1. Introduction
Induction motors (IMs) are considered as the most popular motors in electromechanical energy conversion and various industrial applications. However, various failures in IMs are frequently caused by the operations in harsh environments, continuous wearing, overloads, and unexpected incidents. The broken rotor bar (BRB) fault is one of the most serious failures and needs to be detected on account of its destructiveness. According to the statistics by the Institute of Electrical and Electronics Engineers (IEEE), approximately 9% of IMs faults are due to the BRB, and 8% are caused by BRB faults according to the statistics from the Electric Power Research Institute (EPRI) [
1]. If the BRB fault is left undetected and not resolved in time, it can lead to the occurrence of serious BRB faults or other types of motor faults and can waste the electric power. Consequently, the entire motor may collapse, which can result in higher maintenance costs, lower power transmission efficiency, and even serious accidents.
Currently, researchers have applied considerable efforts to the development of effective IMs fault diagnosis technologies for improving operational reliability and reducing downtime. The most widely applied techniques for BRB faults detection are the motor current signature analysis (MCSA) and the motor vibration signature analysis (MVSA) [
2]. Further, magnetic flux measurements have been used, but a flux sensor needs to be installed that can increase the detection costs. Regarding the diagnosis of BRB faults in IMs, a vibration signature analysis can receive satisfactory detection results as well as the MCSA [
3]. In addition, the vibration diagnosis is a more reliable and standardized analysis since the measured vibration signals from machines contain useful information reflecting the health conditions of machinery systems. The vibration signature analysis is the most widely used technique in condition monitoring and fault diagnosis due to its simplicity, less comparative cost, and relative ease of the application. Further, no structural changes are required for the application of the vibration analysis for the existing machines [
4].
There are a variety of fault diagnosis methods that have been developed based on vibration signal analysis for detecting the BRB faults of IMs. In [
5], the advanced use of wavelet analysis by analyzing an axial vibration signal was introduced for BRB faults detection through removing the effects of the interference frequency components. In [
6], the Zhao-Atlas-Marks (ZAM) distribution was investigated for BRB faults diagnosis based on vibration transient signals. In a recent work, Rangel-Magdaleno [
7] developed a demodulation methodology using discrete wavelet transform (DWT) and autocorrelation methods for the incipient broken bar detection based on MVSA. Delgado-Arredondo [
8] presented the complete ensemble empirical mode decomposition (CEEMD) for BRB faults diagnosis, which solves the drawbacks of empirical mode decomposition (EMD) and the ensemble empirical mode decomposition (EEMD) for obtaining accurate analysis results. The orthogonal matching pursuit algorithm (OMP) in [
9] focused on the decomposition of signals, which demonstrated the detection method is effective. On the other hand, several MCSA-based demodulation techniques such as empirical mode decomposition (EMD) and neural networks [
10], Hilbert-Huang transform (HHT) [
11], multiple signal classification (MUSIC) [
12], and synchrosqueezing transform (SST) [
13] were also developed for BRB fault diagnosis. There are also other techniques such as magnetic field analysis [
14], infrared data analysis [
15], and air-gap torque analysis [
16] that were applied for BRB faults detection.
Despite being powerful signal processing tools and potential applications for BRB fault detection, these diagnosis methodologies mentioned above still have serious shortcomings. They contain substantial computational costs for the exhibition of BRB fault features and lower accuracy for the identification of BRB fault types and severities. For example, solving the intrinsic mode function (IMF) of a signal by using the CEEMD algorithm in [
8] is indeed a complicated calculation process. Similarly, the popular methods based on DWT, EMD, HHT, SST, etc., are also faced with low processing efficiency. The estimated parameters used in [
7,
12] cannot present a clear classification of the BRB fault severities. However, the conventional methods based on a fast Fourier transform (FFT) show its inability to process non-linear and non-stationary vibration signals. Hence, there is an urgent need to develop an accurate and efficient approach for BRB fault diagnosis in IMs.
