# Multiscale Time Series Decomposition for Structural Dynamic Properties: Long-Term Trend and Ambient Interference

Authors: Xiaolei Chu; Wei Cui; Shengyi Xu; Lin Zhao; Hua Guan; Yaojun Ge
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

In structural health monitoring (SHM), variations of structural dynamic properties are paramount to indicate the health status of structures. Structural dynamic properties are time variant due to various long-term effects (e.g., structural deterioration) and periodic effects (e.g., periodic variations of temperature, humidity, and traffic). Sometimes periodic effects will interfere with the quantification of long-term effects. Though important, there are still limited research studies aiming to distinguish these two effects. Given the amount of SHM data, it is possible to solve the issue from a data-driven perspective. This article proposes a time series decomposition methodology to divide the time series of structural dynamic properties into long-term parts, multiscale periodic parts, holiday parts, and error parts. We extract the 10-year dynamic properties of a long-span bridge using the fast Bayesian FFT identification algorithm and choose two dynamic modes of that bridge as examples to explain how our proposed methodology works. We use the long-term parts to extract the rules of structural deterioration. The multiscale periodic parts are utilized to find the relationships with the periodically varying ambient conditions (i.e., temperature and humidity) in different time scales (i.e., yearly, weekly, and daily). Then, the long-term and periodic effects can be distinguished. For the long-term effects, the modal frequencies tend to decrease but the damping ratios seem to increase. For the periodic effects, we find that the increment of temperature will lead to the decrease of both modal frequencies and damping ratios. The variations of modal frequencies induced by deterioration and temperature are of the same amplitudes. The variations of both modal frequencies and damping ratios are not significantly related with humidity. This article could provide references for damage detection and safety assessment for similar bridges.


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## Body

## 1. Introduction

The structural health monitoring (SHM) system of critical infrastructures is very important, which makes it possible for researchers to identify and analyze the structural dynamic properties continuously. Operational modal analysis is prevailing these years because it can analyze SHM data without disturbing the normal service of structures. There are multiple operational modal identification technologies that have been investigated for output-only systems, such as frequency-domain decomposition (FDD) [1], wavelet [2, 3], and stochastic subspace identification (SSI) [4]. Recently, Bayesian system identification [5] has made significant development, which viewed the modal identification as inference with plausible system models. For parameters that need to be inferred, Bayesian system identification can give both the most probable values (MPVs) and corresponding posterior uncertainty. Katafygiotis and Yuen [6] proposed a Bayesian spectral density approach for modal updating, where the power spectral density (PSD) of measured signals was used. Yuen and Katafygiotis [7] and Au [8, 9] developed the Bayesian FFT approach for modal updating, which directly utilized the statistical characteristics of the Fourier transform of measured signals. Both Bayesian spectral density approach and FFT approach have been successfully applied on issues of system identification [10–12]. In this article, we will use the Bayesian FFT approach to extract 10-year dynamic properties of the Xihoumen Bridge in China.Modal frequencies and damping ratios are important for structural integrity and safety. For example, the abnormal reduction of modal frequency is often regarded as an indicator of the loss of structural integrity, and Salawu [13] concluded that 5% reduction of modal frequency was a reliable sign for structural damage. As for damping ratio, it is the key parameter of long-span bridge’s wind-resistant capacity, especially for the performance of vortex-induced vibration [14–16]. The reduction of damping ratio may lead to large-amplitu devortex-induced vibration [14]. In addition, in current engineering practices, the designed service life for critical infrastructures (e.g., long-span bridges) is over hundred years. Considering all reasons listed above, it is paramount for researchers to investigate how dynamic properties will change with time. Nevertheless, recent studies revealed that the fluctuating environmental conditions could also induce the significant changes of dynamic properties, which are not caused by structural damage. For example, Yuen and Kuok [17] monitored a 22-story reinforced concrete building for one year and found that with the increase of ambient temperature, the modal frequency of buildings would increase. For long-span bridges, however, Li et al. [18], Mao et al. [3], Anastasopoulos et al. [19], and Hwang et al. [20] found that the increase of ambient temperature would decrease the modal frequency, which was opposite to the varying rule of buildings. In conclusion, both long-term deterioration (due to structural damage) and varying ambient conditions can influence the modal frequencies and damping ratios. It is important to distinguish the effects of structural deterioration from ambient interference, but few research studies were found since the complicated inherent mechanisms. Thanks to the accumulation of SHM data, however, it is possible to distinguish them from a data-driven perspective. It should be stressed that damping ratio is fairly complicated and related with many factors (not just varying ambient temperature and humidity). For example, damping ratio can also be influenced by aerodynamic effects [18, 21], soil-structure interaction [22], and man-made changes (e.g., maintenance of expansion joints and tuned mass damper [23]). Due to the limitation of the length, this study only focuses on the relationship between structural dynamic properties (modal frequencies and damping ratios) and ambient interference (temperature and humidity).There have been multiple applications of time series methods on the monitoring of civil infrastructures, mainly focusing on the detection of structural damage. For example, Omenzetter and Brownjohn [24] formulated a vector seasonal autoregressive integrated moving average (ARIMA) model to obtain information from the recorded strain signals and found that the changes in the ARIMA coefficients meant potential structural change and damage. Many research studies using the statistical characteristics of SHM data to classify and locate the damage are also investigated [25–28]. Explainable component analysis (i.e., long-term deterioration, seasonal fluctuation, and effects of special events) remains to be a formidable task either for researchers or bridge maintainers. With the proliferation of SHM data, however, it is possible to quantify different components. The Prophet model developed by Facebook [29] is a good tool with interpretable parameters that can be intuitively adjusted by analysts with domain knowledge. The Prophet model used a decomposable time series model [30] with three main components: trend (i.e., structural deterioration in SHM), seasonality (which can be used for the analysis of ambient interference), and holidays (which can be used for explaining the anomaly in special events). In this paper, we used the Prophet model to distinguish and quantify the structural deterioration and ambient interference (temperature and relative humidity [31]) of Xihoumen Bridge, where the recorded acceleration data from 2012 ∼ 2021 were employed.The contents of this paper are organized as follows: In section2, we briefly introduce the theoretical background of the fast Bayesian FFT identification algorithm and the layout of Xihoumen Bridge including the SHM systems; then, 10-year identified structural dynamic properties (modal frequencies and damping ratios) are presented. In section 3, we propose a multiscale time series decomposition methodology to distinguish the long-term part and periodic part of structural dynamic properties. In section 4 and section 5, we decompose the time series so that the effects of structural deterioration and ambient interference (relationships with temperature and relative humidity) are investigated. The amplitudes of the variations caused by structural deterioration and ambient interference are also investigated. In conclusion (section 7), this article employs a data-driven methodology to distinguish the effects of structural deterioration (i.e., structural damage) from ambient interference, which gives insights for future engineering practices.


