CN106441888A - High-speed train rolling bearing fault diagnosis method - Google Patents

Abstract

A high-speed train rolling bearing fault diagnosis method, the steps are as follows: collect the original vibration signal and decompose it using the EEMD method, select the first a IMF components to obtain the energy and energy sum of the components, and obtain the energy feature vector through normalization processing; determine the RBF neural network Structure, determine the number of nodes in the input layer, output layer and hidden layer, determine the training target accuracy and distribution density, select training samples and test samples, use the training samples as input for training, and obtain the preliminary RBF neural network diagnosis model after reaching the target accuracy, The test sample is used as the input of the preliminary model to identify the test sample. If the fault recognition rate reaches the ideal standard, the final model of RBF neural network diagnosis is obtained to diagnose the bearing fault type. The invention provides a new idea for improving the accuracy and real-time performance of the rolling bearing fault diagnosis of the high-speed train, and also provides further guarantee for the performance and driving safety of the high-speed train.

Description

A fault diagnosis method for high-speed train rolling bearings

technical field

The invention relates to a fault diagnosis method for modeling and classification using characteristic signals, in particular to a high-speed train rolling bearing fault diagnosis method based on EEMD and RBF neural network.

Background technique

Rolling bearings are one of the important components of high-speed trains, and their condition is crucial to the safe operation of trains. Speeding up and increasing load is the development trend of railways all over the world, and having trains with full traction is the premise of increasing speed and capacity. At this time, rolling bearings, one of the important components of high-speed trains, deserve more attention. As a mechanical wearing part, a rolling bearing has a remarkable feature that its life span is large and its failure causes are complicated. In practical applications, some rolling bearings have been used for a long time before reaching the design life, but there are various failures, and some are far beyond the design life, but they can still work normally. Therefore, in order to prevent bearing failure, it is necessary to monitor the running state of the bearing.

At present, most of the diagnosis of bearing faults is to analyze its vibration signal, and the vibration signal has the characteristics of nonlinearity and non-stationarity. EEMD is a noise-assisted data processing method, which is suitable for analyzing and processing nonlinear and non-stationary signals. It can be used to obtain information that fully expresses the characteristics of the signal. Bearing fault diagnosis based on RBF neural network is more accurate and faster than BP neural network diagnosis, and is less prone to local minima, so it is more suitable for bearing fault diagnosis.

Contents of the invention

The invention provides a fault diagnosis method for rolling bearings of high-speed trains aiming at the disadvantages of slow model building speed and low fault recognition accuracy in the prior art. It can provide a new idea for the fault diagnosis and condition monitoring research of high-speed train rolling bearings, and also provide a further guarantee for the performance and driving safety of the train.

The present invention aims at the deficiency of prior art, provides a kind of

In order to achieve the above object, the present invention adopts the following technical solutions:

A high-speed train rolling bearing fault diagnosis method, comprising the following steps:

I. Establishment of fault diagnosis model

Step 1: Divide the bearings into four state types: normal bearings, rolling element fault bearings, outer ring fault bearings and inner ring fault bearings. For the above four state types of bearings, collect several sets of original vibration signals for each type;

Step 2: EEMD fault feature vector construction

1) Use EEMD method to decompose each group of original vibration signals, select the first a IMF components obtained from the decomposition, and calculate the energy E _i of each component respectively;

In the formula, C _i t is the i-th IMF component, i=1, 2, 3, L, a, C _i is the amplitude of discrete points, n is the number of sampling points,

2) seek the energy summation E of each IMF component of each group of original vibration signals;

3) Due to the large difference in the vibration amplitude of the bearings in different states, the difference in the value of each IMF component is also large, so the energy is normalized, that is, the energy of each IMF component is compared with the total energy, and a The energy feature vector T of the original vibration signal of the group;

T=E ₁ /E, E ₂ /E, L, E _a /E

Step 3: RBF Neural Network Modeling

1) Determine the RBF neural network structure

Although increasing the number of neurons in the hidden layer can improve the nonlinear mapping ability of the RBF neural network, too many neurons will reduce the network prediction performance, so a three-layer RBF neural network with a single hidden layer is used.

