A novel classification method combining adaptive local iterative filtering with singular value decomposition for fault diagnosis

2.1. The theory of adaptive local iterative filtering

Adaptive local iterative filtering (ALIF) method is the improved version of iterative filtering [21]. The main difference between the two methods is that the former one can locally and adaptively calculate the length of filters and generates moving average line using the solution of Fokker Planck equation, which is so-called FP filter. Adaptive local iterative filtering method is as show in Fig. 1.

Given a signal $f\left(x\right)$ and $x\in R$, we define $L_{w_n,l_n}\left(f\right)\left(x\right)={\int }_{-l_n\left(x\right)}^{l_n\left(x\right)}{f}_n(x+t)w_n(x,t)dt$, which represents the moving average function of the signal $f\left(x\right)$. In the above function, $w_n(x,t)$ is a filter represented by step $n$ and the length is $2l_n\left(x\right)$, in which $t\in [-l_n(x),l_n(x\left)\right]$. $w_n(x,t)$ indicates that $w_n$ changes with $x$.

Fig. 1The flowchart of adaptive local iterative filtering method

According to the definition of the moving average, we can also define an operator $S_{1,n}\left(f_n\right)=f_n-L_{{}_{w_n,l_n}}^{\left(1\right)}\left(f_n\right)=f_{n+1}$ to capture the fluctuation of $f_n$, in which the superscript reflects the number of IMF, thus the first IMF can be represented as $I_1=li{m}_{n\to \infty }{S}_{1,n}\left(f_n\right)$.

There are two loops, an inner loop and an outer loop in the adaptive local iterative filtering algorithm. The inner loop is to capture a single IMF, and the outer loop is to generate all the IMFs of the signal. The iteration equation of the inner loop is shown as follows:

In order to establish the convergence theorem of ALIF algorithm, an equivalent formula $g_n(x,y):[-L,L]\to [-l_n(x),l_n(x\left)\right]$ is used to replace the previous calibration function. The calibration function can be represented as linearized $g_n(x,y)=l_n\left(x\right)y/L$ or cubic form $g_n(x,y)=l_n\left(x\right)y^3/L^3$ or any other function. With the help of $g_n(x,y)$, the above formula can be transformed into:

As a result, Eq. (1) can be written as the following form:

where $W\left(y\right)$($y\in [-L,L]$) is a proper filter. We define a new operator:

Thus, the convergence theorem of the inner loop in ALIF algorithm is to make the$f\left(x\right)$ continuous, in which $x\in R$ and $f\left(x\right)\in L^{\infty }\left(R\right)$. Given that:

When $n$ approaches infinity, if ${\prod }_{i=1}^n{\epsilon }_i\to 0\text{,}$${\prod }_{i=1}^n{\delta }_i\to c>0\text{,}$ accordingly, $\left\{f_n\right(x\left)\right\}$ convergences to an IMF.

The convergence theorem of the outer loop in ALIF algorithm is that $f\left(x\right)$ is continuous and differentiable, in which $x\in R$, and there is a finite number of extreme points in any small time interval of $f\left(x\right)$. Consequently, there are a maximum number of extreme points in $f\left(x\right)$. Given that $x_i$, $i=$ 1, 2,…, $k$, are the extreme points of $f\left(x\right)$. $f\left(x\right)$ is supposed strictly monotone in $[x_i,x_{i+1}]$, and $i=$ 1, 2,…, $k-1$. Then, based on $f_n\left(x\right)$, function $c_n^{\left(1\right)}\left(x\right)$ and $c_n^{\left(2\right)}\left(x\right)$ can be defined as:

Assume that $f\left(x\right)$ is a differentiable function with the above characteristics, it can be known from Eq. (2) that, if the scanning functions are separable, for example, $g_n(x,y)=l_n\left(x\right)h\left(y\right)$, and for any $n\in N$:

If $\underset{n\to \infty }{\int }{f}_n\left(x\right)$ is in existence, the maximum number of extreme points in $f\left(x\right)-\underset{n\to \infty }{\int }{f}_n\left(x\right)$ will be the number of extreme points in$f\left(x\right)$. For this convergence theorem, in the next step of the outer loop, the number of extreme points in $f\left(x\right)-\underset{n\to \infty }{\int }{f}_n\left(x\right)$ is less than that of $f\left(x\right)$, meaning that $f\left(x\right)-\underset{n\to \infty }{\int }{f}_n\left(x\right)$ is smoother than $f\left(x\right)$. If this feature maintains true in each of the outer loop, which means the number of extreme points of the remaining signal continues to decrease, as a result, for the signal$f\left(x\right)$, ALIF algorithm remains convergence.

