## 3 Multi-DoF Coupled Dynamic Model of New TBM Integrated Cutter System
### Structural Dynamic Parameters of the Vibration Model
#### 3.3.3 Equivalent Damping of the Cutter System Components  
This section uses equations to calculate the damping of \(W_{2}\), and uses the empirical formula in the mechanical design manual to calculate the stiffness of \(W_{1}\) and \(W_{3}\), which is shown as Equation (17):  
\[c=2\xi\sqrt{m_{e}k_{e}} \tag{17}\]  
In Equation (17), \(c\) is the damping of the structural parts and \(\xi\) is the damping ratio. In the elastic stage of the steel structure, \(\xi\) is generally 0.02-0.05, so we set it to 0.02 in this paper. \(m_{e}\) and \(k_{e}\) are the equivalent mass and equivalent stiffness of each component, respectively. (1) Equivalent damping of node \(W1\)  
The equivalent radial damping \(c_{yW1}\) and axial damping \(c_{xW1}\) of node \(W1\) can be obtained as Equations (18) and (19):  
\[c_{yW1}=3.25\times 10^{4}(Ns/m) \tag{18}\]  
\[c_{xW1}=1.52\times 10^{4}(Ns/m) \tag{19}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Node** & \(k_{z1}\) & \(k_{z2}\) & \(k_{z3}\) & \(k_{z4}\) & \(k_{z5}\) & \(k_{z6}\) \\ \hline
**Stiffness** & & & & & & \\
**(N/m)** & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Rolling direction equivalent stiffness of each node. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Node** & \(k_{y1}\) & \(k_{y2}\) & \(k_{y3}\) & \(k_{y4}\) & \(k_{y5}\) & \(k_{y6}\) \\ \hline
**Stiffness** & & & & & & \\
**(N/m)** & & & & & & \\ \hline
**Node** & \(k_{y7}\) & \(k_{y8}\) & \(k_{y9}\) & \(k_{y10}\) & \(k_{y11}\) & \(k_{y12}\) \\ \hline
**Stiffness** & & & & & & \\
**(N/m)** & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Vertical direction equivalent stiffness of each node. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Node** & \(k_{x1}\) & \(k_{x2}\) & \(k_{x3}\) & \(k_{x4}\) & \(k_{x5}\) & \(k_{x6}\) \\ \hline
**Stiffness** & & & & & & \\
**(N/m)** & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Lateral direction equivalent stiffness of each node. 2. Equivalent damping of node \(W2\) The equivalent radial damping \(k_{yW2}\) and axial damping \(c_{xW2}\) of node \(W2\) can be obtained as Equations (20) and (21): \[c_{yW2}=6.57\times 10^{3}(Ns/m)\] (20) \[c_{xW2}=6.74\times 10^{3}(Ns/m)\] (21) (3) Equivalent damping of node \(W3\) The equivalent radial damping \(c_{yW3}\) and axial damping \(c_{xW3}\) of node \(W3\) can be obtained as Equations (22) and (23): \[c_{yW3}=9.80\times 10^{3}(Ns/m)\] (22) \[c_{xW3}=1.46\times 10^{4}(Ns/m)\] (23) The vertical and lateral equivalent damping of the cutter at node \(m_{1}\) can be obtained as Equations (24) and (25): \[c_{y1}=c_{y2}=\frac{1}{2}\times\frac{1}{\frac{1}{c_{yW1}}+\frac{1}{c_{yW2}}+ \frac{1}{c_{yW3}}}=1.75\times 10^{3}(Ns/m)\] (24) \[c_{x1}=c_{x2}=\frac{1}{\frac{1}{c_{xW1}}+\frac{1}{c_{xW2}}+\frac{1}{c_{xW3}}}= 3.54\times 10^{3}(Ns/m)\] (25) The rolling direction equivalent damping of the cutter at node \(m_{1}\) can be obtained as Equation (26): \[c_{z1}=c_{z2}=2.71\times 10^{4}(Ns/m)\] (26) The calculated each note's damping results are shown in Tables 7-9. During the rock-breaking process, the cutter will be subjected to the lateral force due to the rotation of the cutterhead and the squeezing of the blade on the rock, resulting in a \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Node** & \(c_{y1}\) & \(c_{y2}\) & \(c_{y3}\) & \(c_{y4}\) & \(c_{y5}\) & \(c_{y6}\) \\ \hline
**Damping** & \(1.75\times 10^{3}\) & \(1.75\times 10^{3}\) & \(1.61\times 10^{4}\) & \(1.61\times 10^{4}\) & \(2.03\times 10^{3}\) & \(2.03\times 10^{3}\) \\ \hline
**Node** & \(c_{y7}\) & \(c_{y8}\) & \(c_{y9}\) & \(c_{y10}\) & \(c_{y11}\) & \(c_{y12}\) \\ \hline
**Damping** & \(1.37\times 10^{4}\) & \(1.37\times 10^{4}\) & \(3.60\times 10^{3}\) & \(3.60\times 10^{3}\) & \(8.11\times 10^{4}\) & \(8.11\times 10^{4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Vertical direction equivalent damping of each node.