## 3 Stochastic Analysis of Ride Comfort Index
### gPC Results for Human-Vehicle Model with Uncertain Parameters
The former is at least 160 times more efficient than the latter. Second, although the average values of the acceleration are the same, the distributions of the probability densities are different depending on the uncertain parameters. Third, when \(k_{1}^{T}\) is uncertain in Figure 8e,f, the probability density of acceleration appears as a solid line instead of adistribution due to a small standard deviation. This is because that \(k_{1}^{T}\) is located at the very position of the hip joint that we calculated. In order to investigate the effect of the variations on vertical accelerations, we calculated the probability densities for the uncertain parameter \(k_{1}\) and \(k_{3}\) in detail. For example, the probability density function of the acceleration at \(t=21.1\) s (red dashed line in Figure 8)  
Figure 8: Variations in the vertical acceleration at the hip joint for several uncertain parameters. Subfigures in (**a,b**), (**c,d**), and (**e,f**) are for uncertain parameters \(k_{1}\), \(k_{3}\), and \(k_{1}^{T}\), respectively. Additionally, figures in (**a,c,e**) are calculated using gPC, and in (**b,d,f**) are calculated using MCS. is shown in Figure 9. Here, the shaded blocks and solid lines represent the MCS and gPC results, respectively. In the case of uncertain parameter \(k_{1}\), a wider distribution is observed than that of \(k_{3}\). From a design point of view, it is crucial that the variation of the acceleration shows a different distribution depending on the uncertainty of the parameter. However, it is not practical to examine the variations for all time responses. Therefore, it is necessary to quantify a specific value that takes into account all of the time responses. Further, we used BS 6841 to quantify subjective ride comfort as a ride index [37]. To calculate the ride index, the acceleration for each axis of the human model is taken from the human-vehicle simulation, as illustrated in Figure 10. Applying Fourier transform, frequency weighting can be imposed using a convolution integral. The inverse Fourier transform of the frequency-weighted acceleration produces the time history of the acceleration, which can be weighted again by multiplying the factor at each axis. Now the root mean square of the time-weighted acceleration on each axis represents the ride index of each axis. Since the developed human model considers two-dimensional motion, in this study, only five accelerations of vertical, horizontal, and rotational directions at the hip joint and vertical and horizontal directions at the back are used in computations. Summing the riding indices of each axis, we can finally quantify the overall ride index of the human model. Figure 10: Procedure to quantify ride comfort based on BS-6841 [46]. Figure 9: Probability density functions of the vertical accelerations when \(t=21.1\) s at hip joint in C-class road for a different uncertain parameter, (**a**) \(k_{1}\) and (**b**) \(k_{1}^{T}\). Where shaded blocks and solid line indicate the MCS and gPC results, respectively. Figure 11 shows the overall ride indices with respect to road roughness for three different uncertain parameters with the discomfort scale represented in BS 6841. We computed all ride indices and standard deviations by the approximate model based on gPC for each uncertain parameter. At each road roughness, the solid circle represents the mean of the ride indices, and the error bar represents 3\(\sigma\), which implies that 99.7% of the data are in this range. The overall ride indices associated with class-A and B road roughness are placed in the 'not uncomfortable' zone, whereas those associated with class-E represent the 'fairly uncomfortable' zone. In general, class-A indicates a smooth surface, while class-C indicates average road roughness (see Table 1). Therefore, it can be predicted easily that the ride indices at E-class would place in a 'fairly uncomfortable' zone. It is also easy to understand that the standard deviation on rough road is greater than on smooth road.