A New Drive-by Method for Bridge Damage Inspection Based on Characteristic Wavelet Coefﬁcient

: The drive-by method has become a popular indirect approach for bridge damage inspection (BDI) because of its simplicity in deployment by evaluating the bridge health status solely via the vehicle dynamic response. Derived from the vehicle dynamic response, the recent proposed contact-point response involves no vibration signal with the vehicle frequency, bearing great potential for drive-by BDI. However, an appropriate methodology for the application of contact-point response in drive-by BDI remains lacking. The present study proposes a novel drive-by method, in which a new damage factor index, i.e., the characteristic wavelet coefﬁcient (CWC), is established for bridge damage identiﬁcation in an efﬁcient and accurate manner. The CWC is obtained by analyzing the contact-point response via the continuous wavelet transform (CWT) and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) techniques. CEEMDAN is introduced to overcome the issue of modal aliasing and pseudo-frequency. First, the general framework of the proposed drive-by BDI method is introduced. Then, a demonstration case study is carried out to examine the effectiveness of the proposed method. Subsequently, a parametric study is carried out to explore the effects of several parameters on the performance of BDI including the scale factor, vehicle speed, environmental noise, and boundary effect. The results indicate that the proposed drive-by BDI method can better eliminate the mode mixing and pseudo-frequency problems during the extraction of the CWC, compared with the traditional ensemble empirical mode decomposition method. The extracted CWC curve is smooth, convenient for damage inspection, and has strong anti-noise performance. After adding white noise with a signal-to-noise ratio of 20, a bridge girder with a damage severity of 20% can be identiﬁed successfully. In addition, the selection of the scale factor is critical for bridge damage inspection based on the extracted CWC. The effective scale factor of the CWC extracted using the proposed method has a wide range, which improves the inspection efﬁciency. Finally, a low vehicle speed is beneﬁcial to alleviate the adverse effect of the boundary effect on the damage inspection of bridge girder ends.


Introduction
Due to the increase in aging infrastructures worldwide, structural system inspection and damage detection have become increasingly important for structural health status assessment. For bridges in particular, it is also crucial to ensure their structural integrity and safety, because bridges usually serve as the critical links in transportation networks. In general, bridge health inspection methods are mainly divided into two categories, i.e., direct inspection methods and indirect inspection methods. Direct inspection methods aim to detect damage by installing sensors on bridges and analyzing vibration data collected by sensors [1][2][3][4]. Although direct inspection methods have been widely used, they often require a large-scale deployment of sensors on the bridge, leading to demanding costs Buildings 2023, 13, 397 3 of 18 attention recently. For example, Abdulkareem et al. [21][22][23] proposed a series of wavelet analysis methods to extract the modal shapes of plate structures and to construct structural damage factors. Zhu and Law [24] employed the continuous wavelet transform (CWT) to identify cracks in bridge beam structures under a moving load. Hester and González [25] successfully detected the damage in a 40 m simply supported bridge beam model by analyzing its acceleration response under a moving load via the WT. Nguyen and Hai [26] proposed an approach for detecting a multi-cracked beam-like structure subjected to a moving vehicle based on the on-vehicle vibration signal and wavelet analysis. By using the continuous wavelet method, Khorram et al. [27] compared the effect of the signal obtained by the bridge mid-span sensor and the sensor installed at the contact point between the mobile vehicle and the bridge for damage inspection. The analysis results showed that the inspection method of the bridge contact sensor was more effective. Tan et al. [28] employed Shannon entropy to select the optimal scale factor in the process of the CWT, which improves the efficiency of bridge damage inspection. The aforementioned WT-based methods are mainly implemented on vehicle dynamic responses, while their performance on the vehicle-bridge contact-point response remains unclear. Yang and Chang [29] employed the EMD technique to decompose the vehicle response into a set of intrinsic mode functions (IMFs), with which the bridge frequencies, especially the higher modes, were successfully extracted. Obrien et al. [30] also utilized EMD to decompose the acceleration of a car body's mass center (vehicle acceleration) to construct the damage index for accurate bridge damage detection. In recent years, ensemble empirical mode decomposition (EEMD) and extreme-point symmetric mode decomposition (ESMD) have been successively used by Zhu and Malekjafarian [31] and Yang et al. [32] to improve the efficiency and accuracy of the frequency inspection of bridges.
It is reported that the above signal decomposition methods such as EEMD still have modal aliasing and pseudo-modal problems [33]. In addition, the contact-point response is proved to be more efficient and accurate than the vehicle dynamic response for bridge damage inspection, yet there is currently a lack of related research in this field [34]. The present study proposes a novel drive-by-based method, in which a new damage factor index, i.e., the characteristic wavelet coefficient (CWC), is established for bridge damage identification in an efficient and accurate manner. The CWC is obtained by analyzing the contact-point response via the CWT and complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) techniques. CEEMDAN is introduced to overcome the issues of modal aliasing and pseudo-frequency. The remainder of this paper is organized as follows. First, the general framework of the proposed drive-by bridge damage inspection methodology is introduced. Then, a demonstration case study is carried out to examine the effectiveness of the proposed drive-by damage inspection methodology. Subsequently, a parametric study is carried out to explore the effects of several parameters on the performance of the proposed damage inspection methodology including the scale factor, vehicle speed, environmental noise, and boundary effect.

