# Transmission Efficiency of Cycloid-Pinion System Considering the Assembly Dimensional Chain
## 4 Transmission Efficiency Model
### Cycloid-Pinion System Friction Model  
Current friction coefficients are mainly based on the basic theory of tribology and empirical values derived from a large number of experiments. However, this empirical formula is predominantly case-oriented with a uniform distribution of contact loads and difficult to adapt to dynamic engagement situations. In the mesh process of the cycloid-pinion system, if the empirical value of the coefficient of friction is directly substituted for the calculation, it causes errors. Under actual operating conditions, the lubrication state of the meshing teeth is constantly changing. More specifically, the friction coefficient changes as the film thickness in the contact position changes. The time-varying coefficient of friction for mixed lubrication is calculated in the following manner. The coefficient of mixed friction of the cycloid-pinion system is as Equation (20). \[f_{mix}=f_{\lambda}^{12}f_{e}+(1-f_{\lambda})f_{b} \tag{20}\]  
where \(f_{e}\) and \(f_{b}\) represent friction coefficient of full film lubrication and boundary lubrication, respectively. \(f_{\lambda}\) denotes the quotient supported by the lubricant film and surface asperities. For the point contact, [PERSON] and [PERSON] [42] proposed the sharing function of Equation (21). Under high-velocity and high-load operating conditions, the general value of boundary lubrication coefficient \(f_{b}\) is 0.1 [43]. \[f_{\lambda}=\frac{1.21\lambda^{0.64}}{1+0.37\lambda^{1.26}} \tag{21}\]  
where \(\lambda\) is the film thickness ratio. This is based on the maximum pressure obtained according to Equation (15). Meanwhile, in previous solutions for elastohydrodynamic lubrication [44; 45], the pressure is usually assumed to be Hertzian distributed and the film thickness is a constant in order to obtain an analytical solution. [PERSON] and [PERSON] [46] obtained the equivalent pressure equation used to represent point contact, as shown in Equation (22). In this paper, a relatively inexpensive mineral-oil-based lubricant is selected. The corresponding effective pressure and friction coefficient are expressed as follows:  
\[p_{\mathit{eff}}=\delta\ln\left[2\delta p_{h}^{-1}e^{na_{0}p_{h}}\left(1- \delta p_{h}^{-1}\right)+2\delta p_{h}^{-2}\right] \tag{22}\]  
\[f_{e}=AW_{L}^{-1}G^{1-n}e^{na_{0}p_{\mathit{eff}}}\left(\eta_{0}\Delta uh_{ \mathrm{min}}^{-1}\right)^{n} \tag{23}\]  
where \(G\) the shear modulus and \(n\) is the power-law exponent with their values determined as \(G\) = 1 MPa and \(n\) = 0.5, respectively [32]; \(h_{\mathrm{min}}\) is the minimum film thickness; \(\eta_{0}\) is absolute viscosity, which is set to 30 mPa\(\cdot\)s; and the coefficient \(\delta=1/na_{0}\). The following assumptions are made for a faster solution. At the moment of start-up of the cycloid-pinion system, the axis of the pinion is parallel to the direction of the axis of the cycloid. At this instant, the contact state corresponding to the minimum film thickness is still a linear contact. Oil film thickness equations are often obtained based on numerical simulations or experimental results, and these formulas have similar expressions for different materials or operating conditions. In the process of data fitting, the results of the fit are often applicable to a range of speed, load, and material parameters because of the limited number of sample data. In this paper, the minimum film thickness model proposed by [PERSON] and [PERSON] was adopted [47]. The reason for choosing this model is that the loading moment is constant during the study of this paper, corresponding to a maximum value of the load parameter of the magnitude of \(1\times 10^{-5}\). According to [PERSON] et al. [48], in this range, the above fitted model will be in better agreement with the results obtained by the improved semi-system approach. The final result is as in Equation (24).