International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011 1
ISSN 2229-5518
A Wavelet based multiresolution analysis for real time condition monitoring of AC machine using vibration analysis
Subhra Debdas, M.F.Quereshi, A.Reddy, D.Chandrakar, D.Pansari
Abstract-Wavelet is a powerful tool used for non stationary signal analysis. It does not change the time information content present in the signal hence it provides a time-frequency representation of the signal. Using the wavelet technique, transients can be decomposed into series of wavelet components, in which each is a time-domain signal that covers a specific frequency band. Disturbances of small intervals are amplified frequency band. In this paper a multi-resolution based pattern recognition technique is used for vibration analysis of angle grinder machine by which different frequencies are analyzed with different resolutions. This method is more reliable as compared to other FFT based techniquesrelative to the rest of the signal when projected to similar size wavelet bases and, thus, they can be easily detected in the corresponding
Index Terms- Fault diagnosis, wavelet transform, multi resolution analysis, pattern recognition, wavelet density estimation.
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ondition monitoring is the process for monitoring any parameter of condition in machinery, such that a signifi-
cant change is indicative of a developing failure. The use of
condition monitoring allows maintenance to be scheduled, or
other necessary actions to be taken to avoid the consequences
of failure, before the failure occurs [1]. Nevertheless, a devia-
tion from a reference value must occur to identify impeding damages in the machinery. Condition monitoring for any ma- chine is much more cost effective than allowing the machinery to fail. The prime aim of vibration monitoring is the detection of changes in the vibration condition of the machine under investigation during its operation.
In industry machines are expected to run continuously with
their full capacity in order to meet the production needs. Any
defects in the machinery must be detected and should be ana-
lyzed at the early stage to avoid its failure. In this case planned
shutdown can be arranged to diagnose the causes of the prob- lem and to make further corrections. In opposite to this condi- tion, the unscheduled shut down of the machinery & equip- ment can cause enormous economic losses and may result high damage of the machine. So condition monitoring of vari- ous machines is gaining importance in every industry since it keeps the plant at healthy condition for maximum production, detecting and diagnosing the fault at very early stage to avoid serious accidents and machine damage and to run the plant economically [2].Most of the defects occurred in the machines give rise to a distinct vibration signature and hence mostly faults can be identified using vibration signature analysis techniques. A similar attempt have been tried out here for the conditioning monitoring of AC machine using wavelet tech- nique for its proper and economical functioning[3]. An AC machine is a handheld power tool used for cutting, grinding and polishing. Angle grinders may be used both for removing excess material from a piece or simply cutting into a piece.AC machines are widely used in metalworking and construction, as well as in emergency rescues. They are commonly found in workshops, service garages and auto body repairs.
1.1 From Fourier Analysis to Wavelet Analysis
Drawbacks of signal processing techniques used in power quality disturbances:
i) RMS is major tool used in signal processing techniques. The RMS of signal is not an analysis technique but it gives some basic information about an electrical system. The main disad- vantages of this algorithm is its dependence on size of sample window[6].As a result of small window RMS parameter be- comes less relevant and loses meaning of mean value of pow- er.
ii) Another most widely used tool in signal processing is
Fourier analysis. It helps in analysis of harmonics and essen-
tial tool for filter design. The DFT and FFT are essential tools for estimation of fundamental amplitude of signal. The DFT importance in area of frequency (spectrum) analysis as it takes a discrete signal in time domain and transforms that signal into the discrete frequency domain representation. A FFT used for transformation of signal from time domain to frequency domain. Speed is main advantage of this technique and also high speed calculations.
iii) In time frequency signal processing, a filter banks is special quadric time frequency distortion (TFD) that represents signal in joint time frequency domain. This technique used for esti-
mation of specific sub-band components.
iv) Another special type of filter is Kalman Filter .Their solu-
tions are based on set of state space equations. These are used
for real time tracking harmonics as proposed in [8], frequency
estimation under distorted signal [9], estimating voltage and
current parameters on power system protection and parame- ter of transient [10].
v) In 1994, use of wavelets was proposed which led to study of non stationary harmonic distortion in power systems. This technique decomposes signals in different frequency sub-
bands and characteristics can be studied separately.
vi) The STFT mainly used in power quality analysis and called
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International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011 2
ISSN 2229-5518
as sliding window version of FFT. The advantage of STFT is its ability to give the harmonic content of signal at every time period specified by defined window.
