# Dynamic Similitude Design Method of the Distorted Model on Variable Thickness Cantilever Plates
## 2 Distorted Scaling Law of Thin Cantilever Plates  
Considering a cantilever plate and the coordinate system \(oxyz\), \(a\) and \(b\) are the length and the width along \(x\) and \(y\) directions, as shown in Figure 1. \(u(x,y,t)\), \(v(x,y,t)\) and \(w(x,y,t)\) represent the displacement of \(x\), \(y\) and \(z\) directions, respectively. The Young's modulus, Poisson's ratio and the density of the plate's material are separately denoted by \(E\), \(\mu\) and \(\rho\).  
The governing equation of the thin plate is [28]  
\[D\
abla^{2}\
abla^{2}w=\rho h\frac{\partial^{2}w}{\partial t^{2}}, \tag{1}\]  
where \(\
abla^{2}\) is the Laplace operator, \(\
abla^{2}=\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y ^{2}}\).  
The cantilever boundary condition can be found in [23] for more details.  
Denote the deflection of the plate by using the equation  
\[w(x,y,t)=W(x,y)\sin(2\pi\omega t+\varphi), \tag{2}\]  
where \(\omega\) is the natural frequency of the plate.  
Substituting Equation (2) into Equation (1) yields:  
\[\frac{\partial^{4}W}{\partial x^{4}}+2\frac{\partial^{4}W}{\partial x^{2} \partial y^{2}}+\frac{\partial^{4}W}{\partial y^{4}}=\alpha^{4}W, \tag{3}\]  
where \(\alpha^{4}=\omega^{2}\frac{\rho h}{D}\).  
Considering that Equation (3) is satisfied by both the model and the prototype:  
\[\frac{\partial^{4}W_{\rm P}}{\partial x_{\rm P}^{4}}+2\frac{\partial^{4}W_{ \rm P}}{\partial x_{\rm P}^{2}\partial y_{\rm P}^{2}}+\frac{\partial^{4}W_{ \rm P}}{\partial y_{\rm P}^{4}}=\alpha_{\rm P}^{4}W_{\rm P}, \tag{4a}\] \[\frac{\partial^{4}W_{\rm m}}{\partial x_{\rm m}^{4}}+2\frac{\partial^{4}W_{\rm m }}{\partial x_{\rm m}^{2}\partial y_{\rm m}^{2}}+\frac{\partial^{4}W_{\rm m}} {\partial y_{\rm m}^{4}}=\alpha_{\rm m}^{4}W_{\rm m}, \tag{4b}\]  
where subscript \(p\) represents the prototype; subscript \(m\) represents the model, which can be rewritten as  
\[\frac{\lambda_{W}}{\lambda_{a}^{4}}\frac{\partial^{4}W_{\rm m}}{\partial x_{ \rm m}^{4}}+2\frac{\lambda_{W}}{\lambda_{a}^{2}\lambda_{b}^{2}}\frac{\partial ^{4}W_{\rm m}}{\partial x_{\rm m}^{2}\partial y_{\rm m}^{2}}+\frac{\lambda_{W }}{\lambda_{b}^{4}}\frac{\partial^{4}W_{\rm m}}{\partial y_{\rm m}^{4}}= \lambda_{\alpha}^{4}\lambda_{W}\alpha_{\rm m}^{4}W_{\rm m}, \tag{5}\]  
where \(\lambda_{j}=\lambda_{j,p}/\lambda_{j,{\rm m}}\) is used to represent the scaling laws, \(j\) represents the symbol of each physical quality, for example, \(j=a,b,E,W\), and so on.  
Figure 1: The cantilever thin plate.  
According to the similitude theory [22]:  
\[\frac{\lambda_{W}}{\lambda_{a}^{4}}=\frac{\lambda_{W}}{\lambda_{a}^{2}\lambda_{b}^ {2}}=\frac{\lambda_{W}}{\lambda_{b}^{4}}=\lambda_{\alpha}^{4}\lambda_{W} \tag{6}\]  
is obtained from Equation (5), and according to \(\alpha^{4}=\omega^{2}\left(\rho h/D\right)\) and \(D=h^{3}/12(1-\mu^{2})\), there is  
\[\lambda_{\alpha}^{4}=\lambda_{\omega}^{2}\frac{\lambda_{\rho}\lambda_{h}}{ \lambda_{D}}=\lambda_{\omega}^{2}\frac{\lambda_{\rho}}{\lambda_{h}^{2}\lambda_ {E}}. \tag{7}\]  
Let \(\lambda_{a}=\lambda_{b}=\lambda\), and  
\[\lambda_{\omega}=\sqrt{\frac{\lambda_{\rm E}}{\lambda_{\rho}}}\frac{\lambda_{ h}}{\lambda^{2}} \tag{8}\]  
can be derived by substituting Equation (7) into Equation (6), where (8) is the distorted scaling laws with respect to the material parameters and the thickness \(h\).  
By using the scaling law (8), the natural frequency of a cantilever thin plate can be predicted by using an MT plate, under the same boundary condition, made of different materials in arbitrary thickness. However, if the prototype is a VT plate, its dynamic properties cannot be predicted by using such a simple scaling law, and an effective approach of designing the scaled down model in predicting the prototype VT plate is needed.