( ) 2! l p. Nonlinear Dynamics for Gear Fault Level. ( ) f ( x) ( ),! = sgn % " p. Open Access. Su Xunwen *,1, Liu Jinhao 1 and Wang Shaoping 2. !

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1 Nonlinar Dynamics for Gar Faul Lvl Su Xunwn Liu Jinhao and Wang Shaoping Snd Ordrs for Rprins o rprins@bnhamscinc.a Th Opn Mchanical Enginring Journal Opn Accss School of Tchnology Bijing Forsry Univrsiy Bijing China School of Auomaion Scinc and Elcrical Enginring Bijing Univrsiy of Aronauics and Asronauics Bijing 009 China Absrac: Sinc hr ar backlash fricion and imvarying siffnss h characrisics of gaair has srong nonlinariy. Whn a gar has fauls such as piing spalling and ooh brakag hr ar ofn som nonlinar characrisics as suprharmonics or subharmonics along wih vibraion signal. Ths characrisics will caus som difficuly for faul diagnosics so i is ncssary o sudy h mchanism of nonlinar vibraion. Fourir xprssion was usd o dscrib imvarying siffnss a facor was usd o dpic gar faul lvl and nonlinar dynamics modl for gar faul lvl was sablishd. Simulaion rsuls show ha wih crain fricion and roaing spd h nonlinar vibraion has dirc rlaionship wih roaing spd and gar fricion and has no dirc rlaionship wih faul yps. Kywords: Dynamics faul diagnosics faul lvl gar nonlinariy.. INTRODUCTION Much work has bn don sinc h firs papr on gar vibraion had bn prsnd in 90 []. Th gar vibraion has srong nonlinariy bcaus of h fricion backlash and imvarying siffnss whn h gaair works normally [6]. As analyic soluions of h nonlinar vibraion quaion canno b obaind Japans rsarchr go h linar approxima quaion by doing som linar work on h quaion and vrifid i by xprimns [7]. Obviously whn a gar has fauls such as piing spalling or a brokn ooh and so on h faul sourc will lad h imvarying siffnss o vary svrly and ha is h mos imporan rason lading o aggrava nonlinar vibraion of gaair. Whn a gar has fauls hr ar ofn som nonlinahnomna such as suprharmonics and subharmonics. Howvr up ill now h rsarch on mchanism of nonlinar vibraion of gar faul is vry fw h faul diagnosics of a gar is mor concnrad on mhods of signal procssing such as im domain analysis frquncy domain analysis imfrquncy join domain analysis c. [8 9]. In [0] h auhors invsigad h phnomna of cycl vibraion and chaos xcid by h picwis coninuous siffnss and dpicd faul gar siffnss using h picwis coninuous funcion. Howvr ha aricl dos no considr h influncs of fricion and backlash which is rahr significan. In addiion hr is no shock phnomnon in h simulaion rsul and i has no bn vrifid by xprimns. Sinc h characrisics of gaair and mshing siffnss hav vas changs so h nonlinar characrisics of h gar faul signal prform mor significanly whn a Addrss corrspondnc o his auhor a h School of Tchnology Bijing Forsry Univrsiy No.35 Tsinghua Eas Road Haidian Disric Bijing China; Tl: ; suxw0703@gmail.com 87455X4 gar has faul. I is hrfor ncssary o sudy h nonlinariy of a faul gar spcially h sadysa rspons of h fauly gar s nonlinar vibraion o hlp h gar s faul diagnosics. Th nonlinar dynamics ingrad modl of normal gars and faul gars is sablishd considring fricion backlash and imvarying siffnss. Fourir sris is usd o dpic varying siffnss a gar faul facor is inducd o dpic gar faul lvl and simulaion rsuls ar also prsnd. Analysis on characrisics of h gar box vibraion shows ha simulaion rsuls ar consisn wih xprimnal rsuls.. NONLINEAR DYNAMICS MODEL FOR A NORMAL GEAR PAIR CONSIDERING FRICTION AND BACKLASH Th nonlinar dynamics modl of a gaair is shown in Fig. ( considring fricion and backlash. Th nonlinar dynamics according o works of [ ] wih fricion and backlash of h gaair ar as follows. c m x k m l p µ c m x k m = T p ( 04 Bnham Opn ( f ( x ( f ( x ( f ( x ( f ( x g r g c m x k m l g µ c m x k m = T g ( Hr l p = an y l g = ( l p h = z p a r g y = k m k a sin ( h ( = a sin h x = r g g f ( x = x b 0 x b ( = sgn p p x > b x b x < b ( l p ( g g l g (

