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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The condition of locomotive bearings, which are essential components in trains, is crucial to train safety. The Doppler effect significantly distorts acoustic signals during high movement speeds, substantially increasing the difficulty of monitoring locomotive bearings online. In this study, a new Doppler transient model based on the acoustic theory and the Laplace wavelet is presented for the identification of fault-related impact intervals embedded in acoustic signals. An envelope spectrum correlation assessment is conducted between the transient model and the real fault signal in the frequency domain to optimize the model parameters. The proposed method can identify the parameters used for simulated transients (periods in simulated transients) from acoustic signals. Thus, localized bearing faults can be detected successfully based on identified parameters, particularly period intervals. The performance of the proposed method is tested on a simulated signal suffering from the Doppler effect. Besides, the proposed method is used to analyze real acoustic signals of locomotive bearings with inner race and outer race faults, respectively. The results confirm that the periods between the transients, which represent locomotive bearing fault characteristics, can be detected successfully.

Economic and social development in most countries has increased considerably the requirement for transportation capability. Railway transportation has played an important role in this development due to its strong transportation capability and high speeds. The continuous operation of trains is crucial in ensuring fluid and efficient traffic circulation. However, failure of train components can result in unexpected breakdowns, which can lead to serious traffic accidents. Hence, both the economy and human safety are at risk if trains have faulty components. Locomotive bearings support the entire weight of a train and they rotate at a high speed when the train is running. The health of these bearings is crucial for the continuous and safe operation of the train. Therefore, the development of an effective technique for monitoring locomotive bearings is profoundly significant [

A bearing usually consists of an inner race, an outer race, a cage, and a few rollers. Once one of these components suffers from a local defect, approximately periodic impacts will be generated when the defective surface comes into contact with the rollers [

The wayside acoustic defective bearing detector (ADBD) system [

Various methods have been developed for bearing fault diagnosis when no relative movement is observed between the bearing and the data acquisition system. Time-frequency analysis, which can extract information from both the time and the frequency domains, was developed for non-stationary signals. Several representative time-frequency distributions [

Matching pursuit is an adaptive approach that selects optimal atoms to approximate a signal through iterations. It is effective for analyzing bearing fault transient signals [

In this paper, a novel technique that combines a Doppler transient model and parameter identification based on the Laplace wavelet and a spectrum correlation assessment is proposed for real locomotive bearing fault detection. The Doppler transient model is constructed by considering the effect of Doppler distortion. Model parameters, including the transient periods, are identified by a correlation assessment between the envelope spectrum of the transient model and the real bearing fault signal. The results obtained through both simulations and real case studies demonstrate the remarkable performance of the technique in identifying locomotive bearing fault types.

The rest of this paper is organized as follows: Section 2 briefly describes the fundamental theory that underlies the Laplace wavelet and the correlation assessment. The proposed method is presented in Section 3, followed by the simulation analysis and the real case study in Section 4. Conclusions are presented in Section 5.

During defective bearing movement, periodic impacts occur in the obtained signals. These transient components can be matched by using elements in a model dictionary. Five representative transient models are usually used to simulate transient components caused by bearing faults, the Morlet wavelet, Harmonic wavelet, Laplace wavelet, single-side Morlet wavelet, and single-side Harmonic wavelet. The Laplace wavelet is a single-sided damped exponential function formulated as the impulse response of a single mode system. It is similar to the waveform feature commonly encountered in bearing fault signal detection tasks [_{d}

A periodic multi-transient model based on the Laplace wavelet is constructed to simulate the waveform characteristics by introducing parameter

In mathematics, the inner product serves as a powerful tool for evaluating the similarity of two time series. Suppose that two time series _{x}_{(}_{n}_{),}_{φγ}_{(}_{n}_{)}, based on the inner product, can be used to assess the degree of correlation between the two time series. Its formula is given by:

In terms of the Cauchy-Schwarz inequality, the correlation coefficient is constrained to:

