Reviews of bearing vibration measurement using fast Fourier transform and enhanced fast Fourier transform algorithms


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1 Advances in Vibration Control of Structures and Machinery  Research Article Reviews of bearing vibration measurement using fast Fourier transform and enhanced fast Fourier transform algorithms Advances in Mechanical Engineering 2019, Vol. 11(1) 1 12 Ó The Author(s) 2019 DOI: / journals.sagepub.com/home/ade HsiungCheng Lin and YuChen Ye Abstract The rolling element bearing is one of the most critical components in a machine. Vibration signals resulting from these bearings imply important bearing defect information related to the machinery faults. Any defect in a bearing may cause a certain vibration with specific frequencies and amplitudes depending on the nature of the defect. Therefore, the vibration analysis plays a key role for fault detection, diagnosis, and prognosis to reach the reliability of the machines. Although fast Fourier transform for time frequency analysis is still widely used in industry, it cannot extract enough frequencies without enough samples. If the real frequency does not match fast Fourier transform frequency grid exactly, the spectrum is spreading mostly among neighboring frequency bins. To resolve this drawback, the recent proposed enhanced fast Fourier transform algorithm was reported to improve this situation. This article reviews and compares both fast Fourier transform and enhanced fast Fourier transform for vibration signal analysis in both simulation and practical work. The comparative results verify that the enhanced fast Fourier transform can provide a better solution than traditional fast Fourier transform. Keywords Vibration, fast Fourier transform, enhanced fast Fourier transform, harmonics, nonstationary Date received: 19 September 2018; accepted: 6 November 2018 Handling Editor: Ali Kazemy Introduction Rolling element bearings have been widely applied in domestic and industrial machinery. These bearings are considered as most critical components, and defects in bearings may cause malfunction or even lead to serious failure of the machinery during operation. The health condition and quality inspection of bearings are directly related to these defects. Therefore, the industrial vibration analysis is regarded as an important measurement tool for identification, prediction, and prevention of failures in rotating machinery. 1 3 For this reason, implementing vibration analysis on the machines can improve the machine efficiency and reliability. Usually, the measurement vibration involves accelerometers to measure the vibration, and then, the data can be collected for further analysis. The plots of vibration signal with time domain or frequency domain may provide sufficient information for the engineers to analyze and Department of Electronic Engineering, National ChinYi University of Technology, Taichung, Taiwan Corresponding author: HsiungCheng Lin, Department of Electronic Engineering, National Chin Yi University of Technology, No.57, Sec. 2, Zhongshan Rd., Taiping Dist., Taichung 41170, Taiwan. Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License ( which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages ( openaccessatsage).
2 2 Advances in Mechanical Engineering determine the machine fault. Many bearings premature malfunction may occur from surface roughness, misalignment, discrete defects, unbalance, contamination and temperature extreme, and geometrical imperfections. Modern machines may produce the vibration frequency range between 20 Hz and 20 khz. 4 6 When a fault of rolling bearings begins to develop, the resulting vibration pulses frequencies may repeat periodically. A band of highfrequency vibration may therefore exist before a rollingelement bearing is burnt out Although there are many kinds of methods such as vibration, acoustic, temperature, and wear debris analysis have been reported for diagnosis and analysis of bearing defects, the vibration spectral measurement is the most widely used approach The discrete Fourier transform (DFT) is a popular tool in spectrum analysis. If the vibration is under the stationary conditions, it may be efficient for measurement. However, the fast Fourier transform (FFT) presents a more efficient computation process. In practice, however, the limitation of the DFT or FFT makes it less efficient in analyzing the signal spectrum from defective rolling element bearings due to cyclostationary and nonstationary characteristics Some techniques such as shorttime Fourier transform (STFT), Wigner Ville distribution (WVD), and continuous wavelet transform (CWT) are also available for signal time frequency analysis based on twodimensional (2D) mapping principle However, STFT is unable to simultaneously improve the time and frequency resolutions. 