21.11.2017  Original Article  Ausgabe 6/2017 Open Access
An Intelligent Harmonic Synthesis Technique for AirGap Eccentricity Fault Diagnosis in Induction Motors
 Zeitschrift:
 Chinese Journal of Mechanical Engineering > Ausgabe 6/2017
Wichtige Hinweise
Supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC), eMech Systems Inc and Bare Point Water Treatment Plant in Thunder Bay, Ontario, Canada.
1 Introduction
Induction motors (IMs) are commonly used in various industrial applications. Furthermore, IMs consume about 50% of the generated electrical energy in the world [
1]. IM defects will lead to low productivity and inefficient energy consumption. Endeavors have been put, for decades, to improve IM operation accuracy and IM driven industrial process efficiency. In industrial maintenance applications, for example, an efficient and reliable IM condition monitor is very useful to detect an IM defect at its earliest stage to prevent malfunction of IMs and reduce maintenance cost.
In general, airgap eccentricity is classified as static eccentricity, dynamic eccentricity, as well as mixed eccentricity of these two types [
2]. In static airgap eccentricity, geometric axis of rotor rotation is not the geometric axis of the stator, and position of the minimal radial airgap length is fixed in space. In dynamic airgap eccentricity, the rotor rotates around the geometric axis of the stator, where the position of the minimum airgap length rotates with the rotor. In a particular case of static airgap eccentricity, rotor geometric axis is not parallel to stator geometric axis; the degree of eccentricity gradually changes along stator axis, which is inclined static eccentricity [
3]. IM airgap eccentricity defects could result in unbalanced magnetic pull, bearing damage, excessive vibration and noise, and even statorrotor rub failure [
4]. Correspondingly, this work will focus on initial IM fault detection of static eccentricity and dynamic eccentricity.
Anzeige
Recently, many research efforts have been undertaken to diagnose IM airgap eccentricity fault using stator current signals due to their low cost and ease of implementation [
5,
6]. For example, Blödt et al. [
7] presented a Wigner distribution method to analyze stator current signals and diagnose IM eccentricity fault. Akin et al. [
8] conducted realtime eccentricity fault detection using reference frame theory. Bossio et al. [
9] employed additional excitation to reveal information about airgap eccentricity fault. Alarcon et al. [
10] applied notch finiteimpulse response filter and WignerVille Distribution to study rotor asymmetries and mixed eccentricities. Faiz et al. [
11] employed instantaneous power harmonics to detect mixed IM eccentricity defect. Huang et al. [
12] applied an artificial neural network for the detection of rotor eccentricity faults. Esfahani et al. [
13] utilized the HilbertHuang transform to detect IM eccentricity fault. Nandi et al. [
14] studied the eccentricity fault related harmonics with different rotor cages. RieraGuasp et al. [
15] applied Gabor analysis for transient current signals to detect eccentricity fault. Park and Hur [
16] analyzed specific frequency patterns of the stator current to detect dynamic eccentricity fault. Mirimani et al. [
17] presented an online diagnostic method for static eccentricity fault detection. Some intelligent tools based on soft computing and pattern classification were also used for motor fault diagnosis in Refs. [
18–
20], in order to explore patterns of the features. These aforementioned techniques, however, cannot thoroughly explore the relations among massive fault harmonic series in the current spectrum, which may degrade fault detection accuracy.
To tackle the aforementioned problems with IM fault detection using current signals, a harmonic synthesis (HS) technique is proposed in this work for incipient IM eccentricity fault detection. The contributions of the proposed HS technique lie in the following aspects: 1) a novel synthesis approach is proposed to integrate several fault harmonic series to recognize fault related features; 2) fault indicators are properly derived from local spectra for IM health condition monitoring. The effectiveness of the proposed HS technique for IM eccentricity defect detection is verified experimentally under different IM conditions.
