# Quasi-Static Load Sharing Characteristics of a Planetary Fear Set with Planet Journal Bearings
## 2 Materials and Methods  
For hydrodynamic journal bearings using an incompressible lubricant of constant viscosity and operating under steady-state conditions, the general two-dimensional form of the Reynolds equation is given by [23]  
\[\frac{\partial}{\partial x}\bigg{(}h^{3}\frac{\partial p}{\partial x}\bigg{)} +\frac{\partial}{\partial z}\bigg{(}h^{3}\frac{\partial p}{\partial z}\bigg{)} =6U\eta\frac{\partial h}{\partial x}, \tag{1}\]  
where, \(p\) is the pressure, \(h\) is the oil film thickness, \(\eta\) is the lubricant viscosity, \(U\) is the tangential speed, \(x\) and \(z\) are the coordinates of journal circumferential direction and axial direction, respectively.  
Figure 2 presents the planet journal bearing inside a planet gear set. The radii of the planet gear shaft and planet gear are \(r\) and \(R\), respectively. The center of the planet shaft \(O_{1}\) is considered as the center of the bearing journal. Due to the loaded force, the center of the planet gear moves to \(O_{2}\), which is also considered as the center of the bearing bushing. The distance between the two centers is defined as the eccentricity, denoted by \(e\). In addition, \(\theta\) is defined as the eccentric angle and \(\omega\) is defined as the relative rotation speed of the planet gear and the planet gear shaft.  
Figure 1: Configuration of a planetary gear set with journal bearings.  
For further analysis of the journal bearing characteristics, the coordinate transformation method is used to make all the parameters in Equation (1) be dimensionless. Equations (2)-(8) give the expressions of the dimensionless parameters:  
\[\varphi=x/r, \tag{2}\]  
\[\lambda=y/(L/2), \tag{3}\]  
\[H=h/c=1+\varepsilon\cos\varphi, \tag{4}\]  
\[c=R-r, \tag{5}\]  
\[\varepsilon=e/c, \tag{6}\]  
\[P=p/p_{0}, \tag{7}\]  
\[p_{0}=6U\eta r/c^{2}, \tag{8}\]  
where, \(\varphi\) is the azimuth angle in the circle, \(L\) is the journal bearing length and \(\lambda\) ranges from \(-1\) to \(1\), \(c\) is the original radial clearance between the bushing and the journal, \(\varepsilon\) is the eccentricity ratio, \(p_{0}\) is the designed pressure.  
Then, the nondimensional Reynolds equation can be expressed as  
\[\frac{\partial}{\partial\varphi}\bigg{(}H^{3}\frac{\partial P}{\partial\varphi }\bigg{)}+(\frac{D}{L})^{2}\frac{\partial}{\partial\lambda}\bigg{(}H^{3}\frac {\partial P}{\partial\lambda}\bigg{)}=\frac{\partial H}{\partial\varphi}. \tag{9}\]  
The finite difference method is used to discretize the dimensionless Reynolds equation. The oil pressure is obtained by the successive over-relaxation iteration method:  
\[A_{i,j}P_{i+1,j}+B_{i,j}P_{i-1,j}+C_{i,j}P_{i,j+1}+D_{i,j}P_{i,j-1}-E_{i,j}P_{ i,j}=F_{i,j}, \tag{10}\]  
where, \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\) are the simplified parameters. They are calculated by Equations (11)-(16):  
\[A_{i,j}=H^{3}_{i+0.5,j}, \tag{11}\]  
\[B_{i,j}=H^{3}_{i-0.5,j}, \tag{12}\]  
\[C_{i,j}=\frac{D}{L}\bigg{(}\frac{\Delta\varphi}{\Delta\lambda}\bigg{)}^{2}H^{3 }_{i,j+0.5}, \tag{13}\]  
Figure 2: Parameters of the planet journal bearing.  
\[D_{i,j}=\frac{D}{L}\left(\frac{\Delta\varphi}{\Delta\lambda}\right)^{2}H_{i,j-0,5}^ {3}, \tag{14}\]  
\[E_{i,j}=A_{i,j}+B_{i,j}+C_{i,j}+D_{i,i}, \tag{15}\]  
\[F_{i,j}=\Delta\varphi(H_{i+0,5,j}-H_{i-0,5,j}). \tag{16}\]  
Reynolds boundary conditions are used in the calculation and adopted as  
\[P=0,\partial P/\partial\varphi=0(\varphi=\varphi_{2}), \tag{17}\]  
where, \(\varphi_{2}\) is the angular starting position of oil film rupture along the circumferential direction [24], which can be obtained when the convergence criterion is reached.  
The convergence criterion to stop the iteration is expressed as  
\[\sum_{i=1}^{m+1}\sum_{j=1}^{n+1}\left|\frac{p_{i,j}^{(k+1)}-p_{i,j}^{(k)}}{p_{ i,j}^{(k+1)}}\right|\leq ERR, \tag{18}\]  
where, \(ERR\) is the specified error, which is \(10^{-4}\) used in the calculation.