## 3 Numerical Example
### Architecture of the Proposed CNN-LSTM  
A CNN-LSTM for the damage localization and severity prediction of hangers is designed. The architecture of the CNN includes four convolutional blocks for extracting the high-dimensional features of accelerations (denoted as CNN features) and three fully connected layers for damage localization or severity recognition. The architecture of LSTM uses the LSTM layer to extract the time-series features of accelerations (denoted as LSTM features). It should be noted that CNN and LSTM share the three fully connected layers. After merging the extracted CNN and LSTM features, the fused features are used as the input of the fully connected layer to predict the damage location or severity.  
The model input adopts the acceleration with dimensions of 1024 \(\times\) 31, where the input height 31 is the number of hangers, and the input width 1024 is the sequence length. However, the input dimension of the LSTM layer is related to the time step and sequence length. This study sets the input dimension of the LSTM layer to 64 \(\times\) 496, where 64 is the time step and 496 is the sequence length. Therefore, the sample dimensions need to be converted according to the following steps. Each sample is divided into 16 parts with  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scenarios**} & \multirow{2}{*}{**Damage Types**} & **Damage** & **Damage** & **Training** & **Validation** & \multirow{2}{*}{**Test Samples**} \\  & & **Severity (\%)** & **Combinations** & **Samples** & **Samples** & **Samples** \\ \hline
1 & & 5 & 31 & 327 & 100 & 100 \\
2 & Single damaged & 15 & 31 & 327 & 100 & 100 \\
3 & hanger & 25 & 31 & 327 & 100 & 100 \\
4 & & 35 & 31 & 327 & 100 & 100 \\
5 & & 45 & 31 & 327 & 100 & 100 \\ \hline
6 & & 5 & 36 & 412 & 100 & 100 \\
7 & Two damaged & 15 & 36 & 412 & 100 & 100 \\
8 & hangers 1 & 25 & 36 & 412 & 100 & 100 \\
9 & & 35 & 36 & 412 & 100 & 100 \\
10 & & 45 & 36 & 412 & 100 & 100 \\ \hline
11 & & 5 & 35 & 395 & 100 & 100 \\
12 & Three damaged & 15 & 35 & 395 & 100 & 100 \\
13 & hangers 1 & 25 & 35 & 395 & 100 & 100 \\
14 & & 35 & 35 & 395 & 100 & 100 \\
15 & & 45 & 35 & 395 & 100 & 100 \\ \hline \(\Sigma\) & / & / & 510 & 5670 & 1500 & 1500 \\ \hline \hline \end{tabular}  
* The damage of the two hangers can only happen to nine randomly selected hangers, i.e., No. 2, 5, 10, 13, 17, 19, 24, 27, and 30; therefore, the number of corresponding damage combination is \(C_{2}^{\prime}\). Moreover, three damages can only occur on hangers No. 5, 10, 13, 17, 19, 24, and 30, and the number of the corresponding damage combination is \(C_{2}^{\prime}\).  
\end{table}
Table 1: Details of damage scenarios and division of samples.  
Figure 6: Part of the pre-processed acceleration data of hanger 15 under scenario 15.  
dimensions of 64 \(\times\) 31. Then, the 16 parts of the data are spliced along the first dimension into a sequence of 496 acceleration data points, so the transformed sample dimension is 64 \(\times\) 496.  
The LSTM layer consists of 64 LSTM units and 496 hidden neurons. Convolutional blocks are composed of a convolutional layer, a BN layer, an activation function, and a pooling layer. Fully connected block 1 is composed of a fully connected layer and a BN layer, and fully connected block 3 is the classification layer. The size of the convolution kernel in convolutional block 1 is 64 \(\times\) 1, and the size of the convolution kernel in other convolutional blocks is 3 \(\times\) 1. The activation function of the convolutional block and the fully connected block 1 selects ReLU. The pooling layer in the convolutional block adopts max pooling with a stride of 2. For the prediction of damage severity, the fully connected block 3 has 31 outputs, corresponding to the number of hangers. For damage localization, each hanger has two states (damaged/undamaged), and the fully connected block 3 has a total of 62 outputs. The architecture of the proposed CNN-LSTM model is shown in Figure 7, and the parameters are shown in Table 2.