# Research on CEEMD-AGA Denoising Method and Its Application in Feed Mixer

Authors: Dong Yang; Yu Sun; Kai Wu
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Rotating shaft is the key part of rotating machinery, which directly affects the performance of the whole machine. Field test is an easy and quick way to obtain the load data in engineering practice. However, because of various reasons, the load data are often mixed with many noise components. Based on the autocorrelation function, the CEEMD (complementary ensemble empirical mode decomposition) denoising method is proposed in this paper. The AGA (adaptive genetic algorithm) is adopted to solve parameter optimization problems in CEEMD. A new similarity function is proposed as the fitness function. Lastly, the proposed denoising method is applied to a feed mixer’s load which is obtained by field test. The result shows that the CEEMD-AGA method has good robustness, noise components of small stress amplitude and large stress mean are removed, and there is a high correlation between the original data and the reconstructed data, which demonstrate that the CEEMD-AGA method can reduce the influence of noise components effectively.


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## Body

## 1. Introduction

Rotating shaft is the key part of rotating machinery, which directly affects the performance of the whole machine. For a high-speed and heavy-duty machine, the shaft endures complex and alternate load and often suffers from fatigue damage. In order to calculate fatigue lifetime in engineering practice, both the deterministic load and the random load should be considered [1]. Compared to the numerical simulation method, field test is an easy and quick way to obtain the load data for many machine equipment in engineering practice and also has a higher accuracy. However, because of the test environment, instruments, human factors, and other reasons, the load data obtained by the field test are often mixed with many kinds of noise components. The testing data are nonstationary and deviated from the real value. These components lead to many false load cycles in the rain-flow counting process [2], which have a great influence on the accuracy of load spectrum compiling and fatigue lifetime prediction. Furthermore, noise can impact the iteration times and accuracy of driving data generated by servo drivers during the bench test [3].In order to obtain more accurate data, the most commonly used denoising processing methods are the Fourier transformation method [4] and the wavelet transformation method [5, 6]. However, the Fourier method, which maps the data from a time domain into a frequency domain, can only give the physically meaningful interpretation of a stationary data’s frequency composition on condition that the noise spectrum is different from the data spectrum. In addition, as for wavelet transformation, wavelet bases, threshold functions, and decomposed scale are often determined by experience, and different parameter values have a great influence on the denoising result. The white noise can be suppressed effectively by the wavelet transform, but the pulse signal cannot [7]. In order to solve the problem of no self-adaptability of the conventional methods, Huang et al. [8] proposed the empirical mode decomposition (EMD) method, which is based on the local characteristic time’s scales of data and can decompose the complex original data into a series of components named intrinsic mode functions (IMFs). The IMFs represent the natural oscillatory mode embedded in the data and work as the basis functions, which are determined by the original data itself. Therefore, the EMD method can be used to process nonlinear and nonstationary data for its self-adaptive. Since proposed, EMD has received a lot of attention in many fields, such as geophysics [9, 10], seismology [11–13], and engineering [14]. Chen et al. [15, 16] proposed a method for removing noise from multiple reflections by using the EMD framework. The result shows that the EMD method can obtain a good performance. However, the EMD has poor robustness and especially the modal aliasing problem which greatly limits its application [17–19]. In order to solve this problem, Wu and Huang [20, 21] presented ensemble empirical mode decomposition (EEMD), which is a noise-assisted data analysis method by adding finite white noise to the investigated data. Although the EEMD method is more effective for signal processing, it still takes a long time to calculate the large ensemble mean. For this reason, Yeh et al. [22] proposed a new novel noise enhanced method as the complementary EEMD (CEEMD). In this method, both positive and negative white noise series are adopted to enhance the performance of EEMD, and the residue of added white noises can completely be removed. The CEEMD has the advantages of EEMD and can also avoid the problem of white noise residual, reduce the computation times, and improve the efficiency.Since proposed, the CEEMD method has received widespread attention from many scholars and has been applied to many fields of practical engineering, for example, energy field [23] and fault diagnosis [24]. It is proved that the CEEMD has the same performance as the EEMD, but the computing time can be greatly reduced [25]. The CEEMD can successfully decompose the original signal into multiple components of different frequencies. The high-frequency components can be considered as interfering noise and should be removed. In order to determine the noise component and obtain the reconstruction signal, Chen et al. [26] proposed a new improved complete ensemble empirical mode decomposition (ICEEMD) algorithm by removing the first decomposed component of each frequency slice, which can solve the problems of spurious artifacts and strong residual noise. Li et al. [27] used the wavelet neural network to reconstruct the IMFs and residuals to obtain the reconstruction signal. However, the value of parameters depends on human judgment and has certain subjectivity. Karatoprak and Seker [28] proposed that an IMF could be considered as high frequency if twice of its frequency is higher than the highest add to the lowest frequency. However, in this method, some IMFs of intermediate frequencies may be removed, which contain real signal components. As a result, useful component may be removed; the reconstruction signal after denoising may have a big difference between the original signal. In view of these problems mentioned above, we proposed a CEEMD denoising method based on the autocorrelation function. According to the characteristic of the autocorrelation function, periodic data’s autocorrelation function is also periodic, and its period is the same as the original data [29]. The autocorrelation function can highlight the periodicity of original data and each IMF component and reflect the differences between noise components and real data components. Therefore, it is feasible to choose IMFs based on the correlation of the autocorrelation functions of original data and each IMF. Moreover, unlike the EMD method, the CEEMD has two additional parameters: the number of ensemble trials M and the amplitude of the added white noise ε. The decomposition result is very sensitive to the choice of parameters. However, there is no general method to set the parameters; the setting of the two parameters mainly depends on the user [30, 31]. In view of these problems, in this paper, we employ an adaptive genetic algorithm to solve parameter optimization problems. Moreover, we combine data to the noise ratio function and the correlation coefficient function and propose a new similarity function to evaluate the optimization results, which brings adaptability to the parameter determination.The rest of this paper is organized as follows. In Section2, the CEEMD denoising method is present. Firstly, the CEEMD algorithm is introduced briefly. Then the denoising method based on the autocorrelation function is proposed. In Section 3, the AGA method is discussed, and the similarity function s is also present. In Section 4, the denoising method proposed in this paper is applied to the load data of a feed mixer.


