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In order to sufficiently capture the useful fault-related information available in the multiple vibration sensors used in rotation machinery, while concurrently avoiding the introduction of the limitation of dimensionality, a new fault diagnosis method for rotation machinery based on supervised second-order tensor locality preserving projection (SSTLPP) and weighted

As one of the most common mechanical equipment classes, rotation machinery occupies an important role in industrial applications such as manufacturing, metallurgy, energy, and transportation. Due to tough working environments, similar materials, and structural properties, rotation machinery can be subject to malfunctions or failures. This can significantly decrease machinery service performance including manufacturing quality and operation safety and cause machinery to break down, which may lead to serious catastrophes [

Large volumes of feature parameters generated by time-domain, frequency-domain, and time-frequency-domain analysis of vibration signals are commonly integrated into a high- dimensional data set to obtain accurate fault diagnostic results [

To further improve the accuracy and the efficiency of fault diagnosis, intelligent classification methods are considered as an indispensable component in the diagnostic procedure. These methods include artificial neural networks (ANN) [

The remainder of this paper is organized as follows. The proposed supervised second-order tensor locality preserving projection based on assembled matrix distance metric (SSTLPP-AMDM) algorithm is discussed in detail in Section

As the tensor extension of LPP, TLPP is essentially equivalent to finding a linear approximation of the eigenfunctions of the Laplace Beltrami operator in a tensor space. The incipient TLPP which was initially presented by He et al. [

Using a series of mathematical derivations, the optimal values for

Finally, the low-dimensional representations of the original data are obtained using

As described in the previous section, there are a certain number of limitations when using the prevailing computation method for the similarity weighting matrix

Firstly, for any two arbitrary matrix formed samples

Secondly, by understanding the class label information of the training samples and the AMDM based distances between samples, the proposed supervised similarity weighting matrix based on AMDM can be defined as

The newly formulated similarity weighting matrix computation equation shown in (

This paper proposes a novel supervised second-order tensor locality preserving projection algorithm with the assembled matrix distance metric (SSTLPP-AMDM) that uses the improvements in both the matrix distances computation of samples in the projection space and the similarity weighting matrix computation expression. In contrast to traditional STLPP, the two transformation matrices

The optimal transformation matrices

In summary, there are two main advantages to the newly proposed SSTLPP-AMDM. (

As stated above, the KNNC method proposed by Cover and Hart in 1967 [

For a given unknown labeled sample

A maximum voting rule is used on all samples in

The above description shows that there are two focus points to KNNC: a similarity measurement method between samples and the establishment of a decision rule. For the first focus point, there have been many similarity measurement methods suggested by previous publications, such as the Euclidean distance, the Manhattan distance, and the cosine angle. However, these vector representations of the data-based metric indexes described above are unsuitable for similarity measurement of the matrix formed data points appearing in this paper. Thus, the AMDM is introduced for the similarity computation of samples in KNNC. It is known that AMDM outperforms common FDM in terms of the similarity presentation between matrix formed samples for classification. Additionally, since selection of neighbors is greatly impacted by the sparsity of the sample distribution, this paper employs a novel assembled matrix distance based on density to efficiently measure the similarity between

Unlike the classical KNNC voting strategy that uses unified weights for neighbors, in this paper, a weighted voting strategy is used to form the weighted

Consequently, the class label of an unknown labeled sample

Additionally, the selection of

Based on the preparations above, this paper proposes a novel multisensor fused fault diagnosis method based on SSTLPP-AMDM and WKNNC-AMDM for rotation machinery. The flow chart for the proposed method is shown in Figure

Implementation process of the proposed fault diagnosis method based on SSTLPP and WKNNC.

Firstly, through prevalent multidomain signal analysis and truncated sampling, a multisensor fused faulty sample set with an

The second step is compression of the high-dimensional

Finally, the low-dimensional projection of the testing sample set and the low-dimensional projection of the training sample set obtained in the previous step are input into WKNNC for fault diagnosis.

The validity of the newly proposed method will now be demonstrated using a fault diagnosis experiment of a single-stage gearbox. As shown in Figure

Specific information of seven sensors.

Number of sensor | Type of sensor | Location | Direction |
---|---|---|---|

1# | Displacement | Input shaft | |

2# | Displacement | Input shaft | |

3# | Accelerometer | Bearing housing of input shaft in the side of motor | |

4# | Accelerometer | Bearing housing of output shaft in the side of motor | |

5# | Accelerometer | Bearing housing of input shaft in the side of load | |

6# | Accelerometer | Bearing housing of output shaft in the side of load | |

7# | Accelerometer | Bearing housing of output shaft in the side of load | |

Gearbox fault simulation test setup: (a) overview and (b) diagram.

