Fully automated (operational) modal analysis ∗

Edwin Reynders†, Jeroen Houbrechts and Guido De Roeck

University of Leuven (KU Leuven), Dept. of Civil Engineering, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium

Abstract

Modal parameter estimation requires a lot of user interaction, especially when parametric system identificationmethods are used and the modes are selected in a stabilization diagram. In this paper, a fully automated, generallyapplicable three-stage clustering approach is developed for interpreting such a diagram. It does not require anyuser-specified parameter or threshold value, and it can be used in an experimental, operational, and combinedvibration testing context and with any parametric system identification algorithm. The three stages of the algorithmcorrespond to the three stages in a manual analysis: setting stabilization thresholds for clearing out the diagram,detecting columns of stable modes, and selecting a representative mode from each column. An extensive validationstudy illustrates the accuracy and robustness of this automation strategy.

Keywords: Modal testing; Operational modal analysis; Automation; Stabilization diagram; Hierarchical clustering;Partitioning method; Modal validation criteria; Mean phase deviation; Structural dynamics.

1 Introduction

1.1 Problem statement

The estimation of modal parameters from measured vibration data involves a substantial amount of user interac-tion. This not only results in a relatively large analysis time and cost, but it also prevents the further spread ofmodal testing to applications such as fault detection at mechanical production lines, or health monitoring of crucialinfrastructure, where a lot of data needs to be processed in a short amount of time. Automating the modal pa-rameter estimation process is therefore the objective of this paper. Modal analysis in the broad sense consists ofdifferent stages: data collection, signal preprocessing, system identification and the determination of a validated setof modal parameters. This work concentrates on the last and often most time-consuming stage, i.e., the estimationof a validated set of modal parameters from a set of identified system models, which is also termed modal analysisin the narrow sense.

In most of the literature on automated modal testing, no clear distinction is made between modal parameterestimation (MPE), which is the estimation of modal parameters from (a single record of) measured data, and modaltracking, i.e., tracking the evolution of the modal parameters of a structure or a group of similar structures throughrepeated MPE. Automated modal tracking algorithms usually need baseline modal parameters to start from, in whichcase they cannot be used for modal parameter estimation based on a single data record. This work is concernedwith automated MPE and as such it is assumed that, if different data records are available, they are processedindependently. Modal tracking is often a necessary second step, but it falls outside the scope of this paper.

Based on the type of excitation, a distinction can be made between experimental, operational, and combinedmodal analysis. In experimental modal analysis (EMA), forces acting on a structure are recorded, and the responsedue to any unmeasured force is regarded as unwanted noise that needs to be removed. For in-situ measurements,operational modal analysis (OMA) is often more appropriate. In such test, the response of the structure to theunmeasured operational loading is recorded, and the modal parameters are extracted from the output-only data,using assumptions on the nature of the unknown forces. A combined approach, where in the identification process,both measured and unmeasured forces are accounted for, yields optimal results, and contains EMA and OMA as

∗Postprint submitted to Mechanical Systems and Signal Processing.Published version: E. Reynders, J. Houbrechts, and G. De Roeck. Fully automated (operational) modal analysis. Mechanical Systems and SignalProcessing, 29:228-250, 2012. http://dx.doi.org/10.1016/j.ymssp.2012.01.007

†Corresponding author. Email: edwin.reynders@bwk.kuleuven.be; tel.: 0032 1632 1677; fax: 0032 1632 19883

1

special cases [39]. This approach is often called operational modal analysis with exogenous inputs (OMAX). In thiswork, a general automation strategy, that is independent from the type of test and the particular (parametric) systemidentification method used, is set up.

All parametric system identification techniques require at least one user-defined integer: the model order n,which equals the number of eigenvalues present in the model, hence, in theory, twice the number of eigenfrequen-cies. From control theory, several model validation techniques are available allowing to choose n in an automatedway, so that the prediction capacity of the identified model is maximized; an overview of such techniques can befound in [26, ch. 16]. However, in modal testing applications, one is not primarily interested in the prediction capacityof an identified model as such, but rather in the physical relevance of the individual modes that constitute the model.An alternative approach has therefore been developed, based on the empirical observation that in a very large num-ber of modal identification problems, the physical modes of the structure appear at nearly the same eigenfrequencywhen the model order is over-specified, while the other, spurious modes, do not [50]. In this approach, parametricmodels are then identified for a wide range of model orders, most of which are larger than the number of modes inthe considered frequency band, and the modes of all these models are plotted in a model order vs. eigenfrequencydiagram, called a stabilization diagram. The physical modes should then show up as vertical lines in this diagram.Although the stabilization diagram has become a key tool in modal testing, see, e.g., the textbooks [1, 19], its inter-pretation, i.e., the selection of physical modes as columns in the diagram, is often not straightforward. The resultsmay depend on the judgement of the analyst, and possible additional validation criteria may be needed.

The goal of this work is to develop a fully automated approach for the interpretation of stabilization diagrams.Naturally, this approach should obey the following five target criteria:

1. not rely on more than one data record or on prior estimates for any of the modal parameters;

2. be as physically intuitive as possible and follow the course of a manual analysis;

3. produce similar results as in a manual analysis;

4. work in an EMA, OMA and OMAX framework and with any parametric system identification algorithm;

5. not contain parameters that need to be specified or tuned by the user.

1.2 Overview of existing approaches

Estimation of modes from identified nonparametric frequency response functions or (positive) power spectral densi-ties is straightforward and physically intuitive: peaks are picked from derived quantities, such as the complex modeindicator function (CMIF) [47] or the averaged normalized power spectral density [32], that are plotted as a functionof frequency. Approaches for automating the peak selection process have been recently proposed [4, 37, 38]. Theyheavily depend on the use of the MAC value, so mode shapes of sufficient spatial resolution are needed, and theymay fail to detect closely spaced modes. Furthermore, the approach of [38] requires initial mode shape estimates,which are most often unavailable, and the approach of [37] needs at least two different data series recorded un-der similar excitation, and is therefore primarily suited for modal tracking. Since simulation studies have confirmedthat modal parameters obtained from identified parametric structural models such as state-space models are farmore accurate than nonparametric estimates [34, 39], most research effort has been spent in the automation ofparametric techniques.

For parametric identification, several strategies have been proposed for the automated interpretation of stabi-lization diagrams. Since analyzing a stabilization diagram boils down to recognizing modes with similar properties,it comes as no surprise that most automation strategies involve clustering, i.e., grouping data points with similarcharacteristics. The following categories can be discriminated:

• Hierarchical clustering is perhaps the most natural approach. It starts by putting all (stable) modes in thestabilization diagram in separate clusters; then, the clusters that are closest together are collected in a singlecluster, one by one, until the distance between all remaining clusters is larger than a user-defined thresholdvalue. The final set of physical modes is then chosen from the clusters containing a minimum number ofelements, corresponding to the minimum number of modes on a stable line in a stabilization diagram. Pappaet al. [31] were perhaps the first to report such an approach, using the eigenfrequency difference and the MACvalue as distance measures. Although the term ‘hierarchical clustering’ is not explicitly coined in this report,they successfully applied it to automate the Eigensystem Realization Algorithm (ERA) [21] for an EMA analysisof the Space Shuttle tail rudder. This particular analysis was later extended, by using genetic algorithms, to

find the ‘optimal’ ERA parameter values (besides n) [7]. Chauhan and Tcherniak [8] present slight variationson the original approach of Pappa et al. [31]. Goethals et al. [16] propose an alternative distance measure,incorporating the eigenfrequency and damping ratio difference. Closely spaced modes are detected throughthe presence of modes with the same model order in the same cluster; they are then separated using theMAC value. Allemang et al. [3] use yet another distance measure, namely the MAC value between extended,pole-weighted mode shape vectors that are identified for each mode instead of the mode shape. Verboven etal. [53] present quite a different approach, where it is assumed that the number of modes in one cluster is apriori known; this is however rarely the case. A successful application of hierarchical clustering is reported byMagalhaes et al. [28], who analyzed more than 2500 high-quality data sets collected on a 280m-span concretearch bridge.

