Novel Ensemble Analytic Discrete Framelet Expansion for Machinery Fault Diagnosis 1

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1 OURNAL OF MECHANICAL ENGINEERING Vol.50 No.17 Sep DOI /ME * , 3 ( ) -- () - TH17 Novel Ensemble Analytc Dscrete Framelet Expanson for Machnery Fault Dagnoss 1 CHEN Bnqang 1 ZHANG Zhousuo 2 ZI Yanyang 2 HE Zhengja 2, 3 (1. School of Physcs and Mechancal & Electrcal Engneerng, Xamen Unversty, Xamen ; 2. School of Mechancal Engneerng, X an aotong Unversty, X an ; 3. State Key Laboratory for Manufacturng Systems Engneerng, X an aotong Unversty, X an ) AbstractAs the celebrated mathematcal scope, the mult-resoluton analyzng capacty of wavelet transform (WT) plays an mportant role n condton montorng and fault dagnoss of mechancal equpment. However, t has proven that the effectveness of WT s hampered by several negatve factors, such as shft-senstveness, sgnfcant energy leaage, and the fxed dyadc frequency-sale pavng. Especally, the dyadc frequency-sale pavng creates nevtable defcency n dentfyng mechancal sgnatures located n transton areas of adjacent wavelet scales. A novel tme-sale analyss methodology, named as derved ensemble analytc framelet (DEAF), based on overcomplete wavelet tght frame, s proposed. The DEAF s developed based on the exstng dual tree complex wavelet transform (DTCWT). The DEAF starts from a selected DTCWT bass, and combnes t wth a hybrd augmented tree-structured flter-ban, whch results n quas analytc wavelet pacet decomposton (QAWPD). Wth the results of QAWPT, an ensemble wavelet pacet generatng strategy s appled such that an unprecedented mplct wavelet pacet tght frame (IWPTF) contanng pseudo dyadc wavelet pacets s obtaned. Wth the combnaton of QAWPD and IWPTF, the proposed DEAF can be derved whch possesses the frequency-sale pavng characterzed by contnued tme-frequency refnement of analyss centers. The proposed technque s appled to the mechancal sgnature analyss of an engneerng applcaton to valdate ts superorty compared wth the exstng methods. * ( )(2010ZX , 2011ZX )

2 Key wordsmechancal fault dagnossdual-tree complex wavelet transformanalytc framelet expansonaugmented tree- structured flter-bantranslaton nvarance [1-3] (Dscrete wavelet transform, DWT) (Fast Fourer transform, FFT) [1, 3] DWT [2, 4-7] Daubeches(DB) DWT [8-9] DB SymmeletCoflet () [10] (MW) [11] (Dual tree complex wavelet transform, DTCWT) [12-13] DWT DWT (Dyadc frequency-scale pavng, DFSP) DFSP [5-6, 14] : (Derved ensemble analytc framelet, DEAF) DFSP DEAF ( )() DEAF DEAF DTCWT (Quas analytc wavelet pacet transform, QAWPT) QAWPT (Implct wavelet pacet tght frame, IWPTF) QAWPT IWPTF DEAF DEAF DTCWT - DEAF DEAF 1 (Quas analytc wavelet transform, QAWT) ( Re Im ) QAWT ( ) QAWT () () () () () { ( t), ( t),, ( t) } (1) 1 1 () Re Im r Im Re () t { () t } (2) {1,2,, r} {} Hlbert Fourer {} Im ˆ ( ) ˆ j ( ) 0 Re j ( ) 0 Re ˆ r (3) j -1 [ π, π]

2014 9 79 () t h (2 t n) (4) () () () 2 0 h Z () t h (2 t n) 1 r (5) () () () 2 h Z () h0 () h r 1, 2 [12-14]

tree-structured flter-ban, ATSFB) 2( ) QAWT DFSP DWT 2 (Quas analytc wavelet pacet transform, QAWPT) QAWT r 1

5) (6) Fourer G ( ) jsgn( ) H ( ) (7) 0 0 H ( ) 0 h0 Z G ( ) 0 g 0 Z H0 ( ) G 0 ( ){ h 0 } { g 0 } H0( )= h0(

