# Fault Detection of Bearing Systems through EEMD and Optimization Algorithm
## 3 Simulation of Bearing Response and Fault Detection Results  
Because of the cyclic nature of impacts between the defect and the surface of bearing elements, vibration responses from different damage locations, such as the inner-race, outer-race, and rolling element, exhibit a unique pattern of spectra. There have been many dynamic models to represent the response of defects in rolling bearing systems. Among them, [PERSON] [26] suggested an improved simulation model of [PERSON] and [PERSON] [27] to produce responses of bearing defects in the inner-race, outer-race, and rolling elements as \(x_{inner}\), \(x_{outer}\) and \(x_{rolling}\), respectively (see Equations (5) and (6)). \[x_{inner}(t)=\sum_{k=-\infty}^{\infty}A(kT_{d})e^{-\xi(t-kT_{d})}\sin(2\pi f_{ s}(t-kT_{d})+\phi_{k1})U(t-kT_{d}) \tag{5}\]  
\[x_{outer}(t)=\sum_{-\infty}^{\infty}e^{-\xi(t-kT_{d})}\sin(2\pi f_{s}\bigg{(} t-kT_{d}-\frac{T_{d}}{2}\bigg{)}+\phi_{k2}\bigg{)}U\bigg{(}t-kT_{d}-\frac{T_{d}}{2} \bigg{)} \tag{6}\]Here, \(f_{s}\) denotes the resonance frequency of the overall system, \(f_{r}\) is the shaft rotation frequency, \(U(t)\) indicates a unit step function, and \(A(t)\) shows the amplitude modulation function, in which the inner-race defect can be expressed as \(A(t)=A_{0}\;max(cos(2\pi f_{r}t),0)cos(2\pi f_{r}t)\). Also, note that as the rolling element collides with the defect surface with the same force, \(A(t)\) remains constant for the case of defects occurring in the outer-race. The rotating frequency of the shaft is defined as \(f_{r}\). Table 2 shows that the defect frequency of the inner-race \(f_{di}\), outer-race \(f_{do}\), rolling element \(f_{dr}\), and damping factor \(\xi\) are also considered for the simulation of vibration response of the bearing system. On the other hand, when the component of the rolling element becomes defective, a portion of the defect collides with both surfaces of the inner and outer ring to create repetitive impact signals. Thus, the combination of the defect signal becomes the summation of inner \(x_{inner}(t)\) and outer ring part, \(x_{outer}(t)\). At this time, since the magnitude of the defect signal generated in the inner ring is smaller than that of the outer ring, the ratio coefficient \(\mu\) is smaller than one. Therefore, the defect signal of rolling element \(x_{rolling}(t)\) can be obtained as shown in Equation (7):  
\[x_{rolling}(t)=x_{outer}(t)+\mu\;x_{inner}(t) \tag{7}\]  
In this study, to verify the technique of bearing fault detection, the defect signal of the bearing was first generated using the Equation (7). The overall process of bearing fault detection and classification is illustrated in Figure 2. Gaussian distribution noise is mixed with the steady-state bearing signal having a defective bearing component. Table 2 shows the constant values of the equation used for generating the bearing signal:  
Figure 3 shows the vibration signals of different bearing conditions reproduced from the previous Equations (5)-(7) and Table 2. Figure 3a shows the normal bearing condition, for which there is no shock signal, but a noise with Gaussian distribution. On the other hand, Figure 3b-d shows that bearings having different defect types produce repeated local-spikes. Figure 4 illustrates the EEMD-performed results from Figure 3. Note that only the first four IMFs are presented for simple comparison. The characteristics of the individual IMF signal differ from each other, depending on the type of bearing defect. For the healthy bearing condition, each IMF is similar to that obtained