## 2 Model and Formulations  
In this work, an LR plate with two-dimensional periodicity is considered, as shown in Figure 1a. The QZS resonators are periodically attached to a thin plate with spatial period \(a_{1}\) and \(a_{2}\) in the \(xy\)-plane. The resonator is composed of two masses, a linear spring \(k_{1}\), and a QZS system, as shown in Figure 1b. It should be noted that, in the static state, the weight of the mass \(m_{2}\) is balanced by the tension force of the spring \(k_{p}\). In addition, the mass \(m_{2}\) is located at the center of the mass \(m_{1}\), and the horizontal springs are in a compressed state with a compression of length \(l_{c}\). Additionally, the horizontal spring \(k_{n}\) is located within limiting sleeves, limiting it to only expand and contract horizontally. The horizontal spring is connected to the mass \(m_{2}\) through a rigid rod, which is hinged at both ends. The terms \(h\), \(E\), \(\rho\), and \(\
u\) denote the plate's thickness, Young's modulus, density, and Poisson's ratio, respectively. Firstly, the transverse vibration of a unit cell of an infinite periodic LR plate is considered. The governing equations of a unit cell can be written as  
\[\Big{(}D\
abla^{4}-\omega^{2}\rho h\Big{)}w_{0}(r)=\sum_{r}k_{1}[w_{1}(R_{\rm s })-w_{0}(R_{\rm s})]\delta(r-R_{\rm s}) \tag{1}\]  
\[\omega^{2}m_{1}w_{1}(R_{\rm s})=k_{1}[w_{1}(R_{\rm s})-w_{0}(R_{\rm s})]+k_{Q}[ w_{1}(R_{\rm s})-w_{2}(R_{\rm s})]=0 \tag{2}\]  
and  
\[\omega^{2}m_{2}w_{2}(R_{\rm s})=k_{Q}[w_{2}(R_{\rm s})-w_{1}(R_{\rm s})] \tag{3}\]  
where \(\
abla^{4}=\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{ \partial y^{2}}\right)^{2}\), and \(D=\frac{El\beta^{3}}{12(1-v^{2})}\) is the bending stiffness of the host plate. The term \(w_{0}(r)\) is the displacement of the host plate at coordinate \(r(x,y)\). In the expression \(R_{\rm s}=R+r_{\rm s}\), the term \(R\) is the position vector of the unit cell and \(r_{\rm s}\) is the vector from the center of the unit cell to the resonator. The term \(w_{0}(R_{\rm s})\) is the displacement of the connection point between the resonator and the plate and \(w_{i}(R_{\rm s})\) is the displacement of the mass \(m_{i}\) (\(i=1,2\)). The term \(\delta(r-R_{\rm s})\) is the two-dimensional Dirac delta function. The term \(k_{Q}\) is the equivalent stiffness of the QZS system, which can be derived as  
\[k_{Q}=k_{p}-2k_{n}\Bigg{[}\frac{l_{c}-d}{\left(d^{2}-\xi^{2}\right)^{3/2}}d^{ 2}+1\Bigg{]} \tag{4}\]  
where \(\xi\) is the relative displacement of mass \(m_{1}\) relative to mass \(m_{2}\). Based on the assumption of small displacements, Equation (4) can be approximated by  
\[k_{Q}=k_{p}-\frac{2l_{c}}{d}k_{n}=k_{p}(1-2\gamma\beta) \tag{5}\]  
where \(\gamma=l_{c}/d\) represents the initial compression ratio of the spring \(k_{n}\) and \(\beta=k_{n}/k_{p}\) is the stiffness ratio of the QZS system. Following the [PERSON]'s theorem [31], the displacement response of the plate can be written as  
\[w_{0}(r)=\sum_{G}W_{0}(G)\mathrm{e}^{-j(\kappa+G)r} \tag{6}\]  
where \(\kappa=(\kappa_{x},\kappa_{y})\) is the Bloch wave vector. The term \(G=mb_{1}+nb_{2}\) denotes the reciprocal-lattice vector, in which \(m\) and \(n\) are integers. The terms \(b_{1}\) and \(b_{2}\) are basis vectors of the reciprocal lattice. For the rectangular array considered in Figure 1, we have \(b_{1}=(2\pi/a_{1},0)\), \(b_{2}=(0,2\pi/a_{2})\). Setting \(m,n=(-N,\cdots,N)\), then the dimension of the reciprocal space is \(M\times M\) with \(M=2N+1\). Figure 1: The schematic view of (**a**) the LR plate and (**b**) the unit cell containing the QZS resonator.