# A Corrected Adaptive Balancing Approach of Motorized Spindle Considering Air Gap Unbalance
## 2 Unbalance and Force of Motorized Spindle
### Electromagnetic Unbalance and Force
#### 2.2.3 Electromagnetic Force of Spindle Motor without Eccentricity  
Firstly, for an asynchronous motorized spindle with three-phase symmetrical windings, the resultant radial magnetic potential can be expressed as  
\[f(\theta,t)=f_{p}(\theta,t)+\sum_{v}f_{v}(\theta,t)+\sum_{\mu}f_{\mu}(\theta,t), \tag{6}\]  
where  
\[f_{p}(\theta,t)=F_{\rm P}\cos(p\theta-\omega_{1}t-\varphi_{0}), \tag{7}\]  
\[f_{v}(\theta,t)=F_{v}\cos(v\theta-\omega_{1}t-\varphi_{v}), \tag{8}\]  
\[f_{\mu}(\theta,t)=F_{\mu}\cos(\mu\theta-\omega_{\mu}t-\varphi_{\mu}), \tag{9}\]  
where \(f_{p}(\theta,t)\), \(f_{v}(\theta,t)\), and \(f_{\mu}(\theta,t)\) are the main magnetic potential, the harmonic magnetic potential of the stator, and the harmonic magnetic potential of the rotor, respectively, while \(v\) and \(\mu\) are the harmonic orders of the stator and rotor, respectively.  
Secondly, the radial magnetic permeance can be approximated as  
\[\lambda(\theta,t)=\Lambda_{0}+\sum_{k_{1}}\lambda_{k_{1}}+\sum_{k_{2}}\lambda _{k_{2}}, \tag{10}\]  
where \(\Lambda_{0}\) is the invariant part of permeance, \(\lambda_{k_{1}}\) is the harmonic permeance caused by the stator slots, \(\lambda_{k_{2}}\) is the harmonic permeance caused by the rotor slots, and both \(\lambda_{k_{1}}\) and \(\lambda_{k_{2}}\) are numerically very small and can be ignored.  
Thirdly, according to Equations (6) and (10), the instantaneous radial magnetic density can be derived as  
\[b(\theta,t)=f(\theta,t)\lambda(\theta,t)=B_{1}\cos(p\theta-\omega_{1}t-\varphi _{0})+\sum_{v}B_{v}\cos(v\theta-\omega_{1}t-\varphi_{v})+\sum_{\mu}B_{\mu}\cos (\mu\theta-\omega_{\mu}t-\varphi_{\mu}), \tag{11}\]  
where \(B_{1}\) is the main magnetic density amplitude, while \(B_{v}\) and \(B_{\mu}\) are the harmonic magnetic density amplitudes of stator and rotor, respectively.  
Finally, the radial electromagnetic force generated by the air gap magnetic field and acting on the unit surface area of rotor is proportional to the square of the radial magnetic flux density in the air gap; the radial electromagnetic force on the unit surface area of spindle rotor can be approximated as  
\[P_{r}(\theta,t)=\frac{b^{2}(\theta,t)}{2\mu_{0}}, \tag{12}\]  
where \(\mu_{0}\) is the vacuum permeability, and \(\mu_{0}=4\pi\times 10^{-7}\) H/m.  
As the electromagnetic force with lower order and larger amplitude plays a major role in the vibration level of the spindle rotor, the higher-order and smaller-amplitude components of the radial  
Figure 3: Electromagnetic force solving process of spindle motor.  
electromagnetic force are omitted, and the constant components are omitted; thus, it can be obtained that the radial electromagnetic force per unit surface area of spindle rotor is  
\[p_{r}(\theta,t)=\frac{1}{2\mu_{0}}\Bigg{\{}\frac{{B_{1}}^{2}}{2}\cos(2p\theta-2 \omega_{1}t-2\varphi_{0r})+\sum_{\mathbb{P}_{2}}\sum_{\mu_{2}}{B_{0r}B_{\mu_{2} }\cos[(\mu\pm v)\theta-(\omega_{\mu}\pm\omega_{1})t-(\varphi_{\mu}\pm\varphi_{ 0})]}\Bigg{\}}. \tag{13}\]  
The integral operation of Equation (13) is obtained as  
\[\left\{\begin{array}{l}F_{\rm emx}=LR\int_{0}^{2\pi}{p_{r}(\theta,t)\cos \theta d\theta}\\ F_{\rm emy}=LR\int_{0}^{2\pi}{p_{r}(\theta,t)\sin\theta d\theta}\\ F_{\rm em}=\sqrt{{F_{x}}^{2}+{F_{y}}^{2}}\end{array}\right., \tag{14}\]  
where \(F_{\rm emx}\) and \(F_{\rm emy}\) are two components of the electromagnetic force along the horizontal and vertical directions in the Cartesian coordinate system, respectively, \(F_{\rm em}\) is their resultant force, and \(L\) and \(R\) are the length and radius of the spindle rotor, respectively. For the case with no eccentricity, according to Equations (13) and (14), the spindle is not subject to UMP, i.e., \(F_{\rm em}=0\).