_{1}

The three-order cumulants’ complex forms of different definitions include different coupling information of signals, and the information can be used to diagnose fault. In the experiment of pressure reducing valve’s fault diagnosis, through these different coupling information, the features of fault signals and normal signals were extracted by wavelet in different directions, then these features were inputted to diagnose the fault. The experiment shows that this method can achieve a satisfactory result.

High-order cumulants can automatically suppress the influence of Gaussian background noise (colored or white), establish a non-Gaussian signal model under Gaussian noise, and extract non-Gaussian signals (including harmonic signals) in Gaussian noise [

Stronach et al. conducted in-depth research on high-order spectra and applied them to various mechanical fault diagnosis [

Let { x ( n ) } be a zero-mean k-order stationary stochastic process, then the k-order cumulant of the process is defined as the k-order joint cumulant of the random variable.

c k x ( τ 1 , τ 2 , ⋯ , τ k − 1 ) = c u m { x ( n ) , x ( n + τ 1 ) , x ( n + τ 2 ) ， ⋯ , x ( n + τ k − 1 ) } (1)

The k-order moment of the process m k x ( τ 1 , τ 2 , ⋯ , τ k − 1 ) is defined as the k-order joint moment of the random variable { x ( n ) , x ( n + τ 1 ) , ⋯ , x ( n + τ k − 1 ) } , which is

m k x ( τ 1 , τ 2 , ⋯ , τ k − 1 ) = m o m { x ( n ) , x ( n + τ 1 ) , ⋯ , x ( n + τ k − 1 ) } (2)

Here, mom() represents the joint moment, and the third-order cumulant is

c 3 x ( τ 1 , τ 2 ) = E { x ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } (3)

According to the literature [

Definition one:

c 3 x ( τ 1 , τ 2 ) = E { x ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } (4)

Definition two:

c 3 x ( τ 1 , τ 2 ) = E { x ∗ ( n ) x ( n + τ 1 ) x ( n + τ 2 ) } (5)

Definition three:

c 3 x ( τ 1 , τ 2 ) = E { x ∗ ( n ) x ( n + τ 1 ) x ∗ ( n + τ 2 ) } (6)

Among them, x ∗ ( n ) is the conjugate complex number of x(n). The complex signal in this paper is obtained by hilbert transform from the original signal collected. Reference [_{1} is 0, in the manner of defining two, its third-order cumulant C_{2} is as shown in Equation (8), and its third-order cumulant C_{3} is defined as Equation (9) shown.

x ( n ) = ∑ i = 1 3 A i exp ( ω i n ) + ϕ i (7)

C 2 = A 1 A 2 A 3 [ exp [ j ( ω 1 τ 1 + ω 2 τ 2 ) ] + exp [ j ( ω 2 τ 1 − ω 1 τ 2 ) ] ] (8)

C 3 = A 1 A 2 A 3 [ exp [ j ( ω 3 τ 1 − ω 1 τ 2 ) ] + exp [ j ( ω 3 τ 1 − ω 2 τ 2 ) ] ] (9)

It can be seen from Equation (8) that in the second way, its third-order cumulant contains only the harmonic signals ω 1 and ω 2 that participate in the coupling, and the third-order cumulant in the third way contains both harmonic signals that participate in coupling ω 1 and ω 2 and also contains the coupled signal.

Wavelet transform can be used for nonlinear, non-stationary mechanical vibration signals. It has the ability to characterize the local features of the signal in both time and frequency domains, it is a time-frequency localization analysis method in which the window size is fixed but its shape is variable, and both the time window and the frequency window are variable. The two-dimensional wavelet function is obtained by tensor product transformation through one-dimensional wavelet function.

The pressure reducing valve in this experiment is a pilot type pressure reducing valve. When the pressure reducing valve has foreign matter in and out of the oil port, the pressure is high or low, which will affect the normal operation of the pressure reducing valve, In order to obtain the operating signal of the pressure reducing valve in the fault state, the experiment conducted in this paper artificially sets the following faults: (

Add ϕ 3 mm core to the pressure inlet of the pressure reducing valve.

Through the experiment, the working condition of the pressure reducing valve can be approximated.

This paper uses LabVIEW software and PCI-6014 data acquisition card and an acceleration sensor to sequentially collect the vibration signal of the pressure reducing valve under normal and fault conditions. In each measurement, the oil pressure is divided into five pressure levels from 1 MPa to 5 MPa. The sampling

frequency is 250 Hz, the reading frequency is 125 Hz, and the sampling process time is about 2 min. In this experiment, 18 groups of 36 sets of data were collected in the normal working state and the fault state of the pressure reducing valve. The number of data used in this experiment is 1536.

