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喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== 喜歡就充值下載吧。。資源目錄里展示的文件全都有，，請放心下載，，有疑問咨詢QQ:414951605或者1304139763 ======================== Contact problem and numeric method of a planetary drivewith small teeth number differenceShuting Li*Nabtesco Co. LTD., Oak-hills No. 202, Heki-cho 7028-2, TSU-shi, Mie-ken 514-1138, JapanReceived 15 July 2007; received in revised form 2 October 2007; accepted 16 October 2007AbstractThis paper deals with a theoretical study on contact problem and numeric analysis of a planetary drive with small teethnumber difference (PDSTD). A mechanics model and finite element method (FEM) solution are presented in this paper toconduct three-dimensional (3D) contact analysis and load calculations of the PDSTD through developing concepts of themathematical programming method T.F. Conry, A. Serireg, A mathematical programming method for design of elasticbodies in contact, Transactions of ASME, Journal of Applied Mechanics 38 (6) (1971) 387392 and finite element methodS. Li, Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear, Transactions ofASME, Journal of Mechanical Design 124 (3) (2002) 511517; S. Li, Finite element analyses for contact strength and bend-ing strength of a pair of spur gears with machining errors, assembly errors and tooth modifications, Mechanism andMachine Theory 42 (1) (2007) 88114 to solve a more complex engineering contact problem. FEM programs are devel-oped through many years efforts. Contact states of teeth, pins and bearing rollers of the PDSTD are made clear throughperforming contact analysis of the PDSTD with the developed FEM programs. It is found that there are only four pairs ofteeth in contact for the PDSTD used as research object when it is loaded with a torque 15 kg m. It is also found that thesefour pairs of teeth are not located in the offset direction of the external gear. They are located at an angular position of 2030? away from the offset direction. Loads shared by teeth, pins and rollers have big difference. The maximum load sharedby the teeth is much greater than the ones shared by pins and rollers. This means that strength calculations of the teeth aremore important than the ones of pins and rollers for the PDSTD. It is also found that all pins share loads while only a partof rollers share loads.? 2007 Elsevier Ltd. All rights reserved.Keywords: Gear; Gear device; Planetary drive; Small teeth number difference; Contact analysis; FEM1. IntroductionIn the latter period of the 20th century, with the development of industry automation, gear devices withlarge reduction ratio found wide applications. Planetary drives with small teeth number difference (PDSTD)was also used widely in automation industry. Though many units of the PDSTD are made every year, strengthdesign calculation of the PDSTD is still a remained problem that has not been solved so far.0094-114X/$ - see front matter ? 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2007.10.003*Tel./fax: +81 059 2566213.E-mail address: shutingnpuyahoo.co.jpAvailable online at Mechanism and Machine Theory xxx (2007) TheoryARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003To perform strength calculations of the PDSTD, it is necessary to know the loads distributed on teeth, pinsand rollers in advance. Since there has not been an effective method available to be able to perform contactanalyses and load calculation of the PDSTD, gear designers have to use ISO standards 46 made for strengthcalculations of a pair of spur and helix gears to perform strength calculations of the PDSTD approximatelyNomenclaturePDSTD planetary drive with small teeth number differenceFEMfinite element methodFEAfinite element analysis3Dthree-dimensionalISOInternational Standard OrganizationFTload on tooth surfaceFPload on pinFRload on rollereeccentricity of the crankshaft.Z1tooth number of the external gearZ2tooth number of the internal gearX1shifting coefficient of the external gearX2shifting coefficient of the internal gearmmodule of gearsB1outside diameter of the internal gearB2inside diameter of the external gearB3diameter of the pin center circle on the external gear(ii0)assumed pair of contact points, also (110), (220), . , (mm0), (aa0), (kk0), (jj0), (bb0), .and (nn0)rused to stand for one elastic body or the external gearsused to stand for the other elastic body or the internal externalekclearance (or backlash) between a optional contact point pair (kk0) before contact. Also, ejFkcontact force between the pair of contact points (kk0) in the direction of its common normalline, also Fjxk, xk0deformations of the assumed pair of contact points (kk0) in the direction of the contact force Fkakj, ak0j0deformation influence coefficients of the contact pointsd0initial minimum