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Wayside acoustic diagnosis of defective train bearings basedon signal resampling and information enhancement

Qingbo He n, Jun Wang, Fei Hu, Fanrang Kong

Department of Precision Machinery and Precision Instrumentation, University of Science and Technology of China, Hefei,

Anhui 230026, PR China

a r t i c l e i n f o

Article history:

Received 8 December 2012

Received in revised form

21 May 2013

Accepted 24 May 2013

Handling Editor: K. ShinAvailable online 25 June 2013

a b s t r a c t

The diagnosis of train bearing defects plays a significant role to maintain the safety of

railway transport. Among various defect detection techniques, acoustic diagnosis is

capable of detecting incipient defects of a train bearing as well as being suitable for

wayside monitoring. However, the wayside acoustic signal will be corrupted by the

Doppler effect and surrounding heavy noise. This paper proposes a solution to overcome

these two difficulties in wayside acoustic diagnosis. In the solution, a dynamically

resampling method is firstly presented to reduce the Doppler effect, and then an adaptive

stochastic resonance (ASR) method is proposed to enhance the defective characteristic

frequency automatically by the aid of noise. The resampling method is based on a

frequency variation curve extracted from the time–frequency distribution (TFD) of an

acoustic signal by dynamically minimizing the local cost functions. For the ASR method,

the genetic algorithm is introduced to adaptively select the optimal parameter of the

multiscale noise tuning (MST)-based stochastic resonance (SR) method. The proposed

wayside acoustic diagnostic scheme combines signal resampling and information

enhancement, and thus is expected to be effective in wayside defective bearing detection.

The experimental study verifies the effectiveness of the proposed solution.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Railway traffic plays a critically significant role in the transport system for rapid development of national economy.

Currently, train speed increase is one of the main trends of railway development. For the speed increase, it is an important

mission to guarantee the safety, the stability and uninterrupted operation of trains for passenger and freight transportation.

There are hundreds of rolling bearings in a train with a significant relation for the train running. As reported, bearing failureis the most common type of train faults [1–4]. Hence it is of great importance to monitor the health conditions and detect

the incipient defects of train bearings, to avoid costly train stoppages or even catastrophic derailments caused by bearing

failure.

In bearing defect diagnosis, increasing attentions have been paid to diagnostic techniques without disassembling the

train bearings, such as oil monitoring, hot-box detection, vibration signal analysis, acoustic emission (AE) and acoustic signal

analysis methods. First, oil monitoring technique is to obtain the lubrication and wear conditions related to the defects

through analyzing the oil sample of lubricant used in train bearings [5–7]. The technique is easy to be operated and effective

to detect incipient fatigue damage of the bearing. However, it is only suitable for oil lubrication bearings, and not suitable for

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/jsvi

Journal of Sound and Vibration

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jsv.2013.05.026

n Corresponding author. Tel.: +86 551 6360 7985.

E-mail address: [email protected] (Q. He).

Journal of Sound and Vibration 332 (2013) 5635–5649

http://www.sciencedirect.com/science/journal/0022460Xhttp://www.elsevier.com/locate/jsvihttp://dx.doi.org/10.1016/j.jsv.2013.05.026mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.jsv.2013.05.026http://dx.doi.org/10.1016/j.jsv.2013.05.026http://dx.doi.org/10.1016/j.jsv.2013.05.026http://dx.doi.org/10.1016/j.jsv.2013.05.026mailto:[email protected]://crossmark.dyndns.org/dialog/?doi=10.1016/j.jsv.2013.05.026&domain=pdfhttp://crossmark.dyndns.org/dialog/?doi=10.1016/j.jsv.2013.05.026&domain=pdfhttp://crossmark.dyndns.org/dialog/?doi=10.1016/j.jsv.2013.05.026&domain=pdfhttp://dx.doi.org/10.1016/j.jsv.2013.05.026http://dx.doi.org/10.1016/j.jsv.2013.05.026http://dx.doi.org/10.1016/j.jsv.2013.05.026http://www.elsevier.com/locate/jsvihttp://www.sciencedirect.com/science/journal/0022460X


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grease lubrication bearings. In addition, the rotating speed of the train bearing should also be slow [8,9]. Second, hot-box

detection method is to judge the degree of bearing wear through detecting the temperature condition of bearing box. There

are two kinds of hot-box detection systems: one is on-vehicle system with a temperature sensor being set on the bearing

housing, the other is wayside infrared detection system. Only in severe faulty conditions will the temperature of the train

bearing raise. Thus, the temperature monitoring method does not have the capability of early failure detection, which is

dangerous for the high speed trains [1,10]. Third, vibration signal analysis is the most popular method for the diagnosis of

rolling bearing defects [2,11]. There has been developing increasing diagnostic approaches based on vibration signals on

account of its sensitivity to most of bearing defects [12–

14]. However, the accelerometers should be attached on the bearinghousing for vibration signal acquisition, which would make the measurement system enormous and expensive as for

hundreds of bearings on a train. Fourth, AE technique is to measure the transient elastic waves generated by the interaction

of bearing elements when a defect appears [11,15–18]. It has been proved that AE could offer an earlier and more reliable

indication of bearing degradation than vibration signal and has the capability of detecting both surface and subsurface

defects [11,16,17]. However, the AE measurement suffers from the drawback of signal attenuation and difficulty of signal

processing, interpreting and classifying [15], as well as the difficulty of detecting inner race defects of a bearing [18]. The AE

sensors also have to be placed as close as possible to the train bearings as the accelerometers do, which is also not

convenient due to the complexity of the monitoring system. Finally, the train bearing acoustic signal with a frequency range

from 3 Hz to 40 kHz can also be monitored by measuring sound pressure [19,20] or sound intensity [3,21] of the bearings.

The sound amplitude is nearly proportional to vibration acceleration in the same direction [19] and is thus sensitive to

incipient defects of the bearing, too. Due to non-contact measurement for acoustic signals [22,23], the acoustic signal

analysis is economical and practicable in wayside real-time bearing defect detection. The acoustic detection systems could

be placed at crucial locations on both sides of the railway track, and each system could monitor thousands of train bearingstraveling through each day [4]. Therefore, this paper focuses on acoustic signal analysis for diagnosis of train bearing defects.

