Shock and Vibration

Volume 2015 (2015), Article ID 708034, 14 pages

http://dx.doi.org/10.1155/2015/708034

## Comparative Study of Time-Frequency Decomposition Techniques for Fault Detection in Induction Motors Using Vibration Analysis during Startup Transient

^{1}HSPdigital-CA Telematica, Procesamiento Digital de Señales, DICIS, Universidad de Guanajuato, Carretera Salamanca-Valle km 3.5+1.8, Palo Blanco, 36700 Salamanca, GTO, Mexico^{2}Department of Electrical Engineering, University of Valladolid (UVa), 47011 Valladolid, Spain^{3}HSPdigital-CA Mecatronica, Facultad de Ingenieria, Universidad Autonoma de Queretaro, Campus San Juan del Rio, Rio Moctezuma 249, 76807 San Juan del Río, QRO, Mexico

Received 15 April 2015; Revised 4 June 2015; Accepted 8 June 2015

Academic Editor: Francesco Franco

Copyright © 2015 Paulo Antonio Delgado-Arredondo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Induction motors are critical components for most industries and the condition monitoring has become necessary to detect faults. There are several techniques for fault diagnosis of induction motors and analyzing the startup transient vibration signals is not as widely used as other techniques like motor current signature analysis. Vibration analysis gives a fault diagnosis focused on the location of spectral components associated with faults. Therefore, this paper presents a comparative study of different time-frequency analysis methodologies that can be used for detecting faults in induction motors analyzing vibration signals during the startup transient. The studied methodologies are the time-frequency distribution of Gabor (TFDG), the time-frequency Morlet scalogram (TFMS), multiple signal classification (MUSIC), and fast Fourier transform (FFT). The analyzed vibration signals are one broken rotor bar, two broken bars, unbalance, and bearing defects. The obtained results have shown the feasibility of detecting faults in induction motors using the time-frequency spectral analysis applied to vibration signals, and the proposed methodology is applicable when it does not have current signals and only has vibration signals. Also, the methodology has applications in motors that are not fed directly to the supply line, in such cases the analysis of current signals is not recommended due to poor current signal quality.

#### 1. Introduction

Induction motors are one of the most used machines in the world. The applications are varied and the advantages of their use are numerous. About half of the electricity consumed by the industry in the U.S. is used by induction motors; in fact, 89% of the engines in manufacturing are electric motors [1]. They are also present in various modes of transportation. As a result, they are basic elements in the modern industrial world. From this arises the need for quick and accurate fault diagnosis for anticipating work stoppage in the processes where these machines are used. Failures in induction motors can occur in any of their three major components: rotor, stator, and bearings [2]. Actually, 38% of failures occur in the stator, 10% are located in the rotor, and around 40% represent mechanical failures including bearing damage, misalignment, eccentricity, and shaft bending [3].

The most popular techniques for fault detection in induction motors are the motor current signature analysis (MCSA) and the vibration analysis. MCSA allows noninvasive fault diagnosis online [4]. This technique uses a Hall-effect sensor to measure signals of stator current and a data acquisition system to acquire the signal [5]. This current signal is then analyzed to determine the signature and characteristic features of components associated with different faults, which can be magnified during the startup transient because the motor operates in stressed conditions [6]. Vibration analysis techniques, on the other hand, are used to make the diagnosis of faults in induction motors using vibration signals from three spatial axes [7]; this technique is particularly suited for determining mechanical faults. The measurement of vibrations is made using accelerometers as primary sensors. The vibration signals are then registered with the data acquisition system [5]. Vibrations in an induction motor are affected by variations in the magnetomotive forces caused by faults in the machine or faults that cause abnormal motor rotation. This technique is noninvasive and does a very feasible job at finding the location of faults [8]. These two techniques can be used both during the startup transient and steady-state operating regimes. Depending on which operating regime is used for monitoring the motor condition, there are certain characteristics associated with different fault conditions.

