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Vibra t ion mon i to r ing o f ro l l inge l e m e n t b e a rin g s b y t h e h ig h -f r e q u e n c y r e s o n a n c e t e c h n i q u ea r e v i e wP. D . M cFadden and J . D . S m i th *V i b r a t i o n m o n i t o r i n g o f r o l l in g e l e m e n t b e a rin g s b y t h e h i g h - f r e q u e n c yr e so n a n c e t e c h n i q u e is r e v ie w e d . I t i s s h o w n t h a t t h e p r o c e d u r e s f o ro b t a i n i n g t h e s p e c t r u m o f t h e e n v e l o p e s ig n al a r e w e l l e s t a b l is h e d , b u t t h a tt h e r e is a n in c o m p l e t e u n d e r s t a n d i n g o f t h e f a c t o r s w h i c h c o n t r o l t h ea p p e a r a n c e o f t h i s s p e c t ru m . U n t i l t h e e n v e l o p e s p e c t r u m c a n be f u l l ye x p l a i n e d , u s e o f t h e t e c h n i q u e is l i m i t e dKeyw ords: v ib ra t io n measurement , ro l l in g bearings , resonanceMany techniques a re ava i lable which use v ibra t ion ana lys isto de t e c t i nc ip i e n t f a u l t s in ro l l i ng e l e m e n t be a r ings . The s et e c hn ique s inc lude s hoc k pu l s e mon i to r ing , c re s t f a c to ra na ly s is , ku r to s i s , s pe c t rum a na lysi s a nd the h igh - f re que n c yr e s o n a n ce t e c h n i q u e , a l so k n o w n a s d e m o d u l a t e d r e s o n a n c ea na lysi s o r e nv e lope pow e r s pe c t ra l de ns i ty a na ly s i s.T h e h i g h - f r e q u e n c y r e so n a n c e t e c h n i q u e , h f r t , i s o n e o f t h eb e s t s u i t ed t o t h e v i b r a ti o n m o n i t o r i n g o f g e a r b o x e s a n dt u r b o m a c h i n e r y b e c a u s e o f i t s a b i l it y t o s e p a ra t e t h ev ib ra t ion ge ne ra t e d by a de fe c t ive be a r ing f rom the v ib ra -t i o n g e n e r a t e d b y t h e o t h e r m a c h i n e e l e m e n t s .T h e h f r t h a s b e e n c h o s e n a s t h e t o p i c f o r s t u d y i n t h i sre po r t , w h ic h c om pr i s e s a r e v ie w o f t he pa pe rs pub l i s he dove r the l a s t de c a d e , c ompa r ing the s imi la r i ti e s a nd d i f fe r -e nc e s , t he s ucc e ss es a nd fa i lu re s o f t he t e c h n ique . Ea c hs ta ge o f t he h f r t w i ll be e xa m ine d in tu rn a nd th ea p p r o a c h e s o f t h e d i f f e r e n t i n ve s ti g a to r s c o m p a r e d .B a s i c p r i n c i p l e sVibrat ion impulsesF u n d a m e n t a l t o t h e h f r t i s t h e c o n c e p t t h a t e a c h t i m e ad e f e c t i n a r o l l in g e l e m e n t b e a r i n g m a k e s c o n t a c t u n d e rloa d w i th a n o the r s u r fa c e in the be a r ing , a n impu l s e o fv ib ra t ion i s ge ne ra t e d . T h i s impu l s e i s o f e x t re m e ly s ho r td u r a t i o n c o m p a r e d w i t h t h e i n t e r v a l b e t w e e n i m p u l s e s ,a nd s o i t s e ne rgy is d i s t r ibu te d a t a ve ry low l e ve l ove r aw ide ra nge o f f r e que nc ie s . I t is th i s w ide d i s t r ibu t ion o fe ne rgy w h ic h m a ke s be a r ing de fe c t s s o d i f f i c u l t t o de t e c tby c o nve n t ion a l s pe c t rum a na lys is i n t he p re s e nc e o fv i b r at i o n f r o m g e ar s a n d o t h e r m a c h i n e e l e m e n t s .Fo r tuna te ly , t he impa c t u s ua l ly e xc i t e s a r e s ona nc e int h e s y s t e m a t a m u c h h i g h e r f r e q u e n c y t h a n t h e v i b r a t i o ng e n e r a t e d b y t h e o t h e r m a c h i n e e l e m e n t s , w i t h t h er e s ul t t h a t s o m e o f t h e e n e r g y is c o n c e n t r a t e d i n t o an a r r o w b a n d w h i c h i s m o r e r e a d i ly d e t e c t e d t h a n t h ew i d e l y d i s t r i b u t ed e n e r g y . T h e r e s o n a n c e c o u l d b e t h ea c t i o n o f t h e r o ll i ng e l e m e n t v i b r a ti n g b e t w e e n t h e i n n e ra nd ou te r r a c e s , t he v ib ra t ion o f t he r a c e s the ms e lve s in

