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A gear fault signal is a typical non-stationary, non-linear signal. How to extract the characteristic parameters that can effectively represent the gear fault state from the signal is the key to realize the gear fault diagnosis.

Wavelet analysis has multi-scale characteristics and mathematical microscopic characteristics. Therefore, wavelet analysis has been widely used in gear fault diagnosis in recent years. However, the traditional wavelet analysis is based on the linear decomposition of frequency, and the analysis of nonlinear and non-stationary signals is not well decomposed.

Morphological wavelet transform is a wavelet transform based on mathematical morphology. It is a direction of nonlinear expansion of wavelet theory. It was first proposed by Goutsias and Heijmans in 2000. It successfully used most linear wavelet sums. Nonlinear wavelets are unified and form a unified framework for multiresolution analysis. As a kind of nonlinear wavelet, the morphological wavelet takes into account the morphological characteristics of mathematical morphology and the multi-resolution characteristics of wavelet, and has good detail retention and anti-noise performance. At present, morphological wavelets have been widely used in image compression, image fusion and power system fault detection, but the application of one-dimensional vibration signals has not been reported.

This paper proposes a method to analyze the gear vibration acceleration signal by using the largest lifting lattice shape wavelet. Firstly, the basic concept and framework of morphological wavelet are introduced. The structure and characteristics of a morphological wavelet using the maximum lifting operator are described in detail. The simulation signal and the actual gear fault signal are used to compare the analysis effect of the shape lifting wavelet with the traditional linear wavelet. The results show that the shape lifting wavelet has stronger pulse extraction ability than the traditional linear wavelet, and can quickly and effectively extract the fault characteristics of the gear. .

2 The concept of morphological wavelet 2.1 morphological wavelet The linear filter in linear wavelet is replaced by nonlinear morphological filter, which can form a kind of nonlinear morphological wavelet, which is mainly divided into dual wavelet and non-dual wavelet. The dual wavelet contains two analysis operators and one synthesis operator. The analysis operator acts on the signal and the other on the detail signal. The non-dual wavelet is a special case of dual wavelet. It contains two synthesis operators: signal synthesis operator and detail signal synthesis operator. Linear orthogonal wavelet belongs to non-dual wavelet, and its signal analysis operator and detail analysis operator are respectively Corresponds to low pass and high pass filters.

Suppose Vj and Wj are two sets, and Vj is the signal space of the jth scale, and Wj is the detail space of the jth scale. The signal is decomposed from the j scale to the j 1 scale and contains 2 analysis operators. The signal analysis is performed by the signal analysis operator j: VjVj 1 and the detail analysis operator w

j: WjWj 1 is composed, and the signal synthesis operator is j: Vj 1Wj 1Vj. The dual wavelet layer decomposition structure is shown as 1.

The above decomposition scheme must satisfy a complete representation of a signal, ie a signal analysis operator (

j,w

j): VjVj 1Wj 1 and the signal synthesis operator j: Vj 1Wj 1Vj must be mutually inverse processes, so the dual wavelet decomposition must satisfy the following two conditions:

j j(x), w j(x))=x, xVj(1) Equation (1) is called an exact reconstruction condition.

j(j(x,y))=x,xVj 1,yWj 1w

j(j(x, y))=y, xVj 1, yWj 1(2) Equation (2) guarantees that the decomposition is a non-redundant decomposition.

For a given signal x0V0, the following recursive analysis is performed: x0{x1, y1}{x2, y2, y1} {xk, yk, yk-1,, y1} (3) where: xj 1=

j(xj)Vj 1.

Yj 1=wj(xj)Wj 1(4) The original signal x0 can be accurately reconstructed by xk and y1, y2, and yk by the following recursive scheme: xj=

j(xj 1, yj 1), j=k-1, k-2,, 0(5) The decomposition scheme determined by equations (3) and (4) is reversible, and 1) The decomposition scheme formed by (5) is called dual wavelet decomposition.

2.2 Maximum lifting lattice shape The lifting method proposed by Sweldens provides a practical and flexible method for designing nonlinear wavelets. It is a flexible wavelet construction method that can use linear, nonlinear or spatially varying lifting operators. And can ensure that the newly constructed wavelet transform is reversible. The promotion plan mainly includes two types of operators: predictive promotion and update promotion. The prediction promotion method improves the detail decomposition operator w and the composition operator w

The update promotion method improves the signal analysis operator and the synthesis operator.