2) Determine the number of nodes in the input layer

The energy feature vector T is used as the input of the RBF neural network, so the number of nodes in the input layer M=a,

3) Determine the number of nodes in the output layer

The ideal output result should be able to directly see the classification of the fault, so 3 binary codes are used, that is, the number of nodes in the output layer is 3, see Table 1,

4) Determine the number of nodes of the hidden layer, determine the training target precision of the RBF neural network and the distribution density SPREAD of the radial basis function,

5) Select group b from the original vibration signals of each bearing state as training samples, and the rest as test samples, and use the training samples as the input of the RBF neural network for training. After the training reaches the target accuracy set in step 4), a preliminary RBF neural network diagnosis model, and then use the test sample as the input of the RBF neural network diagnosis preliminary model to identify the state of the test sample bearing,

6) Performance evaluation of the preliminary model

Firstly, the recognition error is calculated according to the recognition result of the test sample. When the recognition error is within the acceptable range, the recognition result is considered correct, otherwise the recognition result is considered wrong; then the fault recognition rate Q is calculated, Q=correct recognition number/total experimental bearing number, The performance of the RBF neural network is measured by the fault recognition rate Q. If the fault recognition rate Q reaches the ideal standard, the final model of the RBF neural network diagnosis is obtained and used for the bearing fault diagnosis of the unknown state in step II; otherwise, return to step 5), Reselect training samples and test samples for training and testing,

II. Diagnosis of bearing fault types

The original vibration signal of the bearing to be diagnosed in an unknown state is collected and used as the input of the final model of RBF neural network diagnosis, and the state of the bearing is determined by comparing the output of the final model with the output form of the bearing fault.

The specific decomposition steps of the EEMD method are as follows:

(1) Add white noise a _m (t) with a uniform spectrum distribution to the signal R(t) to be decomposed to obtain the signal S(t);

(2) EMD is performed on the signal S(t), and the decomposition process is as follows;

1) Determine all local extreme points on the signal S(t), and the upper and lower envelopes are obtained by connecting all local maximum points and local minimum points with cubic spline curves , namely s(t) _max and S(t) _min ;

2) Calculate the average value of the upper and lower envelopes at each moment, namely

3) get a new signal

Y ₁ (t)=S(t)-u(t)

Judging whether it is symmetrical to the local zero mean, and has the same extreme point and zero crossing point, if yes, record it as C ₁ (t), which is the first IMF component, otherwise repeat steps 1) and 2);

4) Separate C ₁ (t) from S(t) to obtain a difference signal V ₁ (t),

V ₁ (t)=S(t)-C ₁ (t)

5) Take V ₁ (t) as the original data, repeat steps 1) to 3) to get IMF2, and repeat n times to get n IMF components, so we have

When V _n (t) meets the given termination condition, the termination condition is that V _n (t) is a monotone function, and the loop ends,

From Y ₁ (t) and V ₁ (t) can get

That is to say, the original signal is expressed as the sum of the intrinsic mode function components and a residual function V _n (t), and each component C ₁ (t), C ₂ (t), ... C _n (t) covers the original signal from Information in different frequency bands from high to low, and changes with the change of the signal itself,

(3) Repeat process (1) and (2) after adding different white noise each time;

(4) Taking the integrated mean C _i of each IMF component after multiple decompositions as the final result,

Compared with the prior art, the present invention has the beneficial effects:

This patent provides a rolling bearing fault diagnosis method based on EEMD and RBF neural network. Through this method, four kinds of bearing states, such as normal bearing, rolling element fault, outer ring fault and inner ring fault, can be accurately identified. The accuracy and real-time of the diagnosis provide a new idea, and also provide a further guarantee for the performance and driving safety of the train.

Description of drawings

Figure 1 is the specific flow of the EEMD algorithm.

Figure 2 is the structure of the RBF neural network.

Fig. 3 is the original signal diagram of the train rolling element fault bearing.

Fig. 4 is an EEMD exploded view of a train rolling element fault bearing.

Fig. 5 is the normalized energy eigenvector diagram of the train rolling element fault bearing.

Fig. 6 is a diagram of the training process of the RBF neural network.

detailed description

The present invention will be described in detail below in combination with specific embodiments.