There are two main challenges in the wayside acoustic diagnosis of train bearing defects. The first one is the remedy for

Doppler shift in the acquired acoustic signal. A dynamic resampling method based on Hilbert transform and analytical

approach has been proposed to reduce the Doppler effect resulted from the relative motion [24]. However, for the Hilbert

transform based resampling method, a band-pass filter should be first selected for the original signal, which may be

unrealizable when confusion occurs between adjacent frequencies for the high-speed motion, or when the signal is buried

in heavy noise for complex environment; while the analytical approach based resampling method strictly depends on the

spatial and temporal parameters between the microphone and moving vehicle when measuring the acoustic signal. The

second challenge is to enhance the weak defective information from heavy background noise coming from other coupled

train components and measuring environment, especially in the low-frequency region [19], even when the Doppler effect

has been removed, because the heavy noise would submerge some early signatures of bearing defects. In traditional de-

noising solutions, an intuitive one is to remove unwanted noise [25], which, however, may also make the desired signal

distorted simultaneously. Moreover, the blind source separation (BSS) technique has been employed to detect fault-relatedsignal from corrupted machine sound by separating signal and noise [26,27]. In addition, the stochastic resonance (SR) is an

opposite idea to make use of the noise to enhance the weak periodic signal and improve the signal-to-noise ratio (SNR) [28].

Because the defect-induced vibration response generally behaves as periodic transient impulses in a rotating machine, the

SR theory has been developed by many researchers for periodic signature detection in the field of machine fault diagnosis

[29–34] due to its special merit in noise utilization for signal processing. All of the developed SR methods including the SR

with multiscale noise tuning (MST), proposed by our group [32–34], have been confirmed to be effective in vibration signal

analysis. However, none of them has been applied to analysis of acoustic signals, such as the situation considered in the

wayside acoustic diagnosis study.

This paper explores a wayside acoustic analysis solution for train bearing defect diagnosis. It is a combination of signal

resampling and information enhancement. Specifically, the Doppler effect of acoustic signal is first reduced by a new

resampling method according to a frequency variation curve extracted from the time–frequency domain. Then the weak

defective information is enhanced by an adaptive SR (ASR) method. Considering that the noise in the resampled signal is no

longer white noise, the SR with MST method is introduced because of its merit in utilizing colored noise [34]. Since theperformance of SR with MST relies on the value of multiscale tuning parameter, this paper again proposes an intelligent

optimization method based on genetic algorithm to adaptively deliver an optimal system output. The proposed solution

takes advantages of signal resampling for Doppler effect reduction and information enhancement for noise treatment, and

thus is expected to be effective in wayside defective bearing detection. In the following Section 2, a train bearing test rig is

established and the wayside acoustic measurement experiments are carried out to formulate the problem to be studied.

Then the proposed solution is presented in Sections 3 and 4 in detail. The experimental results as provided in Section 5

show that the proposed combination solution can well overcome the difficulties in Doppler effect and noise influence being

involved in the wayside acoustic defective bearing diagnosis problem. Finally, Section 6 draws some concluding remarks.

2. Experimental setup and problem formulation

For the study of wayside acoustic diagnosis of train bearing defects, a train bearing test rig was first made by ourselves toacquire the static acoustic signals of train bearing with outer-race and inner-race defect, as shown in Fig. 1. The static

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defective acoustic signal is then set as the sound source on a moving vehicle which passes by a microphone with a constant

speed to acquire the wayside acoustic signal corrupted by the Doppler effect, as illustrated in Fig. 2.

As seen in Fig. 1(a), the train bearing test rig is made up of a system stand, an induction motor for driving the bearings, a

motor shaft, two supporting bearings, a tested train bearing, a mechanical loader, and an acoustic signal acquisition system.

The motor shaft being supported by two bearings passes through the inner raceway of the tested train bearing. The outerring of the tested bearing is attached to the system stand with a radial force given by the loader. The train bearings used in

this test are all single row cylindrical roller bearings with the type of NJ(P)3226X1. The details of the geometry of this type of

bearing are given in Table 1. Defects in the form of 0.18 mm groove across the outer raceway and the inner raceway were set

separately using electro-discharge machining, as displayed in Fig. 1(b). The acoustic signal acquisition system includes a

microphone (Model: Type 4944-A, Manufacturer: Brüel and Kjær) as the sound pressure sensor being put near the tested

train bearings, an NI data acquisition system (DAS) for signal collection, and a laptop with data acquisition software for data

recording.

The acquired acoustic data are then converted to be audio signals, which are broadcasted by a speaker being put on a

traveling vehicle separately, for acquiring the wayside acoustic signals of defective train bearings, as shown in Fig. 2. The

acoustic signals are collected and preserved by the same acoustic signal acquisition system as above. The microphone is held

in position by a bracket with some vertical distance to the path of vehicle moving.

The wayside acoustic testing can be illustrated in Fig. 2(b), where the notations are explained as follows: V is the vehicle

moving speed, r is the vertical distance between the microphone and the path of vehicle moving, S is the horizontal distancebetween the microphone with the starting point of signal recording on the path of vehicle moving, x(t ) is the horizontal

Laptop

Train bearing

NI data acquisition

system

Induction motor Microphone

Mechanical

loader

Outer-race

defect

Inner-race

defect

Fig. 1. Train bearing test rig: (a) experimental system and (b) tested defective bearings.

Laptop

Microphone

NI data acquisitionsystem

Sound source

V

V t x(t )

S

R(t )r

Microphone

Vehicle

Sound source

(t )θ

Fig. 2. Wayside acoustic signal acquiring experiment: (a) experimental site and (b) testing model.

Table 1

The geometry of the train bearing with the type of NJ(P)3226X1.

