# Fault Diagnosis of Rolling Bearings Based on Improved Fast Spectral Correlation and Optimized Random Forest

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fault Feature Extraction Based on Improved Fast Spectral Correlation

#### 2.1. Brief Introduction of Fast Spectral Correlation

_{n}) be a cyclostationary signal, its spectral correlation is defined as:

_{s}is the sampling frequency, t

_{n}denotes the time instants which can be calculated as t

_{n}= n/F

_{s}, R

_{x}(t

_{n}, τ) represents the cyclic autocorrelation function of x(t

_{n}), τ indicates the time delay, α denotes the cyclic frequency and f represents the frequency.

_{n}) is described as follows:

_{w}represents the window length of STFT, R represents the block shift in STFT; w[n] is the function of time index n; x[n] is the abbreviated form of x(t

_{n}), f

_{k}denotes the k-th discrete frequency and f

_{k}= kΔf, Δf represents the frequency resolution, which has the expression of Δf = F

_{s}/N

_{w}.

_{w}(i, f

_{k}) denotes the complex envelope of signal x(t

_{n}) at iR/F

_{s}, whose center is f

_{k}and bandwidth is Δf. ${\left|{X}_{w}(i,{f}_{k})\right|}^{2}$ represents the energy flow in the frequency band.

_{n}).

_{k}= kΔf and α = pΔf + δ, it can be deduced that f − α = f

_{k}− α ≈ f

_{k}

_{−p}and α ≈ pΔf.

_{n}) is T and its frequency α is 1/T. The energy flow will flow periodically in the band [f

_{k}− Δf/2, f

_{k}+ Δf/2]. When p ≠ 0, X

_{w}(i, f

_{k})X

_{w}(i, f

_{k}

_{−p})

^{*}represents the energy flow between band [f

_{k}− Δf/2, f

_{k}+ Δf/2] and band [f

_{k}

_{−p}− Δf/2, f

_{k}

_{−p}+ Δf/2].

#### 2.2. Improved Fast Spectral Correlation

_{1}(t) is the outer-race impact signal designed based on the rolling bearing outer-race fault model in Reference [47], the natural frequency ${f}_{d}={f}_{n}\sqrt{1-{\xi}^{2}}$, M = 30, D = 1, A = 2, ξ = 0.05, f

_{n}= 3000 Hz and the outer race fault frequency f

_{o}= 120 Hz. x

_{2}(t) is the fundamental frequency interference signal of the frequency f

_{1}= 20 Hz. x

_{3}(t) represents random noise signal. The sampling frequency of the simulated signal is 8192 Hz.

_{1}(t), x

_{2}(t), x

_{3}(t), x(t) and the envelope spectrum of x(t) respectively. Figure 2 and Figure 3 show the analysis results obtained by using fast spectral correlation and improved fast spectral correlation, respectively. Figure 2a shows the fast spectral correlation spectrum of x(t) obtained via the fast spectral correlation method, from which it is hard to identify the fault characteristics. The corresponding enhanced envelope spectrum shown in Figure 2b has peaks at f

_{o}, 2f

_{o}, 3f

_{o}and 4f

_{o}, but many noise components can also be noticed. As shown in Figure 3a, the spectral lines can be identified at f

_{o}, 2f

_{o}, 3f

_{o}and 4f

_{o}, which are more obvious than that reflected in Figure 2a. From the enhanced envelope spectrum obtained via improved fast spectral correlation as shown in Figure 3b, only the spectral lines at f

_{o}, 2f

_{o}, 3f

_{o}and 4f

_{o}are left and the interference amplitudes are inhibited. It is proved that improved fast spectral correlation is more robust to noise and can extract the fault information better than fast spectral correlation.

#### 2.3. Establishment of Fault Feature Vector

_{r}, the characteristic frequency of outer-race fault f

_{o}, the characteristic frequency of inner-race fault f

_{i}and the characteristic frequency of rolling element fault f

_{b}. So, the amplitudes of the four mentioned cyclic frequencies of the enhanced envelope spectrum were selected to form the fault feature vector. The establishment of the fault feature vector using the improved fast spectral correlation approach mainly includes two steps:

- (1)
- The improved fast spectral correlation was utilized for dealing with the fault signal of rolling bearing to obtain the corresponding fast spectral correlation spectrum and enhance envelope spectrum.
- (2)
- Select the amplitudes (i.e., a
_{r}, a_{o}, a_{i}and a_{b}) of the four cyclic frequencies (i.e., f_{r}, f_{o}, f_{i}and f_{b}) in the enhanced envelope spectrum to constitute the fault feature vector**A**= [a_{r}a_{o}a_{i}a_{b}].