The set of techniques for fault diagnosis of induction motors, by analyzing the startup transient vibration signals, is not as widely used compared with those that analyze current signals. Vibration analysis and MCSA give a fault diagnosis focused on the location of spectral components associated with faults, using the Fourier transform which translates a signal from the time domain to the frequency domain, displaying the entire frequency content of a signal, but does not allow observing of the evolution of the signal frequency content over time. This is the reason for extending the Fourier transform capabilities such as linear time-frequency decompositions (short-time Fourier transform and wavelet transform), quadratic time-frequency decompositions, and distributions of time-frequency energy, which allow the evolution of frequency content throughout the duration of the signal in time. This is very useful for nonstationary signals, such as those obtained from the motor startup transient, because signatures associated with faults evolve over time and their frequency content does not remain constant but varies along the startup transient. To locate the signatures associated with motor faults, different tools have been used on MCSA for time-frequency decomposition, allowing tracing of the evolution of such frequencies in time. Examples of these decompositions are the short-time Fourier transform [9–12], discrete wavelet transform [12–15], continuous wavelet transform [16–19], the Hilbert transform [20, 21], the Hilbert-Huang Transform [20, 21], the Wigner-Ville distribution [22–27], the Choi-Williams distribution [26–28], and multiple signal classification (MUSIC) [5]. Some of these tools work together with artificial intelligence classifiers for decision-making about the components or signatures that are present in the signals for identifying faults and their severity, such as artificial neural networks (ANN), fuzzy logic, fuzzy neural networks, and genetic algorithms [6, 10, 14, 16, 17, 24, 29]. Garcia-Perez et al. [5] presented a study for the detection of multiple faults in an induction motor by applying MUSIC to a current signal during the steady-state regime. Afterwards, Garcia-Perez et al. [30] extended the fault detection method of multiple faults in an induction motor with MUSIC including sound signals along with vibration signals, also during the steady-state regime. Rodriguez-Donate et al. [31] developed a method for the identification of multiple faults in an induction motor directly fed to the power grid, based on the discrete wavelet transform (DWT) applied to the startup vibration transient. Pilloni et al. [19] presented a comparative study of different methodologies including the fast Fourier transform (FFT), Hilbert transform (HT), DWT, continuous wavelet transform (CWT), and the Wigner-Ville distribution (WVD) applied to the stator current signal in induction motors for fault detection in both steady-state and transient regimes. Garcia-Perez et al. [32] presented an experimental study of the time-frequency evolution characteristics during the startup transient of the current signal in an induction motor with a partially broken rotor bar when fed directly to the power grid applying MUSIC. Most of these techniques have been used to analyze stator current signals. In the case of vibration signals, there is not much research done to apply time-frequency decomposition techniques for analysis, where mostly the FFT [33] and the Zhao-Atlas-Marks (ZAM) distribution [34] for time-frequency decomposition have been used. Consequently, there is a need to investigate the suitability of time-frequency decomposition techniques to identify motor faults during the startup transient using vibration signals, applying a high-resolution spectral analysis as the MUSIC method.

The contribution of this work is a comparative study of different time-frequency analysis methodologies that can be used for detecting faults in induction motors analyzing vibration signals during the startup transient. The proposed methodologies are the time-frequency distribution of Gabor (TFDG), the time-frequency Morlet scalogram (TFMS), MUSIC, and FFT. The choice of TFDG and TFMS is based on their ability to reduce interferences, also known as cross-terms between parallel evolving harmonics. MUSIC is the technique used in MCSA that has provided the best results, due to its very good frequency resolution. FFT is used as reference for comparison purposes with the other methodologies. The analyzed vibration signals in this paper are associated with the following motor faults: one broken rotor bar, two broken bars, unbalance, and bearing defects.

#### 2. Description of the Treated Faults

Three common faults in induction motors are treated in this paper: unbalance condition (UNB), bearing faults (BDF), and broken rotor bars (BRB).