## 2. Fast Bayesian FFT Identification Algorithm

### 2.1. Theoretical Background

If the acceleration time history measured atn DOFs of a structure is noted as x¨^j∈Rn:j=1,…,N and abbreviated as {x¨^j}, where N is the number of samples per channel, the FFT of {x¨^j} is defined as(1)Fk=Fk+iGk=2∆tN∑j=1Nx¨^jexp−2πik−1j−1N,k=1,…,N,where i2=−1, ∆t is the sampling interval, and Fk=ReFk and Gk=ImFk denote the real and imaginary part of the FFT, respectively. For k=2,3,…,Nq, the FFT corresponds to frequency fk=k−1/N∆t. Here, Nq = int[N/2] + 1 (int ⋅ denotes the integer part) corresponds to the FFT ordinate at the Nyquist frequency. For modal identification, only these Nq−1 FFT values are utilized.In the context of Bayesian inference, the measured acceleration is modeled asx..^j=x..jθ+ϵj, where x..jθ is the acceleration response of the structural model defined by the set of model parameters θ, the subject to be identified, and ϵj is the prediction error that accounts for the deviation between the model response and measured data, possibly owing to measurement noise and modeling error. Yuen and Katafygiotis [7] derived the joint PDF for the augmented FFT vectors {Zk=FkT,GkTT∈R2n:k=2,…,Nq} and applied it to Bayesian modal identification. For a high sampling rate and long duration of data, Zk is a zero-mean Gaussian vector with covariance matrix given by(2)Ck=12ΦReHkΦT−ΦImHkΦTΦImHkΦTΦReHkΦT+σ22I2n,where Φ∈Rn×m is the mode shape matrix confined to the measured DOFs (the i th column gives the i th mode shape), σ2 is the (constant) spectral density level of the prediction error, I2n denotes the 2n×2n identity matrix, and Hk is the spectral density matrix of the model response and its i,j entry is given by(3)Hki,j=Sijβik2−1+i2ζiβik−1βjk2−1−i2ζjβjk−1,where βik=fi/fk = frequency ratio, fi and ζi = natural frequency and damping ratio of the i th mode, respectively, and Sij = cross-spectral density between the i th and j th modal excitation.The set of modal parametersθ consists of modal frequencies, damping ratios, mode shapes, entries {Sij} of the spectral density matrix of modal excitations, and spectral density σ2 of the prediction error. Assuming a noninformative prior distribution, the posterior PDF of θ has given that FFT data are proportional to the likelihood function pZkθ(4)pθZk∝pZkθ=2π−Nq−1/2∏kdetCkθ−1/2×exp−12∑kZkTCkθ−1Zk,where the dependence of Ck on θ has been emphasized [32]. It is convenient to write with the negative log-likelihood function Lθ form(5)pθZk∝exp−Lθ,where(6)Lθ=12∑klndetCkθ+ZkTCkθ−1Zk.Then, the most probable value (MPV)θ^ is(7)θ^=argmaxθpθZk=argminθLθ.People can use simulated annealing [33], MCMC [34], or some other minimum-value searching methods to obtain θ^. Simulated annealing is employed in this paper. The posterior covariance of θ^ can also be quantified by Hessian matrix [32, 35], which is not the main focus of this paper and is not discussed here. Different from the traditional frequency-domain identification methods, the significant digits of the identified parameters here have no concern with the sampling duration. For example, the resolution of frequency in FFT analysis will be higher when the sampling duration is longer. In the Bayesian method, however, all parameters are obtained by optimization, where the probability serves as the metrics but not the sampling duration. So, the significant digits of dynamic properties in this paper can be more than the traditional methods’.


### 2.2. Bridge and Its Layout of SHM Systems

As shown in Figure1, Xihoumen Bridge is a long-span suspension bridge with a 1650-meter central main span located at Zhoushan City, linking Jintang Island and Cezi Island.Figure 1
Location of Xihoumen Bridge.In Xihoumen Bridge’s SHM system shown in Figure2, AC10∼AC18 are the servo accelerometers (ACs). The sampling rate of ACs is 50 Hz. ACs can record accelerations of vertical and lateral directions of the bridge’s cross section. AC12, AC15, and AC18 are for lateral accelerations. The other ACs are for vertical accelerations. Vertical and torsional modes are of primary interests here. As a result, we only utilize the vertical acceleration data but abandon the lateral ones. In this study, we use 6 synchronously measured acceleration data in vertical directions (i.e., AC10, AC11, AC13, AC14, AC16, and AC17) to infer structural dynamic properties. Due to the huge amount of data, we divide the time history of acceleration hour by hour and then modal frequencies and damping ratios are identified hour by hour.Figure 2
Layout of the SHM system of Xihoumen Bridge.


### 2.3. Details of Identification Processing

In this article, the long-term and continuous structural dynamic properties are identified from 2012 to 2021 by hour. Due to the limit of the article length, we only display the results of two dynamic modes to explain the methodology in this study: first, first-order symmetric torsional mode; second, fourth-order symmetric torsional mode [36]. Similar conclusions can also be found in other dynamic modes. Comparing the results of modal frequencies and damping ratios (as shown in section 3.5), we can notice that the damping ratios are much more discrete than modal frequencies. It is also found that the damping ratios from approximately September 2019 to May 2020 are anomalously higher than the others. The anomaly might be caused by some man-made actions (e.g., maintenance of expansion joints). The reason is out of the scope of this study and will not be discussed. For the analysis in section 3, the time series in the anomalous part are not utilized.


