## 3 Model-Updating with Automatic Mode Identification  
### Mode Identification Using Image-Similarity Recognition  
In this study, the rough outline of the experimental mode shape was revealed by the accumulation of fine sand towards the low-amplitude region, which resulted in an inability to obtain accurate displacement information of the mode shape. Nevertheless, the corresponding relationship between the FE and experimental mode shapes can still be distinguished by comparing the mode shape patterns, which facilitates the mode identification using the image-similarity-recognition method. Considering that the FE mode shape contains richer information than the experimental one, it is difficult to directly calculate the similarity between them; thus, it is necessary to unify the mode shape for similarity recognition. The image unification process is depicted in Figure 7. During this process, a displacement threshold is set to extract the low-amplitude region of FE mode shape, and then, median filtering is applied to remove the noise from the experimental mode-shape image.  
Figure 6: Natural frequencies and displacement mode shapes obtained by FE modal analysis.  
After unifying the mode shapes, the binarized images of the FE and experimental mode shapes are obtained. In a broad sense, the binarized image can be regarded as a binarization matrix. Therefore, the image-similarity-recognition problem is transformed into a matrix-similarity-recognition problem. Moreover, since the binarization matrix is composed only of 0 and 1, the effect of the element points in the image on the similarity recognition result is weakened, and the shape of the pattern in the image becomes the main factor that determines the similarity-recognition accuracy. The calculation process of the mode-identification index ISR refers to the MAC equation (proposed by [PERSON] [19]). First, the binarization matrices of the FE and experimental mode shapes are tiled into a vector form. Subsequently, the ISR can be calculated according to the following expression:  
\[\text{ISR}(\Psi_{\text{R}},\Psi_{\text{S}})=\frac{\left(\Psi_{\text{R}}^{\text{T }}\Psi_{\text{S}}\right)^{2}}{\left(\Psi_{\text{S}}^{\text{T}}\Psi_{\text{S}} \right)\left(\Psi_{\text{R}}^{\text{T}}\Psi_{\text{R}}\right)} \tag{1}\]  
where \(\Psi_{\text{R}}\) and \(\Psi_{\text{S}}\) are vectors of the reference and identified mode shapes, respectively. The obtained ISR is a scalar ranging from 0 to 1, and the higher its value, the higher the similarity.  
To verify the validity and accuracy of mode identification using the ISR method, the first 40 mode shapes were extracted from the FE modal analysis and auto-correlation of these mode shapes was performed using the ISR index (Figure 8a). At the same time, the displacement mode shapes were extracted and the same calculation process was followed using the MAC proposed by [PERSON] (Figure 8b).  
According to Figure 8, the colors on the diagonal are all red, indicating that the mode shapes to be identified are the same, and both indices achieved accurate mode identification. However, as shown in Figure 8b, many colors that are not on the diagonal are close to red, indicating that misjudgment is likely to occur when the MAC index is used to identify different mode shapes, while the ISR index can effectively avoid this problem (as shown in Figure 8a, most colors not on the diagonal are closer to blue). Thus, it can be concluded that ISR has higher sensitivity to the difference of mode shapes than MAC.  
Figure 7: Image unification process of the FE and experimental mode shapes.