• Partitioning methods divide the total set of modes into a predefined number of clusters. Possible approachesare K-means clustering, where the clusters are mutually exclusive, and fuzzy C-means clustering, where theclusters overlap. Verboven et al. [54] and Vanlanduit et al. [52] use fuzzy C-means clustering for classifyingthe modes, estimated with frequency-domain Maximum Likelihood Estimation (MLE) for a single model ordern, into two categories (C = 2): physical and spurious modes. Hereto, they choose a set of single-modevalidation criteria, one of them being a measure for the stabilization of a particular mode at lower model orders[52]. Alternatively, Scionti and Lanslots [46] use fuzzy C-means clustering to group the modes, present ina stabilization diagram, directly into a user-defined number of C clusters. The clustering is performed in theeigenfrequency-damping ratio plane. The main drawbacks of this approach are that the number of clusters is auser-defined quantity, and that several non-intuitive enhancements to the basic C-means clustering algorithmand a combination with genetic algorithms are needed to provide reasonable results. Carden and Brownjohn[5] note that, since the coefficient of variation of the damping estimates is in general substantially largerthan that of the eigenfrequency estimates, the analysis is preferably not performed on the eigenvalues in theeigenfrequency-damping ratio plane, but on the eigenvalues in the complex plane: this produces clusters of amore spherical nature.

• Histogram analysis, where the frequency axis in the stabilization diagram is divided into narrow bins, in whichthe number of (stable) modes is counted. Scionti et al. [45] use such a histogram as the basis for an automatedmodal parameter estimation procedure that needs quite a few user-defined parameters, including the bin width.Its performance was evaluated against manually selected modes for in-flight flutter data. A good agreementwas found for the PolyMAX identification method [35], which yields very clear stabilization diagrams but biaseddamping ratio estimates [9], but the performance was less good for the least-squares complex exponential(LSCE) identification method [24].

Approaches that are suited for modal tracking, but not for modal parameter estimation from individual datarecords, include self-learning least-squares support vector machines algorithms [16] and fuzzy C-means clusteringon several data records together [5]. Since modal tracking falls outside the scope of this work, they are not discussedin further detail.

1.3 Methodology and contributions

All automated MPE methods discussed in section 1.2 breach at least one of the five targets set in section 1.1.In particular, they all need user-specified parameter or threshold values, the sole exception being the partitioningapproach of Verboven et al. [54] and Vanlanduit et al. [52]; however, this method applies to the strategy where onlya single model order is considered, while a stabilization diagram contains modes identified at many different modelorders.

In this paper, we present a fully automated approach for the interpretation of stabilization diagrams, involvingclustering at three stages:

1. All modes in the stabilization diagram are classified into two categories: possibly physical and certainly spuri-ous modes. For this purpose, a partitioning method is employed that makes use of as many relevant single-mode validation criteria as possible. The modes that are classified as certainly spurious are removed from thediagram. This first stage automates the setting of the stabilization thresholds, performed by the user to obtaina clear diagram. The diagram may still contain some spurious modes and therefore the modes in the cleareddiagram are termed possibly physical.

2. Similar modes in the cleared stabilization diagram are grouped together. Hereto, hierarchical clustering isemployed, but the cut-off distance is set based on the result of stage 1, not as a user-defined quantity. Thisstage corresponds to the visual inspection of the stabilization diagram by the user, in order to detect verticallines of stable modes.

3. The clusters are grouped into two categories, those containing physical and those containing spurious modes,and a single mode is chosen from each physical cluster. Cluster validation criteria such as the number ofmodes in a cluster are employed. This stage corresponds to the selection by the user of a representativemode from columns of stable modes in the diagram.

The most important contributions of this work are (i) the development of a generally applicable automated (op-erational) modal analysis approach that satisfies all five natural desiderata listed in section 1.1, and (ii) its validationon a large data set of forced and ambient bridge vibration data, by comparing the automated results with those ofmanual analyses by expert users.

The text is organized as follows. The validation criteria that are used in stage one of the automated approachare presented in section 2. The three stages of the automated approach are detailed in section 3. Sections 4and 5 contain validation studies, where a total of 14 real-life OMA and OMAX data sets are both automatically andmanually analyzed. Finally, section 6 concludes the paper.

2 Single-mode validation criteria

This section provides an overview of single-mode validation criteria that may be used in stage 1 of the proposedautomated approach. They are termed single-mode since they are used here for determining whether a particularmode (i.e., point) in a stabilization diagram, is physical or spurious, rather than for assessing the completeness oraccuracy of a modal model, i.e., a set of identified modal parameters for the considered frequency range. Some ofthese are hard criteria, yielding a binary answer (such as stability), while other are soft criteria, yielding a range ofvalues (such as relative frequency difference).

2.1 Eigenfrequency, damping ratio and mode shape distance m easures

Let fuj , ξj and φj denote the undamped eigenfrequency (in Hz), damping ratio (dimensionless), and unscaledmode shape (in any suitable output quantity), respectively, belonging to a particular mode j. Commonly useddimensionless distance measures between two modes j and l are the relative eigenfrequency and damping ratiodifferences:

d(fuj , ful) =|fuj − ful|

max(|fuj |, |ful|)and d(ξj , ξl) =

|ξj − ξl|max(|ξj |, |ξl|)

. (1)

Alternatively, a distance between the continuous-time eigenvalues λcj and λcl of modes j and l, respectively, couldbe used:

d(λcj , λcl) =|λcj − λcl|

max(|λcj |, |λcl|). (2)

This distance incorporates both eigenfrequency and damping ratio information, since [39]

λcj = −|2πfuj|ξj + 2iπfuj

√

1− ξ2j . (3)

For comparing the unscaled mode shapes, the modal assurance criterion (MAC) [2], which is the (dimensionless)correlation coefficient between both mode shapes, is a commonly used tool:

MAC(φj ,φl) ,|φj

∗φl|2||φj ||22||φl||22

, ||φj ||2, ||φl||2 6= 0, (4)

where �∗ denotes complex conjugate transpose. When the mass is approximately equally distributed and the

damping is proportional, one has that, when φj and φl are mode shapes belonging to different modes, their MACvalue should be close to zero. The MAC is commonly used for measuring the distance between eigenvectors in astabilization diagram, for comparing identified and calculated modes, and for validating the identified set of modalparameters. Note that 0 ≤ MAC(φj ,φl) ≤ 1.

As a combined, dimensionless measure of the distance between two modes, the following definition is used inthis work:

d(j, l) = d(λcj , λcl) + 1−MAC(φj ,φl). (5)

The classic stabilization criteria are the distances in eigenfrequency, damping ratio, and mode shape of a modeat a certain model order to the closest mode (in terms of (5)) at the closest lower model order. A low distanceindicates a well-stabilized mode, while a high distance indicates the contrary. These criteria, and the extendedcriterion based on (5), are validation criteria that are attributed to each single mode of the stabilization diagram.

