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by the teeth is much greater than the ones shared by pins and rollers. This means that strength calculations of the teeth are was also used widely in automation industry. Though many units of the PDSTD are made every year, strength design calculation of the PDSTD is still a remained problem that has not been solved so far. * Tel./fax: +81 059 2566213. E-mail address: shutingnpuyahoo.co.jp Available online at Mechanism and Machine Theory xxx (2007) xxxxxx Mechanism and Machine Theory ARTICLE IN PRESS 0094-114X/$ - see front matter C211 2007 Elsevier Ltd. All rights reserved. more important than the ones of pins and rollers for the PDSTD. It is also found that all pins share loads while only a part of rollers share loads. C211 2007 Elsevier Ltd. All rights reserved. Keywords: Gear; Gear device; Planetary drive; Small teeth number dierence; Contact analysis; FEM 1. Introduction In the latter period of the 20th century, with the development of industry automation, gear devices with large reduction ratio found wide applications. Planetary drives with small teeth number dierence (PDSTD) Contact problem and numeric method of a planetary drive with small teeth number dierence Shuting Li * Nabtesco Co. LTD., Oak-hills No. 202, Heki-cho 7028-2, TSU-shi, Mie-ken 514-1138, Japan Received 15 July 2007; received in revised form 2 October 2007; accepted 16 October 2007 Abstract This paper deals with a theoretical study on contact problem and numeric analysis of a planetary drive with small teeth number dierence (PDSTD). A mechanics model and finite element method (FEM) solution are presented in this paper to conduct three-dimensional (3D) contact analysis and load calculations of the PDSTD through developing concepts of the mathematical programming method T.F. Conry, A. Serireg, A mathematical programming method for design of elastic bodies in contact, Transactions of ASME, Journal of Applied Mechanics 38 (6) (1971) 387392 and finite element method S. Li, Gear contact model and loaded tooth contact analysis of a three-dimensional, thin-rimmed gear, Transactions of ASME, 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 and Machine Theory 42 (1) (2007) 88114 to solve a more complex engineering contact problem. FEM programs are devel- oped through many years eorts. Contact states of teeth, pins and bearing rollers of the PDSTD are made clear through performing contact analysis of the PDSTD with the developed FEM programs. It is found that there are only four pairs of teeth in contact for the PDSTD used as research object when it is loaded with a torque 15 kg m. It is also found that these four pairs of teeth are not located in the oset direction of the external gear. They are located at an angular position of 20 30C176 away from the oset direction. Loads shared by teeth, pins and rollers have big dierence. The maximum load shared doi:10.1016/j.mechmachtheory.2007.10.003 Please 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.2007.10.003 2 S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx ARTICLE IN PRESS Nomenclature PDSTD planetary drive with small teeth number dierence To perform strength calculations of the PDSTD, it is necessary to know the loads distributed on teeth, pins and rollers in advance. Since there has not been an eective method available to be able to perform contact analyses and load calculation of the PDSTD, gear designers have to use ISO standards 46 made for strength calculations of a pair of spur and helix gears to perform strength calculations of the PDSTD approximately FEM finite element method FEA finite element analysis 3D three-dimensional ISO International Standard Organization F T load on tooth surface F P load on pin F R load on roller e eccentricity of the crankshaft. Z 1 tooth number of the external gear Z 2 tooth number of the internal gear X 1 shifting coecient of the external gear X 2 shifting coecient of the internal gear m module of gears B 1 outside diameter of the internal gear B 2 inside diameter of the external gear B 3 diameter of the pin center circle on the external gear (ii 0 ) assumed pair of contact points, also (11 0 ), (22 0 ), ...,(mm 0 ), (aa 0 ), (kk 0 ), (jj 0 ), (bb 0 ), ... and (nn 0 ) r used to stand for one