## 3 Dynamic Equation Modeling of the Planetary Gear Train  
A pure torsional dynamic model of planetary gear train is established considering nonlinear factors such as TVMS and meshing clearance. The structural diagram of the 2K-H planetary gear set is shown in Figure 9 and the pure torsional vibration model is shown in Figure 10.  
The connecting shaft of the sun gear is the power input shaft and the cage where the planetary gear is located in the output. Assuming that the number of sun gear is \(Z_{s}\), the number of teeth of planetary gear is \(Z_{p}\), the number of teeth of inner gear ring is \(Z_{r}\), and the transmission ratio of the first stage 2K-H planetary gear train is \(i_{sc}\). \[i_{sc}=1-i_{sr}^{H}=1-\left(-\frac{Z_{r}}{Z_{s}}\right)=1+\frac{Z_{r}}{Z_{s}} \tag{50}\]  
The planetary gear set is different from the traditional fixed shaft gear, there are many meshing points in the transmission process, and the meshing position is constantly changing. The planetary gear has meshed with the ring and the sun gear simultaneously, and there is mutual coupling and restraining effect in the meshing process, so that the mesh frequency component may be affected. Due to the rotation and revolution of the planetary gear, the vibration impact direction caused by planetary gear damage will change constantly. Because the position of the sensor on the planetary gear housing is fixed, the angle between the impact direction and the sensor will constantly change, which makes the fault frequency more difficult to obtain. These characteristics make the traditional signal processing and pattern recognition methods not competent for planetary gear train fault diagnosis and prediction. Figure 10: Pure torsional vibration model of the 2K-H planetary gear set. Figure 9: Structural diagram of 2K-H planetary gear set. For fault mechanism, the dynamic model of the planetary gear train is established to study the influence of damage mode on dynamic response results. The dynamic model includes a lumped parameter model and a distributed parameter model. The lumped parameter model can also be called a multi-degree of freedom spring-mass vibration system. There are two types of coupled planetary vibration models: the pure torsion model and translational torsion model. Among them, the core parameters of the pure torsion model include circumferential torsional stiffness, the moment of inertia of center construction, and mass of planetary gear. The pure torsion model has the advantages of fewer parameters, simpler model and less computation. The translational torsional coupled vibration model considers not only torsion but also translational motion in two orthogonal directions in the vertical plane of the axis. The translational torsional coupled vibration model has more parameters. Therefore, the model is more complex and challenging to solve. When the support stiffness is 10 times or more than the mesh stiffness, the pure torsional stiffness model can also achieve high calculation accuracy [26]. The meshing force of gear is approximately elastic meshing force \(P\) and viscous meshing force \(D\). Where, \(K_{spi}\) and \(K_{rpi}\) denote the mesh stiffness between the sun gear and the \(i\)-th planet gear, the ring gear, and the \(i\)-th planet gear respectively. \(C_{spi}\) and \(C_{rpi}\) represent the meshing damping. \[\left\{ \begin{array}{l}D_{spi}=C_{spi}\bigg{(}\dot{\theta_{s}r_{bs}}-\dot{ \theta_{pi}}l_{rpi}-\dot{\theta_{c}}r_{c}\cos\alpha\bigg{)}\\ D_{rpi}=C_{rpi}\bigg{(}\dot{\theta_{pi}}l_{rpi}-\dot{\theta_{c}}r_{c}\cos \alpha\bigg{)}\\ P_{spi}=K_{spi}(t)f\bigg{(}\theta_{s}r_{bs}-\theta_{pi}r_{pi}-\dot{\theta_{c}} r_{c}\cos\alpha,b_{spi}\bigg{)}\\ P_{rpi}=K_{rpi}(t)f\bigg{(}\theta_{pi}l_{rpi}-\dot{\theta_{c}}r_{c}\cos\alpha,b _{rpi}\bigg{)}\end{array} \right. \tag{51}\]  
where \(C=2\xi\sqrt{\overline{K_{12}}/(1/M_{1}+1/M_{2})}\), \(\xi\) is the damping ratio, \(\overline{K}_{12}\) is the average mesh stiffness of a pair of gear tooth.