During the experiment, the sampling frequency was 5120 Hz, there were 53248 sampling points, the rotation speed of the drive motor was 880 rev/min, and the load was 0.2 A. There were six types of conditions used in the gearbox fault simulation experiment: (

Test signals originating from the seven different sensors under the following conditions: (a) Norm, (b) C_G, (c) B_G, (d) W_P, (e) B_G_C_W_P, and (f) C_G_C_W_P.

Test signals originating from each single sensor under six conditions: (a) 1# sensor, (b) 2# sensor, (c) 3# sensor, (d) 4# sensor, (e) 5# sensor, (f) 6# sensor, and (g) 7# sensor.

50 samples under each condition from a single sensor were subsequently selected, and 30 of these samples were used to train the fault diagnosis model, with the remaining samples used for the testing purposes. The length of each sample was 1024. Furthermore, five time-domain feature parameters and five frequency-domain parameters were calculated to construct a feature set: root mean square, skewness, kurtosis, impulse factor, peak factor, mean frequency, frequency center, root mean square frequency, standard deviation frequency, and kurtosis frequency, as commonly defined in the previous literature [

This subsection validates the effectiveness of the proposed SSTLPP-AMDM algorithm for dimension reduction in fault diagnosis of rotation machinery, as well as its superiority to the traditional STLPP method. Using the calculation procedure described in Section

Scatter plots of the vector-formed dimension reduction result based on different algorithms for the training sample set: (a) SSTLPP-AMDM and (b) STLPP.

For further confirmation of the superiority of the proposed second-order tensor formed faulty samples originating from multisensor fusion over the vector-represented multisensor fused samples and the prevailing vector-formed faulty samples that originated from merely a single sensor, two further groups of experiments were designed. These experiments are the LPP-based dimension reduction of vector expressed multisensor fused samples (LPP-VM) and the LPP-based dimension reduction of faulty samples from any single sensor (LPP-VS) and their purpose is to compare with SSTLPP-AMDM which is input by the proposed second-order tensor-represented faulty samples in terms of the dimension reduction effect shown as Figure

Comparison of scatter parameter values based on ten different dimension reduction methods.

Methods | Notation | The first-dimensional feature | The second-dimensional feature | The third-dimensional feature | ||||||
---|---|---|---|---|---|---|---|---|---|---|

| | | | | | | | | ||

SSTLPP-AMDM for STMD | M1 | | 0.6222 | | 0.0025 | 0.7745 | 307.2982 | 0.0029 | 0.4584 | 158.5695 |

STLPP for STMD | M2 | | 0.1120 | 268.9086 | 0.0085 | 0.0376 | 4.4208 | 0.0054 | 0.0391 | 7.2642 |

LPP for VMD | M3 | | 0.6099 | 1 | 0.0030 | 0.7504 | 251.4014 | 0.0063 | 0.4867 | 77.4677 |

LPP for VSD1 | M4 | | 0.4176 | 535.1605 | | 0.0661 | 101.3214 | 0.0041 | 0.0404 | 9.7608 |

LPP for VSD2 | M5 | 0.0019 | 0.5644 | 296.4655 | 0.0014 | 0.0908 | 64.9652 | 0.0037 | 0.0417 | 11.2092 |

LPP for VSD3 | M6 | 0.0018 | 0.4675 | 262.9062 | 0.0056 | 0.0942 | 16.7078 | 0.0084 | 0.0133 | 1.5848 |

LPP for VSD4 | M7 | 0.0020 | 0.5212 | 262.6050 | 0.0032 | 0.0782 | 24.4434 | 0.0127 | 0.0427 | 3.3715 |

LPP for VSD5 | M8 | 0.0019 | 0.6175 | 328.6004 | 0.0028 | 0.0789 | 28.2364 | 0.0103 | 0.0286 | 2.7673 |

LPP for VSD6 | M9 | 0.0023 | 0.6341 | 281.7623 | 0.0039 | 0.1345 | 34.6284 | 0.0058 | 0.0503 | 8.6715 |

LPP for VSD7 | M10 | | 0.3936 | 371.5203 | | 0.0328 | 34.2578 | 0.0286 | 0.0157 | 0.5488 |

Comparison of fault classification results based on three classifiers and ten types of reduced feature sets.