## 2. CEEMD Denoising Theory

### 2.1. CEEMD Algorithm

The EMD method can successfully decompose the investigated data into a set of IMFs and aRes. The processing steps of EMD can be seen in [32, 33]. On the basis of EMD and EEMD, CEEMD adds the auxiliary noise in the form of positive and negative pairs to the original data, which can not only offset the noise effect of the processed data but also reduce the number of iterations correspondingly and further improve the computing ability. For an original data x(t), the processing steps of CEEMD are as follows:(1)
Generate two reverse white noise series:ωi(t) and −ωi(t), and add them to the original data x(t), then(1)N1tN2t=111−1xtωt.(2)
DecomposeN1(t) and N2(t) by EMD, respectively, and IMFij is the ensemble mean of the corresponding IMF of the data series Ni(t):(2)IMFj=IMF1j+IMF2j2.(3)
Calculate the average of the corresponding IMFij as the result of CEEMD.


### 2.2. CEEMD Denoising Method

In this paper, we consider that, as for the investigated data, the high-frequency IMFs obtained by the CEEMD method can be considered as the noise components, which could be removed. The reconstructed data after denoising processing can be obtained by(3)x′t=∑n=kNIMFi+Res,where N is the number of all IMFs obtained by the CEEMD method, Res is the residual term, kth is the first IMF which is identified as the real component.When using CEEMD for data denoising processing, a key problem is to identify the noise components and real data components. If improper components are saved, it will lead to a poor denoising performance. If useful IMF components are removed, it will lead to a loss of useful information.According to the characteristic of the autocorrelation function, periodic data’s autocorrelation function is also periodic, and its period is the same as the original data. Besides, the autocorrelation function can highlight the periodicity of the primary function. The autocorrelation functions can be obtained by(4)Rnm=1N∑i=1N−1xixi+m,where Rn(m) is the autocorrelation function of IMFn. The correlation coefficients can be calculated by(5)ρj=∑i=1N−1RIMFjiRxi∑i=1N−1RIMFj2i∑i=1N−1Rx2i,where ρj is the correlation coefficient of the original data and the jth IMF. N is the length of data. In this paper, we think that, if ρj ≥ 0.5, it can be considered that the degree of similarity is high, and IMFj is a real data component, which should be preserved and used to reconstruct the data. Otherwise, it is noise component, which should be removed.Because of the periodicity of the load data, the autocorrelation function can highlight the periodicity of original data and each IMF component and reflect the differences between noise components and real data components. Therefore, it is feasible and effective to choose IMFs based on the correlation of the autocorrelation functions of original data and each IMF. The CEEMD denoising process based on the autocorrelation function can be described as follows:(1)
Calculate the autocorrelation functions of the investigated data and each IMF, respectively.(2)
Normalize the autocorrelation functions according to equation (4).(3)
Calculate the correlation coefficients according to equation (5).(4)
Distinguish real data components and noise components.(5)
Remove the noise components and reconstruct the original data according to equation (3).