In this paper, the maximum operator is used as the prediction and update operator to construct the shape lifting wavelet, assuming that the original signal decomposition uses Lazy wavelet decomposition, ie, x1(n)=x(2n), y1(n)=x(2n 1) The prediction and update operators are respectively: (x) (n) = x (n) x (n 1) (6) (y) (n) = - (0y (n-1) y (n) (7)

Then the boosted signal and detail coefficients are:

Y1(=y1(n)-(x1(n)x1(n 1))=x(2n 1)-(x(2n)x(2n 2))(8)x1(=x1(n) (0y1( (n-1) y1((n))=x(2n) (0(x(2n-2)-(x(2n-2)(x(2n))))(x(2n 1)-(x( 2n)(x(2n 2))))(9)

It can be proved that the equation (8)(9) satisfies the condition of complete signal reconstruction, and the maximum lifting method selects the maximum values â€‹â€‹of the two neighborhood values â€‹â€‹x1(n) and x1(n 1) of y1(n) as y1(n) It is predicted that the update maximum also maps the local maximum of x1(n) to the scale signal x1(n) (the signal x has a local maximum definition x(n)) x(n1) at point n. It can be shown that the maximum lifting method constructs a dual wavelet.

3 The simulation signal is analyzed in the vibration signal of the gear transmission process, mainly the gear mesh excitation vibration, and its main frequency component is the meshing frequency and its multiplication component. In the analysis of gear vibration signals, the vibration caused by gear failure is the main purpose of fault diagnosis research, so the signal model caused by gear failure is considered here. When the gear fails (concentration defect or distribution defect), the amplitude and phase of the vibration signal will be modulated, which can be expressed by the following model: xg(t)= mi=1[1 am(t)]cos[m2fmt] bm( t) m] n(t)(10) where: fm is the meshing frequency of the gear, m is the order of the meshing frequency, and the following simulation signal is used for experimental analysis, m is the phase of the mth-order meshing frequency, am(t And bm(t) are the amplitude and phase modulation functions of the mth-order meshing frequency, respectively, which is a function of the period of the rotation of the axis of the faulty gear, and n(t) is white noise.

The simulation signal used assumes that the gear meshing frequency is 200 Hz, the rotating shaft frequency is 10 Hz, the sampling frequency is 2048 Hz, the sampling time is 1 s, and 2 is the simulation signal waveform.

In this paper, the morphological wavelet and the traditional linear wavelet (wavelet function selection commonly used db5 wavelet) are used to decompose the gear simulation signal, and the results of the 3-layer decomposition using db5 wavelet and maximum lifting wavelet respectively. Where ca3 represents the third layer approximation coefficient, and cd3cd1 is the detail coefficient of the third layer to the first layer, respectively. Contrast and contrast, the third layer detail coefficient of db5 wavelet decomposition preserves the impact characteristics of the signal well, but the approximate coefficient amplitude is very small, and almost no impact characteristics are seen; in contrast, the shape lifting wavelet is in the third The approximation coefficient and detail coefficient of the layer well preserve the impact characteristics of the signal, especially the approximation coefficient, which is equivalent to the upper envelope analysis of the signal, and the impact characteristics are very obvious.

And the results of the spectrum analysis of the coefficients of the respective layers. It can be seen that the spectrum of the second and third detail coefficients of the linear wavelet decomposition can reflect the multiples of the modulation frequency, that is, 10 Hz and its multiple frequency. The first layer detail coefficient is very polluted by noise, almost No modulation frequency information can be seen, and the third layer detail coefficient spectrum can hardly see any modulation frequency information. Compared with the morphological wavelet decomposition coefficients shown in the figure, the modulation information of the signal is preserved, and the modulation frequency of 10 Hz and its various frequency doublings can be seen in the spectrum, especially the spectrum of the third layer approximation coefficient, and the linear wavelet. In comparison, the modulation frequency of the signal can be significantly extracted.

4 Gear fault signal analysis To verify the effectiveness of the maximum lifting lattice shape wavelet transform, this paper uses the measured gear fault signal for testing. The gear vibration acceleration data signal comes from the self-built gearbox test bench. The gearbox used is a single-stage spur gear reduction flyer. The number of input shaft gears is 50, the number of output shaft teeth is 30, and the simulated gear failure is the input shaft gear breaking. The input shaft speed of the gearbox is 1340r/min, the frequency conversion is 22.33Hz, the sampling frequency is 2048Hz, and the sampling point is 4096. It is the time domain waveform diagram of the gear broken tooth fault signal.

