## 2 Wear Prediction Model of DPRSs Considering Ambient Temperature
### Rough Surface Contact Analysis of DPRSs
#### 2.1.2 Actual Contact Area and Contact Mechanics  
Under practical conditions, the contact surface of a DPRS pair, taking into account surface roughness, is due to contact between tiny convex bodies, the rest being filled with air or a lubricating medium, as shown in Figure 2. The actual contact area on the surface of a DPRS is therefore less than the macroscopic contact area, and can be decomposed into the sum of the actual areas of many fractal domains using the fractal characteristic.  
When the contact pressure is homogeneous, the dimensionless real contact area on the macroscopic contact surface is as follows:  
\[X^{*}=X^{*}_{r}=\frac{X_{rf}}{L_{1}^{2}}, \tag{4}\]  
Figure 1: Structure of a DPRS.  
Figure 2: Schematic diagram of contact between ideal surfaces and fractal surfaces.  
where \(X_{r}^{*}\) is the dimensionless actual contact area, \(X_{rf}\) is the actual contact area over the fractal domain, and \(L_{i}\) is the upper limit of the sample length.  
Research [13] has shown that, depending on the radius of curvature of a rough DPRS surface and the amount of deformation, the deformation mode can be characterized as completely elastic, elastoplastic, or plastic. The critical area of plastic deformation is as follows:  
\[a_{c}=\frac{H^{2}}{(Y/2e^{\prime})^{2/(F-1)}}, \tag{5}\]  
where \(H\) is the scale factor for the profile amplitude, \(e^{\prime}\) is the equivalent modulus of elasticity of the two contacting bodies, and \(Y\) is the material hardness.  
The area of the largest microcontact in the fractal domain is as follows:  
\[a_{L}=\frac{2-F}{F}\psi^{(F-2)/2}X_{r}^{*}L_{i}^{2}, \tag{6}\]  
where \(\psi\) is the expansion factor related to \(F\).  
When \(a_{L}<a_{c}\), plastic deformation occurs at the DPRS gap, and the actual contact area is as follows:  
\[X_{r}^{*}=\frac{p}{Y}, \tag{7}\]  
where \(p\) is the pressure per unit area on the contact surface.  
The actual contact load of the contact point can be expressed as follows:  
\[F_{r}=K\psi(1+a\mu^{2})^{1/2}A, \tag{8}\]  
where \(K\) is the total contact stiffness and \(A\) is the contact area.  
When \(a_{L}>a_{c}\), simultaneous plastic and elastic deformations at the surface of the differential planetary screw coupling yield the following relationships:  
When \(1<F<1.5\), the actual contact area of the screw coupling is as follows:  
\[X^{*}=\left(\frac{3\sqrt{2\pi}}{4}\right)^{\frac{2}{2-F}}g_{4}(F)\Big{(}\frac {p}{\text{e}^{\prime}}\Big{)}^{\frac{2}{3-F}}\left(\frac{L_{i}}{\text{G}} \right)^{\frac{2F-2}{3-F}}. \tag{9}\]  
When \(1.5<F<2\), the actual contact area of the screw coupling is as follows:  
\[X^{*}=\Big{(}\frac{p}{\text{e}^{\prime}}\Big{)}^{2/F}g_{3}(F)^{-2/F}\Bigg{(} \frac{H}{\text{e}^{\prime}}g_{1}(F)(a_{c}^{*})^{\frac{2-F}{2}}-\frac{4}{3\sqrt {2\pi}}g_{1}(F)\bigg{(}\frac{G}{L_{\text{u}}}\bigg{)}^{F-1}(a_{c}^{*})^{\frac {(3-2F)}{2}}\Bigg{)}^{-2/F}, \tag{10}\]  
in which  
\[\left\{\begin{array}{l}g_{1}(F)=\left[(2-F)/F\right]^{F/2}F/(3-2F)\\ g_{2}(F)=\left[F/(2-F)\right]^{(2-F)/2}\\ g_{3}(F)=2^{F/2}\psi^{(4-4F+F^{2})}\\ g_{4}(F)=\left[g_{1}(F)g_{3}(F)\psi^{(F-2)(3-2F)/4}\right]^{2/(F-3)}\left[F/( 4-2F)\right]^{(3-2F)/(3-F)}\end{array}\right. \tag{11}\]  
The actual contact load at a single point of contact can be expressed as follows:  
\[F_{r}=\frac{e^{\prime}G^{(F-1)}}{3\pi^{1/2}2^{(F/2-3)}}\int_{a_{c}}^{a_{L}}n^ {\prime}(a)(X^{*})^{(3-F)/2}da+K\sigma_{5}\int_{0}^{a_{c}}n^{\prime}(a)X^{*}da, \tag{12}\]  
where \(n^{\prime}(a)\) is the modified statistical distribution function for roughness, and can be expressed as follows:  
\[n^{\prime}(a)=\xi\frac{F}{2}\psi^{(2-F)/2}(a_{L})^{F/2}(X^{*})^{-(F+2)/2}, \tag{13}\]  
where \(\xi\) is the correction factor.