Feature-Based Bearing Fault Classification Using Taylor–Fourier Transform

Abstract

This paper proposes a feature-based methodology for early bearing fault detection and classification in induction motors through current signals using the digital Taylor–Fourier transform (DTFT) and statistical methods. The DTFT allows the application of narrow bandwidth digital filters located in the spurious current signal components, wherewith it is possible to gain information to detect bearing issues and classify them using statistical methods. The methodology was implemented in MATLAB using the digital Taylor–Fourier transform for three fault types (bearing ball damage, outer-race damage, and corrosion damage) at different powering conditions: power grid source at 60 Hz and adjustable speed drive applied (60 Hz, 50 Hz, 40 Hz, 30 Hz, 20 Hz, and 10 Hz) in loading and unloading conditions. Results demonstrate a classification accuracy between 93–99% for bearing ball damage, 91–99% for outer-race damage, and 94–99% for corrosion damage.

1. Introduction

The most commonly used electric machines are induction motors (IM) due to their sturdiness, low cost, and simple installation and maintenance [1,2,3]. IMs are used in several industrial applications such as electric transport, elevators, air compressors, water treatment, wind turbines, distribution, pumps, and fans, among others [4]. Bearing damage is the most common fault in IMs because these are under a large amount of mechanical and electric stress. This kind of damage is inevitable and represents a severe problem for the motor’s condition and the electric systems where it is located. Bearing damage may cause progressive deterioration in the hole system, energetic losses, vibrations, overheating, etc. To prevent catastrophic injury, a lot of measurement devices can be installed in the industry to detect abnormal conditions in electric systems, like phasor measurement units or power quality analyzers, to name a few; specifically for bearing damage, elements like vibrometers, thermographic cameras, microphones, and electric current flow detectors are commonly used to detect and classify abnormal conditions. The effectiveness of the electric measurement devices for fault detection also depends on the algorithm used to process the signal. The works in [5,6] mention that 41% of the IM critical damage is caused by the lack of proper detection of the bearing fault. In this context, bearing fault detection and classification are indispensable.

Several methods have been explored for fault detection based on vibration signals [7,8,9], stray flux signals [10,11,12], and current signals [13,14]. The last one stands out, considering that it is a type of signal that contains information useful for several applications. Thus, it is frequently monitored, so no additional sensor would be required. Techniques based on motor current signal analysis (MCSA) have been proposed to detect and classify bearing damage. Some of them are presented in this section to identify the proposed methodology’s advantages.

One of the most used electrical signal processing techniques is the discrete Fourier transform (DFT), which provides accurate results for steady-state signals [15]. The problem is that the DFT needs to provide acceptable results under transient-state, i.e., when signal amplitude and/or phase are time-varying, which is not a real representation of an electrical system.

In addition to the study of signal processing techniques, detection and classification algorithms based on artificial intelligence or machine learning are popular nowadays [16,17]. Methods like support vector machines, complex decision trees, convolutional neural networks, and deep learning, as presented in [18,19,20,21,22], are widely studied and report results with up to 100% of accuracy. The problem with these methods is that they represent high computational resource consumption and previous training processes. Generally, they depend on databases or dictionaries to detect and classify bearing faults. In most cases, robust equipment is required to process the big data quantities.

For instance, in [18], a technique based on infrared thermography and machine learning is proposed in which the achieved accuracy in most cases is near 100%. In that methodology, the authors ensure that this technique prevents noise-related issues and provides excellent performance for bearing fault diagnosis. Still, detecting faults with noise introduced by an adjustable speed driver needs to be explored. Moreover, it uses methods like training, image processing, complex decision trees, and support vector machines, which are computationally demanding.

In [23], a model-based analysis and quantification of bearing faults in IM is presented based on integrating electrical and mechanical models. In that work, it is possible to detect bearing faults and their severity by estimating the air gap length variation. Furthermore, the authors mention that this methodology can classify and assess the bearing fault severity under any power rate, speed, and load conditions. Still, its real-time implementation represents a technical challenge based on the long processing time.

