^{1}

^{1}

^{2}

^{*}

^{1}

^{1}

^{2}

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/).

Vibration analysis is an effective tool for the condition monitoring and fault diagnosis of rolling element bearings. Conventional diagnostic methods are based on the stationary assumption, thus they are not applicable to the diagnosis of bearings working under varying speed. This constraint limits the bearing diagnosis to the industrial application significantly. In order to extend the conventional diagnostic methods to speed variation cases, a tacholess envelope order analysis technique is proposed in this paper. In the proposed technique, a tacholess order tracking (TLOT) method is first introduced to extract the tachometer information from the vibration signal itself. On this basis, an envelope order spectrum (EOS) is utilized to recover the bearing characteristic frequencies in the order domain. By combining the advantages of TLOT and EOS, the proposed technique is capable of detecting bearing faults under varying speeds, even without the use of a tachometer. The effectiveness of the proposed method is demonstrated by both simulated signals and real vibration signals collected from locomotive roller bearings with faults on inner race, outer race and rollers, respectively. Analyzed results show that the proposed method could identify different bearing faults effectively and accurately under speed varying conditions.

Rolling element bearings are critical mechanical components in rotating machinery and their failure may lead to fatal breakdown and significant economic losses. Hence, the fault detection of rolling element bearing has attracted considerable attention in recent years. Vibration signals collected from bearings carry rich information about their health condition. Therefore, the vibration-based diagnostic methods have received intensive study during the past decade [

When a fault occurs in a bearing, periodic or quasi-periodic impulses will appear in the waveform of the vibration signal, while the corresponding bearing characteristic frequencies (BCFs) and their harmonics emerge in the frequency domain [

Although the above mentioned methods successfully detect fault for bearings to some extents, most of those methods are based on the assumption of constant running speed. However, in practice, almost all of the bearings experience different kinds of speed variations. For instance, the rotating speed (RS) of vehicle bearing is varying with its running speed, the RS of wind turbine supporting bearing is fluctuating with the wind speed [

Several methods have been developed to process non-stationary vibration signals in recent years [

To overcome the shortcomings of conventional methods, a tacholess envelope order analysis technique is established in this paper. In this technique, a tacholess order tracking method is first proposed based on generalized demodulation transform. By using this method, the tacho information of the bearing could be recovered from the vibration signal itself. On this basis, envelope order spectrum is utilized to transform the non-stationary envelope signal in the time-domain into a cyclo-stationary signal in the angular domain. In this way, the smearing problem caused by speed variation is solved effectively. The rest of this paper is organized as follows. The principle and implementation of order tracking and generalized demodulation is briefly reviewed in Section 2. The procedure of proposed technique is given in Section 3. The effectiveness of the proposed method is validated by some simulations and experiments in Section 4 and Section 5, respectively. Finally, conclusions are drawn in Section 6.

As discussed above, order tracking involves resampling the vibration signal at constant increments of shaft angle. Hence, instantaneous phase of shaft,

Generally, the vibration signals from rolling element bearings contain not only the structure resonance frequency excited by impacts, but also the fundamental frequency and its harmonics of bearing shaft. Since all the rotating harmonics are phase-locked with respect to the rotation angle of the bearing shaft, naturally, the _{k}

However, the concept of instantaneous phase has physical meaning only for mono-component signal [_{k}

Generalized demodulation is a novel signal processing tool, which is especially suitable for the separating of non-stationary multi-component signals [

For a signal ^{-}^{j}^{2π}^{s0}^{(}^{t}^{)} is the transform kernel, and _{0}(^{-}^{j}^{2π}^{s0}^{(}^{t}^{)} is considered as the analyzed signal. When _{0}(

If we assume _{G}_{0}), then ^{-}^{j}^{2π[}^{f0}^{+}^{s0}^{(}^{t}^{)]} It implies that a signal with curved IF, _{0}, we simply need to specify a function _{0}(

According to this property of generalized demodulation, the _{k}

Although the definition of GFT is applicable to both real signal and analytic signal, however, the latter one is preferred in practice. It is due to the fact that the interference on the time-frequency plane caused by meaningless negative frequency could be avoided when the signal is analytic [

Estimate the instantaneous frequency of

Construct the kernel function _{0}(_{0} takes the average value of

Apply the GFT to the analytic signal _{k}_{k}_{k}

Apply the inverse GFT to the filtered _{k}_{k}_{k}

To effectively diagnose the faults of rolling element bearings under variable operating conditions without a tachometer, a tacholess envelope order analysis (TLEOA) technique is proposed in this paper. The principle of TLEOA is mainly composed of two parts,

Extract the envelope signal of impulses by using spectral kurtosis and band-pass filtering.

Spectral kurtosis (SK) has been proven efficient in detecting incipient faults from large noise, which provides a means of determining which frequency bands contain a signal of maximum impulsivity [

Estimate the IF of

As discussed in Section 2.2, the accurate instantaneous frequency (IF) of

Recover the instantaneous phase of the shaft via generalized demodulation and Hilbert transform.

Once the IF of _{k}_{k}_{k}

After that, the instantaneous phase of the shaft could be estimated according to

Resample the envelope signal in angular domain.