In this experiment, the 36 sets of data of the normal state and the fault state are first obtained, and the third-order cumulant of each set of data is obtained according to the above three definitions, in this paper, we select the complex third-order cumulant of the normal state and fault state data, let − 127 ≤ τ 1 , τ 2 ≤ 127 , and then when the oil pressure is 1 MPa, 3 MPa, 5 MPa, in the definition of the three modes, the third-order cumulant of the two states is taken as an absolute value, and each of the selected groups is shown in

In the fault identification, if the third-order cumulant data obtained is directly input into the neural network, the calculation amount is too large, and the diagnosis result obtained by the experiment is not satisfactory. Since the various frequency information contained in the original signal must be reflected in its third-order cumulant, and wavelet decomposition can effectively extract these frequency information, therefore, this paper uses two-dimensional wavelet decomposition to extract features that the third-order cumulant of different coupling modes of each measured data, this article uses the db1 wavelet [

and

Mechanical fault diagnosis includes three parts: signal acquisition, signal processing and fault mode classification. Fault mode classification is the core

Normal status | 0.2303 | 0.3309 | 0.0777 | 06620 | 0.4379 | 0.5280 | 0.0139 | 0.3194 |
---|---|---|---|---|---|---|---|---|

0.3131 | 0.3222 | 0.3725 | 0.3740 | 0.3357 | 0.0323 | 0.3156 | 0.0246 | |

0.2426 | 0.4528 | 0.2681 | 0.4695 | 0.5328 | 0.2329 | 0.5135 | 0.5177 | |

Fault status | 0.3236 | 0.3258 | 0.3728 | 0.6658 | 0.4346 | 0.0298 | 0.3191 | 0.6284 |

0.1832 | 0.4721 | 0.5733 | 0.5671 | 0.0350 | 0.5306 | 0.0156 | 0.5230 | |

03644 | 0.3464 | 0.2572 | 0.0669 | 0.5265 | 0.4311 | 0.3144 | 0.3231 |

content of diagnosis. Support Vector Machine (SVM) is proposed to solve the two-class classification problem and has rapidly developed into a powerful tool for classification problems. It has been successfully applied in many engineering fields, such as speech recognition, image classification and so on. LSSVM is an improved algorithm based on SVM method. The LSSVM method uses a least squares linear system as the loss function. Unlike the classic SVM, it has only equality constraints and no inequality constraints, which speeds up the calculation. In this paper, the same LSSVM is established according to the literature [

The figure shows the simulated state of 26 sets of data, in which the open circle indicates the preset state of the training data, for example, the first 13 sets of data are measured normal data, which should be in the 1 state indicated by the upper horizontal line in the figure, and the last 13 sets of fault data should be in the −1 state indicated by the next horizontal line, the red solid circle indicates the classification result of the data after LSSVM simulation., if the open circle of a certain group of data and the red solid circle coincide, it means that the

simulation result of this set of data is correct, otherwise the error.

As mentioned earlier, the high-order cumulant can automatically suppress the influence of Gaussian background noise (colored or white), and the two-dimensional wavelet function decomposition can decompose the low-frequency part of the scale j into four parts. In this paper, the two-dimensional wavelet method is used to extract the features of the normal signal and the fault signal in the high frequency part in three different coupling modes, and use the extracted information for fault diagnosis.

The results show that, because the third-order cumulant contains different frequency information under different coupling modes, the correct rate obtained is also different. The experimental results are shown in

Although the mechanism of various mechanical faults is different, each fault has the most essential difference from the normal state. For example, the fault in this paper is caused by adding a core to the pressure reducing valve in the normal

mode | error numbers |
---|---|

Definition 1 | 5 |

Definition 2 | 7 |

Definition 3 | 3 |

real third-order cumulant | 6 |

state. The essential difference is that the signal in the fault state is more likely to differ most in certain signal characteristics than the signal in the normal state. In this paper, through the different definitions of the complex third-order cumulant, the two-dimensional wavelet extracts the feature values in high frequency part to find the best diagnostic effect, and has achieved a certain degree of success.

This paper is supported by Research Foundation of the Nanchang Normal University for Doctors (NSBSJJ2018014). National Natural Science Foundation of China (61562063) Science and Technology Project of Jiangxi Provincial Education Department (GJJ171113).

The author declares no conflicts of interest regarding the publication of this paper.

Wu, W.B. (2018) Application of Three Order Cumulants in Fault Diagnosis. International Journal of Modern Nonlinear Theory and Application, 7, 97-105. https://doi.org/10.4236/ijmnta.2018.74008