1、92 FTM 10The Influence of the Kinematical Motion Erroron the Loaded Transmission Error of SpiralBevel Gearsby: Claude Gosselin, Louis Cloutier and Q.D. NguyenLaval UniversityAmerican Gear Manufacturers AssociationTECHNICAL PAPERThe Influence of the Kinematical Motion Error on theLoaded Transmission
2、Error of Spiral bevel gearsClaude Gosselin, Louis Cloutier and Q. D. NguyenLaval UniversityThestatementsand opinionscontainedherein arethose of theauthor andshouldnotbe construedasan official action oropinion of the American GearManufacturers Association.ABSTRACT:The optimisation of the kinematicalb
3、ehaviour of spiral bevel gearshas been the focus of muchattentionover the pastyears. The objectives, aimingto optimizethe kinematics of spiral bevel gears underno load, being zeroor close to zeromotion error coupled to a parabolic motion error curvehave been fulfilled by many authors 4,6, 7. However
4、, theseobjectivesbypass the more general caseof minimizingmotion error at a giventransmitted load, which is thedesign goalin aerospace applications where transmissionsoperate over extendedperiods at 80 to 90% of available power. Krenzer6 showed that changing torque significantly modified the shape o
5、f the motion error curve, and that minimum motionerror under load could be achieved only over short torque ranges.This paper presents thebasis of a Loaded ToothContactAnalysis (LTCA) program predictingthe motionerror of spiralbevel gear sets underload and showssome of the influencesof the unloadedmo
6、tionerror curve shape and amplitude onthe kinematical behaviour under load. The effects of tooth composite deflection (bending and shearing), tooth contactdeformation and initial profile separation due to motion error are considered in the development. Due to the complexshape of spiral bevel gear te
7、eth, the tooth bending stiffness is calculated by finite elements. Classical Hertz theory isused to calculate the contact deformation. Numerical examplesarepresented to illustratethe behaviour of motion errorunder load as the unloaded motionerror is modified. Results show that under general circumst
8、ances,low contact ratiogears withparabolic motion error undernoloadmay produceundesirablekinematics underload, while parabolicmotionerror high contact ratio gears are likelyto produce acceptablekinematics under a variety of loads. It is also shown thatthe governing factors, in loaded motion error,ar
9、e the contactratio, thus combined mesh stiffness,and the amplitude ofunloaded motion error curve which is linked to load sharingbetween adjacent tooth pairs.Copyright 1992American Gear Manufacturers Association1500King Street, Suite 201Alexandria, Virginia, 22314October, 1992ISBN: 1-55589-590-5THE I
10、NFLUENCE OF THE KINEMATICAL MOTION ERROR ON THELOADED TRANSMISSION ERROR OF SPIRAL BEVEL GEARSClaude Gosselin, ProfessorLouis Cloutier, ProfessorQ.D. Nguyen, Graduate StudentDepartment of Mechanical EngineeringLaval University, Quebec, QC, Canada, GIK-7P4i. INTRODUCTION Evaluating the load share bet
11、ween meshing teeth inspiral bevel gears is a complex process when moreTo achieve maximum life for a spiral bevel gear than two tooth pairs are simultaneously in contactset, low vibration levels and appropriate bearing and an iterative numerical solution must be used.pattern location and dimensions m
12、ust be present. The This paper presents a general approach to calculatekinematical behaviour of a gear pair (bearing how load is shared between meshing teeth in spiralpattern, line of action and motion error) is bevel gear pairs, taking into account tooth compositetherefore an important consideratio