23 WVD is suitable for nonstationary signals processing, but it may cause misinterpretation in the signal analysis due to the bilinear characteristic. 24 CWT algorithm is superior in analyzing simultaneously both frequency and time information for the vibration event. Its inherent large computational time with fixedscale frequency resolution may discourage practical applications. 25 Hilbert Huang transform (HHT) approach provides vibration signal multiresolution in the instantaneous frequencies using the intrinsic mode functions (IMFs). It is, however, the time resolution may affect the corresponding frequency considerably Alternatively, an envelope spectrum analysis for bearing fault diagnosis is widely used methods such as kurtogram, enhanced kurtogram, improved kurtogram, and sparsogram. The kurtogram is based on the spectral kurtosis that has been used in characterizing nonstationary signals, especially bearing fault signals. 30 However, an analytic bearing fault signal from spectral kurtosis needs to be constructed from either a complex filter or Hilbert transform or filtered by the STFT. Also, its performance efficiency is low in the presence of a low signaltonoise ratio and nongaussian noise. The enhanced kurtogram was therefore developed to find the location of resonant frequency bands, but it needs to remove frequency noise by using autoregressive filtering in advance. 31 Although an improved kurtogram method was reported to overcome the shortcomings of the kurtogram, the wavelet packet transform (WPT) is required to adopt as the filter. 32,33 In addition, a sparsogram algorithm was proposed to quickly determine the resonant frequency bands from the envelopes of wavelet packet coefficients at different wavelet packet decomposition depths. 34 The selection of optimal wavelet packet node, however, usually relies on visually inspecting the largest sparsity value from the wavelet packet coefficients. In a recent work, a spectral kurtosis can be decomposed into squared L2/ L1 norm and spectral Lp/Lq as a general form. 35 When p = 1 and q = 0, the general form was reduced to the reciprocal of the smoothness index. Consequently, the resonant frequency bands may be retained for characterizing bearing fault signals. Principle of DFT and enhanced fast Fourier transform models Background of Fourier transformation The FT analysis is a tool to reconstruct a periodical waveform using series harmonics, where harmonic frequency is defined as a multiple of fundamental. If a waveform i s (t) is periodical with Dirichlet condition satisfied, it can be expressed as i s (t)= X n = i n e j2pft ð1þ where i n =(1=T) Ð T 0 i s(t)e j2pft dt and T(= 1/f) is the period. i 0 is a direct current (DC) component. The DFT is a discrete form from time domain as i s ½nŠ = XN 1 k = 0 I s ½kŠW kn N ð2þ where I s ½kŠ =(1=N) P N 1 n = 0 i s½nšwn kn and W N =exp (j2p=n). Assume i s ½nŠ is periodic with the period T, and the Fourier fundamental angular frequency (Dv) can be defined as Dv = 2p ð3þ T For the waveform sampled using p(p. 1) periods, Dv can be represented as where v 0 = 2p=T. Dv = 2p pt = v 0 p ð4þ
3 Lin and Ye 3 Figure 1. Relation between harmonic frequency and dispersed Energy: 39 (a) smallfrequency deviation and (b) bigfrequency deviation. If the signal is sampled N points using the sampling rate f s, the Fourier fundamental frequency (Df ) can be defined as Df = 1 pt = 1 pn s T s = 1 NT s = f s N ð5þ where N s ¼ D N=p and T s ¼ D 1=f s. The waveform power (P) can be expressed using the Parseval relation as 36,37 P = 1 N X N 1 n = 0 i s ½nŠ 2 = XN 1 I s ½kŠ 2 k = 0 The power at the frequency f k can be expressed as P½f k Š = I s ½kŠ 2 + I s ½N kš 2 = 2I s ½kŠ 2 ð6þ ð7þ where k = 0,1, 2,., N/ The amplitude of the mth harmonic at f k is thus written as p A m ½f k Š = ffiffiffiffiffiffiffiffiffi p P½f k Š = ffiffi 2 Is ½kŠ ð8þ where m = 1,2,., M. When the sampling window is not synchronized