Application of Advanced Parameter Estimators to the ...

APPLICATION OF ADVANCED PARAMETER ESTIMATORS TO THE ANALYSIS OF IN-FLIGHT MEASURED DATA

Antonio Vecchio, Bart Peeters and Herman Van der Auweraer LMS International, Interleuvenlaan 68, B-3001 Heverlee, BELGIUM, e-mail: [email protected]

ABSTRACT Assessing the parameters of a structural model of an aircraft in-flight is of vital importance for validating the flutter models and to clear the flight envelope. But extracting modal parameters from in-flight measured vibration data poses a number of specific challenges. The data can be input/output or output-only data. In most cases, the measured data segments are of short duration and contaminated by significant measurement noise. Several of the critical modes are often closely coupled. The estimate of the damping is of critical importance and hence should be of high quality. Therefore, several advanced parameter estimation schemes are under evaluation in flight data analysis applications. The present paper will discuss results from the application of three methods to a set of flight test data. The investigated methods are: 1. the time domain stochastic subspace method (outputonly data), 2. the total least squares discrete frequency domain method, 3. the frequency domain maximum likelihood method, The research is conducted in the context of the EUREKA project FLITE.

1

INTRODUCTION

Aeroelastic behaviour of aircraft can be estimated with mathematical models. These models can be more or less detailed and take into account effects generated on the aircraft structures by the acting forces resulting from the composition of elastic, inertial, steady and unsteady aerodynamic forces. The goal of numerical models is to have a first estimate of the critical aeroelastic speeds, i.e. the values of airspeed at which critical aeroelastic phenomena as flutter appear. Flutter is a dynamic aeroelastic phenomenon that can show up suddenly with violent self-excited vibrations leading to dynamic instability and loss of structural integrity. Airworthiness regulations require that the flight domain of each new aircraft has to be proven stable, i.e. no critical aeroelastic phenomena should appear in the flight regimes, which the aircraft is certified for. Since numerical models do

not give more than estimates, wind tunnel test on scaled aircraft models or full-scale flight tests are usually carried out in order to achieve better accuracy in predicting critical flutter speed. The airworthiness certification is only granted after the stability throughout the required flight regime is guaranteed through dedicated flight tests also referred to as “flight flutter tests”. Flight tests consist of flying the aircraft at a discrete number of increasing airspeed and analysing the vibration structural response to natural or artificial excitation, as acquired by means of a few sensors placed on the aircraft structure. Used excitation techniques range form natural atmospheric turbulence to artificial systems as input to flight control system and oscillating vanes. In some cases pyrotechnic systems as bonkers or shooting guns placed on the wing tips can be also used. Airspeed is usually increased by steps of about 30 Kts, which represents a good compromise between accuracy needs – which would require a finer step – and costs. For each airspeed value, data are recorded and sent via telemetry to the ground station where a fast analysis has to be performed to the aim of extracting poles and evaluate whether or not self-exciting vibrations show up. Only after the current airspeed has been cleared against flutter, the pilot is given the signal to increase airspeed and the process is iterated up to the maximum airspeed value included in the flight envelope. Such a value is higher (by a safety factor) of the highest flutter-free airspeed, which the aircraft will be certified for. If no problems appear over each of the steps, the whole flight envelope is cleared and the aircraft certified. However flight flutter tests are not carried out as stand alone process. A priori knowledge is usually available with the flight test engineers to allow them identifying and tracking more easily how mode shapes (particularly damping values) evolves during the flight flutter tests. This knowledge is acquired through Ground Vibration Tests (GVT) consisting of vibration tests performed under controlled excitation and aiming at a complete identification of the structural mode shapes of the aircraft [1]. Coupling numerical models and results of accurate GVT sessions allows test engineers to retrieve all mode shapes of the aircraft in the frequency range of interest and to rank

