# Dynamic Similitude Design Method of the Distorted Model on Variable Thickness Cantilever Plates
## 3 Similitude Design of the Variable Thickness (VT) Cantilever Plate
### Equivalence Design Based on Transfer Matrix
#### 3.2.1 Transfer Matrices of the Plate Structures  
For a thin walled plate structure, the transfer matrix, which is used to analytically calculate the dynamic properties, can be derived based on the Kirchhoff hypothesis of the thin walled plate element as below. According to the [PERSON] theory [28], the governing equations of the plate element are:  
\[\frac{\partial M_{x}}{\partial x}+\frac{\partial M_{yx}}{\partial y}-Q_{x}=0, \tag{11a}\] \[\frac{\partial M_{xy}}{\partial x}+\frac{\partial M_{y}}{\partial y}-Q_{y}=0,\] (11b) \[\frac{\partial Q_{x}}{\partial x}+\frac{\partial Q_{y}}{\partial y}=\rho h\frac {\partial^{2}w}{\partial t^{2}}, \tag{11c}\]  
where \(\rho h\left(\partial^{2}w/\partial t^{2}\right)\) is the inertia term; subscripts \(x\) and \(y\) represent the directions along \(x\) and \(y\) axis, respectively. \(M_{x}\), \(M_{y}\) and \(M_{xy}\), \(M_{yx}\) are denoted as bending and twisting moments as:  
\[ \begin{split}& M_{x}=-D\left(\frac{\partial^{2}w}{\partial x^{2}}+ \mu\frac{\partial^{2}w}{\partial y^{2}}\right)\\ & M_{y}=-D\left(\frac{\partial^{2}w}{\partial y^{2}}+\mu\frac{ \partial^{2}w}{\partial x^{2}}\right)\\ & M_{xy}=M_{yx}=-D\left(1-\mu\right)\frac{\partial^{2}w}{\partial x \partial y}\end{split} , \tag{12}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Order \(i_{v}\)** & **VT Plate** & **VT Plate** & **ST Plate** & **ST plate** & **Errors** \\
**Vibration Model** & \(w_{v}\)/Hz** & **Vibration Model** & \(w_{s}\)/Hz & \(\eta_{s}\)/\% \\ \hline
2F & & 1530.40 & & 1477.70 & 3.44 \\  & & \(m=1\), \(n=2\) & & & \\
2T & & 2233.90 & & 2189.40 & 1.99 \\  & & \(m=2\), \(n=2\) & & & \\
1 EB & & 2452.70 & & 2455.40 & 0.11 \\  & & \(m=3\), \(n=1\) & & & \\
3F & & 3810.30 & & 3658.90 & 3.97 \\  & & \(m=1\), \(n=3\) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: _Cont._respectively, where \(D=Eh^{3}/12(1-\mu^{2})\) represents the flexural rigidity of the plate element. Furthermore, an equation of the rotation angle \(\theta_{x}\), which is related to the boundary conditions, is introduced as  
\[\theta_{x}=-\frac{\partial w}{\partial x}, \tag{13}\]  
such that seven equations are established with seven variables of \(\left\{w\,\theta_{x}\,M_{x}\,M_{y}\,M_{xy}\,Q_{x}\,Q_{y}\right\}\). According to governing Equations (11) to (13), denote the variable vector as:  
\[\mathbf{P}\left(x,y,t\right)=\left[ \begin{array}{cccccc}w&\theta_{x}&M_{x}&M_{y }&M_{xy}&Q_{x}&Q_{y}\end{array} \right]^{\mathsf{T}}, \tag{14}\]  
such that Equations (11) to (13) can be written into a matrix form as  
\[\frac{\partial\mathbf{P}(x,y,t)}{\partial x}=\mathbf{U}\mathbf{P}(x,y,t), \tag{15}\]  
where \(\mathbf{U}\) is a \(7\times 7\) matrix given by  
\[\mathbf{U}=\left[ \begin{array}{cccccc}0&U_{12}&0&0&0&0&0\\ U_{21}&0&U_{23}&0&0&0&0\\ 0&0&0&0&U_{35}&U_{36}&0\\ 0&U_{42}&0&0&U_{45}&U_{46}&0\\ 0&0&0&U_{54}&0&0&U_{57}\\ U_{61}&0&0&0&0&0&U_{67}\\ 0&U_{72}&0&0&U_{75}&U_{76}&0\end{array} \right], \tag{16}\]  
with \(U_{12}=-1\); \(U_{21}=\mu\frac{\partial^{2}}{\partial y^{2}}\); \(U_{23}=\frac{12(1-\mu^{2})}{Eh^{3}}\); \(U_{35}=-\frac{\partial}{\partial y}\); \(U_{36}=1\); \(U_{42}=\frac{Eh^{3}}{12}\frac{\partial^{2}}{\partial y^{2}}\); \(U_{45}=-\mu\frac{\partial}{\partial y}\); \(U_{46}=u\); \(U_{54}=-\frac{\partial}{\partial y}\); \(U_{57}=1\); \(U_{61}=\mu\frac{\partial^{2}}{\partial t^{2}}\); \(U_{67}=-\frac{\partial}{\partial y}\); \(U_{72}=\frac{Eh^{3}}{12(1+\mu)}\frac{\partial^{3}}{\partial y^{3}}\); \(U_{75}=-\frac{\partial^{2}}{\partial y^{2}}\), and \(U_{76}=\frac{\partial}{\partial y}\). It is worth noting that the transfer matrix of the flexural vibration can be established by using the one-dimensional method, meaning only two boundary conditions are considered (\(x=0\) and \(x=a\)), to lead a much simpler form than the other two vibration modals (T and EB). Consequently, the flexural vibration is taken as an example in the following work. The other two types of vibrations can be discussed through the same process, and their transfer matrices are shown in references [25; 29].