## 4 Excitation Characteristics of Mistuned Stators and Its Effect in Reducing Excitation Amplitude
### Sectional Non-Uniform Mistuned Stators and Their Effect on Excitation towards Bladed Disks  
A total of six sectional non-uniform mistuned stators were designed, as shown in Table 5.  
Notably, in case IV, the 12 stator blades are divided into six sectors along the circumferential direction, and 1 blade in each sector moves 5 mm away from the rotor along the axial direction. It can be seen from Figure 14a that, compared to the tuned situation, the wake excitation component amplitude of the sectional non-uniform mistuned stator was reduced, whereas the components increased. With sectional non-uniform mistuned stators, the minimum periodic cycle turned to a \(1/M\) circle; thus, the fundamental frequency of excitation changed into \(N_{s}\Omega/x\) instead of \(N_{s}\Omega\). The original fundamental frequency components \(N_{s}\Omega\) still exist, although with a lower amplitude. The components can be written as:  
\[f=k_{s}M\Omega=\frac{k_{s}N_{s}\Omega}{x}\quad k_{s}=1,2,\ldots, \tag{16}\]  
For case III, the minimum symmetric periodic number changed from 12 (\(N_{s}=12\)) to 4 (\(M=4\)); thus, the quadruplicated frequency \(f=4\cdot\Omega\) appears, which is \(EO=4\), and its multiple components in the EO diagram. Similarly, in case IV, EO = 6 and its multiple components appear. Figure 14: Frequency spectrum and circumferential wave spectrum comparison of the tuned and two other types of sectional non-uniform mistuned stators: (**a**) frequency spectrum and (**b**) circumferential wave spectrum. Figure 13: Circumferential wave spectrum of asymmetric mistuned stators: (**a**) \(\text{IGV}=11(10/12)\), (**b**) \(\text{IGV}=12(10/14)\). Figure 14b shows the circumferential wave spectrum of the tuned case and sectional non-uniform mistuned stator case III and IV. Compared to the asymmetric mistuned stator cases in Figure 13, sectional non-uniform mistuned stators bring fewer circumferential wave components. Combining Equation (3) with Equation (16), the circumferential wave spectrum of the sectional non-uniform mistuned stator case can be written as:  
\[n=\pm(k_{s}M-kN_{r})\ \ k=0,1,\ldots,k_{s}=1,2,\ldots, \tag{17}\]  
If \(N_{s}=12\), \(N_{r}=22\), and \(k_{s}=1,2,3,4\), the circumferential wave spectrum of the tuned case can be written as Equation (18). In order to arrange the amplitude from large to small, the formula is arranged from 1 to 4 according to the value of \(k_{s}\):  
\[n=\left\{ \begin{array}{c}N_{r}-N_{s}=10\\ 2N_{s}-N_{r}=2\\ 2N_{r}-3N_{s}=8\\ 4N_{s}-2N_{r}=4\end{array} \right., \tag{18}\]  
Similarly, for mistuned stator case III, \(M=4,k_{s}=1,2,3,4,5,6\), and the circumferential wave spectrum is:  
\[n=\left\{ \begin{array}{c}N_{r}-5M=6M-N_{r}=2\\ M=4\\ N_{r}-4M=6\\ 2M=8\\ N_{r}-3M=10\end{array} \right., \tag{19}\]  
Additionally, for mistuned stator case IV, the circumferential wave spectrum is:  
\[n=\left\{ \begin{array}{c}N_{r}-2M=10\\ M=6\\ N_{r}-3M=4\\ 4M-N_{r}=2\\ 5M-N_{r}=2N_{r}-6M=8\end{array} \right., \tag{20}\]  
According to Table 1, when \(N_{s}=12\), components in wake excitation \(EO=24\) and circumferential wave number \(n=2\), which can excite the eleven-order modal resonance of a bladed disk with a nodal diameter of two. The resonance conditions can be described as follows:  
\[f_{N_{s}=12}=2\cdot N_{s}\cdot\Omega=24\cdot\Omega=\omega_{2,11}, \tag{21}\]  
The amplitude of excitation force is 1313 Pa, as shown in Table 6.  
Compared to the tuned case, excitation components with \(EO=\ N_{s}=12\) as the fundamental frequency still exist; however, the amplitude of each component decreases significantly. Meanwhile, the number of new frequency components added in sectional non-uniform mistuned stator cases is fewer than in asymmetric mistuned stator cases. Table 6 shows the comparison of excitation amplitude at \(EO=24\). Compared to the tuned case, the resonance excitation amplitude of sectional mistuned stator cases III and IV reduced to 40.3% and 72.7%, respectively.