Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors

Mechanical Systems and Signal Processing ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors Reza Tarinejad n, Majid Damadipour Faculty of Civil Engineering, University of Tabriz, Tabriz, East Azerbaijan Province, Iran

a r t i c l e i n f o

abstract

Article history: Received 30 May 2015 Received in revised form 29 August 2015 Accepted 26 October 2015

In this research, a combinational non-parametric method called frequency domain decomposition-wavelet transform (FDD-WT) that was recently presented by the authors, is extended for correction of the errors resulting from asynchronous sensing of sensors, in order to extend the application of the algorithm for different kinds of structures, especially for huge structures. Therefore, the analysis process is based on time-frequency domain decomposition and is performed with emphasis on correcting time delays between sensors. Time delay estimation (TDE) methods are investigated for their efficiency and accuracy for noisy environmental records and the Phase Transform – β (PHAT-β) technique was selected as an appropriate method to modify the operation of traditional FDDWT in order to achieve the exact results. In this paper, a theoretical example (3DOF system) has been provided in order to indicate the non-synchronous sensing effects of the sensors on the modal parameters; moreover, the Pacoima dam subjected to 13 Jan 2001 earthquake excitation was selected as a case study. The modal parameters of the dam obtained from the extended FDD-WT method were compared with the output of the classical signal processing method, which is referred to as 4-Spectral method, as well as other literatures relating to the dynamic characteristics of Pacoima dam. The results comparison indicates that values are correct and reliable. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Frequency domain decomposition-wavelet transform Operational modal analysis Time delay estimation Phase Transform – β 4-Spectral Pacoima dam

1. Introduction Performance evaluation of an arch dam during a large earthquake is a significant engineering challenge. System identification provides the modal characteristics, which are important in the seismic analysis of dams. The accuracy of the results obtained from structural dynamic analysis depends on having a precise mathematical model that is known as the modal model of the system. In fact, the purpose of system identification is to extract information about the mathematical model that characterizes a phenomenon that is observed in the laboratory or the field. Many civil and mechanical structures are difficult to excite artificially due to their physical size, shape or location. In addition, ambient forces, for example, waves, wind or traffic, load civil engineering structures while operating machinery exhibits self-generated vibrations. These natural input forces, which cannot easily be controlled or correctly measured, are used as unmeasured input for Operational Modal Analysis (OMA). OMA is based on measuring only the output of a structure

n

Corresponding author. Tel.: þ98 4133392386; fax: þ98 4133344287. E-mail address: [email protected] (R. Tarinejad).

http://dx.doi.org/10.1016/j.ymssp.2015.10.032 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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and using the ambient and operating forces as unmeasured input. It is used instead of classical modal analysis in order to identify the modal parameters under actual operating conditions and situations where it is difficult or impossible to control an artificial excitation of the structure. In OMA, Dynamic characteristics of a structure can be extracted using techniques such as Peak-picking (PP) and frequency domain decomposition (FDD) in the frequency domain [1–3], Stochastic Subspace Identification (SSI) in the time domain [4] and continuous wavelet transform (CWT) in the time – frequency domain [5–10]. In the previous study, the authors presented a novel approach based on the FDD-WT (frequency domain decompositionwavelet transform) for modal identification of structures [11,12]. In this research, this method is extended for modal identification of large structures considering the correction of errors due to non-synchronicity of records. In large and complex structures, time delays between the recorded responses of seismometers are unavoidable therefore, it is essential to solve the delay problem in order to estimate the modal parameters accurately. Therefore, several TDE (time delay estimation) techniques are introduced and their efficiency and accuracy for noisy environment records are investigated. The appropriate method (PHAT-β method) was selected and applied to modify the cross power spectral density used in the traditional FDD-WT method. In order to show the efficiency and accuracy of the extended method, a theoretical example (3DOF system) was provided to indicate the non-synchronous sensing effects of the sensors on the modal parameters. Besides, a real complicated problem, the Pacoima dam, has been selected as a case study.

