# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 2 Formulations
### Band-Gap of, and Sound Radiation from, an Infinite Periodic Structure  
#### 2.2.1 Band-Gap  
The plane-wave-expansion (PWE) method is used to study the band-gap properties of the LR sandwich plate. For an infinite sandwich plate, there are infinite resonators located periodically in the \(x\) and \(y\) directions. Thus, the total force generated by the periodic resonators on the sandwich plate can be expressed as  
\[f_{Rd}=\sum\limits_{s=-\infty}^{\infty}\sum\limits_{t=-\infty}^{\infty}f_{Rst} \delta(x-x_{s})\delta(y-y_{t}), \tag{7}\]  
where (\(x_{s}=sa_{1}\), \(y_{t}=ta_{2}\)) is the position of the \(st^{\text{th}}\) resonator and \(f_{Rst}\) is the force applied on the sandwich plate by the \(st^{\text{th}}\) resonator. According to the PWE method, the displacements of the LR sandwich plate can be written as  
\[\left\{\begin{array}{l}u(x,y)=\sum\limits_{m=-\infty}^{\infty}\sum\limits_{ n=-\infty}^{\infty}U_{mn}e^{-j(G_{m}+\alpha)x}e^{-j(G_{n}+\beta)y}\\ v(x,y)=\sum\limits_{m=-\infty}^{\infty}\sum\limits_{n=-\infty}^{\infty}V_{mn }e^{-j(G_{m}+\alpha)x}e^{-j(G_{n}+\beta)y}\\ w(x,y)=\sum\limits_{m=-\infty}^{\infty}\sum\limits_{n=-\infty}^{\infty}W_{mn }e^{-j(G_{m}+\alpha)x}e^{-j(G_{n}+\beta)y}\end{array}\right., \tag{8}\]  
where \((\alpha,\beta)\) is a Bloch wave vector and \(\mathbf{G}_{mn}=(G_{m},G_{n})\) is the reciprocal-lattice vector of the \(mn^{\text{th}}\) plane wave, in which \(G_{m}=2m\pi/a_{1}\) and \(G_{n}=2n\pi/b_{1}\). Substitution of Equations (6)-(8) into Equation (5) yields  
\[\left\{\begin{array}{l}(K_{11}U_{mn}+K_{12}V_{mn}+K_{13}W_{mn})-\omega^{2}( M_{11}U_{mn}+M_{12}V_{mn}+M_{13}W_{mn})=0\\ (K_{21}U_{mn}+K_{22}V_{mn}+K_{23}W_{mn})-\omega^{2}(M_{21}U_{mn}+M_{22}V_{mn}+ M_{23}W_{mn})=0\\ \left(K_{31}U_{mn}+K_{32}V_{mn}+K_{33}W_{mm}+\frac{k_{B}}{8}\sum\limits_{p=-M} ^{M}\sum\limits_{q=-N}^{N}W_{pq}-\frac{k_{B}}{5}w_{R0}\right)\\ -\omega^{2}(M_{31}U_{mn}+M_{32}V_{mm}+M_{33}W_{mn})=0\\ \left(-k_{R}\sum\limits_{p=-M}^{M}\sum\limits_{q=-N}^{N}W_{pq}+k_{R}w_{R0} \right)-\omega^{2}(m_{R}w_{R0})=0\end{array}\right., \tag{9}\]  
where \(S=a_{1}b_{1}\) is the element area and the infinite series in Equations (7) and (8) are truncated to \(-M\leq m,s\leq M\) and \(-N\leq n,t\leq N\). The term \(w_{R0}\) is the displacement of the mass block of the resonator located at \((x=0,y=0)\). The expressions of \(K_{ij}\) and \(M_{ij}\) (\(i\), \(j=1\), \(2\), \(3\)) are defined in Equations (A10)-(A27) in Appendix A. After some manipulations, Equation (9) can be written in matrix form as  
\[\left[\mathbf{K}(\alpha,\beta)-\omega^{2}\mathbf{M}(\alpha,\beta)\right] \mathbf{\Phi}=0, \tag{10}\]  
where \(\mathbf{\Phi}=\left[\begin{array}{cccc}\mathbf{U}&\mathbf{V}&\mathbf{W}&w_{R 0}\end{array}\right]^{\text{T}}\), \(\mathbf{U}=\left[U_{mn}\right]_{1\times[(2M+1)\times(2N+1)]}\), \(\mathbf{V}=\left[V_{mn}\right]_{1\times[(2M+1)\times(2N+1)]}\), and \(\mathbf{W}=\left[W_{mn}\right]_{1\times[(2M+1)\times(2N+1)]}\). For each given Bloch wave vector \(\mathbf{k}=(\alpha,\beta)\), the characteristic frequencies can be determined by Equation (10). By sweeping \(\mathbf{k}\) in the directions of \(\Gamma\mathbf{X}\) (\(\alpha:0\rightarrow\pi/a_{1},\beta:0\)), XM (\(\alpha:\pi/a_{1},\ \beta:0\rightarrow\pi/a_{2}\)), and MI (\(\alpha:\pi/a_{1}\to 0,\ \beta:\pi/a_{2}\to 0\)) in the first Brillouin zone (see Figure 1b), the dispersion relationship between wave number and frequency can be obtained, from which the band-gap is then identified.