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Rolling bearings are vital components in rotary machinery, and their operating condition affects the entire mechanical systems. As one of the most important denoising methods for nonlinear systems, local projection (LP) denoising method can be used to reduce noise effectively. Afterwards, high-order polynomials are utilized to estimate the centroid of the neighborhood to better preserve complete geometry of attractors; thus, high-order local projection (HLP) can improve noise reduction performance. This paper proposed an adaptive high-order local projection (AHLP) denoising method in the field of fault diagnosis of rolling bearings to deal with different kinds of vibration signals of faulty rolling bearings. Optimal orders can be selected corresponding to vibration signals of outer ring fault (ORF) and inner ring fault (IRF) rolling bearings, because they have different nonlinear geometric structures. The vibration signal model of faulty rolling bearing is adopted in numerical simulations, and the characteristic frequencies of simulated signals can be well extracted by the proposed method. Furthermore, two kinds of experimental data have been processed in application researches, and fault frequencies of ORF and IRF rolling bearings can be both clearly extracted by the proposed method. The theoretical derivation, numerical simulations, and application research can indicate that the proposed novel approach is promising in the field of fault diagnosis of rolling bearing.

Rolling bearings are of vital importance in rotary machinery systems, and they are prone to failures due to the complex running conditions [

The denoising method based on phase space reconstruction has gradually become one of the most important tools to research nonlinear and nonstationary signals. A time series can be reconstructed into high-dimensional phase space through delay embedding, and the reconstructed phase space is diffeomorphic to original dynamic system, i.e., they have the same dynamic characteristics [

The centroid selection of local neighborhoods is of vital importance in the local projection method to estimate the neighborhood correctly. The first-order statistics and the second-order statistics have been widely used in the analysis of mechanical vibration signals, and these methods are theoretically applicable to the analysis of linear and Gaussian signal [

This paper proposes a novel approach called AHLP denoising method aiming at fault diagnosis of rolling bearings. In the proposed AHLP denoising method, except adopting high-order polynomials to calculate the centroid of neighborhood, optimal orders of dealing with different kinds of rolling bearing faults can be further estimated to achieve a better denoising effect. The AHLP denoising method can better denoise the vibration signal of faulty rolling bearing to extract its fault characteristic frequencies among all disturbing frequencies, which contributes to the field of fault diagnosis of rolling bearing. The organization of this paper is as follows: Section

Assuming that

The algorithm of standard LP denoising method.

(1) Select appropriate embedding dimension |

(2) Determine neighborhood |

(3) Calculate centroid of its neighborhood by fixed neighborhood numbers. |

(4) Calculate covariance matrix _{q} |

(5) Subtract the projection of the phase point in noise subspace by_{11} and ^{3}; |

(6) Get back to step (2) and repeat subsequent steps until all phase points are processed. |

During step (1), embedding dimension

This method may generate certain errors because of local linearization; the centroids of each neighborhood are secants instead of tangents and all centroids (blue circles) are shifted inward regarding the curvature, as shown in Figure

Consider a cloud of points around a curve. (a) The mean of phase points of neighborhood is estimated as centroid of neighborhood. (b) The second-order polynomial is used to estimate the centroid of neighborhood.

The method was proposed by applying high-order polynomials to estimate the centroid of the neighborhood more accurately. Significant achievements have been obtained by high-order algorithms in medical signal processing [

As for any real number _{.}

The effect of the LP denoising method is enhanced by considering using the linear combination of the first and the second orders to estimate the centroid of neighborhood, thus improving the denoising effect. Hence, it can be inferred that

Then for the HLP denoising method, the linear combination of

This formula is used to prove that as for

Obviously, all coefficients determined by (

In this paper, the orders up to 8 are analyzed. In mathematics, binomial coefficients refer to a set of positive integers appear in binomial theorem as coefficients. They are indexed by two nonnegative integers, namely,

The

1 | Order | |||||||||

1 | 1 | 1 | [1] | |||||||

1 | 2 | 1 | 2 | [2, −1] | ||||||

1 | 3 | 3 | 1 | 3 | [3, −3, 1] | |||||

1 | 4 | 6 | 4 | 1 | 4 | [4, −6, 4, −1] | ||||

1 | 5 | 10 | 10 | 5 | 1 | 5 | [5, −10, 10, −5, 1] | |||

1 | 6 | 15 | 20 | 15 | 6 | 1 | 6 | [6, −15, 20, −15, 6, −1 | ||

1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | 7 | [7, −21, 35, −35, 21, −7, 1] | |

1 | 8 | 28 | 56 | 70 | 56 | 27 | 8 | 1 | 8 | [8, −28, 56, −70, 56, −28, −1] |

As mentioned above, as a denoising reduction method, LP method has been verified to be effective in reducing noises existing in vibration signals of rolling bearings after theoretical derivation and extensive tests. Estimation of the centroid of the neighborhood by using high-order polynomials is beneficial to achieve better denoising effect; hence, HLP denoising method is adopted in this paper to deal with vibration signals of faulty rolling bearings. Owing to the reason that vibration signals of ORF and IRF rolling bearings have different nonlinear geometric structures, denoising effects of different orders differ, so the optimal orders of different kinds of faults are different. By choosing the optimal orders before conducting denoising process, a novel AHLP denoising is hereby proposed in this paper, and the scheme of the proposed method is illustrated in Figure

The scheme of AHLP denoising method in fault diagnosis of rolling bearing.

