2. NONLINEAR DYNAMICS MODEL FOR A

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1 Nonlinar Dynamics for Gar Faul Lvl Su Xunwn * Liu Jinhao and Wang Shaoping Snd Ordrs for Rprins o rprins@nhamscinc.a h Opn Mchanical Enginring Journal Opn Accss School of chnology Bijing Forsry Univrsiy Bijing China School of Auomaion Scinc and Elcrical Enginring Bijing Univrsiy of Aronauics and Asronauics Bijing 009 China Asrac: Sinc hr ar acklash fricion and im-varying siffnss h characrisics of gaair has srong nonlinariy. Whn a gar has fauls such as piing spalling and ooh rakag hr ar ofn som nonlinar characrisics as supr-harmonics or su-harmonics along wih viraion signal. hs characrisics will caus som difficuly for faul diagnosics so i is ncssary o sudy h mchanism of nonlinar viraion. Fourir xprssion was usd o dscri im-varying siffnss a facor was usd o dpic gar faul lvl and nonlinar dynamics modl for gar faul lvl was salishd. Simulaion rsuls show ha wih crain fricion and roaing spd h nonlinar viraion 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.. INRODUCION Much work had n don sinc h firs papr on gar viraion was prsnd in 90 []. h gar viraion has srong nonlinariy caus of h fricion acklash and im-varying siffnss whn h gaair works normally [- 6]. As analyic soluions of h nonlinar viraion quaion canno oaind Japans rsarchr go h linar approxima quaion y doing som linar work on h quaion and vrifid i y xprimns [7]. Oviously whn a gar has fauls such as piing spalling or a rokn ooh and so on h faul sourc will lad h im-varying siffnss o vary svrly and ha is h mos imporan rason lading o aggrava nonlinar viraion of gaair. Whn a gar has fauls hr ar ofn som nonlinar phnomna such as supr-harmonics and su-harmonics. Howvr up ill now h rsarch on mchanism of nonlinar viraion of gar faul is vry fw h faul diagnosics of a gar ar mor concnrad on mhods of signal procssing such as im domain analysis frquncy domain analysis im-frquncy join domain analysis c. [8 9]. In [0] h auhors invsigad h phnomna of cycl viraion and chaos xcid y 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 acklash which is mor. In addiion hr is no shock phnomnon in h simulaion rsul and hy hav no n vrifid y xprimns. Sinc h characrisics of gaair and mshing siffnss hav vas changs and h nonlinar characrisics of h gar faul signal prform mor significanly whn a gar has faul. I is ncssary o sudy h nonlinariy of a faul gar spcially h sady-sa rspons of h fauly gar s nonlinar viraion o hlp h gar s faul diagnosics. h nonlinar dynamics ingrad modl of normal gars and faul gars is salishd considring fricion acklash and im-varying 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 ox viraion shows ha simulaion rsuls ar consisn wih xprimnal rsuls.. NONLINEAR DYNAMICS MODEL FOR A NORMAL GEAR PAIR CONSIDERING FRICION AND BACKLASH h nonlinar dynamics modl of a gaair is shown in Fig. () considring fricion and acklash. h nonlinar dynamics according o works [ ] wih fricion and acklash of h gaair ar as follows. f ( x) f ( x) θ p + c m x + k m + l p λµ c m x + k m = p () f ( x) f ( x) θ g r g c m x + k m l g λµ c m x + k m = g () Hr l p = anβ + y l g = l p ω h = z p a + r g y = θ p k m + k a sin( ω h ) ( ) = a sinω h x = θ p r g θ g f ( x) = x 0 x + λ = sgn + θ p x > x x < l p ( ω g + θ g )l g X/4 04 Bnham Opn

