# Optimal Intrinsic Mode Function Based Detection of Motor Bearing Damages
## 4 Measurement and Analysis
### Signal Analysis  
EMD decomposes the current signals to 10 layers of IMFs. Figure 8 illustrates the IMFs and the tendency function of bearing damage signal by using the classical EMD model. The layers \(c_{1}\) to \(c_{9}\) are IMFs from high frequency to low frequency, respectively, and the layer \(c_{10}\) shows the tendency function. The IMFs energy can be calculated to the corresponding detection accuracy by using the BPNN, as shown in Table 2. The selected nodes \(\Delta E_{i}>\ 0\) are \(E_{1}\), \(E_{3}\), \(E_{4}\), \(E_{6}\) and \(E_{10}\), as nodes A, B, C, D, and E and the layers 1, 3, 4, 6, and 10, respectively, are shown in Figure 9. The distance between each of these nodes are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Bearing Model**} & \multicolumn{3}{c}{**Bearing Size (mm)**} \\ \cline{2-4}  & **Inner** & **Outer** & **Thickness** \\ \hline
6001 & 12 & 28 & 8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The bearing specification.  
Figure 6: **(A)** Healthy bearing, **(B)** 1.0 mm holes on both inner and outer race, **(C)** one 1.0 mm hole on inner race, **(D)** one 1.0 mm hole on outer race, **(E)** 0.5 mm holes on both inner and outer race, **(F)** one 0.5 mm hole on inner race, **(G)** one 0.5 mm hole on outer race, and **(H)** seal damage.  
Figure 7: System diagram.  
The shortest path costs are calculated for each selected node. Four rules of the GA are adopted, which are:  
(Rule 1) each node should be the start node,  
(Rule 2) the start node connects to next nearest node,  
(Rule 3) all remaining nodes should be sequentially connected by Rule 2,  
(Rule 4) compare all costs of the shortest close paths formed by each start node in Rule 1.  
For example, assuming node A is the start node, the nearest node will be connected with node A, and the nearest node should be node B; the node distances are shown in Table 3. Then all remaining  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**IMF (\(i\))** & 1 & 2 & 3 & 4 & 5 \\ \hline
**Energy (\%)** & 58.8 & 12.5 & 41.9 & 34.4 & 28.9 \\ \hline
**IMF (\(i\))** & 6 & 7 & 8 & 9 & 10 \\ \hline
**Energy (\%)** & 33.1 & 23.8 & 18.4 & 13.8 & 33.88 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The energy of each IMF of current signals.  
Figure 8: IMFs and the tendency function of bearing damage signal.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Node Number** & **A** & **B** & **C** & **D** & **E** \\ \hline
**IMF Layer** & 1 & 3 & 4 & 6 & 10 \\
**Axis (X,Y)** & (1, 58.8) & (3, 41.9) & (4, 34.4) & (6, 33.1) & (10, 33.8) \\
**A** & 0 & 17.02 & 24.58 & 26.18 & 26.57 \\
**B** & 17.02 & 0 & 7.57 & 9.30 & 10.71 \\
**C** & 24.58 & 7.57 & 0 & 2.39 & 6.03 \\
**D** & 26.18 & 9.30 & 2.39 & 0 & 4.06 \\
**E** & 26.57 & 10.71 & 6.03 & 4.06 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The distance between each of the nodes.  
Figure 9: Selected nodes of \(\Delta E_{i}\)\(>\) 0.  
The costs of the all paths with A, B, C, D, and E as start nodes are shown in Equation (9), in which the lowest costs of the paths formed by the start nodes A, B, and E are obtained. That means the optimal IMF layers 1, 3, and 10 can adequately illustrate the HHT spectrum of the damage type.  
\[\begin{array}{l}\text{Cost(A)}\ =\ 17.02+7.57+2.39+4.06+26.57=\ 57.61\\ \text{Cost(B)}\ =\ 7.57+2.39+4.06+26.57+17.02=\ 57.61\\ \text{Cost(C)}\ =\ 2.39+4.06+10.71+17.02+24.58=\ 58.76\\ \text{Cost(D)}\ =\ 2.39+6.03+10.71+17.02+26.18=\ 62.33\\ \text{Cost(E)}\ =\ 4.06+2.39+7.57+17.02+26.57=\ 57.61\\ \end{array} \tag{9}\]