CN112683392A - ELMD multi-scale fluctuation analysis state monitoring method and device - Google Patents
ELMD multi-scale fluctuation analysis state monitoring method and device Download PDFInfo
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Abstract
The invention discloses an ELMD multi-scale fluctuation analysis state monitoring method and a device, which decompose equipment vibration signals by using an ELMD algorithm, remove noise components and trend terms by using a nonlinear discrimination algorithm, reserve fractal signal components, interpolate local extreme points by using a Lagrange interpolation function, fit envelopes by using a least square method, separate a frequency modulation part, estimate instantaneous frequency by using a TEO algorithm and calculate corresponding instantaneous scale, determine a dereferencing result of the vibration signals according to the analysis scale, calculate a multi-fractal spectrum of a dereferencing signal, extract singular indexes corresponding to a left endpoint, a right endpoint and the extreme points of the multi-fractal spectrum as characteristic parameters of the equipment operation state, identify the equipment operation state, deploy the algorithm to an equipment state monitoring device, can accurately distinguish the equipment operation state, and an equipment state monitoring system has good flexibility and portability, is convenient for engineering application.
Description
Technical Field
The invention relates to the field of equipment state monitoring and fault diagnosis, in particular to an ELMD multi-scale fluctuation analysis state monitoring method and device.
Background
The device vibration signal contains rich fractal features that can describe the operating state of the device. The box dimension, power spectrum analysis and re-standard range method can estimate the single-fractal parameters of stationary signals, and the de-trend fluctuation analysis (DFA) can estimate the single-fractal dimension of non-stationary signals. However, when the device fails, the vibration signal is usually non-stationary and has a multi-fractal characteristic, and the conventional fractal dimension estimation method generates a relatively large error. The multi-fractal detrending fluctuation analysis (MFDF) can estimate multi-fractal parameters of non-stationary signals, but the MFDF method has the problems that the analysis scale needs to be manually determined, the fitting polynomial trend order is difficult to determine, and the data segment is discontinuous. Currently, there is a document that proposes an MFDFA version (MFDFAemd) based on EMD to solve the problem of MFDFA. However, the linear filtering method adopted by mfdfame is easy to destroy the fractal structure of the original signal, and there is a negative frequency phenomenon, and these defects seriously affect the application effect of mfdfame. In summary, in the prior art, it is difficult to accurately extract the multi-fractal features of the device vibration signal, and it is difficult to accurately detect the device operating state.
Disclosure of Invention
The invention provides a method and a device for monitoring the state of ELMD multi-scale fluctuation analysis (MFDFOelmd for short) aiming at the defects. The method provided by the invention is adopted to analyze the equipment vibration signal, can effectively extract the multi-fractal characteristics of the equipment vibration signal, overcomes the problems that the analysis scale of the MFDF method needs to be manually determined, the fitting polynomial trend order is difficult to determine and the data section is discontinuous, solves the phenomena of original signal fractal structure damage and negative frequency existing in the MFDF method, and has the advantages of high accuracy and precision of analysis results, high accuracy of equipment operation state identification results and the like.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: an ELMD multi-scale fluctuation analysis state monitoring method is characterized in that: the method comprises the following steps:
step 1: measuring a device vibration signal x (k) by using an acceleration sensor at a sampling frequency fs, wherein k =1,2, …, N and N are lengths of the sampling signal;
step 2: the signal x (k) is decomposed into the sum of n components and a trend term by using an Ensemble Local Mode Decomposition (ELMD) algorithm, i.e. the sum of n components and a trend term
Wherein c isi(k) Representing the i-th component, r, obtained by the ELMD algorithmn(k) Represents the trend term obtained by the ELMD algorithm, in this example, n = 10;
and step 3: eliminating noise components and trend terms from ELMD decomposition results by adopting a nonlinear discrimination algorithm, and reserving components c containing fractal featuresf(k) F =1,2, …, p, p represents the number of residual components after filtering;
and 4, step 4: determination of cf(k) The local maximum value and the local minimum value of c are respectively compared with c by adopting Lagrange interpolation functionf(k) The local maximum and local minimum are interpolated, and c is fitted by least square methodf(k) The upper envelope u (k) and the lower envelope l (k), then cf(k) Is defined as
The symbol | x | represents taking the absolute value of x;
and 5: repeatedly executing formula
m times, j =1,2, …, m, until
To obtain cf(k) Frequency modulation part FMm(k)，ej(k) Represents cj(k) Envelope of cj(k)=FM(j-1)(k)，c1(k)= cf（k）；
Step 6: calculating FM by using Teager Energy Operator (TEO)m(k) Obtaining the instantaneous frequency of cf(k) Instantaneous frequency instf off(k) To obtain cf(k) Instantaneous scale of
；
And 7: when the scale is s, the detrending result of the vibration signal x (k) is
；
And 8: will be provided withY s (k) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
and step 9: calculate variance of each piece of data:
step 10: calculating a q-th order function:
step 11: changing the value of s, s = sfF =1,2, …, p, repeating the above steps 3 to 10, resulting in a variance function F about q and sq(s);
Step 12: if it is notx(k) Presence of fractal features, thenF q (s) And sizesThere is a power law relationship between:F q (s)~s H q()h (q) represents the generalized Hurst index of x (k);
when in useqWhen the value is not less than 0, the reaction time is not less than 0,H(0) determined by the logarithmic averaging procedure defined by:
step 13: calculating a standard scale index τ (q) = qH (q) -1 for the signal x (k), in this case q is taken in the range (-5, 5);
step 14: calculating the singular index α and the multifractal spectrum f (α) of the signal x (k):
α=H(q)+q H’(q),
f (alpha) = q (alpha-H (q)) +1, wherein H’(q) represents the first derivative of h (q);
step 15: extracting singular indexes corresponding to a left end point, a right end point and an extreme point of the multi-fractal spectrum f (alpha), and describing the running state of the equipment by using the 3 parameters;
step 16: the method in the steps is deployed on a state monitoring device to monitor the state of equipment.
