CN112629849A - FDM multi-scale fluctuation analysis state monitoring method and device - Google Patents

FDM multi-scale fluctuation analysis state monitoring method and device Download PDF

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CN112629849A
CN112629849A CN202011240578.6A CN202011240578A CN112629849A CN 112629849 A CN112629849 A CN 112629849A CN 202011240578 A CN202011240578 A CN 202011240578A CN 112629849 A CN112629849 A CN 112629849A
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寇兴磊
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M13/00—Testing of machine parts
    • G01M13/02—Gearings; Transmission mechanisms
    • G01M13/021—Gearings
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M13/00—Testing of machine parts
    • G01M13/02—Gearings; Transmission mechanisms
    • G01M13/028—Acoustic or vibration analysis
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2218/00—Aspects of pattern recognition specially adapted for signal processing
    • G06F2218/08—Feature extraction
    • G06F2218/10—Feature extraction by analysing the shape of a waveform, e.g. extracting parameters relating to peaks
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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Abstract

The invention discloses a monitoring method and a monitoring device for FDM multi-scale fluctuation analysis state, which decompose a vibration signal by using an FDM algorithm, remove noise components and trend items in decomposition results by using a nonlinear discrimination algorithm, reserve fractal signal components, interpolate local extreme points of the fractal signal components by using a segmented Hermite interpolation function, fit envelopes of the fractal signal components by using a least square method, separate frequency modulation parts of each fractal signal component, estimate instantaneous frequency of the fractal signal components by using a TEO algorithm and calculate instantaneous scale, automatically determine a vibration signal according to the analysis scale to obtain a trend result, calculate a multi-fractal spectrum of a de-trend signal, extract left end point, right end point and extreme point coordinates of the multi-fractal spectrum as characteristic parameters to identify the equipment operation state, have good noise immunity, Robustness and adaptivity.

