An Approach for Time Synchronization of Wireless Accelerometer Sensors Using Frequency-Squeezing-Based Operational Modal Analysis

Wireless sensor networks usually suffer from the issue of time synchronization discrepancy due to environmental effects or clock management collapse. This will result in time delays between the dynamic responses collected by wireless sensors. If non-synchronized dynamic response data are directly used for structural modal identification, it leads to the misestimation of modal parameters. To overcome the non-synchronization issue, this study proposes a time synchronization approach to detect and correct asynchronous dynamic responses based on frequency domain decomposition (FDD) with frequency-squeezing processing (FSP). By imposing the expected relationship between modal phase angles extracted from the first-order singular value spectrum, the time lags between different sensors can be estimated, and synchronization can be achieved. The effectiveness of the proposed approach is fully demonstrated by numerical and experimental studies, as well as field measurement of a large-span spatial structure. The results verify that the proposed approach is effective for the time synchronization of wireless accelerometer sensors.


Introduction
Structural health monitoring (SHM) systems have been widely implemented in a variety of infrastructures to provide continuous and detailed information to decisionmakers [1,2]. The functionalities of SHM systems are mainly composed of acquiring structural responses, extracting structural features, and assessing structural conditions [3][4][5][6]. In the process, SHM-derived knowledge on structural condition assessment will be affected if the structural dynamic properties are extracted from non-synchronized measurements. For example, a 30 µs synchronization error results in a noticeable error in the modal analysis [7]. This non-synchronization-induced misjudgment will impact the subsequent analysis of an SHM procedure. Therefore, the synchronization of dynamic measured data from different sensors should be guaranteed. Synchronization discrepancy hardly occurs or it can be easily eliminated by multiple linked data acquisition units (DAU) in a wired sensor network. However, the traditional wired SHM system may become impractical for large-scale civil structures due to strict power supply conditions and large investments in labor and material resources [8][9][10][11]. With the rapid development of wireless communication techniques, wireless sensor networks (WSNs) have been developed to alleviate these limitations. In WSNs, the data transfer speeds of wireless nodes may be different because of the limited bandwidth and low-power radio transceivers [12]. Furthermore, although modern WSNs use clock-management techniques, non-simultaneity in sensor start-up can also result in the time non-synchronization issue. In particular, for passive and low duty cycle wireless sensor nodes, internal clock validated using the field-measured non-synchronous dynamic data of a cable-net structure subjected to strong wind.

Time Synchronization Approach
Generally, the dynamic behavior of civil structures is described as a linear system with a light and proportional damping assumption. Under this assumption, the mode shapes of structures can be accurately extracted from synchronized measurements. The different degrees of freedom (DOFs) reach the furthest and the equilibrium position simultaneously. Accordingly, the mode shape components of one certain mode extracted between any two synchronous signals lie on the real axis in a complexity plot, i.e., the phase angles are equal to 0 • (in-phase) or 180 • (out of phase) [29]. When these two dynamic response measurements are non-synchronous, there is a mapping relationship between the relative lag and their phase angles. Then, the delays between the signals can be obtained based on the deviation between the actual and ideal phase angles. The framework of the presented synchronization algorithm is shown in Figure 1. by numerical simulation and experimental study. Subsequently, the practicality of the approach is further validated using the field-measured non-synchronous dynamic data of a cable-net structure subjected to strong wind.

