# Achieving Enhanced Sound Insulation through Micromembranes-Type Acoustic Metamaterials
## 2 Basic Calculation of a Square Membrane and Its Cross-Frame
The frame holds the whole membrane and consists of \((n+2)\times(n+2)\) horizontal and vertical beams, where \(n\) is assigned as shown in Figure 2. To calculate the vibration of this arrangement of beams, a load distribution method is introduced. The idea is to separate the force into horizontal (\(x\)-direction) and vertical (\(y\)-direction) components. To calculate how the force is shared in different directions, we suppose that the vertical force is unknown, and then solve the problem by obtaining the balancing force required to produce equal displacement in each direction. Basically, this method, treating the supporting frame of A\(\upmu\)Ms as a grille, is a common approach in the force analysis of a grillage beam. The uniform load on the grille's beams is equivalent to the sum of the loads at the cross-points, while the loads at the cross-points can be further decomposed into its \(x\)- and \(y\)-direction components. Interested readers can find details of this method in Ref. [25]. Here, the horizontal forces are designated \(F_{11}\), \(F_{12}\), \(\ldots\), \(F_{mn}\), as shown in Figure 2. For the point \((\psi,\xi)\) in Figure 2, we obtain [25]:  
\[[U_{1}\ldots U_{\xi}\ldots U_{n}]\left\{ \begin{array}{l}F_{\uptau 1}\\ \ldots\\ F_{\uptau\xi}\\ \ldots\\ F_{\uptau n}\end{array} \right\}+\left[V_{1}\ldots V_{\psi}\ldots V_{n}\right] \left\{ \begin{array}{l}F_{1\xi}\\ \ldots\\ F_{\uptau\xi}\\ \ldots\\ F_{n\xi}\end{array} \right\}=\left[V_{1}\ldots V_{\psi}\ldots V_{n}\right] \left\{ \begin{array}{l}F_{1\xi}^{\prime}\\ \ldots\\ F_{\
u\xi}^{\prime}\\ \ldots\\ F_{n\xi}^{\prime}\end{array} \right\}\, \tag{4}\]  
where \([U_{i}]\) and \([V_{i}]\) are the matrices describing the displacement of a horizontal and a vertical beam under unit load, respectively; \([F_{\uptau i}]\) and \(\left[F_{i\xi}^{\prime}\right]\) are the vectors describing load distribution in the horizontal and vertical directions, respectively; and \(\left[F_{i\xi}^{\prime}\right]\) is a vector representing the force acting on each cross-point. Notice that this adopted method is valid below the first resonant frequency of the supporting frame. Since our investigated A\(\upmu\)Ms for noise insulation effectively work in the low-frequency range, it lies in the valid range of the method presented here. In this paper, all the forces acting on a point are the same and can be written as \(F_{i\xi}^{\prime}=\left(a+r\right)^{2}\cdot P\). Then, by solving the set of equations, the value of the unit force acting horizontally can be obtained. The displacement of every cross-point caused by a horizontal force can be obtained by the following equations  
\[\left\{ \begin{array}{l}L_{c}\leq L_{f},\quad\eta_{j}=\frac{F_{\uptau j}L^{ 3}}{6E_{s}I}\Big{[}3L_{f}-\Big{(}1+2L_{f}\Big{)}L_{\text{C}}\Big{]}\Big{(}1- L_{f}\Big{)}^{2}L_{\text{C}}^{2}\\ L_{c}>L_{f},\quad\eta_{j}=\frac{F_{\uptau j}L^{3}}{6E_{s}I}\Big{[}\Big{(}3-2L_{ f}\Big{)}L_{\text{c}}-L_{f}\Big{]}(1-L_{c})^{2}L_{f}^{2}\end{array} \right.\, \tag{5}\]  
where the meanings of the variables are assigned as shown in Figure 3. In Equation (5), \(F_{\uptau j}\) is the force acting on a cross-point of the beam, \(L_{c}\) is the point to be calculated, \(L_{f}\) is the distance of the force from one side of the boundary, \(L\) is the length of the beam, \(E_{s}\) is the Young's modulus of the frame, and \(I\) is the inertia moment of the beam, for which \(I=\left(ab^{3}\right)/12\). For a certain cross-point \((\Psi,\xi)\), the displacement can be obtained by adding all the \(\eta_{j}\) calculated from the horizontal-acting forces on every cross-point. The entire assembly of A\(\mu\)Ms provides two paths for sound transmission: via the A\(\mu\)Ms and via the frame. Most of the works so far have considered the contribution from the A\(\mu\)Ms [14; 15; 16; 17].