# On the Effect of DLC and WCC Coatings on the Efficiency of Manual Transmission Gear Pairs
## 2 Methodology
### Loaded Gear Frictional Losses Considering Uncoated Surface  
The loaded gears of a manual transmission are considered to operate under the mixed-elastohydrodynamic regime of lubrication due to the high loads they support. The specific operating conditions (especially kinematics) during meshing determine the thickness of the lubricant film, which is calculated by Equation (3) in order to estimate the generated viscous friction [18]:  
\[h_{c0}=\ 2.5\bigg{(}\frac{\pi\eta_{0}U}{4E_{r}R_{x}}\bigg{)}^{0.68}\bigg{(} \frac{2}{\pi}(E_{r}\alpha)\bigg{)}^{0.49}\bigg{(}\frac{\pi W}{2E_{r}R_{x}} \bigg{)}^{-0.073} \tag{3}\]  
where \(\eta_{0}\) is the dynamic viscosity of the lubricant at atmospheric pressure, \(U\) is the entraining speed of the lubricant, \(E_{r}\) is the reduced elastic modulus of the contact, \(R_{x}\) is the radii of curvature along the direction that defines the transverse plane, \(\alpha\) is the pressure-viscosity coefficient, and \(W\) is the applied load. It should be noted that this equation represents line contact. For the helical gears used in the transmission examined, it is assumed that the radius of curvature perpendicular to the direction of entraining speed is much larger than that along the direction of entraining speed. Hence, the contact area of the gear mesh is approximated as a straight line [19]. For the uncoated elastohydrodynamic regime under which the gears operate, the coefficient of friction is given by [20]:  
\[\mu=0.87\alpha\tau_{0}+1.74\ \frac{\tau_{0}}{P_{man}}\ln\bigg{(}\frac{1.2}{ \tau_{0}h_{c0}}\bigg{(}\frac{2\ *\ \dot{K}\ *\ \eta_{0}}{1+9.6\ *\ \dot{\xi}}\bigg{)}^{1/2}\bigg{)} \tag{4}\]  
where \(\alpha\) is the pressure-viscosity coefficient, \(\tau_{0}\) is the Eyring stress of lubricant, \(P_{man}\) is the mean pressure in the apparent area, \(h_{c0}\) is the dimensionless central film thickness, \(\dot{K}\) is the conductivity of the lubricant, \(\eta_{0}\) is the dynamic viscosity of the lubricant at atmospheric pressure, and \(\xi\) is given by [21]:  
\[\xi=\frac{4}{\pi}\frac{K}{h_{c0}/R_{x}}\bigg{(}\frac{P_{man}}{E^{\prime}R_{x} K^{\prime}\rho^{\prime}c^{\prime}U}\bigg{)} \tag{5}\]  
where \(R_{x}\) is the radii of curvature along the direction of entraining motion and \(K^{\prime},\ \rho^{\prime},\ c^{\prime}\) are the thermal conductivity, density, and specific heat capacity of the contacting solids, respectively. The reduced elastic modulus of the contact (\(E_{r}\)) is given by [22]:  
\[r=\frac{\pi}{((1-{\
u_{1}}^{2})/E_{1})+((1-{\
u_{2}}^{2})/E_{2})} \tag{6}\]where \(\
u_{1}\), \(\
u_{2}\) are the Poisson ratios of the two contacting surfaces and \(E_{1}\), \(E_{2}\) are the Young moduli of elasticity, respectively. The viscous friction generated on the gear is then given by  
\[F_{viscous}=\mu W \tag{7}\]  
The aforementioned parameters mainly affect the viscous friction generated on gear teeth during meshing. Yet, loaded gear teeth also suffer from boundary friction as a result of the asperity interactions [16]. Figure 1 presents the distribution of asperities of the uncoated gear teeth as was measured using the Alicona Infinite Focus microscope ([PERSON], Gratz, Austria). As the figure shows the distribution of asperities approaches the Gaussian distribution, which is thereafter assumed. The load that can be carried by the asperities is described as follows, where mixed lubrication regime is assumed due to the high pressures on the loaded gear teeth [16]:  
\[W_{a}=\frac{16\sqrt{2}}{15}\pi(\xi\beta\sigma)^{2}\sqrt{\frac{\sigma}{\beta}} \cdot AF_{\frac{5}{2}}(\lambda) \tag{8}\]  
where \(\beta\) is the average asperity tip radius, \(\sigma\) is the composite RMS surface parameter, \(A\) is the apparent contact area, \({}^{\prime}\) is the reduced elastic modulus of contact, as shown in Equation (9) and \(F_{5/2}(\lambda)\) is a statistical function for Gaussian distribution of asperities affected by the Stribeck oil film parameter \(\lambda\) and is described by Equation (10). \[E^{\prime}=\frac{2E_{r}}{\pi} \tag{9}\]  
\[F_{5/2}=\left\{