Description

FDM multi-scale fluctuation analysis state monitoring method and device
Technical Field
The invention relates to the field of equipment state monitoring and fault diagnosis, in particular to a method and a device for monitoring an FDM multi-scale fluctuation analysis state.
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 an FDM multi-scale fluctuation analysis state (the method is abbreviated as MFDFofdm) 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: a FMD multi-scale fluctuation analysis state monitoring method and device are 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, i.e. into the sum of n components using a Fourier Decomposition Method (FDM) algorithm
Figure 444993DEST_PATH_IMAGE001
Wherein c isi(k) Representing the ith component, r, obtained by the FMD algorithmn(k) Represents the trend term derived from the FMD algorithm, in this example, n = 10; FMD algorithms are well known and are described in Singh P, Joshi S D, Patney R K, et al, The Fourier decomposition method for non-linear and non-stationary time series analysis, Proceedings of The Royal Society A, 2017, 473(2199): 20160871;
and step 3: eliminating noise components and trend terms from FMD 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) Respectively comparing the local maximum value and the local minimum value of c by adopting a segmented Hermite 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
Figure 799751DEST_PATH_IMAGE002
The symbol | x | represents taking the absolute value of x;
and 5: repeatedly executing formula
Figure 469898DEST_PATH_IMAGE003
m times, j =1,2, …, m, until
Figure 164185DEST_PATH_IMAGE004
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
Figure 878063DEST_PATH_IMAGE005
；
And 7: when the scale is s, the detrending result of the vibration signal x (k) is
Figure 113260DEST_PATH_IMAGE006
；
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:
Figure 882370DEST_PATH_IMAGE007
Figure 493480DEST_PATH_IMAGE008
；
step 10: calculating a q-th order function:
Figure 581522DEST_PATH_IMAGE009
；
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:
Figure 661604DEST_PATH_IMAGE010
；
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 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，
Figure 922822DEST_PATH_IMAGE011
，
Figure 697093DEST_PATH_IMAGE012
If e is1And e2All less than 10%, the signal c (k) is discriminated as a noise component or a trend term, c (k) represents the signal component resulting from the FMD algorithm.
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)：
Figure 956036DEST_PATH_IMAGE013
Figure 523414DEST_PATH_IMAGE014
；
x(k) Represents c (k) or c in step 2) of claim 2shuf(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:
Figure 322743DEST_PATH_IMAGE015
Figure 846128DEST_PATH_IMAGE016
；
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:
Figure 587557DEST_PATH_IMAGE017
；
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:
Figure 563603DEST_PATH_IMAGE010
；
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, theThe least squares method in step 4 comprises the steps of: 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
Figure 166623DEST_PATH_IMAGE018
；
3) Let J pair akPartial derivative of
Figure 810094DEST_PATH_IMAGE019
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Figure 958310DEST_PATH_IMAGE020
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:
Figure 687231DEST_PATH_IMAGE021
。
based on the above FMD multi-scale fluctuation analysis state monitoring method, a FMD multi-scale fluctuation analysis state monitoring method and device are provided, where 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 the vertical coordinates of the left end point, the right end point and the extreme point of the multi-fractal spectrum, and taking the three parameters as the characteristic parameters of the running state of the equipment;
16) step: the algorithm is deployed on an equipment state monitoring device to monitor the equipment state.
By adopting the technical scheme, compared with the prior art, the invention has the following advantages:
1) the FDM method is adopted to decompose the vibration signal, noise components and trend items are removed according to the 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 type of the signal trend is automatically determined by utilizing an FDM method, the continuity of the signal trend is ensured, and the defects of the prior art are effectively overcome;
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 MFDFAofdm 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 the MFDF, MFDFEMd and MFDFOFdm 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 graph of analysis results of noisy multi-fractal simulation signals respectively using MFDF, MFDFEMd and MFDFOFdm 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 graph of analysis results of noisy multi-fractal simulation signals respectively using MFDF, MFDFEMd based on correlation filtering, and MFDFOFdm based on correlation filtering in the embodiments 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 MFDFOFdm in an embodiment of the present invention;
FIG. 15 is a diagram showing the classification results of the left end point, the right end point and the extreme point coordinates of the multi-fractal spectrum obtained from MFDFA on the four gear box states in the example of the present invention, where the symbols "circle", "square", "plus" and "diamond" represent the normal, light scratch, heavy scratch and broken tooth gear states, respectively;
fig. 16 is a classification result of coordinates of a left end point, a right end point, and an extreme point of a multi-fractal spectrum obtained from MFDFAemd on 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 results of classifying the four gear box states by the coordinates of the left end point, the right end point and the extreme point of the multi-fractal spectrum obtained from the MFDFAofdm in the embodiment of the present invention, and the symbols "circle", "square", "plus" and "diamond" represent the normal, light scratch, heavy scratch and broken tooth gear states, respectively.
Detailed Description
An embodiment, as shown in fig. 1 and fig. 2, is a method and a device for monitoring FMD multi-scale fluctuation analysis status, including 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, i.e. into the sum of n components using a Fourier Decomposition Method (FDM) algorithm
Figure 828363DEST_PATH_IMAGE022
Wherein c isi(k) Representing the ith component, r, obtained by the FMD algorithmn(k) Represents the trend term derived from the FMD algorithm, in this example, n = 10; FMD algorithms are well known and are described in Singh P, Joshi S D, Patney R K, et al, The Fourier decomposition method for non-linear and non-stationary time series analysis, Proceedings of The Royal Society A, 2017, 473(2199): 20160871;
