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The application of the improved empirical mode decomposition (EMD) theory in gearbox fault diagnosis has been studied in this paper, and the transient features of gearbox vibration signals are shown. Based on using EMD, an improved algorithm of orthogonal empirical mode decomposition (OEMD) is put forward and is applied to extract the fault feature. Finally, fault diagnosis application in a gearbox is used as an example to prove the feasibility of the proposed method.

Gearbox is the most important part of rotating machinery, which covers a broad range of mechanical equipment and plays a significant role in industrial applications. It generally operates under tough working environment and is therefore subject to faults, which may cause machinery to break down and decrease machinery service performance such as manufacturing quality and operation safety. One of the commonly used strategies in fault diagnosis of rotating machinery is adopting effective signal processing techniques to analyze the response signals and to reveal fault characteristics. However, traditional signal processing techniques, including time-domain and frequency-domain analysis, are based on the assumption that the process generating signals is stationary and linear. When transmission component faults occur in machinery, the dynamic behavior of the component is shown as nonlinear. The vibration signals have some characteristics, such as being nonstationary and nonperiodic, and the transient characteristics are especially obvious. Therefore, the need of signal processing techniques fit to the nonstationary and nonlinear signal is increasing in recent years.

Equipment vibration signals not only include vibration signals, but also contain many background signals and noise signals, which contain large amounts of energy and are related with their own properties of each transmission component [

Empirical mode decomposition (EMD) is one of the most powerful time-frequency analysis techniques and is based on the local characteristic of a signal. With EMD, the signal can be decomposed into a set of complete and almost orthogonal components called intrinsic mode function (IMF). EMD has a unique instantaneous ability of identification and the local characteristics scale which is based on signals that can decompose the complex vibration signals into a finite number of IMFs, which can reflect the local characteristics of the nonstationary signals [

However, currently the Hilbert Huang transform (HHT) signal analysis scheme is only an immature method, and it is inevitable that its theoretical basis and theoretical framework need to be further refined [

For the multicomponent signals and the signals containing noise, mode mixing and false components may occur, if these signals are decomposed directly by means of EMD. Therefore, in this paper a method of extracting transient features of a nonstationary signal without mode mixing is proposed based on the combination of orthogonal empirical mode decomposition (OEMD) and Hilbert transform.

EMD is used to decompose the signals into IMF with different scale characteristics, and these functions need to satisfy two conditions [

Flow chart of EMD.

The detailed algorithm of the EMD is designed as below.

Set

The

Let

Find out the local extreme points of

The maximum and minimum value points of

Compute the mean value

Consider

If the judgment criteria of IMFs are contented, then set

Consider

If the number of the extreme points of

From the process above we can conclude that each mode has the identical number of extreme points within the entire length of the entire signals. The local mean value is defined by the upper and lower envelope of the signals. On this basis, the different mode components can be distinguished according to the characteristic scales of the signals (the time span between the adjacent extreme points of the signals). IMFs are no longer limited into the narrow-band signals, and simultaneously they can be presented as amplitude modulations and frequency modulations. Generally, associated with the addition of the order of IMFs, the characteristics scale of IMFs will increase. And consequently the multiorder IMFs sequence, whose time scales are varying from small to large, is achieved. From a spectrum angle, the signals are filtered from high frequency to low frequency.

We compare EMD with classical time-frequency analysis methods, such as short time Fourier transform (STFT) and wavelets as follows.

STFT adopts the same window for the whole signal, which makes it produce constant resolution for all frequencies. Although STFT can overcome the disadvantages of FFT-based methods in processing nonstationary signals, it still cannot obtain a good frequency resolution using wide windows and good time resolution (narrow window) at the same time. Therefore, STFT is suitable for the analysis of stationary signals instead of real nonstationary signals.

Comparing with STFT, wavelets can be utilized to analyze multiscale signals through dilation and translation and extract time-frequency characteristics of the signals effectively. That means wavelets are more suitable than STFT for analyzing nonstationary signals. However, wavelets are nonadaptive and have their own disadvantage that their analysis results depend on the choice of the wavelet base function. This may lead to a subjective and a priori assumption on the characteristics of the signal.

Different from wavelets and short time Fourier transform (STFT), EMD is a self-adaptive signal processing method. It is based on the local characteristic time scales of a signal and could decompose the signal into a set of IMFs. The IMFs represent the natural oscillatory mode embedded in the signal and work as the basis functions, which are determined by the signal itself, rather than predetermined kernels. Although the EMD method shows outstanding performance in processing nonlinear and nonstationary signals, the algorithm itself has some weaknesses. For example, mode mixing sometimes occurs between IMFs; the IMFs are not strictly orthogonal to each other. In conclusion, each time-frequency analysis method suffers from various problems. It is hard to say that one can always exceed the others for any case.

