# Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System
## 3 Results and Discussion
### Static Frequency Analysis of Blades  
Since the natural vibration characteristics of the blades have a direct impact on the coupled vibration of the bladed disk system, the blades of the first and second stage of the bladed disk system were taken as the research objects, and the static frequency analysis of the two stages was carried out to obtain the inherent vibration characteristics. The three-dimensional solid model of the first- and second-stage bladed disk system and the finite element model of the two-stage blade after meshing are shown in Figures 4 and 5.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \begin{tabular}{c} **Methods** \\ **and Errors** \\ **Number of** \\ **Knuckle Diameter** \\ \end{tabular} & \begin{tabular}{c} **Cyclic Symmetric Structure** \\ **Analysis Is Used to** \\ **Harmonize the** **Dimensionless Dynamic** \\ **Frequency of Standard** \\ **Contact Bladed Disk System** \\ \end{tabular} & \
\begin{tabular}{c} **Modal Synthesis Super-element** \\ **Method of Fixed Interface Prestressed** \\ **Method** \\ **Harmonize the** **Dimensionless Dynamic** \\ **Frequency of Standard** \\ **Contact Bladed Disk System** \\ \end{tabular} &
\begin{tabular}{c} **Method** \\ **Harmonizes the Dimensionless** \\ **Error** \\ **Dynamic Frequency of Standard** \\ **Contact Bladed Disk System** \\ \end{tabular} \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 1: Accuracy verification of the modal synthesis super-element method for the prestressed and free interfacial substructures with fixed interfaces.  
Figure 3: First-order dynamic frequency of the substructure method and cyclic commutation method.  
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\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
1 & 667.68 & 31 & 695.3 & 61 & 1813.10 & 91 & 1855.90 & 121 & 2189.90 \\ \hline
2 & 668.27 & 32 & 695.33 & 62 & 1813.50 & 92 & 1870.10 & 122 & 2196.70 \\ \hline
3 & 669.53 & 33 & 695.37 & 63 & 1814.10 & 93 & 1905.80 & 123 & 2198.40 \\ \hline
4 & 671.51 & 34 & 695.40 & 64 & 1814.40 & 94 & 1914.30 & 124 & 2216.40 \\ \hline
5 & 673.57 & 35 & 695.43 & 65 & 1814.70 & 95 & 1942.30 & 125 & 2218.50 \\ \hline
6 & 677.21 & 36 & 695.45 & 66 & 1815.00 & 96 & 1948.30 & 126 & 2255.50 \\ \hline
7 & 678.36 & 37 & 695.46 & 67 & 1815.20 & 97 & 1963.70 & 127 & 2255.80 \\ \hline
8 & 682.41 & 38 & 695.48 & 68 & 1815.40 & 98 & 1968.80 & 128 & 2277.60 \\ \hline
9 & 683.19 & 39 & 1064.10 & 69 & 1815.60 & 99 & 1976.30 & 129 & 2316.50 \\ \hline
10 & 686.42 & 40 & 1208.30 & 70 & 1815.70 & 100 & 1980.80 & 130 & 2317.80 \\ \hline
11 & 687.08 & 41 & 1217.00 & 71 & 1815.80 & 101 & 1984.20 & 131 & 2331.20 \\ \hline
12 & 689.28 & 42 & 1317.70 & 72 & 1815.90 & 102 & 1988.00 & 132 & 2334.40 \\ \hline
13 & 689.90 & 43 & 1331.70 & 73 & 1816.00 & 103 & 1989.40 & 133 & 2396.50 \\ \hline
14 & 691.25 & 44 & 1455.40 & 74 & 1816.10 & 104 & 1992.50 & 134 & 2398.80 \\ \hline
15 & 691.85 & 45 & 1456.30 & 75 & 1\end{tabular}\end{table}
+++  
Through analysis, it can be seen that the low order mode shape of the first-stage bladed disk system is the first order bending vibration of the blade according to the pitch  
Figure 10: Mode pattern of a typical order of the first-stage bladed disk system.  
diameter. With the increase in mode order, the vibration of the disk is excited, and the coupled vibration of the blade and the disk appears. As the modal order continues to increase, the blade begins to transform from a bending vibration to twisting vibration. In addition, it can be found that due to the coupled action of the blade and the wheel, the first-order bending frequency of the blade is increased.