2 488 h Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Sinc dx dτ = dx d d dτ I follows ha x τ = x ω x ( τ ) = x ( ) ω hn from q. (3). w oain x x ω = ω = x ω = x ( τ ) k m ( ) = k + k sinω av a h ω m ω m x ω = x ω = x ω = x ( τ ) = + k sin ω h ω ω = + k sin ωτ Sinc = r g ω g λ = sgn ( + θ p )l p ( ω g + θ g )l g whr Fig. (). Nonlinar dynamics modl of a gaair considring fricion and acklash. Eq. (). I p Eq. (). r g Ig and muliply wih ω w hav x ω + r p + r g + l pλµ + r g + l pλµ + l gλµr g = p ω + gr g ω θ p ω ω + l gλµr g k m c x m ω + f ( x) ω Eq. (). and simplifying i w hav ( ω ) + + l pλµ r p c m x ω + + l pλµ r p k m ω f ( x) Dal wih q(3) and q (4) wih non-dimnsion l x = x y = y m = m m p g ( m p + m g ) m p = m g = r g ω m ω p mp ω = ω h ω τ = ω ζ = c m m ω (3) = m ω (4) ζ = c m ( mp ω ) k = k a kav k = F ω av = p m ω F = a f ( x ) = x 0 x + x > x x < ( + θ p )l p ( ω g + θ g )l g + θ p = ω l p r gω g + r g θ g ω r g ω + y ( τ ) h ω h ω l p l g = ω h + y ( τ ) x ( τ ) F ω cos( ωτ ) Sinc r g l p + l g m p m g r g m λ = a ( + r g ) L g ( τ ) qual q.(5). As known p l p a + r g + m g r g m p m g r g l p m λ (5) = g r g sinc p m p ω + g m g r g ω τ = p + ω m p m g τ = p ( τ ) ω m = F ( τ ) av = a sinω h = a sin ω h ω ω = a sinωτ hn ( τ ) = a sinωτ ( τ ) = aω sinωτ = F ω sinωτ x τ hn q(3) is simplifid a las as low + ζ + g τ = F av + F ω sinωτ g L µ x τ + + g τ µ + k sin( ωτ ) f x ( τ ) = l pλ (7) (6)

3 Nonlinar Dynamics for Gar Faul Lvl h Opn Mchanical Enginring Journal 04 Volum Mshing priod; k a Pak-o-pak of mshing siffnss; k av Man of mshing siffnss Fig. (). Mshing siffnss curv of a normal gaair. Susiuing aov quaion o q.(4). and h ncssary simplificaion w oain y ( τ ) + ζ + g ( τ )µ x ( τ ) m (8) + + g ( τ )µ k ( + k sinωτ ) f ( x ) = F av m Now h non-dimnsional quaions of h gaair nonlinar dynamics ar oaind as low x τ + g τ + ζ + g ( τ ) µ x ( τ ) + µ + k sin ( ωτ ) f ( x ) = F + F av ω sinωτ y ( τ ) + ζ + g ( τ ) µ x ( τ ) + + g ( τ ) µ k ( + k sinωτ ) f ( x ) = F m av m 3. NONLINEAR DYNAMICS MODEL OF A FAUL GEAR PAIR CONSIDERING FRICION AND BACKLASH 3.. im-varying Mshing Siffnss of a Faul Gar Whn a gar has fauls such as piing or rokn ooh h characrisics of mshing siffnss will chang [] in (9) paricular whn a ooh is rokn 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 rducion of mshing siffnss of h corrsponding ooh mshing. Suppos a gar has h numr qual o Z and hr is on ooh rokn h im-varying mshing siffnss coincids wih Fig. (). Suppos h rokn ooh has h sam pako-pak valu as ha of normal h. o k m h Fourir sris of h signal F ( ) in Fig. () is qual ( ) + a 0 + a cos nω + sin nω n h n h + k a πn whr sin nω h n = a 0 = F d = 0 a n = 0 0 F (0) cos( nω )d = 0 Mshing priod; Roaional priod; Z h of driving gar; k a Pak-o-pak valu of mshing siffnss; k av Man of mshing siffnss; k av Man of mshing siffnss of h rokn ooh Fig. (3). Varying of mshing siffnss of a singl rokn ooh.

