## 5 Application Analysis
### Measured Responses and Analysis
Figure 15 presents the fitted results of the simulated spatial power spectral density, indicating a strong level of fitness. Next, based on the simulation results, we discuss the influence of road roughness on dynamic analysis. Assuming a uniform mass density of 10 ton/m and a speed of 10 m/s or 20 m/s for the vehicle group, we calculate the corresponding dynamic responses of the vehicle and the bridge using Equations (10)-(16), considering the impact of road roughness. Figures 16 and 17 illustrate the dynamic response of the vehicle group at different speeds. The comparison of these figures reveals that the increase in speed reduces the response of the vehicle, but the effect is considerably less than that of road roughness. Figure 15 demonstrates that when road roughness is considered, the dynamic responses of the vehicle group, in terms of displacement, velocity and acceleration, significantly increase. Passengers in the vehicle may experience discomfort at certain points in time, as the vertical acceleration approaches the uncomfortable threshold of 0.5 m/s\({}^{2}\). Therefore, it is crucial to consider the precise degree of road roughness and its impact on the vehicle, especially when studying driving comfort. Figure 13: Comparison of interaction of vehicle–bridge under different coupling mass ratios. Figure 16: Vehicle displacement at two different speeds, ignoring road roughness: (**a**) vehicle speed of 10 m/s and (**b**) vehicle speed of 20 m/s. Figure 14: Simulated history of road roughness. Figure 15: Comparison of PSD curves of standard and simulation. On the other hand, the responses of the bridge under different vehicle speeds exhibit periodic fluctuations, as shown in Figures 18 and 19. The dynamic responses of the bridge significantly amplify as the speed of the vehicle group increases. The impact of road roughness on the dynamic responses of the bridge is only slightly greater than that without roughness effects, given the bridge's sufficient stiffness and mass to resist the impacts and disturbances caused by road roughness. Figure 17: Vehicle acceleration at two different vehicle speeds, considering road roughness: (**a**) vehicle speed of 10 m/s and (**b**) vehicle speed of 20 m/s. Figure 18: Bridge displacement at two different vehicle speeds, ignoring road roughness: (**a**) vehicle speed of 10 m/s and (**b**) vehicle speed of 20 m/s. To investigate the combined effect of uniform mass and road roughness on the dynamic response of the bridge, the displacement and acceleration history is calculated assuming a vehicle group of 10 m/s, a coupling mass ratio of 0.15, and mass densities of the main body of 5 ton/m and 10 ton/m. The results are shown in Figures 20 and 21. It is observed that the response amplitudes are mainly determined by the uniform mass of the vehicle group, and heavy vehicle groups may induce excessive deflection and stress, emphasizing the need for the strict monitoring and control of overloading. The rate of increase in structural acceleration caused by surface irregularity generally does not exceed 7%. Figure 11:0 Bridge acceleration at two different vehicle speeds, considering road roughness: (**a**) vehicle speed of 10 m/s and (**b**) vehicle speed of 20 m/s. Figure 11:1 Displacement of bridge with different uniform mass and road roughness. Existing studies have shown that the coupled vibration between different types of vehicles and bridges is complex, and the response does not vary linearly. Therefore, it is necessary to study the typical maximum amplitude under vehicles with different frequencies and speeds to estimate the effective monitoring threshold values. Figures 22 and 23 show family curves of the maximum displacement and acceleration amplitude of the bridge with a uniform mass density of 10 ton/m but different vertical frequencies and speeds. The results indicate that the coupling effect between the vehicle group and the bridge tends to stabilize when the vertical frequency of vehicles is greater than 0.4 Hz.