with the fundamental, the mth harmonic power at f k will disperse over around the f k. Based on the concept of group harmonics, all spilled power around the adjacent harmonics can be collected into a group power as 38,39 P m ½f kš = X+ t Dk = t where t denotes the group bandwidth. (A m ½f k + Dk Š) 2 ð9þ Therefore, the true harmonic amplitude can be retrieved from collecting all dispersed power as qffiffiffiffiffiffiffiffiffiffiffiffi A m ½f kš = P m ½f kš ð10þ Review of the enhanced fast Fourier transform algorithm The enhanced fast Fourier transform (efft) algorithm was developed to improve the FFT for suiting nonstationary vibration signal analysis. The relationship between harmonic frequency and dispersed energy can be classified into smallfrequency deviation and bigfrequency deviation. 39 Case 1: For smallfrequency deviation shown in Figure 1(a), the second larger magnitude (A m ½f k + 1 Š)atf k + 1 is located at the right side of the dominant frequency (f k ), that is, A m ½f k Š.A m ½f k + 1 Š. In equation (11), the actual frequency can be corrected to f k plus the frequency deviation (Df k ). Case 2: For a bigfrequency deviation shown in Figure 1(b), the second larger amplitude (A m ½f k Š)atf k is located at the left side of the dominant frequency (f k +1 ), that is, A m ½f k Š\A m ½f k + 1 Š. Similarly, the actual frequency can be corrected to f k plus the frequency deviation (Df k ). In addition to the frequency correction, the collected energy dispersed around the major harmonic can be used for retrieving the original amplitude, as shown in equation (12). Based on above principle, the mathematical model was deduced from the relation between the frequency deviation amount and dispersed energy distribution. 39 The real frequency can be corrected by the dominant frequency (f k ) plus frequency deviation (Df k ), that is, f k + Df k.
4 4 Advances in Mechanical Engineering The frequency deviation range (FDR) is therefore defined as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P +t Dk = 1 A m ½f k + Dk Š 2 Df k = sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Df P 0 A m ½f k + Dk Š 2 P + +t A m ½f k + Dk Š 2 Dk = t Dk = 1 ð11þ where Df = f s =N and t =0,1, 2, 3,... The energy dispersed around the major harmonic can be collected and used for retrieving the original amplitude. The restored amplitude (RA) can be defined as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X +t RA = t A m ½f k + Dk Š 2 Dk= t ð12þ where t =0,1, 2, 3,... The selection of the group bandwidth (t) is determined by the following rule, where f 1 and f 2 are assumed as two near major harmonics. Note that f 2 is set as zero if only one major component exits jf 1 f 2 j\4df ) t = 1 4Df ł jf 1 f 2 j\6df ) t = 2 6Df ł jf 1 f 2 j\8df ) t = 3 8Df ł jf 1 f 2 j\10df ) t = 4 jf 1 f 2 jø 10Df ) t = 5 The implementation procedure of the efft model to find the major spectrums of vibration signal is shown in Figure 2. System structure Setup of measurement platform In the performance test, the proposed vibration measurement system uses the tool machine (QUASER CNC, MV184) combined with accelerometer, as shown in Figure 3. It mainly consists of five parts, which are described in the following. The major blocks in Figure 3 are described as follows. 1. Microprocessor with FFT and efft model. The microprocessor (PIC18F4520) is used to implement FFT and efft algorithms. It also acts as a communication bridge between the computer (Data Acquisition and Display System) and signal sources, for example, function generator or accelerometer amplifier circuit. Figure 2. Flowchart of the proposed efft model. 39
5 Lin and Ye 5 Php Dynamic Management System MySQL Database Management System Data Acquisition System TCP/IP TCP/IP Microprocessor with FFT & efft Model Accelerometer Amplifier Circuit Tool machine (QUASER CNC, MV184) Figure 3. Experimental platform. Figure 4. Accelerometer amplifier circuit. 