## 2 The Original Dynamic Model of Gear System
### Dynamic Model of the System  
Considering static transmission error, time-varying meshing stiffness, and meshing-in impact, the dynamics model of the gear system was established by using the centralized mass method. Some principles of treatment needed to be adopted. Both gears were treated as rigid discs with radial, axial, and torsional degrees of freedom. Internal excitations of the system acted on the tooth surface along the line of action. The support from the bearings and the shaft was equivalent to the spring fastened to the foundation. The effect of the friction was not taken into account. Based on these principles, the lumped-parameter dynamic model was constructed, as shown in Figure 13.  
Based on the established model above, the displacement vector and the relative displacement of the system are listed in Equations (16) and (17).  
\[\mathbf{X}=[x_{1}\,y_{1}\,z_{1}\,u_{1}\,x_{2}\,y_{2}\,z_{2}\,u_{2}]^{\mathrm{ T}} \tag{16}\]  
\[\delta=[(x_{1}-x_{2})\cdot\sin\psi+(y_{1}-y_{2})\cdot\cos\psi+u_{1}+u_{2}] \cdot\cos\beta_{b}+\,(-z_{1}+z_{2})\cdot\sin\beta_{b}-e_{s} \tag{17}\]  
where \(x_{i}\), \(y_{i}\), and \(z_{i}\) (\(i=1,2\)) represent the radial and axial displacements in the \(x\), \(y\), and \(z\) directions; \(u_{i}\) (\(i=1,2\)) is the torsional displacement around the \(z\)-axis; the subscripts \(1\) and \(2\)  
Figure 11: Approximate location of the original meshing-in point in the case of the meshing-in end of the pinion protruding. (The contact positions on the curve appear in the process from meshing-in to meshing-out for a single tooth).  
Figure 12: Meshing-in impact forces for two cases: (**a**) meshing-in end of the pinion protruding and (**b**) meshing-out end of the pinion protruding.  
refer to the pinion and gear, respectively; \(\psi\) is the angle between the plane of action and the positive \(y\)-axis, as shown in Figure 13, with \(\psi=\alpha_{t}-\varphi\); \(\alpha_{t}\) is the transverse pressure angle; \(\varphi\) is the angle between the center line and the positive \(x\)-axis; \(\beta_{b}\) is the base helix angle; and \(e_{s}\) is the static transmission error.  
Utilizing the Lagrange equation, the dynamic equations of the system are listed as follows:  
\[\left\{\begin{array}{l}m_{1}\ddot{x}_{1}+C_{ss_{1}}\dot{x}_{1}+K_{ss_{1}}x_{1 }+\left(C_{m}\dot{\delta}+K_{m}\delta\right)\cdot\cos\beta_{b}\cdot\sin\psi=-F_ {s}\cdot\cos\beta_{b}\cdot\sin\psi\\ m_{1}\ddot{y}_{1}+C_{sy_{1}}\dot{y}_{1}+K_{sy_{1}}y_{1}+\left(C_{m}\dot{\delta} +K_{m}\delta\right)\cdot\cos\beta_{b}\cdot\cos\psi=-F_{s}\cdot\cos\beta_{b} \cdot\cos\psi\\ m_{1}\ddot{z}_{1}+C_{sz_{1}}\dot{z}_{1}+K_{sz_{1}}z_{1}-\left(C_{m}\dot{\delta} +K_{m}\delta\right)\cdot\sin\beta_{b}=F_{s}\cdot\sin\beta_{b}\\ \frac{l_{1}}{r_{h1}^{2}}\ddot{u}_{1}+\left(C_{m}\dot{\delta}+K_{m}\delta \right)\cdot\cos\beta_{b}=-F_{s}\cdot\cos\beta_{b}+\frac{T_{sat}}{r_{h1}}\end{array}\right. \tag{18}\]  
\[\left\{\begin{array}{l}m_{2}\ddot{x}_{2}+C_{ss_{2}}\dot{x}_{2}+K_{ss_{2}}x_{2 }-\left(C_{m}\dot{\delta}+K_{m}\delta\right)\cdot\cos\beta_{b}\cdot\sin\psi=F_ {s}\cdot\cos\beta_{b}\cdot\sin\psi\\ m_{2}\ddot{y}_{2}+C_{sy_{2}}\dot{y}_{2}+K_{sy_{2}}y_{2}-\left(C_{m}\dot{\delta} +K_{m}\delta\right)\cdot\cos\beta_{b}\cdot\cos\psi=F_{s}\cdot\cos\beta_{b} \cdot\cos\psi\\ m_{2}\ddot{z}_{2}+C_{ss_{2}}\dot{z}_{2}+K_{ss_{2}}z_{2}+\left(C_{m}\dot{\delta} +K_{m}\delta\right)\cdot\sin\beta_{b}=-F_{s}\cdot\sin\beta_{b}\\ \frac{l_{2}}{r_{h2}^{2}}\ddot{u}_{2}+\left(C_{m}\dot{\delta}+K_{m}\delta \right)\cdot\cos\beta_{b}=-F_{s}\cdot\cos\beta_{b}-\frac{T_{sat}}{r_{h2}}\end{array}\right. \tag{19}\]  
where \(m_{i}\) and \(l_{i}\) (\(i=1,2\)) are the mass and rotary inertias of the pinion and gear; \(C_{ssi}\), \(C_{ssi}\), and \(C_{ssi}\) and \(K_{ssi}\), \(K_{syi}\), and \(K_{ssi}\) (\(i=1,2\)) are the supporting damping and supporting stiffnesses; \(K_{m}\) is the time-varying mesh stiffness; \(C_{m}\) is the mesh damping; \(F_{s}\) is the meshing-in impact; and \(r_{h1}\) and \(r_{h2}\) are the base radii of the pinion and gear.