and step 3: eliminating noise components and trend terms from FMD 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) Respectively comparing the local maximum value and the local minimum value of c by adopting a segmented Hermite 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
Figure 326340DEST_PATH_IMAGE002
The symbol | x | represents taking the absolute value of x;
step (ii) of5: repeatedly executing formula
Figure 832408DEST_PATH_IMAGE003
m times, j =1,2, …, m, until
Figure 363140DEST_PATH_IMAGE004
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
Figure 245645DEST_PATH_IMAGE023
；
And 7: when the scale is s, the detrending result of the vibration signal x (k) is
Figure 660446DEST_PATH_IMAGE024
；
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:
Figure 602994DEST_PATH_IMAGE025
Figure 853978DEST_PATH_IMAGE026
；
step 10: calculating a q-th order function:
Figure 274595DEST_PATH_IMAGE027
；
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:
Figure 809482DEST_PATH_IMAGE010
；
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 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，
Figure 657352DEST_PATH_IMAGE028
，
Figure 894167DEST_PATH_IMAGE029
If e is1And e2All less than 10%, the signal c (k) is discriminated as a noise component or a trend term, c (k) represents the signal component resulting from the FMD algorithm.
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)：
Figure 180792DEST_PATH_IMAGE013
Figure 507868DEST_PATH_IMAGE014
；
x(k) Represents c (k) or c in step 2) of claim 2shuf(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:
Figure 605268DEST_PATH_IMAGE015
Figure 17795DEST_PATH_IMAGE016
；
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:
Figure 842531DEST_PATH_IMAGE017
；
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:
Figure 24114DEST_PATH_IMAGE010
；
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
Figure 788021DEST_PATH_IMAGE031
；
3) Let J pair akPartial derivative of
Figure 687844DEST_PATH_IMAGE019
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Figure 316271DEST_PATH_IMAGE032
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:
Figure 352360DEST_PATH_IMAGE033
。
based on the above FMD multi-scale fluctuation analysis state monitoring method, a FMD multi-scale fluctuation analysis state monitoring method and device are provided, where 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.
Experiment 1 the performance of the algorithm of the present invention was verified using a multi-fractal simulation signal generated by a multi-fractal cascade model.
Firstly, a multi-fractal cascade model is adopted
Figure 978514DEST_PATH_IMAGE034
The generated multi-fractal simulation signal verifies the performance of the MFDFA, the MFDFEMd and the MFDFOFdm. In this example, p =0.375N =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 was calculated using the MFDFAofdm, and the result is shown in fig. 5. As can be seen from fig. 5, the instantaneous frequencies calculated by the MFDFAofdm are all positive frequencies, and therefore the MFDFAofdm analysis result conforms to the actual situation. Next, a multi-fractal spectrum of the multi-fractal simulation signal was calculated using MFDFA, MFDFAemd, and MFDFAofdm, respectively, and the result is shown in fig. 6. As shown in fig. 6, it is found through calculation that the mean absolute error of the multi-fractal spectrum obtained from MFDFA and the theoretical value is 0.064, the mean relative error is 12.21%, the mean absolute error of the multi-fractal spectrum obtained from MFDFAemd and the theoretical value is 0.035, the mean relative error is 6.57%, the mean absolute error of the multi-fractal spectrum obtained from MFDFAofdm and the theoretical value is 0.028, and the mean relative error is 5.39%, so that the mean absolute error of the multi-fractal spectrum obtained from MFDFAofdm is reduced by 56.25%, the mean relative error is reduced by 55.86%, the mean absolute error of the multi-fractal spectrum obtained from MFDFAofdm is reduced by 20.00%, and the mean relative error is reduced by 17.96%. Fig. 7 shows the calculation results of two non-linear discrimination parameters in the embodiment of the present invention, and it can be seen that all signal components include 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 noisy multi-fractal simulation signal was calculated using MFDFA, MFDFAemd, and MFDFAofdm, respectively, and the result is shown in fig. 8. According to the results shown in FIG. 8, the multi-fractal spectrum obtained from MFDFA 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 MFDFOFdm from the theoretical value is 0.035, and the mean relative error value is 5.62%, so that the multi-fractal spectrum obtained from MFDFOF is completely deviated from the theoretical valueThe absolute error mean value of the fractal spectrum obtained by dm is reduced by 58.33 percent, and the relative error mean value is reduced by 52.41 percent. It follows that mfdfafofdm is more resistant to noise 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 noisy multi-fractal simulation signals by using MFDFA, mfdfame based on correlation filtering, and mfdfafofdm based on correlation filtering, respectively, in the embodiment of the present invention. As can be seen from fig. 10, the analysis result of the noise-containing multi-fractal simulation signal by the MFDFAemd based on the correlation filtering and the MFDFAofdm based on the correlation filtering completely deviates from the theoretical value, so that the fractal structure of the original signal is easily damaged by the correlation filtering.
Experiment 2 the performance of the algorithm of the invention was verified using gearbox experimental signals.
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 adopting an MFDFOFdm 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 other three gearbox states, 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 MFDFOFdm 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 MFDFAofdm method, and therefore the gear box state identification rate is 100%. It can be seen that the accuracy of the state identification of the gearbox can be improved by 50% by adopting the MFDFOFdm method.
According to the experimental results, after analysis, it is considered that:
1) the FDM method is adopted to adaptively decompose the vibration signal, noise components and trend items are removed according to the 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 type of the signal trend is automatically determined by utilizing an FDM method, the continuity of the signal trend is ensured, and the defects of the prior art are effectively overcome;
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 (8)

