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Spiral bevel gears are basic transmission components which are widely used in mechanical equipment. These components are important elements used in the monitoring and diagnosis of running states for ensuring the safe operations of entire equipment setups. The vibration signals of spiral bevel gears are typically quite complicated, as they present both nonlinear and nonstationary characteristics. In previous studies, multiscale permutation entropy (MPE) has been proven to be an effective nonlinear analysis tool for complexity and irregularity evaluations of complex mechanical systems. Therefore, it is considered that MPE values can be used as the sensitive features for spiral bevel gears fault identifications. However, if the MPEs are used to directly construct the feature vectors, some problems will be encountered, such as large numbers of characteristic quantities, high dimensions, and issues related to diagnosis accuracy and efficiency, which have been proven difficult to obtain at the same time. In order to improve the accuracy and efficiency of fault recognition in spiral bevel gear evaluations, locality preserving projection (LPP) methods can be applied to reduce the high dimensionality feature vectors constructed by MPEs. They have the ability to extract low-dimensional sensitive information from high-dimensional feature data. In order to directly obtain the diagnostic results, classifications are necessary. When compared with traditional neural networks, it has been found that extreme learning machines (ELMs) have the advantages of faster training speeds and stronger learning abilities. In summary, this study proposed the use of MPE values which could be optimized and dimensionality reduced by LPP as the feature vectors, along with ELMs as the classifiers of the fault mode identifications, in order to carry out valuable research of fault diagnosis methods for spiral bevel gears. The proposed method was applied to the diagnoses of four types of fault state spiral bevel gears. Then, the MPE-LPP-ELM results were compared with those obtained using MPE-PCA-ELM and MPE-ELM methods. Their respective diagnostic accuracy is 100%, 98.75%, and 98.75%, and diagnostic time is 0.0023 s, 0.0033 s, and 0.0078 s. It was determined in this study that the results confirmed the accuracy and superiority of the proposed method.

Spiral bevel gears are the core transmission components of power transmissions and speed transformations in modern mechanical equipment. They are widely used in such fields as aerospace, automobile, energy, and so on due to their large meshing surfaces and stable transmission abilities [

When the spiral bevel gear failure occurs, it will be reflected in various aspects, such as vibration [

In order to improve the fault recognition accuracy and efficiency of spiral bevel gears, it was necessary to reduce the dimensionality of high-dimensional characteristic quantities, remove the redundant information, and extract the optimal low-dimensional sensitive characteristic quantities. The traditional dimensionality reduction methods, such as principal component analysis (PCA) [

At the present time, BP networks and PNN networks are considered to be effective neural network fault mode recognition methods. Recently, with the increasing development of neural network theory, Huang et al. [

In conclusion, this study proposed the utilization of MPE values which had been optimized and dimensionally reduced by LPP as the feature vectors, along with ELMs as the classifiers of the fault mode identifications, in order to carry out research regarding the fault diagnosis methods commonly adopted for spiral bevel gears. In addition, in view of the problems related to complex noise and serious noise interference issues in the working environments of spiral bevel gears, the original vibration signals were denoised prior to the construction of the MPE characteristic quantities in this study. The proposed method provides a valuable fault diagnosis way for spiral bevel gears, which combines the superior nonlinear feature extraction ability of MPE, the superior feature dimensionality reduction ability of LPP, and stronger learning ability of ELM.

Aziz W et al. put forward a multiscale permutation entropy theory by studying the permutation entropy and multiscale sample entropy, which is the further optimization of permutation entropy. For the input signals, the multiscale permutation entropy ingeniously implemented multiscale coarse-graining of the signals for the purpose of completing analyses on the basis of the permutation entropy. Then, the permutation entropy values of the coarse-grained results were calculated. In addition, by assuming that the signal length of the spiral bevel gear to be detected was a time series

The detection signal was at first coarse-grained, and the coarse-grained result was as follows:

where

After the phase-space reconstruction of coarse-grained

where ^{th} reconstruction component

After the reconstructed sequence was arranged in ascending order, the following was obtained:

where ^{th} reconstruction component

In the present study, when

In this research investigation, through the detailed analysis of the measured signals and consultations of the relevant related studies, it was determined that the scale factor

As one of the classical manifold learning methods, the local preserving projection (LPP) maps of high-dimensional spatial data can be transformed into low-dimensional spaces through Laplacian feature projection according to the structural characteristics between adjacent points of the original data. LPP has the ability to maintain the local structures and dimensional reductions of the original data. Therefore, it can retain the inherent nonlinear structures and local sensitivity characteristics of the original high-dimensional eigenvectors of the spiral bevel gears and effectively reduce the dimensions of the high-dimensional features and extract the optimal low-dimensional sensitive characteristic quantities.