Recently, a combination of different techniques have been used in order to analyze vibration or current signals from electric machines, which is relevant for researchers because multi-modal analysis could obtain reliable information about the state of the electromechanical system and its dynamics. For instance, in [24], authors analyze the vibration response of the spindle-bearing-housing-belt system with rubbing by the analysis of a theoretical model and finite element method. They analyze the geometry and material parameters in the vibration response, and they also use the Lankarani–Nikravesh contact model to analyze the interaction between the shaft rotor and the front cover stator. Finally, the authors discuss the electromechanical effects of vibrations and the interaction between the elements on the electric machines.

Then, it is noticeable that a great amount of work has been proposed for fault detection relying on computationally complex signal processing algorithms or artificial intelligence-based classifiers. In addition, methods that are based on a classical Fourier-transform approach are constrained to the steady state analysis or require the preprocessing of the signal.

In this regard, this paper proposes an algorithmic implementation of the digital Taylor–Fourier transform (DTFT) for a signal-based methodology to detect and classify bearing faults, with the objective to provide a different alternative for bearing fault detection by using the filtering properties of the DTFT. One of the main advantages of this method is that the filter process only depends on a matrix product that was previously prepared offline, allowing it to be implemented in a real-time system for detection and classification. Other important attributes include its filtering properties, smooth estimates, harmonic rejection capabilities, and an implementation based on the dynamic phasor concept that allows for improved signal approximations from real electric systems. In addition, this methodology could represent a good option considering that, usually, artificial intelligence-based classifiers are more challenging to implement than statistical indicators, such as the ones selected in this work.

2. Fundamentals

2.1. Bearing Damage Signature

Bearing damage may appear in any element that makes up the bearing, either its fixed or mobile parts. The aspects under study in this paper are shown in Figure 1, where rolling elements are highlighted in red, and the guide rings are respectively named as outer race and inner race.

Depending on the damage type, defects cause the appearance of characteristic pulses on the frequency spectrum. These spurious frequencies are located at the sidebands of the primary frequency. Their location depends on the applied load, nature and geometry of the bearing, while its amplitude depends on the severity of the damage [6,25,26]. Figure 2 shows the change of the spurious frequencies regarding the severity of the damage, where $f_m$ is the fundamental system frequency, and $f_d^{-}$ and $f_d^{+}$ are the characteristic spurious frequencies. Figure 2a displays a motor damage behavior in the initial stages, Figure 2b exhibits a motor damage behavior in the intermediate stages, and Figure 2c presents a motor damage behavior of a severe failure. As can be observed, the severity of the damage provokes an increase in the spurious components. Likewise, a high mechanical load can provoke the displacing of the spurious components moving away from the fundamental component.

2.1.1. Localized Defects

A localized defect is a punctual imperfection in a specific part of the bearing. This kind of defect commonly causes characteristic vibrations on the IM, depending on the bearing type. The spurious frequencies in the vibration spectrum for inner race damage, outer race damage, and bearing ball damage can be identified respectively by the Equations (1)–(3) [27].

$$f_o=\frac{N_b}{2}{f}_r[1-\frac{Db}{Dc}cos\left(\beta \right)]$$

(1)

$$f_i=\frac{N_b}{2}{f}_r[1+\frac{Db}{Dc}cos\left(\beta \right)]$$

(2)

$$f_b=\frac{D_b}{D_c}{f}_r[1-\frac{D{b}^2}{D{c}^2}cos{\left(\beta \right)}^2]$$

(3)

where $f_o$ symbolizes the spurious frequency of the outer race damage, $f_i$ stands for the inner race’s damage frequency, $f_b$ represents the bearing ball damage frequency, $f_r$ defines the mechanical rotor frequency, $N_b$ is the number of the balls, $\beta$ is the angle between the ball and the defect, $D_b$ is the diameter of the ball and $D_c$ is the pass diameter. Some of these geometric parameters ($\beta$, $D_b$ and $D_c$) are illustrated in Figure 3. As a consequence of these vibrations, air gap variations are induced, causing variations in the stator current. Then, spurious components in the current spectrum are given by

$$f_{oI}=f_s\pm k{f}_v$$

(4)

where $f_{oI}$ represents the characteristic frequency in the current spectrum, $f_s$ stands for the source frequency, k symbolizes the harmonic, and $f_v$ is the vibration characteristic frequency.