Further, the ELMD algorithm of step 2 includes the following steps:
1) to data x0(k) Adding a white noise sequence produces a new data xj(k) ：
Std[x0(k)]Representative data x0(k) Standard deviation of (Wn)j(k) Representative wnjThe kth data, wn injRepresents the jth randomly generated white noise sequence, wnjThe amplitude is 1, j is more than or equal to 1 and less than or equal to K; x is the number of0(k) Represents x (k) in step 2 of claim 1; in this example, K = 100;
2) for xj(k) Performing local mean decomposition to obtain n components and a trend term
cij(k) Represents a pair xj(k) The i-th component, r, resulting from performing a local mean decompositionnj(k) Represents a pair xj(k) Performing a trend term obtained by local mean decomposition;
3) calculating the average value of the decomposition results of K times
ci(k) Represents a pair x0(k) The ith component, r, obtained by performing a local mean decomposition of the setn(k) Represents a pair x0(k) And performing a trend term obtained by the set local mean decomposition.
Further, the step 3 nonlinear discrimination algorithm includes the following steps:
1) performing rearrangement operation and substitution operation on signals c (k), and using c as data obtained by rearrangement operationshuf(k) Indicating that data obtained after the substitution operation are csurr(k) Represents;
2) for c (k), cshuf(k) And csurr(k) Performing multi-fractal Detrended Fluctuation Analysis (MFDF) respectively to obtain a generalized Hurst index curve, wherein the generalized Hurst index curve of c (k) is represented by H (q); c. Cshuf(k) Generalized Hurst exponential curve of (1) using Hshuf(q) represents; c. Csurr(k) Generalized Hurst exponential curve of (1) using Hsurr(q) represents;
3) two parameters e are defined1And e2，
Further, the data rearrangement operation in the step 1) comprises the following steps: randomly randomizing the order of the components c (k).
Further, the data replacement operation in the step 1) comprises the following steps:
1) performing a discrete fourier transform on component c (k) to obtain the phase of component c (k);
2) replacing the original phase of component c (k) with a set of pseudo-independent identically distributed numbers located in the (-pi, pi) interval;
3) performing inverse discrete Fourier transform on the frequency domain data subjected to phase substitution to obtain data cIFFT(k) To obtain data cIFFT(k) The real part of (a).
Further, the MFDFA method in step 2) includes the following steps:
1) contour of construction x (k), k =1,2, …, NY(i)：
x (k) represents c (k) or c in step 2) of claim 3shuf(k) Or csurr（k）；
2) Signal profileY(i) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
3) fitting a polynomial trend of each section of data by using a least square method, and then calculating the variance of each section of data:
y v (i) Is a first of fittingvTrend of the segment data, if the fitted polynomial trend ismOn the rank, the trend should be recordedThe process is (MF-) DFAm(ii) a In this example, m = 1;
4) calculate the firstqAverage of the order fluctuation function:
5) if it is notx(k) Presence of self-similar features thenqMean value of order fluctuation functionF q (s) And time scalesThere is a power law relationship between:
F q (s)~s H q()；
when in useqIf =0, the formula in step 4) diverges, at which timeH(0) Determined by the logarithmic averaging procedure defined by:
6) taking logarithm of both sides of the formula in step 5) to obtain ln [ 2 ]F q (s)]=H(q)ln(s)+c ，cIs constant, whereby the slope of the straight line can be obtainedH(q)。
Further, the least square method in the step 4 comprises the following steps: for x (t), t =1,2, …, n, x (t) represents the pair c in step 4f(k) A sequence or pair c generated by interpolating the local maxima off(k) The local minima of the sequence, n represents the length of the interpolated sequence,
1) a set of functions r is selected in advancek(t)，k=1,2,…,m，m<n, constructioning function
f(t)=a1r1(t)+a2r2(t)+…+ amrm(t) in which rk(t) represents a second order polynomial, a third order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve or a complex function curve;
2) calculating a least squares metric
；
3) Let J pair akPartial derivative of
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Wherein R isTRepresenting the transposed matrix of R, R-1An inverse matrix representing R;
4）rk(t) respectively selecting a second-order polynomial, a third-order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve and a composite function curve for calculation, then comparing least square indexes J generated by various curve forms, and selecting a curve form r corresponding to the minimum J as the curve formk(t) form (a).
