1. Introduction
With the continuous increase in the number and scale of transportation infrastructure, there is a growing demand for regular inspection and maintenance to ensure its long-term stable operation. Through long-term monitoring of bridge structures, real-time monitoring of the structures can be achieved, aiding in predicting the lifespan of the structures and formulating rational maintenance plans. Simultaneously, the long-term data collected by structural health monitoring (SHM) systems can provide robust support for data-driven decision-making. Combining advanced data analysis techniques, engineers can develop more precise maintenance strategies and engineering plans, enabling the more efficient allocation of resources and enhancing overall operational efficiency.
In SHM, changes in the dynamic characteristics of structures are crucial for assessing structural health. Modal frequency variations are among the key parameters used to evaluate structural conditions [
1,
2,
3,
4,
5]. However, the modal frequencies of real engineering structures are influenced by numerous environmental factors, such as temperature, humidity, ambient noise, and wind, among others. Changes in modal parameters induced by environmental factors are often significant and can even exceed those caused by structural damage or operational loads [
6,
7,
8,
9]. Generally, temperature is recognized as the primary environmental factor affecting structural frequencies [
6]. In recent years, numerous researchers have undertaken studies to assess the impact of temperature on the modal parameters of bridge structures and investigate methods to mitigate its influence.
Several researchers have investigated the reasons behind the influence of temperature changes on structural modal frequencies. Ni et al. [
10], based on field-measured acceleration and temperature data from the Sutong Bridge, studied the variations in natural frequencies and damping ratios with temperature and vibration levels. Their research revealed that the natural frequencies of bridge structures decrease with increasing temperature, possibly due to changes in the elastic modulus of construction materials induced by temperature fluctuations. Cho and Cho [
11] conducted a long-term monitoring study on the dynamic characteristics of two short-span concrete bridges and found that the natural frequencies of the bridges decrease with rising temperatures. They validated this modal change through model updating, attributing it to the temperature-dependent elasticity modulus of concrete. Similar conclusions were drawn by Zhou and Sun [
12], who combined environmental factors with six years of modal frequency data from the East Sea Bridge. They identified cyclic variations in frequency and observed that all modal frequencies of the structure decrease with increasing temperature due to changes in the elastic modulus of structural materials. Cai et al. [
13] analyzed the impact of temperature on the free vibration characteristics of simply supported beams and theoretically deduced that the influence of temperature on the natural frequency of simply supported beams is primarily caused by changes in the elastic modulus of the beam material, with the structural dimensions having a less significant impact. Anastasopoulos et al. [
14] analyzed temperature fluctuations during monitoring periods using modal data collected from a steel arch railway bridge equipped with Fiber Bragg Grating Strain Sensors. They found that temperature can induce changes in the bridge’s overall stiffness, leading to variations in natural frequencies. Sun et al. [
15] studied the relationship between modal frequencies and temperature in a controlled temperature chamber for concrete continuous beam bridge models and steel cable-stayed bridges. They observed that the influence of temperature on structural frequencies is closely related to boundary conditions and that temperature may alter the bridge’s boundary conditions.
Different viewpoints exist among scholars regarding the pattern of modal frequencies decreasing with rising environmental temperatures. Mosavi et al. [
16] investigated the relationship between modal characteristics and temperature in a two-span steel-concrete composite bridge. They found that during the noon hours, the first five frequencies of the main beam were higher than those during the night, with three of the frequencies positively correlated with temperature. A similar phenomenon was observed at the Saint Michel Bridge [
17]. Michal and Katarína [
18] used the Random Subspace Method to identify acceleration data from a steel bridge and processed it in conjunction with temperature data. They observed that the first two frequency modes of the structure increased with rising temperature, and some frequencies showed non-linearity when fitted with linear regression functions against temperature data.
Research has shown that eliminating the influence of a single environmental temperature can provide a more accurate assessment of bridge conditions [
19,
20]. The performance of temperature effects in the presence of multiple environmental factors has also received attention. Mao et al. [
21] employed the Hilbert–Huang Transform method to identify continuous modals over a year for the Sutong Bridge. They analyzed the impact of environmental temperature and traffic volume on modal frequencies and found that temperature effects outweighed traffic volume, being the most significant factor for vertical and torsional modal frequencies. Yang et al. [
22] conducted statistical characterization and probabilistic prediction of environmental load data (wind speed, wind direction, temperature, and humidity) and acceleration response data collected by the SHM system of the Jiashao Multi-Tower Cable-Stayed Bridge. They established mapping relationships between specific load sources and structural responses. Chu et al. [
1] proposed a time-series decomposition method to divide the time series of structural dynamic characteristics into different time scales and explore their relationship with periodic environmental loads. Their research indicated that an increase in temperature leads to a decrease in modal frequencies and damping ratios, while modal frequencies and damping ratios show no significant correlation with humidity. He et al. [
23] considered the influence of temperature and humidity on frequency monitoring data and developed a temperature and humidity elimination method based on BP neural networks, improving the accuracy of bridge condition reliability assessment. He et al. [
24] established a “Frequency-Temperature-Humidity” long-term equilibrium model based on cointegration theory for monitoring data from a three-span concrete bridge model. This model can be used to predict the trend of frequency changes and eliminate the influence of temperature and humidity on frequency. Mu et al. [
25] developed a Bayesian network algorithm to identify the relationship between modal frequencies and various environmental factors. They found that considering the correlation between temperature and relative humidity improved the prediction accuracy when predicting modal frequencies. Meng and Zhu [
26] quantitatively compared the influence of temperature variations, temperature gradients, and material properties on the dynamic performance of large-span suspension bridges through multiscale modeling. They discovered that the time-varying dynamic characteristics of large-span suspension bridges are primarily induced by thermal stress-hardening effects. Wang et al. [
27,
28] proposed an EFR model based on multi-order frequencies and a local thermal-frequency correlation model incorporating Gaussian mixtures to eliminate the variability in modal frequencies caused by temperature, humidity, and wind speed. They verified the effectiveness of these two methods through experiments on cable-stayed bridges.
