COMPARATIVE STUDY OF MODAL ANALYSIS TECHNIQUES FOR BRIDGE DYNAMIC CHARACTERISTICS

Mechanical Systems and Signal Processing (2003) 17(5), 965–988 doi:10.1006/mssp.2002.1568

COMPARATIVE STUDY OF MODAL ANALYSIS TECHNIQUES FOR BRIDGE DYNAMIC CHARACTERISTICS B. Peeters LMS International, Interleuvenlaan 68, B-3001 Leuven, Belgium. E-mail: [email protected]

and C. E. Ventura Department of Civil Engineering, The University of British Columbia, 2324 Main Mall, Vancouver, BC, Canada. V6T-1Z4. E-mail: [email protected] (Received 22 November 2001, accepted 1 October 2002, accepted after revisions 10 October 2002) The purpose of this paper is to present the results of a comparative study of various techniques for evaluating bridge dynamic properties from experimental data. The paper presents a review and synthesis of the work presented in a developed session of the International Modal Analysis Conference of February 2001. Research teams all over the world were invited to participate on a study to compare modal analysis techniques for evaluating the dynamic characteristics of bridges from forced, free and ambient vibration data. The Z24-Bridge, a three-span reinforced concrete bridge in Switzerland, was selected as a case study. The two objectives of the exercise were to compare the modal analysis techniques that are usually employed in the laboratories of the participants and to compare results from typical excitation techniques for large civil engineering structures. A total of six research teams accepted the challenge. The system identification methods that they used ranged from the basic peak-picking method to the advanced subspace identification method. All teams compared at least two excitation types. # 2003 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

Since 1999, the civil engineering community present at the International Modal Analysis Conference (IMAC) is organising special sessions with the objective to compare methods, algorithms and implementations related to civil engineering structural dynamics. About one year before the conference, data sets are made available to the research community. People are invited to use these data and submit papers for the upcoming IMAC (see also the conference website: www.sem.org). In 1999, the topic was damage-detection methods used in vibration-based structural health monitoring. The 2000 theme, operational modal analysis and finite element (FE)-model updating of a building, resulted in a lot of papers. However, we should add that almost all people restricted themselves to the modal analysis part of the study. In view of this experience, the 2001 theme was classical and operational modal analysis applied to a highway bridge. Again, many papers were published in the conference proceedings. This paper intends to be a review and synthesis of these papers. The 2001 material consisted of three data sets, which differ in the type of excitation source applied to the highway bridge: * * *

Ambient sources: traffic, wind, micro-earthquakes. Drop weight: a mass of approx. 100 kg is dropped from a height of 1 m. Shakers: two shakers are exciting the bridge with a flat spectrum from 3 to 30 Hz.

0888–3270/03/+$30.00/0

# 2003 Elsevier Science Ltd. All rights reserved.

966

B. PEETERS AND C. E. VENTURA

Each author was free to choose his preferred method(s) and excitation type(s). This procedure ensures a fair comparison in the sense that all methods are assumed to be applied by experienced users and each of the authors is doing his best to get the most out of his preferred method and/or excitation type. The two objectives of the exercise were to compare the modal analysis techniques that are usually employed in the laboratories of the participants and to compare results from typical excitation techniques for large civil engineering structures. The next section introduces the Z24-Bridge, which happened to be the benchmark object of this study. In Section 3, the modal parameter estimation methods that have been selected by the different research teams are put in perspective. This allows to pinpoint differences and similarities between methods and reveals that identical methods are carrying different names in the literature. Section 4 is the main part of the paper. In that section, the modal parameter estimation results from all teams are reviewed, compared and confronted with earlier results of the same bridge. The paper ends with some conclusions.

2. SIMCES AND THE Z24-BRIDGE

The data used in this comparative study originates from the European Brite-EuRam project SIMCES. The acronym SIMCES stands for System Identification to Monitor Civil Engineering Structures. The project was running from January 1997 to April 1999. Seven partners from six European countries were involved. The Structural Mechanics division of the K. U. Leuven coordinated the project. The work programme consisted of four tasks: (1) data collection, (2) adaptation, application and assessment of stochastic system identification methods, (3) FE modelling of reinforced concrete structures and (4) model-based damage-identification methods. The main efforts were concentrated on one test object: the Z24-Bridge. The Swiss Federal Laboratories for Material Testing and Research EMPA were responsible for all bridge tests. More information can be found on the project’s website (www.bwk.kuleuven.ac.be/bwm/SIMCES.htm) and in De Roeck et al. [1]. The Z24-Bridge was an overpass of the national highway A1 between Bern and Zu. rich, Switzerland. It was a classical post-tensioned concrete box girder bridge with a main span of 30 m and two side-spans of 14 m (Fig. 1). Both abutments consisted of three concrete columns connected with concrete hinges to the girder. Both intermediate supports were concrete piers clamped into the girder. Although there were no known structural problems, the bridge dating from 1963 was demolished at the end of 1998. A new railway adjacent to the highway required a new bridge with one larger side-span. Before complete demolition, the bridge was subjected to progressive damage to study the influence of different realistic damage scenarios on its dynamic properties. The data used in this paper originate from scenario 8. This was a new reference measurement after the induced settlement of one of the piers was removed. It is however possible that some damage remained; at least one of the piers was cut through for the installation of the settlement system. A complete description of the progressive damage tests can be found in Kr.amer et al. [2]. During the night following on the application of a certain damage scenario, an ambient and a shaker test were performed by EMPA. After scenario 8, also a drop weight was used to excite the bridge dynamically. The whole bridge was measured in nine set-ups of 33 accelerometers. In this comparative study, the number of channels was reduced, but it was still high enough to obtain meaningful mode shapes. Three reference sensors were common to every set-up,

COMPARATIVE MODAL ANALYSIS STUDY

967

Figure 1. The Z24-Bridge: longitudinal section and top view [1]. The bridge is slightly skew: the supports are not perpendicular to the longitudinal axis.

enabling to glue the different mode shape parts together in case no force measurements were available. The data were sampled at 100 Hz; the cut-off frequency of the anti-aliasing filter was 30 Hz. The ambient excitation sources acting on the bridge were wind, traffic on the highway and walking of the test crew. The number of data points per channel was 65 536, resulting in a measurement time of 10 min 55 s for each set-up. Two shakers were used for the shaker tests: one was placed on a side-span, and the other was placed at mid-span. The input signal was a band-limited noise (3–30 Hz). The acquisition parameters were the same as in the ambient tests. Finally also a drop weight was used to excite the bridge. The data were also sampled at 100 Hz, but only 8192 data points per channel were measured, resulting in a measurement time of 81.92 s for each set-up. In this time span, four impacts were given. Unfortunately, the input signal was not recorded.

