# Quasi-Static Load Sharing Characteristics of a Planetary Fear Set with Planet Journal Bearings
## 3 Load Sharing Analysis Method  
### Planetary Gear Set Errors  
Manufacturing and assembly errors of the planetary gear set make the center distance change. It is common to allow the sun gear to float in the radical direction in the planetary gear transmissions of vehicles [13]. The sun gear was chosen as the floating component. As shown in Figure 3, in a planetary gear set, \(k_{p}\) and \(k_{s}\) are the support stiffness coefficients of the bearing and the sun gear. \(k_{spi}\) and \(k_{pi}\) are the meshing stiffness coefficients of the \(i\)th sun-planet pair and the \(i\)th ring-planet pair. \(r_{bs}\) and \(r_{bpi}\) are the pitch circle radii of the sun gear and the \(i\)th planet. To study the dynamics of the planet, the equivalent meshing backlash caused by the manufacturing errors [25], the assembly errors [26], and the floating component errors [13] needs to be considered. Therefore, several parts have been taken into account in the cumulative meshing error calculation, as shown in Figure 4.  
Figure 3: Model of a planetary gear set. Because of the manufacturing error of each component of the planetary gear set and the assembly errors in the planet-sun gear pairs and the planet-ring gear pairs, the center of the \(i\)th planet moves to \(O_{i7}\) from position \(O_{i1}\), and the cumulative meshing error is represented by \(e_{spi}\), as shown in Figure 4. The manufacturing errors are composed of four parts: the sun gear, the planet, the carrier, and the ring. As illustrated in Figure 4, the manufacturing errors of the sun gear, the carrier, the ring gear, and the \(i\)th planet are defined by a respective magnitude \(E_{s}\), \(E_{c}\), \(E_{r}\), \(E_{pi}\)[27] and a respective initial orientation angle of the high point \(\beta_{s}\), \(\beta_{c}\), \(\beta_{r}\), \(\beta_{pi}\). The assembly errors consist of the planet pin position error from the sun gear and the planet pin position error from the ring gear. The assembly errors of the sun-planet gear pair and the ring-planet gear pair are defined by their magnitudes \(A_{s}\) and \(A_{r}\) and their initial orientation angles \(\gamma_{s}\) and \(\gamma_{r}\), respectively. The equivalent meshing errors caused by the manufacturing error of each component are given by  
\[e_{Esi}=-E_{s}\sin[(\omega_{s}-\omega_{c})t+\beta_{s}+\alpha_{w}-\varphi_{i}], \tag{19}\]  
\[e_{Epi}=-E_{pi}\sin[(\omega_{p}-\omega_{c})t+\beta_{pi}+\alpha_{w}], \tag{20}\]  
\[e_{Eci}=E_{c}\sin(\beta_{c}-\varphi_{i})\cos\alpha_{w}, \tag{21}\]  
\[e_{Eri}=E_{r}\sin(\omega_{c}t-\beta_{r}+\alpha_{w}+\varphi_{i})\cos 2\alpha_{w}, \tag{22}\]  
where, \(e_{Esi}\), \(e_{Epi}\), \(e_{Eci}\), and \(e_{Eri}\) are the equivalent meshing errors of the \(i\)th planet. \(\omega_{s}\), \(\omega_{c}\), and \(\omega_{p}\) are the rotation speeds of the sun gear, the planetary gear, and the carrier, respectively. \(\alpha_{w}\) is the helix angle. \(t\) is time. \(\varphi_{i}\) is the nominal position angle of the \(i\)th planet relative to the carrier. The equivalent meshing errors, based on two resources of assembly errors, are expressed by  
\[e_{Asi}=-A_{s}\sin(-\omega_{c}t+\gamma_{s}+\alpha_{w}-\varphi_{i}) \tag{23}\]  
\[e_{Ari}=A_{r}\sin(\omega_{c}t-\gamma_{r}+\alpha_{w}+\varphi_{i})\cos 2\alpha_{w}, \tag{24}\]  
where, \(e_{Asi}\) and \(e_{Ari}\) are the equivalent meshing errors of the \(i\)th planet. Therefore, the accumulative meshing error on the meshing line of the \(i\)th planet-sun gear pair can be summed up by  
\[e_{spi}=e_{Esi}+e_{Epi}+e_{Eci}+e_{Eri}+e_{Asi}+e_{Ari}. \tag{25}\]  
Figure 3:0 Schematic of floating component error and cumulative meshing error. The floating component error is assumed as \(\Delta_{si}\), which is a function of time. To calculate the floating error of the sun gear, variables \(x_{s}\) and \(y_{s}\) are introduced to characterize the displacement of the center of sun gear in two directions, shown in Figure 3:1 Thus, the floating component error is  
Figure 3:2  
where,  
Figure 3:3