2. Accelerometer amplifier circuit. The accelerometer (PCB352A25/NC) struck on the tool machine can detect the vibration signal that is then amplified and input to the microprocessor, as shown in Figure 4. The circuit has four parts: Part A is a buffer circuit to provide a highinput impedance, and it can receive the signal from the accelerometer without loss. Part B is an amplification circuit to amplify the detected signal of accelerometer with magnification factor = 158. Part C is a clamp circuit that magnifies the signal 6.8 times with limiting the output signal at the 5 V level. Part D is a secondorder filter to filter out the noise signal and allows only k(hz) signals to pass through the circuit. Note that sensitivity of the accelerometer (PCB352A25/NC) is 2.5 mv/g. 3. Data acquisition system. It is designed to receive the data, that is, realtime results from FFT and efft models, and all data received can be thus transmitted to the data server immediately. 4. MySQL database management system: It works as a data server using the MySQL database to receive the data from the Data Acquisition System. 5. Hypertext preprocessor dynamic management system. Based on hypertext preprocessor (PHP), it can display all realtime vibration data with FFT and efft analyses. All historical records can also be tracked. System execution The system execution block is shown in Figure 5. In the Data Acquisition System, it gives a command to receive the signal spectrum data transmitted from the microprocessor. The received data are then displayed on line and written to MySQL database at the same time. PHP Dynamic Management System can access to MySQL database to upload all realtime vibration data and display dynamic data on line. All historical records can also be tracked. In the Microprocessor with FFT and efft model, the vibration signal is acquired via accelerometer amplifier circuit. Then, the microprocessor performs FFT and efft computation, where the signal will be thus transformed to frequency domain from time domain. All computation results (spectrum) are temporarily stored in the memory before reading out once the command is received from the Data
6 6 Advances in Mechanical Engineering Data Acquisition System Microprocessor with FFT & efft model Command Receive code Acquire vibration signal Receive data Read computation results Computation results temporary storage Perform FFT & efft computation Realtime graphical display Transmit spectrum data Write to database MySQL Database Management System PHP Dynamic Management System Figure 5. System execution block. Figure 6. Spectrum analysis of 120 Hz sine waveform: (a) waveform and (b) spectrum of efft and FFT. Acquisition System. Finally, the spectrum data are transmitted to the Data Acquisition System. This procedure will continue until the system stops. Experimental results The experimental tests are divided into two parts: (1) standard signal calibration and (2) practical performance. The standard signals include sine waveform, integer harmonics signal, noninteger harmonics signal, and SHAKER signal. The practical performance will test the vibratory magnitude from the cutting processing of computer numerical control (CNC) machine. Standard signal calibration The precision of the proposed vibration measurement system is calibrated using preknown signal generated from the function generator. The signal acquisition takes 1024 points using khz sampling rate, where Df = 2 Hz. The signals include 120 Hz sine waveform, 121 Hz sine waveform, 120 square waveform, and 121 square waveform. Note that 2.5 V in the scope scale = 1g (gravity acceleration) after signal amplification adjustment using accelerometer amplifier circuit. 1. Sine waveform (frequency: 120 Hz and ampere: 0.78 V). The results from 120 Hz sine waveform analysis is shown in Figure 6. The amplitude is
7 Lin and Ye 7 Figure 7. Spectrum analysis of 121 Hz sine waveform: (a) waveform and (b) spectrum of efft and FFT. Figure 8. Spectrum analysis of 120 Hz square waveform: (a) waveform and (b) spectrum of efft and FFT. equal to 0.78 V so that the real spectrum magnitude is 0.312g. The results reveal that the implementation for both FFT and efft methods can achieve an accurate spectrum analysis due to no leakage in this case. 2. Sine waveform (frequency: 121 Hz and ampere: 0.78 V). The results from 121 Hz sine waveform analysis is shown in Figure 7. It is clear that efft method can obtain the spectrum magnitude = 0.307g that is very close to the real value (0.312g). Also, the major spectrum frequency is Hz that is very close to the real value (121 Hz). However, the result from the FFT analysis shows that the spectrum magnitude is 0.199g, and the frequency is 122 Hz. It is obvious that the errors occur due to a leakage phenomenon for this case. 3. Square waveform (frequency: 120 Hz and ampere: 0.72 V). The waveform of 121 Hz square signal is shown in Figure 8(a), and its spectrum analysis is shown in Figure 8(b). The major amplitude of spectrum is 0.407g for either efft or FFT due to no spectrum leakage in this case. 4. Square waveform (frequency: 121 Hz and ampere: 0.72 V). The waveform of 121 Hz square signal is shown in Figure 9(a), and its spectrum analysis is shown in Figure 9(b). The major amplitude and respective frequency of
