## 4 Excitation Characteristics of Mistuned Stators and Its Effect in Reducing Excitation Amplitude
### Asymmetric Mistuned Stators and Their Effect on Excitation towards Bladed Disks  
Figure 12 presents a comparison of the wake excitation frequency spectrum on bladed disk systems under the two asymmetric mistuned stator conditions of i: \(\text{IGV}=11(10/12)\) and ii: \(\text{IGV}=12(10/14)\), with tuned conditions: \(N_{s}=11\) and \(N_{s}=12\).  
Figure 11: Two types of mistuned stators: (**a**) asymmetric mistuned stator and (**b**) sectional non-uniform mistuned stator.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case No.** & **Mistuned Type** & **Name** \\ \hline I & Asymmetric & \(\text{IGV}=11(10/12)\) \\ II & & \(\text{IGV}=12(10/14)\) \\ \hline III & \(\text{Sectional non-uniform}\) & \(\text{IGV}=12(4/3,5^{\circ})\) \\ IV & & \(\text{IGV}=12(6/2,5\text{ mm})\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Four different mistuned forms.  
Compared to the tuned stator, the main components of the asymmetric design's wake excitation amplitude were reduced, whereas the components increased. The components can be described in three parts.  
(1) The parts include: fundamental frequency \(EO=N_{1}\), \(EO=N_{2}\), and its multiplication frequency of the fundamental frequency. The amplitudes of these two fundamental frequencies are close, about one-half of the amplitude of the tuned \(EO=N_{\rm s}\) component, namely:  
\[f=\left\{\begin{array}{ll}k_{1}N_{1}\Omega&\quad k_{1}=1,2,\ldots\\ k_{2}N_{2}\Omega&\quad k_{2}=1,2,\ldots\end{array}\right., \tag{13}\]  
(2) The adjacent frequency components of \(EO=N_{1}\), \(EO=N_{1}\), and their multiplication frequency's adjacent components are:  
\[f=\left\{\begin{array}{ll}(k_{1}N_{1}\pm 1)\Omega&\quad k_{1}=1,2,\ldots\\ (k_{2}N_{2}\pm 1)\Omega&\quad k_{2}=1,2,\ldots\end{array}\right., \tag{14}\]  
(3) Odd order excitation components with a small amplitude are:  
\[f=(2k-1)\Omega\quad k=1,2,3,\ldots, \tag{15}\]  
This phenomenon can be explained from the perspective of energy: in the case of tuned stators, the order of flow excitation is dominated by \(EO=N_{\rm s}\) and its multiple components, which is relatively simple, and the energy concentration leads to a higher amplitude. When the stator is mistuned, the main components of the excitation order are transformed into a frequency band related to \(EO=N_{1}\) and \(EO=N_{2}\). Compared to the tuned case, the energy is dispersed, resulting in a significant decrease in amplitude.  
Figure 13 presents a comparison of wake excitation circumferential wave spectrum on bladed disk systems under the two asymmetric mistuned stator conditions of i: \(\mathrm{IGV}=11(10/12)\) and ii: \(\mathrm{IGV}=12(10/14)\), with tuned conditions: \(N_{s}=11\) and \(N_{s}=12\).  
Compared to tuned stators, amplitude of main circumferential wave excitation induced by asymmetric mistuned stators was reduced, although the components increased.  
Combined with the resonance conditions of the Rotor67 bladed disk detailed in Table 1, after mistuning the stator, the amplitude of excitation decreased significantly, but the order excitation that could arouse resonance increased.  
With \(N_{s}=11\), for example, the excitation frequency components that may arouse resonance include the excitations of \(EO=11\) and \(EO=22\) for the tuned case. However, in the case of asymmetric mistuned stators, \(\mathrm{IGV}=11(10/12)\), the amplitude of \(EO=11\) is halved and that of \(EO=22\) becomes zero. However, the new order excitation components that may arouse resonance are as follows: \(EO=5,7,13,21,24,25,27,30,37\). Similar results can be obtained from the analysis of \(\mathrm{IGV}=12(10/14)\).  
Figure 12: Frequency spectrum of asymmetric mistuned stators: (**a**) \(\mathrm{IGV}=11(10/12)\) and (**b**) \(\mathrm{IGV}=12(10/14)\).  
Compared to the tuned cases, the number of components may arouse rapidly increasing resonance; asymmetric mistuned stators are not suitable for application.