2.2 Statistical accuracy of the estimates

Some important system identification methods, such as stochastic subspace identification [43] and maximum like-lihood estimation [36], do not only yield point estimates for the modal parameters, but also the variances of theseestimates and, since the estimates are typically asymptotically normally distributed, even their complete probabilitydensity function. It can be expected that the standard deviations of the eigenfrequency σ(fuj), damping ratio σ(ξj),and mode shape components σ(φjo), with φjo the oth element of φj , are small for physical modes and much largerfor spurious modes. Verboven et al. [54] indeed report that in EMA tests, the standard deviation of identified spu-rious eigenvalues is typically a factor 10 to 100 larger than that of physical eigenvalues. Consequently, wheneveravailable from the MPE algorithm, σ(fuj), σ(ξj), and σ(φjo) are very valuable parameters for distinguishing betweenphysical and spurious modes.

2.3 Measuring mode shape complexity

When a structure is proportionally damped, the mode shape components of a single mode lie on a straight linein the complex plane. For double modes, i.e., two modes with exactly the same eigenfrequency, this is not thecase, but such modes occur very rarely in practice, except for double symmetric structures; for nearly axisymmetricstructures, valid mode shapes that tend to form a circle in the complex plane have been identified [13]. Therefore,although mode shape collinearity is a very powerful single-mode validation criterion, it should be used with care.

The complexity of a mode shape φj ∈ Cny can be measured with the modal phase collinearity (MPC):

MPC(φj) =||Re

(

φj

)

||22 + 1ǫMPC

Re(

φj

T)

Im(

φj

)

(

2(

ǫ2MPC + 1)

sin2 (θMPC)− 1)

||Re(

φj

)

||22 + ||Im(

φj

)

||22(6)

where �T denotes transpose, and

φjo = φjo −∑ny

o=1 φjo

ny

, ǫMPC =||Im

(

φj

)

||22 − ||Re(

φj

)

||222Re

(

φj

T)

Im(

φj

) , and

θMPC = arctan

(

|ǫMPC |+ sign (ǫMPC)√

1 + ǫ2MPC

)

.

A detailed motivation of this expression can be found in [30]. MPC values are dimensionless; they lie between 0(not collinear at all) and 1 (perfect collinearity).

Alternatively, the mean phase (MP) of the mode shape components can be computed, and the mean phasedeviation (MPD), i.e., the (weighted) mean deviation of these components from the mean phase. Approximateexpressions for these are provided in [19], but they fail when a mode shape component has a large imaginary anda small real part, which may occur for instance when a mode shape is well identified in an EMA or OMAX test, butits mass-normalization is of poor quality. Therefore, we derive new expressions.

The mean phase can be computed as the angle of the best straight line fit through the mode shape in thecomplex plane, in the sense that the orthogonal regression is minimal (see Fig. 1). This boils down to the followingtotal least squares problem [51]:

MP(φj) = arg minθ

‖Im(φj)− tan(θ)Re(φj)‖221 + tan(θ)

, (7)

Re(φj)

Im(φj)

MP(φj)Re(φj)

Im(φj)

MP(φj)

∆θo

φjo

Figure 1: The mean phase is determined such that the orthogonal distance of the mode shape compo-nents to the corresponding straight line fit, is minimized (left). The mean phase deviation is determined asa weighted mean of the phase deviations ∆θo of the individual mode shape components φjo from the meanphase MP(φj).

that can be solved as [17]

MP(φj) = arctan

(−V12

V22

)

, USV T =[

Re(φj) Im(φj)]

, (8)

where U ∈ Rny×2, S ∈ R

2×2 and V ∈ R2×2 constitute a singular value decomposition, i.e., S is a diagonal

matrix with decreasing entries along the diagonal, and U and V have orthonormal columns. V12 and V22 denotethe elements (1, 2) and (2, 2) of V , respectively. The deviation of the phase of φjo from the mean phase can be

computed from the scalar product between[

Re(φjo) Im(φjo)]T

and[

V22 −V12

]T. The mean phase deviation is

subsequently obtained as:

MPD(φj) =

∑ny

o=1|φjo|>0

wo arccos

∣

∣

∣

∣

Re(φjo)V22−Im(φjo)V12√V 2

12+V 2

22|φjo|

∣

∣

∣

∣

∑ny

o=1 wo

,

ny∑

o=1

wo 6= 0, (9)

where wo are weighting factors, that may be chosen equal to |φjo| in order to give mode shape components with alarger amplitude a larger weight.

2.4 Modes appearing in complex conjugate pairs

It is well known that, for every physical mode of a structure with continuous-time eigenvalue λcj , mode shape φj ,and continuous-time modal participation vector lcj , the structure has a second mode with parameters λcj , φj , andlcj , where � denotes complex conjugate [1, 14, 19, 29, 39]. The presence of such a complex conjugate mode canbe used as a hard single-mode validation criterion. Also the discrete-time eigenvalues λdj and modal participationvectors ldj appear in complex conjugate pairs.

2.5 Measuring the contribution of a mode to the total respons e

The term modal transfer norm (MTN) was introduced in [40] for denoting a scalar measure for the contribution of aparticular identified mode to the total response. Two variants of the MTN may be discriminated: they are denotedas MTN∞j and MTN2j , respectively. MTN∞j equals the peak gain of a transfer function containing mode j only.In an EMA context, this may be the mobility of mode j, so that, when the measured outputs are velocities, one has:

MTNd∞j = max

ωσ (Hmob,j(ω)) = maxσ

(

φjldjT

z − λdj

∣

∣

∣

∣

∣

z=eiωjT

)

, (10)

where σ(�) denotes the set of singular values, T the sampling period, and ωj the damped circular eigenfrequencyof mode j. The superscript �

d stands for deterministic. When some or all measured outputs are not velocities, thecorresponding components of the mode shape φj should be divided by (in case of displacements) or multiplied by(in case of accelerations) the continuous-time eigenvalues λcj . In an OMA context, the modal contribution to the

positive power spectral density (PSD+), in physical units of velocity, can be measured:

MTNs∞j = max

ωσ(

S+vv,j(ω)

)

= maxσ

(

φjgdjT

z − λdj

∣

∣

∣

∣

z=eiωjT

+φjg

Tdj

2λdj

)

, (11)

where gdj is the discrete stochastic participation vector of mode j. The superscript �s stands for stochastic. When

some or all measured outputs are not velocities, the corresponding components of the mode shape φj and thestochastic participation vector gdj should be divided by (in case of displacements) or multiplied by (in case ofaccelerations) the continuous-time eigenvalues λcj . A derivation of the modal decomposition of the PSD+ that leadsto (11), can be found in appendix A. In an OMAX context, both MTNd

∞j and MTNs∞j can be used; a combined

criterion has also been defined [40], but it might perform less well in some cases since Hmob,j(ω) and S+

vv,j(ω)have different physical units and may therefore be of different orders of magnitude.

As an alternative measure for the contribution of a mode j to the total response, MTN2j equals the root meansquare value of the autocorrelations of the response of mode j to white loading [15]. Since its computation can breakdown when the system identification method used does not guarantee that output correlation matrices synthesizedfrom the identified system model are positive definite, and since experimental results have indicated that it resultsin a less clear separation between physical and spurious modes [15], it is not further considered here.

A stabilization diagram in which only the modes with the highest values of a modal transfer norm are plotted,is usually very clear, as illustrated with simulated and experimental examples in [11, 15, 40, 43]. Since a modaltransfer norm is a positive definite quantity belonging to a particular mode, a relative difference in modal transfernorm between modes,

d(MTN∞j ,MTN∞l) =|MTN∞j −MTN∞l|

max(|MTN∞j |, |MTN∞l)|, (12)

can be used as a stabilization criterion, or to distinguish modes from each other.In [15], it is shown that, when for a certain mode, the poles and the zeros at all outputs coincide, the correspond-

ing modal transfer norm is zero. Since pole-zero cancelation is a typical symptom of spurious modes, this providesanother explanation of why modal transfer norms of spurious modes tend to have a low value.

