1、97FTM15Design, Generation, Stress Analysis andTest of Low-Noise, Increased StrengthFace-Milled Spiral Bevel Gearsby: F.L. Litvin and A.G. Wang, University of Illinois at Chicago,R.F. Handschuh and D.G. Lewicki, NASA Lewis Research Center,Z. Henry, Bell Helicopter TextronTECHNICAL PAPER-=-aReproduced
2、 By GLOBAL- ._ .ENGINEERING DOCUMENTS With The Permission Of AGMA-=;:- Under Royalty AgreementDesign, Generation, Stress Analysis and Test of Low-Noise,Increased Strength Face-Milled Spiral Bevel GearsF.L. Litvin and A.G. Wang, University of Illinois at Chicago, R.F. Handschuhand D.G. Lewicki, NASA
3、Lewis Research Center, Z. Henry, Bell HelicopterTextronThe statements and opinions contained herein are those ofthe author and should not be construed as an official action oropinion of the American Gear Manufacturers Association.AbstractA modified geometry of face-milled spiral bevel gears with uni
4、form and tapered teeth that provides a localized bearingcontact, reduces the level ofnoise and increases the strength ofthe teeth has been developed. The modified geometry isbased on application of specially developed machine-tool settings and the generation of the gears is accomplished byapplicatio
5、n of the commercially available equipment and tools.The main ideas proposed are the followings:(1) The optimization of the geometry is based on the local synthesis of the gear tooth surfaces that provides: (i) alocalized bearing contact in the longitudinal and across the surface directions, (ii) a r
6、educed level oftransmission errorsof a parabolic type, and (iii) the desired magnitude of the major axis of the instantaneous contact ellipse. Theoptimization is achieved by the mismatch of the surfaces of the generating tools.(2) Theincrease in tooth strength is due to the inherent increase in the
7、total tooth contact ratio produced by the optimizedgeometry. The stress analysis is based on solid modeling of the tooth and finite element analysis.(3) The theory developed was tested by a Tooth Contact Analysis (TCA) computer program. Prototypes of the gearswith tapered teeth were manufactured and
8、 vibration, noise and stresses have been measured experimentally. Incomparison with the existing design, the level of noise was reduced by up to 18 decibels, at the spiral bevel meshingfrequencies, and the vibration levels were reduced by 50%. The total stress level for the gear set was reduced enou
9、gh topermit an increase in the maximum operating torque level.Copyright 1997American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314November, 1997ISBN: 1-55589-709-6DESIGN, GENERATION, STRESS ANALYSIS AND TEST OFLOW-NOISE INCREASED STRENGTHFACE-MILLED SPIRAL BEVE
10、L GEARSF. L. Litvin and A. G. WangDepartment of Mechanical EngineeringUniversity of Illinois at ChicagoChicago, IllinoisR. F. Handschuh and D. G. Lewickius Army Research LaboratoryNASA Lewis Research CenterCleveland, Ohioz. HenryBell Helicopter TextronABSTRACTA modified geometry offace-milled spiral
11、 bevel gears withuniform and tapered teeth that provides a localized bearingcontact, reduces the level of noise and increases the strengthof the teeth has been developed. The modified geometry isbased on application of specially developed machine-tool3ettings and the generation of the gears is accom
12、plishedby application of the commerically available equipment andtools.The main ideas proposed are the followings:(1) The optimization of the geometry is based on thelocal synthesis of the gear tooth surfaces that provides: (i)a localized bearing contact in the longitudinal and acrossthe surface dir
13、ections, (ii) a reduced level of transmissionerrors of a parabolic type, and (iii) the desired magnitudeof the major axis of the instantaneous contact ellipse. Theoptimization is achieved by the mismatch of the surfaces ofthe generating tools.(2) The increase in tooth strength is due to the inherent
14、increase in the total tooth contact ratio produced by theoptimized geometry. The stress analysis is based on solidmodelling of the tooth and finite element analysis.(3) The theory developed was tested by a Tooth Con tact Analysis (TCA) computer program. Prototypes of thegears with tapered teeth were
15、 manufactured and vibration,noise and stresses have been measured experimentally. Incomparison with the existing design, the level of noise wasreduced by up to 18 decibels, at the spiral bevel meshingfrequencies, and the vibration levels were reduced by 50%.The total stress level for the gear set wa
16、s reduced enough topermit an increase in the maximum operating torque level.1 INTRODUCTIONThe worst defects of gear drives are: (i) vibration andnoise, and (ii) the shift of the bearing contact to the edgeaccompanied in some cases with the edge contact of geartooth surfaces. Sources of vibration are