The remainder of this paper is organized as follows. The proposed HS technique is discussed in Section
2. Effectiveness of the HS technique for IM airgap eccentricity fault detection is examined experimentally in Section
3. Finally, some concluding remarks of this study are summarized in Section
4.
2 The Proposed HS Technique for IM Eccentricity Fault Detection
The proposed HS technique is composed of two procedures: harmonic series processing (HSP) and local spectra analysis (LSA). The HSP is to synthesize the fault related features in the spectrum, whereas LSA is to extract fault indicators for incipient IM airgap eccentricity fault detection.
Anzeige
2.1 Harmonic Series Processing
In stator current signal based IM fault detection, the characteristic frequency components
f
_{ e } in Hz used to detect static and dynamic eccentricity defects [
21] are given by
where
f
_{ s } is the supply frequency in Hz;
R is the number of rotor slots;
s is the slip;
f
_{ r } is the rotor rotating speed;
k = 1, 2, 3, …;
p is the number of pole pairs;
\(\beta = 1, \, 3, \, 5, \ldots\) is the order of the stator time harmonics;
\(\alpha\) is the eccentricity order:
\(\alpha = 0\) denotes static eccentricity and
\(\alpha = 1\) denotes dynamic eccentricity.
$$\begin{aligned} f_{e} & = f_{s} \left[ {\left( {kR \pm \alpha } \right)\frac{1  s}{p} \pm \beta } \right] \\ & = \left( {kR \pm \alpha } \right)f_{r} \pm \beta f_{s} \\ \end{aligned}$$
(1)
The
Kth order fault harmonic series
f
_{ e_K } will be
where the
Kth order origin frequency component
\(f_{o\_K}\) is
$$f_{e\_K} = f_{o\_K} + \beta f_{s} ,\quad \beta = { 1},{ 3},{ 5}, \, \ldots$$
(2)
$$f_{o\_K} = \left( {kR \pm \alpha } \right)f_{r}$$
(3)
To explore the relationship among different fault harmonic series, the fault harmonic series of interest are synthesized to reveal fault characteristic features. If the first
P fault harmonics (
\(\beta\) = 1, 3, 5, …, 2
P–1) are considered for synthesis, the harmonic frequency band corresponding to the
Kth fault harmonic series will be extracted as
and the bandwidth in Hz is
\(2Pf_{s}\) +1.
$$q_{K} = \left[ {f_{o\_K} , \, f_{o\_K} + 2Pf_{s} } \right]$$
(4)
To synthesize fault harmonic series, the frequency bands are converted to a discretepoint domain representation. The discretepoint closest to
\(f_{o\_K}\) is considered as the first discretepoint in
q
_{ K }, defined as
D
_{ k }{1}. Suppose
u
_{ f } discrete data points represent unit frequency distance in the spectrum, the bandwidth of Eq. (
4) in discretepoint domain will be formulated as
where
\(\left\langle \cdot \right\rangle\) denotes the integer function. Then the harmonic frequency band of Eq. (
4) in the discretepoint domain will be
$$B_{w} = \left\langle {2Pf_{s} + 1} \right\rangle u_{f}$$
(5)
$$D_{K} \{ i\} , \quad i = { 1},{ 2},{ 3}, \, \ldots ,B_{w}$$
(6)
The rootmeansquare (RMS) operation is applied to synthesize harmonic frequency bands. Given the harmonic frequency bands of
M interested fault harmonic series, the
ith component in the synthesized spectrum
v will be derived as
$$v\left\{ i \right\} = \sqrt {\frac{{\sum\nolimits_{j = 1}^{M} {D_{j} \left\{ i \right\}^{2} } }}{M}} \quad i = { 1},{ 2},{ 3}, \, \ldots ,B_{w}$$
(7)
2.2 Local Spectra Analysis
The
mth harmonic (
\(\beta\) = 2
m–1) of the
Kth order fault harmonic series will be derived from Eq. (
2),
$$f_{K,m} = f_{o\_K} + \left( {2m  1} \right)f_{s}$$
(8)
Through the synthesis operation in Eq. (
7), the
mth fault harmonic over all
M harmonic frequency bands is synthesized to form a new fault frequency component. Then the
mth synthesized fault frequency component in the discretepoint domain will be derived in the synthesized spectrum by the following representation:
$$v\left\{ {1 + \left( {2m  1} \right)f_{s} u_{f} } \right\}$$
(9)
The local spectra encompassing the
mth synthesized fault frequency component will be
where
d is the half bandwidth of the local spectra in the frequency domain. The mean value of this local spectrum is the fault indicator regarding the