Inside diameter (mm) Outside diameter(mm) Thickness(mm) Pitch circle diameter (mm) Roller diameter (mm) No. of rolling elements

158 250 80 190 32 14

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distance between the microphone and the vehicle position on the path, R(t ) is the linear distance from the microphone to

the vehicle position, and θ (t ) is the deflection angle of the vehicle moving between its velocity direction and the microphone

direction. Since the microphone is stationary while the sound source is moving with a constant speed, the signal frequencies

as well as the surrounding noise will be corrupted by the Doppler effect. That is to say, the Doppler effect will make an

acquired wayside acoustic signal blurred in the spectrum due to its non-stationary property. Under this situation, we focus

on the problem how to effectively extract the characteristic frequencies of defective bearings from the wayside acoustic

signals with Doppler effect and noise corruption. In the following, this paper will introduce a solution by combining signal

resampling and information enhancement.

3. Signal resampling for reduction of Doppler effect

The Doppler effect involved in the acoustic signals should be firstly removed to achieve a reliable analysis result. Due to

the relative motion, an invariant frequency f 0 contained in the sound source will be time-varying in the acquired acoustic

signal, which can be denoted by f 0(t ). With a constant sampling interval Δt s of DAS, the number of sampling points in one

period, 1/ f 0(t i), will be also time-varying. To rectify the local frequency f 0(t i) to be f 0, which can be called target frequency, the

resampling with variable intervals should be conducted on the acquired data to satisfy that there contains a constant

number n of sampling points in each period as below

1

f 0ðt iÞ⋅Δt i¼ 1

f 0⋅Δt s¼ n (1)

where Δt i represents the resampling interval for f 0(t i). Then Δt i can be obtained as:

Δt i ¼ f 0⋅Δt s f 0ðt iÞ

: (2)

It can be seen in Eq. (2) that the dynamic resampling interval is strictly dependent on the local frequency derived from f 0.

Therefore, the variation of f 0 influenced by the Doppler effect should be first discovered.

For the model in Fig. 2(b), according to the Doppler-shift theory of Morse [35], the time-variation of f 0 can be formulated as

f 0ðt Þ ¼ f 0M ðS −Vt Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðS −Vt Þ2 þ ð1−M 2Þr 2

q

ð1−M 2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiðS −Vt Þ2 þ ð1−M 2Þr 2

q (3)

where M ¼V /c with c being the sound velocity. It can be seen in Eq. (3) that the relationship between f 0(t ) and f 0 is strictlydependent on the spatial and temporal parameters between the microphone and moving vehicle, as well as the sound velocity

in the air. These parameters may be difficult to be measured precisely in practice. For a given signal corrupted by the Doppler

effect without any other information, the instantaneous frequency can be also estimated by the Hilbert transform after a band-

pass filtering for the target frequency [24]. However, it is hard to select a passing band in some cases when adjacent

frequencies blend together resulting from the Doppler effect in high-speed motion condition, or when the signal is buried in

heavy noise in complex environment. To solve this problem, this paper extracts the variation curve of target frequency based

on the time–frequency distribution (TFD) after conducting short-time Fourier transform (STFT) on the acoustic signal.

The STFT is capable of converting the non-stationary signal from 1-D time domain to a 2-D time–frequency domain,

which provides us a direct viewing for the variation of all frequencies contained in the wayside acoustic signal. The TFD, i.e.

the magnitude of STFT on the acoustic signal y(t ), is defined as

TF ðt ; f Þ ¼

Z þ∞−∞

yðτ Þhnðτ −t Þe−i2π f τ dτ

(4)

where h(t ) is a short-time analysis window centered at t ¼0. The width of h(t ) should be large enough to achieve a highfrequency resolution on the TFD, which can distinguish the adjacent frequencies from each other. The TFD reflects the

energy distribution on the time–frequency plane. A frequency variation curve can be regarded as a ridge on the time–

frequency plane, which concentrates the most energy among the local frequency areas. There may lie more than one ridge

on the TFD, but we just need select one of them as the target frequency f 0 because the others follow the same variation rule.

A ridge detection algorithm for continuous wavelet transform has been proposed based on a cost function [36], which is

non-sensitive to noise comparing with the direct maximum method [36,37]. However, this method extracts the ridge from

the time-scale domain. Besides, it is only suitable for single-ridge detection. To extract the variation curve of target

frequency from the time–frequency domain with multiple ridges, a new ridge detection algorithm for STFT is proposed here

by dynamically minimizing the local cost functions, which is described as follows.

To avoid the interference of local strong noise when searching for a ridge on the time–frequency plane, all the points

with large amplitude in the local area should be considered, and the smoothness of the curve should be guaranteed at the

same time. Thus the cost functions are defined as:

CF k ¼ j f kðiÞ− f k−1ðc Þj2−ekjTF ðt k; f kðiÞÞj2; k ¼ 2; 3; …; m (5)



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where m is the length of the column of matrix TF (t , f ), ek is the weight of matrix element amplitude corresponding to the

gradient of ridge, f k(c ) is the frequency minimizing the cost function CF k and is taken as the local frequency at the time t k on

the expectant curve, and f k(i) (i¼1, 2,…) are the candidates for seeking f k(c ). Generally, all the frequencies at t k are taken asthe candidates, but the computational load is heavy. The reference [36] chooses the frequencies corresponding to localmaxima of amplitude among the whole frequency range at each time point, which is still not suitable when multiple ridges

appear on the time–frequency plane. This paper restricts the search range to a small frequency band for one ridge

extraction. And the frequencies that make the value of TF (t k, f k) reach its local maxima among this local frequency area are

selected as the candidates. Therefore, the functions in Eq. (5) are called local cost functions. The frequency band FBk and the

weight factor ek also vary with time as:

FBk ¼ ½ f k−1ðc Þ− f w; f k−1ðc Þ þ f w; k ¼ 2; 3; …; m; (6)

ek ¼ f w

max f k∈FBk

TF ðt k; f kÞ

24

35

2

; k ¼ 2; 3; …; m (7)

with f w as half-width of the frequency band. Given a starting point (t 1, f 1) and a proper value of f w, the other points on theexpectant ridge will be extracted step by step via dynamically minimizing the local cost functions in Eq. (5). The line

connecting these points is precisely the variation curve of the selected target frequency. In the wayside acoustic signal, the

frequency variation curve just follows the function in Eq. (3). In the study, the starting point is chosen as center point of the

frequency band at the beginning of the variation curve. In addition, the value of f w is firstly set as half of the maximum width

of the curve band and finally determined depending on the effect of extracted ridge.

