## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures  
The research primarily employs modal analysis in the design and optimization of mechanical locally resonant metastructures. This analysis is crucial for identifying key vibration characteristics, such as natural frequencies, mode shapes, and modal damping ratios, under specific conditions. These insights enable the engineering of metastructures with tailored mechanical wave propagation behaviors. The study employs a distributed parameter model approach, utilizing partial differential equations (PDEs) to capture the system dynamics more precisely than lumped parameter models. This methodology is particularly applicable to systems where spatial variations are non-negligible, affecting phenomena such as wave propagation, heat transfer, and fluid dynamics. Analytical models are derived using modal analysis through the frequency determinant method, providing a solid theoretical foundation for understanding the intricate behavior of internally coupled resonators within metastructures. The standard distributed model of the metastructure under investigation, subject to base excitation and external forces, is illustrated in Figure 1. Employing Newtonian mechanics and drawing from classical vibration textbooks, the behavior of the metastructure is captured by the following partial differential equation, as detailed in Equation (1). \[\mathcal{L}w(x,t)+C\frac{\partial w(x,t)}{\partial t}+\mathcal{M}\frac{ \partial^{2}w(x,t)}{\partial t^{2}}-\sum_{r=1}^{N_{r}}\bigg{(}k_{r}z_{r}(t)+c _{r}\frac{\partial z_{r}(t)}{\partial t}\bigg{)}\delta(x-x_{r})=\mathcal{F}_{ b_{m}}(x,t) \tag{1}\]  
which includes structural flexibility \(\mathcal{L}\), damping \(\mathcal{C}\), and inertia \(\mathcal{M}\). The interaction with the resonators is represented by the summation term, encompassing the stiffness \(k_{r}\), damping \(c_{r}\), and location \(x_{r}\) of each resonator. The dynamic of the system is a linear homogeneous differential operator, exhibiting orders of \(2p\) and \(2q\), respectively, with \(q\leq p\). The spatial coordinate \(x\) extends over domain \(D\). The function \(w(x,t)\) captures the system's relative transverse vibration compared to the base motion, essentially reflecting the displacement at specific points relative to the base's harmonic movement. On the other hand, \(z_{r}(t)\) denotes the resonator's relative vibration in absolute coordinates, providing insight into its displacement to the overall structure's vibration. The \(\delta(x-x_{r})\) is the Kronecker delta function to pinpoint the resonators' locations on the beam, with \(x_{r}\) specifying the position of the \(r\)-th resonator. Moreover, \(\mathcal{F}\) symbolizes the external force, distributed across \(D\), and incorporates the impact of the base excitation on the beam. Similarly, the governing equation for the resonators, derived from [PERSON]'s second law of motion, is expressed as follows:  
\[m_{r}\frac{\partial^{2}z_{r}(t)}{\partial t^{2}}+c_{r}\frac{\partial z_{r}(t)}{ \partial t}+k_{r}z_{r}(t)+m_{r}\frac{\partial^{2}w(x_{r},t)}{\partial t^{2}}= \mathcal{F}_{b_{r}}(t) \tag{2}\]  
The boundary conditions for the system, as outlined in Equation (1), are defined by Equation (3), where each \(\mathcal{B}_{i}\) is a linear homogeneous differential operator of order no greater than \(2p-1\). \[\mathcal{B}_{i}[w(x,t)]=0,\quad i=1,2,\ldots,p \tag{3}\]  
Proportional damping, a method often used in real-world structures for estimating natural frequencies and mode shapes, relates the damping matrix to the mass and stiffness matrices. This concept allows \(\mathcal{C}\) to be expressed as a combination of mass and stiffness operators \(\mathcal{L}\) and \(\mathcal{M}\), as shown in Equation (4), with \(c_{1}\) and \(c_{2}\) being non-negative constants, determined based on the physical properties of the system. However, engineers usually use experimental modal analysis or fit data from vibration tests to find them.