# Dynamic Similitude Design Method of the Distorted Model on Variable Thickness Cantilever Plates
## 4 The Distorted Model of the VT Plate
### A General Design Process  
According to the discussion in the above sections, a distorted scaled model of the VT plate is designed and the method of reducing the predicted errors is investigated. To be clear, the process of the similitude design is summarized as below and illustrated in Figure 5.  
**Step 1:** Simplify the VT plate by using an ST plate with finite steps \(N\), such that the simplified ST plate can reveal the dynamic properties of the VT plate in a certain order.  
Usually, this process can be facilitated by using a numerical analysis method. For example, in the case study of this paper, the frequency errors of different vibration modals between the VT plate and the ST plate with different \(N\) steps are shown in Figure 6.  
Figure 5: The process of the simplified design.  
Figure 6: The error curves of ST plates.  
Fitting the curves with a four-order polynomials yields:  
\[\left\{\begin{array}{l}\eta_{s,1F}=2.826\times 10^{-4}N^{4}-7.175\times 10^{-3}N^{ 3}+6.908\times 10^{-2}N^{2}-0.3043N+0.537\\ \eta_{s,1T}=1.578\times 10^{-4}N^{4}-3.825\times 10^{-3}N^{3}+3.51\times 10^{-2}N^{2}-0. 1474N+0.2505\\ \eta_{s,1 EB}=2.199\times 10^{-5}N^{4}-6.265\times 10^{-4}N^{3}+6.93\times 10^{-3}N^{ 2}-3.436\times 10^{-2}N+6.448\times 10^{-2}\end{array}\right., \tag{37}\]  
where the adjusted determination coefficient \(R^{2}\) is given as:  
\[R^{2}=1-\left[\frac{n^{\prime}-1}{n^{\prime}-(k+1)}\right]\left(1-\frac{\sum \left(\widehat{Y}_{i}-\overline{Y}\right)^{2}}{\sum\left(Y_{i}-\overline{Y} \right)^{2}}\right)\ (i=1,2,\cdots,n^{\prime}), \tag{38}\]  
where \(n^{\prime}\) is the sample size; \(k\) is the order of the polynomial; \(\widehat{Y}\), \(\overline{Y}\) and \(Y\) are the fitted value, the average value and the actual value (simulation value), respectively.  
Substituting relevant parameters into Equation (38) yields:  
\[R^{2}{}_{1F}=0.9995,\ R^{2}{}_{1T}=0.9846,\ R^{2}{}_{1 EB}=0.9928 \tag{39}\]  
where Equation (39) indicates that the fitted curves can be used to determine \(\eta_{s,1F}\), \(\eta_{s,1T}\) and \(\eta_{s,1 EB}\).  
Assuming \(\eta_{s}\leq 5\%\) is acceptable, the step number \(N\) is solved: \(N=4\). Similarly, calculating the frequency errors of high order vibrations (previous six modals), \(N=5\) is chosen as the number of the steps in the case study.  
**Step 2:** Calculate the equivalent thickness of each order's vibration by using the transfer matrix method, which is obtained as \(h_{e,1}\), \(h_{e,2}\),..., \(h_{e,n}\).  
**Step 3:** Unify the equivalent thickness obtained in Step 2 into any thickness of interest as \(h_{Uni}\), and design the similitude model of the thin plate with the thickness \(h_{Uni}\).  
**Step 4:** Derive the distorted scaling laws (35) according to the equivalent thickness \(h_{e,1}\), \(h_{e,2}\),..., \(h_{e,n}\) and the simplified thickness, as well as the different materials of the model and the prototype.  
**Step 5:** Test the distorted model and predict the dynamic characteristics of the prototype VT plate.  
It is worth noting that the design process summarized above is a general process on designing a distorted similitude model of thin walled structures with continuous thickness variations. Based on the sketch shown in Figure 5, two main issues are addressed in the present study. Firstly, find a simple equivalent structure that has the same dynamic properties of interest to the prototype complex structure. Then, design the similitude model of the simple equivalent structure according to the requirement of the designer, where, in this step, dynamic similitude design approaches that have been studied by the authors can be well applied. The proposed technique in the present study can be extended to dealing with, i.e., thin walled shells, annular plates, etc. that have the characteristic of variable thickness.