clearance between a pair of elastic bodies in the direction of the external forcedrelative displacement of a pair of elastic bodies along the external force under the external force,or angular deformation of the internal gear relative to the external gear under a torque TYslack variables, Y = Y1, Y2, . , Yk, . , YnTXn+1artificial variables, also, Xn+1, Xn+2, Xn+n, . , Xn+n+1Iunit matrix of n n, n is size of the unit matrixZobjective functionSmatrix of the deformation influence coefficientsFarray of contact force of the pairs of contact points, F = F1, F2, . , Fk, . , FnTearray of clearance of the pairs of contact points, e = e1, e2, . , ek, . , enTeunit array, e = 1, 1, .,1, .,1T0zero array, 0 = 0, 0, .,0, .,0Trbradius of the base circle of the internal gearPexternal force applied on a pair of elastic bodiesPGsum of all contact forces between the contact points on tooth surfaces of the PDSTDTtorque transmitted by the PDSTDa0a angle used to express the position of pairs of teeth2S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0037. It has been known that contact problem of the PDSTD is completely different from the one of a pair ofspur and helix gears, so, ISO standards are not suitable for strength calculations of the PDSTD.Manfred and Antoni 8 conducted displacement distributions and stress analysis of a cycloidal drive withFEM. Yang and Blanche 9 also studied design and application guidelines of cycloidal drive with machiningtolerance. Shu 10 conducted study on determination of load-sharing factor of the PDSTD. Chen and Walton11 studied optimum design of the PDSTD.This paper aims to present an effective method to solve contact analysis and load calculation problems ofthe PDSTD. Based on more than 20 years experiences on contact analysis of gear devices and FEM softwaredevelopment, a mechanics model and FEM solution are presented in this paper to conduct contact analysisand load calculations of the PDSTD. Responsive FEM programs are developed through many years efforts.Contact states of the teeth, pins and rollers of the PDSTD are made clear with the developed programs. Loaddistributions on teeth, pins and bearing rollers are also obtained. It is found that there are only four pairs ofteeth in contact for the PDSTD used as research object in this paper when it is loaded under a torque 15 kg m.It is also found that these four pairs of teeth are not located in the offset direction of the external gear.Loads shared by teeth, pins and rollers are compared each other. It is found that the maximum load sharedby teeth is much greater than the ones shared by pins and rollers. It is also found that all pins share loads whileonly a part of rollers share loads. Strength calculations of the PDSTD can be performed easily after loads onteeth, pins and rollers are known.2. Structure and transmission principle introductionsFig. 1 is a simple type of the PDSTD used as research object in this paper. In Fig. 1, this PDSTD consists ofone internal spur gear, one external spur gear, two ball bearings, one input shaft, one output shaft, eight pinsused to transmit torque and 22 rollers used as the center bearing. In order to let teeth of the external gear engagewith the teeth of the internal gear, a radial movement of the external gear relative to the internal gear is needed.This radial movement is realized through rotational movement of a crankshaft. Of course, this crankshaft is acam that can produce offset movement for the external gear (in Fig. 1, when the crankshaft is rotated, a radialmovement of the external gear is produced alternately). The crankshaft is also used as input shaft of the device.Fig. 1 is the position when offset direction of the crankshaft is right up towards to +Y direction. In Fig. 1,O1is the center of the external gear and O2is the center of the internal gear. e is the eccentricity of the crank-shaft. e = O1O2. Gearing parameters and structure parameters of this PDSTD are given in Table 1.Since the PDSTD belongs to K-H-V type of planetary drive and tooth number difference between theinternal spur gear and the external spur gear is small, so this device is often called the planetary drive withsmall teeth number difference. Transmission ratio of this device is equal to Z1/(Z2? Z1) when the internal gearpinsrollersz1z2Input shaftOutput shaftInternal spur gearExternal spur geareAASection A-ACrankshaft (Cam)o1o2pin holeFig. 1. Structure of one kind of planetary drive with small teeth number difference.