*Cambridge University EngineeringDepartment, TrumpingtonStreet, Cambridge CB2 IPZ, UK

a r ing m ode , t he v ib ra t ion o f t he be a r ing hous ing o rma c h ine s t ru c tu re , v ib ra t ion in the t r a ns duc e r , o s c i l la t ioni n t h e i n s t r u m e n t a t i o n s y s t e m , o r m a y b e a c o m b i n a t i o n o ft h e s e . M o r e t h a n o n e r e s o n a n c e m a y b e e x c i t e d , a s i t i sl i ke ly tha t i n p ra c t i c a l s y s t e ms s e vera l na tu ra l f r e que nc ie sw i l l f a l l w i th in the f r e q ue n c y ra nge .Character ist ic defect frequen cyB e c au s e t h e c o n t a c t s b e t w e e n t h e d e f e c t a n d t h e m a t i n gs u r fa c e s in th e be a r ing a re e s s e n t ia l ly pe r iod ic , t he impu l s ew i ll r e c u r a t r e g u la r i n t e r v a l s ; t h e f r e q u e n c y o f o c c u r r e n c eo f the im pu l s e is r e fe r re d to a s t he c ha ra c t e r i s t i c de fe c tf r e q u e n c y .I t i s u s ua l t o c on s ide r t he r e s ona n c e a s be ing a mpl i tudemo du la t e d a t t he c ha ra c t e r i s t i c de fe c t f r e que n c y . Th i s i sno t s inus o ida l mod u la t ion , a s t he l e a d ing e dge c ompr i s e sa ve ry s ha rp r is e c o r re s pond ing to the im pa c t o f t he de fe c t ,w h i l e t h e d e c a y i s a p p r o x i m a t e l y e x p o n e n t i a l , as t h ee ne rgy i s d is s ipa te d by in t e rna l da mping . T he e nd re s u l t ,i n t he s imp le s t t e rms , c ons i s t s o f pe r iod ic bu rs t s o fe xpone n t i a l ly de c a y ing s inus o ida l v ib ra t ion . The f re que nc yo f t h e v i b r a ti o n i s t h e n a t u r a l f r e q u e n c y o f t h e r e s o n a n c e ,w h i l e t h e d e c a y r a t e is d e t e r m i n e d b y t h e d a m p i n g .The c ha ra c t e r i s t i c de fe c t f r e que nc ie s de pe nd on thel o c a t i o n o f t h e d e f e c t a n d a r e d e f i n e d b y e q u a t i o n s o f t h efo rm g ive n in R e fe re nc e s 9 a n d 13 . The s e f re que nc ie s a reo n l y a p p r o x i m a t e a s t h e y w i l l be a f f e c t e d b y s li p pi n g o fthe e l e m e n t s w i th in th e be a r ing a nd s p inn ing o f t he r a c e so n t h e s h a f t o r i n t h e h o u s i n g . U n d e r n o r m a l c o n d i t i o n s ,s l ip i s be l i e ve d to b e re l a t ive ly un im por t a n t . H ow e ve r , a th igh s pe e ds a nd l i gh t l oa ds , s l ip m a y be a s h igh a s 50% 14 .M o d u l a t io n o f t h e r e s o n a n c e b y t h e c h a r a ct e r is t ic d e f e c tf r e q u e n c y m a k e s it p o s si b le n o t o n l y t o d e t e c t t h ep r e s en c e o f a d e f e c t b y t h e e x c i t a t i o n o f t h e r e s o n a n c e ,bu t a l s o to d i a gnos e in w ha t p a r t o f t he be a r ing the de fe c tappears. Where several bearings are c losely group ed in amachine, i t is l ikely that the defect frequencies for eachb e a r i n g w il l b e u n i q u e b e c a u s e o f d i f f e r e n t g e o m e t r ie sb e t w e e n t he be a r ings , s o i t m a y s t il l be pos s ib le t o s e pa ra t ethe v ib ra t ion f rom d i f fe re n t be a r ings e ve n in c l o s ep r o x i m i t y .