## 3. Random Forest Based on Particle Swarm Optimization

#### 3.1. Random Forest

- (1)
- Determine the parameters for training random forest: training sample set, the number of decision tree nTree and the number of random attribute m.
- (2)
- Employ the bootstrap sample method until the sample set is the same as the number of the training samples, which is used as the training sample of a decision tree.
- (3)
- Sample the attribute sets without reusing and extract m attributes, only retaining the data corresponding to the m attributes as training samples.
- (4)
- Train a decision tree using the training samples generated in steps (2) and (3).
- (5)
- Pruning threshold is used to prune the trained decision trees.
- (6)
- If the number of trained decision trees is less than nTree, it will return to step (2) to continue execution. Otherwise, all nTree decision trees will be cascaded through the voting strategy to form random forest.

#### 3.2. Particle Swarm Optimization

_{id}(t) is the particle velocity in the t-th iteration; P

_{id}(t) denotes the optimal position of individual particle in the t-th iteration; P

_{gd}(t) is the global optimal position in the t-th iteration; X

_{id}(t) is the particle position in the t-th iteration; ω represents the inertia weight; d is the population dimension; a

_{1}and a

_{2}are nonnegative constants, they denote the acceleration coefficient; r

_{1}and r

_{2}are random numbers uniformly distributed in the range of [0,1].

#### 3.3. Optimized Random Forest Based on PSO

- (1)
- Initialize the parameters of PSO, including: group size, learning factor, maximum number of iterations, initial location and speed of particles.
- (2)
- The RF of each particle vector is used to predict the learning sample respectively. The prediction error of the current position of each particle is obtained, which is used as the fitness value of each particle. The current fitness value of each particle is compared with the optimal fitness value of that particle. If the current fitness value is smaller, the present position of the particle will be treated as the optimal position of the particle.
- (3)
- Compare the optimal position fitness value of each particle with the fitness value of the optimal position of the group. When the fitness value appears the minimum valve, the optimal position of the particle is taken as the optimal position of the group.
- (4)
- Check whether the end condition of the search is satisfied. If it satisfies, end the optimization and get the best nTree. Otherwise, go back to the step (2) to again.

## 4. The Proposed Method

- (1)
- Collect the fault sample signal of the rolling bearing.
- (2)
- Use the improved fast spectral correlation method to analyze each fault sample to obtain the improved fast spectral correlation spectrum and corresponding enhanced envelope spectrum.
- (3)
- Form the feature vectors based on the enhanced envelope spectrum.
- (4)
- Set up the training set and the test set.
- (5)
- Put the training set and the test set into particle swarm optimization-random forest (PSO-RF) for fault pattern recognition.

## 5. Experimental Results and Analysis

#### 5.1. Experiment 1: Analysis of Ball Bearing Faults

#### 5.2. Experiment 2: Analysis of Cylindrical Roller Bearing Faults

## 6. Conclusions

## Author Contributions

## Fundings

## Conflicts of Interest

## Nomenclature

F_{s} | sampling frequency |

t_{n} | time instants |

R_{x}(t_{n}, τ) | cyclic autocorrelation function of x(t_{n}) |

τ | time delay |

α | cyclic frequency |

f | frequency |

STFT | Short time Fourier transform |

N_{w} | window length of STFT |

N _{0} | central time index of window |

R | block shift in STFT |

w[n] | function of time index n |

x[n] | abbreviated form of x(t_{n}) |

f_{k} | the k-th discrete frequency |

Δf | frequency resolution |

X_{w} (i, f_{k}) | Gabor coefficient at time index i and frequency f_{k} |

L | length of signal x(t_{n}) |

T | period of x(t_{n}) |

p | index of STFT frequency closest to a given cyclic frequency α |

δ | residue |

R_{w}(α) | kernel function |

KA(α) | kurtosis value |

KA’(α) | weighting factor |

${S}_{x}^{Fast}(\alpha ,f)$ | fast spectral correlation |

${S}_{x}^{Fast}{}^{\prime}(\alpha ,f)$ | improved fast spectral correlation |

f_{d} | natural frequency |

M | repeated times of impact |

D | single pulse strength |

A | amplitude |

ξ | system damping ratio |

f_{n} | natural frequency |

f _{o} | outer race fault frequency |

f_{r} | rotating frequency |

f_{i} | inner-race fault frequency |

f _{1} | fundamental frequency |

f_{b} | rolling element fault frequency |

V_{id}(t) | particle velocity in the t-th iteration |

P_{id}(t) | optimal position of individual particle in the t-th iteration |

P_{gd}(t) | global optimal position in the t-th iteration |

X_{id}(t) | particle position in the t-th iteration |

ω | inertia weight |

d | population dimension |

a_{1}, a_{2} | nonnegative constants |

r_{1}, r_{2} | random numbers uniformly distributed in the range of [0,1] |

OOBerror | out-of-bag error |

## Appendix A

- Cage frequency: ${f}_{c}=\frac{{f}_{s}}{2}(1-\frac{d\mathrm{cos}\alpha}{D})$
- Ball passage frequency: ${f}_{bo}=N{f}_{c}$
- Ball to inner race frequency: ${f}_{bi}=\frac{{f}_{s}}{2}(1+\frac{d\mathrm{cos}\alpha}{D})$
- Ball frequency: ${f}_{b}=\frac{D{f}_{s}}{2d}$

_{c}is cage frequency; f

_{s}is shaft frequency; d is nominal ball diameter; α is contact angle; D is pitch circle diameter; N is number of balls in a bearing.