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx3ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003is fixed. Here, Z1is tooth number of the external gear and Z2is tooth number of the internal gear. From Z1/(Z2? Z1), it can be found that when tooth number difference (Z2? Z1) is very small, transmission ratio Z1/(Z2? Z1) shall become very large. For the device as shown in Fig. 1, teeth number difference (Z2? Z1) isequal to 1, so transmission ratio of this device = Z1.Since an internal gear is used in the PDSTD, tip and root interferences with the mating gear must be checkedlikeausualinternalgeartransmissionwheninvoluteprofileisused.Ofcoursethesetipandrootinterferencescanberemovedthroughperformingtoothprofilemodifications,foranexampletipandrootrelieves.Alsootherpro-files such as modified involute curve, arc profile and trochoidal curves can be used to avoid tip and rootinterferences.3. Load analysis and face-contact model of tooth engagement of the PDSTDFig. 2 is an image of loading state of the external gear in the PDSTD. In Fig. 2, it is found that three kindsof loads are applied on the external gear. They are tooth loads FTproduced by tooth engagement, roller loadsFRproduced in center bearing and pin loads FPresulted from the external torque. Tooth loads are along thedirections of the normal lines of the contact points on tooth surfaces of the internal gear. This also means thetooth contact loads shall be along the directions of the lines of action of the contact points on tooth profile ofthe internal gear. Roller loads are along radial directions of the center hole in the external gear. Pin loads areTable 1Gearing parameters and structural dimensions of the PDSTDGearing parametersGear 1Gear 2Structural dimensionsGear typeExternalInternalDiameter B180 mmTooth numberZ1= 49Z2= 50Diameter B236 mmShifting coefficientX1= 0.0X2= 1.0Diameter B341.125 mmFace width12 mm12 mmPin number8Helical angle00Pin diameter4Module (mm)1Roller number22Pressure angle20?Roller diameter3Tooth profileInvolutesCutter tip radius0.375 mOffset direction+YEccentricity, e0.971 mmRoller loadXYnYYkXkXiTooth loadXnYiPin load123456781234578910612345678910111213141516171819202122Pin center circleFTFRFPFig. 2. Load state of the external gear in the planetary drive.4S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003along the tangential directions of the pin center circle. Though these three kinds of loads are shown in Fig. 2,the reality is that we do not know which tooth, pin and roller shall share or not share loads. This is the prob-lem that must be solved in this paper. Contact analysis with the FEM is presented to solve this problem.Before performing contact analysis of the PDSTD, it is necessary to pay an attention to the tooth engage-ment state of this special device. Tooth engagement of the PDSTD is different from a usual internal gear trans-mission in that tooth engagement of a usual internal gear transmission is an engagement of teeth on thegeometrical contact lines and it has been already known in theory how many teeth and which teeth shall comeinto contact in different engagement positions for the usual internal gear transmission while tooth engagementof the PDSTD is not on the geometrical contact lines and it is not known in theory where the teeth shall con-tact on tooth profile, how many teeth shall come into contact and which teeth shall come into contact for thePDSTD. Even, it is not known whether the geometric contact lines exits or not for the PDSTD.The other difference is contact state of one pair of teeth. As it has been stated above, for a usual internalgear transmission, a pair of teeth shall contact on the geometrical contact line. It is called Line-contact of atooth in this paper. But for the PDSTD, the teeth shall contact on a part of face on the profile like the har-monic drive device. It is called Face-contact of a tooth in this paper. Fig. 3 is the real tooth contact states ofthe PDSTD with the parameters as shown in Table 1. From Fig. 3, it is found that the teeth 5, 6, 7, 8 and 9 areface-contact on the most part of tooth profile. So when to perform contact analysis of loaded teeth of thePDSTD with the FEM, a lot of pairs of contact points (ii0), (jj0), (kk0) and (nn0) as shown in Fig. 4 mustbe made between the tooth profiles of the external and the internal gears. These pairs of contact points areassumed to be in contact at first and it shall be made clear finally which pair of points turns out not to bein contact through performing contact analysis of the PDSTD with the FEM presented in this paper.4. Basic principle of elastic contact theory used for contact analysis of a pair of elastic bodies 14.1. Deformation compatibility relationship of a pair of elastic bodiesIn Fig. 5,randsare one pair of elastic bodies which will come into contact each other when an externalforce P is applied. The contact problem to be discussed here is restricted to normal surface loading conditions.InternalgearExternalgear56789Fig. 3. Face-contact of mating teeth.Internal gearExternal gearijknijknFjjFkkFig. 