## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Bandwidth Factor Rectification Method of the ANF  
Generally speaking, the displacement stiffness \(k_{h}\) used to suppress the co-frequency synchronous vibration is a constant value that is measured when the AMB-rotor is statically suspended. However, in fact, \(k_{h}\) fluctuates with changing rotor displacement \(x\). So, the perturbation should be considered on the basis of a constant value. The perturbation of displacement stiffness \(k_{h}\) is compensated adaptively by adjusting the value of bandwidth factor \(\varepsilon\) in this research. The constraint relationship between the notch parameter and the phase angle is calculated below. At the open-loop cutoff frequency \(w_{c}(w_{c}<\Omega)\), the phase characteristic of the ANF exhibits hysteresis. According to (15), the phase-lag angle \(\varphi(\varepsilon)\) (\(\varphi(\varepsilon)<0\)) introduced by an ANF is a function of the bandwidth factor \(\varepsilon\)  
\[\varphi(\varepsilon)=-\arctan\left(\frac{\varepsilon w_{c}}{\Omega^{2}-w_{c}^ {2}}\right)\frac{180}{\pi}. \tag{25}\]  
By constraining \(\varphi(\varepsilon)\) at \(w_{c}\), we ensure that \(|\varphi(\varepsilon)|\) is less than the phase stability margin \(\gamma(\gamma>0)\) of the open-loop frequency characteristics at \(w_{c}\), \(|\varphi(\varepsilon)|<\gamma\). So, according to (25), the constraint condition of \(\varepsilon\) can be obtained  
\[0<\varepsilon<\frac{\Omega^{2}-w_{c}^{2}}{w_{c}}\tan\left(\gamma\frac{\pi}{180 }\right). \tag{26}\]  
Equation (26) is the calculation formula based on the constraint condition of the bandwidth factor \(\varepsilon\). Similarly, the calculation formula based on the constraint condition of the notch width \(\Delta f^{\rm ANF}\) can also be obtained  
\[0<\Delta f^{\rm ANF}<\Delta f^{\rm ANF}_{\rm max}. \tag{27}\]According to (21) and (26), the maximum \(\Delta f^{\rm ANF}_{\rm max}\) of the notch width can be obtained  
\[\Delta f^{\rm ANF}_{\rm max}=\frac{\big{(}\Omega^{2}-w_{c}^{2}\big{)}\tan\big{(} \gamma\frac{\pi}{180}\big{)}\sqrt{4\Omega^{2}+\frac{\big{(}\Omega^{2}-w_{c}^{2 }\big{)}^{2}}{w_{c}^{2}}\tan^{2}\big{(}\gamma\frac{\pi}{180}\big{)}}}{4\pi\Omega w _{c}}. \tag{28}\]  
Although the constraint conditions derived above can ensure the stability of an MSR system with an ANF, the bandwidth factor \(\varepsilon\) and the notch width \(\Delta f^{\rm ANF}\) determined from (26)-(28) are a series of numerical values within a range, not a specific value. However, from the previous analysis in Section 3.1, it is obvious that the notch performance of the ANF is sensitive to changing \(\varepsilon\). Therefore, this research continues to search for an appropriate value of \(\varepsilon\). An appropriate value of the bandwidth factor \(\varepsilon\) needs to be designed in conjunction with the rotor displacement \(x\). The relationship between control current \(i_{\rm X}\), rotor displacement \(x\), and electromagnetic force \(f_{\rm X}\) is shown in Figure 8a. The relationship between control current \(i_{\rm X}\), rotor displacement \(x\), and displacement stiffness \(k_{h}\) is shown in Figure 8b. As can be seen, when the rotor displacement \(x\) changes, the perturbation of displacement stiffness \(k_{h}\) will appear. According to (2), taking the parameter uncertainty of the MSR system itself into account, the dynamic displacement stiffness \(k_{h}\) can be calculated according to the following formula [3]  
\[k_{h}=\frac{\partial f_{\rm X}(i_{\rm X},x)}{\partial x}=\mu_{0}N^{2}A_{p}\cos \big{(}22.5\cdot\frac{\pi}{180}\big{)}\left[\frac{2(i_{0}+i_{\rm X})^{2}}{(p_{ 0}-x)^{3}}+\frac{2(i_{0}-i_{\rm X})^{2}}{(p_{0}+x)^{3}}\right]. \tag{29}\]  
The unilateral protective gap of the auxiliary bearing is defined as \(p\). The stability description of the AMB-rotor system typically encompasses the following two aspects:  
1. There is no pole allocated on the right-half plane in the transfer function of the closed-loop system. 2. The rotor displacement \(x\) is within a safe range that is allowed by the protective gap \(p\).