mth synthesized fault frequency component, denoted by
F
_{ m }.
$$\left[ {v\left\{ {1 + \left( {2m  1} \right)f_{s} u_{f}  du_{f} } \right\}, \, v\left\{ {1 + \left( {2m  1} \right)f_{s} u_{f} + du_{f} } \right\}} \right]$$
(10)
To illustrate the operation of the proposed HSP and LSA processes, a simulated current spectrum with three airgap eccentricity frequency harmonic series is illustrated in Figure
1. It is normalized by deducting mean and then being divided by its standard deviation. In Figure
1, the black arrow points 50 Hz supply frequency; magenta arrows with number 1, green arrows with number 2 and yellow arrows with number 3 denote three fault harmonic series respectively. The origin frequency components
\(f_{o\_K}\) are 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. The harmonics of defect features are computed using Eq. (
2) with
f
_{ s } = 50 Hz and
\(\beta\) = 1, 3, 5, 7, 9. Figure
1 shows that some fault characteristic frequency components are buried in the spectrum, and it is difficult to predict which characteristic frequency components will protrude in the spectrum. The extracted frequency bands represented in Eq. (
6) of three fault harmonic series, and synthesized spectrum represented in Eq. (
7) with fault related local spectra are demonstrated in Figure
2. The dashed vertical lines indicate the boundaries of local spectra encompassing synthesized fault frequency components. Figures
2(a),
2(b) and
2(c) represent three frequency bands computed by Eq. (
4) with staring frequencies
\(f_{o\_K}\) being 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. Figure
2(d) shows results of the synchronization of spectra in Figures
2(a),
2(b) and
2(c) using rootmeansquare calculation using Eq. (
7). It is seen from Figure
2(d) that the fault characteristic features can be highlighted in the spectrum after the HSP operation, which could be used for IM fault detection. In this simulation, the supply frequency
f
_{ s } = 50 Hz;
\(f_{o\_K}\) is the starting point of the spectrum in Figure
2(d). Based on Eqs. (
2) and (
3), the fault frequencies in Figure
2(d) should appear at 50 Hz, 150 Hz, 250 Hz, 350 Hz and 450 Hz. The frequency components outside the boundaries of local spectra are usually caused by the modulation of load variation and supply frequency, and the synchronization of different harmonic series local bands.
×
×
If both static and dynamic eccentricities occur simultaneously, the following characteristic frequency components could also be excited [
22], given by
$$f^{\prime}_{e} = \left {kf_{r} \pm f_{s} } \right,\quad k = 1,{ 2},{ 3}, \, \ldots$$
(11)
The related harmonic series can be expressed as
$$f^{\prime}_{e1} = kf_{r} + f_{s} ,\quad k = { 1},{ 2},{ 3}, \, \ldots$$
(12)
$$f^{\prime}_{e2} = kf_{r}  f_{s} ,\quad k = { 2},{ 3}, \, \ldots$$
(13)
The local spectra of the first
Q harmonics in Eq. (
12) will be transformed to the discretepoint domain, and then synthesized into one spectrum in RMS form using Eq. (
7). The mean value of the synthesized spectrum is considered as a fault indicator denoted by
F
_{ d1}. The fault indicator derived from Eq. (
13) will be denoted by
F
_{ d2}.
3 Performance Evaluations
3.1 Overview
To examine the effectiveness of the proposed HS technique, a series of tests will be conducted for IM airgap eccentricity fault diagnosis using stator current signals. This work focuses on both static and dynamic eccentricity fault detection. The parameters of the proposed HS technique used in the following tests are given in Table
1. To thoroughly analyze fault information, the first 20 fault harmonic series (i.e.,
M = 5) are synthesized and the first eight harmonics (
P = 8) in the synthesized spectrum are utilized to generate fault indicators
F
_{ m },
m = 1, 2, …, 8. The frequency band [0, 8000] Hz is used for analysis. To analyze more informative fault features in harmonic series in Eqs. (
12) and (
13), the first ten harmonics are used to derive fault indicators
F
_{ d1} and
F
_{ d2}. To capture the fault features and exclude the interference nearby, the half bandwidth of the local spectra is selected as 2 Hz.
Table 1
Factors and their levels
f
_{ s }/Hz

