# Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System
## 4 Conclusions  
The whole coupled vibration mode of a two-stage bladed disk system was analyzed. Firstly, the modal analysis of the first- and second-stage bladed disks was carried out, and then the modes and configurations of the first- and second-stage bladed disks were solved, respectively. Finally, the modal analysis of the two-stage bladed disk coupled system was carried out, and the coupled vibration forms of the two-stage bladed disk were analyzed. The following conclusions were drawn. (1) For the single-stage bladed disk system, the low-order mode shape is the first-order bending vibration of the blade according to the pitch diameter. With an increase in the mode order, the vibration of the disk is excited, resulting in the coupled vibration of the blade and the disk. As the modal order continues to increase, the blade begins to transform from a bending vibration to twisting vibration. At the same time, due to the coupling of the blade and the disk, the first-order bending frequency of the blade is increased. (2) For the two-stage bladed disk coupled system, the vibration of the bladed disk has a certain order: the blade vibration is first, followed by the coupled vibration of the bladed disk. For the intersecting position of the two-stage bladed disk frequencies, the interstage coupled vibration of the bladed disk will occur, and the number of pitch diameters or pitch circles of the vibration would be the same. The natural frequencies and mode shapes of the two-stage bladed disk coupled system not only include the natural frequencies and mode shapes of the first- and second-stage bladed disks, respectively, but also have the coupled \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**Level One**} & \multicolumn{2}{c}{**Level Two**} & \multicolumn{2}{c}{**Two Levels of Coupling**} & \multicolumn{1}{c}{**Vibration Mode**} \\ \hline
**Order Number** & **Natural Frequency** & **Order Number** & **Natural Frequency** & **Order Number** & **Natural Frequency** & **Natural Frequency** \\ \hline \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(1\)} & \multirow{2}{*}{650.63} & \multirow{2}{*}{\(1\)} & \multirow{2}{*}{656.78} & \multirow{2}{*}{\begin{tabular}{c} Coupled vibration of the two-stage bladed disk \\ \end{tabular} } \\ \cline{5-5} \cline{7-7}  & & & & & 2–5 & 664.55–668.81 & Level one blade vibration \\ \hline \multirow{2}{*}{\(1\)–\(38\)} & \multirow{2}{*}{667.68–695.48} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{6} & \multirow{2}{*}{674.13} & \multirow{2}{*}{\begin{tabular}{c} Coupled vibration of the two-stage bladed disk \\ \end{tabular} } \\ \cline{5-5} \cline{7-7}  & & & & & 7–39 & 675.66–695.23 & Level one blade vibration \\ \hline \multirow{2}{*}{\(-\)} & \multirow{2}{*}{\(-\)} & \multirow{2}{*}{2-53} & \multirow{2}{*}{706.22–749.50} & \multirow{2}{*}{40-91} & \multirow{2}{*}{720.99–749.30} & \multirow{2}{*}{Level two blade vibration} \\ \cline{5-5} \cline{7-7}  & & & & & 92–103 & 778.35–1426.50 & \multirow{2}{*}{\begin{tabular}{c} Single stage coupled vibration of the vibration of the bladed disk \\ \end{tabular} } \\ \cline{5-5} \cline{7-7}  & & & & & 104–106 & 1471.40–1548.50 & \multirow{2}{*}{\begin{tabular}{c} Coupled vibration of the two-stage bladed disk \\ \end{tabular} } \\ \cline{5-5} \cline{7-7}  & & & & 107–114 & 1555.20–1695.50 & \multirow{2}{*}{\
\begin{tabular}{c} Single-stage coupled vibration of the bladed disk \\ \end{tabular} } \\ \cline{5-5} \cline{7-7}  & & & & 115–116 & 1700.80–1730.40 & \multirow{2}{*}{
\begin{tabular}{c} Coupled vibration of the two-stage bladed disk \\ \end{tabular} } \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Comparison of natural frequencies between the two-stage and single-stage bladed disk coupled systems. modes of the two-stage bladed disk, so the multistage bladed disk model is more accurate to analyze. Conceptualization, Y.Y. and T.Z. ; methodology, Y.Y. and T.Z. ; software, Y.F. ; validation, Y.Y. and T.Z. ; formal analysis, X.J. ; investigation, X.J.