Output Only Modal Identification a Toolkit for Labview

CMNE/CILAMCE 2007 Porto, 13-15 Junho, 2007

© APMTAC, Portugal 2007

OUTPUT-ONLY MODAL IDENTIFICATION

A TOOLKIT FOR LABVIEW

Weihua Hu1*, Elsa Caetano and Álvaro Cunha1

1: Laboratory of Vibration and Monitoring (ViBest)

Faculty of Engineering of University of Porto (FEUP)

Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

e-mail: [email protected], [email protected], [email protected]

web: www.fe.up.pt/vibest

Key words: output-only modal identification, peak-picking, enhanced frequency domain decomposition, toolkit, Labview.

Summary: In this paper, two frequency domain techniques (classical Peak-picking (PP) and more advanced Enhanced Frequency Domain Decomposition (EFDD)) are briefly introduced for modal parameter identification from output-only modal tests. Based on the theory of these frequency domain identification methods, a toolkit in Labview environment is developed for structural modal parameter identification, its potential being illustrated by performing an application to experimental data from Pinhão bridge. By pushing simple buttons the user can easily develop the whole process of structural modal identification: reading data from a database or loading data files from disk, preprocessing the signals, applying modal identification techniques and extracting modal parameters and visualizing mode shapes. With the help of this user friendly toolkit, the structural modal parameters can be identified accurately and rapidly.

Weihua Hu, Elsa Caetano and Álvaro Cunha

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1. INTRODUCTION

In this paper, the theory of modal parameters identification using Peak-picking (PP) and Enhanced Frequency Domain Decomposition (EFDD) methods in frequency domain is summarized, the corresponding procedure to estimate natural frequencies, modal damping ratios and mode shapes being described. Subsequently, a toolkit developed in Labview environment for modal parameters identification is introduced. This toolkit offers the user an easy way to control the complete process for determining the modal parameters of Civil Engineering constructions. Measured acceleration signals can lead in a rapid and accurate way to the eigenfrequencies, damping ratios and mode shapes of the structure, using either the classical Pick-peaking (PP) method or the more advanced Enhanced Frequency Domain Decomposition (EFDD) method. The interesting features of this toolkit are illustrated based on an application to Pinhão bridge. Throughout this example, the user can learn the main functions and detailed conventions of this toolkit.

2. THEORY AND ALGORITHM

2.1 Peak-picking method (PP)

In the output-only modal identification of Civil Engineering structures, the so-called Peak-picking method has been widely used since it seems the simplest approach to estimate the modal parameters of structures. By simply picking the peaks of the spectral estimates of the measured signals, the eigenfrequencies can be identified. In the following part the theoretical justification of this method is described. In ambient vibration tests, only structural responses ),...2,1)(( mityi = are measured. The relation between the unknown inputs)(tx and measured responses )(ty is proposed by Bendat&Piersol [1] as follows:

Txy HGHG )()()()( ωωωω = (1)

where )(ωxG is the rr × spectrum matrix of the unknown excitation, r is the number of inputs, )(ωyG is the mm× spectrum matrix of the response, m being the number of measured responses, and )(ωH is the rm× Frequency Response Function (FRF) matrix (“－” and T denote complex conjugate and transpose, respectively). Assuming the unknown input is white noise, )(ωxG is a constant matrix

CGx =)(ω (2)

Then the FRF matrix can be written in partial fraction as:


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∑= −

+−

=n

k k

k

k

k

j

R

j

RH

1

)(λωλω

ω (3)

where n is the number of modes,kλ is a pole and kR is a residue TkkkR γφ= (4)

where kk γφ , are the mode shape vector and the modal participation vector, respectively. Based on equations (2)-(4), equation (1) becomes:

Hn

k k

k

k

kn

k k

k

k

ky j

R

j

RC

j

R

j

RG

−+

−

−+

−= ∑∑

== 11

)(λωλωλωλω

ω (5)

where H denotes complex conjugate and transpose. After complex modal decomposition, equation (5) can be reduced to a relative simple pole/residue form as follows [2,3]:

∑= −−

+−

+−−

+−

=n

k k

Tk

k

k

k

Hk

k

ky j

A

j

A

j

A

j

AjG

1

)(λωλωλωλω

ω (6)

where kA is the kth residue matrix of the output spectrum matrix )ω(Gy and is given by[2,3]:

