## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Parameters and Characteristic of the ANF  
An ANF is usually used to filter the co-frequency current component to suppress rotor mass imbalanced vibration. However, this may result in an instability of the closed-loop system: an inappropriate bandwidth factor \(\varepsilon\) of the ANF causes a phase-lag angle \(\varphi(\varepsilon)\) at the open-loop cutoff frequency \(w_{c}\), which reduces the phase stability margin \(\gamma\) of the MSR system. So, an ANF bandwidth factor \(\varepsilon\) rectification method based on displacement stiffness \(k_{h}\) perturbation is designed to solve this problem. The perturbation of \(k_{h}\) highlights the innovation of the proposed method: the parameter uncertainty of the MSR system itself is taken into account. Notch parameters and the notch characteristic are quantitatively calculated and qualitatively analyzed, respectively. The block diagram of the ANF is shown in Figure 6a. \(\varepsilon\) is the bandwidth factor and is one of the most important parameters in determining notch performance. The notch frequency \(w\) (with unit rad/s) of the ANF changes synchronously with the rotor speed \(\Omega\) (with unit rad/s). According to [23], the transfer function \(G_{\text{ANF}}(s)\) of the ANF is  
\[G_{\text{ANF}}(s)=\frac{s^{2}+w_{w=\Omega}^{2}}{s^{2}+\varepsilon s +w_{w=\Omega}^{2}}. \tag{15}\]  
A diagram of the definitions of the notch parameters is shown in Figure 7. \(I_{\text{depth}}^{\text{ANF}}\) (with unit dB) is the notch depth, which represents the attenuation of the magnitude at the notch frequency. \(\Delta f^{\text{ANF}}\) (with unit Hz) is the notch width, and the amplitude of vibration in this frequency range is attenuated to different degrees. \(B_{w}^{\text{ANF}}\) (with unit rad/s) is the \(-3\) dB bandwidth, defined as \(B_{w}^{\text{ANF}}=2\pi\Delta f^{\text{ANF}}\). \(Q^{\text{ANF}}\) is the quality factor, defined as \(Q^{\text{ANF}}=\Omega/B_{w}^{\text{ANF}}\). Figure 5: Block diagram of the AMB-rotor control system with the novel integrated AMB controller. Figure 6: Block diagram of the notch filters. (**a**) Block diagram of the ANF. (**b**) Block diagram of the improved DTNFs. The conversion relationship among the notch parameters such as \(\Delta f^{\text{ANF}}\), \(Q^{\text{ANF}}\), and \(\varepsilon\) is calculated below. With the notch frequency \(w=\Omega\) as the center frequency, when the amplitude of the ANF attenuates to \(-3\) dB, the corresponding frequencies \(w\in(w_{1},w_{2})\) are all within the range of the \(-3\) dB bandwidth \(B_{w}^{\text{ANF}}\). \(w_{1}\) is the upper limit frequency, and \(w_{2}\) is the lower limit frequency, so we can obtain  
\[-20\lg\mid G_{\text{ANF}}(jw)\rvert_{w=w_{1}}=-20\lg\mid G_{\text{ANF}}(jw) \rvert_{w=w_{2}}=-3, \tag{16}\]  
\[\left\lvert G_{\text{ANF}}(jw)\right\rvert=\left\lvert\frac{(jw)^{2}+\Omega^ {2}}{(jw)^{2}+\varepsilon\cdot jw+\Omega^{2}}\right\rvert=\frac{\sqrt{2}}{2}. \tag{17}\]  
To obtain the solution of \(w_{1}\) and \(w_{2}\), further calculation of (17) produces  
\[w^{4}+\left(-2\Omega^{2}-\varepsilon^{2}\right)w^{2}+\Omega^{4}=0. \tag{18}\]  
The results of solving (18) are  
\[\left\{ \begin{array}{l}w_{1}^{2}=\frac{2\Omega^{2}+\varepsilon^{2}-\varsigma \sqrt{4\Omega^{2}+\varepsilon^{2}}}{\Omega}\\ w_{2}^{2}=\frac{2\Omega^{2}+\varepsilon^{2}+\varsigma\sqrt{4\Omega^{2}+ \varepsilon^{2}}}{2}\end{array} \right.. \tag{19}\]  
Because \(B_{w}^{\text{ANF}}=w_{2}-w_{1}=2\pi\Delta f^{\text{ANF}}\), and \(\Omega=(w_{1}+w_{2})/2\) are already defined, the following formula can be obtained  
\[\frac{w_{2}^{2}-w_{1}^{2}}{\Omega^{2}}=\frac{4\pi\Delta f^{\text{ANF}}}{ \Omega}. \tag{20}\]  
Figure 7: Magnitude-phase frequency characteristics of the ANF. (**a**) Notch parameters definition diagram. (**b**) \(\Omega\) is a constant and \(\varepsilon\) is varied. (**c**) \(\varepsilon\) is a constant and \(\Omega\) is varied. (**d**) Both \(\varepsilon\) and \(\Omega\) are varied.