Bridge Damage Inspection Methodology
The proposed drive-by bridge damage inspection methodology consists of three parts, i.e., VBI analysis (part I), preliminary damage inspection (part II), and thorough damage inspection (part III), as illustrated in Figure 1. In part I, the VBI analysis is performed, in which the dynamic responses of the vehicle-bridge system are obtained. Subsequently, the dynamic responses of the vehicle-bridge system are analyzed using the CWT, in which the wavelet coefficient (WC) is extracted and used for the preliminary damage inspection of the bridge, as detailed in part II. It is noteworthy that the WC contains a variety of high-order frequency contents of bridge vibration, which has adverse effects on the damage inspection accuracy. To overcome this issue, the WC in part II is further processed by CEEMDAN, with which the CWC is extracted and used for thorough damage inspection, i.e., identifying the damage location and severity of the bridge accurately, as introduced in detail in part III. Each one of the three parts of the proposed drive-by bridge damage inspection methodology will be elaborated in the following subsections.
contains a variety of high-order frequency contents of bridge vibration, which has adverse effects on the damage inspection accuracy. To overcome this issue, the WC in part II is further processed by CEEMDAN, with which the CWC is extracted and used for thorough damage inspection, i.e., identifying the damage location and severity of the bridge accurately, as introduced in detail in part III. Each one of the three parts of the proposed driveby bridge damage inspection methodology will be elaborated in the following subsections.

Input bridge and vehicle information
Calculate the contact position between vehicle and bridge, Generating the equivalent roughness of road surface Initial Initial

Analytical Solution
For illustration purposes, the coupled vehicle-bridge system is simplified as a lumped sprung mass moving on a simply supported beam, as shown in Figure 2. By neglecting the damping effects of the vehicle-bridge system, the corresponding equations of motion can be written as (1) where uv and ub are the vertical displacement of the vehicle body (lumped sprung mass) and the bridge (simply supported beam), respectively; the two dots represent quadratic differentials with respect to time t; mv and mb are the mass of the vehicle and bridge, respectively; Kv is the stiffness matrix of the vehicle; EI is the flexural rigidity of the bridge; x = vt is the position of the moving vehicle on the bridge; v is the moving speed of the vehicle; δ(t) is the Dirac function; and fc(t) is the contact force between the vehicle and bridge, which is given by For illustration purposes, the coupled vehicle-bridge system is simplified as a lumped sprung mass moving on a simply supported beam, as shown in Figure 2. By neglecting the damping effects of the vehicle-bridge system, the corresponding equations of motion can be written as m v ..
where u v and u b are the vertical displacement of the vehicle body (lumped sprung mass) and the bridge (simply supported beam), respectively; the two dots represent quadratic differentials with respect to time t; m v and m b are the mass of the vehicle and bridge, respectively; K v is the stiffness matrix of the vehicle; EI is the flexural rigidity of the bridge; x = vt is the position of the moving vehicle on the bridge; v is the moving speed of the vehicle; δ(t) is the Dirac function; and f c (t) is the contact force between the vehicle and bridge, which is given by where g is the gravitational acceleration. where g is the gravitational acceleration. Based on the modal superposition method, the bridge displacement response ub(x,t) is expressed in terms of modal shapes sin (nπx/L) and generalized coordinates qb,n(t): where L is the length of the bridge.
One can obtain the theoretical solution of the acceleration time histories of the bridge, vehicle, and contact point between the bridge and vehicle as [8,35] where Δst,n =2mvgL 3 /EIπ 4 ; μn = ωb/ωv; Sn = nπv/Lωb; ωbl,n and ωbr,n are the left and right frequencies of the bridge, respectively; ωv is the frequency of the vehicle; and ωd,n is the driving frequency.