2 488 Th Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Sinc dx d = dx d d d i follows ha x ( = x ( x ( = x ( Thn from Eq. (3 w obain: ( x x b = b = x = x ( ( ( x x b = b = x = x ( k m ( = k k sin av a h = k m sin h m = k sin ( ( Sinc = r g g = sgn ( p p l p ( g g l g ( whr Fig. (. Nonlinar dynamics modl of a gaair considring fricion and backlash. Eq. (. I p Eq. (. r g Ig and muliply wih b w hav x b r p r g l pµ r g l pµ l gµr g ( = T p b T gr g b b l gµr g ( k m ( c x m b ( f ( x b Eq. (. and simplifying i w hav ( l pµ ( r p c m x l pµ r p k m ( ( f ( x (3 = T m (4 Dal wih Eq. (3 and Eq. (4 wih nondimnsion l x = x b y = y b m = m m p g ( m p m g m p = m g = r m p mp = h g = = c m ( m = c m mp ( k = k a kav k = p F = a b f ( x = x 0 x x > x x < F av = T p ( m b ( l p ( g g l g ( ( p = l p r g g r g g r g. 0 l p ( y ( bh bh l g ( = b h ( y ( x ( F cos ( Sinc r g l p l g m p m g r g m = a ( ( r g. 0 ( l p a r g. 0 ( ( m g r g m p m g r g l p m (5 L g ( quals o Eq. (5. As known T p = T g sinc r g ( ( T p m p b T g m g r g b b = T p b m p m g ( b = T p ( b m b = F ( av b ( = a sin h = a sin h ( = a sin ( b = a sin x = a sin( Thn Thn Eq. (3 is simplifid a las as blow ( g = F av F sin L ( µ x ( g b = F sin ( µ k sin ( f x g ( = l p (7 ( (6

3 Nonlinar Dynamics for Gar Faul Lvl Th Opn Mchanical Enginring Journal 04 Volum T Mshing priod; k a Pakopak valu of mshing siffnss; k av Man of mshing siffnss Fig. (. Mshing siffnss curv of a normal gaair. Subsiuing abov quaion o Eq. (4 and h ncssary simplificaion w obain y ( g ( µ x ( m (8 g ( µ k ( k sin f ( x = F av m Now h nondimnsional quaions of h gaair nonlinar dynamics ar obaind as blow ( x g k ( g ( µ x ( ( µ sin ( f ( x = F F av sin y ( g ( µ x ( g ( µ k ( k sin f ( x = F m av m 3. NONLINEAR DYNAMICS MODEL OF A FAULT GEAR PAIR CONSIDERING FRICTION AND BACKLASH 3.. TimVarying Mshing Siffnss of a Faul Gar Whn a gar has fauls such as piing or brokn ooh h characrisics of mshing siffnss will chang [] in (9 paricular whn a ooh is brokn h variaion of h mshing siffnss is vry profound. Fig. ( dpics imvarying siffnss of a normal gar supposing ach ooh has h sam varianc [6 9]. Whn a ooh of a gar has faul hr mus b rducion of mshing siffnss of h corrsponding ooh mshing. Suppos a gar has h numbr qual o Z and hr is on brokn ooh h imvarying mshing siffnss coincids wih Fig. (. Suppos h brokn ooh has h sam pakopak valu as ha of normal h. o k m Th Fourir sris of h signal F ( in Fig. ( is qual ( ( a 0 a cos n b sin n n h n h k a n whr n= sin n h n = n= (0 a 0 = T F( T d = 0 a n = T F( 0 T cos( n d = 0 0 T Mshing priod; T Roaional priod; Z Th of driving gar; k a Pakopak valu of mshing siffnss; k av Man of mshing siffnss; k av Man of mshing siffnss of h brokn ooh Fig. (3. Variy of mshing siffnss of a singl brokn ooh.