Time Synchronization Approach
Generally, the dynamic behavior of civil structures is described as a linear system with a light and proportional damping assumption. Under this assumption, the mode shapes of structures can be accurately extracted from synchronized measurements. The different degrees of freedom (DOFs) reach the furthest and the equilibrium position simultaneously. Accordingly, the mode shape components of one certain mode extracted between any two synchronous signals lie on the real axis in a complexity plot, i.e., the phase angles are equal to 0° (in-phase) or 180° (out of phase) [29]. When these two dynamic response measurements are non-synchronous, there is a mapping relationship between the relative lag and their phase angles. Then, the delays between the signals can be obtained based on the deviation between the actual and ideal phase angles. The framework of the presented synchronization algorithm is shown in Figure 1.  Considering two non-synchronous responses y 1 (t) and y 2 (t − τ 12 ) collected by WSNs, where τ 12 denotes the relative time lag between two responses, their Fourier transforms are −∞ e −iωt y 2 (t − τ 12 )dt = e iωτ 12 Y 2 (ω) (2) where ω and i are the circular frequency and the imaginary unit, respectively. The cross power spectral density (CPSD) is a fundamental tool for modal identification, which measures the distribution of power for the pair of signals across a frequency spectrum. Through CPSD, the relationship between these two time-domain signals can also be expressed as where the superscript * refers to the complex conjugate operator. Comparison of Equations (2) and (3) show that the time-delayed response results in a rescaling within the frequency domain by multiplying with e iωτ 12 . Correspondingly, it also describes that the time lags will lead to a shifted phase θ 12 in a polar form where θ 12 = ωτ 12 . Therefore, the relationship between the two outputs can be extended to the frequency spectra for estimation of the relative time lags. For better illustration, dynamic monitoring data of one channel are selected as a reference, then the delays between the referenced channel and the rest channels can be uniquely quantified. The vectors of the rest channels can be written as where τ 12 , · · · , τ 1g are the time lags between the referenced and rest channel, and N is the length of the signal, g is the number of the output channel; Similar to Equation (3), the CPSD matrix is introduced as follows.
Assuming that one mode is dominant at the frequency ω (k) p associated with the resonance of k-th mode. Then, by taking the singular value decomposition (SVD) of the CPSD matrix, the CPSD matrix can be approximated to a 1-rank matrix, rewritten as where the superscript H refers to the complex conjugate transpose operator; and u 1 is the first singular vector representing the estimation of the k-th mode shape where Φ (k) is the k-th mode shape extracted from the ideal synchronization signals. Considering the assumption of proportional damping, the mode shape vectors are real-valued.
Without the loss of generality, suppose that φ and Φ (k) , respectively. Their one-to-one relationship between φ (k) 1n is conducted, given by (8) Accordingly, the time lag τ 1n can be written as follows where sgn means the sign function; and θ (k) 1n denotes the shifted phase in one period (T (k) p ) between the referenced channel and the rest channel n under the k-th mode, that is θ n . Additionally, the phase angle for the k-th mode shape vector can be obtained Considering that the lag τ 1n may exceed the period, Equation (9) is rewritten in a generalized form Similarly, a set of g − 1 equations with (2g − 2) unknown parameters k are obtained: Obviously, Equation (12) is definitely underdetermined because the number of unknowns (2g − 2) exceeds the number of equations (g − 1). Hence, additional information needs to be introduced. Suppose that M (M > 1) modes have been identified by FDD technique [32]. A shifted-phase matrix is built as Accordingly, rewriting Equation (12) in matrix form yields where the superscript in Equation (14) denotes the number of the identified mode; and ω (1) p , · · · , ω (M) p are the selected frequencies associated with the peak of resonance of the identified modes. Since the values k (i) 1j can only be taken in the