# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 2 Research Methodology  
### Torsional Vibration Calculation Model  
The lumped-parameter model remains a highly efficient approach for torsional vibration analysis due to its simplified structure and strong correlation with experimental data [17; 18]. Figure 1 illustrates this methodology applied to an eight-cylinder diesel engine crankshaft, where the system is discretized into 20 rotational degrees of freedom with concentrated masses: \(J_{1}\) represents the combined moment of inertia of the front shaft and main journal, \(J_{20}\) corresponds to the rear shaft and flywheel assembly, and \(J_{19}\) characterizes the rear shaft with crank throws. Intermediate masses \(J_{2}\)-\(J_{18}\) model individual crank throws and pins, incorporating reciprocating masses at the crankpins. System dynamics are further defined by C\({}_{ri}\) (internal crankshaft damping), C\({}_{oi}\) (cylinder-piston assembly damping), and \(K_{i}\) (the torsional stiffness of the shaft segments, \(i\) = 1-19). This configuration balances computational efficiency with mechanical fidelity. This is particularly advantageous for iterative optimization processes that require fast parameter scanning. The internal crankshaft damping is viscous damping, while the cylinder-piston damping constitutes hybrid damping. The computational model in this study is based on an idealized formulation that disregards temperature variations and clearance effects. The torsional vibration dynamic model of the eight cylinder diesel engine is  
\[\overset{\cdot\cdot\cdot}{JX}+\overset{\cdot\cdot}{CX}+\overset{\cdot\cdot}{XX }=T \tag{1}\]  
where \(J\) is the rotational moment of inertia matrix, \(C\) is the damping matrix, \(K\) is the stiffness matrix, and \(T\) is the torque excitation. The torsion angle of the corresponding unit for the shaft system is \(X\). And the above matrix of the simplified crankshaft model can be expressed as:  
\[J= \begin{bmatrix}J_{1}&0&0&0&0\\ 0&J_{1}&0&0&0\\ 0&0&\ddots&0&0\\ 0&0&0&J_{19}&0\\ 0&0&0&0&J_{20}\end{bmatrix} \]  
\[K= \begin{bmatrix}K_{1}&-K_{1}&0&0&0\\ -K_{1}&K_{1}+K_{2}&-K_{2}&0&0\\ 0&0&\ddots&0&0\\ 0&0&-K_{18}&K_{18}+K_{19}&-K_{19}\\ 0&0&0&-K_{19}&K_{19}\end{bmatrix} \]  
Figure 1: Aggregate parameter model diagram. \[C= \begin{bmatrix}C_{\text{r1}}&-C_{\text{r1}}&0&0&0&0\\ -C_{\text{r1}}&C_{\text{r1}}+C_{\text{r2}}&-C_{\text{r2}}&0&0&0\\ 0&0&\ddots&0&0&0\\ 0&0&-C_{\text{r17}}&C_{\text{r17}}+C_{\text{r18}}+C_{\text{o8}}&0&0\\ 0&0&0&0&\ddots&-C_{\text{r19}}\\ 0&0&0&0&-C_{\text{r19}}&C_{\text{r19}}\end{bmatrix} \]  
In the simplified crankshaft model employing the lumped parameter method for dynamic characteristic analysis, a typical crankshaft system primarily consists of stepped shaft segments and periodically distributed crank throws [19]. For a typical hollow cylinder, its torsional stiffness can be expressed as  
\[K=\frac{\pi}{32}G\frac{D^{4}-\text{d}^{4}}{L} \tag{2}\]  
where \(G\) is the shear elastic modulus, \(L\) is the length of shaft end, \(D\) is the outer diameter of the shaft end, and d is the inner diameter of the shaft end. Due to diameter transitions in stepped shafts, the thicker segments exhibit negligible deformation, and their torsional stiffness can be expressed as  
\[K=\frac{G}{32}\frac{D_{1}^{4}D_{2}^{4}}{[D_{2}^{4}(l_{1}+\lambda)+D_{1}^{4}(l_{ 2}-\lambda)]} \tag{3}\]  
where \(l_{1}\) is the length of the thin segments, \(l_{2}\) is the length of the thick segments, \(D_{1}\) is the diameter of the thin segments, \(D_{2}\) is the diameter of the thick segments, and \(\lambda\) is the length of the coarse axis that does not participate in deformation.