8 8 Advances in Mechanical Engineering Figure 9. Spectrum analysis of 121 Hz square waveform: (a) waveform and (b) spectrum of efft and FFT. spectrum using efft model is about 0.4g and 121 Hz, respectively, very close to the real ones. However, the spectrum analysis using FFT shows that the major amplitude and respective frequency of spectrum are 0.263g and 122 Hz, respectively, indicating some errors due to the spectrum leakage. Realtime signal performance In this section, there are three kinds of realtime signals to test: (a) synthetized signal generated from the NI DAQ output using LabVIEW programming, (b) vibration signal generated from SHAKER device, and (c) vibration signal detected from the tool machine. These tests sampled the waveform with 1024 points based on khz sampling rate, that is, Df = 2 Hz. Note that set 1 V in the scope scale = 1g (gravity acceleration) after signal amplification adjustment using accelerometer amplifier circuit in this case. (a) Synthetized signal test. The synthetized signal v(t)=a 1 (2pf 1 t)+a 2 (2pf 2 t)+a 3 (2pf 3 t)+a 4 (2pf 4 t) + a 5 (2pf 5 t)+a 6 (2pf 6 t), containing noninteger harmonics and f 1 =56:8 Hz, a 1 =0:115 V, f 2 =191 Hz, a 2 =0:071 V, f 3 =315 Hz; a 3 =0:225 V, f 4 = 593 Hz, a 4 = 0:09 V, f 5 = 777:2 Hz, a 5 = 0:44 V, f 6 = 989:2 Hz, and a 6 = 0:199 V. The results of spectrum analysis using builtin FFT of LabVIEW, builtin FFT of scope, efft and FFT of microprocessor are shown in Figures 10(a) (c), respectively. (b) Shaker signal test. The shaker can receive the signal from the function generator and then produce the respective vibration signal to be detected by the accelerometer, where the shaker test used MINI SMARTSHAKER (model: K2004E01) shown in Figure Sine waveform (frequency: 200 Hz and ampere: 0.3 V). The spectrum analysis from the shaker and microprocessor is shown in Figure 12. The shaker receives the sine waveform (frequency: 200 Hz, ampere: 0.3 V) generated from the function generator. The results of spectrum analysis from scope using builtin FFT and from microprocessor using efft and FFT are shown in Figures 12(a) and (b), respectively, indicating all results are almost identical due to no leakage. However, it can be seen that the detected vibration strength is 260 mv that has a slight attenuation, where its frequency (200 Hz) has no change. 2. Sine waveform (frequency: 501 Hz and ampere: 0.3 V). The spectrum analysis from the shaker and microprocessor is shown in Figure 13. The shaker receives the sine waveform (frequency: 501 Hz and ampere: 0.3 V) generated from the function generator. The results of spectrum analysis from scope using builtin FFT and from microprocessor using efft and FFT are shown in Figures 13(a) and (b), respectively. Obviously, it can be seen that the result using e FFT model can pursue a more accurate outcome than FFT model form either scope or microprocessor in the both frequency and amplitude measurement. Note that the detected
9 Lin and Ye 9 Figure 10. Synthetized signal analysis: (a) waveform spectrum analysis using builtin FFT in LabVIEW, (b) waveform spectrum analysis from scope using builtin FFT, and (c) spectrum analysis from NIDAQ using efft and FFT. vibration strength from the shaker is 166 mv that has an evident attenuation. (c) Cutting buffeting test. In this test, the tool machine (QUASER CNC, MV184) used 6060 aluminum alloy to implement a cutting process, where sampling rate is khz, sampled points are 1024, that is, Df = 8 Hz; amplitude scale: 0.25 V = 1g after signal amplification adjustment using accelerometer amplifier circuit. Figure 14 indicates that the dominant vibration frequency and spectrum strength using efft is khz and 0.880g, respectively. However, the dominant vibration frequency and spectrum strength using FFT are khz and 0.788g, respectively. It is proved that that efft has a better solution than FFT to restore the original spectrum frequency and amplitude. Figure 11. Profile of SMARTSHAKER (K2004E01).
