## 2 Transient Calculation Theory
### Research on Dynamic Response Methods  
The transient vibration differential equations of the ship propulsion system are usually of multiple degrees of freedom, and the calculation is very extensive, making the solution very costly. To reduce calculation time and improve efficiency, numerical analysis methods are commonly used for calculation. This paper employs the Newmark-\(\beta\) method for step-by-step integration from a mathematical perspective, focusing on solving the system's motion differential equations, solving the initial and boundary value problems of the motion differential equations, ensuring that they satisfy the motion differential equations at each moment, and representing the velocity and acceleration at a certain moment as a linear combination of displacements at adjacent moments. This allows the motion differential equations of the multi-degree-of-freedom vibration system to be transformed into an algebraic equation system represented by the displacement at discrete moments. There is no need to decouple the coupled motion differential equations; instead, they are numerically integrated step by step. Through this process, we can solve the dynamic response of the system at a series of discrete time points. The Newmark method assumes that under the known displacement \(\theta\), velocity \(\dot{\theta}\), and acceleration \(\ddot{\theta}\) at time \(t\), the velocity and displacement at time \(t+\Delta t\) are represented as follows:  
\[\dot{\theta}(t+\Delta t)=\dot{\theta}(t)+[(1-\gamma)\ddot{\theta}(t)+\gamma \ddot{\theta}(t+\Delta t)]\Delta t \tag{2}\]\[\theta(t+\Delta t)=\theta(t)+\dot{\theta}(t)\Delta t+\bigg{[}\bigg{(}\frac{1}{2}- \beta\bigg{)}\ddot{\theta}(t)+\beta\ddot{\theta}(t+\Delta t)\bigg{]}\Delta t^{2} \tag{3}\]  
where \(\beta\) and \(\gamma\) are two constants within the interval [0,1].  
For Equation (3), it can be rewritten as:  
\[\ddot{\theta}(t+\Delta t)=\frac{1}{\beta\Delta t^{2}}(\theta(t+\Delta t)- \theta(t))-\frac{1}{\beta\Delta t}\dot{\theta}(t)-(\frac{1}{2\beta}-1)\ddot{ \theta}(t) \tag{4}\]  
Substituting (4) into (2) yields the following:  
\[\ddot{\theta}(t+\Delta t)=\frac{\gamma}{\beta\Delta t}(\theta(t+\Delta t)- \theta(t))-(1-\frac{\gamma}{\beta})\dot{\theta}(t)+(1-\frac{\gamma}{2\beta}) \ddot{\theta}(t) \tag{5}\]  
Considering the vibration differential equation at the time \(t+\Delta t\) yields the following:  
\[J\ddot{\theta}(t+\Delta t)+c\dot{\theta}(t+\Delta t)+K\theta(t+\Delta t)=T(t +\Delta t) \tag{6}\]  
Substituting (4) and (5) into (6) yields the following:  
\[J\ddot{\theta}(t+\Delta t)+c\dot{\theta}(t+\Delta t)+K\theta(t+\Delta t)=T(t +\Delta t) \tag{7}\]  
In the above formula, the following is calculated:  
\[\left\{\begin{array}{l}\overline{K}=K+\frac{1}{\beta\Delta t^{2}}J+\frac{ \gamma}{\beta\Delta t^{2}}c\\ \overline{T}=T(t+\Delta t)+J\Big{(}\frac{1}{\beta\Delta t^{2}}(\theta(t))- \frac{1}{\beta\Delta t}\dot{\theta}(t)-(\frac{1}{2\beta}-1)\ddot{\theta}(t) \Big{)}+c\Big{(}\frac{\gamma}{\beta\Delta t}(\theta(t))+(\frac{\gamma}{\beta} -1)\dot{\theta}(t)+(1-\frac{\gamma}{2\beta})\ddot{\theta}(t)\Big{)}\\ \end{array}\right. \tag{8}\]  
where \(\overline{K}\) represents the equivalent stiffness matrix, and \(\overline{T}\) denotes the equivalent load vector. From Equation (7), the displacement vector \(\theta(t+\Delta t)\) at time \(t+\Delta t\) can be obtained. Subsequently, the acceleration vector \(\ddot{\theta}(t+\Delta t)\) and velocity vector \(\dot{\theta}(t+\Delta t)\) at a time \(t+\Delta t\) can be determined, respectively, from Equations (4) and (5).  
Consequently, a procedure for analyzing the time domain transient response characteristics of the ship's propulsion shaft system using the Newmark-\(\beta\) method can be established, as illustrated in Figure 1.  
Figure 1: Flowchart for solving the time domain dynamic response Using the Newmark-\(\beta\) method.