## 4 Methods for Calculating the Dynamic Response of Crankshaft System Substructures  
### Crankshaft Substructure Forced Vibration Calculation Method  
The Newmark method is commonly used for dynamic response calculations. However, the substructure model of the crankshaft system often still contains a large number of degrees of freedom. Due to factors such as the selection of master node positions and the distribution of mass and stiffness, significant parameter differences and frequency discrepancies may arise in the dynamic model, leading to coupled responses. This results in poor numerical performance due to stiffness and challenging accurate dynamic response calculations. To address this, the implicit Newmark method is employed for numerical integration, with a prediction-correction mechanism introduced for each time step. By iteratively optimizing the approximate solutions for displacement and velocity, local truncation errors are minimized, resulting in a more adaptable and reliable method for dynamic response calculation. The specific calculation method is as follows. The Newmark-\(\beta\) method computes for each time step within the time interval \([t,t+\Delta t]\) as follows:  
\[\hat{K}=K+a_{0}M+a_{1}C \tag{22}\]  
The effective load at the time \([t,t+\Delta t]\) is  
\[\hat{F}_{t+\Delta t}=F_{t+\Delta t}+M\big{(}a_{0}x_{t}+a_{2}\dot{x}_{t}+a_{3} \ddot{x}_{t}\big{)}+C\big{(}a_{1}x_{t}+a_{4}\dot{x}_{t}+a_{5}\ddot{x}_{t}\big{)} \tag{23}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Coupling Region Size** & **Radial Threshold** & **Axial Threshold** \\ \hline Moderate coupling region & d/3 & d/20 \\ Small coupling region & d/5 & d/50 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter settings for different coupling regions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency** & **Take 1 Master Node and** & **Take 3 Master Nodes and** & **Take 3 Master Nodes** & **ANSYS 19.2** \\
**Order** & **a Moderate Coupling** & **a Moderate Coupling** & **and a Small Coupling** & **Calculations (Hz)** \\  & **Region (Hz)** & **Region (Hz)** & **Region (Hz)** & **Classical \\ \hline
1 & 28.946 & 29.874 & 29.209 & 30.85 \\
2 & 30.186 & 31.238 & 30.461 & 32.38 \\
3 & 77.591 & 80.103 & 79.259 & 80.307 \\
4 & 83.034 & 86.236 & 84.135 & 86.754 \\
5 & 91.382 & 93.684 & 91.407 & 95.053 \\
6 & 140.508 & 145.361 & 141.203 & 143.42 \\
7 & 151.237 & 156.025 & 154.349 & 152.89 \\
8 & 168.614 & 175.562 & 171.268 & 160.93 \\
9 & 224.381 & 233.316 & 226.185 & 229.62 \\
10 & 258.979 & 268.536 & 261.762 & 266.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of crankshaft-free vibration intrinsic frequency results under different modelling conditions. The main journals were defined using one and three master nodes, respectively, and the three master nodes were also selected to calculate smaller coupling regions. The effective displacement at the time \([t,t+\Delta t]\) is  
\[\hat{K}x_{t+\Delta t}=F_{t+\Delta t} \tag{24}\]  
The velocity and acceleration at the time \([t,t+\Delta t]\) are  
\[\dot{x}_{t+\Delta t}=\dot{x}_{t}+a_{6}\ddot{x}_{t}+a_{7}\ddot{x}_{t+\Delta t} \tag{25}\]  
\[\ddot{x}_{t+\Delta t}=a_{0}(x_{t+\Delta t}+x_{t})-a_{2}\dot{x}_{t}-a_{3}\ddot{ x}_{t} \tag{26}\]  
where \(\delta\) and \(\beta\) are adjustable parameters based on accuracy and stability requirements. When \(\delta\geq 1/2\), \(\beta\geq 1/4\big{(}1/2+\delta\big{)}^{2}\), the Newmark-\(\beta\) method is unconditionally stable. Additionally, let  
\[ \begin{array}{l}a_{0}=1/\big{(}\beta\Delta t^{2}\big{)},a_{1}=\delta/(\beta \Delta t),a_{2}=1/(\beta\Delta t),a_{3}=1/2\beta-1,a_{4}=\delta/\beta-1,\\ a_{5}=\Delta t/2(\delta/\beta-2),a_{6}=\Delta t(1-\delta),a_{7}=\delta\Delta t \end{array}  \tag{27}\]  
To address the local truncation error caused by high-frequency vibrations, each time step first computes the predicted displacement and then the corrected displacement.