##### 2.1. Unbalance (UNB)

Mechanical balance in an induction motor involves the entire rotor structure which is made up of a multitude of parts including shaft, rotor laminations, end heads, rotor bars, end connectors, retaining rings, and fans. These many items must be designed and manufactured for an end assembly that achieves stable precision balance. When a motor is properly balanced and aligned, the frequency amplitude associated with the unbalance fault barely changes and remains bounded to a certain level. However, a mechanical unbalance in the induction motor involves a small radial vibration of the stator structure. The vibration level takes its maximum when the rotational speed equals the system natural frequency defined by , where is the stiffness factor and is the unbalance mass [35, 36]. In addition, this vibratory signal oscillates at the rotational frequency, the vibration level is increased as well as the amplitude of the rotation frequency, and the same happens with its harmonics [37]. The analysis of vibration signals can provide a quick and easy way to extract information that permits the diagnosis about the presence of unbalance in an induction motor. The rated speed of the motor, normally expressed in revolutions per minute, is provided by the manufacturer. In asynchronous motors, this speed is slightly below the synchronous speed, , that is related to the electrical supply frequency as follows:where is the number of poles of the motor and the constant “120” is used to express the motor synchronous speed in revolutions-per-minute units.

##### 2.2. Bearing Faults (BDF)

Some authors [38] give a review of the causes and expected frequencies of vibration due to rolling element bearings. A variety of frequencies associated with the rotation of the motor can be calculated from the geometry of the bearing, such as the inner and outer race elements pass frequencies, the frequency of rotation for the cage, and rolling element spin frequency. A defect on the outer race causes an impulse each time rolling elements contact the defect. The rotor speed () is the frequency at which the inner raceway rotates, which must be the frequency of the shaft. The physical phenomenon of the vibration generated in rolling elements such as bearings under the healthy condition can be explained as a combination of different sources such as modulation due to nonuniform loading, flexural bearing modes, and machinery-induced vibrations and noise. The bearing load is assumed to be an unbalanced force. Therefore, the radial load moves around the circumference of the outer ring as the shaft rotates. The single radial load transforms to a distributed load because the inner ring is in contact with more than one ball during the rotation. However, for the bearing fault condition when a defect in one surface of a bearing strikes a mating surface, an impulse is produced which excites resonances in the system. At time , the defect is in contact with one of the rolling elements and lies at the center of the load zone on the line of action of the applied radial load. The mechanical system is symmetrical about the line of the applied load. As the bearing rotates, impacts occur at the ball-pass outer raceway frequency () given by [39]where is the contact angle between the bearing surfaces, is the cage diameter of the bearing and is measured from a ball center to the opposite ball center, is the ball diameter, and is the number of balls in the bearing.

##### 2.3. Broken Rotor Bars (BRB)

In the case of rotor bars, it is known that symmetrical currents in a symmetrical rotor of an induction motor induce a resultant forward rotating magnetic field at synchronous speed with healthy rotor bars. The broken rotor bars result in rotor asymmetries; then there results a backward rotating field at slip frequency with respect to the rotor. Interactions of the rotor backward rotating field with the stator field induce oscillating torque and oscillating velocity, and the frequency of this oscillation is , where this oscillation acts as a frequency modulation on the rotation frequency and a fault frequency () appears around in the vibration spectrum [40]:where is the supply frequency, is the per-unit motor slip, is the rotor speed, and and are positive integers. The slip is defined as the relative mechanical speed of the motor with respect to the motor synchronous speed as follows:

#### 3. Theoretical Background

##### 3.1. Music Algorithm

The subspace methods are known as high-resolution methods that detect frequencies with low signal-to-noise ratio. The subspace methods assume that the discrete-time signal can be represented by complex sinusoids in noise [5] aswithwhere is the number of sample data, is the complex amplitude of the th complex sinusoid, is its frequency, and is a sequence of white noise with zero mean and a variance . This method uses the eigenvector decomposition of to obtain two orthogonal subspaces. The autocorrelation matrix of the noisy signal is the sum of signal and noise autocorrelation matrices ( and , resp.):where is the number of frequencies and the exponent denotes the Hermitian transpose. is the identity matrix, and is the signal vector given byFrom the orthogonality condition of both subspaces, the MUSIC pseudospectrum is given bywhere is the noise eigenvector. This expression exhibits the peaks that are exactly at frequencies of principal sinusoidal components, where .

