589
JournalHERNANDEZ-VARGAS
of Scientific & Industrial
et Research
al: NOVEL METHODOLOGY FOR BROKEN-ROTOR-BAR AND BEARING FAULTS DETECTION
Vol. 71, September 2012, pp. 589-593
Novel methodology for broken-rotor-bar and bearing faults detection
through SVD and information entropy
M Hernandez-Vargas, E Cabal-Yepez*, A Garcia-Perez and R J Romero-Troncoso
HSPdigital Research Group, C A Telematica/Procesamiento Digital de Señales, Division de Ingenierias, Campus IrapuatoSalamanca, Universidad de Guanajuato, Carretera Salamanca – Valle de Santiago km 6.5 + 1.8 km, Comunidad de Palo
Blanco, 36700 – Salamanca, Guanajuato, Mexico
Received 23 April 2012; revised 07 August 2012; accepted 08 August 2012
This study introduces a novel methodology for early detection of broken-rotor-bars and bearing faults on induction motors
by using singular value decomposition and information entropy estimation to detect motor operational condition. Proposed
technique could detect these faults with a certainty greater than 99.7%. A field programmable gate array implementation is
developed for online application of proposed approach in real-time.
Keywords: Broken-rotor-bars (BRB), Faulty bearings (FBR), Field programmable gate arrays (FPGA), Induction motor, Information entropy, Singular value decomposition (SVD)
Introduction
Squirrel-cage induction motors are the most popular
motors in industry1-2 . Studies3-5 are available on induction
motor failure that produces unexpected interruptions on
production lines. At least, 50% of faults in rotating
machines are bearing related, and 10% are rotor faults6 .
Fault identification of induction motors still represents a
big challenge for condition monitoring (CM)7-10. Common
signal processing techniques for CM are fast Fourier
transform (FFT)11-13, and discrete wavelet transform
(DWT)14-16. Singular value decomposition (SVD) and
information entropy are techniques for extracting
significant information from different datasets, but rarely
used for induction motor CM17-19. This study presents a
CM technique for induction motor that combines SVD
with information entropy to detect broken-rotor-bars
(BRB) and faulty bearings (FBR).
Proposed Methodology
Proposed methodology (Fig. 1) consists of a data
acquisition system (DAS), matrix generation with
acquired data in onboard memory, and a hardware
processing unit implemented into an FPGA device for
SVD and information entropy estimation to determine
*Author for correspondence
E-mail: ecabal@hspdigital.org
quantitatively the operational condition of induction motor
as healthy (HLT), BRB and FBR.
Singular Value Decomposition (SVD)
SVD of an m × n matrix A is defined as A = U ΣV T ,
where U and V are m×m and n×n orthogonal matrices,
respectively. UUT=Im , and VVT=In , where I is identity
matrix. Σ is a diagonal matrix such that Σ = diag
(σ1 , σ2 , …, σn ), where σ1 , σ2 , …, σn are singular values
of A20 . For m>n, at least m-n singular values are zero.
If r=rank(A) and r<n, r of singular values are non-zero.
U contains m left singular vectors, and V contains n right
singular vectors. Most SVD algorithms are based on
diagonalizing rotations to form orthogonal transformations
that preserve angles and lengths.
Jacoby Rotations
Usually Jacobi rotation matrix J is used for
diagonalization through rotations 21 . For instance, when
applying J in Eq. (1) to 2×2 symmetric matrix AS in Eq.
(2), rotation angle θ is chosen such that off-diagonal
elements a ij =a ji of AS become zero, thus obtaining diagonal
matrix AD in Eq. (3) as
 cos θ
J = 
 − sin θ
sin θ 

cos θ 
…(1)
590
J SCI IND RES VOL 71 SEPTEMBER 2012
Fig. 1–Block diagram of proposed methodology
 aii
AS = 
 a ji
aij 

a jj 
 a ii
AD = J T AS J = 
 0
Fig. 2–Artificially generated faults: a) One broken-bar; and b)
Bearing defect
…(2)
0 

a jj 
…(3)
If total number of outcomes in random event X is N,
r
probability p(xi ) is given as p ( xi ) = i , where r i
N
represents incidence rate of each possible outcome xi ,
n
and total number of outcomes N is given as N = ∑ ri .