3. OVERVIEW OF MODAL PARAMETER ESTIMATION TECHNIQUES

In this section, an overview of the modal parameter estimation methods that have been used by the different research teams in this comparative study is presented. This overview does not cover all existing methods as only a limited number of methods was actually used by the participants. Many existing textbooks provide an extensive overview of input– output modal parameter estimation methods. See for instance, Heylen et al. [3] and Allemang [4]. An overview and comparison of output-only modal parameter estimation methods can be found in Peeters [5] and Peeters and De Roeck [6]. Typical for the outputonly case is that the lack of knowledge of the input is justified by the assumption that the

968

B. PEETERS AND C. E. VENTURA

input does not contain any information; or in other words, the input is white noise. The theoretical assumption of white noise turns out to be not too strict in practical applications. As long as the (unknown) input spectrum is quite flat, output-only methods will work fine. However, if the input contains harmonics, they cannot be easily separated from the system modes, in general. Another consequence of the lack of knowledge of the input is that the estimated mode shapes cannot be scaled to unity modal ‘a’ or unity modal mass [3, 4]. It is interesting to see how most of the classical input–output methods carry over (after some modifications) to the output-only case. Frequency-response-function (FRF)-driven methods can be converted to spectrum-driven methods; Impulse-response-function (IRF)driven methods are almost identical to output-covariance-driven methods and input– output data-driven methods are very similar to output-only data-driven methods. The main reason to include this section is to point out the similarities and differences between methods and to indicate that essentially the same methods have received different names in the literature. 3.1. FREQUENCY-DOMAIN METHODS Frequency-domain methods use either FRFs or output spectra as primary data. 3.1.1. The peak-picking (PP) method The simplest approach to estimate the modal parameters of a structure is the so-called peak-picking (PP) method [3, 4]. The method is named after its key step: the identification of the eigenfrequencies as the peaks of an FRF plot. The FRF matrix HðjoÞ 2 Clm can be estimated from the input and output data. Theoretically, it can be written in modal form as n X 1 fvi ghliT i ð1Þ HðjoÞ ¼ jo  li i¼1 where o is the circular frequency (rad/s); n is the number of modes; fvi g 2 Cl are the modal vectors; l is the number of outputs; hliT i 2 Cm are the modal participation vectors; m is the number of inputs; KT denotes the transpose of a matrix; and li are the continuous-time eigenvalues, which occur in complex-conjugated pairs and are related to the eigenfrequencies oi and damping ratios xi as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffi li ; lni ¼ xi oi  j 1  x2i oi : ð2Þ Under the conditions of low damping and well-separated eigenfrequencies, the FRFs (1) reach a local maximum around an eigenfrequency oi and can be approximated by Hðjoi Þ

1 fvi ghliT i xi oi

ð3Þ

from which the mode shapes and modal participation factors can be determined once the eigenfrequency and damping ratio are known. In order to obtain damping ratios, it is often suggested to use the half-power bandwidth method, which quantifies the sharpness of a resonance peak. It is however widely accepted that this estimate is not a very accurate one. In the output-only case, the FRF matrix cannot be determined, but it is simply replaced by the output spectrum matrix Sy ðzÞ 2 Cll which can be written as Sy ðjoÞ ¼ HðjoÞRu H T ðjoÞ

ð4Þ

COMPARATIVE MODAL ANALYSIS STUDY

969

mm

where Ru 2 R is the white-noise input covariance matrix. The output-only PP method is widely used because of it ease and convenience of implementation; see for instance Bendat and Piersol [7] and Felber [8]. Under the conditions of low damping and wellseparated eigenfrequencies, spectra (4) reach a local maximum around an eigenfrequency oi and can be approximated by Sy ðjoi Þ ai fvi ghvH i i

ð5Þ

where ai is a scale factor depending on the damping ratio, the eigenfrequency, the modal participation factor and the input covariance matrix; KH denotes the complex conjugate transpose of a matrix. From equation (5), it is clear that each column or row of the spectrum matrix at an eigenfrequency can be considered as an estimate of the mode shape fvi g at that frequency. A violation of the basic assumptions (low damping and well-separated frequencies) leads to erroneous results. In fact, the method identifies operational deflection shapes (ODSs) instead of mode shapes and for closely spaced modes such an ODS will be the superposition of multiple modes. Other disadvantages are that the selection of the eigenfrequencies can become a subjective task if the FRF or spectrum peaks are not very clear and that the eigenfrequencies can only be determined up to the frequency resolution of the discrete Fourier transform. 3.1.2. The complex mode indication function (CMIF) A more advanced method consists of computing the singular value decomposition (SVD) of the output spectrum matrix: Sy ð joÞ ¼ Uð joÞSð joÞU H ðjoÞ

ð6Þ

where U 2 Cll is a complex unitary matrix containing the singular vectors as columns. The diagonal matrix S 2 Rll contains the real positive singular values in descending order. This ‘method based upon the diagonalisation of the spectral density matrix’, as it was called by Prevosto [9], has been in use since the early 1980s to obtain the modes of a vibrating system subjected to natural excitation. Some years later, the method was also applied to FRFs and became known as the complex mode indication function (CMIF). As suggested by the name, the CMIF was originally intended as a tool to count the number of modes present in measurement data. As a useful byproduct, the CMIF also identifies the modal parameters from FRFs; see Shih et al. [10]. Recently, the spectrum-driven method received again attention as an alternative for the PP method in civil engineering applications. In Brincker et al. [11], the old method was reformulated and was given a new name, the frequency-domain decomposition (FDD) method. The method is based on the fact that the transfer function or spectrum matrix evaluated at a certain frequency is only determined by a few modes; see equations (1) and (4). The rank of the spectrum matrix is determined by the number of significantly contributing modes and the number of independent inputs acting on system (4). The SVD is typically used for estimating the rank of a matrix: the number of non-zero singular values equals the rank [12]. The singular values, as a function of frequency SðjoÞ; represent the actual CMIF. Therefore, plotting the CMIF yields the eigenfrequencies as local maxima. The CMIF is also able to detect closely spaced modes: more than one singular value will reach a local maximum around the close eigenfrequencies, provided that the number of uncorrelated stochastic inputs exceeds the mode multiplicity. This seems to be the case for most practical examples [11].