Further, the Teager energy operator method in the step 6 comprises the following steps:
1) for signal c (k), k =1,2, …, N, c (k) = FMm(k) Building a function
ψ(c(k))=c2(k)- c(k+1) c(k-1);
2) Let d (k) = c (k) -c (k-1), instantaneous frequency instf (k) of signal c (k) be defined as:
based on the above-mentioned ELMD multi-scale fluctuation analysis state monitoring method, the device for implementing the method, the state monitoring device in step 16 includes the following parts: the system comprises a data line, an acceleration sensor, a data acquisition card, a case, a notebook computer and signal analysis software, wherein the acceleration sensor is connected with the data acquisition card through the data line, the data acquisition card is installed in the case, the case is connected with the notebook computer through the data line, the signal analysis software is installed on the notebook computer, and the signal analysis software is used for realizing the algorithm.
The method comprises the following steps:
step 1), the following steps: collecting a vibration signal;
step 2), the step of: decomposing an original signal into different component sum forms, wherein some components correspond to noise and trend terms, and some components contain fractal features;
and 3, step 3: removing noise components and trend items in the signal decomposition result by using a nonlinear discrimination algorithm, and only reserving signal components containing fractal features;
4) to 6): separating the frequency modulation part of each fractal signal component, and estimating the instantaneous frequency and the instantaneous scale of each fractal signal component by using TEO;
step 7), the steps of: selecting proper fractal signal components according to the analysis scale, and summing the selected fractal signal components to be used as a signal de-trend result corresponding to the analysis scale;
8) to 14): performing fluctuation analysis on the signal detrending result corresponding to each analysis scale to obtain a multi-fractal spectrum of the original signal;
step 15): extracting singular indexes corresponding to a left end point, a right end point and an extreme point of the multi-fractal spectrum, and taking the three parameters as characteristic parameters of the running state of the equipment;
16) step: deploying the algorithm on an equipment state monitoring device, and monitoring the equipment state;
by adopting the technical scheme, compared with the prior art, the invention has the following advantages:
1) the vibration signal is decomposed by adopting an ELMD method, the noise component and the trend term are removed according to a nonlinear filtering method, the fractal structure of the original signal can be protected, and the damage of the linear filtering method to the fractal structure of the original signal is avoided;
2) the frequency modulation part of the separated signal component estimates the instantaneous frequency of the signal component by using TEO, so that the instantaneous frequency can be ensured to keep a positive value, and the negative frequency phenomenon is avoided;
3) calculating corresponding instantaneous scale according to the instantaneous frequency of the signal component, and performing fluctuation analysis according to the instantaneous scale of the signal component, so that the defect of manually setting the scale is avoided;
4) the ELMD method is utilized to automatically determine the type of the signal trend, ensure the continuity of the signal trend and effectively solve the defects of the prior art;
5) the accuracy and precision of the analysis result are high, and the accuracy of the identification result of the running state of the equipment is high.
The invention is further illustrated with reference to the following figures and examples.
Drawings
FIG. 1 is a flow chart of the method of the present invention in an embodiment of the present invention;
FIG. 2 is a schematic diagram of an apparatus state monitoring device according to an embodiment of the present invention;
fig. 3 is a multi-fractal simulation signal generated by a multi-fractal cascade model in the embodiment of the present invention;
fig. 4 is an instantaneous frequency of a multi-fractal simulation signal obtained by using the MFDFAemd method in the embodiment of the present invention, where the number of signal components is 10;
fig. 5 is an instantaneous frequency of a multi-fractal simulation signal obtained by using the MFDFAoelmd method in the embodiment of the present invention, where the number of signal components is 10;
FIG. 6 is a comparison graph of the multi-fractal simulation signal analysis results respectively using MFDF, MFDFEMd and MFDFaeelmd methods in the embodiment of the present invention;
FIG. 7 is a diagram illustrating the calculation results of two non-linear discriminant parameters, wherein the symbols "circle" and "square" represent e1 and e2, respectively;
FIG. 8 is a comparison diagram of analysis results of noisy multi-fractal simulation signals respectively using MFDF, MFDFEMd, and MFDFaeelmd methods in the embodiment of the present invention;
FIG. 9 is a diagram illustrating correlation coefficients between each signal component and an original signal obtained by EMD according to an embodiment of the present invention;
FIG. 10 is a comparison diagram of analysis results of noisy multi-fractal simulation signals respectively using MFDFA, MFDFAEmd based on correlation filtering, and MFDFAElmd based on correlation filtering in the embodiment of the present invention;
FIG. 11 shows four vibration signals of the gearbox in the embodiment of the invention, wherein (a) - (d) respectively represent normal, light scratch, heavy scratch and broken tooth gear states;
FIG. 12 is a multi-fractal spectrum of the four gearbox vibration signals obtained using MFDFA in an embodiment of the present invention;
FIG. 13 is a multi-fractal spectrum of the four gearbox vibration signals obtained by using MFDFame in the embodiment of the present invention;
FIG. 14 is a multi-fractal spectrum of the four gearbox vibration signals obtained using MFDFaeelmd in an embodiment of the present invention;
fig. 15 is a classification result of singular indexes corresponding to left, right, and extreme points of a multi-fractal spectrum obtained by MFDFA on the four gear box states in the embodiment of the present invention, where the "circle", "square", "plus", and "diamond" symbols represent normal, light, heavy, and broken gear states, respectively;
fig. 16 is a classification result of singular indexes corresponding to left end point, right end point and extreme point of a multi-fractal spectrum obtained by MFDFAemd on the states of the four gearboxes in the embodiment of the present invention, where symbols "circle", "square", "plus" and "diamond" represent normal, light scratch, heavy scratch and broken tooth gear states, respectively;
fig. 17 shows the classification results of the singular indexes corresponding to the left end point, the right end point and the extreme point of the multi-fractal spectrum obtained from MFDFAoelmd according to the embodiment of the present invention on the four gear box states, and the symbols "circle", "square", "plus" and "diamond" represent the normal, light scratch, heavy scratch and broken tooth gear states, respectively.
