## 3 Numerical Results and Discussion
### Sensitivity Analysis Based on Finite Element Model  
Another rotor with rub-impact is discretized using the finite element method (FEM). As shown in Figure 18, the model is made up of 20 equal-length beam elements with the Young's modulus, density and Poisson's ratio equaling to 2 \(\times\) 10\({}^{5}\) MPa, 7.8 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\) and 0.3, respectively. The model is supported at both ends of the shaft. The motion of each element is assumed to be within the plane vertical to axial direction. The multiplication of unbalanced mass and its eccentric distance is equal to 7.0711 \(\times\) 10\({}^{-4}\) kg\(\cdot\)m in this paper and the excitation force is applied based on it. Moreover, the rubbing forces are applied to node 7, with the Coulomb friction model mentioned in Section 3.2. The nominal values of parameters of rubbing interfaces are also prescribed, with stator stiffness \(k_{y}\), the friction coefficient \(\mu\) and the initial gap \(g\) equal to 1.0 \(\times\) 10\({}^{7}\) N/m, 0.5, 0.1 mm, respectively. For the purpose of performance and accuracy, the time sampling number is 1024 and the first  
Figure 14: The sensitivity curves of the x-direction response of the Jeffcott rotor with respect to the three investigated parameters of rub interfaces at \(k_{y}\) = 8 \(\times\) 10\({}^{4}\) N/m, \(\mu\) = 0.7 and \(g\) = 1 mm: (**a**) First-order sensitivities; (**b**) Second-order sensitivities. seven harmonic components are kept to approximate the response. The friction model applied on elements subjected to rubbing forces is exactly the same as that of Section 3.2. Figure 16: The sensitivity curves of the response of the Jeffcott rotor with respect to normal stiffness at \(g=1\) mm and \(\mu=0.7\): (**a**) First-order sensitivities; (**b**) Second-order sensitivities. Figure 17: The sensitivity curves of the response of the Jeffcott rotor with respect to initial gaps at \(k_{y}=8\times 10^{4}\) N/m and \(\mu=0.7\): (**a**) First-order sensitivities; (**b**) Second-order sensitivities. Figure 15: The sensitivity curves of the response of the Jeffcott rotor with respect to friction coefficients at \(g=1\) mm and \(k_{y}=8\times 10^{4}\) N/m: (**a**) First-order sensitivities; (**b**) Second-order sensitivities. Similar conclusions can be drawn based on this model, i.e., the response is much more sensitive to the initial gap \(g\) and with its increase the sensitivity decreases, as in Figure 19a. Outside of the speed range 260.9 rad/s to 298.9 rad/s there is no rub-impact forces so the response is exactly the same with that of separation status. When rubbing forces exist, a multi-solution region emerges because of the nonlinearity of the system. Moreover, the increase of normal stiffness and decrease of friction coefficient results in the expansion of the multi-solution region, as seen in Figure 19b,c. Figure 19: The x-direction response curve of the finite element model with different nominal values of the investigated parameters: (**a**) Response curves with different initial gaps at \(k_{y}=1\times 10^{7}\) N/m and \(\mu=0.5\); (**b**) Response curves with different normal stiffness at \(g=0.1\) mm and \(\mu=0.5\); (**c**) Response curves with different normal friction coefficients at \(k_{y}=1\times 10^{7}\) N/m and \(g=0.1\) mm. Figure 18: Sketch of the investigated finite element model. Figure 20 shows the sensitivities of the frequency-domain response to different parameters of rub-impact interfaces. Similar phenomena have been observed as obtained by the Jeffcott rotor. The responses are more sensitive to the initial gap \(g\) compared to the other investigated parameters. The sensitivity curve near the vicinity of the rotating speeds where the separation-contact status transformation happens is discontinuous, since the sensitivities are equal to zero when no rub-impact exists. An obvious offset of sensitivities is seen under this condition, as a result of the existence of a multi-solution region in the response curve.