## 2 Experimental Method
### Theoretical Background
 \tag{4}\]  
where \(i\) and \(j\) denote the indexes of the discrete grid points, corresponding to node \(i\) in the \(x\) direction and node \(j\) in the \(y\) direction, respectively; \(\varphi_{mn,\ ij}\) denotes the value of the modal displacement at the grid point \((i,j)\) in the mth and nth modal shapes; \(\varphi_{mn,i+1,j}\), \(\varphi_{mn,i-1,j}\), \(\varphi_{mn,\ i,j+1}\)\(\varphi_{mn,\ i,j-1}\), \(\varphi_{mn,i+1,j+1}\), \(\varphi_{mn,i+1,j-1}\), \(\varphi_{mn,i-1,j+1}\), and \(\varphi_{mn,i-1,j-1}\) denote the modal displacements at grid points \((i+1,j)\), \((i-1,j)\), \((i,j+1)\), \((i,j-1)\), \((i+1,j+1)\), \((i+1,j-1)\), \((i-1,j+1)\), and \((l-1,j-1)\), respectively; \(\Delta x\) represents the grid spacing in the \(x\) direction; \(\Delta y\) represents the grid spacing in the \(y\) direction. The undamaged modal displacement is defined as \(\varphi_{mn,\ i,\ j}^{b}\), with the damaged modal displacement as \(\varphi_{mn,\ i,j}^{a}\). The modal displacement difference is subsequently calculated using the following expression:  
\[\Delta\varphi_{mn}(x,y)=\varphi_{mn,i,j}^{a}-\varphi_{mn,i,j}^{b} \tag{5}\]  
Therefore, the corresponding CMD is defined as follows:  
\[ \begin{split}\Delta\kappa_{mn,\ xx,\ ij}&=-\frac{ \Delta\varphi_{mn,i+1,j}-2\Delta\varphi_{mn,\ i,j}+\Delta\varphi_{mn,i-1,j}}{ \Delta x^{2}}\\ \Delta\kappa_{mn,\ yy,\ ij}&=-\frac{\Delta\varphi_ {mn,\ i,j+1}-2\Delta\varphi_{mn,\ i,j}+\Delta\varphi_{mn,\ i,j-1}}{\Delta y ^{2}}\\ \Delta\kappa_{mn,xy,\ ij}&=-\frac{\Delta\varphi_ {mn,i+1,j+1}-\Delta\varphi_{mn,i+1,j-1}-\Delta\varphi_{mn,i-1,j+1}+\Delta \varphi_{mn,i-1,j-1}}{4\Delta x\Delta y}\end{split}  \tag{6}\]  
where \(\Delta\kappa_{mn,\ xx,\ ij}\) denotes the CMD in the \(x\) direction; \(\Delta\kappa_{mn,\ yy,\ i,j}\) denotes the CMD in the \(y\) direction; \(\Delta\kappa_{mn,xy,\ i,j}\) denotes the CMD in torsion. In thin-plate structures, reduced stiffness directly impacts the curvature mode's dynamic behavior and mechanical response. Stiffness serves as a critical parameter determining a thin-plate structure's resistance to bending and torsional deformations [34]. When the stiffness decreases, the deformation characteristics of the plate change, influencing both the distribution and magnitude of the curvature mode. The bending stiffness of a thin-plate is defined as follows:  
\[D=\frac{Eth^{3}}{12(1-\
u^{2})} \tag{7}\]  
where \(E\) represents [PERSON]'s modulus, \(h\) represents the thickness of the sheet, and \(\
u\) represents [PERSON]'s ratio. The stiffness directly impacts the internal forces (bending moments and torques) generated by the curvature mode. The following are the formulas for bending moment and torque:  
\[ \begin{split} M_{mn,x}&=-D\big{(}\kappa_{mn,xx}+\
u \kappa_{mn,yy}\big{)}\\ M_{mn,y}&=-D\big{(}\kappa_{mn,yy}+\
u\kappa_{mn,xx} \big{)}\\ M_{mn,xy}&=-D(1-\
u)\kappa_{mn,xy}\end{split}  \tag{8}\]  
where \(M_{mn,x}\) and \(M_{mn,y}\) represent bending moments in the \(x\) and \(y\) directions, respectively; \(M_{mn,xy}\) denotes torque; \(\upsilon\) denotes damping; \(M_{mn,x}\), \(M_{mn,y}\), and \(M_{mn,xy}\) are all proportional to the stiffness. When the local stiffness decreases, the curvature mode must be increased to maintain the equilibrium and ensure the bending moment and torque equilibrium in the vibration mode.