923

them according to the likelihood to originate flutter phenomena. Once the most important modes (i.e. frequency and damping) and the estimated critical flutter speed as result of numerical models are available, flight test phase can start. The process of coupling numerical models, wind tunnel testing and flight in order to get flutter clearance is also referred to as “opening the flight domain”. It is worth noting that the aim of flutter test is to obtain the maximum of information from the sensor in the minimal laps of time: flight test are quite expensive and reducing time and costs is a major issue. In addition a full list of mode shapes is already available to the test engineers after GVT, therefore the number of sensor to be placed on the aircraft is reduced to the minimum necessary to identify critical behaviour in the fastest way. As a consequence, when flight data are used to identify mode shapes of the aircraft structure, big uncertainties will come up if no a priori knowledge (e.g. GVT) is available. Due to the limited number of sensors used for flight flutter tests, it is not possible to evaluate which of the poles selected in the stabilization diagram corresponds effectively to a true mode shape or it just represents a mathematical pole. Spatial aliasing occurs quite often. The lack of spatial redundancy in the point coverage over the structure, does not allow a correct and meaningful animation of mode shapes, meaning that even though two different poles are selected in the stabilization diagram, their animation does not show any difference because the few points available move equally in both modes. The refinement analysis process consisting of selecting only those poles that represent real mode shapes and discarding complex or not reliable modes is not easy. Functions usually applied to better identify real poles are not always fruitful: MAC matrix computation shows spatial aliasing and does not provide hints to discard bad poles; Modal Participation factors (MPF) and Mode Phase Collinearity (MPC) helps to identify the most clear mode shapes but they provide a little benefit for poles which are less evident.

estimation scheme such as the time-domain Polyreference Least Squares Complex Exponential technique (LSCE) can be applied. The short durations of the test data segments and the high levels of measurement noise however may render the analysis and results interpretation very difficult. Advanced methods such as the Least Squares Complex Frequency Domain method (LSCF) have been investigated [2],[3] and prove to be very robust and leading to an improved estimation process. Both the LSCE and LSCF are non-iterative. Therefore, the use of an iterative approach, based on the Maximum Likelihood estimator, as proposed by Guillaume and Schoukens [4] [5], was also investigated. In case only response data are measurable, or in case no external excitation was applied (turbulence or stickmovement tests), output-only methods must be applied. A modified formulation of the LSCE algorithm also supports the use of output-only data using measured cross-power functions [6]. But more recently, a stochastic subspace identification (SSI) scheme has been proposed which proves superior in analysis power and data quality [7],[8]. This method was hence also used in the present study. The LSCF, MLE and the SSI methods are reviewed below. For the full details, the reader is referred to [2-8]. 2.2

Least Squares Method

Complex

Frequency

Domain

The Least Squares Complex Frequency Domain, or LSCF method, starts from a scalar matrix-fraction description – better known as a common-denominator model – for the measured set of Frequency Response Functions (FRFs) [2]. The FRF between output o and input i is modeled as

Hˆ oi (ω f ) = for

N oi (ω f ) D (ω f )

(1)

i = 1, K , N i and o = 1, K , N o with n

In this paper we address the problem arising when flight test data are used to identify flutter occurrence with no a priori knowledge on the mode shapes. Algorithms available for pole extraction are compared in order to evaluate to what extent an automatic process can be implemented to directly extract modal parameters from on-line time data as measured during flight flutter tests. 2

THEORY

Before starting to describe the data structure and analysis results it is worth introducing the methods used in this work for pole extraction. 2.1

Parameter estimation methods

Depending on the nature of the flight test data, various parameter estimation methods have been used in this study. When input/output data are available, these data can be reduced to frequency response functions and a classical

N oi (ω f ) = ∑ Ω j (ω f ) Boij j =0

the numerator polynomial between output o/ input i and

D (ω f ) =

n

∑ Ω j (ω f ) A j j =0

the common-denominator polynomial. Several choices are possible for the polynomial basis functions Ω j (ω f ) . A discrete-time or a continuous time formulation can be used or even special polynomials such as Orthogonal Forsythe Polynomials can be applied [9]. But for every formulation, the parameters to be estimated are the real-valued coefficients A j and Boij . We used the discrete-time model formulation proposed by Guillaume [2], as this leads to a very stable equation solution. Replacing the model

Hˆ oi (ω f ) in (1) by the measured FRF

H oi (ω f ) gives, after multiplication with the denominator polynomial,

924

n

n

∑ Ω j (ω f )Boij − ∑ Ω j (ω f j =0

for

j =0

)H oi (ω f )A j ≈ 0

l ML (θ) = ∑ ∑ ∑

Note that every equation in (2) can be weighted. As the equations (2) are linear in the parameters, they can be reformulated [2,3] as

0

L

X2

0 0

O 0

X N o Ni

 B  Y1   1  B Y2   2   M ≈0 M     BN o Ni  YN o N i   A 

(3)

