## 3 Results and Discussion
### Effect of Single Uncertainty  
The other three single-uncertainty cases are studied in this section as well. The uncertainty in the bearing's damping, Young's modulus, and density are considered. In the three cases, the deviations of the bearing's damping, Young's modulus, and density are 15%, 5%, and 5%, respectively, which means \(\varepsilon_{c}^{l}=[178.5,241.5]\) Ns/m, \(\varepsilon_{E}^{l}=[199.5,220.5]\) GPa, and \(\varepsilon_{p}^{l}=[7410,8190]\) kg/m\({}^{3}\). Figure 5 shows the uncertain gravity response amplitudes versus rotational speed with single uncertainty in the bearing's damping. As shown in Figure 5, although a larger deviation is considered, the response is slightly changed. There is no local large-amplitude band in each resonance peak. This is because the bearing's damping can only obviously affect the response magnitude near the resonance peak, but it hardly changes the resonance speed. Therefore, tiny bounds near the resonance peaks can be observed while the response amplitudes away from the resonance peak are less influenced. Generally, when compared with the results with single uncertainty in the bearing's stiffness, the influences of the bearing's damping on the response are smaller. Figure 4: The bounds of uncertain gravity response versus rotational speed by the scanning method and Chebyshev interval method (CIM) with a 10% deviation in the bearing’s stiffness. Figure 6 shows the uncertain gravity response amplitudes versus rotational speed with single uncertainty in Young's modulus. Two local large-amplitude bands can be found in the first and the third resonance peaks. The width of the latter is larger than that of the former. Unlike the above-analyzed single-uncertainty cases, an obvious little-amplitude band can be found at the antiresonance position (near 1500 rpm), which shows the high sensitivity of the antiresonance position on the Young's modulus. Besides, the response at a higher rotational speed in Figure 6 is more influenced by the Young's modulus. The origin of this phenomenon is the resonance speed increasing with the increase of the global stiffness of the asymmetric rotor system. The global stiffness is the combination of the stiffness of the rotor and the bearing's stiffness. As the two stiffnesses can be regarded as in series with each other, the global stiffness is more influenced by the smaller stiffness. Thus, the bearing's stiffness can globally influence the response curve (Figure 4), while the Young's modulus has a larger influence on the response curve near 1500 rpm (Figure 5). This is different from Figure 5, where the response at lower rotational speed is more sensitive to the bearing's stiffness. Figure 5: The bounds of uncertain gravity response versus rotational speed by CIM with a 15% deviation in the bearing’s damping. Figure 6: The bounds of uncertain gravity response versus rotational speed by the CIM with a 5% deviation in Young’s modulus. Figure 7 shows the uncertain gravity response amplitudes versus rotational speed with a single uncertainty in density. As the variation in density changes the whirl speed of the asymmetric rotor system, the local large-amplitude bands can be found at the first and the third resonance peaks. A higher sensitivity of the response on the density can be found at a higher rotational speed. The origin of this phenomenon is the resonance speed decreasing with the increase of the density. As the effects of the density are almost opposite to those of the Young's modulus, the density has a larger influence on the response curve at a higher rotational speed as well. However, an obvious difference between the envelope shapes caused by the Young's modulus and density can be observed. For example, the trends of the local large-amplitude bands at the third resonance peak of the two cases are different from each other. A conclusion can be drawn from the above analysis that different types of single uncertainty have different influences on the uncertain gravity response of the generator rotor system.