This paper presents the analysis of the vibration time series of a gear system acquired by piezoelectric acceleration transducer using the detrended fluctuation analysis (DFA). The experimental results show that gear vibration signals behave as double-scale characteristics, which means that the signals exhibit the self-similarity characteristics in two different time scales. For further understanding, the simulation analysis is performed to investigate the reasons for double-scale of gear’s fault vibration signal. According to the analysis results, a DFA double logarithmic plot based feature vector combined with scale exponent and intercept of the small time scale is utilized to achieve a better performance of fault identification. Furthermore, to detect the crossover point of two time scales automatically, a new approach based on the Hough transform is proposed and validated by a group of experimental tests. The results indicate that, comparing with the traditional DFA, the faulty gear conditions can be identified better by analyzing the double-scale characteristics of DFA. In addition, the influence of trend order of DFA on recognition rate of fault gears is discussed.

Generally, the gear transmission systems are characterized with periodic behaviors. However, the defects of gears, bearings, or transmission shafts may cause the nonlinear vibration. The gearbox vibration signals captured by the sensors are complicated, nonlinear, and nonstationary [

In the recent years, the fractal or multifractal time series have been observed in many fields, such as geophysics time series, medical time series, and technical time series [

The DFA was also used in equipment fault diagnosis. de Moura et al. [

However, the detail reasons for multiscales of fault vibration signals were not discussed in abovementioned description. Furthermore, as far as the authors know, the influence of detrend order of DFA on fault recognition was not discussed in previous literatures, and only limited methods for evaluating the crossover points of DFA were developed. In addition, the scale exponents of different time scale intervals were used as the characteristic parameter in previous researches. However, the intercept of double logarithmic plot of the DFA was not utilized. Acutally, the intercept that is used in our research involves a lot of information of vibration signal.

In this paper, the detrended fluctuation analysis (DFA) is employed to analyze the gear vibration signals. According to the double logarithmic plot of the DFA, it is verified that the gear vibration signals exhibit self-similarity in two ranges of time scales. The reason for the double-scale characteristic is discussed through the simulation analysis. Furthermore, the scale exponents and intercepts corresponding to different scale intervals are extracted as the feature vectors to describe the fault condition of gears. It is found that more pieces of information about the gear faults are involved in the small time scale interval. In order to detect the crossover point of two time scales and extract the parameters (scale exponents and intercepts) automatically, a new approach based on the Hough transform is proposed. The experiments were performed with the proposed parameters to classify the gear faults. Combining the Gaussian mixture model (GMM) and Bayesian maximum likelihood classifiers, the classification of gear vibration signals achieved successfully.

The remainder of the paper is organized as follows. Sections

Considering

Map

Divide

For each sub-time series, compute the fluctuation function:

Repeat Steps

The scale exponent

The Hough transform [

Generally, in a Cartesian coordinate plane

The pair of parameters

The outline of the Hough transform consists of the steps shown in Algorithm

Initialize accumulator

For each point

End

Find the values of

The detected lines in the

Different vibration condition signals of gears contain different frequency components and amplitudes. The main frequencies that should be paid more attention include rotation frequency, meshing frequencies, their harmonic frequencies, and sidebands. Usually, the gear’s vibration signal can be regarded as a combination of a series of sinusoid signals with different frequencies and random noise.

Consider the composite signal

The signal-noise-ratio (SNR) of

The logarithm scale fluctuation function maps of

Figure

Moreover, comparing the double logarithmic plot of

For the gear vibration signal, its characteristics analyzed by DFA are correlated with the fault conditions. Different fault patterns will cause different scale exponents. Moreover, a more severe defect will cause a larger vibration intensity, which will cause a larger intercept of double logarithmic plot. In our research, the scale exponent

In this section, the signals corresponding to four gear fault conditions obtained from gearbox experimental facility are analyzed by the DFA. The scale exponent and intercept are extracted as characteristic parameters to describe the gear conditions. Combining Gaussian mixture model (GMM) with Bayesian maximum likelihood classifier, these signals are classified.

The experiment setup is shown in Figure

The experimental setup of the gearbox fault detection.

The schematic diagram of the experimental setup.

The representative vibration signal and

The representative vibration signal, power spectrum, and the DFA curves obtained from the two types of gears with a rotation speed of 985 rpm: (a) Signal from a “Normal” gear. (b) DFA from a “Normal” gear. (c) Signal from a “Toothless” gear. (d) DFA from a “Toothless” gear.

