## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures
This approach, as referenced in [10], maintains consistent mode shapes and similar natural frequencies for both damped and undamped systems. \[\mathcal{C}=c_{1}\mathcal{L}+c_{2}\mathcal{M} \tag{4}\]  
The eigenfunctions \(\phi_{m}(x)\) of the system are derived by solving the eigenvalue problem of the undamped version of Equation (1), presented in Equation (5). \[\mathcal{L}[\phi_{m}(x)]=\lambda_{m}\mathcal{M}[\phi_{m}(x)],\quad m=1,2, \ldots,N_{m} \tag{5}\]  
The symbol \(\lambda_{m}\) represents an eigenvalue associated with the \(m\)th eigenfunction \(\phi_{m}(x)\) of the system. For structures like beams equipped with resonators, the system is defined by coupled differential equations for each resonator and the structure itself. These equations account for the mutual influence of each component on the system's dynamics. The mode shapes of the base structure alone are not the exact mode shapes of the entire metastructure, but using them simplifies the analysis significantly. The solution to Equation (5) is provided in Appendix A. In the case of an Euler beam spanning domain \(D=[0,L]\), assumed to be linearly elastic and homogeneous, the operators \(\mathcal{L}\), \(\mathcal{M}\), \(\mathcal{C}\), \(\mathcal{B}_{1}\), and \(\mathcal{B}_{2}\) are defined in terms of the beam's physical properties: flexural rigidity (\(EI\)), density (\(\rho\)), and cross-sectional area (\(A\)). \[ \begin{split}\mathcal{L}&=EI\frac{\partial^{4}}{ \partial x^{4}},\quad\mathcal{M}=\rho A,\quad\mathcal{C}=c,\\ B_{1}&=1,\quad B_{2}=EI\frac{\partial^{2}}{ \partial x^{2}}\end{split}  \tag{6}\]  
In advancing the understanding of modal expansion in the system, the orthogonality of eigenfunctions is critical for solving Equation (1). The self-adjoint (Hermitian) nature of the eigenvalue problem ensures this orthogonality. For any two eigenfunctions \(\phi_{m}(x)\)  
Figure 1: Example of standard locally resonant metastructures, where \(m\) represents the mass of the resonators, \(c\) is the damping, and \(k\) is the stiffness of the resonators. and \(\phi_{n}(x)\), the problem is self-adjoint if they satisfy the conditions given in Equation (7), as highlighted by [11]. \[ \begin{split}\int_{D}\phi_{m}(x)\mathcal{L}[\phi_{n}(x)]dx& =\int_{D}\phi_{n}(x)\mathcal{L}[\phi_{m}(x)]dx\\ \int_{D}\phi_{m}(x)\mathcal{M}[\phi_{n}(x)]dx&=\int _{D}\phi_{n}(x)\mathcal{M}[\phi_{m}(x)]dx.\end{split}  \tag{7}\]  
When considering unique eigenvalues \(\omega_{m}^{2}\) and \(\omega_{n}^{2}\) with their respective eigenfunctions \(\phi_{m}(x)\) and \(\phi_{n}(x)\), these functions are normalized with respect to \(\mathcal{M}\). This normalization leads to the generalized orthogonality condition outlined in Equations (8) and (9), with \(\delta_{mn}\) being the Kronecker delta function. \[\int_{D}\phi_{m}(x)\mathcal{M}[\phi_{n}(x)]dx=\delta_{mn} \tag{8}\]  
and  
\[\int_{D}\phi_{m}(x)\mathcal{L}[\phi_{n}(x)]dx=\delta_{mn}\omega_{m}^{2} \tag{9}\]  
Assuming proportional damping, the structural damping characteristics are captured by Equation (10). Here, \(\zeta_{m}\) denotes the damping ratio of the \(m\)-th mode, which is precisely defined in Equation (11) utilizing the constants \(c_{1}\) and \(c_{2}\). Equations (8)-(10) are integral to constructing a set of orthonormal eigenfunctions, which together form a complete basis for the solution space pertinent to the eigenvalue problem. \[\int_{D}\phi_{m}(x)\mathcal{C}[w_{m}(x)]dx=c_{1}\delta_{mn}\omega_{m}^{2}+c_{2 }\delta_{mn}=2\delta_{mn}\zeta_{m}\omega_{m} \tag{10}\]  
with  
\[\zeta_{m}=\frac{1}{2\omega_{m}}\Big{(}c_{1}\omega_{m}^{2}+c_{2}\Big{)} \tag{11}\]  
Modal decomposition is a method used to describe the structure's vibration across a domain \(D\) by representing it as a sum of modal shapes in one direction. This method assumes that the behavior of the structure can be accurately captured using a finite set of modes. For instance, the Euler-Bernoulli beam theory, commonly used in these analyses, may not provide sufficient accuracy in high-frequency situations.