# Wavelet-Based Optimum Identification of Vehicle Axles Using Bridge Measurements
## 6 Field Test of Fifth Wushui Bridge, China
### Wavelet Analysis of Test Results for Single Truck Crossing  
When an axle arrives or passes mid-span, the weighing sensors attached to the soffits of the girders exhibit a discontinuity of slope. The FAD sensors, because they are attached to the slab, have a much more 'peaked' response, ideally showing a significant peak when an axle passes. However, the peak is highly dependent on the transverse position of the wheel in the lane and whether it is over the girder web or the slab. If the vehicle deviates from the center line of the lane, the FAD sensors may not show clear peaks in response to passing axles [28]. For example, Figure 9 gives the (unprocessed) response to Truck A crossing the bridge in Lane 3, at Sensor 13 (labeled here as L3-FAD-1) and Sensor 17 (labeled as L3-FAD-2). It is noted that there was no filtering during data acquisition. Each FAD signal should have two peaks corresponding to the times when the two axles passed each sensor. However, there is only one obvious peak in each case. This could be because one of the axles had double tires and the other did not. Lesser peaks, before and after the dominant one, can be seen, but it is not clear which peaks correspond to passing axles and which are caused by vibration in the slab. To improve axle detection, the L3-FAD-1 signal of Figure 9 is subjected to the proposed approach associated with the Daubechies wavelet family (_db1-db9_). All the computation is analyzed using the Matlab software, Version 2018b. The processor (CPU) is AMD Ryzen7 3700x 8-Core with 16 GB RAM. The operating system is Win 10. In this case, the computation time including calculation of wavelet transform, Shannon entropy, and correlation factor is 0.132s So far, the proposed method was not embedded in the BWIM system. But the computation time associated with all wavelet analysis is short, so the proposed method has the potential to be implemented in BWIM system for real-time application. As shown in Figure 10, the maximum correlation coefficient is again achieved with _db2_ for which the minimum Shannon entropy is at a scale of 8. The wavelet coefficients for this optimum mother wavelet and scale are plotted in Figure 11a. Two peaks are now clearly visible although the second is less pronounced than the first. Alternative transforms are illustrated in Figure 11 and show the effects of adopting sub-optimal scales (Figure 11b,c) and mother wavelets (Figure 11d-f). \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Axle 1** & **Axle 2** \\ \hline \multirow{2}{*}{Truck A} & Axle spacing (m) & & 4.7 & \\ \cline{2-4}  & Axle weight (tonnes) & 5.8 & 24.5 \\ \hline \multirow{2}{*}{Truck B} & Axle spacing (m) & & 4.7 & \\ \cline{2-4}  & Axle weight (tonnes) & 7.4 & 21.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Vehicle characteristics. Figure 9: The strain signal of free-of-axel-detector (FAD) sensors in Lane 3. The same approach is applied to the L3-FAD-2 signal in Figure 9. The same mother wavelet, db2, and scale of 8 are found to be optimal for this signal. Figure 12 shows the transformed signals for both sensors. Unlike the raw signal of Figure 9, there are no other significant peaks that could be easily mistaken for axles. The time stamps for the L3-FAD-1 signal peaks are 1.880 s and 2.490 s for the first and second peaks respectively. The corresponding respective time stamps for L3-FAD-2 are 2.905 s and 3.495 s. The interval between the main peaks in the two signals, with the known 8 m spacing between the sensors, provides the vehicle speed. This, with the 1 st to 2 nd peak (within-signal) intervals, is used to determine the axle spacing. The result is inferred spacings of 4.80 and 4.60 m from signals L3-FAD-1 and L3-FAD-2 respectively, just 2.1% above and below the statically measured spacing of 4.7 m. As shown, the identified axle spacing, which affects the axle weight recognition accuracy, is calculated based on the vehicle speed. Therefore, the vehicle speed is crucial for the BWIM system. In addition, there are additional 9 runs of single truck (No.