2. Methodology 2.1. FDD-WT method The frequency domain decomposition (FDD) technique is a non-parametric operational modal analysis technique introduced by Brincker et al. [3,13]. It is very similar to the Peak-peaking (PP) method and uses the power spectral density (PSD) of the signals instead of just the Fourier transform of the impulse response function (FRF). If the power spectral density matrix of system frequency can be decomposed into its singular values and vectors in the desired frequency, peaks of the first singular value will be equal to the natural frequency of the system and singular vectors corresponding to the peaks of the first singular values approximate mode shape vectors. In this method, the half-power bandwidth or logarithmic decrement technique can be used to obtain the damping coefficient. Both techniques use the singular values spectrum for extracting damping ratios, but the values would have significant errors because of leakage errors in these spectra. Therefore, using the CACF (cyclic averaging of correlation functions) technique and also, timefrequency domain methods (e.g., wavelet transform) is necessary [11,12]. The continuous wavelet transform is a linear transform, which is defined as the convolution of the free decay signal and the dilated one. In this research, the modified complex Morlet wavelet function is selected as the mother wavelet function. Use of the CACF technique avoids the preprocessing of ambient responses with the random decrement technique (RDT) [14–16]. The length of time segments that are extracted from the response time histories is important for the RD technique. In the frequency domain modal identification methods, it is important that the RD functions are evaluated with sufficient time length to have a complete decay within that length, so that the effects of leakage and noise are reduced [17]. For short signals, it is difficult or impossible to achieve this aim and the use of RDT is not proper. The use of averaged correlation functions is a powerful technique for reducing the leakage and random errors. The averaging technique reduces the leakage bias error by digitally filtering the data to eliminate the frequency information that cannot be described by the FFT [18]. On the other hand, the number of analytical signals (correlation functions) is decreased from n2 to n signals, and therefore, the computational time is also decreased. In the work reported here, the accuracy and efficiency of the TDE methods are investigated and the proper one was utilized in order to eliminate the undesirable effects of the time delays between sensors installed in the structure. For this purpose, based on the works of [19], the CPSD matrix used in the FDD-WT algorithm was extended and modified to determine the modal parameters of huge and complex structures (e.g., hydraulic and marine structures), accurately. 2.1.1. Extended FDD-WT algorithm The FDD method can extract natural frequencies and non-scaled mode shapes. The parameters can be estimated by improving the CPSD matrix. In this work, the PHAT-β technique based on a filter or weighting function was used to modify the CPSD. On the other hand, the WT method has the ability to decouple the measured multicomponent signal to monocomponent signals via the modified Morlet wavelets, and is thus able to estimate the damping ratios [11,12]. The modal identification process of the extended FDD-WT method is presented in Fig. 1. 2.2. 4-Spectral method The classical signal processing method, referred to as 4-spectral, was used to re-compute the modal parameters of the dam in order to verify the results obtained using the extended FDD-WT method. The 4-spectral method is an easy and straightforward tool to process ambient records and extraction of modal parameters that is described below [20–22]: Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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Fig. 1. Extended FDD-wavelet algorithm.

The four spectrums, PSD (Power spectral density), CPSD (Cross power spectral density), CS (Coherency spectrum) and PS (Phase spectrum), related to the records of different channels are obtained using the Welch method. The frequencies corresponding to the peaks of the PSD that also have a peak in the CPSD and have a coherency close to one are selected as the natural frequencies. Mode shapes are determined as follows: 1. The root of the PSD amplitude is calculated at the natural frequency. 2. The phase angle of the CPSD that specifies the sign of mode shapes is determined at the same natural frequency. Mode shapes and damping ratios obtained from the 4-spectral method are not reliable in the cases of modal interference occurrence, therefore in the present research, the 4-spectral method was utilized for comparison of the natural frequencies obtained from the FDD-WT method.

3. Time delay estimation (TDE) 3.1. TDE methods During the last 40 years, the problem of estimating the time delay between signals received at two spatially separated microphones in the presence of noise has been considered for a variety of applications; such as in acoustics, radar communication and speech recognition [23]. This physical problem is shown in Fig. 2. The received signal at the two microphones can be modeled by: r 1 ðtÞ ¼ sðt Þ þ n1 ðtÞ; r 2 ðtÞ ¼ sðt DÞ þ n2 ðtÞ

0 rt rT

ð1Þ

where r 1 ðtÞ and r 2 ðtÞ are the outputs of two spatially separated microphones, sðt Þ is the source signal, n1 ðtÞ and n2 ðtÞ represent additive noises, T denotes the observation interval, and D is the time delay between the two received signals [23]. However, the issue has been not well investigated in earthquake engineering and operational modal analysis (OMA) of civil structures. Sensors installed on large structures such as dams often cannot synchronically record vibration responses. The problem is shown in Fig. 3. The time delays affect the estimation of modal parameters, especially mode shapes, and accounting for time delays between the received signals is essential. Four well-known methods of time delay estimation are described in this paper. These methods are the cross-correlation (CC), phase transform-β (PHAT-β), average square difference function (ASDF) and average magnitude difference function (AMDF). 3.1.1. Cross-correlation (CC) method In this method, the time lag at which the peak of the cross correlation between the two measured signals occurs is estimated. The cross-correlation function is:

and

Rr1r2 ðτÞ ¼ E½r 1 ðtÞr 2 ðtτÞ

ð2Þ


 ðτ r1r2 ÞCC ¼ τ:Rr1r2 ðτÞismaximized

ð3Þ

If the signals r i ðtÞ have N samples then the cross-correlation function has 2N þ1 values. 3.1.2. Phase transform-β (PHAT-β) method A way to sharpen the peak of the cross correlation is to whiten the input signals by using a weighting function, which leads to the so-called generalized cross-correlation technique (GCC). Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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Fig. 2. Time-delay associated with two microphones [23].

Fig. 3. Time-delay associated with two seismographs installed on a dam.