Detailed Description
Embodiment, as shown in fig. 1 and fig. 2, an ELMD multi-scale fluctuation analysis state monitoring method is characterized in that: the method comprises the following steps:
step 1: measuring a device vibration signal x (k) by using an acceleration sensor at a sampling frequency fs, wherein k =1,2, …, N and N are lengths of the sampling signal;
step 2: the signal x (k) is decomposed into the sum of n components and a trend term by using an Ensemble Local Mode Decomposition (ELMD) algorithm, i.e. the sum of n components and a trend term
Wherein c isi(k) Representing the i-th component, r, obtained by the ELMD algorithmn(k) Represents the trend term obtained by the ELMD algorithm, in this example, n = 10;
and step 3: eliminating noise components and trend terms from ELMD decomposition results by adopting a nonlinear discrimination algorithm, and reserving components c containing fractal featuresf(k) F =1,2, …, p, p represents the number of residual components after filtering;
and 4, step 4: determination of cf(k) The local maximum value and the local minimum value of c are respectively compared with c by adopting Lagrange interpolation functionf(k) The local maximum and local minimum are interpolated, and c is fitted by least square methodf(k) The upper envelope u (k) and the lower envelope l (k), then cf(k) Is defined as
The symbol | x | represents taking the absolute value of x;
and 5: repeatedly executing formula
m times, j =1,2, …, m, until
To obtain cf(k) Frequency modulation part FMm(k)，ej(k) Represents cj(k) Envelope of cj(k)=FM(j-1)(k)，c1(k)= cf（k）；
Step 6: calculating FM by using Teager Energy Operator (TEO)m(k) Obtaining the instantaneous frequency of cf(k) Instantaneous frequency instf off(k) To obtain cf(k) Instantaneous scale of
；
And 7: when the dimension is s, the detrending of the vibration signal x (k)The potential result is
；
And 8: will be provided withY s (k) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
and step 9: calculate variance of each piece of data:
step 10: calculating a q-th order function:
step 11: changing the value of s, s = sfF =1,2, …, p, repeating the above steps 3 to 10, resulting in a variance function F about q and sq(s);
Step 12: if it is notx(k) Presence of fractal features, thenF q (s) And sizesThere is a power law relationship between:F q (s)~s H q()h (q) represents the generalized Hurst index of x (k);
when in useqWhen the value is not less than 0, the reaction time is not less than 0,H(0) determined by the logarithmic averaging procedure defined by:
step 13: calculating a standard scale index τ (q) = qH (q) -1 for the signal x (k), in this case q is taken in the range (-5, 5);
step 14: calculating the singular index α and the multifractal spectrum f (α) of the signal x (k):
α=H(q)+q H’(q),
f (alpha) = q (alpha-H (q)) +1, wherein H’(q) represents the first derivative of h (q);
step 15: extracting singular indexes corresponding to a left end point, a right end point and an extreme point of the multi-fractal spectrum f (alpha), and describing the running state of the equipment by using the 3 parameters;
step 16: the method in the steps is deployed on a state monitoring device to monitor the state of equipment.
The ELMD algorithm of the step 2 comprises the following steps:
1) to data x0(k) Adding a white noise sequence produces a new data xj(k) ：
Std[x0(k)]Representative data x0(k) Standard deviation of (Wn)j(k) Representative wnjThe kth data, wn injRepresents the jth randomly generated white noise sequence, wnjThe amplitude is 1, j is more than or equal to 1 and less than or equal to K; x is the number of0(k) Represents x (k) in step 2 of claim 1; in this example, K = 100;
2) for xj(k) Performing local mean decomposition to obtain n components and a trend term
cij(k) Represents a pair xj(k) The i-th component, r, resulting from performing a local mean decompositionnj(k) Represents a pair xj(k) Performing a trend term obtained by local mean decomposition;
3) calculating the average value of the decomposition results of K times
ci(k) Represents a pair x0(k) The ith component, r, obtained by performing a local mean decomposition of the setn(k) Represents a pair x0(k) And performing a trend term obtained by the set local mean decomposition.