∑=

−−+

−−=

n

s sk

Tsk

sk

Hsk

k

CRRCRRA

1 λλλλ (7)

When sλ approaches kkkkk j ωξωξλ 212)1( −+−= ( ξω, mean eigenfrequency and damping ratio, respectively), the contribution of the k-th mode is given by

kk

Hkk

k

CRRA

ωξ2= (8)

Around the k-th natural frequency, the summation of the two corresponding terms in equation (3) is dominating, assuming low damping ratios, and thus, equation (6) can be simply expressed by:

( ) ( )22)(kk

Hkk

Tkk

kk

Hkk

ky

CCRRG

ωξφγγφ

ωξω =≈ (9)

By defining the scalar constant kα as [2,3]:

( )2kk

kTk

k

C

ωξγγα = (10)

the approximated spectrum at resonance can be expressed as:


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HkkkkyG φφαω ≈)( (11)

From the viewpoint of Peak-picking method, the interpretation of equation (11) is that at the resonance frequency, each column (or equivalent row) of the spectrum matrix can be considered as an estimate of the observed mode shape up to some scaling factor. An appropriate systematization and practical implementation of the Peak-picking method was firstly developed by Felber[4]. In order to ensure a large number of eigenfrequencies of a structure are identified from a single spectrum, he suggested considering the averaged normalized power spectrum density (ANPSD) of all measured locations. For determining damping ratios, Bendat&Piersol [1] suggest to use the half-power bandwidth method. However, this procedure is not enough accurate. Peak-picking method is the most widely used method of identification in civil engineering because of its simplicity. Many civil engineering cases exist where the method is successfully applied [5]. Despite its popularity, the disadvantages of Peak-picking method are obvious: this method relies on the assumption that damping is low and the modes are well-separated, and so violation of these assumptions will lead to erroneous results. Instead of identifying mode shapes, only operational deflection shapes are determined. For closely-spaced modes, such an operational deflection shape will be the superposition of multiple modes. For structures like towers the Peak-picking method may become problematic, since the bending modes along any of the 2 principle axes modes are likely to have closely-spaced frequencies. Another disadvantage is that the selection of the eigenfrequencies can become a subjective task if the spectrum peaks are not very clear and the estimation of eigenfrequencies is restricted by the frequency resolution. For overcoming these drawbacks of Peak-picking method, a more advanced technique named EFDD (Enhanced Frequency Domain Decomposition) was developed as an alternative [3,6]. The theory of EFDD method is simply introduced in the following section.

2.2 Enhanced Frequency Domain Decomposition method (EFDD)

Enhanced Frequency Domain Decomposition method consists of computing the Singular Value Decomposition (SVD) of the spectrum matrix. The SVD is typically used for counting the rank of a matrix because the number of non-zero singular values equals the rank of the matrix. Another fact is that the spectrum matrix evaluated at a certain frequency is only determined by a few modes and the number of these modes coincides with the rank of the spectrum matrix. Originally, this method was used as a tool to count the number of modes a certain frequency. Subsequently, it was extended to identify the modal parameters from spectrum matrices [3,6]. When applying the EFDD method for modal parameters identification, the first step is to estimate the spectral matrix in equation (1). Then, by taking the SVD (singular value


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decomposition) of the output spectral matrix at discrete frequencies kωω = , the matrix can be decomposed as follows:

)()()()( kHkkkkkky USUG ωωωω = (12)

where the matrix [ ])(),...(),()( 21 kmkkkk jujujuU ωωωω = is an unitary matrix holding the singular vectors )( kk ju ω ,and )( kkS ω is a diagonal matrix holding the singular values

)( kkks ω . By plotting diagonal singular value versus discrete frequency, eigenfrequencies can be identified by the peaks of this plot. If only one mode is dominating at a certain resonance frequency kω ,the spectrum matrix can be approximately decomposed as

)()()()( 1111 kH

kkkyy uusG ωωωω = (13)

By comparing this equation with equation (11) it is concluded that the first singular vector at resonance frequency is an estimate of the modal shape kφ of that eigenfrequency. The corresponding singular value is the auto power spectral density function of the SDOF system. To such system, single-DOF modal parameters estimation technique could be applied to extract modal frequency and modal damping. In case of mode multiplicity at a certain eigenfrequency, every singular vector corresponding to a non-zero singular value yields a mode shape estimate, if the mode shapes are orthogonal. The detailed identification process of frequency and damping of structures are referred in [3,6].