According to the theories above, a flowchart of signal resampling for the Doppler effect reduction can be provided in

Fig. 3. It includes the following several steps. (1) The STFT is conducted on the acoustic signal to obtain the TFD TF (t , f ) of the

wayside acoustic signal y(t ). (2) A target frequency f 0 is chosen and its variation curve (ridge) f 0(t ) is extracted on the TFD

according to the local cost functions in Eq. (5). (3) The dynamic resampling interval series {Δt i} is calculated by Eq. (2). (4)

The acoustic signal is resampled according to {Δt i}, and the corrected signal yr (t ) is finally achieved.

To verify the effectiveness of the proposed method of Doppler effect reduction for wayside acoustic signal of a defective

bearing, a simulated signal is constructed to simulate the defective static acoustic signal of train bearing as below:

yðt Þ ¼ e−α ⋅modðt ;1= f dÞ sin ð2π f r t Þ (8)

where α ¼1200, f d¼160 as the characteristic frequency of train bearing and f r ¼1500 being the central frequency of theresonance band. As shown in Fig. 4(a), the simulated signal (sampling frequency: 6 kHz) is in the form of amplitude

modulation with f d as the modulation frequency. The simulated signal plus a Gaussian white noise (SNR: −20 dB) is taken as

the sound source in the model shown in Fig. 2(b). With the speed of the vehicle as 20 m/s, the signal acquired by the

wayside microphone is shown in Fig. 4(b) according to the acoustic theory of sound emission from moving sources [35]. As

seen in the spectrum in Fig. 4(b), the frequencies contained in static signal in Fig. 4(a) are completely contaminated by the

Doppler effect and heavy noise, which makes it impossible for the Hilbert transform based resampling method. Usually,

envelope spectral analysis is a general approach to identify the characteristic frequency. However, as displayed in Fig. 4(c),

the modulation frequency at 160 Hz also suffers Doppler shift in the envelope spectrum. Moreover, a low-frequency

component with enormous amplitude is also introduced in the envelope spectrum because of the Doppler effect, which has

been confirmed through plenty of simulations. It is related to the signal amplitude attenuation derived from the variation of sound level when the vehicle passes by the microphone. Furthermore, the TFD of simulated wayside acoustic signal is

Fig. 3. Flowchart of signal resampling for the Doppler effect reduction.



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shown in Fig. 4(d). When conducting the STFT of simulated signal, the Hamming window size and the overlap between

adjacent segments are set to be 511 and 510 points respectively. Although the TFD is full of noise, a curve around the localfrequency band centered at 1500 Hz is still faintly visible. Therefore, 1500 Hz is selected as the target frequency to extract its

variation curve on the TFD by the proposed ridge detection algorithm.

By dynamically minimizing the local cost functions, the variation curve of target frequency is successfully extracted, as

shown on the TFD in Fig. 5(a). According to this curve, the wayside acoustic signal in Fig. 4(b) is resampled with dynamic

intervals in Eq. (2). As seen from the spectrum of the resampled signal in Fig. 5(b), the frequencies in the resonance band

have been well recovered as comparing to Fig. 4(a) and (b). It can also be found from the TFD of the resampled signal in Fig. 5

(c) that the variation curve has been straightened at 1500 Hz, which is just the target frequency. This confirms the

effectiveness of the proposed signal resampling method for reduction of Doppler effect.

The envelope spectrum of the resampled signal is also shown in Fig. 5(d), where the modulation frequency at f d¼160 Hzcan be found now. However, there are also many other frequency components with high amplitude, such as f 1 and f 2appearing near f d, and a dominant low-frequency component as illustrated in Fig. 5(d), though the Doppler shift has been

resolved. This makes the characteristic frequency become a kind of weak information and be not easy to be distinguished

from the resampled result.

4. Weak defective information enhancement

After signal resampling, the noise involved in wayside signal acquisition is not white noise, but colored noise.

The defective characteristic frequency becomes a kind of weak information embedded in the colored noise, as demonstrated

in Fig. 5(d). At this situation, the SR with MST is capable to enhance the characteristic frequency by well dealing with the

multiscale noise, and is thus employed in this study for weak defective information enhancement.

4.1. Theory of SR with MST

The SR has been theoretically developed in a non-linear bistable system which is defined by the Langevin equation as:

d yodt

¼ ayo−by3o þ A0 sin ð2π f dt þ φÞ þ ffiffiffiffiffiffiffi

2Dp ξðt Þ (9)

0 0.2 0.4 0.6

0

500

1000

1500

2000

2500

0.5

1

1.5

10 500 1000 1500 2000 2500 30000

0.1

0.2

0 0.2 0.4 0.6 0.8-2

0

2

0 500 1000 1500 2000 2500 30000

5

10

0 0.2 0.4 0.6 0.8-2

0

2

0 500 1000 1500 2000 2500 30000

0.2

0.4

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spectrum Spectrum

Waveform Waveform

Frequency (Hz)

A m p l i t u d e

A m p l i t u d e

Time (s)

Frequency (Hz)

A m p l i t u d e

A m p l i t u d e

Time (s)

F r e q u e n c y ( H z )

0 500 1000 1500 2000 2500 30000

10

20

30Envelope Spectrum

Envelope Spectrum

1.5 50

150 2000

0.1

0.2

Fig. 4. Simulated signal: (a) static acoustic signal; (b) wayside acoustic signal; (c) envelope spectrum of wayside acoustic signal (the whole spectrum and

the spectrum in [10,3000] Hz); (d) TFD of wayside acoustic signal.



