# Dynamic Similitude Design Method of the Distorted Model on Variable Thickness Cantilever Plates
## 4 The Distorted Model of the VT Plate  
### Distorted Models and the Scaling Law  
It has been discussed in the previous sections that for each vibration modal, an equivalent thickness of a VT plate can be calculated as \(h_{e,n}\) via the simplified ST plate by using the transfer matrix method. However, in practice, it is obviously impossible to design an MT plate with a specific thickness \(h_{e,n}\) for each modal of the prototype VT plate. Usually, a unified thickness, \(h_{Umi}\), is chosen in the design of the scaled model. In order to address this issue, denote the ration of the thickness as  
\[\Xi=\frac{h_{e,n}}{h_{Umi}}. \tag{32}\]  
According to the frequency scaling law (8), the frequency relationship between the prototype equivalent thin plate and its scaled down model is  
\[\lambda_{\omega,\mathrm{p-m}}=\sqrt{\frac{\lambda_{E}}{\lambda_{\rho}}}\frac{ \lambda_{h_{e}}}{\lambda^{2}}, \tag{33}\]  
while the scaling law between the scaled model thin plate and the distorted model plate with the unified thickness \(h_{Umi}\) is calculated as  
\[\lambda_{\omega,\mathrm{m-u}}=\frac{h_{(e,n),\,\mathrm{m}}}{h_{Umi}}=\Xi \lambda_{h_{e}}, \tag{34}\]  
where \(h_{(e,n),\,\mathrm{m}}\) represents the model thickness against the \(n\)th modal of the thin plate. Consequently, the scaling law between the unified distorted model and the prototype equivalent thin plate, as well as the VT plate, is derived as  
\[\lambda_{\omega}=\lambda_{\omega,\mathrm{p-m}}\lambda_{\omega,\mathrm{m-u}}= \sqrt{\frac{\lambda_{E}}{\lambda_{\rho}}}\frac{\lambda_{h_{e}}}{\lambda^{2}} \Xi\lambda_{h_{e}}=\sqrt{\frac{\lambda_{E}}{\lambda_{\rho}}}\Xi, \tag{35}\]  
when \(\lambda_{a}=\lambda_{b}=\lambda_{h_{e}}=\lambda\). Next, a similitude design case study is provided to illustrate the design process of the scaled down plate. In this case study, the simplified steps of the ST plate are given as \(N=5\), and the material and the geometrical parameters are shown in Table 2, where in the ANSYS simulation (ANSYS 14.0, ANSYS, Pittsburgh, PA, USA), the element Solid 186 is used. The equivalent thickness of the previous fourth flexural vibration modal defined as \(h_{eiF}\), \(i=1,\ldots,4\), and the relative prediction error is calculated by  
\[\eta_{e}=\frac{|\omega_{e}-\omega_{s}|}{\omega_{s}}\times 100\%, \tag{36}\]  
as shown in Table 4, where the width, length and material parameters of the equivalent thin plate are the same as the ST plate as shown in Table 2, and \(\omega_{e}\) and \(\omega_{s}\) are the natural frequencies of the equivalent thickness thin plate and the ST plate, respectively. The results in Table 4 show that the ST plate has the same order and shape of the vibration modal as the equivalent thin plate, and the natural frequencies of each order's vibration are also close to a small relative error of \(\eta_{e}<2\%\). This indicates that the design algorithm on calculating the equivalent thickness is applicable, and, therefore, the connections between the VT plate, ST plate and the thin plate are established. Consider that, in a scaled down model, the material of the model plate is NO. 45 steel and the geometrical parameters of the plate are shown in Table 5 with \(\lambda=2\). The unified thickness of the model plate is \(h_{Uni}=2\,\mathrm{mm}\), and the predicted natural frequencies obtained by using the scaling law (35) are shown in Table 6, where \(\eta_{ave}\) is the relative error of predictive values. Table 6 shows accurate predicted results that in the same vibration modal, the predicted natural frequencies are close to that of the prototype with a relative error \(\eta_{ave}<5\%\), indicating that the method of predicting a VT plate's vibration characteristics by using an