3. DESCRIPTION OF THE TOOKLIT USING PINHÃO BRIDGE C ASE STUDY

In this section, experimental modal analysis of Pinhão bridge using the toolkit is performed. Throughout it, the basic functions of this toolkit are reviewed. They fall into four categories: reading file, preprocessing, system identification and visualization. Different interface windows for desired tasks and the functions of embedded buttons are introduced. The modal analysis of Pinhão bridge is presented and user will get acquainted with the various possibilities of this toolkit.

3.1 Bridge description

The Pinhão bridge is a multiple-span metallic arch bridge connecting Régua with Pinhão. The entire bridge consists of three spans with a total length of 6.207 m. Since the three spans are identical, only the second span was analyzed. Fig.1 shows a side view of the arch span of the bridge. The entire bridge and the general arrangement drawings of the second span are shown in Fig.2.


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Fig.1 Side view of Pinhão Bridge

Elevation of the bridge

Elevation of span 2

Section through arch span

Fig. 2 General arrangement of the entire bridge and the second span

69.2m Pier 3 Pier 2 Span 2

Roadway

4.6m

6m

Pavement

0.7m

Pavement

0.7m


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Pier 2 Pier3

1 2 3 4 5 6 7

17 16 15 14 13 12 11

Span 2

3.2. Output-only modal test

The output-only modal test of the Pinhão bridge was carried out using the method of ambient vibration [7]. The equipment used to measure the response consisted of triaxial seismographs duly synchronized using GPS sensors. All measurements were taken by placing the instruments near the pavement as shown in Fig. 3.

Fig.3 Instrumentation of the bridge with four seismographs

Fig.4 Measurement points used in ambient vibration test

Tab.1 Test setups

Setup Moveable points Reference points

1 1,11 3,13

2 2,12 3,13

3 4,14 3,13

4 5,15 3,13

5 6,16 3,13

6 7,17 3,13

Seismographs


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The location of measurement points is shown in Fig.4. Six test setups were conceived to cover the testing area of the second span of the bridge with a total of 14 locations (7 points per side). Each setup consists of 2 reference and 2 moveable tri-axial seismograph points. Tab.1 shows the different points arrangement of each setup. During the test of ambient vibration, normal traffic flow was permitted at normal speed. Testing was started from Pier 3, once the data was collected in one setup, the moveable seismographs were moved to the next points while the reference seismographs remained fixed. Each setup yielded 32× reference-points and 32× moveable-station data sets (channels). The same sequence was repeated six times to get measurements on all stations. The sampling frequency was 100Hz. The ambient vibration measurement was simultaneously recorded for 13 minutes at all accelerometers, which result in about 78000 data points per channel.

3.3 Modal analysis using the toolkit for Labview

The functions in main panel shown in Fig.5 consist of four categories: reading file, preprocessing, system identification and visualization. To perform each task, a new window is open and after finishing every desired task, the window is closed, returning to main panel.

Fig.5 Main panel


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Fig.6 Data loading menu

3.3.1Read file

There are two options for importing data files: reading file from a database developed with SQL (Structured Query Language) program and loading file from local disk in user’s computer. For the field vibration test, the sampled signals are saved in the local disk of the computer. By pushing button ‘Load from disk’ in ‘Main panel’ (Fig.5), user can open the ‘Data loading menu’ as shown in Fig.6. User can click ‘Load files’ button and load the sampled signals into the toolkit from local disk directly. After that, the names of selected signals will be shown in ‘File name’ listbox. Then, user can chose the wanted signals by selecting corresponding names. By pushing ‘Define sampling frequency’ button, user can define the sampling frequency used in ambient test. Information such as length of signal, maximum / minimum values and the corresponding time, mean value and variance of the time series are computed automatically. User has also the choice to display the waveform graph of the selected signals in time domain by choosing ‘Plot’ button. ‘Return’ button is used to close current menu and return the ‘Main panel’.

