# Ride Comfort Prediction on Urban Road Using Discrete
## 3 Numerical Simulation
### Road Profile  
To guarantee the reliability of the simulation results, we collected 108 real urban road profiles with an IRI range of 0.82-12.95 and a distance range of 58-1282 m. The elevations of the road profiles were captured by a laser-based profiler in Shanghai, China. The profiler was equipped with five laser sensors, and thus we could easily capture the road profiles of both the right and left wheel paths. During the measurement, the moving speed of the profiler was not fixed, meaning that the profile data were discrete and had an unequal space  
Figure 3: Scheme of the full-car model.  
interval (about 0.1-0.3 m). Thus, a cubic spline interpolation method was first applied to obtain a uniform road profile, with a space interval of 0.01 m, which corresponded to the spatial frequency of 100 m\({}^{-1}\).  
Prior to the simulation and further analysis, the original road profile data needed to be filtered because the raw data usually contained surface irregularities and cracks. To this end, [PERSON] [25] applied a 250 mm moving average filter to smooth the road profile, which was widely used in the current studies. Although this filter is suitable for the IRI calculation, it is not applicable to calculate the DRI because the moving average filter can hardly eliminate some sudden changes in the road profile. The DRI calculation is very sensitive to the emergent changes in the road profile, but the vertical acceleration of the seat is hardly influenced by such emergent changes. Since the road profile can be considered as the system excitation, and the vertical acceleration of the seat is regarded as the system response, we can obtain the corresponding frequency response function (FRF) at various speeds, as shown in Equation (17),  
\[H(\omega_{s})=\frac{a_{seat}(\omega)}{z_{road}(\omega_{s})} \tag{17}\]  
where \(a_{seat}(\omega)\) and \(z_{road}(\omega_{s})\) are the frequency spectrums of acceleration on the seat and the road profile, respectively. Note that the \(\omega_{s}\) is spatial frequency, whose unit is m\({}^{-1}\). By solving the response excited by a unit impulse of the road profile, FRF curves at different speeds are obtained (see Figure 4).  
It is seen that the magnitudes of the frequency response function are mainly distributed in the spatial frequency range of [0, 6] m\({}^{-1}\), which indicates a slight effect of high-frequency parts (over 10 m\({}^{-1}\)) on the vertical acceleration of the seat. Therefore, we adopted a wavelet-based filter to smooth the original road profile. The original profile data were firstly decomposed using a five-level wavelet, and then the fifth component corresponding to the frequency bands of [0, 6.25] was selected as the smoothed data. Figure 5 shows the original profile and two smoothed road profiles using the moving average filter and wavelet-based filter, respectively. It is seen that the wavelet-based filter can effectively eliminate the abrupt change at around 17 m.  
Next, the DRI and IRI values of the road profiles are computed and compared, as illustrated in Figure 6. It is noteworthy that the box plot illustrates the distribution of the DRI across each road section. It can be observed that road sections with higher IRI values exhibit a more scattered distribution of the DRI, indicating the presence of numerous localized irregularities in those sections. Moreover, even in road sections with lower IRI values, there are still some localized irregularities present. This suggests that the DRI is more sensitive to localized abnormal irregularities compared to the IRI.  
Figure 4: Frequency response functions at different speeds.