## 4 Investigation of Reduction of Gear Meshing Power Loss  
### Gear Meshing Power Loss Solution  
Meshing efficiency, a key performance indicator for high-performance gear transmissions, is affected by tooth surface friction. This friction, influenced by both topography and load conditions, leads to a portion of the input power being lost. Gear meshing power loss quantifies the energy consumed by tooth surface friction. This instantaneous meshing power loss is defined as follows:  
\[P_{z}(i)=\mu_{mz}(i)\cdot F_{n}(i)\cdot v_{s}(i) \tag{18}\]  
The meshing power loss is related to three key parameters: the instantaneous friction coefficient \(\mu_{mz}\) of the meshing surfaces, the total normal force \(F_{n}\), and the relative sliding speed \(v_{s}\) at the instant meshing position. These parameters can be determined using Equations (19)-(21), respectively. \[\mu_{mz}(i)=0.048\cdot\left(\frac{w_{i}^{n}}{v_{e}(i)\cdot\rho_{t}(i)}\right) ^{0.2}\cdot Ra^{0.25}\cdot\eta_{oil}^{-0.05}\cdot X_{L} \tag{19}\]  
\[F_{n}(i)=\sum_{j}^{M}w_{ij}^{n} \tag{20}\]  
\[v_{e}(i)=v_{g1}(i)+v_{g2}(i) \tag{21}\]  
\[\rho_{t}=\rho/\cos\beta_{b} \tag{22}\]  
Among these equations, \(Ra\) demotes the average surface roughness of a gear pair, \(v_{e}\) is the entrainment speed, \(\rho_{t}\) is the transverse relative curvature radius, and the instantaneous relative sliding speed \(v_{s}\) is calculated by Equation (17). If a meshing cycle (an angular pitch, \(2\pi/\text{z}\)) is divided into \(N\) equal parts, the rotation angle of each part is \(\theta\), unit in rad. \[\theta=2\pi/(z_{1}\cdot N) \tag{23}\]  
The meshing time of each equal part of the angle is \(t\), in seconds. \[t=\theta/\omega_{1} \tag{24}\]  
The meshing loss energy of one meshing cycle is \(Q_{z}\), in joules. \[Q_{z}=\sum_{i}^{N}P_{z}(i)\cdot t_{i} \tag{25}\]  
Since the energy lost in each meshing cycle is equal, the average meshing power loss of the gearset is \(P_{mz}\), in watts. \[P_{mz}=Q_{z}/(N\cdot t) \tag{26}\]The instantaneous meshing efficiency and average meshing efficiency of a gear pair are Equations (27) and (28), respectively. \[\eta_{z}(i)=\frac{T_{1}\omega_{1}-P_{z}(i)}{T_{1}\omega_{1}} \tag{27}\]  
\[\mu_{mz}(i)=0.048\cdot\left(\frac{w_{i}^{n}}{v_{e}(i)\cdot\rho_{t}(i)}\right) ^{0.2}\cdot Ra^{0.25}\cdot\eta_{oil}^{-0.05}\cdot X_{L} \tag{19}\]0  
The results below present the average meshing power loss and average efficiency for each gearset under a single load condition. Verification of the Effectiveness of Modification Methods Aimed at Reducing Flash Temperature in Improving Meshing Efficiency  
This section will verify whether the previous modification scheme optimized for tooth surface flash temperature can also reduce meshing power loss. First, the gear LTCA based on FE is used to calculate the instantaneous load distribution on the tooth surface, and the normal force distribution \(w_{n}\) is obtained. Take the geometric parameters of gears into the Equations (14)-(17) to obtain local curvature radius and relative sliding speed of the meshing tooth surface. The average meshing power loss and average meshing efficiency can then be obtained by Equations (18)-(28). Taking the analysis results of gear pair G1735 as an example, Figure 11(a) is the theoretical tooth surface meshing power loss colormap; the maximum power loss is 4279 W, which occurs under full load conditions; and the average meshing power loss under the load spectrum is 1266.9 W. Figure 11(b) is the colormap of the meshing power loss for optimized tooth, the maximum loss also occurs under full load conditions, which is 3176 W, and the average power loss is 924.1 W. Compared to the theoretical tooth surface, the maximum meshing power loss of the optimized tooth surface is reduced by 1103 W, a reduction rate of 25.8%, and the average meshing power loss is reduced by 342.8 W, a reduction rate of 27.1%. Figure 12(a) is a colormap of meshing efficiency of the theoretical tooth surface of gear pair G1735.