# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 2 Research Methodology
### Forced Vibration Calculation  
In the dynamic analysis of mechanical systems, solving forced vibration responses necessitates incorporating the coupling effects of periodic external excitation loads and damping dissipation mechanisms. This process requires the application of the Fast Fourier Transform (FFT) to decompose the time-varying cylinder excitation torque into discrete harmonic components. Subsequently, based on the crankshaft's structural modal parameters, the coupled multi-degree-of-freedom differential equations are decoupled into independent single-degree-of-freedom equations through modal coordinate transformation. The harmonic responses from each order are then superimposed to obtain the torque and additional stresses on the crankshaft system components, enabling subsequent investigations into parameter matching optimization for crankshaft vibration dampers.  
The differential equation for forced torsional vibration of the crankshaft system is:  
\[\dot{X}+\mathbf{C}\dot{X}+\mathbf{K}X=M_{e} \tag{12}\]  
Per classification society standards [19], the first 12 harmonics of each cylinder's output torque must be considered. Through engine excitation analysis, it is observed that the excitation torque diminishes with increasing harmonic order. Consequently, forced vibration analysis typically focuses on the vibrational responses of low-order dominant harmonics. For the eight-cylinder diesel engine case study, the single-cylinder excitation torque is expanded into a Fourier series:  
\[M_{e}=\sum_{n=0}^{\infty}A_{n}\sin(nwt+\phi_{n}) \tag{13}\]  
where \(n\) is the harmonic order, \(\omega\) is the circular frequency corresponding to the fundamental frequency, \(\phi_{n}\) is the phase angle of the \(n\)-th harmonic, and \(A_{n}\) is the amplitude of the \(n\)-th harmonic.  
The comprehensive response of nodes is the superposition of various harmonic responses. It can be expressed as:  
\[X(t)=\sum_{K=1}^{m}A_{K}\sin(\omega_{k}t+\phi_{k}) \tag{14}\]  
where \(A_{k}\) is the amplitude of the \(K\)-th mode, \(\omega_{k}\) is the circular frequency of the \(K\)-th mode, and \(\phi_{k}\) is the phase angle of the \(K\)-th mode.  
By calculating the comprehensive response of nodes and combining them with the corresponding torsional stiffness, the torque of each unit in the crankshaft system can be obtained. The magnitude of the torque of each unit in the crankshaft system is related to the displacement difference of its corresponding node. Therefore, the expressions for torque and additional stress can be expressed as:  
\[T=\frac{K_{e}}{2}(|R_{\text{Max}}|-|R_{\text{Min}}|) \tag{15}\]  
\[\tau_{e}=\frac{16T}{\pi D_{e}^{3}} \tag{16}\]  
In the above equation, \(K_{e}\) is the torsional stiffness of the unit, \(R_{\text{Max}}\) is the maximum value of the time-domain response of the node, \(R_{\text{Min}}\) is the minimum value of the time-domain response of the node, and \(D_{e}\) is the diameter of the circular cross-section where the unit is located.  
To validate the accuracy of the proposed model and algorithm, comparative analyses were conducted using AVL Excite commercial software for torsional vibration calculations in the crankshaft system. As shown in Figure 3a,b, the results for dominant harmonic responses and torque amplitudes exhibit deviations of 4.04% and 1.78%, respectively, between the proposed method and Excite simulations. These minor discrepancies demonstrate high computational reliability, confirming the methodology's suitability for optimizing torsional damper configurations in multi-cylinder engine systems. The close alignment with industry-standard software underscores the robustness of the numerical framework in balancing computational efficiency with predictive accuracy--a critical requirement for iterative design optimization under complex dynamic loading conditions.