_{1}

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The high-order spectrum can effectively remove Gaussian noise. The three-spectrum and its slices represent random signals from a higher probability structure. It can not only qualitatively describe the linearity and nonlinearity of vibration signals closely related to mechanical failures, Gaussian and non-Gaussian Performance, and can greatly i mprove the accuracy of mechanical fault diagnosis. The two-dimensional slices of trispectrum in normal and fault states show different peak characteristics. 2-D wavelet multi-level decomposition can effectively compress 2-D array information. Least squares support vector machine can obtain the global optimum under limited samples, thus avoiding the local optimum problem, and has the advantage of reducing computational complexity. In this paper, 2-D wavelet multi-level decomposition is used to extract features of trispectrum 2-D slices, and input LSSVM to diagnose the fault of the pressure reducing valve, which has achieved good results.

High-order spectra can effectively remove Gaussian noise. The trispectrum and its slices can not only qualitatively describe the linearity and nonlinearity of vibration signals closely related to mechanical faults, and the performance of Gaussian and non-Gaussian, but also greatly improve the diagnosis accuracy of mechanical faults [

The software virtual instrument software development tool LabVIEW [

Let the high-order cumulant c k x ( τ 1 , τ 2 , ⋯ , τ k − 1 ) be absolutely summable:

∑ τ 1 = − ∞ ∞ ⋯ ∑ τ k − 1 = − ∞ ∞ | c k x ( τ 1 , ⋯ , τ k − 1 ) | < ∞

Then the k-order cumulant spectrum is defined as the k − 1 dimensional Fourier transform of the k-order cumulant, that is:

S k x ( ω 1 , ⋯ , ω k − 1 ) = ∑ τ 1 = − ∞ ∞ ⋯ ∑ τ k − 1 = − ∞ ∞ c k x ( τ 1 , ⋯ , τ k − 1 ) exp [ − j ∑ i = 1 k − 1 ω i τ i ]

Then trispectrum is:

T x ( ω 1 , ω 2 , ω 3 ) = ∑ τ 1 = − ∞ ∞ ∑ τ 2 = − ∞ ∞ ∑ τ 3 = − ∞ ∞ c 4 x ( τ 1 , τ 2 , τ 3 ) e − j ( ω 1 τ 1 + ω 2 τ 2 + ω 3 τ 3 )

After the AR model is established, the three spectrum can also be expressed in the form of frequency characteristic function:

T ( ω 1 , ω 2 , ω 3 ) = γ 4 a H ( ω 1 ) H ( ω 2 ) H ( ω 3 ) H ∗ ( ω 1 + ω 2 + ω 3 )

where H ( ω ) = 1 1 + ∑ i = 1 p ϕ i e − j i ω is transfer Function,

γ 4 a = E { a 4 ( n ) } − 3 [ E { x 2 ( n ) } ] 2

is tosis, a(n) is an independent and identically distributed non-Gaussian stochastic process.

The triple spectrum can reflect the steepness of the system's deviation from the Gaussian process, and can be used to detect and characterize the cubic phase coupling of the system. The mode of the trispectrum is a function of three frequency variables, which needs to be described in a four-dimensional space.

In order to obtain the trispectrum graph, a moving ball and movable surface can usually be used to display the amplitude of each point of the three-spectrum in the four-dimensional space, as shown in

Fix a certain frequency coordinate of the three-frequency ( ω 1 , ω 2 , ω 3 ) in the trispectrum, and then obtain a 2-D slice of the trispectrum. Let ( ω 1 , ω 2 ) be variables and ω 3 = ω 0 be a constant to get a 2-D slice of the trispectrum

T S 2 D ( ω 1 , ω 2 , ω 0 ) = γ 4 a H ( ω 1 ) H ( ω 2 ) H ( ω 0 ) H ∗ ( ω 1 + ω 2 + ω 0 )

The 2-D slice of the normalized trispectrum is

T c S 2 D ( ω 1 , ω 2 , ω 0 ) = T S 2 D ( ω 1 , ω 2 , ω 0 ) | P ( ω 1 ) P ( ω 2 ) P ( ω 0 ) P ( ω 1 + ω 2 + ω 0 ) |

The normalized amplitude expression is

| T A R ( ω 1 , ω 2 , ω 0 ) | = | K | | 1 + ∑ i = 1 p γ i e − j i ( ω 1 + ω 2 + ω 0 ) | | 1 + ∑ i = 1 p γ 4 a e − j i ω 1 | | 1 + ∑ i = 1 p γ 4 a e − j i ω 2 | | 1 + ∑ i = 1 p γ 4 a e − j i ω 0 |

Wavelet transform is a signal time-scale analysis method. It has the characteristics of multi-resolution analysis, and it has the ability to characterize the local characteristics of the signal in both the time and frequency domains. It is a kind of window whose size is fixed but its shape can be changed. Time-frequency localization analysis method with variable time window and frequency window.