The vibration signal model of faulty rolling bearing used hereinafter was proposed by Randall et al. [

Define

Based on the vibration signal model of faulty rolling bearing in Section

The parameters of the vibration signal model of ORF rolling bearing.

3 | 0 | 100 | 0.01 | 2000 | 1 | 0 | 0 | 800 |

Time and frequency domain plots of simulated vibration signal of ORF rolling bearing.

Time and frequency domain plots of simulated noisy signal of ORF rolling bearing.

It can be observed from the time and frequency domain plots in Figure

The SNRs of denoised signal with different orders.

Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

SNR | 8.92 | 9.73 | 10.21 | 8.50 | 5.61 | 5.09 | 4.43 |

It can be seen from Table

Time and frequency domain plots of denoised signal of ORF rolling bearing.

As shown in Figure

The SNRs of denoised signal with different orders.

Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|---|

SNR | 1 | 8.92 | 9.73 | 10.21 | 8.50 | 5.61 | 5.09 | 4.43 | |

2 | 10.17 | 12.42 | 12.46 | 10.30 | 8.89 | 7.06 | |||

3 | 11.65 | 12.45 | 14.90 | 14.70 | 13.97 | 10.45 | 7.90 | ||

4 | 12.76 | 14.34 | 18.25 | 16.80 | 13.46 | 12.38 | 10.66 | ||

5 | 12.83 | 13.78 | 14.67 | 14.38 | 13.78 | 11.79 | 11.51 |

When the SNR is bigger than 5, which implies there are not too much additive noises in the signal, the HLP can achieve proper denoising effect in most cases. Hence, the above simulations are conducted to find optimal orders aiming at vibration signal of ORF rolling bearing. As shown in Table

Based on the vibration signal model of faulty rolling bearing in Section

The parameters of the vibration signal model of IRF rolling bearing.

3 | 20 | 100 | 0.01 | 2000 | 1 | 0 | 0 | 800 |

Time and frequency domain plots of simulated vibration signal of IRF rolling bearing.

Time and frequency domain plots of simulated noisy signal of IRF rolling bearing.

It can be observed from the time and frequency domain plots in Figure

The SNRs of denoised signal with different orders.

Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

SNR | 6.91 | 7.28 | 7.62 | 7.20 | 6.72 | 5.07 | 4.52 |

It is clear from Table

Time and frequency domain plots of the denoised signal of IRF rolling bearing.

As shown in Figure

The SNRs of denoised signal with different orders.

Order | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |
---|---|---|---|---|---|---|---|---|---|

SNR | 1 | 6.91 | 7.28 | 7.62 | 7.20 | 6.72 | 5.07 | 4.52 | |

2 | 7.68 | 8.04 | 8.30 | 6.62 | 6.61 | 5.93 | 5.12 | ||

3 | 8.14 | 8.25 | 8.41 | 8.12 | 6.55 | 6.53 | 5.89 | ||

4 | 8.73 | 8.81 | 8.70 | 8.44 | 7.50 | 7.17 | 6.81 | ||

5 | 8.82 | 9.25 | 8.59 | 8.15 | 8.06 | 7.73 | 7.01 |

As shown in Table

Through numerical simulations, the proposed novel AHLP denoising method is applied to deal with vibration signals of ORF and IRF rolling bearings successfully, and there are certain optimal orders for different faults of rolling bearing. By choosing optimal order of HLP, the best denoising effect can be obtained. Here, two cases of test data are used to validate the effectiveness of the proposed AHLP denoising method in this paper.

To verify the effectiveness of the proposed method in application to fault diagnosis of rolling bearing, the bearing data from Intelligent Maintenance Systems (IMS) Center of University of Cincinnati [

The schematic diagram and a picture of the apparatus [

Rolling element bearing parameters.

Rolling element bearing parameters of ZA-2115 (diameter/cm) | |||
---|---|---|---|

Ball number |
Contact angle |
Ball diameter |
Pitch diameter |

16 | 15.17 | 0.331 | 2.815 |

Bearing characteristic frequencies.

Fault type | Failure frequency |
---|---|

Defect on inner race | |

Defect on outer race |

Characteristic frequencies of Rexnord ZA-2115 rolling bearing.

Fault type | Fault frequency (Hz) |
---|---|

Outer ring fault | |

Inner ring fault |

At the end of the no. 2 experiment, ORF happened on the rolling bearing. The time and frequency domain plots of the collected vibration signal of the faulty rolling bearing (772nd data) are shown in Figure

Time and frequency domain plots of collected vibration signal of ORF rolling bearing.

