# 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
#### 3.2.2 Equivalent Thickness of the Thin Plate  
Assume that the simplified ST plate contains \(N\) steps with each length of \(a_{si_{0}}\) and the thickness of \(h_{si_{0}}\) (\(i_{0}=1,2,\cdots,N\)), as shown in Figure 4.  
Each step of the plate, by applying the transfer matrix method, can be divided into \(r\) sections, such that, in total, \(R=N\times r\) subsections are applied in analyzing the ST plate, where the boundary of each section is defined as \(\xi_{i}\), \((i=0,1,\cdots,R)\). For the ST plate, Equation (22) can be written as:  
Figure 4: The sub-sections of the stepped thickness (ST) plate. \[\left[ \begin{array}{c}\widetilde{w}\\ \widetilde{\theta}_{x}\\ \widetilde{M}_{x}\\ \widetilde{M}_{y}\\ \widetilde{M}_{xy}\\ \widetilde{Q}_{x}\\ \widetilde{Q}_{y}\end{array} \right]_{\xi_{(n-1)r}}=\left[ \begin{array}{c}T_ {11}&T_{12}&T_{13}&T_{14}&T_{15}&T_{16}&T_{17}\\ T_{21}&T_{22}&T_{23}&T_{24}&T_{25}&T_{26}&T_{27}\\ T_{31}&T_{32}&T_{33}&T_{34}&T_{35}&T_{36}&T_{37}\\ T_{41}&T_{42}&T_{43}&T_{44}&T_{45}&T_{46}&T_{47}\\ T_{51}&T_{52}&T_{53}&T_{54}&T_{55}&T_{56}&T_{57}\\ T_{61}&T_{62}&T_{63}&T_{64}&T_{65}&T_{66}&T_{67}\\ T_{71}&T_{72}&T_{73}&T_{74}&T_{75}&T_{76}&T_{77}\end{array} \right]_{n}\left[ \begin{array}{c}\widetilde{w}\\ \widetilde{\theta}_{x}\\ \widetilde{\theta}_{x}\\ \widetilde{M}_{x}\\ \widetilde{M}_{y}\\ \widetilde{Q}_{x}\\ \widetilde{Q}_{y}\end{array} \right]_{\xi_{nr}}, \tag{23}\]  
for the \(n\) th step, where \(T_{ij}\) (\(i=1,2,\cdots,7\); \(j=1,2,\cdots,7\)) are the elements of the transfer matrix. According to Equation (21), the \(n\) th transfer matrix in (23) is calculated as:  
\[T_{n}=\mathrm{e}^{(\widetilde{U}_{n}a_{m})}, \tag{24}\]  
where  
\[T_{n}=\left[ \begin{array}{cccccc}T_{11}&T_{12}&T_{13}&T_{14}&T_{15}&T_{16}&T _{17}\\ T_{21}&T_{22}&T_{23}&T_{24}&T_{25}&T_{26}&T_{27}\\ T_{31}&T_{32}&T_{33}&T_{34}&T_{35}&T_{36}&T_{37}\\ T_{41}&T_{42}&T_{43}&T_{44}&T_{45}&T_{46}&T_{47}\\ T_{51}&T_{52}&T_{53}&T_{54}&T_{55}&T_{56}&T_{57}\\ T_{61}&T_{62}&T_{63}&T_{64}&T_{65}&T_{66}&T_{67}\\ T_{71}&T_{72}&T_{73}&T_{74}&T_{75}&T_{76}&T_{77}\end{array} \right]_{n}.\]  
Combining all \(N\) steps of the ST plate by using the transfer matrix (23) yields  
\[\left[ \begin{array}{c}\widetilde{w}\\ \widetilde{\theta}_{x}\\ \widetilde{M}_{x}\\ \widetilde{M}_{y}\\ \widetilde{M}_{xy}\\ \widetilde{Q}_{x}\\ \widetilde{Q}_{y}\end{array} \right]_{0}=\prod\limits_{n=1}^{N}\mathbf{T}_{n} \left[ \begin{array}{c}\widetilde{w}\\ \widetilde{\theta}_{x}\\ \widetilde{M}_{x}\\ \widetilde{M}_{y}\\ \widetilde{Q}_{x}\\ \widetilde{Q}_{y}\end{array} \right]_{Nr}=\prod\limits_{n=1}^{N}\mathrm{e}^{( \widetilde{U}_{n}a_{m})}\left[ \begin{array}{c}\widetilde{w}\\ \widetilde{\theta}_{x}\\ \widetilde{M}_{x}\\ \widetilde{M}_{y}\\ \widetilde{M}_{xy}\\ \widetilde{Q}_{x}\\ \widetilde{Q}_{y}\end{array} \right]_{Nr}, \tag{25}\]  
where \(\overline{T}=\prod\limits_{n=1}^{N}\mathrm{e}^{(\widetilde{U}_{n}a_{m})}\) represents the transfer matrix of the ST plate. Substituting the boundary condition of the cantilever plate,  
\[\widetilde{w}=\widetilde{\theta}_{x}=0\text{ for }x=0,\] (26a) and \[\widetilde{M}_{x}=\widetilde{M}_{y}=\widetilde{M}_{xy}=\widetilde{Q}_{x}= \widetilde{Q}_{y}=0\text{ for }x=a\] (26b) into Equation ( 25 ), yields \[\left[ \begin{array}{c}0\\ 0\end{array} \right]_{0}=\left[ \begin{array}{cc}\overline{T}_{11}& \overline{T}_{12}\\ \overline{T}_{21}&\overline{T}_{22}\end{array} \right]\left[ \begin{array}{c} \widetilde{w}\\ \widetilde{\theta}_{x}\end{array} \right]_{Nr}, \tag{27}\]  
where \(\overline{T}_{ij}\) are the elements of the transfer matrix \(\overline{T}\). According to Equation (27), the natural frequency of the ST plate can be obtained by letting  
\[\det\left(\left[ \begin{array}{cc}\overline{T}_{11}&\overline{T}_{12}\\ \overline{T}_{21}&\overline{T}_{22}\end{array} \right]\right)=0. \tag{28}\]  
The specific algorithm of calculating Equation (28) is discussed in [19] in details. Similarly, the natural frequency of a thin plate can be calculated as  
\[\det\left(\left[