Pole-zero cancelation detection methods are classic tools in control theory for model order selection [48], andthey have also been used in an EMA setting [54]. These tools interpret pole-zero cancellation in terms of overlappingconfidence bounds of pole and zero estimates, but the availability of such bounds depends on the particular MPEmethod that is used. Other quantities for measuring modal contributions have been proposed as well. The modeparticipation of Heylen et al. [19], that is based on the modal FRF residues, can be used in an EMA setting. Whena nonparametric transfer function is used as primary data for an MPE method, its singular values at an identifiedeigenfrequency could be used to indicate physical modes, as illustrated for transmissibility functions in [12].

All modal contribution criteria discussed above should be applied with care, as weakly excited or highly dampedphysical modes may exhibit low values.

2.6 Stability and damping ratio range

In normal operating conditions, structures are strictly stable, hence the damping ratios of the physical modes shouldbe positive. Stability is therefore a very useful hard single-mode validation criterion. On the other side, modes thatare very highly damped are rarely encountered in practice. In nearly all modal testing applications, damping ratioslarger than 20% are not physically realistic.

In some MPE methods, notably the poly-reference least squares complex exponential (pLSCE) method [55,56] and its frequency-domain counterpart, the poly-reference least squares complex frequency-domain (pLSCF)method [18], the mathematical poles, that arise when noiseless data are available but the model order is over-estimated, can be forced to have negative damping with a proper choice of constraints. This property, which wasdiscussed in the early 1980s in the single-input single-output (SISO) signal processing literature [23, 22], has beenapplied for multiple-input multiple-output (MIMO) analysis as well [6]; a firm proof for the MIMO case can be foundin [39, sec. B.7]. The main problem, however, of the pLSCE and pLSCF estimators is that they yield systematicerrors on the identified modal parameters, especially on the damping ratios, as discussed in more detail in [9, 39].

2.7 EMA and OMAX criteria

Some additional validation criteria can be used in an EMA or OMAX setting, i.e., when measured forces are avail-able. When in that case, a driving point measurement is made, absolute scaling of mode shapes is possible, e.g.,

to unit modal mass [19]. The least-squares estimate of the ratio of two mass-normalized modes is called the modalscale factor (MSF) [2]:

MSF(φj ,φl) =φl

∗φj

||φl||22= φl

†φj , ||φl||2 6= 0. (13)

When the MSF is real and close to ±1, both mass-normalized mode shapes have nearly the same amplitude andphase. A useful stabilization criterion is then

d(qj , ql) =

∣

∣

∣

∣

ln

(

MSF(φj ,φl)

sign (max (Re (MSF(φj ,φl)) , Im (MSF(φj ,φl))))

)∣

∣

∣

∣

, (14)

where qj and ql denote the modal scaling factors of modes j and l, respectively. Thanks to the absolute value ofthe logarithm in this expression, it does not depend on the mode order: d(qj , ql) = d(ql, qj) [44].

If the structure is proportionally damped and mass-normalized mode shapes are available, each mode j shouldbe purely real. For an EMA analysis, a good validation criterion is therefore

|MP(φj)|, (15)

where the mean phase can be computed as in (8).The mode shapes and modal participation vectors should obey reciprocity [19]. When more than one driving

point is available, the MAC(ldpj ,φdpj), where �dp selects the driving point degrees of freedom (DOFs), provides auseful validation criterion.

The mode over complexity value (MOV) proposed by Heylen et al. [19, sec. A.4.4] measures whether the eigen-frequency of a mode has the correct, negative sensitivity with respect to mass addition. It can only be computedwhen deterministic modal participation vectors are available. When the structure is proportionally damped, the MOVprovides similar information as the MPC and MPD (section 2.3).

2.8 Other criteria

Some other validation criteria have been proposed, most of which can be used in combination with a particularsystem identification algorithm only, such as the Consistent Mode Indicator (CMI) for the Eigensystem RealizationAlgorithm [30], or the Modal Confidence Factor (MCF) for use with the Ibrahim Time Domain or pLSCE methods[20, 25]. Although only criteria that have been discussed earlier will be used further in this text, the proposedautomation strategy that is to be discussed now is general enough to include additional or alternative useful single-mode validation criteria.

3 Automated interpretation of a stabilization diagram in 3 c lustering stages

In this section, the three stages of the proposed approach for the automated interpretation of stabilization diagramsare described in detail. The first stage automates the setting of the stabilization thresholds, performed by the userfor obtaining a clear diagram. The second stage corresponds to the visual inspection of the stabilization diagram bythe user, in order to detect vertical lines of stable modes. The third and final stage corresponds to the selection bythe user of a representative mode from columns of stable modes in the diagram.

3.1 Automated clearing of a stabilization diagram

When all identified modes would be plotted in a stabilization diagram, it would often look very busy and complex,and it would be very difficult to select a set of physical modes from the diagram, see, e.g., Figs. 8(a,c,e), 9(a,c,e)and 10(a,c,e). In a manual analysis, the user therefore chooses a set of threshold values for the stabilization criteriaand possibly other single-mode validation criteria discussed in section 2. Modes that do not pass the thresholds areclassified as certainly spurious and removed from the diagram. Setting these threshold values judiciously is a taskthat is hard to automate, especially when many validation criteria are considered, because different sets of datarequire different sets of threshold values in order to obtain a clear diagram in which all relevant modes are present.

For clearing out the stabilization diagram, the algorithm therefore uses a slightly different approach, which con-sists of the following steps:

1. compute as many relevant single-mode validation criteria as possible;

2. classify the modes as certainly spurious or possibly physical, using the soft validation criteria and a clusteringalgorithm, in an automated way;

3. apply the hard validation criteria to the set of possibly physical modes.

Suppose that nvs soft and nvh hard validation criteria have been computed for each mode in step 1. In step 2,each mode is represented by an nvs-dimensional vector containing its soft validation criteria, i.e., the modes arerepresented by points in Rnvs , in which the clustering will take place. Table 1 shows the soft validation criteria thatare used in this paper. In order to give each criterion equal weight, the variables VS5, VS9 and VS11 are subtractedby their minimum and divided by their range, so that they produce values in the interval [0, 1]. A partitioning methodis then employed for classifying the modes into two clusters: one of them contains the certainly spurious modes andthe other one contains the possibly physical modes. In this work, a k-means clustering algorithm with k = 2 clustersis used, but alternative partitioning methods may be employed as well.

criterion value ideal physical ideal spuriousVS1 d(λj , λl) 0 1VS2 d(fuj , ful) 0 1VS3 d(ξj , ξl) 0 1VS4 MAC(φj ,φl) 1 0VS5 MTNs

∞j large (1) 0VS6 d(MTNs

∞j ,MTNs∞l) 0 1

VS7 MPC(φj) 1 0VS8 MPD(φj)/90

◦ 0 1

VS9 MTNd∞j large (1) 0

VS10 d(MTNd∞j ,MTNd

∞l) 0 1VS11 d(qj , ql) 0 large (1)

Table 1: List of soft validation criteria that are used in this paper, and the values they take for an ideal physicaland an ideal spurious mode. The values between brackets are the ideal values after re-scaling, i.e., subtractingthe parameters by their minimum value and dividing them by their range.