TRIBO LO GY in ternational 03 01-6 79 X/84 /010 003 -08 ~03.00 1984 But terwor th & Co (Pub li shers ) L td 3

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M cFadden and Sm i th - v i b ra t ion m o n i t o r i ng o f r o l li ng bea ri ngs by h f r tInstrumentationIn p r ac t i ce , t he v i b r a t i on i s s ensed by a t r ansduce r mount edon t he ex t e r i o r o f t he mach i ne cas i ng c l ose t o t he bea r i ngt o be moni t o r ed . Usua l l y t he t r ansduce r used i s a p i ezo-e l ec t r i c acce l e romet e r , a s t hese have good r e sponse a t t heh i ghe r f r equenc i es where t he r e sonances o f t en occur .A cha rge amp l i f i e r i s t hen used , poss i b l y i nc l ud i ng an i n t e -g r a l l ow-pass f i l te r t o a t t enu a t e t he t r ansdu ce r r e sona nce .T h e o u t p u t o f t h e a m p l i fi e r m a y b e r e c o r d e d o n m a g n e t i ct ape fo r o f f - li ne p rocess i ng 13 , o r a l t e rna t i ve l y , be ana l ysedon- l ine 7 as fo l lows.T he s i gna l i s bandpass f i l t e r ed a round t he r e sonan tf r e q u e n c y t o e l im i n a t e t h e o t h e r f r e q u e n c ie s g e n e r a t e d b ys h a f t i m b a l a n c e , g e ar m e s h i n g , b a c k g r o u n d n o i se o r o t h e rr e sonances . T he s i gna l whi ch r ema i ns cons i s t s o f an a r r o w b a n d c a r ri e r a t t h e r e s o n a n t f r e q u e n c y , a m p l i t u d em o d u l a t e d a t t h e c h a r a c te r i st i c d e f e c t f r e q u e n c y .In gene ra l , t he f r equency and t he dampi ng cha rac t e r i s t i c sof t he r e sonance i t s e l f ar e no t o f i n t e r e s t , bu t t he cha rac -t e r i s t i c de f ec t f r equency i s . I t can be ex t r ac t ed by t hes i m p l e te c h n i q u e o f e n v e l o p e d e t e c t i o n , i n w h i c h t h es i gna l i s r ec t i f i ed and t hen smoot hed t o e l i mi na t e t her e m a i n in g c o m p o n e n t s a t t h e c a r ri e r fr e q u e n c y a n d i t sha rm oni cs , l eav i ng on l y t he en ve l ope s i gna l.Signal analysisT h e f i n a l a n d m o s t i m p o r t a n t s t e p i s t o a n a l y s e t h eenve l ope s i gna l . T he re a r e s eve ra l poss i b l e me t ho dsd e p e n d i n g o n t h e c o m p l e x i t y o f th e m e c h a n i c a l sy s t e m ,v a r y in g f r o m a s i m p l e r m s m e a s u r e o f t h e p o w e r a t t h ec h a r a c t er i s ti c d e f e c t f r e q u e n c y , t h e n a r r o w b a n d s p e c t r u mana l ys i s o f t he en ve l ope s i gna l , c r e s t f ac t o r an a l ys i s ,c e p s t r u m a n a l y si s , m i n i m u m p h a s e r e c o n s t r u c t i o n , o r o n eo r m o r e o f t h e s e f o l l o w e d b y s o p h i s t ic a t e d p a t t e r n r e c og -n i t i on t ech n i ques . A l l o f t hese me t h ods o f si gna l ana l ys ishave been used i n t he pas t w i t h va ry i ng degrees o f success .T r a n s d u c e rPiezoelectr ic accelerometersAl l i nves t i ga t o r s have used p i ezoe l ec t r i c acce l e rom et e r sfo r s ens i ng v i b r a t i on 1 -1s . T h i s is t o b e e xpe c t ed , becauseo f t h e g o o d p e r f o r m a n c e o f t h e p i e z o e l e c t ri c a c c e l e r o m e t e ra t t he h i gh f r equenc i es where t he r e sonances o f t en occur .T h e m e c h a n i c a l c o n s t r u c t io n o f a n a c c e l e r o m e t e r h a s as t rong i n f l uence on i t s cha rac t e r i s t i c s . Acce l e romet e r s o ft h e s i m p l e a x i a l -c o m p r e s s i o n t y p e a r e r o b u s t , w i t h a v e r ygood h i gh- f r equen cy r e spon se and r e l a t ive l y l ow cos t 16.T h e m o r e s o p h i s t i c a t e d d e l t a -s h e a r ty p e h a s a l o w e rc ross -ax i s s ens i t i v i ty bu t a l ower f r eque ncy r ange andh i ghe r cos t 16 . G i ven t ha t t he acce l e r a t i on l eve ls i n t hev i c in i t y o f a d a m a g e d b e a r i n g c a n e x c e e d 1 0 0 g3 , th er o b u s t a x i a l - c o m p r e s si o n t y p e m a y b e p r e f e r a b l e .Frequency rangeT h e f r e q u e n c y r a n g e o f t h e a c c e l e r o m e t e r u s e d v a ri e sw i d e l y a m o n g t h e i n v e s t i g at i o n s. S o m e e x p e r i m e n t s h a v econs i de red f r equenc i es a s h i gh a s 250 kH z 7-9 , bu t t h i sm u s t b e c o n s i d e r e d e x t r e m e . M o r e t y p i c a l l y , t h e f r e q u e n c yrange exam i ned l i e s be t w een abo u t 10 and 50 k Hz 1-6,12 ,1s .U l t i m a t e l y , t h e f r e q u e n c i e s t o b e e x a m i n e d a r e d e t e r m i n e db y t h e r e s o n a n c e s g e n e ra t e d i n t h e b e a r i n gs a n d m a c h i n es t ruc t u r e . W hi le t he r e h ave been seve ra l a t t em pt s t o ca l cu-

