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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (

A fault diagnosis strategy based on the wayside acoustic monitoring technique is investigated for locomotive bearing fault diagnosis. Inspired by the transient modeling analysis method based on correlation filtering analysis, a so-called Parametric-Mother-Doppler-Wavelet (PMDW) is constructed with six parameters, including a center characteristic frequency and five kinematic model parameters. A Doppler effect eliminator containing a PMDW generator, a correlation filtering analysis module, and a signal resampler is invented to eliminate the Doppler effect embedded in the acoustic signal of the recorded bearing. Through the Doppler effect eliminator, the five kinematic model parameters can be identified based on the signal itself. Then, the signal resampler is applied to eliminate the Doppler effect using the identified parameters. With the ability to detect early bearing faults, the transient model analysis method is employed to detect localized bearing faults after the embedded Doppler effect is eliminated. The effectiveness of the proposed fault diagnosis strategy is verified via simulation studies and applications to diagnose locomotive roller bearing defects.

Bearing defects are the dominant type of fault in railway vehicles, which leads to serious accidents and significant costs for the rail transport industry [

However, the effectiveness of the wayside acoustic monitoring-based technique is decreased when vehicles pass by at high speeds. One of the problems caused by the high relative movement is the Doppler effect, as it can lead to obvious frequency shifts, frequency band expansion, and amplitude modulation for the recorded acoustic signal, which reduces the diagnostic performance [

Aside from the wayside acoustic monitoring system, the Doppler effect also widely exists in the signal from a moving acoustic source. For example, in the areas of underwater acoustic communication and acoustical holography for moving vehicles, the Doppler effect contained in the recorded acoustic signal is also a barrier that could significantly diminish the effectiveness of signal processing. Stojanovic

Dybała [

In this paper, a fault diagnosis strategy is invented for locomotive bearing fault diagnosis based on the wayside acoustical monitoring technique. Through the proposed strategy, the Doppler effect embedded in the recorded bearing acoustic signal can be eliminated by a Doppler effect eliminator. In the Doppler effect eliminator, a so-called Parametric-Mother-Doppler-Wave (PMDW) is constructed based on the kinematic model parameters. The kinematic model parameters, including the moving speed and sound speed of the railway vehicle, as well as geometric parameters of the model are then identified via correlation filtering analysis. All of the parameters can be identified based on the signal itself. A time domain signal resampler is invented and employed to eliminate the Doppler effect using the identified kinematic model parameters. After the embedded Doppler effect is eliminated, the transient model parameters of the Doppler-free signal are identified to detect the localized bearing faults.

The rest of this paper is organized as follows: Section 2 introduces the proposed locomotive bearing fault diagnosis strategy. The construction of the PMDW, kinematic model parameter identification based on correlation filtering analysis, Doppler effect elimination based on the resampling method, and fault feature extraction based on transient model analysis are all introduced in this section. A simulation case study is provided in Section 3. An experimental verification test using defective locomotive roller bearings with outer race defect and inner race defect is discussed in Section 4. Finally, Section 5 presents the concluding remarks.

In the wayside acoustic bearing monitoring system, microphones are fixed by the wayside to record the acoustic signals emitted by the bearings of a passing vehicle. The basic kinematic model (

Given the high relative speed between the railway vehicle and the microphone, the recorded signal is distorted by the Doppler effect, which causes the signal frequency to shift and the frequency band to expand. This condition is a barrier to further analysis, especially for the methods based on frequency domain analysis.

Two steps are implemented for the proposed strategy to detect the localized faults of the bearing of the fast moving railway vehicle. In the first step, the embedded Doppler effect is eliminated. The frequential structure disturbance is eliminated, and the amplitude is demodulated. In the next step, the transient model parameters of the Doppler-free signal are identified to detect the localized bearing faults. A flowchart of the proposed strategy that includes two signal processing modules is presented in

As shown in _{dop}_{0}_{0}

When the different structures of the passing vehicles are considered, the parameter _{0}, _{0}

A data-driven Doppler effect eliminator is invented in this paper. Through this eliminator, all of the kinematic model parameters can be identified based on the recorded signal itself. A flowchart of the eliminator is shown in _{k}

In the following subsection, the correlation filtering analysis method and the two signal processing modules (

The construction of the PMDW is introduced in this section. As illustrated in _{e}_{c}t_{e}_{e}_{s}_{e}_{e}_{r}_{r}_{r}

According to the Morse acoustic theory [

The detailed procedure of the PMDW construction is presented in _{c}

These parameters belong to the subsets _{fc}_{c}_{r}_{X0}_{V0}_{a}

Given an initialized model parameter set of γ_{Ψ}_{c}_{0}_{0}

_{c}_{e}_{e}

_{r}

_{r}_{r}^{p}_{r}_{r}

^{p}_{d}_{e}_{0}/c_{0} represents the distance between the starting point

Two PMDWs are generated through the preceding steps. The PMDW shown in

Correlation filtering analysis aims to measure the strength and direction of the linear relationship between two signals by calculating the Pearson's correlation coefficient between them [

