## 2 Dynamic Modeling
### Dynamic Equation  
Figure 1 shows a schematic view of the smart plate of this study. A homogeneous plate (dimensions: \(L_{p,x}\times L_{p,y}\times h_{p}\)) was considered. The PZT actuator and sensor have the dimensions of \(L_{a,x}\times L_{a,y}\times h_{a}\) and \(L_{s,x}\times L_{s,y}\times h_{s}\), respectively. The vibration of the plate was excited by both a sinusoidal external disturbance \(F(t)\) and a sinusoidal actuating voltage \(y(t)\). The displacement of the plate under pure bending is represented by \(\xi(x,y,t)\). For simplicity, the parameters of \(\xi(x,y,t)\) in the brackets will be omitted in the following derivation. The sensing voltage of PZT is denoted as \(e(t)\). The Cartesian coordinate system for the smart plate is shown in Figure 1, and its origin is taken at one vertex of the plate. When the plate vibration is subjected to both disturbance \(F(t)\) and actuating voltage \(y(t)\), the dynamic equation of the plate [18; 19] is:  
\[D_{p}\Big{[}\frac{\partial^{4}\xi}{\partial x^{4}}+\frac{\partial^{4}\xi}{ \partial y^{4}}+2\frac{\partial^{4}\xi}{\partial x^{2}\partial y^{2}}\Big{]}+ \rho_{p}h_{p}\ddot{\xi}=F(t)\delta(x-x_{F})\delta(y-y_{F})-\chi_{a}y(t)\Big{(} \frac{\partial^{2}R_{a}}{\partial x^{2}}+\frac{\partial^{2}R_{a}}{\partial y ^{2}}\Big{)} \tag{1}\]  
where \(D_{p}=E_{p}I_{p}\big{(}1{+}\eta_{p}\big{)}\) is the complex flexural rigidity of the plate; \(E_{p}\), \(I_{p}\), \(\rho_{p}\), \(h_{p}\), and \(\eta_{p}\) represent the Young's modulus, the inertia moment, the density, the thickness, and the loss factor of the plate, respectively; \(\chi_{a}\) is a constant determined by the converse piezoelectric effect of PZT [20]; \((x_{F},y_{F})\) is the location of the external disturbance \(F(t)\); and \(R_{a}\) is the generalized location function of PZT. In Equation (1), the additional mass and stiffness of PZT are neglected, since the dimensions of the piezoelectric patches are small compared with those of the plate [21].  
Utilizing the separation variable method, the transverse displacement of the plate is denoted as:  
\[\xi(x,y,t)=\mathbf{\Phi T} \tag{2}\]  
where \(\mathbf{\Phi}=[\varphi_{11}(x,y),\varphi_{12}(x,y),\cdots,\varphi_{MN}(x,y)]\); \(\mathbf{T}=[T_{11}(t),T_{12}(t),\cdots,T_{MN}(t)]^{\mathrm{T}}\); \(\varphi_{mn}(x,y)\) is the mode shape function satisfying the geometric boundary conditions; \(T_{mn}(t)\) is the modal response of the first \((m,n)\) mode; \(m\) and \(n\) are the number of half-waves in the longitudinal (\(x\)) and lateral (\(y\)) directions; and \(M\) and \(N\) are the total number of mode shapes in the \(x\) and \(y\) directions, respectively.  
Based on the piezoelectric effect, the sensing signal of PZT is generated by the plate vibration. As shown in Figure 2, the amplified sensing signal, \(e(t)\), is denoted as [22]:  
\[e(t)=\frac{\chi_{s}}{H_{c}}\mathbf{\Phi}_{s}^{\prime\mathrm{T}}\mathbf{T} \tag{3}\]  
where \(\chi_{s}\) is a constant determined by the piezoelectric effect of PZT; \(H_{c}\) represents the transfer function of the charge amplifier circuit; \(\mathbf{\Phi}_{s}^{\prime\prime}=L_{s,x}L_{s,y}\chi_{s}[\phi_{11}(x_{s},y_{s}),\phi_{12}(x_{s},y_{s}),\cdots,\)\(\phi_{MN}(x_{s},y_{s})]\); and \((x_{s},y_{s})\) is the center coordinate of PZT. Meanwhile, \(\phi_{mn}(x_{i},y_{i})\) represents the modal strain of the plate, which is given by:  
\[\phi_{mn}(x_{i},y_{i})=\left.\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{ \partial^{2}}{\partial y^{2}}\right)\varphi_{mn}\right|_{x=x_{i},y=y_{i}} \tag{4}\]  
As shown in Equation (3), the modal strain has a direct influence on the sensing voltage of PZT.