3 () t h (2 t n) (4) () () () 2 0 h Z () t h (2 t n) 1 r (5) () () () 2 h Z () h0 () h r 1, 2 [12-14] KINGSBURY DTCWT ( r 1) [12] SELESNICK [13] QAWT [14] r 2 DFSP r DFSP QAWT( r 2 ) Mallat (Augmented tree-structured flter-ban, ATSFB) 2( ) QAWT DFSP DWT 2 (Quas analytc wavelet pacet transform, QAWPT) QAWT r 1 DTCWT 1 QAWPT QAWPT 2.1 SELESNICK DTCWT - Re () t Im () t h0 g 0 g n h n (6) ( ) 0 0 ( 0.5) (6) Fourer G ( ) jsgn( ) H ( ) (7) 0 0 H ( ) 0 h0 Z G ( ) 0 g 0 Z H0 ( ) G 0 ( ){ h 0 } { g 0 } H0( )= h0( )exp( j ) Z (8) G0( )= Z g0( )exp( j ) (9) DTCWT QAWPT DTCWT [15] DTCWT DTCWT ( 2) [15]DB9MV8 (Common factorzaton, CF) CF18 [16] [15] 18 DTCWT s() t

4 s() t 0.5cos(2π30 t) cos(2π85 t) 1.5cos(2π170 t) cos(2π340 t) 0.5cos(2π700 t) t[0,1.024] f 2000Hz L 2048 s 85 Hz170 Hz340 Hz 4 D 2 D 3 D 4 s () t cos(2πf t) t [0, 1.024] (10) s() t D Cd {, s} D Cd {, s} ( d ( ) d ) ( s ( ) s ) Z 2 2 [ d ( ) d] [ s ( ) s] Z Z f 2 fs /2 2 ˆ ˆ 0 f ED { } D( f) d f D( f) df (11) (12) (11)(12) Symmlet-18 h10 Re h11 Re Symmlet-18 h Im 01 h11 Im Im Re h10 h0 ( n 1) (13) Im Re h11 h11 ( n 1) (14) DTCWT DTCWT Q-Shft18 Re Re Im Im { h0, h1, h0, h1 } 2 DTCWT DTCWT QAWPT Q-Shft18 3 Re Re Im Im { h0, h1, h0, h1 } Q -Shft18 DB9 MV8 CF Q-Shft18 DB9 MV8 CF Q-Shft18 ([15] 23 ) 2.2 DTCWT [14] QAWPT 13 4 ( 2) QAWPT 3 DTCWT DTCWT { f 0, f 1 ( n )} Re 0 (15) f h 0 f n h n (16) ( ) Re 1 1 ( ) Z F 0 ( ) F 1 ( ) 4 4 QAWPT

5 QAWPT QAWPT 2 DTCWT QAWPT (1) 5 3 QAWPT () 4 QAWPT DTCWT (3) DTCWT [12] DTCWT QAWPT 7 7 QAWPT DTCWT 6 QAWPT DTCWT 7 QAWPT QAWPT (2) 6 ( 3 )QAWPT DTCWT 2 DFSP QAWPT DFSP 3

6 (1) QAWPT QAWPT { x } ( ) QAWPT x 2 P ( x) { D 1 2} (17) 4 P (2) (1) ( x) 1) D B { 0, 1,, 2, 1 log n n nm nm m 2 + 1} (18) B m1 m n (19) 0 2) B { 0, 1,, 2, 1 log n n n m n m m 2 + 1} (20) B B nm m1 nm mod( nm nm 1,2) 0,1,, 2 3) B (21) P( x) { D 1 2} (22) m1 m n (23) 0 (3) P( x) D 1 D 2 P( x) D (24) { } (Implct wavelet pacet, IWP) 21 2 I D D (25) 2 I ( x) { I () n 1 2 1} (26) 3 IWPTF QAWPT IWPTF DFSP 3 IWPTF QAWPT IWPTF QAWPT 0.5 f s f s 2 f 1 f 1 s 2 IWPTF 8 25 QAWPT I ( x) 8 IWPTF(25 )- 4 3 IWP QAWPT - IWPTF ( )() IWPT (1) QAWPT ( ) ( ) ( ) ( ) ( ) x n D n D n D n D n D D D D D D D D D D D D D I D (27) IWPTF QAWPT s

2014 9 83 (2) 9 3 IWPTF IWP e, () t IWP IWPTF

QAWPT - 10 I 10 QAWPT IWPTF QAWPT IWPTF 1)

(28) 2,1 3,2 4,4 2,2 fs /4 10 DEAF - 2) IWPTF I

1 2050 11 ( 65/22) 4 11 4 ( 11 14) 281.3 r/mn(4.

