## 3 Multi Objective Optimization of SFD Parameters
### Analysis Results of Optimization Parameters of Rotor System with the Misalignment Fault
According to the design parameters, the numerical simulation of the SFD- rotor system is carried out again, and the dynamic response spectrum of the system is obtained, as shown in Figure 13, and compared with the spectrum diagram in Figure 10a before optimization. It can be concluded that after optimization, the amplitude of fundamental frequency of system dynamics response decreases from 0.6916 to 0.3913, the amplitude decreases by about 43.4%; the amplitude of 2\(\times\) fundamental frequency decreases from 0.0766 to 0.0555, the amplitude decreases by about 27.5%; the amplitude of 3\(\times\) fundamental frequency decreases from 0.2591 to 0.0863, and the amplitude decreases by about 66.7%. Analysis Results of Optimization Parameters of Rotor System with the Misalignment-Rubbing Coupling Fault  
In this section, the SFD optimization parameters of the rotor system with the fault of misalignment referred in front section are used as the initial calculated parameters of the rotor system SFD, and the dynamic characteristics of the rotor system with misalignment-rubbing coupling faults are analyzed. Figure 14 shows the system response characteristic of the system with misalignment-rubbing coupling faults. The fundamental frequency, 2\(\times\) fundamental frequency, and 3\(\times\) fundamental frequency are mainly shown in spectrum. Compared with a spectrum diagram with a misalignment fault, the rubbing coupling faults made the amplitude of fundamental frequency of the system increase a lot, and made the amplitude of the 2\(\times\) fundamental frequency, 3\(\times\) fundamental frequency change a small amount, and brought about an emergency of the continuous spectrum. However, the impact is modest, and the energy is low, meaning the calculated result obtained in this paper is in agreement with the experimental result of reference [18]. As seen in Figure 14a,b the phase diagram is a net 8-shaped figure, which means the system is in quasi periodic motion. The energy of the misalignment-rubbing coupling faults is mainly concentrated in fundamental frequency, 2\(\times\) fundamental frequency, and 3\(\times\) fundamental frequency components, thus the amplitude of the fundamental frequency, 2\(\times\) fundamental frequency, and 3\(\times\) fundamental frequency of optimization objective function of the system responses is minimal. Figure 14: Dynamic behavior of system with the misalignment-rubbing coupling fault. (**a**) Spectrum diagram; (**b**) Phase diagram. Figure 13: Spectrum diagram of SFD optimized system with misalignment fault. Figure 15 is a comparison of the calculated results of 10 sets of design parameters which selected by using optimal Latin hypercube method under misalignment fault and the predictive results of approximate model which is built by using the radial basis function (RBF). Here, the average error of fundamental frequency is 0.56%, and the maximum error is 1.44%; the average error of 2\(\times\) fundamental frequency is 1.47%, and the maximum error is 4.10%, the average error of 3\(\times\) fundamental frequency is 0.47%, and the maximum error is 1.2%. The error of the approximate model output variables can be controlled within 10%, which shows that the approximate model has high accuracy and can be used to optimize the SFD parameters of the rotor system with misalignment-rubbing coupling faults. Figure 16 shows the optimization parameters of the oil film gap \(C_{\mathrm{S}}\), elastic support stiffness \(K_{\mathrm{a}}\) and the impact of the viscosity of the lubricating oil \(\mu_{\mathrm{s}}\) on the amplitude of the fundamental, 2\(\times\) fundamental and 3\(\times\) fundamental frequency. From Figure 16, it can be seen that the impact of the three parameters on the system is nonlinear. Here, \(C_{\mathrm{S}}\) has a great influence on 2\(\times\) fundamental frequency, and basically it increases linearly in the early stage.