2.1.2. Nonlocalized Defects

The nonlocalized defects are scattered in the bearing, meaning that the abnormal conditions are in many places and cannot be considered punctual. Thus, there are no feature vibrations, and there are no punctual spurious frequencies. An example is oxide damage, a condition under study in this paper. To analyze the current signals of the bearing under these defects, the DTFT is employed; its theoretical background is given next.

2.2. Digital Taylor–Fourier Transform

To estimate the amplitude of fault components of an induction motor, current signals are filtered using the digital Taylor–Fourier transform. The DTFT is defined as an expansion of the traditional Fourier model through the Taylor polynomial based on the dynamic phasor concept. Starting on the classical continuous-time signal model expressed as

$$s\left(t\right)=Re\left\{a\left(t\right)e^{\phi \left(t\right)}{e}^{j2\pi f_{1}t}\right\}$$

(5)

where $p\left(t\right)=a\left(t\right)e^{\phi \left(t\right)}$ represents the dynamic phasor, $a\left(t\right)$ is the amplitude and $\phi \left(t\right)$ is the phase, both are time varying, and $f_1$ represents the primary signal frequency [28].

The dynamic phasor $p\left(t\right)$ can be better approximated by a closed Taylor polynomial centered at $t_0$ and truncated at K, which is represented as

$$p_K\left(t\right)=p\left(t_0\right)+\dot{p}\left(t_0\right)(t-t_0)+\dots +p^K\left(t_0\right)\frac{{(t-t_0)}^K}{K!}$$

(6)

where $t_0=-\frac{T}{2}\le t\le t_0+\frac{T}{2}$ and T is the signal period.

Then, the traditional Fourier model in dynamic phasor terms can be written as follows,

$$X\left[k\right]=\sum _{n=0}^{N-1}p\left(t\right)e^{j2\pi f_{1}t}{e}^{-j\frac{2\pi }{N}kn}$$

(7)

where N is the number of samples, n is the sample number and $k=0,1,\dots ,N-1$ is the harmonic order.

Finally, the Taylor–Fourier subspace is a linear combination represented by

$${\widehat{s}}_{CN}={\mathbf{B}}_{\mathbf{CN}}{\widehat{\xi }}_{CN}$$

(8)

where the Taylor–Fourier matrix ${\mathbf{B}}_{\mathbf{CN}}$ comprises the results of modulating the Taylor terms in (6) by the Fourier coefficients in (7), ${\widehat{s}}_{CN}$ is the estimated signal and ${\widehat{\xi }}_{CN}$ represents the estimated Taylor–Fourier coefficients.

$${\mathbf{B}}_{\mathbf{CN}}=\frac{1}{N}\left(\begin{array}{cccc}\mathbf{I}& {\mathbf{T}}_1& \dots & \mathbf{T}{}_1^K\\ \mathbf{I}& {\mathbf{T}}_2& \dots & \mathbf{T}{}_2^K\\ ⋮& ⋮& \ddots & ⋮\\ \mathbf{I}& {\mathbf{T}}_C& \dots & \mathbf{T}{}_C^K\end{array}\right)\left(\begin{array}{cccc}{\mathbf{W}}_N& 0& \dots & 0\\ 0& {\mathbf{W}}_N& \dots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \dots & {\mathbf{W}}_N\end{array}\right)$$

(9)

where $\mathbf{T}{}_i^K$ is composed of $N\times N$ diagonal submatrices conformed by the time sequence ${\mathbf{t}}_{\mathbf{n}}=-(K+1)T_s(n_s/2)$ to $(K+1)T_s(n_s/2)$ with $n_s$, being each sample of Taylor’s interpolating polynomial at each sampling frequency $F_s$, N is the number of samples per cycle, ${\mathbf{W}}_N$ represents the Fourier matrix formed by harmonic phase factors ${\omega }_N^h=e^{j2\pi h/N}$ at each vector $h=0,\dots ,N-1$, and $\mathbf{I}$ is the identity matrix.

