Broken Bearings Fault Detection for a Permanent Magnet Synchronous Motor under non-constant working conditions by means of a Joint Time Frequency Analysis J. Rosero, J. Cusido, A. Garcia Espinosa, J. A. Ortega, L. Romeral
Motion Control and Industrial Applications Group, Universitat Politecnica de Catalunya. C/ Colom. 08222 Terrassa. Catalonia. Spain. e-mail:
[email protected] Abstract- This work is an approach to develop new and reliable tools in order to diagnose mechanical faults in PMSM motors under non constant working conditions. These kinds of faults (especially damaged bearings) are more present in the industry. The paper presents a theoretical overview of the different techniques for a joint time frequency analysis and an experimental study of detection and diagnosis of damaged bearings on a Permanent Magnet Synchronous Motor (PMSM). The experiments were performed with variable rotor speed in such a way that the conventional methods such as MCSA do not work well. The stator current is analysed by means of STFT and Gabor Spectrogram. Both results are presented and discussed.
Index Terms - PMSM drives, fault detection, bearing damage, STFT, Gabor Spectrogram. I. INTRODUCTION.
Many attempts have been made in order to detect and diagnose electrical and mechanical faults in electrical machines, such as shorted turns, broken bars, bearing damage and misalignment [1]. Nearly all of these methods are focused to obtain the electrical signature, known as MCSA (Motor Current Signature Analysis). The MCSA objective is to determine the harmonics contained in the stator current, so that each set of harmonics corresponds to a sort of failure. Nevertheless this method is not so successful in low power machines, or under non stationary conditions of load or velocity. The temporal change of the working conditions results in a shift of the harmonic frequencies. The FFT transform performs the average and the fault condition is missed. This can be seen in Fig. 1. The best way to analyze the frequencies of non stationary signals is by means of a joint time-frequency analysis [2]. This kind of transforms shows how the frequency content varies with time and then fault condition is easier shown. STFT is one of the main signal transforms used to perform a time-frequency analysis. It is a simply and intuitive method, but the time and frequency resolution depends on the spread in time and frequency of the selected window (Rectangle, Hamming, Gaussian, Hanning, etc.), that has to be selected prior to start the analysis. The frequency and time resolution remains constant for the full range of times and frequencies.
1-4244-0755-9/07/$20.00 (C2007 IEEE
The Wigner Ville Distribution (WVD) presents the best joint time and frequency resolution [3], [4]. Nevertheless, WVD is a quadratic transform and if the signal has several components it presents the problem of cross-term interference. For the fault detection purpose, this is a great drawback. There are several ways in order to mitigate the cross-term interference. Due to the oscillating nature of the cross-terms, it is possible to smooth the WVD both in time or in frequency domains, obtained the Pseudo Wigner Ville Distribution (PWVD). This kind of transform effectively suppresses cross-terms that correspond to a pair of autoterms with different time or frequency centres, but not both simultaneously [5]. All the cross-terms could be effectively suppressed by means of a Gabor expansion, obtaining the Gabor Spectogram [6], [7]. In this work we present the results obtained analysing non stationary current signals by means of the time-frequency transforms STFT and Gabor Spectogram. Two PMSM motors have been tested, both with nominal load while decreasing the velocity from 3000 rpm to 2000 rpm. One motor has a bearing damage that consists in a destroyed ball in the DE, whereas the other is healthier than the first one.
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10 0
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-40. 60
100
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i
i
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i
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mF0 1 0 100 2000 250 303
*
Frequek~(Hz)
I
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350 400X 450 5
Fig 1. FFT of the PMSM stator current under non stationary working conditions.