This study analyzes nearly five months of vibration acceleration and temperature and humidity data collected from four representative vertical acceleration sensors located inside the main beam of the Jintang Hantan Twin-Island Bridge in Chengdu, China, using the Covariance-Driven Stochastic Subspace Identification method (SSI-COV). It employs clustering algorithms to automatically identify modal frequencies from steady-state diagrams and utilizes Support Vector Machine Regression (SVR) analysis to model the identified frequencies in relation to environmental temperature and humidity. Finally, based on the fitted model, the influence of temperature on the identified frequencies is eliminated, and the post-temperature-adjusted frequency fluctuations are observed. Finally, the main conclusions of the article were summarized, and potential directions for further in-depth research in the future were discussed.
4. Temperature-Frequency Nonlinear Model Analysis
The Support Vector Machine (SVR) regression model was employed to analyze the relationship between modal frequency and environmental temperature. Since humidity was found to have a minor impact on modal frequencies through correlation analysis in the previous sections, this study only established a model for modal frequencies and environmental temperature, excluding humidity as a predictor. It should be noted that, due to the limited sample data collected, the model established in this study is only applicable to the range of normal atmospheric temperature variations. The data used were collected during the spring and summer seasons in Chengdu, with temperature variations ranging from 14.8 to 38.2 °C. Effective monitoring data from April to August (i.e., no missing data for frequency and temperature) were selected for this study. The total number of collected samples amounted to 547 sets. Following a ratio of 0.85:0.15, the collected samples were randomly divided into training and testing sets. Among these, 465 sets of data were selected as training samples to build the model, denoted as and . Subsequently, 82 sets of data were employed as testing samples to assess and verify the predictive capability of the model, labeled as and .
4.1. Nonlinear Model Based on SVR Regression Analysis
Support Vector Regression (SVR) is a non-linear regression analysis method derived from Support Vector Machine (SVM) technology. The core of SVR lies in mapping complex non-linear problems to a new linear space using a kernel function. The linear equation in the new space can be expressed as follows:
where
is the weight vector,
is a constant,
represents the error, and
is the transformation function with respect to the input vector
. If there is a training sample
, then the SVR model can be transformed into an optimization problem for solving [
33]. This results in a system of linear equations,
where
and
are Lagrange multipliers,
is the regularization parameter.
is the kernel matrix computed based on the kernel function. Each element in
can be calculated using Equation (6),
Thus, by solving Equation (4), the required fitting model can be obtained. Observing Equation (5), it is not difficult to find that in the SVR method, the kernel matrix is determined by the kernel function . Therefore, we do not need to know the exact expression of the transformation function ; we only need to determine the kernel function to determine the regression model. There are many types of kernel functions to choose from, depending on the specific circumstances. In this study, the commonly used Gaussian radial basis function (RBF) was selected for modeling and analysis.
In this study, 6th-order recognition frequency was selected for investigation. The key model parameters of the SVR model are listed in
Table 5, and the calculation results of the regression model are shown in
Figure 7. It can be seen that both the fitting and prediction results closely match the curves of the measured samples. Therefore, it is evident that the SVR regression method based on the radial basis kernel function can also establish a reasonably effective model for the temperature-frequency relationship. As a contrast, the random forest model is also used to train and predict the same frequency data; the number of decision trees constructed is 10. The recognition results are shown in
Figure 8. The accuracy of the random forest model will be discussed in a subsequent section.
4.2. Model Quality Analysis
To assess and compare the fitting quality of the models, an analysis of the model’s residuals was conducted. If the fitting model is denoted as
, then the fitting residuals are defined as
, and the prediction residuals as
. Based on the fitting residuals and prediction residuals, four evaluation metrics were calculated: root mean square error (RMSE), coefficient of determination (R
2), kurtosis, and skewness. The root mean square error (RMSE) is the square root of the second sample moment of the difference between predicted values and observed values, reflecting the accuracy of the model. The kurtosis of the residuals is defined as:
The skewness of the residuals can be defined as,
The root mean square error (RMSE) can be defined as,
where
is the length of
or
.