integer domain, this greatly narrows the scope of the solution. For each row of the matrix in Equation (14), through a series trial of k (i) 1j where j ∈ [1, · · · , g] and i ∈ [1, · · · , M], then, a candidate pool for the actual time lags can be easily determined aŝ Γ = τ 12 , · · · ,τ 1g (15) Obviously, there exists more than one possible candidate for each relative time lag. Therefore, it necessitates a solution to determine the optimal time lags. The final time lag can be estimated based on the lowest standard deviation of Equation (15). In other words, the lags are often around the expectation of the set of final lags, which yields the lowest standard deviation. The actual lags τ 12 , · · · , τ 1g can be estimated as follows It is noteworthy that the accuracy of the identified natural frequencies and nonsynchronous mode shapes play a primary role in lag estimation. However, in the process of estimating the CPSD by FDD, the sampled signal duration is limited and accompanied by various noises. When the SNR is low, there are multiple potential candidate peaks caused by the noise, which increases the uncertainty in the peak selection. As a result, the presence of noise affects the estimation of the modal parameter, especially in weak excitation. Thus, it is necessary to minimize the influence of the noise.
Learning from the stabilization diagram [32], the actual modes can be identified from alignments of stable poles since the spurious modes tend to be more scattered when increasing model orders. Given this, this study introduces frequency-squeezing to improve the readability of the power spectrum/singular value (SV) spectrum representation. The FSP is based on shifting the local spectrum shape to its nearby natural frequency without changing its magnitude [33]. A schematic diagram of FSP is depicted in Figure 2, which consists of three main steps detailed as follows.  Step 1: Amplitude pretreatment of the first-order singular spectrum. First, the spectrum amplitude is normalized to [0, 1]. Then, the amplitude is "shaped" by taking the mth power of the normalized amplitude for subsequent processing. The amplitude pretreatment can be written as Step 1 ：Amplitude pre-treatment Step 2 ：Frequency-squeezing Step 3 ：Amplitude restoration Step 2 ：Frequency-squeezing  Step 1: Amplitude pretreatment of the first-order singular spectrum. First, the spectrum amplitude is normalized to [0, 1]. Then, the amplitude is "shaped" by taking the m-th power of the normalized amplitude for subsequent processing. The amplitude pretreatment can be written as α(ω i ) = α(ω i )/max(α(ω i )) i = 1, 2, . . . , K where α(ω i ) andα(ω i ) are the spectral amplitudes before and after processing, respectively; ω i is the sampling frequency point from a vector ω = [ω 1 , ω 2 , · · · , ω K ]; m can be set as integer multiples of 10 to reduce rapidly the amplitude of the peak nearby.
Step 2: Frequency-squeezing for the pretreated signal. Considering the continuous 2p + 1 (p ≥ 1) spectral linesα(ω i−p ), · · · ,α(ω i ), · · · ,α(ω i+p ), the frequencyω i is replaced by the centroid coordinate of a graph, which is composed of these spectral lines and the frequency ω i−p , · · · , ω i , · · · , ω i+p , given bŷ where p and K are the user-specified step and the signal length, respectively. Then, repeat the step until the convergence criterion is satisfied. The convergence criterion is defined as.
where s is the number of iteration.
Step 3: Amplitude restoration and zero settings. Since the magnitude of the amplitude is normalized in Step 1, the accurate amplitude information should be retained. The original amplitude vector is assigned to the newly generated frequency vector in the order of subscripts, and the amplitude between the edges and the cluster of aggregated frequency points is set to zero, which can be written aŝ where Ω is the set of frequency subscripts corresponding to the set of zero amplitude. δ ω < ∆ω = ω i+1 − ω i is the indicator to determine the abnormal frequency, which is suggested to be set as 0.01 or 0.001 times ∆ω.
In conclusion, the FSP technique artificially changes the orthogonality characteristics of the basis vector after the Fourier transform of the signal. It highlights the natural frequency, which can serve as the referenced frequency for peak selection in the FDD method.

