## 2 Experimental Equipment Construction and Thermal Deformation Measurement
### Thermal Deformation Theory  
In [PERSON]'s law, the structural deformation can be derived from the relation between the elastic constant and the elastic modulus in elasticity theory. The thermal deformation caused by the temperature variations of the material can be obtained by the superposition principle from the effects of stress and temperature variation.  
During dynamic operation of a machine tool, heat generated by the friction causes the temperature to rise and a temperature difference inside a structure, followed by a thermal deformation. From a microscopic perspective, thermal deformation occurs after the temperature rise because the molecules gain greater kinetic energy and vibrate at larger amplitudes. Thus, the intermolecular average distance is increased, which causes the deformation. Considering a homogeneous material, as shown in Figure 6, when the difference temperature \(\Delta T\) rises, according to thermal expansion theory [25], the elongations in all directions are  
\[\Delta a=\alpha_{x}\Delta T\cdot a, \tag{1}\]  
\[\Delta b=\alpha_{y}\Delta T\cdot b, \tag{2}\]  
\[\Delta c=\alpha_{z}\Delta T\cdot c, \tag{3}\]  
Figure 5: Spindle power measurement process.  
where \(\Delta a\), \(\Delta b\), and \(\Delta c\) represent the elongation in the \(x\) direction, \(y\) direction, and \(z\) direction, respectively; \(\alpha\) is the coefficient of thermal expansion; and \(\Delta T\) is the temperature difference. Therefore, the strains caused by temperature changes in all directions are given as  
\[\left(\varepsilon_{xx}\right)_{T} =\frac{\Delta a}{a}=\alpha_{x}\cdot\Delta T, \tag{4}\] \[\left(\varepsilon_{yy}\right)_{T} =\frac{\Delta b}{b}=\alpha_{y}\cdot\Delta T,\] (5) \[\left(\varepsilon_{zz}\right)_{T} =\frac{\Delta c}{c}=\alpha_{z}\cdot\Delta T, \tag{6}\]  
where \(\varepsilon\) is the strain due to temperature change, \(\left(\varepsilon_{xx}\right)_{T}\) is the strain in the \(x\) direction due to temperature change, \(\left(\varepsilon_{yy}\right)_{T}\) is the strain in the \(y\) direction, and \(\left(\varepsilon_{zz}\right)_{T}\) is the strain in the \(z\) direction.  
Assuming that the material is isotropic, i.e., \(\alpha_{x}=\alpha_{y}=\alpha_{z}\), the following equation is obtained:  
\[\left(\varepsilon_{xx}\right)_{T}=\left(\varepsilon_{yy}\right)_{T}=\left( \varepsilon_{zz}\right)_{T}=\alpha\cdot\Delta T. \tag{7}\]  
Therefore, under the stress and temperature, the strains in different directions can be obtained by the superposition principle as  
\[\varepsilon_{xx}=\frac{\sigma_{xx}-\
u\!\left(\sigma_{yy}+\sigma_{zz}\right)} {E}+\alpha\cdot\Delta T, \tag{8}\]  
\[\varepsilon_{yy}=\frac{\sigma_{yy}-\
u\!\left(\sigma_{xx}+\sigma_{zz}\right)} {E}+\alpha\cdot\Delta T, \tag{9}\]  
\[\varepsilon_{zz}=\frac{\sigma_{zz}-\
u\!\left(\sigma_{xx}+\sigma_{yy}\right)} {E}+\alpha\cdot\Delta T, \tag{10}\]  
where \(E\) is [PERSON]'s modulus (E-modulus), \(\
u\) is [PERSON]'s ratio, and \(\sigma\) is the thermal stress, which can be obtained from \(\sigma=E\varepsilon\). In Equations (8)-(10), under the effects of stress and temperature, the strains in all directions are obtained by the superposition principle, \(\varepsilon_{xx}\) is the \(x\)-direction strain, \(\varepsilon_{yy}\) is the \(y\)-direction strain, and \(\varepsilon_{zz}\) is the \(z\)-direction strain.  
Figure 6: Thermal deformation schematic diagram.