modal identification of ambient vibration structure using frequency ...

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

MODAL IDENTIFICATION OF AMBIENT VIBRATION STRUCTURE USING FREQUENCY DOMAIN DECOMPOSITION AND WAVELET TRANSFORM Thai-Hoa Le1 and Yukio Tamura2 Global COE Associate Professor, 2 Professor and Director, Wind Engineering Research Center, Tokyo Polytechnic University, 1583 Iiyama, Atsugi, Kanagawa 243-0297, Japan [email protected], [email protected]

1

ABSTRACT Frequency domain decomposition (FDD) has been widely used for output-only system identification due to ambient excitations in the frequency domain. FDD, however, usually requires a prior knowledge on natural frequencies, and also has troublesome in modal damping ratio identification as well as its strict hypothesis on uncorrelated white noise excitation and lightly-damped structures. Recently, wavelet transform (WT) has been developed most recently for the output-only system identifications in the time-frequency plane, especially its unique advantages on dealing with non-stationary, transient and non-linear inputs and outputs as well as dynamic system information in both time and frequency domains. This paper aims to present FDD and WT used for the output-only system identification of the ambient vibration structures. Modal parameters will be estimated from full-scale ambient vibration measurements of 5-storey steel structure. KEYWORDS: AMBIENT VIBRATION, FREQUENCY DOMAIN DECOMPOSITION, WAVELET ANALYSIS

Introduction System identification of ambient vibration structures using output-only identification techniques has become a key issue in structural health monitoring and assessment of engineering structures. Modal parameters of the ambient vibration structures consist of natural frequencies, mode shapes and modal damping ratios. So far, a number of mathematical models on the output-only identification techniques have been developed and roughly classified by either parametric methods in the time domain or nonparametric ones in the frequency domain. Each identification method in either the time domain or the frequency one has its own advantage and limitation. Generally, parametric methods such as Ibrahim time domain (ITD), Eigensystem realization algorithm (ERA) or Random decrement technique (RDT) are preferable for estimating modal damping but difficulty in natural frequencies, mode shapes extraction, whereas nonparametric ones such as Peak-picking (PP), Frequency domain decomposition (FDD) or Enhanced frequency domain decomposition (EFDD) advantage on natural frequencies, mode shapes extraction, but uncertainty in damping estimation [He and Fu (2001), Tamura et al. (2005)]. Recently, new approach based on Wavelet transform (WT) and Hilbert-Huang transform (HHT) have been developed for output-only identification techniques in the time-frequency plane. Among the nonparametric methods in the frequency domain, FDD has been very widely used recently for output-only system identification through the ambient vibration measurements due to its reliability, straightforward and effectiveness [Brincker et al. (2001)], applied for wind-excited structures [Tamura et al. (2005), Carasalle and Percivale (2007)]. FDD is also powerful for closed natural frequencies extraction. However, FDD always requires the prior-selected natural frequencies as well as its strict hypotheses of uncorrelated

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

white noise excitations and lightly structural damping. Under these strict hypotheses, the output PSD matrix can be expressed similarly as form of conventionally matrix decompositions, consequently, first-order linear approximation of the output PSD matrix is used for estimating mode shapes and damping. Output PSD matrix can be decomposed via fast-decayed decomposition methods such as QR decomposition and almost Singular value decomposition (SVD), but use of the Proper orthogonal decomposition (POD) is here discussed. WT has invented in early 90s last century, and the wavelet analysis has been exploited most recently in the output-only system identifications with new concept of the timefrequency analysis [Ladies and Gouttebroze (2002)]. WT advantages in dealing with nonstationary, transient and non-linear inputs and outputs. However, WT has been most used so far for estimating the natural frequencies and the modal damping ratio [Slavic et al. (2003), Kijewski and Kareem (2003)], but not for the mode shapes extraction. Moreover, the use of WT for output-only system identifications of ambient vibration structures needs more clarification and discussion in comparison with other methods. In this paper, the core backgrounds of FDD and WT for the output-only identification of the ambient vibration structures are presented with the numerical example of the ambient vibration measurements of 5-storey steel structure at a test site. Modal parameters of the structure are identified in comparison. Frequency Domain Decomposition-based Identification Relationship between excitation inputs F(t) and output response X(t) can be expressed in the frequency domain through the Frequency response function (FRF) matrix as follows: ( )= ( ) ( ) ∗ ( ) (1) ( ), ( ): PSD matrices of where *, T denote conjugate and transpose operations; inputs and outputs, respectively; ( ): FRF matrix. Normally, FRF matrix can be expressed under a form of residues/poles [He and Fu (2001), Brincker et al. (2001)]: ( )=∑