And the result of three-layer decomposition of the gear broken tooth fault signal by linear db5 wavelet and maximum lifting lattice shape wavelet. Similar to the simulated signal, the amplitude of the 3rd layer approximation coefficient of the linear wavelet decomposition is very small, and the impact characteristic information of the gear broken tooth fault is hardly seen, and the shape wavelet has the characteristics of local extremum retention, and the approximation coefficient of the third layer The impact characteristic information is better retained.

0 and 1 are the results of the spectrum analysis of the coefficients of each layer. It can be seen from 0 that the spectrum of the first and second detail coefficients of the linear wavelet decomposition can reflect the frequency doubling of the modulation frequency, that is, 22 Hz and its frequency doubling, and the spectrum of the third layer approximation coefficient is hardly seen. Any modulation frequency information. Compared with 0, the morphological wavelet decomposition of each layer coefficient in 1 retains the modulation information of the signal, and the modulation frequency of 22 Hz and its various frequency doublings can be seen in the spectrum, especially the spectrum of the third layer approximation coefficient, and Compared with the linear wavelet, the periodic shock characteristics of the gear broken tooth fault can be extracted significantly.

5 Conclusions This paper proposes a new method for feature extraction of gear fault signals using maximum lifting lattice shape wavelet transform. The maximum lifting shape wavelet takes into account the morphological characteristics of mathematical morphology and the multi-resolution characteristics of wavelets, has good detail retention and anti-noise performance, and has a simple and flexible construction scheme, and because it only involves addition, subtraction and enlargement. Take small calculations, so the calculation is very simple and fast.

The processing results of the simulated signal and the measured gear broken tooth fault signal show that the maximum lifting shape wavelet transform not only suppresses the noise but also fully highlights the impact characteristics of the fault signal. Compared with the traditional linear wavelet transform, it has better fault feature extraction effect and The computational efficiency provides a new fast and efficient method for gear fault feature extraction.

A gear fault signal is a typical non-stationary, non-linear signal. How to extract the characteristic parameters that can effectively represent the gear fault state from the signal is the key to realize the gear fault diagnosis.

Wavelet analysis has multi-scale characteristics and mathematical microscopic characteristics. Therefore, wavelet analysis has been widely used in gear fault diagnosis in recent years. However, the traditional wavelet analysis is based on the linear decomposition of frequency, and the analysis of nonlinear and non-stationary signals is not well decomposed.

Morphological wavelet transform is a wavelet transform based on mathematical morphology. It is a direction of nonlinear expansion of wavelet theory. It was first proposed by Goutsias and Heijmans in 2000. It successfully used most linear wavelet sums. Nonlinear wavelets are unified and form a unified framework for multiresolution analysis. As a kind of nonlinear wavelet, the morphological wavelet takes into account the morphological characteristics of mathematical morphology and the multi-resolution characteristics of wavelet, and has good detail retention and anti-noise performance. At present, morphological wavelets have been widely used in image compression, image fusion and power system fault detection, but the application of one-dimensional vibration signals has not been reported.

This paper proposes a method to analyze the gear vibration acceleration signal by using the largest lifting lattice shape wavelet. Firstly, the basic concept and framework of morphological wavelet are introduced. The structure and characteristics of a morphological wavelet using the maximum lifting operator are described in detail. The simulation signal and the actual gear fault signal are used to compare the analysis effect of the shape lifting wavelet with the traditional linear wavelet. The results show that the shape lifting wavelet has stronger pulse extraction ability than the traditional linear wavelet, and can quickly and effectively extract the fault characteristics of the gear. .

2 The concept of morphological wavelet 2.1 morphological wavelet The linear filter in linear wavelet is replaced by nonlinear morphological filter, which can form a kind of nonlinear morphological wavelet, which is mainly divided into dual wavelet and non-dual wavelet. The dual wavelet contains two analysis operators and one synthesis operator. The analysis operator acts on the signal and the other on the detail signal. The non-dual wavelet is a special case of dual wavelet. It contains two synthesis operators: signal synthesis operator and detail signal synthesis operator. Linear orthogonal wavelet belongs to non-dual wavelet, and its signal analysis operator and detail analysis operator are respectively Corresponds to low pass and high pass filters.