## 2.1. CEEMD Algorithm

The EMD method can successfully decompose the investigated data into a set of IMFs and aRes. The processing steps of EMD can be seen in [32, 33]. On the basis of EMD and EEMD, CEEMD adds the auxiliary noise in the form of positive and negative pairs to the original data, which can not only offset the noise effect of the processed data but also reduce the number of iterations correspondingly and further improve the computing ability. For an original data x(t), the processing steps of CEEMD are as follows:(1)
Generate two reverse white noise series:ωi(t) and −ωi(t), and add them to the original data x(t), then(1)N1tN2t=111−1xtωt.(2)
DecomposeN1(t) and N2(t) by EMD, respectively, and IMFij is the ensemble mean of the corresponding IMF of the data series Ni(t):(2)IMFj=IMF1j+IMF2j2.(3)
Calculate the average of the corresponding IMFij as the result of CEEMD.


## 2.2. CEEMD Denoising Method

In this paper, we consider that, as for the investigated data, the high-frequency IMFs obtained by the CEEMD method can be considered as the noise components, which could be removed. The reconstructed data after denoising processing can be obtained by(3)x′t=∑n=kNIMFi+Res,where N is the number of all IMFs obtained by the CEEMD method, Res is the residual term, kth is the first IMF which is identified as the real component.When using CEEMD for data denoising processing, a key problem is to identify the noise components and real data components. If improper components are saved, it will lead to a poor denoising performance. If useful IMF components are removed, it will lead to a loss of useful information.According to the characteristic of the autocorrelation function, periodic data’s autocorrelation function is also periodic, and its period is the same as the original data. Besides, the autocorrelation function can highlight the periodicity of the primary function. The autocorrelation functions can be obtained by(4)Rnm=1N∑i=1N−1xixi+m,where Rn(m) is the autocorrelation function of IMFn. The correlation coefficients can be calculated by(5)ρj=∑i=1N−1RIMFjiRxi∑i=1N−1RIMFj2i∑i=1N−1Rx2i,where ρj is the correlation coefficient of the original data and the jth IMF. N is the length of data. In this paper, we think that, if ρj ≥ 0.5, it can be considered that the degree of similarity is high, and IMFj is a real data component, which should be preserved and used to reconstruct the data. Otherwise, it is noise component, which should be removed.Because of the periodicity of the load data, the autocorrelation function can highlight the periodicity of original data and each IMF component and reflect the differences between noise components and real data components. Therefore, it is feasible and effective to choose IMFs based on the correlation of the autocorrelation functions of original data and each IMF. The CEEMD denoising process based on the autocorrelation function can be described as follows:(1)
Calculate the autocorrelation functions of the investigated data and each IMF, respectively.(2)
Normalize the autocorrelation functions according to equation (4).(3)
Calculate the correlation coefficients according to equation (5).(4)
Distinguish real data components and noise components.(5)
Remove the noise components and reconstruct the original data according to equation (3).