Structural Description
A linear time-invariant model of a four-story building ( Figure 3) is used as tested. Each floor is represented as masses m i (i ∈ (1, · · · , 4)) interconnected with springs k i and dampers c i . The weight of each mass, the constants of lateral shear stiffness k i , and the damping coefficients c i between adjacent floors are 10 kg, 1000 kg/m, and 10 N·s/m, respectively. The mass matrix M, stiffness matrix K, and damping matrix C can be expressed as respectively. The mass matrix M, stiffness matrix K, and damping matrix C can be expressed as    Each floor is excited by a stationary, zero-mean, Gaussian white noise. By adding state noise N(0, 0.01) and output noise N(0, 0.001) into the structure, then the structural responses under white noise excitation are simulated (SNR = 20 dB). All the responses have a duration of 100 s and the sampling rate is 50 Hz. Suppose that each floor has an independent acquisition unit for response collection. Four sets of dynamic responses with different lags are set intentionally to assess the impact of the non-synchronization on modal identification. The first channel is set as the referenced channel, and the relative time delays of other channels are shown in Table 1, where the positive sign indicates that the time is behind the reference timeline and vice versa.  Figure 4 shows the unprocessed acceleration response measurements. For showing the impact of time delay on the mode shapes, theoretical mode shapes computed with synchronous data are introduced. The modal phase angles obtained from the theoretical mode shapes and the ones identified by FDD using non-synchronous data are plotted in the polar form shown in Figure 5. It is noteworthy that the phase of the theoretical mode shapes (red dash lines) lies on nearly straight lines, as expected. The phases of the 1st mode are moving in phase whereas the rest of modes are moving out of phase. However, the mode shapes identified from non-synchronous data are highly complex, which could lead to wrong conclusions such as high levels of nonlinearities or large damping. Meanwhile, by a complex-to-real conversion of mode shapes, it is found that the amplitude of identified mode shapes is smaller than the theoretical results. One primary reason for it is that the amplitudes are rescaled by a factor caused by phase shift. Therefore, the time delay in dynamic measurement greatly affects the identification of the modal parameters.
In order to find actual lags, Figure 6 displays the first-order singular value obtained from FDD-FSP. Recalling Equation (20), the computational parameters are as follows: the step size (2p + 1) is set as 121, the order of exponentiation (m) is set as 50, the frequency convergence threshold (δ) is set as 1 × 10 −6 , and the total iteration number is 1000. The spectrum is concentrated at the true position of natural frequency. The advantage of FSP is the reduction of distortion in the target frequency pickup and the improved estimation accuracy of the delay (Equation (14)). To exemplify this, a zoomed-in view of Figure 6 is shown. It is clear that the original first-order spectral line moves to the target peak where the 1st to 4th order frequencies are well-reflected.
After performing the FSP-FDD, the candidate pool of lags can be easily conducted. Taking an explanatory example of the relative delay between channel 1 and channel 2, the candidate lags can be written as 12 through trial computation, respectively, the relative delay τ 12 is −0.2350, −0.2379, −0.2382, −0.2403, which has the lowest standard deviation of the relative delay set. And the mean of this set (−0.2378) is nearly equal to the preset delay (−0.2400). Considering that the relative time lag should satisfy the integral multiple of sampling interval (0.0200 s), the time lag is obtained as −0.2400 s. Similarly, the detected lag of the rest of channels is also solved and shown in Figure 7. Finally, the modal parameters are re-identified using the realigned dynamic response (Figure 8). The modal assurance criterions (MACs) between the mode shapes obtained from the realigned and synchronous data can be written as where Φ t i and Φ m j refers to the mode shape vector extracted from the realigned and the previous synchronous responses, respectively. The MACs of the mode shapes are near 1 (Figure 8), which shows that the mode shapes obtained by the processed data match well with the theoretical ones. (0, 0.001) N into the structure, then the structural responses under white noise excitation are simulated (SNR = 20 dB). All the responses have a duration of 100 s and the sampling rate is 50 Hz. Suppose that each floor has an independent acquisition unit for response collection. Four sets of dynamic responses with different lags are set intentionally to assess the impact of the non-synchronization on modal identification. The first channel is set as the referenced channel, and the relative time delays of other channels are shown in Table 1, where the positive sign indicates that the time is behind the reference timeline and vice versa.  Figure 4 shows the unprocessed acceleration response measurements. For showing the impact of time delay on the mode shapes, theoretical mode shapes computed with synchronous data are introduced. The modal phase angles obtained from the theoretical mode shapes and the ones identified by FDD using non-synchronous data are plotted in the polar form shown in Figure 5. It is noteworthy that the phase of the theoretical mode shapes (red dash lines) lies on nearly straight lines, as expected. The phases of the 1st mode are moving in phase whereas the rest of modes are moving out of phase. However, the mode shapes identified from non-synchronous data are highly complex, which could lead to wrong conclusions such as high levels of nonlinearities or large damping. Meanwhile, by a complex-to-real conversion of mode shapes, it is found that the amplitude of identified mode shapes is smaller than the theoretical results. One primary reason for it is that the amplitudes are rescaled by a factor caused by phase shift. Therefore, the time delay in dynamic measurement greatly affects the identification of the modal parameters.  In order to find actual lags, Figure 6 displays the first-order singular value obtained from FDD-FSP. Recalling Equation (20), the computational parameters are as follows: the step size (2p + 1) is set as 121, the order of exponentiation (m) is set as 50, the frequency convergence threshold (δ) is set as 1 × 10 −6 , and the total iteration number is 1000. The spectrum is concentrated at the true position of natural frequency. The advantage of FSP is the reduction of distortion in the target frequency pickup and the improved estimation accuracy of the delay (Equation (14)). To exemplify this, a zoomed-in view of Figure 6 is shown. It is clear that the original first-order spectral line moves to the target peak where the 1st to 4th order frequencies are well-reflected. After performing the FSP-FDD, the candidate pool of lags can be easily conducted. Taking an explanatory example of the relative delay between channel 1 and channel 2, the candidate lags can be written as