 B1   B   2  or [J ] M  ≈ 0 (4)    B No Ni   A   A0   Boi 0  A  B  1 oi1   , A=  with Bk =  M   M       An   Boin   X k (ω1 )   Yk (ω1 )      Xk =  M M  , Yk =    X (ω )  Y (ω )  k Nf   k Nf  X k ( ω f ) = Woi ( ω f )[ Ω 0 (ω f ), Ω1 (ω f ), K , Ωn (ω f )] Yk ( ω f ) = − X k ( ω f ) ⋅ H oi ( ω f ) and

The ML estimate of

H

Re( J J ) as this results in a faster implementation. As a

Obviously, the optimisation approach leads to an iterative procedure, which increases the duration of the estimation process. A proper choice of the starting values (preferably by a first step linear parameter estimation process) is important to ensure rapid convergence [11]. But the highquality results, the capabilities to let the method perform a semi-automated analysis and the capability to complement the estimation results with uncertainties [12], give the ML approach unique features. 2.4

Stochastic Subspace Identification Method

The Stochastic Subspace Identification method is a time domain approach. As measurement data, the response signals from a set of transducers are first processed into cross-correlations with respect to a pre-selected set of reference signals. From these correlation functions, the parameters of a state-space model are estimated. The following stochastic discrete time state space model is considered: {x k +1} = [ A]{x k } + {wk } (6) {yk } = [C]{x k } + {vk }

{x k } represents the state vector of dimension n and {wk } , {v k } are zero-mean, white vector sequences,

where

respectively representing the process noise and measurement noise. The matrices [ A] and [C ] are respectively the state space matrix and the output matrix.

[ ]

Along with this model, the observability matrix O p and the

[ ]

controllability matrix C p of order p are defined:

[O p ]

Maximum Likelihood Method

Starting from the same scalar matrix-fraction model formulation as in (1), Guillaume et al. propose a different parameter estimation approach based on a maximum likelihood approach [4],[5]. Assuming the different FRFs to be uncorrelated, the (negative) log-likelihood function reduces to [10]

θ = [ B1T , K , B NT o N i , AT ]T is given by

Re( J mH J m ) and Re( J mH rm ) ) in a time efficient way.

discrete time-domain model – which generally leads to a well-conditioned Jacobian matrix J – has been used, the explicit calculation of the normal equations is justified. 2.3

(5)

minimizing (5). This can be done by means of a GaussNewton optimization algorithm, which takes advantage of the quadratic form of the cost function (5). The Jacobian matrix J m has the same structure as the one given in (3). Also here it is possible to form the normal equations (i.e.,

k = (o − 1) N i + i = 1, K , N o N i .

The Least Squares estimate can be computed efficiently via a “structured” QR decomposition of the Jacobian matrix [J] of eq. (3). However, most estimators used in modal analysis form the normal equations explicitly, i.e. they compute

2

var{H oi (ω f )}

o =1 i =1 f =1

i = 1, K , N i , o = 1, K , N o and f = 1,K , N f .

X1 0   M   0

Hˆ oi (θ, ω f ) − H oi (ω f )

N o Ni N f

(2)

 [C ]   [C][ A]  ; C = [G ] =   p M  p −1  [C ][ A] 

where

[ ] [

[

[G] = E {x k +1 }{yk }

[ A][G] T

].

L

]

[ A] p −1[G]

(7)

E[.] denotes the expected

[ ] and [C p ]

value operator. The matrices O p

are assumed

to be of rank n. The problem considered here is the estimation of the matrices [ A] and [C ] in equation (9), up to a similarity transformation, using only the output measurements {yk} . This identification problem is also known as the stochastic realization problem [13] - [15].

925

The key element in the approach is the establishment and decomposition of a block Hankel matrix of the measured correlation functions. Defining the empirical correlation matrix of the measured output vector

[ Rk ]

=

1 M T yt + k }{ yt } { ∑ M t=0

(8)

where M is the number of observations, the following block Hankel matrix can be defined and decomposed into its singular values;

[H ] p, p

 [ R1 ]   [ R2 ] =  M   R p

[ ] [ S1 ]   [0]

[[U1 ] [U2 ]]

[

[ R2 ]

L

[ R3 ]

L

M

O

R p+1

]

L

[ ] [ ]