4 490 h Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Fig. (4). Pak-o-pak of h mshing siffnss of a rokn ooh mshing. n = 0 F sin( nω )d = 4k a πn n = ω = π h h low is o oain h paramrs of h Fourir sris k m ( ) of h signal F ( ) in Fig. (). Fig. (4) dpics h pak-o-pak of h mshing siffnss of a rokn ooh mshing. I can sn ha h Fig. (4) is h diffrnc of Figs. ( 3). L. Sinc F( ) = a 0 + a n cosn + n sinn a 0 = F 0 d = F 0 Z a n = n = 0 0 F F cos( nω )d = F 0 nπ sin ( nπ Z ) sin( nω )d = F 0 πn cos nπ Z hn h Fourir ranslaion of Fig. (4) is oaind as low k( ) = k Z + k nπ sin ( nπ Z )cosn + k πn cos ( nπ Z )sinn () hn h xprssion of h mshing siffnss of a gar pair wih a rokn ooh faul or h Fourir ransmission of Fig. (3) is qual o k mf ( ) = k m ( ) k( ) + k a sinnω h n = πn k Z () k + nπ sin ( nπ Z )cosn + k πn cos ( nπ Z )sinn h curvs in Figs. (5-7) ar h simulaion rsuls of q (0) o q() rspcivly using Mala. h mshing priod is = 0 04 Sc ; h of h gar is Z = 6 ; man of mshing siffnss is k av = 0 ; pak-o-pak of mshing siffnss is k a = 0.5 ;wih h maximum Fourir ordr of /Sc /Sc Fig. (5). im-varying mshing siffnss of a normal gar /Sc /Sc Fig. (6). im-varying mshing siffnss of h rokn ooh mshing.

5 Nonlinar Dynamics for Gar Faul Lvl h Opn Mchanical Enginring Journal 04 Volum Fig. (7). im-varying mshing siffnss of a gar wih faul of a rokn ooh. n max = 0. From Fig. (7) i can sn ha h curv of Fig. (7) can xprss h im-varying siffnss of a faul gar in Fig. (3). approximaly. 3.. Nonlinar Dynamics Ingrad Modl Sinc hr will varying whn a gar has fauls such as piing and spalling and so on and h variaion of h mshing siffnss is lss han ha of ooh rokn faul ransmi q.(). as low k mf ( ) = k m ( ) αk ( ) k + a sinnω h n = k av πn αk av k sin nπ k nπ k Z av cosn + Zk av + k cos nπ πnk Z av sinn + k sinnω h n = πn k 3 k αk 3 av Z + nπ sin ( nπ Z )cosn + k 3 πn cos ( nπ Z )sinn whr (3) k = k a k av k 3 = k k av (45) α [ 0]is gar faul facor whn a gar is in normal sausα = 0 ; whn a gar has srious faul of ooh rokn α = ; ohrwis whn a gar has ligh fauls such as crack piing and spalling 0 < α <. Dividing q (3) yω m hn simplify i ino is nondimnsional form w hav ( ) k mf = + k sin n ω h ω ω m πn ω n = k 3 α k 3 Z + nπ sin ( nπ Z )cosn + k 3 πn cos ( nπ Z )sinn = + k sin( nωτ )n = πn k 3 α k 3 Z + nπ sin ( nπ Z )cos nτ ω + k 3 πn cos ( nπ Z )sin nτ ω = + k sin( nωτ )n = πn k 3 α k 3 Z + nπ sin ( nπ Z )cos( nτ k ) + k 3 πn cos ( nπ Z )sin( nτ k ) L k = k πn sin( nωτ )n = (4) k = k 3 Z + k 3 nπ sin ( nπ Z )cosnτ + k 3 πn cos ( nπ Z )sinnτ

