# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 2 Research Methodology
### Selection of Optimization Parameters for Shock Absorbers  
Traditional torsional damper configuration, rooted in the dual-pendulum model, faces limitations in accurately characterizing multi-cylinder engine dynamics [22; 23]. Nevertheless, empirical guidelines derived from decades of practice--such as maintaining a damper-to-crankshaft inertia ratio between 0.1 and 0.3--provide critical baseline parameters for initial optimization scoping. Given that excitation torque at the first-order natural frequency predominantly induces resonant stresses, damper tuning prioritizes the suppression of this dominant mode. In the dual-pendulum framework, the crankshaft system is simplified as a lumped inertial-elastic unit, where equivalent rotational inertia \(J_{\text{e}}\) and torsional stiffness \(K_{\text{e}}\) can expressed as:  
\[J_{e}=\sum J_{i}a_{i}^{2} \tag{17}\]  
\[k_{e}=\sum J_{e}w_{i}^{2} \tag{18}\]  
Among them, \(J_{i}\) is the moment of inertia of the \(i\)-th node of the crankshaft system without torsional vibration dampers, \(\alpha_{i}\) is the torsional vibration amplitude of the \(i\)-th node of the crankshaft system without torsional vibration dampers, and \(\omega_{i}\) is the first-order natural frequency of the original crankshaft system.  
According to Equations (17) and (18), the equivalent moment of inertia and equivalent torsional stiffness of the crankshaft system of an eight cylinder diesel engine without torsional vibration dampers are 22.20 kg\(\cdot\)m\({}^{2}\) and 10.34 (MN\(\cdot\)m/rad), respectively.  
According to the calculation of the double torsion pendulum model, when the inertia ratio is constant, the optimal tuning ratio \(\lambda\) is the damper-to-system frequency ratio. It can be expressed as  
\[\lambda=\frac{1}{1+\mu} \tag{19}\]  
Figure 3: Comparison of calculation results for eight cylinder shaft system: (**a**) diagram of primary harmonic response of shaft system; (**b**) diagram of maximum torque of shaft system.  
In the above equation, \(\mu\) is the damper to crankshaft inertia ratio. When the optimal tuning ratio is chosen for the double pendulum model, the vibration amplitude of the crankshaft system is minimized, and the corresponding optimal damping ratio \(\xi\) can be expressed as  
\[\xi=\sqrt{\frac{3\mu}{8(1+\mu)^{3}}} \tag{20}\]  
Given the correlation between tuning ratio selection and critical rotational speeds, conventional optimal tuning ratios may not universally apply across a diesel engine's full operating range. To address this, parameter optimization was conducted at both upper and lower bounds of the recommended inertia ratio (2.22-6.66 kg\(\cdot\)m\({}^{2}\)). The search domain was expanded by \(\pm\)5% around baseline tuning ratios and corresponding damping coefficients, yielding stiffness and damping optimization ranges of 0.771-2.025 (MN\(\cdot\)m\(/\)rad) and 483-2160 (N\(\cdot\)m\(\cdot\)s\(/\)rad), respectively. A dataset of 10,000 randomly generated parameter combinations was systematically evaluated using the developed algorithm to quantify torsional responses and stress concentrations. This multi-variable screening approach enables the identification of Pareto-optimal damper configurations that simultaneously minimize vibrational amplitudes and fatigue-inducing stresses, particularly critical for resonant frequency mitigation under transient operating conditions.