The two-dimensional wavelet function is obtained through the tensor product transformation of the 21D wavelet function. The 2-D wavelet function decomposition is to decompose the low-frequency part of scale j into four parts: the low-frequency part a_{j} of scale j + 1 and the high frequency part in three directions (horizontal, horizontal, Vertical, diagonal) d j 1 , d j 2 , d j 3 .

Suppose the original image has a total of N × N pixels, the scale parameter j = 0, the recursive decomposition is started from the original image, and the wavelet coefficient matrix a_{j}, d j 1 , d j 2 , d j 3 is decomposed at the j-th level, with ( 2 − j N ) 2 pixels respectively. (The resolution is reduced 2 − j ). Since the factor j represents the decomposition level and is related to the resolution of the image, j is often called the resolution or decomposition scale. Taking the three-level wavelet decomposition of a 2-d digital image as an example, the wavelet coefficient distribution is shown in

First, the normalized amplitude-frequency characteristic diagram of the three-spectrum 2-d slice of the normal signal and the fault signal is obtained from the experimental data. The slice diagram is actually a 3-d figure of a 2-d array. In this paper, 1024 data are used to analyze each set of data. The 2-d slice is a 2-d array of 255 * 255. Now three vibration states measured when the oil pressure is 1 MPa, 3 MPa, and 5 MPa are selected. For each of the 3 sets of data below, the 2-d slices and their corresponding wavelet packet coefficients are shown in Figures 6-9. The horizontal axis in the figure represents the frequency in HZ, and the vertical axis represents the normalized amplitude, which is dimensionless. The 3-d graphics are shown in

It can be seen from

It can be seen from

Based on the symmetry of the 2-d slice spectrum, this article uses the db1 wavelet, and uses the compressed low-frequency coefficients as the input of LSSVM for fault identification. The compressed low-frequency coefficients are a 4 * 4 two-dimensional array, and the graph is shown below the two-dimensional spectrum of the data, as shown in

Normal state | 0.3751 | 0.0255 | 0.0207 | 0.0196 | 0.0255 | 0.0113 | 0.0012 | …… |
---|---|---|---|---|---|---|---|---|

0.1459 | 0.0646 | 0.0665 | 0.0209 | 0.0646 | 0.0603 | 0.0111 | …… | |

0.2420 | 0.0424 | 0.0483 | 0.0380 | 0.0424 | 0.0324 | 0.0061 | …… | |

…… | ||||||||

Fault state | 0.1394 | 0.0683 | 0.0622 | 0.0383 | 0.0683 | 0.0476 | 0.0139 | …… |

0.1477 | 0.0751 | 0.0577 | 0.0336 | 0.0751 | 0.0436 | 0.0104 | …… | |

0.1066 | 0.0604 | 0.0775 | 0.0424 | 0.0604 | 0.0616 | 0.0127 | …… | |

…… |

The results show that there are only 2 sets in the normal data, and only 1 set in the fault data, a total of 3 sets of data which recognition errors, and the total recognition accuracy rate is close to 90% (excluding 10 groups of training data), indicating that the fault diagnosis method proposed in this paper is effective. Compared with bispectrum, trispectrum and its slices characterize random signals from a higher probability structure and provide an effective basis for analyzing non-Gaussian and nonlinear signals. It can not only qualitatively describe the linearity and nonlinearity of vibration signals, Gaussian and non-Gaussian performance, and can greatly improve the accuracy of mechanical fault diagnosis.

The high-order spectrum can effectively remove Gaussian noise, and the tri-spectrum and its slices represent random signals from a higher probability structure, providing an effective basis for analyzing non-Gaussian and nonlinear signals, and can greatly improve the accuracy of mechanical fault diagnosis. This paper uses two-dimensional wavelet multilevel decomposition to compress the two-dimensional array information of the three-spectrum two-dimensional slice, and then uses the least squares support vector machine to obtain the advantages of the global optimum under limited samples, and uses the compressed low-frequency coefficients as the input of LSSVM is used to diagnose the fault of the speed control valve, which has achieved good results. The author of this article will compare this fault diagnosis method with other methods in the follow-up work to prove the advantages of this fault diagnosis method and conduct theoretical discussions.

This paper is supported by Research Foundation of the Nanchang Normal University for Doctors (NSBSJJ2018014). Key R & D Projects of Jiangxi Provincial Department of Science and Technology: 20192BBHL80002, 20192BBEL50040.

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

Wu, W.B. (2020) Fault Diagnosis of Overflow Valve Based on Trispectrum. World Journal of Engineering and Technology, 8, 765-773. https://doi.org/10.4236/wjet.2020.84055