## 2.1. Theoretical Background

If the acceleration time history measured atn DOFs of a structure is noted as x¨^j∈Rn:j=1,…,N and abbreviated as {x¨^j}, where N is the number of samples per channel, the FFT of {x¨^j} is defined as(1)Fk=Fk+iGk=2∆tN∑j=1Nx¨^jexp−2πik−1j−1N,k=1,…,N,where i2=−1, ∆t is the sampling interval, and Fk=ReFk and Gk=ImFk denote the real and imaginary part of the FFT, respectively. For k=2,3,…,Nq, the FFT corresponds to frequency fk=k−1/N∆t. Here, Nq = int[N/2] + 1 (int ⋅ denotes the integer part) corresponds to the FFT ordinate at the Nyquist frequency. For modal identification, only these Nq−1 FFT values are utilized.In the context of Bayesian inference, the measured acceleration is modeled asx..^j=x..jθ+ϵj, where x..jθ is the acceleration response of the structural model defined by the set of model parameters θ, the subject to be identified, and ϵj is the prediction error that accounts for the deviation between the model response and measured data, possibly owing to measurement noise and modeling error. Yuen and Katafygiotis [7] derived the joint PDF for the augmented FFT vectors {Zk=FkT,GkTT∈R2n:k=2,…,Nq} and applied it to Bayesian modal identification. For a high sampling rate and long duration of data, Zk is a zero-mean Gaussian vector with covariance matrix given by(2)Ck=12ΦReHkΦT−ΦImHkΦTΦImHkΦTΦReHkΦT+σ22I2n,where Φ∈Rn×m is the mode shape matrix confined to the measured DOFs (the i th column gives the i th mode shape), σ2 is the (constant) spectral density level of the prediction error, I2n denotes the 2n×2n identity matrix, and Hk is the spectral density matrix of the model response and its i,j entry is given by(3)Hki,j=Sijβik2−1+i2ζiβik−1βjk2−1−i2ζjβjk−1,where βik=fi/fk = frequency ratio, fi and ζi = natural frequency and damping ratio of the i th mode, respectively, and Sij = cross-spectral density between the i th and j th modal excitation.The set of modal parametersθ consists of modal frequencies, damping ratios, mode shapes, entries {Sij} of the spectral density matrix of modal excitations, and spectral density σ2 of the prediction error. Assuming a noninformative prior distribution, the posterior PDF of θ has given that FFT data are proportional to the likelihood function pZkθ(4)pθZk∝pZkθ=2π−Nq−1/2∏kdetCkθ−1/2×exp−12∑kZkTCkθ−1Zk,where the dependence of Ck on θ has been emphasized [32]. It is convenient to write with the negative log-likelihood function Lθ form(5)pθZk∝exp−Lθ,where(6)Lθ=12∑klndetCkθ+ZkTCkθ−1Zk.Then, the most probable value (MPV)θ^ is(7)θ^=argmaxθpθZk=argminθLθ.People can use simulated annealing [33], MCMC [34], or some other minimum-value searching methods to obtain θ^. Simulated annealing is employed in this paper. The posterior covariance of θ^ can also be quantified by Hessian matrix [32, 35], which is not the main focus of this paper and is not discussed here. Different from the traditional frequency-domain identification methods, the significant digits of the identified parameters here have no concern with the sampling duration. For example, the resolution of frequency in FFT analysis will be higher when the sampling duration is longer. In the Bayesian method, however, all parameters are obtained by optimization, where the probability serves as the metrics but not the sampling duration. So, the significant digits of dynamic properties in this paper can be more than the traditional methods’.


## 2.2. Bridge and Its Layout of SHM Systems

As shown in Figure1, Xihoumen Bridge is a long-span suspension bridge with a 1650-meter central main span located at Zhoushan City, linking Jintang Island and Cezi Island.Figure 1
Location of Xihoumen Bridge.In Xihoumen Bridge’s SHM system shown in Figure2, AC10∼AC18 are the servo accelerometers (ACs). The sampling rate of ACs is 50 Hz. ACs can record accelerations of vertical and lateral directions of the bridge’s cross section. AC12, AC15, and AC18 are for lateral accelerations. The other ACs are for vertical accelerations. Vertical and torsional modes are of primary interests here. As a result, we only utilize the vertical acceleration data but abandon the lateral ones. In this study, we use 6 synchronously measured acceleration data in vertical directions (i.e., AC10, AC11, AC13, AC14, AC16, and AC17) to infer structural dynamic properties. Due to the huge amount of data, we divide the time history of acceleration hour by hour and then modal frequencies and damping ratios are identified hour by hour.Figure 2
Layout of the SHM system of Xihoumen Bridge.


## 2.3. Details of Identification Processing

In this article, the long-term and continuous structural dynamic properties are identified from 2012 to 2021 by hour. Due to the limit of the article length, we only display the results of two dynamic modes to explain the methodology in this study: first, first-order symmetric torsional mode; second, fourth-order symmetric torsional mode [36]. Similar conclusions can also be found in other dynamic modes. Comparing the results of modal frequencies and damping ratios (as shown in section 3.5), we can notice that the damping ratios are much more discrete than modal frequencies. It is also found that the damping ratios from approximately September 2019 to May 2020 are anomalously higher than the others. The anomaly might be caused by some man-made actions (e.g., maintenance of expansion joints). The reason is out of the scope of this study and will not be discussed. For the analysis in section 3, the time series in the anomalous part are not utilized.


## 3. Multiscale Time Series Decomposition for Structural Dynamic Properties

The Prophet model was developed by Facebook team [29], which was an easily interpretable tool for time series analysis. For the real-world application, it is believed that the time series are periodically fluctuating due to the periodic ambient interference [30]. The Prophet model utilizes a decomposable and interpretable time series model with three main model components: trend, seasonality, and holidays. These three components are combined as follows:(8)yt=gt+st+ht+ϵt,where gt is the trend term which describes nonperiodic changes, st models periodic changes (e.g., daily, weekly, and yearly seasonality), ht represents the effects of holidays that occur on specific days, and the error term ϵt means any idiosyncratic changes which are not properly accommodated by the model.


### 3.1. The Trend Model

Usually, there are two kinds of trend models that cover many real-world applications [29]: the saturating growth model and the piecewise linear model. For the application of a saturating growth model on SHM, it requires the domain knowledge for the rules of deterioration, which is usually case-specific and not general. As a result, a piecewise linear model is utilized in this study for convenience.Because the growth rate of the trend model is not constant, there should be changepoints to incorporate a varying rate while fitting the historical data. GivenS changepoints at times sj, j=1,…,S, a vector of rate adjustments δ∈RS is defined, where δj is the change in rate that occurs at time sj. The rate at time t is the base rate k plus all of the adjustments up to that point: k+∑j:t>sjδj. Alternatively, we can define a vector at∈0,1S for the sake of concision:(9)ajt=1,ift≥sj,0,otherwise.Then, the rate at timet is k+atTδ. The offset parameter m will be adjusted accordingly with varying k in order to connect the endpoints of the segments. The correct adjustment at changepoint j can be given as follows:(10)γj=sj−m−∑l<jγl1−k+∑l<jδlk+∑l≤jδl.Lastly, the trend model with linear trend is obtained as follows:(11)gt=k+atTδt+m+atTγ,where k is the growth rate, δ has the rate adjustments, m is the offset parameter, and γj is set to −sjδj to make the function continuous.With sufficient domain knowledge, the changepointssj can be specified by experts. Again, due to the limitation of domain knowledge, automatic selection of changepoints sj is more practical, which is accomplished by putting a sparse prior on δ.