elastic body or the external gear s used to stand for the other elastic body or the internal external e k clearance (or backlash) between a optional contact point pair (kk 0 ) before contact. Also, e j F k contact force between the pair of contact points (kk 0 ) in the direction of its common normal line, also F j x k , x k 0 deformations of the assumed pair of contact points (kk 0 ) in the direction of the contact force F k a kj , a k 0 j 0 deformation influence coecients of the contact points d 0 initial minimum clearance between a pair of elastic bodies in the direction of the external force d relative 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 T Y slack variables, Y=Y 1 , Y 2 , ..., Y k , ..., Y n T X n+1 artificial variables, also, X n+1 , X n+2 , X n+n , ..., X n+n+1 I unit matrix of n n, n is size of the unit matrix Z objective function S matrix of the deformation influence coecients F array of contact force of the pairs of contact points, F=F 1 , F 2 , ..., F k , ..., F n T e array of clearance of the pairs of contact points, e=e 1 , e 2 , ..., e k , ..., e n T e unit array, e = 1, 1, ...,1,...,1 T 0 zero array, 0 = 0, 0, ...,0,...,0 T r b radius of the base circle of the internal gear P external force applied on a pair of elastic bodies P G sum of all contact forces between the contact points on tooth surfaces of the PDSTD T torque transmitted by the PDSTD a 0 a angle used to express the position of pairs of teeth Please 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.2007.10.003 7. It has been known that contact problem of the PDSTD is completely dierent from the one of a pair of spur 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 with FEM. Yang and Blanche 9 also studied design and application guidelines of cycloidal drive with machining tolerance. Shu 10 conducted study on determination of load-sharing factor of the PDSTD. Chen and Walton 11 studied optimum design of the PDSTD. This paper aims to present an eective method to solve contact analysis and load calculation problems of the PDSTD. Based on more than 20 years experiences on contact analysis of gear devices and FEM software development, a mechanics model and FEM solution are presented in this paper to conduct contact analysis and load calculations of the PDSTD. Responsive FEM programs are developed through many years eorts. Contact states of the teeth, pins and rollers of the PDSTD are made clear with the developed programs. Load distributions on teeth, pins and bearing rollers are also obtained. It is found that there are only four pairs of teeth 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 oset direction of the external gear. Loads shared by teeth, pins and rollers are compared each other. It is found that the maximum load shared by teeth is much greater than the ones shared by pins and rollers. It is also found that all pins share loads while Since the PDSTD belongs to K-H-V type of planetary drive and tooth number dierence between the S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx 3 ARTICLE IN PRESS internal spur gear and the external spur gear is small, so this device is often called the planetary drive with small teeth number dierence. Transmission ratio of this device is equal to Z 1 /(Z 2 C0 Z 1 ) when the internal gear pins rollers z1 z2 Input shaft Output shaft Internal spur gear External spur gear e A A Section A-A Crankshaft (Cam) o1 o2 pin hole only a part of rollers share loads. Strength calculations of the PDSTD can be performed easily after loads on teeth, pins and rollers are known. 2. Structure and transmission principle introductions Fig. 1 is a simple type of the PDSTD used as research object in this paper. In Fig. 1, this PDSTD consists of one internal spur gear, one external spur gear, two ball bearings, one input shaft, one output shaft, eight pins usedtotransmittorqueand22rollersusedasthecenter bearing.Inordertoletteethoftheexternalgearengage with 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 a cam that can produce oset movement for the external gear (in Fig. 1, when the crankshaft is rotated, a radial movement of the external gear is produced alternately). The crankshaft is also used as input shaft of the device. Fig. 1 is the position when oset direction of the crankshaft