l a t e t he r e sonan t f r equenc i es o f t he bea r i ng r aces ~ '2 '4 ,t he va l ues ob t a i ned bea r l i t t l e r e l a t i on t o t hose found bye x p e r i m e n t . S o m e o f t h e d i s c r e p a n c y m a y b e a t t r i b u t e dt o t he u nce r t a i n t i e s i n t he co r r ec t i ons app l i ed t o com pen -~sa t e fo r t he t i gh t nes s o f f i t o f t he r ace i n it s hous i ng o r ont he sha f t 4 . E ven i f accura t e r e sonan t f r equenc i es cou l d beca l cu l a t ed , t he r e a r e a t p r e sen t no means o f de t e rmi n i ngwhi ch o f t he g r ea t m any poss i b l e r e sonances i s l i ke l y t ob e d o m i n a n t . C o n s e q u e n t l y , t h e a p p r o p r i a t e f r e q u e n c yrange is bes t s e lec t ed a f t e r exam i n i ng t he fu ll spec t rum oft he m ach i ne v i b r a t i on . T h e re fore i t i s e s sen t ia l i n t he ea r l ys t ages o f t he ap p l i ca t i on o f t he h f r t t o have a su f f ic i en t l ybroad f r equency r ange t o enab l e t he fu l l spec t rum t o bee x a m i n e d . A f t e r d e c id i n g w h i c h r e s o n a n c e t o s t u d y ,a s e c o n d a c c e l e r o m e t e r w i t h a l o w e r f r e q u e n c y r a n g e c a nbe subs t i t u t ed i f des i r ed .Location and orientationT h e s e l e c t io n o f th e m o s t s u i ta b l e m o u n t i n g l o c a t io n a n do r i e n t a t i o n o f t h e a c c e l e r o m e t e r a r e v e r y i m p o r t a n t . T h eacce l e romet e r shou l d be l oca t ed a s c l ose t o t he bea r i ng a sp o s s ib l e , w i t h a g o o d m e c h a n i c a l p a t h w a y b e t w e e n t h ebea r i ng and t he acce l e rom et e r t o a s s is t i n the t r ansmi s s i onof v i b r a t i on . I t has even been sugges t ed t ha t su i t ab l e pa t h -ways o r wavegu i des shou l d be de l i be r a t e l y i ncorpora t ed i na mach i ne a t t he des i gn s tage i s . F or l a rge o r co mp l exm a c h i n e s , i t m a y b e n e c e s sa r y t o u s e a n u m b e r o f a c c e le r o -me t e r s l oca t ed nea r t he d i f f e r en t bea r i ngs . T he acce l e ro -me t e r shou l d be f i xed secure l y t o t he cas i ng t o ensure goodt r ansmi ss i on o f t he h i gh- f r equency v i b r a t i on . Hand-he l dprobe s a r e no t gene ra l l y su i t ab l e a t h i gh f r equenc i es .T he e f f ec t o f acce l e rom et e r o r i en t a t i on i s bes t i l lus t r a t edi n Refe r ence 1 . An acce l e romet e r was mount ed r ad i a l l ynea r an ax i a l l y l oaded bea r i ng . W hen a de f ec t on t he i nne rr ace s t ruck a ba l l c l ose t o t he acce l e romet e r l oca t i on as t rong s i gna l was r eg i s t e r ed . W hen t he de f ec t s t r uc k a ba l ld i ame t r i ca l l y oppo s i t e t he acce l e rom et e r l oca t i on , t he s ignalp rod uced was l ower i n amp l i t ude . T he s i gnal l evel wasl o w e s t w h e n t h e i m p a c t s o c c u r r e d a t t h e o r i e n t a t io n spe rpend i cu l a r t o t he acce l e rom et e r l oca t i on .