10 10 Advances in Mechanical Engineering Figure 12. Spectrum analysis from shaker at 200 Hz: (a) spectrum analysis from scope using builtin FFT and (b) spectrum analysis from microprocessor using efft and FFT. Figure 13. Spectrum analysis from shaker at 501 Hz: (a) spectrum analysis from scope using builtin FFT and (b) spectrum analysis from microprocessor using efft and FFT. Conclusion Although FFT is still widely applied to signal analysis in industry, it may suffer from incorrect outcomes if the vibration frequency is noninteger due to spectrum leakage. In efft model, the relation between vibration frequency and dispersed leakage caused from FFT can be induced. The spectrum of individual frequency and amplitude can be thus accurately calculated. This article has reviewed and compared both efft and FFT models in vibration measurement. Before practical analysis, the models were calibrated with preknown sine and triangle waveforms. The group bandwidth (t) in efft model is determined between 1 and 5 according to the selection rule of the efft to retrieve the original amplitude. Also, the spectrum frequency can be rectified by the dominant frequency (f k ) plus frequency deviation (Df k ). Consequently, the efft model can efficiently restore the dispersed power caused by FFT and thus achieve a correct spectrum analysis. As can be seen from experimental results, it is shown that the efft model is better than the usual FFT. However, the group bandwidth (t) should be appropriately selected especially in case of close harmonics.
11 Lin and Ye 11 Figure 14. Spectrum analysis from tool machine. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article. ORCID id HsiungCheng Lin References 1. Saruhan H, Sandemir S, Cxicxek A, et al. Vibration analysis of rolling element bearings defects. J Appl Res Technol 2014; 12: Cong F, Chen J, Dong G, et al. Vibration model of rolling element bearings in a rotorbearing system for fault diagnosis. J Sound Vib 2013; 332: Patil AM, Kadam PB and Mithari RS. Vibration analysis of bearings using FFT analyzer. Int J Adv Technol Eng Sci 2014; 2, ijates.com/images/short_ pdf/ _vibration_analysis_of_bear INGS_USING_FFT_ANALYZER_5968.pdf 4. Pan M and Tsao W. Using appropriate IMFs for envelope analysis in multiple fault diagnosis of ball bearings. Int J Mech Sci 2013; 69: Li R, He D and Zhu J. Investigation on full ceramic bearing fault diagnostics using vibration and AE sensors. In: Proceedings of the 2012 IEEE conference on prognostics and health management (PHM), Denver, CO, June 2012, pp New York: IEEE. 6. Lacey SJ. An overview of bearing vibration analysis. Maint Asset Manage 2008; 23: Fan Z and Li H. A hybrid approach for fault diagnosis of planetary bearings using an internal vibration sensor. Measurement 2015; 64: Orhan S, Aktu rk N and Cxelik V. Vibration monitoring for defect diagnosis of rolling element bearings as a predictive maintenance tool: comprehensive case studies. NDT&E Int 2006; 39: Garad A, Sutar KB, Shinde VJ, et al. Analysis of vibration signals of rolling element bearing with localized defects. Int J Curr Eng Technol 2017; 7: Patel VN, Tandon N and Pandey RK. Vibrations generated by rolling element bearings having multiple local defects on races. Proc Tech 2014; 14: Yan R, Gao RX and Chen X. Wavelets for fault diagnosis of rotary machines: a review with applications. Signal Process 2014; 96: Shah DS and Patel VN. A review of dynamic modeling and fault identifications methods for rolling element bearing. Proc Tech 2014; 14: Randall RB and Antoni J. Rolling element bearing diagnostics. Mech Syst Signal Process 2011; 25: Aherwar A and Khalid MS. Vibration analysis techniques for gearbox diagnostic: a review. Int J Adv Eng Technol 2012; 3: Slavic J, Brkovic A and Boltezar M. Typical bearingfault rating using force measurementsapplication to real data. J Vib Control 2012; 17: Lee DH, Lee JH and Ahn JW. Mechanical vibration reduction