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A Study on the Exponential Decay for the Bearing Defect Diagnosis 沈 毓 泰 南台科技大學 機械系 教授 1. Introduction Signal processing for bearing defect vibration typical of amplitude modulation 1.5 1 Amplitude 0.5 0 -0.5 -1 -1.5 0 0.005 0.01 Time (Sec) 0.015 0.02 A single mode vibration 2007/ 5/ 19 2 南台科大 訊號處理與機械診斷實驗室 2. High frequency resonance technique The traditional signal processing band-pass filter rectifier low-pass filter 1 0.9 0.8 Amplitude 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.005 0.01 0.015 0.02 Time (Se c) 0.025 0.03 0.035 The demodulated single 2007/ 5/ 19 3 南台科大 訊號處理與機械診斷實驗室 3.Mechanical vibration analysis The vibration signal with amplitude modulation for the bearing vibration L md l 1 m 1 v(t ) ( d m ( ) qm ( ) alm ( ) e l (t ) cos(2 f l (t ) lm ) d nr t + g n ( ) qn ( ) aln ( ) e l ( t ) cos(2 fl (t ) ln ) d ) n 1 2007/ 5/ 19 t 4 南台科大 訊號處理與機械診斷實驗室 The vibration signal for the lth mode vibration would be i i vl (i ) l ( j ) cos(2 fl ) l ( j )sin(2 f l ) fs fs 2007/ 5/ 19 i : the sampling point of the bearing vibration f s : the sampling frequency j : the interval number of the envelope signals 5 南台科大 訊號處理與機械診斷實驗室 4.The envelope estimation the number of data points is h over the jth half period i i cos(2 f ) sin(2 f ) 1 1 fs fs v(i ) v(i 1) i 1 i 1 cos(2 f ) sin(2 f ) 1 1 fs fs v(i 2) v(i h 1) cos(2 f i h 1) sin(2 f i h 1) h1 1 1 fs fs 1 ( j ) w(i ) ( j) w(i 1) 1 2 ( j ) w(i 2) L ( j ) w(i h 1) h1 2 L1 2007/ 5/ 19 6 i 1 sin(2 f L ) fs i h 1 sin(2 f L ) h2 L fs sin(2 f L i ) fs 南台科大 訊號處理與機械診斷實驗室 The solution obtained in a linear least squares sense would be X 2 L1 M (21L 1)(2 L 1)V2 L1 , T 1 T X ( M M ) M h2 L h2 L h2 LVh1 , 2 L1 for h=2L for h>2L The envelope signal would be el ( j ) l ( j )2 l ( j )2 2007/ 5/ 19 7 南台科大 訊號處理與機械診斷實驗室 -4 0.02 0 -0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Amplitude 0.007 0.009 Amplitude 0 6 -3 0x 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 (d) 3 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.02 0.1 (e) 0 -0.02 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 2007/ 5/ 19 1000 2000 3000 4000 5000 6000 7000 (b) 1 -4 0x 10 1000 2000 3000 4000 5000 6000 2 7000 (c) 1 -4 0x 10 1000 2000 3000 4000 5000 6000 2 7000 (d) 1 0 8 -6 0x 10 1000 2000 3000 4000 5000 6000 7000 (e) 4 0 8 -4 0x 10 2 0 Time (sec) (a) 1 0 0.018 (c) 2 0 Amplitude Amplitude 0.02 (b) 0 Amplitude 0.01 0.014 Amplitude Amplitude 0 x 10 Amplitude (a) Amplitude Amplitude The inner-race defect case 0 1000 2000 3000 4000 5000 6000 7000 南台科大 訊號處理與機械診斷實驗室 The advantages 2007/ 5/ 19 the envelope estimation algorithm could theoretically retrieve several envelope signals corresponding to their mode vibrations and the envelope signals are less distorted. the envelope signals would automatically become low-sampling-rate signals. 9 南台科大 訊號處理與機械診斷實驗室 2007/ 5/ 19 it is easy in calculation to retrieve the envelope signals from a modulated vibration or even to reconstruct the modulated vibration from the envelope signals. 10 南台科大 訊號處理與機械診斷實驗室 5. The estimation of exponential decay The envelope of an impulse response in the time interval, Dt, could be expressed as elm (t ) Clm e l t with 0 t Dt and would be rewritten as ln(elm (1)) 1/ 2 fl 1 ln(e (2)) 1/ 2 f 1 l lm l l ln( C ) lm ln( e ( h )) 1/ 2 f 1 lm l h1 h2 2007/ 5/ 19 11 南台科大 訊號處理與機械診斷實驗室 The solution for l and ln(Clm) l 1 M h2 E 21 , for h=2 ln(Clm ) l ( M T M )1 M T E , for h>2 ln(C ) h2 h2 h2 h1 lm 2007/ 5/ 19 where ln(elm (1)) ln(e (2)) lm E h1 ln(elm (h)) h1 12 南台科大 訊號處理與機械診斷實驗室 5、Experimental Result The exponential decay for the envelope signals at running speed 800 rpm 2007/ 5/ 19 Resonance mode Bearing type First resonance mode Second resonance mode Third resonance mode Normal 202 575 1430 Outer-race defect 587 1127 2332 Inner-race defect 594 1219 2721 Roller defect 486 1159 2054 Multiple defect 441 1216 1910 13 南台科大 訊號處理與機械診斷實驗室 The exponential decay for the envelope signals at running speed 1600 rpm 2007/ 5/ 19 Resonance mode Bearing type First resonance mode Second resonance mode Third resonance mode Normal 116 399 1033 Outer-race defect 582 1115 2934 Inner-race defect 587 1244 2798 Roller defect 581 1221 2640 Multiple defect 438 1287 2278 14 南台科大 訊號處理與機械診斷實驗室 6、Conclusion 2007/ 5/ 19 an envelope estimation algorithm is carried out to retrieve the envelope signals from the corresponding resonance modes. the diagnosis index based on the exponential decay is proposed and proved effective in monitoring the bearing running condition. 15 南台科大 訊號處理與機械診斷實驗室 Thank you 2007/ 5/ 19 16 南台科大 訊號處理與機械診斷實驗室