## 2 Perturbation Analysis of the CMS
Equation (3) is rewritten as  
\[\left[1+\Delta f_{mn}+2(1+\Delta f_{cn})R^{2}-\lambda\right]U_{n}-(1+\Delta f _{cn})R^{2}U_{n-1}-(1+\Delta f_{cn})R^{2}U_{n+1}=0 \tag{4}\]  
Simplify Equation (4) as  
\[(\mathbf{A}-\lambda\mathbf{I})\{\mathbf{U}\}=\mathbf{0} \tag{5}\]  
where \(\lambda=\omega^{2}/\omega_{m}^{2}\), \(\mathbf{I}\) is the unit matrix, \(\{\mathbf{U}\}\) is the displacement vector, and the expression for matrix \(\mathbf{A}\) is given by  
\[\mathbf{A}=\begin{bmatrix}A_{11}&-(1+\Delta f_{c1})R^{2}&0&\ldots&-(1+\Delta f_{ c1})R^{2}\\ -(1+\Delta f_{c2})R^{2}&A_{22}&-(1+\Delta f_{c2})R^{2}&0&\ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ \ldots&\ldots&\ldots&\ldots&\ldots\\ -When the structure is mistuned, the matrix \(\mathbf{A}\) is decomposed into a base system matrix \(\mathbf{A}_{0}\) and a perturbation matrix \(\Delta\mathbf{A}_{1}\):  
\[\mathbf{A}=\mathbf{A}_{0}+\Delta\mathbf{A}_{1} \tag{7}\]  
For a weakly coupled structure (where the coupling strength is less than the mistuning strength), the components associated with the coupling strength are considered as small perturbations. Consequently, each matrix in Equation (7) can be individually expressed as follows:  
\[\mathbf{A}_{0}= \begin{bmatrix}1+\Delta f_{m1}&0&\dots&\dots&0\\ 0&1+\Delta f_{m2}&0&\dots&\dots\\ \dots&\dots&\dots&\dots&\dots\\ \dots&\dots&\dots&\dots&\dots\\ 0&\dots&\dots&0&1+\Delta f_{mN}\end{bmatrix}  \tag{8}\]  
\[\Delta\mathbf{A}_{1}= \begin{bmatrix}2(1+\Delta f_{c1})R^{2}&-(1+\Delta f_{c1})R^{2 }&0&\dots&-(1+\Delta f_{c1})R^{2}\\ -(1+\Delta f_{c2})R^{2}&2(1+\Delta f_{c2})R^{2}&-(1+\Delta f_{c2})R^{2}&0&\dots \\ \dots&\dots&\dots&\dots&\dots\\ -(1+\Delta f_{cN})R^{2}&0&\dots&-(1+\Delta f_{cN})R^{2}&2(1+\Delta f_{cN})R^{ 2}\end{bmatrix}  \tag{9}\]  
When the structure is tuned (\(\Delta f_{mn}=\Delta f_{cn}=0\)), the eigenvectors are expressed as follows [32]:  
\[ \begin{split}\{\mathbf{U}_{n}^{c}\}&=\{1,\ \cos\!\alpha_{n},\ \dots,\ \cos(N-1)\alpha_{n}\}^{t},\,n=1,\ 2,\ \dots,\ \frac{N}{2}+1\\ \{\mathbf{U}_{n}^{s}\}&=\{0,\ \sin\!\alpha_{n},\ \dots,\ \sin(N-1)\alpha_{n}\}^{t},\,n=2,\ 3,\ \dots,\ \frac{N}{2}\end{split}  \tag{10}\]  
where \(\alpha_{k}=2\pi(k-1)/N\). The eigenvalues are  
\[\lambda_{n}=1+2R^{2}\bigg{[}1-\cos\frac{2\pi(n-1)}{N}\bigg{]}\,\,n=1,\ 2,\ \dots,\ \frac{N}{2}+1 \tag{11}\]  
When \(N\) is an even number, except for the cases where \(n=1\) or \(n=N/2+1\), double eigenvalues occur; when \(N\) is an odd number, the range of \(n\) extends to \((N+1)/2\), and there is only one simple eigenvalue for \(n=1\). Representing the eigenvectors as unperturbed eigenvectors and multi-order perturbed eigenvectors,  
\[\{\mathbf{U}_{n}\}=\left\{\delta^{0}\mathbf{U}_{n}\right\}+\left\{\delta^{1}\mathbf{U}_{ n}\right\}+\left\{\delta^{2}\mathbf{U}_{n}\right\} \tag{12}\]  
Similarly, the expression for the eigenvalues can be expressed as  
\[\lambda_{n}=\delta^{0}\lambda_{n}+\delta^{1}\lambda_{n}+\delta^{2}\lambda_{n} \tag{13}\]  
When there is an absence of coupling between the resonant units, the unperturbed eigenvectors and eigenvalues in Equations (12) and (13) are as follows:  
\[\left\{\delta^{0}\mathbf{U}_{n}\right\}=\{0,\ \dots,\ 1_{n-th},\ \dots,\ 0\}^{t},\,n=1,\ 2,\ \dots,\ N \tag{14}\]  
\[\delta^{0}\lambda_{n}=1+\Delta f_{mn},\,n=1,\ 2,\ \dots,\ N \tag{15}\]The first-order perturbation eigenvectors and eigenvalues can be derived as follows [31,33]:  
\[\left\{ \begin{array}{l}\left\{\delta^{1}\mathbf{U}_{n}\right\}_{n-1}=\frac{-\left(1 +\Delta f_{cn}\right)\right)^{2}\mathrm{\mathstrut}^{2}}{\delta^{\partial_{ \mathbf{\lambda}_{n}}}-\delta^{\partial_{\mathbf{\lambda}_{n-1}}}},\,n=2,\,\ldots,\,N\\ \left\{\delta^{1}\mathbf{U}_{n}\right\}_{n+1}=\frac{-\left(1+\Delta f_{cn}\right) \right)^{2}\mathrm{\mathstrut}^{2}}{\delta^{\partial_{\mathbf{\lambda}_{n}}}- \delta^{\partial_{\mathbf{\lambda}_{n+1}}}},\,n=1,\,\ldots,\,N-1\\ \left\{\delta^{1}\mathbf{U}_{n}\right\}_{\mathrm{other}}=0\end{array} \right.