-m
"lfm n (t) = a 2
"lf(
t -n b0 am
) (4)
0
Where a = ao o o
m and b = nb am
The wavelet transform is representation of signal as sum of wavelets at different location and scales. The main advantage of wavelet transform is its varying length window. The wave- let transform can be classified in three different ways. The con- tinous wavelet Transform possesses ability to construct a time- frequency representation of signal that offers very good time and frequency realization. The second type of transform known as wavelet series which maps function of continous variables into sequence of coefficients. The third is Discrete wavelet in which wavelets discretely sampled and has advan- tage of temporal resolution as it captures both frequency and location information.
2.1 Why Discrete Wavelet Transforms
The continous wavelet transform was developed to overcome resolution problem to short time Fourier transform. It is corre- lation between wavelets at different scales and signal with scale being used as measure of similarity.DWT are applied to discrete data sets and produce discrete outputs. The DWT is special case of wavelet transform that provides a compact re- presentation of signal in time and frequency that can be com- puted efficiently. When compared to Fourier transform, wave- let can obtain both time and frequency information of signals while frequency information obtained by Fourier transform [3,
4, 16, 18, and 19]. The signal can be represented in terms of
both the scaling and wavelet functions as follows:
The DWT analysis can be performed using fast pyramidal al- gorithm related to multirate filter banks.
A multiresolution analysis is design method of most of the practically relevant discrete wavelet transform. A low pass approximation and high pass details can be obtained from original signal .The original signal divided into different scales of resolution whereas in Fourier transform it is divided in dif- ferent frequencies. A low pass filter removes high frequency componentsand high pass filter helps for picking high fre- quency components [3, 4, 16, 18, and 19].
The synthesized signal is finally decomposed into subbands of continous wavelets where each band represents part of origi- nal signal at particular frequency.
Following steps are taken to identify time interval of distur- bances of wavelet:
a) Actual data signal is generated,
b) Application of different wavelet transformation hav- ing suitable mother wavelet.
c) Identification of disturbance intervals with help of wavelet coefficients.
f(t) =∑n Cj (n)C(t − n) + ∑n ∑J=o dj (n)2 "lf(2J t − n)
(1)
J- 2J
Where cj is the J level scaling coefficient,
dj is the j level wavelet coefficient,
j(t) is the scaling function,
y(t) is wavelet function,
J is the highest level of wavelet transform,
t is time.
For practical applications and for efficiency reasons one pre-
fers continuously differentiable function with compact sup-
port as mother wavelet.
Wavelet theory can be expressed by continous wavelet trans- formation as,
00
Various power quality disturbances for small scale signal de- composition can be detected by use of choice of analysis of mother wavelet. Daub 4 and Daub 6 wavelets are useful for fast and short transient disturbances. Daub 8 and 10 are suita- ble for slow and long transient disturbances [3, 4, 16, 18, and
19]. At scale 1, mother wavelet localized in time and oscillates
more rapidly in short spam of time. As wavelet reaches higher
scale analysing wavelets become less localized in time and
oscillations, so as a result of high scale signal decomposition, fast and short transient disturbances detected st lower scales and for high scales, slow and long transient disturbances will be detected.
CWTx(a,b)=Wx(a,b) =∫-00 x(t)"lfa,b (t)dt
Where Ψa, a (scale) and b (translation) are real numbers.
(2)
Both time domain & frequency domain methods can be used
to analyze vibration signals. The time domain refers to a dis- play or analysis of the vibration data as a function of time. The
The discretization of this equation necessary for practical ap- plication.