The PHAT is a GCC procedure that has received considerable attention due to its ability to avoid causing flattening of the peak of the correlation function [24]. Recently, a modified version of PHAT was presented that investigates the effect of weighting function to parametrically control the level of whitening influence on the magnitude spectrum [25]. This transform referred to as PHAT-β and defined as: Z þ1 ψ ðf ÞGr1r2 ðf Þej2πf τ df ð4Þ Rr1r2 ðτÞ ¼ 1

where 1  Gr1r2 ðf Þβ

ψ ðf Þ ¼ 

ð5Þ


 ðτ r1r2 ÞPHAT ¼ τ:Rr1r2 ðτÞismaximized

ð6Þ

and

where Gr1r2 ðf Þ is the cross-spectrum of the received signal, ψ ðf Þ is the PHAT-β weighting function and β is the additional parameter that controls the extend of spectral whitening and can take values in the range 0 r β r 1. When β ¼1, Eq. (4) becomes the conventional PHAT. Therefore, by varying β between 0 and 1, different levels of spectral normalization are achieved. Based on the research work presented in Ramamurthy's thesis, the values of β suggested to use under noisy conditions is 0:65 r β r0:8 [25]. In this paper β ¼0.8 has been considered for evaluating the time delays between records of the case study (Pacoima dam). The role of the filter or weighting function in the PHAT-β method is to ensure a large sharp peak is obtained in the crosscorrelation function thus ensuring good resolution in estimating the delay. In a noisy environment, the PHAT is the best choice because of its excellent performance in sharpening the correlation at the correct time delay and its small SNR threshold [23].

3.1.3. Average square difference function (ASDF) method The ASDF method is based on finding the position of the minimum squared error between the two received noisy signals and considering this position value as the estimated time delay. i.e., RASDF ðτÞ ¼ E½r 1 ðt Þ  r 2 ðt þ τÞ2

ð7Þ

Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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and






τ ASDF ¼ τ:RASDF ðτÞisminimized

5

ð8Þ

The favorable performance of the ASDF-based estimator is due to the fact that it gives perfect estimation in the absence of noise while the direct correlation does not [26]. Also, this technique does not require knowledge of the input spectra. In the simulated noisy environment, ASDF has a simple computation, easy detection and small SNR threshold. However, when the SNR is below a specified threshold, the performance of all the three methods rapidly deteriorates due to large anomalous or ambiguous estimates [23]. 3.1.4. Average magnitude difference function (AMDF) method The AMDF estimator does not involve multiplications and it can be considered in those applications where low computational complexity (fast estimation) is required, [27]. It considers a correlation function as RAMDF ðτÞ ¼ Efjr 1 ðt Þ  r 2 ðt þ τÞjg and

ð9Þ


 τ AMDF ¼ τ:RAMDF ðτÞ is minimized

ð10Þ

The AMDF method has lower precision than ASDF and both of them are based on minimum error criteria. 3.2. Correction of errors due to non-synchronous sensing In OMA, various modal identification algorithms can be applied to the output response data of the structure in order to identify modal parameters. In this study, the FDD-WT and 4-spectral methods are used to identify the dam and results verification, respectively. These algorithms, like the other OMA methods suffer from errors when using the nonsynchronous samples directly. These errors affect only slightly the identified frequencies but affect the mode shapes much more [19]. 3.2.1. Error elimination In order to eliminate the errors due to non-synchronous sensing of sensors, Feng and Katafygiotis developed a correction approach to recover the true spectral density using non-synchronous samples in the frequency domain. This approach is based on the spectral relationship of synchronous data and non-synchronous data. Because only spectral densities or correlation functions are needed for most of modal identification algorithms and raw synchronous time histories are not needed, reconstruction of the signal in the time domain is unnecessary [19]. Consider two time histories x and y0 . Note that y0 has a constant time shift τ relative to x. The cross power spectral density of signals x and y0 would be as follows: ^ 0 ðjωi Þ ¼ X ðωi Þ:Y 0  ðωi Þ Gyy

ð11Þ

^ 0 ðjωi Þ needs to be modified, because it has where X ðωi Þ and Y ðωi Þ are Fourier transform of signals x and y . The CPSD Gyy been constructed from signal y0 that has a constant time shift. Modification of the CPSD function is performed as follows: 0

0

^ ðjωi Þ ¼ X ðωi Þ:Y 0  ðωi Þ:expðj2π f t Þ Gyy i d

ð12Þ

or ^ 0 ðjωi Þ:expðj2π f t Þ ^ ðjωi Þ ¼ Gyy Gyy i d

ð13Þ 0

where t d is the time delay between signals x and y . In accordance with the above relationship, multiplying the initial CPSD ^ 0 ðjωi Þ by expðj2π f t Þ yields the correct cross power spectral density function the FDD-WT and 4-spectral algorithms. Gyy i d

4. Modal parameters identification The time delays that occur between the records of installed sensors on large structures such as dams affect their dynamic characteristics. To study the effects of non-synchronous sensing and its correction, two cases of the power spectral density functions are considered: original (non-modified) and modified. As previously mentioned, all OMA methods suffer from errors while using non-synchronous records (original records) directly. In this section, in order to determine the dynamic characteristics of structures, extraction procedure of the modal parameters is described based on the techniques used in the FDD-WT method. 4.1. Natural frequencies The modal identification process based on an earthquake response is performed according to the algorithm presented in Fig. 1. Generally, natural frequencies are determined from the largest peaks of the first singular value spectrum. The second Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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singular values spectrum should be used if some natural frequencies are not visible from the first singular value. Natural frequencies of dynamic systems can be extracted by using the records of accelerometers located at the different stations of the structure. The natural frequencies are selected so that they are visible from the spectra of records at different points. Ambient excitation forces can yield spurious peaks or modes in the power spectra. There are some criteria for detection and elimination of these modes, such as the MAC (Modal Assurance Criterion), ACS (Averaged Coherence Spectrum) and stabilization diagram.