Embodiment 1 This embodiment is completed in Matlab R2010a software.

A high-speed train rolling bearing fault diagnosis method, comprising the following steps:

I. Establishment of fault diagnosis model

Step 1: Divide the bearings into four state types: normal bearings, rolling element fault bearings, outer ring fault bearings, and inner ring fault bearings.

The specific method of this embodiment is as follows: First, the rolling element faults, outer ring faults and inner ring faults are arranged on the rolling bearing by using electric discharge machining technology. Under the working conditions, the acceleration sensor is used to collect the vibration signal, which is collected by the 16-channel DAT recorder. A total of 40 sets of data are used, that is, normal bearings, bearings with rolling element faults, bearings with outer ring faults, and bearings with inner ring faults, each with 10 sets. Taking the data of rolling element faulty bearings as an example, the number of sampling points is 12,000. The original signal waveform is shown in Figure 3 Show.

Step 2: EEMD fault feature vector construction

1) Utilize the EEMD method to decompose each group of original vibration signals, select the first a IMF components obtained by the decomposition, a=8 in this embodiment, as shown in Figure 4, and obtain the energy E _i of each component respectively;

In the formula, C _i (t) is the i-th IMF component, i=1, 2, 3, L, a, C _i is the amplitude of discrete points, n is the number of sampling points,

2) seek the energy summation E of each IMF component of each group of original vibration signals;

3) Due to the large difference in the vibration amplitude of the bearings in different states, the difference in the value of each IMF component is also large, so the energy is normalized, that is, the energy of each IMF component is compared with the total energy, and a The energy eigenvector T of group original vibration signal, as shown in Figure 5;

T=[E ₁ /E, E ₂ /E, L, E _a /E]

Repeat steps 1) to 4) of step 2 to perform EEMD on each set of data to obtain 40 sets of energy eigenvectors corresponding to bearings in different states.

Step 3: RBF Neural Network Modeling

1) Determine the RBF neural network structure

Although increasing the number of neurons in the hidden layer can improve the nonlinear mapping ability of the RBF neural network, too many neurons will reduce the network prediction performance, so a three-layer RBF neural network with a single hidden layer is used.

2) Determine the number of nodes in the input layer

The energy feature vector T is used as the input of the RBF neural network, so the number of nodes in the input layer M=8,

3) Determine the number of nodes in the output layer

The ideal output result should be able to directly see the classification of the fault, so 3 binary codes are used, that is, the number of nodes in the output layer is 3, see Table 1,

Table 1 Bearing fault output form

4) Determine the number of nodes of the hidden layer, determine the training target precision of the RBF neural network and the distribution density SPREAD of the radial basis function,

This embodiment adopts the method of combining method experience and experiment, tries different numbers of nodes in 4:24 one by one, selects the number of nodes with the best performance 22 as the number of hidden nodes of the model, and at the same time, the training of RBF neural network The target is set to 10 ^-5 , the distribution density SPREAD of the radial basis function takes the default value of 1,

5) Select group b from the original vibration signals of each bearing state as training samples, and the rest as test samples, and use the training samples as the input of the RBF neural network for training. After the training reaches the target accuracy set in step 4), a preliminary RBF neural network diagnosis model, and then use the test sample as the input of the RBF neural network diagnosis preliminary model to identify the state of the test sample bearing,

In this embodiment, 6 groups are selected as training samples from each bearing state, and 4 groups are used as test samples, so there are 24 groups of training samples in four bearing states, and the rest are test samples. These 24 groups of training samples are used as RBF neural network The training process of the RBF neural network is shown in Figure 6. It only needs to go through 18 steps of training to reach the set accuracy, and obtain the preliminary RBF neural network diagnosis model, and then use the remaining 16 test samples as the RBF neural network diagnosis model. input to the preliminary model, to identify the state of the test sample bearings,

6) Performance evaluation of the preliminary model

Firstly, the recognition error is calculated according to the recognition result of the test sample. When the recognition error is within the acceptable range, the recognition result is considered correct, otherwise the recognition result is considered wrong; then the fault recognition rate Q is calculated, Q=correct recognition number/total experimental bearing number, The performance of the RBF neural network is measured by the fault recognition rate Q. If the fault recognition rate Q reaches the ideal standard, the final model of the RBF neural network diagnosis is obtained and used for the bearing fault diagnosis of the unknown state in step II; otherwise, return to step 5), Reselect training samples and test samples for training and testing,

