Gear System Time-varying Reliability Analysis Based on Elastomer Dynamics

Transcription

1 A publcaon of CHEMICAL ENGINEERING TRANSACTIONS VOL. 33, 013 Gues Edors: Enrco Zo, Pero Barald Copyrgh 013, AIDIC Servz S.r.l., ISBN ; ISSN The Ialan Assocaon of Chemcal Engneerng Onlne a: DOI: /CET Gear Sysem Tme-varyng Relably Analyss Based on Elasomer Dynamcs Zhy Ma a, Janguo Zhang*,a, Jng Sun b, Peng Gao b, Me L a a Scence & Technology Laboraory on Relably& Envronmenal Engneerng,Bejng ,Chna; School of Relably and Sysems Engneerng,Behang Unversy,Bejng,100191,Chna b Bejng Spacecrafs zjg@buaa.edu.cn Ths paper esablshes nonlnear dynamc equaon based on he consderaon of elasc deformaon of shaf whch happens n gear meshng process. In addon, he sffness ncenve, error ncenve and exernal ncenves caused by eccenrc phenomenon durng he process are analysed. The Newmark algorhm s also been used. Then he dynamc load coeffcen of he gear could be goen. Based on hs, a me-varyng relably algorhm s used o calculae he degree of relably. Mone Carlo algorhm s used o prove he accuracy of hs mehod. Fnally he numercal example s gven. 1. Inroducon A presen, he hgh speed machne and hgh power generang se mosly use flexble roor. The gear dynamcs model ha does no consder axs ransverse vbraon s no applcable. (Shao e al, 005) In addon, he gear sysem relably analyss and srengh calculaon mosly use he mehod ha nqures he mechancal desgn manual o ge dynamc load coeffcen. So he subjec facor canno be avoded. (L,e al, 011). The orson vbraon model s used o esablsh he gear dynamc model. Bu hs model does no consderng he effec of he gear bendng. (Bague e al, 010) he fne elemen dynamcs mehod s used o esablsh gear dynamc model. Alhough he nfluence beween he shaf and gear s consdered, he mehod s oo complcaed o be used o solve he engneer problem. (Mejr e al,010) PHI mehod s used. Bu canno provde he accurae me-varyng relably. Moreover, he falure rae whch s mporan n relably analyss s no goen. (Huang e al, 003) Sress maxmum and srengh mnmum values are seleced n he me process o esablsh lm sae funcon. Then he momen mehod s used o ge relably calculaon resul. As a resul, he change rule of relably could no be goen. Ths paper deduces he dynamc equaons of gear-shaf sysem whch ncludes elasc vbraon model and plane vbraon model of spur gear. The me-varan excaon of gear s offered by FEM. Then he dynamc load coeffcen of spur gear under hs crcumsance would be goen. Fnally, he me-varyng falure rae and relably level s obaned by a new relably calculaon mehod.. Dynamc Model I s ceran ha here wll be a radal force when gears mesh wh ohers. Ths nfluence can be gnored f he gear shaf s shor enough. Ignorng he nfluence of a long and flexble shaf deformaon wll resul n a bg error, as shown n fgure 1. So he nfluence of shaf flexbly mus be aken no consderaon when dynamc equaons are deduced. Dfferenal equaons can be deduced for he dynamc analyss on elasc body n hs model. Bu s oo complcaed o be used o solve an engneerng problem. To deal wh hs problem, Fne Elemen Mehod (FEM) would be used. Please ce hs arcle as: Ma Z., Zhang J., Sun J., Gao P., L M., 013, Gear sysem me-varyng relably analyss based on elasomer dynamcs, Chemcal Engneerng Transacons, 33, DOI: /CET

2 Fgure 1Gear roang sysem dagram.1 Dynamc equaons of shaf n FEM Gear shaf wll have a vbraon n he roaonal and horzonal drecon under radal force and bendng momen. Jus as Tmoshenko beam elemen, he shape funcon of horzonal and roaonal dsplacemen can be nerpolaed ndependenly. Because deflecon curve s cubc when beam s endured concenraed load, he shape funcon can be formulaed n cubc equaon: 3 u ( x, ) = c0 + c1x + c x + c3x (1) Subsuon boundary condon u 4 ( x, ) ϕ ( x) u ( ) = = 1 Analogy n Tmoshenko beam elemen Angle shape funcon can use lnear nerpolaon θ (, ) = a + a x x 0 1 Subsuon boundary condon θ ( x, ) = ϕ ( ) = 1 θ Geng beam elemen poenal energy () (3) (4) ( x, ) 1 1 ( x, ) dx + GI dx 1 1 u θ v() = EI P 0 x 0 x (5) The knec energy T of beam elemen: 1 u( x, ) 1π 4 θ ( x, ) ρ ρ 0 (6) T() = A dx+ R dx Subsuon Lagrange equaon can ge beam elemen vbraon equaon: { } + { } = { } M y K y F shaf shaf shaf (7). Gear dynamc equaon Sragh gear conac can be descrbed by wo dmensonal vbraon model sysem as fgure showng, sad end dsplacemen of beam elemen ha connec wh gear. m1, I1, r1 T 1 Y 1 θ 1 e θ m, I, r T Y Fgure Sragh gear plane drve model dagram 476

