## 1 Introduction  
Transmission efficiency is one of the most important performance indexes of the transmission system that has a straight connection with the power losses, temperatures, noise, and wear. As the energy crisis intensifies, transmission efficiency improvement is becoming more significant and valuable. Therefore, many researchers have already studied transmission efficiency of gear pairs in order to find the rules and improve gear transmission efficiency. For gear transmission efficiency, the research based on elasto-hydrodynamic lubrication (EHL) theory for both line contact and point contact has been conducted for many years. [PERSON] et al. [1] studied the instantaneous meshing efficiency at any contact point of spur gear, considering mixed EHL conditions. The influence of the working conditions and design parameters on the gear efficiency had also been analyzed to guide parameter design. [PERSON] et al. [2] proposed a new method to calculate gear transmission efficiency considering the 3D micro-topography of the tooth surface. [PERSON] et al. [3] explored the influences of addendum modification on spur gear efficiency and determined the best modification parameters to balance load capacity, safety against wear, vibration and efficiency. [PERSON] et al. [4] built an optimal model for the balance of gear mass, gear mesh efficiency, and transmission errors, in which the best values of the addendum and dedendum were determined. [PERSON] et al. [5] proposed a kind of non-standard gears to improve gear efficiency by minimizing sliding and rolling power losses, in which the influence of gear and drive parameters were discussed, including gear ratio, pressure angle, teeth number, addendum, etc. [PERSON] and [PERSON] et al. [6; 7] studied helical gear efficiency and spur gear efficiency by experimental method, respectively, in which the influences of load, oil, and rotational speed on transmission efficiency were discussed. [PERSON] et al. [8] presented an algorithm to determine the transmission efficiency of a simple planetary mechanism based on numerical calculation. [PERSON] et al. [9] presented a new method for efficiency modeling of transmission efficiency and tested it on a six-speed dual-clutch transmission in a passenger car application, which could determine transient loss and the motion of a single part as well as the whole system. [PERSON] et al. [10] derived the formula transmission efficiency formula of planetary gear/star gear and studied its effect on the closed differential double helical gear train, in which the results showed that the difference is 2%. [PERSON] et al. [11] studied the influencing factors of gear train efficiency in GTF and built the efficiency calculation model of the gear train. Recently, with regards to spiral level gear (SBG) efficiency studies, researchers have also carried out some studies that focus on friction, lubrication, and efficiency. [PERSON] et al. [12] built a thermo-elastic finite element model to study thermal characteristics and transient temperature fields and optimized the temperature field that would be beneficial for performance and life. [PERSON] et al. [13] established a quasi-analytical flow model and predicted windage power loss for spiral level gears. The influence of specific parameters of windage power loss was discussed and results were verified by experiment. [PERSON] et al. [14] proposed a coupling thermo-elasto-hydrodynamic model for the friction coefficient and wear of the spiral level gears and calculated the pressure, flash temperature, friction coefficient, and wear rate of spiral level gears. [PERSON] et al. [15] established a mixed EHL model and studied the transient friction, temperature, and contact fatigue behaviors for spiral level gears in different contact trajectories. [PERSON] et al. [16] proposed a transient thermal elasto-hydrodynamic calculation model of spiral level gears built on the generalized Reynolds equation, considering mass-conserving cavitation, non-Newtonian flow, mixed friction, and micro-hydrodynamics.