Numerical Simulation
Generally, the bridge can be discretized as several beam elements, and the vehicle is idealized as a spring-mass model. To consider the vehicle-bridge interaction, it is common to formulate the governing equation of the coupled vehicle-bridge system as [8]  Based on the modal superposition method, the bridge displacement response u b (x,t) is expressed in terms of modal shapes sin (nπx/L) and generalized coordinates q b,n (t): where L is the length of the bridge.
One can obtain the theoretical solution of the acceleration time histories of the bridge, vehicle, and contact point between the bridge and vehicle as [8,35] .. ..
in which the amplitude coefficients A v,n , A d,n , A br,n , and A bl,n are where ∆ st,n = 2m v gL 3 /EIπ 4 ; µ n = ω b /ω v ; S n = nπv/Lω b ; ω bl,n and ω br,n are the left and right frequencies of the bridge, respectively; ω v is the frequency of the vehicle; and ω d,n is the driving frequency.

Numerical Simulation
Generally, the bridge can be discretized as several beam elements, and the vehicle is idealized as a spring-mass model. To consider the vehicle-bridge interaction, it is common to formulate the governing equation of the coupled vehicle-bridge system as [8] where in which l is the length of the bridge element. The numerical solution of the coupling between the vehicle and the bridge system can be realized by the separation iteration between the vehicle and the bridge system. In the solving process of the vehicle-bridge interaction program, the geometric and mechanical coupling relationship between the vehicle and bridge systems is satisfied through the Newmark-β method and separation iteration, so as to realize the independent solution of the bridge and vehicle motion equations ( Figure 1, part I). For more information about the numerical solution system in this article, see [36][37][38].

Preliminary Damage Inspection Based on CWT
The WT is a mathematical approach that enables the extraction of the prominent characteristics of the original dataset, such as discontinuities, trends, and breakdown points, which has been widely used for signal processing applications. The WT is divided into the CWT and discrete WT (DWT). In the present study, the CWT is employed to process the acceleration response of the vehicle-bridge system, which is given by where f (t) is the signal (the acceleration response in the present study); ψ a,b (t) is the wavelet basis function; ψ * (t) represents the conjugate function of the basic wavelet function; t is the time; a is the scale factor; and b is the time shift factor. When the parameters t, a, and b are continuous variables, the above transformation is the CWT, and WT f (a,b) is the wavelet coefficient under scale factor a and shift factor b. The WC of the signal at a certain scale can be obtained by applying the CWT to the acceleration response of the vehicle-bridge system obtained in the previous section, which is subsequently used for the damage inspection of the bridge girder.

