# Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System
## 3 Results and Discussion
### Modal Analysis of the Two-Stage Bladed Disk Coupled System  
For the interstage coupled vibration analysis of multi-stage bladed disk system, the two-stage bladed disk system composed of the first- and second-stage bladed disk systems is selected firstly, and the overall model of the two-stage bladed disk system is taken as the analysis object. Figure 14a shows the overall three-dimensional solid model of the two-stage bladed disk system. Figure 14b shows the overall finite element model of a two-stage bladed disk system with meshing and boundary conditions considered. Through modal analysis, the first 195 th order natural frequencies and vibration shapes of the two-stage bladed disk system were calculated and solved, as shown in Tables 8 and 9. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Order** & **Natural** & **Order** & **Natural** & **Order** & **Natural** & **Order** & **Natural** & **Order** & **Natural** \\
**Number** & **Frequency** & **Number** & **Frequency** & **Number** & **Frequency** & **Number** & **Frequency** & **Number** & **Frequency** \\ \hline
21 & 747.39 & 51 & 749.43 & 81 & 2091.90 & 111 & 2135.00 & 141 & 2483.00 \\ \hline
22 & 747.52 & 52 & 749.46 & 82 & 2092.40 & 112 & 2135.40 & 142 & 2491.20 \\ \hline
23 & 747.75 & 53 & 749.50 & 83 & 2102.90 & 113 & 2135.40 & 143 & 2493.90 \\ \hline
24 & 747.92 & 54 & 797.09 & 84 & 2103.50 & 114 & 2135.50 & 144 & 2500.10 \\ \hline
25 & 748.02 & 55 & 835.36 & 85 & 2110.40 & 115 & 2230.70 & 145 & 2502.50 \\ \hline
26 & 748.21 & 56 & 999.34 & 86 & 2111.20 & 116 & 2232.80 & 146 & 2507.30 \\ \hline
27 & 748.27 & 57 & 1002.60 & 87 & 2115.80 & 117 & 2233.00 & 147 & 2509.30 \\ \hline
28 & 748.42 & 58 & 1305.60 & 88 & 2116.60 & 118 & 2235.90 & 148 & 2513.10 \\ \hline
29 & 748.46 & 59 & 1306.80 & 89 & 2119.90 & 119 & 2238.20 & 149 & 2514.90 \\ \hline
30 & 748.55 & 60 & 1500.20 & 90 & 2120.70 & 120 & 2247.20 & 150 & 2517.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order Number** & **Natural Frequency/Hz** & **Vibration Mode** \\ \hline
1 & 650.63 & The vibration of the blade and the wheel is coupled with 0 pitch diameter \\ \hline
2–53 & 706.22–749.50 & \multicolumn{2}{c}{Blade first-order bending vibration} \\ \hline
54–78 & 797.09–2048.40 & The blade and the wheel vibrate together \\ \hline
79–131 & 2075.10–2364.90 & First-order torsional vibration of blades \\ \hline
132–150 & 2396.30–2517.80 & Blade bending-torsion coupled vibration \\ \hline \hline \end{tabular}
\end{table} 
Table 7: First 150 order natural frequencies and mode shapes of stage 2 blading disk systems. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Order Number** & **Natural Frequency/Hz** & **Vibration Mode** \\ \hline
1 & 650.63 & The vibration of the blade and the wheel is coupled with 0 pitch diameter \\ \hline
2–53 & 706.22–749.50 & Blade first-order bending vibration \\ \hline
54–78 & 797.09–2048.40 & The blade and the wheel vibrate together \\ \hline
79–131 & 2075.10–2364.90 & First-order torsional vibration of blades \\ \hline
132–150 & 2396.30–2517.80 & Blade bending-torsion coupled vibration \\ \hline \hline \end{tabular}
\end{table} 
Table 8: First 195 order natural frequencies of a two-stage bladed disk coupled system. The first 160 th order mode shapes of the two-stage bladed disk coupled system are shown in Table 9:  
The mode diagram of the typical order of the two-stage bladed disk coupled system is as follows:  
1. As shown in Figures 15-19, coupled mode shape when the first-stage bladed disk vibration is dominant. 2. As shown in Figures 20-24, coupled mode shapes of the second-stage bladed disk when the vibration is dominant. 3. As shown in Figures 25-29, coupled mode of two stages of the bladed disk with the same vibration amplitude. According to the modal and mode shape analysis of the two-stage bladed disk coupled system, in the low-order mode, the first-order bending vibration of the first-order blade and the second-order blade is firstly manifested.