1、02FTMS1Design and Stress Analysis of NewVersion of Novikov-WildhaberHelical Gearsby: I. Gonzalez-Perez and L. Carnevali, University ofIllinois at ChicagoTECHNICAL PAPERAmerican Gear Manufacturers AssociationDesign and Stress Analysis of New Version ofNovikov-Wildhaber Helical GearsI. Gonzalez-Perez
2、and L. Carnevali, University of Illinois at ChicagoThestatementsandopinionscontainedhereinarethoseoftheauthorandshouldnotbeconstruedasanofficialactionoropinion of the American Gear Manufacturers Association.AnewversionofNovikov-Wildhabergeardriveisconsidered. Thecontentsofthepapercoverdesign,generat
3、ion,TCA(Tooth Contact Analysis), and stress analysis of a new type of Novikov-Wildhaber helical gear drive. The greatadvantages of the developed gear drive in comparison with the previous ones are:(1) Reduction of noise and vibration caused by errors of alignment.(2) The possibility of grinding and
4、application of hardened materials.(3) Reduction of stresses.These achievements are obtained by application of:(i) newgeometry (based on application of parabolicrack-cutters),(ii) double-crowning of pinion, and (iii) parabolic type of transmission errors. The manufacture of gears is based onapplicati
5、onofgrindingorcuttingdisks,andgrindingorcuttingworms. Theadvantagesofthedevelopedgeardrivehavebeenconfirmedbysimulationofmeshingandcontact,analysisofstressesandformationofbearingcontact.Computerprograms for such analysis have been developed by the authors of the paper. Helical gears of new geometry
6、can beapplied in high-speed transmissions. The developed theory is illustrated with numerical examples.Copyright 2002American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 2002ISBN: 1-55589-812-21Design and Stress Analysis of New Version of Novikov-Wild
7、haber Helical Gearsby Ignacio Gonzalez-Perez and Luca CarnevaliGear Research CenterDepartment of Mechanical and Industrial EngineeringUniversity of Illinois at Chicago842 W. Taylor StreetChicago, Illinois, 60607-7022Tel: 312-996-2866E-mail: flitvinuic.edu1 IntroductionWildhaber 16 and Novikov 14 hav
8、e proposedhelical gears based on generation by circular arcrack-cutters. The difference between the twoinventions is that the gear tooth surfaces are in linecontact of Wildhaber gears and in point contact ofNovikov gears. Figures 1 and 2 show the first andsecond versions of Novikov gears with one an
9、d twozones of meshing, respectively. Point contact ofNovikov gears has been achieved by application oftwo mismatched rack-cutters for generation of thepinion and the gear, respectively. Reduction ofcontactstress hasbeen provideddue toapplicationof small relative curvature of tooth surfaces.The exist
10、ing design of Novikov-Wildhaber geardrives has the following disadvantages:(i) The function of transmission errors of a misa-ligned gear drive is a discontinuous linear oneand the transfer of meshing between the teethis accompanied by high acceleration and noise10.(ii) Bending stresses of Novikov ge
11、ars, especiallyof the first design (Fig. 1), are of largemagnitude.The manufacturing of Wildhaber-Novikov gears isbased by application of two mating hobs that areconjugated to the respective mismatchedrack-cutters.ConjugationofgeartoothsurfacesofNovikovgearsis achieved by running of the gears in its
12、 ownhousing and lapping. This is the reason whyNovikov gears of existing designhave beenappliedfor low-speed transmissions only, and hardenedmaterials and grinding of tooth surfaces have notbeen applied.Figure 1: Novikov gears of previous design withone zone of meshing.Novikov gears have been a subj
13、ect of intensiveresearch 2, 3, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17.The new tendencies of design of gear drives arebased now on application of a double-crownedpinion tooth surface. Crowning in profile direction2enablestolocalizethe bearingcontact. Crowninginlongitudinal direction enables to provid
14、e apredesigned parabolic function of a limited value ofmaximal transmission errors 12 and reducevibration and noise.Application of only profile crowning, but notdouble-crowning, has been applied in the initiallyproposed Novikov gears (Figs. 1 and 2).Therefore,the noise of such gears was inevitable.T
15、he contents of this paper cover a new version ofNovikov-Wildhaber gears that is based on thefollowing ideas:(1) Two mismatched parabolic rack-cutters areapplied instead of rack-cutters of circular arcprofiles. This enables to increase tooth rigidityand decrease bending stress. The reduction ofcontac
16、t stresses is obtained due to the smallmagnitude of relative curvature.(2) The tooth surface of the pinion is doublecrowned (in profile and longitudinal directions)whileinconventionalNovikov-Wildhabergearsonly profile crowning is provided. Crowning inlongitudinal direction is obtained by plunging of
