1、06FTM15Optimal Tooth Modifications in Spiral BevelGears Introduced by Machine ToolSetting Variationby: V. Simon, Budapest University of Technology and EconomicsTECHNICAL PAPERAmerican Gear Manufacturers AssociationOptimal Tooth Modifications in Spiral Bevel GearsIntroduced by Machine Tool Setting Va
2、riationVilmos Simon, Budapest University of Technology and EconomicsThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractA method for the determination of optimal tooth
3、modifications in spiral bevel gears based on improved loaddistribution, minimized tooth root stresses and reduced transmission errors is presented. The modificationsareintroducedintothepiniontoothsurfacebythevariationofmachinetoolsettingsinpiniontoothprocessing.The applied load distribution and tran
4、smission error calculations are based on a new idea: As the toothsurface modifications are relatively small and the conjugation of the mating surfaces is relatively good, it isassumed that the point contact under load spreads over a surface along the whole or part of the “potential”contactline,which
5、lineismadeupofthepointsofthematingteethsurfacesinwhichtheseparationsofthesesurfacesareminimal,insteadofassuminganellipticalcontactpattern.Thismethodincludesthebendingandshearing deflections of gear teeth, local contact deformations of mating surfaces, gear body bending andtorsion,deflectionsofthesup
6、portingshafts,andthemanufacturingandalignmenterrorsofmatingmembers.A computer program implements the method.By using the computer program that was developed the influence of machine tool settings for pinion toothmanufacture on load distribution, stresses and transmission errors is investigated. On t
7、he basis of theobtainedresultstheoptimalmachinetoolsettingsaredeterminedintroducingtheoptimaltoothmodifications.By applying this optimal set of machine tool setting parameters the maximum tooth contact pressure isreduced by 5.42%,the tooth fillet stresses in the pinion by 8.07% and the angular posit
8、ion error of the drivengearby48.43%,inregardtothespiralbevelgearpairmanufacturedbymachinetoolsettingsdeterminedbythe commonly used method.Copyright 2006American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2006ISBN: 1-55589-897-11Optimal Tooth Mod
9、ifications in Spiral Bevel GearsIntroduced by Machine Tool Setting VariationVilmos Simon, Budapest University of Technology and EconomicsIntroductionDuring the last decades manyresearch workshavebeen directed towards the synthesis, tooth contactanalysis and manufacture of spiral bevel gears.Handschu
10、h1givesaverygoodreviewofprogressfor the analysis of spiral bevel gears. Local synthe-sis of spiral bevel gears with localized bearing con-tact and predesigned parabolic function of a con-trolled level for transmission errors is proposed byLitvinandZhang2.InthepaperpublishedbyArgy-ris et al. 3 a comp
11、uterized method of local synthe-sis and simulation of meshing of spiral bevel gearswith pinion tooth surface generated by applyingmodifiedrollispresented.ByHustonandCoy4ananalysis of the surface geometry of spiral bevelgears formed by a circular cutter with involute,straight and hyperbolic profile i
12、s presented. Kawa-sakietal.5dealwithspiralbevelgearsinKlingeln-berg cyclo-palloid system. The paper contains thedesign method, calculation of paths of contact andtransmission errors and the investigation of influ-ence of assembly errors on meshing characteris-tics. Based on grinding mechanism and ma
13、chine-tool settings for the Gleason modified roll hypoidgrinder, a mathematical model for the toothgeome-try of spiral bevel and hypoid gears is developed byLin and Tsay 6. The computer algorithm, present-ed by Gosselin et al. 7, uses the sensitivity of the“errorsurface”toselectedmachinesettingchang
14、esto calculate new machine settings to match atheoretical tooth surface to measurement data,within specified tolerances. In the paper publishedby Lin et al. 8 the sensitivities of tooth surface duetothevariationsofmachinesettingsisinvestigated.Thecorrectivemachine-toolsettings,calculatedbyusingthese
15、nsitivitymatrixandthelinearregressionmethod, are used to minimize the tooth-surfacedeviations. By Stadtfeld 9 the use of a six-axisfree-form machine for processing spiral bevelgears is discussed. A numerical method for thetooth contact analysis of uniform tooth height epi-cyclical spiral bevel gears
16、 stemming from the Klin-gelnbergs cyclo-palloid system is proposed byLelkes et al. 10. Longitudinal settings of contactpatterns or contact across the surfaces from toothroot to tooth top were obtained as a function of ma-chine-settingsand theinfluence ofeach cuttingpa-rameterwasisolatedand discussed
17、.Gosselin etal.11proposedanalgorithmwhichallowstheuseofadifferent cutter, either in diameter and/or pressureangle,toobtainthesametoothflanksurface.Signif-icant cost reductions may be obtained with the ap-plication of the method. In Ref. 12, published byLitvin et al., different cutter profiles were u
18、sed tointroduce the optimal tooth modifications in order toreduce the noise and vibration levels and to in-crease endurance.Papers13-19 dealwith loadedtooth contactanal-ysisandstressanalysisinspiralbevelgears.Wilcox13 in his paper outlines the general theory for cal-culating stresses in bevel and hy
19、poid gears usingflexibility matrix method in combination with the fi-nite element method. The loaded tooth contactanalysis predicting the motion error of spiral bevelgear sets, by applying influence matrices, is pre-sented by Gosselin et al 14. Handschuh and Bibel15 analytically and experimentally r
20、olled throughmesh a spiral bevel gearset to investigate the toothbending stress by finite element method. Paperpublished by Falah et al. 16 summarizes the ex-perimental and numerical results of the meshing ofspiral bevel gears underload, fromwhich theactualcontact ratio is evaluated. Linke et al. 17
21、 calculatetheloaddistributionbasedona methodfor usingin-fluence coefficients and also calculate the corre-sponding tooth root and contact stresses. ByFuentes et al. 18 the FEM was used for stressanalysis in spiral bevel gears. Fang and Wei 19consider the edge contact in loaded tooth contactanalysis.