The 2-means clustering algorithm minimizes the sum of the squared Euclidian distances between each modej, represented by a point pj ∈ Rnvs , and the nearest cluster centroid pck, i.e., the centroid of the cluster to whichthe mode belongs. In other words, the centroid of the cluster of possibly physical modes, denoted as pc1, and thecentroid of the cluster of certainly spurious modes, denoted as pc2, are computed as

{pc1,pc2} = args minpck

2∑

k=1

nm(k)∑

j=1

‖pj,c − pck‖22, (16)

where the number of modes in each cluster k is denoted as nm(k), the modes belonging to the cluster of possiblyphysical modes are members of the set {pj,1} = {pj| ‖pj − pc1‖ ≤ ‖pj − pc2‖}, and the modes belonging to thecluster of certainly spurious modes are members of the set {pj,2} = {pj|pj /∈ {pj,1}}. The objective function in(16) is locally minimized in an iterative optimization process, where the starting points for pc1 and pc2 are chosenaccording to their ideal values listed in table 1. Each iteration consists of two steps:

1. The Euclidian distance between each point pj and each cluster center pck is computed, and point pj isassigned to the set {pj,1} or {pj,2}.

2. The centroids are computed so as to minimize the total squared Euclidian distance within each cluster, i.e.,they are computed as in (16), but for fixed sets {pj,1} and {pj,2}. As such, each centroid is computed as themean of the points in the corresponding fixed set.

Initially, step 1 is performed for all points at once; this enhances the speed of the minimization process, but it mayprevent its full convergence. Therefore, near the end of the minimization process, step 1 is performed only for onepoint at a time. The modes that belong to {pj,2} after convergence of the partitioning method are classified ascertainly spurious, and they are removed from the stabilization diagram.

In the third and final step of the automated clearing stage, the stabilization diagram is cleared out further byapplying the nvh hard validation criteria. Only the modes that meet all hard criteria are retained in the stabilizationdiagram as possibly physical modes. It is important to perform steps 2 and 3 in the correct order, since the appli-cation of the hard validation criteria (step 3) may remove all of the spurious modes, and as a result physical modesmay be lost in the partitioning (step 2). Table 2 lists the hard validation criteria that are used in this paper.

criterion test possibly physical certainly spuriousVH1 ξj > 0 1 0VH2 ξj < 20% 1 0VH3 complex conjugate mode present? 1 0

Table 2: List of hard validation criteria that are used in this paper, and the boolean values they take for apossibly physical and a certainly spurious mode.

As a first example, figs. 8 to 10 show the full stabilization diagrams, obtained during 9 different operational modalanalysis setups on a post-tensioned concrete bridge, as well as the ones obtained after automated clearing. In theclearing step, many to nearly all of the spurious modes are removed from the diagram. A full description of thedata and results will be given in section 4; here we illustrate, for the first measurement setup, the clearing step inmore detail. The soft validation criteria VS1 to VS8 of table 1, and the hard criteria VH1 to VH3 of table 2 havebeen used. Fig. 2 shows the possibly physical and certainly spurious modes in two-dimensional projections of theeight-dimensional space of soft validation criteria in which the classification took place. The initial centroids, chosenaccording to table 1, and the final centroids, obtained after convergence of the partitioning method, are shown aswell. The plots illustrate that, for each validation criterion, the range of values is continuous rather than concentratedaround two values; while this would make manual classification a difficult task, the automated approach results in aclassification that is optimal according to (16).

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Figure 2: Z24 bridge, first setup, possibly physical modes (black crosses) and certainly spurious modes (greydots), obtained after applying the stabilization diagram clearing strategy: (a) frequency distance vs. dampingratio distance; (b) mean phase deviation vs. damping ratio distance. �: initial centroids; ⋄: final centroids.

As a second example, Fig. 12 shows the full stabilization diagrams, obtained during an OMAX analysis on a steelfootbridge, before and after automated clearing using all validation criteria from Tables 1 and 2. A full descriptionof the data and results will be given in section 5; here we illustrate the clearing step in more detail. Fig. 3 showsthe possibly physical and certainly spurious modes in two-dimensional projections of the 11-dimensional space ofsoft validation criteria in which the classification took place, as well as the initial and final centroids. The plots lookdifferent as in the previous example, as it concerns data from a different type of test (OMAX vs. OMA) on a differenttype of structure (steel footbridge vs. concrete bridge). However, in both cases, the clustering is performed with thesame algorithm, without any user interaction.

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Figure 3: Wetteren footbridge, second setup, possibly physical modes (black crosses) and certainly spuriousmodes (grey dots), obtained after applying the stabilization diagram clearing strategy: (a) frequency distancevs. damping ratio distance; (b) mean phase deviation vs. damping ratio distance. �: initial centroids; ⋄: finalcentroids.

3.2 Grouping similar modes in a cleared stabilization diagr am

After a stabilization diagram has been cleared out as described in the previous section, similar modes in the diagramare grouped together with a hierarchical clustering approach. Hierarchical clustering of stabilization diagrams is nota novel idea (cfr. section 1.2), but the main difference with previous work is that the current approach does notcontain any parameter that needs to be specified by the user. The different steps of this stage are:

1. All modes from the cleared stabilization diagram are put in separate clusters, and the mutual distance betweenany two clusters k and l is computed according to (5):

d(k, l) = d(λck, λcl) + 1−MAC(φk,φl).

2. The two clusters that are closest together are collected in a single cluster, and the mutual distance betweenall clusters is recomputed as the average distance between their elements.

3. Step 2 is continuously repeated until the distance between the two closest clusters k and l exceeds the allowedmaximal value d, that depends on the results obtained in the previous clustering stage:

d(k, l) ≥ d = µp1+ 2σp1

, (17)

where µp1and σp1

are the sample mean and sample standard deviation, respectively, of the distance betweena possibly physical mode in the analyzed data set, and the closest mode at a lower model order, accordingto the distance measure (5). In other words, µp1

and σp1are the sample mean and standard deviation of

pj,1(1) + 1 − pj,1(4) over the set {pj,1}, with pj,1(1) and pj,1(4) the first and fourth elements of the possiblyphysical mode j, represented by the vector of soft validation criteria pj,1, according to table 1.

As a result, the hierarchical clustering stage yields a set of similar mode sets from the cleared stabilization diagram.Additional constraints could be introduced, for example, an upper bound on d could be imposed so that the largestmode set cannot contain more modes than the number of different model orders in the stabilization diagram. How-ever, no further fine-tuning has proved necessary in this paper.

The hierarchical clustering stage is first illustrated for the cleared stabilization diagram shown in Fig. 8b. Itcontains 769 possibly physical modes. Fig. 4 visualizes the clustering result in a hierarchical tree, consisting of Π-shaped lines that connect the clusters in a hierarchical way. The height of each Π represents the distance betweenthe two clusters being connected. The figure also shows the ‘cut-off distance’, computed as in (17); at this distance,the tree is ‘cut’, resulting in 62 sets of similar modes.

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Figure 4: Z24 bridge, first setup: dendrogram of the hierarchical clustering step (full lines), and automaticallydetermined cut-off distance (bold dashed line).

As a second example of the hierarchical clustering stage, Fig. 5 shows the hierarchical tree for the 3624 possiblyphysical modes in the cleared stabilization diagram of Fig. 12b. The automatically computed cut-off distance issubstantially smaller than in the previous example, which illustrates that also the second stage of the fully automatedstrategy is able to handle very different data sets. Here, the hierarchical clustering stage results in 250 sets of similarmodes.

3.3 Selecting a final set of physical modes

The sets of modes obtained in stage 2 are split into two clusters: one containing the sets of physical modes, and onecontaining the sets of spurious modes. Since in contrast to spurious modes, physical modes are ideally identified ateach model order from a certain model order on, it can be expected that the sets of physical modes contain manyelements, while the sets of spurious modes do not. In order to avoid that a threshold number of elements needsto be specified by the user, again a partitioning method, e.g., k-means clustering with k = 2, is employed for theclassification. Since stage 2 may yield physical mode sets only, an additional number of empty sets is added, equalto the number of mode sets containing more than one fifth of the number of modes in the largest set. As a result,there are nh mode sets in total, some of which are empty.