P

u
_{ f }

Q

M

d/Hz


50

8

3

10

5

2

To derive
F
_{ m } for eccentricity fault detection, the coefficient
\(\alpha\) = 0 in Eq. (
3) is set for static eccentricity detection and
\(\alpha\) = 1 for dynamic eccentricity analysis. To evaluate the effectiveness of the HS technique, the power spectral density (PSD) is used for comparison. To implement PSD, the mean values of ten local bands are used as fault indicators, whose half bandwidth is set as 2 Hz. Eight of ten center frequencies of the local bands are calculated using Eq. (
1) with
k = 1 and
\(\beta\) = 1, 2, …, 8. The other two center frequencies are computed using
\(f^{\prime}_{e1}\) with
k = 1 and
\(f^{\prime}_{e2}\) with
k = 2. In addition, a variant of the HS technique is employed for comparison, in which the center characteristic frequencies in the synthesized local spectra rather than the mean values are utilized as fault indicators, and is denoted by CHS. The frequency components in Eq. (
9) with
m = 1, 2, …, 8, and the two center frequencies of the synthesized local spectra derived from Eqs. (
12) and (
13), respectively, are used as fault indicators of the CHS method. The HilbertHuang transform (HHT) based fault indicators [
13] will be used for comparison. Since this work focuses on the analysis of the current signal based fault indicators, only the current signal related fault indicators provided in Ref. [
13] will be used for comparison. The HHT is applied to current signal analysis to generate fault indicators. The mean values of four local bands centered at
\(f^{\prime}_{e1}\) with
k = 1, 2, and
\(f^{\prime}_{e2}\) with
k = 2, 3, and averaged Hilbert marginal spectrum in local bands centered at
\(f^{\prime}_{e1}\) with
k = 1, and
\(f^{\prime}_{e2}\) with
k = 2, in the first two intrinsic mode functions (IMF), and averaged instantaneous amplitudes of the first two IMFs are used as fault indicators. The half bandwidth of these local bands in HHT is set as 2 Hz.
The HHT, CHS and the proposed HS technique generate fault indicators from the PSD spectrum. To examine if the fault indicators could be used for different classifiers, both the support vector machine (SVM) [
23] and linear discriminant analysis (LDA) [
24] are utilized to test the accuracy of different fault detection techniques. A series of tests have been conducted for this verification; however, the test results corresponding to five load conditions (i.e., 0, 20%, 50%, 70% and 100% load levels) will be used for demonstration. In data preparation, 200 data sets are collected for each IM condition (healthy, static airgap eccentricity fault, and dynamic airgap eccentricity fault) in each of five load conditions. The sampling frequency is
f
_{ p } = 20000 Hz and the time span of each data set is 3 s. The fault indicators of PSD, HHT, CHS and the proposed HS are extracted from these totally 3000 data sets.
3.2 Experiment Setup
Figure
3 shows the experiment setup used in this test. The tested IMs are 3phase, 1/3 hp motors made by Marathon Electric. Its speed is controlled by a speed controller (VFDB) with output frequency 0.1–400 Hz. A magnetic clutch system (PHC50) is used as the dynamometer to provide external loading, with torque ranging from 1 to 40 N·m. A gearbox (Boston Gear 800) is used to adjust the speed ratio of the dynamometer. An encoder (NSN10242 MF) is used to measure the shaft speed. Phase current signals are measured by the use of current sensors (LTS 6NP). A data acquisition board (Quanser Q4) is used for signal measurement. Motor specifications are given in Table
2. The static airgap eccentricity is implemented by introducing a 0.025 inch horizontal travel of the bearing housing of end bells in the IM. The dynamic airgap eccentricity is implemented by intentionally bending rotor in the center 0.0127 cm to 0.0254 cm, while making sure the rotor does not bind the stator in the IM [
25]. Based on the motor specification, the introduced airgap eccentricity faults are considered to be at the early stage of the IM fault.
Table 2
Motor specifications
Phase

