## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Parameters and Characteristic of Improved DTNF  
Multi-stage-blade mode suppression is the prerequisite for MSR system stability. Multi-stage-blade modal vibration will occur when the turbo rotor is subjected to a current step signal or an imbalanced exciting force, as shown in Figure 4a,b. Although rotor mass imbalance vibration (co-frequency synchronous vibration) is the main source of disturbances in MSTMPs, it is also important and indispensable to suppress multi-stage-blade modal vibration (multi-frequency mode vibration) by cascaded notch filters. The transfer function \(G_{\text{BNF}}(s)\) of the traditional biquadratic notch filter (BNF) is  
\[G_{\text{BNF}}(s)=\frac{w_{a}^{2}}{w_{b}^{2}}\frac{s^{2}+2\xi_{b}w_{b}s+w_{b}^ {2}}{s^{2}+2\xi_{a}w_{a}s+w_{a}^{2}}=\frac{\lambda s^{2}+\eta s+\
u}{\lambda s^ {2}+\tau s+u} \tag{30}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \(x\) **(m)** & \(k_{h}\) **(N m\({}^{-1}\))** & \(w_{c}\) **(Hz)** & \(\gamma\) **(deg)** & \(\varepsilon_{\max}\) & \(\Delta f_{\max}^{\text{ANF}}\) **(Hz)** & \(Q_{\min}^{\text{ANF}}\) \\ \hline \(6\times 10^{-5}\) (\(0.3p\)) & \(9.7831\times 10^{5}\) & 92.42 & 21.91 & 160.194 & 25.639 & 4.680 \\ \(8\times 10^{-5}\) (\(0.4p\)) & \(1.0836\times 10^{6}\) & 91.06 & 21.75 & 168.146 & 26.927 & 4.457 \\ \(1\times 10^{-4}\) (\(0.5p\)) & \(1.2336\times 10^{6}\) & 89.10 & 21.52 & 179.662 & 28.796 & 4.167 \\ \(1.5\times 10^{-4}\) (\(0.75p\)) & \(1.9177\times 10^{6}\) & 79.39 & 20.22 & 236.038 & 38.024 & 3.156 \\ \(2\times 10^{-4}\) (\(p\)) & \(3.5507\times 10^{6}\) & 47.48 & 13.94 & 398.951 & 65.680 & 1.827 \\ \hline \end{tabular}
\end{table} 
Table 1: Design process of ANF notch parameters based on the perturbation of displacement stiffness \(k_{h}\) (With \(\Omega=120\) Hz). Figure 9: Open-loop frequency characteristics without ANF. (**a**) Magnitude-phase frequency characteristics of the open-loop system. (**b**) Phase stability margin \(\gamma\) at the open-loop cutoff frequency \(w_{c}\). where \(\lambda=w_{a}^{2}/w_{b}^{2}\), \(\tau=2\zeta_{a}w_{a}^{3}/w_{b}^{2}\), \(\eta=2\zeta_{b}w_{a}^{2}/w_{b}\), \(u=w_{a}^{4}/w_{b}^{2}\), and \(v=w_{a}^{2}\). As shown in Figure 10, the disadvantage of the traditional BNF is that the notch width is not adjustable, and the amplitude of the positive and negative resonance peaks cannot be adjusted separately. In order to facilitate the calculation of multiple parameters and the adjustment of notch performance (such as notch depth \(l_{\text{depth}}^{\text{DTNF}}\), notch width \(\Delta f^{\text{DTNF}}\), phase-lag angle, etc. ), the structure of the traditional BNF is improved in this research. The block diagram of the improved DTNF is shown in Figure 6b. If we let \(\lambda=1/w_{f}^{2}\), \(\tau=k/w_{f}\), \(\eta=\zeta k/w_{f}(0<\zeta<1)\), and \(u=v=1\), the transfer function \(G_{\text{DTNF}}(s)\) of the improved DTNF can be obtained as follows  
\[G_{\text{DTNF}}(s)=\frac{\lambda s^{2}+\eta s+v}{\lambda s^{2}+\tau s+u}=\frac {s^{2}+\zeta kw_{f}s+w_{f}^{2}}{s^{2}+kw_{f}s+w_{f}^{2}} \tag{31}\]  
where \(w_{f}\) is the notch frequency (with unit rad/s), \(\zeta\) is the depth factor, and \(k\) is the width factor. The definitions of the notch parameters of the DTNF are similar to those of the ANF, and are shown in Figure 7. The conversion relationship between the notch parameters, such as \(l_{\text{depth}}^{\text{DTNF}}\), \(\Delta f^{\text{DTNF}}\), \(Q^{\text{DTNF}}\), and \(k\) and \(\zeta\), are calculated below.