The coefficient of determination (R
2) can be defined as,
where
is the sum of squares of residuals,
is the total sum of squares.
Root Mean Square Error (RMSE) reflects the accuracy of the model, and its value is always non-negative, with a value of 0 indicating a perfect fit to the data. In general, a lower RMSE is better than a higher one. The coefficient of determination (R2) is used to assess the goodness of fit of a regression model, with a value closer to 1 indicating a higher degree of fit. The kurtosis and skewness of residuals are both measures of the normality of their amplitude distribution. Kurtosis is used to assess the steepness of the residual distribution: indicates that the residual distribution is flatter than the normal distribution, while indicates that the residual distribution is sharper than the normal distribution. Skewness is used to evaluate the asymmetry of the residual distribution: indicates that the majority of residual values (including the median) are located to the right of the mean, while indicates that the majority of residual values (including the median) are located to the left of the mean. Therefore, when is closer to 3 and is closer to 0, it indicates that the residual distribution is closer to normal. A residual distribution closer to normality indicates a higher model quality.
Figure 9 shows the residual distribution of the support vector machine model, and
Table 6 shows the error evaluation parameters of the SVR model and random forest model for identifying the 6th-order frequency. Comparing the residual distribution with the normal distribution curve, it is evident that the distribution of fitting residuals is closer to normal than that of prediction residuals, and the parameters of the residual distribution perform relatively well. Both the coefficient of determination R
2 for fitting and prediction are close to 1, and the root mean square error values are small. This indicates that the fitting and prediction have achieved a high level of accuracy, effectively reflecting the true state of the training samples. Compared to the SVR model, the Random Forest model performs better in terms of RMSE for fitting results. However, both the R
2 for fitting results and the RMSE and R
2 for prediction results are worse than those of the SVR model. This indicates that the SVR model demonstrates superior overall performance in both fitting and prediction.
4.3. Eliminate Temperature Effects
To mitigate the influence of temperature on the identified frequencies, the temperature-frequency model proposed by Fan et al. [
34] and validated on the Tingjiu Bridge was adopted. The formula for this model is shown in Equation (11).
where
represents the expected value of the
order frequency, and
is the frequency value identified from measured data.
Figure 10 shows the time history of the 6th-order frequency after eliminating the temperature effect. The frequency after eliminating the effect of temperature had a maximum change of 1.6% and an average change of 0.4% compared to the frequency before elimination. It can be observed that after removing the temperature influence, the frequency fluctuations with a 24 h periodicity have been significantly reduced. Therefore, this obtained frequency can better characterize changes in structural properties in structural health monitoring. Additionally, this result also demonstrates the excellent capability of the support vector machine nonlinear model in effectively modeling the relationship between modal frequency and environmental temperature.
5. Discussion and Conclusions
Based on four months of acceleration and temperature-humidity data from the Jintang Hantan Twin Island Bridge health monitoring system, this study successfully employed the Covariance-Driven Stochastic Subspace Identification (SSI-COV) method and clustering algorithm to identify acceleration data, obtaining stable higher-order frequencies. Subsequently, a correlation analysis was conducted to investigate the influence of environmental factors on these higher-order frequencies. It also employed Support Vector Machine Regression (SVR) for modeling analysis and analyzed the residuals of the SVR model. Finally, the real structural frequencies, free from environmental influences, were obtained. The specific conclusions are as follows:
Linear regression was used to observe the relationship between identified frequencies and environmental temperature. The results indicated a significant negative correlation between the bridge’s frequency and environmental temperature, while humidity showed no significant correlation with modal frequencies.
A nonlinear Support Vector Machine (SVM) model was employed for fitting and modeling. Residual analysis of the model revealed a clear linear relationship between frequency and temperature. The SVM model effectively fit the sample data and provided reliable predictive results.
After obtaining temperature-induced frequency data from the SVR model, the original structural frequencies were smoothed, and temperature influences were eliminated. It was observed that, after eliminating temperature effects, the frequency fluctuations with a 24 h period significantly decreased.
In this study, the influence of temperature and humidity on the modal frequencies of large-span cable-stayed bridges was analyzed, and the higher-order modal frequencies after eliminating the temperature influence were obtained. It can serve as a reference for the health monitoring of similar large-span cable-stayed bridge structures in practical engineering and can also serve as a foundation for further research on the impact of other environmental factors on bridge structural modal parameters.
Due to the complexity of the relationship between non-uniform temperature fields and structural frequencies and the limited number of existing bridge measurement points, this study only included uniform temperature effects without considering the non-uniform temperature field caused by structural spatial distribution and sunlight factors. Additionally, the impact of other environmental and operational factors (such as wind, vehicle loads, and noise interference) on the modal frequencies of the cable-stayed bridge should be addressed in future research phases.