7 (2) 9 3 IWPTF IWP e, () t IWP IWPTF DTCWT QAWPT IWPTF ( ) 5 9 IWPTF (3 ) (3) - 4 QAWPT - 10 I 10 QAWPT IWPTF QAWPT IWPTF 1) IWPTF QAWPT 10 QAWPT IWPTF IWP I, I, I,, I, (28) 2,1 3,2 4,4 2,2 fs /4 10 DEAF - 2) IWPTF I I ( 1) I ( 65/22) ( 11 14) r/mn(4.688 Hz) Hz DEAF DEAF L 4 xn ( ) n0.05l x 4 L 1 x Kx [] 3 (29)

8 s Z nterval (s) (30) Z pnon f pnon 12 L x x x x x ANTONI [17-18] DEAF 13a 240 Hz 160 Hz 13b 13c s Hz Hz

9 (1) (2) DTCWT QAWPT IWPTFIWPTF IWPTF DTCWT (3) QAWPT IWPTF DEAF DEAF QAWPT IWPTF [1]. [M]2007. HE ZhengjaZI YanyangZHANG Xnng. Modern sgnal processng and engneerng applcaton[m]. X an X an aotong Unversty Press2007 [2] GAO R XYAN R Q. WaveletsTheory and applcatons for manufacturng[m]. HedelbergLondonSprnger [3] MALLAT S G. A wavelet tour of sgnal processng[m]. MassachusettsAcademc Press1999. [4] WANG S BHUANG W GZHU Z KTransent modelng and parameter dentfcaton based on wavelet and correlaton flterng for rotatng machne fault dagnoss[]. Mechancal Systems and Sgnal Processng (4) [5] CHEN B QZHANG Z SSUN Cet al. Fault feature extracton of gearbox by usng overcomplete ratonal dlaton dscrete wavelet transform on sgnals measured from vbraton sensors[]. Mechancal Systems and Sgnal Processng [6] HE W PZI Y YHE Z et al. Tunable Q-factor wavelet transform denosng wth neghborng coeffcents and ts applcaton to rotatng machnery fault dagnoss[m]. Scence Chna Technologcal Scences201356(8) [7] WANG GAO R XYAN R Qet al. Current envelope analyss for defect dentfcaton and dagnoss n nducton motors[m]. ournal of Manufacturng Systems (4) [8] PENG Z KACKSON R RCHU F Let al. On the energy leaage of dscrete wavelet transform[]. Mechancal Systems and Sgnal Processng200923(2) [9] WANG Y XHE Z ZI Y Y. Enhancement of sgnal denosng and multple fault sgnatures detectng n rotatng machnery usng dual-tree complex wavelet transform[]. Mechancal Systems and Sgnal Processng (1) [10] SWDLDENS W. The lftng schemea constructon of second generaton wavelets[]. SIAM ournal on Mathematcal Analyss199629(2) [11] STRELA VHELLER P NSTRANG Get al. The applcaton of multwavelet flterbans of mage processng[]. IEEE Trans. Sgnal Processng19998(4) [12] KINGSBURY N G. Complex wavelets for shft nvarant analyss and flterng of sgnals[]. Appled and Computatonal Harmonc Analyss [13] SELESNICK I WBRARNIUK R GKINGBURY N G. The dual tree complex wavelet transform[]. IEEE Sgnal Processng Magazne200522(6) [14]. [] (9) CHEN Bnqang ZHANG Zhousuo HE Zhengja. Enhancement of wea feature extracton n machnery fault dagnoss by usng double densty dual tree complex wavelet transform[]. ournal of Mechancal Engneerng (9) [15]. [] (3)7-12. CHEN BnqangZHANG ZhousuoGUO Tnget al. Tme-frequency doman constructon of dual tree complex wavelets for assembly clearance detecton of gear

10 chans[]. ournal of X an aotong Unversty (3)7-12. [16] SELESNICK I W. The desgn of approxmate Hlbert transform pars of wavelet bases[]. IEEE Transactons on Sgnal Processng200250(5) [17] ANTONI. Fast computaton of the urtogram for the detecton of transent faults[]. Mechancal Systems and Sgnal Processng200721(1) [18] WANG DPETER T WTSUI K L. An enhanced urtogram method for fault dagnoss of rollng element bearngs[]. Mechancal Systems and Sgnal Processng (1-2) ()1986 E-malmung.cbq@stu.xjtu.edu.cn 1964 E-malzzs@mal.xjtu.edu.cn 2012/2013 * (E-mallangl@tsnghua.edu.cn) (ESP) 30% 2012 ESP ESP ESP 70% ESP (VSEKF) ESP 2. (1) (2) ESP * (ICFDM2014)

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