In this embodiment, 1, 2, 3 and 4 are used to represent normal bearings, rolling element fault bearings, outer ring bearings and inner ring bearings respectively, and the recognition error ε is calculated after the RBF neural network outputs the results. This embodiment stipulates that if |ε|≤0.5 , indicating that the recognition result is correct, otherwise the recognition result is wrong. The diagnosis results of EEMD combined with RBF neural network are shown in Table 2. It can be seen that the recognition output result of the preliminary model of RBF neural network diagnosis is very close to the output result of the target test, so it can be known that the bearing fault diagnosis model Can accurately identify the type of bearing failure,

Further, statistics are made on each test bearing type and the fault recognition rate of the model is calculated. After calculation, the correct recognition rate of normal bearings, rolling element faults, outer ring faults, and inner ring faults is 100%, so the total recognition rate is 100%, so it can be clearly explained that EEMD combined with RBF neural network model is used for fault diagnosis of rolling bearings, and the effect of model classification and recognition is very ideal, that is, it can be used as the final model of RBF neural network diagnosis,

II. Diagnosis of bearing fault types

The original vibration signal of the bearing to be diagnosed in unknown state is collected and used as the input of the final model of RBF neural network diagnosis. The output of the final model is compared with the output form of the bearing fault to determine the state of the bearing. After comparison, this implementation can correctly diagnose the type of bearing fault.

Table 2 EEMD combined with RBF neural network fault diagnosis

In order to verify that the method of the present invention is better than other methods, two methods of EMD and EEMD are used to extract the eigenvectors of bearing data in different states respectively, and then four kinds of EMD and BP, EMD and RBF, EEMD and BP and EEMD and RBF are respectively established. The bearing fault diagnosis model uses the extracted feature vectors to classify and identify bearing faults. The simulation results are shown in Table 3. It can be seen from the table that the EEMD method has advantages over EMD in extracting fault feature vectors; BP neural network is not suitable for fast modeling of large amounts of data, and the training convergence speed is slow. The RBF neural network model has higher precision and is more suitable for pattern recognition of bearing faults than BP neural network. Therefore, EEMD and RBF methods have their unique advantages in the fault diagnosis of train rolling bearings.

Table 3 Comparison of results of four bearing fault diagnosis models

in conclusion

1) In view of the problems of modal aliasing in EMD method in bearing fault diagnosis, the EEMD method is proposed, which can more accurately obtain the IMF energy of the bearing vibration signal as the eigenvector of the bearing in different states.

2) Simulation results show that the present invention is a nonlinear approximation network with good performance, which can identify bearing fault types very accurately. In the network training process, under the condition of the same expected error sum of squares and equal input nodes and output nodes, the convergence speed of the present invention is obviously higher than that of the BP neural network, which not only shortens the learning time of samples and reduces the complexity, but also is not easy to A local minimum occurs.

3) Through comparative analysis, it can be known that it is feasible to use the present invention to carry out fault diagnosis on train rolling bearings, and the present invention is more efficient and more accurate than BP neural network diagnosis, and is more suitable for fault diagnosis.

Therefore, the method proposed in this paper can not only be used for the fault diagnosis of rolling bearings, but also can be completely applied to the fault diagnosis of gearboxes, large rotating equipment, etc., and has a very wide application prospect.

The full name of EEMD is Ensemble Empirical Mode Decomposition (ensemble empirical mode decomposition), which is an improved algorithm of EMD (empirical mode decomposition). However, this method has the defect of modal mixing, and the EEMD method can overcome the above defects, and effectively solve the mixing phenomenon of EMD. The specific decomposition steps of the EEMD method are as follows:

(1) Add white noise a _m (t) with a uniform spectrum distribution to the signal R(t) to be decomposed to obtain the signal S(t);

(2) EMD is performed on the signal S(t), and the decomposition process is as follows;

1) Determine all local extreme points on the signal S(t), and the upper and lower envelopes are obtained by connecting all local maximum points and local minimum points with cubic spline curves , namely S(t) _max and S(t) _min ;