+

∗ ∗

=∑

+

∗

∗ ∗ ∗

(2)

where i: index of mode; N: number of modes; , : residue and pole in which = with light damping and = − + 1 − ; : modal damping ratio; , : mode shape vector and scaling factor. If inputs are uncorrelated white noise inputs, input PSD matrix is diagonal constant ( )= one ( , , … ) and damping is light, one can obtains the output PSD matrix at evaluated frequency decomposed modally as follows: ( )=∑

+

∗

∗

∗ ∗

(3)

where : scalar constant. Expression of the output PSD matrix in Eq.(3) is similar one of some matrix decompositions in the complex domain, thus these can be used to decompose the output PSD matrix. The output PSD matrix ( )has been orthogonally decomposed using the Proper Orthogonal Decomposition (POD) to obtain spectral eigenvalues and spectral eigenvectors: ( ) = Φ( )Θ( )Φ∗ ( ) = ∑ ( ) ( ) ∗ ( ) (4) where Φ( ), Θ( ): spectral eigenvectors and spectral eigenvalues matrices; k, M: index and number of decompositions. Because POD is a fast-decayed decomposition, thus the output PSD matrix can be estimated by first-order approximation as follows: ( ) = (ω)θ (ω)φ∗ (ω) (5) Due to the first-order spectral eigenvalue and eigenvector are dominant in a term of energy contribution, thus the first spectral eigenvalue contains full information of dominant frequencies to be used for extracting natural frequencies, whereas the first spectral eigenvector

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

brings information of mode shapes at each dominant frequency. The ith mode shape can be estimated from the first spectral eigenvector at certain dominant frequencies ( : ith natural frequency) as follows: = ( ) (6) As can be seen from Eq.(6) that FDD extracts the mode shape from the first spectral eigenvector at selected natural frequency, thus prior knowledge of the natural frequencies must be required for this identification technique. Accuracy of estimated mode shapes can be evaluated via correlation criteria between estimated mode shapes and analytical ones , moreover, among these criteria Modal assurance criterion (MAC) is preferably used: ( %) =

(7)

In the damping estimation, single-degree-of-freedom PSD function of each mode is identified using the first spectral eigenvalues and based on its frequency peak and selected MAC value. From identified uni-variate PSD function of the SDOF, logarithmic decrement and damping ratio of each mode can be estimated via inverse Fourier transform from univariate PSD function back to time series [Brincker et al. (2002), Tamura et al. (2005)]. Wavelet Transform-based Identification WT is defined in a continuous form (also called continuous wavelet transform) as convolution operator of a time series X(t) and a oscillatory wavelet function , ( ) as follows: ∞ ( , )= ∫∞ ( ) ∗( ) (8) √

where

∗

(∙) : complex conjugate of wavelet function

∗ ,

(∙) as

( )=

(

√

) ; s, :

wavelet scale (inverse of frequency) and translation parameters. The complex Morlet wavelet is the most commonly used for CWT, the Morlet wavelet and its Fourier transform can be denoted as: ( ) = (2 ) / exp ( 2 )exp (− /2) (9a) / ( ) = (2 ) exp (2 − ) (9b) where ( ): Fourier transform of complex Morlet wavelet; : , : Fourier frequency and central wavelet frequency. Complex Morlet wavalet

Fourier transform

0.4

0.4 Real Imaginary

0.3

0.35

0.2

0.3

0.1

0.25

0

0.2

-0.1

0.15

-0.2

0.1

-0.3

0.05

-0.4 -4

-3

-2

-1

0 Time (sec.)

1

2

3

4

0 -4

-3

-2

-1

0 st

1

2

3

4

Figure 1: Complex Morlet Wavelet and its Fourier Transform Conversion between the wavelet scale and the Fourier frequency can be established: = (10) where : Sampling frequency.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