### 3.2. Seasonality

Time series that occur in the real-world often have multiscale seasonality as a result of periodically fluctuating ambient interference and the human behaviors. For example,a 24-hour duration can be a period because of the natural daily variation; one-year duration can also be a period since the condition of natural environment (e.g., temperature and humidity) varies by year. Another example of the human behaviors is that a 7-day work week can make effects on the time series that repeat each week since people’s periodic work often counts week by week. To fit and forecast these effects, we usually specify seasonality models that are periodic functions of t.Fourier series are employed here to provide a flexible model of periodic effects [37], which have both computational efficiency and interpretability. Assume P is the regular period we expect the time series to have (e.g., if the timescale is defined by day, P=365.25 for yearly data and P=7 for weekly data). For an arbitrary period P, its seasonal effects are obtained by the following equation:(12)st=∑n=1Nancos2πntP+bnsin2πntP.Fitting seasonality requires estimating the2N parameters β=a1,b1,…,aN,bNT. This is done by constructing a matrix of seasonality vectors for each value of t in our historical and future data, for example, with weekly seasonality and N=3:(13)Xt=cos2π1t7,…,sin2π3t7.The seasonal component is then obtained as follows:(14)st=Xtβ,where β can be obtained by putting a sparse prior.


### 3.3. Holidays and Events

Holidays and events may make impacts on time series to some extent. For example, the structural dynamic properties may vary on holidays due to the heavier traffic condition [38]. In this study, we assume the effects of holidays and events are observed each year on New Year’s Day, Chinese New Year, Tomb-Sweeping Day, Labor Day, Dragon Boat Festival, Mid-Autumn Festival, and National Day, as shown in Table 1.Table 1
Examples of holidays in China.
HolidayCountryYearDateNew Year’s DayThe whole world201501.01–01.03Chinese New YearChina201502.19–02.24Tomb-Sweeping DayChina201504.04–04.06Labor DayChina201505.01–05.03Dragon Boat FestivalChina201505.20–05.22Mid-Autumn FestivalChina201509.26–09.27National DayChina201510.01–10.07New Year’s DayThe whole world201601.01–01.03⋯⋯⋯⋯For each holidayi, let Di be the set of past and future dates for that holiday. We add an indicator function representing whether time t is during holiday i, and assign each holiday a parameter κi which is the corresponding change in the forecast. This is done in a similar way as seasonality by generating a matrix of regressors(15)Zt=1t∈D1,…,1t∈DL,and taking(16)ht=Ztκ,where κ can be obtained by putting a sparse prior.


### 3.4. Performance Metrics

We can evaluate the performance of our model by comparing the similarity of the estimated data with the historical data. In this study, we use the correlation coefficient as the metric(17)M=n∑i=1nyiy^i−∑i=1nyi⋅∑i=1ny^t2n∑i=1ny^i2−∑i=1ny^t2n∑i=1ny^i2−∑i=1ny^i2,where n is the number of time series, yi is the real value (i.e., historical value), y^i is the estimated value, and M is the performance metric. The closer M is to 1, the better the performance of the model is.


### 3.5. Identified Results

In this subsection, we choose the fourth-order symmetric torsional mode and the first-order torsional mode as examples to explain how the Prophet model works and to present the results of different timescale components. Modal frequency, damping ratio, and mode shapes are identified hour by hour. However, we do not find any obvious varying patterns for mode shapes. We only discuss modal frequencies and damping ratios here.


#### 3.5.1. Decomposition for Modal Frequency

As shown in Figure3, the black points are the identified time series of the modal frequency from the SHM system. The red points are the estimated time series, which are reconstructed by the Prophet model. The modal frequency exhibits an obvious yearly fluctuating phenomenon. Meanwhile, a general decreasing trend is also found.Figure 3
Fourth-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.7999.As shown in Figure4, the yearly fluctuation and general decreasing trend are also found in the first-order symmetric torsional mode.Figure 4
First-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.5997.As shown in Figure5, we use the Savitzky–Golay smoothing and differentiation filter [39] to delineate the trend of modal frequencies more prominent, where the data of estimated time series are used.Figure 5
Filtered time series of modal frequencies: (a) Fourth-order symmetric torsional frequency. (b) First-order symmetric torsional frequency.
(a)(b)The Prophet model has the capacity to decompose the time series. In Figure6, we decompose the times series of the modal frequency to multiple timescales. The yearly seasonality is prominent, where the modal frequency got to the maximal value around February (winter) and to the minimal value around August (summer) each year. It is strongly believed that the yearly seasonality has something to do with the yearly variation of ambient interference (variation of the temperature is the reason, which is investigated in section 5). Similarly, the daily seasonality can also be noticed due to the daily variation of ambient interference, where the maximal value is observed around 1 am and the minimal value is observed around 15 pm (the reason is similar to the yearly seasonality). The weekly seasonality has no obvious rule, which is related with the traffic condition on the bridge and varies with each individual. For most cases, the holiday effects seem to increase the modal frequencies. It is obvious that the general trend of the modal frequency is continuously decreasing. In addition, compared with the yearly seasonality, the amplitudes of the other type seasonality and holiday effects are negligible. The daily seasonality and the holiday effects are of the same amplitudes (about 10−4 Hz). The amplitude of the weekly seasonality is 10−5 Hz. The yearly seasonality has the largest amplitude, which is 10−3 Hz.Figure 6
Fourth-order symmetric torsional frequency: multitimescale component.As shown in Figure7, the similar rules for yearly and daily seasonality are also found in the other mode. The general trend of the modal frequency is also decreasing.Figure 7
First-order symmetric torsional frequency: multitimescale component.


