1、11FTM19AGMA Technical PaperConvoloid GearingTechnology - TheShape Of The FutureBy B.E. Berlinger Jr., andJ.R. Colbourne, GenesisPartners, LPConvoloid Gearing Technology - The Shape Of The FutureBernard E. Berlinger Jr., and Dr. John R. Colbourne, Genesis Partners, LPThe statements and opinions conta
2、ined herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractSincethe1694inventionoftheinvolutecurvebytheFrenchscientist,PhilippedelaHire,andtheapplicationthereoftogearingbytheprolificSwissmathematicianLeonardE
3、uler,theworldhasembracedanddevelopedthis type of gear tooth form to a very high degree of engineering and manufacturing excellence.Improvements in recent years have been relatively modest, since this form has been so rigorously studiedandapplied. Thelong-termadoptionoftheinvoluteisrootedinlargepartt
4、othesimplicityofitstoolsandfieldoperation. Straightsidedtoolsandconjugacy,evenwithlimitedchangesincenterdistance,wereconsistentwith the industrial revolution of the 18th,19th, and 20thcenturies, and the mechanically based machine toolsand tolerance capabilities of these ages. The recent ubiquitous n
5、ature of computers and CNC machineryenhances the cost-effective freedom to optimize many parameters affecting our everyday life including geartooth forms.Convoloidisanewgeartoothformcapableofincreasingtorques20%to35%overthoseofconventionallydesignedinvolutepairs.Theformiscomputeroptimized,iscompatib
6、lewiththeworldsexistingcapitalassetinfrastructure, and mirrors the manufacturing sequences, processes and basic production costs of involutegears. The result is a major enhancement in gear drive system power density and cost reduction for a givenpower requirement. Convoloid gearing is totally scalab
7、le and is used in parallel axis helical, planetary, andother configurations.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-018-63 11FTM19Convoloid Gearing Technology - The Shape Of The FutureBernard E. Berl
8、inger Jr., and Dr. John R. Colbourne, Genesis Partners, LPIntroductionSincethe1694inventionof theinvolutecurveby theFrenchscientist, PhilippedelaHire, andtheapplicationthereoftogearingbytheprolificSwissmathematicianLeonardEuler,theworldhasembracedanddevelopedthis type of gear tooth form to a very hi
9、gh degree of engineering and manufacturing excellence. Improve-mentsinrecentyearshavebeenrelativelymodest,sincethisformhasbeensorigorouslystudiedandapplied.The long-term adoption of the involute is rooted in large part to the simplicity of its tools and field operation.Straight sided tools and conju
10、gacy, even with limited changes in center distance, were consistent with theindustrialrevolutionofthe18th,19th,and20thcenturies,andthemechanicallybasedmachinetools andtoler-ance capabilities of these ages. The recent ubiquitous nature of computers and CNC machinery enhancesthe cost-effective freedom
11、 to optimize many parameters affecting our everyday life including gear toothforms.Convoloidis anewgear toothform capableof increasingtorques 20% to35% over those of conventionallydesignedinvolutepairs. Theformis computeroptimized, iscompatiblewiththeworldsexistingcapitalassetinfrastructure, and mir
12、rors the manufacturing sequences, processes and basic production costs of involutegears. The result is amajor enhancement ingear drivesystem power density andcost reductionfor agivenpower requirement. Convoloid gearing is totally scalable and is used in parallel axis helical, planetary, andother con
13、figurations. See Figure 1.Primary objectives of the power transmission industryThe major quest of power transmission system designers and manufacturers today is twofold:S Increasing power (torque) density, i.e., maximum power through the smallest possible enclosedvolume.S Maintaining lowest cost per
14、 unit torque or power.Figure 1. 12 tooth, 10.4 module (2.44 NDP) Convoloid pinion4 11FTM19These two parameters are heavily interdependent for general economic viability.Convoloid gearing technology can provide these advantages. Convoloid gear tooth forms are computeroptimized and characterized by co
15、ncave/convex meshing surfaces. Involute tooth forms are, in contrast,convex/convex resulting in surface stresses on involute forms considerably higher than those of Convoloidforms. Theconcavetoothform intheConvoloidgear dedendum(lower partof thetooth) isconducivetohightooth beam strength which adds
16、to the Convoloid advantages.S Increases torque density from 20% to 35% over equivalent involute forms.S Is totally scalable for gearing from a few millimeters to several meters in diameter.S Is compatible with todays capital asset infrastructure so that the existing world gearing facilities canmanuf
17、acture and inspect this gearing.S Reflects the same manufacturing sequences and processes as now commonly practiced.S Is easily retrofitted intoexisting housings or other support structures by matchinggear center distances,ratios, and face widths.S Has manufacturing cost structures closely approxima