970

B. PEETERS AND C. E. VENTURA

If only one mode is important at a certain eigenfrequency oi ; the spectrum approximates a rank-one matrix and can be decomposed as [equation (6)] Sy ðjoi Þ s1 ðjoi Þfu1 ðjoi ÞghuH 1 ðjoi Þi:

ð7Þ

By comparing equation (7) with equation (5), it is evident that the first singular vector at resonance is an estimate of the mode shape at that frequency. In case of mode multiplicity at a resonance frequency, every singular vector corresponding to a non-zero singular value is considered as a mode shape estimate. The input–output variant of the method is very similar. In this case, the SVD is applied to the FRF matrix. For additional details see [10, 3, 4]. 3.1.3. Rational fraction polynomial (RFP) The RFP method is a ‘classical’ modal parameter estimation method. The method was introduced by Richardson and Formenti [13] and was extended to the multiple input (or polyreference) case by Van der Auweraer and Leuridan [14]. It can be shown that the FRF matrix can be parameterised as an RFP model: HðjoÞ ¼ ½ðjoÞp bp þ ðjoÞp1 bp1 þ    þ b0 ½ðjoÞp I þ ðjoÞp1 ap1 þ    þ a0 1 mm

ð8Þ

lm

where ai 2 R are the denominator matrix coefficients and bi 2 R are the numerator matrix coefficients. The polynomial order p is related to the number of modes as pm ¼ n: By applying linear least squares (LS), the matrix coefficients can be estimated from FRF measurements. Afterwards, it is straightforward to find the modal parameters from the matrix coefficients. Reformulating the matrix polynomials in terms of orthogonal polynomials improves the numerical properties of the method. Without giving a sound mathematical derivation, Schwarz and Richardson [15] recently proposed a modification of the RFP method so that it also works in cases where no input information is available and by consequence it is not possible to determine FRFs. Instead of traditional FRFs, the so-called ‘ODS FRFs’ are used in the curve-fitting process. An ODS FRF is synthesised from the magnitudes of the power spectrum and the phases of the cross-spectrum between the sensor and a reference sensor. 3.2. TIME-DOMAIN METHODS Input–output time-domain methods use IRFs or directly input–output time histories as primary data. Output-only time-domain methods use output covariances or directly output time histories as primary data. 3.2.1. Two-stage least squares (2LS) method It can be shown that the so-called ARMA model can represent a structure excited by white noise: yk þ a1 yk1 þ    þ ap ykp ¼ ek þ g1 ek1 þ    þ gp ekp

ð9Þ

where yk 2 Rl is the sampled output vector (the measurements) at time instant k; ek 2 Rl is a white-noise vector sequence. The left-hand side is called the autoregressive (AR) part and the right-hand side the moving average (MA) part, hence, the name of the model. The matrices ai 2 Rll are the AR matrix parameters; matrices yi 2 Rll are the MA matrix parameters. Sometimes, as in the present case of multiple outputs, one speaks of ARMAV models as to stress their multivariable character. The ARMA model order p is related to the number of modes as pl ¼ n: The classical approach to identify an ARMA model is the prediction error method; see Ljung [16]. However, this results in a highly non-linear optimisation problem with related

COMPARATIVE MODAL ANALYSIS STUDY

971

problems as convergence not being guaranteed, local minima, and sensitivity to initial values and, especially in the multivariable case, in an almost unreasonable computational load. The 2LS method avoids iterative search procedures by identifying an ARMA model in two linear steps. In a first step, a high-order AR model is identified together with its residual sequence. Next, an ARX model is identified in which the residuals of the first step serve as inputs. Details of the method can be found in Ljung [16, p. 337]. 3.2.2. The ibrahim time-domain (ITD) method The ITD method was first presented by Ibrahim and Mikulcik [17] and was originally intended to identify the modal parameters from free decay responses. Later, it evolved to a polyreference IRF-driven modal parameter estimation method. The IRF is obtained as the inverse Fourier transforms of the FRF (1) and can be written in discrete-time modal form as hk ¼

n X fvi ghliT iðeli Dt Þk1 ¼ VLdk1 LT

ð10Þ

i¼1

where hk 2 Rlm is the IRF at discrete time instant k; the matrix V 2 Cln contains the mode shapes; the matrix LT 2 Cnm contains the modal participation factors; the matrix Ld 2 Cnn is a diagonal matrix containing the discrete-time eigenvalues eli Dt where Dt is the sampling time. The IRFs are gathered in the following block Hankely matrix Hpq;1 2 Rplqm : 0

Hpq;1

h1 Bh B 2 ¼B @ 0

hp

h2 h3

 





hpþ1 1



V C B B VLd C T CðL B ¼B C @  A p1 VLd

1 hq hqþ1 C C C  A hpþq1

T Ld LT    Lq1 d L Þ ¼ Op;m Gq;m :

ð11Þ

The subscripts of Hpq;1 indicate the number of block rows p; the number of block columns q; and the time instant of the IRF on the first position in the Hankel matrix. The second equality of equation (11) follows from equation (10) and shows how the block Hankel matrix decomposes into the so-called modal observability matrix Op;m 2 Cpln and modal controllability matrix Gq;m 2 Cnqm ; defined by the third equality of equation (11). In the ITD method, the matrix W 2 Rplpl is found as the LS solution of the following equation: Hpq;2 ¼ WHpq;1

ð12Þ

where Hpq;2 is similar to Hpq;1 ; but starts with h2 instead. Both Hankel matrices can be constructed from IRF estimates. It is now easy to see from equations (10)–(12) that the modal controllability matrix Op;m and the discrete eigenvalue matrix Ld are found by y

A Hankel matrix is a matrix that is constant along its anti-diagonals.

972

B. PEETERS AND C. E. VENTURA

applying the eigenvalue decomposition to W: WOp;m ¼ Op;m Ld :

ð13Þ

The mode shapes are equal to the first l rows of Op;m : The eigenfrequencies and damping ratios are found from Ld as indicated by equations (2) and (10). 3.2.3. IRF-driven/covariance-driven subspace identification It can be shown that a vibrating structure can be represented by a discrete-time deterministic/stochastic state-space model (see for instance [5] for a detailed derivation) given as xkþ1 ¼ Axk þ Buk þ wk yk ¼ Cxk þ Duk þ vk l