13、n at the design deflection (bending, shearing), tooth contactstage. The size of the bearing pattern and the deformation and initial profile separation due tolocation of the line of action (LOA) both influence profile modification.the contact deformation and bending displacement ofa gear tooth while
14、kinematical motion errordetermines whether unloaded motion will transfer 2. LIST OF SYMBOLSsmoothly between adjacent tooth pairs 5. However,as teeth bend under load motion error changes and the _-i, _0, _+i load share vectors for tooth pairspossibility of excessive motion error and contact -i, 0 and
15、 +I respectivelyentry interference under heavy loads arises, which AS-I, AS0, AS+I initial tooth pair profilemay generate vibrations and noise 10. The analysis separation vectorsof motion error of gears under load is therefore a _H-1, 8H0, _H+I contact deformation vectorsnecessary step in the design
16、 of a gear train. (Hertz)_c-z, _c0, _=+z radius of contact vectorsOne of the recent advances in gear train design is _-1, _0, _+1 torque vectorsthe increase in the number of simultaneous tooth _p total applied torque vectorpairs in contact: if two to three tooth pairs, as in _-11, _00, =K+ll stiffne
17、ss vectors at contacthigh .contact ratio gears (HCR), are in simultaneous pointscontact instead of one to two pairs, as in low K-11x,y,z stiffness vector x, y and zcontact ratio gears (LCR), the overall mesh stiffness components at contact pointscan be increased and the load share carried by each _-
18、ii, _00, _+11 compliance vectorstooth pair can be significantly reduced. Although _pij,_sij primary and secondary displacementactual load share values can differ substantially, it vectorsis possible under certain conditions to increase the Coo, Cl0, C01, C11 weighting coefficientsload carrying capac
19、ity of a gearset without _c, _c, _c contact point reference frameincreasing its weight 13. HCR gears can be _#3 motion error without loadobtained in several ways: I) by decreasing the _3L motion error under loadpressure angle, 2) by increasing the addendum, 3) by m_ speed ratiousing teeth of finer p
20、itch and 4) by increasing biason the LOA of spiral bevel gears.3. GENERAL ASSUMPTIONS where (subscript t indicates that the tangentialdirection is considered, e.g. normal to the radius atIn order to somewhat simplify the complexity of the contact point - see figure 2):the problem, the following gene
21、ral assumptions aremade: Fappthere is neither friction in the contact zone, norpresence of lubricant that can affect the A S_1 A S 0 A S +idistributionof contact pressure andcreate anoil _ _ _film of varying thickness between the meshingteeth; -1 0 +1displacements due to tooth deflection, shearing o
22、rcontact deformation are sufficiently small that F-IT FO _ F+I_the tooth surface normal at the contact point isnot affected; K-lp K0p K+Ipa TCA computer program, described in referencesi, 2, 3, is used to obtain the kinematicalcharacteristics of the analysed gear sets, such as Ktooth surface coordin
23、ates, contact points along -ig Kog K+1the line of action, bearing pattern, and contactstreeses; _ _ _the contact is of hertzian type, and there are noboundary corrections made for edge contact since, Tooth -I Tooth 0 Tooth +1in most cases, under the applied torques themotion error curves remain of p
24、arabolic shape and Figure I: Load Share Componentsthe loads at contact entry are minimal;at this point, tooth coupling effects areneglected, e.g. the movement of one tooth due toa loadappliedon an adjacenttooth,althoughsucheffects may significantly alter load sharing andare under consideration.4. PI
25、NION BLANK ROTATION COMPATIBILITY CONDITIONSpiral bevel gear teeth are theoretically point pointscontact surfaces. In practice however, because of Direction 0_I._ 1_._ _the low relative longitudinal curvature between the _ire_on_pinion and gear tooth flanks, the instant contactpoint is close to a li