In order to identify natural frequencies, WT coefficients have been extracted by parallel slide to the frequency axis at so-called wavelet ridges in the time-frequency plane where WT coefficients take maximum values at instantaneous frequency and time. Thus, positions of the wavelet ridges reveal corresponding natural frequencies in the frequency axis. In many cases, however, WT coefficients reach peaks at different time intervals due to dominant spectral components do not simultaneously occur. Output response of MDOF system can be decomposed in the structural normalized coordinates as follows: ( )=∑

exp (− 2

)sin ( 1 −

2

+

)

(11)

where , , , : Amplitude, modal damping ratio, undamped natural frequency and initial phase lag of ith structural mode, respectively. Replacing Eqs.(9a,9b), Eq.(11) into Eq.(8), the WT coefficient of output response can be expressed in case of a zero phase lag as follows: ( , ) = √ ∑ { exp(− 2 ) exp[− 2 −2 ] exp (i2π τ)} (12) where: : damped modal damping ratio of j-th structural mode ( = 1 − ) For a previous selected (i.e. fixed ), the mode shape can be estimated via the wavelet transforms of output response at point k and reference point as follows: =

( , )

(13)

( , )

In term of modal damping ratios, the WT envelopes are extracted by parallel slide windows in the time axis at identified natural frequencies. These WT envelopes are used to estimate the corresponding damping ratios. Ambient Vibration Measurements Ambient vibration measurements have been carried out on a 5-storey steel structure at the test site of the Disaster Prevention Research Institute (DPRI), Kyoto University (see Figure 2). Cross sections of steel members are shown in Fugure 2. Ambient data were recorded at all 5 floor levels and ground as reference, by tri-axial velocity sensors with output velocity signals (VCT Corp., Models UP255S/UP252) with A/D converter, amplifier and laptop computer. All data were sampled for period of 30 minutes per floor (5 minutes per a set-up) with sampling rate of 100Hz [Kuroiwa and Iemura (2007)]. Velocity sensors arrangement also is indicated in Figure 2. Ch.11(X), Ch.12(Y)

Floor 5 Ch.9(X), Ch.10(Y)

Floor 4 Ch.7(X), Ch.8(Y)

Floor 3 Ch.5(X), Ch.6(Y)

Floor 2 Ch.3(X), Ch.4(Y)

Ch.1(X), Ch.2(Y)

Floor 1

Z X

Figure 2: 5-storey Steel Structure and Velocity Sensors Arrangement

Ground

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

2

x 10

-3

Floor5

1

0

-1

-1 0

50

x 10

100

-3

150

200

250

0 -0.5

4

0 x 10

0

Floor4

50

100

-4

150

200

250

50

x 10

1

0.5

-1

-2

300

Floor3 Amplitude (m)

Amplitude (m)

1

-3

1

0

-2

x 10

2

100

-3

150

200

250

300

200

250

300

200

250

300

Floor2

0

-1

300

0

Floor1

50

x 10

5

100

-4

150 Ground

2 0

0

-2 -4 0

50

100

150 Time (s)

200

250

-5

300

0

50

100

150 Time (s)

Figure 3: Integrated Displacement Time Series in Y direction Because dynamic behavior of the steel structure is sensitive in Y direction, thus only outputs sensors and modal parameters in the Y direction have been solved and discussed in this paper. All output sensors were velocity time series, moreover, a single integration in the time domain using a trapezoic integration approach has been required to obtain output displacements which are necessary for estimating mode shapes in next steps. A drift and unknown initial condition of displacements during the time integration have been treated. Output displacements after integarting from the output velocities are shown in Figure 3. Modal Parameters Identification and Discussions In FDD identification, prior knowledge on natural frequencies in the output displacement time series have been determined using PSD analysis with high resolution of 0.0015Hz. Thus, basic information on dominant natural frequencies can be obtained from the PSD functions of the outputs (see Figure 4). Ground