The conventional bearing fault detection methods have been developed for situations with no relative movement between the signal acquisition system and the defective bearing, and thus the acquired signal is not affected by the Doppler effect, however, locomotive bearing signals suffer from high frequency shifts, frequency band expansions, and amplitude modulations due to the Doppler effect. The fault-related impact intervals are not identical in this situation. Hence, the conventional detection methods are not applicable in the diagnosis of real locomotive bearing faults. In this study, a Doppler transient model based on the Laplace wavelet and a spectrum correlation assessment is proposed to address the inability of traditional methods to handle Doppler-distorted acoustic signals in real locomotive bearing fault detection. The correlation coefficient in the frequency domain does not need to consider the transient model's time delay in this method, reducing the computation time required for parameter identification and thus improving the computational efficiency. A flowchart of the proposed scheme is shown in

The proposed method follows the steps of transient model construction, Doppler distortion, parameter identification through the assessment of the envelope spectrum correlation, and bearing fault type identification through the recognized impact periods. Each step is discussed in detail in the following subsections.

The Doppler effect was first proposed in 1842 by Austrian physicist Christian Doppler [_{s}

The Doppler effect makes traditional techniques unsuitable for processing locomotive bearing signals. To address this problem, the Doppler transient model is constructed for further analysis. The Doppler effect is embedded manually into the conventional transient model so that the constructed model is under the same distortion environment as the real locomotive bearing signal. According to acoustic theory, the following formula and procedures can be proposed:

Calculating the emission and reception time instants: The reception time instants {_{R}_{0}, _{0} + 1/_{s}_{0} + 2/_{s}_{0} + _{s}_{s}_{0}_{sw}_{e}

Interpolation: The periodic transient model _{e}_{e}_{e}

Amplitude modulation: The amplitudes of the waveform are modulated during transmission from the moving sound source to the microphone. As introduced by Morse acoustic theory [_{s}_{sw}_{s}_{sw}

As shown in ^{2}) is the amplitude modulation function and _{sw}

The Doppler effect is thus embedded in the constructed transient model to ensure that the Doppler transient model experiences the same distortion as the real locomotive bearing signal.

To simulate the characteristics of the received waveform in the fault signal of the locomotive bearing, the parameters of the constructed Doppler transient model must be adjusted to match the actual periodic impacts in the locomotive bearing signal. A suitable criterion must be established to optimize the parameters from the subsets, as shown in

The Hilbert transforms of the periodic Doppler transient model and the real locomotive bearing fault signal are obtained [_{A}(t)

Frequency spectrum analysis is performed by:

The degree of correlation of the envelope spectrum is assessed by:

The number and the diameter of the rolling elements in the locomotive bearings are represented by _{m}_{n}

If the surface of the outer race suffers a defect, then every time the rolling element passes through the crack, periodic impulses will be created with interval

Similarly, the ball pass frequency in the inner race (BPFI) is given by:

Therefore, the inner race fault characteristic frequency is equivalent to

A simulated signal contaminated by noise is investigated to confirm the effectiveness of the proposed Doppler transient model in recognizing the impact intervals embedded in fault signals. The simulated signal is represented as:

The sampling frequency is 50,000 Hz and the impact interval embedded in the simulated signal is 0.016 s. The number of data points is 12,401. A randomly distributed noise

To simulate the actual Doppler distortion caused by the relative movement between the moving sound source and the receiver, Doppler distortion is added to the simulated signal according to the procedures specified in Section 3.1. The parameters in _{s}_{sw}_{0}_{0}= 0.0124 s. Thus the reception time instants {_{R}

The proposed detection method is applied to the Doppler-distorted signal. The transient model is first constructed according to

When the impact interval of the Doppler transient model is determined to be 800/50,000 = 0.016 s, the maximal correlation coefficient of the envelope spectrum between the Doppler transient model and the simulated Doppler distorted signal can be obtained. The optimal parameters

