Vibration analysis is the most used technique for defect monitoring failures of industrial gearboxes. Detection and diagnosis of gear defects are thus crucial to avoid catastrophic failures. It is therefore important to detect early fault symptoms. This paper introduces signal processing methods based on approximate entropy (ApEn), sample entropy (SampEn), and Lempel-Ziv Complexity (LZC) for detection of gears defects. These methods are based on statistical measurements exploring the regularity of vibratory signals. Applied to gear signals, the parameter selection of ApEn, SampEn, and LZC calculation is first numerically investigated, and appropriate parameters are suggested. Finally, an experimental study is presented to investigate the effectiveness of these indicators and a comparative study with traditional time domain indicators is presented. The results demonstrate that ApEn, SampEn, and LZC provide alternative features for signal processing. A new methodology is presented combining both Kurtosis and LZC for early detection of faults. The results show that this proposed method may be used as an effective tool for early detection of gear faults.

Gearboxes play an important role in industrial applications and unexpected failures often result in significant economic losses. Numerous papers considering gear condition monitoring through vibration measurements were published over the years. Compared to classical techniques such as statistical time indicators or Fast Fourier Transform, advanced signal processing techniques like time-frequency analysis (STFT, Wigner-Ville) [

Various researches are focused on the understanding of dynamic gears behaviour. Parey and Tandon [

Randall [

A large number of statistical metrics have been applied for monitoring gear defects, such as the root mean square, Crest Factor, Kurtosis, FM0, FM4, FM

Therefore, it is interesting to use others techniques as nonlinear estimations of parameters which may be a good alternative to monitoring hidden faults in the measured signals. Yan and Gao [

As said before, the effects of various nonlinearities cause a variation in the dynamic behaviour of the gear. As the default develops, more contact pressure between two mating parts changes. The contact pressure variation results in amplitude and frequency modulations and hence more frequency will appear in the frequency domain with their harmonics and an increase of the noise can be observed. Therefore, the fault progress leads to complexity (randomness) changes because of a variation of the contact pressure between two mating parts. The complexity of a signal increases when more frequency components exist. On the other hand, when the defect increases, we observe more frequency components in the signal and we can use the complexity of the signal as a quantitative measurement to evaluate the severity of the defect. The complexity of the signal can be described by the approximate entropy (ApEn), the sample entropy (SampEn), and the Lempel-Ziv complexity (LZC). These parameters are nonlinear parameters which can be used to characterise the regularity of the signal.

The Lempel-Ziv complexity (LZC), sample entropy (SampEn), and approximate entropy (ApEn) thus present alternative tools for signal analysis involving nonlinear dynamics. These methods are becoming more popular and have found wide applications in various disciplines, especially in the field of biomedical engineering. ApEn has recently received more attention. Yan and Gao [

On the other hand, Yan and Gao [

The use of SampEn was introduced by Dennis Wong et al. [

As illustrated above, LZC, SampEn, and ApEn are becoming more and more attractive in the field of detection and fault diagnosis. However, no work has been found to apply ApEn, LZC, or SampEn measurement for diagnosis of gear faults. Therefore, in this paper, ApEn, SampEn, and LZC are introduced to analyse vibration signals from gear and investigate their efficiency for the defect detection and severity evaluation of gears faults.

Consider a time series

For each vector

A finite time series consisting of

Sample entropy was developed because it has better representation of the entropy in the analysed signals as comparing with the original approximate entropy (ApEn) [

For a given

The distance between such vectors is defined as

Similarly, we define another function:

The symbol

The complexity analysis is based on the Lempel-Ziv definition [

The flow chart of LZC algorithm.

For generality sake, normalized complexity

From the above described algorithm of ApEn and SampEn, it can be seen that the calculated ApEn and SampEn values depend on two parameters which are the embedded dimension

However, no guidelines exist for optimizing their values. In order to simulate the vibratory signals of gearbox, a gear multiplicative model whose meshing is modulated in amplitude has been used. The gear model as defined in [

Table

Geared system data of the simulated signal.

Frequency (Hz) | Amplitude | |
---|---|---|

Pinion: number of teeth = 20 | 15 | 0.15 |

Gear: number of teeth = 21 | 14.28 | 0.15 |

Gear mesh: the first harmonic | 300 | 1 |

Gear mesh: the second harmonic | 600 | 0.6 |

Gear mesh: the third harmonic | 900 | 0.3 |

Figures

The simulated signal.

Spectrum of the simulated signal.

For a given dimension

The selections of

From Figure

The calculated ApEn values by different parameters (

Like ApEn, sample entropy (SampEn) depends on the two parameters

The calculated sample entropy values by different parameters (

However, a larger

By this investigation,

The relationship between the ApEn, SampEn, and LZC values and the data length are illustrated in Figure

The calculated ApEn values by different parameters (length of the data and the sample frequency).