3.3.2 Preprocessing

Processing comprises two parts: main ‘preprocessing’ and appended ‘specify channel’. Preprocessing refers to data treatment before system identification and it highly influences the identification results. The interface window is shown in Fig.7. In the section of ‘signal condition’, three main options are implemented: 1. Decimate: the data is low-pass filtered and re-sampled at a lower rate. The identification


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concentrates on a limited low frequency band. 2. Detrend: the linear trend of the data will be removed. 3. Time window: only a portion of a large data record is selected for further manipulation. There is also an ‘original signal’ option, which allows user to retrieve the original signal. After clicking ‘apply’ button, the effect of signal condition can be seen immediately in both time and frequency domain. Windowlength, overlap and type of windows button allow user to define different parameters for frequency spectral analysis.

Fig.7 Preprocessing menu


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(a) single channel mode

(b) multiple channels mode

Fig.8 Specification of channel with physical location and direction

(a) single channel mode; (b) multiple channels mode

For later visualization of the mode shapes, it is essential to define the physical location (DOFs) of the measurement points associated to different signal channels and the direction of measurement. There are two options for specification: single channel mode and multiple channel mode, which depends on the numbers of channels selected by the user. Single channel and multiple channels modes are shown in Fig.8.

3.3.3 System identification

In this toolkit, two approaches for system identification in frequency domain are implemented: the PP (Peak-picking) method and the EFDD (Enhanced Frequency Domain Decomposition). If the ‘PP’ option is chosen, the user will see a window with ‘ANPSD graph’, ‘PMR plot’ and ‘Phase plot’ as shown in Fig.9 (detailed explanation these concepts can be found in [4]). User can fill proper values in ‘coarse peaks position’ control and push ‘apply’ button. The peaks of the ANPSD curve around these values (eigenfrequencies) will be searched automatically and actual eigenfrequencies corresponding to the peaks are shown in ‘peak frequency’ indicator. For Peak-picking method, the reference node must be defined. So, user should indicate the reference channel in ‘define reference node’ listbox. Then, user will view the ‘PMR plot’ and the ‘Phase plot’ at different eigenfrequencies by selecting different frequency values. Finally, by pushing ‘save and return’ button, the mode parameter file will be saved in the local disk for further visualization of modal shape.


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Fig.9 Peak-picking menu

The second method is EFDD method. The main interface window of modal parameter identification is shown in Fig.10.The identification process of EFDD technique can be divided into three steps as follows: 1. The first step is to view the results of SVD of spectrum matrix as shown in Fig.10 and

select different peak frequencies in different SVD curves. User can make this by choosing the values for ‘coarse peak’, ‘width’ and ‘MAC value’ controls and selecting different SVD curves in ‘plot number’ control.

2. In the second step, the window shown in Fig.11 for further selection of SVD curve will be open after user pushing ‘apply’ button (Fig.10). By clicking ‘return’ button in Fig.11 after the free decay curve is available, this window will be closed.

3. Finally, user should select ‘identification of modal parameters’ button (Fig.10). The window for accurate identification of natural frequency and modal damping ratio will be open as shown in Fig.12. After application of ‘spline interpolation’ on the free decay curve, smoother plot is obtained for curve fitting. By defining proper peaks in ‘peak1’ and ‘peak2’ control, a free decay curve will fit all the peaks between these selected two peaks, and accurate frequency and damping ratio can be identified. Similarly to PP method, it is also necessary to define a reference channel in EFDD method. Clicking ‘save and return’ button, the mode shape file will be saved for future visualization of global mode shape.


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Fig.10 Singular values plot for peak frequency selection in EFDD method

Fig.11 Selected singular value around certain peak frequency and

corresponding free decay curve


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Fig.12 Identification of natural frequency and modal damping ratio

Fig.13 Visualization of mode shape

3.3.4 Visualization

The identified mode shapes are graphically represented as global deformation of the structure. The DOFs were already attributed to the channels in the previous step (Fig.8). Before visualization, user should upload all identified mode shape components for different parts of the structure. The loaded mode shape files are organized by the toolkit automatically. For


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visualization of the mode shape, the ‘node file’ for node number and location, ‘beam file’ and ‘deck file’ for connection of these nodes are also necessary. These three files can be easily generated by pushing buttons or loaded the pre-defined Excel files. Finally, the mode shape is visualized as shown in Fig.13. This window offers different possibilities: scrolling through all modes, representation of undeformed structure and 3D-view with possibility to change the viewpoint. The identified modal parameters of Pinhão bridge deck, using the two different methods, are shown in Fig.14-15.