26、ne, which we will call theinstan line of contact (LOC). Under the action of .of actionan applied load, the gear teeth in contact displace _l_ _ Iby bending and shearing and their surfaces deform atthe contact point to form an elongated ellipse. 0The load sharingcompatibilityconditionis based _onon t
27、he fact that 6, II, 12: /i) the pinion and gear blank rotation due to thedisplacement of any tooth under load must bethe same for all simultaneously meshing toothpairs; this includes tooth bending deflectionand shearing displacement, contact deformationand profile separation due to profilemodificati
28、on or motion error (figure i); Figure 2: Tooth Numbering Conventionii) the sum of the torque contribution of eachmeshing tooth pair must equal the total appliedtorque.Equations (i) and (2) state these conditions fora HCR gear set with three tooth pairs in contact:_-I +nS_1+_ll _-5-+AS + _Hl =K+n -t(
29、i)_-n, _oo, _-I_+11: is the combined stiffness vector,at a contact point, for tooth_a_ _ _-1 + _0 + _+1 (2) pairs -i-I, 00 and +i+irespectively; in figure 2, toothpair -I-I is going out of mesh, where compliance Ci is the inverse of stiffness K_.tooth pair 00 is the main tooth Equations (9) and (i0)
30、 are rearranged as:pair in centact, and tooth pair (AS0 . AS_I) + F0 Co+i+iiscomingintomesh; F_I = (ii)_S_I,AS0, AS+I : is the initial separation vector C-Ibetween tooth pairs -I-i, 00 and+I+I respectively, due to profile F+I = (AS “ AS+I)+ F Co (12)modification or motion error; the C+Iinitial separ
31、ation is obtainedfrom the motion error curve, andis in the tangential direction Combined with equation (8), equations (ii) and (12)(figure2); yield:_H_I, 5H0, _H+I : is the contact deformation vector (AS0 _ AS_I) + (Fapp _ F_I _ F+I) Coof tooth pairs i-i, 00 and +I+I F_I = (13)respectively; contact
32、is assumed C-Ihertzian and is calculated as perreferences 8, 14; (AS0_ AS+I) + (Fapp _ F_I _ F+I)Co_-I,_0,_+i : is the individualloadvector for F+I = (14)each tooth pair; C+I_c-l, _c0, _c+l : is the contact radius vector onthe pinion tooth; Extracting F_1 from equations (13) and (14) and_-i, _0, _+i
33、 : is the individual vectorial subtracting gives:contribution to total appliedtorque; C_ 1 (AS0 _ AS_I ) + Co (AS_ 1 _ AS+I ) + Fapp Co C_ 1_app : is the total applied torque vector. F+I =C+I C-z + C+I Co + CO C-IThe individual torque vectors are defined by the (15)following cross products: (AS0 _ A
34、S_I) + (Fapp _ F+I) COF_I = (16)_-I = _-i X _0-1 (3) C-1 + C0_0 = _o x _c0 (4) Fo = Fa_ - F+I - F-I (17)Equations (15) to (17) can now be used to closelySince contact deformations do not vary linearly estimate the initial load shares of each tooth pair,with the applied load, equation (i) cannot be s
35、olved and help speeding up convergence and preventdirectly and a Newton-Raphson based iterative process numerical divergence.is used. To speed up convergence and to preventnumerical divergence, close initial load share valuescan be obtained by modifying equation (i) where only 5. BENDING AND SHEARIN
36、G STIFFNESS VECTORSbending deflection and initial separation vectorsbetween meshing tooth pairs are considered: Combined stiffness vectors _-II, _00 and _K+II arecalculated from the individual pinion and gear toothI_-i + AS_I _0_+A_S 0 _+I +AS+I seriesStiffnesscomponents which are combined as spring
37、sin(figureI):II_o-11 II_oo II_o+_rl l = _l + 1 (ze)(6) K-n x,y,z K-lp x,y K-I_ x.y,z1 1 1Assuming that the modules of contact radii vectors = - + (19)_o-l, _c0 and _c+l are close, equations (6) and (2) can K00 x,y,z K0p x,y,z K0s x,y.zbe further simplified to:i i I_ - _ + (20)_-i+AS-I = + As0 = + as
