## 2 Experimental Equipment Construction and Thermal Deformation Measurement
### Heating Capacity of Spindle  
For the bearing heating capacity calculation, SKF proposed the bearing heat generating rate computing formulas in 2003. According to the friction types of rolling bearing, the effect of different friction sources is calculated [27] as  
\[M=\varphi_{ish}\varphi_{rs}M_{rr}+M_{sl}+M_{sall}+M_{dmg}, \tag{13}\]  
where \(M\) is the total frictional moment of bearing, \(M_{rr}\) is the rolling frictional moment, \(M_{sl}\) is the sliding frictional torque, \(M_{sall}\) is the friction torque of the seals, \(M_{drag}\) is the friction torque due to drag, \(\varphi_{ish}\) is the inlet shear heating reduction factor, and \(\varphi_{rs}\) is the kinematic replenishment reduction factor. The rolling frictional moment can be given as  
\[M_{rr}=G_{rr}(\
u n)^{0.6}, \tag{14}\]  
where \(G_{rr}\) is the rolling frictional moment variable, \(\
u\) is the kinematic viscosity of lubricating oil at the operating temperature, and \(n\) is the rotation speed of the bearing. The sliding frictional torque can be given as  
\[M_{sl}=G_{sl}\mu_{sl}, \tag{15}\]where \(G_{sl}\) is the variable depending on the bearing type and its dimensions, and \(\mu_{sl}\) is the sliding friction coefficient. The friction torque of the seals can be expressed as  
\[M_{seal}=K_{s1}d_{s}^{y}+K_{s2}, \tag{16}\]  
where \(M_{seal}\) is the friction torque of the seals; \(K_{s1}\) is the constant by the bearing model depending on the seal type, bearing type and size; \(K_{s2}\) is the constant by the bearing model depending on the seal type, bearing type, and bearing size; \(d_{s}\) is the seal counter face diameter; \(\gamma\) is the constant by the bearing model depending on the seal type, bearing type, and bearing size. The inlet shear heating reduction factor can be given as  
\[\varphi_{ish}=\frac{1}{1+1.84\times 10^{-9}{(nd_{m})}^{1.28}v^{0.64}} \tag{17}\]  
where \(\varphi_{ish}\) is the inlet shear heating reduction factor, \(n\) is the rotation speed of the bearing, \(d_{m}\) is the pitch-circle diameter of the bearing, and \(v\) is the kinematic viscosity of the lubricating oil at the operating temperature. The kinematic replenishment reduction factor can be expressed as  
\[\varphi_{rs}=\frac{1}{e^{K_{rs}vm(d+D)\sqrt{\frac{K_{s2}}{2(D-d)}}}} \tag{18}\]  
where \(\varphi_{rs}\) is the kinematic replenishment reduction factor, \(K_{rs}\) is the replenishment constant, and \(K_{Z}\) represents the geometric constants for the roller bearing type. The friction torque due to the drag of ball bearing can be expressed as  
\[M_{drag}=V_{M}K_{ball}{d_{m}}^{5}n^{2} \tag{19}\]  
where \(M_{drag}\) is the friction torque due to the drag of the ball bearing, \(V_{M}\) is the drag loss factor, \(K_{ball}\) is the ball bearing correlation coefficient, \(n\) is the rotation speed of the bearing, and \(d_{m}\) is the pitch-circle diameter of the bearing. The heat generating rate Q can be given as  
\[ \begin{array}{l}\text{Q}=0.105\times 10^{-6}M\cdot n\ (N)\\ =1.03\times 10^{-6}M\cdot n\ (kgf)\end{array}  \tag{20}\]  
where \(M\) is the friction torque (\(N\cdot mm\ or\ kgf\cdot mm\)), and \(n\) is the rotational speed (rpm) of the bearing. The unit of the heat generating rate is kW. In addition, the motor heating capacity calculation and SKF torque computing equations can be compared with the measurement value of the spindle by the power meter. Because the copper and iron losses of the motor can be converted to heat, the copper loss and core loss of the motor shall be obtained. The copper loss is related to the motor resistivity, where a higher motor resistivity represents a larger copper loss. The FANUC motor model for the built-in spindle motor in this study is \(\alpha 112\text{L}/1500_{iB}\), which is detailed in FANUC motor specification [28]. The resistance values are 91 m\(\Omega\) and 40 m\(\Omega\) at the rotational speed of the motor below 3500 rpm and the range 3500 rpm-15,000 rpm, respectively.