## 2 Experimental Method  
### Theoretical Background  
The curvature mode represents the unique curvature distribution of a structure in its bending vibration state for each order of displacement mode. Curvature mode is closely related to the displacement mode and its corresponding curvature distribution. This investigation conducts a modal test of the HFGCWs by uniformly dividing the glass panel into several small units. The corresponding modal parameters, including natural frequency, modal displacement, and damping, are obtained by striking the nodes of these small units with a force hammer. Subsequently, modal identification is analyzed in depth using advanced algorithms. The HFGCW panel exhibits a significantly larger length and width compared to its thickness, classifying it as a typical thin-plate structure. Thus, the curvature modal theory is investigated using a thin-plate structural model. By adopting the thin-plate structure as the research subject, the modal displacement is derived based on [PERSON]'s thin-plate theory as follows:  
\[\varphi_{mn}(x,y)=A_{mn}sin\Big{(}\frac{m\pi x}{a}\Big{)}sin\Big{(}\frac{n\pi y }{b}\Big{)} \tag{1}\]  
Among them, \(A_{mn}\) represents the modal amplitude, \(sin\Big{(}\frac{m\pi x}{a}\Big{)}\) represents the mode function along the \(x\) direction, \(m\) is the modal orders along the \(x\) direction, and \(a\) is the width of the \(x\) direction; \(sin\Big{(}\frac{n\pi y}{b}\Big{)}\) represents the mode function of the \(y\) direction, where \(n\) is the mode orders of the \(x\) direction and \(b\) is the width of the \(y\) direction. The curvature modes are obtained by solving for the second-order partial derivatives through the modal displacements as follows:  
\[\kappa_{mn,xx}=-\frac{\partial^{2}\varphi_{mn}}{\partial x^{2}};\ \kappa_{mn,yy}=-\frac{\partial^{2}\varphi_{mn}}{\partial y^{2}};\ \kappa_{mn,xy}=-\frac{\partial^{2}\varphi_{mn}}{ \partial x\partial y} \tag{2}\]  
where \(\kappa_{mn,\ xx}\) denotes the \(x\) direction curvature mode; \(\kappa_{mn,\ yy}\) denotes the \(y\) direction curvature mode; \(\kappa_{mn,xy}\) denotes the torsional curvature mode. The negative sign ensures that the curvature is in the same direction as the sheet's bending or torsion direction. The curvature modes in the \(x\) direction, \(y\) direction, and torsional direction are mathematically formulated as follows:  
\[ \begin{array}{l}\kappa_{mn,xx}=-\frac{\partial^{2}\varphi_{mn}}{\partial x^{ 2}}=-\Big{(}-(\frac{m\pi}{a})^{2}\varphi_{mn}\Big{)}=(\frac{m\pi}{a})^{2} \varphi_{mn}(x,y)\\ \kappa_{mn,yy}=-\frac{\partial^{2}\varphi_{mn}}{\partial y^{2}}=-\Big{(}-( \frac{n\pi}{b})^{2}\varphi_{mn}\Big{)}=(\frac{n\pi}{b})^{2}\varphi_{mn}(x,y)\\ \kappa_{mn,xy}=-\frac{\partial^{2}\varphi_{mn}}{\partial x\partial y}=-\frac{m \pi}{a}\frac{m\pi}{b}\cos(\frac{n\pi y}{a})cos(\frac{n\pi y}{b})\end{array}  \tag{3}\]where \(\kappa_{mn,\ xx}\) and \(\kappa_{mn,\ yy}\) are proportional to the \(\varphi_{mn}(x,y)\), with scaling factors of \(\left(\frac{m\pi}{a}\right)^{2}\) and \(\left(\frac{m\pi}{b}\right)^{2}\), respectively, which reflect the bending characteristics along the \(x\) and \(y\) directions; \(\kappa_{mn,xy}\) exhibits a cosine distribution. Directly obtaining a curvature mode is often challenging. However, modal testing techniques are relatively well-developed. Thus, utilizing the characteristic that modal displacements are provided in a grid form within discrete systems, modal tests are conducted to acquire the modal displacements at the collected grid nodes. The curvature modes are then approximated using the central difference method:  
\[ \begin{split}\kappa_{mn,xx,j,j}&=-\frac{\varphi_{ mn,i+1,j}-2\varphi_{mn,i,j}+\varphi_{mn,i-1,j}}{\Delta x^{2}}\\ \kappa_{mn,yy,i,j}&=-\frac{\varphi_{mn,i,j+1}-2 \varphi_{mn,i,j}+\varphi_{mn,i,j-1}}{\Delta y^{2}}\\ \kappa_{mn,xy,i,j}&=-\frac{\varphi_{mn,j+1,j+1}- \varphi_{mn,i+1,j-1}-\varphi_{mn,i-1,j+1}+\varphi_{mn,j-1,j-1}}{4\Delta x \Delta y}\end{split}