## 3 Multi Objective Optimization of SFD Parameters
### Single Objective Parameter Optimization Analysis of the Normal Rotor System  
Figure 6 shows the system response characteristic of the system in the state of no fault, it can be found that spectrum diagram mainly manifested as the fundamental frequency vibration caused by the unbalance eccentricity. The motion of rotor is an ellipse, which means the system completes a single-period movement.  
\begin{table}
\begin{tabular}{c c c} \hline
**Design Parameters** & **Initial Value** & **The Changing Range/mm** \\ \hline \(C_{\text{S}}\) (mm) & \(8\times 10^{-5}\) & \((4\times 10^{-5},2\times 10^{-4})\) \\ \(K_{\text{a}}\) (N-m) & \(3\times 10^{6}\) & \((6\times 10^{5},8\times 10^{7})\) \\ \(\mu_{\text{s}}\) (Pa-s) & 0.02 & (0.005, 0.01) \\ \hline \end{tabular}
\end{table}
Table 1: SFD Scheme structure variable.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Design Parameter** & **1** & **2** & **3** & **80** \\ \hline \(C_{\text{S}}\) (mm) & \(1.4\times 10^{-4}\) & \(1.1\times 10^{-4}\) & \(1.5\times 10^{-4}\) & \(1.8\times 10^{-4}\) \\ \(K_{\text{a}}\) (N-m) & \(6\times 10^{6}\) & \(8.7\times 10^{6}\) & \(3.9\times 10^{6}\) & \(4.8\times 10^{6}  
Because of the vibration energy of normal system is mainly concentrated in the fundamental frequency, the optimization objective function is aiming at the minimum of its amplitude.  
Figure 7 shows the calculation results of the 10 sets of normal system design parameters selected by the optimal Latin hypercube method and the comparison of the predicted results with the RBF approximation model. Because the fundamental frequency which caused by the rotating speed is the mainly response, which can get the average error value of the amplitude of one time frequency is 0.9%, the maximum error value is 2.7%. The error of the approximate model output variables can be controlled within 10%, which shows that the approximate model has high precision, and can be used to optimize the SFD parameters of the normal rotor system.  
Figure 8 shows the optimization parameters of oil film gap \(C_{\text{S}}\), elastic support stiffness \(K_{\text{a}}\), and the influence of the oil viscosity of \(\mu_{\text{s}}\) on the amplitude of the fundamental frequency. The effect of three parameters to the system is strongly nonlinear, the \(C_{\text{S}}\) has the most obvious effect.  
Figure 8: Effects of SFD design parameters on the amplitude of the fundamental frequency of the normal system.  
Figure 6: Dynamic behavior of the normal rotor system. (**a**) Spectrogram; (**b**) Phase Diagram.  
Figure 7: Comparison between calculated results and approximate model with no fault.  
Sequential quadratic programming method (NLPQL) is introduced to optimize SFD parameters of normal system, the objective function is the amplitude of the fundamental frequency. The optimal solution for the design parameters of SFD can be solved as: \(C_{S}=4\times 10^{-5}\); \(K_{a}=\) 8,008,000; \(\mu_{s}=\) 0.038.  
According to the design parameters, the numerical simulation of the normal SFD- rotor system is made, and the dynamic response spectrum of the system is obtained. After comparing Figures 6 and 9, it can be seen that, after optimization, the amplitude of the fundamental frequency of the system dynamics response decreases from 0.8913 to 0.6858, while the amplitude is reduced by nearly 23%.