An envelope detection method based on the first-vibration-mode of bearing vibration Yuh-Tay Sheen * Department of Mechanical Engineering, Southern Taiwan University, 1 Nan-Tai Street, Yung Kang City, 710 Tainan County, Taiwan, ROC Received 29 March 2007; received in revised form 12 September 2007; accepted 23 November 2007 Available online 4 December 2007 Abstract In this paper, the resonance frequency in the first-vibration-mode of mechanical system is studied and applied in the envelope detection for the bearing vibration. The vibration signal of a bearing system is a typical vibration with amplitude modulation. Under the assumption of a stepwise function for the envelope signal, the modulated signal could be decom- posed into a sinusoidal function basis at the first-vibration-mode resonance frequency. According to the vibration spec- trum, the first-vibration-mode resonance frequency could be initially designated. By applying a recursive estimation algorithm, the resonance frequency could be derived more precisely. Thus, the envelope signal could be retrieved by esti- mating the coefficients of the function basis with the linear least squares analysis. In addition, the vibration signal with noise rejection could be directly reconstructed from the envelope signal. According to the experimental study, the envelope detection method for the first-vibration-mode resonance frequency could be effectively applied in the signal processing for the bearing defect diagnosis. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Resonance frequency; Amplitude modulation; Envelope detection; Bearing defect 1. Introduction The vibration signal of a bearing system is a typ- ical vibration with amplitude modulation. The high- frequency resonance technique [1] is usually app lied to the mechanical vibrations with amplitude modu- lation. When applying the high-frequency resonance technique to detect the envelope signal, the vibra- tion signal is operated through a bandpass filter to derive a single mode vibration and then the envelop- ing transformation is applied to retrieve an envelope signal. In the range of the system resonance, this technique takes advantage of the absence of low-fre- quency mechanical noise to demodulate a vibration signal and, therefore, provides a low-frequency envelope signal with a high signal-to-noise ratio. In order to implement the high-frequency resonance technique, the Hilb ert transform is often applied in vibration signal demodulation to provide a complex signal. Accordingly, the envelope signal could be 0263-2241/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.measurement.2007.11.007 * Tel.: +886 6 2533131x3522; fax: +886 6 2425092. E-mail address: syt@mail.stut.edu.tw Available online at www.sciencedirect.com Measurement 41 (2008) 797–809 www.elsevier.com/locate/measurement obtained from the absolute value of the complex sig- nal. There are many papers [2–5] applying this method in the envelope detection of vibration signals. Although the high-frequency resonance tech- nique is widely applied in the envelope detection of bearing vibration with amplitude modulation, it is difficult to properly designate the bandpass filter through which a complete mode vibration could be filtered. There is no guarantee a complete mode vibration of bearing vibrations could be filtered through such a bandpass filter. Thus, it could seri- ously distort the envelope signal. In addition, when applying the Hilbert transform in the high-fre- quency resonance technique for the envelope detec- tion of vibration signals, the computing burden would be very high. In this paper, a signal process ing method and a recursive estimation algorithm are proposed to apply in the vibration signal with amplitude modulation. Based on the resonance frequency in the first-vibration-mode of a mechanical system, the envelope detection for the vibration signal with amplitude modulation could be easily and fast achieved by the linear least squares analysis. According to the vibration spectrum, the resonance frequency in the first-vibration-mode would be ini- tially designated. A recursive estimation algorithm would be then proposed to improve the precision of estimation for the resonance frequency. Under the assumption of a stepwise function for the enve- lope signal, the vibration signal could be decom- posed into a sinusoidal function basis at the resonance frequency. Thus, the envelope signal could be directly retrieved by estimating the coeffi- cients of the function basis with the linear least squares analysis. Besides, the vibration signal with noise rejection could be easily reconstructed from the envelope signal. In the experimental study, the effectiveness of the envelope detection method with the first-vibration-mode resonance frequency would be invest igated and applied in the bearing defect diagnosis. 