The step 3 of the nonlinear discriminant algorithm comprises the following steps:
1) performing rearrangement operation and substitution operation on signals c (k), and using c as data obtained by rearrangement operationshuf(k) Indicating that data obtained after the substitution operation are csurr(k) Represents;
2) for c (k), cshuf(k) And csurr(k) Performing multi-fractal Detrended Fluctuation Analysis (MFDF) respectively to obtain a generalized Hurst index curve, wherein the generalized Hurst index curve of c (k) is represented by H (q); c. Cshuf(k) Generalized Hurst exponential curve of (1) using Hshuf(q) represents; c. Csurr(k) Generalized Hurst exponential curve of (1) using Hsurr(q) represents;
3) two parameters e are defined1And e2，
The data rearrangement operation in the step 1) comprises the following steps: randomly randomizing the order of the components c (k).
The data replacement operation in the step 1) comprises the following steps:
1) performing a discrete fourier transform on component c (k) to obtain the phase of component c (k);
2) replacing the original phase of component c (k) with a set of pseudo-independent identically distributed numbers located in the (-pi, pi) interval;
3) performing inverse discrete Fourier transform on the frequency domain data subjected to phase substitution to obtain data cIFFT(k) To obtain data cIFFT(k) The real part of (a).
The MFDF method in the step 2) comprises the following steps:
1) contour of construction x (k), k =1,2, …, NY(i)：
x (k) represents c (k) or c in step 2) of claim 3shuf(k) Or csurr（k）；
2) Signal profileY(i) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
3) fitting a polynomial trend of each section of data by using a least square method, and then calculating the variance of each section of data:
y v (i) Is a first of fittingvTrend of the segment data, if the fitted polynomial trend ismOrder, then note the de-trending process as (MF-) DFAm(ii) a In this example, m = 1;
4) calculate the firstqAverage of the order fluctuation function:
5) if it is notx(k) Presence of self-similar features thenqMean value of order fluctuation functionF q (s) And time scalesThere is a power law relationship between:
F q (s)~s H q()；
when in useqIf =0, the formula in step 4) diverges, at which timeH(0) Determined by the logarithmic averaging procedure defined by:
6) taking logarithm of both sides of the formula in step 5) to obtain ln [ 2 ]F q (s)]=H(q)ln(s)+c ，cIs constant, whereby the slope of the straight line can be obtainedH(q)。
The least square method in the step 4 comprises the following steps: for x (t), t =1,2, …, n, x (t) represents the pair c in step 4f(k) A sequence or pair c generated by interpolating the local maxima off(k) The local minima of the sequence, n represents the length of the interpolated sequence,
1) a set of functions r is selected in advancek(t)，k=1,2,…,m，m<n, constructioning function
f(t)=a1r1(t)+a2r2(t)+…+ amrm(t) in which rk(t) represents a second order polynomial, a third order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve or a complex function curve;
2) calculating a least squares metric
；
3) Let J pair akPartial derivative of
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Wherein R isTRepresenting the transposed matrix of R, R-1An inverse matrix representing R;
4）rk(t) respectively selecting a second-order polynomial, a third-order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve and a composite function curve for calculation, then comparing least square indexes J generated by various curve forms, and selecting a curve form r corresponding to the minimum J as the curve formk(t) form (a).
The Teager energy operator method in the step 6 comprises the following steps:
1) for signal c (k), k =1,2, …, N, c (k) = FMm(k) Building a function
ψ(c(k))=c2(k)- c(k+1) c(k-1);
2) Let d (k) = c (k) -c (k-1), instantaneous frequency instf (k) of signal c (k) be defined as:
based on the above-mentioned ELMD multi-scale fluctuation analysis state monitoring method, the device for implementing the method, the state monitoring device in step 16 includes the following parts: the system comprises a data line, an acceleration sensor, a data acquisition card, a case, a notebook computer and signal analysis software, wherein the acceleration sensor is connected with the data acquisition card through the data line, the data acquisition card is installed in the case, the case is connected with the notebook computer through the data line, the signal analysis software is installed on the notebook computer, and the signal analysis software is used for realizing the algorithm.
The performance of the algorithm of the present invention was verified using drive gearbox vibration data.