## 3. Parameter Optimizing in CEEMD by AGA

Unlike the EMD method, there are two additional parameters in the CEEMD method: the number of ensemble trialsM and the amplitude of the added white noise ε. The standard deviation of the error εn (the difference between the corresponding IMF and the input data) has a relationship with M and ε [21], which is shown in the following equation:(6)εn=εM.It is evident that the added noise data with small amplitude will lead to a small error. However, if the amplitude is too small, it is not enough to cause the change of extreme point in the original data. It is true especially when the data set has a large gradient. In order to obtain a better denoising result,ε should not be too small. But by only increasing M, the effect of the added white noise may be reduced to a small level and can be neglected. Research studies [34] show that increasing noise amplitudes and ensemble numbers alters the decomposition little as long as the added noise has moderate amplitude and the ensemble has a large enough number of trials.However, there is no universal method for determining the proper amplitude until now. In order to determine the appropriate value ofM and ε, a genetic algorithm is employed to optimize the denoising process. Because the traditional GA has the problem of premature convergence and slow evolution [35, 36], the adaptive genetic algorithm (AGA) is adopted, in which the crossover rate and the mutation rate can be changed with the colony fitness degree of each generation automatically [37, 38]. When the individual fitness converges to the local optimum, the crossover rate and the mutation rate can be increased. Otherwise, when the individual fitness scattered, the crossover rate and the mutation rate should be decreased. Meanwhile, the individuals above the average fitness value should be protected to enter the next generation. And the individuals below the average fitness value can be eliminated. The crossover rate Pc and the mutation rate Pm can be obtained by equations (7) and (8), respectively.(7)Pc=P1−P1−P2f−favgfmax−favg,f≥favg,P1,f<favg,(8)Pm=P3−P3−P4fmax−ffmax−favg,f′≥favg,P3,f′<favg,where fmax is the maximum fitness value in the current generation, favg is the average fitness value, f is the larger fitness value of the two crossover-individuals, f’ is the fitness value of the mutation individual, and P1, P2, P3, and P4 are constants.In order to evaluate the denoising result, signal-to-noise ratio (SNR) and correlation coefficient (ρ) are the most common used methods, as shown in equations (9) and (10), respectively.(9)SNR=10∗logPsPn,where Ps is the effective power of the investigated data and Pn is the effective power of the noise component. The unit of SNR is dB. It is believed that a higher SNR indicated a better denoising result and a lesser residual noise in the reconstructed data.(10)ρ=covxt,x′tσxt⋅σx′t,where x(t) are the original data and x′(t) are the reconstructed data obtained by CEEMD denoising processing. The value of ρ shows the correlation coefficient of the original data and the reconstructed data. The closer ρ to 1, the better the denoising result.However, neitherSNR nor ρ can only reflect the partial characteristic of the reconstructed data and the original data, which is incomplete. In this paper, a comprehensive function is proposed to evaluate the CEEMD denoising result, and we call it as the similarity function, denoted as ξ, which combined SNR and ρ together, as shown in equation (11). And the plot of the function is shown in Figure 1. Because of the different dimensions, the arc-tangent function is employed for normalized process.(11)ξ=ρ∗2πarctanSNR,where ξ is the similarity of the reconstructed data and the original data, ξ∈0,1. It is obvious that the higher the similarity value, the better the denoising result. Furthermore, the similarity function is used as the fitness function in AGA optimization. The processing steps of CEEMD parameter optimization by AGA are shown in Figure 2.Figure 1
Relationship of similarity functionξ, SNR, and ρ.Figure 2
Flowchart of the CEEMD-AGA denoising algorithm.