Thorough Damage Inspection Based on CWC
As mentioned previously, the WC contains a variety of high-order frequency contents of bridge vibration, which has adverse effects on the damage inspection accuracy. To improve the identification accuracy, the CEEMDAN technique is applied to the WC to obtain the CWC for thorough damage inspection.
Huang et al. [39] first developed the EMD method to examine the nonlinear and nonstationary signals in an adaptive time-frequency-amplitude space. With the EMD method, the signal can be decomposed into several intrinsic mode functions (IMFs) and a residue. Despite its wide applications, EMD experiences some problems, such as the presence of oscillations of very disparate amplitudes in a mode, or mode mixing. To overcome these problems, ensemble EMD (EEMD) has been proposed, which performs the EMD over an ensemble of the signal plus Gaussian white noise [40]. Based on EEMD, the CEEMDAN method has recently been proposed, in which a particular noise is added at each stage of the decomposition and a unique residue is computed to obtain each mode [41]. This helps to further reduce the modal aliasing and pseudo-modal problems as well as enhancing the anti-noise ability. The flowchart of CEEMDAN is illustrated in Figure 1. Given a set of measured data X(t), the process of the CEEMDAN algorithm decomposing each order's IMF is described as follows: (1) The Gaussian white noise is added to the signal X(t) to obtain a new signal W N n , and the new signal is decomposed through EMD to obtain the first- order IMF and remainder (Equation (12)). The above process is repeated N times, and N first-order IMFs are obtained because the white noise added is different.
where E(·) refers to the EMD process; n ∑ n=1 W N n refers to the white noise; n is the number of times the white noise is added; C j 1 (t) is the first-order IMF; r j (t) is the remaining margin; and j is the number of decompositions.
(2) The average of the resulting N first-order IMFs yields the first IMF C 1 (t) decomposed by CEEMDAN: (3) The residual of the original signal X(t) is calculated after removing the first modal component: (4) The signal obtained in the previous step is treated as a new signal, which is processed by the first three steps to obtain the second-order IMF. Subsequently, the procedure is applied repeatedly to extract the third-and fourth-order IMF components (if they exist), until the residual signal is a monotonic function. Finally, assuming the number of extracted IMFs is K, the original signal X(t) can be expressed as the summation of all IMF components and the residual signal, as given by

Numerical Case
A demonstration case study is carried out to examine the effectiveness of the proposed drive-by damage inspection methodology.

VBI of Simply Supported Bridge Beam
The demonstration case involves a single-degree-of-freedom (DOF) sprung mass moving along a simply supported bridge beam model, as shown in Figure 3. The total length of the bridge beam is 15 m, which is divided into 30 elements with an equal length of 0.5 m. The damage of the bridge girder is simulated by means of a reduction in the elastic modulus. The time step during the VBI analysis is set to 0.005 s. In addition, the speed of the vehicle is set to 4 m/s, and the key parameters of the vehicle-bridge system are shown in Table 1. The VBI is realized using a self-compiled program in the MATLAB software. For validation purposes, the simulated dynamic responses of the vehicle-bridge system are compared with the analytical solutions, as shown in Figure 4. It is clearly shown in Figure 4 that the vertical acceleration of the bridge mid-span, the vertical acceleration of the vehicle body, and the vertical acceleration of the contact point obtained from the numerical simulations are in good agreement with those of the analytical solutions.    First-order frequency of bridge 6.623 Hz w b2 Second-order frequency of bridge 26.492 Hz Figure 3. Single-DOF sprung mass moving along a simply supported bridge beam model. First-order frequency of bridge 6.623 Hz wb2 Second-order frequency of bridge 26.492 Hz In total, 14 cases are considered, as listed in Table 2. In cases 1~5, the damage is assumed to be located at seventh element with various degrees of severity, i.e., 0%, 5%, 10%, 20%, and 40%. The vehicle mass and vehicle speed remain unchanged in cases 1~5, i.e., mv = 1000 kg and V = 4 m/s. To examine the influence of the damage location on the feasibility of the proposed damage inspection methodology, the damage location is shifted from the seventh element to the second element of the bridge in cases 6~8, with a damage In total, 14 cases are considered, as listed in Table 2. In cases 1~5, the damage is assumed to be located at seventh element with various degrees of severity, i.e., 0%, 5%, 10%, 20%, and 40%. The vehicle mass and vehicle speed remain unchanged in cases 1~5, i.e., mv = 1000 kg and V = 4 m/s. To examine the influence of the damage location on the feasibility of the proposed damage inspection methodology, the damage location is shifted from the seventh element to the second element of the bridge in cases 6~8, with a damage severity of 0%, 20%, and 40%. The vehicle mass and vehicle speed in cases 6~8 are the same as those in cases 1~5. To further investigate the influence of the vehicle speed, the vehicle speed is reduced from 4 m/s to 1 m/s in cases 9~11. The damage in cases 9~11 is located at the second element of the bridge with a damage severity of 0%, 20%, and 40%. It is noted that cases 1~11 only contain one damage location, while cases 12~13 assume two damage locations. To be specific, in case 12, elements No. 4 and No. 14 are damaged with a severity of 10% and 30%, while in case 13, elements No. 4 and No. 14 are damaged with a severity of 30% and 10%.

