# A Fast Wavelet-Based Bridge Condition Assessment Approach Using Only Moving Vehicle Measurements
## 4 Numerically Verification
### The Effect of Road Roughness
Both signals are applied to wavelet transform and the wavelet energies are calculated as shown in Figure 10. As shown, both of the energies have a peak around scale 493, which means that, based on Equation (25), the wavelet pseudo frequency  
Figure 6: Vehicle responses and its spectrums with road roughness. (**a**) the axle acceleration \(\ddot{u}_{a}\); (**b**) spectrum for \(\ddot{u}_{a}\); (**c**) the vehicle body acceleration \(\ddot{u}_{s}\); (**d**) spectrum for \(\ddot{u}_{s}\). is 0.51 Hz (the central frequency of 'Mexican hat' is 0.25 Hz). This pseudo frequency is close to the frequency of the vehicle \(f_{v1}\), which is consistent with the findings in Figure 6d. It is worth noting that the bridge's 1 st natural frequency changes to 2.08 Hz from 2.17 Hz due to the damage, with a change ratio of 3.85%. This change may be hard to detect in the frequency domain using the indirect measurements because of the short data in the VBI system. As shown in Figure 10, there is also a clear drop in the peak of the wavelet energy due to the bridge damage. The change of the wavelet energy can be used to reflect the bridge conditions. Figure 8: Accelerations for different bridge conditions. Figure 7: Identifying the bridge damage. (**a**) scalogram without road roughness; (**b**) particular wavelet coefficient without road roughness; (**c**) scalogram with road roughness; (**d**) particular wavelet coefficient with road roughness. Once the bridge damage is detected, the proposed damage location index is then used to localize the damage. As mentioned above, the vehicle accelerations are evenly divided into ten parts and the corresponding component wavelet energies are plotted in Figure 11. It should be noted that the bridge damage is located at 13 m from the left end, namely located at the component signal of No. 7. As shown, there are no clear changes in the wavelet energies for the first few component signals until close to the damage location. The wavelet energy of No. 6 starts to vary slightly and that of No. 7 and No. 8 varies more. Based on Equation (28), the damage location index is calculated, and the results are shown in Figure 12. Obviously, the damage location index has a maximum value where the bridge is damaged. It is demonstrated that it is feasible to approximately identify the damaged location of the bridge using the proposed approach. To further verify the effectiveness of the proposed approach, another case is simulated where the bridge damage is set at 8 m from the left end, namely located at the component signal of No. 5. Other properties of the VBI system are kept the same. The results are illustrated in Figure 13. Similarly, a drop in the peak of the wavelet energy can be clearly observed due to the change in the bridge's health conditions. In addition, the proposed damage location index can accurately indicate the true damage location of the bridge. Figure 10: Wavelet energy for different bridge conditions. Figure 9: Wavelet scalogram on the acceleration under damaged conditions. Figure 10.0 Wavelet energy comparison of component signals. Figure 10.1 Damage location index. Figure 10.2 Bridge damage identification. (**a**) wavelet energy; (**b**) damage index.