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Article I D: 100721172(2005)0320303204 Numerical Analysis of Plastic Gear Stiffness X IE W en2bo 13 (谢文博 ), IIJ IM A K iichiroh 2( 饭岛喜一郎 ), LU H ao1(陆 皓) (1. School ofM aterials Sci . 2.Fuji Xerox Co. L td. , Japan) Abstract: This paper established practical 32D gear models to study the stiffness influencing factors of a loaded gear by finite elementmethod, such as friction parameters, materialproperties, and gear structures . The research show s that, in elastic deformation, gear stiffness increaseswhen sliding friction ability of contact pair decreases; meanwhile, the gear structure, especially asymmetric design in gears shaft direction w ill also decrease gear stiff2 ness . Key words: plastic gear; stiffness; structure; friction; numerical analysis Document code: A Received date: 2004211210 Foundation item: Cooperated w ith Fuji Xerox Co.L td. , Japan 3E-mail:wenbo. xiefujixerox. co. jp Introduction Gear stiffness is one of the key influencing factors on geartransm ission performance.For years, overseas and native scholars carried out lots of researches on gear stiffness.Especially, in re2 cent years, w iththe development of numerical method and computer technology,gear stiffness research and prediction can be performed by nu2 merical si mulation. Up to now , the research of gear stiffness is al2 most about the gearsmeshing. The research about gear structure is relatively li m ited,and in most cases models w ere 22D.M eanw hile,current re2 searches mostly focus on metal gear; how ever the plastic gear w hich is low cost and is frequently used in light transm ission process(such as laser printer or copymachine)is seldom studied. There2 fore, about plastic gear, in order to direct gear de2 sign w ithout trial manufacture, it is necessary to establish 32D model and predict gear stiffness of various structures .So, this paper performed finite element analysis about this point, and then verified si mulation results by experi ment. The tooth deformation of plastic gear includes elastic and plastic parts. The plastic deformation is small,w hich helps tooth contact accuracy and tooth running2in 1. So this paper does not discuss plastic deformation,only discusses the stiffness and its influencing factors w hich are based on the tooth contact deformation and tooth torsional de2 formation 2. 1 Si mulation and Experi mental Con2 ditions A series of gears are studied by FEM.Figure 1 show s the layout of gears,namely standard gear,symmetry gear and offset gear. Symmetry gear and offset gear both have two kinds of thick2 ness, w hich are“thickness 2”and“thickness 4”. Gear is fixed by jigs onto shaft. A pressure of 5 kN is applied to jigs, thus gear can not rotate freely due to the friction force betw een gear body and jigs.A driving force of 10 50 Nis applied to tooth through a hitter.The contact point betw een tooth and hitter is on pitch circle(r= 33. 5 mm ). Because w e only w anted to study the tooth contact deformation and tooth torsional deformation, for si mplification,w eusedsingle2tooth trapezium gear, rather than the usual involute gear. The material of gears is polyacetals .O ther parts(such as shaft,jigs, hitter)w ere made of steel SU S304. Because the deformation of SU S304 Journal of ShanghaiJiaotong U niversity(Science),Vol . E210,No. 3, 2005, 303306 Fig. 1 M odel layouts (a)Standard, (b)Symmetry 2, (c)Symmetry 4, (d)Offset 2, (e)Offset 4, (f)Trapezium tooth and Hitter is much smaller than polyacetals,it is flexible2 rigid contact.In si mulation, w e can define SU S304 parts as rigid bodies .W henrotating angleis small, by know ing the relation of“load2rotation angle”, w e are able to calculate gear stiffness 3. M eanw hile,insi mulation, the movementand boundaries of rigid bodies can be si mply controlled by a pilot node(rigid node ). In this paper,regarding material properties, the influences of Friction Coefficient(M U ), Stat 2 ic?Dynam ic Friction Ratio(R), and YoungsM od2 ulus(E)on gear stiffness w ere discussed; regard2 ing structure, the influence of various gear shapes on gear stifrnessw as discussed. Especially, because of the unavoidable manu2 facture defects by machining, there are defects at the tooth root, as show n in Fig.2.In si mulation, the same defectsw ere also considered. 2 Equation of Stiffness Calculation In experi ment, because the practical shaft is not absolute rigid body, w hen hitter is driving gear at the tooth w ith an angle, shaftw ill also rotate w ith an angle .If the radius of pitch circle isr, then U Y=r(- )(1) Here,U Ymeans the node displacement on pitch circle. So, the experi ment stiffness can be calculat2 Fig. 2 Defect at tooth root ed as K= F U Y = F r(- ) (2) In si mulation, because shaft is considered as an absolute rigid body, = 0. So, the si mulation stiffness can be calculated as K= F U Y = F r (3) 3 The Gear Stiffness Influencing Factors 3. 