is right up towards to +Y direction. In Fig. 1, O 1 is the center of the external gear and O 2 is the center of the internal gear. e is the eccentricity of the crank- shaft. e = O 1 O 2 . Gearing parameters and structure parameters of this PDSTD are given in Table 1. Fig. 1. Structure of one kind of planetary drive with small teeth number dierence. Please 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.2007.10.003 is fixed. Here, Z 1 is tooth number of the external gear and Z 2 is tooth number of the internal gear. From Z 1 / (Z 2 C0 Z 1 ), it can be found that when tooth number dierence (Z 2 C0 Z 1 ) is very small, transmission ratio Z 1 / (Z 2 C0 Z 1 ) shall become very large. For the device as shown in Fig. 1, teeth number dierence (Z 2 C0 Z 1 )is equal to 1, so transmission ratio of this device = Z 1 . Sincean internal gear is used in the PDSTD, tip and root interferences with the mating gear must be checked likeausualinternalgeartransmissionwheninvoluteprofileisused.Ofcoursethesetipandrootinterferencescan beremovedthroughperformingtoothprofilemodifications,foranexampletipandrootrelieves.Alsootherpro- files such as modified involute curve, arc profile and trochoidal curves can be used to avoid tip and root Table 1 Gearing parameters and structural dimensions of the PDSTD Gearing parameters Gear 1 Gear 2 Structural dimensions Gear type External Internal Diameter B 1 80 mm Tooth number Z 1 =49 Z 2 = 50 Diameter B 2 36 mm Shifting coecient X 1 = 0.0 X 2 = 1.0 Diameter B 3 41.125 mm Face width 12 mm 12 mm Pin number 8 Helical angle 0 0 Pin diameter 4 Module (mm) 1 Roller number 22 Pressure angle 20C176 Roller diameter 3 Tooth profile Involutes Cutter tip radius 0.375 m Oset direction +Y Eccentricity, e 0.971 mm 4 S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx ARTICLE IN PRESS interferences. 3. Load analysis and face-contact model of tooth engagement of the PDSTD Fig. 2 is an image of loading state of the external gear in the PDSTD. In Fig. 2, it is found that three kinds of loads are applied on the external gear. They are tooth loads F T produced by tooth engagement, roller loads F R produced in center bearing and pin loads F P resulted from the external torque. Tooth loads are along the directions of the normal lines of the contact points on tooth surfaces of the internal gear. This also means the tooth contact loads shall be along the directions of the lines of action of the contact points on tooth profile of the internal gear. Roller loads are along radial directions of the center hole in the external gear. Pin loads are Yn Y Yk Xk Tooth load Xn Yi 2 3 4 1 2 3 4 5 7 8 9 10 6 3 4 5 6 7 8 9 FT Roller load X Xi Pin load 15 6 7 8 1 210 11 12 13 14 15 16 17 18 19 20 21 22 Pin center circle FR FP Fig. 2. Load state of the external gear in the planetary drive. Please 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.2007.10.003 S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx 5 ARTICLE IN PRESS along 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. Internal gear External gear 5 6 7 8 9 Fig. 3. Face-contact of mating teeth. Internal gear External gear i j k n i j k n Fj j Fk k Fig. 4. Pairs of contact points on tooth surfaces of the internal gear and the external gear. 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 dierent from a usual internal gear trans- mission in that tooth engagement of a usual internal gear transmission is an engagement of teeth on the geometrical contact lines and it has been already known in theory how many teeth and which teeth shall come into contact in dierent engagement positions for the usual internal gear transmission while tooth engagement of 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 the PDSTD. Even, it is not known whether the geometric contact lines exits or not for the PDSTD. The other dierence is contact state of one pair of teeth. As it has been stated above, for a usual internal gear transmission, a pair of teeth shall contact on the geometrical contact line. It is called Line-contact of a tooth 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 of the PDSTD with the parameters as shown in Table 1. From Fig. 3, it is found that the teeth 5, 6, 7, 8 and 9 are face-contact on the most part of tooth profile. So when to perform contact