## 3 Results and Discussion
### Principal Component Analysis Results
(**a**) 2D plot using PC1 an PC3; (**b**) 2D plot using PC2 and PC3; (**c**) 3D plot using largest PC1, PC2, and PC3. In Figure 12 and Equation (8), area (A) represents the lower area of the frequency response graph curve; \(X_{i}\) denotes the magnitude at each point of the frequency response data:  
\[\text{A}=\int f(x)dx=\sum X_{i}. \tag{8}\]  
Root mean square (RMS) can be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values in a cycle:  
\[RMS=\sqrt{\frac{1}{n}\sum_{i=1}^{n}X_{i}^{2}}. \tag{9}\]  
The geometrical moment of area is obtained by first dividing the shape of area A into n small areas; refer to each small random area as \(a_{i}\); hold the arbitrary orthogonal \(x\) for this shape, and adjust the small area \(a_{i}\) (\(x_{i}\), \(y_{i}\)). The sum of \(a_{i}\times y_{i}\) for the entire shape is referred to as the primary moment of section with respect to the x-axis, and is given as follows:  
Figure 9:0  
The centroid pertains to the coordinates at which the geometrical moment of area of the cross-section for the orthogonal coordinate axis is zero at any given section. To obtain the distance from the orthogonal coordinate axis to the centroid, divide the geometrical moment of area of the section by the area of the shape, as follows:  
Figure 9:1  
Because the analysis using time data cannot identify the cap damage and internal damage, an analysis of the frequency response data limited to cristobalite materials was performed. Through PCA, the vector with the greatest contribution of data variances was identified as the moment of real value (98.4%), and the second was the moment of imaginary value (0.10%). The x-axis coordinates were calculated by PC1 (moment of real value) of the largest variance, and the y-axis coordinates were calculated by PC2 (moment of imaginary value) of the second largest variance; these were plotted two-dimensionally, as shown in Figure 9:2  
Figure 9:3 Features extracted from frequency response of experimental data. The analysis based on two principal component vectors showed that normal data were clustered within the range of red ellipse; the damage data showed the following differences. Because porcelain damage data were located in the negative direction of PC1 and the positive direction of PC2, it was possible to distinguish these from intact data. Moreover, cap damage data were found to be distributed in the negative direction of PC2 compared with intact data; hence, it was possible to distinguish these from intact data. In the case of internal damage, it was presumed that the interior of the porcelain was damaged because one data point was located in the negative direction of PC1; it was presumed that the interior of the cap was damaged because two data points were located in the negative direction of PC2. However, because one of the cap damage data points was close to the distribution range of normal data, a third PC vector (area of real value) was employed to perform a three-dimensional analysis, as shown in Figure 9:4  
The three-dimensional graph analysis of the distribution differences between cap damage data and intact data showed that the former were distributed in the negative direction of PC2 and the latter in the positive direction of PC3. Figure 9:5 Three-dimensional plot using three eigenvectors extracted from frequency response data. Figure 9:6 Two-dimensional plot using two eigenvectors extracted from frequency response data. Using the frequency response data, the two-dimensional and three-dimensional analyses of the newly extracted principal component enabled the distinction among the three types of damage that were initially set. Data of internal damage could be anticipated as porcelain and cap damage depending on the distribution location of data.