## 7 Conclusions  
A full rigid body dynamics model and a rigid-flexible coupled model for the flexible frame of an inner axle-box bogie of a battery track engineering vehicle were established. Nonlinear critical speed analysis, curving performance theory analysis, and stability optimization were performed for the rigid-flexible coupled model of a battery track engineering vehicle.  
We can draw the following conclusions:  
(1) Correlation and sensitivity analyses of the primary suspension design variables of the bogie by the optimal Latin hypercube design methodology revealed that lateral acceleration correlates the most with Longitudinal stiffness and is most affected by Longitudinal stiffness, while vertical acceleration correlates the most with Longitudinal stiffness and is most affected by Longitudinal stiffness.  
(2) Using the NSGA-II and AMGA algorithms, the primary suspension parameters were multi-objectively optimized by taking the minimum values of the lateral acceleration and vertical acceleration as the optimization objectives. The optimizations reduced the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Suspension Parameter** & **Pre-Optimization** & **Post-Optimization** & **Rate of Change/\%** \\ \hline longitudinal stiffness & 500,000 & 585,706 & 17.141 \\ \(x_{2}\)/(N\(\cdot\)m\({}^{-1}\)) & 10,000 & 11,938 & 19.380 \\ lateral damping & 25,000 & 28,874 & 15.496 \\ \(x_{5}\)/(N\(\cdot\)s\(\cdot\)m\({}^{-1}\)) & 100,000 & 86,778 & \(-\)13.222 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of various suspension parameters before and after optimization using the AMGA algorithm.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dynamics Index** & **Pre-Optimization** & **Post-Optimization** & **Rate of Change/\%** \\ \hline disengage rail factor & 0.770 & 0.770 & 0 \\ wheel axle lateral force/N & 19,338 & 22,182.100 & 14.707 \\ wheel load reduction rate & 0.270 & 0.210 & 22.222 \\ lateral acceleration & & & \\  /m\(\cdot\)s\({}^{-2}\) & 1.800 & 1.380 & \(-\)23.333 \\ vertical acceleration & & & \\  /m\(\cdot\)s\({}^{-2}\) & 0.715 & 0.690 & \(-\)3.497 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Comparison of the dynamic indexes before and after optimization using the AMGA algorithm.  
vertical acceleration of the vehicle from 0.715 to 0.69 during driving, which improved the stability of the vehicle by 3.497%. Furthermore, they reduced its lateral acceleration from 1.8 to 1.38, which improved its stability by 23.333%, and the optimization effect was evident.  
Combining the dynamics model with the optimization algorithm for multi-objective optimization, the vertical and lateral stability of a battery track engineering vehicle navigating a minimum radius curve was significantly improved. This approach not only improved the performance of the vehicle but also demonstrated the effectiveness of the combined model and algorithm in enhancing its stability. The results of this study lay a theoretical foundation for the improvement of vehicle dynamics performance and the operational safety of rail vehicles.  
Conceptualization, J.Z. and Y.S.; methodology, J.Z. and M.Z.; software, J.Z.; validation, J.Z.; formal analysis, J.Z. and C.W.; investigation, C.W.; resources, C.W.; data curation, J.Z.; writing--original draft preparation, J.Z.; writing--review and editing, J.Z. and M.Z.; visualization, J.Z.; supervision, M.Z.; project administration, Y.S.; funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Hubei Province, a major project of the Indian broad gauge ballast shaping car key technology research and manufacturing, grant number 2022 EHB022.  
Not applicable.  
Data Availability StatementData are available on request from the authors.  
The authors declare no conflicts of interest.