## 3 Dynamic Equation Modeling of the Planetary Gear Train
\(M_{1}\) and \(M_{2}\) is the equivalent gear mass of the gear pair, for gear components in planetary gear, \(M=l/r_{b}\)2, for planet gear carrier, \(M_{c}=l_{c}/r_{bc}^{2}+\sum\limits_{i=1}^{N}m_{pi}/\cos^{2}\alpha\), where \(\alpha\) is the pressure angle. \(r_{bs}\) and \(r_{bpi}\) are the pitch circle radius of the sun gear and the planet gear respectively. \(b_{spi}\) and \(b_{rpi}\) are the meshing clearance constant between the sun gear and the \(i\)-th planet gear and the gear ring respectively, \(f(x,b)\) is a nonlinear clearance function, which can be specifically expressed as follows:  
\[f(x,b)=\left\{ \begin{array}{ll}x-b&(x>b)\\ 0&(-b\leq x\leq b)\\ x+b&(x<-b)\end{array} \right. \tag{52}\]  
where \(b\) is the gap constant. The dynamic simulation model of the planetary gear set can be shown as follows:  
\[\left\{ \begin{array}{l}I_{s}\ddot{\theta_{s}}+\sum\limits_{i=1}^{N}\Big{(}D _{spi}+P_{spi}\Big{)}r_{bs}=T_{D}\\ I_{pi}\ddot{\theta_{pi}}-\Big{(}D_{spi}-D_{rpi}+P_{spi}-P_{rpi}\Big{)}r_{bpi}=0\\ \Bigg{(}l_{c}+\sum\limits_{i=1}^{N}m_{pi}r_{bc}^{2}\Big{)}\ddot{\theta_{c}}- \sum\limits_{i=1}^{N}\Big{(}D_{spi}+D_{rpi}+P_{spi}+P_{rpi}\Big{)}r_{bc}=-T_{L} \end{array} \right. \tag{53}\]  
The above equation is a system of positive semidefinite nonlinear second order differential equations with \(N+2\) degrees of freedom. The angular displacement of the rigid body in the system is transformed into relative linear displacement. The relative displacements between the sun gear and the planet gear and between the sun gear and the carrier are defined as follows:  
\[\left\{ \begin{array}{l}x_{spi}=\theta_{s}r_{bs}-\theta_{pi}l_{pi}-\theta_{c}r _{c}\cos\alpha\\ x_{sc}=\theta_{s}r_{bs}-2\theta_{c}r_{c}\cos\alpha\end{array} \right. \tag{54}\]Substituting Formulas (52) and (54) into the Formula (51), we can get the following results:  
\[\left\{ \begin{array}{l}\overline{D}_{spi}=C_{spi}\dot{x}_{spi}\\ \overline{D}_{rpi}=C_{rpi}\big{(}\dot{x}_{sc}-\dot{x}_{spi}\big{)}\\ \overline{p}_{spi}=K_{spi}(t)f\big{(}x_{spi},b_{spi}\big{)}\\ \overline{p}_{rpi}=K_{rpi}(t)f\big{(}x_{sc}-x_{spi},b_{rpi}\big{)}\end{array} \right. \tag{55}\]  
Substituting Formulas (54) and (55) into the Formula (53), we can get the following results:  
\[\left\{ \begin{array}{l}\ddot{x}_{spi}+\sum\limits_{i=1}^{N}\Big{[}\Big{(} \frac{1}{M_{\rm s}}+\frac{1}{M_{\rm c}}\Big{)}C_{spi}\dot{x}_{spi}+\frac{1}{M_ {\rm c}}C_{rpi}\big{(}\dot{x}_{sc}-\dot{x}_{spi}\big{)}\Big{]}+\frac{1}{M_{\rm rpi }}\big{[}C_{spi}\dot{x}_{spi}-C_{rpi}\big{(}\dot{x}_{sc}-\dot{x}_{spi}\big{)} \Big{]}\\ +\sum\limits_{i=1}^{N}\Big{[}\Big{(}\frac{1}{M_{\rm s}}+\frac{1}{M_{\rm c}} \Big{)}K_{spi}(t)f\big{(}x_{spi},b_{spi}\big{)}+\frac{1}{M_{\rm c}}K_{rpi}(t)f \big{(}x_{sc}-x_{spi},b_{rpi}\big{)}\Big{]}\\ +\frac{1}{M_{\rm rpi}}\big{[}K_{spi}(t)f\big{(}x_{spi},b_{spi}\big{)}-K_{rpi}(t) f\big{(}x_{sc}-x_{spi},b_{spi}\big{)}\Big{]}=\frac{T_{D}}{M_{\rm rpi}}+\frac{T_{L}}{M_{ \rm c}r\cos\alpha}\\ \ddot{x}_{sc}+\Big{(}\frac{1}{M_{\rm s}}+\frac{2}{M_{\rm c}}\Big{)}\sum\limits _{i=1}^{N}C_{spi}\dot{x}_{spi}+\frac{2}{M_{\rm c}}\sum\limits_{i=1}^{N}C_{rpi} \big{(}\dot{x}_{sc}-\dot{x}_{spi}\big{)}+\Big{(}\frac{1}{M_{\rm c}}+\frac{2}{ M_{\rm c}}\Big{)}\sum\limits_{i=1}^{N}K_{spi}(t)f\big{(}x_{spi},b_{spi}\big{)}\\ +\frac{2}{M_{\rm c}}\sum\limits_{i=1}^{N}K_{rpi}(t)f\big{(}x_{sc}-x_{spi},b_{ rpi}\big{)}=\frac{T_{D}}{M_{\rm rpi}}+\frac{2T_{L}}{M_{\rm c}r\cos\alpha} \end{array} \right. \tag{56}\]