4.1.1. Modal Assurance Criterion The Modal Assurance Criterion (MAC) essentially compares the relationship between two complex mode shape vectors φ and ψ by comparing their linearity. MAC ¼

jψT φj2 ðψT ψÞðφT φÞ

ð14Þ

In fact, calculating the correlation between the first singular vector at the peak and the first singular vector at neighboring points defines the above modal coherence. If the modal coherence is close to unity, then the first singular value at the neighboring point corresponds to the same modal coordinate, and therefore, the same mode is dominating [28]. In general, if the MAC value is above a user-specified MAC rejection level, the frequency corresponding to that peak is the modal frequency. It is important to note that the MAC criterion is not only used at a defined peak and its neighboring points, but can also be used to calculate the MAC between two different peaks (two modes).

4.1.2. Averaged coherence spectrum criterion The ACS criterion is based on averaging the coherence spectra calculated between different response channels: n P n P

γ¼

i¼1j¼1 n2

γ 2ij ð15Þ

where

γ2ij ¼

jSij ðfÞj2 ; i and j ¼ 1; 2; 3; :::; n Sii ðf ÞSjj ðfÞ

ð16Þ

In Eqs. (16) and (17), n is the number of response channels, Sij and γ2ij are the cross power spectral density and the coherence spectrum corresponding to the ith and jth channels of the recorded response, respectively; and γ is the Averaged Coherence Spectrum. The ACS intensifies the modal frequencies and plotting it shows a peak around the dominant frequency. Therefore, peaks of the ACS that correspond to the peaks of the first singular value spectrum and which also have their spectral domain near unity, have a strong assurance of corresponding to natural frequencies.

4.1.3. Stabilization diagram The stabilization diagram is widely used as a robust tool to distinguish true modes from spurious modes. The technique is based on subspace methods. The stabilization procedure is performed sequentially for all system orders up to a userspecified maximum. At each step, poles identified at the current system order are compared with poles identified at the previous system order. The criteria for defining a pole as stable were chosen as a change of less than 1% difference in frequency and 5% difference in damping with a pole identified with the previous model order. There are small differences in modal parameters identified at different model orders [29]. Because this paper focuses on non-parametric methods (e.g., FDD and WT), the background on the stabilization diagram, which is a parametric method, is not presented here. In this paper, MAC, ACS and stabilization criteria are used to determine and separate true and spurious modes. The preference is for those modes that satisfy the above criteria and have strong and clear peaks. Initially, the peaks of singular value spectra are determined in accordance with the stabilization diagram. Then the selected peaks are evaluated using the MAC and ACS criteria. 4.2. Mode shapes Singular vectors corresponding to the peaks of the first singular values are used to approximate the mode shape vectors. The mode shapes obtained from the FDD-WT are un-scaled and although some methods have been developed to modify the unscaled mode shapes in recent years [30], they are not investigated in the present study. The scaled and unscaled mode shapes are not different in phase or the general shape and only differ in the amplitude of the mode shape vectors. Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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4.3. Damping ratios As previously seen in Fig. 1, the FDD-WT method can calculate damping ratios using both the HPB (half-power bandwidth) and the LD (logarithmic decrement) techniques. In this paper, the logarithmic decrement technique (LDT) is applied to single frequency signals (skeletons) obtained from a continuous wavelet transform. The FDD method works in the frequency domain and has a high frequency resolution. Therefore, the FDD is an appropriate method for the evaluation and selection of ridges in the continuous wavelet transform.

5. Numerical example and engineering applications In this section, a numerical example is presented to verify the results obtained from the extended FDD-WT method in comparison with traditional methods. Also a real world case study (Pacoima dam) is considered as an application of the extended method. 5.1. Numerical example To study the effects of synchronization errors on modal identification, a 3-DOF structural model subjected to the El Centro accelerogram, shown in Fig. 4 was simulated. The time increment chosen corresponded to a sampling frequency of 100 Hz and the length of the record was 40 s. The distribution of mass and stiffness in the structure was assumed to be regular, with m ¼ 10t and k ¼ 1200KN m. The damping ratios corresponding to the modes of 3-DOF system are as follows: 8 9 > < 0:05 > = ξ ¼ 0:03 ð17Þ > > : ; 0:02 The damped response of the system subjected to El Centro earthquake excitation was calculated by using the fourthorder Runge–Kutta procedure. In order to study the performance of the TDE methods, two cases of output response were considered: 1. No time shift (synchronous sensing).

Fig. 4. Analytical 3DOF model.

500 Mass 1

0

Acceleration (cm/s2)

-500 0

5

10

15

20

25

30

35

40

500 Mass 2

0 -500 0

5

10

15

20

25

30

35

500

40 Mass 3

0 -500 0

5

10

15

20

25

30

35

40

Time (sec) Fig. 5. Responses of 3DOF system without time delays.

Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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2. Constant time shift (non-synchronous sensing). The simulation of non-synchronous sensing was assumed as follows: 2

3 2 3 τ1;2 0:07 6τ 7 6 7 4 1;3 5 ¼ 4 0:15 5 s τ2;3 0:08

ð18Þ

where τi;j is the time delay between channels i and j. Note that τi;j is zero for the case of synchronous sensing. The responses of 3DOF system for two cases of without and with time delays are shown in Figs. 5 and 6. The results indicate that responses compared to each other have a trivial time shift (In a few hundredths of a second) so that general form of them are similar.

