## 3 Excitation Characteristics of Tuned Stators and Blade Period Design
### Effect of Stator Blade Number on Excitation towards the Bladed Disk  
First, \(N_{s}=11\) is taken as an analysis example. Figure 6 shows the frequency spectra of the monitoring points at the leading edge, basin, back, and railing edges of the rotor blade. In order to show the multiple relationships between the frequency of unsteady aerodynamic force and the fundamental frequency of the rotor speed, EO is adopted as an abscissa. The excitation components at the root, middle, and tip of the blade are distinguished by different colors. Figure 4: The (**a**) cloud picture of the pressure field behind the rotor blade; (**b**) cloud picture of static pressure field distribution at 50% blade height. Figure 5: The (**a**) pressure time signals (one rotating period) at each monitoring point; (**b**) circumferential pressure distribution at each monitoring point (at one moment). Figure 6 shows that the main frequency components of pressure pulsation (with the mean pressure removed) at each point correspond to the order excitation of stator wake, i.e., the order of excitation is the number of stator period 11 and its multiple, and the amplitude at the leading edge is the largest. Except for the trailing edge, the relative amplitude of frequency components at each point basically meets the law that the higher the order, the smaller the amplitude. Generally speaking, the fluid pressure pulsation amplitude is the highest at the blade root, followed by that at the middle of the blade, with the smallest at the blade tip. The frequency characteristics of excitation at the trailing edge have something in common with the leading edge and the basin--the distributions are the same--but the amplitude of each order is not consistent. The amplitude of EO = 22 = \(N_{r}\) and the excitation component at the blade root are the largest, indicating that the excitation at the trailing edge is more sensitive to the number of rotor blades. However, its overall amplitude is much smaller than that of the leading edge and the basin, and the excitation components at this place have little contribution to the pressure pulsation excitation components of the blade disk system. Pressure pulsation at the leading edge of the rotor blade is much larger than that at other monitoring points, which has significant representativeness. Figure 7 shows the spectrum of primitive-level leading edge monitoring points in moving blades corresponding to three different values (\(N_{s}=10,12,13\)) in working conditions. Figure 6: Frequency spectra of monitoring points on the rotor (\(N_{s}=11\)): (**a**) leading edge; (**b**) basin; (**c**) back; and (**d**) trailing edge. By analyzing the excitation spectra generated by the above four conditions with different static sub-periods (Figures 6 and 7), the following conclusions can be drawn:  
1. The excitation generated by the stator with the period number of \(N_{\rm s}\) on the downstream rotor blade includes each frequency component corresponding to the wake excitation order, namely, \(f_{k}=k_{\rm s}N_{\rm s}\Omega\ \ k=1,2,\ldots\);
2. Among all amplitudes of the frequency components, order \(N_{\rm s}\) is the largest, and the amplitude of the other component (\(k_{\rm s}N_{\rm s}\)) decreases with the increase in \(k_{\rm s}\);
3. In general, the amplitude of EO = \(3N_{\rm s}\) is less than half that of the amplitude of the main excitation (EO =\(N_{\rm s}\)), while the higher-order amplitude becomes smaller. The Circumferential Wave Distribution Characteristics of Excitations and the Influence of the Stator Blade Number  
Firstly, take \(N_{\rm s}=11\) as an example. For a bladed disk system with \(N_{\rm r}\) blades, the maximum nodal diameter number of its nodal mode vibration is:  
\[n_{\rm max}=\left\{ \begin{array}{ll}\frac{N_{\rm r}}{2}&,N_{\rm s}\mbox{ is even}\\ \frac{N_{\rm r}-1}{2}&,N_{\rm r}\mbox{is odd}\end{array} \right. \tag{9}\]  
When \(N_{\rm r}=22\), the maximum nodal path number is 11.