17、the tool that generates the pinion. Plunging ofthe pinion tool is executed by a parabolic func-tion and enables application of a predesignedparabolic function of transmission errors.(3) Analternativemethodofobtainingofaparabol-ic function of transmission errors is based onapplication of modified rol
18、l (see section 6).(4) The generation of pinion and gear tooth sur-faces may be accomplished by a grinding diskoragrindingworminadditiontogenerationbyahob. The possibility of grinding has opened thepossibilityforapplicationofhardened toothsur-faceswithpotentialofdistortionfreesurfaces.Absorption of t
19、ransmission errors caused bymisalignment (due to existence of a proposedpredesigned parabolic function of transmissionerrors) enables reduction of noise.Figure 2: Profiles of rack-cutter for Novikov gears with two zones of meshing.3Fig. 3 shows new version of Novikov-Wildhabergears. Fig. 4(a) shows
20、the pitch cylinders (rollingcylinders) of the pinion and the gear. Plane istangent to the pitch cylinders. A skew rack-cutterfor the new design is shown in Fig. 4(b).The contents of the paper cover: (i) variousmethods for generation of pinion-gear toothsurfaces of new design; (ii) avoidance ofunderc
21、utting; (iii) stress analysis.The developed theory is illustrated with severalnumerical examples.Figure 3: New version of Novikov-Wildhabergears in 3D.2 Parabolic Rack-CuttersThe geometry of rack-cutters represented in thissection is the basis for the design of tools (grindingdisks, hobs and worms)
22、for the generation ofdiscussed helical gears.Normal and Transverse Sections. The normalsectiona-aoftherack-cutterisobtainedbyaplanethat is perpendicular to plane and whichorientation is determined by angle(Fig. 4(b).Thetransverse section of the rack-cutter is determinedas a section by a plane that h
23、as the orientation ofb-b (Fig. 4(b).Figure 4: Pitch cylinders and skew rack-cutter:(a) pitch cylinders of pinion and gear (b) toothsurface of rack-cutter.Mismatched Parabolic Rack-Cutters. It wasmentioned above that two mismatchedrack-cutters are applied for separate generation ofthe pinion and gear
24、 of the new version of helicalgears. Figure 5(a) shows the profiles of the normalsections of the rack-cutters. Figures 5(b) and 5(c)show the profiles of the pinion and gearrack-cutters, respectively. Dimensions s1and s2arerelatedbymodulemandparameterbasfollowss1+s2= m(1)b =s1s2(2)Here, b is chosen i
25、n the process of optimization,relatespinionandgeartooththicknessanditcanbe4varied to modify the relative rigidity. In aconventional case of design, we have b=1.Theprofilesoftherack-cuttersareparaboliccurvesthat are in internal tangency. Points Q and Q* (Fig.5(a) are the points of tangency of the nor
26、malprofiles of driving and coast sides of the teeth,respectively. The common normals to the profilespasses through point P that belongs to theinstantaneous axis of rotation P1P2(Fig. 4(a).Figure 5: Normal sections of pinion and gear rack-cutters: (a) mismatched profiles; (b) profiles of pinionrack-c
27、utter in coordinate systems Saand Sb; (c) profiles of gear rack-cutter in coordinate systems Seand Sk.5A parabolicprofile ofa rack-cutteris representedinparametric form in an auxiliary coordinate systemSi(xi,yi) as follows (Fig. 6):xi= ui, yi= aiu2i(3)where aiis the parabola coefficient. The origin
28、of Sicoincides with Q.Figure 6: Parabolic profiles of rack-cutter innormal section.PinionParabolicRack-Cutter.Thesurface oftherack-cutter is designated by cand is derived asfollows:(i) The mismatched profiles of pinion and gearrack-cutters are represented in Figure 5(a).The pressure angles are dfor
29、the driving pro-file andcfor the coast profile. The locations ofpoints Q and Q* are designated by|QP| = ldand |Q *P| = lcwhere ldand lcare defined asld=m1+bsindcosdcoscsind+c(4)lc=m1+bsinccosccosdsind+c(5)(ii) Coordinate systems Saand Sbare located inthe plane of the normal section of the rack-cut-t
30、er(Fig.5(b).Thenormalprofileisrepresentedin Sbby the matrix equationrbuc= Mbarauc= Mbaucacu2c01T(6)(iii) The rack-cutter surface cis represented incoordinate system Sc(Fig. 7) wherein the nor-mal profile performs translational motion alongc-c. Then we obtain that surface cis deter-mined by vector fu
31、nctionrcuc,c= Mcbcrbuc= McbcMbarauc(7)Figure 7: For derivation of pinion rack-cutter.Gear Parabolic Rack-Cutter. We applycoordinate systems Seand Sk(Fig. 5(c) andcoordinate system St. The coordinatetransformation from Skto Stis similar totransformation from Sbto Sc(Fig. 7). The gearrack-cutter surfa
32、ce is represented by the followingmatrix equationrtut,t= MtktMkereut(8)Rack-Cutters for Modified Involute HelicalGears. The idea of mismatched rack-cutters hasbeen extended by the authors for the design ofmodified involute helical gears as follows:(i) The rack-cutter for the pinion is applied as apa
33、rabolicone,buttherack-cutterforthegearisaconventionaloneandhasstraightlineprofilesin the normal section.(ii) In addition to profile crowning, the pinion iscrowned in longitudinal direction to provide aparabolic function of transmission errors (seesections 5, 6 and 7).(iii) Theprincipleofnewdesignof