22、Conjugated spiral bevel gears are theoretically inline contact. In order to decrease the sensitivity ofthe gear pair to errors in tooth surfaces and to themutual position of the mating members, carefullychosenmodificationsareusuallyintroducedintotheteeth of one or both members. As a result of thesem
23、odifications, the spiral bevel gear pair becomes“mismatched,” and a point contact of the meshed2teeth surfaces appears instead of line contact. Inpractice these modifications are usually introducedby applying the appropriate machine tool setting forpinion and gear manufacture. It is usually assumedt
24、hatthetheoreticalpointcontactunderloadspreadsover an elliptical area. In this papera newapproachfor the computerized simulation of load distributionin mismatched spiral bevel gears with point contactispresented.Themainfeatureofthismethodisthatthe tooth deflections of the pinion and the gear arecalcu
25、lated by FEM, and the tooth contact is treatedin a special way instead of using the usuallyappliedtheoryofellipticalcontactarea.Themainideais:Asthe tooth surface modifications are relatively smalland the conjugation of the mating surfaces is rela-tively good, thus the point contact under loadspreads
26、overasurfacealongthewholeorpartofthe”potential” contact line, which line is made up of thepointsofthematingtoothsurfacesinwhichthesep-arationsofthesesurfacesalongthetoothfacewidthare minimal. The load distribution calculationsmade for hypoid 20 and worm gears 21 haveshownthatthisapproachgivesamorere
27、alisticcon-tact pattern and contact pressure than the ellipticalone.The method is implemented by a computer pro-gram. By using this program the influence of ma-chine tool settings for pinion tooth manufacture onload distribution, stresses and transmission errorsis investigated. On the basis of the o
28、btained resultsthe optimal machine tool settings are determinedintroducing the optimal tooth modifications.Theoretical BackgroundAGleasontypespiralbevelgearpairwiththegener-ated pinion and gear is treated. The pinion is thedriving member. The convex side of the gear toothand the mating concave side
29、of the pinion tooth arethe drive sides. The modifications are introducedinto the pinion teeth, therefore, only the generationof the pinion tooth surface will be presented in thispaper. The processing of gear teeth is described inRef. 22.Generation of the Pinion Tooth SurfaceThe machine tool setting
30、used for the generation ofpinion teeth is given in Fig. 1.y10ym111xxm1xT101f1k1O11OT1m1yy10yT1zT1rtm1zMM(c)bfz10gcpT1Om1O10O10O1OrT110Oc(1)z10z1x10x10x11uf1T1zepm1OCutter T1pFigure 1. Machine tool setting for pinion tooth surface finishing.3The surface of the tool used for the generation ofpinion te
31、eth is in the coordinate system KT1(at-tached to tool T1) defined by the following equation(based on Fig. 1)r(T1)T1= u(rT1+ utg1) cos(rT1+ utg1) sin1(1)The generation of the pinion tooth surface is de-scribed mathematically by the equationv(T1,1)m1e(T1)m1= 0(2)wherev(T1,1)m1= relative velocity vecto
32、r of tool T1to thepinione(T1)m1= unit normal vector of the tool surfaceOnthebasisofFig.1andEq.(1),fortherelativeve-locityvectorandfortheunitnormalvectorofthetoolsurface, it followsv(T1,1)m1= (c)igpz(T1)m1+ gcos1z(T1)m1 igpz(T1)m1+ gsin1igpy(T1)m1sin1x(T1)m1 ccos1 y(T1)m1(3)d(T1)m1= M1pe(T1)T1= M1psi
33、n1cos1coscos1sin0(4)where=10 0 00sincp- cos cpepcoscp0coscpsincp- epsincp00 0 1r (T1)T1r(T1)m1= M1pr(T1)T1(5)MatrixM1pprovidesthe coordinatetransformationsfrom the movable coordinate system KT1(rigidlyconnectedtothecradleandhead-cutterT1)intothefixed coordinate system Km1rigidly connected tothe cutt
34、ing machine.TheconcavesideofpinionteethisinthecoordinatesystemK1(attachedtothepinion,Fig.1)definedbythe following system of equationsv(T1,1)m1e(T1)m1= 0r(1)1= M3pM2pM1pr(T1)T1(6)where matrix M2pdescribes the installment of thepinionatthecuttingmachineandprovidesthecoor-dinatetransformationfrom syste
35、m Km1intothesta-tionary system K10rigidly connected to Km1;matrixM3pdefines the transition from the stationary sys-tem K10into the system K1rigidly connected to thebeing generated pinion. Stated mathematically(based on Fig. 1)r10= M2prm1=cos1- sin 10-ccos1sin1cos10f csin1001g0001rm1(7)r1= M3pr10= co
