CN112697474A - NMD multi-scale fluctuation analysis state monitoring method and device - Google Patents
NMD multi-scale fluctuation analysis state monitoring method and device Download PDFInfo
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Abstract
The invention discloses a NMD multi-scale fluctuation analysis state monitoring method and a device, which decompose equipment vibration signals by using an NMD algorithm, remove noise components and trend terms by using a nonlinear discrimination algorithm, reserve fractal signal components, interpolate local extreme points by using a segmented Hermite interpolation function respectively, fit an envelope 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 vibration signal detrending result according to the analysis scale, calculate a multi-fractal spectrum of a detrended 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 a method and a device for monitoring an NMD 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 the state of NMD multi-scale fluctuation analysis (the method provided by the invention is abbreviated as MFDFommd) 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 NMD 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 by using a Nonlinear Mode Decomposition (NMD) algorithm, i.e.
Wherein c isi(k) Representing the i-th component, r, obtained by the NMD algorithmn(k) Represents the trend term derived by the NMD algorithm, in this example, n = 10;
NMD algorithms are well known and are described in the literature
Dmytro Iatsenko, Peter V. E. McClintock, Aneta Stefanovska. Nonlinear mode decomposition: A noise-robust, adaptive decomposition method, PHYSICAL REVIEW E, 2015, 92: 032916；
And step 3: eliminating noise components and trend terms from NMD 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
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 11, 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 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 2shuf(k) Or csurr（k）；
2) Signal profileY(i) Divided into non-overlappingN s Length of segment beingsDue to data lengthNIn generalCan not be removed completelysSo 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)。
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 NMD multi-scale fluctuation analysis state monitoring method, the device for realizing the method 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.
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 NMD method, the noise component and the trend item 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 NMD method is used for automatically determining the type of the signal trend, 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 MFDFAonmd 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 MFDFOMd 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 MFDFOMd 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, MFDFEMd based on correlation filtering, and MFDFOMd 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 MFDFOMd 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 MFDFAonmd 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
In an embodiment, as shown in fig. 1 and fig. 2, a method for monitoring a state of NMD multi-scale fluctuation analysis includes 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 a Nonlinear Mode Decomposition (NMD) algorithm, i.e.
Wherein c isi(k) Representing the i-th component, r, obtained by the NMD algorithmn(k) Represents the trend term derived by the NMD algorithm, in this example, n = 10;
NMD algorithms are well known and are described in the literature
Dmytro Iatsenko, Peter V. E. McClintock, Aneta Stefanovska. Nonlinear mode decomposition: A noise-robust, adaptive decomposition method, PHYSICAL REVIEW E, 2015, 92: 032916；
And step 3: eliminating noise components and trend terms from NMD 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
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 11, 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 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 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:
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 NMD multi-scale fluctuation analysis state monitoring method, the device for realizing the method 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.
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 the MFDFA, the MFDFAemd and the MFDFOnmd. 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 using the MFDFAonmd, and the result is shown in fig. 5. As can be seen from fig. 5, the instantaneous frequencies calculated by the MFDFAonmd are all positive frequencies, and therefore the MFDFAonmd analysis result conforms to the actual situation. Next, a multi-fractal spectrum of the multi-fractal simulation signal is calculated using MFDFA, MFDFAemd, and MFDFAonmd, respectively, and the result is shown in fig. 6. From the results shown in fig. 6, it can be found by calculation that the average of absolute errors between the fractal spectrum obtained from MFDFA and the theoretical value is 0.064, the average of relative errors is 12.21%, the average of absolute errors between the fractal spectrum obtained from mfdfame and the theoretical value is 0.035, the average of relative errors is 6.57%, the average of absolute errors between the fractal spectrum obtained from mfdfamond and the theoretical value is 0.027, and the average of relative errors is 5.17%, so that the fractal spectrum obtained from mfdfamond is decreased by 57.81%, the average of relative errors is decreased by 57.66%, and the fractal spectrum obtained from mfdfamond is decreased by MFDFAemd, the absolute error mean value of the obtained multi-fractal spectrum is reduced by 22.86%, and the relative error mean value is reduced by 21.31%. 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 noise-containing multi-fractal simulation signal was calculated by using MFDFA, MFDFAemd, and MFDFAonmd, 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 mfdfaamd 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 mfdfaomd from the theoretical value is 0.036, and the mean relative error value is 5.41%, so that the multi-fractal spectrum obtained from mfdfaomd is reduced by 57.14% and the mean relative error value is reduced by 54.19% compared to the mean absolute error value of the multi-fractal spectrum obtained from mfdfaamd. It follows that MFDFAonmd 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 noisy multi-fractal simulation signals by using MFDFA, mfdfame based on correlation filtering, and MFDFAonmd 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 mfdfame based on the correlation filtering and the MFDFAonmd 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.
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 MFDFOMad 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 MFDFONmd 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 gearbox 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 MFDFAonmd method, and therefore the gearbox state recognition rate is 100%. It can be seen that the MFDFONGd 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 NMD 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 NMD method is used for automatically determining the type of the signal trend, 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 understood by those skilled in the art that the foregoing specific embodiments are merely exemplary for better understanding of the present invention, and are not to be construed as limiting the scope of the present invention, as long as the modifications are made according to the technical scheme of the present invention.
Claims (8)
1. A NMD 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 by using a Nonlinear Mode Decomposition (NMD) algorithm, i.e.
Wherein c isi(k) Representing the i-th component, r, obtained by the NMD algorithmn(k) Represents a trend term derived from the NMD algorithm;
and step 3: eliminating noise components and trend terms from NMD 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
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 11, 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 NMD multi-scale fluctuation analysis status monitoring method according to claim 1, characterized in that: 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，
3. The NMD multi-scale fluctuation analysis status monitoring method according to claim 2, characterized in that: the data rearrangement operation in the step 1) comprises the following steps: randomly randomizing the order of the components c (k).
4. The NMD multi-scale fluctuation analysis status monitoring method according to claim 2, characterized in that: 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. The NMD multi-scale fluctuation analysis status monitoring method according to claim 2, characterized in that: 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 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:
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 useqWhen =0, the formula in step 4) diverges, which isTime of flightH(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)。
6. The NMD multi-scale fluctuation analysis status monitoring method according to claim 1, characterized in that: 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).
7. The NMD multi-scale fluctuation analysis status monitoring method according to claim 1, characterized in that: 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:
8. apparatus for implementing a method for monitoring the condition of NMD multiscale fluctuation analysis according to any of claims 1 to 7, 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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