If the number of modes in set j is denoted as nhj, j = 1, . . . , nh, the 2-means algorithm computes the centroidof the cluster of physical mode sets nhc1, and the cluster of spurious mode sets nhc2, as

{nhc1, nhc2} = args minnhck

2∑

k=1

nS(k)∑

j=1

(nhj,c − nhck)2, (18)

where the number of mode sets in each cluster k is denoted as nS(k), the modes numbers {nhj,1} correspond tophysical modes, i.e., {nhj,1} = {nhj| (nhj − nhc1)

2 ≤ (nhj − nhc2)2}, and the modes numbers {nhj,2} correspond to

spurious modes, i.e., {nhj,2} = {nhj |nhj /∈ {nhc1}}. The objective function in (18) is minimized in an iterative localminimization process, as explained in section 3.1, and the starting points are chosen to be nhc1 = maxj(nhj) andnhc2 = 0.

Finally, a representative element is chosen from each set of similar physical modes that results from the parti-tioning. In this work, the mode with the median damping value is chosen. When real normal modes are expected,the mode with the highest MPC or MPD value can be alternatively chosen as the representative element.

As an illustration, Figs. 6a and 6b show the number of modes in each set resulting from the hierarchic clusteringin Figs. 4 and 5, respectively. For the Z24 bridge example, 8 mode sets are classified as physical and 54 as spurious

250 500 750 1000 1250 1500 1750 2000 2250 2500 2750 3000 3250 35000

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Figure 5: Wetteren footbridge, second setup: dendrogram of the hierarchical clustering step (full lines), andautomatically determined cut-off distance (bold dashed line).

with the automated partitioning approach. The physical mode sets contain between 42 and 80 modes, the spuriousbetween 1 and 28 modes. For the Wetteren footbridge example, 35 mode sets are classified as physical and 215 asspurious. The physical mode sets contain between 38 and 93 modes, the spurious between 1 and 35 modes.

Fig. 7 shows all modes in the physical mode sets for the first setup of the Z24 bridge, as well as their repre-sentative elements, in a damping ratio vs. eigenfrequency diagram. It can be observed that the relative variation indamping ratio is large and that outliers exist. However, choosing the mode with the median damping ratio value ineach mode set as its representative, ensures that this representative is not sensitive w.r.t. the damping ratio variationwithin each mode set.

4 OMA validation example: the Z24 bridge

4.1 The structure

The Z24 bridge was part of the road connection between the villages of Koppigen and Utzenstorf, Switzerland, over-passing the A1 highway between Bern and Zurich. It was a classical post-tensioned concrete two-cell box-girderbridge with a main span of 30 m and two side spans of 14 m. The bridge, that dated from 1963, was demolishedat the end of 1998, because a new railway adjacent to the highway required a new bridge with a larger side span.Before complete demolition, the bridge was subjected to a short-term progressive damage test, and after eachapplied damage scenario, a full forced and an ambient operational vibration test were performed.

The data from one of these scenarios (no. 8) were presented as benchmark data for assessing the performanceof system identification methods for (operational) modal analysis. In this test, 291 degrees of freedom have beenmeasured in total: three acceleration components on the pillars, and mainly vertical and lateral accelerations onthe bridge deck. The data were collected in 9 different setups using 5 channels that were common to all setups. Ineach setup, 65536 data samples were collected at a sampling rate of 100 Hz, using an analog anti-aliasing filter withcut-off frequency of 30 Hz.

More information on the structure, the experimental setup, the short-term progressive damage tests and a long-term vibration monitoring test that was performed as well, and reported benchmark results, can be found in [41].

4.2 OMA identification results

The data from each of the 9 setups have been processed with the reference-based covariance-driven stochasticsubspace identification (SSI-cov/ref) algorithm [33]. The 5 channels that were common to each setup were chosen

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Figure 6: Automatic classification of the sets of modes, obtained from hierarchical clustering, into physical andspurious sets, based on the number of modes in each set. (a) Z24 bridge, first setup; (b) Wetteren footbridge,second setup. Black bars: physical sets; white bars: spurious sets.

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mode µ(fuj,a) σ(fuj,a) µ(fuj,m) σ(fuj,m) µ(ξj,a) σ(ξj,a) µ(ξj,m) σ(ξj,m) MPCa MPCm MAC[Hz] [Hz] [Hz] [Hz] [%] [%] [%] [%] [−] [−] [−]

1 3.86 0.01 3.86 0.01 0.8 0.1 0.8 0.1 1.00 1.00 1.002 4.90 0.01 4.89 0.01 1.4 0.2 1.4 0.1 0.99 1.00 0.993 9.76 0.02 9.76 0.02 1.4 0.2 1.4 0.2 0.97 0.97 1.004 10.30 0.09 10.30 0.08 1.3 0.3 1.3 0.2 0.93 0.94 1.005 12.41 0.19 12.39 0.18 2.8 0.4 2.9 0.4 0.97 0.95 0.986 13.22 0.15 13.35 0.15 3.4 1.1 3.6 1.7 0.90 0.96 0.94

Table 3: Z24 bridge: mean values (µ) and standard deviations (σ) of the undamped eigenfrequencies (fuj)and damping ratios (ξj), computed from the values obtained for each setup with a manual ( m) or automated( a) analysis of the stabilization diagrams. The MPC values of the merged mode shapes are also tabulated, aswell as the MAC values between the mode shapes obtained in the manual and the automated analysis.

as reference channels, ı = 50 was chosen as half the number of block rows in the data Hankel matrix, and a modelorder range from 2 to 160 in steps of 2 was chosen for the construction of the stabilization diagrams. These diagramswere then interpreted with the fully automatic three-stage clustering approach proposed in section 3, for which nouser-defined parameters are needed.

Figures 8, 9 and 10 show the full stabilization diagrams before and after clearing according to step 1 of theautomated approach. The automatically selected eigenfrequencies are plotted as vertical lines on top of the clearedstabilization diagrams. They coincide with columns of stabilized modes, as in a manual analysis.

In the automated analysis, 6 modes are found in all 9 setups. They all have eigenfrequencies below 14 Hz.It appears that the higher modes are not always well excited by the ambient forces. This is in agreement withbenchmark results reported in the literature, where, based on a manual stabilization diagram analysis, only the first5 modes are used for damage assessment [27, 49]. As a total of 5 response degrees of freedom are common toall setups, it is possible to apply a scaling factor to the partial mode shapes obtained in each setup so as to fit thecommon DOFs to the same values in a least squares sense. The merged global mode shapes obtained in this wayare shown in Fig. 11. The corresponding eigenfrequency and damping ratio values, obtained through averagingover all setups, are also listed.

In order to assess the performance of the automated analysis, a classic manual analysis, where the stabilizationcharts are cleared out by means of user-defined thresholding, and stable modes are picked from the diagram, wasperformed as well. It should be noted that, as the manual interpretation of a stabilization diagram depends onthe experience and engineering judgement of the analyst, the result of such analysis is user-dependent; only onesuch result is reported here, but it agrees very well with results reported by other expert users that have analyzedthe same benchmark data, cfr. [41] and the references therein. In the manual analysis, only 6 modes were foundin all 9 setups, so the automated analysis was able to retrieve all relevant modes. Table 3 offers a quantitativecomparison between the results obtained from the automated and manual analyses. The quality of the identifiedeigenfrequency and damping ratio values, measured in terms of the sample standard deviation over all 9 setups, isvery similar, except for mode 6, where the damping ratio estimate is more accurate in the automated analysis; this isprobably due to the fact that, in the last step of each automated analysis, the mode with the median damping valueis chosen as the representative from each set of similar physical modes. The MPC values for the merged modeshapes are also plotted; here the manual analysis yields a slightly better value for mode 6. Finally, the merged modeshapes are compared by means of the MAC; it can be concluded that mode shapes 1 tot 5 are identical while modeshape 6 is almost identical.

