## 2 Numerical Methods
### Vibration Response Analysis Method  
Based on the Mindlin model, the dynamic response analysis of the friction damping contact surface of the compressor stator blade is realized. The contact surface between the blade and the outer ring is connected by a spring damping element to construct the finite element model [27]. It should be noted here that the equivalent stiffness coefficient \(K_{\rm eq}\) and equivalent damping coefficient \(C_{\rm eq}\) are defined on the model contact surface according to the geometric position of the element. By averaging the equivalent stiffness and damping  
Figure 1: Mindlin microslip model. coefficients under a certain normal load N, tangential stiffness and damping coefficients of each element matrix are obtained as shown in Figure 2.  
The elastic and kinematic responses of the spring-damping element are determined by the stiffness coefficient of the stiffness matrix element and the damping coefficient of the damping matrix element. The stiffness matrix **K** and damping matrix **C** of the spring damping element are 12 \(\times\) 12 matrices. Suppose \(x\), \(y\) and \(z\) are the directions on the node, then the stiffness matrix **K** and damping matrix **C** are:  
\[\textbf{K}=\left[ \begin{array}{cccccccccccc}k_{x}&0&0&0&0&0&-k_{x}&0&0&0&0&0 \\ 0&k_{\tau y}&0&0&0&0&0&-k_{\tau y}&0&0&0&0\\ 0&0&k_{\tau z}&0&0&0&0&0&-k_{\tau z}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots\\ -k_{x}&0&0&0&0&0&k_{x}&0&0&0&0\\ 0&-k_{\tau y}&0&0&0&0&0&k_{\tau y}&0&0&0\\ 0&0&-k_{\tau z}&0&0&0&0&0&k_{\tau z}&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots&\vdots\\ 0&0&0&0&0&0&0&0&0&0&0\\ \end{array} \right] \tag{8}\]  
\[\textbf{C}=\left[ \begin{array}{cccccccccccc}c_{x}&0&0&0&0&0&-c_{x}&0&0&0&0 \\ 0&c_{\tau y}&0&0&0&0&0&-c_{\tau y}&0&0&0&0\\ 0&0&c_{\tau z}&0&0&0&0&-c_{\tau z}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ -c_{x}&0&0&0&0&c_{x}&0&0&0&0&0\\ 0&-c_{\tau y}&0&0&0&0&c_{\tau y}&0&0&0&0\\ 0&0&-c_{\tau z}&0&0&0&0&0&c_{\tau z}&0&0&0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots& \vdots&\vdots\\ 0&0&0&0&0&0&0&0&0&0&0&0\\ \end{array} \right] \tag{9}\]  
\[k_{x}=\frac{K_{n}}{n};\,k_{\tau y}=\frac{K_{eq}}{n};\,k_{\tau z}=\frac{K_{eq} }{n} \tag{10}\]  
\[c_{x}=0;\,c_{\tau y}=\frac{C_{eq}}{n};\,c_{\tau z}=\frac{C_{eq}}{n} \tag{11}\]  
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. For the friction-damping model studied in this paper, the response obtained by the numerical solution of the harmonic method is also harmonic when subjected to harmonic excitation. In order to reduce the calculation workload, only the first harmonic is considered in the friction force. Assuming that the relative displacement is \(A\)cos\(\theta\), the friction interface  
Figure 2: Spring damping element. in the contact region is simplified into a spring damping model. Take the elastic force and damping together to obtain the friction force. \[f=f_{c}(A)\cos\theta+f_{s}(A)\sin\theta=K_{eq}A\cos\theta-C_{eq}\omega A\sin\theta \tag{12}\]  
The characteristic parameters of system vibration include displacement, frequency and phase. In the formula, they are represented by \(A\), \(\omega\) and \(\theta\), respectively. The equivalent coefficient of contact surface can be obtained by the following formula. \[K_{eq}=\frac{f_{c}(A)}{A}=\frac{1}{\pi A}{\int_{0}^{2\pi}}f(A,\theta)\cos \theta d\theta \tag{13}\]  
\[C_{eq}=-\frac{f_{s}(A)}{\omega A}=-\frac{1}{\omega\pi A}{\int_{0}^{2\pi}}f(A, \theta)\sin\theta d\theta \tag{14}\]  
The calculation method of the blade vibration response is given in Figure 3. It should be noted that the calculation in this study requires a large number of iterative processes to obtain the final result. A brief summary includes the following steps: (1) First, the initial normal load should be defined for the contact surface.