In the current paper, the kinematic model parameter set described by

Given an input Doppler-shifted signal, _{dop}

The discrete-time analytic signal of the parametric wavelet is obtained:

The Pearson's correlation coefficients are calculated as follows:
_{γ}_{γ}_{dop}_{dop}

The criterion of the inherent linear relationship between

After the kinematic model parameters are identified, the Doppler-shifted signal and the identified kinematic model parameters are inputted into the signal resampler, through which the embedded Doppler effect can be clearly eliminated.

If the identified kinematics model parameters are as follows:
_{r}^{dop}^{dop}

Emit-time-vector:
^{dop}

Receive-time-vector:
^{dop}^{dop}

According to ^{dop}

and cos^{dop}

The detailed procedure of the signal resampler is shown in

Given a certain identified kinematic model parameter set of _{k}^{opt}^{opt}_{0}^{opt}_{0}^{opt}^{opt}

_{d}^{dop}_{0}^{dop}/c^{opt}_{0}^{dop}/c^{opt}_{s}_{0} ^{dop}/c^{opt}^{dop}-1)/f_{s}^{d}_{d}^{dop}

_{r}^{dop}^{d}_{r}^{dop}

_{r}_{e}^{dop}

The Doppler effect is eliminated through the aforementioned data-driven Doppler effect eliminator. Conventional fault feature extraction methods can be employed to analyze the Doppler-free signal, extract features, and then make a maintenance decision. In the past decades, numerous methods have been proposed to extract features for bearing signals, such as time-domain analysis [

A method of transient modeling by wavelet and parameter identification based on correlation filtering is first introduced and applied on bearing fault diagnosis by Wang

The rolling element bearing typically consists of an inner race, an outer race, a number of rolling elements, and a cage. Once a localized fault is formed on the surface of the inner or outer race, a transient with an exponential decay is generated by the roller striking the localized fault.

The Laplace wavelet, a single-sided damped exponential function formulated as the impulse response of a single mode system, is highly similar to the waveform feature commonly encountered in bearing fault signal detection tasks.

The results reported in reference [_{d}

The speed variation has been removed during the procedure of Doppler effect elimination, however, the train of transients is not a strict periodic phenomenon when considering the “jitter” [

If the surface of the outer race of the bearing suffers a single defect based on the bearing geometries and rotation speed, _{r}_{m}_{n}

Similarly, if a single defect occurs on the surface of the inner race of the bearing, the ball pass frequency over the inner race defect (BPFI) can be obtained by the following:

Every time the rolling element passes through the defect, periodic impulses are created with time interval,

After the Doppler effect embedded in the acoustic signal of the bearing is eliminated. The time interval can be identified through the improved correlation filtering analysis introduced in Section 2.1.2 between the periodic multi-transient model shown in

In this section, a simulated Doppler-shifted bearing signal is analyzed to verify the effectiveness of the investigated diagnosis strategy. The source signal of the bearing without the Doppler effect can be described as follows:
_{0}, and frequency, _{0}, are set at 0.05 and 1000 Hz, respectively. The impact interval embedded in the simulated signal is 0.025

A randomly distributed noise,

Through the PMDW generator introduced in Section 2.1.1, a sine wave can be embedded with the Doppler effect. If the sine wave is replaced with the aforementioned simulated signal, a Doppler-shifted bearing signal can be obtained with the same procedure and with the following kinematic parameters: _{0}_{0}_{1}^{2}. The wave form and FFT spectrum of the simulated Doppler-shifted bearing signal are illustrated in

The proposed diagnosis strategy is then applied to the Doppler-shifted bearing signal. The Doppler-shifted bearing signal is first inputted to the Doppler effect eliminator introduced in Section 2.1 to eliminate the embedded Doppler effect. Through the eliminator, the kinematic parameters are first identified through the correlation filtering analysis introduced in Section 2.1.2 between the input Doppler-shifted signal and the PMDWs constructed with parameter subsets described by

The selection of the parameter subsets is crucial. On the one hand, the larger interval range and the smaller step of the parameter subset obtain a more accurate result. On the other hand, the larger interval range and the smaller step of the parameter subset cost excessive computation and decrease the efficiency of the method. When both accuracy and efficiency are considered, the subset, _{fc}_{c}_{r}_{X0}_{V0}_{1}_{a}

A searching grid of the model parameters is constructed based on the aforementioned six parameter subsets. Once a group of parameters is determined, the parameters are inputted to the PMDW generator introduced in Section 2.1.1 to generate a PMDW. The correlation filtering analysis introduced in Section 2.1.2 is then performed between the PMDW and the simulated Doppler-shifted signal.

