## 3 Suppression Strategy of Synchronous Vibration and Multi-Stage-Blade Modal Vibration
### Parameters and Characteristic of Improved DTNF
To obtain the solution for the notch depth \(l_{\text{depth}}^{\text{DTNF}}\), if the frequency \(w\) of the input signal is equal to the notch frequency \(w_{f}\) of the improved DTNF, \(\left|G_{\text{DTNF}}(jw)\right|_{w=w_{f}}\) will represent the value of the notch depth \(l_{\text{depth}}^{\text{DTNF}}\), so we can obtain  
\[\left|G_{\text{DTNF}}(jw)\right|_{w=w_{f}}=\left|\frac{\left(jw_{f}\right)^{2 }+\zeta kw_{f}\cdot jw_{f}+w_{f}^{2}\right|}{\left(jw_{f}\right)^{2}+kw_{f} \cdot jw_{f}+w_{f}^{2}}\right|=\zeta, \tag{32}\]  
\[l_{\text{depth}}^{\text{DTNF}}=-20\lg\left|G_{\text{DTNF}}\left(jw_{f}\right) \right|=-20\lg\zeta. \tag{33}\]  
Figure 10: Magnitude-phase frequency characteristics of the traditional BNF and the improved DTNF. (**a**) BNF, \(w_{a}=300\) Hz, \(w_{b}=150\) Hz. (**b**) BNF, \(w_{a}=150\) Hz, \(w_{b}=300\) Hz. (**c**) DTNF, \(k\) is a constant and \(\zeta\) is varied. (**d**) DTNF, \(\zeta\) is a constant and \(k\) is varied. With the notch frequency \(w_{f}\) as the center frequency, when the amplitude of improved DTNF attenuate to \(-3\) dB, the corresponding frequencies \(w\in(w_{3},w_{4})\) are all within the range of the \(-3\) dB bandwidth \(B_{w}^{\rm DTNF}\). \(w_{3}\) is the upper limit frequency, and \(w_{4}\) is the lower limit frequency, so we can obtain  
\[-20\lg\mid G_{\rm DTNF}(jw_{3})\mid=-20\lg\mid G_{\rm DTNF}(jw_{4})\mid=-3, \tag{34}\]  
\[\left|\frac{(jw)^{2}+\zeta kw_{f}\cdot jw+w_{f}^{2}}{(jw)^{2}+kw_{f}\cdot jw+w _{f}^{2}}\right|=\frac{\sqrt{2}}{2}. \tag{35}\]  
To obtain the solution of \(\Delta f^{\rm DTNF}\) and \(Q^{\rm DTNF}\), further calculation of (35) produces  
\[\left(\frac{w}{w_{f}}\right)^{4}+\left(-2+2\zeta^{2}k^{2}-k^{2}\right)\left( \frac{w}{w_{f}}\right)^{2}+1=0. \tag{36}\]  
The results of solving (36) are  
\[\left\{ \begin{array}{l}\left(\frac{w_{3}}{w_{f}}\right)^{2}=\frac{\left(-2+ 2\zeta^{2}k^{2}-k^{2}\right)-\sqrt{\left(-2+2\zeta^{2}k^{2}-k^{2}\right)^{2} -4}}{2}\\ \left(\frac{w_{4}}{w_{f}}\right)^{2}=\frac{\left(-2+2\zeta^{2}k^{2}-k^{2} \right)+\sqrt{\left(-2+2\zeta^{2}k^{2}-k^{2}\right)^{2}-4}}{2}\end{array} \right.. \tag{37}\]  
Because \(B_{w}^{\rm DTNF}=w_{4}-w_{3}=2\pi\Delta f^{\rm DTNF}\) and \(w_{f}=(w_{3}+w_{4})/2\) are already defined, the following formula can be obtained  
\[\frac{w_{4}^{2}-w_{3}^{2}}{w_{f}^{2}}=\frac{4\pi\Delta f^{\rm DTNF}}{w_{f}}. \tag{38}\]  
According to (37) and (38), the solution of the notch width \(\Delta f^{\rm DTNF}\) and the quality factor \(Q^{\rm DTNF}\) can be obtained  
\[\Delta f^{\rm DTNF}=\frac{w_{f}\sqrt{\left(-2+2\zeta^{2}k^{2}-k^{2}\right)^{2 }-4}}{4\pi}, \tag{39}\]  
\[Q^{\rm DTNF}=\frac{w_{f}}{B_{w}^{\rm DTNF}}=\frac{w_{f}}{2\pi\Delta f^{\rm DTNF }}=\frac{2}{\sqrt{\left(-2+2\zeta^{2}k^{2}-k^{2}\right)^{2}-4}}. \tag{40}\]  
When the value of the depth factor \(\zeta\) is very small, we can obtain \(\Delta f^{\rm DTNF}\approx w_{f}k\sqrt{k^{2}+4}/(4\pi)\) and \(Q^{\rm DTNF}\approx 2/\left(k\sqrt{k^{2}+4}\right)\). To obtain the solution of the width factor \(k\), according to (37) and (38), we can further obtain  
\[\left(-2+2\zeta^{2}k^{2}-k^{2}\right)^{2}-4=\frac{16\pi^{2}\Delta f^{2}}{w_{ f}^{2}}. \tag{41}\]  
According to (41), the solution of the width factor \(k\) is  
\[k=\sqrt{\frac{2-\sqrt{\frac{16\pi^{2}\left(\Delta f^{\rm DTNF}\right)^{2}}{w_ {f}^{2}}+4}}{2\times\left(10^{-\frac{\rm DTNF}{200}}\right)^{2}-1}}. \tag{42}\]When the value of the depth factor \(\zeta\) is very small, we can obtain  
\[k{\approx}\sqrt{\sqrt{16\pi^{2}(\Delta f^{\text{DTNF}})^{2}/w_{f}^{2}+4-2}}. \tag{43}\]  
To obtain the solution of the depth factor \(\zeta\), according to (33), the solution of the depth factor \(\zeta\) is  
\[\zeta=10^{-\frac{\lambda^{\text{DTNF}}}{20}}. \tag{44}\]  
In summary, (33), (39), (40), (42), and (44) are the core formulas of the improved DTNF, which are important foundations of the multi-parameter optimization algorithm.