WC Extraction Based on CWT
Previous studies have shown that abnormal changes in the WC under a certain scale factor can be used to identify the bridge damage effectively. Figure 5a shows the timefrequency diagram after applying the CWT to the vehicle acceleration under case 5. The ordinate is the scale factor (Scale) of the CWT, and the abscissa is the time shift factor (Shift) of the CWT, which can be understood as the relative position of the bridge. The color shade indicates the magnitude of the WC. It is observed that the color of the WC varies significantly at the position of 6.5/30 = 0.21 (the center of the damaged No.7 element), and the scale factor ranges from approximately 30 to 60. To clearly observe the variation in the WC, the WC under the scale factor of 35 in case 5 is displayed in Figure 5b. For comparison purposes, the WC in the undamaged case (case 1) is also plotted in Figure 5b. As shown in Figure 5b, there is an obvious peak of the WC at the center of the damaged No.7 element, which suggests that the significant variation in the WC under an effective scale factor can help to identify the damage location of the bridge. It is also worth mentioning that the wavelet basis function could affect the damage identification results. The commonly used wavelet basis functions include the Haar wavelet, Morlet wavelet, Mexican hat wavelet, and Daubechies wavelet. A preliminary analysis was conducted to examine the effectiveness of the type of wavelet basis function on the damage identification results, where the Mexico hat wavelet outperformed the other selected wavelet basis functions and was selected for the subsequent analysis. In addition to the WC of vehicle acceleration, the WC of the bridge mid-span acceleration and the contact-point acceleration in cases 2~5 is also extracted, as shown in Figure  6. It is observed from Figure 6a,b that although the trend of the WC shows a certain degree of increase at the damaged location, such an increase is not obvious, especially when the damage severity is less than 40%. This is because the overall trend of the WC is largely contaminated by its fluctuations, i.e., the high-frequency contents. In addition to the WC of vehicle acceleration, the WC of the bridge mid-span acceleration and the contact-point acceleration in cases 2~5 is also extracted, as shown in Figure 6. It is observed from Figure 6a,b that although the trend of the WC shows a certain degree of increase at the damaged location, such an increase is not obvious, especially when the damage severity is less than 40%. This is because the overall trend of the WC is largely contaminated by its fluctuations, i.e., the high-frequency contents. In addition to the WC of vehicle acceleration, the WC of the bridge mid-span acceleration and the contact-point acceleration in cases 2~5 is also extracted, as shown in Figure  6. It is observed from Figure 6a,b that although the trend of the WC shows a certain degree of increase at the damaged location, such an increase is not obvious, especially when the damage severity is less than 40%. This is because the overall trend of the WC is largely contaminated by its fluctuations, i.e., the high-frequency contents.

Extracting CWC Based on WC Using EEMD or CEEMDAN
The previous discussion indicates that the extracted WC after applying the CWT to the acceleration response of the bridge mid-span and contact point has a poor damage inspection performance, due to the contamination of the high-frequency content of the WC. In order to improve the identification accuracy, it is natural to filter out the highfrequency contents of the WC to make the trend of the WC visible. In view of this, the extracted WC is further analyzed using two signal decomposition techniques, i.e., EEMD and CEEMADAN, in which the extracted WC is decomposed into a series of IMF components with different frequencies. Figures 7 and 8 show the IMFs and the associated FFT spectra after processing the WC (case 5) with EEMD and CEEMDAN, respectively. It is observed that the IMFs help to separate the first few bridge natural frequencies or the vehicle driving frequency. In addition, obvious mode mixing can be observed in IMF3 and IMF4 extracted using the EEMD method, as shown in Figure 7. Furthermore, our analysis showed that the IMFs with a dominant frequency higher than the first vertical natural frequency (6.623 Hz) of the bridge have adverse effects on the bridge damage identification results, while the opposite result is found for the IMFs with a frequency lower than

