# Edge Effect Analysis and Edge Defect Detection of Titanium Alloy Based on Eddy Current Testing
## 2 Edge Effect Evaluation
### Model Fitting and Verification  
After factor screening, effects which were significant to edge effect were obtained, and could be used to preliminary construct the first-order regression model. For \(D_{half}\), it can be written as:  
\[D_{half}=\beta_{0}+\beta_{1}*B+\beta_{2}*C+\beta_{3}*D+\beta_{4}*E+\beta_{24}*CE \tag{5}\]  
For \(D_{30\%}\), it can be written as:  
\[D_{30\%}=\gamma_{0}+\gamma_{1}*B+\gamma_{2}*C+\gamma_{3}*D+\gamma_{4}*E \tag{6}\]  
Figure 6: Half-normal probability plot of the effects for the \(2^{5}\) factorial (\(D_{half}\)).  
where \"\(B\)\", \"C\", \"D\", and \"\(E\)\" represent the linear effects of factors in Table 2. The term \"CE\" is the interaction of the linear lift-off with outside diameter. Other effects which were not significant were removed from the first-order model. \(\beta_{0}\), \(\beta_{1}\),..., \(\gamma_{0}\),...,\(\gamma_{4}\) are model coefficients.  
All the experimental data were from previous factorial design, and the Least-square method was used to obtain model coefficient estimates. Suppose that n observations on the response are available. The response \(D_{halfi}\) and variables \(B_{i},\ C_{i},\ D_{i},\ E_{i},\) and \(CE_{i}\) denote the \(i\)th observation of the response and regressor variables. The model Equation (5) can be written in terms of observations as  
\[D_{halfi}=\beta_{0}+\beta_{1}*B_{i}+\beta_{2}*C_{i}+\beta_{3}*D_{i}+\beta_{4}* E_{i}+\beta_{24}*CE_{i}+\varepsilon_{i} \tag{7}\]  
where \(\varepsilon_{i}\) is the error term. The least square function is  
\[L=\sum_{i=1}^{n}{\varepsilon_{i}}^{2} \tag{8}\]  
The method of least squares chooses the \(\beta s\) in Equation (7) so that the sum of the squares of the errors \(L\) is minimized.  
The fitted model of response \(D_{half}\) using Least-square method is  
\[D_{half}=1.0286+0.0636B+0.3318C-0.003206D+0.1768E-0.03805 CE \tag{9}\]  
The fitted model of response \(D_{30\%}\) can be obtained using the same method  
\[D_{30\%}=1.8535+0.1160B+0.3625C-0.008577D+0.11883E \tag{10}\]  
The result of testing for lack of fit shown in Table 5 indicates that there was no strong evidence of lack of fit, so there was no need to construct a higher order regression model.  
To validate the predicting ability of this regression model, five additional simulations were conducted. The coil parameters used in these five experiments were out of range of the original parameters in factorial design. It turned out that all these five observations fell inside the prediction interval on the response at that point, which provided some assurance that the regression models can  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source of Variation** & **Degree of Freedom** & **Sum of Squares** & **Mean Square** & \(\mathbf{F}_{0}\) & \(\boldsymbol{p}\)**-Value** \\ \hline Model & 15 & 2.09262 & 0.13951 & 27 & 0 \\ Linear & 4 & 2.00297 & 0.50074 & 96.93 & 0 \\ \(B\) & 1 & 0.09177 & 0.09177 & 17.76 & 0.001 \\ \(C\) & 1 & 0.17522 & 0.17522 & 33.92 & 0 \\ \(D\) & 1 & 1.12384 & 1.12384 & 217.54 & 0 \\ \(E\) & 1 & 0.61214 & 0.61214 & 118.49 & 0 \\ \(BC\) & 1 & 0.01243 & 0.01243 & 2.41 & 0.14 \\ \(BD\) & 1 & 0.00409 & 0.00409 & 0.79 & 0.387 \\ \(BE\) & 1 & 0.01457 & 0.01457 & 2.82 & 0.112 \\ \(CD\) & 1 & 0.00001 & 0.00001 & 0 & 0.964 \\ \(CE\) & 1 & 0.04245 & 0.04245 & 8.22 & 0.011 \\ \(DE\) & 1 & 0.00499 & 0.00499 & 0.97 & 0.34 \\ \(BCD\) & 1 & 0.00001 & 0.00001 & 0 & 0.973 \\ \(BCE\) & 1 & 0.01045 & 0.01045 & 2.02 & 0.174 \\ \(BDE\) & 1 & 0.00065 & 0.00065 & 0.13 & 0.728 \\ \(CDE\) & 1 & 0 & 0 & 0 & 0.986 \\ \(BCDE\) & 1 & 0 & 0 & 0 & 0.992 \\ Error & 16 & 0.08266 & 0.00517 & & \\ Total & 31 & 2.17528 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Analysis of Variance for \(D_{half}\) in \(B\), \(C\), \(D\), and \(E\).  
describe the quantitative relationship between coil parameters and the edge effect. Such results can be applied to sensor design and optimization especially for edge defect detection.