# Parameter-Matching Multi-Objective Optimization for Diesel Engine Torsional Dampers
## 2 Research Methodology
### Free Vibration Calculation  
The current free vibration analysis of crankshaft systems predominantly employs the Holzer method, which involves laborious tabular computations. To enhance computational efficiency and adaptability, this study adopts numerical approaches for solving free vibration equations. Firstly, the differential equation of free vibration [20] is  
\[J\overset{\cdot\cdot}{X}+KX=0 \tag{5}\]  
Let its general solution be  
\[X(t)=X_{0}\cos(\omega t+\varphi) \tag{6}\]where \(X_{0}\) is the n-order mode array vector, which is called the main array of the system; \(\omega\) is the circular frequency; and \(\varphi\) is the initial phase. Substituting the solution of the equation into Equation (5) yields  
\[KX_{0}=\omega^{2}JX_{0} \tag{7}\]  
The symmetric positive definite mass matrix \(J\) can be decomposed into  
\[J=LL^{T} \tag{8}\]  
due to being a lower triangular matrix. The generalized eigenvalue eigenvector problem (4) can be transformed into a standard eigenvalue eigenvector problem  
\[NY=\omega^{2}Y \tag{9}\]  
where the eigenvalue of the matrix \(N\) is the square of the natural circular frequency of the system, and its corresponding eigenvector is \(Y\). The matrix is represented as  
\[N=L^{-1}K\Big{[}L^{-1}\Big{]}^{T} \tag{10}\]  
The symmetric positive-definite property of the matrix facilitates efficient eigenvalue and eigenvector computation. However, the multi-degree-of-freedom crankshaft model adopted in this study, while improving analytical accuracy, introduces numerical challenges due to significant disparities between rotational inertia and torsional stiffness across shaft segments. Such mechanical heterogeneity results in ill-conditioned matrices, where high condition numbers degrade computational precision, causing eigenvalue distortion or even divergence in eigenvector solutions. This inherent instability underscores the necessity of robust numerical algorithms tailored for systems with extreme parameter variations, particularly when balancing computational efficiency against fidelity in torsional vibration analysis. For systems with limited degrees of freedom and potential zero-frequency modes, the Jacobi iterative method remains practical due to its simplicity. While the Lanczos method offers superior precision and flexibility for large-scale structural vibration analyses, its higher computational overhead renders it less suitable for crankshaft-specific applications. To address systems with limited torsional degrees of freedom and poor numerical conditioning, this study enhances the Jacobi Screening method by incorporating dynamic threshold mechanisms. The specific iterative process is shown in Figure 2.  
These adaptations systematically reduce computational redundancy and oscillation artifacts--common pitfalls in ill-conditioned crankshaft models--thereby improving both convergence rates and solution accuracy. The refinements strike an optimal balance between algorithmic robustness and computational economy, particularly critical when resolving closely spaced eigenvalues or mitigating round-off errors in systems exhibiting extreme stiffness-to-inertia ratios. This tailored approach circumvents the excessive resource demands of Lanczos methods while surpassing conventional Jacobi iterations in precision, making it uniquely suited for torsional vibration analyses where model fidelity and numerical stability outweigh sheer computational scale. The formula is  
\[X_{i}^{(k)}=\frac{\left(b_{i}-\sum\limits_{j=1,j\
eq 1}^{n}a_{ij}x_{j}^{(k)}- \sum\limits_{j=j+1}^{n}a_{ij}x_{j}^{(k-1)}\right)}{a_{ii}} \tag{11}\]  
where \(b_{i}\) is the constant term of the \(i\)-th equation, \(a_{ij}\) represents the element in the \(i\)-th row and \(j\)-th column of the coefficient matrix, \(x_{j}^{(k)}\) indicates the value of the \(j\)-th element after the \(k\) iteration, \(x_{j}^{(k-1)}\) is the old value from the \(k-1\)-th iteration of the element, and \(a_{ii}\) represents the diagonal elements of the coefficient matrix.