As discussed, after EMD, a complex signal

Theoretically, EMD is perfect tool to separate monocomponents. But when high frequency components in signal create intermittence, the decomposition by EMD produces mode mixing [

There are two causes of the problem:

For the signals formed by

Time-domain waveform of simulation signal

EMD effect and Hilbert spectrum of IMF1–IMF4.

When the frequencies ratio of the two signal components varies from 0.5 to 2, it can easily be interpreted that the signal is made up of some IMFs with certain degrees of modulation according to the standard algorithms of EMD, which deviate from the nature of the original signals. Since the frequency ratio 90/50 = 1.8 < 2 lies within the same octave, which means normal EMD cannot separate the two modes, the mode mixing clearly appears in the IMF component IMF2. IMF2 contains not only 90 Hz component but also part of 50 Hz component during the intermittent period. Although IMF3 contains the intermittent 50 Hz component, the result of IMF2 is severely distorted.

Essentially, empirical mode decomposition is the nonstationary signals that are treated axisymmetrically, separated the intrinsic mode functions according to the frequency in descending order. The intrinsic mode functions need to satisfy the two following conditions:

It is difficult to meet the second condition mentioned above, so various criteria are proposed by different researchers to achieve ideal results. Aimed at the second condition, the criterion is proposed as follows: the ratio of the signals’ local mean curve and the energy of the signal is smaller than a threshold, where the threshold is determined by the characteristics of the components contained in the signals and the decomposition accuracy of the signals.

Generally, the signals need to filter the IMF by EMD. Actually, the empirical mode decomposition is a process of adaptive filtering, obtaining the intrinsic mode functions by the adaptive band-pass filter [

First, let the signal be detected

It can be seen that when obtaining the

When

For any two intrinsic modes function components

Obviously, any two components

All components, obtained by the decomposition mentioned above, are mutually orthogonal, so the decomposition process above is called OEMD. Meanwhile, formula (

It can be seen from the decomposition process above that every IMF can be obtained by applying OEMD to the signal; meanwhile, IMF is a result which can be obtained by filtering the original signal. Therefore, the resulting problem is not very serious and the previous components’ boundary effects will not have a later effect on the obtained component. The phenomenon of divergence and inward pollution, which appear in the data at both ends when the general EMD is adopted, will not happen, overcoming the end effect problem that exists in EMD.

For an arbitrary time series,

In principle, there are infinitely many ways of defining the imaginary part, but the Hilbert transform provides a unique way of defining the imaginary part so that the result is an analytic function. A brief tutorial on the Hilbert transform with the emphasis on its physical interpretation was proposed by Bendat and Piersol. Essentially (

After obtaining the intrinsic mode function components, we will have no difficulties in applying the Hilbert transform to each component and computing the instantaneous frequency according to (

Function

Then the analytical signal is constructed as follows:

Thus, the amplitude function and phase function are all obtained and can be expressed by the following formulae (

At the same time, the instantaneous frequency can be obtained as follows:

Therefore, the instantaneous frequency and amplitude are used to depict the frequency of the signal instead of the power spectrum. So the Hilbert spectrum can be denoted as follows:

However, Hilbert marginal spectrum

The method mentioned in this paper is applied to analyze the aforementioned simulated signal. The IMFs components and Hilbert spectrum are shown in Figure

Main IMF and Hilbert spectrum of IMFs obtained by OEMD.

As can be seen from Figure

The experiments are conducted by means of a gearbox failure detection experimental device in the Mechanical Transmission National Key Laboratory of Chongqing University. The gearbox is constituted by two pairs of gear pairings. The meshing frequencies of gear 1 and gear 2 are both 372 Hz, and the meshing frequencies of gear 3 and gear 4 are 228.5 Hz. Gear 1 with rational

When the gearbox is in the fault state, its time-domain waveform is shown as Figure

Time-domain signals of gearbox and description of gearbox signal spectrums.

Main components of IMF achieved by means of EMD.

The results obtained after processing the gearbox signals by means of EMD directly are shown as Figure

The signals are dealt with in accordance with the method mentioned in this paper; the results of OEMD are shown in Figure

IMF1–4 obtained after OEMD processing.

The mode mixing appearing in the process of EMD is brought about by the IMF containing various time scales. The interference signals are eliminated by means of OEMD, avoiding the mode mixing and reducing the influence of the false frequency and noise. The results achieved by processing the gearbox vibration signals show that the method is practical and effective.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors acknowledge the National Science Foundation (nos. 61304104 and 61004118) and the Program for Excellent Talents of Chongqing Higher School (no. 2014-18) and the Natural Science Foundation Project of CQ CSYC (cstc2011jjA30010).