Firstly, a multi-fractal cascade model is adopted
The generated multi-fractal simulation signal verifies the performance of MFDFA, MFDFAemd and MFDFAelmd. In this example, p =0.375 and n =14, the resulting multi-fractal simulation signal is shown in fig. 3. The instantaneous frequency of the multi-fractal simulation signal is calculated by using the MFDFAemd, and the result is shown in fig. 4. As can be seen from fig. 4, the instantaneous frequency calculated by mfdfame has many negative frequencies, and the mfdfame analysis result has a large error because the negative frequencies have no physical significance. The instantaneous frequency of the multi-fractal simulation signal is calculated by using MFDFAoelmd, and the result is shown in fig. 5. As can be seen from fig. 5, the instantaneous frequencies calculated by MFDFAoelmd are all positive frequencies, so the MFDFAoelmd analysis result conforms to the actual situation. Next, a multi-fractal spectrum of the multi-fractal simulation signal is calculated using MFDFA, MFDFAemd, and mfdfaeelmd, respectively, and the result is shown in fig. 6. According to the results shown in fig. 6, it can be found through calculation that the average of the absolute errors between the multi-fractal spectrum obtained from MFDFA and the theoretical value is 0.064, the average of the relative errors is 12.21%, the average of the absolute errors between the multi-fractal spectrum obtained from MFDFAemd and the theoretical value is 0.035, the average of the relative errors is 6.57%, the average of the absolute errors between the multi-fractal spectrum obtained from mfdfaeelmd and the theoretical value is 0.023, and the average of the relative errors is 5.20%, so that the average of the absolute errors between the multi-fractal spectrum obtained from mfdfaeelmd is 64.06% smaller than that obtained from MFDFAemd, the average of the relative errors is 57.41%, the average of the absolute errors between the multi-fractal spectrum obtained from MFDFAemd is 34.29% smaller than that obtained from MFDFAemd, and the average of the relative errors is 20.85%. FIG. 7 shows two non-linear discriminant parameters in an embodiment of the present inventionFrom the calculation, it can be seen that all the signal components contain fractal signal components. Then, a noisy signal with a signal-to-noise ratio of 20dB is constructed by a method of adding white Gaussian noise to the multi-fractal simulation signal. The multi-fractal spectrum of the noise-containing multi-fractal simulation signal was calculated by using MFDFA, MFDFAemd, and mfdfaeelmd, respectively, and the result is shown in fig. 8. According to the results shown in fig. 8, the multi-fractal spectrum obtained from mfdfaamd completely deviates from the theoretical value, the mean absolute error value of the multi-fractal spectrum obtained from MFDFAemd from the theoretical value is 0.084, the mean relative error value is 11.81%, the mean absolute error value of the multi-fractal spectrum obtained from mfdfaeelmd from the theoretical value is 0.038, and the mean relative error value is 5.43%, so that the multi-fractal spectrum obtained from mfdfaeelmd is reduced by 54.76% and the mean relative error value is reduced by 54.02%. It follows that MFDFAoelmd has better noise immunity than MFDFA and MFDFAemd. Fig. 9 shows the correlation coefficient between each signal component and the original signal obtained by EMD in the embodiment of the present invention, and it can be seen that the 7 th component has the weakest correlation with the original signal and should be removed from the original signal. Fig. 10 is a comparison diagram of analysis results of a noisy multi-fractal simulation signal by using MFDFA, mfdfame based on correlation filtering, and mfdfaeelmd based on correlation filtering, respectively, in the embodiment of the present invention. As can be seen from fig. 10, the analysis result of the MFDFAemd based on the correlation filtering and the mfdfaeelmd based on the correlation filtering on the noisy multi-fractal simulation signal completely deviates from the theoretical value, so that the fractal structure of the original signal is easily damaged by the correlation filtering method.
The invention relates to a gearbox fault simulation experiment table for gearbox vibration data. This experiment simulates the process of a tooth from normal to failure by making different degrees of scoring on the root of a tooth until the tooth is finally completely destroyed. The gear box used in the experiment is in two-stage gear transmission, the number of gear teeth from the input end to the output end is respectively 25, 40, 22 and 55, the fault gear teeth are positioned on the input shaft gear, the rotating speed of the driving motor is 2000RMP, and the vibration signal of the gear box is measured by an acceleration sensor positioned on the shell of the input end. The collected vibration signals comprise four fault states of normal, light scratch, heavy scratch and broken tooth, and the vibration signals represent the process of the gear tooth from normal to failure to a certain extent. The vibration signal sampling frequency was 16384Hz, and 20 pieces of data with a length of 10000 points were collected in each gearbox state, and these four gearbox vibration signals are shown in fig. 11. Firstly, the four gearbox vibration signals are analyzed by using an MFDF method, and the multi-fractal spectrums corresponding to the four gearbox vibration signals are obtained as shown in FIG. 12, so that the multi-fractal spectrums corresponding to the light scratches and the heavy scratches are seriously overlapped. Then, the four gearbox vibration signals are analyzed by using an MFDFame method, and the multi-fractal spectrums corresponding to the four gearbox vibration signals are obtained as shown in FIG. 13, so that the multi-fractal spectrums corresponding to the four gearbox states are seriously overlapped. Finally, the four gearbox vibration signals are analyzed by using an MFDFooelmd method, and the multi-fractal spectrums corresponding to the four gearbox vibration signals are obtained as shown in fig. 14, so that the multi-fractal spectrums of the vibration signals in the tooth breaking state are obviously different from the multi-fractal spectrums corresponding to the states of the other three gearboxes, and the multi-fractal spectrums corresponding to the normal, light-scratch and heavy-scratch gear states can be clearly separated when alpha is less than 0.4. Singular indexes corresponding to a left end point, a right end point and an extreme point of a multi-fractal spectrum obtained by the MFDF, MFDFEMd and MFDFOelmd methods are respectively extracted to classify the four states of the gearbox, and the results are respectively shown in FIGS. 15-17. As can be seen from fig. 15, the normal state and the tooth breakage state can be correctly distinguished by using the singular indexes corresponding to the left end point, the right end point, and the extreme point of the multi-fractal spectrum obtained by the MFDFA method, but the light scratch state and the heavy scratch state cannot be distinguished, so that the gear box state recognition rate is 50%. As can be seen from fig. 16, the normal state and the tooth breakage state can be correctly distinguished by using the singular indexes corresponding to the left end point, the right end point and the extreme point of the multi-fractal spectrum obtained by the MFDFAemd method, but the light scratch state and the heavy scratch state cannot be distinguished, so that the gear box state recognition rate is 50%. As can be seen from fig. 17, the four gear box states can be correctly distinguished by using the singular indexes corresponding to the left end point, the right end point and the extreme point of the multi-fractal spectrum obtained by the MFDFAoelmd method, and therefore the gear box state recognition rate is 100%. It can be seen that the MFDFooelmd method can improve the accuracy of the state identification of the gearbox by 50%.