ð14Þ

m

where yk 2 R is the measured output vector; uk 2 R is the measured input vector; xk 2 Rn is the discrete state vector; wk 2 Rn is the process noise due to disturbances and modelling inaccuracies; vk 2 Rl is the measurement noise due to sensor inaccuracy. A 2 Rnn is the state transition matrix describing the dynamics of the system (as characterised by its eigenvalues); B 2 Rnm is the input matrix; C 2 Rln is the output matrix, which is describing how the internal state is transferred to the outside world in the output measurements yk : D 2 Rlm is the direct transmission matrix. The noise vectors comprise unmeasurable vector signals assumed to be zero-mean, white and with covariance matrices: " # ! ! wp Q S T T E ð15Þ ðwq vq Þ ¼ dpq ST R vp where E is the expected value operator and dpq is the Kronecker delta. Such state-space models will be identified when using the so-called subspace identification methods. In a second step, the modal parameters are obtained from the matrices A; B and C: The derivation starts with the eigenvalue decomposition of A: A ¼ CLd C1

ð16Þ

where C 2 Cnn is the eigenvector matrix and Ld 2 Cnn is a diagonal matrix containing the discrete-time eigenvalues; as defined in equations (10) and (2). The modal participation matrix and mode shape matrix, introduced in equation (10), are obtained as LT ¼ C1 B V ¼ CC:

ð17Þ

An important property of the (noise-free) state-space model (14) is that the IRFs can be written as h0 ¼ D hk ¼ CAk1 B:

ð18Þ

This relation originates from the well-known paper by Ho and Kalman [18]. This property is the basis for many so-called realization methods, belonging to the class of subspace identification methods. The best-known variant in the modal analysis world is the eigensystem realization algorithm (ERA) by Juang and Pappa [19] and Juang [20]. Deterministic (input–output) realization algorithms start by gathering the IRFs in the

COMPARATIVE MODAL ANALYSIS STUDY

973

ilim

following block Hankel matrix Hii;1 2 R ; similar as in equation (11): 0 1 0 1 C h1 h2    hi Bh B C h3    hiþ1 C B 2 C B CA C Hii;1 ¼ B C¼B CðB AB    Ai1 BÞ ¼ Oi Gi : @    A @  A hi

hiþ1



h2i1

ð19Þ

CAi1

The second equality of equation (19) follows from equation (18) and shows how the block Hankel matrix decomposes into the so-called observability matrix Oi 2 Riln and controllability matrix Gi 2 Rnim ; defined by the third equality of equation (19). If the number of block rows and columns i is chosen large enough, and if the system is observable and controllable, the rank of the li  mi Hankel matrix equals n; since it is the product of a matrix with n columns and a matrix with n rows. The actual algorithm consists of estimating the IRFs hk ; computing the SVD of Hii;1 ; truncate the SVD to the model order n; estimating Oi and Gi by splitting the SVD into two parts and finally estimating A; B; and C from Oi and Gi : Finally, the modal parameters are found as indicated in equations (16) and (17). Implementation details can be found in [3, 4, 19, 20]. In the stochastic (output-only) case, a vibrating structure excited by white noise can be represented by a stochastic state-space model; i.e. equation (14) without the terms in uk : Although no IRFs can be computed, a similar algorithm can be used that is based on output covariances instead. Output covariances are defined as Ri ¼ E½ykþi yTk  ¼ lim

N!1

1 X 1N ykþi yTk N k¼0

ð20Þ

where the second equation follows from the ergodicity assumption. Of course, in practice, only a finite number N of data points is available and a covariance estimate is simply obtained by dropping the limit in equation (20). By defining the next state–output covariance matrix G 2 Rnl as G ¼ E½xkþ1 yTk 

ð21Þ

it can be shown (see for instance [21, 22]) that the output covariances of a stochastic statespace system can be written as Ri ¼ CAi1 G:

ð22Þ

By comparing this property with the decomposition property of the IRFs (18), it is clear that the same algorithm as in the deterministic case can be applied. The use of covariancedriven subspace identification for operational modal analysis is for instance demonstrated by Benveniste and Fuchs [23], Hermans and Van der Auweraer [24], and Peeters [5] and Peeters and De Roeck [6]. 3.2.4. Data-driven subspace identification Recently, a significant amount of research effort in the system identification community has been devoted to data-driven subspace identification, as evidenced by the book of Van Overschee and De Moor [22] and the second edition of Ljung’s book [16]. Subspace methods identify state-space models from (input and) output data by applying robust numerical techniques such as QR factorisation, SVD and LS. As opposed to the subspace methods of Section 3.2.3, data-driven subspace identification methods directly work with the measured time histories, without having to convert them to IRFs or output covariances. A general overview of subspace identification (both deterministic and

974

B. PEETERS AND C. E. VENTURA

stochastic) is provided in [22]. The use of subspace identification for input–output modal analysis has been demonstrated and illustrated by Abdelghani et al. [25]. They demonstrated that the data-driven algorithms outperform an optimal version of the ERA. The use of data-driven stochastic subspace identification for output-only modal analysis has been demonstrated in Peeters et al. [26] and Peeters and De Roeck [27]. It is beyond the scope of this paper to explain the data-driven methods in detail. The interested reader is referred to the above-cited literature. 3.3. NAMES AND ACRONYMS At this point, it is perhaps useful to synthesise the names and acronyms that are commonly used to indicate the modal parameter estimation methods. A general description is given in Table 1.

4. COMPARATIVE STUDY OF THE MODAL ANALYSIS RESULTS

4.1. EXPECTED RESULTS Before presenting an overview of the results provided by each participant, the results that could be expected are presented in this section. These results were obtained from a data analysis as described in [5, 28]. Input–output subspace identification was applied to the shaker data and output-only subspace identification was applied to the drop weight and ambient data. A typical problem in (parametric) system identification is the determination of the model order. When trying to estimate the modal parameters from real data, it is generally a good idea to over-specify the model order considerably; i.e. to try to fit high-order models that contain much more modes than present in the data. Afterwards, interpreting a so-called stabilisation diagram helps separate the true physical modes from the spurious

Table 1 Overview of the system identification methods that have been applied by the different research teams Acronym

Alternative acronyms

PP CMIF

Peak-picking method. Complex mode indication function. Recently also the name frequency domain decomposition method was proposed. OP Rational fraction polynomial, which is in fact the name of the model. Sometimes also the name orthogonal polynomials (OP) is used, as it indicates how the numerical problems of the basic RFP method can be tackled. Ibrahim time domain method. ERA, CVA, Subspace identification methods. The term subspace basically means BR that the method identifies a state-space model and that it involves an SVD truncation step. Many variants exist in this class of methods including eigensystem realization algorithm, canonical variate analysis (CVA) and balanced realization (BR). ARMAV Two-stage Least Squares method. In [34], it is called ARMAV technique, since the method identifies a multivariable ARMA model by two linear least squares steps.

RFP

ITD SUBSP

2LS

Names

FDD

An acronym and a name designate the methods. Alternative names, which can be found in the literature for basically the same methods, are also represented in the table.