4 490 Th Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Fig. (4. Pakopak valu of h mshing siffnss of a brokn ooh mshing. b n = T F( T sin( n d = 4k a 0 n n = = h T Th blow is o obain h paramrs of h Fourir sris k m ( of h signal F ( in Fig. (. Fig. (4 dpics h pakopak valu of h mshing siffnss of a brokn ooh mshing. I can b sn ha h Fig. (4 is h diffrnc of Figs. ( 3. L. Sinc F( = a 0 ( a n cosn b n sinn T n= ( a 0 = F T 0 d = F 0 Z ( a n = T F( T 0 cos( n d = F 0 n sin n Z ( b n = T F( T 0 sin( n d = F 0 n cos n Z ( Thn h Fourir ranslaion of Fig. (4 is obaind as blow k( = k b Z. n= ( k b n sin ( n Z cosn p k b n cos ( n Z sinn p ( Thn h xprssion of h mshing siffnss of a gar pair wih a brokn ooh faul or h Fourir ransmission of Fig. (3 is qual o k mf ( = k m ( k( k a sinn h n = ( n n= k b Z ( k b n sin ( n Z cosn p k b n cos ( n Z sinn p. ( n= Th curvs in Figs. (57 ar h simulaion rsuls of Eq. (0 o Eq. ( rspcivly using Malab. Th mshing priod is T = 0 04 Sc ; h of h gar is Z = 6 ; man of mshing siffnss is k av = 0 ; pakopak valu of mshing siffnss is k a = 0.5 ;wih h maximum Fourir ordr of Sc Sc Fig. (5. Timvarying mshing siffnss of a normal gar Sc Sc Fig. (6. Timvarying mshing siffnss of a brokn ooh mshing.

5 Nonlinar Dynamics for Gar Faul Lvl Th Opn Mchanical Enginring Journal 04 Volum Fig. (7. Timvarying mshing siffnss of a gar wih faul of a brokn ooh. n max = 0. From Fig. (7 i can b sn ha h curv of Fig. (7 can xprss h imvarying siffnss of a faul gar in Fig. (3. approximaly. 3.. Nonlinar Dynamics Ingrad Modl Sinc hr will b variy of gar fauls such as piing and spalling and so on and h variaion of h mshing siffnss is lss han ha of brokn ooh faul ransmi Eq. ( as blow k mf ( = k m ( k ( k a ( ( sinn h n = k av n n= k av k b sin n k b n k Z av ( cosn p ( ( Zk av k b. cos n n= nk Z av ( sinn p 0 k ( ( sinn h n = n n= k 3 k k 3 av Z n sin ( n Z cosn p ( ( k 3. n cos ( n n= Z sinn p 0 whr (3 k = k a k av k 3 = k b k av (45 [ 0]is gar faul facor whn a gar is in normal saus = 0 ; whn a gar has srious faul of brokn ooh = ; ohrwis whn a gar has ligh fauls such as crack piing and spalling 0 < <. Dividing Eq. (3 by m hn by simplifying i ino is nondimnsional form w g ( k mf = k sin n h m n (n = n= k 3 3 k 3 Z n sin ( n Z cosn k 3 4 n cos ( n 00 n= Z sinn = k sin( n8 n = n n= k 3 3 k 3 Z n sin ( n Z cos n8 ( k n= 3 4 n cos ( n Z sin n ( : 7 = k sin( n8 n = n n= k 3 3 k 3 Z n sin ( n Z cos( n8 k k 3 4 n cos ( n 00 n= Z sin( n8 k L k = k n sin( n n = n= (4 k = k. 3 Z ( k 3 n sin ( n Z cosn k 3 n cos ( n Z sinn n=