Extracting CWC Based on WC Using EEMD or CEEMDAN
The previous discussion indicates that the extracted WC after applying the CWT to the acceleration response of the bridge mid-span and contact point has a poor damage inspection performance, due to the contamination of the high-frequency content of the WC. In order to improve the identification accuracy, it is natural to filter out the high-frequency contents of the WC to make the trend of the WC visible. In view of this, the extracted WC is further analyzed using two signal decomposition techniques, i.e., EEMD and CEEMADAN, in which the extracted WC is decomposed into a series of IMF components with different frequencies. Figures 7 and 8 show the IMFs and the associated FFT spectra after processing the WC (case 5) with EEMD and CEEMDAN, respectively. It is observed that the IMFs help to separate the first few bridge natural frequencies or the vehicle driving frequency. In addition, obvious mode mixing can be observed in IMF3 and IMF4 extracted using the EEMD method, as shown in Figure 7. Furthermore, our analysis showed that the IMFs with a dominant frequency higher than the first vertical natural frequency (6.623 Hz) of the bridge have adverse effects on the bridge damage identification results, while the opposite result is found for the IMFs with a frequency lower than the first vertical natural frequency of the bridge. Therefore, only the IMFs with a dominant frequency lower than the first vertical natural frequency of the bridge are selected to construct the new damage index, which is termed as the CWC in the present study.
The CWC decomposed by EEMD and CEEMDAN is shown in Figure 9. It is observed in Figure 9 that the CWC extracted by both EEMD and CEEMDAN shows an abrupt increase at the damage location of the bridge girder, and such an abrupt increase is more obvious under severer damage conditions. To be specific, the peak of the CWC extracted by EEMD is 2.13 × 10 −4 , 2.87 × 10 −4 , and 4.72 × 10 −4 under damage of 0%, 20%, and 40%, while the peak of the CWC extracted by CEEMDAN is 1.41 × 10 −4 , 3.35 × 10 −4 , and 6.08 × 10 −4 under damage of 0%, 20%, and 40%. This indicates that the CWC can be used to detect the damage location and to quantify the damage severity of the bridge girder. In addition, it is found that the CWC extracted using the CEEMDAN method is smoother than that extracted using the EEMD method. Therefore, the CEEMAN method is adopted for the subsequent analysis. by EEMD is 2.13 × 10 −4 , 2.87 × 10 −4 , and 4.72 × 10 −4 under damage of 0%, 20%, and 40%, while the peak of the CWC extracted by CEEMDAN is 1.41 × 10 −4 , 3.35 × 10 −4 , and 6.08 × 10 −4 under damage of 0%, 20%, and 40%. This indicates that the CWC can be used to detect the damage location and to quantify the damage severity of the bridge girder. In addition, it is found that the CWC extracted using the CEEMDAN method is smoother than that extracted using the EEMD method. Therefore, the CEEMAN method is adopted for the subsequent analysis.

CWC for Multi-Damage Case
In this section, the performance of the proposed drive-by damage inspection methodology in a multi-damage case is further evaluated. Two damage scenarios of the bridge girder with two damage locations are investigated. In the first damage scenario (case 12), elements No. 4 and No. 14 are damaged with a severity of 10% and 30%, while in the second damage scenario (case 13), elements No. 4 and No. 14 are damaged with a severity of 30% and 10%. Figure 10 shows the CWC by analyzing the bridge mid-span acceleration, vehicle acceleration, and contact-point acceleration. As shown in Figure 10a, the CWC extracted from the bridge mid-span acceleration has a poor damage inspection performance. In contrast, as for the CWC extracted from the vehicle acceleration and contactpoint acceleration, two noticeable peaks can be observed at the two damaged elements, as shown in Figure 10b,c. In addition, for damage scenario one, the peak at element No. 4 with a 10% damage severity is less than that at element No. 14 with a 30% damage