##### 3.2. Time-Frequency Distribution of Gabor

An extended version of the STFT (short-time Fourier transform) is the time-frequency distribution of Gabor (TFDG), which uses a Gaussian window type and a FT (Fourier transform) to achieve the time-frequency analysis [41]. The TFDG has a tradeoff drawback as the STFT, caused by the fixed width of the window, but it has better resolution in frequency than the STFT.

The TFDG is described by the following equation:As in the continuous case, the discrete TFDG is identical to discrete STFT, with the particular characteristic of the Gaussian window. The discrete TFDG can expand as a linear combination of Gabor coefficients and basic functions [42, 43]. For a finite set of data , the Gabor expansion is obtained fromwhere the array is periodic in and with period . The sequence is known as the synthesis window. The array of Gabor coefficients can be found via the TFDG:where the sequence is the analysis window. The Zak transform is used to obtain the Gabor coefficients. The discrete Zak transform of a periodized window is defined as one-dimensional discrete Fourier transform of the sequence :where and are adjustment parameters, is the period, and is the sample length.

##### 3.3. Morlet Scalogram

Since the continuous wavelet transform behaves like an orthonormal basis decomposition, it can be shown that it preserves energy:where is the energy of . This leads us to define the scalogram of as the squared modulus of the continuous wavelet transform. It is an energy distribution of the signal in the time-scale plane, associated with the measure .

As for the wavelet transform, time and frequency resolutions of the scalogram are related via the Heisenberg-Gabor principle: time and frequency resolutions depend on the considered frequency [44].

The frequency resolution is clearly a function of the frequency as it increases with . The interference terms of the scalogram are restricted to those regions of the time-frequency plane where the corresponding autoscalograms (signal terms) overlap. Hence, if two signal components are sufficiently far apart in the time-frequency plane, their cross-scalogram is essentially zero.

The Morlet wavelet is the most popular complex wavelet used in practice, whose mother wavelet is defined aswhere is the central frequency of the mother wavelet. Note that the term is used for correcting the nonzero mean of the complex sinusoid, and it can be negligible when . Therefore, in some research the mother wavelet definition of the Morlet wavelet is given bywhere the central frequency . The Morlet wavelet has a form very similar to the Gabor transform. The important difference is that the window function is also scaled by the scaling parameter, while the size of window in Gabor transform is fixed [45].

#### 4. Validation of the Proposed Techniques

To validate the proposed methodology a synthetic signal is generated as stated in (17) with the aim to emulate some vibration harmonics, present in real signals of an electric motor. The synthetic signal has three pure sinusoidal signals with constant frequency at 80, 670, and 700 Hz, plus a sinusoidal signal with variable frequency ranging from 0 Hz to 56.7 Hz from 0 to 2 s and then remaining with constant frequency. Normally distributed random noise is also added to the signal. The constant frequency components located at 670 and 700 Hz are used to evaluate the method performance to discriminate close components. The sinusoid with variable frequency emulates a startup transient reaching the steady state at 56.7 Hz, which is closely located to a constant frequency component at 80 Hz. Finally, Gaussian noise () is added to evaluate the behavior of the treated method to low signal-to-noise ratio signals. The synthetic signal is quantized at a sampling frequency of 1.5 kHz, comprising 4096 samples for a total running time of 2.73 s. Also the synthetic signal has a signal-to-noise ratio equal to SNR = −3.6 dB:where is the synthetic signal, , , , andFigure 1 shows the results of the validation process for the synthetic signal in the time domain and the time-frequency decomposition obtained with the treated methods. In Figure 1(a) the time-domain synthetic signal is depicted. The true (theoretical) time-frequency decomposition is shown in Figure 1(b). The time-frequency decompositions of the treated methods are depicted in Figure 1(c) for the STFT, Figure 1(d) for the TFDG, Figure 1(e) for the TFMS, and Figure 1(f) for MUSIC.