4 show the first seven natural frequencies, with good accuracy for both excitationseven if some frequencies are close (6th and 7th). However, we must notice that a random excitation produces a highervariance for the identification, particularly at these two close frequencies (Fig. 4b and Table 1). The identification ofdamping is good for both excitations except at the 7th mode, where the result is erroneous due to the proximity of the 6thfrequency.

Table 1 compares the first seven modal parameters of the plate as simulated by Ansys and those identified at efficientorder 5, when no noise is added. A good agreement is found between simulated and identified modal parameters. However,a larger variance of the identified damping ratios is observed when the excitation is random.

4.2. Real structure testing

The same plate has been experimentally excited with an uncontrolled shock, and the excitation forces (amplitudes andlocations) have not been considered. The responses at the six locations have been simultaneously acquired (Fig. 5) andsampled at a frequency of 1280 Hz. We can see from the figure that the transient response has been perturbed by a secondexcitation. Despite this unexpected perturbation, the entire signal of 4000 samples has been considered. Furthermore,several additive random white noises (1–400% of the signal r.m.s.) have been numerically added to each measurementchannel for the assessment of noise effect.

4.2.1. Model order selection and noise rate estimation

In order to select an efficient model order for the system at different noise rates, the method described earlier has beenused. Fig. 6 presents the evolution of the NOF factor at different noise rates. It can be observed that the efficient order of thesystem can be set close to 6 whatever the noise rate.

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441040

To check the efficiency of the NOF method, the results have been compared to the MDL criterion [28,34]. Fig. 7 exhibitsthe comparison of NOF and MDL at very low noise rate (Fig. 7a, NSR=1%) and very high noise rate (Fig. 7b, NSR=400%).It can be noticed that MDL gives the order between 4 and 11, respectively, with the noise rate while the NOF is found atorder 6, regardless the noise rate.

In fact, there is a good agreement between NOF and MDL under a moderate noise environment, but the MDL mayoverestimate the order with noisy-free data and underestimate the order at very high noise levels. Since we are looking fora criterion which is stable regardless the noise level, the proposed NOF factor revealed thus better than MDL for orderselection in a wide range of noise environment. Table 2 shows the comparison of model orders, computed from MDL andNOF methods and the estimation of noise at different additive noises levels, as computed from Eq. (23).

The computational order pcom shown in Table 2 may be selected greater or equal to the efficient order peff. The pcom hasbeen selected from a threshold of the uncertainty on modal parameters as it is described later. As expected, it is shown thatthe computational order increases with the noise level.

4.2.2. Selection of modes and modal parameters identification

Consider the case with a noise rate of 100%. For an assessment of model orders, a frequency stabilization diagram isconstructed from orders 2 to 30 (Fig. 8). It is seen in agreement with NOF that a model order of 6 is the smallest value

Fig. 10. OMAC diagram (NSR=100%).

Table 3Identified modal parameters with their 95% confidence intervals.

Mode Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7

NSR=0% (pcom=9)Frequency/confidence 38.6/0.04 74.6/0.03 107.7/0.14 162.9/0.78 208.2/0.02 273.0/0.06 288.2/0.01

Damping/confidence 1.29/0.11 0.27/0.05 0.40/0.13 2.19/0.47 0.36/0.01 0.26/0.02 0.18/0.005

NSR=1% (pcom=12)Frequency/confidence 38.6/0.09 74.6/0.2 107.8/0.14 163.9/0.51 208.2/0.018 273.0/0.06 288.2/0.13

Damping/confidence 1.47/0.24 0.27/0.03 0.4/0.13 1.24/0.31 0.37/0.01 0.26/0.02 0.21/0.05

NSR=10% (pcom=12)Frequency/confidence 38.6/0.11 74.6/0.03 107.8/0.14 163.9/0.52 208.2/0.018 273.0/0.07 288.2/0.18

Damping/confidence 1.54/0.3 0.27/0.04 0.4/0.13 1.26/0.32 0.37/0.01 0.27/0.03 0.23/0.06

NSR=100%(pcom=12)Frequency/confidence 38.6/0.12 74.6/0.02 107.8/0.1 163.9/0.4 208.2/0.02 273.0/0.07 288.2/0.19

Damping/confidence 1.44/0.3 0.27/0.03 0.36/0.09 0.89/0.25 0.37/0.01 0.27/0.03 0.24/0.06

NSR=200% (pcom=14)Frequency/confidence 38.6/0.12 74.6/0.02 107.8/0.1 163.9/0.4 208.2/0.02 273.0/0.05 288.2/0.13

Damping/confidence 1.44/0.31 0.27/0.03 0.36/0.09 0.89/0.25 0.37/0.01 0.27/0.02 0.21/0.04

NSR=400% (pcom=14)Frequency/confidence 38.6/0.12 74.6/0.02 107.8/0.1 163.9/0.4 208.2/0.02 273.0/0.05 288.2/0.13

Damping/confidence 1.44/0.31 0.27/0.03 0.36/0.1 0.89/0.25 0.37/0.01 0.27/0.02 0.21/0.04

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1041

required for the stabilization of all natural frequencies. Seven natural frequencies of Table 1 can distinctly be identified onthe stabilization diagram. However, one can wonder if the frequency at 120 Hz is a real frequency or not.

