## 3 Numerical Results and Discussion
### Sensitivity Analysis of a Jeffcott Rotor  
Since the accuracy and capability of the methods adopted are demonstrated in Section 3.1, in this section the sensitivities of a rotor subjected to rubbing forces are investigated by adopting a lumped model, which is also named a Jeffcott rotor. A schematic diagram of the Jeffcott rotor is illustrated in Figure 11. It's assumed that the periodic excitation force is induced by the unbalanced mass \(m_{\text{ub}}\) and the eccentric distance is \(e\). The structural damping coefficient, \(c\), is assumed to be proportional to the stiffness \(k\) and the mass \(m\). And the relevant parameters are listed in Table 1.  
The influences of other parameters of rubbing interfaces, such as normal stiffness \(k_{y}\) and initial gap \(g\), are investigated. The normal values of normal stiffness \(k_{y}\), initial gap \(g\) and friction coefficient \(\mu\) are \(8\times 10^{4}\) N/m, 1 mm, and 0.7, respectively. The friction model applied is a classical Coulomb friction model, as shown in Figure 12. It should be noted that the model is established based on local rotating frame. The analytical expressions for tangential rubbing force \(f_{\tau}\) and normal rubbing force \(f_{n}\) in Figure 12 can be derived in time domain easily. As for \(f_{n}\), since the values of force along normal direction will always be positive, it can be expressed by the following formula:  
\[f_{n}=\left\{ \begin{array}{cc}0&\eta<g\\ k_{y}(\eta-g)&\eta\geq g\end{array} \right. \tag{35}\]  
Then the tangential friction force can also be determined in the following form:  
\[f_{\tau}=\left\{ \begin{array}{cc}0&f_{n}=0\\ -\mu f_{n}&f_{n}>0\end{array} \right. \tag{36}\]  
where \(\mu\) is the friction coefficient at the interfaces;\(f_{\tau}\) is always negative considering that the direction of friction force is opposite to the direction of relative velocity. Based on this friction model, firstly, the response of the Jeffcott rotor with different initial parameters of rubbing interfaces is illustrated. These parameters include the normal stiffness of stator, the friction coefficient and the initial gap. The first seven harmonic components are kept to approximate the periodic response. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Unit** & **Value** \\ \hline Eccentric distance \(e\) & mm & 0.4 \\ Stiffness \(k\) & N/m & \(5\times 10^{4}\) \\ Mass \(m\) & kg & 2 \\ Damping coefficient \(c\) & (N\(\cdot\)s)/m & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Constant parameters of the Jeffcott rotor. Figure 11: Schematic diagram of the Jeffcott rotor. Firstly, the sensitivity of the response to different parameters is investigated using the once-at-a-time (OAT) method [32], which implies changing an individual variable slightly to investigate its influence on the response. According to Figure 13a, the variation of initial gap \(g\) has a significant influence on the response curve of the Jeffcott rotor. It can not only influence the maximum displacements, but also determines the range of the speed region subjected to rubbing forces. Furthermore, with the increase of initial gap \(g\), the influence of initial gap gradually becomes less significant since the variation of response curve becomes smaller. Although normal stiffness \(k_{y}\) and friction coefficient \(\mu\) don't influence the response as the way initial gap does, they can change the characteristics of multi-solution region. As shown in Figure 13b,c, with the decrease of \(k_{y}\) or increase of \(\mu\), the multi-solution region narrows. The influence of \(ky\) and \(\mu\) is much smaller than that of the initial gap \(g\), according to the variation of the response curve. Figure 12: The Coulomb friction model in local frame.