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**Figure 1.**The simulated signal: (

**a**) time waveform of x

_{1}(t); (

**b**) time waveform of x

_{2}(t); (

**c**) time waveform of x

_{3}(t); (

**d**) time waveform of x(t); (

**e**) envelope spectrum of x(t).

**Figure 2.**The processing results of x(t) via fast spectral correlation: (

**a**) the fast spectral correlation spectrum; (

**b**) the corresponding enhanced envelope spectrum.

**Figure 3.**The processing results of x(t) via improved fast spectral correlation: (

**a**) the improved fast spectral correlation spectrum; (

**b**) the corresponding enhanced envelope spectrum.

**Figure 7.**Three fault conditions of the faulty bearings: (

**a**) Inner-race fault; (

**b**) outer-race fault; (

**c**) rolling element fault.

**Figure 8.**The waveform of (

**a**) the normal; (

**b**) inner-race fault; (

**c**) outer-race fault; (

**d**) rolling element fault.

**Figure 9.**(

**a**) The time waveform of an inner-race fault signal; (

**b**) the improved fast spectral correlation spectrum; (

**c**) the corresponding enhanced envelope spectrum; (

**d**) the fault feature vector.

**Figure 10.**The fault feature vectors of: (

**a**) the normal; (

**b**) inner-race fault; (

**c**) outer-race fault; (

**d**) rolling element fault.

**Figure 11.**Identification results of: (

**a**) random forest (RF); (

**b**) particle swarm optimization-random forest (PSO-RF).

**Figure 14.**Three fault conditions of the faulty bearings: (

**a**) Inner-race fault; (

**b**) outer-race fault; (

**c**) rolling element fault.

**Figure 15.**The waveform of (

**a**) the normal; (

**b**) inner-race fault; (

**c**) outer-race fault; (

**d**) rolling element fault.

**Figure 16.**The fault feature vectors of: (

**a**) the normal; (

**b**) inner-race fault; (

**c**) outer-race fault; (

**d**) rolling element fault.

**Figure 18.**The recognition results of: (

**a**) empirical mode decomposition-singular value decomposition (EMD-SVD); (

**b**) multi-scale permutation entropy (MPE).

**Figure 19.**The classification results of (

**a**) extreme learning machine (ELM); (

**b**) support vector machine (SVM).

Bearing Type | Roller Diameter | Pitch Diameter | Number of Roller | Contact Angle |
---|---|---|---|---|

LYC6205E | 7.94 mm | 39 mm | 9 | 0° |

Bearing Fault | Rotating Frequency | Inner-Race Fault | Outer-Race Fault | Rolling Element Fault |
---|---|---|---|---|

Characteristics frequency (Hz) | 24.5 | 133 | 88 | 115 |

Number of Training/Test Sets | Fault Type | Speed of Training/Test Samples (rpm) | Label of Class |
---|---|---|---|

10/30 | Normal | 1470/1470 | 0 |

10/30 | Inner-race fault | 1470/1470 | 1 |

10/30 | Outer-race fault | 1470/1470 | 2 |

10/30 | Rolling element fault | 1470/1470 | 3 |

Bearing Type | Roller Diameter | Pitch Diameter | Number of Roller | Contact Angle |
---|---|---|---|---|

N205 | 7.5 mm | 39 mm | 12 | 0° |

Bearing Fault | Rotating Frequency | Inner-Race Fault | Outer-Race Fault | Rolling Element Fault |
---|---|---|---|---|

Characteristics frequency (Hz) | 24 | 172 | 116 | 118 |

Number of Training/Test Sets | Fault Type | Speed of Training/Test Samples (rpm) | Label of Class |
---|---|---|---|

10/30 | Normal | 1440/1440 | 0 |

10/30 | Inner-race fault | 1440/1440 | 1 |

10/30 | Outer-race fault | 1440/1440 | 2 |

10/30 | Rolling element fault | 1440/1440 | 3 |

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## Share and Cite

**MDPI and ACS Style**

Tang, G.; Pang, B.; Tian, T.; Zhou, C.
Fault Diagnosis of Rolling Bearings Based on Improved Fast Spectral Correlation and Optimized Random Forest. *Appl. Sci.* **2018**, *8*, 1859.
https://doi.org/10.3390/app8101859

**AMA Style**

Tang G, Pang B, Tian T, Zhou C.
Fault Diagnosis of Rolling Bearings Based on Improved Fast Spectral Correlation and Optimized Random Forest. *Applied Sciences*. 2018; 8(10):1859.
https://doi.org/10.3390/app8101859

**Chicago/Turabian Style**

Tang, Guiji, Bin Pang, Tian Tian, and Chong Zhou.
2018. "Fault Diagnosis of Rolling Bearings Based on Improved Fast Spectral Correlation and Optimized Random Forest" *Applied Sciences* 8, no. 10: 1859.
https://doi.org/10.3390/app8101859