5.1.1. Evaluation of TDE methods Figs. 7 and 8 illustrate the estimation of time delays using CC, AMDF, ASDF and PHAT methods for responses obtained from masses 1 and 2. Without noise environment is assumed and therefore conventional PHAT, in which β ¼ 1, is used. The results obtained from four TDE methods are presented in Table 1. The results indicate that the PHAT- β algorithm has the best ability to estimate time delays amongst the four methods in noisy environments (e.g., El Centro earthquake). 500 Mass 1

0

2

Acceleration (cm/s )

-500

0

5

10

15

20

25

30

35

40

500 Mass 2

0 -500

0

5

10

15

20

25

30

35

500

40 Mass 3

0 -500 0

5

10

15

20

25

30

35

40

Time (sec)

Fig. 6. Responses of 3DOF system with time delays.

Cross-correlation function

Average Magnitude Difference Function 4000

1000

3500

AMDF

CC

500 0 -500

2500

-1000 -100

-50

0

50

2000

100

Average Square Difference Function

6000

PHAT

ASDF

4000 3000 0

100

200

Time lags

300

400

0

1

5000

2000

3000

100

200

300

400

Phase Transform Function

0.5 0 -0.5 -100

-50

0

50

100

Time lags

Fig. 7. Time lags estimated by CC, AMDF, ASDF and PHAT methods for responses of masses 1 and 2 without time shifts.

Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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Cross-correlation function

1000

Average Magnitude Difference Function 4000 3500

AMDF

CC

500 0 -500

3000 2500

-1000 -100

-50

0

50

2000

100

Average Square Difference Function

6000

PHAT

ASDF

4000 3000 0

100

200

300

0

1

5000

2000

9

200

300

400

Phase Transform Function

0.5 0 -0.5 -100

400

100

-50

Time lags

0

50

100

Time lags

Fig. 8. Time lags estimated by CC, AMDF, ASDF and PHAT methods for responses of masses 1 and 2 with time shifts. Table 1 Estimated time delays related to 3-DOF model. Method

Time delay (s) No time shift

CC AMDF ASDF PHAT-β

Constant time shift

τ1;2 0

τ1;3 0

τ2;3 0

τ1;2 0.07

τ1;3 0.15

τ2;3 0.08

0.15 0.16 0.15 0

0 0 0 0

0.12 0.13 0.12 0

0.22 0.23 0.22 0.07

0.15 0.15 0.15 0.15

0.20 0.21 0.20 0.08

Table 2 Mode shapes obtained from various types of system responses. Modes

1th (0.776 Hz)

2th (2.174 Hz)

3th (3.142 Hz)

Natural frequencies (Hz) Exact

Unimproved

Improved with PHAT results

1.000 0.801 0.445 1.000  0.555  1.247 1.000  2.247 1.801

1.000 0.801 0.441 1.000  0.532 1.191 1.000  2.280  1.813

1.000 0.801 0.441 1.000  0.532  1.191 1.000  2.280 1.813

5.1.2. Investigation of time delay effects on mode shapes The first channel is considered the reference for identifying mode shapes and all values were normalized by the value of the first channel. The values of mode shapes obtained from two cases of with and without correction of the errors resulting from asynchronous sensing, are presented in Table 2. The results indicate that time delays between responses (errors resulting from asynchronous sensing) are mainly resulted in phase changes of mode shapes. Figs. 9 and 10 show mode shapes obtained from the system responses, for two cases of without and with correction of errors resulting from asynchronous sensing, respectively. The results indicate that considering the effects of nonsynchronous sensing of sensors leads to accurate estimation of mode shapes. Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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10

Modal shape # 1

Modal shape # 2

3

3

Exact Identified

Exact Identified

Exact Identified

2.5

2.5

2

2

2

1.5

number of stories

2.5

number of stories

number of stories

Modal shape # 3

3

1.5

1.5

1

1

1

0.5

0.5

0.5

0 -1

-0.5

0

0.5

0 -2

1

-1

0

1

0

2

-2

0

2

Fig. 9. Mode shapes for unimproved case.

Modal shape # 1

Modal shape # 2

3

3

Exact Identified

Exact Identified

Exact Identified

2.5

2.5

2

2

2

1.5

number of stories

2.5

number of stories

number of stories

Modal shape # 3

3

1.5

1.5

1

1

1

0.5

0.5

0.5

0 -1

0 -0.5

0

0.5

1

0 -1

0

1

-2

0

2

Fig. 10. Mode shapes for improved case.

5.2. Case study (Pacoima dam) The Pacoima dam is a 113 m high concrete arch dam with a thickness varying from about 3 m at the crest to 30 m at the base and is located north of Los Angeles, California. The 17-channel array of accelerometers located on the dam body and abutments is shown in Fig. 11. Since the FDD-WT and 4-spectral methods are based on output-only, the accelerometer records in the radial channels of the dam body were used for modal identification. On January 13, 2001, a magnitude 4.3 earthquake occurred, with an epicentral distance of about 6 km from the Pacoima dam and a focal depth of about 9 km. The water level of the reservoir was about 41 m below the crest at the time of the earthquake. No damage was observed during the earthquake and the response of the dam was assumed to be linear. The acceleration records of this earthquake were used to carry out the modal identification and calibration of the finite element

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Fig. 11. Locations of 17 accelerometers on the Pacoima dam [31].