#### 3.5.2. Decomposition for Damping Ratio

As shown in Figure8, the black points are the identified time series of the damping ratio from the SHM system. The red points are the estimated time series reconstructed by the Prophet model. It can be noticed that from July 2019 to April 2020, the damping ratio increased abruptly (which are found in most modes). The latent reason might be that there were some temporary measures to suppress the vibration (e.g., maintenance of expansion joints). This anomaly should be confirmed with the bridge maintainers, which is out of the scope of this paper. As a result, we only use the data from January 2012 to June 2019 for the analysis of the damping ratio. Compared with the modal frequency, the damping ratio is much more discrete. The damping ratio is also yearly fluctuating but less obvious than the modal frequency. The general trend is not significant at the first glance.Figure 8
Fourth-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.1874.As shown in Figure9, the similar regularity is also found in the first-order symmetric torsional mode.Figure 9
First-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.2068.Figure10 shows the trend of damping ratios more prominently with the data of estimated time series.Figure 10
Filtered time series of damping ratio: (a) Fourth-order symmetric torsional damping ratio. (b) First-order symmetric torsional damping ratio.
(a)(b)Figure11 shows the decomposed time series of damping ratios. The yearly seasonality attains the maximal value around January (winter) and the minimal value around September (summer) each year, which might be caused by the yearly fluctuating ambient interference. The daily seasonality attains the maximal value around 5 am and the minimal value around 13 pm. The weekly seasonality varies with each individual, amplitude of which is negligible compared with other types of seasonality. The holiday effects seem to decrease the damping ratio. The general trend seems to be increasing. But the damping ratio is more complicated and can be changed by man-made actions. As a result, it is hard to conclude that the natural process of damping ratio is to increase with structural usage. For the damping ratio, the amplitude of yearly seasonality is usually more significant. But the holiday effects is often of the same amplitude as the yearly seasonality. The amplitude of daily seasonality is ambiguous, which may be of the same amplitude (Figure 11) or the lower amplitude (Figure 12) as the yearly seasonality. Furthermore, the performance metric ((17)) of the damping ratio is relatively lower than that of the modal frequency, which means the daily seasonality may also be in numerical errors.Figure 11
Fourth-order symmetric torsional damping ratio: multitimescale component.Figure 12
First-order symmetric torsional damping ratio: multitimescale component.


## 3.1. The Trend Model

Usually, there are two kinds of trend models that cover many real-world applications [29]: the saturating growth model and the piecewise linear model. For the application of a saturating growth model on SHM, it requires the domain knowledge for the rules of deterioration, which is usually case-specific and not general. As a result, a piecewise linear model is utilized in this study for convenience.Because the growth rate of the trend model is not constant, there should be changepoints to incorporate a varying rate while fitting the historical data. GivenS changepoints at times sj, j=1,…,S, a vector of rate adjustments δ∈RS is defined, where δj is the change in rate that occurs at time sj. The rate at time t is the base rate k plus all of the adjustments up to that point: k+∑j:t>sjδj. Alternatively, we can define a vector at∈0,1S for the sake of concision:(9)ajt=1,ift≥sj,0,otherwise.Then, the rate at timet is k+atTδ. The offset parameter m will be adjusted accordingly with varying k in order to connect the endpoints of the segments. The correct adjustment at changepoint j can be given as follows:(10)γj=sj−m−∑l<jγl1−k+∑l<jδlk+∑l≤jδl.Lastly, the trend model with linear trend is obtained as follows:(11)gt=k+atTδt+m+atTγ,where k is the growth rate, δ has the rate adjustments, m is the offset parameter, and γj is set to −sjδj to make the function continuous.With sufficient domain knowledge, the changepointssj can be specified by experts. Again, due to the limitation of domain knowledge, automatic selection of changepoints sj is more practical, which is accomplished by putting a sparse prior on δ.


## 3.2. Seasonality

Time series that occur in the real-world often have multiscale seasonality as a result of periodically fluctuating ambient interference and the human behaviors. For example,a 24-hour duration can be a period because of the natural daily variation; one-year duration can also be a period since the condition of natural environment (e.g., temperature and humidity) varies by year. Another example of the human behaviors is that a 7-day work week can make effects on the time series that repeat each week since people’s periodic work often counts week by week. To fit and forecast these effects, we usually specify seasonality models that are periodic functions of t.Fourier series are employed here to provide a flexible model of periodic effects [37], which have both computational efficiency and interpretability. Assume P is the regular period we expect the time series to have (e.g., if the timescale is defined by day, P=365.25 for yearly data and P=7 for weekly data). For an arbitrary period P, its seasonal effects are obtained by the following equation:(12)st=∑n=1Nancos2πntP+bnsin2πntP.Fitting seasonality requires estimating the2N parameters β=a1,b1,…,aN,bNT. This is done by constructing a matrix of seasonality vectors for each value of t in our historical and future data, for example, with weekly seasonality and N=3:(13)Xt=cos2π1t7,…,sin2π3t7.The seasonal component is then obtained as follows:(14)st=Xtβ,where β can be obtained by putting a sparse prior.


## 3.3. Holidays and Events

Holidays and events may make impacts on time series to some extent. For example, the structural dynamic properties may vary on holidays due to the heavier traffic condition [38]. In this study, we assume the effects of holidays and events are observed each year on New Year’s Day, Chinese New Year, Tomb-Sweeping Day, Labor Day, Dragon Boat Festival, Mid-Autumn Festival, and National Day, as shown in Table 1.Table 1
Examples of holidays in China.
HolidayCountryYearDateNew Year’s DayThe whole world201501.01–01.03Chinese New YearChina201502.19–02.24Tomb-Sweeping DayChina201504.04–04.06Labor DayChina201505.01–05.03Dragon Boat FestivalChina201505.20–05.22Mid-Autumn FestivalChina201509.26–09.27National DayChina201510.01–10.07New Year’s DayThe whole world201601.01–01.03⋯⋯⋯⋯For each holidayi, let Di be the set of past and future dates for that holiday. We add an indicator function representing whether time t is during holiday i, and assign each holiday a parameter κi which is the corresponding change in the forecast. This is done in a similar way as seasonality by generating a matrix of regressors(15)Zt=1t∈D1,…,1t∈DL,and taking(16)ht=Ztκ,where κ can be obtained by putting a sparse prior.


## 3.4. Performance Metrics

We can evaluate the performance of our model by comparing the similarity of the estimated data with the historical data. In this study, we use the correlation coefficient as the metric(17)M=n∑i=1nyiy^i−∑i=1nyi⋅∑i=1ny^t2n∑i=1ny^i2−∑i=1ny^t2n∑i=1ny^i2−∑i=1ny^i2,where n is the number of time series, yi is the real value (i.e., historical value), y^i is the estimated value, and M is the performance metric. The closer M is to 1, the better the performance of the model is.


## 3.5. Identified Results

In this subsection, we choose the fourth-order symmetric torsional mode and the first-order torsional mode as examples to explain how the Prophet model works and to present the results of different timescale components. Modal frequency, damping ratio, and mode shapes are identified hour by hour. However, we do not find any obvious varying patterns for mode shapes. We only discuss modal frequencies and damping ratios here.