2) Calculate the average value of the upper and lower envelopes at each moment, namely

3) get a new signal

Y ₁ (t)=S(t)-u(t)

Judging whether it is symmetrical to the local zero mean, and has the same extreme point and zero crossing point, if yes, record it as C ₁ (t), which is the first IMF component, otherwise repeat steps 1) and 2);

4) Separate C ₁ (t) from S(t) to obtain a difference signal V ₁ (t),

V ₁ (t)=S(t)-C ₁ (t)

5) Take V ₁ (t) as the original data, repeat steps 1) to 3) to get IMF2, and repeat n times to get n IMF components, so we have

When V _n (t) meets the given termination condition, the termination condition is that V _n (t) is a monotone function, and the loop ends,

From Y ₁ (t) and V ₁ (t) can get

That is to say, the original signal is expressed as the sum of the intrinsic mode function components and a residual function V _n (t), and each component C ₁ (t), C ₂ (t), ... C _n (t) respectively covers the original signal Information in different frequency bands from high to low, and changes with the change of the signal itself,

(3) Repeat process (1) and (2) after adding different white noise each time;

(4) Taking the integrated mean C _i of each IMF component after multiple decompositions as the final result,

The Radial Basis Function (RBF) neural network is a three-layer feed-forward neural network consisting of an input layer, a hidden layer and an output layer, as shown in Figure 2. The mapping from the input layer to the output layer is nonlinear, while the mapping from the hidden layer to the output layer is linear. The essence of RBF neural network is to make linearly indistinguishable problems in low-dimensional space linearly distinguishable in high-dimensional space. The variance of the basis function, the center of the basis function and the weight from the hidden layer to the output layer are the parameters that the RBF neural network algorithm needs to solve. The most commonly used radial basis function in the RBF neural network is the Gaussian function, so the activation function of the RBF neural network It can be expressed as:

In the formula, X _m is the input variable; σ is the variance of the Gaussian function; ||X _{m -ci} || is the Euclidean norm; _ci is the center of the Gaussian function.

The network output obtained by the structure of the RBF neural network shown in Figure 2 is:

In the formula, ω _ij is the connection weight from the hidden layer to the output layer; i=1, 2, L, N is the number of nodes in the hidden layer; y _j is the output network of the jth node corresponding to the input network.

Claims (2)