## 4. Application of the CEEMD-AGA Denoising Method

### 4.1. The Load Test of the Feed Mixer

Feed mixer is one of the four main machines of the feed processing machinery, which largely determines the production of the feed industry. With the development of the automation and the intelligent level of pellet feed production line, the reliability level of the mixer becomes more and more important. As one of the most important components, the performance of the shaft largely determines the reliability level of the whole machine.As for the feed mixer, there are several kinds of powder materials in the motion space, which are discontinuous and not full, and the shaft undergoes spatial large overall motion. All these problems bring difficulties to finite element analysis and numerical calculation. In this case, load test is an executable and convenient way to obtain the load data.In order to obtain the load data, field testing was conducted for a type of feed mixer (Jiangsu Muyang Co. Ltd., Yangzhou, China). The testing was performed by the wireless measurement system (BeeData Technology Co. Ltd., Beijing, China). Experimental equipment includes strain gauges, wireless strain node (SG401, BeeData Technology Co. Ltd., Beijing, China), wireless gateway (BS901, BeeData Technology Co. Ltd., Beijing, China), and BeeData analysis software. Due to the limited space, a bracket was designed to place the wireless node. The test site is shown in Figure3. The load data in a working cycle are shown in Figure 4. As shown in Figure 4, due to the test environment, instrument, human factors, and other reasons, the load data are mixed with noise components, and the load data are nonstationary. To obtain the true value and reduce the impact of noise, it is necessary to denoise process the data obtained by test.Figure 3
The load test site of the feed mixer.Figure 4
The original load data obtained by test.


### 4.2. CEEMD-AGA Denoising of the Feed Mixer

To research the denoising of feed mixer load data, the CEEMD-AGA method is encoded in MTALAB 2016a. In the CEEMD-AGA algorithm, the population size is 100, the terminal iteration generation is 200,P1 = 0.9, P2 = 0.3, P3 = 0.1, and P4 = 0.001. By using the method proposed in Section 3, we obtain the optimal parameters: M = 93, ε = 0.0493, SNR = 39.0656 dB, ϕ = 0.9864, and similarity function ξ = 0.9702. The evolution performance of the optimization algorithm is shown in Figure 5. Furthermore, thirteen IMFs and one Res are obtained after the ensemble empirical mode decomposition of the original data. The CEEMD result is shown in Figure 6.Figure 5
Evolution performance of the AGA optimization algorithm.Figure 6
Results of CEEMD-AGA for load data.According to the CEEMD-AGA result, IMF1–IMF3 are noise components and should be removed, and the rest of IMFs are real data components which could be used to reconstruct the data. Comparison of the original data and the reconstructed data is shown in Figure 6. The figure shows that a large number of high-frequency and low-amplitude load cycles are removed. Figure 7 also reveals that the reconstructed data are periodic, and their period is equal to the rotation period of the shaft, which shows the reconstructed data have a high SNR and a high correlation coefficient with the original data. The data after denoising processing have a high similarity with the original data. As a word, the denoising method proposed above is feasible and effective for the load data of the feed mixer and achieves a good result.Figure 7
Comparison of the original data and the reconstructed data.


### 4.3. Load Spectrum Compiling

In order to study the effectiveness of the CEEMD-AGA method, the influence of the method on fatigue life prediction is conducted. In general, the cumulative damage theory is a common method to evaluate fatigue life. In this theory, to predict the fatigue life, a load spectrum is necessary to be established first. The key to load spectrum compiling is using statistical theories to transform the load-time data into a series of full-cycle or half-cycle sequences. The rain-flow counting method is a two-parameter statistical method which can transform the load-time series to a distribution matrix of stress mean and amplitude by calculating the peak and valley values in load-time series [39]. The rain-flow counting method is proposed by Matsuiski and Endo in 1950s, and it can take full cycle and half cycle in to consideration at the same time and use these to describe the stress effect of cyclic loading [40]. Since proposed, the rain-flow counting method has gained more and more attention in load spectrum compilation [41, 42] and fatigue reliability assessment [43, 44]. Many new counting models are proposed based on the rain-flow method, such as, four-peak valley rain counting method, three metamorphoses counting method [45]. The basic principle and implementation of the rain-flow counting method are not introduced in this paper; it can be seen in [45]. The rain-flow counting method is adopted to count the number of load cycles for the reconstructed data and the original data, respectively. The rain-flow counting results are shown in Figure 8(a).Figure 8
Comparison of rain-flow counting results. (a) Rain-flow counting result of the reconstructed data after CEEMD-AGA denoising. (b) Rain-flow counting result of the original data. (c) Rain-flow counting result of the reconstructed data after wavelet denoising.
(a)
(b)
(c)Furthermore, the traditional wavelet denoising is applied to the investigated data, and the rain-flow counting result of reconstructed data is shown in Figure8(c) and Table 1. In addition, to study the sensitivity of the CEEMD-AGA denoising method, different levels of white noise (SNR = 5, 10, and 15) are added to the original signal. Then, the method is applied to the corresponding series. In order to ensure the accuracy, each level is computed twice. The rain-flow counting result is compared with the result of the original signal, as shown in Table 2.Table 1
Result of different processing methods.
Load data
Number of load cycles
Similarity
The original data
2442
1
CEEMD-AGA method
1267
0.9703
Wavelet method
883
0.8271Table 2
Result of sensitive analysis of different input noise levels.
SNR
Number of load cycles
Error (%)
Correlation coefficient
5
1236
2.44
0.9965
1292
1.97
0.9974
10
1306
3.08
0.9921
1317
3.95
0.9895
15
1331
5.05
0.9860
1352
6.71
0.9823From Figure7 and Table 1, it is illustrated that the CEEMD-AGA method can remove many noise components of small stress amplitude and large stress mean, preserve more real load components, and greatly reduce the number of load cycles. The CEEMD-AGA method can effectively eliminate the influence of noise and improve the similarity.As shown in Table2, compared with the reconstructed signal shown in Figure 6, with the increasing of SNR, the error of cyclic load numbers increases, and the correlation coefficient between the corresponding reconstruction data decreases. But they all change very little. Therefore, it is illustrated that the CEEMD-AGA denoising method has a poor sensitivity to the level of input noise level, and it has good robustness.


