# Demand Analysis of Lubricating Oil in Spur Gear Pairs
## 2 Theory
### Oil for Lubrication
#### 2.1.1 Analysis of Meshing Process  
Figure 1 illustrates the meshing principle of an involute spur gear pair. The instantaneous meshing point \(K\) is located at the common tangent of the base circle of the two gears, \(N_{1}N_{2}\) is the theoretical meshing line and \(P\) is the pitch point. The symbols \(r_{a1}\), \(r_{a2}\), \(\alpha\), \(\alpha_{1}^{\prime}\) and \(\alpha_{2}^{\prime}\) represent the radius of the pinion base circle, radius of the gear base circle, pressure angle of the reference circle, pressure angle of the pinion addendum circle and pressure angle of the gear addendum circle, respectively. The intersection point \(B_{1}\) between the addendum circle of a gear and the meshing line is the starting point of meshing, intersection point \(B_{2}\) between the addendum circle of the pinion and the meshing line is the end point of meshing and \(B_{1}B_{2}\) is the actual meshing line. When the instantaneous meshing point is located on line \(B_{1}C\), two pairs of teeth mesh simultaneously and the meshing point of another pair of teeth is located on line \(B_{2}D\). The distance between two meshing points on the meshing line is the base pitch \(P_{b}\). The line \(CD\) is the single tooth meshing area. A rectangular coordinate system is established with pitch point \(P\) as the origin and the direction from the start to the end of meshing as the positive direction. The coordinate of the instantaneous meshing point is \(s\). The range of \(s\) is \([-B_{1}P,B_{2}P]\) in a meshing period.  
In accordance with the geometric character, the distance from the pitch point to the start of meshing and the that from the pitch point to the end of meshing can be described as [23]  
\[\mathrm{B_{1}P}=\mathrm{N_{2}B_{1}-N_{2}P}=\mathrm{r_{a2}}(\tan\alpha_{2}^{ \prime}-\tan\alpha)=\frac{\mathrm{m_{2}}}{2}\mathrm{cos}\,\alpha(\tan\alpha_{2}^ {\prime}-\tan\alpha), \tag{2}\]  
\[\mathrm{B_{2}P}=\mathrm{N_{1}B_{2}-N_{1}P}=\mathrm{r_{a1}}(\tan\alpha_{1}^{ \prime}-\tan\alpha)=\frac{\mathrm{m_{2}}}{2}\mathrm{cos}\,\alpha(\tan\alpha_{1 }^{\prime}-\tan\alpha), \tag{3}\]  
The relative radius of curvature and entrainment velocity can be written as  
\[\mathrm{R}=\frac{(\mathrm{r_{a1}}\tan\alpha+\mathrm{s})(\mathrm{r_{a2}}\,\tan \,\alpha-\mathrm{s})}{(\mathrm{r_{a1}}+\mathrm{r_{a2}})\mathrm{tan}\,\alpha}, \tag{4}\]  
\[\mathrm{u_{E}}=\frac{\mathrm{u_{1}}+\mathrm{u_{2}}}{2}, \tag{5}\]  
where \(\mathrm{u_{1}}\) and \(\mathrm{u_{2}}\) denote the relative velocity between tooth surfaces and the meshing point, respectively, which can be written as  
\[\mathrm{u_{1}}=\frac{\mathrm{m_{1}}}{30}(\mathrm{r_{a1}}\tan\alpha+\mathrm{s}), \tag{6}\]  
\[\mathrm{u_{2}}=\frac{\mathrm{\pi m_{1}}}{30}(\mathrm{r_{a1}}\tan\alpha-\mathrm{ s}) \tag{7}\]  
The load on the tooth surface is calculated using the load distribution coefficient, which can be described as follows:  
\[\varsigma=\left\{\begin{array}{l}\frac{1}{3}\big{(}1+\frac{\mathrm{B_{1}P}+ \mathrm{s}}{\mathrm{B_{1}P}-\mathrm{PC}}\big{)}\,(-\mathrm{PB_{1}}\leq\mathrm{ s}\leq-\mathrm{PC})\\ 1\,(-\mathrm{PC}<\mathrm{s}<\mathrm{PD})\\ \frac{1}{3}\big{(}1+\frac{\mathrm{B_{2}P}-\mathrm{s}}{\mathrm{B_{2}P}-\mathrm{ PC}}\big{)}\,(-\mathrm{PB_{1}}\leq\mathrm{s}\leq-\mathrm{PC})\end{array}\right. \tag{8}\]  
where  
\[\mathrm{PD}=\mathrm{B_{1}B_{2}-B_{1}P-B_{2}D},\,\,\mathrm{PC}=\mathrm{B_{1}B_{2 }-B_{2}P-B_{1}C},\,\,\mathrm{B_{1}C}=\mathrm{DB_{2}}=(\varepsilon-1)\mathrm{P_ {b}}\]  
\[\mathrm{CD}=(2-\varepsilon)\mathrm{P_{b}},\,\,\mathrm{P_{b}}=\mathrm{\pi mcos }\alpha.\]  
Figure 1: Meshing model of involute spur gear.