# Vibro-Acoustic Performance of a Sandwich Plate with Periodically Inserted Resonators
## 2 Formulations
### Vibro-Acoustic Formulation of a Finite Periodic Structure  
An LR sandwich plate with a simply-supported boundary condition is placed in an infinite rigid baffle, which is shown in Figure 2. A harmonic point force is applied on the plate. As the acoustic medium chosen in this paper is air (a light fluid), the effects of fluid load by the acoustic medium on the structure are neglected. For an LR sandwich plate with a simply-supported boundary condition, the displacements can be written as  
\[\left\{ \begin{array}{l}u(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{ \infty}\overline{U}_{mn}\cos(k_{m}x)\sin(k_{n}y)\\ v(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\overline{V}_{mn} \sin(k_{m}x)\cos(k_{n}y)\\ w(x,y)=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}\overline{W}_{mn} \sin(k_{m}x)\sin(k_{n}y)\end{array} \right., \tag{17}\]  
where \(k_{m}=m\pi/a\) and \(k_{n}=n\pi/b\). Substituting Equation (17) into Equations (5) and (6), and considering the orthogonality of trigonometric functions yield  
\[\left\{ \begin{array}{l}\left(\overline{K}_{11}\overline{U}_{mn}+\overline{K }_{12}\overline{V}_{mn}+\overline{K}_{13}\overline{W}_{mn}\right)-\omega^{2} \big{(}\overline{M}_{11}\overline{U}_{mn}+\overline{M}_{12}\overline{V}_{mn} +\overline{M}_{13}\overline{W}_{mn}\big{)}=0\\ \left(\overline{K}_{21}\overline{U}_{mn}+\overline{K}_{22}\overline{V}_{mn} +\overline{K}_{23}\overline{W}_{mn}\big{)}-\omega^{2}\big{(}\overline{M}_{21} \overline{U}_{mn}+\overline{M}_{22}\overline{V}_{mn}+\overline{M}_{23}\overline {W}_{mn}\big{)}=0\end{array} \right.\\ \left\{ \begin{array}{l}\left(\begin{array}{l}\overline{K}_{31}\overline{U}_{mn }+\overline{K}_{32}\overline{V}_{mn}+\overline{K}_{33}\overline{W}_{mn}\\ -\frac{4k_{n}}{ab}\sum\limits_{p=1}^{M}\sum\limits_{q=1}^{S}\sum\limits_{s=1} ^{T}\sum\limits_{t=1}^{T}\chi_{mn}\chi_{pq}\overline{W}_{pq}\\ -\frac{4k_{n}}{ab}\sum\limits_{s=1}^{S}\sum\limits_{t=1}^{T}\chi_{mn}w_{Rst} \end{array}\right)-\omega^{2}\big{(}\overline{M}_{31}\overline{U}_{mn}+ \overline{M}_{32}\overline{V}_{mn}+\overline{M}_{33}\overline{W}_{mn}\big{)}=F _{mn}\\ \left(-k_{R}\sum\limits_{p=1}^{M}\sum\limits_{q=1}^{N}\chi_{pq}\overline{W}_{ pq}+k_{R}w_{Rst}\right)-\omega^{2}(m_{R}w_{Rst})=0\end{array} \right. \tag{18}\]  
where \(\chi_{mn}=\sin(k_{m}x_{s})\sin(k_{n}y_{t})\), \(\chi_{pq}=\sin\!\left(k_{p}x_{s}\right)\sin\!\left(k_{q}y_{t}\right)\), and \(w_{Rst}\) is the displacement of the mass block of the \(st^{\text{th}}\) resonator. For a point force excitation applied at position (\(x_{0}\), \(y_{0}\)), the force term is written as \(F_{mn}=\frac{4F_{0}}{ab}\sin(k_{m}x_{0})\sin(k_{n}y_{0})\). The expressions of \(\overline{K}_{ij}\) and \(\overline{M}_{ij}\) (\(i\), \(j=1\), \(2\), \(3\)) are defined in  
Figure 2: Schematic of a finite locally resonant sandwich plate (LR-SP) baffled in an infinite rigid baffle. Equations (A49)-(A66) in Appendix A. After further manipulations, Equation (18) can be written in matrix form as  
\[\left[\overline{\mathbf{K}}-\omega^{2}\overline{\mathbf{M}}\right]\overline{ \mathbf{\Phi}}=\mathbf{F}, \tag{19}\]  
where \(\overline{\mathbf{\Phi}}=\left[ \begin{array}{cccc}\overline{\mathbf{U}}& \overline{\mathbf{V}}&\overline{\mathbf{W}}&\overline{\mathbf{W}}_{\mathbf{R} }\end{array} \right]^{\mathrm{T}}\), \(\overline{\mathbf{U}}=\left[\overline{U}_{mn}\right]_{\mathrm{I\times(M\times N )}}\), \(\overline{\mathbf{V}}=\left[\overline{V}_{mm}\right]_{\mathrm{I\times(M\times N )}}\), \(\overline{\mathbf{W}}=\left[\overline{W}_{nm}\right]_{\mathrm{I\times(M\times N )}}\), \(\overline{\mathbf{W}}=\left[\overline{W}_{nm}\right]_{\mathrm{I\times(M\times N )}}\), \(\overline{\mathbf{W}}_{\mathbf{R}}=\left[\overline{W}_{Rst}\right]_{\mathrm{I \times(S\times T)}}\), \(\mathbf{F}=\left[ \begin{array}{cccc}0&0&\mathbf{F}_{w}&0\end{array} \right]\), and \(\mathbf{F}_{w}=\left[F_{nm}\right]_{\mathrm{I\times(M\times N)}}\). The infinite series in Equation (17) are truncated to \(1\leq m\leq M\) and \(1\leq n\leq N\).