## 4 Methods for Calculating the Dynamic Response of Crankshaft System Substructures
### Crankshaft Substructure Forced Vibration Calculation Method
The predicted displacement is given by  
\[q_{n+1}^{pred}=q_{n}+\Delta t\dot{q}_{n}+\Delta t^{2}\bigg{(}\frac{1}{2}- \beta\bigg{)}\ddot{q}_{n} \tag{28}\]  
The corrected displacement is obtained using iterative correction as follows:  
\[q_{n+1}^{corr}=q_{n+1}^{pred}+\beta\Delta t^{2}\ddot{q}_{n+1} \tag{29}\]  
The error assessment quantity \(E\) is defined as the difference between the two:  
\[E=\parallel q_{n+1}^{corr}-q_{n+1}^{pred}\parallel \tag{30}\]  
The error reflects the truncation error or the lack of iterative correction due to high-frequency modes in the current time step. The time step is adjusted according to the comparison of the error \(E\) with the preset tolerance error \(\mathcal{E}\). The following update formula is used:  
\[\Delta t_{new}=\Delta t(\frac{\epsilon}{E})^{\frac{1}{\beta}}. \tag{31}\]  
where \(p\) is the order of integration; for the Newmark method take \(p=2\); \(\mathcal{E}\) is the set error tolerance--take \(\mathcal{E}=0.05\). When \(E\geq\epsilon\), it indicates that the local truncation error is large and \(\Delta t\) should be reduced,  
When \(E\ll\epsilon\), \(\Delta t\) can be increased appropriately to improve the computational efficiency. In the nonlinear iterative solution at each time step, a dynamic relaxation factor \(\omega_{k}\) is added to correct the updating formulae to prevent instability caused by too large corrections:  
\[q_{n+1}^{(i+1)}=q_{n+1}^{(i)}+\omega_{k}\Delta q_{n+1}^{(i)} \tag{32}\]  
where \(\Delta q_{n+1}^{(i)}\) is the displacement correction obtained in the \(i\)-th iteration. This treatment effectively suppresses excessive corrections caused by high-frequency modes, ensuring iterative convergence. The dynamic relaxation factor \(\omega_{k}\) is defined as  
\[\omega_{k}=\frac{1}{1+\alpha\Delta t} \tag{33}\]where the parameter \(\alpha\) is related to the system's stiffness and damping characteristics. When the time step is small, \(\omega_{k}\) tends to have a smaller value, thereby reducing the correction magnitude. When the time step is large, \(\omega_{k}\) approaches 1, allowing for more thorough iterative corrections. At each time step, to solve for \(\tilde{q}_{n+1}\) and update \(q_{n+1}\), it is usually necessary to construct the \"effective stiffness matrix\". For classical problems, it can be represented as  
\[K_{eff}=\frac{1}{\beta\Delta t^{2}}M+\frac{\tau}{\beta\Delta t}C+K \tag{34}\]  
To perform order reduction, corrections need to be made, and the iterative update formula is  
\[q_{n+1}^{(i+1)}=q_{n+1}^{(i)}+\omega_{k}K_{eff}R_{n+1}^{(i)} \tag{35}\]  
where the residual \(R_{n+1}^{(i)}\) is defined as  
\[R_{n+1}^{(i)}=f_{n+1}-\left(M\tilde{q}_{n+1}^{(i)}+C\tilde{q}_{n+1}^{(i)}+KCq_ {n+1}^{(i)}\right) \tag{36}\]  
By adjusting the dynamic relaxation factor, the correction amount for each iteration can be controlled, ensuring the convergence of the iteration process and improving computational efficiency. Building on the improved Newmark method, a dynamic calculation tool was designed and developed to perform numerical computations for the crankshaft system's substructure dynamic model. This tool can compute the diesel engine crankshaft system's torsional, longitudinal, and lateral vibration characteristics under various operating conditions. It can also assess vibration responses under conditions such as changes in rotational speed and load fluctuations, making it valuable for evaluating and studying the crankshaft system's vibration behaviour.