## 3 Efficiency Analysis of Herringbone Planetary Gear
### Power Loss Due to Gear Meshing
#### 3.2.1 Calculation of Gear Frictional Power Loss  
The internal and external gears affect relative sliding speeds of tooth surfaces during operation. As shown in Figure 5, the theoretical meshing segment is N\({}_{1}\)N\({}_{2}\) and the actual meshing segment is \(B_{1}B_{2}\). The point of intersection of the centerline and mesh line when two gears are meshing is P. At contact point K, the sliding speed of gear 1 relative to gear 2 is [29]  
\[V_{12}=(\omega_{1}+\omega_{2})l=\omega_{1}z_{1}\bigg{(}\frac{1}{z_{1}}+\frac{ 1}{z_{2}}\bigg{)}l \tag{15}\]  
where \(\omega\) is angular velocity; \(z\) is number of gear teeth; subscripts 1 and 2 denote gear 1 and gear 2, respectively; and \(l\) is the distance from meshing point to node \(P\). Because \(V_{12}=0\) at node \(P\), \(P_{N2}\) represents the positive direction. Power before and after the node, \(P_{f+}\) and \(P_{f-}\), are not the same. Friction loss near node \(P\) is [29]  
Figure 5: Schematic diagram of external helical gear transmission. \[P_{f\pm}(I)=F_{f}\cdot V_{12}=\pm fF_{N}\omega_{1}z_{1}\left(\frac{1}{z_{1}}+ \frac{1}{z_{2}}\right)I \tag{16}\]  
where \(f\) is the tooth surface sliding friction coefficient and \(F_{N}\) is normal force of the tooth surface. Gear drives typically have three lubrication states:  
1. Under the EHL (Elastohydrodynamic lubrication) state, friction surfaces of the meshed teeth are completely separated by a viscous fluid film. The film thickness is determined by relative sliding speed of the tooth surfaces. Pressure generated by the fluid film balances the external load. 2. With boundary lubrication, gear teeth must be separated by boundary lubrication in order to separate the tooth flanks. However, absolutely smooth tooth flanks do not exist, and for this reason, almost all gears work in a hybrid lubrication state. 3. In the hybrid lubrication state, the elastic oil film will either be in a mixed or boundary-lubricated state. Mixed lubrication refers to the combination of fluid lubrication and boundary lubrication, that is, the normal load is shared by the fluid lubrication film and the boundary film, and the friction force includes fluid damping and boundary friction components. Accurately estimating the minimum oil film thickness between tooth profiles is an important basis for analyzing the lubrication state of gears. The calculation formula of the minimum oil film thickness is as follows:  
\[h_{min}=1.6\alpha^{0.6}(uv_{e})^{0.7}E_{e}{}^{0.03}R_{e}{}^{0.43}\delta^{-0.13} \tag{17}\]  
where \(v_{e}\) is the the suction speed of the meshing point, \(R_{e}\) is the equivalent radius of curvature of the meshing point, and \(E_{e}\) is the equivalent elastic modulus. The viscosity of lubricating oil is affected by temperature and pressure; the relationship is as follows:  
\[\mu=\mu_{0}e^{\alpha p} \tag{18}\]  
\[\text{loglog}(u+a)=b-\text{logT} \tag{19}\]  
where \(p\) is lubricating oil pressure and \(a\), \(b\), and \(c\) are constants related to lubricating oil. The film thickness ratio is an important parameter of the lubrication state of the gear friction pair. When the film thickness ratio coefficient \(\lambda<1.4\), it is the boundary lubrication state; when \(1.4<\lambda<3\), it is the mixed lubrication state; and when \(\lambda>3\), it is the elastohydrodynamic lubrication state. \[\lambda=\frac{h_{min}}{\sqrt{R_{a1}^{2}+R_{a2}^{2}}} \tag{20}\]  
Considering the empirical formula proposed by [PERSON] and [PERSON] [29], which involves more comprehensive factors and better practical application effect, this empirical formula is used for the friction factor in this paper. \[f=0.0127\left(\frac{50}{50-39.37s_{f}}\right)\text{lg}\left(\frac{29.666}{uV_ {s}V_{r}^{2}}\right) \tag{21}\]  
where \(s_{f}\) is the gear surface roughness (root mean square roughness); \(\delta\) is the unit load and \(V_{s}\) and \(V_{r}\) are the instantaneous sliding speed and rolling speed of the meshing point, respectively.