For Discrete time system,
00
frequency domain approach allows both the amplitude &
phase spectrum to be an identified and are more useful for
vibration analysis. The Fourier transform is a frequency do-
main approach which converts a continuous time signal into
frequency domain. Fourier representation X (f) which is calcu-
DWTψx(m,n)= ∫-00 x(t)"lfm,n (t)dt
(3)
lated by the Fourier transforms integral shown by:
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International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011 3
ISSN 2229-5518
X(f) = ∞ x(t)e -i2nft dt (5)
The disadvantage of frequency-domain analysis approach is
that a significant amount of information (transients, non repe-
titive signal components) may be lost during the transforma-
tion process. This information is non retrievable unless a per- manent record of the raw vibration signal has been made. The problem of Fourier transform is overcome up to some extent using Short Term Fourier Transform. STFT is simply the result
translation parameter; and (t) is called mother wavelet. The wavelet function is given by
= t-b (10)
The Discrete Wavelet Transform (DWT) coefficients are usual-
ly sampled from the CWT on a dyadic grid parameters of
translation b = n*2m and scale a = 2m and is defined as,
t-n2m
of multiplying the time series by a short time window and
performing a discrete Fourier transform. Mathematically for a
m,n (t) = √2m
2m (11)
signal, it is written as
∞
It is not strictly a time-frequency representation but rather a
time-scale representation of the signal. WT can give a time-
frequency analysis if the centre frequency of the wavelet is
STFT{x(t)} ≡ X(r, w) = ∫ x(t)w(t − r)e -Jwt dt
(6)
estimated for each scale [6].
For discrete signals, this transform is known as Short Term
Discrete Fourier Transform (STDFT) expressed mathematical-
ly with signal x[n]& window w[n] as
5 PROPOSED DATA ACQUISITION SYSTEM FOR SIGNAL
ACQUISITION
The very first step of vibration monitoring of any equipment
STFT{x[n]} ≡ X(m, w) = ∑∞
x[n]w[n − m] e -Jwn (7)
or machine is the acquisition of the vibration signal. This sys- tem uses Muratha piezoelectric shock & vibration sensor (model no.PKS1-4A1) for measuring angle grinder vibration signals which is mounted on the surface of the AC machine. Vibration signals are amplified using pre-amplifier circuit & fed to PC using its audio port. Finally the vibration signal uses MATLAB environment for further processing using wavelet based multi resolution technique
Fig.1 Multi-resolution wavelet decomposition.
Application of STFT have been used to for analyzing different vibration signals for different application but having problem that time resolution is same for all spectral components. This problem is overcome by using the wavelet transform [4]. It is a technique which allows the time-frequency place to be divided in a more flexible way such that a smaller time is user for higher frequencies & larger time is used for lower frequencies. It is calculated by convolving the wavelet with the original signal, multiply the shifted wavelet with the original signal, then sum the result to produce a single value [5].
The continuous wavelet transform is defined as the convolu-
tion between the original signal s (t) and a wavelet a, b (t).
W (a, b) = ∫ ∞ S(t) (t)dt (8)
Fig.2. Vibration Signature of Angle Grinder machine
6 PROPOSED NEW METHOD USING MRA & DENSITY ESTIMATION
The multi-resolution analysis algorithm decomposes a signal into scales with different time and frequency resolution [7]. The fundamental concept involved in MRA is to find the av-
erage features and the details of the signal via scalar products
√a ∫-00 s(t)rp(
t-b
a
)dt (9)
with scaling signals and wavelets.MRA is designed to give good time resolution and poor frequency resolution at
Where s(t) is the input signal; ‘a’ is the scaling factor; ‘b’ is the
high frequencies and good frequency resolution and poor
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International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011 4
ISSN 2229-5518
time resolution at low frequencies[16]. The algorithm of wavelet signal decomposition is illustrated in Fig 1 where
f (t)
S (k)2j / 2(2j t k)
d j (k)2 j /2 (2 j t k)
down sampling operation is performed. In Fig 1, h represents
j
k k
low pass decomposition filter, g is the high pass decomposi-
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
tion filter. A
(t), A
(t) are approximation coefficient of origi- V
2 j / 2
2 j
1 2
j S p a n
t k
nal signal at level 1, 2 and D1 (t), D2 (t) are the detailed coeffi- cients at levels 1 & 2.wavelet Filter Banks are used to
represent the vibration signal in the next lower scale [17,
18].