For the effective selection of computational order and structural modes, stabilized diagrams of the correspondingprospective frequencies and damping rates with their 95% confidence interval are constructed in Fig. 9. The OMAC for thosefrequency candidates are plotted in Fig. 10.

Several discussions may be made:

�

On the frequency stabilization diagram, a natural frequency must be stable from the minimum order, since theidentified part of the signal does not change significantly above this order value. � A stable frequency belongs to a mode if its damping rate takes a non-zero positive stable value on the diagram and if

the OMAC is closed to unity above the efficient order. Otherwise, it must belong to a harmonic excitation (zerodamping) or to a computational frequency. The destabilizations of OMAC, frequency and damping reveal that thedoubtful frequency at 120 Hz (Fig. 9d) was not a natural frequency (and can be due to electrical perturbation orcomputational mode).

� Even by considering an NSR of 100%, the results show that the confidence intervals converge to the good value at high

orders, and give us the uncertainty of the estimation (such as at the 5th frequency (Fig. 9e) that presents a relativelylarge uncertainty). By considering an acceptable uncertainty (threshold), a computational order pcom may thus beselected for the modal parameter identification. Similar conclusions were found for other simulated noise rate cases(Table 1).

Fig. 11. Mode shape identification at NSR=100% and by FEA.

Fig. 11. (Continued)

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–10441042

Table 3 shows the results of modal parameter identification (frequency in Hz and damping rate in %) of the sevenstructural modes selected at the corresponding computational order with their 95% confidence interval, for different noiserates ranging from 0% to 400%. It is found from Table 3 that:

�

Whatever the noise rate, the modal parameters are accurately identified at the computational order. � When the noise rate changes, the modal parameters and their confidence intervals are considered to be stable which

proves the performance of the proposed method even in a noisy environment.

� The confidence intervals of frequencies are relatively much smaller than those of damping ratios. This explains the

reliability on identification of frequency compared to the damping.

Fig. 11 plots the seven scaled mode shapes identified with a noise rate NSR=100% versus the mode shapes numericallycomputed by finite element method and the corresponding MAC factors (after having removed the perturbation mode at

V.H. Vu et al. / Mechanical Systems and Signal Processing 25 (2011) 1028–1044 1043

120 Hz). In the left figures, the mode shapes are represented and interpolated from the deformation of the six sensorlocations (on grid lines) while the confidence intervals are displayed by the darkness. The MAC is computed for each modein order to compare with numerical results. It is found that even if the signal is perturbed by a very high noise rate andsome modes are close, each identified mode shape is corresponding to an analytical one. As expected, the seventh andhigher mode shapes are not well identified because of the limitation of the sensor number.

5. Conclusions

An operational modal analysis based on a multivariable autoregressive model is presented with the ability of computingthe uncertainty on modal parameters identification of selected modes, after updating and selecting efficient model orders.The least squares implemented in the form of QR factorization of the data matrix produces a fast, conditioned andconvenient algorithm for updating. The Noise rate Order Factor (NOF) is effective for selecting the efficient model orderfrom which the modal parameters converge regardless the noise rate. It is seen that, compared to the MDL method, theNOF is more stable with respect to noise and hence the modal parameters start to be stable from this order in thestabilization diagrams. The new orderwise version of the correlation factor called OMAC derived between two successivemodel orders is useful for the selection of structural modes. The confidence intervals of each natural frequency anddamping ratio are added to the modal parameters on the stabilization diagram, which decrease as the model orderincreases. Computational model order can be chosen from acceptable confidence intervals for automatic identification ofmodal parameters, and hence varies with the noise rate. Simulations and experiments on a steel plate show that this newmethod is a good technique for operational modal analysis even in noisy environments ranging from 0% to 400%. Furtherstudies are conducted to develop the algorithm for online identification in the time domain and for nonstationary systems.

Acknowledgements

The support of NSERC (Natural Sciences and Engineering Research Council of Canada), through Research Cooperativegrants was gratefully acknowledged. The authors would like to thank Hydro-Quebec Research Institute for theircollaboration.

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