Once the Taylor–Fourier subspace is established, the filter algorithm is obtained through the least square solution [29], resulting in the analysis equation as follows,

$$\widehat{\xi }={\mathbf{B}}^{†}s,$$

(10)

where $\widehat{\xi }$ represents the estimated Taylor–Fourier coefficients for each multiple of the base frequency $f_1$, that is, the harmonic frequencies. ${\mathbf{B}}^{†}$ is the pseudoinverse of $\mathbf{B}$ and s stands for the measured signal.

Considering that the Taylor–Fourier coefficients converge like a sinc function, their spectra will converge like an ideal filter. Their algorithmic implementation lets users easily center and modulate the filter’s bandwidth, which is especially useful for analyzing spurious frequencies introduced by a damaged bearing.

Figure 4 shows the filter response for different numbers of Taylor polynomials, i.e., K values. As this parameter increases, a flatter response at the central frequency is given, as well as a better harmonic rejection in the sidebands of the filter.

This investigation uses the fifth-order bandpass filter associated with the frequency response of the estimated Taylor–Fourier filters $\widehat{\xi }$ to extract the amplitude per phase of current signals for each damage condition. Notice that the flatter response of the filters allows capturing the energy of spurious components in a better way. This is due to its similarity to an ideal filter, by rejecting the other spectral components and reducing the sideband lobes.

Figure 5 shows both the Taylor–Fourier of third order and Fourier filters behavior. In comparison with the Fourier filter, the Taylor–Fourier filter shows some advantages, like a flatter response of the filter and the sideband lobes reduction that provokes better harmonic rejection properties [30].

Finally, the amplitude $\widehat{a}$ of the signal can be estimated with the following expression:

$$\widehat{a}\left(t_0\right)=2\left|\widehat{\xi }\right|$$

(11)

It is also worth noticing that in other works where the Taylor–Fourier theory is applied, such as the one presented in [31], the computational complexity for the DTFT is declared as ${\left(CN\right)}^2$ products, whereas an empirical mode decomposition algorithm requires $41SNlo{g}_2\left(N\right)$, where S is the number of sifting operations and N is the length of the signal. For a wavelet packet decomposition approach, the computational complexity is $2^{(L+1)}Nlo{g}_2\left(N\right)$. Therefore, the DTFT is less complex, especially for implementation purposes.

3. Methodology

The proposed methodology is presented in Figure 6, and it is entirely implemented in Matlab 2018b, Mathworks, Natick, MA, USA. This allows for the analysis of signals to detect and classify faults associated with three types of damage. Signal acquisition was performed with one damaged bearing at a time.

In the data acquisition block, the stator current measurements, for four IM tests are prepared for processing: synthetic bearing damage (bearing ball, outer race, and corrosion), and healthy motor conditions. Notice that the length of the acquired signal depends on the size of the bandwidth of the filter and the size of the sliding window. The minimum number of samples for estimating a valid result is given by the number of samples for the sliding window: at the same time, the size of the sliding window L is determined by the number of signal cycles C required to perform the analysis; therefore, $L=CN$ where $C=K+1$ and N is the number of samples per cycle of the signal. In this regard, the system requires at least L samples of the signal to have valid results for amplitude estimates. A segment of the signals under study is shown in Figure 7. Notice that the identification of a fault by means of a visual inspection is quite difficult. That is the reason why further signal processing is required in order to detect a fault in current signals. Its analysis is only carried out in one phase per test at a time since the bearing faults affect the three phases in the same proportion [32].