COMPARATIVE MODAL ANALYSIS STUDY

975

numerical ones. The poles corresponding to a certain model order are compared to the poles of a one-order-lower model. If differences in eigenfrequencies, damping ratios and related mode shapes (or modal participation factors) are within pre-set limits, the pole under consideration is labelled as a stable one. The spurious numerical poles will not stabilise at all during this process and can be sorted out of the modal parameter data set more easily [3–5]. Typical stabilisation diagrams for the three excitation types are shown in Figs 2–4. In the frequency range from 0 to 30 Hz, at least 10 modes are present. The identified eigenfrequencies and damping ratios are given in Table 2. There are three pairs of rather close modes: 9.77–10.5, 12.4–13.2, and 19.3–19.8 Hz. Although these modes are not too closely spaced, some of the research teams failed to identify both modes in such a pair (see Section 4.2). From the shaker stabilisation plot, it is seen that the sixth mode only stabilises at a high model order. This mode was probably not very well excited by the shakers. The ninth mode is not clear from the drop weight stabilisation diagram. The ambient stabilisation diagram is the most difficult to interpret. Mode 7 is missing; and also modes 8–10 are not very stable within the specified model order band and with the pre-set stabilisation criteria. The frequency differences between the excitation types are generally small (Table 2). The variances of the modal parameters and the differences between the excitation types can partly be explained by temperature changes during the measurement period. Measuring

Figure 2. Stabilisation diagram obtained by applying subspace identification to the shaker data. The criteria are 1% for frequencies, 5% for damping ratios, and 2% for the mode shape correlations. The used symbols are: ‘’ for a stable pole; ‘.v’ for a pole with stable frequency and vector; ‘.d’ for a pole with stable frequency and damping; ‘.f’ for a pole with stable frequency and ‘.’ for a new pole.

976

B. PEETERS AND C. E. VENTURA

Figure 3. Stabilisation diagram obtained by applying subspace identification to the drop weight data. The criteria are 1% for frequencies, 5% for damping ratios, and 2% for the mode shape correlations. The used symbols are: ‘’ for a stable pole; ‘.v’ for a pole with stable frequency and vector; ‘.d’ for a pole with stable frequency and damping; ‘.f’ for a pole with stable frequency and ‘.’ for a new pole.

nine set-ups for all three excitation types took almost 1 day. As shown by Peeters and De Roeck [29], temperature changes had a significant influence on the value of the eigenfrequencies of this bridge. Although it is not shown in Table 2, it was determined that the standard deviations of the ambient results were somewhat larger. Taking into account their higher uncertainty, the damping ratios seem to be consistently identified from all three data sets. The mode shapes are graphically represented in Figs 5 and 6. The first mode is a vertical bending mode. The second mode is a transverse bending mode, combined with torsion of the girder. The third and fourth modes are combining vertical bending with torsion, which is typical for skew bridges. The fifth mode is a vertical symmetric bending mode. The sixth mode is a vertical anti-symmetric bending mode with an important vertical movement of the piers. Mode 7 is a highly complex mode, with a phase difference of 458 between the motions of both side-spans. When computer animation is used, a complex mode has typically the appearance of a travelling wave. The eighth and tenth modes are torsion modes and, finally, the ninth mode is a bending mode. 4.2. PARTICIPANTS’ RESULTS In this section, a description of the methods and results of each participant will be given in the order of appearance in the IMAC proceedings in which these were published. The comparison of the results is given in Section 4.3.

977

COMPARATIVE MODAL ANALYSIS STUDY

Figure 4. Stabilisation diagram obtained by applying subspace identification to the ambient data. The criteria are 1% for frequencies, 5% for damping ratios, and 2% for the mode shape correlations. The used symbols are: ‘’ for a stable pole; ‘.v’ for a pole with stable frequency and vector; ‘.d’ for a pole with stable frequency and damping; ‘.f’ for a pole with stable frequency and ‘.’ for a new pole.

Table 2 Comparison of identified eigenfrequencies and damping ratios between the three excitation types Mode

1 2 3 4 5 6 7 8 9 10

Eigenfrequencies (Hz)

Damping ratios (%)

Shaker

Drop weight

Ambient

Shaker

Drop weight

Ambient

3.87 4.82 9.77 10.5 12.4 13.2 17.2 19.3 19.8 26.6

3.85 4.81 9.74 10.4 12.2 13.2 16.9 19.2 } 26.7

3.86 4.90 9.77 10.3 12.5 13.2 } 19.0 19.9 }

0.9 1.7 1.5 1.6 3.1 4.6 5.0 2.5 4.9 3.2

0.8 1.6 1.7 1.8 3.8 4.1 4.9 2.3 } 3.3

0.9 1.4 1.3 1.4 2.5 3.0 } 2.0 2.3 }

4.2.1. Schwarz and Richardson In [30], the authors apply the multiple reference RFP method [13] to the multiple-input– multiple-output (MIMO) FRF matrix determined from the shaker data. Two shakers were

978

B. PEETERS AND C. E. VENTURA

Figure 5. 3D representation of mode shapes 1–6 of the Z24-Bridge, ordered from left to right from top to bottom. All 6 modes are (almost) real.

z

z y

x

y

x

z z y

x

y

x

Figure 6. 2D representation of mode shapes 7–10 of the Z24-Bridge, ordered from left to right from top to bottom. A full line represents the real part, whereas a dotted line represents the imaginary part. Mode 7 is a highly complex mode.

used and 75 responses were measured, resulting in a 75  2 FRF matrix. This classical input–output modal analysis approach yielded a very complete set of modal parameters. Nine modes were identified in the 0–30 Hz range. This means that this group only did not identify the mode at 19.8 Hz, which is close to the mode at 19.3 Hz. For the drop weight and ambient tests, no input measurements are available, so these researchers applied the PP method to the output power spectra to estimate the eigenfrequencies. No damping was estimated. The mode shapes were found from the power spectra amplitudes and the phases of the cross-spectra between all outputs and a reference output. Since peak picking becomes difficult if the peaks at resonance are not too clear, it is no surprise that less modes could be identified when compared with the shaker data analysed with the RFP method.

