Aiming at the nonlinear and nonstationary feature of mechanical fault vibration signal, a new fault diagnosis method, which is based on a combination of empirical mode decomposition (EMD) and 1.5 dimension spectrum, is proposed. Firstly, the vibration signal is decomposed by EMD and the correlation coefficient between each intrinsic mode function and original signal is calculated. Then these intrinsic mode function components, which have a big correlation coefficient, are selected to estimate its 1.5 dimension spectrum. And this method uses 1.5 dimension spectrum of each intrinsic mode function to reconstruct its power spectrum. And these power spectrums are summed to obtain the primary power spectrum of gear fault signal. Finally, the information feature of fault is extracted from the reconstructed 1.5 dimension spectrum. A model to reconstruct 1.5 dimension spectrum is established, and the principle and steps of the method are presented. Some simulated and measured gear fault signals have been processed to demonstrate the effectiveness of new method. The result shows that this method can greatly inhibit the interference of Gauss noise to raise the SNR and recognize the secondary phase coupling feature of the signal. The proposed method has a good real-time performance and provides an effective method to determine the early crack fault of gear root.

A common method for extracting the mechanical fault is assuming that vibration signals are stable with Gaussian distribution [

Empirical mode decomposition (EMD) is also called Huang transform, which can effectively separate various frequency components of the signal from the time curve in the form of intrinsic mode function (IMF) [

After

Ideally,

The above procedure is repeated till

The residue

At the end of the decomposition,

In order to verify the reconfigurable property of EMD, a simulated signal is processed. Firstly, the original signal is decomposed with EMD method, and then the obtained IMFs are summed to reconstruct signal. Figure

Error between the reconstructed signal and original signal. (a) Original signal. (b) EMD decomposition of original signal. (c) Reconstructed signal. (d) Error.

Higher order statistics is a mathematical tool which is used to describe the higher order (second-order or above) statistical feature of a random process [

For a random variable

If

Assuming that the higher order accumulation

For a random variable

1.5 dimension spectrum holds several excellent characteristics and provides a convenient way for the signal processing [

(a) Supposing

The properties show that the fundamental frequency component of signal is strengthened when the 1.5 dimension spectrum is used to analyze harmonic signal. It provides a good method to extract the fundamental frequency component of the signal.

(b) In case that

(c) Assuming that

1.5 dimension spectrum is a special case of the higher order spectrum. It has the ability to suppress Gauss noise and symmetrical distributed noise and has the minimum amount of computation which is similar to the amount of Fourier power spectrum in the high order spectrum [

Figure

In order to eliminate the interference of cross-term which is introduced by different frequency components, the fault signal

The correlation analysis is applied between the obtained IMF and the original signal, and then the pseudocomponent with small correlation coefficient is eliminated.

The 1.5 dimension spectrum is calculated from the selected IMF components.

The 1.5 dimension spectrum of each order of IMF is added up to get the 1.5 dimension spectrum of original signal.

Finally the diagnostic results are obtained from the reconstructed 1.5 dimension spectrum.

Algorithm flow chart of fault diagnosis based on EMD-1.5 dimension spectrum.

The effectiveness of the fault diagnosis method based on EMD decomposition and 1.5 dimension spectrum will be verified by analyzing the simulated signal. Simulated signal is described as follows:

Time series of simulated signal.

Fourier spectrum of simulated signal.

Figure _{5}” are the dominant components, and “Residual” is the remaining amount or a weak item because its amplitude is 1/200 of original signal. Thus, the EMD is an adaptive process of decomposition from high frequency to low frequency.

EMD decomposition and spectrum of IMFs of simulated signal.

