# Study on Influence of Multi-Parameter Variation of Bladed Disk System on Vibration Characteristics
## 4 The Influence of Detuning Parameters on Bladed Disk System Is Analyzed  
### Finite Element Analysis of the Disc System  
On the edge of the disk 18 blades were evenly installed, with two adjacent blades having an installation angle of 20\({}^{\circ}\), and the edge of the blade root and disk along the disk radial rigid connection with the two rivets. The thickness of each blade was 6 mm, width was 18 mm and height was 162 mm. The blade root was closely matched with the outer edge of the disc. The blade root height was 10 mm and the thickness of both sides of the blade root was 2 mm. A keyway with a diameter of 6 mm was opened at a distance of 32 mm from the top of the blade, with a length of 80 mm and a diameter of 6 mm at both ends of the keyway. The main purpose of this keyway is to add the counterweight bolts of different mass on it, and the position of the counterweight bolts can be changed, with the maximum displacement of 80 mm. We added 18 blades to the disk to form the bladed disk system, and the geometric model of the bladed disk system is shown in Figure 4. The blade shape and blade installation diagram are shown in Figure 5. In the figure, the small counterweight bolts on the blades are the schematic diagram for adding different mass and different positions on the blades. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **Section Line** & **Section Line** & **Section Line** & **Section Line** & **Section Line** \\  & **m = 1** & **m = 2** & **m = 3** & **m = 4** & **m = 5** \\ \hline test sand frequency/Hz & — & 194.5 & 462.1 & 837.9 & 1329.6 \\ the finite element calculation frequency/Hz & 88.998 & 200.251 & 467.163 & 848.601 & 1343 \\ finite element and experimental error/\% & — & 2.96 & 1.10 & 1.28 & 1.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The result comparison of simulation impeller sand frequency and finite element calculation. Figure 4: Bladed disk geometric model. By restricting the position of the central hole of the bladed disk system the first 100 vibration frequencies of the disc system test device were obtained by finite element calculation, as shown in Figure 6. It can be seen from Figure 6 that a frequency steering phenomenon occurs in the modal changes of the bladed disk system, that is, the system characteristic value trajectory first converges with some system characteristic parameters, but does not intersect, and then separates. It can be seen that with the increase of order, the natural frequency is roughly divided into six approximately horizontal regions and five climbing regions, which is more obvious at high frequency than at low frequency. The frequency in the horizontal region is highly dense, and the dominant blade vibration mode is mainly in this region. Five climbing regions are frequency steering regions, and the disk leads the vibration mode. As an axial symmetric system, the bladed disk system must have a heavy root, which is not considered here. The typical mode of vibration in the first 100 climbing zones is extracted, as shown in Figure 7. In the figure, m represents the pitch number of the disk system and n represents the pitch number of the disk system. With the increase of the order, the vibration frequency of the bladed disk system increases, and the number of pitch diameter and pitch circle increases gradually. (A) is a 1 st-order vibration mode with a vibration frequency of 65.517 Hz. The bladed disk system has only one pitch path. (B) is the mode of 3 rd order vibration with the vibration frequency of 86.308 Hz. The disc system has no pitch diameter and circle, and the disc expands radially. (C) is the mode of 4 th order vibration with the vibration frequency of 117.22 Hz. (D) is the mode of 6 th order vibration with the vibration frequency of 180.05 Hz. The bladed disk system has no pitch diameter and circle. (E) is the mode of 24 th order vibration with the vibration frequency of 241.34 Hz. (F) is the mode of 28 th vibration with the vibration frequency of 331.02 Hz. Figure 5: Blade shape and blade installation schematic.