The optimal order is chosen as 4 among the proposed AHLP denoising method. To illustrate the effectiveness of the proposed method, the frequency domain plots of the original signal and signal processed by WPD are also presented. The wavelet function db15 with 11 layers is adopted in the WPD. To observe the fault characteristic frequency clearly and conduct a comparative analysis to verify the effectiveness of the proposed method, the partial enlarged frequency plots of frequency domain of collected signal and WPD denoised signal are shown in Figure

(a) Frequency domain plot of collected signal. (b) Frequency domain plot of denoised signal by WPD (red circles denote characteristic frequency, and green circles denote interfering frequency, as same hereinafter).

(a) Frequency domain plot of denoised signal by LP denoising method. (b) Frequency domain plot of denoised signal by AHLP denoising method.

It can be seen from Figure

To further verify the effectiveness of the proposed method in the application of weak fault diagnosis, the experiment is conducted on Drivetrain Diagnostics Simulator. Here, ORF and IRF rolling bearings are used during the experiment, and fault signals are collected to be processed. The experimental apparatus is produced by SQI Company, United States. The experiment is aimed at collecting the signal of ORF and IRF rolling bearings. The sensor is placed on the rolling bearing end plate to collect its acceleration signal. The experimental apparatus is composed of a variable speed drive, a torque transducer and encoder, a parallel shaft gearbox which includes two parallel shaft rolling bearing, and a programmable magnetic brake. The schematic diagram of the experimental apparatus and its photo are shown in Figure

The schematic diagram and the picture of the experimental apparatus. 1: variable speed drive, 2: torque transducer and encoder, 3: parallel shaft gearbox, 4: test point, and 5: programmable magnetic brake.

(a) IRF of rolling bearing, (b) ORF of rolling bearings, and (c) location of sensor.

The experiment was conducted to collect the vibration signal of weak fault rolling bearing. During the experiment, the sampling frequency is 8192 Hz, and the rotational frequency

Characteristic frequencies of FAFNIR deep groove rolling bearing.

Fault type | Fault frequency (Hz) |
---|---|

Inner ring fault | |

Outer ring fault |

The time and frequency domain plots of the collected signal of fault rolling bearing are shown in Figure

Time and frequency domain plots of collected vibration signal of IRF rolling bearing.

The vibration signal of IRF rolling bearing is used here to verify the effectiveness of proposed AHLP denoising method; thus, the optimal order can be chosen as 3 among the proposed AHLP denoising methods. To better analyze the denoising effect, the frequency range of 0–300 Hz is analyzed. To observe the fault characteristic frequency clearly and conduct a comparative analysis to verify the effectiveness of the proposed method, the partially enlarged frequency plots of frequency domain of collected signal and WPD denoised signal are shown in Figure

(a) Frequency domain plot of collected signal. (b) Frequency domain plot of denoised signal by WPD.

(a) Frequency domain plot of denoised signal by standard LP denoising method. (b) Frequency domain plot of denoised signal by AHLP denoising method.

It can be seen from Figure

In this paper, the research work elaborates the validity and effectiveness of the proposed novel AHLP denoising approach in fault diagnosis of rolling bearing, through theoretical deviation, numerical simulations, and practical applications. In the field of fault detection and isolation, it is of great significance to reduce noise in vibration signal of faulty rolling bearing, to extract fault characteristic frequencies correctly and effectively. By adopting high-order polynomials to the estimate centroid of neighborhood among LP denoising method, the proposed HLP method can achieve better noise reduction effect. By choosing the optimal order among HLP denoising method aiming at vibration signals of ORF and IRF rolling bearings, the optimal denoising effect can be obtained in the proposed AHLP denoising method. The proposed method has exhibited good performance during the numerical simulations and application researches, containing simulated and practical vibration signal processing of ORF and IRF rolling bearings. In application researches, the fault characteristic frequencies can be both well extracted for the rolling bearing with severe fault corresponding to IMS bearing data processing and the rolling bearing with incipient fault corresponding to our own conducted experiment on Drivetrain Diagnostics Simulator. The displayed results and comparative analysis indicate that the proposed method can achieve good denoising results towards different degrees of faults. To sum up, the research work in this paper can demonstrate the significance and superiority of the proposed novel approach in fault diagnosis of rolling bearing.

Furthermore, this paper mainly deals with vibration signal of ORF and IRF rolling bearing, while in the field of rotary machinery, most of the denoising methods cater to various fault detections, such as gear faults and rotor faults. In our researches, gear faults have been dealt with and proper denoising effect can be achieved, so the proposed novel approach holds potential for wide employment in practical engineering applications. For the practical application in real industrial situation, the additional overhead costs concerning hardware, software, installation, and other factors should be also considered. This aspect of the proposed approach would be researched in our future work to explore its suitable application fields.

The authors also appreciate the free download of the original bearing failure data and one photo picture provided by Intelligent Maintenance Systems (IMS) Center of University of Cincinnati.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research project is supported by the National Natural Science Foundation of China under Grant no. 51475339 and no. 51105284, the Natural Science Foundation of Hubei Province under Grant no. 2016CFA042, and the Postgraduate Overseas Visiting Scholar Fund of Wuhan University of Science and Technology.