# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 6 Experiments and Analysis
### Rolling Element Bearing Fault Data Set  
We use the real rolling element bearing fault data provided by the University of Cincinnati [27] to verify effectiveness of proposed method in fault diagnosis application. The mechanical device used in this experiment is shown in Figure 5.  
Four Rexnord ZA-2115 double bearings are mounted on the support frame, and the speed is maintained at 2000 r/min. We analyze the health of rolling element bearing by collecting the vibration  
Figure 4: The dimensionality reduction effect of M-LLE algorithm in noisy artificial data set. Figure 5: The schematic diagram of experimental bearing structure [27]. data of bearing. Among them, vibration signals are obtained by corresponding acceleration sensor, and the frequency of data sampling is 20 kHz. Figure 6 shows the time domain waveform of different bearing states. The states they depicted are normal state, inner race fault, outer race fault and ball fault. Figure 6a is a diagram of the vibration signal waveform under the normal state, as can be seen from the Figure 6a, it is covered with a lot of noise burr. Figure 6b is the inner fault waveform, affected by bearing surface damage, the vibration signal appeared obvious pulse. Figure 6c is the diagram of outer race fault. Due to the excessive noise, the pulse signal is submerged. Figure 6d is the time domain waveform when the fault occurred to the ball. Because of the rotation of the ball, pulse is periodically distributed, and it is also mixed with noise. Feature extraction is an important stage of fault diagnosis. It is necessary to extract the corresponding effective features to determine whether the data is normal or faulty. From the time-domain waveforms of different states of rolling element bearing, it can be seen that the amplitude and probability distribution of time-domain signal will change when bearing fails. Besides, frequency components, energy of the different frequency components, and main energy peak position will also change [28]. Therefore, when we collect the vibration signal of rolling element bearing, vibration signal should be extracted from effective features to facilitate the fault identification and diagnosis. In this paper, the following 10 time-domain characteristic parameters and 10 frequency-domain characteristic parameters are adopted to extract useful information of rolling element bearing. Table 2 is the information of extracted features. Figure 6: The time-domain diagram in rolling element bearing. The x axis is the time series and \(y\) axis is the data which is collected by the accelerators. (**a**) The waveform of normal state; (**b**) the waveform of inner race fault; (**c**) the waveform of outer race fault; (**d**) the waveform of ball fault. In Table 2, \(x(n)\) represents the time series of signals, \(n=1,\ 2,\ldots,N\), \(N\) is the number of samples, \(s(k)\) represents the spectrum of \(x(n)\), \(k=1,2,\ldots,K\), \(K\) is the number of lines, \(f_{k}\) is the frequency of the \(k\)-th line. The time-domain characteristic parameters \(c_{1}\) and \(c_{3}\sim c_{5}\) represent the amplitude and energy of the time-domain signal; \(c_{2}\) and \(c_{6}\sim c_{10}\) reflect the time series distribution of the time-domain signal. The frequency domain characteristic parameter \(c_{11}\) indicates the frequency of vibration energy in the frequency-domain. \(c_{12}\sim c_{13},c_{15}\) and \(c_{18}\sim c_{20}\) represent the dispersion or concentration of the spectrum. \(c_{14}\) and \(c_{16}\sim c_{17}\) reflect the changes in the main band position. The above 20 characteristic parameters can reflect the information of fault data, but they may contain redundant features. The existence of these redundant features can not only increase the complexity of state recognition, but also reduce the accuracy of fault diagnosis. In order to eliminate redundant information in feature selection, it is necessary to further process these features and extract the intrinsic features. It is a benefit to get true state of system and perform corresponding fault diagnosis.