4. Pairs of contact points on tooth surfaces of the internal gear and the external gear.S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx5ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003Discrete forces can be taken to represent distributed pressures over finite areas. The following assumptions aremade: (1) deformations are small; (2) two bodies obey the laws of linear elasticity; and (3) contact surfaces aresmooth and have continuous first derivatives. With above assumptions, contact analysis of this pair of elasticbodies can be made within the limits of the elasticity theory.In Fig. 5, contact of this pair of elastic bodies is handled as contact of many pairs of points on both sup-posed contact surfaces ofrandslike gears contact as shown in Fig. 4. These pairs of contact points arePP123makjbqnk123makjbqn0Supposedcontact faceFjFjFjFj(a) Three-dimensional view PPakjbakjbakjbakjbkkkBefore contactAfter contact(b) Section view Fig. 5. Model of a pair of elastic bodies: (a) three-dimensional view and (b) section view.6S. Li / Mechanism and Machine Theory xxx (2007) xxxxxxARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.003expressed as (110), (220), . , (mm0), (aa0), (kk0), (j-j0), (bb0), . and (nn0). n is the total number of con-tact point pairs assumed. Fig. 5b is a section view of Fig. 5a in the normal plane of the contact bodies. InFig. 5, ekis a clearance (or backlash) between a optional contact point pair (kk0) before contact. Fkis contactforce between the pair of contact points (kk0) in the direction of its common normal line when k contacts withk0under the load P (It is assumed that all the common normal lines of the contact point pairs are approxi-mately along the same direction of the external force P in this paper because a contact area is usually verynarrow. This assumption is reasonable in engineering, but we shall use the real direction of the contact pointpairs in this paper). xk, xk0are deformations of the points k and k0in the direction of the force Fkafter con-tact. d0is the initial minimum clearance betweenrandsand d is displacement of the points O1relative tothe point O2(the loading points in Fig. 5b).For the optional contact point pair (kk0), if (kk0) contacts, (xk xk0 ek), the amount of the deforma-tions and clearance on the point pair (kk0), shall be equal to the relative displacement quantity d, and if (kk0)does not contact, (xk xk0 ek) shall be greater than d. Eqs. (1) and (2) can be used to express these relation-ships in the following. Eq. (3) is used to sum Eqs. (1) and (2):xk xk0 ek? d 0Not contact1xk xk0 ek? d 0Contact2Then,xk xk0 ek? d P 0k 1;2;.;n3According to Hertzs theory, contact deformation under the external force P has the relationship with outlinesof the contact surfaces and the external force P. This means the contact deformation is determined by twofactors, geometry of the contact surfaces and the external force P. When the external force P is changed, con-tact area of a pair of elastic bodies is also changed correspond. This change of the contact area makes it a non-linearity, the relationship between contact deformation and the external force P. But since this non-linearity isonly resulted from increase and decrease in contact areas, this non-linearity is the so-called geometric non-linearity, not the so-called material non-linearity. So, for the pairs of points assumed to be in contact, rela-tionship between deformation and contact force (force on contact point pairs, not the external force P) is stilllinearity when elastic deformations are considered. Then the elastic deformations xkand xk0of the pairs ofpoints in contact can be expressed with Eq. (4) by using deformation influence coefficients akjand ak0j0,xkXnj1akjFj;xk0Xnj1ak0j0Fj4where Fjis contact force between the point pair (jj0). If Eq. (4) is substituted into Eq. (3), (5) can be obtainedand if Eq. (5) is expressed in a form of matrix expression, Eq. (6) can be obtained,Xnj1akj ak0j0Fj ek? d P 05S?fFg feg ? dfeg P f0g6whereS? Skj? akj ak0j0?fFg fF1;F2;.;Fk;.;FngTfeg fe1;e2;.;ek;.;engTfeg f1;1;.;1;.;1gTf0g f0;0;.;0;.;0gTk;j 1;2;.;n; k0;j0 10;20;.;n0S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx7ARTICLE IN PRESSPlease cite this article in press as: S. Li, Contact problem and numeric method of a planetary drive ., Mech. Mach.Theory (2007), doi:10.1016/j.mechmachtheory.2007.10.0034.2. Force equilibrium relationship of elastic contact bodiesInthispaper,itisassumedthatallthecontactforcesbetweenthecontactpointpairsarealongthedirectionoftheexternalforceP.Sinceacontactareaisusuallyverysmall,thisassumptionisreasonableinengineering.Withtheassumption,itcanbethoughtthattheexternalforcePisequaltothesumofallthecontactforceFj(j = 1ton).Then Eq. (7) can be obtained. If the Eq. (7) is written in a form of matrix expression, Eq. (8) can be obtained,P Xnj1Fj7fegTfFg P84.3. Mat