-5

Floor1

-5

10

10

1.73Hz 5.34Hz 8.85Hz 13.66Hz 11.43Hz

-10

-10

10

19.76Hz 18.05Hz

PSD

PSD

10

-15

-15

10

10

-20

-20

10

0

5

10

15 Frequency (Hz)

20

25

10

30

Floor2

-5

10

0

5

30

25

30

25

30

19.76Hz 18.05Hz

PSD

PSD

11.43Hz 8.85Hz 13.66Hz

-10

10

19.76Hz 18.05Hz

-15

-15

10

10

-20

-20

10

0

5

10

15 Frequency (Hz)

20

25

30

Floor4

-5

10

25

5.34Hz

8.85Hz 13.66Hz

10

20

1.73Hz

5.34Hz -10

15 Frequency (Hz) Floor3

10

1.73Hz

10

10

-5

0

5

10

15 Frequency (Hz)

20

Floor5

-5

10

1.73Hz

1.73Hz

5.34Hz

-10

8.85Hz

8.85Hz

-10

13.66Hz 11.43Hz

10

11.43Hz 11.65Hz 13.66Hz

19.76Hz

19.76Hz

PSD

PSD

10

5.34Hz

-15

-15

10

10

-20

-20

10

0

10 5

10

15 Frequency (Hz)

20

25

30

0

5

10

15 Frequency (Hz)

20

Figure 4: High-resolved PSD Functions of Output Displacements

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

10

0

1.73Hz

Normalized eigenvalues

Mode 1

Mode 2

Eigenvalue1 Eigenvalue2 Eigenvalue3 Eigenvalue4 Eigenvalue5 Eigenvalue6

5.35Hz

Mode 3 10

8.84Hz Mode 4 10.16Hz 11.45Hz13.69Hz

-5

Mode 519.75Hz 18.12Hz

10

-10

0

5

10

15 Frequency (Hz)

20

25

30

Figure 5: Normalized Spectral Eigenvalues 99.90%

0.07%

0.01%

0.00%

Figure 6: First Four Spectral Eigenvectors and their Energy Contributions The output PSD matrix ( )has been established from the output displacements ( ) = { ( ), ( ), … ( )} (here M: number of sensors). Spectral eigenvalues and eigenvectors have been determined via POD of the output PSD matrix. All six normalized spectral eigenvalues and first four eigenvectors are shown in Figure 5 and Figure 6. Energy contributions of each spectral eigenvector and its corresponding eigenvalue can be estimated roughly based on its summed eigenvalues on analyzed frequency band (here 0-30Hz bandwidth). Concretely, first four eigenvalues and associated eigenvectors roughly contribute 99.9%, 0.07%, 0.01%, 0.00% respective to the dynamically structural system. As a results, the first spectral eigenvalues and eigenvectors characterize modal parameters of the structure. As can be seen from the Figure 5, the first eigenvalue contains dominant natural frequencies of the structural system. These estimated natural frequencies are compared with results from the analytical FE model [Kuroiwa and Iemura (2007)] in order to identify nutural frequencies associated with an order of mode shapes and vibration direction. Frequency identification of first five modes in the Y direction has also been indicated in the Figure 5. Bending modes in the Y direction corresponding the estimated natural frequencies can be determined via the first spectral eigenvector at these frequencies following the Eq.(6). MAC values of the first five mode shapes evaluated by Eq.(7) are respectively 100%, 99.76%, 99.8%, 98.95% and 99.3%. Thus, there is a good agreement between the FDD-based mode shapes and FE-based ones.

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

WT has been carried out with all output displacements using the complex Morlet wavelet. WT coefficients due to Eq.(8) of the output displacements are shown in Figure 7. Some wavelet ridges at the instantaneous frequencies and time intervals have been observed. The natural frequencies can be estimated at the wavelet ridges, for example Figure 8 shows sliding windows along the frequency axis of WT coefficients of the outputs at floor 1 and floor 5 at certain time point = 80 . Some dominant frequencies can be identified at these sliding windows.

Figure 7: WT Coefficients of Output Displacements -3

3.5

x 10

tau=80second

Floor 1

-3

2

1.72Hz

x 10

tau=80s

Floor 5

1.77Hz

Wavelet coefficient

Wavelet coefficient

3 2.5 2 1.5 1

1.5

1

0.5

0.5 0

2.5 Frequency(Hz)

5

7.5 1012.515 1 7.5 20

0

2.5 Frequency(Hz)

5

7.5 1012.515 1 7.5 20

Figure 8: WT Coefficient Slides along Frequency Axis at Wavelet Ridges Comparison in identified natural frequencies of the first five bending modes in the Y direction between FE model and FDD, WT is indicated in Table 1. It can be seen Table 1: Comparison in natural frequencies that FDD seems better than WT in term Mode FE FDD Diff.(%) WT Diff.(%) of natural frequencies identification Mode 1 1.69 1.73 2.3 1.77 4.5 because limitation in frequency resolution Mode 2 5.22 5.34 2.2 5.91 11.0 for analyzing WT coefficients at low and Mode 3 9.26 8.85 4.6 9.12 1.5 high frequency bandwidths. Moreover, the natural frequency of the 5 th mode Mode 4 13.60 13.66 0.9 14.02 2.9 hasn’t been identified via WT (see Table Mode 5 17.80 18.05 1.7 1).