2. A study on the bearing vibrations 2.1. Amplitude modulation of bearing vibration In a bearing system, the carrier signal could be a combination of the resonant frequencies of the bear- ing or even of the mechanical system, and thus the vibration signal with amplitude modulation could be represented as [6–8] where w(t) is the low-frequency mechanical noise, and the first two summation terms describe the vibrations of defect components and normal compo- nents. In the first term of Eq. (1), md is the number of defect, d m (t) depicts the defect impulse train of con- tact and is the modulating signal, q m (t) describes the dimension informat ion of defect and the sensitivity of striking energy, and a lm (t) is the charact eristic function of transmission path. In the other term, nr is the number of roller, g n (t) describes the surface functions of normal bearing components with respect to roller n and could also be a modulating signal, q n (t) is the equivalent stiffness of roller n and is a function of the structure stiffness and the oil film stiffness, and a ln (t) is the transmission path function which describes the vibration strength excited by roller n. For the other variables, r l and f l are respectively the exponential damping fre- quency and the carrier frequency, and would be the intrinsic characteristics of the system. L is the quantity of vibration mode of the system. h lm (t) and h ln (t) are the initial angles for the amplitude modulation. It should be noted that the frequencies of modulating signal d m (t) and g n (t) would be always much smaller than that of the carrier signal, and the frequencies of modulating signal d m (t) and g n (t) are higher than those of q m (t), a lm (t), q n (t)anda ln (t). Accordingly, the modulated signal would be expanded in frequency band whose center frequency vðtÞ¼ X L l¼1 X md m¼1 Z t À1 d m ðsÞq m ðsÞa lm ðsÞe Àr l ðtÀsÞ cosð2pf l ðt À sÞþh lm Þds þ X nr n¼1 Z t À1 g n ðsÞq n ðsÞa ln ðsÞe Àr l ðtÀsÞ cosð2pf l ðt À sÞþh ln Þds ! þ wðtÞð1Þ 798 Y T. Sheen / Measurement 41 (2008) 797–809 would be at the frequencies of carrier signal. This phenomenon is named as amplitude modulation. Suppose that the impulse responses due to impacts d m (t) are completely died out in a time interval between two consecutive impact contacts, and the resonance frequency f l is high. The impact energy due to the surface irregularity g n (t)q n (t) would be much smaller than that due to the bearing defects. Thus, the second term of Eq. (1) could be neglected. Accordingly, the envelope signal of the lth mode vibration could be written as e l ðtÞ¼ X md m¼1 u m ðtÞq m ðtÞa lm ðtÞþw l ðtÞ; ð2Þ with u m ðtÞ¼e Àr l t 0 and t 0 = mod(t,1/f dm ), where w l (t) is the low-frequency mechanical noise occurring in thelth vibration mode, mod(t,1/f dm ) represents a residue of t,andf dm is the frequency of impulse train d m (t). The spectrum for the envelope signal would show a pattern of the modulating signal frequency and its harmonics with equal frequency spacing sidebands which are induced by the frequencies of q m (t) and a lm (t). When applying the high-frequency resonance technique to detect the envelope signal, the vibration signal of defect bearing is operated through a bandpass filter to derive a single vibration mode and then taking the enveloping transforma- tion to retrieve an envelope signal. However, there is no guarantee such a signal processing could de- rived a complete mode vibration. Thus, it could seri- ously distort the envelope signal. 2.2. Envelope detection for bearing vibrations In general, the resonant vibration modes of a mechanical system would be more than one and the structure should be strong enough to stably sup- port the bearing system. Thus, the first-vibration- mode resonance frequency could be high and away from the range of low-frequency noise. In this paper, the first mode vibration is suggested to be fil- tered out in order to reduce the computing burden. According to Eq. (1), the vibration signal decom- posed into a sinusoidal function basis at the reso- nance frequency f 1 in the first-vibration-mode for a defect bearing could be represented as v 1 ðtÞ¼ X md m¼1 u m ðtÞq m ðtÞa 1m ðtÞ cosð2pf 1 t þ h 1m Þþw 1 ðtÞ ¼ aðtÞ cosð2pf 1 tÞþbðtÞ sinð2pf 1 tÞþw 1 ðtÞð3Þ where a(t)andb(t) are the coefficients for the vibra- tion signal v 1 (t) mapping to a