Calculating the correlation coefficients between each IMF and the original signal, respectively (in other words, the correlation coefficient between “imf_{1}” and the original signal is calculated, and then the correlation coefficient between “imf_{2}” and the original signal is calculated, and so on), they are 0.761, 0.753, 0.624, 0.617, 0.178, and 0.162. It can be seen that “_{4}” have much larger correlation coefficient than the following two components. Their value is not in the same order of magnitude. Thus, the components after “imf_{4}” can be deduced as the false component. In the analysis of original signal, the 1.5 dimension spectrum of “_{4}” is calculated. And 1.5 dimension spectrum of original signal can be obtained after superposition. Figure

EMD-1.5 dimension spectrum of simulated signal.

Figure

In order to further verify the ability of suppressing of Gauss white noise, Figure

EMD-1.5 dimension spectrum (left) and Fourier spectrum (right) of simulated signal with different SNR. (a) SNR = 10 dB. (b) SNR = 0 dB. (c) SNR = −10 dB.

Table

Run-time comparison of three methods under different SNR conditions.

Sampling points | SNR/dB | Running time/s | ||
---|---|---|---|---|

Fourier spectrum | 1.5 dimension spectrum | Bispectrum | ||

1024 | −20 | 0.819 | 0.968 | 3.87 |

−10 | 0.736 | 0.975 | 2.93 | |

0 | 0.489 | 0.761 | 2.07 | |

10 | 0.314 | 0.653 | 1.97 | |

20 | 0.205 | 0.542 | 1.86 |

From the above simulation experiments, we come to a conclusion that the 1.5 dimension spectrum can effectively extract the nonlinear coupling features of the signal while restraining the Gauss white noise and it is faster than other higher order spectrum. 1.5 dimension spectrum is suitable for the real-time feature extraction of fault signal under mass data and low SNR condition.

In order to further verify the effectiveness of the proposed method in the fault feature extraction of gears, the gear with gear tooth crack fault is tested and analyzed. A comprehensive fault simulation test bench is used to do gear fault diagnosis test. The test bench is composed of motor, a mechanical transmission device, sensor, the hardware circuit, computer, and related software. Figure

Comprehensive fault simulation test bench.

Time series of measured signal.

Fourier spectrum of measured signal.

The fault signal is decomposed by EMD method. The IMF components and their corresponding power spectrum are shown in Figure

EMD decomposition and spectrum of IMFs of measured signal.

EMD-1.5 dimension spectrum of measured signal.

Partial enlarged drawing of EMD-1.5 dimension spectrum of measured signal (80 Hz–200 Hz).

Partial enlarged drawing of EMD-1.5 dimension spectrum of measured signal (200 Hz–300 Hz).

In Figure

Combining the advantage of empirical mode decomposition and 1.5 dimension spectrum, a united mechanical fault diagnosis method is proposed. Simulated and measured signal analysis shows that EMD decomposition is a complete adaptive decomposition process for signal. And the decomposed IMF can be used to reconstruct the original signal, which lays the foundation for the selection of IMF with high correlation degree. 1.5 dimension spectrum is a special case of the high order spectrum. It has the ability to suppress Gauss noise and symmetrical distributed noise and has the minimum amount of computation which is similar to the amount of power spectrum. The analysis performance of bispectrum for quadratic phase coupling can be retained by 1.5 dimension spectrum. Based on empirical mode decomposition and 1.5 dimension spectrum, the nonlinear coupling characteristics can be extracted and the Gauss white noise can be effectively suppressed by the proposed method; EMD-1.5 dimension spectrum method provides the conditions for accurately extracting fault feature information.

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

The authors wish to acknowledge the assistance and support of all those who contributed to their effort to enhance and develop the described system. The authors express their appreciation for the financial support provided by the National Natural Science Foundation of China (Project no.: 41304098), Hunan Provincial Natural Science Foundation of China (Project no.: 12JJ4034), Young Scientific Research Fund of Hunan Provincial Education Department, PRC (Project no.: 13B076), Fund of the 11th Five-Year Plan for Key Construction Academic Subject (Optics) of Hunan Province, Optoelectronic information technology Hunan Province Talent training base between School and enterprise, and Hunan Province Key Laboratory of Photoelectric Information Integration and Optical Manufacturing Technology.