## 3 Multi-DoF Coupled Dynamic Model of New TBM Integrated Cutter System
### Structural Dynamic Parameters of the Vibration Model
#### 3.3.2 Equivalent Stiffness of the Cutter System Components  
This section mainly studies the vibration characteristics of the TBM new cutter system, and the components which have the greatest influence on the cutter system will be regarded as a whole, such as a cutter ring, bearing, and cutter shaft. The other components' influence is too small to be ignored. The stiffness of the \(W1\) cutter body can obtain the stiffness of the whole cutter, the \(W2\) bearing (including the bearing outer ring and bearing half roller), and the \(W3\) cutter shaft (including the cutter shaft and bearing the inner ring) in series.  
The stiffness \(k\) of the whole disc cutter can be expressed in Equation (7):  
\[\frac{1}{k}=\frac{1}{k_{W1}}+\frac{1}{k_{W2}}+\frac{1}{k_{W3}} \tag{7}\]  
This section uses equations to calculate the stiffness of \(W2\), and uses the FEA method to calculate the stiffness of \(W1\) and \(W3\), considering the low accuracy in using equations to calculate the nonstandard parts.  
(1) Equivalent stiffness of node \(W1\)  
Fix the cutter body's internal surface, apply a vertical load and lateral load, respectively, on the cutter ring in finite element software, and calculate the cutter body's deformation results as shown in Figure 6. From the definition of the stiffness and FEA method, the equivalent radial stiffness \(k_{yW1}\) and axial stiffness \(k_{xW1}\) of node \(W1\) can be obtained as Equations (8) and (9):  
\[k_{yW1}=\frac{F_{ymax}}{\delta_{ymax}}=9.06\times 10^{9}(N/m) \tag{8}\]  
\[k_{xW1}=\frac{F_{xmax}}{\delta_{xmax}}=1.98\times 10^{9}(N/m) \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node** & \(m_{1}\) (kg) & \(m_{2}\) (kg) & \(m_{3}\) (kg) & \(m_{4}\) (kg) & \(m_{5}\) (kg) \\ \hline
**Equivalent Mass** & \(141.17\) & \(77.19\) & \(4.57\) & \(77.19\) & \(4.57\) \\ \hline
**Node** & \(m_{6}\) (kg) & \(m_{7}\) (kg) & \(m_{8}\) (kg) & \(m_{9}\) (kg) & \(m_{10}\) (kg) \\ \hline
**Equivalent Mass** & \(5.51\) & \(156.74\) & \(5.51\) & \(156.74\) & \(0.96\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Nodes equivalent mass of cutter system components.  
2. Equivalent stiffness of node \(W2\)  
The bearing is a standard part, and the equivalent radial stiffness \(k_{yW2}\) and axial stiffness \(k_{xW2}\) of node \(W2\) can be obtained by Equations (10) and (11):  
\[k_{yW2}=1.08\times 10^{9}(N/m) \tag{10}\]  
\[k_{xW2}=2.07\times 10^{9}(N/m) \tag{11}\]
3. Equivalent stiffness of node \(W3\)  
Fix both ends of the cutter shaft and apply a vertical load during the vertical deformation analysis. Since the lateral load will transfer to the cutter shaft through the end cover, fix the cutter shaft through the side with no end cover and apply the lateral load to the side near the end cover; the calculated deformation results of the cutter body are shown in Figure 7. From the definition of the stiffness and FEA method, the equivalent radial stiffness \(k_{yW3}\) and axial stiffness \(k_{xW3}\) of node \(W3\) can be obtained as Equations (12) and (13):  
\[k_{yW3}=\frac{F_{ymax}}{\delta_{ymax}}=3.02\times 10^{9}(N/m) \tag{12}\]  
\[k_{xW3}=\frac{F_{xmax}}{\delta_{xmax}}=6.69\times 10^{9}(N/m) \tag{13}\]  
Figure 6: Calculated deformation results of the cutter ring.  
Figure 7: Calculated deformation results of the cutter shaft.  
The vertical and lateral equivalent stiffness of the cutter at node \(m_{1}\) can be obtained as Equations (14) and (15):  
\[k_{y1}=k_{y2}=\frac{1}{2}\times\frac{1}{\frac{1}{k_{yW1}}+\frac{1}{k_{yW2}}+\frac {1}{k_{yW3}}}=3.66\times 10^{8}(N/m) \tag{14}\]  
\[k_{x1}=k_{x2}=\frac{1}{\frac{1}{k_{xW1}}+\frac{1}{k_{xW2}}+\frac{1}{k_{xW3}}}=1.82\times 10^{8}(N/m) \tag{15}\]  
The rolling direction equivalent stiffness of the cutter at node \(m_{1}\) can be obtained as Equation (16):  
\[k_{z1}=k_{z2}=3.24\times 10^{9}(N/m) \tag{16}\]  
The equivalent stiffness of other nodes can be calculated by using the above method, as shown in Tables 4-6.