Acceleration records for channels 1-2-5-6-7-8

Acceleration (cm/s 2)

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Time (sec) Fig. 12. Recorded accelerations for Jan 13, 2001 earthquake.

model of the dam–reservoir–foundation system. The acceleration records corresponding to the radial channels on the dam body during the earthquake are shown in Fig. 12. Eight thousand and two hundred points were recorded with a sampling period Δt ¼ 0:005 s in a 41 s time interval. These records were ambient seismic responses. The responses were divided into segments of 2048 points with a 50 percent overlap in order to calculate the power spectral density matrix and extract modal parameters. Considering the results of Section 3.1, the PHAT- β method was used to determine the time delays between the sensors of the Pacoima dam at the crest and at 80% height levels. The estimated delays are listed in Table 3. The time delays were calculated using the records of channels 1, 2 and 5 along the crest level and channels 6, 7 and 8 located at the 80% height level. For this, the time histories from the 13 Jan 2001 earthquake were used. The time delays shown in Table 3 are used to modify the power spectral density in order to eliminate the errors due to non-synchronous sensing. 5.2.1. Identification of frequencies The singular values spectrum and stabilization diagram related to seismic data from channels 1, 2 and 5 are presented in Figs. 13 and 14, respectively. The natural frequencies extracted from the first singular value spectrum and averaged coherence spectrum of the responses are shown in Fig. 15. As mentioned previously, the peaks of the first singular value spectrum were investigated by using the stabilization diagram. The peaks identified by the green and red points in Fig. 14 correspond to true and spurious modes, respectively. Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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Table 3 Time delays estimated by the PHAT-β for Pacoima dam. Time delay (s) Crest level

80% height level τ1;5  0.03

τ1;2  0.01

τ2;5  0.01

τ6;7  0.01

τ6;8  0.05

SV of PSD matrices

40

singular value 1 singular value 2 singular value 3

20 Singular Values [dB]

τ7;8  0.005

0 -20 -40 -60 -80

0

5

10

15

20 25 30 frequency (Hz)

35

40

45

50

Fig. 13. Singular values of the cross power spectral density matrix for the responses recorded by channels 1, 2 and 5.

Stabilization Diagram

100 90

Order of Model

80 70 60 50 40 30 20 Unstable Pole Freq. Stabilization (Semi-Stable Pole) Freq. & Damping Stabilization (Stable Pole)

10 0

0

1

2

3

4

5 6 Frequency (Hz)

7

8

9

10

[ACS]

2 2

SV1 [(cm/s ) /Hz]

Fig. 14. Stability diagram obtained from records of channels 1, 2 and 5; blue circles are the stable poles, green circles are the semi-stable poles and red points are unstable poles. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article).

First Singular Value 150 100 50 0

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6

7

8

9

10

7

8

9

10

Averaged Coherence Spectrum 0.9 0.8 0.7 0

1

2

3

4

5

6

Frequency(Hz) Fig. 15. First singular value (up) and averaged coherence (down) spectra of channel 1, 2 and 5 records; green points are true peaks according to stabilization diagram. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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The singular values spectrum and stabilization diagram related to the records from sensors at the 80% height level are shown in Figs. 16 and 17. First singular value and averaged coherence spectra are presented in Fig. 18. Natural frequencies extracted from the responses at the two different levels using the uncorrected and corrected CPSDs are presented in Tables 4 and 5, respectively. Good agreement is obtained for the natural frequencies extracted from the records at both levels. Only the first six frequencies appeared in the singular value spectra of the records from both levels. Therefore, the two last frequencies (6.177 and 6.714 Hz) related to the crest level were neglected. The results for corrected case are presented in Table 5. Comparison of the results presented in Tables 4 and 5 indicate that modification of the power spectral density function does not significantly affect the natural frequencies. SV of PSD matrices 20

singular value 1 singular value 2 singular value 3

Singular Values [dB]

0 -20 -40 -60 -80 -100

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frequency (Hz) Fig. 16. Singular values of the cross power spectral density matrix for responses recorded by channels 6, 7 and 8.

Stabilization Diagram 100 90 80

Order of Model

70 60 50 40 30 20 Unstable Pole Frequency Stabilization (Semi-Stable Pole) Frequency-Damping Stabilization (Stable Pole)

10 0

0

1

2

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5

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7

8

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Frequency (Hz)

[ACS]

2 2

SV1 [(cm/s ) /Hz]

Fig. 17. Stability diagram obtained from records of channels 6 and 7; blue circles are the stable poles, green circles are the semi-stable poles and red points are unstable poles. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

First Singular Value

15 10 5 0

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5

6

Frequency(Hz) Fig. 18. First singular value (up) and averaged coherence (down) spectra of channels 6, 7 and 8 records; green points are true peaks according to stabilization diagram. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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Table 4 Frequencies of Pacoima dam for uncorrected case. Natural frequencies (Hz) Modes

Crest

80% H

jErrorj

Freq:ave:

MACave:

1 2 3 4 5 6 7 8

3.442 3.687 4.150 4.419 4.871 5.212 6.177 6.714

3.357 3.650 4.126 4.382 4.834 5.188 – –

0.085 0.037 0.024 0.037 0.037 0.024 – –

3.399 3.668 4.138 4.400 4.853 5.200 – –

Surious 99.967 Spurious Spurious 98.676 99.776 – –

Table 5 Frequencies of Pacoima dam for corrected case. Natural frequencies (Hz) Modes

Crest

80% H

jErrorj

Freq:ave:

MACave:

1 2 3 4 5 6 7 8

3.442 3.687 4.150 4.419 4.871 5.212 6.177 6.714

3.357 3.650 4.114 4.382 4.834 5.188 – –

0.085 0.037 0.036 0.037 0.037 0.024 – –

3.399 3.668 4.132 4.400 4.853 5.200 – –

Spurious 99.964 Spurious Spurious 98.707 99.751 – –

+ Fig. 19. Positive and negative directions of mode shapes on the dam body.

Uncorrected

Corrected

Modal shape # 1

Modal shape # 2

Modal shape # 3

Modal shape # 4

Modal shape # 5

Modal shape # 6

Fig. 20. Estimated mode shapes for six vibration modes from crest level recordings (Modes 2, 5 and 6 are true).

5.2.2. Identification of mode shapes The positive and negative directions for mapping the mode shapes on the dam body are indicated in Fig. 19. The constant time shift (time delay) obtained from PHAT-β method removes the errors from the extracted mode shapes. Mode shapes extracted from the records at the crest and 80% height levels are illustrated in Figs. 20 and 21 for both the uncorrected and corrected cases, respectively. Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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The figures indicate that correcting for non-synchronous effects leads to considerable changes in some extracted mode shapes (modes 4–6). When no correction is performed, mode shapes extracted from records at both levels are in phase.

5.2.3. Identification of damping ratios Figs. 22–24 indicate the approach of damping extraction in the FDD-WT method by LD technique. The average damping ratio obtained from channels 1, 2 and 5 records was 2.70% for the first true mode. As previously seen, the natural frequencies obtained from the original and corrected CPSD functions were equal or very close to each other and it can be assumed that the damping ratios obtained from these two cases would also be similar. Therefore, the damping ratios were estimated only for the corrected case. The damping ratios corresponding to the six identified modes are presented in Table 6. According to Table 6, the average of damping ratio for six vibration modes is 2.64%.

6. Results verification 6.1. Based on 4-spectral method In this section, the classical 4-spectral method is used to re-identify the modal parameters of the arch dam. This method is not able to extract the mode shapes and damping ratios when modal interference exists. Therefore, the 4-spectral method is only used to extract the natural frequencies in order to compare with results of FDD-WT method. The records of channels 1, 2, 5 and 9 were considered as output data and the record of channel 9 was selected as the reference. The responses were divided into segments of 2048 points with a 50 percent overlapping. The results related to this method are shown in Figs. 25–27.

Uncorrected

Corrected

Modal shape # 1

Modal shape # 2

Modal shape # 3

Modal shape # 4

Modal shape # 5

Modal shape # 6

Fig. 21. Estimated mode shapes for six vibration modes from 80% height level recordings (Modes 2, 5 and 6 are true).

Percentage of energy for each wavelet coefficient

-4

x 10

3.96

18

Mirror

Frequency (Hz)

3.86

16

Unstable region

End effect

3.76

14 12 10

3.66

8

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6 4

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-6

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8

10

Time (sec) Fig. 22. CWT of autocorrelation function related to crest level channels and surveys of ridges by FDD.

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Real [W(a,t)]

5

single frequency signal: 3.668Hz

x 10

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Ln [Amplitude]

Fig. 23. The skeleton obtained from the autocorrelation function of channels 1, 2 and 5 for the first true mode (Mode2: 3.668 Hz). 65.5

zeta : 2.70 %

65 64.5

line equation : -0.622 t + 66.703

64 63.5 2.5

3

3.5

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4.5

5

time (sec) Fig. 24. Linear regression of the autocorrelation amplitude related to the channels 1, 2 and 5 (Mode2: 3.668 Hz).

Table 6 Damping ratios extracted using the LDT. Damping ratio (%) Channels 1, 2 and 5

Channels 6, 7 and 8

Average damping

1 2 3 4 5 6

2.85 2.70 2.66 2.60 2.47 2.41

2.92 2.78 2.72 2.62 2.48 2.40

2.89 2.74 2.69 2.61 2.48 2.41

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Fig. 25. Acceleration record of channel 1; SXX: PSD amplitude of channel 1, SXY: CPSD amplitude of channel 1 relative to the reference channel (channel 9), CXY: CS amplitude related to channels 1 and 9, PXY: CPSD phase of channels 1 and 9.

The peaks illustrated by solid lines in the figures are clear and strong in all spectra and are selected as the natural frequencies. The other peaks that are indicated by dashed lines are weak or invisible in some of the spectra; nonetheless, they may be natural frequencies of the structure. The natural frequencies obtained from the 4-spectral and FDD-WT methods are presented in Table 7. There is a good agreement between the values obtained from the 4-spectral and the FDD-WT. In Figs. 25–27, the peaks related to the frequencies 6.0–6.2 Hz and 6.6–6.8 Hz are weak in most of the spectra and are not presented in Table 7. Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

SXX

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Fig. 26. Acceleration record of channel 2; SXX: PSD amplitude of channel 2, SXY: CPSD amplitude of channels 2 relative to the reference channel (channel 9), CXY: CS amplitude related to channels 2 and 9, PXY: CPSD phase of channels 2 and 9.