1. A FMD multi-scale fluctuation analysis state monitoring method is characterized by comprising the following steps: 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, i.e. into the sum of n components using a Fourier Decomposition Method (FDM) algorithm
Figure 165127DEST_PATH_IMAGE001
Wherein c isi(k) Representing the ith component, r, obtained by the FMD algorithmn(k) Represents a trend term derived from the FMD algorithm;
and step 3: eliminating noise components and trend terms from FMD 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) Respectively comparing the local maximum value and the local minimum value of c by adopting a segmented Hermite 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
Figure 144584DEST_PATH_IMAGE002
The symbol | x | represents taking the absolute value of x;
and 5: repeatedly executing formula
Figure 924321DEST_PATH_IMAGE003
m times, j =1,2, …, m, until
Figure 634045DEST_PATH_IMAGE004
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
Figure 448417DEST_PATH_IMAGE006
；
And 7: when the scale is s, the detrending result of the vibration signal x (k) is
Figure 333197DEST_PATH_IMAGE008
；
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 data, the data is read fromThe reverse direction of the data is segmented with the same length, thus obtaining 2 in totalN s Segment data;
and step 9: calculate variance of each piece of data:
Figure 600230DEST_PATH_IMAGE010
Figure 346600DEST_PATH_IMAGE011
；
step 10: calculating a q-th order function:
Figure 15479DEST_PATH_IMAGE012
；
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:
Figure 336739DEST_PATH_IMAGE013
；
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 FMD multi-scale fluctuation analysis state monitoring method of claim 1, wherein: 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，
Figure 91068DEST_PATH_IMAGE014
，
Figure 139664DEST_PATH_IMAGE015
If e is1And e2All less than 10%, the signal c (k) is discriminated as a noise component or a trend term, c (k) represents the signal component resulting from the FMD algorithm.
3. A FMD multi-scale fluctuation analysis status monitoring method according to claim 2, wherein: the data rearrangement operation in the step 1) comprises the following steps: randomly randomizing the order of the components c (k).
4. A FMD multi-scale fluctuation analysis status monitoring method according to claim 2, wherein: 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).
5. A FMD multi-scale fluctuation analysis status monitoring method according to claim 2, wherein: the MFDF method in the step 2) comprises the following steps:
1) contour of construction x (k), k =1,2, …, NY(i)：
Figure 663050DEST_PATH_IMAGE016
Figure 155211DEST_PATH_IMAGE017
；
x(k) Represents c (k) or c in step 2) of claim 2shuf(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:
Figure 396836DEST_PATH_IMAGE018
Figure 485009DEST_PATH_IMAGE019
；
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:
Figure 128480DEST_PATH_IMAGE020
；
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:
Figure 260384DEST_PATH_IMAGE013
；
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)。
6. An FMD poly as defined in claim 1The scale fluctuation analysis state monitoring method is characterized by comprising 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
Figure 989306DEST_PATH_IMAGE022
；
3) Let J pair akPartial derivative of
Figure DEST_PATH_IMAGE023
K =1,2, …, m, when (a)1,a2,…,am)T =(RTR)-1RTX， X=(x(1),x(2),…,x(n)) T，
Figure DEST_PATH_IMAGE025
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).
7. The FMD multi-scale fluctuation analysis state monitoring method of claim 1, wherein: 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:
Figure DEST_PATH_IMAGE027
。
8. apparatus for performing a method of monitoring the condition of a FMD multi-scale fluctuation analysis according to any of claims 1 to 7, wherein: 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.
CN202011240578.6A 2020-11-09 2020-11-09 FDM multi-scale fluctuation analysis state monitoring method and device Withdrawn CN112629849A (en)

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