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where a and b are the barrier parameters with positive real numbers; A0, f d, and φ are the amplitude, the driving frequency,

and the initial phase of input periodic signal of the system, respectively; D is the noise intensity and ξ(t ) represents

a Gaussian white noise with zero mean and unit variance; and yo(t ) is the final output signal of the bistable system. It can be

seen that Eq. (9) is a non-linear ordinary differential equation, which can be solved by the Runge–

Kutta method [29]. The SR model describes the overdamped motion of a Brownian particle in a bistable potential in the presence of noise and periodic

forcing. The weak low-frequency driving forcing will be enhanced greatly (with an improved SNR) by the white noise

through this non-linear bistable SR system. However, it is hard to realize the SR phenomenon in the traditional SR model

when the driving frequency is in the high-frequency area (⪢1 Hz), or the background noise is not white but colored, due to

the adiabatic approximation or linear response theory [38–40], such as the detection of weak characteristic frequency in

Fig. 5(d).

The MST was proposed to realize a large-parameter SR at a fixed intensity of white noise as well as colored noise in the

measured signal by transferring the noise at multiple scales to be distributed in an approximate 1/ f form [32–34]. The SR

with MST has been proved to be insensitive to noise intensity, active of multiscale noise, and capable of detecting high

frequency [34]. The strategy of MST is briefly described as follows:

(1) Multiscale information acquisition: Conduct discrete wavelet transform (DWT) on the noisy periodic signal yr (t ) and

obtain the approximation coefficients a j and the detail coefficients d j, respectively, as

a jðkÞ ¼Z

yr ðt Þϕ j;kðt Þdt (10)

d jðkÞ ¼Z

yr ðt Þψ n j;kðt Þdt (11)

where ϕ and ψ are the scaling function and the primary wavelet function, respectively, and j is the decomposition scale.

The cut-off decomposition level J should meet the following condition:

f s

2 J þ1 ≤ f d ≤

f s

2 J (12)

where f d is the driving frequency and f s is the sampling frequency. Then we get the set of detail coefficients at multiple

scales as

Φ¼ fd1; d2; …; d J g: (13)

0 0.2 0.4 0.60

500

1000

1500

2000

2500

0.5

1

1.5

2

0 0.2 0.4 0.60

500

1000

1500

2000

2500

The variation curve of target frequency

0.5

1

1.5

0 500 1000 1500 2000 2500 30000

10

20

30

10 500 1,000 1,500 2,000 2,500 3,0000

0.2

0.4

0 0.2 0.4 0.6 0.8-2

0

2

0 500 1000 1500 2000 2500 30000

1

2

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Time (s)


Time (s)

F

r e q u e n c y ( H z )

Frequency (Hz)

A m p l i t u d e

A m p l i t u d e Envelope Spectrum

1.5 50

Envelope Spectrum

150 2000

0.10.2 f d f 1 f 2

Fig. 5. Reduction of the Doppler effect for simulated signal: (a) variation curve of target frequency lying on the TFD; (b) resampled signal; (c) TFD of

resampled signal; (d) envelope spectrum of resampled signal (the whole spectrum and the spectrum in [10,3000] Hz).



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(2) Multiscale information redistribution: Redistribute the spectrum with multiscale information to be approximate 1/ f form by

varð ~d jÞ ¼ 2α þ j; j ¼ 1; 2; …; J (14)where α is a tuning parameter. Then a new set of detail coefficients is obtained as

~Φ ¼ f ~d1; ~d2; …; ~d J g: (15)

(3) Modified signal reconstruction: With the original approximation coefficients a J and the adjusted detail coefficients set ~Φ,

the new signal yn(t ) is reconstructed as

ynðt Þ ¼ ∑k∈ Z

a J ðkÞϕ J ;kðt Þ þ ∑ J

j ¼ 1∑

k∈ Z

~d jðkÞψ j;kðt Þ: (16)

The new reconstructed signal with approximate 1/ f distribution in spectrum is then sent to the non-linear bistable

system as expressed in Eq. (9) with the normalized parameters a ¼1 and b ¼1, by replacing the signal plus noise part in themodel [33,34]. For contributions of the multiscale noise to the SR effect, there would exist the best condition according to

the tuning parameter α to reach the balance among the driving force, noise, and the non-linear system for SR.

4.2. ASR method based on genetic algorithm

Since the performance of the output depends on the tuning parameter α in the SR system with MST, an adaptive SR with

MST method based on genetic algorithm is proposed in this paper to search for the optimal parameter α to enhance the

defective characteristic frequency as much as possible. Genetic algorithm is an adaptive heuristic search algorithm that

mimics the process of natural evolution, which is used to generate useful solutions to optimization and search problems

[41]. First pioneered by John Holland in the 1960s, the genetic algorithm has been widely studied, experimented and applied

in many fields such as bioinformatics, computational science, engineering and economics. A simple form of genetic

algorithm works as follows:

(1) The algorithm starts with a set of solutions called population initially generated randomly. One solution is called an

individual.

(2) The algorithm then creates a sequence of new populations with the following steps:a. Calculate the fitness value of each current individual in the fitness function. The fitness function is the objective

function to be optimized.

b. Choose some of the individuals in the current population that have lower fitness values as elite children, which will

be passed directly to the next population.

c. Select specific number of individuals from the current population based on their fitness to form the parents. The

more suitable the individual is the more chances it has to be selected, which is just the evolutionary idea of natural

selection and genetics.

d. Apply genetic operators to the parents to produce children. Besides the elite children, the other children are

produced either by combining the vector entries of a pair of parents—crossover—or by making random changes to a

single parent—mutation.

e. Replace the current population with the children to form the next generation.(3) The algorithm stops when one of the stopping criteria (such as the number of generations or the time limit or the

improvement of the best solution) is satisfied.