analysis of loaded teeth of the PDSTD with the FEM, a lot of pairs of contact points (ii 0 ), (jj 0 ), (kk 0 ) and (nn 0 ) as shown in Fig. 4 must be made between the tooth profiles of the external and the internal gears. These pairs of contact points are assumed to be in contact at first and it shall be made clear finally which pair of points turns out not to be in 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 1 4.1. Deformation compatibility relationship of a pair of elastic bodies In Fig. 5, r and s are one pair of elastic bodies which will come into contact each other when an external force P is applied. The contact problem to be discussed here is restricted to normal surface loading conditions. Please 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.2007.10.003 6 S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx ARTICLE IN PRESS P a q n Discrete forces can be taken to represent distributed pressures over finite areas. The following assumptions are made: (1) deformations are small; (2) two bodies obey the laws of linear elasticity; and (3) contact surfaces are smooth and have continuous first derivatives. With above assumptions, contact analysis of this pair of elastic bodies 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 of r and s like gears contact as shown in Fig. 4. These pairs of contact points are P 1 2 3 m k j b k 1 2 3 m a k j b q n 0 Supposed contact face F j F j F j F j (a) Three-dimensional view P P a k j b a k j b a k j b a k j b k k k Before contact After contact (b) Section view Fig. 5. Model of a pair of elastic bodies: (a) three-dimensional view and (b) section view. Please 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.2007.10.003 express tact point Fig. 5 force k 0 unde mately tact. r s the poi For tions does k k k k k k Then, linearity when elastic deformations are considered. Then the elastic deformations x k and x k 0 of the pairs of points x k a kj F j ; x 0 a 0 0F j 4 where F j is contact force between the point pair (jj ). If Eq. (4) is substituted into Eq. (3), (5) can be obtained S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx 7 ARTICLE IN PRESS and if Eq. (5) is expressed in a form of matrix expression, Eq. (6) can be obtained, X n j1 a kj a k 0 j 0F j e k C0 dP0 5 SC138fFgfegC0dfegPf0g6 where SC138S kj C138a kj a k 0 j 0C138 fFgfF 1 ;F 2 ; ...;F k ; ...;F n g T fegfe 1 ;e 2 ; ...;e k ; ...;e n g T fegf1;1; ...;1; ...;1g T T 0 0 0 0 0 Please Theory j1 k j1 k j 0 X n X n in contact can be expressed with Eq. (4) by using deformation influence coecients a kj and a k 0 j 0, x k x k 0 e k C0 dP0 k 1;2; ...;n 3 According to Hertzs theory, contact deformation under the external force P has the relationship with outlines of the contact surfaces and the external force P. This means the contact deformation is determined by two factors, 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 is only 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 still x k x k 0 e k C0 d 0 Contact 2 ships in the following. Eq. (3) is used to sum Eqs. (1) and (2): x x 0 e C0 d 0 Not contact1 d 0 is the initial minimum clearance between and and d is displacement of the points O 1 relative to nt O 2 (the loading points in Fig. 5b). the optional contact point pair (kk 0 ), if (kk 0 ) contacts, (x k x k 0 e k ), the amount of the deforma- and clearance on the point pair (kk 0 ), shall be equal to the relative displacement quantity d, and if (kk 0 ) not contact, (x x 0 e ) shall be greater than d. Eqs. (1) and (2) can be used to express these relation- narrow. This assumption is reasonable in engineering, but we shall use the real direction of the contact point pairs in this paper). x k , x k 0 are deformations of the points k and k 0 in the direction of the force F k after con- ed as (11 0 ), (22 0 ), ...,(mm 0 ), (aa 0 ), (kk 0 ), (j-j 0 ), (bb 0 ), ...and (nn 0 ). n is the total number of con- pairs assumed. Fig. 5b is a section view of Fig. 5a in the normal plane of the contact bodies. In , e k is a clearance (or backlash) between a optional contact point pair (kk 0 ) before contact. F k is contact between the pair of contact points (kk 0 ) in the direction of its common normal line when k contacts with r the load P (It is assumed that all the common normal lines of