## 2 Generalized Dynamic Model
### Establishment of Global Dynamics Model
#### 2.1.4 Planetary Carrier Element  
As a very important component, the planetary carrier not only plays the role of supporting the planetary gear, but also serves as the output component of the system in the third gear set and is the input component of the system in the second and fourth gears. In addition, the planetary carrier in the second gear set serves as a coupling component in the middle of the first gear set and the third set. Figure 5 shows the general planetary carrier element. The deformation of planet carrier \(\delta_{\rm c}\) is expressed as Equation (11). \[\delta_{\rm c}=V_{\rm c}q_{\rm c} \tag{11}\]  
Figure 5: Planetary carrier element. where \(q_{\rm c}=\{x_{\rm c},y_{\rm c},\theta_{\rm c},x_{\rm pi},y_{\rm p},\theta_{\rm pi}\}\) represents the displacement column vectors of each node of the planetary carrier element, \(V_{\rm c}\) is the projection vector of the displacements in all directions in the global coordinate system of the planetary carrier, which can be expressed by Equations (12) and (13):  
\[V_{\rm c}^{\rm x}=[\cos\phi_{\rm i},\sin\phi_{\rm i},0,-1,0,0] \tag{12}\]  
\[V_{\rm c}^{\rm y}=[-\sin\phi_{\rm i},\cos\phi_{\rm i},r_{\rm c},0,-1,0] \tag{13}\]  
where \(r_{\rm c}\) is the radius of the planetary carrier, \(\phi_{\rm i}\) is the mounting position angle of the i-th planetary gaer. For the planetary transmission system with uniform distribution of planetary gears, \(\phi_{\rm i}=2\pi({\rm i}-1)/n\), where, i = 1, 2,..., n, n is the number of planetary gears. The motion differential equation of the planetary carrier element can be obtained in the matrix form as Equation (14). \[M_{\rm c}\ddot{q_{\rm c}}+K_{\rm c}q_{\rm c}+C_{\rm c}\dot{q_{\rm c}}=F_{\rm c} \tag{14}\]  
where \(M_{\rm c}\), \(C_{\rm c}\) and \(K_{\rm c}\), represent mass matrix, damping matrix and stiffness matrix, respectively, \(F_{\rm c}\) is the column vector of the force applied to the planetary carrier. The system dynamic model diagram by discrete element matrix is shown in Figure 6, in which B1-B3 are three brakers. \(K_{\rm{sip}}\) and \(K_{\rm{rip}}\) (j = 1, 2, 3) represent the external and internal meshing stiffness of the three planetary gear sets, respectively. The 25 3-dof mass nodes include the input shaft node in, the coupling disc node LP, the sun gear node Si in i-th set (i = 1, 2, 3), the inner ring gear node Ri, the planetary carrier node Ci, and the planetary gear node pij in the i-th set. The meanings of each connection stiffness in Figure 6 are shown in Table 1. The matrix form of the connection stiffness is \(diag(k_{\rm x},k_{\rm y},k_{\rm 0})\), \(k_{\rm x}\), \(k_{\rm y}\) are the bending stiffnesses, and \(k_{\rm 0}\) is the torsional stiffness. The dynamics models of the first and second planetary gear set and the third planetary gear set in the dynamics model of the three planetary gear set transmissions are established in two coordinate systems, respectively, as shown in Figure 7a,b. XY in the two figures is the fixed coordinate system of the center gear, and its coordinate origin is located at the theoretical center of the center gear. Horizontally to the right is the positive direction of X, and the positive direction of X rotated 90\({}^{\circ}\) counterclockwise is the positive direction of Y. \(x_{\rm i}y_{\rm i}\) (i = 1, 2,..., N, N is the number of planetary gears) in two figures for each gear set of planetary gear in the local coordinate system. With the speed of the planet carrier uniform rotation for the moving coordinate system, the origin of the theory of center in the case of a planet gear, center gear's theory and the theory of the planet g  
Figure 6: Dynamics model of three planetary gear set transmissions. connection point to the direction of the positive direction for \(x_{i}\), the positive direction of \(x_{i}\) rotated by 90 degrees counterclockwise is the positive direction of \(y_{i}\).