## 3 Results
### Vehicle Suspension Simulation
Comparing the frequency components of the acceleration using the power spectral density (PSD), the maximum frequency component has been reduced from 2.42 (m/s\({}^{2}\))\({}^{2}\)/Hz in the passive suspension  
Figure 9: Simulated road profiles: (**a**) Road bump, (**b**) Class A road according to ISO 8608.  
to 1.60 (m/s\({}^{2}\))\({}^{2}\)/Hz for the semi-active suspension. Furthermore, it is evident that the maximum suspension deflection does not exceed \(z_{max}=0.05\) m under any circumstances, thereby substantiating the viability of implementing the proposed methodology without compromising the integrity of the vehicle suspension system. In every instant, the results demonstrate that the suspension deflection is observed to be lower over time for the semi-active suspension in comparison with the passive suspension. The RMS values for the suspension deflection are 0.014, 0.012, and 0.011 m for the passive, semi-active, and active vehicle suspensions, respectively. It is evident that the normalized dynamic tire load (NDTL) never surpasses the value of 1, which serves to confirm that all vehicles are consistently maintaining contact with the road. It has been demonstrated that the semi-active suspension system has the capacity to enhance the vehicle's performance in terms of comfort when compared to the passive suspension system. Moreover, the system does not compromise driving safety, as it ensures that the tires maintain consistent contact with the road surface and that the vehicle suspension does not deflect beyond its mechanical limits. Although active suspensions are naturally superior in terms of performance, due to the fact that the MR damper force is limited by the sign of the term \(\dot{z}_{def}+\frac{V_{0}}{X_{0}}z_{def}\) in (1) and the maximum allowable MR damper current, it has been demonstrated that the proposed semi-active suspension system is an effective mean of enhancing vehicle performance with its inherent benefits when compared to active suspensions, which are that they do not add energy to the vehicle system, and they have a lower energy cost. Additional simulation tests for the road bump under various vehicle speeds are detailed in Table 3. \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Vehicle Speed** & **Passive** & **Semi-Active** & **Active** \\ \hline
15 km/h & 0.789 m/s\({}^{2}\) & 0.722 m/s\({}^{2}\) & 0.510 m/s\({}^{2}\) \\
20 km/h & 1.079 m/s\({}^{2}\) & 0.959 m/s\({}^{2}\) & 0.635 m/s\({}^{2}\) \\
25 km/h & 1.133 m/s\({}^{2}\) & 0.998 m/s\({}^{2}\) & 0.683 m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: RMS results for the vertical acceleration of the vehicle under different bump simulations. Figure 10: Simulation results under a road bump: (**a**) Vertical acceleration, (**b**) Vertical acceleration PSD, (**c**) Suspension deflection, (**d**) Normalized dynamic tire load, (**e**) Input force, (**f**) MR damper current. The Class A road simulation results are presented in Figure 11. Once again, it is demonstrated that the semi-active suspension can enhance the vehicle behavior compared to the passive suspension. The lower power spectral density (PSD) of the vertical acceleration for the semi-active suspension indicates that it has a more smooth and comfortable drive experience rather than the passive suspension. With regard to the suspension deflection, a notable enhancement is observable over time, with RMS values of 0.0025 m, 0.0015 m, and 0.0013 m for the passive, semi-active, and active suspensions, respectively. The results of this test demonstrate that safety is not compromised during typical highway driving conditions. Additionally, the vertical acceleration of the vehicle is reduced, which enhances ride comfort. Additional simulation tests for the Class A road under various vehicle speeds are detailed in Table 4.  
To demonstrate the interest of the proposed methodology, Table 5 provides a comparison with alternative approaches.