1 wavelet packet decomposition wavelet analysis is a branch of mathematics that has been gradually developed in the past 20 years. It is currently used in signal processing, image processing, speech processing, mechanical fault diagnosis, seismic wave analysis and many other fields. Wavelet packet analysis is a widely used wavelet decomposition method. Wavelet packet decomposition can not only decompose the low frequency of the signal, but also decompose the high frequency part of the signal. It can adaptively select the corresponding frequency band according to the characteristics of the analyzed signal, thereby improving the time-frequency resolution of the signal.
The two-scale equation of wavelet packet transform is:
W2n(x)=2kh(k)wn(2x-k)(1)
W2n 1(x)=2kg(k)wn(2x-k)(2)
The sequence {wn(x)} is called a wavelet packet determined by the basis function w0(x) = (x).
Fast wavelet packet transform can be implemented using tower decomposition. The wavelet packet decomposition coefficient can be given by the following discrete convolution equation: Cj 1,2m(k)=nCj,m(n)h(n-2k)(3)
Cj 1,2m 1(k)=nCj,m(n)g(n-2k)(4)
The reconstruction algorithm of the wavelet packet decomposition coefficient can be described as:
Cj,m(k)=nCj 1,2m(n)h(k-2n)-Cj 1,2m 1(n)g(k-2n)(5)
The following briefly describes the process of wavelet packet noise reduction. The basic model for setting a noisy signal is of the form:
s(n)=f(n) e(n)(6)
In the formula: time n is equal time interval.
In the simplest model, e(n) is considered to be Gaussian white noise N(0,1), and the noise level is considered to be equal to 1. The purpose of noise reduction is to reduce the value of the noise portion to recover the signal f.
The noise can be eliminated in the following three steps: (1) selecting the level of wavelet and wavelet packet decomposition, and calculating the wavelet packet decomposition of the signal s to the Nth layer; (2) for each layer from the first to the Nth layer, A threshold is selected, and the high frequency coefficient is processed by a soft threshold; (3) the coefficient after the noise reduction processing is restored by the wavelet packet reconstruction to restore the original signal.
2 Fractal Classification Theory 2.1 Fractal Classification Principle Fractal theory reveals the unity of order, disorder and unity, certainty and randomness in nonlinear systems. The fractal dimension can reflect the operating state of mechanical equipment and mechanical components as well as the irregularity and instability of the signal. By means of the fractal dimension, it is helpful to classify and identify the characteristic signals of the mechanical equipment fault state. .
In this paper, the fractal classification principle is adopted: when the vibration signal is used to diagnose the rotating machinery, the signals of different states of the system can be segmented, the correlation dimension of each segment is calculated, the feature mode is established and the model space is formed, and the features are similar. The distances in the pattern space are similar, so the established model space point samples can be used to establish a distance function for the mode to be detected, and the mathematical relationship of the distances is used to determine which state the mode to be detected belongs to.
Let the signal be divided into p segments, and the correlation dimension Djk, k=1, 2, and p of each segment is calculated and used as the feature quantity of the jth state. For the signal to be tested, the correlation dimension Dk, k=1, 2, and p are also calculated, and the defined state distance is:
Rj=pk=1|Dk-Djk|/p(7)
This function reflects the close distance between the signal to be tested and the jth state. The smaller the Rj, the closer the signal to be measured is to the jth state.
2.2 Calculation method of correlation dimension For any set of vibration signal data, it can be considered to constitute a real sequence {xt} (t=1, 2). Firstly, the phase space reconstruction is carried out, and the embedding dimension of the reconstructed phase space is m, then the delay sampling of the signal is delayed, and the m-dimensional space of M=N-(m-1) points is obtained. : X1, X2, X3,, XN-(m-1).
For a pair of phase points in m-dimensional space: Xi={xi,xi 1,,xi m-1}Xj={xj,xj 1,,xj m-1} Let the Euclidean distance between them be: rij(m )=Xi-Xj=max|xi k,xj k|k=0,1,2,,m-1(8)
The distance between all spatial phasors constitutes a matrix, and the maximum value rmax and the minimum value rmin of the spatial distance matrix elements are found, so that the critical distance e is a real number in the interval [rmin, rmax], and the distance matrix is ​​smaller than e. The ratio of the number of elements to the total number of elements is denoted as N(e): N(e)=1M2i, jH(e-xi-xj), ij(9)
Where: M is the total number of points in the phase space, and H satisfies the following conditions: H(x)=1, x>00, x<0(10)
Select the appropriate e, the following relationship exists in the scale-free area: N (e) = eD (m), D (m) = lime 0lnN (e) lne (11)
Where: D(m) is the associated dimension.
The following gives the selection of each parameter in the formula: (1) Delay time: It is selected by autocorrelation function method, and its autocorrelation function can be expressed as: C=ni=1(xi-x)(xi -x)ni=1( In the formula xi-x)2(12): x is the average of the time series. At the time of selection, C is a specific threshold. Research shows that the corresponding threshold C when selecting a value is: when the value of C is 0.5 for the first time; or when the first time is 0; or the first inflection point of C. This article uses the value when the first time is 0.
(2) Embedding dimension m: The embedding dimension m2d1 can be obtained by the Takens principle. Where d is the dynamic dimension of the system, it is difficult to select the embedding dimension because the size cannot be predicted. If the m selection is too small, the geometry of the power system cannot be fully opened, so a larger embedding dimension should be chosen. At present, the trial and error method is generally used in engineering, that is, m is sequentially increased from 3 until the obtained correlation dimension tends to be saturated, and m is a reasonable embedding dimension at this time.
(3) Critical distance e: The global traversal method is adopted. First calculate the distance of all points in the phase space, obtain the maximum value rmax and the minimum value rmin, and then determine the number of points s to be inserted, then the step size l = (rmax - rmin) / s, e is taken as {rmin, rmin l, Rmin 2l,,rmax}.
3 system composition and experimental verification system based on wavelet packet decomposition and fractal theory design, the process is shown as 1.
The process of gear fault diagnosis is as follows: the collected signal is denoised and decomposed by wavelet packet to obtain p segment, the correlation dimension of each segment is calculated, and the distance is compared with the correlation dimension of the signal in different states obtained in advance. In comparison, the smaller the distance, the closer the state of the signal to be tested is to the state of the analog signal, thereby judging the state of the signal to be tested.
The experimental equipment adopts JZQ250-1000 type reducer to simulate two gear fault states, namely tooth root crack and gear broken tooth. When collecting signals, the sampling frequency is 8 kHz, the number of sampling points is 2048, and the speed of the reducer pulley is 1000 r/min. Under the same experimental conditions, an unknown state signal is measured as the signal to be tested. The signals of the four states use the above theoretical method to calculate the respective correlation dimensions. The correlation dimensions obtained by calculation are listed in 1.
1 After the correlation dimension in different states is obtained from the correlation dimension in the table, the state distance between the signal to be tested and the other three states is obtained, and 2.
2 The state distance between the signal to be tested and the set state can clearly identify the state of the signal to be tested from the calculation result, that is, there is a root crack failure of the signal to be tested.
4 Conclusions (1) In this paper, the gear signal is detected by the combination of wavelet packet noise reduction, decomposition and fractal classification theory, and the gear detection system is established accordingly. It is feasible to verify this method through experiments; (2) in the calculation of correlation In the process of dimension, it is difficult to determine the embedding dimension m and delay time, which has a great influence on the experimental results, and requires a wealth of experimental experience and a large amount of experimental data.
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