## 2 Description of the Models
### Dynamic Model of HGRS
#### 2.2.1 Element Models  
Introducing spring and clearance, gear mesh element is used to simulate instantaneous engagement state of gear pairs. It consists of two nodes, each node has 6 DOF, as shown in Figure 6. The spring stiffness \(k_{m}\) refers to the LPMS, and the clearance \(e_{m}\) denotes the CMS. The two gears rotate around \(O_{p}\) and \(O_{g}\). \(\beta_{b}\) and \(\alpha\) are the base helix angle and mesh angle. \(\psi\) is the phase angle.  
The generalized coordinate vector of gear mesh element can be defined as  
\[u_{m}=\left\{x_{p},y_{p},z_{p},\theta_{xp},\theta_{yp},\theta_{2p},x_{g},y_{g},z_{g},\theta_{xg},\theta_{yg},\theta_{zg}\right\}^{T} \tag{6}\]  
The relative displacement of driving and driven gear along normal line of action can be calculated using  
\[\tau_{m}=Tu_{m} \tag{7}\]  
where \(T\) can be expressed as  
\[\begin{array}{l}T=\left[\cos\beta_{b}\sin\varphi,\cos\beta_{b}\cos\varphi,- \sin\beta_{b},\ r_{p}\sin\beta_{b}\sin\varphi,r_{p}\sin\beta_{b}\cos\varphi,r_{ p}\cos\beta_{b},\right.\\ \left.-\cos\beta_{b}\sin\varphi,-\cos\beta_{b}\cos\varphi,\sin\beta_{b},\ r_{g}\sin\beta_{b}\sin\varphi,r_{g}\sin\beta_{b}\cos \varphi,r_{g}\cos\beta_{b}\right]\end{array} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Teeth** & **30/30** & **Addendum Coefficient** & **1** \\ \hline Normal module/mm & 6 & Bottom clearance coefficient & 0.25 \\ Pressure angle/\({}^{\circ}\) & 20 & Width/mm & 40 \\ Helix angle/\({}^{\circ}\) & \(\pm\)18 & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of helical gear pair.  
Figure 6: Dynamic model of gear mesh element.  
Based on the Newton motion law, introducing LPMS and CMS, the motion equations for gear mesh element can be written as  
\[\begin{cases}\begin{array}{l}m_{p}\ddot{x}_{p}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_ {m}-e_{m})]\cos\beta_{b}\sin\varphi=0\\ m_{p}\ddot{y}_{p}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]\cos\beta_{b}\cos \varphi=0\\ m_{p}\ddot{z}_{p}-[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]\sin\beta_{b}=0\\ I_{xp}\ddot{\theta}_{xp}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{p}\sin \beta_{b}\sin\varphi=0\\ I_{yp}\ddot{\theta}_{yp}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{p}\sin \beta_{b}\cos\varphi=0\\ I_{zp}\ddot{\theta}_{zp}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{p}\cos \beta_{b}=0\\ m_{g}\ddot{x}_{g}-[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]\cos\beta_{b}\sin \varphi=0\\ m_{g}\ddot{y}_{g}-[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]\cos\beta_{b}\cos \varphi=0\\ m_{g}\ddot{z}_{g}^{2}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]\sin\beta_{b}= 0\\ I_{xg}\ddot{\theta}_{xg}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{g}\sin \beta_{b}\sin\varphi=0\\ I_{yg}\ddot{\theta}_{yg}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{g}\sin \beta_{b}\cos\varphi=0\\ I_{zg}\ddot{\theta}_{zg}+[c_{m}\dot{\tau}_{m}+k_{m}(\tau_{m}-e_{m})]r_{g}\cos \beta_{b}=0\end{array}\end{cases} \tag{9}\]  
where \(\varphi=\alpha-\psi\), \(K_{m}=k_{m}V^{T}V\) and \(C_{m}=c_{m}V^{T}V\).  
The matrix form of Equation (8) can be expressed as  
\[M_{m}\ddot{u}_{m}(t)+C_{m}\dot{u}_{m}(t)+K_{m}(t)[u_{m}(t)-e_{m}(t)]=0 \tag{10}\]  
where \(M_{m}\) and \(e_{m}(t)\) are mass matrix and CMS vector, respectively.  
Timoshenko beam element is used to establish the dynamic model of shaft element, which includes 2 nodes with 12 DOF [15]. Ignoring the housing flexibility, bearing element is established using bearing stiffness and damping which connects shaft with foundation. In general, owing to the number change of loaded rollers, the stiffness of rolling bearing is usually time-dependent, whereas the published work shows that the numerical simulation results of a geared rotor system agree well with the experimental results when constant bearing stiffness is used. Thus, the time-dependent characteristic of the rolling bearing is neglected in this study. The stiffness matrix, damping matrix, and motion equation of bearing element can be found in [15].