Suppose Vj and Wj are two sets, and Vj is the signal space of the jth scale, and Wj is the detail space of the jth scale. The signal is decomposed from the j scale to the j 1 scale and contains 2 analysis operators. The signal analysis is performed by the signal analysis operator j: VjVj 1 and the detail analysis operator w

j: WjWj 1 is composed, and the signal synthesis operator is j: Vj 1Wj 1Vj. The dual wavelet layer decomposition structure is shown as 1.

The above decomposition scheme must satisfy a complete representation of a signal, ie a signal analysis operator (

j,w

j): VjVj 1Wj 1 and the signal synthesis operator j: Vj 1Wj 1Vj must be mutually inverse processes, so the dual wavelet decomposition must satisfy the following two conditions:

j j(x), w j(x))=x, xVj(1) Equation (1) is called an exact reconstruction condition.

j(j(x,y))=x,xVj 1,yWj 1w

j(j(x, y))=y, xVj 1, yWj 1(2) Equation (2) guarantees that the decomposition is a non-redundant decomposition.

For a given signal x0V0, the following recursive analysis is performed: x0{x1, y1}{x2, y2, y1} {xk, yk, yk-1,, y1} (3) where: xj 1=

j(xj)Vj 1.

Yj 1=wj(xj)Wj 1(4) The original signal x0 can be accurately reconstructed by xk and y1, y2, and yk by the following recursive scheme: xj=

j(xj 1, yj 1), j=k-1, k-2,, 0(5) The decomposition scheme determined by equations (3) and (4) is reversible, and 1) The decomposition scheme formed by (5) is called dual wavelet decomposition.

2.2 Maximum lifting lattice shape The lifting method proposed by Sweldens provides a practical and flexible method for designing nonlinear wavelets. It is a flexible wavelet construction method that can use linear, nonlinear or spatially varying lifting operators. And can ensure that the newly constructed wavelet transform is reversible. The promotion plan mainly includes two types of operators: predictive promotion and update promotion. The prediction promotion method improves the detail decomposition operator w and the composition operator w

The update promotion method improves the signal analysis operator and the synthesis operator.

In this paper, the maximum operator is used as the prediction and update operator to construct the shape lifting wavelet, assuming that the original signal decomposition uses Lazy wavelet decomposition, ie, x1(n)=x(2n), y1(n)=x(2n 1) The prediction and update operators are respectively: (x) (n) = x (n) x (n 1) (6) (y) (n) = - (0y (n-1) y (n) (7)

Then the boosted signal and detail coefficients are:

Y1(=y1(n)-(x1(n)x1(n 1))=x(2n 1)-(x(2n)x(2n 2))(8)x1(=x1(n) (0y1( (n-1) y1((n))=x(2n) (0(x(2n-2)-(x(2n-2)(x(2n))))(x(2n 1)-(x( 2n)(x(2n 2))))(9)

It can be proved that the equation (8)(9) satisfies the condition of complete signal reconstruction, and the maximum lifting method selects the maximum values â€‹â€‹of the two neighborhood values â€‹â€‹x1(n) and x1(n 1) of y1(n) as y1(n) It is predicted that the update maximum also maps the local maximum of x1(n) to the scale signal x1(n) (the signal x has a local maximum definition x(n)) x(n1) at point n. It can be shown that the maximum lifting method constructs a dual wavelet.

3 The simulation signal is analyzed in the vibration signal of the gear transmission process, mainly the gear mesh excitation vibration, and its main frequency component is the meshing frequency and its multiplication component. In the analysis of gear vibration signals, the vibration caused by gear failure is the main purpose of fault diagnosis research, so the signal model caused by gear failure is considered here. When the gear fails (concentration defect or distribution defect), the amplitude and phase of the vibration signal will be modulated, which can be expressed by the following model: xg(t)= mi=1[1 am(t)]cos[m2fmt] bm( t) m] n(t)(10) where: fm is the meshing frequency of the gear, m is the order of the meshing frequency, and the following simulation signal is used for experimental analysis, m is the phase of the mth-order meshing frequency, am(t And bm(t) are the amplitude and phase modulation functions of the mth-order meshing frequency, respectively, which is a function of the period of the rotation of the axis of the faulty gear, and n(t) is white noise.

