## 2 Nonlinear Dynamic Modeling of Spur Gear Systems
### Derivation of the Dynamic Model
\begin{array}{c}F\sin\varphi\\ F\cos\varphi\\ 0\\ 0\\ 0\\ Fd\cos\varphi\\ Fd\sin\varphi\\ Fr_{b2}-T_{c}\end{array} \right] \tag{26}\]  
Substitute Equations (24)-(26) into [PERSON]'s equation, which is  
\[\frac{d}{dt}\bigg{(}\frac{\partial T}{\partial\dot{z}_{i}}\bigg{)}-\frac{ \partial T}{\partial z_{i}}+\frac{\partial U}{\partial z_{i}}+\frac{\partial V} {\partial\dot{z}_{i}}=Q_{i} \tag{27}\]  
The dynamic equation of the gear system is obtained as follows:\[\left\{ \begin{array}{l}m_{1}\ddot{x}_{1}+k_{b1}(x_{1}-\delta_{1}x_{b2}-\delta_{2 }x_{b1})+c_{b1}(\dot{x}_{1}-\delta_{1}\dot{x}_{b2}-\delta_{2}\dot{x}_{b1})=-F \sin\varphi\\ m_{1}\dot{y}_{1}+k_{b1}(y_{1}-\delta_{1}y_{b2}-\delta_{2}y_{b1})+c_{b1}(\dot{y }_{1}-\delta_{1}\dot{y}_{b2}-\delta_{2}\dot{y}_{b1})=-F\cos\varphi\\ m_{b1}\ddot{x}_{b1}-k_{b1}\delta_{2}(x_{1}-\delta_{1}x_{b2}-\delta_{2}x_{b1})+k_ {bxy}x_{b1}+c_{bxy}\dot{x}_{b1}-c_{b1}\delta_{2}(\dot{x}_{1}-\delta_{1}\dot{x }_{b2}-\delta_{2}\dot{x}_{b1})=0\\ m_{b1}\dot{y}_{b1}-k_{b1}\delta_{2}(y_{1}-\delta_{1}y_{b2}-\delta_{2}y_{b1})+k_ {bxy}y_{b1}+c_{bxy}y_{b1}-c_{b1}\delta_{2}(\dot{y}_{1}-\delta_{1}y_{b2}-\delta _{2}y_{b1})=0\\ m_{b2}\dot{x}_{b2}-k_{b1}\delta_{1}(x_{1}-\delta_{1}x_{b2}-\delta_{2}x_{b1})+k_ {bxy}x_{b2}+c_{bxy}\dot{x}_{b2}-c_{b1}\delta_{1}(\dot{x}_{1}-\delta_{1}\dot{x }_{b2}-\delta_{2}\dot{x}_{b1})=0\\ m_{b2}\dot{y}_{b2}-k_{b1}\delta_{1}(y_{1}-\delta_{1}y_{b2}-\delta_{2}y_{b1})+k_ {bxy}y_{b2}+c_{bxy}y_{b2}-c_{b1}\delta_{1}(\dot{y}_{1}-\delta_{1}\dot{y}_{b2}- \delta_{2}\dot{y}_{b1})=0\\ I_{x1}\dot{\theta}_{x1}=-Fd\cos\varphi\\ I_{y1}\dot{\theta}_{y1}=-Fd\sin\varphi\\ I_{x1}\dot{\theta}_{z1}=T_{s}-Fr_{b1}\\ m_{2}\ddot{x}_{2}+k_{b1}(x_{2}-\delta_{3}x_{b4}-\delta_{4}x_{b3})+c_{b1}(\dot{ x}_{2}-\delta_{3}\dot{x}_{b4}-\delta_{4}\dot{x}_{b3})=F\sin\varphi\\ m_{2}\ddot{y}_{2}+k_{b1}(y_{2}-\delta_{3}y_{b4}-\delta_{4}y_{b3})+c_{b1}(\dot{ y}_{2}-\delta_{3}\dot{y}_{b4}-\delta_{4}\dot{y}_{b3})=F\cos\varphi\\ m_{b3}\ddot{x}_{b3}-k_{b1}\delta_{4}(x_{2}-\delta_{3}x_{b4}-\delta_{4}x_{b3}) +k_{bxy}x_{b3}+c_{bxy}\dot{x}_{b3}-c_{b1}\delta_{4}(\dot{x}_{2}-\delta_{3} \dot{x}_{b4}-\delta_{4}\dot{x}_{b3})=0\\ m_{b3}\ddot{y}_{b3}-k_{b1}\delta_{4}(y_{2}-\delta_{3}y_{b4}-\delta_{4}y_{b3}) +k_{bxy}y_{b3}+c_{bxy}\dot{y}_{b3}-c_{b1}\delta_{4}(\dot{y}_{2}-\delta_{3} \dot{y}_{b4}-\delta_{4}\dot{y}_{b3})=0\\ m_{b4}\dot{x}_{b4}-k_{b1}\delta_{3}(x_{2}-\delta_{3}x_{b4}-\delta_{4}x_{b3}) +k_{bxy}x_{b4}+c_{bxy}\dot{x}_{b4}-c_{b1}\delta_{3}(\dot{x}_{2}-\delta_{3} \dot{x}_{b4}-\delta_{4}\dot{x}_{b3})=0\\ m_{b4}\ddot{y}_{b4}-k_{b1}\delta_{3}(y_{2}-\delta_{3}y_{b4}-\delta_{4}y_{b3}) +k_{bxy}y_{b4}+c_{bxy}\dot{y}_{b4}-c_{b1}\delta_{3}(\dot{y}_{2}-\delta_{3} \dot{y}_{b4}-\delta_{4}\dot{y}_{b3})=0\\ I_{x2}\dot{\theta}_{x2}=Fd\cos\varphi\\ I_{y2}\dot{\theta}_{y2}=Fd\sin\varphi\\ I_{z2}\ddot{\theta}_{z2}=Ft_{b2}-T_{c}\end{array} \right. \tag{28}\]  
In order to simplify the 9 th and the 18 th expression in Equation (28), a new variable, \(x=r_{b1}\theta_{z1}-r_{b2}\theta_{z2}-e(t)\), is introduced to represent the relative displacement of the gears along the LOA, caused by torsional vibration and gear error. Then, let \(m_{3}={l_{z1}}/{r_{b1}}^{2}\), \(m_{4}={l_{z2}}/{r_{b2}}^{2}\), \(F_{1}=T_{s}/{r_{b1}}\), \(F_{2}=T_{c}/{r_{b2}}\), and \(m=m_{3}m_{4}/(m_{3}+m_{4})\). The simplified equations can be obtained as, i.e.,  
\[\lambda=(x_{1}-x_{2})\sin\varphi-(y_{1}-y_{2})\cos\varphi+x \tag{29}\]  
\[m\ddot{x}+k(t)f(\lambda)+c\dot{\lambda}=\frac{F_{1}m}{m_{3}}+\frac{F_{2}m}{m_{ 4}}-m\ddot{e}(t)=F(t) \tag{30}\]  
where \(F(t)\) denotes the sum of the equivalent external error excitation and equivalent internal error excitation. Let \(F_{m}=\frac{F_{1}m}{m_{3}}+\frac{F_{2}m}{m_{4}}\), and \(F_{h}(t)=-m\ddot{e}(t)\). \(F_{m}\) denotes the equivalent external excitation, and \(F_{h}(t)\) denotes equivalent internal error excitation. Meanwhile, in order to facilitate nonlinear analysis, the above dynamic equations of the gear system should be nondimensionalized.