Furthermore, IMs are typical rotary machines, so that the measured vibration signal from the motor surface has strong periodic modulation characteristics, and these periodically-modulated vibration signals can be regarded as the second-order cyclostationary (CS2) signals. The CS2 signal is the advancement of the first-order cyclostationary (CS1) signal characterized by the mean or expected value, and both CS2 and CS1 signals belong to the category of cyclostationarity [
17]. Cyclostationarity refers to a promising signal processing method that has been subverting the field of mechanical signature analysis in recent decades. The conventional Fourier transform (FT) has been proven to be effective in dealing with CS1 components, but it fails to describe CS2 signals because the CS2 signals are complexly modulated. In order to remedy this deficiency, a cyclic spectral analysis was proposed in the 1990s by Gardner and applied to the analysis of CS2 signals [
18]. It effectively reconstructs the CS2 signals from a new perspective and extracts the useful information related to mechanical failures. Briefly stated, a cyclic spectral analysis represents the CS2 contents by two frequency variables: Spectral frequency (
ƒ) and cyclic frequency (
α). The spectral frequency is employed to exhibit the carrier frequency contents of the signal, and the cyclic frequency aims to indicate the modulation components leading to the faults [
19]. Cyclic spectral analysis is therefore effective in processing non-stationary and non-linear signals, which can reveal the periodic behaviour associated to mechanical faults hidden in the CS2 signals.
According to the advantages of vibration-based fault diagnosis in terms of the cyclic spectral analysis, an improved cyclic spectral analysis method based on the vibration signature analysis was investigated to detect the BRB faults of IMs. One of the cyclic spectral analysis approaches, cyclic modulation spectrum (CMS), that has been described in [
20] is referred to and improved on in this paper. The CMS algorithm contains the calculation of the short-time Fourier transform (STFT) method, and it can fully present its value in the diagnosis and identification of a specific mechanical fault by optimizing the use of the STFT. The focus of the present work is to improve the CMS algorithm to get higher accuracy in the classification of BRB fault types and severities and larger computational gains by optimizing the window function, window length and step size for the application of the STFT. Firstly, an optimal window function used in the STFT was selected according to the specific signal form, which can increase the accuracy of identifying the BRB fault types and the severities. Subsequently, the window length and step size were optimized based on the selected window function, which can receive a better computational gain. According to the analysis results in [
21], the amplitude of characteristic frequency can be considered as the indicator for the identification of BRB faults. In this paper, the amplitudes of BRB fault characteristic frequencies serve as a criterion for judging BRB fault types and severities. The effectiveness and performance of the proposed method is validated through processing the vertical vibration signals issued from healthy motor and damaged motors with 1 BRB and 2 BRB faults under 0%, 20%, 40%, 60%, and 80% load conditions. Compared to the power spectral density (PSD) [
2,
3,
5,
22,
23,
24,
25] that has been widely explored for BRB fault detection and other cyclic spectral analysis estimators, such as the cyclic power spectrum (CPS) [
26], averaged cyclic periodogram (ACP) [
27], and fast spectral correlation (FSC) [
28], the improved CMS can provide better diagnosis capability.
This paper is organized as follows:
Section 2 briefly presents the information in connection with BRB faults which need to be analyzed and reviews the brief principle of the CMS algorithm.
Section 3 introduces the procedures of the improved CMS method. The performance of the improved CMS algorithm for BRB fault feature extraction is validated by simulation studies in
Section 4.
Section 5 provides the introduction of test rigs and the experimental verification. The conclusions are finally drawn in
Section 6.