## 4.1. The Load Test of the Feed Mixer

Feed mixer is one of the four main machines of the feed processing machinery, which largely determines the production of the feed industry. With the development of the automation and the intelligent level of pellet feed production line, the reliability level of the mixer becomes more and more important. As one of the most important components, the performance of the shaft largely determines the reliability level of the whole machine.As for the feed mixer, there are several kinds of powder materials in the motion space, which are discontinuous and not full, and the shaft undergoes spatial large overall motion. All these problems bring difficulties to finite element analysis and numerical calculation. In this case, load test is an executable and convenient way to obtain the load data.In order to obtain the load data, field testing was conducted for a type of feed mixer (Jiangsu Muyang Co. Ltd., Yangzhou, China). The testing was performed by the wireless measurement system (BeeData Technology Co. Ltd., Beijing, China). Experimental equipment includes strain gauges, wireless strain node (SG401, BeeData Technology Co. Ltd., Beijing, China), wireless gateway (BS901, BeeData Technology Co. Ltd., Beijing, China), and BeeData analysis software. Due to the limited space, a bracket was designed to place the wireless node. The test site is shown in Figure3. The load data in a working cycle are shown in Figure 4. As shown in Figure 4, due to the test environment, instrument, human factors, and other reasons, the load data are mixed with noise components, and the load data are nonstationary. To obtain the true value and reduce the impact of noise, it is necessary to denoise process the data obtained by test.Figure 3
The load test site of the feed mixer.Figure 4
The original load data obtained by test.


## 4.2. CEEMD-AGA Denoising of the Feed Mixer

To research the denoising of feed mixer load data, the CEEMD-AGA method is encoded in MTALAB 2016a. In the CEEMD-AGA algorithm, the population size is 100, the terminal iteration generation is 200,P1 = 0.9, P2 = 0.3, P3 = 0.1, and P4 = 0.001. By using the method proposed in Section 3, we obtain the optimal parameters: M = 93, ε = 0.0493, SNR = 39.0656 dB, ϕ = 0.9864, and similarity function ξ = 0.9702. The evolution performance of the optimization algorithm is shown in Figure 5. Furthermore, thirteen IMFs and one Res are obtained after the ensemble empirical mode decomposition of the original data. The CEEMD result is shown in Figure 6.Figure 5
Evolution performance of the AGA optimization algorithm.Figure 6
Results of CEEMD-AGA for load data.According to the CEEMD-AGA result, IMF1–IMF3 are noise components and should be removed, and the rest of IMFs are real data components which could be used to reconstruct the data. Comparison of the original data and the reconstructed data is shown in Figure 6. The figure shows that a large number of high-frequency and low-amplitude load cycles are removed. Figure 7 also reveals that the reconstructed data are periodic, and their period is equal to the rotation period of the shaft, which shows the reconstructed data have a high SNR and a high correlation coefficient with the original data. The data after denoising processing have a high similarity with the original data. As a word, the denoising method proposed above is feasible and effective for the load data of the feed mixer and achieves a good result.Figure 7
Comparison of the original data and the reconstructed data.


