## 2 Temperature Field and Thermal Error Modeling of Nut-Rotating Ball Screw Transmission Mechanism
### Analysis of Temperature Field of the Nut-Rotating Ball Screw Pair under the Action of Heat Source
#### 2.3.1 Constant Heat Source Temperature Response  
Using one side of the screw as a constant power heat source and the boundary condition, the screw temperature field caused by the constant heat source is studied. Assuming that the initial temperature of the system is consistent with the environment, the initial condition is \(T(x,0)=T_{f}\); If the temperature of a constant heat source is \(T_{m}\), then the boundary condition is \(T(0,t)=T_{m}\); The screw transient temperature field distribution is obtained by solving Equation (3):  
\[T_{t}(x,t)=\Gamma_{\max}\bigg{[}1-erf(\frac{x}{2\sqrt{\alpha_{C}t}})\bigg{]}+ T_{f} \tag{4}\]  
where \(T_{t}(x,t)\) stands for the screw's transient temperature value; \(\Gamma_{\max}=T_{m}-T_{f}\) represents the screw's temperature elevation after reaching thermal stability; \(erf(x)\) is defined in mathematics as an error function:  
\[erf(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-\delta^{2}}d\delta \tag{5}\]  
Equation (4) indicates that after a sufficiently long period of time, when the screw is insulated from the air, the steady-state temperature values at each point on the screw are equal to the heat source temperature \(T_{m}\). However, this situation is not in line with actual working conditions, thus it is necessary to establish a steady-state temperature field model for the screw and to solve the screw's steady-state temperature values at each point while taking into account convective heat transfer. When the screw temperature field reaches the steady state, the temperature will not change with time at each point, that is, \(\partial T(x,t)/\partial t=0\). The screw steady-state heat conduction equation can be obtained from Equation (1):  
\[\frac{\partial^{2}T(x,t)}{\partial x^{2}}=\frac{4h}{\kappa d_{0}}\Big{[}T(x,t )-T_{f}\Big{]} \tag{6}\]The screw steady-state temperature field distribution acquired through solving Equation (6) is:  
\[T_{\rm s}(x,t)=C_{1}e^{-\sqrt{\frac{4\omega}{\omega_{0}}}x}+C_{2}e^{\sqrt{\frac{4 \omega}{\omega_{0}}}x}+T_{f} \tag{7}\]  
where \(T_{\rm s}(x,t)\) stands for the screw's steady-state temperature value; \(C_{1}\) and \(C_{2}\) represent coefficients for the screw's temperature rise, in degrees, and temperature distribution, respectively.  
When taking convective heat transfer into account, the screw's steady-state temperature rise value is \(\Gamma_{\rm max}=T_{\rm s}(x,t)-T_{f}\). The theoretical model of the screw temperature field under a constant heat source on one side can be obtained by combining Equations (4), (5) and (7):  
\[T_{c}(x,t)=\left(C_{1}e^{-\sqrt{\frac{4\mu}{\kappa d_{0}}}x}+C_{2}e^{\sqrt{ \frac{4\mu}{\kappa d_{0}}}x}\right)\left(1-\frac{2}{\sqrt{\pi}}\int_{0}^{2} \frac{x}{2\sqrt{\pi t}}\;e^{-\beta^{2}}d\delta\right)+T_{f} \tag{8}\]  
The first of the two terms in Equation (8) represents the screw's temperature rise distribution after steady state, while the second term represents the temperature changes at different points on the screw at different times.