## 3 Numerical Model  
### Finite Element Model of Test Blade  
In this paper, the test bench is built by using a structure similar to the actual compressor stator blade ring. The vibration laws of the blade ring in the presence of contact friction characteristics are studied experimentally and numerically. The structure of the whole test bench is relatively large, but the main experimental objects are concentrated in the compressor stator blade and the contacted holding ring. Therefore, the model is reasonably simplified in the following numerical research process and mostly retains the required core computational simulation structure. As mentioned earlier in the testbed introduction, the outer holding ring's contact with the top of the compressor blade is stably fixed and supported by the platform. At the same time, the test blade can be raised using a hydraulic jack at the bottom to meet the requirements of experimental installation. The hydraulic jack can then be removed, and the entire test structure resembles a cantilever beam.  
Based on the experimental bench used in this study, a numerical contact friction model of a test stator blade with a contacted outer holding ring is established for the studied object to assess the nonlinear vibration problem. It should be noted that some bolt holes and through holes are reasonably simplified in the subsequent finite element model. Figure 6 shows a schematic diagram of the established numerical model. The top outer holding ring is fixed through the platform and is in contact with the T-barb of the test blade model. The contact interface has adjustment bolts, and normal loads can be applied using a torque wrench. The blade surface is mechanically stimulated by an exciter to cause the blade to vibrate. Then, the vibration laws of test model blade under contact friction can be analyzed. The boundary conditions of the test model are briefly summarized below. The top surface of the outer holding ring imposes a fixed constraint. The spring-damping element is established on the contact interface between top of test blade model and the contacted outer holding ring. Some surfaces are represented by blue lines, which define the degrees of freedom of the coupling. The boundary conditions of the exciting force are applied at the position where the exciter acts on the blade surface.  
The finite element model for the experimental blade is given in Figure 7. The whole model mainly consists of an outer holding ring, an inner ring, a blade and a support block at the bottom. The height of the blade part is 210 mm, while the overall height of the structure, including each part, is 420 mm. The axial length and maximum thickness of the blade are 92 mm and 11 mm, respectively. [PERSON]'s modulus in 217 GPa, Poisson's ratio is 0.3 and the density is 7800 kg/m\({}^{3}\). The parameters used here correspond to the material parameters of the experimental model. A finite element model of the test bench is established using solid185 elements in ANSYS. Additionally, a small number of solid186 elements are also used in the transition area. The finite element model of the experimental piece consists of 7270 nodes. In addition, 12 spring-damping elements are established between the contact interfaces at the top of the blade. The solution process is a complicated iterative calculation. The criterion for calculating convergence is that the iteration error is less than 0.5%. In the end, a response curve for test blade is obtained using the above model.  
Figure 6: Experimental stator blade contact friction model.