The optimal Doppler transient model and the simulated Doppler distorted signal are shown in

Real locomotive bearing fault signals suffering from the Doppler effect are analyzed to further validate the performance and applicability of the proposed method. Two sequential experiments are conducted indoors and outdoors to obtain a Doppler-distorted acoustic signal. In the first experiment, the acoustic signals of locomotive bearings with an inner race defect and an outer race defect are acquired through the microphone. The collected acoustic signals are embedded with the Doppler effect in the second experiment. The test rigs for these experiments are illustrated in

As shown in

_{s}_{sw}

To detect the characteristic interval embedded in the fault signal, a periodical transient model with parameters adjustable using

The conventional method, which conducts the correlation assessment in the time domain, is applied to the problem for comparison. A transient model with parameters requiring optimization from sets

An outer race fault signal under a different loading, 1 t, is analyzed.

The conventional method in the time domain is again used for a comparative analysis.

The actual inner race fault signal shown in

A transient model with optional parameters is established to recognize the locomotive bearing fault. The Doppler distortion is added into the constructed model. The maximal correlation coefficients for every selected impact period after parameter optimization are shown in

A comparative analysis between the proposed method and the conventional method is also conducted on the inner race fault signal processing.

In this study, a new Doppler transient model based on the Laplace wavelet and a spectrum correlation assessment is proposed for diagnosing locomotive bearing faults. The proposed scheme includes Laplace wavelet transient model construction, Doppler distortion, spectrum correlation assessment, and parameter optimization. After implementing the proposed method, the fault-related impact interval can be successfully determined using on the optimal Doppler transient model.

The Laplace wavelet is used as the impact base function due to its superior ability to match actual bearing fault impulses. A periodical transient model based on the Laplace wavelet is constructed. The parameters of the model require optimization to properly match the real locomotive bearing fault impact interval.

Through acoustical theoretical analysis, a procedure for adding the Doppler effect to the constructed periodical transient model is proposed to simulate the Doppler distortion experienced by real locomotive bearing fault signals.

A new criterion is established to choose proper parameters during Doppler transient model construction. Correlation analysis is conducted between the envelope spectrum of the established Doppler transient model and the locomotive bearing fault signal. The parameters for obtaining the maximal correlation coefficient are found to be the optimal parameters for the model. Hence, the impact interval in the optimal Doppler transient model is recognized as the fault-related impact interval.

The results obtained by investigating both simulated signals and locomotive bearing fault signals indicate that the proposed method exhibits satisfactory performance in analyzing Doppler-distorted locomotive bearing acoustical fault signals. The proposed method could be developed further for use in a wayside train condition monitoring system.

The work described in this paper was partly supported by the Natural Science Foundation of China (Grant No. 51075379 and 51375322) and a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 122011). The authors also would like to appreciate the two anonymous reviewers for their constructive comments and suggestions.

The authors declare no conflict of interest.

(

Flowchart of the proposed locomotive bearing fault diagnosis.

Depiction of a single sound source moving in a straight line.

Simulated signals. (

The emission and reception time instants.

Maximal correlation coefficients for different elements from the set

Results obtained by the proposed method: (

Experimental setup for acoustic signal acquisition with the Doppler effect: (

Artificial defects on the (

Outer race fault signal of the locomotive bearing under the loading of 3 t: (

Maximal correlation coefficients for different elements from set

Results obtained using the proposed method: (

Maximal correlation coefficients for different elements from set

Outer race fault signal of the locomotive bearing under the loading of 1 t: (

Maximal correlation coefficients for different elements from set

Results for a loading of 1

Maximal correlation coefficients for different elements from set

Inner race fault signal of the locomotive bearing: (

Maximal correlation coefficients for different elements from set

Results obtained by using the proposed method: (

Maximal correlation coefficients for different elements from set

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 mm |

Parameters used in the first experiment.

3 t, 1 t | |

1,430 rpm | |

50 kHz |