# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 4 Discussion  
The findings of this study highlight critical advancements in torsional damper optimization for multi-cylinder diesel engines while revealing nuanced trade-offs between vibration suppression and stress mitigation. The following points synthesize the implications of these results within the broader context of torsional vibration control. The modified Jacobi sweep algorithm, enhanced with dynamic thresholds, resolves ill-conditioned matrices in multi-degree-of-freedom crankshaft models, achieving <1% frequency deviations from ANSYS 19.2 benchmarks. Coupled with a Bayesian-optimized neural network and NSGA-II derived Pareto frontiers, this framework provides actionable engineering guidelines. High-inertia dampers are optimal for vibration-sensitive applications, while low-inertia solutions demand stiffness compensation to reconcile lightweight design with fatigue risks. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Parameters of Torsional**} & \multicolumn{2}{c}{**Pareto Optimization**} & \multicolumn{2}{c}{**Numerical Calculation**} & \multicolumn{2}{c}{**Error Value**} \\
**Vibration Damper** & \multicolumn{2}{c}{**Solution Parameters**} & \multicolumn{2}{c}{**Solution**} & \multicolumn{2}{c}{**Solution**} & \\ \hline
**Moment of** & **Torsional** & **Twist** & **Additional** & **Twist** & **Additional** & **Twist** & **Migig** \\
**Inertia** & **Stiffness** & **Angle** & **Stress** & **Angle** & **Stress** & **Angle** & **(\%)** & **Angle** \\ /(kg\(\cdot\)m\({}^{2}\)) & /(N\(\cdot\)m/rad) & /(mrad) & /(Mpa) & /(mrad) & /(Mpa) & /(Mpa) & /(\%) & /(\%) \\ \hline
2.22 & 1,309,201 & 3.44112 & 18.92056 & 3.44377 & 18.85 & 0.077 & 0.373 \\
2.46 & 1,508,795 & 3.43292 & 19.15328 & 3.41851 & 19.29 & 0.420 & 0.714 \\
5.25 & 1,790,494 & 3.05977 & 22.94072 & 3.05135 & 23.11 & 0.275 & 0.738 \\
6.46 & 1,849,238 & 2.88952 & 25.25194 & 2.89001 & 25.14 & 0.017 & 0.443 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Pareto optimization parameters. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Theoretical Shock**} & \multicolumn{2}{c}{**Pareto-Optimized Damper**} & \multicolumn{2}{c}{**Theoretical Shock**} & \multicolumn{2}{c}{**Change in Optimization**} \\
**Absorber Parameters** & \multicolumn{2}{c}{**Parameters**} & \multicolumn{2}{c}{**Absorber Response**} & \multicolumn{2}{c}{**Objectives**} \\ \hline
**Moment of** & **Torsional** & **Moment of** & **Torsional** & **Twist** & **Additional** & **Angle** & **Stress** \\
**Inertia** & **Stiffness** & **Inertia** & **Stiffness** & **Angle** & **Stress** & **Increase** & **Mitigation** \\ /(kg\(\cdot\)m\({}^{2}\)) & /(N\(\cdot\)m/rad) & /(kg\(\cdot\)m\({}^{2}\)) & /(N\(\cdot\)m/rad) & /(mrad) & /(Mpa) & /(\%) & /(\%) \\ \hline
2.22 & 865,311.4 & 2.22 & 1,309,201 & 3.4180 & 19.91 & 0.754 & 5.323 \\ \hline
2.46 & 940,286.6 & 2.46 & 1,508,795 & 3.3900 & 20.76 & 0.841 & 7.081 \\ \hline
5.25 & 1,619,536 & 5.25 & 1,790,494 & 3.0416 & 23.60 & 0.320 & 2.076 \\ \hline
6.46 & 1,828,091. & 6.46 & 1,849,238 & 2.8886 & 26.21 & 0.049 & 4.082 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison parameters between Pareto optimization and theoretical shock absorbers. The current study presents several inherent limitations stemming from its idealized framework, which assumes constant stiffness and damping characteristics while neglecting critical physical factors such as friction and thermal effects. The linearized model formulation may restrict the analysis to steady-state conditions and potentially obscures transient operational dynamics that could influence system stability. Future work should integrate multi-physics models (thermal-structural coupling) and validate Pareto-optimized parameters experimentally. This integrated methodology bridges theoretical simplifications and industrial demands, offering a systematic pathway for precision damper design in modern multi-cylinder diesel engines.