## 4 Simulation Results and Analysis
### Effects of Static Displacement Eccentricity  
Static displacement eccentricity always exists in rotating machinery due to many factors, such as the manufacturing tolerances and incorrect assembly. This kind of eccentricity can produce the UMP and vibrations. Therefore, it is necessary to study the effects of static displacement eccentricity on dynamic characteristics. The static displacement eccentricity in the \(x\)-direction is selected for analysis in this paper. The rotor shaft orbits and bearing shaft orbits for different eccentricities are as shown in Figure 6. It is noted that the orbits of the rotor and bearings are similar and they are center circles when the static displacement eccentricity is zero. When the eccentricity exists, the orbits become irregular and larger. Since the static displacement eccentricity in the \(x\)-direction is selected in this study, the air-gap becomes short in the \(x\)-direction and the UMP pulls the rotor moving towards the stator in this direction. As a result, the orbits deviate from the origin with the increase in the eccentricity. From Figure 6b,c, we can see that the effects of the eccentricity on the left bearing are greater than the right due to the offset rotor, and this indicates that the bearing closer to the rotor can be affected more easily. It also can be seen that even a small eccentricity can cause a great variation for the orbits and this can cause vibration and motor noise, and a large eccentricity may produce a rub between the rotor and stator. Therefore, to avoid the appearance of rub, the static displacement eccentricity must to be monitored and controlled as much as possible. The dynamic responses of the system for different eccentricities are displayed in Figure 7. With the increase of eccentricity, the dynamic responses increase nonlinearly due to the effects of UMP and bearing forces. In Figure 7a, the results show that both the displacement responses (\(x\) and \(y\)) and angle responses (\(\theta_{x}\) and \(\theta_{y}\)) increase when the eccentricity increases, and this phenomenon indicates that the displacement responses and angle responses are interactive. It can be seen clearly that the change regulations of \(x\) and \(\theta_{y}\) are similar, and the change regulations of \(y\) and \(\theta_{x}\) are also similar. The results are in agreement with the reference [23]. The reference also indicated that the averages of \(x\) and \(\theta_{y}\) within one period deviated from zero and the averages of \(y\) and \(\theta_{x}\) within one period remained zero when the eccentricity existed. However, in our investigation, the averages of both the dynamic displacement responses and angle responses within one period are not zero. This indicates \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Outer race radius (\(R_{o}\)) & 63.9 mm \\ Inner race radius (\(R_{l}\)) & 40.1 mm \\ Number of balls (\(N_{b}\)) & 9 \\ Radial clearance (\(\gamma\)) & 10 μm \\ Hertzian contact stiffness (\(C_{b}\)) & 1.334 \(\times\) 10\({}^{10}\) N/m\({}^{3/2}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Main parameters of the ball bearings. Figure 6: Rotor shaft orbits and bearing shaft orbits for different eccentricities. (**a**) Rotor shaft orbits; (**b**) Left-bearing shaft orbits; (**c**) Right-bearing shaft orbits. that the static displacement eccentricity in the \(x\)-direction can also affect the dynamic responses in the \(y\)-direction in our model. As Figure 7b shows, the dynamic responses of the bearings are similar in the same direction and the change regulations are similar with the displacement responses of the rotor. It is noted that the dynamic responses of the left bearing are clearly larger than the right when the eccentricity is small. If the eccentricity becomes a little larger, there will be little difference between the left and the right. The main cause of this phenomenon is the bearing forces. When the eccentricity is small, the dynamic responses and the radial deformations of the bearings are small.