# A Corrected Adaptive Balancing Approach of Motorized Spindle Considering Air Gap Unbalance
## 2 Unbalance and Force of Motorized Spindle
### Electromagnetic Unbalance and Force
#### 2.2.6 Quantitative Relationship between Unbalanced Mechanical and Electromagnetic Forces  
According to Table 2, for the same order formula of electromagnetic forces, two first-order 1\(\times\) rotation frequency components are obtained; thus, Equation (17) can be simplified as Equation (20).  
\[p_{n}(\theta,t)=\frac{\epsilon^{\prime}({B_{1}}^{2}+\sum\limits_{v}{{B_{v}}^{2}+ \sum\limits_{\mu}{{B_{\mu}}^{2}}})}{2\mu_{0}}\cos(\theta-\omega_{r}t). \tag{20}\]  
Substituting Equation (20) into Equation (14), we get  
\[\left\{\begin{array}{l}F_{\text{em}x}=\frac{{LR({B_{1}}^{2}+\sum\limits_{\mu }{{B_{\mu}}^{2}+\sum\limits_{v}{{B_{v}}^{2}})\pi\epsilon^{\prime}}}}{{2\mu_{0}}} \cos\omega_{r}t\\ F_{\text{em}y}=\frac{{LR({B_{1}}^{2}+\sum\limits_{\mu}{{B_{\mu}}^{2}+\sum \limits_{v}{{B_{v}}^{2}})\pi\epsilon^{\prime}}}}{{2\mu_{0}}}\sin\omega_{r}t\\ F_{\text{em}}=\sqrt{{F_{x}}^{2}+{F_{y}}^{2}}=\frac{{LR({B_{1}}^{2}+\sum \limits_{\mu}{{B_{\mu}}^{2}+\sum\limits_{v}{{B_{v}}^{2}})\pi\epsilon^{\prime}} }}{{2\mu_{0}}}\end{array}\right.. \tag{21}\]  
For the squirrel-cage motor, its rotor has a compensation effect for the uneven magnetic field; thus, the UMP shown in Equation (21) is corrected by adding an empirical coefficient \(\chi\)[35; 36] as follows:  
\[F_{\text{em}}=\frac{{\chi{LR({B_{1}}^{2}+\sum\limits_{\mu}{{B_{\mu}}^{2}}+ \sum\limits_{v}{{B_{v}}^{2}})\pi\epsilon^{\prime}}}}{{2\mu_{0}}}. \tag{22}\]  
Therefore, in the case of dynamic air gap eccentricity, the rotor is pulled by an unbalanced electromagnetic force with the frequency of 1\(\times\) rotating frequency, the direction of UMP from the rotation center of the rotor to the minimum air gap position is changing, and the amplitude of force is proportional to the dynamic air gap eccentricity.  
According to Equations (2) and (22), we can define a coefficient \(k\).  
\[k=\frac{{F_{\text{em}}}}{{F+F_{\text{em}}}}, \tag{23}\]  
where \(k\) represents the quantitative relationship between the unbalanced mechanical and electromagnetic forces, which is used for the correction of the adaptive balancing method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Order of EF**} & \multirow{2}{*}{\begin{tabular}{c} **First Order of** \\ **MFD** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Second Order** \\ **of MFD** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Order Number** \\ **of EF** \\ \end{tabular} } & \multirow{2}{*}{**Frequency**} \\ \hline \multirow{2}{*}{\(p\pm(p\pm 1)\)} & \multirow{2}{*}{2} & 1 & 1 & 0.5\(f_{1}\) \\ \cline{3-5}  & & 3 & 1 & 0.5\(f_{1}\) \\ \hline \multirow{2}{*}{\(v\pm(v\pm 1)\)} & \multirow{2}{*}{\(v\)} & \(v-1\) & 1 & 0.5\(f_{1}\) \\ \cline{3-5}  & & \(v+1\) & 1 & 0.5\(f_{1}\) \\ \hline \multirow{2}{*}{\(\mu\pm(\mu\pm 1)\)} & \multirow{2}{*}{\(\mu\)} & \(\mu-1\) & 1 & 0.5\(f_{1}\) \\ \cline{3-5}  & & \(\mu+1\) & 1 & 0.5\(f_{1}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: The 1\(\times\) rotation frequency components of electromagnetic forces under dynamic air gap eccentricity. MFD—magnetic flux density.