The optimal tuning parameter α should be the one that results in the biggest SNR in the output of SR system. The

optimization problem in this paper can be described as to minimize the fitness function defined by

fit ¼−SNR ðα Þ (17)

where SNR(α ) is the function of SNR corresponding to the variable α . For a specific value of α , the power spectrum of the

output signal yα oðt Þ is obtained as Y α oð f Þ ¼ ½Y α 1; Y α 2; …; Y α N , where N is the frequency point number for conducting fast Fouriertransform. Since the frequency components mainly focus in the low-frequency area for the SR output, the SNR can be

calculated by the following formula:

SNR ðα Þ ¼ 10 log10Y α k0

∑2k0k ¼ 1Y

α k−Y

α k0

(18)



9/15

where k0¼⌊ f d N / f s⌋ with ⌊ ⌋ representing the largest integer less than or equal to the given numeric expression. Eq. (18)defines a local SNR, which is appropriate for our study to judge the optimum output based on the concentration of energyaround the driving frequency.

For a genetic algorithm, the encoding method should be first determined to represent the individuals by chromosomes.

To reduce the computational complexity and improve the computational efficiency, this paper takes the real-number

encoding method which does not need decoding. That is to say, the gene in a chromosome is just the value of α and there is

only one gene in a chromosome in this study.

As talked above, the genetic algorithm produces three types of children to make up the next population: elite children,

crossover children and mutation children, as illustrated in Fig. 6. The elite children are the individuals in the current

population with their fitness values on the top list. These individuals are remained to be members of the next population.

The crossover children are created by selecting the genes from pairs of parents randomly and recombining them. Given a

pair of parents α p1 and α p2 in the tuning parameter optimization algorithm, the crossover child is taken as α p1 or α p2 with the

same selecting probability. The mutation children are generated by applying random changes to the genes of a single parent.

If a parent α p is used for mutation, the mutation child will be α p′¼α p+ε, where ε is a random number taken from a Gaussiandistribution (the standard derivation decreases linearly at each new generation).

The parameters which should be pre-set in the genetic algorithm include the population size, the elite count to specify

the number of elite children, the crossover fraction to specify the fraction of crossover children in the population excluding

the elite children, and the generation to specify the maximal iteration when the algorithm stops. In this paper, the

population size is set to be 20, the elite count is 2, the crossover fraction is 0.8, and the generation is 50. Then the number of

each type of children in each step can be obtained as follows: the number of elite children is 2, the number of crossover

children is 14 by rounding the computation 0.8 (20−2)¼14.4, and the remaining 4 individuals in the current populationare to produce mutation children. Totally, the number of parents for creating crossover and mutation children is calculated

to be 2 14+4¼32. It can be seen that the number of parents is not equal to the population size. The following describes therule for parent selection.

The parents are selected from current individuals based on their fitness. However, the fitness should be converted into a

more suitable range of values at first. This paper scales the raw fitness values based on the position of individuals in their

sorted fitness values. The scaled value of an individual with the nth position is given by 1= ffiffiffi

np

. The parents are then selected

according to the scaled values. Next, the algorithm first draws a line and allocates each individual a section with the lengthproportional to its scaled value. Then, it moves along the line with equal step size for the times of the parents number.

At each step, the algorithm chooses the individual belonging to the section it lands on as one parent. In this strategy, the

individual with a lower fitness value may be selected more than once as a parent, so sufficient number of parents can be

obtained finally. To explain the parent selection method clearly, an example is provided as follows. A schematic diagram for

parent selection is also presented in Fig. 7. Suppose the current individuals are CI ¼[a, b, c , d, e]. The position of each currentindividual in the vector CI is n1¼1, 2, 3, 4, 5 from left to right. The corresponding fitness values of current individuals areFV ¼[10,8,12,15,6]. Thus the position of each individual in the ascendingly sorted fitness values is n2¼3, 2, 4, 5, 1. In thefollowing, the scaled value corresponding to the individual in CI at the same position is given as SV [n1]¼1=

ffiffiffiffiffin2

p , resulting in

SV ¼½1= ffiffiffi

3p

; 1= ffiffiffi

2p

; 1= ffiffiffi

4p

; 1= ffiffiffi

5p

; 1= ffiffiffi

1p

. However, the sequence of SV should be reordered according to the formula SV ′[n2]¼1= ffiffiffiffiffin1p , delivering the reordered scaled values SV ′¼½1= ffiffiffi5p ; 1= ffiffiffi2p ; 1= ffiffiffi1p ; 1= ffiffiffi3p ; 1= ffiffiffi2p . According to the relationshipbetween CI and SV , the individuals are reordered as CI′¼[d, b, e, a, c ] corresponding to SV ′. Next, the probabilities of theindividuals to be selected in CI′ are calculated to be PB¼[0.1384, 0.2188, 0.3094, 0.1787, 0.1547] based on SV ′. And thecumulative probabilities are CPB¼[0.1384, 0.3572, 0.6666, 0.8453, 1]. Draw a line with the length of 1 and mark the valuesof CPB on it, as displayed in Fig. 7, then the section for each individual is allocated. It can be seen that the individual with a

Elite child

Crossover child

Mutation child

α p

αe αe

α p1

α p2

α p1or α p2

α p'

Fig. 6. Illustration of three types of children produced in genetic algorithm: (a) elite children; (b) crossover children; (c) mutation children.



10/15

lower fitness value has a longer length on the line, which indicates that it has a greater likelihood to be selected as a parent.

The next step is to move along the line and select parents. If the required parent number is 7, the moving step size will be

1/7. Take the first step position as a random number less than 1/7, then a possible position group of all steps are SP¼[0.0042,0.1472, 0.2900, 0.4329, 0.5757, 0.7186 and 0.8615], which is also drawn above the line of CPB in Fig. 7. Since 0.0042 lands on

the section of d, the individual d is selected as a parent. 0.1472 and 0.2900 are both in the range of [0.1384, 0.3572], so the

individual b is selected twice. The remaining parents are selected in the same way. Finally, the parents are obtained as PR ¼[d, b, b, e, e, a , c ].