Structural Description
As shown in Figure 9, the shake table test model of a five-floor steel frame is utilized to further demonstrate the effectiveness of the proposed approach. The geometrical and material properties of this structure are: the floor height is h = 300 mm, the cross-section of the columns is A = 50 × 5 mm, the elastic modulus E = 206 GPa, the Poisson's ratio υ = 0.31, and the mass density is 7850 kg/m 3 . Each floor consists of two steel plates with a size of 300 × 300 × 20 mm, connected to the columns by eight angle-iron brackets. The layout of the wireless acceleration sensors (WASs) is also depicted in Figure 9. The channel number of these WASs is the same as the floor numbers. The DAU contains two 3-channels and a 24-bit analog-to-digital conversion (ADC). This frame was excited by the Hollister earthquake [34]. The sampling frequency was set to 128 Hz. The data collected during the

Structural Description
As shown in Figure 9, the shake table test model of a five-floor steel frame is utilized to further demonstrate the effectiveness of the proposed approach. The geometrical and material properties of this structure are: the floor height is h = 300 mm, the cross-section of the columns is A = 50 × 5 mm, the elastic modulus E = 206 GPa, the Poisson's ratio υ = 0.31, and the mass density is 7850 kg/m 3 . Each floor consists of two steel plates with a size of 300 × 300 × 20 mm, connected to the columns by eight angle-iron brackets. The layout of the wireless acceleration sensors (WASs) is also depicted in Figure 9. The channel number of these WASs is the same as the floor numbers. The DAU contains two 3-channels and a 24-bit analog-to-digital conversion (ADC). This frame was excited by the Hollister earthquake [34]. The sampling frequency was set to 128 Hz. The data collected during the

Structural Description
As shown in Figure 9, the shake table test model of a five-floor steel frame is utilized to further demonstrate the effectiveness of the proposed approach. The geometrical and material properties of this structure are: the floor height is h = 300 mm, the cross-section of the columns is A = 50 × 5 mm, the elastic modulus E = 206 GPa, the Poisson's ratio υ = 0.31, and the mass density is 7850 kg/m 3 . Each floor consists of two steel plates with a size of 300 × 300 × 20 mm, connected to the columns by eight angle-iron brackets. The layout of the wireless acceleration sensors (WASs) is also depicted in Figure 9. The channel number of these WASs is the same as the floor numbers. The DAU contains two 3-channels and a 24-bit analog-to-digital conversion (ADC). This frame was excited by the Hollister earthquake [34]. The sampling frequency was set to 128 Hz. The data collected during the warming up of the shake table is discarded, and total 15,360 discrete data during the earthquake excitation were acquired. The main measurement responses are shown in Figure 10. warming up of the shake table is discarded, and total 15,360 discrete data during the earthquake excitation were acquired. The main measurement responses are shown in Figure  10.