Where
k
2 k N 1
S j ( k )
h ( m
2 k ) S
j 1 ( m )
d j ( k )
m2 k 2 Nk 1
m 2 k
g ( m
2 k ) S
j 1 ( m )
The sum of number of coefficients in sequence {sj(k)} and {dj(k)} is almost equal to the number coefficients in the sequence {sj+1(k)}
a7nd CnoOinNfoCrmLUatiSonIOisNloSst in the splitting of frequency bands of t
Six levels of decomposition have been selected in this paper to estimate density and detail coefficient of AC machine. This wavelet based decomposition technique identifies the vibra- tion fluctuation due to overloading on single phase fault con- dition, more easily and efficient way. An attempt has been carried to provide the necessary information of abnormal con- dition of an AC machine .The implementation technique can process vibration data with low consumption cost and in reli- able manner decomposed to other signal processing tech- niques.
Fig.3. Decomposition upto six level
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
[1] Salem A. Ansari and Rauf Baig, “A PC-Based Vibration Analyzer for Condition Monitoring of Process Machinery”, IEEE Transactions on Instrumentation and Measurement, VOL.47, No. 2, April 1998.
[2] D.Mori and Takeo Ishikawa, “Force and Vibration Analysis of Induc-
tion Motors”, IEEE Transactions on Magnetics, Vol. 41, No. 5, May
2005.
[3] David Hart, David Uy, Damir Novosel, Steven Kunsman, Carl
LaPlace and Marco Tellarini, “Improving Power Quality”, ABB Re-
V j
S p a n 2 j / 2
2 j
t k
view, 4/2000.
[4] Adly Girgis, Bi Chang and Elham Makram, “A Digital Recursive
k
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Measurement for On Line Tracking of Power Systems Harmonics”, IEEE transactions on Power Delivery, Vol. 6, No.3 July 1991.
V j
S p a n 2 j / 2
2 j
t k
[5] S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Appli-
k
cations. Englewood Cliffs, NJ: Prentice-Hall, 1996.
[6] Stefanos K. Goumas, Michael E. Zervakis, G. S. Stavrakakis , “Classi-
f (t)V
f (t) s k
( j1)/22j1(t k)
fication of Washing Machines Vibration Signals Using Discrete Wave-
j1
j1( )2
k
(12)
let Analysis for Feature Extraction”, IEEE Transactions On Instrumen-
tation And Measurement, vol. 51, No. 3, June 2002.
[7] S Mallet,A Theory for Multiresolution Signal Decomposition: The
Wavelet Representation, IEEE Trans.Patt. Recog. and Mach. Intell.,
The decomposition of signal is Vj+1 which is sum of two
11(7), July 1989, pp.674-693.
seuences with dj(k).
IJSER © 2011 http://www.ijser.org
International Journal of Scientific & Engineering Research Volume 2, Issue 10, October-2011 5
ISSN 2229-5518
[8] Irene Gu, Math Bollen, “Time Frequency and Timescale Domain Analysis of Voltage Disturbances”, IEEE Transactions on PowerDe- livery, Vol. 15, No.4 October 2000.
[9] Tongxin Zheng, Elham Makran and Adly Girgis, “Power System Transient and Harmonic Studies Using Wavelet Transform”, IEEE Transactions on Power Delivery, Vol. 14, N0.4, October 1999.
[10] Gerald Heydt and A. Galli, “Transient Power Quality Problems Ana- lyzed using Wavelets”, IEEE Transactions on Power Delivery, Vol.
12, No.2, April 1997.
