## 2 Definition of Assumed Mode Functions  
In Figure 2, the natural frequencies and natural modes of the single plate and the connecting plate structures were confirmed through the finite element analysis (FEA) program. As a result, it can be confirmed that the first and second natural modes of the main structure among the connected plate structures are similar to the analysis results of the single plate, and the natural frequency is in the middle of the fixed-simple support and fixed-fixed support conditions of the single plate. Therefore, the mode function of the combined state of the two conditions was defined and the result was confirmed. First, as a method to effectively assume the plate structure, a method of assuming a linear combination of simple beam mode functions in the \(x\)-axis and \(y\)-axis directions is commonly used. In other words,  
\[w(x,y,t)=\sum_{m=1}^{p}\sum_{n=1}^{q}A_{mn}(t)\cdot X_{m}(x)\cdot Y_{n}(y) \tag{1}\]  
where \(X_{m}(x)\), \(Y_{n}(y)\) are the mode functions of the beam in the \(x\)-axis and \(y\)-axis directions, p and q are the terms of the mode function and \(A_{mn}(t)\) is the unknown coefficient for each term. Among the assumed mode functions of a beam, the polynomial of the first-order mode function that satisfies the fixed-fixed boundary condition (\(\Psi\)), the fixed-simple boundary condition (\(\Phi\)) and the simple-fixed boundary condition (\(\gamma\)) is Equations (2)-(4) and it can be obtained respectively. Figure 2: Finite element analysis (FEA) results of single and connected plates. \[\psi_{1}(\eta)=A_{1}(\eta+1)^{2}(\eta-1)^{2}\ -1\leq\eta\leq 1 \tag{2}\]  
\[\varphi_{1}(\eta)=B_{1}\big{(}\eta^{4}-\eta^{3}-3\eta^{2}+\eta+2\big{)} \tag{3}\]  
\[\gamma_{1}(\eta)=C_{1}\big{(}\eta^{4}+\eta^{3}-3\eta^{2}-\eta+2\big{)} \tag{4}\]  
The coefficients \(A_{1}\), \(B_{1}\) and \(C_{1}\) are implemented using the orthogonal formula of the beam function. \[\int_{-1}^{1}\psi_{i}\psi_{j}d\eta=\delta_{ij} \tag{5}\]  
\[\int_{-1}^{1}\varphi_{i}\varphi_{j}d\eta=\delta_{ij} \tag{6}\]  
\[\int_{-1}^{1}\gamma_{i}\gamma_{j}d\eta=\delta_{ij} \tag{7}\]  
where \(i,j\) is the vibration order and \(\delta_{ij}\) is Kronecker delta. \[A_{1}=\frac{\int_{-1}^{1}\psi_{1}^{2}d\eta}{\sqrt{\int_{-1}^{1}\left(\eta^{4} -2\eta^{2}+1\right)^{2}}d\eta} \tag{8}\]  
\[B_{1}=\frac{\int_{-1}^{1}\varphi_{1}^{2}d\eta}{\sqrt{\int_{-1}^{1}\left(\eta^ {4}-\eta^{3}-3\eta^{2}+\eta+2\right)^{2}}d\eta} \tag{9}\]  
Figure 2:0  
The waveform function of the second mode or more can be implemented from the following Equations (11)-(13). Figure 2:1  
Figure 2:2  
Figure 2:3  
The coefficients \(a_{ki}\), \(b_{ki}\), \(c_{ki}\) can be obtained from the orthogonal relation of the complementary function and the Equations (11)-(13) above. In Figure 3, the y-axis of the \(w_{1}(x,y)\) plate is constrained by the structure between the deck and the deck, so the mode function is assumed only in the fixed-fixed support condition, and in the case of the x-axis, the fixed-fixed (\(\psi\)) and fixed-simple (\(\phi\)) mode functions are expressed as a combination. The case of \(w_{2}(x,y)\) was also defined in a similar way to the above, but the mode function of the x-axis was summarized using a simple-fixed (\(\phi\)) support assume mode function. Figure 2:4Figure 2:5  
Figure 2:6  
Figure 2:7  
Figure 2:8  
Figure 2:9  
where \(p_{Ai}(t),q_{Ai}(t),p_{Bj}(t),p_{CK}(t),q_{\rm{CK}}(t),p_{DI}(t)\) are the general coordinate system in the assumed mode function of beam and \(f_{A11}(t),\cdots,f_{Amm}(t),g_{A11}(t),\cdots,g_{Amm}(t),f_{B11}(t),\cdots,f_ {Bsu}(t),g_{B11}(t),\cdots,g_{Bsu}(t)\) are the general coordinate system of the assumed mode function that makes up the wall structure.