Method Validation
As outlined in Table 2, three cases of relative time delays were artificially injected into the acceleration data, then the relative percentage error (RPE) between estimated time warming up of the shake table is discarded, and total 15,360 discrete data during the earthquake excitation were acquired. The main measurement responses are shown in Figure  10.

Method Validation
As outlined in Table 2, three cases of relative time delays were artificially injected into the acceleration data, then the relative percentage error (RPE) between estimated time

Method Validation
As outlined in Table 2, three cases of relative time delays were artificially injected into the acceleration data, then the relative percentage error (RPE) between estimated time lags and exact time lags also were calculated for evaluating the accuracy of time delay estimation.
Time delays in each case are estimated by using the proposed time synchronization approach. Although affected by random measurement noise, the first four modes can be easily identified through the reference peak position by FSP (Figure 11). The first-order spectrum is smoothly concentrated at the target frequencies. Then, the relationship between the candidate pool of relative time lags and its standard deviation is obtained by minimizing the standard deviation (Equation (16)), as is depicted in Figure 12. Time delays in each case are estimated by using the proposed time synchronizat approach. Although affected by random measurement noise, the first four modes can easily identified through the reference peak position by FSP (Figure 11). The first-or spectrum is smoothly concentrated at the target frequencies. Then, the relationship tween the candidate pool of relative time lags and its standard deviation is obtained minimizing the standard deviation (Equation (16)), as is depicted in Figure 12. As is shown in Figure 13, the value of the MAC matrix indicates that the mode sha obtained from the realigned responses are very similar to those obtained from the s chronous data. In particular, in the synchronous case (Case 3), the estimated lag is n zero. Hence, the relative lags of the experimental data are precisely estimated by the p posed approach, which validates the effectiveness of the proposed approach.  As is shown in Figure 13, the value of the MAC matrix indicates that the mode shapes obtained from the realigned responses are very similar to those obtained from the synchronous data. In particular, in the synchronous case (Case 3), the estimated lag is near zero. Hence, the relative lags of the experimental data are precisely estimated by the proposed approach, which validates the effectiveness of the proposed approach.

Description of NSSO and Its Monitoring System
To further investigate the performance of the presented approach, the field-measured data of the National Speed Skating Oval (NSSO) is adopted. The NSSO (Figure 14), located in the Beijing Olympic Park, China, was built for hosting the speed skating events during the 2022 Beijing Winter Olympics, with a span of 220 m × 153 m. It comprises four main parts: the saddle-shaped cable net, the mega ring truss, the concrete stand columns, and the stay cables. As shown in Figure 15

Description of NSSO and Its Monitoring System
To further investigate the performance of the presented approach, the field-measured data of the National Speed Skating Oval (NSSO) is adopted. The NSSO (Figure 14), located in the Beijing Olympic Park, China, was built for hosting the speed skating events during the 2022 Beijing Winter Olympics, with a span of 220 m × 153 m. It comprises four main parts: the saddle-shaped cable net, the mega ring truss, the concrete stand columns, and the stay cables. As shown in Figure 15, the cable net consists of stable cables and load-bearing cables and has a span of 200 m × 130 m.
A customized wireless SHM system designed by Zhejiang University Space Structure Center is implemented on the structure [35,36]. This wireless SHM system consists of more than 300 sensors. Each WAS, composed of a tri-axis accelerometer and a wireless unit, is deployed at the cables to obtain the modes of interest. An idle-wakeup mechanism is used in this wireless SHM system to reduce energy consumption. The measured data from all WAS is transmitted to the sink nodes by Long Range Transmission (LoRa), which is a proprietary low-power wide-area network modulation technique. Although this wireless system promotes high flexibility and less implementation cost, it also brings the time synchronization challenge. A customized wireless SHM system designed by Zhejiang University Space Structu Center is implemented on the structure [35,36]. This wireless SHM system consists of mo than 300 sensors. Each WAS, composed of a tri-axis accelerometer and a wireless unit, deployed at the cables to obtain the modes of interest. An idle-wakeup mechanism is us in this wireless SHM system to reduce energy consumption. The measured data from WAS is transmitted to the sink nodes by Long Range Transmission (LoRa), which is proprietary low-power wide-area network modulation technique. Although this wirele system promotes high flexibility and less implementation cost, it also brings the time sy chronization challenge.

