# Dynamics Analysis of a Double-Plunger-Type Turbine Overspeed Protection Mechanism
## 3 Dynamic Modeling of Plungers
### Analysis of Plunger Motion States
 \tag{6}\]  
It is clear that matrix \(\dot{A}A^{T}\) is the skew-symmetric matrix of the vector \(\mathbf{\omega}=[\omega_{x}\,\omega_{y}\,\omega_{z}]^{\mathrm{T}}\), denoted as \(\widetilde{\omega}\). Substituting this into Equation (3) yields  
\[\dot{\mathbf{r}}=\widetilde{\omega}\mathbf{r}+A\dot{\mathbf{\rho}} \tag{7}\]  
where \(\dot{\rho}\) is the velocity vector of moving point C in the relative coordinate system and \(A\dot{\mathbf{\rho}}\) is its velocity vector in the absolute coordinate system. Then, the absolute velocity of the center of mass of the inner plunger is  
\[\dot{\mathbf{r}}=\mathbf{\omega}\times\mathbf{r}+A\dot{\mathbf{\rho}} \tag{8}\]  
To find the acceleration of the movement of moving point C, Equation (8) is derived for time:  
\[\ddot{\mathbf{r}}=\dot{\mathbf{\omega}}\times\mathbf{r}+\mathbf{\omega}\times\dot{\mathbf{r}}+ \frac{d}{dt}(A\dot{\mathbf{\rho}}) \tag{9}\]  
Defining the rate of change of the vector in time with respect to the fixed coordinate system as the absolute derivative and the rate of change in time with respect to the dynamic coordinate system as the relative derivative, the last term in Equation (9) becomes the absolute derivative of the relative velocity vector. Since the absolute derivative of a vector is equal to its relative derivative plus the cross product of the angular velocity of the dynamic system and this vector [8]; therefore,  
\[\frac{d}{dt}(A\dot{\mathbf{\rho}})=\frac{d\dot{\mathbf{\rho}}}{dt}+\mathbf{\omega}\times( A\dot{\mathbf{\rho}}) \tag{10}\]  
Substitute Equations (8) and (10) into Equation (9) yields  
\[ \begin{split}\ddot{\mathbf{r}}&=\dot{\mathbf{\omega}}\times\mathbf{r}+ \mathbf{\omega}\times(\mathbf{\omega}\times\mathbf{r}+A\dot{\mathbf{\rho}})+\frac{d\dot{\mathbf{ \rho}}}{dt}+\mathbf{\omega}\times(A\dot{\mathbf{\rho}})\\ &=\dot{\mathbf{\omega}}\times\mathbf{r}+\mathbf{\omega}\times(\mathbf{\omega} \times\mathbf{r})+2\mathbf{\omega}\times A\dot{\mathbf{\rho}}+\frac{d\dot{\mathbf{\rho}}}{dt} \\ &=\mathbf{a}_{e}+\mathbf{a}_{r}+\mathbf{a}_{c}\end{split}  \tag{11}\]  
where \(\mathbf{a}_{e}=\dot{\mathbf{\omega}}\times\mathbf{r}+\mathbf{\omega}\times(\mathbf{\omega}\times \mathbf{r})\) is the acceleration of the dynamic system with respect to the fixed system, i.e., the convective acceleration; \(\mathbf{a}_{r}=2\mathbf{\omega}\times A\dot{\mathbf{\rho}}\) is the Coriolis acceleration [11]; \(\mathbf{a}_{c}=\frac{d\dot{\mathbf{\rho}}}{dt}\) is the acceleration of the moving point with respect to the moving system, i.e., the relative acceleration. It can be seen that the acceleration of a plunger in motion is a vector sum of the above three accelerations. The complex motion of an object in space can be described as a linear combination of basis vectors by establishing multiple basis vectors [10], each of which usually represents a certain characteristic direction of the object during its motion. In this study, the basis vectors \(\mathbf{e}_{\theta}\) and \(\mathbf{e}_{\rho}\) are established along the tangential and radial directions of the main shaft, respectively. Then, the convective acceleration for the inner plunger is  
\[\mathbf{a}_{e2}=\ddot{\theta}\rho_{2}\mathbf{e}_{\theta}-\dot{\theta}^{2}\rho_{2}\mathbf{e }_{\rho} \tag{12}\]  
the relative acceleration is  
\[\mathbf{a}_{e2}=\ddot{\rho}_{2}\mathbf{e}_{\rho} \tag{13}\]the Coriolis acceleration is  
\[\mathbf{a}_{c2}=2\dot{\rho}_{2}\dot{\theta}\mathbf{e}_{\mathbf{\theta}} \tag{14}\]  
Similarly, the acceleration of the outer plunger during the motion is  
\[\mathbf{a}_{e1}=\ddot{\theta}\rho_{2}\mathbf{e}_{\mathbf{\theta}}-\dot{\theta}^{2}\rho_{2} \mathbf{e}_{\mathbf{\rho}}\ \ \ \ \ \mathbf{a}_{r1}=\ddot{\rho}_{2}\mathbf{e}_{\mathbf{\rho}}\ \ \ \ \ \mathbf{a}_{c1}=2\dot{\rho}_{2}\dot{\theta}\mathbf{e}_{\mathbf{\theta}} \tag{15}\]