[11] S. Santoso, J. P. Edward, W. M. Grady, and A. C. Parsons, “Power Quality Disturbance Waveform Recognition Using Wavelet-Based Neural Classifier—Part 1: Theoretical foundation,” IEEE Trans. Pow- er Delivery, Vol.15, pp. 222–228, Feb.2000.
[12] V.Wijayakulasooriya, G. A. Putrus, and P. D. Minns, “Electric Power
Quality Disturbance Classification Using Self-Adapting Artificial Neural Networks,”Generation, Transmission and Distribution, IEE Proceedings , Vol. 149, No. 1, pp.98–101, Jan. 2002.
[13] John Williams and Kevin Amaratunga "Introduction To Wavelets in Engineering", International Journal for Numerical Methods in Engi- neering, Vo1.37. pp. 2365-2388, 1994.
[14] A.W.Galli, G.T. Heydt, & P.F.Ribeiro, “Exploring the power of wave-
let analysis”, IEEE Compute. Applicat. Power, pp. 37–41, Oct. 1996.
[15] Aparna Karwal, “Vibration analysis of rotating machines: A Compet- itive analysis of different techniques”, International J. of Engg. Research
& Indu. Appls. (IJERIA).Vol.3, No. II (May 2010), pp 15-24.
[16] Dennis Lee RE Morrison, “Multiresolution based Pattern Recognition
Approach for Condition Monitoring of Switchgear”.
[17] L. L. Scharf, Statistical Signal Processing. Detection, Estimation, and Time
Series Analysis. Reading, MA: Addison-Wesley, 1991.
[18] S. Mallat and W. Hwang, “Singularity detection and processing with wavelets,” IEEE Trans. Inform. Theory, vol. 38, pp. 617–643, 1992.
First S.Debdas was born in Naihati, West Bengal, India, on November,
1978. He graduated in Electrical Engineer- ing in 2001 from the Bengal Engineering College Shibpour, Howrah, West Bengal, India and M.Tech in Electrical Power System from the Bengal Engineering and Science University Shibpour, Howrah, West Bengal, In- dia. The author is now a research scholar of NIMS Uni-
versity Jaipur, Rajasthan India. His special field of re- search includes power quality, harmonic detection and real time condition monitoring system. Now he is work- ing as Reader in Disha Institute of Management and Technology, Raipur, India. Mr.S. Debdas became a Mem-
ber (M) of IACSIT and IAENG.
Second M.F.Qureshi was born in Raipur, Chhattisgarh on 07th of July
1958.He received his B.E. in Electrical Engineering from GGDU, Bilaspur in the year 1984 and M.E in Electrical High Voltage from RDU Jabalpur in the year of 1998 and PhD Electrical Engineering from GGDU in 2004. His special field of research includes fuzzy type two, high voltage; power quality, harmonic detection and real time condition
monitoring system. Now he is working as Principal in
Govt. Polytechnic College, Janjgir-Champa, and Chhat- tisgarh, India.
Third A.Reddy was born in Raipur, Chattisgarh on 10th July 1989.She received her B.E.in Electrical and Electronics Engineering from Disha Institute of Management and Technology Raipur, Chattisgarh, India in year 2011 and currently she is working as Lecturer in Disha Institute of Management and Technology, Raipur, India.Her special field of interest is power quality.
Fourth D.Chandrakar was born in Raipur, Chhattis- garh on 28th of October 1988.She received her B.E. in Electrical and Electronics Engineering from Govern- ment Engineering college Raipur, Chhattisgarh,India in the year 2010 and currently she is a M-tech student in
Disha institute o management and technology, Raipur, Chhattisgarh. Her special field of interest includes power quality and power electronics.
Fifth D. Pansari was born in Raipur, Chhattisgarh on 9th of June 1988.She received her B.E. in Electrical and Elec- tronics Engineering from Government Engineering col- lege Raipur, Chhattisgarh ,India in the year 2010 and currently she is a M-tech student in Disha institute o management and technology, Raipur, Chhattisgarh. Her
special fields of interest include power quality and wave- let transformation.
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