5 OMAX validation example: the Wetteren footbridge

5.1 The structure

Because the construction of a roundabout at the N42 national road on top of the E40 highway at Wetteren, Belgium,created a potentially dangerous situation for cyclists and pedestrians due to numerous connections between theroundabout and the highway, a new footbridge was built in 2003 to the west of the roundabout in order to separatethe bicycle track and the footpath completely from the road traffic. This steel bridge has two spans, a short spanwhich measures 30.33 m and a large span of 75.23 m. At the large span, the bridge is of the bow-string type, with avertical inclination of 13.8◦ of the bows.

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Figure 8: Z24 bridge: (a-c-e) full stabilization diagrams and (b-d-f) automatically cleared stabilization diagramsfor (a-b) setup 1, (c-d) setup 2 and (e-f) setup 3. The automatically selected eigenfrequencies are plotted asvertical lines on top of the cleared diagrams.

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Figure 9: Z24 bridge: (a-c-e) full stabilization diagrams and (b-d-f) automatically cleared stabilization diagramsfor (a-b) setup 4, (c-d) setup 5 and (e-f) setup 6. The automatically selected eigenfrequencies are plotted asvertical lines on top of the cleared diagrams.

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Figure 10: Z24 bridge: (a-c-e) full stabilization diagrams and (b-d-f) automatically cleared stabilization dia-grams for (a-b) setup 7, (c-d) setup 8 and (e-f) setup 9. The automatically selected eigenfrequencies areplotted as vertical lines on top of the cleared diagrams.

mode 1 - 3.86Hz - 0.8% mode 2 - 4.90Hz - 1.4% mode 3 - 9.76Hz - 1.4%

mode 4 - 10.30Hz - 1.3% mode 5 - 12.42Hz - 2.8% mode 6 - 13.22Hz - 3.4%

Figure 11: Z24 bridge: eigenfrequencies, damping ratios and mode shapes, obtained through an automatedstabilization diagram analysis.

In October 2007, an OMA and an OMAX test were performed on the footbridge in order to identify its modes inthe frequency range 0 − 20 Hz [42]. In the OMAX test, a pneumatic artificial muscle (PAM) was used as actuator[10]. As force signal, a swept logarithmic sine between 0.1 and 10 Hz was chosen, which makes that the ambientresponse dominates above 10 Hz, while below 10 Hz, the forced and ambient response have a similar amplitude[42]. Two additional OMAX tests were performed in January 2008, with a drop weight and an impact hammeras actuator, respectively. The OMAX-PAM test data will be used here for validating the three-stage clusteringapproach to automated modal analysis. In this test, 72 degrees of freedom have been measured in total: verticalaccelerations of the bridge deck at 44 locations, horizontal accelerations of the bridge deck at 23 locations, and outof plane accelerations of the bows at 5 locations. The data were collected in 5 different setups, using an NI PXI-1050front-end with built-in anti-aliasing filter. 7 channels were common to all setups and the measurement duration was600 s for each setup. The measured signals were digitally low pass filtered with an eighth-order Chebychev type Ifilter with a cutoff frequency of 20 Hz in both the forward and the reverse direction to remove all phase distortion,and then resampled at 50 Hz. Subsequently, the signals were high-pass filtered with a fourth-order Butterworth filterwith a cutoff frequency of 0.2 Hz, again in both the forward and the reverse direction.

More information on the structure and the different types of test performed, and a detailed comparison of manualidentification results obtained in the OMA and OMAX-PAM test, as well as additional OMAX tests performed later,can be found in [42].

5.2 OMAX identification results

The data from each of the 5 setups have been processed with the reference-based combined deterministic-stochasticsubspace identification (CSI/ref) algorithm [40]. The 7 channels that were common to all setups were chosen asreference channels, ı = 40 was chosen as half the number of block rows in the data Hankel matrix, and a modelorder range from 2 to 200 in steps of 2 was chosen for the construction of the stabilization diagrams. These di-agrams were then interpreted with the fully automatic three-stage approach developed in section 3, for which nouser-defined parameters are needed. Fig. 12 shows, for one of the setups, the full stabilization diagram before andafter clearing according to step 1 of the automated approach. The soft validation criteria VS1 to VS11 of table 1, andthe hard criteria VH1 to VH3 of table 2 have been used. The automatically selected eigenfrequencies are plottedas vertical lines on top of the cleared stabilization diagrams. They coincide with columns of stabilized modes, as ina manual analysis. Many of the identified modes are closely spaced.

With the automated approach, 27 modes are found in all 5 setups. Figs. 13 and 14 show the global mode shapes,obtained after merging the partial mode shapes obtained in all setups in a least-squares sense. The correspondingeigenfrequency and damping ratio values, obtained through averaging over all setups, are also plotted. For most ofthe mode shapes, there is a strong interaction between the bows and the bridge deck, which makes these modeshapes truly three-dimensional. Some modes, like mode 9, consist mainly of deformation of the bows while the

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bridge deck remains relatively undeformed.

mode 2 - 1.669 Hz - 0.26 % mode 3 - 1.761 Hz - 0.77 % mode 4 - 2.188 Hz - 0.48 %

mode 5 - 3.731 Hz - 0.55 % mode 6 - 3.836 Hz - 0.52 % mode 7 - 3.947 Hz - 0.70 %

mode 9 - 5.157 Hz - 0.46 % mode 10 - 5.591 Hz - 0.54 % mode 11 - 5.626 Hz - 0.64 %

mode 12 - 6.117 Hz - 0.28 % mode 13 - 6.318 Hz - 0.50 % mode 14 - 6.605 Hz - 0.58 %

mode 15 - 7.242 Hz - 2.40 % mode 16 - 7.490 Hz - 0.67 % mode 17 - 7.574 Hz - 1.30 %

Figure 13: Wetteren footbridge: eigenfrequencies, damping ratios and mode shapes (top: top view, middle:side view from the southeast side, bottom: three-dimensional view from the south side), obtained throughautomated stabilization diagram analysis, for modes 1− 17.

mode 18 - 8.297 Hz - 1.21 % mode 19 - 8.565 Hz - 0.69 % mode 20 - 9.565 Hz - 0.74 %

mode 21 - 9.969 Hz - 1.12 % mode 22 - 10.476 Hz - 0.66 % mode 24 - 11.811 Hz - 1.63 %

mode 25 - 12.726 Hz - 0.36 % mode 29 - 14.808 Hz - 0.42 % mode 30 - 15.215 Hz - 0.50 %

mode 31 - 16.500 Hz - 0.57 % mode 32 - 17.509 Hz - 0.32 % mode 33 - 17.832 Hz - 0.30 %

mode 34 - 22.796 Hz - 0.38 %

Figure 14: Wetteren footbridge: eigenfrequencies, damping ratios and mode shapes (top: top view, middle:side view from the southeast side, bottom: three-dimensional view from the south side), obtained throughautomated stabilization diagram analysis, for modes 18− 34.

