# 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
In a flexural vibration, the displacement functions and the corresponding stresses are assumed as follows:  
\[\mathbf{P}\left(x,y,t\right)=\left[ \begin{array}{c}\widetilde{w}\left(x\right) \sin\left(my\right)\\ \widetilde{\theta}_{x}\left(x\right)\sin\left(my\right)\\ \widetilde{M}_{x}\left(x\right)\sin\left(my\right)\\ \widetilde{M}_{y}\left(x\right)\sin\left(my\right)\\ \widetilde{M}_{xy}\left(x\right)\cos\left(my\right)\\ \widetilde{Q}_{x}\left(x\right)\cos\left(my\right)\\ \widetilde{Q}_{y}\left(x\right)\cos\left(my\right)\end{array} \right]\mathrm{e} ^{\mathrm{i}\omega t}, \tag{17}\]  
where denotation of the wave number along the \(y\) direction is \(m=1\), and \(\left\{\widetilde{w}\left(x\right)\,\widetilde{\theta}_{x}\left(x\right)\, \widetilde{M}_{x}\left(x\right)\,\widetilde{M}_{y}\left(x\right)\right.\)\(\widetilde{M}_{xy}\left(x\right)\,\widetilde{Q}_{x}\left(x\right)\,\widetilde{Q}_{y}\left(x\right)\right\}\) are undefined functions only related to \(x\). Substituting Equation (17) into Equation (15) yields:  
\[\frac{\mathrm{d}\widetilde{\mathbf{P}}(x)}{\mathrm{d}x}=\widetilde{\mathbf{U}}\widetilde {\mathbf{P}}(x), \tag{18}\]where  
\[\widetilde{\mathbf{P}}\left(x\right)=\left[ \begin{array}{cccccc}\widetilde{w}& \widetilde{\partial}_{x}&\widetilde{M}_{x}&\widetilde{M}_{y}&\widetilde{M}_{xy}& \widetilde{Q}_{x}&\widetilde{Q}_{y}\end{array} \right]^{\mathsf{T}}, \tag{19}\]  
and  
\[\widetilde{\mathbf{U}}=\left[ \begin{array}{cccccc}0&\widetilde{U}_{12}&0&0&0&0&0\\ \widetilde{U}_{21}&0&\widetilde{U}_{23}&0&0&0&0\\ 0&0&0&0&\widetilde{U}_{35}&\widetilde{U}_{36}&0\\ 0&\widetilde{U}_{42}&0&0&\widetilde{U}_{45}&\widetilde{U}_{46}&0\\ 0&0&0&\widetilde{U}_{54}&0&0&\widetilde{U}_{57}\\ \widetilde{U}_{61}&0&0&0&0&0&\widetilde{U}_{67}\\ 0&\widetilde{U}_{72}&0&0&\widetilde{U}_{75}&\widetilde{U}_{76}&0\end{array}  \right], \tag{20}\]  
where \(\widetilde{U}_{12}=-1\); \(\widetilde{U}_{21}=-m^{2}\mu\); \(\widetilde{U}_{23}=\frac{12(1-\mu^{2})}{Eh^{3}}\); \(\widetilde{U}_{35}=m\); \(\widetilde{U}_{36}=1\); \(\widetilde{U}_{42}=-\frac{Eh^{3}}{12}m^{2}\); \(\widetilde{U}_{45}=m\mu\); \(\widetilde{U}_{46}=\mu\); \(\widetilde{U}_{54}=-m\); \(\widetilde{U}_{57}=1\); \(\widetilde{U}_{61}=-\rho h\omega^{2}\); \(\widetilde{U}_{67}=m\); \(\widetilde{U}_{72}=-m^{3}\frac{Eh^{3}}{12(1+\mu)}\); \(\widetilde{U}_{75}=m^{2}\); \(\widetilde{U}_{76}=-m\), and \(m=1\). Considering that the thin plate is divided into \(K\) sub-sections along the \(x\) direction, for the \(k\)th sub-section, the solution of (18) can be written as [30]  
\[\widetilde{\mathbf{P}}(a_{k})=T(a_{k})\widetilde{\mathbf{P}}\left(a_{k-1}\right), \tag{21}\]  
where \(a_{k}\) is the length of the \(k\)th sub-section, and \(T(a_{k})=\mathbf{e}^{(\widetilde{U}a_{k})}\) is the transfer matrix of the plate element. By combining Equation (21) along the length of the plate, there is  
\[\widetilde{\mathbf{P}}\left(a\right)=\mathbf{T}_{th}\widetilde{\mathbf{P}}\left(0\right), \tag{22}\]  
where \(T_{th}=\prod\limits_{k=1}^{K}e^{\widetilde{\mathbf{U}}a_{k}}=e^{\widetilde{\mathbf{U} }a}\) is the transfer matrix of the thin plate.