## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Multi-Objective Constrained Optimization Method of Cascaded Improved DTNFs  
Taking the notch depth \(\lambda^{\text{DTNF}}_{\text{depth}}\) at the notch frequency \(w_{f}\), the notch width \(\Delta f^{\text{DTNF}}\) corresponding to \(-3\) dB, and the phase-lag angle \(\phi\) introduced by the DTNF at the open-loop cutoff frequency \(w_{c}\) as the multi-objective constraints, the multi-parameter optimization algorithm of cascaded multi-frequency improved DTNFs is as follows. As shown in Figure 10, the phase-frequency characteristic of the improved DTNF at the open-loop cutoff frequency \(w_{c}(w_{c}<w_{f})\) exhibits hysteresis. According to (31), the phase-lag angle \(\phi<0\) introduced by the improved DTNF is a function of the depth factor \(\zeta\) and width factor \(k\)  
\[\phi(\zeta,k)=\left(\arctan\frac{\zeta kw_{f}w}{w_{f}^{2}-w^{2}}-\arctan\frac {kw_{f}w}{w_{f}^{2}-w^{2}}\right)\frac{180}{\pi}. \tag{45}\]  
The relationship between the depth factor \(\zeta\), width factor \(k\), and phase-lag angle \(\phi\) is shown in Figure 10(a). Obviously, the phase lag angle \(\phi\) decreases with increasing depth factor \(\zeta\), and the phase-lag angle \(\phi\) decreases with the decreasing width factor \(k\). Figure 11: Diagram of phase-lag angle \(\phi\). (**a**) Relationship between depth factor \(\zeta\), width factor \(k\), and phase-lag angle \(\phi\). (**b**) Phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})\) is affected by depth factor \(\zeta_{i}\) and width factor \(k_{i}\). However, the multi-objective optimization should consider multiple constraints, so the effects of the depth factor \(\zeta\) and width factor \(k\) on the notch depth \(l_{\text{depth}}^{\text{DTNF}}\) and the notch width \(\Delta f^{\text{DTNF}}\) should also be taken into account, instead of just considering the effects of depth factor \(\zeta\) and width factor \(k\) on the phase-lag angle \(\phi\). To consider the effect of the depth factor \(\zeta\) on the notch depth \(l_{\text{depth}}^{\text{DTNF}}\) and the phase-lag angle \(\phi\), an analysis is carried out. For the cascade \(i\)-stage improved DTNFs, \(l_{\text{depth}_{i}}^{\text{DTNF}}\) can be determined by the peak amplitude \(E_{\text{amp}_{i}}\) at multi-stage-blade modal frequencies, as shown in Figure 4a. According to (44), the constraint condition of \(\zeta_{i}\) can be obtained  
\[0<\zeta_{i}\leq 10^{-\frac{E_{\text{amp}_{i}}}{2\lambda}}. \tag{46}\]  
If the depth factor \(\zeta_{i}\) is too large, the notch depth \(l_{\text{depth}_{i}}^{\text{DTNF}}\) provided by the improved DTNFs will not be sufficient, thus, the amplitude attenuation will be incomplete. If the depth factor \(\zeta_{i}\) is too small, not only will the amplitude attenuation be excessive but also a larger phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})\) will be introduced by the improved DTNFs. So, to ensure that \(\phi_{i}(\zeta_{i},k_{i})\) introduced by \(\zeta_{i}\) is minimized, \(\zeta_{i}\) is designed to be the maximum value \(10^{-E_{\text{amp}_{i}}/20}\). Furthermore, the maximum depth factor \(\zeta_{i}=10^{-E_{\text{amp}_{i}}/20}\) is calculated when the notch depth \(l_{\text{depth}_{i}}^{\text{DTNF}}\) is exactly equal to the peak amplitude \(E_{\text{amp}_{i}}\) of the multi-stage-blade modal vibration. This guarantees that the incomplete attenuation phenomenon will not occur. To consider the effect of the width factor \(k\) on the notch width \(\Delta f^{\text{DTNF}}\) and the phase-lag angle \(\phi\), an analysis is carried out. As shown in Figure 11b, varying \(\zeta_{i}\) in a small range \(\zeta_{i}\in(0.01,0.10)\) mostly has very little impact on the phase-lag angle \(\phi_{i}(\zeta_{i},k_{i})\). Especially when the value of the width factor \(k_{i}\) is smaller, such as \(k\in(0,7)\).