Contact problem and numeric of a planetary drive with small teeth number difference,Shuting Li.pdf

Contact problem and numeric method of a planetary drive with small teeth number diff erence 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 diff erence (PDSTD). A mechanics model and fi nite 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 fi nite 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 modifi cations, Mechanism and Machine Theory 42 (1) (2007) 88114 to solve a more complex engineering contact problem. FEM programs are devel- oped through many years eff orts. 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 off set direction of the external gear. They are located at an angular position of 20 30? away from the off set direction. Loads shared by teeth, pins and rollers have big diff erence. The maximum load shared by the teeth is much greater than the ones shared by pins and rollers. This means that strength calculations of the teeth are 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. ? 2007 Elsevier Ltd. All rights reserved. Keywords: Gear; Gear device; Planetary drive; Small teeth number diff erence; 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 diff erence (PDSTD) 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. 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.jp Available online at Mechanism and Machine Theory xxx (2007) xxxxxx Mechanism and Machine Theory ARTICLE IN PRESS 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 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 eff ective 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 Nomenclature PDSTD planetary drive with small teeth number diff erence FEM fi nite element method FEA fi nite element analysis 3Dthree-dimensional ISOInternational Standard Organization FTload on tooth surface FPload on pin FRload on roller eeccentricity of the crankshaft. Z1tooth number of the external gear Z2tooth number of the internal gear X1 shifting coeffi cient of the external gear X2 shifting coeffi cient of the internal gear mmodule of gears B1outside diameter of the internal gear B2inside diameter of the external gear B3diameter 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) 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 ekclearance (or backlash) between a optional contact point pair (kk0) before contact. Also, ej Fkcontact force between the pair of contact points (kk0) in the direction of its common normal line, also Fj xk, xk0deformations of the assumed pair of contact points (kk0) in the direction of the contact force Fk akj, ak0j0 deformation infl uence coeffi cients of the contact points d0initial minimum clearance between a pair of elastic bodies in the direction of the external force drelative 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 Yslack variables, Y = Y1, Y2, . , Yk, . , YnT Xn+1 artifi cial variables, also, Xn+1, Xn+2, Xn+n, . , Xn+n+1 Iunit matrix of n n, n is size of the unit matrix Zobjective function S matrix of the deformation infl uence coeffi cients Farray of contact force of the pairs of contact points, F = F1, F2, . , Fk, . , FnT earray of clearance of the pairs of contact points, e = e1, e2, . , ek, . , enT eunit array, e = 1, 1, .,1, .,1T 0zero array, 0 = 0, 0, .,0, .,0T rbradius of the base circle of the internal gear Pexternal force applied on a pair of elastic bodies PGsum of all contact forces between the contact points on tooth surfaces of the PDSTD Ttorque transmitted by the PDSTD a0a angle used to express the position of pairs of teeth 2S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx ARTICLE IN PRESS 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 diff erent 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 eff ective 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 eff orts. 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 off set 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 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 used to transmit torque and 22 rollers used as the center bearing. In order to let teeth of the external gear engage 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 off set 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 off set 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 diff erence between the internal spur gear and the external spur gear is small, so this device is often called the planetary drive with small teeth number diff erence. Transmission ratio of this device is equal to Z1/(Z2? Z1) when the internal gear pins rollers z1 z2 Input shaftOutput shaftInternal spur gearExternal spur gear e A A Section A-A Crankshaft (Cam)o1 o2 pin hole Fig. 1. Structure of one kind of planetary drive with small teeth number diff erence. S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx3 ARTICLE IN PRESS 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 fi xed. 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 diff erence (Z2? Z1) is very small, transmission ratio Z1/ (Z2? Z1 ) shall become very large. For the device as shown in Fig. 1, teeth number diff erence (Z2? Z1) is equal 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 checked likeausualinternalgeartransmissionwheninvoluteprofi leisused.Ofcoursethesetipandrootinterferencescan beremovedthroughperformingtoothprofi lemodifi cations,foranexampletipandrootrelieves.Alsootherpro- fi les such as modifi ed involute curve, arc profi le and trochoidal curves can be used to avoid tip and root 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 FTproduced by tooth engagement, roller loads FRproduced in center bearing and pin loads FPresulted 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 profi le of the internal gear. Roller loads are along radial directions of the center hole in the external gear. Pin loads are Table 1 Gearing parameters and structural dimensions of the PDSTD Gearing parametersGear 1Gear 2Structural dimensions Gear typeExternalInternalDiameter B180 mm Tooth numberZ1= 49Z2= 50Diameter B236 mm Shifting coeffi cientX1= 0.0X2= 1.0Diameter B341.125 mm Face width12 mm12 mmPin number8 Helical angle00Pin diameter4 Module (mm)1Roller number22 Pressure angle20?Roller diameter3 Tooth profi leInvolutes Cutter tip radius0.375 m Off set direction+Y Eccentricity, e0.971 mm Roller load X Yn Y Yk Xk Xi Tooth load Xn Yi Pin load 1 2 3 4 5 6 7 8 1 2 3 4 5 7 8 9 10 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 Pin center circle FT FR FP Fig. 2. Load state of the external gear in the planetary drive. 4S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx ARTICLE IN PRESS 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 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. 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 diff erent 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 diff erent 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 profi le, 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 diff erence 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 profi le 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 profi le. So when to perform contact analysis of loaded teeth of the PDSTD with the FEM, a lot of pairs of contact points (ii0), (jj0), (kk0) and (nn0) as shown in Fig. 4 must be made between the tooth profi les of the external and the internal gears. These pairs of contact points are assumed to be in contact at fi rst and it shall be made clear fi nally 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,randsare 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. 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. S. Li / Mechanism and Machine Theory xxx (2007) xxxxxx5 ARTICLE IN PRESS 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 Discrete forces can be taken to represent distributed pressures over fi nite 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 fi rst 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 ofrandslike gears contact as shown in Fig. 4. These pairs of contact points are P P 1 2 3 m a k j b q n k 1 23 m ak j b q n 0 Supposed contact face Fj Fj Fj Fj (a) Three-dimensional view P P a k j b a k j b a kj b a kj b k k k Before 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) xxxxxx ARTICLE IN PRESS 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 expressed 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. In Fig. 5, ekis a clearance (or backlash) between a optional contact point pair (kk0) before contact. Fkis contact force between the pair of contact points (kk0) in the direction of its common normal line when k contacts with k0under 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 very narrow. This assumption is reasonable in engineering, but we shall use the real direction of the contact point pairs 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 to the point O2(the loading points in Fig. 5b).