# Fault Detection of Bearing Systems through EEMD and Optimization Algorithm
## 3 Simulation of Bearing Response and Fault Detection Results
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Description** & **Value** \\ \hline \(f_{r}\) & Resonance frequency & 800 Hz \\ \(f_{r}\) & Shaft rotating frequency & 16.67 Hz \\ \(f_{di}\) & Inner-race fault frequency & 82.02 Hz \\ \(f_{do}\) & Outer-race fault frequency & 51.34 Hz \\ \(f_{dr}\) & Rolling element fault frequency & 34.34 Hz \\ \(\mu\) & Ratio coefficient & 0.55 \\ \(F_{s}\) & Sampling frequency & 4 kHz \\ \(t_{max}\) & Simulation time & 2 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System parameters and frequency values for generating bearing signals. Figure 2: A schematic diagram of overall process for bearing fault detection. by simply decomposing a white noise. Obviously, it is difficult to find the characteristic portion of impacts in EEMD from the healthy bearing signal (see Figure 4a). However, in the case of EEMD performed on the bearing signal with a defect, the impact nature due to the defect is mostly dominant in the first three IMFs. When a bearing element rotates, the impulse is repeated at a predetermined interval. The signal characteristics, such as the interval at which the impact occurs, and the intensity of the generated signal, are unique, depending on the type of defect: an inner ring defect, outer-ring defect, or rolling element defect. Figure 4: _Cont_. Figure 3: Simulation signals of a bearing system: (**a**) healthy condition (**b**) inner-race defect (**c**) outer-race defect, and (**d**) rolling element defect. Again, we employed EEMD to decompose the time-series of the bearing signals. For each damage case, 16 time-segments having data points of 0.125 s duration were decomposed into eight IMFs. In order to develop a feature vector, five parameters, such as the kurtosis, root mean square, standard deviation, zero-crossing rate, and entropy, are extracted from each IMF. Thus, an individual time-segment produced a vector having 40 feature elements. In summary, 16 time-segments were collected from each of the three cases of bearing defects and one healthy condition. All of the parameters were normalized, to reduce systematic irregularities and biases. Because each segment represented a state of the current bearing condition, it could be visualized as a point in three-dimensional space through a PCA or Isomap-based dimensional reduction. Although the feature-extracted parameter vector itself represented a unique signature of a bearing condition, each element of those parameters may have a different level of contribution in classifying damage types or categories. Thus, it is reasonable to assign a proper weight to each element of the parameter vector to achieve the best classification performance. We can find optimal sets of parameter weights that balance both goals, of maximizing the effect of separation from different defect types, and by minimizing the Euclidian distance among the points (parameter vectors), from the group of identical bearing defects. Here, a point or parameter vector represents a single state of bearing condition in multi-dimensional space from a segment of time-series data. This study employed the PSO algorithm to find weightings for the parameter vector. Having developed parameter vectors, PSO iteratively searched for the best set of weights in all 40 feature-elements, while minimizing the Euclidian distances among points, and maximizing the separations between damage groups. Hence, the objective function Q to determine parameter weightings was the Dunn index (Equation (8)) for assessing the performance of grouping in classification problems [16]:  
\[\text{Q}=\frac{\min_{1\leq i<j\leq m}d(i,j)}{\max_{1\leq k\leq\text{m}}d^{ \prime}_{k}} \tag{8}\]  
Here, \(d(i,j)\) is the distance between the \(i\)-th and \(j\)-th group in a total \(m\) number of groups. The largest distance between any members in the \(k\)-th group is denoted as \(d^{\prime}_{k}\). Thus, a larger value of Q implies that each group becomes more successfully separated in multi-dimensional space.