## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Multi-Objective Constrained Optimization Method of Cascaded Improved DTNFs
Therefore, after determining the depth factor \(\zeta_{i}\), in order to select width factor \(k_{i}\), the effect of \(k_{i}\) on the notch width \(\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})\) and phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})\) is balanced and optimized by employing a linear normalization method. The normalization of the notch width \(\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})_{\text{normal}}\) is designed as follows  
\[\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})_{\text{normal}}=\frac{\Delta f^{ \text{DTNF}}_{i}(\zeta_{i},k_{i})-\min\bigl{[}\Delta f^{\text{DTNF}}_{i}( \zeta_{i},k_{i})\bigr{]}}{\max\bigl{[}\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{ i})\bigr{]}-\min\bigl{[}\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})\bigr{]}} \tag{47}\]  
where \(\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})=w_{f_{i}}\sqrt{\bigl{(}-2+2\zeta_{ i}^{2}k_{i}^{2}-k_{i}^{2}\bigr{)}^{2}-4}/4\pi\). Similarly, the normalization of the phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})_{\text{normal}}\) is designed as follows  
\[\phi_{i}(\zeta_{i},k_{i})_{\text{normal}}=\frac{\phi_{i}(\zeta_{i},k_{i})- \min\bigl{[}\phi_{i}(\zeta_{i},k_{i})\bigr{]}}{\max\bigl{[}\phi_{i}(\zeta_{i}, k_{i})\bigr{]}-\min\bigl{[}\phi_{i}(\zeta_{i},k_{i})\bigr{]}} \tag{48}\]  
where \(\phi_{i}(\zeta_{i},k_{i})=\bigl{[}\arctan\zeta_{i}k_{i}w_{f_{i}}w_{i}/(w_{f_{ i}}^{2}-w_{i}^{2})-\arctan k_{i}w_{f_{i}}w_{i}/(w_{f_{i}}^{2}-w_{i}^{2}) \bigr{]}180/\pi\). To balance the effect of the width factor \(k_{i}\) on the notch width \(\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})\) and the phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})\), \(P_{i}\) is introduced as an adjustable weight. According to (46) and (47), the objective function \(M_{\text{optimal}}\) is designed as follows  
\[M_{\text{optimal}}=P_{i}\Delta f^{\text{DTNF}}_{i}(\zeta_{i},k_{i})_{\text{ normal}}+(1-P_{i})\phi_{i}(\zeta_{i},k_{i})_{\text{normal}}. \tag{49}\]  
When the MSTMP works at the rated speed of 24,000 rpm, it can be seen from the analysis in Section 2.2 that there are six different frequencies (350 Hz, 463 Hz, 530 Hz, 628 Hz, 745 Hz, 980 Hz) of multi-stage-blade modal vibration. At six different blade mode frequencies, comprehensive normalized design results for the width factor \(k\) with multi-objective constraints are shown in Figure 12. Designed values of the notch depth \(l_{\text{depth}_{i}}^{\text{DTNF}}\), the depth factor \(\zeta_{i}\), the width factor \(k_{i}\), and the phase-lag angle \(\phi_{i}(w_{c})\) are shown in Table 2. In actual operation, parameters are designed according to different mode frequencies at different speed stages. Framework implementation of the integrated AMB controller and the proposed algorithm is shown in Figure 13. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(w_{f}\) (Hz) & \(I_{\text{depth}}^{\text{DTNF}}\) (dB) & \(\zeta\) & \(P\) & \(k_{\text{span}}\) & \(k\) & \(\phi(w_{c})\) (deg) \\ \hline
350 & 2 & 0.7943 & 0.85 & \((0,5)\) & 1.4290 & \(-2.3306\) \\
463 & 14 & 0.1995 & 0.84 & \((0,7)\) & 0.7143 & \(-3.9647\) \\
530 & 51 & 0.0028 & 0.83 & \((0,7)\) & 0.2857 & \(-1.0263\) \\
628 & 29 & 0.0355 & 0.82 & \((0,7)\) & 0.4286 & \(-3.6685\) \\
745 & 5 & 0.5623 & 0.81 & \((0,6)\) & 0.8571 & \(-2.7702\) \\
980 & 11 & 0.2818 & 0.80 & \((0,7)\) & 0.4286 & \(-1.7279\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Designed values of improved DTNF based on multi-objective constrained optimization algorithm (With \(\Omega=\) 24,000 rpm). Figure 12: Normalized results of the width factor \(k\) based on multi-objective constraint with multi-stage-blade modal frequencies. (**a**) \(w_{f}=\) 350 Hz. (**b**) \(w_{f}=\) 463 Hz. (**c**) \(w_{f}=\) 530 Hz. (**d**) \(w_{f}=\) 628 Hz. (**e**) \(w_{f}=\) 745 Hz. (**f**) \(w_{f}=\) 980 Hz. Figure 13: Framework implementation of the proposed multi-parameter optimization algorithm.