The identified kinematic model parameters and the Doppler-shifted bearing signal are then inputted to the signal resampler introduced in Section 2.1.3 to eliminate the embedded Doppler effect. First, the amplitude vector of the input Doppler-shifted signal is matched with the delayed-time-vector, _{d}^{dop}_{0}^{dop}/c^{opt}_{0}^{dop}/c^{opt}_{s}_{0}^{dop}/c^{opt}^{dop}_{s}^{d}_{d}^{dop}_{r}^{dop}_{e}^{dop}_{s}_{s}^{d}_{r}^{dop}_{r}^{dop}_{r}^{dop}_{e}^{dop}

The wave form of the obtained Doppler-free signal is plotted as an overlay on the original bearing signal, shown in

After the Doppler effect is eliminated, the transient model analysis method introduced in Section 2.2 is applied to the Doppler-free signal to detect the defect. The transient model is constructed according to _{c}

As a comparison, the simulated signal before the Doppler effect elimination is also analyzed via the transient model analysis method with the same model parameter subsets. The maximal correlation coefficients for the different elements from set

Two separate experiments were implemented to achieve the Doppler-shifted acoustic signals from defective locomotive roller bearings and verify the effectiveness of the proposed method. The obtained Doppler-shifted bearing acoustic signals are then analyzed by the proposed diagnosis strategy.

In the first experiment, the acoustic signal of a locomotive roller bearing with a single localized defect was acquired by the test bench in

The second experiment can be referred to the model illustrated in

A single artificial crack with 0.18 mm width was set by the wire-electrode cutting machine on the outer and inner race, as shown in

The wave form of the Doppler-shifted bearing signal with the out-race defect is shown in

The proposed diagnosis strategy is then applied to the filtered Doppler-shifted bearing signal. The signal is first inputted to the Doppler effect eliminator introduced in Section 2.1 to identify the kinematic model parameters and eliminate the embedded Doppler effect. During the kinematic parameter identification procedure, the subset _{fc}_{c}

The value of sound speed

The parameter sets _{r}_{X0}_{1}_{V0}_{a}

PMDWs are then generated with different parameters from the previously determined parameter subsets by the PMDW generator introduced in Section 2.1.1 to generate a PMDW. The correlation filtering analysis introduced in Section 2.1.2 is then performed between the generated PMDW and the Doppler-shifted signal.

The signal resampler introduced in Section 2.1.3 is then employed to eliminate the embedded Doppler effect with the identified kinematic parameters in _{d} ^{dop}_{0}^{dop}/c^{opt}_{0}^{dop}/c^{opt}_{s}_{0}^{dop}/c^{opt}^{dop}_{s}_{0}^{dop}^{d}_{d} ^{dop}_{r}^{dop}_{e}^{dop}_{s}_{s}^{d}_{r}^{dop}_{r}^{dop}_{e}^{dop}

The wave form of the obtained Doppler-free signal is shown in

The transient model analysis method introduced in Section 2.2 is then applied to detect the characteristic interval of the Doppler-free fault signal. A periodical transient model with parameters adjustable using

The outer race characteristic frequency is 138.74 Hz as calculated by

The signal before the Doppler effect elimination is also analyzed via transient model analysis method with the same model parameter subsets. The maximal correlation coefficients for the different elements from set

The obtained Doppler-shifted bearing signal (

The proposed Doppler effect eliminator is first applied to identify the kinematic model parameters and eliminate the embedded Doppler effect. The subset _{fc}_{c}_{r}_{X0}_{V0}_{a}

The embedded Doppler effect is then eliminated through the signal resampler introduced in Section 2.1.3. First, by matching the amplitude vector of the input Doppler-shifted signal with the delayed-time-vector, _{d}^{dop}_{0}^{dop}/c^{opt}_{0}^{dop}/c^{opt}_{s}_{0}^{dop}/c^{opt}^{dop}_{s}_{0}^{dop}^{d}_{r}^{dop}_{e}^{dop}_{s}_{s}^{d}_{r}^{dop}_{r}^{dop}_{e}^{dop}

The wave form of the obtained Doppler-free signal is illustrated in

The transient model analysis method introduced in Section 2.2 is then applied to detect the characteristic interval embedded in the Doppler-free fault signal. A periodical transient model with parameters adjustable using

The periodical impact interval is 0.0052 s, calculated by

The Doppler-shifted signal is then directly analyzed by the transient model analysis method with the same model parameter subsets. The maximal correlation coefficients for the different elements from set