CWC for Multi-Damage Case
In this section, the performance of the proposed drive-by damage inspection methodology in a multi-damage case is further evaluated. Two damage scenarios of the bridge girder with two damage locations are investigated. In the first damage scenario (case 12), elements No. 4 and No. 14 are damaged with a severity of 10% and 30%, while in the second damage scenario (case 13), elements No. 4 and No. 14 are damaged with a severity of 30% and 10%. Figure 10 shows the CWC by analyzing the bridge mid-span acceleration, vehicle acceleration, and contact-point acceleration. As shown in Figure 10a, the CWC extracted from the bridge mid-span acceleration has a poor damage inspection performance. In contrast, as for the CWC extracted from the vehicle acceleration and contact-point acceleration, two noticeable peaks can be observed at the two damaged elements, as shown in Figure 10b,c. In addition, for damage scenario one, the peak at element No. 4 with a 10% damage severity is less than that at element No. 14 with a 30% damage severity, while the opposite result is found for damage scenario two. The above observations clearly suggest that the CWC extracted from the vehicle acceleration and contact-point acceleration is also effective for damage identification in multi-damage scenarios. severity, while the opposite result is found for damage scenario two. The above observations clearly suggest that the CWC extracted from the vehicle acceleration and contactpoint acceleration is also effective for damage identification in multi-damage scenarios.

Parametric Analysis
In this section, a parametric study is carried out to explore the effects of several parameters on the performance of the proposed damage inspection methodology including the scale factor, vehicle speed, environmental noise, and boundary effect.

Effect of Scale Factor
As discussed earlier, the scale of the CWT has a significant effect on the performance

Parametric Analysis
In this section, a parametric study is carried out to explore the effects of several parameters on the performance of the proposed damage inspection methodology including the scale factor, vehicle speed, environmental noise, and boundary effect.

Effect of Scale Factor
As discussed earlier, the scale of the CWT has a significant effect on the performance of the proposed drive-by damage inspection methodology. In practice, it is usually necessary to find a better scale factor through trial and error. Figure 11 shows the CWCs at scale factors of 15, 35, and 55, which are extracted from the contact-point accelerations in case 1, case 3, and case 5, respectively. It is observed that the CWC under a scale factor of 35 outperforms that under scale factors of 15 and 55.

Effect of Environmental Noise
Vehicle signals are inevitably polluted by environmental noise during the acquisition process, resulting in signal distortion or some key information being missing. Therefore, in this section, the effect of noise on the performance of the proposed damage identification framework is investigated. During the analysis, Gaussian white noise with three different signal-to-noise ratios (SNR = 15, 20, and 50) is added to the vehicle acceleration response in cases 1~5. Figure 12 shows the CWC under SNR = 15, 20, and 50. It is observed that appreciable fluctuations are induced all along the bridge girder due to the Gaussian white noise. As such, the CWC can only identify the damage locations under a severity of 40%, below which it becomes quite difficult to successfully carry out damage identification.

Effect of Environmental Noise
Vehicle signals are inevitably polluted by environmental noise during the acquisition process, resulting in signal distortion or some key information being missing. Therefore, in this section, the effect of noise on the performance of the proposed damage identification framework is investigated. During the analysis, Gaussian white noise with three different signal-to-noise ratios (SNR = 15, 20, and 50) is added to the vehicle acceleration response in cases 1~5. Figure 12 shows the CWC under SNR = 15, 20, and 50. It is observed that appreciable fluctuations are induced all along the bridge girder due to the Gaussian white noise. As such, the CWC can only identify the damage locations under a severity of 40%, below which it becomes quite difficult to successfully carry out damage identification.

Vehicle Speed
The previous studies indicate that the performance of the conventional damage inspection method under a low vehicle speed is better than that under a high vehicle speed. In this section, the performance of the proposed damage inspection methodology in cases 1, 4, and 5 under two vehicle speeds, i.e., 8 m/s and 12 m/s, is investigated. It is observed in Figure 13 that the CWC is able to identify the damage location under both vehicle speeds, indicating that the proposed damage inspection methodology has a good performance under a relatively high vehicle speed.

