## 2 Methods
### Sensitivity Analysis  
In this paper, a local sensitivity analysis is performed by evaluating the local gradients of maximum displacements with respect to parameters of rubbing interfaces. Typically, what we need is the sensitivity of the maximum displacement in time domain, \(x_{max}\), with respect to different parameters, i.e., the frictional coefficient, the normal stiffness \(k_{y}\), etc. The local sensitivities, defined as slight changes of the response caused by a shift of individual variable, are computed as follows:  
\[s=x_{\max,0}^{+}-x_{\max,0}=\frac{\partial x_{\max}}{\partial p}\left(p_{0}^{+} -p_{0}\right)+\frac{1}{2}\frac{\partial^{2}x_{\max}}{\partial p^{2}}\left(p_{0 }^{+}-p_{0}\right)^{2}+o\left(\left(p_{0}^{+}-p_{0}\right)^{3}\right) \tag{15}\]  
where \(s_{1}=\frac{\partial x_{\max}}{\partial p}\) and \(s_{2}=\frac{\partial^{2}x_{\max}}{\partial p^{2}}\) are the first-order and second-order sensitivities of time-domain displacements with respect to rubbing parameter \(p\). They need to be derived based on the results obtained in frequency domain. By differentiating Equation (3) with respect to \(p\), the analytical expression of first-order derivatives of harmonic components with respect to the investigated parameter is obtained as:  
\[\frac{d\mathbf{R}}{dp}=\frac{\partial\mathbf{Z}}{\partial p}\mathbf{Q}+ \mathbf{Z}\frac{\partial\mathbf{Q}}{\partial p}+\frac{\partial\mathbf{F}_{n}} {\partial p}+\frac{\partial\mathbf{F}_{n}}{\partial\mathbf{Q}}\frac{\partial \mathbf{Q}}{\partial p}+\frac{\partial\mathbf{F}_{u}}{\partial p}=0 \tag{16}\]  
Thus the first-order sensitivity in frequency domain can be expressed as:  
\[\frac{\partial\mathbf{Q}}{\partial p}=-\left(\mathbf{Z}+\frac{\partial\mathbf{ F}_{n}}{\partial\mathbf{Q}}\right)^{-1}\left(\frac{\partial\mathbf{Z}}{\partial p} \mathbf{Q}+\frac{\partial\mathbf{F}_{n}}{\partial p}+\frac{\partial\mathbf{F}_ {u}}{\partial p}\right) \tag{17}\]  
Equation (17) is a generalized form of first-order sensitivity. If \(p\) is selected as a parameter in the friction model, considering that both \(\mathbf{Z}\) and \(\mathbf{F}_{u}\) are linear parts independent of nonlinear forces, then the corresponding terms \(\frac{\partial\mathbf{Z}}{\partial p}\) and \(\frac{\partial\mathbf{F}_{u}}{\partial p}\) are equal to zero. Thus Equation (17) can be rewritten as:  
\[\frac{\partial\mathbf{Q}}{\partial p}=-\mathbf{J}^{-1}\left(\frac{\partial \mathbf{F}_{n}}{\partial p}\right) \tag{18}\]  
where \(\mathbf{J}=\mathbf{Z}+\frac{\partial\mathbf{F}_{n}}{\partial\mathbf{Q}}\) is the Jacobian matrix. Since the nonlinear force is usually explicitly derived in most cases, the term \(\frac{\partial\mathbf{F}_{n}}{\partial p}\) can be calculated easily with a Fourier expansion described in Section 2.1. The derivatives of harmonic coefficients of rubbing force with respect to harmonic coefficients of system DOFs can be formulated by:  
\[\frac{\partial\mathbf{F}_{n}}{\partial\mathbf{Q}}^{ij}=\frac{\partial\mathbf{F}_ {n}^{i}}{\partial\mathbf{Q}^{j}} \tag{19}\]  
where the superscripts \(i,j=1,\cdots N\) refer to the \(i\)-th harmonic component of \(\mathbf{F}_{n}\) and the \(j\)th component of \(\mathbf{Q}\), respectively. To derive the detailed analytical expressions in Equation (19), the analysis in time domain is conducted at first. By differentiating nonlinear forces with respect to the displacements in local coordinate frame, one can obtain that:  
\[\mathbf{D}=\left[ \begin{array}{cc}\frac{\partial f_{\tau}}{\partial \mathbf{Q}}&\frac{\partial f_{\tau}}{\partial\eta}\\ \frac{\partial f_{\tau}}{\partial\xi}&\frac{\partial f_{\tau}}{\partial\eta} \end{array} \right] \tag{20}\]  
so the expression for derivatives of rubbing force with respect to global displacements can be derived as:  
\[\mathbf{D}^{*}=\mathbf{T}\mathbf{D}\mathbf{T}^{-1} \tag{21}\]  
where \(\mathbf{T}\) is the transformation matrix between local frame and global frame. For fixed local frame, it's the identity matrix and for rotating local frame it has the following form:  
\[\mathbf{T}=\left[