17、 the transmissionerrors caused by gear misalignment and deflection of gearshafts due to variation of load. Gear misalignment affectsthe conditions of transfer of meshing when one pair of teethis changed for another one. The transfer of meshing is ac companied by piecewise almost linear functions and
18、 causevibrations at the gear meshing frequency determined by1 =;.(fig. 1).The main goals of this paper are: (i) to change the shapeof function of transmission errors as shown in fig. 2 andreduce the level of transmission errors; (ii) to localize andstabilize the bearing contact and avoid the edge co
19、ntact ofgear tooth surfaces.A more favorable shape of function of transmission isbased on application of a predesigned parabolic functionof transmission errors that is able to absorb the almost lin ear functions of transmission errors caused by misalignment9.The localization of bearing contact requi
20、res the crowningof surfaces to substitute an instantaneous line of tangencyby point tangency. However, if crowning of surfaces is notaccompanied with the transformation of the shape of trans mission errors (fig. 2) and reduction of the magnitude ofsuch errors, vibration and noise will not be reduced
21、.A popular way of crowning of surfaces is based on appli cation of two imaginary generating surfaces Epand Egthatsatisfy the following conditions:(i) The generating surfaces Epand Egare rigidly con nected to each other and they have a common line of tan gency.(ii) The pinion and the gear tooth surfa
22、ces E1and E2are generated separately: E1is generated by Epand E2( a )( b )FIGURE 1: Transmission function and transmission errorsof a misaligned gear driveIdeal lransmission funclion( a )( b )FIGURE 2: Modification of transmission errors by applica tion of a parabolic functionby Eg The analysis of m
23、eshing of generated surfaces Eand E2shows: (a) the bearing contact is localized, (b) anideal transmission function is provided only for aligned geardri,s;misalignffitnt causes unfavorable transn:ission errorsshown in fig. 1. Therefore, such a method for modificationof gear tooth surfaces should not
24、be used for high speedgear drives for avoidance of vibrations. Examples of thedescribed method for generation are as shown in figs. 3-5.The imaginary surfaces of rack-cutters applied for genera tion of double circular arc Novikov-Wildhaber helical gears(fig. 3) have two common straight lines (lines
25、of tangency)since two zones of meshing are provided. Fig. 4 showsthe schematic of rack-cutters that might be applied for gen eration of modified involute helical gears. The couple ofgenerating surfaces has a common straight line, the line oftangency. In both cases (shown in figs. 3 and 4) the bearin
26、gcontact is localized since the generated surfaces are in pointtangency at every instant, but the shape of transmissionerrors is unfavorable. Fig. 5 shows the generating surfacesthat might be applied for the localization of bearing contactof spiral bevel gears with uniform teeth. In the first case(f
27、ig. 5a) the generating surfaces are two cones and theyare in tangency along a common generatrix of the cones.The path of contact is directed across the gear tooth sur face. In the second case (fig. 5(b) one of the generatinl!surfaces is a cone, and the other one is a surface of rev olution. The comm
28、on line of the generating surfaces is acircle, and the path of contact of the generated surfaces isdirected in the longitudinal direction, along the surface. Inboth cases shown in fig. 5, the transmission function is anideal linear function only if the gear drive does not havemisalignment. The misal
29、ignment of the gear drive causestransmission errors of such a shape (fig. 6) that will causehigh vibrations.In this paper, generation of spiral bevel gears with twogenerating surfaces that are in point tangency (fig. 7) isproposed. The generating surfaces do not have a commonline of tangency as show
30、n in fig. 5. The surfaces are mis matched and the magnitude of the mismatch is a powerfultool used for synthesis of the generated pinion-gear toothsurfaces. Fig. 7(a) shows that the generating surfaces aretwo cones. A cone and a surface of revolution may be ap plied as shown in fig. 7(b). A generati
31、ng cone is formed bystraight line blades of the head-cutter. A surface of revolu tion is formed when curved line blades of the head-cutter areapplied. In the case of generation of. spiral bevel gears withtapered teeth, the generating cones interfere each other inthe neighborhood of point M (fig. 7(a
32、) since the axes ofthe cones are not collinear. However, it will not cause inter ference of the generated surfaces due to the propercontroof the mismatch of the generating surfaces (by the properchoose of radii Rpand Rgshown in fig. 7(a).PlINUal eo g.1If uv-;“11.(a)(a,DolSR;“1“10.11R;“(bI Rack-cutte
33、r E,(bl“(cI Rack-cutter Ep (e)FIGURE 3: Schematic of rack-cutters applied for genera tion of double-circular arc Novikov-Wildhaber helical gearsFIGURE 4: Schematic of rack-cutters applied for genera tion of modified helical involute gearsThe contents of the paper covers:(i) Generation of pinion-gear