Sensor Attitude Adjustment for Modal Identification
Acceleration measurements were automatically recorded by the monitoring syste during strong wind on 18 May 2021. The acceleration WAS-1, shown in Figure 16, w chosen as the reference. Note that these measured 3-dimensional accelerations conta gravity information. The mean components in the x-axis, y-axis, and z-axis of WAS-1 a −0.094 g, −0.105 g, and 1.003 g, respectively. However, the ideal components should be g, 0 g, 1 g when the sensor coordinate system coincides with the Earth coordinate system This indicates that the sensor attitude is changed from the instrumentation plan due  A customized wireless SHM system designed by Zhejiang University Space Structure Center is implemented on the structure [35,36]. This wireless SHM system consists of more than 300 sensors. Each WAS, composed of a tri-axis accelerometer and a wireless unit, is deployed at the cables to obtain the modes of interest. An idle-wakeup mechanism is used in this wireless SHM system to reduce energy consumption. The measured data from all WAS is transmitted to the sink nodes by Long Range Transmission (LoRa), which is a proprietary low-power wide-area network modulation technique. Although this wireless system promotes high flexibility and less implementation cost, it also brings the time synchronization challenge.

Sensor Attitude Adjustment for Modal Identification
Acceleration measurements were automatically recorded by the monitoring system during strong wind on 18 May 2021. The acceleration WAS-1, shown in Figure 16, was chosen as the reference. Note that these measured 3-dimensional accelerations contain gravity information. The mean components in the x-axis, y-axis, and z-axis of WAS-1 are −0.094 g, −0.105 g, and 1.003 g, respectively. However, the ideal components should be 0 g, 0 g, 1 g when the sensor coordinate system coincides with the Earth coordinate system. This indicates that the sensor attitude is changed from the instrumentation plan due to

Sensor Attitude Adjustment for Modal Identification
Acceleration measurements were automatically recorded by the monitoring system during strong wind on 18 May 2021. The acceleration WAS-1, shown in Figure 16, was chosen as the reference. Note that these measured 3-dimensional accelerations contain gravity information. The mean components in the x-axis, y-axis, and z-axis of WAS-1 are −0.094 g, −0.105 g, and 1.003 g, respectively. However, the ideal components should be 0 g, 0 g, 1 g when the sensor coordinate system coincides with the Earth coordinate system. This indicates that the sensor attitude is changed from the instrumentation plan due to the curvature changes of cable or installation deviation. The sensor attitudes can be corrected into the earth coordinate system by applying the coordinate transformation matrix [37] to improve the accuracy of the identified mode shapes. The proposed measurement responses after the sensor attitude correction are depicted in Figure 17. the curvature changes of cable or installation deviation. The sensor attitudes can be corrected into the earth coordinate system by applying the coordinate transformation matrix [37] to improve the accuracy of the identified mode shapes. The proposed measurement responses after the sensor attitude correction are depicted in Figure 17.  the curvature changes of cable or installation deviation. The sensor attitudes can be corrected into the earth coordinate system by applying the coordinate transformation matrix [37] to improve the accuracy of the identified mode shapes. The proposed measurement responses after the sensor attitude correction are depicted in Figure 17.