0   V1  [ S2 ]  V2T T

[ Rp ]  [ Rp+1 ]  =

[

  R2 p−1   M

]

(9)

(

)

with σ n +1 〉〉σ n

(10)

p is a user defined parameter chosen so that p>2Nm. where Nm represents the number of physical modes. [ S1 ] and [ U1 ] contain respectively the n first singular values and the corresponding left singular vectors. From the stochastic

[

]

realization theory, can be factored out as H p, p ) ) H p, p = Op C p , and hence an estimate of the ) 1 observability matrix is given by Op = [U1 ][ S1 ] 2 . The

[

] [ ][ ]

[ ]

system matrices [A] and [C] are then estimated up to similarity transformation, using the shift structure of O$

[ ] p

[15]. The dynamics of the system are completely characterized by the eigenvalues and the observed parts of the eigenvectors of the [ A] matrix. Note that the extracted mode shapes can not be massnormalized as this requires the measurement of the input force. Finally, it should be noted that also formulations using the LSCF and Maximum Likelihood approach to output-only data have been proposed [16]. A full review of stochastic system identification methods for experimental modal analysis is presented in [17], [18]. 3

The pilot applied an excitation to the aircraft structures by means of an electronically controlled signal introduced in the aircraft control system. The excitation signal was a narrow band (0-60 Hz) white noise that generated a periodically varying deflection of the aircraft ailerons. The deflection was measured and recorded on the right aileron. Responses were acquired through 12 accelerometers placed in 9 different locations on the aircraft. Most of them were distributed on the wingspan; some sensors were also placed on the rudder and the fuselage nose. Data collected are output only data, though the measurement of the deflection induced in the aileron could be considered as a kind of input. Time data were then saved in digital format and stored on a CD-ROM.

   

where S1 = diag(σ 1Lσ n ), σ 1 ≥ σ 2 Lσ n ≥ 0

S2 = diag σ n +1Lσ pNresp

in four steps of 30 Kts each. The last flight test was performed at the maximum airspeed but with aircraft’s tanks fully loaded.

DATA STRUCTURE AND ANALIYIS

The FLITE consortium provided a dataset of in-flight flutter tests executed on a prototype aircraft. Data were acquired during several flight tests sessions carried out to the aim of getting flutter clearance. Data refer to 5 different flight sessions corresponding to different flight conditions. In a first test session the aircraft configuration (namely mass distribution and load) was kept constant and the only changed parameter was the airspeed, which was increased

Using Throughput Monitor of CADA-X, data were replayed as they were acquired on-line and several functions were calculated. First FRFs were estimated using the aileron deflection as input signal and all other channels as output. This way, it was possible to obtain a set of input-output measurements to be used for a classical Modal analysis (i.e. LSCE method). The aim was to evaluate to what extent it is possible to use classical methods for mode shape identification of “difficult data“. In the case under study the difficulty comes not only from the low number of sensors available but also from the hypothesis that deflection is a reliable input signal for a classical input-output analysis. Successively the different pole estimators were applied: first the classical LSCE method, then the new LSCF (Least Square Complex Frequency) and the MLE (Maximum Likelihood). Finally the algorithm in use for operational modal Analysis (OMA) as the Stochastic Sub-space Methods (namely the Balanced Realization Method) was directly applied to time data. The results of such comparison are reported in the following paragraph. 3.1

LSCE based Analysis

Classical modal analysis methods require FRF or impulse response data (obtained as the inverse Fourier transforms of FRFs), therefore a set of input-output data was obtained by considering the deflection as input signal and calculating FRF of all responses. A first analysis of the input signal shows that the excitation can be considered as white noise in the frequency range of interest: 0- 50 Hz (fig.1). When using LSCE method on difficult data, a problem arises regarding the poles to be selected in a stabilization diagram. A helpful tool is the MIF, which gives some hints on which poles to select. In figure 2 the summed spectrum and the MIF are superimposed.

926

Figure 4: MAC Matrix, LSCE pole selection

Figure 1: Auto power spectrum of aileron excitation

Other validation tools as mode complexity and modal phase collinearity do not provide more insight. 3.2

Figure 2: Sum Block (dark line) and MIF (light line)

New modal parameter estimators

This section presents the results of the comparison among some new methods for modal parameters extraction. The so-called Least Squares Complex Frequency Domain method (LSCF) and MLE (Maximum Likelihood Estimator) are applied to FRFs as calculated in CADA-X Throughput Monitor in replay mode. A correlation-driven stochastic subspace identification method is applied to the inverse Fourier Transform of the cross spectra. All used parameter estimation methods are implemented in the LMS Cada-X software [19].