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Section II is devoted to a signal processing overview, section III presents the experimental results, and section IV is for conclusions. Finally acknowledgments and references are also provided. II. SIGNAL PROCESSING OVERVIEW Fourier analysis separates a signal into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a mathematical transform for convert our view of the signal from time-based to frequency-based. In transforming to the frequency domain, time information is lost. When looking at a Fourier transform of a signal, it is impossible to tell when a particular event took place. If the frequency contents do not change over time - that is, if it is what is called a stationary signal - this drawback is not very important. In our case, we are interested in analyzing current signals provided by motors that work under variable conditions, i.e. speed. This results in non-stationary signals. Fourier analysis is not suitable for this kind of signals and a Joint Time Frequency Analysis (JTFA) has to be performed. Short Time Fourier Transform (STFT) a) A simple way to introduce time in Fourier Transform (FT), consist in windowing the signal around a particular time and then apply FT, with this process we obtain the so called Short Time Fourier Transform (STFT). Mathematically speaking, it is defined by: 00
Fx (t,v; h) Jx(u)h* (u t)e-j27TvudU -
(1)
_00
Where x(t) is the signal to analyse and h(t) is the selected window. We can also evaluate the STFT-based spectrogram as the square of the STFT. Fourier transform is applied to each data block to indicate the frequency contents of each if them. STFT roughly reflects how frequency contents change over time. The duration of the window, or data block size, determines the time accuracy: the smaller the block size, the better the time resolution. However, frequency resolution is inversely proportional to the size of a block. While the small block yields good time resolution, it also deteriorates the frequency resolution and vice versa. This phenomenon is known as the window effect.
b)
Wigner-Ville Distribution (WVD) The Wigner-Ville Distribution [3] is defined as:
Wx (t,v) =2x(t±v)x (t _
-2IvTdr
The Wigner-Ville Distribution shows the best jointed frequency-time resolution, but unfortunately it is severely affected by the cross-term interference [8]. As the WVD is a bilinear function of the signal x, the quadratic superposition principle applies:
WX+Y (t,v) = Wx (t, v) + WY (t,v) + 291W ly (ti v);
(2)
(4)
Where
±+)Q( -
Wx y (tj v) =
4 -j2rv dr
(5)
This is the cross-WVD of x and y and results in the crossterm interference. This result is easily generalized to N components. Pseudo Wigner-Ville Distribution c) One way to mitigate the effects of the cross-term interferences consists in windowing the Wigner Ville Distribution in time or smoothing in frequency domains [4], obtaining the Pseudo Wigner-Ville Distribution, PWx(t,v), as it is shown in (6):
PWx (t, v) = h(t - r)Wx (r,v)dr JH(v - E)WX (r, )de (6) -00
-GO
The filter process results in a partial reduction of the crossterm interference. For the temporal windowing, are suppressed only the cross-terms with different time location. On the other hand, for the frequency smoothing the crossterms suppressed are those with different frequency location.
d)
Gabor Spectrogram An effective way to suppress all the cross-terms is to expand the signal by means of the Gabor Expansion [9]. The signal is decomposed in a series of shifted and modulated Gaussian functions weighted for coefficients, as it is shown in (7).
x(t)
or equivalently as:
Wx (t,v) = X(v±+ )x(v-
Where X(w) is the Fourier Transform of x(t). It is well known that the power spectrum of a given signal, P(w), could be calculated as the Fourier Transform of the signal autocorrelation, R(r). Taking this into account, we can consider the Wigner-Ville Distribution as the Fourier Transform of the instantaneous autocorrelation, and thus the instantaneous power spectrum of the signal, or the signal energy distribution in the joint time-frequency plane.
00
=
00
inft ZCm,nh(t-mT)e m=-oo n=-oo
(7)
where Cm,n are the Gabor coefficients and h(t) is a Gaussian Function defined as:
h(t) (
(3)
3416
a
Ye -at2
(8)
where a is a constant value. Taking into account the Gabor Expansion, the Wigner- Ville Decomposition could be rewrited as:
mf (t)
WVDx (t, v) = =
00
00
Z
Z
00
m=-oon=-oom
00
ZCm,nC mt,n'WVDh,h'(t,V)
Z
where WVDh,h '(t,v) is the cross-WVD of two time frequency- shifted Gaussian functions.
-
and
WVDh,h '(t,V) could be considered as an energy atom, that is, it is well localized both in time and in frequency. In (9) it is indicates that the energy of the signal can be thought of as the sum of an infinite number of energy atoms. The highly oscillated atoms are directly associated with cross-term interference, but have negligible influence to the signal energy. The greater rm-m'r+jn-n'j the more interference is the atom. Taking this into account, the Gabor Spectogram, based in the Gabor Expansion [10] is defined as: 00
GSD (t,v)
Cm,nc ml,n,WVDh,h'(t,V) (10)
|m+m'+|n+n'|