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Fig. 27. Acceleration record of channel 5; SXX: PSD amplitude of channel 5, SXY: CPSD amplitude of channel 5 relative to the reference channel (channel 9), CXY: CS amplitude related to channels 5 and 9, PXY: CPSD phase of channels 5 and 9. Table 7 Comparison of natural frequencies identified by the 4-spectral and FDD-WT methods. Vibration mode

Natural frequency (Hz)

1 2 3 4 5 6

4-spectral

FDD-WT

3.3–3.4 3.6–3.7 4.1–4.2 4.4–4.5 4.8–5.0 5.1–5.3

3.399 3.668 4.132 4.400 4.853 5.200

Table 8 Natural frequencies of Pacoima Dam–reservoir–foundation system. Modes

1 2 3

Natural frequencies (Hz) FDD-WT

MODE-ID (by Alves)

3.668 4.853 5.200

– 4.73 5.06

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Modal shape # 2: 4.853 Hz

Modal shape # 3: 5.200 Hz

Fig. 28. Mode shapes estimated by extended FDD-WT method for second and third true modes.

Fig. 29. Symmetric and antisymmetric mode shapes estimated by MODE-ID [32].

Fig. 30. Symmetric and antisymmetric mode shapes computed from the linear SCADA calibrated model corresponding to the MODE-ID results [32].

Table 9 Experimental and analytical frequencies of the Pacoima dam. Mode

1 2 3

Natural frequencies (Hz) FDD-WT

4-spectral

MODE-ID (by Alves)

SCADA (by Alves)

3.668 4.853 5.200

3.6–3.7 4.8–5.0 5.1–5.3

– 4.73 5.06

– 4.82 5.02

6.2. Based on results of the other literatures In this study, the natural frequencies obtained from the FDD-WT method are compared with the results of MODE-ID, a program developed by James Beck [32], in Table 8. Documentation on MODE-ID can be found in Beck and Jennings [33] and Werner et al. [34]. The results indicate that two identified modes by the MODE-ID program correspond to the fifth and sixth modes obtained from the FDD-WT method (second and third true modes), whereas the first true mode identified by the FDD-WT (3.668 Hz) has been not determined by Alves (based on MODE-ID program) [32]. This problem can be due to the use of different techniques in the FDD-WT (output-only) and MODE-ID (input-output) programs or arbitrary selection of the user. On the other hand, according to the results of the 4-spectral method, the fifth and sixth modes have a higher certainty than the other modes and therefore these two modes were considered to be the conclusive modes. Damping values of the overall dam–reservoir–foundation system obtained using the MODE-ID were 6.2% and 6.6% for the first symmetric and anti-symmetric vibration modes, respectively (see Fig. 29) and the average damping ratio was 6.4%. On the other hand, the average damping ratio obtained from the FDD-WT method was 2.64% for the dam structure. When combined with the damping ratio of 4% for the rock [32], damping value of the dam–water–foundation rock system is assumed to be 6.64% for all vibration modes. The mode shapes obtained from the FDD-WT, MODE-ID and SCADA are presented in Figs. 28–30, respectively. In general, the comparison of mode shapes presented in Figs. 28–30 indicates good agreement between the results of the different programs. Experimental and analytical frequencies of the updated model from the SCADA program are given in Table 9. Comparison of the results indicates that values obtained from extended FDD-WT are correct and reliable. The SCADA program has determined only the two fundamental modes and the first natural frequency is not obtained. The reason of this discrepancy can be as follows: Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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 Smeared Crack Arch Dam Analysis (SCADA) is a finite element analysis program for modeling concrete arch dams. The smeared crack method used in the program does not model the discrete interface behavior of joints and cracks [32].

7. Conclusions An effective approach was used to estimate the modal parameters of an arch dam with emphasis on correction of errors due to non-synchronicity sensing of sensors. This work is an extension of the FDD-WT algorithm that was recently published in the journal of sound and vibration by the authors [12]. The purpose of the extension is correction of the errors resulting from asynchronous sensing of sensors, in order to extend the application of the algorithm for different kinds of structures especially for huge structures. In this research, ability of the various TDE methods was evaluated by using a numerical example (3DOF system). Finally, the PHAT- β technique that is able to avoid spreading of the peak of the correlation function was selected in order to ensure a large sharp peak in the calculated cross-correlation function. The time delays obtained from the technique were used to modify the CPSD function used in system identification method. The extended FDD-WT method that is based on output only was used for identifying the modal parameters of the case study (Pacoima dam). Comparison of the values obtained from extended FDD-WT method with results from the MODE-ID, SCADA and classical 4-spectral method indicates that the agreement is good. From these comparisons, it was concluded that the proposed approach is able to determine the modal parameters accurately. The results presented in Section 5 revealed that disregarding non-synchronous effects (time delay) leads to the errors in the extracted modal parameters, especially in mode shapes. Therefore, time delays obtained from the PHAT- β function of the various records were used to estimate the parameters more accurately.

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Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i

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Please cite this article as: R. Tarinejad, M. Damadipour, Extended FDD-WT method based on correcting the errors due to non-synchronous sensing of sensors, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.10.032i