From that figure it is possible to identify two frequency ranges where the poles are present. However, although some hint is provided, data remain difficult and the interpretation of the stabilization diagram (fig.3) is not easy.

Figure 5: Stabilization diagram with LSCF method

Figure 3: Stabilization diagram (LSCE method)

Only 3 up to 5 existing poles in the frequency range 0-20 Hz show a stable behaviour, for the missing poles a long refinement process is required. Fig. 4 shows the MAC matrix for a selection of 7 poles. Spatial aliasing occurs but since geometry is too poorly defined, assessing the quality of mode shape based on geometry animation is not beneficial.

Using the LSCF algorithm on the same data the stabilization diagrams (fig. 5) is much easier to interpret. Only few and very stable pole are in diagram, furthermore the 5 poles identified with LSCE methods are all correctly represented. In some of the dataset, when measurements were noisier or when the frequency resolution was not appropriate, some pole was missing. In this respect the outcome of the analysis is that for flight flutter test a frequency resolution of 0.05 Hz is to be recommended. However, when performing the second step with the MLE optimisation, all the poles present in the frequency band show up correctly.

927

MLE performs very well and is very well suited for an automated analysis. The algorithm is such that all the stable poles are extracted from data with no needs of manual selecting them in the stabilization diagram. However, as some mathematical poles are also selected, a second step is needed in order to eliminate spurious poles. In this respect, using LSCF method it is possible to facilitate the process. Starting from the MLE pole estimation one can begin a LSCF estimation process and select in the stabilization diagram also the poles extracted by MLE. Calculating MAC matrix between the two methods allows easily identifying only the physical poles. Fig. 6 and fig.7 show the result of such comparison. The 5 physical stable poles are highly correlated, while all spurious modes show poor correlation and can be easily identified. By removing lowly correlated poles the MAC matrix values are strongly improved (fig.7).

better performance. Fig. 8 and fig. 9 show respectively the stabilization diagram resulting from SSI and the MAC matrix obtained comparing SSI and MLE. Although the high values on the diagonal indicate that SSI methods retrieves all physical modes, the MAC shows that spatial aliasing occurs and it would be difficult to identify only the mode shapes based on the SSI results. Further validation is required.

Figure 8: Stochastic Sub Space Identification: stabilization diagram

Figure 6: MAC Matrix: LSCF vs. MLE method, first step

Figure 9: MAC matrix between MLE and SSI methods

3.3

Figure 7: MAC Matrix: LSCF vs. MLE method after spurious pole removal

The comparison with Stochastic Sub space method shows that despite the stabilization diagram is easier to interpret than in the LSCE case, correlation with other methods is not excellent. Some pole is missing while some new and spurious pole appears in the stabilization diagram. Finally, the method requires more computation time. In this comparison the SSI algorithm was used on the original time data and all sensors were used as reference. A more in depth analysis showed that the sensor placed on the rudder does not correlate well with the others. When this sensor was not considered in the analysis, SSI showed a

The flight envelope

The analysis of all datasets allowed extracting frequency and structural damping as a function of airspeed. Using the coupled technique MLE and LSCF it was possible to dramatically reduce time requested for the analysis. MLE performs very well in fast extracting poles from different datasets, while LSCF and MAC allowed easily tracking the mode shapes along with the increasing of airspeed. The results are illustrated in fig. 10. 4

CONCLUSIONS

This paper presented the result of a comparison carried out on different new methods for modal parameter estimation. The aim was to identify a procedure for making it easier to extract modal parameters from difficult data as time recording acquired through few sensors in flight flutter testing. Using the classical stabilization diagram approach, the LSCF clearly outperforms the classical LSCE method. When no a priori information is available it is recommended

928

to use MLE estimator as valuable method to automatically extract poles. A fine-tuning process is then possible by using LSCF and MAC matrix. This allows fast discarding of spurious poles as it is easier to interpret the stabilization diagrams of LSCF method. When more than one dataset have to be analyses in short time as it is the case in flight flutter tests, coupling LSCF and MLE gives a possibility to speed up the process and achieve a faster tracking of the modal parameters of structures under changing flight conditions.

