A R, H ilbert2h uang T ran sfo rm and A R M odel

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1 : (004) H ilberth uang A R,, (, 41008) : H ilberth uang A R H ilberth uang IM F ( Intrinsic M ode Function), IM F, IM F A R, M ahalanobis,, : H ilberth uang ; ; ; A R ; ; : TH 165; TH 113; TH 133 : A A Fau lt D iagno sis A pp roach fo r Ro ller Bearings Based on H ilberth uang T ran sfo rm and A R M odel CH EN G Junsheng, YU D ejie, YAN G Yu (Co llege of M echan ical and A u tom o tive Engineering, H unan U n iversity, Changsha 41008, Ch ina) Abstract: A fault diagno sis app roach fo r ro ller bearings based on H ilberth uang transfo rm and A R model is p ropo sed. T he H ilberth uang transfo rm is used to decompo se the vibration signal of a ro ller bearing into a num ber of IM F components and the instantaneous amp litudes and frequencies of each IM F component are obtained. T hen the A R model of each instantaneous amp litude and frequency sequence is estab lished. T he m ain au toregressive param eters and the variances of rem nan t are regarded as the featu re vecto rṡ T hu s, the M ahalanob is distance criterion function is estab lished to iden tify the condition and fault pattern of a ro ller bearing. P ractical examp les demonstrate that the app roach based on H ilberth uang tran sfo rm and A R m odel can be app lied to the ro ller bearing fau lt diagno sis effectively. Key words: H ilberth uang transfo rm; instantaneous amp litude; featu re vecto r; distance criterion function instantaneous frequency A R model; 1 3 :, ;, ;,,,,,,, H ilberth uang A R, : : ( ); ( ) : (1968- ),,,,, Em ail: 163. net

2 H ilberth uang A R 93 M ahalanob is A R,, A R [ 1,,A R ], A R, A R,,, A R NA R, [ 3 ], A R H ilberth uang H ilberth uang H uang [ 4 ], EM D (Em p iricalm ode D ecom po sition) H ilbert EM D, ( In trin sicm ode Function, IM F),, IM F,, [ 4 IM F ] IM F H ilbert IM F, -, IM F,,, IM F, EM D IM F, IM F H ilbert, IM F, IM F A R, A R M ahalanob is,,,, H ilberth uang A R, H ilbert-huang [ 4, 5 ] H ilberth uang EM D H ilbert EM D, ( IM F) :, ;, IM F x (t) : 1),,, m 1, x (t) - m 1 = h 1. (1), h 1 IM F, h 1 x (t) ) h 1 IM F, h 1, (1), m 11, h 11 = h 1 - m 11 IM F,, k, h 1 (k- 1) - m 1k = h 1k, h 1k IM F c1 = h 1k, c1 x (t) IM F 3) c1 x (t), r1 = x (t) - c1. () r1 1) ), x (t) IM F c, n, x (t) n IM F r1 - c = r, g rn- 1 - cn = rn. rn IM F, () (3) (3)

3 x (t) = 6 n j= 1 cj + rn. (4), x (t) n rn,, c1, c,, cn,, rn x (t) (4) ci (t) H ilbert H [ci (t) ] = Π 1 - ci (Σ) dσ. (5) t - Σ z i (t) = ci (t) + jh [ci (t) ] = a i (t) e j5 i (t), (6) a i (t) = c i (t) + H [ci (t) ], (7) 5 i (t) = arctan H [ci (t) ], (8) ci (t) f i (t) = 1 d5 i (t). (9) Π dt 1 1, EM D, 16 IM F rn 5 IM F H ilbert ( : m s - ) ( : H z), a i f i ( i = 1,, 5 ) i IM F, EM D, IM F, IM F, 3 H ilbert-huang AR EM D x (t) n IM F c1 (t), c (t),, cn (t), IM F H ilbert a 1 (t), a (t),, a n (t) f 1 (t), f (t),, f n (t),, a i (t) f i (t) i IM F ci (t) H ilberth uang, x (t), a i (t) f i (t), x (t) s (t) A R ( m ) [ 6 ], Υk (k = 1,,, m ) s (t) + 6 m k= 1 Υk s (t - k) = e (t), (10) s (t) A R ( m ) ; e (t) m, Ρ i IM F a i (t) f i (t)