a d e d z o n e

F i g 1 L o a d d i s t r i b u t i o n i n a r o l l in g e l e m e n t b e a r i n g ~7F = A p p l i e d r a d i a l l o a d ; ~ k 1 = A n g u l a r e x t e n t o f l o a d e dz o n e . F o r b e a r i n g w i t h c l e a r a n c e > 0 t h e n 0 < q J1 < 1 8 0d e g r e e s

4 Feb rua ry 1984 Vo l 17 N o 1

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M cFadden and Sm i th - v i b ra t ion m on i t o r i ng o f r o l li ng bear ings by h f r tI n a s t u d y o f a h e l i c o p t e r g e a r b o x c o n t a i n i n g b a l l a n dro l l e r bea r ings 4 , i t was foun d t ha t f o r r o l l e r bea r i ngs i tw a s i m p o r t a n t t o a r ra n g e t h e a x i s o f t h e a c c e l e r o m e t e rpe rpend i cu l a r t o t he ro l l e r s t o ob t a i n a r epresen t a t i vev i b ra t i on s i gna l. T h i s is t o be exp ec t ed a s the ro l le r s cant r ansm i t on l y a r ad i a l l oad . C onver se l y , f o r ba ll bea r ingsi t w a s f o u n d t h a t t h e o r i e n t a t i o n o f t h e a c c e l e r o m e t e rwas le s s c r i t ica l , p r e sum abl y becau se t he be a r i ng cou l dsuppor t bo t h r ad i a l and ax i a l l oads and so cou l d t r ansmi tv i b r a t i on i n bo t h o f t hese d i r ec t i ons . I n gene ra l , t he bes tr e su l ts w e r e o b t a i n e d w h e n t h e a c c e l e r o m e t e r w a s l o c a t e dc lo s e to t h e l o a d e d z o n e o f t h e b e a r i n g . T h e f o r m o f t h el oaded zone fo r a bea r i ng under r ad i a l l oad i s i l l us t r a tedi n F i g 117 . By l oca t i ng t he acce l e romet e r nea r t he l oadedz o n e , t h e p a t h l e n g t h t h r o u g h w h i c h v i b r a t i o n m u s t t r a v e lf r o m t h e p o i n t w h e r e i t is g e n e r a te d t o t h e a c c e l e r o m e t e ri s r educed . A ny v i b r a t i on o f the i nne r r ace o r r o l l inge l e m e n t s c a n o n l y b e t r a n s m i t t e d t o t h e c a si n g t h r o u g ht he ro l li ng e l emen t s and o u t e r r ace , and so i t is li ke l y t ob e d e t e c t e d b e s t a t t h e l o c a t i o n w h e r e g o o d c o n t a c tbe t w een r aces and ro l l ing e l emen t s is a s sured , t ha t i s ,w i t h i n t h e l o a d e d z o n e .A c c e l e r o m e t e r e s o n a n ceI n n o r m a l v i b r a t i o n m e a s u r e m e n t s w i t h a c c e l e r o m e t e r s ,i t i s usua l t o con t r i ve t o keep t he f r equency r e sponse o ft h e m e a s u r e m e n t s y s t e m f l a t b y r e s tr i ct i n g th e f r e q u e n c yrange t o a sma l l f r ac t i on o f t he acce l e romet e r r e sonan tf r equency , and us i ng an ampl i f i e r w i t h an i n t egra l l ow-passf i l t e r t o a t t enua t e t he acce l e romet e r r e sonance . An a l t e r -n a t iv e a p p r o a c h w h i c h h a s b e e n t r ie d w i t h t h e h f r t i s t oomi t t he l ow-pass f i l t e r and a l l ow t he acce l e romet e rreson anc e to am pl i fy th e vibra t ion s ignal 1'11 . The accelero -m e t e r c o u l d b e c o n s i d e r e d a s p a r t o f t h e s y s t e m w h i c hresona t es . An adva n t age i s t ha t am pl i f i ca t i on is p rov i deda t t he acce l e romet e r r e sonan t f r equency , g i v i ng ani mp roved s i gna l -t o -no ise r a t i o . A d i sadvan t age i s t ha ti m p u l s e s f r o m o t h e r m a c h i n e e l e m e n t s , s u c h a s th em e s h i n g o f g e a r s , m a y a ls o e x c i t e t h e a c c e l e r o m e t e r