Consider the double logarithmic plot as a binary image on which the values of pixels at a given coordinate,

As Figure

Except for the “Circular pitch error”, there is a small overlap between the “Normal,” “Scratched,” and “Toothless” gears. In the large time scale, the maps of “Scratched” and “Toothless” gears are overlapped completely, which indicates that the clustered results are better in the small time scale than those in large time scale. In theory, even if the gear has a tiny defect or the gear fault condition changes slightly, the vibration condition caused by the reduction of meshing stiffness will change. However, these changes may be so subtle that only local fluctuation in signals is affected. The local fluctuation of signals corresponds to signals’ morphological characters in small time scales interval or high frequency components rather than the large time scales interval or low frequency components. Only when the severity of defects reaches a lever or the fault conditions change a lot will the variation of large fluctuation in vibration signals and the difference in large time scale intervals be observed. Therefore, comparing to large time scales, there are more useful pieces of diagnostic information in small scales. As a contrast, the characteristic parameter maps of four kinds of gear faults by traditional DFA without considering linear relationship in different time range are shown in Figure

Considering we have a training dataset and a testing dataset which consist of a series of gearbox vibration signals, the main steps of the proposed algorithm for gear fault classification are described as follows.

For a kind of fault condition, employ DFA to plot the double logarithm graphs of all training signals and extract the feature vector

Build the Gaussian mixture model (GMM) of training space corresponding to this kind of fault by expectation maximum algorithm. The GMM is defined by

Repeat Steps

Before a testing signal is classified, the feature vector

In our classification experiment, for each gear condition, 100 signals were selected to constitute training dataset and 50 signals were selected to constitute testing dataset to verify the proposed approach. The minimum and maximum window sizes are 8 and 512 sampling points, respectively, and the mixture number of GMM is four. In order to evaluate the influence of the trend order of DFA which may change the position of crossover point, the classification experiments, when detrend order of DFA ranges from one to six, are conducted and the classification results are listed in Tables

Classification results with proposed method (DFA1).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 49 | 0 | 1 | 0 | 98% |

SCR | 0 | 46 | 4 | 0 | 92% |

TL | 5 | 3 | 42 | 0 | 84% |

CPE | 0 | 0 | 0 | 50 | 100% |

NOR: “Normal”; SCR: “Scratched”; TL: “Toothless”; CPE: “Circular pitch error.”

Classification results with proposed method (DFA2).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 50 | 0 | 0 | 0 | 100% |

SCR | 0 | 45 | 5 | 0 | 90% |

TL | 6 | 2 | 42 | 0 | 84% |

CPE | 0 | 0 | 0 | 50 | 100% |

Classification results with proposed method (DFA3).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 49 | 0 | 0 | 1 | 98% |

SCR | 0 | 48 | 2 | 0 | 96% |

TL | 7 | 1 | 42 | 0 | 84% |

CPE | 0 | 0 | 0 | 50 | 100% |

Classification results with proposed method (DFA4).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 46 | 0 | 3 | 0 | 92% |

SCR | 0 | 49 | 1 | 0 | 98% |

TL | 10 | 1 | 39 | 0 | 78% |

CPE | 0 | 0 | 0 | 50 | 100% |

Classification results with proposed method (DFA5).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 45 | 1 | 3 | 1 | 90% |

SCR | 0 | 47 | 3 | 0 | 94% |

TL | 10 | 1 | 39 | 0 | 78% |

CPE | 0 | 0 | 0 | 50 | 100% |

Classification results with proposed method (DFA6).

Gear fault | Diagnosis results | Recognition rate | |||
---|---|---|---|---|---|

NOR | SCR | TL | CPE | ||

NOR | 46 | 0 | 4 | 0 | 92% |

SCR | 1 | 46 | 3 | 0 | 92% |

TL | 14 | 4 | 32 | 0 | 64% |

CPE | 1 | 0 | 0 | 49 | 98% |

The plotting of recognition rate versus trend order of DFA.

Tables

In this study, the detrended fluctuation analysis (DFA) which can deal with nonstationary signals was employed to analyze the gear vibration signals acquired by piezoelectric acceleration transducer and the experimental results showed that the characteristics of all the gear condition signals turned out to be double-scale. To further understand the phenomenon of the double-scale, the experimental signals were analyzed by simulation. The simulation results show that the double-scales correspond to high and low frequency components of signal, respectively, and the intercepts are determined by signal intensity. A feature vector which employed an exponent

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

The reported research was partially based upon work supported by National Natural Science Foundation of China (NSFC) awards (Grant nos. 51105284, 51475339, 51375154, and 51405353).