With the value of parameter α optimized by the genetic algorithm, the weak defective characteristics will be greatly

enhanced in the output of SR system with MST. The flowchart of the ASR method for weak defective information

enhancement is presented in Fig. 8. The main steps are subsequently implemented as follows. (1) The envelope signal ye(t ) is

extracted from the resampled wayside acoustic signal. (2) The genetic algorithm is applied to the signal ye(t ) to obtain the

optimal tuning parameter value α 0. (3) The envelope signal ye(t ) is adjusted to be a new signal yn(t ) with its spectrumdistributed in an approximate 1/ f form with the selected optimal parameter. (4) The modified signal yn(t ) is sent to the

Fig. 8. Flowchart of the weak defective information enhancement by the ASR method.

a(10) b(8) c(12) d (15) e(6)

1

3

1

2

1

4

1

5

1

1

1

1

1

2

1

3

1

4

1

5

1

5

1

2

1

1

1

3

1

4

d b e a c

CI(FV):

CI :

2

1

n:

1

1

n:

SV[n1] =

1

1

n :SV [n2] =

0 0.1384 0.3572 0.6666 0.8453 1

CPB:d b e a c

0.1472 0.4329 0.7186

0.8615SP:

0.0042 0.2900 0.5757

d b e a cPR: b e

n1 (position of current individual in CI): 1, 2, 3, 4, 5.

n2 (position of current individual in sorted FV): 3, 2, 4, 5, 1.

Fig. 7. Schematic diagram for parent selection.

0 0.2 0.4 0.6 0.8

-20

0

20

0 200 400 600 800 10000

200

400

600

0 0.2 0.4 0.6 0.8-100

0

100

0 200 400 600 800 10000

1

2

x 104

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Frequency (Hz)

Time (s)

A m p l i t u d e

A m

p l i t u d e

Spectrum

Waveform

f d

f 1 f 2

f d f 1

f 2

Fig. 9. The waveforms and spectra of the output of SR system with MST for resampled simulated signal with different parameter values: (a) α ¼38.7;(b) α ¼30.



11/15

bistable SR system in Eq. (9) with a ¼1 and b ¼1, and the output signal yo(t ) is obtained. Finally, the defective characteristicfrequency f d will be exactly identified by spectral analysis of the signal yo(t ).

To verify the effectiveness of the proposed ASR method, the envelope signal of the resampled acoustic signal in Fig. 5(b)

is taken for analysis. No frequency components need to be filtered out in the envelope spectrum including the great low-

frequency one shown in Fig. 5(d), because they are free energy for use in SR model. According to the genetic algorithm, the

value of parameter α is chosen as 38.7. The corresponding output result is shown in Fig. 9(a). As seen in the spectrum of

Fig. 9(a), the characteristic frequency f d is prominent while the low-frequency component with great amplitude in Fig. 5(d)

is largely suppressed in Fig. 9(a). Furthermore, it’s easy to distinguish f d from f 1 and f 2, which are interfering frequencies in

Fig. 5(d). However, with an improper value of α , the characteristic frequency may be still not easy to be identified in the

spectrum of the output signal. For example, the result with α ¼30 is also shown in Fig. 9(b). It can be seen that the frequency f 1 has the highest amplitude in the spectrum, misleading the final judgment of the characteristic frequency. Therefore, the

proposed ASR method based on genetic algorithm is effective in enhancing the weak defective information.

5. Experimental results

As presented above, this paper proposes a feasible wayside acoustic diagnostic scheme of defective bearings by

combining wayside signal resampling and weak information enhancement. In the following, the proposed scheme is applied

to analyze the experimental signals acquired in Section 2 to further validate its effectiveness.

As described in Section 2, train bearings with outer-race and inner-race defects were separately used to acquire the static

and wayside defective acoustic signals for analysis. The static signals were acquired with the radial load of tested bearingbeing 30 kN, the rotating speed of motor shaft being 1430 rev/min and the sampling frequency being 50 kHz. According to

the geometry of the bearing and the rotating speed, the defective characteristic frequencies, i.e. the ball passing frequencies

over the outer-race and inner-race defects, are calculated to be f BPFO¼138.7 Hz, and f BPFI¼194.9 Hz, respectively. Thewayside acoustic signals were acquired by the microphone with a vertical distance of 2 m to the path of vehicle moving. The

sampling frequency was also set as 50 kHz. And the speed of the vehicle was 30 m/s (108 km/h).

The static and wayside acoustic signals of the bearing with outer-race defect are shown in Fig. 10(a) and (b). As seen in

the spectrum of Fig. 10(a), the significant frequencies concentrate lower than 3 kHz. To reduce the amount of calculation and

the disturbance of noise in high-frequency area, the wayside acoustic signal is downsampled with a new sampling

frequency of 6 kHz, as shown in Fig. 10(c). The TFD of the downsampled signal is displayed in Fig. 10(d). For STFT of the

0 0.1 0.2 0.30

500

1000

1500

2000

2500

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0 500 1000 1500 2000 2500 30000

0.005

0.01

0 0.1 0.2 0.3 0.4-0.5

0

0.5

0 5 10 15 20 250

0.05

0.1

0 0.1 0.2 0.3 0.4-0.4

-0.2

00.2

0.4

0 5 10 15 20 250

2

4

Frequency (kHz)

Time (s)

A m p l i t u d e

A m p l i t u d e Waveform

Frequency (kHz)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Time (s)

Time (s)


0 1000 2000 3000 4000 50000

2

4

Frequency (Hz)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Time (s)

Spectrum

Fig. 10. Acoustic signal of train bearing with outer-race defect: (a) static signal; (b) wayside signal; (c) downsampled result of wayside signal; (d) TFD of downsampled signal.



12/15

downsampled signal, the Hamming window size is 311 points and the overlap between adjacent segments is 310 points.

It can be seen that there are four obvious curves lying on the TFD. The curve around 1247 Hz is selected to be extracted

based on the local cost functions, as shown on the TFD in Fig. 11(a). The resampled signal according to this frequencyvariation curve is shown in Fig. 11(b), as well as its TFD displayed in Fig. 11(c). It can be seen from the spectrum and TFD of

the resampled signal that four prominent frequency components contained in the static acoustic signal shown in Fig. 10(a)

are well rectified from the wayside acoustic signal. And then, the envelope signal of resampled signal is shown in Fig. 11(d).