sinusoidal function basis. It is noted that the envelope signal would be- come smoother, and the envelope signal could thus be approximated by a stepwise function. Moreover, it is assumed that every time period of the carrier signal could be divided into two intervals for the envelope detection. Accordingly, the above signal could be expressed in a discrete mode v 1 ðiÞ¼a j cos 2pf 1 i f s  þ b j sin 2pf 1 i f s  þ w j ; ð4Þ where i, f s and w j are the sampling point, the sam- pling frequency of the vibration signal, and the vibration noise induced by the vibration noise w l (t), respectively. j is the interval number of the envelope signal. Accordingly, the sampling fre- quency f s must be at least 6 times higher than the resonance frequency f 1 . In addition, an anti-aliasing low-pass filter with the cutoff frequency between f 1 and 0.5f s Hz should be adopted. As shown in Fig. 1, there are more than three equations to solve the unknown coefficients a j , b j and w j . If the number of data points is h over the kth half period of the vibration signal v 1 (t), these points could be expressed in the matrix form v 1 ðhðk À 1Þþ1Þ v 1 ðhðk À 1Þþ2Þ . . . v l ðhkÞ 2 6 6 6 6 4 3 7 7 7 7 5 ¼ cos 2pf 1 hðkÀ1Þþ1 f s  sin 2pf 1 hðkÀ1Þþ1 f s  1 cos 2pf 1 hðkÀ1Þþ2 f s  sin 2pf 1 hðkÀ1Þþ2 f s  1 . . . . . . . . . cos 2pf 1 hk f s  sin 2pf 1 hk f s  1 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 a k b k w k 2 6 4 3 7 5 ð5Þ Y T. Sheen / Measurement 41 (2008) 797–809 799 The above equation could be rewritten as a simpli- fied expression ½V k  hÂ1 ¼½M k  hÂ3 a k b k w k 2 6 4 3 7 5 ð6Þ The solution of the matrix func tion in three un- knowns could be obtained in a linea r least squares sense by simple equations a k b k w k 2 6 4 3 7 5 ¼½M k  À1 hÂ3 ½V k  hÂ1 ; forh ¼ 3 a k b k w k 2 6 4 3 7 5 ¼½M k  T hÂ3 ½M k  hÂ3  À1 ½M k  T hÂ3 ½V k  hÂ1 ; forh > 3 8 > > > > > > > > > > > < > > > > > > > > > > > : ð7Þ where T denote the transpose of a matrix. Accord- ingly, the envelope signal in Eq. (2) could be derived from the following equation: e 1 ðjÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 j þ b 2 j q : ð8Þ Moreover, based on the envelope signal, a recon- structed vibration signal with noise rejection could be directly derived from v 0 1 ðiÞ¼a j cos 2pf 1 i f s  þ b j sin 2pf 1 i f s  : ð9Þ Accordingly, Eq. (9) is the vibration signal corre- sponding to the first mode vibration for the vibra- tion signal. 2.3. The designation of the first-vibration-mode resonance frequency for bearing vibrations However, the most important problem in the implementation of the above envelope detection method is how to designate the resonance frequency in the range of the first-vibration-mode for a vibra- tion signal. In general, an easy and feasible way to designate the resonance frequency is to choose a peak at the frequency ^ f 1 in the range of the first-vibration- mode. Thus, the peak frequency ^ f 1 would be an esti- mation of the resonance frequency in the range of the first mode vibration. It is very possible that an error Df occurred in the designation of resonance fre- quency. Thus, the actual resonance frequency would be f 1 ¼ ^ f 1 þ Df . The vibration signal v 0 1 ðiÞ of first- vibration-mode decomposed into a sinusoidal func- tion basis at the estimated resonance frequency could be an estimation of Eq. (9) and would be written as v 0 1 ðiÞ¼a 0 j cos 2p ^ f 1 i f s  þ b 0 j sin 2p ^ f 1 i f s  ¼ a 0 j cos 2pðf 1 À Df Þ i f s  þ b 0 j  sin 2pðf 1 À Df Þ i f s  ð10Þ Time Amplitude v l (k 1 )= α 1 cos(2 π f l k 1 /f s )+ β 1 sin(2pi f l k 1 /f s ), k 1 =1, 2, ,5 v l (k 1 )= α 2 cos(2 π f l k 1 /f s )+ β 2 sin(2 π f l k 1 /f s ), k 1 =6, 7, ,10 v l (k 1 )= α 3 cos(2 π f l k 1 /f s )+ β 3 sin(2 π f l k 1 /f s ), k 1 =11, 12, ,15 v l (k 1 )= α 4 cos(2 π f l k 1 /f s )+ β 4 sin(2 π f l k 1 /f s ), k 1 =16, 17, ,20 v l (k 1 )= α 3 cos(2 π f l k 1 /f s )+ β 3 sin(2 π f l k 1 /f s ), k 1 =21, 22, ,25 Fig. 1. The modulated signal for a vibration impact decomposes into a sinusoidal function at its resonant frequency. 