# Dynamics Analysis of a Double-Plunger-Type Turbine Overspeed Protection Mechanism
## 3 Dynamic Modeling of Plungers  
### Analysis of Plunger Motion States  
In the dynamics of multibody systems, to describe the motion of each object accurately, it is necessary to establish a suitable coordinate system based on its connections and constraints [9]. As shown in Figure 4, the absolute coordinate system, i.e., the fixed coordinate system \(\alpha\)\(\mathrm{v}\)\(\mathrm{y}\)\(\mathrm{y}\)\(\mathrm{z}\)\(\mathrm{z}\)\(\mathrm{z}\), and the relative coordinate system, i.e., the dynamic coordinate system oxyz, which is attached to the main shaft, are established. The x-axis of the dynamic coordinate system coincides with the axis of the turbine main shaft and is defined as the axial direction, and the z-axis of the dynamic coordinate system is defined as the radial direction. The basis vectors along the z and y axes in the dynamic coordinate system are \(\mathbf{e}_{\boldsymbol{\rho}}\) and \(\mathbf{e}_{\boldsymbol{\theta}}\), respectively. In this study, the spindle only has rotational motion around its axis; relative rotation around the x-axis between the relative and absolute coordinate systems occurs; and the angle between the y\({}_{0}\)-axis in the absolute coordinate system and the y-axis in the relative coordinate system is \(\theta\). The angular velocity of the spindle rotation is \(\omega=\theta\)  
Figure 3: (**a**) Center of mass distribution; (**b**) hitting the leveraging mechanism. After analyzing the constraints between each component, it can be seen that the inner and outer plungers have two degrees of freedom, and the compound motion of the inner plunger consists of two parts: the rotation around the axis along with the spindle, denoted as \(\theta\), and the radial movement along the y-axis of the dynamic coordinate system in the outer plunger cavity, denoted as \(\rho_{2}\). Similarly, the compound motion of the outer plunger consists of two parts: the rotation following the spindle, which is at the same angle as the rotation of the inner plunger and therefore also \(\theta\), and the radial movement inside the spindle bore cavity, denoted as \(\rho_{1}\). The moving points C\({}_{1}\) and C\({}_{2}\) are defined as the centers of mass of the outer and inner plungers, respectively; their radius vectors to the origin O in the absolute coordinate system are defined as \(r_{1}\) and \(r_{2}\), and the radius vectors to the origin in the relative coordinate system are defined as \(\rho_{1}\) and \(\rho_{2}\), respectively. The coordinate transformation matrix from the fixed coordinate system to the dynamic coordinate system is \(A\). Then, the equations of motion of the plungers can be expressed as  
\[r=A(t)\rho(t)\quad A= \begin{bmatrix}1&0&0\\ 0&-\cos\theta(t)&\sin\theta(t)\\ 0&\sin\theta(t)&\cos\theta(t)\end{bmatrix} \quad\rho= \begin{bmatrix}\rho_{1}(t) &\rho_{2}(t)\end{bmatrix}  \tag{1}\]  
where \(r\) and \(\rho\) are matrices with three rows and two columns. It is clear that matrix \(A\) satisfies \(AA^{T}=I\), i.e., \(A^{-1}=A^{T}\), which means that it is an orthogonal matrix [10]. Therefore, \(\rho\) can be expressed as  
\[\rho=A^{-1}r=A^{T}r \tag{2}\]  
To find the velocity of motion of the center of mass, Equation (1) is derived with respect to time:  
\[\dot{\mathbf{r}}=\dot{A}\rho+A\dot{\mathbf{\rho}}=\dot{A}A^{T}r+A\dot{\mathbf{\rho}} \tag{3}\]  
Matrix \(\dot{A}A^{T}\) is discussed next, with derivatives for time t on both sides of the equation \(AA^{T}=I\):  
\[\frac{d}{d\text{t}}(AA^{T})=\dot{A}A^{T}+A\dot{A}^{T}=\dot{A}A^{T}+\left(A\dot {A}\right)^{T}=0 \tag{4}\]  
which means  
\[\dot{A}A^{T}=-\left(A\dot{A}\right)^{T} \tag{5}\]  
Figure 4: Coordinate systems: (a) absolute coordinate system; (b) relative coordinate system. This shows that matrix \(\dot{A}A^{T}\) is an antisymmetric matrix whose diagonal elements are all zero and whose only three elements are independent. Therefore, it can be written in the following form:  
\[\dot{A}\mathbf{A}^{T}= \begin{bmatrix}0&-\omega_{z}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{bmatrix}