## 2 Model and formulations  
### Forced Response of the Finite LR Plate with Elastic Boundary Conditions  
In this subsection, the displacement is derived by solving the forced response of the LR plate with elastic boundary conditions. Shown in Figure 1 is a finite rectangular LR plate with multiple spring-mass resonators periodically attached on the base plate, of which all four edges are applied with elastic boundary conditions. The base plate has dimensions of thickness \(h_{p}\), length \(a\), and width \(b\), and material parameters of Young's modulus \(E\), Possion's ratio \(\
u\), and mass density \(\rho_{p}\). A harmonic point force \(F_{0}e^{i\omega t}\) is applied at the position of \((x_{0},y_{0})\) on the base plate. The differential equation of motion for a forced vibration of the base plate is given by:  
\[D\
abla^{4}w(x,y)-\rho_{p}h_{p}\omega^{2}w(x,y)\ =\ F, \tag{1}\]  
where \(D\ =\ Eh_{p}^{3}/12\Big{(}1-\
u^{2}\Big{)}\) is the bending stiffness of the plate, \(\
abla^{4}\ =\ \partial^{4}/\partial x^{4}+2\partial^{4}/\partial x^{2} \partial y^{2}+\partial^{4}/\partial y^{4}\), and \(F\) includes both the external force and the forces from the resonators. In the case of the LR plate attached with two periodic resonators of \((m_{\text{R1}},\,k_{\text{R1}})\) and \((m_{\text{R2}},\,k_{\text{R2}})\) along the \(x-\)direction, \(F\) can be expressed as:  
\[F\ =\ F_{0}\delta(x-x_{0})\delta(y-y_{0})+\sum\limits_{s=1}^{S}\sum\limits_{t \,=\,1}^{T}F_{1 st}\delta(x-x_{1s})\delta(y-y_{t})+\sum\limits_{s=1}^{S}\sum \limits_{t\,=\,1}^{T}F_{2 st}\delta(x-x_{2s})\delta(y-y_{t}), \tag{2}\]  
where \(F_{0}\) is the amplitude of the external force and \(\delta(x-x_{0})\delta(y-y_{0})\) is the 2-D Dirac delta function. On the base plate \(2S\times T\) resonators are attached. The terms \(F_{1 st}\) and \(F_{2 st}\) are the amplitudes of forces from the vibrating resonators of \((m_{R1},\,k_{\text{R1}})\) and \((m_{\text{R2}},\,k_{\text{R2}})\), respectively. The terms \((x_{1s},y_{t})\) and \((x_{2s},y_{t})\) represent the coordinates of the individuals in both kinds, where \(s\ (s\in[0,S])\) is an odd number, \(t\ (t\in[0,T])\) is an integer, and \(x_{2s}\ =\ x_{1s}+a_{L}\). The transverse displacement of a thin rectangular plate can be expressed as [27; 28]:  
\[w(x,y)\ =\ \sum\limits_{m\ =\ 0\ \,n\ =\ 0}^{\infty}A_{mn}\cos(\lambda_{am}x) \cos(\lambda_{lm}y)+\sum\limits_{l\,=\,1}^{4}\bigg{(}\xi_{b}^{l}(y)\sum \limits_{m\ =\ 0}^{\infty}c_{m}^{l}\cos(\lambda_{am}x)+\xi_{a}^{l}(x)\sum \limits_{n\ =\ 0}^{\infty}d_{n}^{l}\cos(\lambda_{lm}y)\bigg{)}. \tag{3}\]  
Let the infinite series be truncated to \(m\ =\ M\) and \(n\ =\ N\) in the following calculation, and then Equation (3) can be written as:  
\[w(x,y)\ =\ \sum\limits_{m\ =\ 0}^{M}\sum\limits_{n\ =\ 0}^{N}A_{mn}\cos(\lambda_{ am}x)\cos(\lambda_{lm}y)+\sum\limits_{l\,=\,1}^{4}\bigg{(}\xi_{b}^{l}(y)\sum \limits_{m\ =\ 0}^{M}c_{m}^{l}\cos(\lambda_{am}x)+\xi_{a}^{l}(x)\sum \limits_{n\ =\ 0}^{N}d_{n}^{l}\cos(\lambda_{lm}y)\bigg{)}. \tag{4}\]  
Figure 1: The finite locally resonant plate with elastic boundary conditions. where \(\lambda_{am}\ =\ m\pi/a\) and \(\lambda_{bm}\ =\ n\pi/b\). The terms \(A_{mm}\), \(c_{m}^{l}\), and \(d_{n}^{l}\) are unknown expansion coefficients that need to be determined by both boundary equations and governing differential equations. The terms \(\xi_{a}^{l}(x)\) and \(\xi_{b}^{l}(y)\) are given in the form of trigonometric functions and are defined as:  
\[\left\{ \begin{array}{l}\xi_{a}(x)\ =\ \left[\begin{array}{cccc}\xi_{a}^{1}(x) &\xi_{a}^{2}(x)&\xi_{a}^{3}(x)&\xi_{a}^{4}(x)\end{array}\right]^{T}\ =\ \mathbf{A}_{a}\mathbf{\varphi}_{a},\\ \xi_{b}(y)\ =\ \left[\begin{array}{cccc}\xi_{b}^{1}(y)&\xi_{b}^{2}(y)&\xi_{b}^{3}(y)& \xi_{b}^{4}(y)\end{array}\right]^{T}\ =\ \mathbf{A}_{b}\mathbf{\varphi}_{b}\end{array} \right., \tag{5}\]  
where \(\mathbf{A}_{a}\) and \(\mathbf{A}_{b}\) are two arbitrary \(4\times 4\) matrixes of full rank, and:  
\[\left\{