### 3.5.1. Decomposition for Modal Frequency

As shown in Figure3, the black points are the identified time series of the modal frequency from the SHM system. The red points are the estimated time series, which are reconstructed by the Prophet model. The modal frequency exhibits an obvious yearly fluctuating phenomenon. Meanwhile, a general decreasing trend is also found.Figure 3
Fourth-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.7999.As shown in Figure4, the yearly fluctuation and general decreasing trend are also found in the first-order symmetric torsional mode.Figure 4
First-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.5997.As shown in Figure5, we use the Savitzky–Golay smoothing and differentiation filter [39] to delineate the trend of modal frequencies more prominent, where the data of estimated time series are used.Figure 5
Filtered time series of modal frequencies: (a) Fourth-order symmetric torsional frequency. (b) First-order symmetric torsional frequency.
(a)(b)The Prophet model has the capacity to decompose the time series. In Figure6, we decompose the times series of the modal frequency to multiple timescales. The yearly seasonality is prominent, where the modal frequency got to the maximal value around February (winter) and to the minimal value around August (summer) each year. It is strongly believed that the yearly seasonality has something to do with the yearly variation of ambient interference (variation of the temperature is the reason, which is investigated in section 5). Similarly, the daily seasonality can also be noticed due to the daily variation of ambient interference, where the maximal value is observed around 1 am and the minimal value is observed around 15 pm (the reason is similar to the yearly seasonality). The weekly seasonality has no obvious rule, which is related with the traffic condition on the bridge and varies with each individual. For most cases, the holiday effects seem to increase the modal frequencies. It is obvious that the general trend of the modal frequency is continuously decreasing. In addition, compared with the yearly seasonality, the amplitudes of the other type seasonality and holiday effects are negligible. The daily seasonality and the holiday effects are of the same amplitudes (about 10−4 Hz). The amplitude of the weekly seasonality is 10−5 Hz. The yearly seasonality has the largest amplitude, which is 10−3 Hz.Figure 6
Fourth-order symmetric torsional frequency: multitimescale component.As shown in Figure7, the similar rules for yearly and daily seasonality are also found in the other mode. The general trend of the modal frequency is also decreasing.Figure 7
First-order symmetric torsional frequency: multitimescale component.


### 3.5.2. Decomposition for Damping Ratio

As shown in Figure8, the black points are the identified time series of the damping ratio from the SHM system. The red points are the estimated time series reconstructed by the Prophet model. It can be noticed that from July 2019 to April 2020, the damping ratio increased abruptly (which are found in most modes). The latent reason might be that there were some temporary measures to suppress the vibration (e.g., maintenance of expansion joints). This anomaly should be confirmed with the bridge maintainers, which is out of the scope of this paper. As a result, we only use the data from January 2012 to June 2019 for the analysis of the damping ratio. Compared with the modal frequency, the damping ratio is much more discrete. The damping ratio is also yearly fluctuating but less obvious than the modal frequency. The general trend is not significant at the first glance.Figure 8
Fourth-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.1874.As shown in Figure9, the similar regularity is also found in the first-order symmetric torsional mode.Figure 9
First-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.2068.Figure10 shows the trend of damping ratios more prominently with the data of estimated time series.Figure 10
Filtered time series of damping ratio: (a) Fourth-order symmetric torsional damping ratio. (b) First-order symmetric torsional damping ratio.
(a)(b)Figure11 shows the decomposed time series of damping ratios. The yearly seasonality attains the maximal value around January (winter) and the minimal value around September (summer) each year, which might be caused by the yearly fluctuating ambient interference. The daily seasonality attains the maximal value around 5 am and the minimal value around 13 pm. The weekly seasonality varies with each individual, amplitude of which is negligible compared with other types of seasonality. The holiday effects seem to decrease the damping ratio. The general trend seems to be increasing. But the damping ratio is more complicated and can be changed by man-made actions. As a result, it is hard to conclude that the natural process of damping ratio is to increase with structural usage. For the damping ratio, the amplitude of yearly seasonality is usually more significant. But the holiday effects is often of the same amplitude as the yearly seasonality. The amplitude of daily seasonality is ambiguous, which may be of the same amplitude (Figure 11) or the lower amplitude (Figure 12) as the yearly seasonality. Furthermore, the performance metric ((17)) of the damping ratio is relatively lower than that of the modal frequency, which means the daily seasonality may also be in numerical errors.Figure 11
Fourth-order symmetric torsional damping ratio: multitimescale component.Figure 12
First-order symmetric torsional damping ratio: multitimescale component.


## 3.5.1. Decomposition for Modal Frequency

As shown in Figure3, the black points are the identified time series of the modal frequency from the SHM system. The red points are the estimated time series, which are reconstructed by the Prophet model. The modal frequency exhibits an obvious yearly fluctuating phenomenon. Meanwhile, a general decreasing trend is also found.Figure 3
Fourth-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.7999.As shown in Figure4, the yearly fluctuation and general decreasing trend are also found in the first-order symmetric torsional mode.Figure 4
First-order symmetric torsional frequency: monitored time series and estimated time seriesM=0.5997.As shown in Figure5, we use the Savitzky–Golay smoothing and differentiation filter [39] to delineate the trend of modal frequencies more prominent, where the data of estimated time series are used.Figure 5
Filtered time series of modal frequencies: (a) Fourth-order symmetric torsional frequency. (b) First-order symmetric torsional frequency.
(a)(b)The Prophet model has the capacity to decompose the time series. In Figure6, we decompose the times series of the modal frequency to multiple timescales. The yearly seasonality is prominent, where the modal frequency got to the maximal value around February (winter) and to the minimal value around August (summer) each year. It is strongly believed that the yearly seasonality has something to do with the yearly variation of ambient interference (variation of the temperature is the reason, which is investigated in section 5). Similarly, the daily seasonality can also be noticed due to the daily variation of ambient interference, where the maximal value is observed around 1 am and the minimal value is observed around 15 pm (the reason is similar to the yearly seasonality). The weekly seasonality has no obvious rule, which is related with the traffic condition on the bridge and varies with each individual. For most cases, the holiday effects seem to increase the modal frequencies. It is obvious that the general trend of the modal frequency is continuously decreasing. In addition, compared with the yearly seasonality, the amplitudes of the other type seasonality and holiday effects are negligible. The daily seasonality and the holiday effects are of the same amplitudes (about 10−4 Hz). The amplitude of the weekly seasonality is 10−5 Hz. The yearly seasonality has the largest amplitude, which is 10−3 Hz.Figure 6
Fourth-order symmetric torsional frequency: multitimescale component.As shown in Figure7, the similar rules for yearly and daily seasonality are also found in the other mode. The general trend of the modal frequency is also decreasing.Figure 7
First-order symmetric torsional frequency: multitimescale component.