4. Simulation Study
In this section, a synthetic cyclostationary signal was established to validate the performance of the proposed method. The simulated signal is similar to the vibration signal obtained from the IMs with BRB faults at a steady-state operation. When the BRB fault occurs, the fault characteristic frequency of the rotor is a low frequency component, the rotation frequency of the rotating shaft is a high frequency component, and the low frequency related to the BRB fault is modulated on the rotating shaft frequency. The synthetic cyclostationary signal consists of a carrier frequency
fr showing the rotation frequency of rotation shaft, and a modulation frequency
α0 denoting the frequency component related to the BRB faults. The simulated signal x(t) consists of a carrier signal x1(t), a modulation signal x2(t), and a Gaussian white noise signal n(t) with a specific signal-to-noise ratio (SNR) of −2 dB. The length of the samples
L and sampling frequency
Fs were 10
6 and 10 kHz, respectively. Where
t =
n/
Fs (
n = 0, 1, ...,
L − 1) denotes the discrete time, and M1, M2 represent the amplitude of a carrier signal and a modulation signal, respectively. M1 and M2 can indicate the energy of the carrier signal and the modulated signal and the different fault levels can be built by setting their values. In this study, M1 was set to 1, and M2 was set to 1 and 0.5 to represent 2 BRBs fault and 1 BRB fault cases. Moreover, the carrier frequency
fr was set to 40 Hz, and the modulation frequency
α0 was set to 4.27 Hz and 3.27 Hz to denote 2 BRBs fault and 1 BRB fault, respectively. Therefore, the simulation signals can be expressed as:
4.1. The Improvement of Window Function
In order to determine the optimal window function, the different window functions were applied to the analysis with the same simulated data. The objective is to select the most suitable window function by analyzing the diagnosis results.
In this processing, the
A1 and
A2 respectively represent the amplitude of the characteristic frequencies
α0 of simulated 1 BRB and 2 BRBs faults. In addition,
η is to show the failure level of 2 BRBs to 1 BRB. The higher the failure level is, the better of the selected window function can be in the classification of the BRB faults.
As can been seen from
Table 2, the flat top function is able to receive a higher BRB detection capability based on the simulated BRB fault signals. Obviously, its diagnostic power is more than four times that of the rectangular window function. As a result, the CMS method using the flat top function can get more accurate BRB diagnosis performance based on the simulated signals.
4.2. The Improvement of the Window Length and Step Size
To get a better computational gain of the CMS algorithm, the flat top function with the window length of 2
6, 2
7, 2
8, and 2
9 was respectively used in the simulation. Additionally, the step size was selected as 25%, 30%, 35%, 40%, 45%, and 50% of the flat top window length, respectively, which ensures that the overlap between two adjacent moving windows is between 50% and 75% to reduce spectrum leakage and the effect of window length and step size on BRB fault identification. The step size was selected to be an integer according to the rounding principle. For example, if the window length is 2
6, then the step size is chosen as follows: 16, 19, 22, 26, 29, and 32. The simulated results are shown in
Figure 2.
Thus, in order to ensure time efficiency, the window length of 2
8 and the step size of 40% of the window length were selected for the CMS operation. The appropriate step size used for the simulation study was 102 and the required simulation time was only 0.5540 s as shown in
Figure 2. All simulations and their experimental data processing were performed on a computer with a i5-7400 CPU processor 3.00 GHz. The parameter values used in the improved CMS algorithm and calculation time for all algorithms utilized in synthetic simulation are presented in
Table 3. The improved CMS not only ensures the optimal BRB fault identification accuracy based on the existing window functions, but also improves the calculation efficiency.
The optimized window length and step size make only 0.5540 s is taken by the improved CMS method, in which its computational efficiency is only second to conventional PSD method as shown in
Table 3. In the synthetic simulation, the characteristic frequency can be clearly extracted by several algorithms and the order of computational efficiency is PSD > CMS > FSC > CPS > ACP. However, it is worth mentioning that the simulation time of the ACP algorithm takes approximately
s, which is far from the computational efficiency of other algorithms. This is the reason why this algorithm is greatly limited in practical applications. Considering the higher computational cost and lower value of the ACP algorithm in practical applications, in the following study, ACP will not be applied to the processing of the experimental data.
According to the simulation Equation (10), the improved CMS algorithm with the flat top window function can extract the fault characteristic frequency from the simulated signals with a SNR of −30 dB as shown in
Figure 3. This offers an excellent immunity to noise in the measurements. Therefore, the CMS method can be improved to give optimal analysis results based on the flat top function, with a window length of 2
8, and step size of 40% of the window length.