## 2 Wear Prediction Model of DPRSs Considering Ambient Temperature
### Temperature Rise of DPRSs and Thermal Deformation Analysis
#### 2.2.2 Temperature Prediction Models  
The first step is to establish the thermal network. When a DPRS operates, heat is generated by friction at the contact surfaces and is transferred to the neighboring parts. The outer surfaces of these parts, then, come into contact with the environment and transfer heat. Temperature nodes are selected at the necessary locations and connected to form a thermal network model, as shown in Figure 4. As the DPRS structure is symmetric with respect to the centerline, one quarter of the mechanism was chosen to model the thermal resistance network. A thermal network model with 39 nodes was developed to reduce the computational time by using a smaller number of temperature nodes while simultaneously completely characterizing the temperature rise of the entire structure. The numbers and locations of the nodes are shown in Table 1.  
Figure 3: Thermal resistance diagram for raceway contact.  
The second step is to establish the thermal balance equations of the system. In a thermal network, there is thermal resistance between the temperature nodes, and the energy of any node is conserved. As shown in Figure 5, in a temperature system comprising four temperature nodes, node A takes the heat flow from the outside and outputs the heat flow to the other nodes. According to the first law of thermodynamics, the transient thermal balance equation for temperature node A can be expressed as follows:  
\[H_{in}=\left(\frac{T_{A}-T_{B}}{R_{A-B}}+\frac{T_{A}-T_{C}}{R_{A-C}}+\frac{T_{A }-T_{D}}{R_{A-D}}\right)+m_{A}c_{A}\frac{\partial T_{A}}{\partial t}, \tag{20}\]  
where \(T_{x}\) is the temperature of each node; \(x=A,B,C\) and \(D\); and \(R_{x-x}\) is the difference in thermal resistance between two nodes.  
\begin{table}
\begin{tabular}{c c c} \hline
**Temperature Node Number** & **Corresponding Location** & **Temperature Variables** \\ \hline
1–9 & Temperature of outer surface of nut & \(T_{1}\)–\(T_{8}\) \\ \hline
10–17 & Thread tooth temperature on nut side & \(T_{9}\)–\(T_{15}\) \\ \hline
18 & Temperature of the cage & \(T_{16}\) \\ \hline
19 & Roller end temperature & \(T_{17}\) \\ \hline
20–32 & Roller thread tooth temperature & \(T_{18}\)–\(T_{30}\) \\ \hline
33–39 & Screw side thread tooth temperature & \(T_{31}\)–\(T_{37}\) \\ \hline \end{tabular}
\end{table}
Table 1: Temperature node locations of a DPRS.  
Figure 4: Thermal network model of a DPRS.  
The thermal balance equation for each node can be expressed in the form of the following matrix:  
\[[H]+[T][R]^{-1}=[m][C]\frac{\partial[T]}{\partial t}. \tag{21}\]  
The thermal balance equation for the whole system can be derived as follows:  
\[R_{39\times 39}\cdot T_{39\times 1}+H_{39\times 1}=m_{39\times 39}\cdot C_{39 \times 39}\cdot\frac{\partial T_{39\times 1}}{\partial t}, \tag{22}\]  
where \(R_{39\times 39}\) is the thermal resistance matrix of the temperature node, \(T_{39\times 1}\) is the vector to be solved at the temperature node, \(H_{39\times 1}\) is the heat flow matrix at the temperature node, \(m_{39\times 39}\) is the temperature node heat flow matrix, and \(C_{39\times 39}\) is the specific heat matrix of the temperature nodes.