According to the experimental results, after analysis, it is considered that:
1) the vibration signal is adaptively decomposed by adopting an ELMD method, the noise component and the trend term are removed according to a nonlinear filtering method, the fractal structure of the original signal can be protected, and the damage of the linear filtering method to the fractal structure of the original signal is avoided;
2) the frequency modulation part of the separated signal component estimates the instantaneous frequency of the signal component by using TEO, so that the instantaneous frequency can be ensured to keep a positive value, and the negative frequency phenomenon is avoided;
3) calculating corresponding instantaneous scale according to the instantaneous frequency of the signal component, and performing fluctuation analysis according to the instantaneous scale of the signal component, so that the defect of manually setting the scale is avoided;
4) the ELMD method is utilized to automatically determine the type of the signal trend, ensure the continuity of the signal trend and effectively solve the defects of the prior art;
5) the accuracy and precision of the analysis result are high, and the accuracy of the identification result of the running state of the equipment is high.
It should be appreciated by those skilled in the art that the foregoing embodiments are merely exemplary for better understanding of the present invention, and should not be construed as limiting the scope of the present invention as long as the modifications are made according to the technical solution of the present invention.
Claims (9)
1. An ELMD multi-scale fluctuation analysis state monitoring method is characterized in that: the method comprises the following steps:
step 1: measuring a device vibration signal x (k) by using an acceleration sensor at a sampling frequency fs, wherein k =1,2, …, N and N are lengths of the sampling signal;
step 2: using Ensemble local mean decomposition (Ensemble Lo)The cal Mode Decomposition, ELMD) algorithm decomposes the signal x (k) into the sum of n components and a trend term, i.e.
Wherein c isi(k) Representing the i-th component, r, obtained by the ELMD algorithmn(k) Represents a trend term derived from the ELMD algorithm;
and step 3: eliminating noise components and trend terms from ELMD decomposition results by adopting a nonlinear discrimination algorithm, and reserving components c containing fractal featuresf(k) F =1,2, …, p, p represents the number of residual components after filtering;
and 4, step 4: determination of cf(k) The local maximum value and the local minimum value of c are respectively compared with c by adopting Lagrange interpolation functionf(k) The local maximum and local minimum are interpolated, and c is fitted by least square methodf(k) The upper envelope u (k) and the lower envelope l (k), then cf(k) Is defined as
The symbol | x | represents taking the absolute value of x;
and 5: repeatedly executing formula
m times, j =1,2, …, m, until
To obtain cf(k) Frequency modulation part FMm(k)，ej(k) Represents cj(k) Envelope of cj(k)=FM(j-1)(k)，c1(k)= cf（k）；
Step 6: calculating FM by using Teager Energy Operator (TEO)m(k) Obtaining the instantaneous frequency of cf(k) Instantaneous frequency instf off(k) To obtain cf(k) Instantaneous scale of
；
And 7: when the scale is s, the detrending result of the vibration signal x (k) is
；
And 8: will be provided withY s (k) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
and step 9: calculate variance of each piece of data:
step 10: calculating a q-th order function:
step 11: changing the value of s, s = sfF =1,2, …, p, repeating the above steps 3 to 10, resulting in a variance function F about q and sq(s);
Step 12: if it is notx(k) Presence of fractal features, thenF q (s) And sizesThere is a power law relationship between:F q (s)~s H q()h (q) represents the generalized Hurst index of x (k);
when in useqWhen the value is not less than 0, the reaction time is not less than 0,H(0) determined by the logarithmic averaging procedure defined by:
step 13: calculating a standard scale index τ (q) = qH (q) -1 for the signal x (k);
step 14: calculating the singular index α and the multifractal spectrum f (α) of the signal x (k):
α=H(q)+q H’(q),
f (alpha) = q (alpha-H (q)) +1, wherein H’(q) represents the first derivative of h (q);
step 15: and extracting singular indexes corresponding to a left end point, a right end point and an extreme point of the multi-fractal spectrum f (alpha), and describing the running state of the equipment by using the 3 parameters.
2. The method of claim 1, wherein the method comprises the following steps: the ELMD algorithm of the step 2 comprises the following steps:
1) to data x0(k) Adding a white noise sequence produces a new data xj(k) ：
Std[x0(k)]Representative data x0(k) Standard deviation of (Wn)j(k) Representative wnjThe kth data, wn injRepresents the jth randomly generated white noise sequence, wnjThe amplitude is 1, j is more than or equal to 1 and less than or equal to K; x is the number of0(k) Represents x (k) in step 2 of claim 1;
2) for xj(k) Performing local mean decomposition to obtain n components and a trend term
cij(k) Represents a pair xj(k) Performing local mean decomposition to obtaini components, rnj(k) Represents a pair xj(k) Performing a trend term obtained by local mean decomposition;
3) calculating the average value of the decomposition results of K times
ci(k) Represents a pair x0(k) The ith component, r, obtained by performing a local mean decomposition of the setn(k) Represents a pair x0(k) And performing a trend term obtained by the set local mean decomposition.
