## 2 Numerical Methods  
### Frictional Contact Model  
Generally speaking, models for dry friction can be divided into two types. One is the macroscopic slip model, which is easy to understand and solve, but its scope of application is relatively narrow. Generally, macroscopic models can only be applied to dry friction structures with small normal load or large relative displacement. The other is the micro-slip model, which is characterized by the consideration of elastic deformation between contact surfaces, so it has a wider application and can simulate dry friction models more accurately. At present, the contact state between blade-damping structures usually includes point to surface contact and surface to surface contact. In fact, the friction damping models applicable to different contact states are also very different, which cannot be characterized by a unified one. However, in the process of blade machining, there are some errors in machining accuracy. As a result, the blade-damping contact surface is usually characterized by point-to-surface contact. In this part, the Mindlin model, a refuting slip model that describes the point-to-surface contact state, is used to establish the contact relationship between blade-damping contact surfaces. The micro-slip model is also called the local slip model. Different from the macroscopic slip model, it assumes that the stress and strain states of each node on the contact surface are not identical under the action of external forces. Under a large normal load, part of the contact area slides macroscopically and the rest is in a viscous local slip state. Compared with the macroscopic slip model, the micro-slip model can more accurately describe the friction-damping characteristics between contact surfaces. Considering the elastic contact deformation of contact surfaces, the Mindlin microslip model is proposed to describe the frictional damping characteristics of a contact surface [26]. As shown in Figure 1, the relationship of friction with respect to displacement can be expressed as:  
\[f(u)=\left\{ \begin{array}{ll}\mu N-2\mu N\bigg{\{}1-\left(\frac{1}{2}\right) ^{\frac{3}{2}}\left[1+\frac{2}{3A_{0}}\cdot\left(u-A+\frac{3}{2}A_{0}\right) \right]^{\frac{3}{2}}\bigg{\}}&0\leq\theta\leq\theta^{*}\\ -\mu N&\theta^{*}\leq\theta<\pi\\ -\mu N+2\mu N\bigg{\{}1-\left(\frac{1}{2}\right)^{\frac{3}{2}}\left[1-\frac{2 }{3A_{0}}\cdot\left(u+A-\frac{3}{2}A_{0}\right)\right]^{\frac{3}{2}}\bigg{\}}& \pi\leq\theta\leq\pi+\theta^{*}\\ \mu N&\pi+\theta^{*}\leq\theta<2\pi\end{array} \right. \tag{1}\]  
where \(f\)--friction force/N; \(A\)--displacement amplitude of the relative motion/\(\mu\)m; \(A_{0}\)--critical displacement amplitude of the relative motion/\(\mu\)m; \(u\)--displacement of the relative motion/\(\mu\)m; \(\mu\)--dynamic friction coefficient and \(N\)--normal forces of contact surfaces/N. It is assumed that the critical displacement \(A_{0}\) is the ratio of the critical friction \(\mu N\) and tangential stiffness \(K_{\rm d}\) on the contact surface, which can be described by the following formula. \[A_{0}=\frac{\mu N}{K_{\rm d}} \tag{2}\]  
When the material parameters of two touching objects are the same, the contact radius \(a\) is described by the following formula. \[a=\left[\frac{4}{3}\pi\times\frac{N\times(k_{1}+k_{2})\times R_{1}\times R_{2 }}{R_{1}+R_{2}}\right]^{1/3} \tag{3}\]  
\[k_{1}=\left(1-v_{1}^{2}\right)/(\pi E_{1}) \tag{4}\]  
\[k_{2}=\left(1-v_{2}^{2}\right)/(\pi E_{2}) \tag{5}\]  
where \(a\) is the radius of the contact area/m; \(N\) is the positive contact pressure over \(N\); \(R_{1}\) is the radius of the first contact pair/m; \(R_{2}\) is the radius of the second contact pair over m; \(v_{1}\) is the first contact to Poisson ratio; \(v_{2}\) is the second contact to Poisson ratio; \(E_{1}\) is the elastic modulus of the first contact pair/Pa and \(E_{2}\) is the elastic modulus of the second contact pair/Pa. The tangential contact stiffness \(K_{\rm d}\) of the contact pair can be calculated by the following formula.