As discussed previously, the envelope signal of the impulses is non-stationary due to speed variation, which in turn leads to the smearing effect in the envelope spectrum. To deal with this problem, the envelope signal is resampled in angular domain according to the instantaneous phase information recovered from the vibration signal. In this way, the non-stationary envelope signal in time-domain can be transformed into a cyclo-stationary signal in the angular domain [

Detect the bearing fault by identifying the BCFs in envelope order spectrum.

Since the envelope signal has already been resampled (or order tracked), the traditional FFT-based methods can be effectively applied to that signal. By identifying of the BCFs in envelope order spectrum, the fault type of bearing can be determined.

To validate the proposed method, a simulated signal is generated according to the vibration model for rolling element bearings [

The simulated signal _{i}_{i}_{n}_{n}

In this simulation, we assume the outer race is keep fixed and the inner race is rotating with shaft. There is a local fault on the outer race, and the fault excited impulse is simulated by an exponentially decaying sinusoid as follows:
_{r}_{1} = 0.3, _{2} = 0.5, _{3} = 0.4, _{1} = π/6, _{2} = −π/3, _{3} = π/2, respectively. The bearing experiences a speed up and coast down process during the measurement, and the speed curve is given by

White noise is added to obtain a noisy signal with SNR of −3 dB. The sampling frequency is 10 k Hz and the time length of data is 4 s. The impulse signal, harmonics of shaft, noise signal and mixed signal are shown in

The impulses generated by fault can hardly be observed in the mixed signal due to heavy noise. Moreover, the impulses are not equally spaced in time domain due to speed variation, which will bring difficulties to the conventional diagnostic methods based on constant speed. To demonstrate this problem, the conventional envelope analysis [

For comparison, the proposed TLEOA technique is performed on the same simulated signal by the following steps.

Firstly, in order to find out the optimal frequency-band for demodulation, a fast-kurtogram [

Subsequently, the time-frequency representation of the simulated signal is obtained by using ASTFT.

Once the IF of 2nd harmonic is estimated, generalized demodulation is further applied to extract its waveform from the raw signal. The exacting procedure is presented intuitively in _{0}(_{0}(

Finally, the envelope signal is resampled according to the instantaneous phase recovered from the vibration signal. The spectral analysis is performed on this resampled signal and the corresponding envelope order spectrum is presented in

In this section, the vibration signals collected from a locomotive bearing will be used to demonstrate the effectiveness of the proposed method.

The overview of the bearing test bench is given in

A tri-axial PCB accelerometer with the model number 356A12 is mounted on the shaft end, as shown in

Firstly, SK and band-pass filtering are performed on the raw signal to extract the envelope signal. Then the time-frequency representation of the raw vibration signal is obtained by ASTFT and the corresponding spectrogram is illustrated in

The IF of 19th harmonic is estimated by ridge searching in the ASTFT spectrogram. After that, the 19th harmonic shown in

The envelope order spectrum obtained by the proposed method is illustrated in

To further confirm the reliability of the proposed technique, the vibration signals from inner race and roller faults as shown in

A technique has been proposed to extract the fault information of rolling element bearings in the variable speed case. Some conclusions are drawn in this work as follows:

Generalized modulation is an effective non-stationary signal analysis tool, which is capable of extracting one particular harmonic of bearing shaft rotating frequency from the raw signal. By using the instantaneous phase information of the extracted harmonic, order tracking can be performed on the envelope signal without tachometer. Envelope order spectrum could exploit the cyclic feature from non-stationary envelope signals, thus the smearing problem encountered in conventional methods could be addressed effectively. By combining those advantages, the proposed tacholess envelope order analysis technique could not only extend the conventional envelope analysis to speed varying case, but also be carried out without the use of additional sensors, such as tachometer or encoder. The effectiveness of the proposed method is demonstrated by simulation and experimental results. It is shown that this method gives a more reliable diagnostic result than conventional technique and therefore it is a promising method for bearing fault diagnosis.

The work is supported by the National Key Science & Technology Specific Projects (2010ZX04014-016), which is highly appreciated by the authors.

Time-frequency distribution and frequency-bands of harmonics: (

Flow chart of the proposed tacholess envelope order analysis technique.

Simulated signal: (

The conventional envelope spectrum of simulated signal.

Kurtogram of simulated signal.

(

(

Extracting the 2nd harmonic by generalized demodulation: (

(

Envelope order spectrum obtained by proposed TLEOA technique.

(

The raw vibration signal.

The envelope spectrum of the bearing signal.

The ASTFT spectrogram of the bearing signal: (

The ASTFT spectrogram of the extracted 19th harmonic.

The envelope order spectrum obtained by proposed technique.

The spall fault on the outer race.

(

Comparison with conventional method: (

Geometric parameters of the roller bearing.

180 | 23.775 | 9 | 20 |

BCFs of the bearing.

BPFO | 57.97 | 8.695 |

BPFI | 75.37 | 11.305 |

BSF | 24.81 | 3.721 |

The enhancement in amplitude by proposed TLEOA technique.

BPFI | 113.7% (6.6 dB) | 194.0% (9.4 dB) | 154.0% (8.1 dB) |

BSF | 60.5% (4.1 dB) | 113.8% (6.6 dB) | 207.1% (9.7 dB) |