3 The gear conac dynamcs equaon: { } + { } + { } = { } M y C y K y F gear gear gear gear (8).3 Drvng sysem dynamc equaon Sackng all he elemens vbraon equaon and gear vbraon equaon, he force of varous elemens and beween gear and elemen as nernal force s offse, he sysem dynamc equaon as follows:. [ M]{ y} [ C]{ y} [ K]{ y} { F} + + = (9) 3. Gear drve ncenve analyss 3.1 Sffness ncenve Durng he gear meshng, due o he meshng ooh logarhmc change and elasc deformaon from op o roo, mesh comprehensve sffness would be perodc changed by me. I leads gear ooh meshng force perodc o change.i s no praccal o use analycal mehod o calculae he me-varyng sffness of gear. Fne elemen smulaon mehod o calculae elasc deformaon of he gear ooh s easer. Afer usng FEM o ge dsplacemen and force, he calculae gear meshng of me varyng sffness can be calculaed by equaon 10 (S. Bague e al, 010). p n F = 1 K = δ + δ n By usng ANSYS APDL language programmng o esablsh he model of gear engagemen, as shown n fgure 3.Then he gear meshng of me-varyng sffness can be obaned, as shown n fgure 4. (10) Fgure 3 :Gear meshng fne elemen model dagram Fgure 4:Gear meshng me-varyng sffness dagram All of perodc funcons can be regarded as superposon of sne funcon. Fourer seres s suable o f smulaon daa. Then he me varyng sffness expresson of gear meshng can be expressed by equaon 11. n K () = Km + k sn ( ) 1 a + b = (11) 3. Error ncenve Gear meshng error s devaon beween ooh profle acual poson and deal poson durng he gear meshng caused by gear machnng error and nsallaon error. Smlarly, Gear error ncenve s also a knd of cyclcal ncenve, s suable for usng sne funcon o descrbe (M. Mejr, e al,010). π e () = e0 + esn + ϕ T (1) 3.3 Exernal ncenve Gear ransmsson sysem wll be nevable no manufacurng error. I wll lead o he roor eccenrcy. Eccenrc roor wll produce he dynamc load ncenve, he expresson as followng: 477

4 Fw = mrω (13) 4. Dynamc load coeffcen calculaon We can ge he dynamc equaon of gear ransmsson sysem whch s shown by equaon 9 : [ M]{ y} + [ C]{ y} + K(){} y = K() e() + [ F ] + T () + T () w n ou (14) From above, he concluson ha he gear ransmsson sysem sffness marx, nernal and exernal ncenves are me-varyng and nonlnear s goen. The radonal ransfer marx mehod and vbraon mode Superposon mehod are no suable o hs suaon. The paper uses he Newmark algorhm shown by equaon 15 MY + CY + K +Δ Y K + e + F T + T + = ( ) ( ) ( ) ( ) ( ) 0 (15) +Δ +Δ +Δ W n ou Y+ = Y + ( 1+ α) Y+ αy+δ (16) 1 Y+ = Y + Y+ βy+ βy+ (17) Usng above hree equaons can oban Y +, Y +, Y +Δ As long as Y = 0 Y = 0 Y = 0 a nal me are gven, by ncreasng Δ he dsplacemen, velocy and acceleraon can be goen a each me. Thus, he gear meshng lne dsplacemen can be obaned. If α = 1, β = 14, can ensure o ge uncondonal sably soluon. By gear dynamc load calculaon formula 18 and 19 dynamc load coeffcen can be goen. As followng: Fd = K y (18) Fd + F Kv = (19) F 5. Tme Varan Relably Analyss of Gear Logarhmc normal dsrbuon or We Bull dsrbuon can be used o represen falure probably n radonal relably analyss of gear sysem. Bu hs mehod s no suable o solve he problem falure mechansm. In addon, he load on he gear s me varan, as shown n fg 6, s unscenfc o analyze he relably n dynamc load coeffcen sable value. Some documens used ou-crossng rae mehod whch can jus esmae upper and lower boundary of breakdown rae. To he me varan and nonlnear characer of dynamc loadng coeffcen, hs paper uses FORM mehod o calculae relably ndex β () and normal vecor α ().and uses relably calculaon mehod ha based on falure rae and uses Mone Carlo mehod o check he accuracy of hs mehod. 5.1 Relably calculaon based on gear bendng sress Lm sae funcon F G = σf σflm = YSYbKv σflm (0) bm 5. Tme varyng relably calculaon mehod Order X ( ω, ) as mechancal problem random varables, represen me course, ω represen a sample pon n sample space Ω.herefore, gear me-varyng lm sae funcon as followng: G( )=0 hen he relably funcon can use R() = prob( G(, X ( ω, ) )) 0 o descrbe. A he same me falure rae funcon can be defned: λ () ( ) ( ) lm Δ Δ prob( B) prob A B prob A B = lm = Δ 0 Δ 0 (1) 478

