## 3 Numerical Model
### Vibration Response Calculation Method Considering Contact Friction  
In this study, a numerical analysis of the nonlinear vibration response of a test stator blade considering its contact friction characteristics is carried out. The specific theoretical and numerical methods related to this research are described in detail in reference [26]. According to the literature, the nonlinear vibration of the stator blade is analyzed by introducing the Mindlin model to deal with the frictional damping effect of the contact surface. At the same time, a spring-damping element is used to establish the finite element model contact relationship between the stator blade and the outer holding ring. In this paper, only the key models and iterative methods used in the calculation are given. The Mindlin model is used to deal with the contact friction action of the interface, and the vibration response of the test blade is studied. The key aim of establishing a finite element model is to construct a spring-damping element. They are used to simulate the interface between the test blade and the outer holding ring. The methods of obtaining the key coefficients at the contact interface, including the equivalent stiffness coefficient \(K_{\text{eq}}\) and the equivalent damping coefficient \(C_{\text{eq}}\), are briefly summarized. That is, the coefficient of each element is obtained by averaging the coefficient of the contact interface under a certain normal force. The response characteristics of the spring-damping element depend on the corresponding parameters of the stiffness and damping matrix. \(\mathbf{K}\) and \(\mathbf{C}\) are 12 \(\times\) 12 matrices that define the stiffness and damping characteristics of the spring-damping element. Their formulas are given here, where \(x,y\) and \(z\) are the directions:  
\[\mathbf{K}=\left[ \begin{array}{cccccccccc}k_{x}&0&0&0&0&0&-k_{x}&0&0&0&0&0\\ 0&k_{\tau y}&0&0&0&0&0&-k_{\tau y}&0&0&0&0\\ 0&0&k_{\tau z}&0&0&0&0&-k_{\tau z}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ -k_{x}&0&0&0&0&0&k_{x}&0&0&0&0&0\\ 0&-k_{\tau y}&0&0&0&0&0&k_{\tau y}&0&0&0\\ 0&0&-k_{\tau z}&0&0&0&0&0&k_{\tau z}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ \end{array} \right] \tag{1}\]  
Figure 7: Experimental stator blade finite element model. \[\mathbf{C}=\left[ \begin{array}{cccccccccccc}c_{x}&0&0&0&0&0&-c_{x}&0&0&0&0&0\\ 0&c_{\mathrm{ry}}&0&0&0&0&0&-c_{\mathrm{ry}}&0&0&0&0\\ 0&0&c_{\mathrm{rz}}&0&0&0&0&0&-c_{\mathrm{rz}}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ -c_{x}&0&0&0&0&c_{x}&0&0&0&0\\ 0&-c_{\mathrm{ry}}&0&0&0&0&c_{\mathrm{ry}}&0&0&0\\ 0&0&-c_{\mathrm{rz}}&0&0&0&0&0&c_{\mathrm{rz}}&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ 0&0&0&0&0&0&0&0&0&0&0\end{array} \right] \tag{2}\]  
\[k_{x}=\frac{K_{n}}{n};\,k_{\mathrm{ry}}=\frac{K_{\mathrm{eq}}}{n};\,k_{ \mathrm{rz}}=\frac{K_{\mathrm{eq}}}{n} \tag{3}\]  
\[c_{x}=0;\,c_{\mathrm{ry}}=\frac{C_{\mathrm{eq}}}{n};\,c_{\mathrm{rz}}=\frac{C _{\mathrm{eq}}}{n} \tag{4}\]  
where \(K_{n}\) is the normal contact stiffness between contact surfaces/N\(\cdot\)m\({}^{-1}\) and \(n\) is the number of spring-damping units between contact surfaces. Figure 8 shows the specific steps for solving the above model. Due to the existence of some nonlinear characteristics, solving the vibration response of the test model requires a lot of iterative calculation to obtain the convergence result. The main steps of numerical analysis are summarized below: (1) First, the initial normal force at the interface is given. (2) Specify the initial frequency of the calculation. (3) Next, assume a value as the initial displacement at the interface. (4) Then, calculate the equivalent coefficients of the interface elements. (5) Based on the parameters obtained above, calculate the displacement using the finite element model of the test blade.