## 3.5.2. Decomposition for Damping Ratio

As shown in Figure8, the black points are the identified time series of the damping ratio from the SHM system. The red points are the estimated time series reconstructed by the Prophet model. It can be noticed that from July 2019 to April 2020, the damping ratio increased abruptly (which are found in most modes). The latent reason might be that there were some temporary measures to suppress the vibration (e.g., maintenance of expansion joints). This anomaly should be confirmed with the bridge maintainers, which is out of the scope of this paper. As a result, we only use the data from January 2012 to June 2019 for the analysis of the damping ratio. Compared with the modal frequency, the damping ratio is much more discrete. The damping ratio is also yearly fluctuating but less obvious than the modal frequency. The general trend is not significant at the first glance.Figure 8
Fourth-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.1874.As shown in Figure9, the similar regularity is also found in the first-order symmetric torsional mode.Figure 9
First-order symmetric torsional damping ratio: monitored time series and estimated time seriesM=0.2068.Figure10 shows the trend of damping ratios more prominently with the data of estimated time series.Figure 10
Filtered time series of damping ratio: (a) Fourth-order symmetric torsional damping ratio. (b) First-order symmetric torsional damping ratio.
(a)(b)Figure11 shows the decomposed time series of damping ratios. The yearly seasonality attains the maximal value around January (winter) and the minimal value around September (summer) each year, which might be caused by the yearly fluctuating ambient interference. The daily seasonality attains the maximal value around 5 am and the minimal value around 13 pm. The weekly seasonality varies with each individual, amplitude of which is negligible compared with other types of seasonality. The holiday effects seem to decrease the damping ratio. The general trend seems to be increasing. But the damping ratio is more complicated and can be changed by man-made actions. As a result, it is hard to conclude that the natural process of damping ratio is to increase with structural usage. For the damping ratio, the amplitude of yearly seasonality is usually more significant. But the holiday effects is often of the same amplitude as the yearly seasonality. The amplitude of daily seasonality is ambiguous, which may be of the same amplitude (Figure 11) or the lower amplitude (Figure 12) as the yearly seasonality. Furthermore, the performance metric ((17)) of the damping ratio is relatively lower than that of the modal frequency, which means the daily seasonality may also be in numerical errors.Figure 11
Fourth-order symmetric torsional damping ratio: multitimescale component.Figure 12
First-order symmetric torsional damping ratio: multitimescale component.


## 4. Long-Term Trend

According to the decomposed time series of dynamic properties, we get the long-term trend. In this section, we give the evolving regularity of modal frequencies and damping ratios.


### 4.1. Modal Frequency

As shown in Figure13, the modal frequencies are prone to decrease in the long term. For the first-order symmetric torsional mode, the modal frequency decreases 0.16% (from 0.2297 to 0.2293) in 10 years according to the equation of linear regression. For the fourth-order symmetric torsional mode, the deterioration of modal frequency in 10 years is about 0.21% (from 0.3271 to 0.3264).Figure 13
Long-term trend of modal frequency: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


### 4.2. Damping Ratio

Figure14 indicates that the damping ratio is likely to increase in the long term. Again, because the damping ratio can be changed by man-made action, the extracted long-term trend is not necessary the natural evolving regularity. For the first-order symmetric torsional mode, the damping ratio increases 14% (from 0.0026 to 0.0030). The damping ratio of the fourth-order symmetric torsional mode increases 5% (from 0.0028 to 0.0029).Figure 14
Long-term trend of damping ratio: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 4.1. Modal Frequency

As shown in Figure13, the modal frequencies are prone to decrease in the long term. For the first-order symmetric torsional mode, the modal frequency decreases 0.16% (from 0.2297 to 0.2293) in 10 years according to the equation of linear regression. For the fourth-order symmetric torsional mode, the deterioration of modal frequency in 10 years is about 0.21% (from 0.3271 to 0.3264).Figure 13
Long-term trend of modal frequency: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 4.2. Damping Ratio

Figure14 indicates that the damping ratio is likely to increase in the long term. Again, because the damping ratio can be changed by man-made action, the extracted long-term trend is not necessary the natural evolving regularity. For the first-order symmetric torsional mode, the damping ratio increases 14% (from 0.0026 to 0.0030). The damping ratio of the fourth-order symmetric torsional mode increases 5% (from 0.0028 to 0.0029).Figure 14
Long-term trend of damping ratio: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5. Ambient Interference

The ambient interference affects the dynamic properties greatly. In this section, we conduct the correlation analysis between the time-variant environmental conditions and the yearly seasonality of the dynamic properties, where the effects of long-term trend are excluded. The temperature and relative humidity at the 2 meter height above the sea surface are chosen as the representative ambient interference. The database of the ambient interference that we employ is from the ERA5 global reanalysis [31]. Due to the limit of spatial resolution of the database, the location of the measured data is not the same as that of the bridge. As shown in Figure 1, we choose the closest point 30°N,121.9°E to the bridge in the database, which will not affect the results much.


### 5.1. Temperature

Figure15 shows the varying temperature at the 2-meter height above the sea surface from 2012 to 2022 at the location of 30°N,121.9°E.Figure 15
Temperature at the 2-meter height above the sea surface30°N,121.9°E.


#### 5.1.1. Modal Frequency

Figure16 indicates that the temperature and the yearly seasonality of the modal frequency have a strong correlation. The frequency will decrease if the temperature increases. For the first-order symmetric torsional mode, the modal frequency will approximately decrease by 0.82×10−4 Hz if the temperature increases by 1°C. For the fourth-order symmetric torsional mode, the modal frequency will approximately decrease by 1.49×10−4 Hz if the temperature increases by 1°C. The conclusions above are valid when the temperature is 0−25°C. Recalling the general deterioration of modal frequencies (Figure 13), it is found that the variations caused by natural deterioration and temperature are of the same amplitude, both 10−4.Figure 16
Relationship between temperature and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


#### 5.1.2. Damping Ratio

Figure17 shows that the damping ratio will decrease with the increment of temperature in general. Compared with the modal frequency, however, the temperature and damping ratio are not one-to-one relationship. For example, Figure 17(a) shows that both 3°C and 10°C correspond the variation of the damping ratio by 2×10−4.Figure 17
Relationship between temperature and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


### 5.2. Relative Humidity

Figure18 shows the varying relative humidity at the 2 meter height above the sea surface from 2012 to 2022 at the location of 30°N,121.9°E.Figure 18
Relative humidity at the 2 meter height above the sea surface30°N,121.9°E.


#### 5.2.1. Modal Frequency

For the modal frequency, the relative humidity seems to have nothing to do with the yearly seasonality of the modal frequency, as shown in Figure19.Figure 19
Relationship between relative humidity and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


#### 5.2.2. Damping Ratio

As shown in Figure20, there is no significant correlation between the relative humidity and the yearly seasonality of the damping ratio as well.Figure 20
Relationship between relative humidity and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.1. Temperature

Figure15 shows the varying temperature at the 2-meter height above the sea surface from 2012 to 2022 at the location of 30°N,121.9°E.Figure 15
Temperature at the 2-meter height above the sea surface30°N,121.9°E.