control of twomass permanent magnet synchronous motor using adaptive notch filter with fast Fourier transform analysis. IET Electr Power App 2012; 6: Singh KM and Sumathi P. Vibration parameter estimation methods for ultrasonic measurement systems a review. IET Sci Meas Technol 2015; 9: Singleton RK, Strangas EG and Aviyente S. Extended Kalman filtering for remainingusefullife estimation of bearings. IEEE T Ind Electron 2015; 62: Saxena V, Chowdhury N and Devendiran S. Assessment of gearbox fault detection using vibration signal analysis and acoustic emission technique. J Mech Civil Eng 2013; 7: Patil SS and Gaikwad JA. Vibration analysis of electrical rotating machines using FFT: a method of predictive maintenance. In: Proceedings of the 4th international conference on computing, communications and networking technologies (ICCCNT), Tiruchengode, India, 4 6 July 2013, pp.1 6. New York: IEEE. 21. Rai VK and Mohanty AR. Bearing fault diagnosis using FFT of intrinsic mode functions in Hilbert Huang transform. Mech Syst Signal Process 2007; 21: Atoui I, Meradi H, Boulkroune R, et al. Fault detection and diagnosis in rotating machinery by vibration monitoring using FFT and wavelet techniques. In: Proceedings of the 8th international workshop on systems, signal processing and their applications (Wosspa), Algiers, Algeria, May 2013, pp New York: IEEE. 23. Satish L. Shorttime Fourier and wavelet transform for fault detection in power transformers during impulse tests. IEEE P Sci Meas Technol 1998; 145: Staszewski WJ, Worden K and Tomlinson GR. Timefrequency analysis in gearbox fault detection using the
12 12 Advances in Mechanical Engineering WignerVille distribution and pattern recognition. Mech Syst Signal Process 1997; 11: Qin Z, Chen L and Bao X. Continuous wavelet transform for nonstationary vibration detection with phaseotdr. Opt Express 2012; 20: Ouelaa DN, Benchaabane C and Laefer DF. Application of the wavelet multiresolution analysis and Hilbert transform for the prediction of gear tooth defects. J Meccanica 2012; 47: Yan R and Gao RX. Hilbert Huang transformbased vibration signal analysis for machine health monitoring. IEEE T Instrum Meas 2006; 55: Feldman M. Hilbert transform in vibration analysis. Mech Syst Signal Process 2011; 25: Li H, Zhang Y and Zheng H. HilbertHuang transform and marginal spectrum for detection and diagnosis of localized defects in roller bearings. J Mech Sci Technol 2009; 23: Sun P, Liao Y and Lin J. The shock pulse index and its application in the fault diagnosis of rolling element bearings. Sensors 2017; 17: Wang D, Tse PW and Tsui KL. An enhanced Kurtogram method for fault diagnosis of rolling element bearings. Mech Syst Signal Process 2013; 35: Lei Y, Lin J, He Z, et al. Application of an improved kurtogram method for fault diagnosis of rolling element bearings. Mech Syst Signal Process 2011; 25: Chen X, Feng F and Zhang B. Weak fault feature extraction of rolling bearings based on an improved kurtogram. Sensors 2016; 16: Tse PW and Wang D. The design of a new sparsogram for fast bearing fault diagnosis. Mech Syst Signal Process 2013; 40: Wang D. Spectral L2/L1 norm: a new perspective for spectral kurtosis for characterizing nonstationary signals. Mech Syst Signal Process 2018; 104: Oppenheim AV and Schafer RW. Discretetime signal processing. London: Prentice Hall, Press WH, Flannery BP, Teukolsky SA, et al. Numerical recipes: the art of scientific computing. Cambridge: Cambridge University Press, 1986, pp IEC : Electromagnetic compatibility (EMC) Part 4: testing and measurement techniques Section 7: general guide on harmonics and interharmonics measurements and instrumentation for power supply systems and equipment connected thereto. 39. Lin HC, Ye YC, Huang BJ, et al. Bearing vibration detection and analysis using enhanced fast Fourier transform algorithm. Adv Mech Eng 2016; 8: 1 14
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