The Seventh Asia-Pacific Conference on Wind Engineering, November 8-12, 2009, Taipei, Taiwan

Mode 2

Mode 1

Floor5

Mode 3

Floor5

Mode 4

Mode 5

Floor5

Floor5

Floor5

Floor4

Floor4

Floor4

Floor4

Floor3

Floor3

Floor3

Floor3

Floor3

Floor2

Floor2

Floor2

Floor2

Floor2

Floor1

Floor1

Floor1

Floor4

FEM Identified

FEM Identified

Ground 0

0.25

0.5

0.75

1

Ground -1

-0.5

0

0.5

1

Ground -1

Floor1

Floor1 FEM Identified -0.5

FEM Identified

FEM Identified 0

0.5

1

Ground -1

-0.5

0

0.5

1

Ground -1

-0.5

0

0.5

1

Figure 9: Comparison of First Five Mode Shapes WT has difficulty for extracting high modal and non-dominant natural frequencies. High-resolution in frequency is required for these cases. Comparison of the first five estimated mode shapes with corresponding analytical FE ones is shown in Figure 9. Good agreement between the estimated mode shapes by FDD and analytical ones can be observed. Conclusions Modal parameters identification with emphasis on the natural frequencies and mode shapes of the 5-storey steel structure based on both FDD and WT has been presented in comparisons between them as well as with FE model results. Identified natural frequencies and mode shapes from FDD and WT are quite good agreement with the FE results. It seems that FDD expresses better than WT in the natural frequencies extraction not the low-order modes but also high-order ones. FDD can extract natural frequencies at arbitrary frequency resolution, where WT is favorable for actually dominant frequencies. WT also requires more localized high-resolution analysis for extracting natural frequencies of high-order and nondominant mode shapes. Acknowledgement. This study was funded by the Ministry of Education, Culture, Sport, Science and Technology (MEXT), Japan through the Global Center of Excellence (Global COE) Program, 2008-2012. Ambient vibration data have been provided and permitted for use by Dr.Kuroiwa, T. and Prof.Iemura, H. at Structural Dynamics Laboratory, Kyoto University for which the first author would express the special thank.

References Brincker, R., Zhang, L., Andersen, P. (2001), “Modal identification of output-only systems using frequency domain decomposition”, Smart Materials and Structures, 10, 441-445. Brincker, R., Ventura, C.E., Andersen, P. (2001), “Damping estimation by frequency domain decomposition”, Proceedings of the 19th Int’l Modal Analysis Conference (IMAC), 698-703. Carassale, L., Percivale, F. (2007), “POD-based modal identification of wind-excited structures”, Proceedings of the Twelfth International Conference on Wind Engineering, Cairns, Australia, 1239-1246. He, J., Fu, Z.F. (2001), Modal analysis, Butterworth-Heinemann. Kijewski, T.L., Kareem, A. (2003), “Wavelet transforms for system identification: considerations for civil engineering applications”, Journal of Computer-aided Civil and Infrastructure Engineering, 18, 341-357. Kuroiwa, T., Iemura, H. (2007), “Comparison of modal identification of output-only systems with simultaneous and non simultaneous monitoring”, Proceedings of the World Forum on Smart Materials an Smart Structures Technology (SMSST07), Chongqing & Nanjing, China. Ladies, J., Gouttebroze, S. (2002), “Identification of modal parameters using wavelet transform”, Int’l Journal of Mechanical Science, 44, 2263-2283. Slavic, J., Simonovski, I., Boltezar, M. (2003), “Damping identification using a continuous wavelet transform: application to real data”, Journal of Sound and Vibration, 262, 291-307. Tamura, Y., Yoshida, A., Zhang, L. (2005), “Damping in buildings and estimation techniques”, Proceedings of the 6th Asia-Pacific Conference on Wind Engineering (APCWEVI), Seoul, Korea, 193-213.