## 3 Numerical Analysis and Simulation Calculation
### Thermodynamic Model Parameter Calculation of the Nut-Rotating Ball Screw Transmission Mechanism
#### 3.1.1 Heat Generation Calculation  
According to a previous analysis, two heat source types are available for the nut-rotating ball screw transmission mechanism: (1) calculation of heat production from the screw support bearing and the bearing in the nut assembly; (2) Calculation of heat production at the screw-nut contact point.  
1. Heat generation calculation for rolling bearings  
Rolling bearings mainly generate heat through friction. The friction moment method is used to calculate heat produced through bearing under a given speed and friction moment. Its calculation formula is:  
\[Q_{B}=1.047\times 10^{-4}n_{B}M_{B} \tag{52}\]  
where \(Q_{B}\) indicates heat produced through bearing \((\mathrm{W})\); \(n_{B}\) represents bearing speed (RPM); \(M_{B}\) stands for frictional torque acting on the bearing \((\mathrm{N}\cdot\mathrm{mm})\).  
Friction torque acting on rolling bearings is calculated based on the Palmgren empirical formula:  
\[M_{B}=M_{1}+M_{2} \tag{53}\]  
\[M_{1}=f_{1}Pd_{m} \tag{54}\]  
\[f_{1}=\begin{cases}0.0013\times(P_{0}/C_{0})^{0.33}&\text{Single-row installation}\\ 0.001\times(P_{0}/C_{0})^{0.33}&\text{Double-row installation}\end{cases} \tag{55}\]  
\[P=\begin{cases}P_{a}+0.1P_{r}&\text{Single-row installation}\\ 1.44P_{a}-0.1P_{r}&\text{Double-row installation}\end{cases} \tag{56}\]  
\[M_{2}=\begin{cases}10^{-7}f_{0}(v_{0}n_{B})^{2/3}d_{m}^{3}&v_{0}n_{B}\geq 2000 \\ 160\times 10^{-7}f_{0}d_{m}^{3}&v_{0}n_{B}<2000\end{cases} \tag{57}\]  
where \(M_{1}\) is the friction torque during startup and low-speed operation \((\mathrm{N}\cdot\mathrm{mm})\); \(M_{2}\) represents friction torque associated with the bearing model, lubricating grease, and speed \((\mathrm{N}\cdot\mathrm{mm})\); \(f_{1}\) is a coefficient associated with the load and bearing type; \(P\) represents the external load of bearing \((\mathrm{N})\); \(d_{m}\) accounts for the pitch diameter of bearing \((\mathrm{mm})\); \(P_{0}\) suggests equivalent static load of bearing \((\mathrm{N})\); \(C_{0}\) stands for the basic rated static load of bearing \((\mathrm{N})\); \(P_{a}\) indicates axial load on bearing \((\mathrm{N})\); \(P_{r}\) presents radial load on bearing \((\mathrm{N})\); \(v_{0}\) is lubricant viscosity (cst or mm\({}^{2}\)/s); \(f_{0}\) is an empirical coefficient associated with bearing model and lubricant, with \(f_{0}=4\) for double-row installation and \(f_{0}=2\) for single-row installation.  
2. Calculation of heat generation at the contact point of the screw and the nut  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Values** & **Unit** \\ \hline Ball screw diameter \(d_{0}\) & 16 & mm \\ Axial force \(F_{a}\) & 500 & \(\mathrm{N}\) \\ Thermal conductivity \(\kappa\) & 50 & \(\mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\) \\ Thermal diffusivity \(\alpha_{C}\) & \(1.04\times 10^{-5}\) & \(\mathrm{m}^{2}/\mathrm{s}\) \\ Lubricant kinematic viscosity \(v_{0}\) & 40 & cst or mm\({}^{2}/\mathrm{s}\) \\ Average linear expansion coefficient of screw \(\alpha_{T}\) & \(11.8\times 10^{-6}\) & mm/ \({}^{\circ}\mathrm{C}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Thermodynamic parameters of the nut-rotating ball screw precision transmission mechanism.  
The frictional heat produced through the nut-rotating ball screw pair can be determined by adding the heat generated by the rotating nut and that generated by the screw:  
\[Q_{NS}=1.047\times 10^{-4}(n_{N}M_{N}+n_{S}M_{S}) \tag{58}\]  
where \(Q_{NS}\) is heat produced at the nut screw pair contact point \((\mathrm{W})\); \(n_{N}\) stands for the nut rotational speed (RPM); \(n_{S}\) indicates the screw rotational speed (RPM); \(M_{N}\) indicates the mut friction torques \((\mathrm{N\cdot mm})\), which can be obtained from kinematic analysis; \(M_{S}\) stands for the screw friction torque \((\mathrm{N\cdot mm})\), which can also be obtained from kinematic analysis.