If A is not symmetric, off-diagonal elements can still
be annihilated by applying a symmetrizing rotation as
AS = J T A and a diagonalization rotation as
AD = J T AS J .
i =1
For entropy calculation, Mitchell algorithm24 is considered
in this study because of its advantages during hardware
implementation.
Experimental Setup and Instrumentation System
Hestenes-Jacobi Method
Hestenes 22 observed equivalence between
annihilation of a matrix element and orthogonalization of
two vectors applying Jacobi rotations. If an m×n nonsymmetric matrix A is multiplied by an orthogonal matrix
U, a matrix B whose rows are orthogonal is obtained. B
is normalized by calculating squares of row norms, then
B=SS -1B=SV, where S is a diagonal matrix containing
square norm of each row in B, and V is an orthonormal
matrix obtained by dividing each row in B by its
corresponding square norm. From this, and because U is
an orthogonal matrix, B=UA =SV and A=UTSV, where S
= diag(s1 , s2 , …, sn ) are squares of singular values (σ1 ,
σ2 , …, σn ) of A.
Information Entropy
Shannon23 defined entropy as a measure of average
information contents associated with a random outcome.
Considering a random event X with n possible outcomes
x1 , x2 , x3 ,…, xn , and every xi (i=1, ..., n) with a probability
p(xi ), then information entropy H(X) of a random event
X is given as
n
H ( X ) = − ∑ p ( xi ) log 2 [ p ( xi )]
i =1
…(4)
Proposed methodology based on SVD computation
and information entropy estimation for BRB and FBR
identification in real-time is used to classify induction
motor condition online. Under experiment setup, different
1-hp three-phase induction motors (model WEG
00136APE48T) are used. Tested motors have 2 poles,
28 bars, and receive a power supply of 220 V AC, at 60
Hz. Applied mechanical load is that of an ordinary
alternator. One phase of startup transient current is
acquired using an AC current clamp (model Fluke i200s).
A 16-bit serial-output analog-to-digital converter (ADC)
(ADS7809) 25 is used in data acquisition system (DAS).
A sampling frequency f 0 = 1.5 kHz is used to obtain
4096 samples during induction motor startup transient,
acquiring up to the tenth harmonic of fundamental
frequency, and beyond. Start of motor is controlled by a
relay in order to synchronize the data acquisition with
the motor switch on. Sampled data are arranged into a
32×128 matrix. Obtained matrix is processed by SVD
computation algorithm implemented in FPGA device along
with information entropy unit, which provides induction
motor condition based on the information contents from
obtained singular values.
HERNANDEZ-VARGAS et al: NOVEL METHODOLOGY FOR BROKEN-ROTOR-BAR AND BEARING FAULTS DETECTION
591
Fig. 3–Power-supply-current to induction motors under different operational conditions: a) Healthy; b) Motor with a broken bar; and c)
Motor with faulty bearing
Fig. 4–Density functions defining detectability zones obtained by proposed methodology to identify a healthy motor (HLT), a motor with
a broken-rotor-bar (BRB), and a motor with a faulty bearing (FBR)
Case Study 1: Broken Rotor Bars (BRB)
BRB condition is produced artificially by drilling one
hole (diam 7.938 mm) without harming rotor shaft
(Fig. 3a).
Case Study 2: Faulty Bearing (FBR)
FBR is produced artificially by drilling a hole (diam
1.191 mm) on its outer race using a tungsten drill bit
(Fig. 3b).
Hardware Implementation
Proposed methodology was implemented in a low
cost FPGA device Cyclone- II EP2C35F672C6 from
Altera, embedded on DE2 development board from
Terasic 26 with maximum operational frequency of 65.928
MHz. FPGA implementation figures (used and available,
respectively) in Altera device for various resources were
as follows: MULT 18×18, 21 of 35; logic elements, 7632
of 33216; and RAM bits, 434,643 of 483,840.
Results and Discussion
Utilizing FPGA implementation of proposed
methodology, startup transient current supply to a healthy
motor (Fig. 3a), a motor with one BRB (Fig. 3b), and a
motor with a FBR (Fig. 3c) is processed by computing
SVD of generated matrix. Then, information entropy from
obtained singular values is estimated. This process is
applied on 20 different trials for each motor condition.