6 49 h Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. hn q (6) can simplifid ino k mf ( ) = + k ω αk. m hrfor from q() and q() h nonlinar dynamics ingrad modl of a normal gar and a faul gar considring acklash and fricion is saisfid I p θ p + r c m x + k mf I g θ g r g c m x + k mf f ( x) f ( x) + l pλµ c m x + k mf l pλµ c m x + k mf h Non-dimnsional form is qual o x τ + g τ + k αk f ( x) f ( x) + ζ + g ( τ )µ x ( τ ) + µ [ ] f ( x ) = F av + F ω sinωτ y ( τ ) + ζ + g ( τ )µ x ( τ ) + m + g ( τ )µ k ( + k αk ) f ( x ) = F av m 4.SOLUION = p = g (5) (6) Eq. (8) is a scond ordr diffrnial quaion which can solvd y 4-5 ordr Rung-Kua mhod. In ordr o supply horical rfrnc of gar fauls diagnosis h sady-sa rspons of a normal gar and a faul gar viraion is focusd as wll on h frquncy domain characrisics. For simpliciy h h of h driving gar is z p = 6 and h oharamrs ar as shown in al. Pars of hm ar according o rfrnc []. Givn µ ω andα h oharamrs can solvd using paramrs in al. Non-dimnsional frquncy f nd = ω π. al. Paramr Symols Paramr valus. Valu Paramr Symols Valus z 6 ζ 0.0 a 0 mm ζ 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 ω = 0.3 hn non-dimnsional frquncy is f nd = ; non-dimnsional priod is nd = 0.9 ; non-dimnsional roaional priod is nd = 5 ; non-dimnsional roaional frquncy is f nd = Sady-sa rspons viraions ar shown in Figs. (8-0). wih h frquncy rsoluion of Δf = α = 0 his mans h gar is in normal saus. And Fig. (8) dpics h sady sa viraion rspons of a normal gar. α = 0. his mans h gar has som sligh fauls such as iniial crack piing or spalling. Fig. (). dpics h sady-sa viraion rspons of a gar wih faul of iniial crack piing or spalling. α = his mans h gar has srious fauls as ooh rokn. Fig. (0) dpics h sady-sa viraion rspons of a gar wih faul of rokn ooh. From h im domain signal of Fig. (0). i can sn ha h shock phnomnon is mor and mor clar along wih h incras of h faul facor α wih shock priod aou 5. Morovr in h frquncy domain signal of Fig. (0) h sidands of x x and high frquncy muliplicaion of mshing frquncy wih an inrval aou ar mor and mor clar. () µ = 0 ω = 0.7 wih non-dimnsional mshing frquncy f nd = 0.5 non-dimnsional mshing priod nd = 9.0 non-dimnsional roaional priod nd = 54 and nondimnsional roaional frquncy f nd = h sady-sa viraion signals ar shown as in Figs. ( ) wih frquncy rsoluion Δf = α = 0 mans h gar is in normal saus. Fig. () dpics h sady sa viraion rspons of a normal gar. α = mans h gar has srious fauls as rokn ooh. Fig. () dpics h sady-sa viraion rspons of a gar wih faul of ooh rokn. From Fig. () i can sn ha h normal gar viraion signal also has sidands howvr whn α = 0 hr is no modulaion frquncy in h dynamics q(8) so h sidands should suprharmonics. In Fig. () h signal is aou a gar wih srious faul such as rokn ooh i is ovious ha hr ar sidands a h harmonics of mshing frquncy wih inrval of roaional frquncis. hn w conclud ha whn a gar is in nonlinar viraion h supr-harmonics ar rlad wih mshing frquncyω and hav nohing o do wih gar fauls. From Fig. () i rvals ha h ampliuds of supr-harmonics ar much lowr han ha of sidands of mshing frquncy wih gar fauls. CONCLUSION h nonlinar dynamics ingrad modl of normal gars and faul gars is salishd considring fricion

7 Nonlinar Dynamics for Gar Faul Lvl h Opn Mchanical Enginring Journal 04 Volum Fig. (8). Sady-sa rspons of a normal gar viraion Fig. (9). Viraion rspons of a gar wih faul of arlir crack piing or spalling. acklash and im-varying mshing siffnss. Fourir sris is usd o dpic varying siffnss ha is mor suial o solv h dynamics quaion. h gar faul facorovids convninc for h ingrad modl o dpic normal gar viraion and faul gar viraion. Simulaions show ha

8 494 h Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. Fig. (0). Sady-sa viraion rspons of a gar wih rokn faul of ooh. Fig. (). Sady-sa rspons of a normal gar viraion.

