## 3 Dynamic Load Generation
### Pavement Roughness Generation  
There are two major indexes for pavement roughness, including the international roughness index (IRI) and the power spectral density (PSD). The IRI was proposed by the World Bank and is widely used in transportation departments as a direct reflection of ride quality; the index measures the overall relative velocity between the axle and the sprung mass of the quarter car. PSD is widely used in the vehicle industry and the research community to analyze the dynamic interaction between the vehicle and the pavement, suspension optimization, and energy consumption of the vehicle. The PSD method is used in this study to generate the pavement roughness, as it is more mechanically related to the pavement behavior. A large number of measurements show that the pavement roughness is a uniform random field obeying Gaussian distribution with zero mean value, the characteristics of which can be described by a PSD function \(G_{q}(n)\) as shown in Equation (1). PSD is defined as the limiting mean-square value of a signal per unit frequency bandwidth, according to the definition of the PSD of vertical road profile displacement in ISO 8608 [20]. \[G_{q}(n)=\left\{ \begin{array}{ll}G_{q}(n_{0})\left(\frac{n}{n_{0}}\right)^{ -nv}&n_{l}<n<n_{u}\\ 0&other\end{array} \right. \tag{1}\]  
where \(n\) represents the spatial frequency in m\({}^{-1}\); \(n_{0}\) is the reference spatial frequency, taken as \(0.1\) m\({}^{-1}\); \(w\) is the frequency index, taken as \(2\)[20]; \(G_{q}(n_{0})\) is the PSD of pavement roughness at the reference spatial frequency \(n_{0}\), also known as the pavement roughness coefficient, in m\({}^{3}\). In this paper, the dynamic load is calculated by using a quarter-car model which is in point contact with the pavement. By referring to [21], if the width of the transverse joint in concrete pavement is small and no faulting exists, the influence of the transverse joint on the roughness is minor. Therefore, the influence of transverse joint or the slab length on thegeneration of the pavement roughness is ignored in this study. Due to the vehicle vibration suspension system, the displacement or acceleration response of the vehicle to a certain spatial frequency of pavement roughness excitation is extremely small; it is not necessary to consider all spatial frequency when generating the pavement roughness. Therefore, the spatial frequency is intercepted, and the lower and the upper limits of the effective frequencies are set to be \(n_{l}\) and \(n_{u}\), respectively, as shown in Equation (1). The lower and upper limits of \(n_{l}\) and \(n_{u}\) are related to the lower and upper limits of the natural frequency range (\(f_{l}\),\(f_{u}\)) of the vehicle vibration, respectively:  
\[n_{l}=\frac{f_{l}}{v},\ n_{u}=\frac{f_{u}}{v} \tag{2}\]  
where \(v\) is the speed of the vehicle in m/s. In general, the natural frequency of the body part of a vehicle is 1-2 Hz, and the natural frequency of the wheel part is 10-15 Hz [20]. The frequency range of [0.033 m\({}^{-1}\), 1.5 m\({}^{-1}\)] is calculated according to Equation (2). For engineering application, [0.011 m\({}^{-1}\), 2.83 m\({}^{-1}\)] was proposed by ISO 8608 as the effective spatial frequency range which covers the natural frequency of most vehicle vibrations when the vehicle speed ranges 36-108 km/h. The discrete Fourier transform method is used to construct the discrete pavement roughness data from Equation (1), which is used as the input for the dynamic load calculation for the quarter-car model, as illustrated in Section 2.