## 3 Dynamic Load Generation
### Pavement Roughness Generation
The sampling space and the sampling length for the discrete Fourier transform should be controlled by satisfying Equation (3) to avoid PSD confounding and to ensure a meaningful lower limit of the effective spatial frequency \(n_{l}\):  
\[\Delta l\leq\frac{1}{2n_{u}},\ L\geq\frac{1}{n_{l}} \tag{3}\]  
where \(\Delta l\) is the sampling spacing distance of the pavement roughness in m; \(L\) is the total sampling length of the pavement roughness in m.  
In this paper, \(n_{l}\) is taken as 0.01 m\({}^{-1}\), \(n_{u}\) is taken as 3 m\({}^{-1}\), \(\Delta l\) is taken as 0.1 m, \(L\) is taken as 1000 m. Referring to the classification method by ISO 8608 [20] and the literature [6], four pavement roughness levels ranging from level A to D were selected, as shown in Table 1. It can be found that \(G_{q}(n_{0})\) increases as the pavement roughness deteriorates, the average value of which becomes four times the original when the roughness level develops to the next level, and the average \(G_{q}(n_{0})\) values of levels A, B, C, and D are taken as 16, 64, 256, and 1024, respectively. The discrete roughness is then generated following the steps below:  
(1) Let \(x_{m}\) (\(m\) = 0, 1, 2, \(\ldots\), \(N-1\), \(N\) is an even number) be the pavement discrete roughness data. \(X_{k}\) is the result of the discrete Fourier transform of \(x_{m}\). The relationship between \(X_{k}\) and the pavement roughness PSD function \(G_{q}(n)\) is shown in Equations (4)-(6):  
\[X_{k}=\sqrt{\frac{N}{2\Delta l}G_{q}(n_{k})}e^{i\varphi_{k}}\bigg{(}k=0,1, \ldots,\frac{N}{2}\bigg{)} \tag{4}\]  
\[n_{k}=k\Delta n \tag{5}\]  
\[\Delta n=\frac{1}{L} \tag{6}\]  
where \(X_{k}\) is the result of Fourier transform at spatial frequency \(n_{k}\); \(\varphi_{k  
on Equation (4). Then, the discrete Fourier inverse transform of \(X_{k}\) gives the discrete data of the pavement roughness (\(x_{m}\)) in the spatial domain, as shown in Equation (7):  
\[x_{m}=\frac{1}{N}\sum_{k=0}^{N-1}X_{k}e^{\frac{2x_{m}y_{m}}{N}}(m=0,1\ldots,N-1) \tag{7}\]  
where \(x_{m}\) is the roughness information at sampling location \(m\Delta l\) in m.  
The generated pavement roughness curves in the spatial domain for different roughness levels are shown in Figure 1a. The statistical distribution information is shown in Figure 1b and Table 2. It can be seen that the pavement roughness fluctuation increases from level A to level D, and the mean and median values remain near 0. Therefore, the generated roughness data are consistent with the assumption of a Gaussian distribution with a mean value of 0. In particular, when \(G_{q}(n_{0})\) expands by a factor of 4, the standard deviation of pavement roughness expands by about a factor of 2, which is consistent with Equation (  
To validate the accuracy of the generated discrete roughness data in the spatial domain, their PSD functions are calculated by Equations (8)-(10). The PSD functions of the generated discrete roughness data for different pavement roughness levels are the black curves shown in Figure 2. The straight lines are the theoretical PSD functions of level A, B, C, and D pavement roughness, obtained based on Equation (1). It can be observed that the PSD functions based on the generated discrete roughness data fluctuate around the theoretical PSD functions, which indicates that the generated discrete roughness data represent very well the different roughness levels as shown in Table 1. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Pavement Level** & **A** & **B** & **C** & **D** \\ \hline Standard deviation (mm) & 3.21 & 6.30 & 12.61 & 29.60 \\ Mean value (mm) & 0.04 & 0.07 & 0.21 & 0.02 \\ Minimum (mm) & \(-\)10.91 & \(-\)19.78 & \(-\)34.94 & \(-\)84.22 \\
25\% value (mm) & \(-\)2.05 & \(-\)4.22 & \(-\)8.30 & \(-\)20.70 \\ Median value (mm) & 0.28 & 0.14 & 0.04 & \(-\)1.17 \\
75\% value (mm) & 2.29 & 4.37 & 8.57 & 21.86 \\ Maximum (mm) & 9.61 & 15.86 & 49.76 & 100.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The statistical distribution information for different roughness conditions.