## 3 Multi Objective Optimization of SFD Parameters
### Analysis Results of Optimization Parameters of Rotor System with the Misalignment Fault  
In this section, the SFD optimization parameters of the rotor system with no fault are used as the initial calculated parameters of the SFD of rotor system under misalignment fault, to analyze the dynamic characteristics of the rotor system with misalignment, in which the misalignment value is \(\mathrm{e}_{0}\) = \(3\times 10^{-4}\) m.  
Figure 10 shows the response characteristic of the system under the misalignment fault, from Figure 10a, it can be found that the fault makes the system produce 2\(\times\) fundamental frequency, 3\(\times\) fundamental frequency and other high frequency components, the VC frequency of rolling bearing has growth, while the frequency spectrum is mainly fundamental frequency, 2\(\times\) fundamental frequency, and 3\(\times\) fundamental frequency. The calculated results are in agreement with those in reference [12]. From Figure 10b, the phase diagram of the system is shown as a net 8-shaped figure, which means the system is in quasi periodic motion. The energy of misalignment fault system is mainly concentrated in fundamental frequency, 2\(\times\) fundamental frequency, 3\(\times\) fundamental frequency components, the amplitude of fundamental frequency, 2\(\times\) fundamental frequency, 3\(\times\) fundamental frequency of the optimization objective function for the system response is minimal. Figure 11 is a comparison of the calculated results of 10 sets of design parameters which were selected by using the optimal Latin hypercube method under misalignment fault and the predictive  
Figure 10: Dynamic behavior of system with misalignment fault. (a) Spectrum diagram; (b) Phase Diagram. Figure 9: Spectrum diagram of optimized system with no fault. results of approximate model which is built by using the radial basis function (RBF). In this figure, the average error of fundamental frequency is 1.2%, and the maximum error is 6.8%; the average error of 2\(\times\) fundamental frequency is 1.8%, and the maximum error is 8.4%; the average error of 3\(\times\) fundamental frequency is 0.6%, and the maximum error is 2.1%. The error of the approximate model output variables can be controlled within 10%, which shows that the approximate model has high accuracy and that it can be used to optimize the SFD parameters of the rotor system with misalignment-rubbing coupling faults. Figure 12 shows the optimization parameters of the oil film gap \(C_{\text{S}}\), elastic support stiffness \(K_{\text{a}}\) and the impact of the viscosity of the lubricating oil \(\mu_{\text{s}}\) on the amplitude of fundamental frequency, 2\(\times\) fundamental frequency and 3\(\times\) fundamental frequency. From Figure 12, it can be seen that the effect of the three parameters on the system is nonlinear. As a result, fundamental frequency, 3\(\times\) fundamental frequency mainly influences the oil film gap \(C_{\text{S}}\); and the 2\(\times\) fundamental frequency is mainly influenced by the support stiffness \(K_{\text{a}}\). The SFD parameters of rotor system under the misalignment fault is optimized by using NSGA-II, the parameter configuration of the algorithm is as follows: the population size is 40, algebra is 200, the crossover rate is 0.9, cross distribution index is 10, variation distribution index is 20. On these grounds, the optimal solution of SFD is \(C_{\text{S}}=2\times 10^{-4}\); \(K_{\text{a}}=6\times 10^{5}\); \(\mu_{\text{s}}=5\times 10^{-3}\). Figure 11: Comparison between calculated results and approximate model with misalignment fault. (**a**) Fundamental frequency; (**b**) 2\(\times\) fundamental frequency; (**c**) 3\(\times\) fundamental frequency. Figure 12: Effect of SFD design parameters on the amplitude of the fundamental, 2\(\times\) fundamental and 3\(\times\) fundamental frequency of the system with misalignment fault.