## 4 Results and Discussion
### Blade Vibration under Different Exciting Forces  
Similarly, an experimental study that involved changing the excitation force was further carried out while the contact surface normal load was fixed. The normal force for the test model contact interface was fixed at 20 N, and the vibration response curves of test blade with excitation forces of 1 N, 2 N, 3 N and 4 N were measured. The measuring point of the experimental condition was still set at 10% of the height of the exhaust side of the blade, and the test data were compared with the numerical calculation results of the corresponding working condition. Figures 10, 13, 18 and 19 show the test results under various exciting forces. It is not hard to see from the figures that the vibration response of the blade changes significantly, while the exciting force is different. The maximum amplitude is 27 \(\upmu\)m at 1 N, 40 \(\upmu\)m at 2 N, about 50 \(\upmu\)m at 3 N and 61 \(\upmu\)m at 4 N. The corresponding numerical results are 26 \(\upmu\)m, 39 \(\upmu\)m, 52 \(\upmu\)m and 63 \(\upmu\)m. It can also be found that the relative error between the numerical and experimental results is still at a good level, and it can be clearly seen that the vibration response increases as the exciting force increases. Careful analysis of this working condition shows that the relationship between the increase in the exciting force and the change in the vibration response is not linear. That is to say, the characteristics of nonlinear vibration also appear at the contact interface. Figure 19: Response curve under 20 N normal load with 4 N exciting force. Figure 18: Response curve under 20 N normal load with 3 N exciting force. While test model's normal force is fixed, the state of the model's contact surface is basically viscous or develops fretting slip under a small exciting force. When the exciting force gradually increases, the state of the contact surface changes abruptly at a certain level of the exciting force, and then, changes to a macroscopic slip state. At this time, the relative motion of the contact surface consumes vibration energy. The blade has nonlinear vibration characteristics, and it can be seen in the corresponding Figure 20 that the amplitude no longer increases linearly an the increase in exciting force. Regarding the test stator blade constructed in this study, the most notable vibration characteristics parameters are mainly resonance frequency and maximum response amplitude. The measured frequency values and vibration response amplitude data under all test conditions are compared with the numerical results in detail in Table 1. It can be seen from the table that the frequency and maximum response amplitude obtained using the vibration analysis method constructed are in good agreement with the measured values. This shows that the numerical method is well validated by the experiments. Of course, the above studies were carried out and analyzed on experimental small blades. In reality, the structure of a stator ring in the compressor of industrial equipment is  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Contact Surface** & **Blade** & **Resonance** & **Resonance** & & **Maximum** & **Maximum** & \\
**Surface** & **Exciting** & **Frequency** & **Frequency** & **Resonance** & **Response** & **Response** & **Relative** \\
**Normal** & **Force** & **Measured** & **Calculated** & **Error** & **Amplitude** & **Amplitude** & **Error** \\
**Load** & **(N)** & **Value** & **Value** & **Measured** & **Calculated** & **Value** (\(\upmu\)m) & \\
**/N** & & **(Hz)** & **(Hz)** & & **Value** (\(\upmu\)m) & & \\ \hline
10 & & 28.7 & 28.0 & 2.50\% & 17.8 & 18.5 & 3.78\% \\
20 & 1 & 29.1 & 28.0 & 3.93\% & 26.5 & 26.0 & 1.92\% \\
30 & & 28.1 & 28.0 & 0.36\% & 27.  
more complicated, and the airflow force on the blade and the normal load on the contact surface will be more variable. In this study, the nonlinear vibration laws of a test blade, considering contact friction when the top of the outer holding ring is fixed, are analyzed.