In the preprocessing stage, a frequency analysis of the current measurements in Figure 7 is conveyed at one phase of the induction machine. DTFT is very suitable for oscillatory signals in a transient state, and given that in this work, the analysis is performed based on real signals, it is expected that the signals have certain amplitude variations. The amplitude estimation would have less error than when using only the Fourier transform, for example. In addition, in this particular case, the fault introduces components of low amplitude in the current signal; therefore, the DTFT allows the amplitude estimation with an excellent resolution thanks to its filtering characteristics.

This analysis is performed for each fault and the healthy condition in order to identify the spurious components, design the filter bandwidth, and locate its position. The bearing damage signature can be identified relying on Equations (1)–(3) together with a visual inspection of the spectrum.

Since the filter is designed considering the bandwidth and the location of the spurious components, the following parameters are established according to the fundamentals in Section 2.2. The central frequency $f_1$ is selected in the DTFT filter design block, and the bandwidth is modulated by using the number of samples per cycle N defined by $Fs$ and the value of K.

The K value is selected based on the precision-computational cost ratio and considering the filter response around the spurious frequency. The selection of K is completely based on the signal characteristics and the type of implementation where it is required. The theory states that the estimates have less and less error as K increases; however, the selection cannot be arbitrary as K is considered when constructing the sliding window, and at the same time, the size of the window L should not surpass the length of the signal [33]. In addition, as K increases, more coefficients are calculated, restricting also the memory that can be stored when working in the implementation of the system. Hence, based on the waveforms under study, the selected value for this work is $K=5$.

Then, once the number of Taylor polynomials is defined, the Taylor–Fourier coefficients can be calculated. Current signals are individually filtered by applying a windowing process, where each window comprises $K+1$ cycles, i.e., six cycles for the signals used in this work. The amplitude estimation for each filtered signal is computed as stated in (11), for detection and classification purposes. At this point, the feature extraction is accomplished by the current signals filtering; theoretically, amplitudes of filtered signals should be different for each motor condition (healthy and faulted).

In the detection block, a vector of estimated amplitudes is created, containing the maximum amplitudes of the filtered signals for each test of each condition. In this case, 100 tests for each fault condition are processed, and then 100 maximum amplitude values are stored in each fault vector. Based on this information, a statistical analysis is performed, where the median and standard deviation are calculated to obtain a boxplot. The median is used as the main discriminant factor in the performance of the selection process. It is considered a good detection if the healthy condition, in the upper adjacent, and the fault condition, in the lower adjacent, are not overlapped, and the distance between them is at least three standard deviations according to the Chebyshev inequality, as shown in Figure 8 [34].

Once the detection is achieved for all the frequencies at the different fault conditions (bearing ball damage, outer-race damage, and corrosion damage) for loaded and unloaded motors, in the classification block, the amplitudes are compared to select the type of fault.

A threshold value is selected from the boxplot graphs based on the median value of the healthy distribution plus three standard deviations more. A different threshold is selected for each type of fault. Then, each signal of the data set is processed, and the amplitude is estimated and compared by means of an $if-else$ sentence. If the amplitude lies within certain threshold values, the signal corresponds to a certain type of fault or to a healthy motor, as is shown in Figure 9. Given that the data set is labeled during the characterization process, i.e., the health of the motor is previously known, the classification percentages are assessed based on the ratio between the number of signals classified in a certain class versus the actual class to which the signal belongs.

Finally, in the last block, the classification information is processed to obtain an accuracy percentage to quantify the proposed method’s accuracy. In the next section, experimental results are presented.

4. Results

4.1. Experimental Setup

The experimental setup for current signal analysis (showed in Figure 10) is based on a three-phase IM WEG 00118ET3EM143TW, 1HP, 1800RPM, 220VAC, and 2.98A. The load of the IM is mechanically generated, where the IM nominal current is used as the maximum value of the applied load, and then the loaded condition used in this paper is assumed when the IM is consuming 60% of the nominal current.

