## 2 Perturbation Analysis of the CMS  
A spring-oscillators model with a cyclic distribution is established. The oscillators are connected to the center ring through main springs, and coupling springs are set up between adjacent oscillators, as illustrated in Figure 1. Assuming that the oscillator moves solely in one direction within the \(x\)-\(o\)-\(y\) plane, the model can be regarded as a single-degree-of-freedom model. When multiple degrees of freedom are considered, it can be further extended based on the single-degree-of-freedom model [31]. Figure 1: Spring-oscillators model with a cyclic distribution. Assuming that the direction of the main spring stiffness and damping is perpendicular to the spring tension direction, let \(k_{m}\) and \(c_{m}\) be the stiffness and damping coefficient of the main spring, respectively. The direction of the coupling spring stiffness and damping is along the spring tension direction, let \(k_{c}\) and \(c_{c}\) be the stiffness and damping coefficient of the coupling spring, respectively. Let \(N\) be the total number of the resonant units and \(n\) be the serial number of the resonant units, with \(n=1\), \(2\), \(3\), \(\ldots\), \(N\). Let the displacement of the \(n\)-th resonant unit be \(u_{n}\). For the periodic cyclic structure, there are \(u_{0}\equiv u_{N}\) and \(u_{N+1}\equiv u_{1}\). Define \(\omega_{mn}=\sqrt{k_{m}/m_{n}}\) and \(\omega_{cn}=\sqrt{k_{c}/m_{n}}\) as the main frequency and the coupled frequency of the \(n\)-th resonant unit when the structure is mistuned, respectively, where \(m_{n}\) is the mass of the \(n\)-th resonant unit. Define \(\omega_{m}=\sqrt{k_{m}/\overline{m}}\) and \(\omega_{c}=\sqrt{k_{c}/\overline{m}}\) as the standard main frequency and standard coupled frequency of the \(n\)-th resonant unit when the structure is tuned, respectively, where \(\overline{m}\) is the average mass of all resonant units. The main frequency mistuning can be defined as \(\Delta f_{mn}=\left(\omega_{mn}^{2}-\omega_{m}^{2}\right)/\omega_{m}^{2}\); the coupled frequency mistuning can be defined as \(\Delta f_{cn}=\left(\omega_{cn}^{2}-\omega_{c}^{2}\right)/\omega_{c}^{2}\); the coupling strength can be defined as \(R^{2}=\omega_{c}^{2}/\omega_{m}^{2}\). The frequency mistuning strength can be calculated as follows:  
\[\varepsilon=\sqrt{\frac{1}{N-1}\sum\limits_{n=1}^{N}\left|\Delta f_{n}-\Delta \overline{f}\right|^{2}} \tag{1}\]  
where \(\Delta\overline{f}=\sum\limits_{n=1}^{N}\Delta f_{n}/N\) is the average frequency mistuning. The model is partitioned into distinct regions according to different spring oscillators, and the \(n\)-th region is analyzed. The motion state and coupling relationship of the resonant unit are simplified, so the motion differential equation for the \(n\)-th resonant unit can be approximately formulated as [31]:  
\[m_{n}\ddot{u}_{n}+k_{m}u_{n}+k_{c}(2u_{n}-u_{n-1}-u_{n+1})+c_{m}\dot{u}_{n}+c_ {c}\big{(}2\dot{u}_{n}-\dot{u}_{n-1}-\dot{u}_{n+1}\big{)}=0 \tag{2}\]  
Assuming the oscillators undergo simple harmonic motion with an angular frequency \(\omega\), when the structure is undamped, Equation (2) can be rewritten as follows:  
\[-\omega^{2}U_{n}+\omega_{mn}^{2}U_{n}+2\omega_{cn}^{2}U_{n}-\omega_{cn}^{2}U_ {n-1}-\omega_{cn}^{2}U_{n+1}=0 \tag{3}\]  
where \(U_{n}\) is the complex amplitude of the \(n\)-th resonant unit.