### 5.1.1. Modal Frequency

Figure16 indicates that the temperature and the yearly seasonality of the modal frequency have a strong correlation. The frequency will decrease if the temperature increases. For the first-order symmetric torsional mode, the modal frequency will approximately decrease by 0.82×10−4 Hz if the temperature increases by 1°C. For the fourth-order symmetric torsional mode, the modal frequency will approximately decrease by 1.49×10−4 Hz if the temperature increases by 1°C. The conclusions above are valid when the temperature is 0−25°C. Recalling the general deterioration of modal frequencies (Figure 13), it is found that the variations caused by natural deterioration and temperature are of the same amplitude, both 10−4.Figure 16
Relationship between temperature and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


### 5.1.2. Damping Ratio

Figure17 shows that the damping ratio will decrease with the increment of temperature in general. Compared with the modal frequency, however, the temperature and damping ratio are not one-to-one relationship. For example, Figure 17(a) shows that both 3°C and 10°C correspond the variation of the damping ratio by 2×10−4.Figure 17
Relationship between temperature and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.1.1. Modal Frequency

Figure16 indicates that the temperature and the yearly seasonality of the modal frequency have a strong correlation. The frequency will decrease if the temperature increases. For the first-order symmetric torsional mode, the modal frequency will approximately decrease by 0.82×10−4 Hz if the temperature increases by 1°C. For the fourth-order symmetric torsional mode, the modal frequency will approximately decrease by 1.49×10−4 Hz if the temperature increases by 1°C. The conclusions above are valid when the temperature is 0−25°C. Recalling the general deterioration of modal frequencies (Figure 13), it is found that the variations caused by natural deterioration and temperature are of the same amplitude, both 10−4.Figure 16
Relationship between temperature and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.1.2. Damping Ratio

Figure17 shows that the damping ratio will decrease with the increment of temperature in general. Compared with the modal frequency, however, the temperature and damping ratio are not one-to-one relationship. For example, Figure 17(a) shows that both 3°C and 10°C correspond the variation of the damping ratio by 2×10−4.Figure 17
Relationship between temperature and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.2. Relative Humidity

Figure18 shows the varying relative humidity at the 2 meter height above the sea surface from 2012 to 2022 at the location of 30°N,121.9°E.Figure 18
Relative humidity at the 2 meter height above the sea surface30°N,121.9°E.


### 5.2.1. Modal Frequency

For the modal frequency, the relative humidity seems to have nothing to do with the yearly seasonality of the modal frequency, as shown in Figure19.Figure 19
Relationship between relative humidity and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


### 5.2.2. Damping Ratio

As shown in Figure20, there is no significant correlation between the relative humidity and the yearly seasonality of the damping ratio as well.Figure 20
Relationship between relative humidity and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.2.1. Modal Frequency

For the modal frequency, the relative humidity seems to have nothing to do with the yearly seasonality of the modal frequency, as shown in Figure19.Figure 19
Relationship between relative humidity and modal frequency’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 5.2.2. Damping Ratio

As shown in Figure20, there is no significant correlation between the relative humidity and the yearly seasonality of the damping ratio as well.Figure 20
Relationship between relative humidity and damping ratio’s yearly seasonality: (a) first-order symmetric torsional mode; (b) fourth-order symmetric torsional mode.
(a)(b)


## 6. Distribution of Model Errors

Model errors are the residuals between the estimated values and the measured values, which are mainly caused by improper model assumptions.For the modal frequency, the normal distribution is used to fit model errors as shown in Figure21 (Table 2).(18)fx=1σ2πe−1/2x−μ/σ2,where σ is the standard variance, μ is the mean value, and f⋅ is the PDF.Figure 21
Histogram of model error for frequencies. (a) Fourth-order symmetric torsional mode. (b) First-order symmetric torsional mode.
(a)(b)Table 2
Distribution parameters for modal frequencies.
Modeσμ4-ST7.44×10−4−2.38×10−81-ST6.74×10−4−2.43×10−8For the damping ratio, we use the bimodal normal distribution to fit model errors as shown in Figure22 (Table 3).(19)fx=pAσ12πexp−12x−μ1σ12+1−pAσ22πexp−12x−μ2σ22,where μi and σi are the mean value and standard variation i=1,2; p and 1−p are the weights of bimodal models; and A is the normalized coefficient.Figure 22
Histogram of model error for damping ratios. (a) Fourth-order symmetric torsional mode. (b) First-order symmetric torsional mode.
(a)(b)Table 3
Distribution parameters for damping ratios.
Modeσ1μ1σ2μ2pA4-ST−1.99×10−49.12×10−51.65×10−49.96×10−50.43911.00531-ST−1.96×10−41.05×10−41.62×10−49.86×10−50.45301.0160


## 7. Conclusions

In order to distinguish the long-term and periodic effects of structural dynamic properties, this article proposes an interpretable methodology to decompose the time series into long-term parts, multiscale periodic parts, holiday parts, and model errors. By analyzing the long-term parts, we can conclude that the modal frequencies are decreasing with the structural usage (about0.20% each decade), which reflect the effects of structural deterioration. The damping ratios seem to increase in the long term. However, the damping ratios can be influenced by exterior conditions (e.g., structural maintenance, tuned mass damper, and wind velocity). So, the long-term varying rules of the structural damping still need to be investigated in the future. By analyzing the relationship between multiscale periodic parts and ambient environmental conditions, we find that the increment of temperature can decrease the modal frequencies and damping ratios. For modal frequencies, the variations caused by natural deterioration and temperature are of the same amplitudes 10−4. Differently, for buildings, the increment of temperature can increase the modal frequencies, which means the underlying mechanics between the modal frequencies and temperatures can be complicated. For the varying reasons of long-span bridges, a possible explanation is that the higher temperature will loosen the cables and the loose cables decrease the stiffness (i.e., the modal frequencies). For buildings, the varying phenomenon may be caused by temperature-sensitive material properties. Due to the limit of data acquisition, we did not discuss the relationship among cable forces, cable temperatures, and modal frequencies, which needs to be investigated further in future researches. As for the humidity, this paper does not find its reasonable relationship with structural dynamic properties.In a nutshell, this article investigates to what extent the structural deterioration and the ambient environmental conditions can change the structural dynamic properties, which provide a reference for damage detection and safety assessment of similar long-span bridges.



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*Source: 6485040-2023-02-09.xml*