## 3 Establishment of the Dynamic Model of Three-Support Shafting
### Analysis of the Influence of Support Parameters on Shafting Response  
Based on the simulation platform, the vibration response of the shaft system could be obtained by solving the differential equation of the rotor system. The modules provided in Simulink were used to build the block diagram of the numerical model. In the process of establishing the numerical model of three-support Shafting, the From and Goto modules in signal routing were used to avoid the connection problem of complex block diagrams. When building the model, we only needed to build a separate solving model according to a certain determinant, and each solving model was connected by From and Goto. In Section 4.1, the first order critical speed of three-support Shafting was obtained. In this paper, the stiffness, damping and mass of support II were changed when the rotor system was at transcritical speed and after the critical speed under the eccentric load of the shaft system. The dynamic response of the rotor system under different support parameters was analyzed. The specific simulation parameter values are shown in Table 5.  
(1) Analysis of the influence of stiffness on the dynamic response of the shafting  
The influence of the support stiffness of the rotor system on the dynamic response of the shafting at transcritical and subcritical speeds is shown in Figures 9 and 10. Figure 9 shows the vibration response of the disc and the support when the rotor shaft system is in steady-state operation with \(\Omega=400\) rad/s and the different stiffness values of the smart spring support. It can be seen from Figure 9 that, with the increase of the smart spring support stiffness, the steady-state vibration response of the disc and the support could be effectively suppressed. The stiffness increased from \(3.8\times 10^{5}\) N\(\cdot\)m \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Synchronous Positive Whirl**} & \multicolumn{2}{c}{**Synchronous Counter Whirl**} \\ \(\boldsymbol{\omega}_{\text{Fc}}\)**/rad**\(\mathbf{s^{-1}}\) & \(\boldsymbol{n}_{\text{Fc}}\)**/r**\(\mathbf{min^{-1}}\) & \(\boldsymbol{\omega}_{\text{BC}}\)**/rad**\(\mathbf{s^{-1}}\) & \(\boldsymbol{n}_{\text{BC}}\)**/r**\(\mathbf{min^{-1}}\) \\ \hline
410.6 & 3923 & \(-\)410.4 & \(-\)3921 \\
673.1 & 6431 & \(-\)673.1 & \(-\)6431 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Critical angular velocity and critical speed. Figure 8: The relationship between rotation speed and whirl frequency. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **Stiffness/N m\({}^{-1}\)** & **Damping/N s m\({}^{-1}\)** & **Mass/kg** \\ \hline \⃝1 & 3.8e5 & 0 & 4 \\ \⃝2 & 3.8e6 & 75 & 8 \\ \⃝3 & 3.8e8 & 150 & 12 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Different support parameters of the rotor shafting simulation. to \(3.8\times 10^{8}\) N\(\cdot\)m\({}^{-1}\), and the peak amplitude of the disc decreased by 45.45%. As can be seen from Figure 10a, when the rotor shaft system operated stably at \(\Omega=600\) rad/s, the change of vibration amplitude at the disc with the smart spring support stiffness was not obvious, but the steady-state response was the minimum when the smart spring support stiffness was \(\Omega=600\) rad/s. It can be seen from Figure 10b that the steady vibration response of the smart spring was smaller than that of \(3.8\times 10^{6}\) N\(\cdot\)m\({}^{-1}\) when the support stiffness was \(3.8\times 10^{5}\) N\(\cdot\)m\({}^{-1}\). Compared with Figures 9 and 10, it can be seen that when the vibration response under a steady state was large, the vibration response of the disc and the support decreased with the increase of the support stiffness of the smart spring. When the vibration response was small in the steady state, the change of the support stiffness of the smart spring had no obvious effect on the vibration at the disc.