36、s10sin1001 0 -p- sin10cos100001r10(8)while 1= igpcp cp0Tooth Contact of Mismatched Spiral BevelGearsIn this paper a modified spiral bevel gear pair withgeneratedpinionandgearistreatedwhosegeome-tryisfullydescribedin22.Astheresultofthemodi-ficationsintroducedintothepinionteeth,theoreticalpoint contac
37、t appears between the mating toothsurfaces. In this contact point the common normalvector of the pinion and gear tooth surfaces existsand thegeneration equationsof thepinion andgeartoothsurfacesaresatisfied22,23.Therefore,foraparticular position of the pinion and the gear, theinstantaneous contact p
38、oint is defined by thefollowing system of equationsv(T2,2)m2e(T2)m2= 0r(1)02= r(2)02e(1)02= e(2)02v(T1,1)m1e(T1)m1= 0(9)where r(1)02and r(2)02are the position vectors of toothsurface points, e(1)02and e(2)02are the unit normal vec-torsinthesamesurfacepointsforthepinionandthe4gear, respectively, e(T1
39、)m1is the unit normal vector ofthe tool surface T1for pinion tooth generation, e(T2)m2is the unit normal vector of the tool surface T2forgear tooth generation, v(T1,1)m1is the relative velocityvectoroftoolT1tothepinionand v(T2,2)m2istherelativevelocity vector of tool T2to the gear.The system of vect
40、orial Eqs. (9) is equivalent tosevenscalar equations. Thesolution of this systemof equations, for a prescribed value of the angularposition of the pinion, 1, gives the instantaneouscontact point of the mating tooth surfaces and theactual angular position of the gear 22.Because of the mismatch of the
41、 gear pair, only inone point of the path of contact, called as the initialcontact point, the basic mating equation of the con-tacting tooth surfaces is satisfied, producing thecorrectvelocityratiobasedonthenumbersofteeth.Usually, the middle point of the gear tooth flank ischosen for the initial cont
42、act point. The machine-tool setting is always determined due to this initialcontact point.The assumption in this paper is made that the pointcontactunderloadspreadsoverasurfacealongthewhole or part of the “potential” contact line made upof the points of the mating tooth surfaces in whichthe separati
43、ons of these surfaces along the toothfacewidthareminimal.Theseparationsaredefinedas the distances of the corresponding surfacepointsthataretheintersection-pointsofthestraightline parallel to the common surface normal in theinstantaneous contact point, with the pinion andgear tooth surfaces. Mathemat
44、ically it means theminimization of the functions =xc(2)02 xc(1)022+yc(2)02 yc(1)022+zc(2)02 zc(1)022(10)where rc(1)02and rc(2)02are the position vectors of thecorresponding points on the pinion and gear toothsurfaces. The method for thedetermination ofmini-mal separations and the corresponding “pote
45、ntial”contact lines is fully described in Ref. 22.Transmission ErrorsThetotaltransmissionerrorconsistsofthekinemat-ical transmission error due to the mismatch of thegear pair and eventual tooth errors and misalign-ments of the meshing members, and of the trans-mission error caused by the deflection
46、of teeth.It is assumed that the pinion is the driving memberand that is rotating at a constant velocity. As the re-sult of the mismatch of gears, a changing angularvelocity ratio of the gear pair and an angular dis-placementofthegearmemberfrom thetheoretical-ly exact position based on the ratio of t
47、he numbersof teeth occur. This angular displacement of thegear can be expressed as(k)2= 2 20 N1(1 10)N2+ 2s(11)where10and20aretheinitialangularpositionsofthe pinion and the gear, 2is the instantaneous an-gular position of the gear calculated by the systemofEqs.(9)foraparticularangularpositionofthepi
48、n-ion, 1;N1and N2are the numbers of pinion andgearteeth,respectively,and2sistheangulardis-placement of the gear due to edge contact in thecase of misalignments of the mating memberswhen a “negative” separation occurs on a toothpairdifferent from the tooth pair for which the angularposition by Eq. (9
49、) is calculated 22.The angular displacement of the gear, 2(d),caused by the variation of the compliance of con-tacting pinion and gear teeth rolling through mesh,will be determined in the load distributioncalculation.Therefore, the total angular position error of thegear is defined by the equation2= (k)2+ (d)2(12)Load DistributionThe load di