18、ting those of involute gears.S Can be designed into various configurations of parallel axis helicals including planetaries and otherepicyclics.Economic implicationsThereis asignificant potentialimprovement inthe cost and performanceof gear drive systems in many gearmarkets through the use of Convolo
19、id technology. Primary advantages of this new gearing concept are:RetrofittingA balanced gear drive design retrofitted to a given involute pairs center distance, with the same face width,sameratio, material, andheat treatment, yet canrateat 20-35%higher torqueandpower ratingat thesameorlowerstressle
20、velsinbothbendingandsurfacedurability. Here,assumingadequatebearingdesign,hous-ingstrength, etc., thegear system canbeupratedwithinacost indexat, orclosetothat benchmarkedbytheinvolute pair.Higher gear pair ratiosSinceConvoloidgeardesignsarenotdependentonthebasecircletheory,butarecraftedfromtheConvo
21、loidparameters free from involute-related constraints, smaller numbers of teeth in pinions can be designedwithout undercut to provide single pair ratios considerably higher than that commonly found with involutedesigns, and yet, are considered “balanced” viz-a-viz bending strength and surface durabi
22、lity stress levels.Result - potentially fewer gear pairs in a multistage transmission of given overall ratio, with the associatedreduction in gears themselves, bearings, housing machining, enclosed volume, etc.Process conversionLower stress characteristics of Convoloidgearingcanincreasethe opportuni
23、ty for process conversion inthemanufactureof thegears themselves. Dependingonstress levels, life, andother parameters, wrought steelgears can now be substituted with powdered metal, ground gears with hobbed gears, and other lower costconversions. Result - lower cost, higher productivity for a given
24、gear system of specified performancecapability.General power density increasesConvoloid designs that are specifically calculated to achieve a given performance level can be up to 20%smaller in enclosed volume compared to their involute stress level “equivalent.” Result - lower costs, lessenclosed vo
25、lume, less weight, and - lower cost.Design of Convoloid gearingThe design protocols of Convoloid gearing incorporate the following precepts which are crafted to maximizethe load carrying capacity per unit volume (maximum power density) of gear drive systems.5 11FTM19S Conjugacy. The Euler Savary equ
26、ation for conjugacy must be met.S The preponderance of the tooth meshing surfaces are conformal, i.e., convex/concave to lower Hertzstresses as far as possible.S Nearly constant relative radii of curvatures for essentially constant Hertz stresses up the entire toothflank.S Optimized relative curvatu
27、re ratios for lowest possible Hertz stresses.S Carefullycraftedconcavededendumandrootfilletforbesttoothbeamstrengthcharacteristics. ThisfactisespeciallyimportantsinceConvoloidpairsrunningatconsiderablylowersurfacestressescomparedtoinvolutes can run at increased torques requiring higher beam strength
28、 to affect a balanced design.S Inaddition, manufacturingandinspectionrequirements of the toothform must fit wellinto todays capitalasset infrastructure. The combination of Convoloid forms and the plethora of CNC machinery in usetoday permits this new technology to be efficiently and economically man
29、ufactured and inspected.Special considerationsFirst a caveat. The more constraints that are placed on the design of Convoloid pairs the more difficult itbecomes to affect the highest load carrying Convoloid design. Although it is made relatively easy to designConvoloidformsthroughcomputerprograms,th
30、ebasicdifferencesinconstructionofConvoloidformsrequireas much freedom as possible to gain the most economic advantage.No base circleConvoloid forms have no base circle and are therefore not subject to undercutting phenomena. A smoothcraftedconcavededendum androot fillet maximizes toothbeam strengthc
31、haracteristics evenwithlownum-bers of teeth.Tip reliefConvoloid tip relief protocols are especially tailored to the unique characteristics of the form. These aremodeled after AGMA standards but the higher beam strength characteristics of Convoloid forms create astiffer tooth with less deflection und
32、er load compared to involutes. This fact reduces the amount of tip reliefrequired of Convoloids under unit loads experienced with involutes.Contact ratiosProfilecontactratiosofConvoloidpairsvarywithdesignbutaregenerallyinthe1.1to1.3range. Facecontactratiopracticeuses integers of facecontact ratios w
33、ithasmalloveragetoallowfor crowningeffects andotheranomalies, e.g., 1.05, 2.05, 3.05, etc.Center distance tolerance practiceUsingmodernCNCmachinetools center distancetolerances adequatefor mountingConvoloidpairs canbeproduced. Stiffness of bearings, housings andgear shafts must alsobeconsidered. Usi