COMPARATIVE MODAL ANALYSIS STUDY

979

In a companion paper [15], the authors proposed an extension to the RFP method to make it also work in cases where no input information is available, so that it is not possible to determine FRFs. Instead of traditional FRFs, so-called ‘ODS FRFs’ are used in the curve-fitting process. An ODS-FRF is synthesised from the magnitudes of the power spectrum and the phases of the cross-spectrum between the sensor and a reference sensor. The authors’ conclusion is that, as expected, the shaker data yielded the best results, but that drop weight or ambient excitation testing is more representative of tests that can be performed under a wider set of circumstances [30]. They also state that good agreement was achieved between the first three modes obtained from the three excitation types. The authors’ opinion is that the best explanation for the failure of both the impact and ambient tests to match the shaker results is that the higher frequency modes were simply not excited during those tests [15]. 4.2.2. Luscher, Brownjohn, Sohn and Farrar In [31], the authors apply both the CMIF method and a subspace method to all three data sets. Since the output-only variants of the methods have been used, the shaker force data is neglected when analysing the first data set. Using the CMIF method, this group identified six modes from the shaker data and five modes from the drop weight and ambient data. Apparently, not all modes showed a peak in the singular values plot. Hence, it was not possible to identify these with the CMIF method. Damping ratios were determined by the rather unreliable half-power bandwidth method. A good agreement between all three excitation types was achieved for the first and third modes. The authors found that the estimation results from the subspace method had a high variance. They concluded that, due to memory limitations of their PC, the dimensions of the Hankel matrix containing the output covariances could not be chosen large enough to stabilise the results. This dimension parameter is indeed crucial for the parameter estimation quality. 4.2.3. Womack and Hodson In [32], the authors analysed the shaker data with the PP method. As in Section 4.2.1, the 75  2 MIMO FRF matrix was determined from the data. The authors visually inspected plots of these 150 FRFs to find the maxima, which are indications of eigenfrequencies. By doing so, almost all modes could be identified, except the sixth and ninth modes, which were only visible as a peak in a few FRFs and, therefore, easily missed by the PP method. The ambient data were analysed using the CMIF method. For the first five eigenfrequencies, the authors achieved a good agreement between the PP and the CMIF results. In both analyses, no damping was estimated. The authors indicated that the mode shapes identified with the CMIF method were not reliable. 4.2.4. Reynolds and Pavic In [33], the authors applied a stochastic subspace identification method to the output covariances obtained from both the drop weight as the ambient test. Instead of analysing the covariances between all outputs and the three references at once, the authors performed three separate analyses to the three columns of the covariance matrices. The drop weight data yielded six modes, whereas the ambient data yielded only four modes (among others, the fourth mode was not identified in either analysis). The authors expressed little confidence in their ambient results, because the mode shapes did not look too ‘logical’ and the results were not consistent. They suggested, however, that the results of the ambient measurements could have been improved if longer duration

980

B. PEETERS AND C. E. VENTURA

of data sets had been obtained and analysed [33]. This statement is however not confirmed by the other research teams, which were able to extract complete results from the same data. More likely, the authors encountered the same problem as explained in Section 4.2.2, namely that the dimension of their covariance Hankel matrix was not sufficiently large. 4.2.5. Fasana, Garibaldi, Giorcelli and Sabia In [34], the authors analysed the drop weight data. They used both the ITD method, which was originally developed to analyse free responses [17], and a 2LS method [16]. The 2LS method identifies an ARMA model (In [34], the method is called the ‘ARMAV technique’). The classical approach to identify an ARMA model is the Prediction Error Method. However, this results in a highly non-linear optimisation problem with related problems such as the convergence not being guaranteed, local minima, sensitivity to initial values and, especially in the multivariable case, in an almost unreasonable computational load. The 2LS method avoids iterative search procedures by identifying an ARMA model in two linear steps; details of the method can be found in [16, p. 337]. The ITD method yielded six modes, while the 2LS method yielded seven modes. The fourth mode is missing in both analyses. The authors did not compare the mode shapes resulting from the analyses of both excitation types. 4.2.6. Marchesiello, Piombo and Sorrentino In [35], the authors applied a stochastic subspace method to all three data sets. This means that they opted to neglect the input information in the shaker data. For all three excitation types, they obtained a very complete set of modal parameters. Their good results can be attributed to the large number of covariances used in the Hankel matrix and, as a consequence, the high-order models that could be properly identified. Models from order 2 to 300 (!) were fitted to the data. By using the stabilisation diagram, the stable physical poles could be separated from the unstable numerical ones. The eigenfrequencies, damping ratios and mode shapes had a good agreement between the three data sets. 4.3. SUMMARY OF RESULTS This section contains a tabular overview of the applied system identification methods and a graphical overview of the parameter estimation results reported by each of the participating groups. A number is assigned to each research team as follows: 1. 2. 3. 4. 5. 6. 7.

Schwarz and Richardson [30,15] Luscher, Brownjohn, Sohn and Farrar [31] Womack and Hodson [32] Reynolds and Pavic [33] Fasana, Garibaldi, Giorcelli and Sabia [34] Marchesiello, Piombo and Sorrentino [35] Peeters, Maeck and De Roeck [28]

Table 3 indicates which methods have been applied to each of the three data sets. From this table, it is clear that the most popular method was the stochastic subspace identification. These results suggest there is an evolution in modal civil engineering application from the simple PP and CMIF methods to the more sophisticated subspace identification methods. Interestingly, when analysing the shaker data, most research teams neglected the force measurements by applying an output-only method. This may suggest that most of the teams were comfortable analysing output-only data. Probably, also the civil engineering background of most of the teams plays a role in this matter.

981

COMPARATIVE MODAL ANALYSIS STUDY

Table 3 Overview of the system identification methods that have been applied by the different research teams to the three data sets Shaker data PP

i/o o/o

3

CMF

o/o

2

REP

i/o o/o

1

Drop weight data

Ambient data

1

1

2

2, 3

1

1

ITD

o/o

5

2LS

o/o

5

SUBSP

i/o o/o

7 2, 6, 7

2, 4, 6, 7

2, 4, 6, 7

The methods are designated as indicated in Table 1. The label ‘i/o’ means that both input as output data have been used; ‘o/o’ means that only output data have been used.

Figure 7. Relative eigenfrequencies extracted by the different identification methods and research teams from the shaker data.

In Figs 7–12, the eigenfrequency and damping ratio identification results of the research teams are compared. The relative values between teams are shown in these figures; i.e. the values obtained by the participants divided by the results represented in Table 2, Section 4.1. The ‘method and team’ combinations ‘RFP 1’ and ‘SUBSP 6’ obtained the most consistent shaker results, although also in ‘PP 3’ most eigenfrequencies were accurately determined. When looking at a typical shaker stabilisation diagram (Fig. 2), it does not come as a surprise that the sixth mode was the most difficult to find. The most complete drop weight results were obtained by the combination ‘SUBSP 6’. Most research teams had difficulties extracting modes 4 and 5 from the drop weight data. In the ambient data set, it is observed that all methods and teams agreed very well on the first three modes.