1st vertical mode(f1=2.799Hz) 2nd vertical mode( f2=5.491Hz)

3rd vertical mode(f3=8.150Hz) 4th vertical mode(f4=10.614Hz)

5th vertical mode( f5=12.661Hz) 6th vertical mode(f6=14.052Hz)


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Fig.14 (continuing)

7th vertical mode(f7=15.597Hz) 1st torsion mode(f1=5.940Hz)

1st transversal mode(f1=1.709Hz) 2nd transversal mode(f2=4.224Hz)

Fig.14 Typical modal parameters obtained from field test by Peak-picking method


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1st vertical mode(f1=2.804Hz %15.1, 1 =ξ ) 2nd vertical mode( f2=5.491Hz %32.0, 2 =ξ )

3rd vertical mode(f3=8.152Hz %20.0, 3 =ξ ) 4th vertical mode( f4=10.620Hz %33.0, 4 =ξ )

5th vertical mode( f5=12.660Hz %20.0, 5 =ξ ) 6th vertical mode(f6=14.052Hz %50.0, 6 =ξ )

Fig.15 (continuing)


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7th vertical mode(f7=15.597Hz %33.0, 7 =ξ ) 1st torsion mode(f1=5.942Hz %33.0, 1 =ξ )

1st transversal mode(f1=1.716Hz %21.1, 1 =ξ ) 2nd transversal mode(f2=4.224Hz %41.0, 2 =ξ )

Fig.15 Typical modal parameters obtained from field test by

Enhanced Frequency Domain Decomposition method

It can be observed that the two identification methods led to similar natural frequencies and mode shapes, but modal damping ratios were also estimated by EFDD method.

4. CONCLUSIONS

In this paper, the theory of the classical Peak-picking (PP) method and the more advanced Enhanced Frequency Domain Decomposition (EFDD) is summarized. Based on these two methods, a toolkit in Labview environment was developed for modal identification of structures. The basic functions and the detailed conventions of this toolkit are briefly described based on the ambient vibration test of Pinhão bridge. It was shown that natural


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frequencies, modal damping ratios and mode shapes can be identified rapidly and accurately by simply pushing buttons with this user friendly toolkit.

ACKOWLEDGEMENTS

The authors acknowledge the financial support provided by the Portuguese Foundation for Science and Technology (FCT) in the context of the Research Project “Vibrations in Cable-Stayed Bridges” and Basic Funding of CEC / FEUP, as well as the Ph.D. Scholarship provided to the first author.

REFERENCE

[1] J.S.Bendat, A.G.Piersol, Engineering application of correlation and spectral analyisis, John Wiley&Sons,1993.

[2] Peeters.B, System identification and damage detection in civil engineering, Ph.D.thesis,Department of civil engineering, Katholieke unversiteit Leuven, Belgium, 2000.

[3] Brinker R. Zhang L.,and Andersen P, “Modal identification from ambient response using frequency domain decomposition”. Proceedings of IMAC 18, the international Modal Analysis Conference, San Antonio, TX,USA, Feb, 2000.

[4] Felber A.J, Development of a hybrid bridge evaluation system ,PhD thesis, University of British Columbia, Vancouver, Canada, 1993.

[5] A.Cunha, E. Caetano, F. Magalhães, C. Moutinho. “From input-output to output-only modal identificaiton of civil engineering structures”, 1St international operational modal analysis conference (IOMAC), Copenhagen, Denmark, 2005.

[6] Brincker.R, C.E.Ventura, Andersen.P, “Damping estimation by frequency domain decomposition”, Proceedings of IMAC 19, the international Modal Analysis Conference, Kissimmee, USA, 2001.

[7] Filipe Magalhães, Bruno Costa, Álvaro Cunha and Elsa Caetano, “Experimental validation of the Finite Element Modeling of Pinhão Bridge, III European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering, Lisboa, Portugal, 2006.

Output Only Modal Identification a Toolkit for Labview

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