The calculated SampEn values by different parameters (length of the data and the sample frequency).

The calculated LZC values by different parameters (length of the data and the sample frequency).

However, the sampling frequency and the data length are linked. So we must take consideration of the component defining the signal. The sample frequency must be more than 2 times the highest frequency presented in the signal, for Shannon respect. In our case, the sample frequency must be 2 times greater than the third meshing frequency (900 Hz). The simulation was conducted with

Figure

Time computation of both LZC and ApEn for different length of data.

It is clear that the LZC is the best method for time computing. The sample entropy algorithm is simpler than the ApEn algorithm, requiring approximately one-half time to calculate.

It is well known that a white noise contains most abundant frequency components compared with other kinds of signal. If a signal is contaminated by a white noise, calculated ApEn, SampEn, and LZC values will also be. Using the simulated signal, the ApEn, SampEn, and LZC values corresponding to different SNRs are calculated, as listed in Table

ApEn, SampEn, and LZC for different SNR of gear signals.

Cases | ApEn | LZC | SampEn |
---|---|---|---|

The simulated signal | 0.3529 | 0.1294 | 0.2644 |

SNR = 80 dB | 0.3547 | 0.1364 | 0.2656 |

SNR = 60 dB | 0.3648 | 0.1692 | 0.2765 |

SNR = 40 dB | 0.4495 | 0.3056 | 0.3500 |

SNR = 0 dB | 1.3163 | 0.8909 | 1.2190 |

It may be noticed that the ApEn, SampEn, and LZC values increase as the SNR decreases, which corresponds to a degradation of the data quality. As discussed in [

From Table

The recordings of vibration signal were carried out at CETIM, France, on a gear system with a train of gearing, with a ratio of 20/21 functioning continuously until its destruction. The sample frequency is equal to 20 kHz. Table

Geared system data.

Parameter | Pinion | Gear |
---|---|---|

Number of teeth | 20 | 21 |

Speed (rpm) | 1000 | 952.38 |

Drive torque (Nm) | 200 | |

Face width (m) | 0.015 | 0.03 |

Module (m) | 0.01 | 0.01 |

Pressure angle | 20° | 20° |

Addendum coefficient | 1.0 | 1.0 |

Dedendum coefficient | 1.4 | 1.4 |

Daily mechanical appraisal.

Day | Observation |
---|---|

1 | No acquisition |

2 | No anomaly |

3 | No anomaly |

4 | // // |

5 | // // |

6 | // // |

7 | Chipping teeth 1/2 |

8 | No evolution |

9 | Tooth 1/2: no evolution; tooth 15/16: start chipping |

10 | Evolution of the chipping of the teeth 15/16 |

11 | // // |

12 | // // |

13 | Chipping across the full width of the tooth |

Evolution of the chipping.

Pareya et al. [

The evolutions of these descriptors are displayed in Figures

RMS (m/s²) value during the test for all days.

Peak (m/s²) value during the test for all days.

Kurtosis value during the test for all days.

Crest Factor value during the test for all days.

Skewness value during the test for all days.

Evolution of acceleration signal with wear.

It can be seen that the all indicators increase greatly after day 11. After day 11 the signal becomes impulsive. The Crest Factor (CF) observes a little growth after day 9 (17% of evolution compared to day 9).

The same observations were observed for the Peak values (26%), the

Consequently, the classical time domains are ineffective for early detection of gear defects. These measures are appropriate for detection and diagnosis when mechanical faults are in advanced state of the degradation. As discussed in [

Therefore, ApEn, SampEn, and LZC of the experimental signal were computed from day 2 to day 13 (on day 1, no signal was taken). The values of ApEn and SampEn are plotted in Figure

ApEn and SampEn value during the test for all days.

LZC value during the test for all days.

Spectrum of the experimental signal.

Figure

Kurtosis versus LZC value during the test for all days.

The last region is marked by an increase of Kurtosis and LZC stays in the same level as the second region. At this stage, the signal becomes impulsive and the gear is damaged.

Using this representation, we can easily distinguish the different steps of the process of the degradation. This representation gives both information on the impulsiveness of the signal and on the effect of the number of frequency components and noise into the signal. Consequently, this representation combining Kurtosis and LZC may be used as an efficient tool for early detection of faults.

This paper introduces ApEn, SampEn, and LZC metrics to analyse the vibration signal recorded from defected gears. With respect to gear signals, the parameter selection of ApEn and SampEn was investigated and the results show that

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

The financial support of NSERC (Natural Sciences and Engineering Research Council of Canada), FQRNT (Fonds Québecois de la Recherche sur la Nature et les Technologies), MITACS Canada, Pratt & Whitney Canada, and CETIM which provided the experimental results are gratefully acknowledged.