The model of the bearings used in this work was 6204 with the following specifications: $N_b=9,R_b=4$ mm, $R_c=19.5$ mm. The studied damages were artificially created with a roller tool of 1.58 mm ≈ 1.6 mm diameter, the perforated elements were one ball and the outer-race of each bearing, corresponding to bearing ball damage and outer-race damage, respectively. Finally, the nonlocalized defect condition was emulated with a corroded bearing.

The tests were performed under different bearing damage conditions: bearing ball damage, outer-race damage, and generalized damage for oxide. The damage cases were artificially created aiming to characterize the fault and compare their information with a healthy bearing in order to apply Equations (1)–(3) and visual inspection to identify the fault signature of each case. Faulted bearings are presented in Figure 11. In this work, only drive end bearings were considered.

4.2. Experimental Results

To implement the detection and classification methodology, 100 stator current measurements for different scenarios were analyzed, giving a total of 4800 tests where each test represents the input signals. Current measurements for each damage condition were acquired at different sample rates and different operation frequencies (see

Table 1. Classification percentages for loaded motor.

Loaded Motor
	Frequency	BBD	ORD	OD	Out of Class Range
BBD		96%	0%	3%	1%
ORD	60 Hz	0%	98%	0%	2%
OD		4%	0%	95%	1%
BBD		95%	0%	0%	5%
ORD	50 Hz	0%	100%	0%	0%
OD		0%	0%	99%	1%
BBD		99%	0%	0%	1%
ORD	40 Hz	0%	99%	0%	1%
OD		0%	0%	100%	0%
BBD		99%	0%	0%	1%
ORD	30 Hz	0%	95%	1%	4%
OD		0%	1%	97%	2%
BBD		97%	0%	0%	3%
ORD	20 Hz	0%	98%	1%	1%
OD		0%	0%	94%	6%
BBD		93%	5%	0%	2%
ORD	10 Hz	0%	97%	0%	3%
OD		0%	0%	98%	2%

and

Table 2. Classification percentages for unloaded motor.

Unloaded Motor
	Frequency	BBD	ORD	OD	Out of Class Range
BBD		97%	0%	0%	3%
ORD	60 Hz	0%	98%	0%	2%
OD		0%	0%	98%	2%
BBD		98%	0%	0%	2%
ORD	50 Hz	0%	100%	0%	0%
OD		0%	0%	98%	0%
BBD		97%	0%	0%	3%
ORD	40 Hz	0%	97%	0%	3%
OD		0%	0%	99%	1%
BBD		99	0%	0%	1%
ORD	30 Hz	0%	95%	0%	5%
OD		0%	0%	100%	0%
BBD		97%	0%	0%	3%
ORD	20 Hz	0%	98%	0%	2%
OD		0%	0%	98%	2%
BBD		98%	0%	0%	2%
ORD	10 Hz	0%	91%	0%	9%
OD		0%	1%	98%	1%

) to obtain the same number of samples per cycle of the mentioned induction machine.

By analyzing the spectrum of a particular case just as an example, the fault components are identified to set the central frequency of the filter at 89.6 Hz, as depicted in Figure 12 [32] (the spurious components of the different faults will not necessarily be at the same frequency like this). Therefore, this value is selected as the central frequency for the bandpass filter, as shown in Figure 13. Notice the narrow bandwidth, which can be modulated depending on the fault component.

Below, the results for each fault condition are presented, where accuracy percentages are the ratio between the estimated fault prediction and the actual prediction.

4.3. Bearing Ball Damage (BBD)

A total of 1400 measurements are filtered to characterize the signal in order to get the estimated amplitude for detection and classification purposes.

According to Section 3, the statistical analysis is performed to obtain the discriminant for detection and classification. To obtain the motor condition, the detection boxplot is constructed and presented in Figure 14, where label H represents the healthy motor condition, D represents a bearing damage condition, BBD is the bearing ball damage, ORD is the outer-race damage, and OD is the oxide damage. The tests were performed at 60, 50, 40, 30, 20, and 10 Hz. The classification accuracy percentage is presented in Table 1 and Table 2 for both loaded and unloaded motor conditions.

