## 4 Test Results and Analysis
### Vibration Attenuation Law and Safety Distance Calculation  
According to the existing research theories [8; 9; 10], vibration attenuation is affected by the dual effects of wavefront geometrical diffusion and medium absorption. In actual engineering, the formation is not an ideal linear elastic medium, during the vibration transmission process, the formation absorbs part of the vibration energy, resulting in the vibration energy in the form of exponential decay. On the other hand, the excitation of the vibration source is related to the travel speed diffused in the horizontal direction, with the increase of the propagation distance, its energy density decays according to the power function law.  
Based on previous research findings, it has been observed that the vibration velocity decreases exponentially as the distance from the vibration source increases [47]. Additionally, the vibration velocity increases as the vehicle speed increases, following a power function [48]. Therefore, a mathematical equation Equation (1) can be formulated to represent the relationship between vibration velocity and distance from the vibration source. Notably, the findings supporting Equation (1) have been validated by the author's team in environmental vibration studies and analogous investigations.  
\[v=k\cdot\frac{v_{0}^{2}}{r^{b}}\exp(-c\cdot r) \tag{1}\]  
Here, \(v\) is the surface vibration velocity; \(r\) is the distance of the measuring point from the vibration source; \(v_{0}\) is the travel speed; \(k\) is the equivalent coefficient, which is related to the impact energy and site medium conditions; \(a\) is the index related to the travel speed; \(b\) is the geometric damping attenuation coefficient; and \(c\) is the energy absorption coefficient of the soil, which is related to the engineering properties of the soil.  
To confirm the suitability and precision of Equation (1), the collected data from various vehicle speeds at the Chengshange Village site were subjected to fitting and analysis. Specifically, Equation (1) was input into the origin software to determine its functional form, and the parameter values were obtained as follows: k = 0.152, a = 0.74, b = 0.338, and c = 0.015. R2 represents the variance of the fit. Figure 7 displays the attenuation curves of vibration velocity as a function of distance from the vibration source for various vehicle speeds.  
As can be seen from Figure 7, the attenuation of surface vibration intensity is in good agreement with the field-measured data, indicating that Equation (1) can better describe the propagation law of traffic load vibration, and the specific functional relationship can be expressed as follows.  
\[v=0.152\cdot\frac{v_{0}^{0.74}}{r_{0}.338}\mathrm{e}^{-0.015r} \tag{2}\]  
Referring to the \"Technical specifications for protection of historic buildings against man-made vibration\" (GB/T 50452-2008) and the research results of related literature [38], the critical value of the vibration velocity is taken as 0.15 mm/s. For a given travel speed, the safety distance can be back-calculated by Equation (2), and the calculation results are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Serial Number** & **Travel Speed (km/h)** & **Safety Distance (m)** \\ \hline
1 & 10 & 34.6 \\
2 & 20 & 57.4 \\
3 & 30 & 72.2 \\
4 & 40 & 83.2 \\
5 & 50 & 92.0 \\
6 & 60 & 99.3 \\
7 & 70 & 105.5 \\
8 & 80 & 111.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Calculation of safety distance under various conditions.  
From the calculation results in Table 4, it can be seen that with the increase in travel speed, the safety distance also increases, and the safety distance is positively correlated  
Figure 7: Decay relationships between vibration velocity and distance. (**a**) Travel speed 10 km/h; (**b**) Travel speed 20 km/h; (**c**) Travel speed 40 km/h; (**d**) Travel speed 60 km/h.  
with the travel speed. The safe area of cultural relics can be drawn by fitting curves, as shown in Figure 8.