## 2 Dynamic Model Analysis  
### Dynamic Model of Spur Gear Pair Considering the Meshing State of Multiple Pairs of Teeth  
In this work, the dynamic transmission system consists of a pair of gears installed on properly aligned shafts. In many previous literatures, the dynamic model of a single pair of teeth was usually used to describe the dynamic behavior of teeth in the whole meshing cycle, as shown in Figure 1. Here, \(\theta_{1}\) and \(\theta_{2}\) are the rotation angles of the driving and driven wheels, respectively; \(R_{b1}\) and \(R_{b2}\) are the radius of the base circle of driving and driven wheels, respectively; \(T_{1}\) and \(T_{2}\) are torques of driving and driven wheels, respectively; \(k(t)\) is comprehensive meshing stiffness \((\mathrm{N}/\mathrm{m})\); \(c_{m}(t)\) is comprehensive damping \((\mathrm{N}\cdot\mathrm{s}/\mathrm{m})\); \(e(t)\) is comprehensive transmission error (comprehensive meshing error) \((m)\); \(b(t)\) is comprehensive backlash \((m)\). In the process of gear transmission, the meshing state of each pair of teeth may be different due to the influence of tooth surface modification, manufacturing error, backlash and so on. When two or more pairs of teeth participate in meshing at the same time, the established dynamic model with all teeth being treated as a pair of teeth will not fully reflect the meshing state of each pair of teeth in the meshing process. Therefore, by analyzing the dynamic behavior of each pair of teeth involved in meshing, the authors developed a single degree of freedom nonlinear dynamic model considering the meshing state of multiple pairs of teeth [41], as shown in Figure 2. Here, we set the \(j-1\)th, \(j\)th and \(j+1\)th pairs of teeth participating in meshing and the meshing stiffness, meshing damping, meshing error and backlash of each pair of teeth are functions of time. For example, the meshing stiffness, meshing damping, meshing error and backlash of the \(j\)th pair of teeth are \(k^{(j)}(t)\), \(c^{(j)}(t)\),\(e^{(j)}(t)\) and \(b^{(j)}(t)\), respectively. Figure 1: Traditional SDOF dynamic model for a gear system. Assuming that three pairs of gear teeth participate in meshing at the same time, the corresponding dynamic equations can be obtained as follows:  
\[ \begin{array}{ll}I_{1}\dot{\theta}_{1}+c^{(j-1)}(t)&R_{1}\Big{(}\dot{\theta}_{ 1}R_{b1}-\dot{\theta}_{2}R_{b2}-\dot{e}^{(j-1)}(t)\Big{)}+c^{(j)}(t)R_{b1}\Big{(} \dot{\theta}_{1}R_{b1}-\dot{\theta}_{2}R_{b2}-\dot{e}^{(j)}(t)\Big{)}\\ &+c^{(j+1)}(t)R_{b1}\Big{(}\dot{\theta}_{1}R_{b1}-\dot{\theta}_{2}R_{b2}-\dot{e }^{(j+1)}(t)\Big{)}+k^{(j-1)}(t)R_{b1}f^{(j-1)}\Big{(}\dot{\theta}_{1}R_{b1}- \theta_{2}R_{b2}-e^{(j-1)}(t)\Big{)}\\ &+k^{(j)}(t)R_{b1}f^{(j)}\Big{(}\dot{\theta}_{1}R_{b1}-\theta_{2}R_{b2}-e^{(j) }(t)\Big{)}+k^{(j+1)}(t)R_{b1}f^{(j+1)}\Big{(}\theta_{1}R_{b1}-\theta_{2}R_{b2 }-e^{(j+1)}(t)\Big{)}=T_{1}\\ I_{2}\dot{\theta}_{2}-c^{(j-1)}(t)&R_{b2}\Big{(}\dot{\theta}_{1}R_{b1}-\dot{ \theta}_{2}R_{b2}-\dot{e}^{(j-1)}(t)\Big{)}-c^{(j)}(t)R_{b2}\Big{(}\dot{\theta }_{1}R_{b1}-\dot{\theta}_{2}R_{b2}-\dot{e}^{(j)}(t)\Big{)}\\ &-c^{(j+1)}(t)R_{b2}\Big{(}\dot{\theta}_{1}R_{b1}-\dot{\theta}_{2}R_{b2}-\dot{ e}^{(j+1)}(t)\Big{)}-k^{(j-1)}(t)R_{b2}f^{(j-1)}\Big{(}\theta_{1}R_{b1}-\theta_{2}R_{b2 }-e^{(j-1)}(t)\Big{)}\\ &-k^{(j)}(t)R_{b2}f^{(j)}\Big{(}\dot{\theta}_{1}R_{b1}-\theta_{2}R_{b2}-e^{(j) }(t)\Big{)}-k^{(j+1)}(t)R_{b2}f^{(j+1)}\Big{(}\theta_{1}R_{b1}-\theta_{2}R_{b2 }-e^{(j+1)}(t)\Big{)}=-T_{2}\end{array}  \tag{1}\]  
Let dynamic transmission error \(x(t)=\theta_{1}R_{b1}-\theta_{2}R_{b2}\), we can obtain:  
\[