4.4. Outer-Race Damage (ORD)

In the same way, the measurements are filtered aiming to estimate the amplitudes and hence, to obtain the statistical information based on the boxplot, as illustrated in Figure 15, to detect and classify the fault.

Accuracy percentages are presented in Table 1 and Table 2. For that fault in the loaded motor condition, the worst accuracy percentage is 95%, and the best is 100%.

4.5. Corrosion Damage (OD)

Finally, for the oxide damage case, the statistical analysis (see Figure 16) is performed based on these damage measurements. As in the previous cases, the accuracy results are presented in Table 1 and Table 2.

Notice that the classification is successful for both load and unload conditions, considering that the statistical distribution is non-overlapping.

The filter location in some cases is different for detection and classification to avoid overlapping at the same frequency, aiming to replicate a boxplot similar to Figure 9 where the classification process is straightforward.

In

Table 3. Comparative review of proposed methods for bearing fault detection.

Year	Technique for Feature Extraction	Type of Fault	Classifier	Classifier Accuracy
2017 [35]	Adaptative impulse modelling based wavelet	Ball, inner and outer race	Statistical analysis with neural network-based classifier	100%
2020 [36]	Discrete Wavelet transform	Inner and outer race	Random forest and extreme gradient boosting	99.3%
2021 [37]	Refined composite generalized multiscale dispersion entropy	Ball, inner and outer race	Multiclass adaptative neuro-fuzzy classifier	89.62–99.27%
2021 [38]	k-optimized adaptive local iterative filtering and improved multiscale permutation entropy	Ball, inner and outer race	Back-propagation neural network	91.57–99.98%
2021 [19]	Continuous Wavelet transform	Ball, inner and outer race	Convolutional neural network and support vector machine	98.75–98.89%
This work	Amplitude estimation based on the Taylor–Fourier transform	Ball, inner and outer race and corrosion	Statistical analysis	93–100%

, the proposed method is compared with other works based on the techniques used for diagnosis, the classifier, the type of faults considered, and the accuracy of the classification. One of the most prominent advantages of using the Taylor–Fourier transform is the high detection accuracy without requiring complex methods such as neural networks. It is important to mention that the accuracy percentages were reported in the literature and are used in this work only as references.

5. Discussion

In this work, three different types of faults were studied for six different operation frequencies for loaded and unloaded IMs. Based on the current signal characteristics of each motor condition, the Taylor–Fourier filters were designed, and amplitudes were estimated. Finally, faults were detected and classified according to the statistical distributions of the amplitude estimates.

The main difficulties encountered during the elaboration of this work were related to the design of the filter, given that the nature of the fault-related components demanded a fine filter. In this regard, the bandwidth had to be small enough for good amplitude estimates, so a larger number of samples in time had to be processed in each sliding window. This process made the computing time slow; however, once the statistical indicators were calculated and the threshold values were selected, the classification time was negligible in comparison with the signal processing time. In practice, once the motor is characterized, only the classification time is considered.

6. Conclusions

This paper uses a methodology for current-based analysis for bearing damage detection and classification through the DTFT. Exhibited results show that the filtering characteristics of the DTFT provide an excellent accuracy ratio, highlighting its capability to process signals in a transient state. Results show that the lowest accuracy percentage is 93%, but overall reaches 99%.

Classification accuracies for the unloaded condition are mostly lower than for the loaded condition; however, both cases presented an excellent performance. Notice that for 50 Hz and 60 Hz, which are the grid frequencies in Europe and America, respectively, the classification percentages go from 95% to 100%.

Those results could be useful for motor diagnosis in industrial applications and to give solutions to the challenges that the actual society demands, like the implementation of alternative power sources, electric transportation, the inclusion of smart devices in daily life, among others based on the low computational complexity of the algorithmic implementation. This implies the potential to develop a non-intrusive online detection and classification device based on the accurate results provided by the DTFT.

In future works, authors will describe the DTFT algorithm in hardware to analyze the real computational complexity and explore the possibilities of silicon implementation for an online device.