6 49 Th Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Thn Eq. (6 can b simplifid ino k mf ( = k k. m Thrfor from Eq. ( and Eq. ( h nonlinar dynamics ingrad modl of a normal gar and a faul gar considring backlash and fricion is saisfid ( I p r b c m x k mf I g g r g c m x k mf ( f ( x ( f ( x l pµ c m x k mf l pµ c m x k mf Th Nondimnsional form is qual o x g k (k ( f ( x ( f ( x ( g ( µ x ( ( µ [ ] f ( x = F av F sin y ( g ( µ x ( m g ( µ k ( k (k f ( x = F av m 4 SOLUTION = T p = T g (5 (6 Eq. (8 is a scond ordr diffrnial quaion which can b solvd by 45 ordr RungKua mhod. In ordr o supply horical rfrnc of gar fauls diagnosis h sadysa rspons of a normal gar and a faul gar vibraion is focusd as wll on h frquncy domain characrisics. For simpliciy h h of h driving gar ar z p = 6 and h oharamrs ar as shown in Tabl. Pars of hm ar according o rfrnc []. Givn µ and h oharamrs can b solvd using paramrs in Tabl. Nondimnsional frquncy f nd =. Tabl. Paramr Symbols Paramr valus. Valu Paramr Symbols Valus z a 0 mm 0.0 b 0. mm k mm k 3 0. r g mm F av 0. r ag 63 mm F kg.m k av kg.m ( L µ = 0.05 = 0.3 hn nondimnsional frquncy is f nd = ; nondimnsional priod is T nd = 0.9 ; nondimnsional roaional priod is T nd = 5 ; nondimnsional roaional frquncy is f nd = Sadysa rspons vibraions ar shown in Figs. (80. wih h frquncy rsoluion of f = = 0 his mans h gar is in normal saus. And Fig. (8 dpics h sadysa vibraion rspons of a normal gar. = 0. his mans h gar has som sligh fauls such as iniial crack piing or spalling. Fig. (. dpics h sadysa vibraion rspons of a gar wih faul of iniial crack piing or spalling. = his mans h gar has srious fauls as brokn ooh. Fig. (0 dpics h sadysa vibraion rspons of a gar wih faul of brokn ooh. From h im domain signal of Fig. (0 i can b sn ha h shock phnomnon is vn mor clar along wih h incras of h faul facor wih shock priod abou 5. Morovr in h frquncy domain signal of Fig. (0 h sidbands of x x and high frquncy muliplicaion of mshing frquncy wih an inrval of abou ar mor and mor clar. ( µ = 0 = 0.7 wih nondimnsional mshing frquncy f nd = 0.5 nondimnsional mshing priod T nd = 9.0 nondimnsional roaional priod T nd = 54 and nondimnsional roaional frquncy f nd = Th sadysa vibraion signals ar shown in Figs. ( wih frquncy rsoluion f = = 0 mans h gar is in normal saus. Fig. ( dpics h sadysa vibraion rspons of a normal gar. = mans h gar has srious fauls as brokn ooh. Fig. ( dpics h sadysa vibraion rspons of a gar wih faul of brokn ooh. From Fig. ( i can b sn ha h normal gar vibraion signal also has sidbands. Howvr whn = 0 hr is no modulaion frquncy in h dynamics Eq. (8 so h sidbands should b suprharmonics. In Fig. ( h signal is abou a gar wih srious faul such as brokn ooh i is obvious ha hr ar sidbands a h harmonics of mshing frquncy wih inrval of roaional frquncis. Thn w conclud ha whn a gar is in nonlinar vibraion h suprharmonics ar rlad wih mshing frquncy and hav nohing o do wih gar fauls. Fig. ( rvals ha h ampliuds of suprharmonics ar much lowr han ha of sidbands of mshing frquncy wih gar fauls. CONCLUSION Th nonlinar dynamics ingrad modl of normal gars and faul gars has bn sablishd considring

7 Nonlinar Dynamics for Gar Faul Lvl Th Opn Mchanical Enginring Journal 04 Volum Fig. (8. Sadysa rspons of a normal gar vibraion Fig. (9. Vibraion rspons of a gar wih faul of arlir crack piing or spalling. fricion backlash and imvarying mshing siffnss. Fourir sris is usd o dpic varying siffnss ha is mor suiabl o solv h dynamics quaion. Th gar faul facor provids convninc for h ingrad modl o dpic normal gar vibraion and faul gar vibraion. Simulaions show ha

8 494 Th Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Fig. (0. Sadysa vibraion rspons of a gar wih faul of brokn ooh. Fig. (. Sadysa rspons of a normal gar vibraion. a. Wih h incras of h faul facor which mans h faul lvl is growing from ligh lvl o srious lvl h shock phnomnon is mor and mor clar. Mshing frquncy and is mulipl frquncy of h

9 Nonlinar Dynamics for Gar Faul Lvl Th Opn Mchanical Enginring Journal 04 Volum Fig. (. Vibraion rspons of a gar vibraion wih faul of a ooh brokn. vibraion signal ar modulad by h roaional frquncy clarly ha mans h sidbands wih inrval of roaional frquncy ar mor obvious. b. Faul lvl of a gar is no h roo caus lading o subharmonics or suprharmonics of gar vibraion whos rason is rlad wih roaional spd and fricion cofficin of gaair. NOMENCLATURE = Momns of inria of conacing ooh pairs g = Roaional angls of conacing ooh r g l p l g T p T g m p m g pairs = Bas radiuss of conacing ooh pairs = Fricion arms of conacing ooh pairs = Torqus of conacing ooh pairs = Masss of conacing ooh pairs g = Angular vlociis of conacing ooh O p O g z p z g x pairs = Origin poins of conacing ooh pairs = Th of conacing ooh pairs = Rlaiv displacmn bwn mshing poins y = Linar displacmn of mshing poin of driving gar = Dircion cofficin of fricion µ = Cofficin of kinic fricion f ( x = Backlash funcion c m k m = Mshing damping cofficin; ( = Mshing siffnss; = Prssur angl; a ( = Saic rror b k av k a a = Cnr disanc of conacing ooh pairs = Mshing im = Half of backlash = Man of varying mshing siffnss = Pakopak of varying mshing siffnss ( = Ampliud of saic rror h = Mshing frquncy m = Equivaln inrial mass of gaair = Equivaln naural frquncy of gar pair