34、modifiedinvolutehelical gears enables to localize the bearing6contact, avoid edge contact and reduce trans-mission errors.An example of modified involute helical gears isrepresented in section 9 for comparison ofstressesof new version of Novikov-Wildhaber helical gearsand modified involute helical g
35、ears.3 Profile Crowned Pinion and Gear ToothSurfacesThe profile crowned pinion and gear tooth surfacesare designated as and 2, respectively, wherein1indicates the pinion double-crowned one.Profilecrownedpiniontoothsurfaceisgeneratedas the envelope to the pinion rack-cutter surfacec. We consider for
36、derivation coordinate systemsScand Sthat are rigidly connected to cand .The sought for surface is determined by theequationsruc,c,= Mc() rcuc,c(9)Ncv(c)c= fcuc,c,= 0(10)Here: vector function rc(uc, c) represents pinionrack-cutter surface c;matrixMcrepresentscoordinate transformation from coordinate
37、systemScto coordinate system S; vector functionr(uc, c, ) represents the family of rack-cuttersurfaces cin coordinate system S; equation (10)is the equation of meshing.Equations (9) and (10) considered simultaneouslydetermine profile crowned pinion tooth surface .Surface is determined by three relat
38、edparameters (uc, c, ) where (uc, c)arethesurface parameters of the pinion rack-cutter c,and is the parameter of motion in meshing of cand .Similarly we derive gear surface 2generated bygear rack-cutter surface t.Pinion and gear tooth surfaces and 2are inpointtangencyintheprocessofmeshingsincetheyar
39、e profile crowned.4 TCA (Tooth Contact Analysis) ofPinion-Gear Profile Crowned ToothSurfacesThe algorithm of TCA (Tooth Contact Analysis) forsimulation of meshing provides conditions ofcontinuoustangencyofcontactingtoothsurfacesofthe pinion and the gear 9. The meshing andcontact is simulated in the
40、paper for two cases: (i)thepinionofthegeardriveisprofilecrownedand(ii)the pinion is double crowned (see sections 5, 6 and7). Comparison of the output for both cases(sections 4 and 7) shows that double-crowning ofthe pinion enables to reduce transmission errorsand noise and vibration of the gear driv
41、e. Piniondouble-crowning is favorable as well for avoidanceof edge contact.DrawingsofFig.8illustrateinstantaneoustangencyofsurfaceand2inafixedcoordinatesystemSf.The surfaces have to be represented in Sftakinginto account the errors of alignment. It is supposedthat and 2are profile crowned and theref
42、oretheyareinpointtangency.Tangencyofand2atcommon point M means that they have at M thesame position vector and the surface normals arecollinear. Then we obtain the following system ofvector equations 9:r()fuc,c,r(2)fut,t,2,2= 0(11)N()fuc,N(2)fut,2,2= 0(12)fcuc,c,= 0 (13)ft2ut,t,2= 0(14)Figure 8: Ill
43、ustration of continuous tangency ofcontacting tooth surfaces and 2.Here: fc=0,ft2= 0 are the equations of meshingofthe pinion and gear with the respective generating7rack-cutters cand t; and 2are the angles ofrotation of the profile crowned pinion and gear; 0isascalarfactorintheequationofcollinearit
44、yof surface normals.One of the parameters, say , is chosen as theinput one. The Jacobian D of the system of scalarequations obtained from equations (11) - (14) hasto differ from zero as the precondition of pointtangency of surfacesand 2. In accordance to the theorem of implicitfunction system existe
45、nce 5, observation ofinequality D 0 enables to solve the system ofequations (11) - (14) by functionsuc,c,ut,t,2,2 C1(15)Solutionofsystemofnonlinearequations(11)-(14)is an iterative computerized process based onapplication of computer programs for an iterativeprocess of Newton-Raphson rules 5.The com
46、putational procedure provides the paths ofcontact on pinion and gear tooth surfaces and thefunction of transmission errors.An example of meshing of profile crowned pinionand gear tooth surfaces has been investigated forthe following data: N1= 17, N2= 77, m = 5.08 mm,b =0.7,=20,d=c=25, ac=0.016739 mm
47、- 1,at=0.0155 mm- 1.The followingerrors ofalignmenthave been simulated: (i) change of center distanceE=70mm;(ii)error=2arcminoftheleadangle;(iii) change of shaft angle = 2 arcmin.Results of computation are as follows:(1) Thepathofcontactisorientatedindeedlongitu-dinally (Fig. 9(a) and (b).(2) Error
48、E of center distance does not causetransmission errors, but causes the shift of thebearing contact.(3) However, errors of alignment and causea discontinuous linear function of transmissionerrors2()(Fig.9(c).Therefore,thetrans-fer of meshing from one pair of teeth to theneighboring one is accompanied with high ac-celeration, and vibration and noise become in-evitable.Figure 9: Output of TCA in a profile crowned geardrive: (a) path of contact with error E; (b) path ofcontact with error ; (c) functions oftransmission errors with error .Noise an