Classifier | Classification accuracy (%) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | |

SVM | 99.8 | 97.5 | 96.0 | 88.67 | 89.17 | 92.22 | 84.5 | 86.8 | 87.5 | 83.33 |

MLP neural network | 83.33 | 66.67 | 66.67 | 65.0 | 64.17 | 63.33 | 65.0 | 63.5 | 63.33 | 65.0 |

SVDD | 98.5 | 97.22 | 94.33 | 90.33 | 94.17 | 67.5 | 83.33 | 76.67 | 82.22 | 82.5 |

Scatter plots of dimension reduction results based on LPP with different input data: (a) LPP-VM and (b) LPP-VS.

Comparison of scatter parameter values based on different methods: (a)

The results shown in the scatter distribution diagrams in Figure

The previously-mentioned training sample set is used, which contains six-class faulty condition data for the gearbox as the input. Ten groups of experiments are performed to calculate the corresponding scatter parameters of the first three-dimensional features of the vectored dimension reduction results: (

As mentioned earlier, in order to acquire direct evidence of the superiority of the proposed SSTLPP-AMDM algorithm as well as the multisensor data fusion, three frequently-used intelligent classifiers (SVM, MLP neural network, and SVDD), respectively, acted on the first three-dimensional features of the vectored dimension reduction results of the ten methods (M1~M10), which are marked as F1~F10. Each experiment is carried out ten times. For the SVM classifier, this paper employs a radial basis kernel function and the value of the kernel parameter is 1. For the MLP neural network, the commonly used three layers structure is employed: input layer, hidden layer, and output layer, and the numbers of nodes in the input and output layers are set to 3 and 6. These values depend on the number of input features and output classes. The geometric pyramid rule determines that the number of hidden layer nodes is 5. The Gaussian kernel function is used for the SVDD model, and the corresponding kernel parameter is set to 3. The classification results of the three models which are applied to each of the ten types of feature sets originating from the previous experiment are listed in Table

As shown in Table

The following experiments and analysis were also employed to verify the superiority of the proposed WKNNC-AMDM method, as well as the overall fault diagnosis approach proposed by this paper. Using the implementation procedure for the proposed fault diagnosis method shown in Figure

Fault diagnosis results of three different classifiers with the reduced low-dimensional tensor formed multisensor fused samples.

Classifier | Cum. number of FCS | Norm | C_G | B_G | W_P | B_G_C_W_P | C_G_C_W_P | Total testing accuracy (%) |
---|---|---|---|---|---|---|---|---|

Number of FCS within Class 1 | Number of FCS within Class 2 | Number of FCS within Class 3 | Number of FCS within Class 4 | Number of FCS within Class 5 | Number of FCS within Class 6 | |||

KNNC-FDM | 35 | 16 | 15 | 0 | 3 | 0 | 1 | 70.83 |

WKNNC-FDM | 13 | 0 | 13 | 0 | 0 | 0 | 0 | 89.17 |

WKNNC-AMDM | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 100 |

It can be seen from Table

This paper has presented a novel multisensor fused fault diagnosis approach for rotation machinery based on SSTLPP-AMDM and WKNNC-AMDM. Based on significant experimental analysis and comparisons that were performed, the main conclusions can be summarized as follows.

In contrast with traditional STLPP, the proposed SSTLPP-AMDM algorithm can obtain better dimension reduction effects for the original high-dimensional second-order tensor-represented samples. This was achieved by the addition of the class label information and improvement of the similarity evaluation method for matrix formed samples by AMDM. Furthermore, it was also verified that SSTLPP-AMDM based dimension reduction of multisensor fused second-order tensor formed samples is superior to LPP-based dimension reduction of multisensor fused vector-formed samples and LPP-based dimension reduction of vector-formed samples from a single sensor in terms of the clustering performance of samples of different classes after reduction.

The proposed WKNNC-AMDM can obtain higher classification accuracy than WKNNC-FDM and KNNC-FDM due to the introduction of weighted voting strategy and assembled matrix distance metric for similarity representation of second-order tensor formed samples.

Using the advantages of second-order tensor formed multisensor fused faulty sample representation, SSTLPP-AMDM for efficient dimension reduction, and WKNNC-AMDM for rapid fault classification, the proposed fault diagnosis approach achieves higher classification accuracy for rotation machinery than the other homogenous methods.

In summary, the proposed fault diagnosis approach has the following strengths: more adequate fault information, lower calculation complexity, and higher fault recognition accuracy. Therefore, it is extremely suited to engineering applications for fault diagnosis of rotation machinery.

The authors declare that they have no competing interests.

This research is supported by National Natural Science Foundation of China (no. 51575143).