## 2 Temperature Field and Thermal Error Modeling of Nut-Rotating Ball Screw Transmission Mechanism
### Heat Conduction Analysis of Nut-Rotating Ball Screw Pairs  
Thermal conductivity analysis on the nut-rotating ball screw feed system ultimately focuses on the screw itself. Due to the fact that the machine tool's feed accuracy is mostly affected by thermal deformation in the screw axis direction, assuming the uniform temperature distribution on the screw cross-section, with the screw being regarded as a one-dimensional thermal conductor with only temperature gradients in the axis direction. Therefore, the thermal conductivity equation of the one-dimensional rod is as follows.  
\[\frac{\partial^{2}T(x,t)}{\partial x^{2}}=\frac{1}{\alpha_{C}}\frac{\partial T (x,t)}{\partial t}+\frac{4h}{\kappa d_{0}}\Big{[}T(x,t)-T_{f}\Big{]} \tag{1}\]  
where \(T(x,t)\) stands for temperature function on the screw, representing temperature change at time \(t\) at position \(x\) from the heat source; \(\alpha_{C}\) is the thermal diffusivity, \(\alpha=\kappa/\rho c\); \(\kappa\) indicates thermal conductivity; \(\rho\) represents screw density; \(c\) indicates specific heat capacity; \(h\) suggests a convective heat transfer coefficient between the external environment and the screw surface; \(d_{0}\) stands for lead screw's nominal diameter; \(T_{f}\) indicates ambient temperature.  
As shown in Figure 2, there are four major heat sources in the nut-rotating ball screw feed system: heat source \(Q_{B1}\) near the motor end screw bearing, heat source \(Q_{B2}\) at the nut component bearing, heat source \(Q_{N}\) for the screw nut pair, and heat source \(Q_{B3}\) far away from the motor end screw bearing. The screw motor is connected to the screw through a diaphragm coupling, and an insulation pad is installed inside the coupling. Therefore, the heat source at the screw motor has a negligible impact on the system temperature field. In addition, due to the fact that the nut motor transmits motion to the nut through a synchronous belt and has a large transmission distance, the heat source at the nut motor has no effect on the screw's thermal deformation. According to the superposition principle, the heat source of the nut bearing and that of the screw nut pair can be superimposed to form \(Q_{NB2}=Q_{N}+Q_{B2}\). During operation, the nut component shows forward and backward movements along the screw axis direction within the effective travel range of the screw. Therefore, \(Q_{NB2}\) is regarded as a ring-shaped fixed heat source on the screw's cylindrical surface.  
In the case of combined action of multiple heat sources, as Equation (1) belongs to the second-order linear partial differential equation, based on the linear superposition principle, the screw's temperature response at a given time and position can be determined by adding screw's temperature responses under each single heat source, which is displayed below:  
\[T_{total}(x,t)=\Sigma T_{i}(x,t) \tag{2}\]  
Figure 2: Schematic diagram of heat source distribution in the transmission system.  
in which \(T_{1}(x,t)\) stands for temperature response resulting from heat source \(Q_{B1}\); \(T_{2}(x,t)\) represents temperature response resulting from heat source \(Q_{NB2}\); \(T_{3}(x,t)\) represents temperature response resulting from heat source \(Q_{B3}\); \(T_{total}(x,t)\) represents the sum of the screw's temperature response under multiple heat sources.