r e s o n a n c e . T h is w o u l d d e f e a t t h e m a i n p u r p o s e o f t h ehf r t , wh i ch a s desc r i bed ea r l ie r , it t o avo i d t he i n t e r f e r encef r o m m e s h i n g g e a rs a n d o t h e r c o m p o n e n t s o f t h e m a c h i n e .B a n d p a s s f i l t e rC e n t re fr e q u e n c yT he cen t r e f r equency shou l d be se l ec t ed t o co i nc i de wi t ht he r e sonance t o be s t ud i ed , whi ch a s desc r i bed ea r l i e r ,gene ral ly fa l l s in the range 10 to 50 k Hz 1-6 '12 '1s .F o r e x a m p l e , F ig 2 s h o w s t h e s p e c t r u m o f t h e u n p r o c e s s e dvibra t ion s ignal f rom a bear in g l i fe tes t r ig 12 . A smal l bu tb roa d r e sonance i s v i s ib l e i n the v i c i n i t y o f 11 t o 13 kHz ,so t he cen t r e f r eq uen cy o f t he bandpass f i l te r was se t t o1 2 k H z .B a n d w i d t hT h e b a n d w i d t h u s e d v a r i es c o n s id e r a b l y b e t w e e n s t u d ie s ,bu t shou l d be a t l ea s t doub l e t he h i ghes t cha rac t e r i s t i cde fec t f r equency i n o rde r t ha t t he f i l t e r can pas s t heca r r i e r f r equency and a t l ea s t one pa i r o f modul a t i ons i d e b a n d s 4 . I n p r a c t ic e , t h e b a n d w i d t h s h o u l d b e s o m e -w h a t g r e a t e r t o a c c o m m o d a t e t h e f ir s t t w o p a i r s o fm o d u l a t i o n s i d e b a n d s a r o u n d t h e c a r ri e r fr e q u e n c y , a st hese add i t i ona l componen t s may as s i s t d i agnos i s . A band-wi d t h o f a t l ea s t f our t i mes t he h i ghes t cha rac t e r i s t i c de f ec tf r e q u e n c y w o u l d t h e r e f o r e s e e m a d v i sa b l e .A b a n d w i d t h w h i c h h a s b e e n u s e d f r e q u e n t l y i s + -5 % o ft he cen t r e f r equ ency 1 '4 -6 . I t appe a r s t o b e sa t i s f ac t o ryb u t i t s h o u ld b e b o r n e i n m i n d t h a t t h e + - 5 % b a n d w i d t hwas f i r s t desc r i bed i n Refe r ence 1 , and subsequen t l yf o l lo w e d b y R e f e r e n c e s 4 - 6 . I t s p o p u l a r i t y d o e s n o tnecessa r i l y i nd i ca t e t ha t i t i s t he m os t su i t ab l e va l ue , bu ti t does p rov i de a conven i en t gu i de i n t he absence o f anyb e t t e r e s t i m a t e s .T h e v i b r a t i o n s p e c t r u m c a n b e u s e d t o g o o d e f f e c t in t h es e l e ct i o n o f a s u i ta b l e b a n d w i d t h . F o r e x a m p l e , i n R e f e r -e n c e 1 2 th e c e n t r e f r e q u e n c y i s t u n e d t o 1 2 k H z a n d t h eb a n d w i d t h s e t t o + 1 k H z t o m a t c h t h e r e s o n a n c e s h o w nin F ig 2.