[5] Guillaume, P., Identification of multi-input multi-output systems using frequency-domain models. Ph.D. Dissertation, Vrije Universiteit Brussel, Belgium, 1992. [6] James III, G., Carne, T., and Laufer, J., The natural excitation technique (NExT) for modal parameter extraction from operating structures, Int. J. of Analytical and Experimental Modal Analysis, 10(4), 260-277, 1995. [7] Van Overschee, P., De Moor, B., Subspace algorithm for the stochastic identification problem, Automatica, 29(3), 649-660, 1993.

14

[8] Hermans, L., Van der Auweraer, H., Modal testing and analysis of structures under operational conditions: industrial applications, Mechanical Systems and Signal Processing, 13(2), 193-216, 1999.

12

10

8

[9] Van der Auweraer, H., Leuridan, J., Multiple input orthogonal polynominal parameter estimation, Mechanical Systems and Signal Processing, 1(3), 259272, 1987.

6

4

2

0

1

2

3

4

5

0 0,005 0,01

[10] Guillaume, P., Parametric identification of multivariable systems in the frequency-domain – a survey, In Proc. 21-st ISMA, Leuven (B), Sept. 18-20, 1996, Vol.2, 1069-1082.

0 , 0 15 0,02 0,025 0,03 0,035 0,04 0,045 0,05

Figure 10: Structural frequency (top) and damping (bottom) vs. airspeed

ACKNOWLEDGEMENT The discussed research results were obtained in the context of the EUREKA project E!2419 “FLITE”. The support of the Flemish institute for scientific and technological innovation in industry (IWT) is gratefully acknowledged. REFERENCES [1] Fargette P.et Alii Tasks for improvements in Ground Vibration Testing of large aircraft. In Proceedings of IFASD 2001, Madrid, June 5-7, 2001. [2] Guillaume, P., Verboven, P., Vanlanduit, S., Frequency domain maximum likelihood identification of modal parameters with confidence intervals, In Proc. 23-rd ISMA, Leuven (B), Sept. 16-18, 1998, 359-366. [3] Van der Auweraer, H, Guillaume, P., Verboven P., and Vanlanduit, S., Application of a fast-stabilizing frequency domain parameter estimation method, ASME J. of Dyn. Systems, Meas. And Control, 123(4), 2001 (in press). [4] Guillaume, P., Pintelon R., and Schoukens, J., Robust parametric transfer function estimation using complex logarithmic frequency response data, IEEE Trans. on Automatic Control, 40(7), 1180-1190, 1995

[11] Verboven, P., Guillaume, P., and Van Overmeire, M., Modal parameter identification: estimation of starting values for MLE-like algorithms, In Proc. 23-rd ISMA, Leuven (B), Sept. 16-18, 1998. [12] Schoukens, J., and Pintelon, R., Identification of linear systems: a practical guideline to accurate modeling. Pergamon Press, 1991. [13] Desai, U., Debajyoti, P., Kirkpatrick, R., A realization approach to stochastic model reduction, Int. J. Control, vol. 42, no. 4, pp. 821-838, 1985. [14] Van Overschee, P., and De Moor, B., Subspace identification for linear systems: theory, implementation, applications. Kluwer Academic Publishers, 1996. [15] Basseville, M., Benveniste, A., Goursat, M., Hermans, L., Mevel, L., Van Der Auweraer, H., Output-Only Subspace-Based Structural Identification: From Theory to Industrial Testing Practice, ASME J. of Dyn. Systems, Meas. And Control, 123(4), 2001 (in press). [16] Hermans, L., Van der Auweraer, H., and Guillaume, P., A frequency-domain maximum likelihood approach for the extraction of modal parameters from output-only data, In Proc. 23-rd ISMA, Leuven (B), Sept. 1618,1998, 367-376. [17] Peeters B. System Identification and Damage Detection in Civil Engineering. PhD thesis, Department of Civil Engineering, K.U.Leuven, Belgium, [www.bwk. kuleuven.ac.be/bwm], December 2000. [18] Peeters B. and De Roeck G. Stochastic system identification for operational modal analysis: a review. ASME Journal of Dynamic Systems, Measurement, and Control, 123(4), 2001 (in press). [19] LMS INTERNATIONAL. Cada-X Modal Analysis Manual Rev 3.5.D, Leuven, Belgium, [www.lmsintl.com], 2001.

929