9 Nonlinar Dynamics for Gar Faul Lvl h Opn Mchanical Enginring Journal 04 Volum Fig. (). Viraion rspons of a gar viraion wih faul of a ooh rokn. a. Wih h incras of h faul facor which mans h faul lvl growing from ligh lvl o srious lvl h shock phnomnon is mor and mor clar. Mshing frquncy and is mulipl frquncy of h viraion signal ar modulad y h roaional frquncy clarly ha mans h sidands wih inrval of roaional frquncy ar mor ovious.. Faul lvl of a gar is no h roo caus of lading o su-harmonics or supr-harmonics of gar viraion whos rason is rlad wih roaional spd and fricion cofficin of gaair. NOMENCLAURE θ p θ g = Momns of inria of conacing ooh pairs = Roaional angls of conacing ooh pairs x y λ = Rlaiv displacmn wn mshing poins = 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 β a = Mshing damping cofficin; ( ) = Mshing siffnss; = Prssur angl; ( ) = Saic rror = Cnr disanc of conacing ooh pairs r g l p l g p g m p m g ω g O p O g z p z g = Bas radiuss of conacing ooh pairs = Fricion arms of conacing ooh pairs = orqus of conacing ooh pairs = Masss of conacing ooh pairs = Angular vlociis of conacing ooh pairs = Origin poins of conacing ooh pairs = h of conacing ooh pairs k av k a a ω h m = Mshing im = Half of acklash = Man of varying mshing siffnss = Pak-o-pak of varying mshing siffnss ( ) = Ampliud of saic rror = Mshing frquncy = Equivaln inrial mass of gaair

10 496 h Opn Mchanical Enginring Journal 04 Volum 8 Xunwn al. ω ω p τ ξ ξ r ag F av F g ( ) g = Equivaln naural frquncy of gar pair = Naural frquncy of driving gar = Non-dimnsional im = Damping raio = Damping raio = Pich Radius of drivn gar = Man of non-dimnsional xciing = Non-dimnsional ampliud of ransmission rror = Funcions rlad fricion dircion o simplify dynamic quaion k k k 3 k k = Siffnss funcions ε = Conac raio x yω = Corrsponding non-dimnsional symols of x yω k av = Man of mshing siffnss F ( ) = Picwis coninuous funcion of imvarying mshing siffnss Z n max f nd nd nd f nd = Mshing priod of gaair = Roaion priod of driving gar = h of driving gar = h maximum numr of Fourir sris = Non-dimnsional mshing frquncy = Non-dimnsional mshing priod = Non-dimnsional roaion priod of driving gar = Non-dimnsional roaion frquncy Δf = Frquncy rsoluion CONFLIC OF INERES W dclar ha his aricl conn has no conflic of inrs wih ohopl or organizaions ha can inapproprialy influnc our work. ACKNOWLEDGEMENS h auhors would lik o acknowldg h suppor from h Fundamnal Rsarch Funds for h Cnral Univrsiis (YX04-04). REFERENCES [] A. Pary and N. andon "Spur gar dynamic modls including dfcs: A rviw" h Shock and viraion digs vol. 35 pp [] W. Sanmin S. Yunwn and D. Haijun "Chaos and ifurcaion analysis of a spur gaair wih comind fricion and claranc" Chins Journal of Mchanical Enginring pp [3] W. Jianjun L. Qihan and L. Runfang "Rsarch advancs for nonlinar viraion 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 vl gar wih claranc" Journal of Viraion and Shock vol. pp [5] J. Wang and. C. Lim. "Effc of ooh msh siffnss asymmric nonlinariy for driv and coas sids on hypoid gar dynamics" Journal of Sound and Viraion vol. 39 pp [6] F.Elasha C. Ruiz-Cárcl D. Ma G. Kia I. Nz and G. Yra "Piing dcion in worm garoxs wih viraion analysis" Enginring Failur Analysis vol. 4 pp [7] Y. Cai and. Hayashi "h Linar approximad quaion of viraion 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 garoxs using novl wavl chnology" Insigh vol. 5 pp [9] W. Barlmus F. Chaari R. Zimroz and M. Haddar "Modlling of garox dynamics undr im-varying nonsaionary load for disriud faul dcion and diagnosis" Europn Journal of Mchanics A-solids 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 havior in a high conac raio spur gar ooh and is ffc on msh siffnss" Enginring Failur Analysis vol. 34 pp Rcivd: Spmr 0 04 Rvisd: Novmr 5 04 Accpd: Novmr 5 04 Xunwn al.; Licns Bnham Opn. his is an opn accss aricl licnsd undr h rms of h Craiv Commons Ariuion Non-Commrcial Licns (hp://craivcommons.org/licnss/ y-nc/3.0/) which prmis unrsricd non-commrcial us disriuion and rproducion in any mdium providd h work is proprly cid.