The simulation signal used assumes that the gear meshing frequency is 200 Hz, the rotating shaft frequency is 10 Hz, the sampling frequency is 2048 Hz, the sampling time is 1 s, and 2 is the simulation signal waveform.

In this paper, the morphological wavelet and the traditional linear wavelet (wavelet function selection commonly used db5 wavelet) are used to decompose the gear simulation signal, and the results of the 3-layer decomposition using db5 wavelet and maximum lifting wavelet respectively. Where ca3 represents the third layer approximation coefficient, and cd3cd1 is the detail coefficient of the third layer to the first layer, respectively. Contrast and contrast, the third layer detail coefficient of db5 wavelet decomposition preserves the impact characteristics of the signal well, but the approximate coefficient amplitude is very small, and almost no impact characteristics are seen; in contrast, the shape lifting wavelet is in the third The approximation coefficient and detail coefficient of the layer well preserve the impact characteristics of the signal, especially the approximation coefficient, which is equivalent to the upper envelope analysis of the signal, and the impact characteristics are very obvious.

And the results of the spectrum analysis of the coefficients of the respective layers. It can be seen that the spectrum of the second and third detail coefficients of the linear wavelet decomposition can reflect the multiples of the modulation frequency, that is, 10 Hz and its multiple frequency. The first layer detail coefficient is very polluted by noise, almost No modulation frequency information can be seen, and the third layer detail coefficient spectrum can hardly see any modulation frequency information. Compared with the morphological wavelet decomposition coefficients shown in the figure, the modulation information of the signal is preserved, and the modulation frequency of 10 Hz and its various frequency doublings can be seen in the spectrum, especially the spectrum of the third layer approximation coefficient, and the linear wavelet. In comparison, the modulation frequency of the signal can be significantly extracted.

4 Gear fault signal analysis To verify the effectiveness of the maximum lifting lattice shape wavelet transform, this paper uses the measured gear fault signal for testing. The gear vibration acceleration data signal comes from the self-built gearbox test bench. The gearbox used is a single-stage spur gear reduction flyer. The number of input shaft gears is 50, the number of output shaft teeth is 30, and the simulated gear failure is the input shaft gear breaking. The input shaft speed of the gearbox is 1340r/min, the frequency conversion is 22.33Hz, the sampling frequency is 2048Hz, and the sampling point is 4096. It is the time domain waveform diagram of the gear broken tooth fault signal.

And the result of three-layer decomposition of the gear broken tooth fault signal by linear db5 wavelet and maximum lifting lattice shape wavelet. Similar to the simulated signal, the amplitude of the 3rd layer approximation coefficient of the linear wavelet decomposition is very small, and the impact characteristic information of the gear broken tooth fault is hardly seen, and the shape wavelet has the characteristics of local extremum retention, and the approximation coefficient of the third layer The impact characteristic information is better retained.

0 and 1 are the results of the spectrum analysis of the coefficients of each layer. It can be seen from 0 that the spectrum of the first and second detail coefficients of the linear wavelet decomposition can reflect the frequency doubling of the modulation frequency, that is, 22 Hz and its frequency doubling, and the spectrum of the third layer approximation coefficient is hardly seen. Any modulation frequency information. Compared with 0, the morphological wavelet decomposition of each layer coefficient in 1 retains the modulation information of the signal, and the modulation frequency of 22 Hz and its various frequency doublings can be seen in the spectrum, especially the spectrum of the third layer approximation coefficient, and Compared with the linear wavelet, the periodic shock characteristics of the gear broken tooth fault can be extracted significantly.

5 Conclusions This paper proposes a new method for feature extraction of gear fault signals using maximum lifting lattice shape wavelet transform. The maximum lifting shape wavelet takes into account the morphological characteristics of mathematical morphology and the multi-resolution characteristics of wavelets, has good detail retention and anti-noise performance, and has a simple and flexible construction scheme, and because it only involves addition, subtraction and enlargement. Take small calculations, so the calculation is very simple and fast.

The processing results of the simulated signal and the measured gear broken tooth fault signal show that the maximum lifting shape wavelet transform not only suppresses the noise but also fully highlights the impact characteristics of the fault signal. Compared with the traditional linear wavelet transform, it has better fault feature extraction effect and The computational efficiency provides a new fast and efficient method for gear fault feature extraction.

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