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R o l l o f f r a teOne aspe c t o f bandp ass f i l te r i ng whi ch has so f a r r ece ivedno a t t en t i on i n the l i t e r a t u r e i s t he qu es t i on o f t he r a t eof f i l t e r r o l l o f f . I t i s p rob ab l e t ha t m os t i nves t i ga t o r s a r es i m p l y m a k i n g u s e o f a v a i la b l e c o m m e r c i a l l y -b u i l t b a n d -pass f il t e rs and s i mpl y acc ep t i ng t he i r r o l l o f f a s su it ab l e .Y e t t h e r a te o f r o l l o f f c o u l d h a v e a p r o f o u n d i n f lu e n c eon t he success o r o t he rwi se o f t he h f r t i f app l i ed t o amach i ne i n whi ch i n t e r f e r ing pe r i od i c s ignal s such a sgea r mesh i ng a r e l oca t ed a t f r equenc i es nea r t he se l ec t edr e s o n a n c e .

Im p u l s e re s p o n s eAn ot he r a spec t o f bandpa ss f i lt e r i ng no t cons i de red i s t hei mpul se r e sponse o f the f i l te r . T he r e sponse o f t hec o m m o n l y - u s e d B u t t e r w o r t h f i lt e r t o a h i g h - f re q u e n c yi m p u l s e c a n c o n t a i n h i g h - f r e q u e n c y r in g in g c o m p o n e n t s ;t he i mpul s i ve na t u r e o f t he bea r i ng de fec t s i gna l maycon t r i bu t e t o r i ng i ng o f t he f i l t e r .E n v e l o p e d e t e c t o rT he en ve l ope d e t ec t o r cons i s t s o f a f u l l-wave o r ha lf -waverec t i f ie r t o co nver t t he b i po l a r i npu t s i gna l t o a un i po l a r