In this paper, a fault diagnosis strategy based on the wayside acoustic monitoring technique is invented for locomotive bearing fault diagnosis. A parametric wavelet called PMDW is introduced and employed to identify the kinematic model parameters based on correlation analysis. A time domain signal resampler is introduced and employed to eliminate the embedded Doppler effect in the recorded bearing acoustic signal. The transient model analysis method is also employed to detect the localized bearing faults after the Doppler effect is eliminated. One of the best benefits of the proposed strategy is that all the kinematic model parameters, including the sound speed and the moving speed of the vehicle, as well as the geometric parameters of the model, can be identified based on the recorded signal itself. Thus, the proposed strategy overcomes the difficulties of kinematic model parameter measurement and is adjustable to different types of passing vehicles. Besides, the embedded Doppler effect can be eliminated through the proposed strategy, paving the way for the conventional invented signal processing methods and feature extraction methods. The performance of the proposed strategy has been evaluated by both simulated and practical Doppler-shifted bearing signals carrying fault information. Given the merits revealed in this study, the proposed fault diagnosis strategy can be widely used in wayside health monitoring systems, particularly in situations when vehicles pass by at high moving speeds and kinematic model parameters are difficult to estimate. The proposed data-driven Doppler effect eliminator is also hopeful to be used in other areas such as acoustic communication techniques and sound field holography for moving vehicles.

This work is supported by the National Natural Science Foundation of China under Grant 51075379 and 51005221 and partly by the Natural Science Major Project of Education Department of Anhui Province (No.KJ2013A010).

The authors declare no conflict of interest.

Basic kinematic model of the wayside acoustic bearing monitoring system.

Flowchart of the proposed wayside bearing fault diagnosis strategy.

Flowchart of the Doppler effect eliminator.

Procedure of the construction of the PMDW.

Illustration of the procedure of the construction of a PMDW and procedure of the resampler.

Two PMDWs generated by the invented PMDW generator. (

Procedure of the signal resampler.

Procedure of the transient model analysis method.

(

(

Maximal correlation coefficients for the different elements from a specified PMDW model parameter subset for the simulated signal.

(

Comparison between the obtained Doppler-free signal and the original simulated bearing signal (

Results for simulated Doppler-shifted bearing signal using the transient model analysis method. (

Experimental setup for signal acquisition with Doppler effect (

Artificial defects on the components of the bearing (

(

Maximal correlation coefficients for the different elements from a specified PMDW model parameter subset for the Doppler-shifted signal with the out-race defect.

(

Results for Doppler-shifted bearing signal with the out-race defect using the transient model analysis method (

(

Maximal correlation coefficients for the different elements from a specified PMDW model parameter subset for the Doppler-shifted signal with the inner-race defect.

(

Results for Doppler-shifted bearing signal with the inner-race defect using the transient model analysis method (

Kinematic model parameters subset and the optimal kinematic model parameters of the simulated signal.

_{c} |
_{0} |
_{0} |
_{0}^{2}] | |||
---|---|---|---|---|---|---|

Range | [800,1200] | [320,360] | [0.5,1.5] | [2,6] | [20,40] | [35,45] |

Step | 10 | 1 | 0.1 | 0.1 | 0.1 | 0.5 |

Optimal value | 1000 | 340 | 1 | 4 | 30 | 40 |

True value | 1000 | 340 | 1 | 4 | 30 | 40 |

Error | 0% | 0% | 0% | 0% | 0% | 0% |

Comparison of the transmit model analysis results between the signals before and after Doppler effect elimination.

_{res} |
||||
---|---|---|---|---|

Before Doppler effect elimination | 0.195 | 1060 | 0.019 | 0.028 |

After Doppler effect elimination | 0.780 | 1000 | 0.050 | 0.025 |

Specifications of the testing bearing.

Diameter of the outer race | 250 mm |

Diameter of the inner race | 130 mm |

Pitch diameter ( |
190 mm |

Diameter of the roller ( |
32 mm |

Number of the roller ( |
14 |

Kinematic model parameter subsets and the optimal kinematic model parameters of the bearing signal with the out-race defect.

_{c} |
_{0} |
_{0} |
_{0}^{2}] | |||
---|---|---|---|---|---|---|

Range | [800:1600] | [320:360] | [1.5:2.5] | [2:6] | [20:40] | [0:10] |

Step | 10 | 1 | 0.1 | 0.1 | 0.1 | 1 |

Optimal value | 1250 | 339 | 2 | 3.9 | 30.5 | 3 |

Kinematics model parameters subset and the optimal kinematics model parameters of the bearing signal with the inner-race defect.

_{c} |
_{0} |
_{0} |
_{0}^{2}] | |||
---|---|---|---|---|---|---|

Range | [1400:2000] | [320:360] | [1.5:2.5] | [2:6] | [20,40] | [0,10] |

Step | 10 | 1 | 0.1 | 0.1 | 0.1 | 1 |

Optimal value | 1750 | 342 | 1.9 | 4.1 | 32 | 5 |