2.2. The definition of singular value energy (SVE)

Singular value decomposition (SVD) is a typical orthogonal method. For a real matrix $A\in R^{m\times n}$, there must be orthogonal matrix:

which will meet the following Eq. (7):

where $D=\left(diag\left({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_q\right),0\right)$ and its transpose depends on the value of $m$ and $n$, $m<n$ or $m>n$, $D\in R^{m\times n}$, 0 represents the zero matrix, $q=\mathrm{m}\mathrm{i}\mathrm{n}(m,n)$, ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_q>$ 0, and they are singular values of matrix $A$.

Commonly, matrix $A$ can also be written as:

where $u_i$, $v_i$ are respectively correspond to the column vector of $U$ and $V$.

Thus, the singular value energy of the signal can be expressed as:

2.3. Feature extraction scheme based on adaptive local iterative filtering and singular value decomposition

In this paper, a fault classification method based on ALIF for gear and rolling bearing is proposed. After experimental studies, it can be concluded that ALIF method has an obvious effect on denoising towards non-linear and non-stationary signal, etc. Thus, the ALIF method can be well applied to a variety of mechanical fault signals. As for gear and bearing parts, different fault types correspond to different fault degrees. From all above, ALIF method can be employed to decompose gear and bearing fault signals, and then apply SVE to conduct classification of different faults. The flowchart of the proposed method is illustrated in Fig. 2.

Fig. 2The flowchart of proposed method based on adaptive local iterative filtering and singular value decomposition

4.1. Application to processing of case western reserve university bearing data

Through numerical simulations, the proposed method based on adaptive local iterative filtering and singular value decomposition can be applied to the bearing signals. The fault bearing data from Case Western Reserve University Bearing Data Center Website [25] is used to verify the proposed method. The website provides a large collection of experimental bearing data that are related to faulty bearings, and among those the data for faulty bearings are acquired by using electric spark machining. In this paper, inner race, ball, and outer race fault data are both adopted. The sampling frequency in the experiment is 12000 Hz, and the rotational speed is 1730 rpm. Experimental fault bearing type is 6205-2RS JEM SKF, and its specific parameters are shown in Table 1.

The test equipment of Case Western Reserve University includes two motors, a coupling (containing torque sensor and encoder), and other related devices. Accelerometer is placed on the testing site to collect acceleration signal that can be used for bearing fault analysis. The schematic diagram of a signal acquisition device and a picture are shown in Fig. 7.

Generally, the collected signals such as rolling bearings of inner race, ball, and outer race contain many noise components. Then, the time-domain plots of one set of bearing signals are shown in Fig. 8. It is no doubt that the time-domain analysis of these groups of signals cannot clearly identify the difference between different kinds of faults.

Fig. 7The schematic diagram and a picture of the device: 1, 3 – Hp motor, 2 – torque transducer and encoder

Fig. 8Time-domain waveform of rolling bearing vibration signals with four states: a) inner race fault, b) outer race fault, c) roller fault, d) normal state

a)

b)

c)

d)

Empirical mode decomposition, ensemble empirical mode decomposition and adaptive local iterative filtering method are applied to decompose 15 groups of signals, and singular values of decomposed vibration signals are calculated, so as to conduct fault classification. The results are shown in Fig. 9, Fig. 10 and Fig. 11 respectively.

Fig. 9The singular value power of original signals decomposed by EMD

Fig. 10The singular value power of original signals decomposed by EEMD

Based on the previous several mode decomposition algorithms, the decomposed components will constitute an initial feature matrix to express the signal energy and then the singular values of initial feature matrix are calculated. Through the mode decomposition and feature reconstruction of the original signal, it can be shown from the Fig. 9 that the SVE value of the working condition is not different from the other working state in the result obtained by the EMD. Inside, the inner race and the rolling element already have some aliasing. Similarly, the EEMD method was used to decompose the vibration signals and the SVE values were shown in Fig. 10. It can discriminate the inner race fault and rolling element defect, however, the SVE values of the outer race fault and the normal is not clearly. However, the bearing vibration signals are decomposed by the ALIF, the SVE of bearing signals of four kinds of working conditions are distinguished very clearly from the Fig. 11.

Fig. 11The singular value power of original signals decomposed by ALIF

In order to explain the classification effect of faults more accurately, the fault accuracy is used to evaluate the classification results of several methods. From the result graph of EMD decomposition, we can see that 2 groups of rolling element and outer race have been mixed, and the total sample number is 15 groups, so the classification accuracy is 86.67 %. From the decomposition effect of EEMD, we can see that the boundary between the inner ring and the rolling body is obvious, as shown in Fig. 10, and the boundary between the normal and outer race is very vague. One of the groups has been coincident, so the classification accuracy is 93.33 %. However, the results of ALIF decomposition can be divided into several kinds of faults, and the boundary between them is obvious, and the classification accuracy is close to 100 %.