The original high-dimensional eigenvectors of the spiral bevel gears can be expressed as the high-dimensional space matrixes of _{i} is the vector of

In order to obtain the projection matrixes, LPP finds the minimum distance between y_{i} and y_{j} by optimizing the function (

Therefore, in the present study, the following formula was obtained by the projection matrix via solving the following generalized eigenvalue decomposition problem:

Then, the projection matrix formed by the eigenvectors corresponding to the

In recent years, with the further development of neural networks, ELM algorithms have emerged as times have required. These theories have been improved on the basis of single-hidden layered feedforward neural networks (SLFNs), and the previous shortcomings of the SLFNs have been greatly overcome, such as time-consuming training and difficulties in global optimization. Therefore, the methods have become simple and efficient learning methods. Since only the number of network hidden layer nodes and excitation function are required to be set, the connection weights of the input layers and hidden layers can be effectively generated, as well as the random generations of the thresholds of the input layers, as the composition consists of input layers, hidden layers, and output layers, and the structure of ELM is detailed in Figure

The structure diagram of ELM.

The mathematical modular form of ELM is as follows:_{i} are the input and output vectors, respectively.

For

The input weight _{i} were set randomly and remained unchanged after initialization,

The output matrix

The output weight values

The flow of the spiral bevel gear fault diagnosis method proposed in this study is shown in Figure

The diagnostic flowchart.

The vibration signals under the S1, S2, and S3 rotation speed conditions of the spiral bevel gears were collected. Then, taking the rotation speed condition S1 as an example, the normal state was set as A; one-third broken tooth state as B; two-thirds broken tooth state as C; and serious scratch state as D. There were a total of

CEEMDAN (Complete Ensemble Empirical Mode Decomposition with Adaptive Noise) decomposition of the state samples was implemented under the S_{1} rotation speed condition as detailed in Step (1). The correlation coefficients were used to select two intrinsic mode function reconstructions with the highest level of correlation in order to obtain the reconstructed signals containing the main information components. The corresponding reconstruction signals of the four fault states were denoted as _{c}, _{c}, _{c}, and _{c}, respectively.

A multiscale permutation entropy method was used for the calculations of the reconstructed signals, as detailed in Step (2), and the high-dimensional eigenvectors which could preliminarily characterize the different fault states of spiral bevel gears were obtained as follows:

The popular dimension reduction features of the LPP were applied to the obtained high-dimensional eigenvectors. Some of the redundant eigenvectors were eliminated, and the optimal low-dimensional sensitive eigenvectors among the high-dimensional eigenvectors were effectively extracted as follows:

The optimal sensitive eigenvectors were input into the ELM for training and testing purposes, and the diagnosis results were obtained.

In this study, the effectiveness of the proposed diagnosis method based on MPE-LPP and ELM was proven through fault diagnosis experiments of spiral bevel gears. The experimental data were obtained from the fault test bench for spiral bevel gear box as shown in Figure

The spiral bevel gearbox test equipment.

The auxiliary equipment included a drive motor, governor, coupler, and so on. Then, the PULSE system of the B&K Company was used as the signal acquisition system, and the driving gear at the input shaft was taken as the test gear, with the number of gear teeth set as 10. The fault states of normal gears; gears with one-third broken tooth; gears with two-thirds broken tooth; and gears with severe scratches were simulated in this study, as shown in Figure

Gears with different fault states. (a) Normal status. (b) 1/3 broken tooth. (c) 2/3 broken tooth. (d) Severe scratches.

During the experimental testing processes, the input shaft speed was controlled by the speed governor in order to maintain constant speeds of 900 r/min, 1,200 r/min, and 1,500 r/min for the simulations of three rotation speed conditions of the spiral bevel gears denoted as S_{1}, S_{2}, and S_{3}, respectively. A vibration acceleration sensor was installed at the bearing seat of the input shaft, and the sampling frequency was set as 16384 Hz.

This study selected the vibration signals with an input shaft speed of 900 r/min for further analysis. Figure

The collected vibration signal. (a) Normal status. (b) 1/3 broken tooth. (c) 2/3 broken tooth. (d) Severe scratches.

A CEEMDAN system was used to decompose the original vibration signals and select the two intrinsic mode function reconstructions with the largest correlation coefficients in order to obtain the reconstructed signals after noise elimination. The meaningless low-frequency components within the reconstructed signals were also eliminated. Meanwhile, the middle- and high-frequency parts containing the main information were retained, which effectively filtered out the interference effects of the background noise and increased the proportion of impactful components in the reconstructed signals.

The entropy values under the best scale factor _{k}. It was found that the larger the _{k} value was, the better the discrimination degree of the scale factor _{C}, _{C}, _{C}, and _{C}, it was confirmed that the MPE values of the different scale factors had varied under the same state. The MPE values of the reconstructed signals of the four states under the same scale factor were also different, and the larger _{k} values were observed to be mainly distributed in the front several scale factors, as shown in Table

The MPE and S_{k} of the reconstructing signal.

