# Review of Vibration-Based Structural Health Monitoring Using Deep Learning
## 3 Application of Machine Learning to System Identification by Vibration Responses
For this purpose, various features were calculated. During the clamping process, the axial force and the boundary stiffness of the bolt increased. With the increasing axial force, a shift in the resonance frequency occurs in the bolt vibration responses. The increasing bolt resonant responses are transmitted to the nut-runner. The accelerometer was used to measure the nut-runner vibration responses. The cepstrum analysis was used to analyze the vibration characteristics of the nut-runner after exclusion of the harmonic motor rotation components generated during the tightening process. During the fastening process, the nut-runner responses related to the bolt vibration occur in a relatively high frequency band, as visualized in Figure 2. It is not straightforward to explicitly determine the difference according to the change of the clamping force in the transient responses. This is more clearly presented through the cepstrum analysis. The real cepstrum was used as:  
\[\hat{x}_{r}(\tau)=DCT\big{[}\log\big{(}|X(f)|\big{)}\big{]} \tag{3}\]where \(DCT\) is the discrete cosine transform and _X(f)_ is the Fourier transform of the transient signal. The discrete cosine transform is defined as:  
\[DCT(y(k))=\sqrt{\frac{2}{N}}\sum_{n=1}^{N}y(n)\frac{1}{\sqrt{1+\delta}}\cos( \frac{\pi}{2N}(2n-1)(k-1)) \tag{4}\]  
where \(k\) = 1, 2, \(\cdots\), N, \(y\) is the measured signal, \(N\) is length of signal and \(\delta\) is Kronecker delta. Figure 3 shows the identified clamping force from the distance between the data formed by each clustering algorithm using the data with the largest clamping force as a reference. As can be seen from the identified results, the classification of the vibration responses using machine learning was performed with high accuracy that was not possible by the other approaches. Although the initial setting parameters of each method are different and the empirical method should be used to derive the optimal value, the suggested algorithms still can give a precise estimation with simple vibration measurements. In addition to k-means clustering algorithm introduced through [72], the results in Figure 3 discriminate features derived using PCA, agglomerative clustering and DBSCAN. The k-means algorithm and the PCA confirm that the distribution of the identified results is relatively small. The PCA confirms that there is only one outlier. The agglomerative clustering shows that the dispersion of the discrimination increases as the tightening force increases. The DBSCAN shows an even spread in all tightening forces. The accuracy of the identified results may increase significantly with the use of deep learning algorithms, as discussed in the following sections. Figure 1: Vibration responses of the nut-runner during bolt tightening for clamping force of (**a**) 47 kN, (**b**) 37 kN, (**c**) 27 kN and (**d**) 17 kN. Figure 3: Comparison of Euclidean distance for determination of bolt clamping force by vibration responses calculated by different by machine learning algorithms: (**a**) k-means, (**b**) principal component analysis (PCA), (**c**) agglomerative clustering, (**d**) density-based spatial clustering of application with noise (DBSCAN). Figure 2: Short-time cepstral coefficients for clamping force of (**a**) 47 kN, (**b**) 37 kN, (**c**) 27 kN and (**d**) 17 kN.