## 4 Simulation Results and Analysis
### Effects of Static Displacement Eccentricity
In this situation, the bearing forces will be very small and the dynamic responses of the bearings are mainly affected by the rotor. Thus, the bearing closer to the rotor will be affected more greatly and the dynamic responses will be larger. However, when the eccentricity becomes large, the bearing forces will be large with the increase of the radial deformation of the bearing. The effect of the bearing forces on the left bearing will gradually become larger and it will slow down the increase of dynamic responses. Through the above analysis, we can see the bearing forces can affect the dynamic responses of the system in some way. The spectral characteristics of the dynamic responses are as shown in Figure 8. The results of the studied model in reference [13] indicated that the frequency components contained the rotating frequency, the double power frequency and the double power frequency minus the rotating frequency. It is noted that the main frequencies in Figure 8a,b are 0 Hz, 25 Hz and 50 Hz. Since the rotating frequency and power frequency are both 25 Hz in our research, the results can be seen as consistent with the results from reference [13]. The frequency components in Figure 8a also contain the integer multiples of rotating frequency when the eccentricity exists, which means that the eccentricity can produce large dynamic responses and excite high integer multiples of rotating frequency. It is commonly accepted that the varying compliance (VC) vibration from a ball bearing is an inevitable source of excitation to the rotor bearing system [33], and the VC frequency (\(f_{\rm{vc}}\)) can be calculated by BN \(\times\)\(\Omega\), where BN = \(R_{i}\times N_{b}/(R_{i}+R_{o})\)[34]. The value of BN depends on the dimensions of the bearing and is 3.47 in this study. It can be observed that the components of VC frequency minus integral multiples of rotating frequency exist in Figure 8a, which indicates that the bearing forces have an effect on the rotor system. Figure 8b displays the spectra of angle response in the \(x\)-direction and also contains the integral multiples of rotating frequency and the combined frequencies. It can be noticed that the frequency components of angle response are more complex than displacement response, which indicates that the eccentricity has a greater effect on the angle response. The combined frequencies \(f_{\rm{vc}}-3\Omega,f_{\rm{vc}}+3\Omega,f_{\rm{vc}}+5\Omega,f_{\rm{vc}}+7\Omega\) and \(f_{\rm{vc}}+9\Omega\) are more obvious than other combined frequencies. From Section 3.1, we can see that the frequency components of radial Maxwell stress are the even multiples of power frequency. Thus, the combined frequencies \(f_{\rm{vc}}-3\Omega,f_{\rm{vc}}+3\Omega,f_{\rm{vc}}+5\Omega\), \(f_{\rm{vc}}+7\Omega\) and \(f_{\rm{vc}}+9\Omega\) can be seen as the combination of VC frequency, rotating frequency and even multiples of power frequency, from which it can be inferred that the coupling effects of bearing forces, unbalanced mass force and UMP obviously exist. Although the amplitudes of these combined frequencies are not large, the coupling interaction can also cause some unwanted vibrations and noises. The spectra of dynamic displacement and angle responses in the \(y\)-direction are similar to the spectra in the \(x\)-direction and the analysis will no longer be illustrated. The spectra of the dynamic displacement response of the left bearing (\(x_{A}\)) are displayed in Figure 8c. It can be noticed that the frequency components of the bearing are more complex when the static displacement eccentricity exists, while the frequency component is \(\Omega\) when the static displacement eccentricity is zero. The bearing shaft orbit in Figure 6 indicates that the dynamic displacement responses of bearings are small when the eccentricity is zero and the bearing forces cannot be formed in this case. As the eccentricity increases, the dynamic displacement responses of bearings become large and the bearing forces are then formed.