Vehicle Speed
The previous studies indicate that the performance of the conventional damage inspection method under a low vehicle speed is better than that under a high vehicle speed. In this section, the performance of the proposed damage inspection methodology in cases 1, 4, and 5 under two vehicle speeds, i.e., 8 m/s and 12 m/s, is investigated. It is observed in Figure 13 that the CWC is able to identify the damage location under both vehicle speeds, indicating that the proposed damage inspection methodology has a good performance under a relatively high vehicle speed.

Vehicle Speed
The previous studies indicate that the performance of the conventional damage inspection method under a low vehicle speed is better than that under a high vehicle speed. In this section, the performance of the proposed damage inspection methodology in cases 1, 4, and 5 under two vehicle speeds, i.e., 8 m/s and 12 m/s, is investigated. It is observed in Figure 13 that the CWC is able to identify the damage location under both vehicle speeds, indicating that the proposed damage inspection methodology has a good performance under a relatively high vehicle speed.

Boundary Effect
It is noted that in the process of the CWT, the WC is derived from the window function and wavelet convolution. When the window is located at the end of the signal, the signal is inevitably distorted because the signal is forced to zero at its end. This is referred to as the boundary effect of the CWT. In the time-frequency diagram, the quality of the signal becomes poor with the decrease in the frequency (scale a), as shown in Figure 14a. The boundary effect could inundate the damage information of the beam element near the bridge girder end. For example, when the vehicle speed is 4 m/s, it is impossible to extract the damage information of element 2, as shown in Figure 14a. The influence of the boundary effect is illustrated in the influence cone, in which the signal inside the influence cone is not affected by the boundary effect, while the signal outside the influence cone is significantly affected by the boundary effect. It is also noted that as the vehicle speed reduces from 4 m/s to 1 m/s, the influence cone becomes flatter and the influence of the boundary effect becomes less significant. As shown in Figure 14b, when the vehicle speed is 1 m/s, the second element can be successfully identified. Therefore, a low-speed test vehicle is expected to alleviate the adverse boundary effect. However, it remains impossible to identify the damage at the girder ends (i.e., 1st and 30th elements), even with a low-speed test vehicle, and more advanced algorithms may be considered in the future to resolve this limitation.

Concluding Remarks
The present study proposes a novel drive-by-based method, in which a new damage factor index, i.e., the CWC, is established for bridge damage identification in an efficient and accurate manner. The following conclusions were obtained: (1) Compared with the EEMD method, the CEEMAN algorithm can better eliminate the mode mixing and pseudo-frequency problems during the extraction of the CWC. The introduction of this method also makes the CWC curve smooth, convenient for damage inspection, with strong anti-noise performance. After adding white noise with a signal-to-noise ratio of 20, a bridge girder with a damage severity of 20% can be identified. (2) The selection of the scale factor is critical for bridge damage inspection based on the extracted CWC. The effective scale factor of the CWC extracted using the proposed method has a wide range, which improves the inspection efficiency. (3) A low vehicle speed is beneficial to alleviate the adverse effect of the boundary effect on the damage inspection of bridge girder ends.
It should be noted that, as with most existing studies on drive-by methods for bridge health inspection, the proposed method is numerically verified, demonstrating the potential for actual practice. However, there remain great challenges in achieving a satisfactory accuracy of bridge health inspection in real-world environments. In particular, the collected vibrational signals for damage identification are usually contaminated due to the interference of random traffic flow, environmental noises, etc. It is therefore recommended to carry out such a practical test when there is less adverse traffic excitation. Meanwhile, a specialized test vehicle with fewer internal noises (e.g., from motor vibration) is also beneficial to the performance of damage identification. To further improve the drive-by method, one may present further investigations on eliminating the common boundary effect through innovative algorithms and/or novel experimental designs. Data Availability Statement: All data, models, or codes that support the findings of this study are available upon reasonable request.

Conflicts of Interest:
The authors declare no conflict of interest.