## 2 Wear Prediction Model of DPRSs Considering Ambient Temperature
### Temperature Rise of DPRSs and Thermal Deformation Analysis
#### 2.2.1 Thermal Resistance Model  
Modeling the closed thermal network of a DPRS requires the selection of suitable temperature nodes and the determination of the thermal resistance between the nodes. Ref. [13] shows that the main types of thermal resistance of DPRSs are contact, conduction, and convection thermal resistance.  
The contact thermal resistance of the DPRS is discussed first.  
The contact thermal resistance of the joint surface is one of the important sources of the contact thermal resistance. Combining a rough surface contact area and multiscale mechanics, the following expression for the contact thermal resistance of the bond surface, considering roughness, is obtained:  
\[R_{c}=\frac{1}{h_{c}A}=1/\left(\frac{A}{L_{g}}\bigg{(}\frac{A_{r}}{A}\frac{2k _{1}k_{2}}{k_{1}+k_{2}}+\frac{A_{v}}{A}k_{f}\bigg{)}\right), \tag{14}\]  
where \(A\) is the total contact area, \(h_{c}\) is the total heat transfer coefficient, \(A_{r}\) is the actual contact area, \(A_{v}\) is the void area, \(L_{g}\) is the void thickness, and \(k_{f}\) is the thermal conductivity within the void.  
The raceway contact thermal resistance is also a non-negligible source of thermal resistance in DPRS systems. In elliptical coordinates, the combined boundary conditions \(\left\{\begin{array}{l}x=0,T=T_{0}\\ x\rightarrow\infty,T\to 0\end{array}\right.\) yield the temperature distribution in the hemispheric space outside the contact surface as follows:  
\[T(x)=\frac{Q}{4\pi k_{smo}}\int_{x}^{\infty}\frac{dx}{\sqrt{(a^{2}+x)(b^{2}+x) x}}, \tag{15}\]  
where \(a\) and \(b\) are the long and short semi-axes of the contact ellipse, respectively; \(k_{smo}\) is the thermal conductivity of the object on a smooth surface; and \(Q\) is the heat flow leaving the contact surface.  
The contact thermal resistance can be expressed as follows:  
\[R=\frac{T_{0}-T_{x\rightarrow\infty}}{Q}=\frac{1}{4\pi k_{smo}}\int_{0}^{ \infty}\frac{dx}{\sqrt{(a^{2}+x)(b^{2}+x)x}}=\frac{Z(a/b)}{4k_{smo}a}=\frac{ \left[2K_{1}(e,\pi/2)/\pi\right]}{4k_{smo}a}, \tag{16}\]  
where \(e\) is the contact ellipse eccentricity, \(Z\) is the geometric factor determined by the contact area, and \(K_{1}(e,\frac{\pi}{2})\) is the first type of complete elliptic integral.  
The contact thermal resistance of the threaded teeth of a single roller to the screw and nut is shown in Figure 3 and can be expressed as follows:  
\[\left\{\begin{array}{l}R_{e\_s}=\frac{1}{n}\bigg{(}\frac{Z(a_{s}/b_{s})}{4k _{r}a_{s}}+\frac{Z(a_{s}/b_{s})}{4k_{s}a_{s}}\bigg{)}\\ R_{e\_n}=\frac{1}{n}\bigg{(}\frac{Z(a_{n}/b_{n})}{4k_{r}a_{n}}+\frac{Z(a_{n}/b _{n})}{4k_{s}a_{n}}\bigg{)}\end{array}\right., \tag{17}\]  
where \(R_{e\_s}\) is the equivalent contact thermal resistance of a single threaded tooth on a single roller with a screw race, \(R_{e\_n}\) is the equivalent contact thermal resistance of a single threaded tooth on a single roller with a nut race, and \(n\) is the number of rollers.  
The conduction thermal resistance and convection thermal resistance in DPRSs is discussed next. The nut of a DPRS can be simplified to a hollow cylinder, whose axial and radial thermal resistance can be expressed as follows:  
\[\left\{\begin{array}{l}R_{z}=\dfrac{\delta}{\lambda S_{z}}\\ R_{j}=\dfrac{\ln(d_{w}/d_{n})}{2\pi\lambda\delta}\end{array}\right., \tag{18}\]  
where \(R_{z}\) is the axial thermal resistance, \(R_{j}\) is the radial thermal resistance, \(\delta\) is the axial length, \(S_{z}\) is the axial cross-sectional area of the cylinder, \(d_{w}\) and \(d_{n}\) are the inner and outer diameters of the nut, respectively.  
In a DPRS, the convective heat transfer is generalized forced convection, and the convective thermal resistance can be expressed as follows:  
\[R_{dh}=\dfrac{1}{h_{f}S_{dh}}=\dfrac{1}{(N_{n}\lambda/D)S_{dh}}, \tag{19}\]  
where \(S_{dh}\) is the convective heat transfer area, \(h_{\rm f}\) is the convective heat transfer coefficient, \(N_{n}\) is the Nussle coefficient, \(\lambda\) is the thermal conductivity of the fluid, and \(D\) is the geometry of the heat exchange.