34、 tooth surfaces by generationof spiral bevel gears with tapered and uniform teeth.(ii) Synthesis of pinion-gear surfaces that provide im proved conditions of meshing achieved by: (a) applicationof a predesigned parabolic function of transmission errors;(b) observation of the desired magnitude of the
35、 major axisof the instantaneous contact ellipse; (c) observation of theprescribed direction of the path of contact (direction of thebearing contact).(iii) Development of TeA computer program for the sim ationof meshing and contact.(iv) Determination of pinion tooth stresses by applicationof finite e
36、lement method.(v) Detection and avoidance of interference of the filletsurface of one of the gear with the addendum surface of themating gear.The developed theory is illustrated with numerical exam ples.2 SYSTHESIS AND GENERATION FOR SPIRALBEVEL GEARS WITH TAPERED TEETHLocal SynthesisThe main goals
37、of the proposed method of synthesis are:(i) reduction of gear noise, (ii) localization and stabilizationof the bearing contact, and (iii) reduction of stresses in thefillet. While performing the synthesis of the gear tooth sur faces, we consider that the tooth surfaces will be generatedas follows:(a
38、) Both gear tooth sides are generated simultaneouslyby straight-line blades of the head-cutter (see Appendix 1).The machine-tool settings and the design parameters of thegear head-cutter are known (such data, for instance, maybe adopted from the summary calculation information).(b) Each pinion tooth
39、 side is generated separately bystraight-line or curved blades (see Appendix 2). The pro file angle of the pinion blade is considered as given. Thepinion head-cutter settings (including the cutter point ra dius of the pinion) must be determined by application ofthe proposed synthesis method that wil
40、l guarantee that thepinion-gear tooth surfaces are in point contact, the toothsurfaces are in tangency of the second order at the chosenmean contact point (designated as M), and the followingrequirements at M are observed: the chosen direction ofthe tangent to the path of contact, the magnitude of m
41、ajoraxis of the instantaneous contact ellipse, and the derivativeof the predesigned parabolic function of transmission errors.The method developed for local synthesis is based on theFIGURE 6: Transmission errors for a misaligned spiralbevel gear drive with uniform teeth(“Y= 3 arc. min.)Surfaces E1an
42、d E2are therefore in tangency and mesh ing at M and within the neighborhood of M. Equation (1)permits the determination of angleO)for orientation ofcoordinate system S2 with respect to St at the initial po sition of S2. Coordinate system Sl coincides initially withSt.To provide the second order tang
43、ency of E1and E2, theprincipal cmvatures and directions of E1and E2at point Mare related, taking into account the following requirements:(a) Smface E1and E2must be in point contact at pointM, (b) the conditions of meshing and contact at point Mmust provide: (i) the prescribed direction of the tangen
44、t tothe contact path, (ii) the prescribed magnitude of the majoraxis of the instantaneous contact ellipse, and (iii) the mag nitude of the derivative of the parabolic function of trans mission errors. Then at point M the locally determined27r-11,-50 -40 -30 -20 -10 0 10 20 30 40 50(Deg) P,321o = lEA
45、l The magnitude and sign of . can bedetermined by equation (4). Interference occurs if . . LOW-liSe design willi Toprem7550Transmission input torque, % of maximumOL-“-“_2516542 3Frequency, kHz100090 I80CD 70“t:I-CD600Q.“t:I50c:II0enI: t0FIGURE 24: Noise spectrum, tests at 100% speed andtorqueFIGURE
46、26: Vibration test results; input bevel housinp;accelerometer60L-:15025 50 75 100 125Transmission input torque, % of maximum110100Q.CD 90“t:II 80“t:Ic:I70o 8aseline desagno High-s1nlngth designl:. Low-lise design with Toprem0 Baseline design150 0 HigIHtrengtII designl:.Low-noiSe design with Toprem10
47、0iii,0.5,(1T/NIl(n)-8 -8Length 2a of major axis of contact ellipse at At (mm)12.0 12.0Radius R, of surface of revolution (mm)1200.Output ParametersCutter point radius R. (mm)92.927 93.356Radial setting 5. (mm)96.971 92.438Installment angle ql541545“ -535453“Machine center to backXD,(mm)-5.43 I -7.46
48、 ISliding baseXB,(mm)0.712 ; 1.292Blank offsetEm,(mm)4.176 -0.188Ratio of cutting mI.3.159 3.017iCenter of surface of revolution.XC;1(mm)- -996.040Center of surface of revolution.Zc;,(mm)- 503.204,TABLE 4: Bending Stresses at the fillet of the pinion fromFEAInput DataGear Ratioi19/62Applied Pinion T
49、orqueI686.77 Nm (6,076 in-IbfJModule!4.33 mmMaterialIAlS11060Yield Stressi1,828 Mpa (265.000 psi)Output DataMagnitude of radius of blade fillet I Stresses m Mpa (in psi) atI Toe I Center I Heel1.04 mm (0.04“) ! 697. (101.050) I 1070. (155.128) I 877. (127.14712.08 mm (0.08“) I 640. (92.787) I 963. (139.615) I 694. 1100.616)