38、+l r_n_,y._ K+i_,y._ K+IBx,y,z_-ll t _00 _ -_K+n (7)where Kip x.y,z and Kis x,y,z are the individual pinionand and gear tooth bending stiffness components, given inthe pinion fixed reference frame XlX2XS (figure 3),_app = _-i + _o + _+i (8) defined by:where the _i are the applied normal loads on the
39、 Ki z.y,z = Fi x.y,z (21)teeth. To further simplify the solution, equation _i x.y,z(7) is split in two parts and rewritten in scalarform as: and where Fi x,y,z are the load components and _i x,y,zare the bending deflection components in the X1, XzF.i C_1 + AS_I = F0 CO + AS0 (9) and X3 directions re
40、spectively (figure 3).F+i C+i + AS+i - F0 CO + ASo (i0)v 3Contact Xc Tooth _p11 _slz .- _slnNormal _ Neutral X _s21 _s22 “ _s2nii “XXS _ = BiasFbgure 6: Geometry I (5,038 Bias) Figure 8: Geometry 3 (4,998 Bias)tooth -I barely touches the leftmost end of themotion error curve of tooth +I, which indic
41、ates a. profile contact ratio close to 2. The loaded motionerror curves 6_3L of geometries I and 2 show aflattening of the unloaded motion error d_3 fromgeometry 1 to geometry 2, with peak to valley loadedmotion error decreasing from 9“ to 4“, which is inline with the decrease in unloaded motion err
42、or 653;this trend is reversed between geometries 2 and 3where peak-to-valley loaded motion error increasesfrom 4“ to 5“ although the unloaded motion error atthe transfer point (_3) actually decreased from 7“to 3,5“.Figure 9: Geometry 4 (2,499 Bias)_ransfer Points_o- /. I / _i_, _6“!2 .-2 “/12o- -z d
43、 +_- _3L-160-Motion-_-200 _50 _.0 15.0 35.0 55.0 75.0Figure i0: Geometry 5 (21500 Bias) 83)Figure Ii: Geometry 1 (5,038 Bias)A 150 in-lbs torque was applied to geometries i to 83(.) 7“5, which produced maximum contact pressures of 156 0- 8+3_ ,=_.%f._i_7-17-f -000 psi on geometry 1 where the contact
44、 area major _ / “ k“ _,/_/_ “_ _ _T 28,axis is 0,140“, and maximum contact deformation of ., -_.=-0,00025“ on geometries i and 4 which present the _0- :Ilargest longitudinal pinion to gear tooth relative - _ 4“curvatures due to pinion and gear cutter radius -80- difference (table i). =Figures II to
45、13 present the unloaded and loaded -1204 8_3L -1 O +1motion error curves, _3 and 6_3L, for geometries 1 to3 respectively. In each graph, three consecutive -160-motion error curves are presented, depicting motionover three tooth engagements. Motion proceeds fromleft to right, first on tooth -i, then
46、proceeding -200 onto tooth 0 at the first transfer point (figure II), -250 _.0 15.0 35.0 55.0 75.0and finally onto tooth +I at the next transfer point. 03()The horizontal axis is angle 03, the pinion rotation Figure 12: Geometry 2 (5,012 Bias)angle in degrees, and the vertical axis is _3, thegear mo
47、tion error in arc-seconds. Unloaded motion 8_3_) 8_3 3,5“transfers from one tooth to the next at transferpoints, e.g. where motion error curves intersect. _ “.“:i_J._“In all cases, the unloaded motion error curve is -40- . “-I“of parabolic shape, and the gear is lagging the _ 5“pinion. The compariso
48、n of the unloaded motion error -80-curves (_3) on figures ii to 13 shows that as thepinion to gear tooth longitudinal relative curvature - 83Lis reduced by an increase in pinion cutter radius, -120- -i O +1motion error at the transfer point decreases from 20“in geometry I to 3,5“ in geometry 3 as bi
49、as is kept -180-constant at - 5 o and while the overall amplitude of Concavitythe motion error curve drops from 76“ (geometry i,figure ii) to 13“ (geometry 3, figure 13). In each -200 case, the rightmost end of the motion error curve of -25.0 _0 15.0 35.0 55.0 75.0e3(-)Figure 13; Geometry 3 (4,998 * Bias)7in geometry 2, a fact again improved in geometry 3B3_ B_ _ _ _“ “_“_10“ (figure 18). While an improvement in loaded motionerror _L may be seen between geometry i and