4 H ilberth uang A R 95 5 IM F (10) A R,, a i (t) f i (t) A R i IM F A i = [Υia1, Υia, Ρia, Υif 1, Υif, Ρ if ],, Υia1 Υia Ρ ia a i (t) ; Υif 1 Υif Ρ if f i (t),, ( Υia1 Υia Υif 1 Υif ) ( Ρ ia Ρ if ) A i 3, EM D, H ilbert, IM F ci (t) a i (t) f i (t) A R, Υia1 Υia Υif 1 Υif Ρ ia Ρ if A i = [Υia1, Υia, Ρia, Υif 1, Υif, Ρ if ], M ahalanob is ( M ) [ 7 ] 3 H ilberth uang A R : 1), f s N, 3N ; ) EM D, IM F n 1, n,, n3n, n1, n,, n 3N n, IM F nk < n (k = 1,,, 3N ), n c1 (t), c (t),, cn (t), ci (t) = {0} (i = n k + 1, nk +,, n) ; f i (t) ; 3), (5) - (9) IM F ci (t) a i (t) 4), a i (t) f i (t) a δ i (t) f δ i (t) a δ a i (t) i (t) =, f δ f i (t) i (t) =. (11) a i (t) dt - f i (t) dt -

5 5) a δ i (t) f δ i (t) A R, FPE [ 6 ] m, Υiak (k = 1,,, m ) Υif k (k = 1,,, m ) Ρ ia Ρ if,, Υiak Υif k a i (t) f i (t) k ; 6) N Υiak (k = 1, ) Ρ ia Υif k (k = 1, ) Ρif λ Υ iak (k = 1, ) Ρ λ ia λ Υ if k (k = 1, ) Ρ λ if V ar (Υiak) V ar (Ρ ia) V ar (Υif k) V ar (Ρ if ), i IM F A ϖ j, i = [Υ λ ia1, Υ λ ia, Ρ λ ia, Υ λ if 1, Υ λ if, Ρ λ if ], (1) j = 1,, 3 ; 7) x (t), EM D x (t), IM F n, n c1 (t), c (t),, cn (t) ; IM F a i (t) f i (t), (11) a i (t) f i (t) a δ i (t) f δ i (t), a δ i (t) f δ i (t) A R, Υx, iak (k = 1, ) Υx, if k (k = 1, ) Ρ x, ia Ρ x, if, x (t) i IM F A x, i = [Υx, ia1, Υx, ia, Ρ ia, Υx, if 1, Υx, if, Ρ x, if ]; (13) 8) x (t) i IM F A x, i i IM F A ϖ j, i M ahalanob is d j, i = 6 k= 1 Υx, iak - λ Υiak V a r (Υiak) + Ρ x, ia - Ρ λ ia V ar (Ρia ) + 6 k= 1 Υ λ if k V ar (Υif k) Υx, if k - + Ρ x, if - Ρ λ if V ar (Ρ if ) 1g, (14), j = 1,, 3 ; i (i = 1,,, n) i IM F ; ) a 1, a,, an, x (t) d j = a13 d j, 1 + a3 d j, + + an3 d j, n = 6 n, a1, a,, a n : 6 n a i = 1, j = 1,, 3 ; i= 1 i= 1 a i3 d j, i, (15) 10) d 1, d, d 3, x (t), ,,, 0. 15mm, 0. 13mm, 0, 4096H z, 10,,,, IM F, FPE m, a i (t) f i (t) m, m 13 Υiak (k = 1, ) Υif k (k = 1, ) Ρ ia Ρ if, 6, 1 IM F ( V ar (Υiak) V ar (Ρ ia) V ar (Υif k) V ar (Ρ if ) ), A ϖ j, i ( j = 1,, 3, i = 1,, IM F ) j i IM F a1, a, a 3 d j ( j = 1,, 3, ),, a 1 = 0., a = 0. 6, a3 = 0.

6 H ilberth uang A R 97,,, IM F 6,, 30, 1 Υ λ λ ia1 Υia Ρ λ λ ia Υ λ if 1 Υif Ρ λ if A ϖ 1, A ϖ 1, A ϖ 1, A ϖ, A ϖ, A ϖ, A ϖ 3, A ϖ 3, A ϖ 3, ( a 1 = 0., a = 0. 6, a3 = 0. ) d 1 d d A R,,, A R,, A R H ilberth uang,,, A R, A R,, M ahalanob is,,,,,, H ilberth uang A R ( 109 )

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