In order to assess the performance of the automated analysis, its results are compared with independently ob-tained, previously reported results from a manual analysis, in which exactly the same system identification algorithmwith exactly the same parameters has been used [42]. Table 4 offers a quantitative comparison between the resultsobtained from the automated and manual analyses. In the manual analysis, 30 modes were found, 6 of which werenot found in the automated analysis because they are not well stabilized, although all but one of them were foundin some to nearly all setups. Retrieving these modes automatically in all setups would require, as in the manualanalysis, a modal tracking approach, and this falls outside the scope of this paper. On the other hand, the auto-mated analysis yielded 3 modes that were not reported in the manual analysis. The accuracy of the eigenfrequency,damping ratio, and mode shape estimates is similar for both cases. The MAC value between the mode shapes thatwere found in both the automated and the manual analyses, are also listed: the very high values confirm that themode shapes are nearly identical. The sometimes high values for the mean phase indicate that the correspondingmodes are not well excited by the actuator; nevertheless, they are identified since in an OMAX setting, also theoperational loading is taken into account during the identification process.

nr. Manual analysis Automated analysisµ(fuj) σ(fuj) µ(ξj) σ(ξj) MPC MP µ(fuj) σ(fuj) µ(ξj) σ(ξj) MPC MP MAC[Hz] [Hz] [%] [%] [−] [◦] [Hz] [Hz] [%] [%] [−] [◦] [−]

1 0.693 0.005 1.05 0.80 0.98 252 1.669 0.001 0.23 0.11 0.99 13 1.669 0.001 0.26 0.10 0.99 43 1.003 1.758 0.003 0.76 0.11 0.98 1 1.761 0.003 0.77 0.14 0.98 13 1.004 2.195 0.002 0.50 0.09 0.99 1 2.188 0.009 0.48 0.08 0.99 0 1.005 3.731 0.010 0.55 0.15 0.98 9 3.731 0.010 0.55 0.15 0.98 15 1.006 3.838 0.007 0.49 0.17 0.98 8 3.836 0.006 0.52 0.12 0.99 8 1.007 3.947 0.008 0.70 0.13 0.96 28 4.480 0.041 0.76 0.50 0.96 789 5.154 0.010 0.44 0.16 0.97 8 5.157 0.009 0.46 0.12 0.96 22 1.0010 5.591 0.006 0.54 0.09 0.96 8411 5.626 0.004 0.64 0.07 0.90 612 6.117 0.003 0.27 0.05 0.99 4 6.117 0.003 0.28 0.05 0.98 2 1.0013 6.321 0.007 0.50 0.09 0.99 79 6.318 0.006 0.50 0.09 0.99 78 1.0014 6.605 0.003 0.58 0.09 0.99 1 6.605 0.003 0.58 0.09 0.99 3 1.0015 7.238 0.006 2.34 0.16 0.79 3 7.242 0.008 2.40 0.14 0.78 4 1.0016 7.488 0.011 0.70 0.04 0.88 4 7.490 0.015 0.67 0.05 0.86 3 0.9917 7.577 0.015 1.29 0.37 0.93 74 7.574 0.015 1.30 0.36 0.92 75 1.0018 8.307 0.013 1.18 0.06 0.97 2 8.297 0.018 1.21 0.07 0.95 1 1.0019 8.565 0.006 0.68 0.07 0.95 0 8.565 0.007 0.69 0.07 0.94 1 1.0020 9.565 0.002 0.74 0.03 0.96 1 9.565 0.003 0.74 0.03 0.97 1 1.0021 9.967 0.013 1.10 0.11 0.94 73 9.969 0.016 1.12 0.12 0.94 72 1.0022 10.475 0.004 0.64 0.04 0.97 5 10.476 0.006 0.66 0.05 0.96 8 1.0023 11.214 0.049 0.78 0.09 0.92 7424 11.821 0.033 1.68 0.11 0.97 55 11.811 0.032 1.63 0.07 0.97 82 1.0025 12.728 0.013 0.35 0.09 0.85 3 12.726 0.011 0.36 0.08 0.85 5 0.9926 12.863 0.024 0.72 0.10 0.65 6527 13.530 0.014 0.72 0.13 0.76 6528 13.606 0.009 0.39 0.09 0.92 529 14.810 0.005 0.41 0.09 0.95 85 14.808 0.007 0.42 0.09 0.95 74 1.0030 15.213 0.020 0.46 0.10 0.87 61 15.215 0.020 0.50 0.10 0.91 79 0.9831 16.502 0.039 0.53 0.05 0.96 5 16.500 0.041 0.57 0.03 0.96 3 0.9932 17.508 0.012 0.31 0.26 0.90 45 17.509 0.013 0.32 0.26 0.88 46 0.9933 17.833 0.009 0.28 0.12 0.88 0 17.832 0.013 0.30 0.12 0.88 13 1.00

Table 4: Wetteren footbridge: mean values (µ) and standard deviations (σ) of the undamped eigenfrequencies(fuj) and damping ratios (ξj), computed from the values obtained for each setup with a manual and automatedanalysis of the stabilization diagrams. The MPC and MP values of the merged mode shapes are also tabulated,as well as the MAC values between the mode shapes obtained in the manual and the automated analysis.

Finally, it can be noted that an additional torsional mode at 22.796 Hz is shown in Fig. 14. The eigenfrequency ofthis mode lies outside the frequency of interest and above the low-pass filtering frequency, and therefore it was notconsidered in the manual analysis. However, the fact that it was automatically identified indicates that the CSI/refalgorithm is able to retrieve modes that are only very weakly present in the data.

6 Summary and conclusions

In this paper, a fully automated, three-stage clustering approach is developed for interpreting stabilization diagrams,that obeys the following five target criteria:

1. not rely on more than one data record or on prior estimates for any of the modal parameters;

2. be as physically intuitive as possible and follow the course of a manual analysis;

3. produce similar results as in a manual analysis;

4. work in an EMA, OMA and OMAX framework and with any parametric system identification algorithm;

5. not contain parameters that need to be specified or tuned by the user.

A total of fourteen real-life benchmark operational bridge vibration data sets were analyzed in two validation studies,and the resulting modal parameters were compared with those obtained from a manual analysis by an expert user.The automated approach was able to find all stabilized modes and the resulting modal parameters have an accuracythat is similar to the manual analysis results.

Acknowledgements

This research was partially supported by the Fund for Special Research of K.U.Leuven and the Research Founda-tion - Flanders, Belgium (Postdoctoral Research Fellowships provided to ER), and by the Ministerio de Fomento,Spain (research project P8/08 HLX-C0263). The financial support from these institutes is gratefully acknowledged.The Z24 bridge tests were performed in a joint measurement campaign by members from EMPA, Switzerland andK.U.Leuven and LMS International, Belgium. The Wetteren footbridge tests were performed in a joint measurementcampaign by members from K.U.Leuven, KaHo Sint-Lieven, and V.U.B., Belgium.

A Modal decomposition of positive power spectral densities .

In [39] it was shown that

S+

vv,j(ω) =φjgdj

T

z − λdj

∣

∣

∣

∣

z=eiωjT

+Λ0,j

2, (19)

where Λ0 represents the zero-lag output correlation matrix of a structure under discrete white loading. The velocityresponse of such structure at time instance k, yk, can be written in modal state-space form as

yk =

n∑

j=1

yj,k (20)

xj,k+1 = λd,jxj,k + wj,k (21)

yj,k = φjxj,k, (22)

where (wj,k) is a discrete white noise sequence. Using the fact that xj,k and wj,k are statistically independent [39],we then have that

Λj,0 = E(

yj,kyj,kT)

= φjΣsjφj

T (23)

gj = E(

xj,k+1yj,kT)

= λd,jΣsjφj

T , (24)

where E denotes the expectation operator and Σsj = E (xj,kxj,k). Solving the last equation for Σs

jφjT and substituting

the result into the first one yields

Λ0j =φjgj

λd,j

. (25)

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