A low-frequency component is also involved as seen in the envelope spectrum, just like the simulation case in Fig. 5(d). The

characteristic frequency f BPFO can be found in the envelope spectrum. However, it seems so weak as compared to the

inevitable low-frequency component. Applying the proposed ASR method to the envelope signal, the final output is shown

in Fig. 12. It can be seen that the output waveform indicates a dominant periodic signal. The power spectrum further

demonstrates that the defect frequency f BPFO is just the principal component contained in the output signal. Therefore, the

effectiveness of the proposed wayside acoustic diagnostic scheme is verified in the diagnosis of train bearing with outer-race

defect.

Next, the train bearing acoustic signal with inner-race defect is analyzed by the proposed method. The static, wayside

and downsampled acoustic signals are shown in Fig. 13(a–c), respectively. The TFD of the downsampled signal is displayed

in Fig. 13(d) using 311-points Hamming window and 310-points overlap between adjacent segments. And the brightestfrequency variation curve is extracted from the TFD, as illustrated in Fig. 14(a). The downsampled signal is resampled

according to this variation curve, as shown in Fig. 14(b). Comparing the TFD of resampled signal in Fig. 14(c) with the TFD in

Fig. 14(a), it can be seen that the Doppler shift has been largely reduced in the resampled signal. As seen in the envelope

spectrum of resampled signal in Fig. 14(d), the inner-race characteristic frequency f BPFI is revealed, but still very weak. Apart

from the involved large low-frequency component because of the Doppler effect, another frequency at f r ¼23.8 Hz is alsoobvious. This is just the rotating frequency of the shaft of train test rig shown in Fig. 1(a). Furthermore, there are also some

side-band frequencies around f BPFI with 23.8 Hz as the frequency interval. Apparently, the characteristic frequency is

modulated by the rotating frequency. This is because that the defect in the inner raceway of the train bearing is rotating

while the mechanical loading point is fixed, which makes the radial force on the defect periodically change. Utilizing

the proposed ASR method to analyze the envelope signal, the result is shown in Fig. 15. It can be seen that, although the

envelope spectrum in Fig. 14(d) is complex, the characteristic frequency f BPFI can be enhanced greatly by transferring

the most energy in low-frequency area to the area around f BPFI. Therefore, the defect frequency can be easily identified in the

spectrum of the output signal. As a result, the proposed wayside acoustic diagnostic method is exactly effective for movingbearing defect identification.

0 0.1 0.2 0.30

500

1000

1500

2000

2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.1 0.2 0.30

500

1000

1500

2000

2500


0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4-0.4

-0.2

0

0.2

0 500 1000 1500 2000 2500 30000

0.05

0.1

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Time (s)

F

r e q u e n c y ( H z )

Time (s)

F r e q u

e n c y ( H z )

0 0.1 0.2 0.3 0.4

0

0.2

0.4

0 200 400 600 800 10000

0.1

0.2

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spec t rum

Waveform

0 5 100

0.1

0.2

f BPFO

Fig. 11. Reduction of Doppler effect for wayside acoustic signal of train bearing with outer-race defect: (a) variation curve of target frequency lying on the

TFD; (b) resampled signal; (c) TFD of resampled signal; (d) envelope signal of resampled signal.



13/15

6. Conclusions

This paper proposes an effective solution for wayside acoustic diagnosis of defective train bearings. Two problems in the

wayside acoustic signal, including Doppler shift and heavy noise interference, are solved successively by the proposed

method. To reduce the Doppler shift of frequencies contained in the original acoustic signal, a new ridge extraction method

based on the local cost functions is proposed for dynamically resampling the wayside acoustic signal. The local cost

functions concern the points with relative large amplitude inside local frequency band on the TFD, as well as the gradient of

ridge, which makes the extracted curve smooth and accurate. To enhance the weak defective information embedded in

colored noise in the resampled signal, the SR method with MST is introduced by transferring the multiscale noise energy to

the area around the characteristic frequency. An ASR method is proposed based on the genetic algorithm to produce the

optimal output of SR systems automatically, which can enhance the weak defective information as much as possible. To

verify the effectiveness of the proposed diagnostic scheme based on the combination of signal resampling and informationenhancement, a train bearing test rig is established and experiments simulating real train passing by a wayside microphone

0 0.1 0.2 0.3 0.4-100

0

100

0 200 400 600 800 10000

1

2 x 10

4

Frequency (Hz)

Time (s)

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

f BPFO

Fig. 12. The SR output result for resampled outer-race defective acoustic signal by the ASR method.

0 0.1 0.2 0.30

500

1000

1500

2000

2500

0.05

0.1

0.15

0.2

0.25

0.3

0 0.1 0.2 0.3 0.4-0.5

0

0.5

0 500 1000 1500 2000 2500 30000

2

4x 10

-3

0 0.1 0.2 0.3 0.4

-0.5

0

0.5

0 5 10 15 20 250

0.02

0.04

0 0.1 0.2 0.3 0.4-1

0

1

0 5 10 15 20 250

0.1

0.2

Frequency (kHz)

Time (s)

Waveform

Frequency (kHz)

A m p l i t u d e

A m p l i t u d e

A m p l i t u d e

A m p l i t u d e

A m p l i t u d e

A m p l i t u d e

Spectrum

Waveform

Time (s)

Time (s)


Spectrum

Frequency(Hz)

Spectrum

Waveform

Time (s)

Fig. 13. Acoustic signal of train bearing with inner-race defect: (a) static signal; (b) wayside signal; (c) downsampled result of wayside signal; (d) TFD of

downsampled signal.



14/15

are implemented to acquire the wayside acoustic signals. Two defective bearings with outer-race and inner-race defects arerespectively used for experiments and method verification. The simulation and experimental results show that the proposed

solution is effective in wayside acoustic diagnosis of defective train bearings, and is thus hoped to be applicable to practical

wayside acoustic bearing detection system to enhance the diagnostic performance.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (51075379 and 51005221). The valuable

comments from anonymous reviewers are sincerely appreciated for their help to further improve the paper.

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0 0.1 0.2 0.30

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0 0.1 0.2 0.3 0.4-0.6-0.4

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0

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Frequency (Hz)

Time (s)

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Waveform

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