Analysis Results
As mentioned above, the time synchronicity cannot be secured with long-distance and multi-hop communication in the WSN system. The proposed approach is used to detect the relative time delays between different response channels.
The FE model of the NSSO was built to analyze the modal parameters, and the modal frequencies calculated from the FE model are listed in Table 3. As can be seen, this structure has a large number of closely-spaced modes, which makes it difficult to identify the modal parameters. To highlight the modes of interest, the acceleration data were down-sampled from 15.625 Hz to 2 Hz. There were a total of 2304 samples in each measurement. The identified frequencies of the first three dominant modes were 0.55 Hz, 0.71 Hz, 0.91 Hz, respectively, as shown in Figure 18. To better show the effectiveness of this proposed approach, two orthogonal vertical projections in the north-south and east-west directions of these mode shapes are introduced in Figure 20. The results show that the mode shapes extracted from the processed data are closer to the theoretical results than those obtained from the unprocessed data, which further demonstrates the practicality of the proposed time synchronization approach. The corresponding mode shapes were extracted from the proposed measurement responses. It is found that there is a difference between the theoretical and the identified modal frequencies, which may be caused by the stiffness degradation induced by the cable relaxation. Based on the prior knowledge of the WSNs, the relative lag ranges from −5 s to 5 s. The estimated time delay between the first output and the rest outputs was calculated in sequence as −0.6814 s, 2.8492 s, 3.7431 s, 1.1204 s, 2.2698 s, −0.7102 s, −0.9444 s, −0.2650 s, 0.9082 s by the presented approach. Then the time axis of the WASs was shifted according to the estimated time lags. Ideally, the mode shape components at a symmetric location of sensor placement should have approximately symmetric or anti-symmetric properties. Although a previous synchronization measurement response is best to serve as a reference for comparison, none of the responses are guaranteed to be synchronous due to an inborn deficiency of non-synchronization in such long-range transmission by WSNs. Therefore, the mode shapes calculated from the FE model were set as the reference. For comparison, the modes shapes extracted from the responses before and after the shifted time axis are plotted in Figure 19, along with the mode shapes obtained from the FE model. It can be seen that the mode shape seems to be erratic before the time axis shifts compared to the FE result. The MAC between the reference mode shapes and the mode shapes extracted from non-synchronous data is calculated to quantify the consistency (Figure 19a,c,e). Among the first three dominant modes, the maximum MAC is no more than 0.25, which indicates that the mode shapes identified by non-synchronous data are not correct. On the contrary, the mode shapes obtained from the data after shifting the time axis appear in a symmetric or anti-symmetric manner (Figure 19b,d,f), and the maximum MAC increases to about 0.9. To better show the effectiveness of this proposed approach, two orthogonal vertical projections in the north-south and east-west directions of these mode shapes are introduced in Figure 20. The results show that the mode shapes extracted from the processed data are closer to the theoretical results than those obtained from the unprocessed data, which further demonstrates the practicality of the proposed time synchronization approach. To better show the effectiveness of this proposed approach, two orthogonal vertical projections in the north-south and east-west directions of these mode shapes are introduced in Figure 20. The results show that the mode shapes extracted from the processed data are closer to the theoretical results than those obtained from the unprocessed data, which further demonstrates the practicality of the proposed time synchronization approach.

Conclusions
This study proposes a new time synchronization approach by extending the frequency domain decomposition (FDD) technique. When fed with asynchronous vibration measurements, this data-driven approach that is only based on output fulfills integrated

Conclusions
This study proposes a new time synchronization approach by extending the frequency domain decomposition (FDD) technique. When fed with asynchronous vibration measurements, this data-driven approach that is only based on output fulfills integrated estimation of time lags and identification of modal properties. The relative time lag identified by using lower modes can be regarded as a conservative estimate of the true relative time lag. The Frequency-squeezing processing (FSP) is used in the modal identification by FDD technique to reduce the influence of noise and to improve the readability of the power spectrum representation. A candidate pool of the lags is obtained, and the lags can be further determined by minimizing their standard deviation. Three cases of simulation, experimental test, and field measurement are employed to demonstrate and validate this approach, including the non-synchronous output of a four-story building subjected to white noise excitation, the misaligned acceleration measurements of a five-floor steel frame struck by the Hollister earthquake, and the non-synchronous dynamic record of the National Speed Skating Oval caused by a strong wind.
The application of this time synchronization approach presupposes that at least two modes need to be identified so that the relative time delay can be uniquely quantified. The accuracy of the time delay estimation is incrementally related to the higher modes obtained through the non-synchronous dynamic measurement responses. The analysis results of the presented three cases show that the proposed time synchronization approach is effective and helps improve the performance of modal identification in WSNs applications.