10 496 Th Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. p = Naural frquncy of driving gar = Nondimnsional im = Damping raio = Damping raio r ag F av F g ( g = Pich Radius of drivn gar = Man of nondimnsional xciing = Nondimnsional ampliud of ransmission rror ( = Funcions rlad fricion dircion o simplify dynamic quaion k k k 3 k k = Siffnss funcions = Conac raio x y = Corrsponding nondimnsional symbols of x y k av = Man of mshing siffnss F ( = Picwis coninuous funcion of imvarying mshing siffnss T T Z n max f nd T nd T nd f nd = Mshing priod of gaair = Roaion priod of driving gar = Th of driving gar = Th maximum numbr of Fourir sris = Nondimnsional mshing frquncy = Nondimnsional mshing priod = Nondimnsional roaion priod of driving gar = Nondimnsional roaion frquncy CONFLICT OF INTEREST W dclar ha his aricl conn has no conflic of inrs wih ohopl or organizaions ha can inapproprialy influnc our work. ACKNOWLEDGEMENTS Th auhors would lik o acknowldg h suppor from h Fundamnal Rsarch Funds for h Cnral Univrsiis (YX0404. REFERENCES [] A. Pary and N. Tandon Spur gar dynamic modls including dfcs: A rviw Th Shock and Vibraion Digs vol. 35 pp [] W. Sanmin S. Yunwn and D. Haijun Chaos and bifurcaion analysis of a spur gaair wih combind fricion and claranc Chins Journal of Mchanical Enginring pp [3] W. Jianjun L. Qihan and L. Runfang Rsarch advancs for nonlinar vibraion of gar ransmission sysms Advancs in Mchanics vol. pp [4] Y. Xianyong Z. Xiaojun and L. Yong Bifurcaion and chaos of nonlinar sysm of hlical bvl gar wih claranc Journal of Vibraion and Shock vol. pp [5] J. Wang and T. C. Lim. Effc of ooh msh siffnss asymmric nonlinariy for driv and coas sids on hypoid gar dynamics Journal of Sound and Vibraion vol. 39 pp [6] F. Elasha C. RuizCárcl D. Mba G. Kia I. Nz and G. Ybra Piing dcion in worm garboxs wih vibraion analysis Enginring Failur Analysis vol. 4 pp [7] Y. Cai and T. Hayashi Th Linar approximad quaion of vibraion of a pair of spur gars (hory and xprimn Journal of Mchanical Dsign vol. 6 pp [8] K. C. Gryllias L. Glman B. Shaw and M. Vaidhianahasamy Local damag diagnosis in garboxs using novl wavl chnology Insigh vol. 5 pp [9] W. Barlmus F. Chaari R. Zimroz and M. Haddar Modlling of garbox dynamics undr imvarying nonsaionary load for disribud faul dcion and diagnosis Europn Journal of Mchanics Asolids vol. 9 pp [0] G. Liak and M. I. Friswll Dynamics of a gar sysm wih fauls in mshing siffnss Nonlinar Dynamics vol. 4 pp [] L. Runfang and W. Jianjun Dynamics of gar sysm Bijing: Scinc Prss 997. [] Y. Pandya and A. Pary Crack bhavior in a high conac raio spur gar ooh and is ffc on msh siffnss Enginring Failur Analysis vol. 34 pp f = Frquncy rsoluion Rcivd: Spmbr 0 04 Rvisd: Novmbr 5 04 Accpd: Novmbr 5 04 Xunwn al.; Licns Bnham Opn This is an opn accss aricl licnsd undr h rms of h Craiv Commons Aribuion NonCommrcial Licns (hp:craivcommons.orglicnss bync4.0 which prmis unrsricd noncommrcial us disribuion and rproducion in any mdium providd h work is proprly cid.