34、ngFEA studies andangular transmission analysis a general rule is shown in Table 1.Rack offsetsRackoffsetvariationsarenotusedwithConvoloidgearing. Convoloidgearsaredesignedtonominalspecific-ations for both pinion and gear notwithstanding the design being a speed decreasing or speed increasingdrive.Ta
35、ble 1. CD tolerance0.005 X transverse m oduleTransverse module Transverse DP CD tolerance +/-, mm2.0 12.7 0.014.0 6.35 0.026.0 4.23 0.036 11FTM19Graphical depiction of Convoloid vs involute characteristicsFundamental but advantageous differences in the Convoloid tooth form compared to the classical
36、involuterequire different design approaches in order to optimize the power density and cost of Convoloid Gear Sys-tems. These differences are not complicated but emphasize the basic stress advantages of the Convoloidform.Figure 2compares allowabletransmissibletorquevalues as afunctionof thenumber of
37、 pinionteethof apairof gears - one involute and the other Convoloid - with the common data shown in Table 2.Table 2. Common data parametersDescription Data parameterCenter distance 300 mmFace width 75 mmRatio 3.0to1Helix angle for face contact ratio 1.01Stress cycle factor For 108cycles: ZN= 0.87900
38、8,YN= 0.928346Dynamic factor 1.1Load distribution factor 1.2Factored surface stress allowable 1363.6 MPaFactored fillet stress allowable 448.1 MPaFigure 2. Computer generated tooth profile7 11FTM19Figure 3shows that thebest theoretical number of teethfor theinvolute pair is about 25 whereit is 13 fo
39、r theConvoloid. Thebasicreasonforthisadvantageisthat Convoloidgearshavenobasecircle. Assuch, designconcerns centering on undercut are unnecessary. The conformal tooth forms are designed at every point inmesh to maintain:S nearly constant relative curvature;S optimized relative curvature values;S con
40、jugacy.These advantages must be utilized in the design procedure to obtain the best power density and cost of agivenConvoloiddesign. ItisgenerallyNOTbesttosimplyreplaceaninvolutesystemwithitsdirectConvoloidreplacement using approximately the same numbers of teeth in the system. Doing so will show li
41、ttle gain inlowerstressnumbersorasmallerenclosedvolumeat stressnumbersequaltothoseprevailingintheinvolutesystem.There are generally two alternatives for gear system power density/cost improvements when redesigningusing Convoloid technology:1. Highersinglepairgearreduction ratioswhenusingthesamecente
42、rdistance,facewidths,materials,heattreatment,etc.,withthishigherratiogenerallybeingabletobeenclosedinthesamevolumeasitsinvolutecounterpart with stress levels at or less than those of its involute counterpart.2. A significant reduction in enclosed volume while maintaining ratio, stress levels, torque
43、s and speedscompared to the involute system the Convoloid gears replace.Convoloid torque advantage parametersAgeneraltreatmentofthevariablesthataffectthetorqueadvantagesofConvoloidgearingoverinvolutegear-ingisshowninFigure 4. Oneofthemajorparameterstoconsideristheratiooffacewidthtocenterdistanceofth
44、e Convoloid pair. There is an area where little or no advantage exists for Convoloid over involute designs.This area generally is at gear face width to center distance ratios of 0.2 and lower. General tendencies are:S Higher gear ratio increases TRat same F/CrS Higher helix angle (i.e., 20 to 30) in
45、creases TRFigure 3. Allowable transmissible torque values8 11FTM19Figure 4. Convoloid torque advantage chartS Lower helix angle (i.e., 30 for either Convoloid or Involute, reduce helix angle to lower value and increase totalnumber of teeth in the pair accordingly- ForConvoloidpairs,thelowesttotalnum
46、berofteethinthepairshouldbeusedconsistentwithacceptableresulting helix angles- Derating factors for both cases:Load distribution factor 1.2Dynamic factor 1.1L1life factor for 108cycles- Torque of each pair is increased until first limiting parameter is reached - bending stress or surfacedurability s
47、tress- AGMA rating practice used for Involutes with conventional Hertz Theory and test rig developed stressconcentration factors for the Convoloids.9 11FTM19RatingThe rating protocols and practice governing Convoloid gearing mirror the large majority of the practices fol-lowed by AGMA, ISO, and othe
48、rs.Bending stressThe bending stress is calculated initially using cantilever beam theory and then modified using a stress con-centration factor of 1.5 through FEA. Substantiation of this value was confirmed with test data.Surface durabilityContact stresses are calculated using conventional Hertz the
49、ory.Derating factorsDeratingfactors for Convoloidgearingareappliedusingacceptedandestablishedcriteria. Dynamicfactors,lifefactors, stress concentrationfactors, andloaddistributionfactors areappliedinthesamemanner aswithinvolute practice.Materials and heat treatmentAllowablestresslimitsforvariousmaterialsandtheirheattreatmentareapplieddirectlytoConvoloiddesignsas they would be for involutes.Flash temperature and scuffing analysisItisinthisareathattheanalysisofConvoloidpairsisdifferentfromthatofinvolutes. TheConvoloidformmustbeanalyzedusingitsownparticul
