## 4 Conclusions  
The torsion beam suspension in vehicles is known to exhibit significant responses, even to small vibrations, and is difficult to adjust, unlike in independent suspension systems. Because of these drawbacks, the torsion beam suspension provides poor ride comfort. In this study, to enhance the structural stability of the rear torsion beam suspension in small SUVs, various shapes of cross-members with additional reinforcements were investigated, and structural and stiffness analyses were performed. Driving stability analyses were conducted for cross-member shapes with different reinforcement curvatures, and the following conclusions were drawn:  
1. To achieve structural stability in a torsional beam, it is crucial to minimize both bending and torsional stresses. This study revealed that the reinforcements with concave shapes exhibited lower stresses than those with convex shapes.
2. As the torsional rigidity increased, torsional deformation caused by the external forces decreased, indicating stronger stiffness. The analysis results showed that,  
\begin{table}
\begin{tabular}{c c c c} \hline
**Evaluation Factor** & **Type** & **Max(\({}^{\circ}\))** & **Min(\({}^{\circ}\))** \\ \hline Smooth Road Shake & Type 1 & 3.226 & \(-\)3.566 \\ Type 4 & 3.156 & \(-\)3.502 \\ \hline Impact Hardness and Road Shake & Type 1 & 3.258 & \(-\)3.397 \\ Type 4 & 3.250 & \(-\)3.461 \\ \hline \end{tabular}
\end{table}
Table 13: Best SVSA angle in roll analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Evaluation Factor** & **Type** & **Max(\({}^{\circ}\))** & **Min(\({}^{\circ}\))** \\ \hline Smooth Road Shake & Type 1 & 3.389 & \(-\)3.740 \\ Type 4 & 3.389 & \(-\)3.740 \\ \hline Impact Hardness and Road Shake & Type 1 & 3.536 & \(-\)3.673 \\ Type 4 & 3.536 & \(-\)3.673 \\ \hline \end{tabular}
\end{table}
Table 12: Best SVSA angle in ride analysis.  
among the concave shapes, Type 4, with higher torsional rigidity, exhibited the most stable structure.
3. Based on a multibody dynamics simulation, the SVSA angle data for different cross-member shapes were analyzed, and the results showed that the Type 4 shape provided the most stable ride. Although comfort perception based on the driving habits of the drivers may differ significantly, adopting a Type 4 cross-member shape could lead to noticeable overall improvements in driving stability.
4. SRS, impact hardness, and memory shake represent the longitudinal and vertical accelerations experienced by the driver at the steering wheel, seat, and foot positions, respectively, during vehicle operation. Based on the analysis results, Types 1 and 4 exhibited the most comfortable ride characteristics for each of these factors.  
Generally, ride comfort evaluation is performed based on the vibration caused by acceleration under dynamic conditions. In contrast, the analysis in this study was based on the geometry factor called SVSA under static conditions, because SVSA is a kinematic factor and does not perform equally well under dynamic conditions.  
[PERSON], [PERSON] and [PERSON] conceived and designed the data analysis, analyzed the data, and wrote the paper; [PERSON] and [PERSON] performed the analysis and produced the analytic data; [PERSON] researched the paper and proposed the direction. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The authors declare no conflict of interest.