800 Y T. Sheen / Measurement 41 (2008) 797–809 The above equation could be also rewritten as a sinusoidal function with fundamental frequency f 1 , and be expressed as v 0 1 ðiÞ¼ a 0 j cos 2pDf i f s  À b 0 j sin 2pDf i f s   cos 2pf 1 i f s  þ b 0 j cos 2pDf i f s  þa 0 j sin 2pDf i f s  sin 2pf 1 i f s  ð11Þ In comparison between Eqs. (9) and (11),ifv 0 1 ðiÞ¼ v 1 ðiÞ the following equations could be derived: a j ¼ a 0 j cos 2pDf i f s  À b 0 j sin 2pDf i f s  ð12:aÞ b j ¼ b 0 j cos 2pDf i f s  þ a 0 j sin 2pDf i f s  ð12:bÞ If substituting Eq. (12) into Eq. (8), the envelope signal could also be written as e 1 ðjÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 02 j þ b 02 j q : ð13Þ It could prove that the envelope signal could be accurately retrieved, even though, by applying an estimation of the resonance frequency. Thus, an estimation of the resonance frequency f 1 could be easily derived from the peak in the first-vibration- mode of the spectrum for the vibrat ion signal. Thus, the proposed envelope detection method would be capable of applying in practice. Nevertheless, the estimation of the resonance fre- quency should be accurate enough to apply in Eq. (4) to derive the envelope signal. In the following, a recursive estimation algorithm for deriving the first- vibration-mode resonance frequency is proposed: 1. Accor ding to the vibration spectrum of the mea- sured signal, a peak in the first vibration mode would be initial ly designated as the resonance frequency. 2. Applying the resonan ce frequency, the envelope signal could be derived from the linear least square analysis, as shown in Eqs. (7) and (8). 3. The vibration signal with noise rejection could be reconstructed from the envelope signal, as shown in Eq. (9). 4. Accor ding to the vibration spectrum of the reconstructed signal derived in Step 3, the peak would be designated as the resonance frequency . 5. Repeat Steps 2–4, until the resonance frequency is converged. Theoretically, a resonance frequency must exist within the range of first vibration mode on the vibration spectrum. The above algorithm should not diverge in estimating the resonance frequency. 3. Experimental study In the following, the applications of the proposed method on the vibration signals of tapered roller bearings (SKF type 32208) are studied. The electri- cal-discharge machining method is applied to pro- duce artificial defect on the surface of bearing components whi ch are roller, outer race and inner race. The defect sizes are described in the Table 1. The vibration signals are measured on the housing of the test bearing. The measured direction is radial to the shaft. Under different running speeds, the pro posed envelope detection method is applied and the results are shown in Figs. 2–4, respectively. According to Figs. 2a, 3a and 4a, the passband frequencies are similar under different running speeds except that the amplitude is increased, especially at the high-fre- quency band, with the running speed. There could be four resonance modes with the passband fre- quencies from 1 to 3 kHz, 3 to 5 kHz, 6 to 8 kHz and 8 to 10 kHz, respectively. In addition, it is noted that the four resonance modes are also similar under different types of bearing defect. Thus, the cutoff frequency of the low-pass filter for deriving the first-vibration-mode of bearing vibrations could be designated to be 3 kHz and a sampling rate 18 kHz are applied. According to Figs. 2b, 3b and 4b, the initial values designated for the first-vibra- tion-mod resonance frequencies are 2386, 2487 and 1671 Hz, respectively. By applying the recursive estimation method in the estimation of the reso- nance frequencies, the vibration spectra are shown in Figs. 2c, 3c and 4c with at the first-vibration- mod resonance frequencies at 2386, 2487 and 2519 Hz, respectively. It is noted that the resonance frequency shown in Fig. 4(b), is converged to 2519 Hz, as shown in Fig. 4c, in three times of Table 1 Defect sizes of defective bearings Defect type Defect size (length  width  depth) Roller defect 16 mm  0.15 mm  0.1 mm Outer-race defect 14 mm  0.15 mm  0.1 mm Inner-race defect 18.5 mm  0.15 mm  0.1 mm Y T. Sheen / Measurement 41 (2008) 797–809 801 recursive iterations. In addition, the vibration spec- tra shown in Figs. 2c, 3c and 4c for the reconstructed signals are almost the same as those shown in Figs. 2b, 3b and 4b, respectively, in the passband from 1.5 to 3 kHz for the measured signals. From the theoretical study, it is assumed that the first mode vibration should be filtered through an anti-aliasing low-pass filter. According to the above experimental study, it would prove that the first vibration mode of vibration signal could be filtered 0 2000 4000 6000 8000 10000 12000 0 0.1 0.2 0.3 Amplitude Frequency (Hz) a 1st vibration mode 2nd vibration mode 3rd vibration mode 4th vibration mode 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 0.1 0.2 Amplitude * 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 0.1 0.2 Frequency (Hz) Amplitude * :the initial designated resonance frequency b c * :the first-vibration-mode resonance frequency * Fig. 2. The vibration spectra for the roller