3. The method of claim 1, wherein the method comprises the following steps: the step 3 of the nonlinear discriminant algorithm comprises the following steps:
1) performing rearrangement operation and substitution operation on signals c (k), and using c as data obtained by rearrangement operationshuf(k) Indicating that data obtained after the substitution operation are csurr(k) Represents;
2) for c (k), cshuf(k) And csurr(k) Performing multi-fractal Detrended Fluctuation Analysis (MFDF) respectively to obtain a generalized Hurst index curve, wherein the generalized Hurst index curve of c (k) is represented by H (q); c. Cshuf(k) Generalized Hurst exponential curve of (1) using Hshuf(q) represents; c. Csurr(k) Generalized Hurst exponential curve of (1) using Hsurr(q) represents;
3) two parameters e are defined1And e2，
4. The method of claim 3, wherein the ELMD multi-scale fluctuation analysis state monitoring method comprises the following steps: the data rearrangement operation in the step 1) comprises the following steps: randomly randomizing the order of the components c (k).
5. The method of claim 3, wherein the ELMD multi-scale fluctuation analysis state monitoring method comprises the following steps: the data replacement operation in the step 1) comprises the following steps:
1) performing a discrete fourier transform on component c (k) to obtain the phase of component c (k);
2) replacing the original phase of component c (k) with a set of pseudo-independent identically distributed numbers located in the (-pi, pi) interval;
3) performing inverse discrete Fourier transform on the frequency domain data subjected to phase substitution to obtain data cIFFT(k) To obtain data cIFFT(k) The real part of (a).
6. The method of claim 3, wherein the ELMD multi-scale fluctuation analysis state monitoring method comprises the following steps: the MFDF method in the step 2) comprises the following steps:
1) contour of construction x (k), k =1,2, …, NY(i)：
x (k) represents c (k) or c in step 2) of claim 3shuf(k) Or csurr（k）；
2) Signal profileY(i) Divided into non-overlappingN s Length of segment beingsDue to data lengthNUsually cannot be removedsSo that a segment of data remains unavailable; in order to fully utilize the length of the data, the data is segmented with the same length from the reverse direction of the data, so that 2 is obtainedN s Segment data;
3) fitting a polynomial trend of each section of data by using a least square method, and then calculating the variance of each section of data:
y v (i) Is a first of fittingvTrend of the segment data, if the fitted polynomial trend ismOrder, then note the de-trending process as (MF-) DFAm(ii) a 4) Calculate the firstqAverage of the order fluctuation function:
5) if it is notx(k) Presence of self-similar features thenqMean value of order fluctuation functionF q (s) And time scalesThere is a power law relationship between:
F q (s)~s H q()；
when in useqIf =0, the formula in step 4) diverges, at which timeH(0) Determined by the logarithmic averaging procedure defined by:
6) taking logarithm of both sides of the formula in step 5) to obtain ln [ 2 ]F q (s)]=H(q)ln(s)+c ，cIs constant, whereby the slope of the straight line can be obtainedH(q)。
7. The method of claim 1, wherein the method comprises the following steps: the least square method in the step 4 comprises the following steps: for x (t), t =1,2, …, n, x (t) represents the pair c in step 4f(k) A sequence or pair c generated by interpolating the local maxima off(k) The local minima of the sequence, n represents the length of the interpolated sequence,
1) a set of functions r is selected in advancek(t)，k=1,2,…,m，m<n, constructioning function
f(t)=a1r1(t)+a2r2(t)+…+ amrm(t) in which rk(t) represents a second order polynomial, a third order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve or a complex function curve;
2) calculating a least squares metric
；
3) Let J pair akPartial derivative of
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Wherein R isTRepresenting the transposed matrix of R, R-1An inverse matrix representing R;
4）rk(t) selecting a second order polynomial, a third order polynomial, a hyperbolic curve, an exponential curve, a logarithmic curve and a complex function curve respectivelyCalculating, comparing the least square indexes J generated by various curve forms, and selecting the curve form r corresponding to the minimum J ask(t) form (a).
8. The method of claim 1, wherein the method comprises the following steps: the Teager energy operator method in the step 6 comprises the following steps:
1) for signal c (k), k =1,2, …, N, c (k) = FMm(k) Building a function
ψ(c(k))=c2(k)- c(k+1) c(k-1);
2) Let d (k) = c (k) -c (k-1), instantaneous frequency instf (k) of signal c (k) be defined as:
9. apparatus for performing a method for monitoring the state of an ELMD multi-scale fluctuation analysis according to any of claims 1 to 8, characterized in that: the device comprises the following parts: the system comprises a data line, an acceleration sensor, a data acquisition card, a case, a notebook computer and signal analysis software, wherein the acceleration sensor is connected with the data acquisition card through the data line, the data acquisition card is installed in the case, the case is connected with the notebook computer through the data line, the signal analysis software is installed on the notebook computer, and the signal analysis software is used for realizing the algorithm.
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