## 3 Friction-Induced Vibration
### Friction-Induced Bifurcations  
Consider a very small domain of \(S\in\mathbb{R}^{5}\) and divide \(S\) into two subdomains as \(S_{1}\) and \(S_{2}\). Define two vector fields of \(F_{1}(x)\) and \(F_{2}(x)\), where \(F_{1}\) and \(F_{2}\) are smooth in the rotor/stator nonlinear system. According to Equation (3), the switching boundary between \(S_{1}\) and \(S_{2}\) is \(H(x)=0\). The switching boundary is characterized with discontinuity and refers to curve hypersurfaces (\(\Sigma\)). The corresponding subregions and switching manifold are as follows:  
\[ \begin{cases}S_{1}:=\left\{x\in\mathbb{R}^{5}:\ H(x)=\Omega R_{d}+x_{1}x_{4}>0 \right\}\\ S_{2}:=\left\{x\in\mathbb{R}^{5}:\ H(x)=\Omega R_{d}+x_{1}x_{4}<0\right\}\\ \sum:=\left\{x\in\mathbb{R}^{5}:\ H(x)=\Omega R_{d}+x_{1}x_{4}=0\right\}\end{cases}  \tag{4}\]  
In the friction-involved rotor/stator system, the oscillation route of the system either runs across the switching manifold (\(H(x)=0\)) or slides along the boundary. So, we further separate the switching manifold into two domains: the crossing domain with \(\Sigma_{c}\) and the sliding domain with \(\Sigma_{s}\), where \(\Sigma_{s}\cup\Sigma_{c}=\Sigma\). In the case of \(F_{1}(x)\) and \(F_{2}(x)\) pointing to \(\Sigma\), some oscillation routes may be limited by the hypersurface of the switching manifold. So, the boundary of the sliding domains can be identified by the gradient of the vector fields by [PERSON]'s and [PERSON]'s methods [3,4]. Here, the sliding domain (\(\Sigma_{\text{s}}\)) and the corresponding boundary (\(\partial\Sigma_{\text{s}}\)) are defined as follows:  
\[ \begin{cases}\Sigma_{\text{s}}=\left\{x\in\Sigma:\ \mathcal{L}_{F_{1}}H(x) \cdot\mathcal{L}_{F_{2}}H(x)<0\right\}\\ \partial\Sigma_{\text{s}}=\left\{x\in\Sigma:\ \mathcal{L}_{F_{1}}H(x) \cdot\mathcal{L}_{F_{2}}H(x)=0\right\}\end{cases}  \tag{5}\]  
where \(\mathcal{L}_{F}\) stands for the Lie derivative along the \(F(x)\) flow:  
\[\mathcal{L}_{F}H(x)=\frac{\partial H(x)}{\partial x}F(x)\]  
\(\frac{\partial H(x)}{\partial x}\) represents the gradient of \(H(x)\). \(\mathcal{L}_{F_{1}}H(x)\) and \(\mathcal{L}_{F_{2}}H(x)\) are the normal components of \(F_{1}(x)\) and \(F_{2}(x)\) on the manifold (\(\Sigma\)), respectively. \(\mathcal{L}_{F_{1}}H(x)\) and \(\mathcal{L}_{F_{2}}H(x)\) are given as follows:  
\[\left\{ \begin{array}{l}\mathcal{L}_{F_{1}}H(x)=\Omega^{2}\sin(\theta-x_{3} )-2\zeta x_{1}x_{4}-x_{2}x_{4}+\gamma x_{1}-\mu(x_{1}-R_{0})\\ \mathcal{L}_{F_{2}}H(x)=\Omega^{2}\sin(\theta-x_{3})-2\zeta x_{1}x_{4}-x_{2}x _{4}+\gamma x_{1}+\mu(x_{1}-R_{0})\end{array} \right. \tag{6}\]  
Based on [PERSON]'s method, we define the following:  
\[F_{\text{s}}=\frac{-\mathcal{L}_{F_{2}}H(x)}{\mathcal{L}_{F_{1}} H(x)-\mathcal{L}_{F_{2}}H(x)}F_{1}(x)+\frac{\mathcal{L}_{F_{1}}H(x)}{ \mathcal{L}_{F_{1}}H(x)-\mathcal{L}_{F_{2}}H(x)}F_{2}(x) \tag{7}\] \[=\left( \begin{array}{l}x_{2}\\ \Omega^{2}\cos(\theta-x_{3})-2\zeta x_{2}-(\beta+1)x_{1}+x_{1}{x_{4}}^{2}+R_{ 0}\\ x_{4}\\ -\frac{x_{2}x_{4}}{x_{1}}\\ \Omega\end{array} \right)\]  
as the sliding flow of a such rotor/stator system. \(F_{\text{s}}\) points orthogonal to the \(H(x)\) and tangential to the \(\Sigma\). Since the deflection is greater than the clearance (\(R>R_{0}\)) when backward whirling oscillation from dry friction happens, the condition should be guaranteed:  
\[\mathcal{L}_{F_{2}}H(x)-\mathcal{L}_{F_{1}}H(x)=2\mu(R-R_{0})>0 \tag{8}\]  
The condition of (8) guarantees that the sliding domain (\(\Sigma_{\text{s}}\)) is attracting and stable in the normal direction when backward whirling oscillation from dry friction happens.