1 Friction Parameters Itissurface2surface contactbetw eengear body and jigs, and the same betw een tooth and hit2 ter. The friction status of these contact pair has significant influence on gear stiffness .The friction status can be described byM UandR.In this pa2 per,M U1andR1are used for tooth2hitter contact pair;M U2andR2are used for gear body2jigs con2 tact pair. Firstly, w e discuss the influence ofM U1and M U2onstiffness of standard gear.W hen dis2 cussingM U1(0. 0250. 3),M U2is set to + (bondedcontact pair ); w hendiscussingM U2 (0. 0250. 3),M U1keeps 0. 15. M eanw hile, R1, R2andEare all set to common values, w hich are 1. 1, 1. 1, and 2. 744 GPa. Si mulation results are show n in Fig.3, from w hich w e can see thatM U1has linear influence on gear stiffness . W henM U1increases, the difficulty 403 X IE W en2bo(谢文博 ), IIJ IM A K iichiroh(饭岛喜一郎 ), et al of the slide friction betw een hitter and tooth in2 creases,thus the torsional deformation of tooth becomes smaller, leading to an increase of stiffness (refer to Eq.(3). But w e should pay attention that, if theM Uis too large, it may make gear ro2 tation difficult, and decrease the transm ission effi2 ciency.Therefore, gear transm ission performance is decreased. Fig. 3 The influence ofM U on stiffness The influence ofM U2is a bit more complex thanM U1. W henM U2is small, gear stiffness is very sensitive toM U2s variation.IncreasingM U2 can enhance gear stiffness remarkably. W henM U2 is larger than 0. 15, the influence becomes smooth. This phenomenon is because slipping occurs be2 tw een gear body and jigs by driving force w hen M U2is low , therefore it is show n as stiffness de2 crease. W henM U2is high enough,this slipping can be restrained w ell,so only a small slipping may occur.So, at this moment, increasingM U2 cannot enhance gear stiffness too much, show n as the flat area of the curve in Fig. 3. Secondly,regarding Symmetry 4, the influ2 ence ofRon gear stiffness is studied.Calculation condition isM U1=M U2= 0. 15,E= 2. 744 GPa. Si mulation results are show n in Fig. 4, from w hich w e can see that w henRranges from 1. 0 to 3. 0, increasingR1can help enhance gear stiffness to some degree, about increasing 5. 4%. But increas2 ingR2does not help enhance gear stiffness .To2 gether w ith Fig. 3w ew ill know that, w henM U2= 0. 15, the slipping betw een gear body and jigs is very small,so increasingR2cannot help enhance gear stiffness . Fig. 4 The influence ofR on stiffness 3. 2 YoungsM odulus Youngs M odulus has very remarkable influ2 ence on plastic gear stiffness .Regarding all five kinds of gears, Fig. 5 show s the influence of differ2 entEon gear stiffness .The influence is linear, and the increase percentage is same among all 5 kinds of gears.That means, w henE increases 10% , stiffness w ill increase about 9%. A nd this linear influence is structure2independent. Fig. 5 The influence ofEon stiffness 3. 3 Geometry Structure In Fig.6, w e can see that in both experi ment and si mulation, gear stiffness is influenced by gear geometry structure remarkably.The stiffness se2 quence is: Standard Symmetry 2 Offset 2 Symmetry 4 Offset 4. Fig. 6 Si mulation stiffness - Experi ment stiffness 503 N umericalA nalysis of Plastic Gear Stiffness Thismeans that as the gear thickness decreas2 es, tooth torsional deformation becomes larger, thus stiffness becomes low er. W hen thickness is same,symmetry gears have higher stiffness than offset gears .This is because that in offset gear, the concave part continuously locates at one side of the thickness, how ever in symmetry gear,it lo2 cates at the both sides of thickness, therefore the accumulated deformation of offset gear is larger than symmetry one,so offset gears stiffness is low er than symmetry gear. Besides, for offset gear, hitter has two rela2 tive directions to drive tooth.But si mulation re2 sults show that the relative position betw een con2 cave and hitter doesnt influence gear stiffness . 4 Experi mental Results Figure 6 show s the relationship betw een si m2 ulation stiffness and experi ment stiffness, w hen M U= 0. 15,R= 1. 1, andE= 2. 744 GPa.From this figure, among these five kindsof gears, a high linearity betw een si mulation results and experi2 ment results can be seen.This linearity verifies that the si mulation is correct. But si mulation stiff2 ness is higher than experi ment value; there is a lin2 ear difference betw een them.It may be because the material parameters in si mulation are not accu2 rate;finite element algorithm 4; plastic vis2 cosity is not considered in si mulation;the de2 fects at tooth root in si mulation model are not ac2 curate enough. 5 Conclusion (1)It is feasible to predict gear stiffness by establishing 32D model and using finite lement method. (2)Friction parametersM UandRboth influ2 ence gear stiffness . D ifficulty of sliding friction in2 creases w henM Uincreases, w hich enhances gear stiffness . But if Friction Co