982

B. PEETERS AND C. E. VENTURA

Figure 8. Relative eigenfrequencies extracted by the different identification methods and research teams from the drop weight data.

Figure 9. Relative eigenfrequencies extracted by the different identification methods and research teams from the ambient data.

However, the results for the higher modes start to deviate with the mode order, probably because these modes are not so well excited and, by consequence, can only be found by methods that can deal with noisy data. Not all research teams compared mode shape estimates between the excitation types. For the ones that did, the results are shown in Figs 13–15. These figures represent the MAC-values between mode shape estimates. MAC stands for modal assurance criterion and is nothing more than the (squared) correlation between two modal vectors [36]. By definition, the MAC is a value between 0 and 1. Values below 0.8 indicate that the two mode shapes are already deviating considerably. Consequently, with the exception of ‘SUBSP 4’, it is only the first mode shape that was consistently identified by each research team, independent of the data set used, and of the applied parameter estimation method.

COMPARATIVE MODAL ANALYSIS STUDY

983

Figure 10. Relative damping ratios extracted by the different identification methods and research teams from the shaker data.

Figure 11. Relative damping ratios extracted by the different identification methods and research teams from the drop weight data.

For ‘SUBSP 6’ and ‘SUBSP 7’, and a bit depending on the data set, the first six modes could consistently be identified.

5. CONCLUSIONS

The results of the comparisons presented in this paper clearly indicated that the quality of the identification depends significantly on the identification method and the excitation type. Next to these two factors, other variables affect the quality of the results: the experience of the modal analyst, the time spent to analyse the data and even the memory and capacity of the analysis computer. Overall, we can conclude that the RFP method

984

B. PEETERS AND C. E. VENTURA

Figure 12. Relative damping ratios extracted by the different identification methods and research teams from the ambient data.

Figure 13. MAC values between mode shapes extracted from shaker data and drop weight data.

applied to the shaker FRFs (see Section 4.2.1 and [30]) and the subspace methods applied to all data sets (see Section 4.2.6 and [35], and Section 4.1 and [28]) yielded the most complete and consistent modal parameter estimates. We should add that with the subspace method, it is also possible to obtain less complete and consistent results (see Section 4.2.2 and [31], and Section 4.2.4 and [33]) if the number of covariances used in the estimation process is not chosen large enough. This is however not an explicit deficiency in the method, but related to the experience of the user and the memory of the computer (although that any state-of-the-art computer would do). Another observation is that the stabilisation diagram is a very powerful tool and it generally does not get the attention it deserves. It is a classical tool in input–output modal analysis that is easily ported to output-only modal analysis as long as a method is used in

COMPARATIVE MODAL ANALYSIS STUDY

985

Figure 14. MAC values between mode shapes extracted from shaker data and ambient data.

Figure 15. MAC values between mode shapes extracted from drop weight data and ambient data.

which a parametric model is fitted to the data. The main benefit of the stabilisation diagram is that it allows the analyst to objectively select the eigenfrequencies, without having to find often-unclear peaks in spectrum or frequency-domain singular value plots as in the PP or CMIF method. Not using the stabilisation diagram is probably the main reason for the many missing modes in this comparative study. Related to the excitation type, the following observations can be made: *

*

If mass-normalized mode shapes are required, one cannot use ambient excitation. To obtain the correct scaling of the mode shapes, the applied force has to be known. If the cost of testing is a major concern, the use of shakers can be excluded. The price of a shaker and the additional manpower needed to install it on a structure may make the test not very cost-effective; see Kr.amer et al. [37].

986 *

*

*

*

B. PEETERS AND C. E. VENTURA

If a structure has low-frequency (below 1 Hz) modes, it may be difficult to excite them with a shaker, whereas this is generally no problem for a drop weight or ambient sources. The high-frequency modes on the other hand are not always well excited by ambient sources. By adjusting the settings of the damper on which the mass of the drop weight system falls, the frequency content of the excitation can be controlled in some sense. The level of excitation can be determined by the initial height of the mass. Above a certain frequency, the frequency content and the level of excitation of the shaker can also be controlled. This is evidently not the case for the ambient sources. The use of artificial excitation only makes sense when the generated response surpasses the ambient response that is always present. For very large structures, e.g. long-span cable-stayed bridges, this becomes almost impossible. If the purpose of the tests is continuous monitoring, only ambient excitation can be used. For intermittent monitoring, also the use of a drop weight can be considered: it is cheap, fast and easy to install.

The accelerations of a structure associated with ambient excitations are typically very small, generally in the order of tens to hundreds of milli-g’s, and can vary considerably during acquisition; for instance depending on whether a truck, a car or no traffic is passing at a certain speed. This causes challenges to the sensors, the acquisition system and the identification algorithms that must be able to extract weakly excited modes from noisy data. The developments of the last years both on the acquisition side as on the identification side (i.e. the development of subspace identification methods) greatly enhanced the use of ambient vibration testing to estimate the modal parameters of a large civil engineering structure. Next to hardware and software, the judgement and experience of the modal analyst plays a role in the success of the modal parameter estimation. Research into the automation of this process, so that any user interaction could be excluded, is certainly useful.

ACKNOWLEDGEMENTS

The data for this research were obtained in the framework of the BRITE-EURAM Programme CT96 0277, SIMCES with a financial contribution by the European Commission. Partners in the project were: K.U. Leuven, Aalborg University, EMPA, LMS International, WS Atkins, Sineco, T.U. Graz. The authors would like to express their sincere thanks to all the research groups that participated in this study. Their enthusiasm and willingness to share their knowledge and expertise on system identification of large Civil Engineering Structures is acknowledged. Without the participation of these groups, this project could have not been completed and the advancement of the state of the art on modal identification techniques for large structures would have hampered. The support of the organising committee of IMAC to proceed with the session leading to the papers discussed here is also acknowledged.

REFERENCES 1. G. de Roeck, B. Peeters and J. Maeck 2000 Proceedings of IASS-IACM 2000, Computational Methods for Shell and Spatial Structures, Chania, Greece. Dynamic monitoring of civil engineering structures.