Poles

HP

Connection

Rotor bars

Stator slots


3

2

1/3

Y

34

24

×
3.3 Static Airgap Eccentricity Fault Detection
Static airgap eccentricity fault detection is considered as a binary classification problem; the data sets collected from a healthy IM and an IM with the dynamic airgap eccentricity defect belong to one class; the data sets collected from the IM with the static airgap eccentricity defect belong to the other class. The averaged successful rate of ten runs threefold cross validation in terms of static airgap eccentricity fault diagnosis using SVM and LDA for classification is summarized in Tables
3 and
4, respectively. It is seen from Tables
3 and
4 that the proposed HS technique has the highest averaged successful rates in both training and testing. The proposed HS technique outperforms PSD under all five load levels, because it could explore more fault information and enhance fault features using HSP. The HS outperforms the CHS in all five scenarios, because the mean value of the local band is a more representative feature for fault detection. HS is superior to the HHT in that more effective fault features are extracted and employed for fault detection.
Table 3
Averaged successful rates (SR) of ten runs threefold cross validation in terms of static airgap eccentricity fault diagnosis using SVM (%)
Load level

0

20

50

70

100


PSD training SR

69.71

65.84

68.23

65.38

60.46

PSD test SR

65.97

61.49

65.52

61.66

53.57

HHT training SR

99.26

72.30

53.20

52.20

77.27

HHT test SR

98.41

71.59

48.76

51.20

74.99

CHS training SR

96.18

81.47

96.68

96.14

96.17

CHS test SR

94.54

78.92

95.79

94.73

95.77

HS training SR

100

100

99.56

100

97.87

HS test SR

100

100

99.20

100

97.28

Table 4
Averaged successful rates (SR) of ten runs threefold cross validation in terms of static airgap eccentricity fault diagnosis using LDA (%)
Load level