defect bearing running at 800 rpm. (a) For the vibration signal with a passband from 0 to 12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal. 802 Y T. Sheen / Measurement 41 (2008) 797–809 through an anti-aliasing low-pass filter with the cut- off frequency at 3 kHz. Thus, the characteristics of the vibration signal in the first vibration mode could be very well represented by the reconstructed signal in Eq. (9). In the following, Fig. 5 shows the effectiveness of the proposed method on the envelope detection for a roller defect bearing at the running speed 1600 rpm. Fig. 5a–c shows the measured vibration signal, the retrieved envelope signal from Fig. 5a, 0 2000 4000 6000 8000 10000 12000 0 1 2 3 4 5 Amplitude Frequency (Hz) a 1st vibration mode 2nd vibration mode 3rd vibration mode 4th vibration mode 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 0.5 1 1.5 Amplitude 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 0.5 1 1.5 Frequency (Hz) Amplitude * :the initial designated resonance frequency * b * :the first-vibration-mode resonance frequency * c Fig. 3. The vibration spectra for the roller defect bearing running at 1600 rpm. (a) For the vibration signal with a passband from 0 to 12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal. Y T. Sheen / Measurement 41 (2008) 797–809 803 and the reconstructed signal derived from Fig. 5b, respectively. For the purpose of comparison, the envelope signal derived from the high-frequency resonant technique with the passband from 1 to 3 kHz is shown in Fig. 5d. In comparison with Fig. 5d, Fig. 5b shows a smoother signal with lower noise. In addition, Fig. 6a and b show the envelope spectra for the envelope signal in Fig. 5b and d, respectively. The envelope spectra are also similar and could obviously show the characteristic 0 2000 4000 6000 8000 10000 12000 0 3 6 9 Amplitude Frequency (Hz) a 1st vibration mode 2nd vibration mode 4th vibration mode 3rd vibration mode 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 Amplitude * 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0 1 2 3 Frequency (Hz) Amplitude * :the initial designated resonance frequency b * :the first-vibration-mode resonance frequency c * Fig. 4. The vibration spectra for the roller defect bearing running at 2400 rpm. (a) For the vibration signal with a passband from 0 to 12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal. 804 Y T. Sheen / Measurement 41 (2008) 797–809 frequency at 154 Hz for the roller defect. Thus, it would prove that the proposed method could be effective in the envelope detection for the bearing vibration. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -2 0 2 Amplitude 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.5 1 Amplitude 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -2 0 2 Amplitude 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.5 1 Amplitude Time (sec) a b c d Fig. 5. (a) The measured vibration signal for the roller defect bearing running at 1600 rpm, (b) the envelope signal of (a) in the first vibration mode, (c) the reconstructed signal from (b), (d) the envelope signal derived from the high-frequency resonance technique. 0 100 200 300 400 500 600 0 2 4 6 8 Amplitude 0 100 200 300 400 500 600 0 3 6 9 12 Amplitude Frequency (Hz) a b Fig. 6. The envelope spectra for a roller defect bearing running at 1600 rpm. (a) By applying the envelope detection with the first- vibration-mode resonance frequency, (b) by applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz. Y T. Sheen / Measurement 41 (2008) 797–809 805 Similar to Figs. 6–8 show the envelope spectra with the characteristic frequency at 190 Hz and 266 Hz for the outer-race defect bearing and the inner-race defect bearing, respectively. It is found 0 100 200 300 400 500 600 0 2 4 6 8 10 Amplitude 0 100 200 300 400 500 600 0 2 4 6 8 10 Amplitude Frequency (Hz) a b Fig. 7. The envelope spectra for a outer-race defect bearing running at 1600 rpm. (a) By applying the envelope detection with the first- vibration-mode resonance frequency. (b) By applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz. 0 100 200 300 400 500 600 0 1 2 Amplitude 0 100 200 300 400 500 600 0 1 2 Amplitude Frequency (Hz) a b Fig. 8. The envelope spectra for a inner-race defect bearing running at 1600 rpm. (a) By applying the envelope detection with the first- vibration-mode resonance frequency. (b) By applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz. 806 Y T. Sheen / Measurement 41 (2008) 797–809 [...]