Obtained entropy values on each condition are sent to a
PC for presentation purposes. Mean µ(H) and standard
deviation σ(H) are computed. A 3σ rule that ensures a
99.7% of certainty on detecting motor condition is used
(Fig. 4). Detectability zones defined by proposed
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J SCI IND RES VOL 71 SEPTEMBER 2012
Table 1Statistical study on obtained current signal and singular vales from generated 32×128 matrix
Statistical parameter
Current signal
Generated-matrix
Singular values
Mean
Standard deviation
RMS
Entropy
Mean
Standard deviation
RMS
Entropy
Motor condition
HLT
BRB
FBR
-0.0233
10.7055
10.7042
9.3858
22.4587
120.1740
121.7812
1.6276
-0.0236
10.9270
10.9257
9.2949
23.7738
122.5029
124.3065
2.0094
-0.0233
7.3456
7.3443
9.0368
24.7252
127.7475
127.9070
2.7170
Table 2Comparison chart of advantages and disadvantages of proposed methodology against earlier studies for
induction motor fault detection
Method (ref)
Detected faults
Applied analyses
Implementation
On/Off-line
[7]
Faulty bearings, 3
broken bars
Noise statistics, bias analysis,
FFT, statistical reliability test,
signature identification
Qualitative analysis in
software
Off-line
[8]
Shaft defect, faulty
bearings
Analysis of a laser beam of
reflection through installation
a robust optical sensor
Infrared thermography, 2D
Qualitative analysis by
voltage signal
observation
Off-line
[9]
Misalignment,
faulty bearing, mass
unbalance
wavelet transform, mahalanobis
distance, relief algorithm,
support vector machine,
linear discriminant
Quantitative analysis in
software
Off-line
[12]
1 Broken bar
FFT, wavelet transform, peak
measurement, and multivariate
control charts
Qualitative analysis in
software
Off-line
[14]
Voltage supply,
broken bars, and
locked rotor
Motor model, wavelet multi
resolution, artificial neural
networks
Qualitative analysis in
software
Off-line
[15]
Faulty bearings
Wavelet transform, support
vector machine, artificial neural
networks, mean square error
Qualitative analysis in
software
Off-line
[16]
Short circuit, 5
broken bars, and
faulty bearings
Wavelet packet, mean, variance,
standard deviation, skewness,
kurtosis
Qualitative analysis in
software
Off-line
[19]
1 broken bar
Hilbert transform, symbol tree
sliding window, information
entropy
Quantitative analysis in
software
Off-line
Proposed
1 broken bar, faulty
bearings
SVD, information entropy
Quantitative analysis in
Hardware (FPGA)
On-line
methodology are shown as density functions. From results
(Table 1) from distinct statistical analyses on obtained
startup-transient current data and singular values of
generated 32×128 matrix for each treated condition, it is
HERNANDEZ-VARGAS et al: NOVEL METHODOLOGY FOR BROKEN-ROTOR-BAR AND BEARING FAULTS DETECTION
clear that proposed methodology ensures 99.7% of
effectiveness on BRB or FBR detection in a quantitative
way, since corresponding detectability zones are clearly
and completely separated. From comparison chart
(Table 2), it is clear that proposed methodology offers a
simple approach optimized for hardware implementation
through SVD and information entropy estimation,
different from earlier studies that usually require
combination of standard, and non-standard techniques
executed off-line for single or multiple fault detection.
FPGA-based hardware implementation of proposed
technique is used for online analysis in real-time
applications. It employs 24.3152 ms to identify induction
motor condition, whereas a 2.4GHz Intel Core Duo
elapses 680.6819 ms to carry out this task.
Conclusions
This study introduced a novel methodology for early
detection of BRB and FBR in induction motors. Proposed
approach fuses singular value decomposition with
information entropy. Results show high effectiveness of
proposed technique utilizing a 3σ rule, which ensures
99.7% of certainty on estimating induction motor condition
in a quantitative way. FPGA-based hardware
implementation allows online analysis utilizing proposed
methodology in real-time applications.
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9
10
11
12
13
14
15
16
17
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