0

20

50

70

100


PSD training SR

73.09

69.41

68.37

71.07

68.47

PSD test SR

71.63

67.95

66.60

69.88

66.32

HHT training SR

93.13

72.94

66.87

67.92

77.96

HHT test SR

92.42

71.62

66.05

66.50

77.15

CHS training SR

95.98

80.79

96.69

94.47

93.57

CHS test SR

95.78

79.05

95.90

93.97

92.83

HS training SR

99.99

100

99.41

100

95.73

HS test SR

99.83

100

99.10

100

95.17

The score plots of SVM and LDA with respect to the HS static eccentricity fault indicators for static eccentricity diagnosis are illustrated in Figure
4. Totally 600 samples are shown in each plot. The scores of SVM classification and LDA classification are normalized over [–1, 1]. In each axis (
xaxis or
yaxis), –1 and 1 denote two different classes respectively. The closer the sample is to the –1 or 1, the more probably the sample belongs to the class denoted by –1 or 1. The zerovertical line and zerohorizontal line are the separating hyperplanes to differentiate two classes using SVM and LDA respectively. From Figures
4(a)–
4(d),
4(g), and
4(h), it is seen that the two classes could be separated using either SVM or LDA under 0, 20% and 70% load levels. It is seen in Figures
4(e) and
4(f) that there is one sample misclassification in both training and testing in 50% load state. The samples in Figures
4(i) and
4(j) are a bit scattered because heavy load condition weakens fault features, however, the samples could still be correctly classified mostly in 100% load level. Therefore, the proposed HS technique is a useful tool to detect IM static airgap eccentricity fault in different load conditions.
×
3.4 Dynamic Airgap Eccentricity Fault Detection
For dynamic airgap eccentricity fault diagnosis, the data sets corresponding to healthy IM condition and static airgap eccentricity condition will belong to one class; the data sets corresponding to dynamic airgap eccentricity condition belong to the other class. The averaged successful rates of ten runs threefold cross validation in terms of dynamic airgap eccentricity fault diagnosis using SVM and LDA for classification is given in Tables
5 and
6, respectively. From Tables
5 and
6, it is seen that the proposed HS technique has the highest accuracy in both training and testing. HS technique outperforms PSD in all five scenarios because of the extensive fault information exploration and fault features enhancement. The proposed HS is superior to the CHS in all five cases because of its effective local fault information analysis. HS outperforms HHT because more effective fault features are extracted for fault analysis.
Table 5
Averaged successful rates (SR) of ten runs threefold cross validation in terms of dynamic airgap eccentricity fault diagnosis using SVM (%)
Load level

0

20

50

70

100


PSD training SR

84.14

63.21

60.68

65.44

61.28

PSD test SR

81.82

60.13

57.11

60.94

58.41

HHT training SR

82.60

64.22

58.73

56.52

60.38

HHT test SR

79.99

60.23

53.67

52.66

59.30

CHS training SR

94.02

91.14

85.97

78.27

75.92

CHS test SR

92.30

88.66

85.71

75.23

74.66

HS training SR

99.47

97.69

98.50

94.74

93.72

HS test SR

98.96

97.13

97.73

94.53

92.47

Table 6
Averaged successful rates (SR) of ten runs threefold cross validation in terms of dynamic airgap eccentricity fault diagnosis using LDA (%)
Load level

0

20

50

70

100


PSD training SR

83.95

66.93

67.17

68.22

66.40

PSD test SR

82.73

64.75

65.02

66.87

63.45

HHT training SR

79.40

67.89

67.57

68.24

68.06

HHT test SR

78.75

60.30

65.73

65.42

65.20

CHS training SR

93.51

89.00

89.45

80.79

79.59

CHS test SR

92.88

87.73

88.82

79.67

78.17

HS training SR

99.02

95.99

96.27

94.67

91.78

HS test SR

98.98

95.43

95.47

93.87

91.35

The score plots of SVM and LDA with respect to the HS dynamic eccentricity fault indicators for dynamic eccentricity diagnosis are shown in Figure
5. From Figures
5(a)–
5(d), it is seen that the two classes could be separated using either SVM or LDA under 0% and 20% load levels. It can be seen from Figure
5(f) that there is one sample misclassification in testing by LDA in 50% load condition. The samples in Figures
5(g)–
5(j) have larger variance due to interference of the heavy load condition; however, most samples could still be correctly classified in 70% and 100% load conditions. It demonstrates that the proposed HS technique is a useful tool to detect IM dynamic airgap eccentricity fault in different load conditions.
×
4 Conclusions
A harmonic synthesis (HS) technique is proposed in this work for initial IM airgap eccentricity fault detection. In the HS, the fault harmonic series are synthesized to enhance fault characteristic features. The local spectra statistical analysis is employed to extract representative features. The SVM classifier and the LDA classifier are utilized to evaluate the performance of different fault detection techniques. The effectiveness of the proposed HS technique is verified by a series of experimental tests under five load conditions: 0%, 20%, 50%, 70% and 100%. Test results show that the proposed HS technique is an effective fault detection tool, and it outperforms the other techniques in all five load conditions. It is able to process massive fault related information, analyze fault features statistically, and enhance representative features for static eccentricity and dynamic eccentricity fault diagnosis based on the current signals. Future research will be undertaken on classification of health state, static eccentricity state and dynamic eccentricity state altogether, as well as the analysis of stochastic resonance based fault detection.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (
http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.