... resonance frequency From the above, it would prove the effectiveness of the proposed envelope detection method in both the resonance frequency estimation and the envelope detection Moreover, it would be capable of applying in the bearing defect diagnosis In comparison with the high-frequency resonance frequency technique, the proposed envelope detection method could be applied in the envelope detection. .. calculation to derive the envelope signal or even to reconstruct the vibration signal from the envelope signal Based on the envelope signal, the vibration signal with noise rejection could be easily reconstructed It is noted that the envelope detection method is to retrieve the corresponding envelope signal by directly decomposing the vibration signal into the sinusoidal function basis at the resonance frequency,... Conclusions According to the first -vibration- mode resonance frequency of the mechanical system, the signal processing of envelope detection is easily carried out with the linear least squares analysis Under the assumption of a stepwise function for the envelope signal, the vibration signal could be decomposed into a sinusoidal function basis at the resonance frequency Accordingly, the envelope signal could be... resonance frequency Fortunately, the resonance frequency in the first vibration mode would be low Thus, the computer performance is not required to be very high In general, the sampling rate could be less than 30 kHz for most cases in mechanical systems The implementation of the proposed envelope detection method for the vibration signal analysis could be easily achieved 5 Conclusions According to the. .. with the firstvibration -mode resonance frequency (b) By applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz -4 x 10 a Amplitude 8 6 4 2 0 0 x 10 10 0 200 300 400 500 -3 b 1 Amplitude 600 0.8 0.6 0.4 0.2 0 0 10 0 200 300 400 500 600 Frequency (Hz) Fig 10 The envelope spectra for a normal bearing running at 16 00 rpm (a) By applying the envelope detection with the first-vibrationmode... estimating the coefficients of the function basis with the linear least squares analysis It is noted that the resonance frequency could be recursively estimated from the peak within the range of first vibration mode of the spectrum for the vibration signal Thus, it would be convenient to apply in practice From the experimental study, it is shown that the proposed method could be effectively applied in the envelope. .. envelope detection of defect bearing vibrations Therefore, it would be useful to apply the signal processing method in the bearing defect diagnosis In addition, such a signal processing method could be easily reversed for the signal reconstruction with noise rejection In comparison with the high-frequency resonance technique, the proposed method possesses better properties in both envelope detection and... fault detection of tapered roller bearing: frequency domain analysis, Journal of Sound and Vibration 15 5 (19 92) 75–84 [3] N.G Nikolaou, I.A Antoniadis, Rolling element bearing fault diagnosis using wavelet packets, NDT&E International 35 (2002) 19 7–205 [4] Y.-T Sheen, C.-K Hung, Constructing a wavelet -based envelope function for vibration signal analysis, Mechanical Systems and Signal Processing 18 (1) ... signal reconstruction of vibration signals Acknowledgements This study is financially supported by the National Science Council of Taiwan (NSC 96-22 21- E 218 -043) References [1] P.D McFadden, J.D Smith, Vibration monitoring of roller element bearing by the high-frequency resonance technique – A review Tribology International, 19 84, p 3 10 Y.-T Sheen / Measurement 41 (2008) 797–809 [2] Y.T Su, S.J Lin, On initial... the wavelet decomposition and reconstruction could not accurately determine the carrier frequency because the wavelet is a sub-band function with a range of frequencies Thus, the above analytical expression is difficult to be accomplished by the wavelet transform However, this method theoretically has a limitation that the sampling rate of the vibration signal must be at least six times higher than the . An envelope detection method based on the first -vibration- mode of bearing vibration Yuh-Tay Sheen * Department of Mechanical Engineering, Southern Taiwan University, 1 Nan-Tai Street, Yung Kang. frequency in the first -vibration- mode of mechanical system is studied and applied in the envelope detection for the bearing vibration. The vibration signal of a bearing system is a typical vibration with. bearing vibrations 2 .1. Amplitude modulation of bearing vibration In a bearing system, the carrier signal could be a combination of the resonant frequencies of the bear- ing or even of the mechanical