## 2 Model and formulations
### Forced Response of the Finite LR Plate with Elastic Boundary Conditions
\begin{array}{l}\mathbf{\varphi}_{a}\ =\ \left[\begin{array}{cccc}\sin(\pi x /2a)&\cos(\pi x/2a)&\sin(3\pi x/2a)&\cos(3\pi x/2a)\end{array}\right]^{T},\\ \mathbf{\varphi}_{b}\ =\ \left[\begin{array}{cccc}\sin(\pi y/2b)&\cos(\pi y/2b)&\sin(3\pi y /2b)&\cos(3\pi y/2b)\end{array}\right]^{T}\end{array} \right. \tag{6}\]  
The first- to fourth-order derivatives used in the boundary equations and governing differential equations can be determined as:  
\[\left\{\begin{array}{l}\xi_{a}{}^{(1)}(x)\ =\ \mathbf{A}_{a}\mathbf{B}_{a1}\mathbf{B}_{a0} \mathbf{\varphi}_{a},\ \xi_{a  
\(Q_{y}\), are defined in Equation (A4) in Appendix A. Substituting Equation (4) into Equation (11) and applying the orthogonality of trigonometric functions give:  
\[\left\{ \begin{array}{l}\sum\limits_{l=1}^{4}\sum\limits_{m=0}^{M}g_{i,1}c_{m}^ {l}+\sum\limits_{l=1}^{4}g_{i,2}d_{n}^{l}\ =\ \sum\limits_{m=0}^{M}g_{i,3}A_{mn}\ (n\ =\ 0,1,\cdots,N,\ i\ =\ 1,2,3,4),\\ \sum\limits_{l=1}^{4}g_{i,1}c_{m}^{l}+\sum\limits_{l=1}^{4}\sum\limits_{n=0}^{ N}g_{i,2}d_{n}^{l}\ =\ \sum\limits_{n=0}^{N}g_{i,3}A_{mn}\ (m\ =\ 0,1,\cdots,M,\ i\ =\ 5,6,7,8)\end{array} \right., \tag{12}\]  
where \(g_{i,1}\), \(g_{i,2}\), and \(g_{i,3}\) are given by Equation (A5) in Appendix A. Equation (12) represents \(4(M+1)+4(N+1)\) boundary equations, and they can be rewritten in a matrix form as:  
\[\mathbf{Hp}\ =\ \mathbf{Qa}, \tag{13}\]  
where \(\mathbf{H}\) and \(\mathbf{Q}\) are coefficient matrixes represented with the elements \(g_{i,1}\), \(g_{i,2}\), and \(g_{i,3}\). The terms \(\mathbf{p}\) and \(\mathbf{a}\) are defined as:  
\[\mathbf{p}\ =\ \Big{[}c_{0^{\prime}}^{1}\qquad c_{1^{\prime}}^{1},\ \cdots,\ c_{M^{\prime}}^{1},\ c_{0^{\prime}}^{2},\ c_{1 ^{\prime}}^{2},\ \cdots,\ c_{M^{\prime}}^{2},c_{0^{\prime}}^{3},\ c_{1^{ \prime}}^{3},\ \cdots,\ c_{M^{\prime}}^{3},\ c_{0^{\prime}}^{4},\ c_{1^{ \prime}}^{4},\ \cdots,\ c_{M^{\prime}}^{4}, \tag{14}\]  
\[\mathbf{a}\ =\ \big{[}A_{00},\ A_{01},\ \cdots,\ A_{0N},\ A_{10},\ A_{11},\ \cdots,A_{1N},\ \cdots,\ A_{mn},\ \cdots,\ A_{M0},\ A_{M1},\cdots,\ A_{MN} \big{]}^{T}, \tag{15}\]  
respectively. Thus, \(p\) can be expressed by \(a\), namely:  
\[\mathbf{p}\ =\ \mathbf{H}^{-1}\mathbf{Qa}. \tag{16}\]  
In Equation (4), a total of \((M+1)(N+1)+4(M+1)+4(N+1)\) unknown expansion coefficients can be obtained. Except for \(4(M+1)+4(N+1)\) boundary equations supported by Equation (13), \((M+1)(N+1)\) governing differential equations are provided in the following. Furthermore, express the force terms found in Equation (2) in Fourier series with a cosine form as:  
\[F^{\prime}\delta(x-x^{\prime})\delta(y-y^{\prime})\ =\ \sum\limits_{m=0}^{M}\sum \limits_{n=0}^{N}F_{mn}\cos(\lambda_{am}x)\cos(\lambda_{bn}y), \tag{17}\]  
where \(F_{mn}\) is the modal force. By using the orthogonality of trigonometric functions, multiplying \(\cos(\lambda_{am}x)\cos(\lambda_{bn}y)\) on both sides of Equation (17), and integrating in the area of \((x,y)\in([0,a]\times[0,b])\), \(F_{mn}\) can be given by:  
\[F_{mn}\ =\ \frac{F^{\prime}}{L_{am}L_{bn}}\cos(\lambda_{am}x^{\prime})\cos( \lambda_{bn}y^{\prime}), \tag{18}\]  
where \(L_{am}\ =\ \left\{ \begin{array}{cc}a,&m\ =\ 0\\ a/2,&m\
eq 0\end{array} \right.\) and \(L_{bn}\ =\ \left\{ \begin{array}{cc}b,&n\ =\ 0\\ b/2,&n\
eq 0\end{array} \right..\)  
Hence, the external force in Equation (2) can be expressed as:  
\[F_{0}\delta(x-x_{0})\delta(y-y_{0})\ =\ \sum\limits_{m=0}^{M}\sum\limits_{n=0}^{N}F_{mn 0}\cos(\lambda_{am}x)\cos(\lambda_{bn}y), \tag{19}\]  
where \(F_{mn0}\ =\ (F_{0}/L_{am}L_{bn})\cos(\lambda_{am}x_{0})\cos(\lambda_{bn}y_{0})\).