COMPARATIVE MODAL ANALYSIS STUDY

987

2. C. Kr A. mer, C. A. M. de Smet and G. de Roeck 1999 Proceedings of IMAC 17, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 1023–1029. Z24 Bridge damage detection tests. 3. W. Heylen, S. Lammens and P. Sas 1995 Department of Mechanical Engineering, Katholieke Universiteit Leuven, Belgium. Modal analysis theory and testing. 4. R.J. Allemang 1999 Vibrations: Experimental Modal Analysis, Course Notes, 7th edn, Structural Dynamics Research Laboratory, University of Cincinnati, OH, U.S.A. [http:// www.sdrl.uc.edu/course info.html]. 5. B. Peeters 2000 Ph.D. thesis, Department of Civil Engineering, Katholieke Universiteit Leuven, Belgium. System identification and damage detection in civil engineering [http:// www.bwk.kuleuven.ac.be/bwm]. 6. B. Peeters and G. de Roeck 2001 ASME Journal of Dynamic Systems, Measurement, and Control 123. Stochastic system identification for operational modal analysis: a review. 7. J.S. Bendat and A.G. Piersol 1993 Engineering Applications of Correlation and Spectral Analysis, 2nd edn.New York, U.S.A.: John Wiley & Sons. 8. A.J. Felber 1993 Ph.D. thesis, University of British Columbia, Vancouver, Canada. Development of a hybrid bridge evaluation system. 9. M. Prevosto 1982 Ph.D. thesis, Universit!e de Rennes I, France. Algorithmes d’ Identification des Caract!eristiques Vibratoires de Structures M!ecaniques Complexes. 10. C. Y. Shih, Y. G. Tsuei, R. J. Allemang and D. L. Brown 1988 Mechanical Systems and Signal Processing 2, 367–377. Complex mode indication function and its application to spatial domain parameter estimation. 11. R. Brincker, L. Zhang and P. Andersen 2000 Proceedings of IMAC 18, the International Modal Analysis Conference, San Antonio, TX, U.S.A., 625–630. Modal identification from ambient responses using frequency domain decomposition. 12. G. H. Golub and C. F. van Loan 1996 Matrix Computations. 3rd edn. Baltimore, MD, U.S.A.: The Johns Hopkins University Press. 13. M. Richardson and D. L. Formenti 1982 Proceedings of IMAC 1, the International Modal Analysis Conference, Orlando, FL, U.S.A., 167–182. Parameter estimation from frequency response measurements using rational fraction polynomials. 14. H. van der Auweraer and J. Leuridan 1987 Mechanical Systems and Signal Processing 1, 259–272. Multiple input orthogonal polynomial parameter estimation. 15. B. Schwarz and M. Richardson 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 1017–1022. Modal parameter estimation from ambient response data. 16. L. Ljung 1999 System Identification: Theory for the User, 2nd edn. Upper Saddle River, NJ, USA: Prentice-Hall. 17. S. R. Ibrahim and E. C. Mikulcik 1977 Shock and Vibration Bulletin 47, 183–198. A method for the direct identification of vibration parameters from the free response. 18. B. L. Ho and R. E. Kalman 1966 Regelungstechnik 14, 545–548. Effective construction of linear state-variable models from input/output data. 19. J.-N. Juang and R. S. Pappa 1985 Journal of Guidance, Control, and Dynamics. An eigensystem realization algorithm for modal parameter identification and model reduction, 8, 620–627. 20. J.-N. Juang 1994 Applied System Identification. Englewood Cliffs, NJ, U.S.A.: Prentice-Hall. 21. H. Akaike 1974 IEEE Transactions on Automatic Control 19, 667–674. Stochastic theory of minimal realization. 22. P. van Overschee and B. de Moor 1996 Subspace Identification for Linear Systems: Theory}Implementation}Applications. Dordrecht, The Netherlands: Kluwer Academic Publishers. 23. A. Benveniste and J.-J. Fuchs 1985 IEEE Transactions on Automatic Control 30, 66–74. Single sample modal identification of a nonstationary stochastic process. 24. L. Hermans and H. van der Auweraer 1999 Mechanical Systems and Signal Processing 13, 193–216. Modal testing and analysis of structures under operational conditions: industrial applications. 25. M. Abdelghani, M. Verhaegen, P. van Overschee and B. de Moor 1998 Mechanical Systems and Signal Processing 12, 679–692. Comparison study of subspace identification methods applied to flexible structures. 26. B. Peeters, G. de Roeck, T. Pollet and L. Schueremans 1995 Proceedings of New Advances in Modal Synthesis of Large Structures: Nonlinear, Damped and Nondeterministic Cases, Lyon,

988

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

B. PEETERS AND C. E. VENTURA

France. 151–162. Stochastic subspace techniques applied to parameter identification of civil engineering structures. B. Peeters and G. de Roeck 1999 Mechanical Systems and Signal Processing 13, 855–878. Reference-based stochastic subspace identification for output-only modal analysis. B. Peeters, J. Maeck and G. de Roeck 2000 Proceedings of the European COST F3 Conference on System Identification and Structural Health Monitoring, Madrid, Spain, 341–350. Excitation sources and dynamic system identification in civil engineering. B. Peeters and G. de Roeck 2001 Earthquake Engineering and Structural Dynamics 30, 149–171. One-year monitoring of the Z24-Bridge: environmental effects versus damage events. B. Schwarz and M. Richardson 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 829–835. Post-processing ambient and forced response bridge data to obtain modal parameters. D. J. Luscher, J. M. W. Brownjohn, H. Sohn and C. R. Farrar 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 836–841. Modal parameter extraction of Z24 Bridge data. K. Womack and J. Hodson 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 842–845. System identification of the Z24 Swiss bridge. P. Reynolds and A. Pavic 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 846–851. Comparison of forced and ambient vibration measurements on a bridge. A. Fassana, L. Garibaldi, E. Giorcelli and D. Sabia 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 852–856. Z24 Bridge dynamic data analysis by time domain methods. S. Marchesiello, B. A. D. Piombo and S. Sorrentino 2001 Proceedings of IMAC 19, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 864–869. Application of the CVA-BR method to the Z24 Bridge vibration data. R. J. Allemang 2002 Proceedings of IMAC 20, the International Modal Analysis Conference, Los Angeles, CA, U.S.A., 397–405. The modal assurance criterion (MAC): twenty years of use and abuse. C. KrA. mer, C. A. M. de Smet and B. Peeters 1999 Proceedings of IMAC 17, the International Modal Analysis Conference, Kissimmee, FL, U.S.A., 1030–1034. Comparison of ambient and forced vibration testing of civil engineering structures.

APPENDIX A: LIST OF ACRONYMS

A list of acronyms used throughout this paper is given in the following. The acronyms for the identification methods can be found in Table 1. FE FRF IRF LS MAC MIMO ODS PC SIMCES SVD

finite element frequency response function impulse response function lease squares modal assurance criterion multiple-input–multiple-output operational deflection shape personal computer system identification to monitor civil engineering structures singular value decomposition