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M cFadden and Sm i th - v i b ra t ion m o n i t o r i ng o f r o l li ng bea ri ngs by h f r ts i gna l, f o l l owed by a smoo t h i ng c i r cu i t ; a s i mpl e enve l opede t ec t o r i s i l l us t ra t ed i n F i g 3 . F ew de t a i l s o f t he env e l opede t ec t o r s used a r e g i ven in t he l i t e r a t u r e . O nl y one pap e r 4s t a t e s exp l i c i t l y t ha t ha l f - wave r ec t i f ica t i on has be en used ,a l t hough i t can be deduced f r om t he i l l us t r a t i ons i n oneo t he r 14 t ha t ha l f - wave r ec t i f ica t i on has p r ob ab l y b een used .S i mi l a r l y , on l y Ref e r ence 14 spec if i e s t ha t a pea k- ho l dsmo ot h i ng c i r cu i t is used .I t s h o u l d b e e m p h a s i z e d t h a t t h e d e c a y p o r t i o n o f t h eenve l ope s i gna l is o f l i tt l e use i n t he ana l ys i s o f t he e nve -l ope s i gna l , because i t i s t he r epe t i t i on r a t e o r f r eq uenc yof t he de f ec t s whi ch i s mos t o f t en used f o r t he d i agnos i so f t h e o r i g i n o f t h e d e f e c t .T he r ec t i f i e r c i r cu it mu s t be des i gned wi t h ca r e i f a mea n-i ngf u l ou t p u t s i gnal i s t o be ob t a i n ed , a s r ea l d iodes don o t h a v e c h a r a c te r i st ic s w h i c h m a t c h t h o s e o f t h e i d e a ld i ode l a . I ns t ead , t he r e ex i s t s a b i a s vo l t age whi ch t hei npu t vo l t age mus t exceed be f or e a cur r en t w i l l f l ow .T h e p r o b l e m i s re a d i ly o v e r c o m e b y a n o p e r a t i o n a l a m p l i f i e rw i t h a s e c o n d d i o d e i n th e f e e d b a c k c i r cu i t t o c o m p e n s a t ef o r th e n o n i d e a l b e h a v i o u r . F i g 4 s h o w s t h e f o r m o f th ec i r cu i t used g i v ing a good a ppr o x i m a t i on t o t he i dea l d i ode i s .T h e s m o o t h i n g c i r c u it i n t h e e n v e l o p e d e t e c t o r i s c o m -m on l y conf i gur e d a s a pea k- ho l d c i r cu i t 14 , i n wh i ch acapa c i t o r i s cha r ged b y t he r ec t i f i ed enve l ope s i gna l anda l l owed t o d i scha r ge t h r oug h a g i ven r e s is t ance . T he se l ec -t i o n o f a su i ta b l e t i m e c o n s t a n t f o r t h e s m o o t h i n g c i r c ui ti s v e r y i m p o r t a n t . I f t o o l o n g , t h e s m o o t h i n g c i r c u i t w il ld i s t o rt t h e s h a p e o f t h e e n v e l o p e b y m a k i n g t h e d e c a y o ft h e p e a k s t o o s l o w , s o t h a t t h e s m a l l p e a k s m a y b e b u r i e di n t h e d e c a y o f a p r e c e d i n g la r ge p e a k , a s s h o w n i nF i g 5 ( a ) 18 . I f t oo shor t , ca r r i e r r i pp l e w i l l r ema i n i n t hes i gnal a s shown i n F i g 5 ( b ) ~a , neces s i t a t i ng f u r t he r sm oo t h-i n g. F o r t u n a t e l y t h e e x p e c t e d s h a p e o f t h e e n v e l o p e s ig n ali s a r ap i d r is e a t t he s t a r t o f the i m pac t , f o l l ow ed by anexp onen t i a l de cay o f t he env e l ope s i gnal a s the osc i l l a t i onsubs i des . I t i s pos s i b l e to r e pr od uce t h i s shape by t he peak-h o l d s m o o t h i n g c i r c u it p r o v i d e d t h a t t h e t i m e c o n s t a n t i sc o r r e c t l y c h o s e n .At t he h i gh f r equenc i es a t whi ch bea r i ng r e sonances a r el i k e ly t o o c c u r , i t w o u l d b e e x p e c t e d t h a t t h e d a m p i n g i nt he m ach i ne s t r uc t u r e i s h i gh and t he d i s s i pa t i on o f ther e s o n a n c e r a p i d . U n d e r e s t i m a t i o n o f t h e d a m p i n g le a d s t os e l e c ti o n o f a t i m e c o n s t a n t w h i c h i s t o o l o n g f o r t h emechan i ca l sys t em under t e s t . T h i s i s we l l i l l us t r a t ed by t heenve l op e si gna ls i n Ref e r en ce 13 . I t wo ul d a ppea r t ha t t het i m e c o n s t a n t o f th e e n v e l o p e d e t e c t o r w h i c h p r o d u c e dt hese s ignal s was t oo l ong , w i t h t he r e su l t t ha t t he e xpo -n e n t i a l d e c a y o f t h e p e a k s d o m i n a t e s t h e s i g n a l. E v e n t h es m a l l p e a k s in t h e b a c k g r o u n d n o i se s h o w s t r o n g e x p o -n e n t i a l d e c a y .S i g n a l a n a l y s i sN a r r o w b a n d f i l t e r in gT h e s i m p l e s t m e t h o d o f a n a l y si s w h i c h h a s b e e n p r o p o s e d ~ '2i s t o pas s t he enve l op e s igna l t o a na r r ow band pass F d t erw h i c h c a n b e t u n e d i n t u r n t o t h e c h a r a c t e ri s t ic d e f e c tf r e q u e n c ie s o f t h e i n n e r a n d o u t e r r a c es a n d t h e r o ll in ge l eme nt s . T he ou t pu t o f the F i lt er is con nec t e d t o a t h r e s -h o l d d e t e c t o r a n d a l a r m s y s t e m w h i c h a n n o u n c e s w h e n ap r e s e t l ev e l h a s b e e n e x c e e d e d .A k e y f a c t o r i n t h is s c h e m e w h i c h i s n o t s u f f i c ie n t l ys t r e s sed 2 is t ha t t he bea r i ng on w hi ch t he s chem e was

BnOoss I I I n V e ' pi l t e re d H a l f - w a v e R e c t if ie d P e a k - h o l d d e t e c t e do r s i g n a l s m o o t h i n g s i g n a ls i gna l _ f u l l w av e - ~ -

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