# Study on Influence of Multi-Parameter Variation of Bladed Disk System on Vibration Characteristics
## 2 Modal Calculation of Bladed Disk System System Based on Group Theory
### Results Comparative Analysis  
The geometrical model of the disk section was established, and the mesh was divided. According to the measurement of density and stiffness of solid disc materials, the unit properties of simulated disc materials are defined. The simulated disc material is steel no. 45, the elastic modulus E = 2.1 \(\times\) 10\({}^{11}\) pa, Poisson's ratio \(\mu\) = 0.3, and density \(\rho\) = 7850 kg/m\({}^{3}\). According to the installation conditions of the test device, the boundary conditions are defined as fixed axial and circumferential displacement of the nodes at the center of the simulated disk, and free radial displacement at the same time. The overall geometric model of the simulated disk and grid division are shown in Figure 2.  
Sprinkle fine sand evenly on the horizontally placed disk, adjust the frequency of the signal generator, slowly increase from 20 Hz, so that the excit  
Figure 1: Experiment device of the modal impeller test.  
Figure 2: Mesh and geometric model of whole simulation impeller.  
a certain excitation frequency, until a clear and stable sand shape appeared on the disk (before the orderly motion of the fine sand was seen, and the amplitude of the motion gradually increased), and recorded the corresponding vibration mode of the sand at this time, the frequency of the corresponding signal generator and the waveform corresponding to the oscilloscope. The output frequency of the signal generator continued to increase until the vibration mode of multi-set vibrating sand was obtained, and the corresponding frequency of different sand forms and the waveform on the oscilloscope were also recorded. This sand shape was formed by the nodal line or nodal circle formed by flowing into the minimum amplitude region or static region of the vibrating fine sand in the large amplitude region. The comparison between the four types of sand type m = 2, 3, 4, 5 and the finite element analysis results is shown in Figure 3.  
Figure 3: The sand pattern is compared with the results of finite element analysis. (**A**) m = 2, (**B**) m = 3, (**C**) m = 4, (**D**) m = 5.  
The modal frequencies corresponding to the 1 st, 4 th, 7 th, 12 th and 16 th finite element analysis results are extracted. Comparison of sand shape frequency between the simulated disk test and finite element calculation frequency is shown in Table 2.  
The experimental results are consistent with the finite element calculation, which verifies the accuracy of the finite element analysis. Only the linear mode of m = 2, 3, 4, 5 joints can be measured in the test. Because the test adopts the single point excitation method, and the simulated disk thickness is larger than the simulated disk diameter, and the simulated disk strength is large, it is difficult to achieve the phenomenon of pitch circle. Therefore, the optimal thickness and shape of the disk can be calculated in a targeted manner. When the simulated disk is in joint line vibration, the amplitude of the simulated disk will be consistent on the symmetric side. Therefore, the open balance holes of the disk are usually not even but odd (5 or 7) to avoid cracking and damage to the simulated disk caused by the concentrated stress. When the simulated disk is in the mode frequency, the amplitude of the symmetrical part on the edge of the simulated disk increases obviously and the amplitude of the connected blade on the disk increases obviously, which will cause the dynamic and static collision of the unit, and the fatigue deformation and even damage the disk system if the vibration continues for a long time.