1. a high-speed train rolling bearing fault diagnosis method, is characterized in that, comprises the following steps: I. Establishment of fault diagnosis model Step 1: Divide the bearings into four state types: normal bearings, rolling element fault bearings, outer ring fault bearings and inner ring fault bearings. For the above four state types of bearings, collect several sets of original vibration signals for each type; Step 2: EEMD fault feature vector construction 1) Use EEMD method to decompose each group of original vibration signals, select the first a IMF components obtained from the decomposition, and calculate the energy E _i of each component respectively; ${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{<span\; class="notranslate">\; \&Integral;</span>}_{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; \&infin;</span>}}^{<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&infin;</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}\mathrm{<span\; class="notranslate">\; d</span>}\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$ In the formula, C _i (t) is the i-th IMF component, i=1, 2, 3, L, a, C _i is the amplitude of discrete points, n is the number of sampling points, 2) seek the energy summation E of each IMF component of each group of original vibration signals; $\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; a</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}$ 3) Due to the large difference in the vibration amplitude of the bearings in different states, the difference in the value of each IMF component is also large, so the energy is normalized, that is, the energy of each IMF component is compared with the total energy, and a The energy feature vector T of the original vibration signal of the group; T=[E ₁ /E, E ₂ /E, L, E _a /E] Step 3: RBF Neural Network Modeling 1) Determine the RBF neural network structure Although increasing the number of neurons in the hidden layer can improve the nonlinear mapping ability of the RBF neural network, too many neurons will reduce the network prediction performance, so a three-layer RBF neural network with a single hidden layer is used. 2) Determine the number of nodes in the input layer The energy feature vector T is used as the input of the RBF neural network, so the number of nodes in the input layer M=a, 3) Determine the number of nodes in the output layer The ideal output result should be able to directly see the classification of the fault, so 3 binary codes are used, that is, the number of nodes in the output layer is 3, see Table 1, Table 1 Bearing fault output form 4) Determine the number of nodes of the hidden layer, determine the training target precision of the RBF neural network and the distribution density SPREAD of the radial basis function, 5) Select group b from the original vibration signals of each bearing state as training samples, and the rest as test samples, and use the training samples as the input of the RBF neural network for training. After the training reaches the target accuracy set in step 4), a preliminary RBF neural network diagnosis model, and then use the test sample as the input of the RBF neural network diagnosis preliminary model to identify the state of the test sample bearing, 6) Performance evaluation of the preliminary model Firstly, the recognition error is calculated according to the recognition result of the test sample. When the recognition error is within the acceptable range, the recognition result is considered correct, otherwise the recognition result is considered wrong; then the fault recognition rate Q is calculated, Q=correct recognition number/total experimental bearing number, The performance of the RBF neural network is measured by the fault recognition rate Q. If the fault recognition rate Q reaches the ideal standard, the final model of the RBF neural network diagnosis is obtained and used for the bearing fault diagnosis of the unknown state in step II; otherwise, return to step 5), Reselect training samples and test samples for training and testing, II. Diagnosis of bearing fault types The original vibration signal of the bearing to be diagnosed in an unknown state is collected and used as the input of the final model of RBF neural network diagnosis, and the state of the bearing is determined by comparing the output of the final model with the output form of the bearing fault. 2. high-speed train rolling bearing fault diagnosis method as claimed in claim 1, is characterized in that, described EEMD method concrete decomposition step is as follows: (1) Add white noise a _m (t) with a uniform spectrum distribution to the signal R(t) to be decomposed to obtain the signal S(t); (2) EMD is performed on the signal S(t), and the decomposition process is as follows; 1) Determine all local extreme points on the signal S(t), and the upper and lower envelopes are obtained by connecting all local maximum points and local minimum points with cubic spline curves , namely S(t) _max and S(t) _min ; 2) Calculate the average value of the upper and lower envelopes at each moment, namely $\mathrm{<span\; class="notranslate">\; u</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; S</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; a</span>}\mathrm{<span\; class="notranslate">\; x</span>}}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; S</span>}{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>}_{\mathrm{<span\; class="notranslate">\; m</span>}\mathrm{<span\; class="notranslate">\; i</span>}\mathrm{<span\; class="notranslate">\; no</span>}}}{\mathrm{<span\; class="notranslate">\; 2</span>}}$ 3) get a new signal Y ₁ (t)=S(t)-u(t) Judging whether it is symmetrical to the local zero mean, and has the same extreme point and zero crossing point, if yes, record it as C ₁ (t), which is the first IMF component, otherwise repeat steps 1) and 2); 4) Separate C ₁ (t) from S(t) to obtain a difference signal V ₁ (t), V ₁ (t)=S(t)-C ₁ (t) 5) Take V ₁ (t) as the original data, repeat steps 1) to 3) to get IMF2, and repeat n times to get n IMF components, so we have $\left.\begin{array}{c}{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ <span\; class="notranslate">\; .</span>\\ {\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>\end{array}\right\}$ When V _n (t) meets the given termination condition, the termination condition is that V _n (t) is a monotone function, and the loop ends, From Y ₁ (t) and V ₁ (t) can get $\mathrm{<span\; class="notranslate">\; S</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\underset{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; no</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; V</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; t</span>}<span\; class="notranslate">\; )</span>$ That is to say, the original signal is expressed as the sum of the intrinsic mode function components and a residual function V _n (t), and each component C ₁ (t), C ₂ (t), ... C _n (t) covers the original signal from Information in different frequency bands from high to low, and changes with the change of the signal itself, (3) Repeat process (1) and (2) after adding different white noise each time; (4) Taking the integrated mean C _i of each IMF component after multiple decompositions as the final result, ${\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; 1</span>}}{\mathrm{<span\; class="notranslate">\; m</span>}}\underset{\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{\overset{\mathrm{<span\; class="notranslate">\; m</span>}}{<span\; class="notranslate">\; \&Sigma;</span>}}{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; i</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; L</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; ;</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; L</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; .</span>$