## 4.3. Load Spectrum Compiling

In order to study the effectiveness of the CEEMD-AGA method, the influence of the method on fatigue life prediction is conducted. In general, the cumulative damage theory is a common method to evaluate fatigue life. In this theory, to predict the fatigue life, a load spectrum is necessary to be established first. The key to load spectrum compiling is using statistical theories to transform the load-time data into a series of full-cycle or half-cycle sequences. The rain-flow counting method is a two-parameter statistical method which can transform the load-time series to a distribution matrix of stress mean and amplitude by calculating the peak and valley values in load-time series [39]. The rain-flow counting method is proposed by Matsuiski and Endo in 1950s, and it can take full cycle and half cycle in to consideration at the same time and use these to describe the stress effect of cyclic loading [40]. Since proposed, the rain-flow counting method has gained more and more attention in load spectrum compilation [41, 42] and fatigue reliability assessment [43, 44]. Many new counting models are proposed based on the rain-flow method, such as, four-peak valley rain counting method, three metamorphoses counting method [45]. The basic principle and implementation of the rain-flow counting method are not introduced in this paper; it can be seen in [45]. The rain-flow counting method is adopted to count the number of load cycles for the reconstructed data and the original data, respectively. The rain-flow counting results are shown in Figure 8(a).Figure 8
Comparison of rain-flow counting results. (a) Rain-flow counting result of the reconstructed data after CEEMD-AGA denoising. (b) Rain-flow counting result of the original data. (c) Rain-flow counting result of the reconstructed data after wavelet denoising.
(a)
(b)
(c)Furthermore, the traditional wavelet denoising is applied to the investigated data, and the rain-flow counting result of reconstructed data is shown in Figure8(c) and Table 1. In addition, to study the sensitivity of the CEEMD-AGA denoising method, different levels of white noise (SNR = 5, 10, and 15) are added to the original signal. Then, the method is applied to the corresponding series. In order to ensure the accuracy, each level is computed twice. The rain-flow counting result is compared with the result of the original signal, as shown in Table 2.Table 1
Result of different processing methods.
Load data
Number of load cycles
Similarity
The original data
2442
1
CEEMD-AGA method
1267
0.9703
Wavelet method
883
0.8271Table 2
Result of sensitive analysis of different input noise levels.
SNR
Number of load cycles
Error (%)
Correlation coefficient
5
1236
2.44
0.9965
1292
1.97
0.9974
10
1306
3.08
0.9921
1317
3.95
0.9895
15
1331
5.05
0.9860
1352
6.71
0.9823From Figure7 and Table 1, it is illustrated that the CEEMD-AGA method can remove many noise components of small stress amplitude and large stress mean, preserve more real load components, and greatly reduce the number of load cycles. The CEEMD-AGA method can effectively eliminate the influence of noise and improve the similarity.As shown in Table2, compared with the reconstructed signal shown in Figure 6, with the increasing of SNR, the error of cyclic load numbers increases, and the correlation coefficient between the corresponding reconstruction data decreases. But they all change very little. Therefore, it is illustrated that the CEEMD-AGA denoising method has a poor sensitivity to the level of input noise level, and it has good robustness.


## 5. Conclusion

This paper proposes a CEEMD-AGA denoising method to eliminate the noise components in load data obtained by field test. With the scientific problem illustrated, the denoising algorithm based on the autocorrelation function and a similarity function is put forward to guarantee the effectiveness of the CEEMD-AGA method.A case of feed mixer is studied. In comparison to the traditional method, the results illustrate that the CEEMD-AGA method can remove many load cycles with small stress amplitude and large stress mean, preserve more real load components, and greatly reduce the number of load cycles.The denoising method has been proved to be effective and feasible to eliminate the influence of noise in load spectrum compiling. The CEEMD-AGA method can not only be used to the feed mixer but also to many other machines with complex working conditions.



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