5 { } { } Noe : A= G( +Δ, X ( ω, +Δ) ) 0 ; B = G( +Δ, X ( ω, +Δ) ) 0 ; As we know ( ) expressed by () prob B ( ) φ( β) prob B can be = (3) Inroduce wo dmensonal normal dsrbuon funcon φ and correlaon coeffcen, so prob( A B) = φ β( ), β( +Δ ), ρ(, +Δ ) (4) Noe: (, ) ρ(, ) α( ). α( ) ( ) ρ +Δ can expressed by lm sae surface normal vecor +Δ = +Δ (5) φ calculaon mehod as follows φ β, β +Δ, ρ, +Δ = ( () ( ) ( )) β() β( +Δ) ( ) ( ()) ( ( )) ϕ xy,, ρ dxdy= φ β. φ β +Δ ρ exp () ( ) () ( ) ( ) ( β + β +Δ + zβ β +Δ ) dz 0 π 1 z 1 z T λ() d 0 Usng relably funcon R() = e can oban relably of he whole me hsory (6) x x GX (, ( ω,)) = 0 G ( +Δ, X( ω, +Δ )) = 0 α() β () α ( +Δ) β ( +Δ) x 1 x 1 Fgure 5Lm sae surface a dfferen me 6. Calculaon example Gear relaed parameers as s shown n Table 1: Table 1 Gear relaed parameers Number of eeh Modulus Tooh wdh Mass (kg) Pressure Angle Cener Dsance Small gear Bg gear Shaf relaed parameers as shown n Table : Table Shaf relaed parameers Lengh Radus Elasc modulus(gpa) Eccenrcy Dsance Shaf Gear dynamc load coeffcen calculaon resuls and Gear sysem relably evaluaon Usng above mehod o wre correspondng calculaon program can ge ou load coeffcen calculaon resuls. As shown n fgure 6Tme sep s 0.0 seconds, due o he nfluence of dampng gradually afer 6 s become sable. The mean and varance of varous parameers n formula 0 are as follows F N ( 34644, ), Ys1 N (.6,0.01 ), Yb1 N ( 1.595,0.011 ), Ys N (.14,0.015 ), Y N 1.83,0.01, σ N 1300,156. ( ) ( ) b F lm 479

6 Fgure 6 Dynamc load coeffcen dagram Fgure 7Relably ndex a dfferen me Tme dscree relably ndex can' reflec he relaonshp beween me pons. Use formula can ge me-varyng falure rae curve he mehod of numercal negraon for me-varyng relably and Mone Carlo smulaon comparson resuls was shown n Fgures 8 and 9: Fgure 8Falure rae Fgure 9Comparson for dfferen mehod The maxmum error s less han 0.01, bu s calculaon me sgnfcanly less han Mone Carlo algorhm. 7. Concluson Ths paper consders axs elasc deflecon and gear me-varyng sffness o esablsh he nonlnear dynamc equaon whch s solved by Newmark mehod. hen he dynamc load coeffcen s goen. Fnally a mproved me-varyng relably calculaon mehod s used o calculae relably of he gear sysem. In addon, Mone Carlo algorhm s used o verfy he accuracy and he feasbly of hs mehod. The mproved me-varyng relably calculaon mehod has he advanage ha can ge me-varyng falure rae. A he same me, can also be appled a he suaon ha he lm sae funcon s hghly nonlnear. I s expeced o have a ceran gudng sgnfcance for falure and lfe predcon. References Bague S., Jacqueno G., 010, Nonlnear couplngs n a gear-shaf-bearng sysem [J] Mechansm and Machne Theory,45: Huang Wenbo, Zhang Shengkun, 003, Hull me-varyng relably analyss [J], journal of Shangha ranspor unversy, 115:1-6 L Gu Xan, L Yong Qang. 011, Marne wo sage double row helcal gear planeary gear sysem he esablshmen of dynamcs equaons [J].journal of shp mechancs, Mejr M., Cazuguel M., Cognard J..Y, 010, A me-varan relably approach for ageng marne srucures wh non-lnear behavour, Journal Compuers and Srucures,5: Renpng Shao, 005, Mechancal sysem dynamcs [M]. Bejng: mechancal ndusry publshng company. 480