## 5 Bolt Looseness Detection Results and Discussion
### Comparison for Different Input Patterns and Other Neural Network Models  
To provide a comprehensive comparison, alternative input patterns, namely time series and Fourier amplitude spectra, along with different neural network architectures, specifically CNN and CNN-LSTM, were evaluated for their effectiveness in detecting bolt looseness. For the time series input, the received signals from each wave propagation path (\(\mathrm{y_{mn}(t)}\)) were filtered using a bandpass filter within the range of 60 kHz to 100 kHz. Signal segments in the time duration [0, 350 us] were obtained and concatenated to create the input pattern, which has a dimension of 4 \(\times\) 350. Meanwhile, the Fourier transform was applied to \(\mathrm{y_{mn}(t)}\), yielding Fourier amplitude spectra spanning from 0 to 194 kHz. These spectra were utilized to construct the frequency-domain training pattern, sized 4 \(\times\) 200. These input patterns, derived from time series, Fourier amplitude spectra, and time-frequency spectra, were fed into CNN, CNN-LSTM, and CNN-LSTM-MHSA models to assess their accuracy in estimating bolt looseness conditions. Given the dimensions of the above two types of training patterns, the kernel sizes for two convolutional layers, and the max-pooling layer were set to 1 \(\times\) 4 and 1 \(\times\) 1, respectively, with other hyperparameters fine-tuned for optimal performance. The outcomes of bolt looseness localization and severity estimation are listed in Table 4. Among the models tested, the CNN-LSTM-MHSA model, when supplied with time-frequency spectrum inputs, demonstrated superior performance.  
The proposed method was also compared with the wave energy-based method as in the author's previous work [44]. The wave energy reflection coefficient (\(I_{REF,ij}\)) on the wave path \(\mathrm{S_{i}}\)-\(\mathrm{S_{j}}\) is defined based on the wave energies from the directly incoming (\(E_{ij}^{inc}\)) and end-reflected (\(E_{ij}^{ref}\)) wave packets as follows:  
\[I_{REF,ij}=E_{ij}^{ref}/E_{ij}^{inc} \tag{11}\]  
The method of calculating wave energies can refer to [44]. \(I_{REF,ij}\) was normalized using the wave energy reflection coefficient in the undamaged case (\(I_{REF,ij}^{0}\)) as  
\[I_{REF,ij}^{nor}=I_{REF,ij}^{0}/I_{REF,ij} \tag{12}\]  
A BPNN was designed, taking as input the four \(I_{REF,ij}^{nor}\) values from the four wave paths, with the outputs representing the damage severities of each row (\(r_{i}\)). There were two hidden layers with six neural nodes in each layer. \(Error^{local}\) and \(Error^{overall}\) were calculated as 6.3% and 2.5%, respectively, and the Acc value was 99.20%.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Neural Network and**} & \multicolumn{1}{c}{**Time**} & \multicolumn{1}{c}{**Fourier Amplitude**} & \multicolumn{1}{c}{**Time–Frequency**} \\ \multicolumn{2}{c}{**Evaluation Index**} & \multicolumn{1}{c}{**Series**} & \multicolumn{1}{c}{**Spectrum**} & \multicolumn{1}{c}{**Spectrum**} \\ \hline \multirow{3}{*}{CNN} & Acc & 97.39\% & 99.48\% & 99.74\% \\  & \(Error^{local}\) & 13.40\% & 6.84\% & 7.53\% \\  & \(Error^{overall}\) & 7.70\% & 5.09\% & 3.94\% \\ \hline \multirow{3}{*}{CNN-LSTM} & Acc & 99.74\% & 100\% & 100\% \\  & \(Error^{local}\) & 7.44\% & 5.85\% & 4.88\% \\  & \(Error^{overall}\) & 4.44\% & 3.27\% & 2.50\% \\ \hline \multirow{3}{*}{CNN-LSTM- MHSA} & Acc & 100\% & 100\% & 100\% \\  & \(Error^{local}\) & 5.96\% & 5.98\% & 3.96\% \\ \cline{1-1}  & \(Error^{overall}\) & 3.79\% & 2.83\% & 2.09\% \\ \hline \hline \end{tabular} Note: \(Error^{local}\) is the average value of \(Error^{local}_{i}\).  
\end{table}
Table 4: Comparison for bolt looseness detection using different input patterns and neural network models.