As shown in Table 2, the rolling element energy value is the smallest, which the energy value varies from 2.42 to 3.15 and the average value is 2.62. Furthermore, the energy value of fault vibration signal about normal bearing varies from 11.56 to 13.42 and the average value is 12.56. The energy value with the inner fault vibration signal is changed from 20.29 to 24.28 and the average value is calculated as 21.98. Moreover, the energy value about outer race fault vibration signal is from 4.25 to 5.94 and the average value is calculated as 5.51. It is illustrated that different fault types correspond to different energy average values and ranges, which can be used to classify different types of mechanical fault modes. Therefore, it can be concluded that several groups of bearing signals under different types of operating conditions can be classified by the proposed method.

4.2. Application to drivetrain diagnostics simulator

Through the simulation analysis, the proposed fault classification method based on adaptive local iterative filtering (ALIF) and singular value decomposition (SVD) is applied to the experimental gear signals. Owing to the unqualified manufacture or improper manipulation, different kinds of gear faults would occur during the operation process. In the studies of this paper, the collected different fault signals of Drivetrain Diagnostics Simulator are adopted to verify the effectiveness of the proposed method. The experimental apparatus has a single stage of gear transmission. The small gear is installed on the input shaft with tooth $z_1=$ 20, and the big gear is installed on the output shaft with tooth $z_2=$ 37. The module of gears is determined as 3. The load is generated by the magnetic powder brake, and the acceleration sensor is installed on bearing pedestal of the input shaft in a vertical direction. Under different working conditions, several group signals as normal gear, broken tooth, wearing and circular pitch error gear vibration signals, were collected. There is a significant crack in the root of the broken tooth, the depth account tooth thickness of 2/5. The wearing tooth is produced by a distinct wear on the gear surface. The specific parameters of test conditions are described as follows: the speed of high-speed shaft is 363 r/min, and the load is 0 when the apparatus is idle, the sampling frequency and time are set as 2000 Hz and 3 s respectively. Thus, each set of data has 6000 sampling points. The experimental apparatus and its schematic diagram are shown in Fig. 12. The apparatus contains a variable speed drive, a single stage gear transmission, and a magnetic powder brake.

Fig. 12The schematic diagram and a picture of the experimental apparatus: 1 – magnetic powder brake, 2 – single stage gear transmission, 3 – variable speed drive

Fig. 13Time-domain waveform of gear vibration signals with four states: a) broken tooth, b) wearing, c) circular pitch error, d) normal state

a)

b)

c)

d)

Since this apparatus is small and has a high rigidity, the collected signals such as normal gear, broken tooth, wearing and circular pitch error signals are all affected by vibration of the foundation bolts. The time-domain plots of one set of gear signals are shown in Fig. 13. The spectral analysis of these groups of signals cannot demonstrate obvious differences between different kinds of faults.

The 15 groups of signals correspond to each faulty mode are decomposed by the above methods. A set of mode decomposition result provided by ALIF is plotted in the following Fig. 14. Then, the energy value of the decomposed signal is calculated, so as to conduct the different fault classification. The results of classification are shown in Fig. 17. In order to verify the effectiveness of the proposed method, the EMD and EEMD methods are also used to decompose the above signals, and the results are shown in Fig. 15 and Fig. 16 respectively.

By decomposing the original fault signal into several modes by the above methods, the obtained mode components are formed into an initial characteristic matrix. Then the singular value of the initial characteristic matrix is used to calculate the signal energy. As shown in Fig. 15, the SVE value of the working condition is not different from the other working state in the result obtained by the EMD. Inside, the Pitch error and the normal already have some aliasing.

Fig. 14A set of mode decomposition result provided by ALIF: a) broken tooth, b) wearing, c) circular pitch error, d) normal state

a)

b)

c)

d)

The result of EEMD is better than that of EMD. We can see from the Fig. 16 that it can distinguish the pitch error, the normal and the broken teeth, but there are still 2 sets of aliasing between the broken teeth and the wear. However, the gear vibration signals are decomposed by the ALIF, and the SVE of the gear signal under four working conditions with the energy division has obvious diversity from the Fig. 17.

Just like the former, in order to explain the classification results more accurately, the classification accuracy of EMD and EEMD methods is 60 % and 86.67 % respectively. It is worth mentioning that the classification accuracy of ALIF decomposition is close to 100 %.

Fig. 15The singular value power of original signals decomposed by EMD

Fig. 16The singular value power of original signals decomposed by EEMD

Fig. 17The singular value power of original signals decomposed by ALIF

As shown in Table 3, the SVE of vibration signals of pitch error are the largest, ranging from 1079.1 to 1257.1, and their mean value is 1176.3. The SVE of normal gear varies from 291.77 to 350.57, and the average value is 320.24. The SVE of broken gear is changed from 241.44 to 292.26, and their mean value is 268.40. Moreover, the SVE of wearing gear are smallest, ranging from 235.91 to 197.59, and their average value is 216.02. The different singular value energy with each fault types can be performed as criterion of fault classification. Therefore, the proposed method can be used to classify different faulty modes under different working conditions.