# Analytical Investigation of Vertical Force Control in In-Wheel Motors for Enhanced Ride Comfort
## 3 Results and Discussion
### Frequency-Weighted Acceleration  
The frequency-weighted acceleration, \(a_{w}\), is an indicator used to evaluate ride comfort; lower values of the frequency-weighted acceleration correspond to better ride comfort. The frequency-weighted acceleration is calculated from Equations (39) and (40), based on the Power Spectral Density (PSD) of the sprung mass vertical acceleration, \(\bar{z}_{b}\), shown in Figure 13. Comparisons of the frequency-weighted acceleration for each type of road profile are presented in Tables 3-5. Table 3 shows the frequency-weighted acceleration  
Figure 13: PSD of the sprung mass vertical acceleration: (**a**) PSD of the open-loop system, (**b**) PSD of the closed-loop system for road profile classes A to B, (**c**) PSD of the closed-loop system for road profile classes B to C, and (**d**) PSD of the closed-loop system for road profile classes C to D.  
for road profile classes A to B, Table 4 for road profile classes B to C, and Table 5 for road profile classes C to D.  
Tables 6-8 show that as the proportional gain (\(K_{p}\)) increases, the frequency-weighted acceleration (\(a_{w}\)) decreases, leading to enhanced ride comfort. This indicates that the method of controlling the driving force from the IWM using the proportional ride-blending controller, as discussed in Section 2.2, effectively reduces the vehicle's vibration. The decrease in \(a_{w}\) is directly correlated with the increase in \(K_{p}\). \begin{table}
\begin{tabular}{c c} \hline
**Proportional Gain, \(K_{p}\)** & **RMS of \(\tilde{z}_{b}\) (m/s\({}^{2}\))** \\ \hline
0 (Open-Loop) & 4.130 \\
100 & 4.014 \\
200 & 3.904 \\
300 & 3.800 \\
400 & 3.702 \\
500 & 3.609 \\ \hline \end{tabular}
\end{table} 
Table 6: The root mean square (RMS) of the vertical acceleration of the sprung mass for the road profile classes A to B. \begin{table}
\begin{tabular}{c c c} \hline
**Proportional Gain, \(K_{p}\)** & \
\begin{tabular}{c} **Frequency-Weighted** \\ **Acceleration (\(a_{w}\),m/s\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Percentage Decrease** \\ **(Compared with Open-Loop)** \\ \end{tabular} \\ \hline
0 (Open-Loop) & 0.3721 & 0\% \\
100 & 0.3622 & 2.6641\% \\
200 & 0.3528 & 5.1856\% \\
300 & 0.3439 & 7.5756\% \\
400 & 0.3355 & 9.8441\% \\
500 & 0.3275 & 12.000\% \\ \hline \end{tabular}
\end{table} 
Table 3: Proportional gain and frequency-weighted acceleration for road profile classes A to B. \begin{table}
\begin{tabular}{c c c} \hline
**Proportional Gain, \(K_{p}\)** & \
\begin{tabular}{c} **Frequency-Weighted** \\ **Acceleration (\(a_{w}\),m/s\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Percentage Decrease** \\ **(Compared with Open-Loop)** \\ \end{tabular} \\ \hline
0 (Open-Loop) & 0.7349 & 0\% \\
50 & 0.7250 & 1.3583\% \\
100 & 0.7153 & 2.6789\% \\
150 & 0.7058 & 3.9632\% \\
200 & 0.6966 & 5.2127\% \\
250 & 0.6877 & 6.4288\% \\ \hline \end{tabular}
\end{table} 
Table 4: Proportional gain and frequency-weighted acceleration for road profile classes B to C. \begin{table}
\begin{tabular}{c c c} \hline
**Proportional Gain, \(K_{p}\)** & \
\begin{tabular}{c} **Frequency-Weighted** \\ **Acceleration (\(a_{w}\),m/s\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Percentage Decrease** \\ **(Compared with Open-Loop)** \\ \end{tabular} \\ \hline
0 (Open-Loop) & 1.4884 & 0\% \\
25 & 1.4783 & 0.6800\% \\
50 & 1.4683 & 1.3506\% \\
75 & 1.4585 & 2.0119\% \\
100 & 1.4488 & 2.6641\% \\
125 & 1.4392 & 3.3073\% \\ \hline \end{tabular}
\end{table} 
Table 5: Proportional gain and frequency-weighted acceleration for road profile classes C to D.  
### Frequency-Weighted Acceleration  
Based on the IWM L1500 information in Figure 14, the wheel forces are limited. Figures 15-17 illustrate examples of the front wheel forces based on the proportional ride-blending controller as defined in Equation (28). These figures show the time response of the front wheel forces over the first 30 s for each type of road profile (classes A to B, B to C, and C to D) and for a variety of proportional gains.