## 3 Excitation Characteristics of Tuned Stators and Blade Period Design
### Stator Number Design to Avoid Bladed Disk Resonance  
In this section, the Rotor67 bladed disk system is taken as an example to illustrate how to determine the period number of the stator (IGV) in order to avoid resonance.  
For bladed disks with 22 blades, as in this study, the main frequency components of wake excitation generated by the stator with the number of structural periods are:  
\[f=\text{EO}\cdot\Omega=k_{s}N_{s}\Omega,\;k_{s}=1,2,3 \tag{11}\]  
The main excitation components from the stator wake are:  
\[\left.\begin{array}{c}n=N_{s}\\ n=N_{r}-N_{s}=22-N_{s}\\ n=2N_{s}-N_{r}=2N_{s}-22\\ n=2N_{r}-3N_{s}=44-3N_{s}\end{array}\right\}n\leq\frac{N_{r}}{2}=11, \tag{12}\]  
Consider three options, \(N_{s}=13\), \(14\), \(15\). Tables 3 and 4 give the frequency component and circumferential wave components, respectively, in the excitation of the flow field of rotational static interference corresponding to these three different choices.  
Figure 9: Circumferential wave spectrum of monitoring point pressure on rotors with different stator blade numbers at: (**a**) the leading edge; (**b**) the basin; (**c**) the back; and (**d**) the trailing edge.  
By comparing Tables 2 and 3 with the resonance analysis results of the bladed disk structure (Table 1), it can be seen that:  
1. When \(N_{s}=13\), the excitation component with excitation order 13 and circumferential wave number 9 will excite the modal resonance with nodal diameter \(n=9\);
2. When \(N_{s}=15\), the excitation component with the excitation order of 30 and the circumferential wave number of 8 will excite the modal resonance with nodal diameter \(n=8\);
3. When \(N_{s}=14\), the excitation does not contain the frequency component or circumferential wave component that causes resonance. Therefore, for the Rotor67 blade disk with 22 moving blades, resonance can be avoided with an upstream stator \(N_{s}=14\).  
Figure 10 presents the frequency spectrum and circumferential wave spectrum of excitation at the primary-level leading edge monitoring point in the moving blade under the working condition of \(N_{s}=14\), based on three-dimensional unsteady flow field analysis. Their components are completely consistent with the values in Tables 2 and 3, and the relative amplitude of each component is also consistent with the above analysis results.  
Using the same method, the rationality of the stator number \(N_{s}=16\sim 20\) (i.e., whether or not it will cause resonance) can be judged. Through numerical analysis, it has been shown that this method can determine the rationality of the stator-structural period  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(N_{s}\) & 22\(-N_{s}\) & 2\(N_{s}-\)22 & 144\(-\)3\(N_{s}\)1 \\ \hline
13 & 13 & 26 & 39 \\ \hline
14 & 14 & 28 & 42 \\ \hline
15 & 15 & 30 & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Excitation harmonic components in the wake of the stator with respective periods.  
Figure 10: Wake at the leading edge of rotor blades (\(N_{s}=14\)): (**a**) frequency spectrum; (**b**) circumferential wave spectrum.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(N_{s}\) & \multicolumn{3}{c}{EO} \\ \cline{2-4}  & \(k_{s}\)=1 & \(k_{s}\)=2 & \(k_{s}\)=3 \\ \hline
13 & 13 & 26 & 39 \\ \hline
14 & 14 & 28 & 42 \\ \hline
15 & 15 & 30 & 45 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Stator wake excitation frequency components with respective periods.  
without the need to perform numerical simulations of three-dimensional unsteady flow induced by rotor/stator interactions.