## 2 Nonlinear Dynamic Model of the Planetary Gearbox Considering Nonlinear Mesh Stiffness  
A planetary gear stage mainly consists of a carrier, a ring gear, a sun gear, several planet gears and bearings. The structure of a planetary gear stage with three planets is shown in Figure 1.  
In the improved rigid multibody model in [29], the reference frame for the carrier, the ring and the sun gear is the inertial frame, and the reference frame of each planet gear is a moving frame. The origin of the moving frame is at the center of the planet. The frame revolves around the sun gear and rotates with the same speed, and the revolution speed is equal to the instantaneous rotational speed of the carrier. Equation (1) shows the dynamic model in matrix form,  
\[M\ddot{q}+C(q)\dot{q}+K(q)q=F(t), \tag{1}\]  
in which matrices \(M\), \(C\), \(K\) can be obtained using the element assembly method given in [29]. Depending on the mesh stiffness of all of the gear pairs and the stiffness of all of the bearings, the total stiffness matrix \(K\) is time varying. The stiffness of bearings is modeled as six independent linear springs in this paper. The instantaneous mesh stiffness of a gear tooth pair \(k\) is determined by many factors, including manufacturing errors, assembly errors, the instantaneous motions, etc. The simplified dynamic time-varying mesh stiffness model in [29] is used in this paper, in which the transmission error is omitted and mesh phase relationships among the sun-planet and ring-planet meshes are considered. A brief description of the mesh stiffness model is presented below for completeness of the paper. To describe the mesh state of a meshing gear tooth pair, a relative contact position \(X\) for a meshing tooth pair is defined as,  
\[X=\left(\phi-\phi_{A}\right)/\left(\phi_{B}-\phi_{A}\right), \tag{2}\]  
where \(\phi\) is one of the gears' instantaneous rotation angular displacements, \(A\) and \(B\) represent the position where the tooth steps into and steps out of the current mesh process and \(\phi_{A}\) and \(\phi_{B}\) represent their corresponding rotation angular displacements. For the active gear in a spur gear pair, the mesh starting point \(A\) is near the root of the tooth, while for the passive gear, it is located at the top of the tooth. For a helical gear pair, the relative contact position is proportional to the length of the contact line. Transmission error is omitted in this paper, i.e., points \(A\) and \(B\) of each tooth keep the same among all  
Figure 1: Structure diagram of a planetary gear stage, where \(1\) represents the carrier, \(2\) represents the ring, \(3\) represents the sun, \(4\) represents one planet, \(5\) represents the carrier bearing, \(6\) represents the sun bearing, \(7\) represents the planet bearing, \(8\) represents the inner mesh between the ring and the planet, \(9\) represents the outer mesh between the sun and the planet, \(10\) represents the ring bearing and \(11\) represents the housing. of its mesh process, and thus, the nonlinear mesh stiffness of a tooth pair \(k\) only depends on its contact position. For a healthy gear tooth pair, a parabola is used to fit its stiffness \(k(X)\), shown as,  
\[k(X)=aX^{2}+bX+c,\] \[a=-4(k_{max}-k_{min})/e^{2},\] \[b=4(k_{max}-k_{min})/e,\] \[c=k_{min}, \tag{3}\]  
where \(e\) is the contact ratio, \(k_{mean}\) represents the mean mesh stiffness and \(k_{max}\) and \(k_{min}\) represent the maximum and minimum mesh stiffness, respectively. In this paper, the \(k_{max}\) and \(k_{min}\) are estimated by Equation (4),  
\[k_{C}=k_{mean}/(e(2/3+m/3)),\] \[k_{min}=mk_{C}, \tag{4}\]  
where \(m=0.67\) for an external spur gear pair, \(m=0.6\) for an internal spur gear pair and \(m=0\) for a helical gear pair. Since differences may exist among the teeth of the same gear, the teeth should be numbered, and the number of teeth in meshing should be considered in the mesh stiffness \(k\). Denote the tooth pair mesh stiffness as \(k(X)|(\kappa_{1},\kappa_{2})\), where \(\kappa_{i}\,(i=1,2)\) are the number of the two teeth in meshing.