# Dynamics Analysis of a Double-Plunger-Type Turbine Overspeed Protection Mechanism
## 3 Dynamic Modeling of Plungers
### Analysis of the Plungers Force Situation  
During the entire motion, the inner plunger may be subjected to the contact force \(F_{t2}\) along the tangential direction of the spindle, the contact force \(F_{r2}\) along the radial direction of the spindle, the frictional force \(f_{2}\), and the elastic force \(S_{2}\) generated by the inner spring and the outer plunger. These forces always satisfy the following equations:  
\[F_{r2}+f_{2}+S_{2}=m\ddot{\rho}_{2}\mathbf{e}_{\mathbf{\rho}}-m\dot{\theta}^{2}\rho_{ 2}\mathbf{e}_{\mathbf{\rho}} \tag{16}\]  
\[F_{t2}=\ddot{\theta}L_{2}\mathbf{e}_{\mathbf{\theta}}+2m\dot{\rho}_{2}\dot{\theta}\bm {e}_{\mathbf{\theta}} \tag{17}\]  
Similarly, the outer plunger may be subjected to a contact force \(F_{t1}\) along the tangential direction of the spindle, a contact force \(F_{r1}\) along the radial direction of the spindle, a frictional force \(f_{1}\), an elastic force \(S_{1}\) generated by the outer spring, and an elastic force \(S_{2}\) generated by the inner spring. These forces always satisfy the following equations:  
\[F_{r1}+f_{1}+S_{1}+S_{2}=m\ddot{\rho}_{1}\mathbf{e}_{\mathbf{\rho}}-m\dot{\theta}^{2} \rho_{1}\mathbf{e}_{\mathbf{\rho}} \tag{18}\]  
\[F_{t1}=\ddot{\theta}L_{1}\mathbf{e}_{\mathbf{\theta}}+2m\dot{\rho}_{1}\dot{\theta}\bm {e}_{\mathbf{\theta}} \tag{19}\]  
The force on the plungers is different when they are in different positions. As shown in Figure 5, there are three main cases for an inner plunger, as follows.  
When the inner plunger is located in the initial position, at which time \(\dot{\rho}_{2}=\ddot{\rho}_{2}=0\), Equations (16) and (17) can be rewritten as  
\[F_{r2}+f_{2}+S_{2}=m\dot{\theta}^{2}\rho_{2}\mathbf{e}_{\rho} \tag{20}\]  
\[F_{t2}=\ddot{\theta}L_{2}\mathbf{e}_{\mathbf{\theta}} \tag{21}\]  
When the inner plunger moves inside the outer plunger cavity and its upper and lower end surfaces are not in contact with the inner wall of the outer plunger, the outer plunger  
Figure 5: Three different force conditions of the inner plunger.  
does not produce a contact force along the radial direction of the main shaft, \(F_{r2}=0\). At this time, Equation (16) is written as follows:  
\[f_{2}+S_{2}=m\ddot{\rho}_{2}\boldsymbol{e_{\rho}}-m\dot{\theta}^{2}\rho_{2} \boldsymbol{e_{\rho}} \tag{22}\]  
When the inner plunger moves to the outermost end, its upper surface comes into contact with the top surface of the inner wall of the outer plunger, and there is no relative motion between the inner and outer plunger. At this time, the outer plunger and inner plunger can be seen as a whole and Equation (16) can be written as follows:  
\[F_{r2}+S_{2}=m\ddot{\rho}_{2}\boldsymbol{e_{\rho}}-m\dot{\theta}^{2}\rho_{2} \boldsymbol{e_{\rho}} \tag{23}\]  
Similarly, as shown in Figure 6, there are two cases of forces on the outer plunger, as follows.  
When no action occurs, similar to the inner plunger, \(\dot{\rho}_{1}=\dot{\rho}_{1}=0\), and Equations (18) and (19) are written as  
\[F_{r1}+f_{1}+S_{1}=m\dot{\theta}^{2}\rho_{1}\boldsymbol{e_{\rho}} \tag{24}\]  
\[F_{t1}=m\dot{\theta}L_{1}\boldsymbol{e_{\theta}} \tag{25}\]  
With the outer plunger action, due to the outer plunger action and the adjustable bushing out of contact, the adjustable bushing on the radial contact force along the spindle disappears. At this time, \(F_{r1}=0\), and Equation (18) is rewritten as follows:  
\[f_{1}+S_{1}=m\ddot{\rho}_{1}\boldsymbol{e_{\rho}}-\dot{\theta}^{2}\rho_{1} \boldsymbol{e_{\rho}} \tag{26}\]