The scale factor | Failure mode | _{k} | |||
---|---|---|---|---|---|

1 | 0.989 | 0.991 | 0.974 | 0.959 | 0.111 |

2 | 0.991 | 0.986 | 0.980 | 0.939 | 0.162 |

3 | 0.988 | 0.967 | 0.953 | 0.940 | 0.158 |

4 | 0.975 | 0.967 | 0.958 | 0.996 | 0.122 |

5 | 0.982 | 0.985 | 0.989 | 0.995 | 0.043 |

… | … | … | … | … | … |

12 | 0.982 | 0.984 | 0.968 | 0.976 | 0.054 |

13 | 0.975 | 0.989 | 0.984 | 0.972 | 0.060 |

14 | 0.987 | 0.973 | 0.979 | 0.985 | 0.048 |

15 | 0.989 | 0.986 | 0.980 | 0.981 | 0.032 |

The MPE values of the denoised signals of the spiral bevel gears under the scale factor

The 3D map of MPE-LPP.

In order to verify the effectiveness of the proposed MPE-LPP feature extraction method, the results were compared with those obtained using the feature extraction methods MPE-PCA and MPE. The original vibration signals were denoised by CEEMDAN. The three-dimensional diagrams constructed by the first three-dimensional eigenvectors of the MPE-PCA and MPE are shown in Figures

The 3D map of MPE-PCA.

The 3D map of MPE.

In the present study, twenty groups of data in each of the four states of the spiral bevel gears were selected to construct eighty samples. The eigenvectors were extracted as training samples using the feature extraction methods MPE-LPP, MPE-PCA, and MPE. ELM training was then performed. In addition, twenty groups of data were taken in the same way from each of the four states of the spiral bevel gears and eighty groups of testing samples for classification and diagnosis purposes, and the results are shown in Figure

ELM recognition results of three feature extraction methods. (a) MPE-LPP-ELM. (b) MPE-PCA-ELM. (c) MPE-ELM.

ELM diagnostic results and identification time.

Eigenvectors | Dimension | Diagnostic accuracy rate (%) | Time (s) | ||
---|---|---|---|---|---|

Categories | Average | ||||

MPE-LPP | 3 | 1 | 100 | 100 | 0.0023 |

2 | 100 | ||||

3 | 100 | ||||

4 | 100 | ||||

MPE-PCA | 3 | 1 | 100 | 98.75 | 0.0033 |

2 | 100 | ||||

3 | 95 | ||||

4 | 100 | ||||

MPE | 15 | 1 | 100 | 98.75 | 0.0078 |

2 | 95 | ||||

3 | 100 | ||||

4 | 100 |

In order to further verify the effectiveness of this study’s proposed method, the vibration signals of the spiral bevel gears with input shaft speeds of 1,200 r/min and 1,500 r/min were selected for further case analysis purposes. Figures

The 3D map at speed of 1,200 r/min. (a) The 3D map of MPE-LPP. (b) The 3D map of MPE-PCA. (c) The 3D map of MPE.

The 3D map at speed of 1,500 r/min. (a) The 3D map of MPE-LPP. (b) The 3D map of MPE-PCA. (c) The 3D map of MPE.

Fault diagnosis result of spiral bevel gears at different speeds.

Shaft speeds r (min) | Eigenvectors | Dimension | Diagnostic accuracy rate (%) | Time (s) |
---|---|---|---|---|

1200 | MPE-LPP | 3 | 100 | 0.0041 |

MPE-PCA | 3 | 98.75 | 0.0031 | |

MPE | 15 | 97.5 | 0.0081 | |

1500 | MPE-LPP | 3 | 100 | 0.0033 |

MPE-PCA | 3 | 97.5 | 0.0037 | |

MPE | 15 | 97.5 | 0.0124 |

In this study, a fault diagnosis method for spiral bevel gears was proposed based on MPE-LPP and ELM. The MPE values which had been optimized and dimensionality reduced by LPP are used as the feature vectors, along with ELMs that are used as the fault classifiers for the fault mode identifications. The case analysis results of the vibration signals of spiral bevel gears in four states under three rotating speeds were obtained. Then, the results of the proposed system were compared with those of the MPE-PAC-ELM and MPE-ELM recognition methods. It was determined that the application of the proposed MPE-LPP and ELM method in the fault diagnosis of spiral bevel gears had resulted in obvious advantages related to diagnostic accuracy and diagnostic speed, when compared to the other examined methods. The proposed MPE-LPP-ELM recognition method can be used in other diagnostic requirements, such as the diagnosis of spur gears, bearings, and so on. It can even be used in other pattern recognition requirements. Therefore, further popularization and application of this method should be considered in the future research works.

The data are available upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (Grant no. 11872022) and the National Aeronautical Science Foundation of China (Grant no. 20200033116001).