## 2 Dynamic Modeling
### Three-Dimensional Variable Positive Pressure Hysteresis Coulomb Friction Contact Model  
The contact pressure changes with the movement of the contact pair and a three-dimensional variable positive pressure friction model are shown in Figure 3. \(N_{0}\) represents the normal initial positive pressure acting on the contact element; \(\mathrm{s}_{u}(t)\) and \(\mathrm{s}_{v}(t)\) represent the displacements of the mass in two directions; \(k_{\mathrm{u}}\) and \(k_{\mathrm{v}}\) represent the shear spring stiffness in two directions, \(k_{\mathrm{n}}\) represents the normal spring stiffness; \(w_{u}(t)\) and \(w_{\mathrm{v}}(t)\) represent the displacements of the friction damper in two directions; \(Q_{\mathrm{u}}\) and \(Q_{\mathrm{v}}\) represent the external excitations along the two directions, and \(Q_{\mathrm{n}}\) represents the normal external excitation. \(\mu\) represents the friction coefficient between the contact surfaces. \begin{table}
\begin{tabular}{c c c c c} \hline
**Mode Order** & \begin{tabular}{c} **Original Model** \\ **Frequency/Hz** \\ \end{tabular} & \
\begin{tabular}{c} **Reduced Model** \\ **Frequency/Hz** \\ \end{tabular} &
\begin{tabular}{c} **Frequency** \\ **Difference/\%** \\ \end{tabular} & **MAC** \\ \hline
1 & 506.83 & 506.83 & \(2.17\times 10^{-3}\) & 0.99 \\
2 & 526.37 & 526.37 & \(1.42\times 10^{-3}\) & 0.99 \\
3 & 1230.70 & 1230.68 & \(2.28\times 10^{-3}\) & 0.99 \\
4 & 1325.70 & 1325.72 & \(6.11\times 10^{-4}\) & 0.99 \\
5 & 2567.10 & 2567.17 & \(3.27\times 10^{-4}\) & 0.99 \\
6 & 3206.60 & 3206.68 & \(9.67\times 10^{-3}\) & 0.99 \\
7 & 3559.80 & 359.83 & \(8.71\times 10^{-3}\) & 0.99 \\
8 & 3575.50 & 3575.56 & \(1.12\times 10^{-2}\) & 0.99 \\
9 & 4802.10 & 4802.29 & \(4.16\times 10^{-2}\) & 0.99 \\
10 & 5449.50 & 5449.80 & \(5.87\times 10^{-2}\) & 0.99 \\ \hline \end{tabular}
\end{table} 
Table 2: Correlation data of simplified blade model with under-platform damper before and after CB reduction. Figure 3: Three-dimensional variable positive pressure friction model. In this model, in view of the normal displacement of the mass block, pressure \(N\) is not constant. Therefore, \(N\) can be expressed as:  
\[N=\left\{ \begin{array}{ll}N_{0}+k_{\rm n}\times n,&n\geq N_{0}/k_{\rm n}\\ 0,&n<-N_{0}/k_{\rm n}\end{array} \right., \tag{1}\]  
In the formula, \(n\) represents the distance of the normal motion of the mass block. Since the normal action of the mass block has a simple harmonic excitation \(Q_{\rm n}\), the normal displacement \(n(t)\) of the mass block will change harmonically, and its value also has positive and negative values. Therefore, when the normal displacement of the mass block is greater than \(-N_{0}/k_{\rm n}\), \(N_{0}\) may be greater than 0, and the contact surfaces make contact with each other; otherwise, the contact surfaces separate, and \(N_{0}\) is 0. The friction force is judged in the same way as in the two-dimensional friction model, Hence, the total friction force generated by the mass block can be expressed as:  
\[f=\left\{ \begin{array}{ll}\sqrt{\left[k_{\rm u}(s_{u}-w_{\rm u})\right]^{2}+ \left[k_{\rm v}(s_{v}-w_{\rm v})\right]^{2}},&\sqrt{\left[k_{\rm u}(s_{u}-w_{ \rm u})\right]^{2}+\left[k_{\rm v}(s_{v}-w_{\rm v})\right]^{2}}<\mu N\\ \mu N&\sqrt{\left[k_{\rm u}(s_{u}-w_{\rm u})\right]^{2}+\left[k_{\rm v}(s_{v} -w_{\rm v})\right]^{2}}\geq\mu N\end{array} \right., \tag{2}\]  
In the friction model shown in Figure 3, when the mass block is in contact with the plane, due to the existence of the shear springs \(k_{\rm u}\) and \(k_{\rm v}\), the displacement of the friction device along the \(u\) and \(v\) directions is different. Then, the friction device in the plane produces the following four motions:  
1. The velocity of the friction damper along the \(u\) and \(v\) directions is zero;
2. The velocity of the friction damper along the \(u\) direction is zero, and the velocity along the \(v\) direction is not zero;
3. The velocity of the friction damper along the \(v\) direction is zero, and the velocity along the \(u\) direction is not zero;
4.