1、11FTM09AGMA Technical PaperStandardization of LoadDistribution Evaluation:Uniform Definition ofKHfor Helical GearsBy K. Nazifi, Zollern DorstenerAntriebstechnik GmbH however,thesesuggestionsdiffertoalargedegree. ThestandardisationoftheloaddistributionevaluationandauniformdefinitionofKHforhelicalgear
2、senableasaferdesignforthemanufacturersandaneasiercomparabilityoftheresultsforthecustomers.ThepapercomparesthedifferentsuggestionstotheKHdefinitionandwillderiveasuitabledefinitionforthecalculation methods in DIN 3990 and ISO 6336.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax St
3、reet, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-008-73 11FTM09Standardization of Load Distribution Evaluation: Uniform Definition of KHforHelical GearsDr.-Ing. Khashayar Nazifi, Zollern Dorstener Antriebstechnik GmbH ZHis zone factor;ZEis elasticity factor, N/mm2;Zis helix angl
4、e factor;Zis contact ratio factor;Ftis transverse tangential load at reference cylinder per mesh, N;d1is reference diameter of pinion, mm;b is face width, mm;u is gear ratio.Theapproachinthecalculationstandards is always thesame; thedrivingtorqueis calculatedtoaloadatthereference or operating cylind
5、er and is converted by geometrical, material and influence factors to a nominalcontact stress at the reference or operating circle as to be seen in equation 1.In an ideal stiff environment andgears withno deviations, the loaddistribution alongthe facewidth wouldbeuniform, see Figure 1 light upper bo
6、x. However in the real world the load dependant deformations and flankform deviations duetomanufacturingresult inanunevenloaddistribution, seeFigure 1dark lowerbox. Thisuneven load distribution leads to high local loads and therefore to high local contact stresses which canexceed the material streng
7、th limit.4 11FTM09Figure 1. Load distribution on a spur gear for an ideal stiff gear and under realisticcircumstancesInequation2, theseinfluences arereflectedwithK factors and theconversion of the contact stress from thereferenceor operatingcircletothepointsB orDisdonethere,too. This resultsinatwodi
8、mensionalproblemwhichhastobesolvedbyaonedimensionalformula. ForreflectionoftheunevennessoftheloaddistributionthefaceloaddistributionfactorKHisusedasinISO6336-1:20084,MethodCandDIN3990-13MethodBthat are virtually the same. ISO 6336-1:2008, Method B, uses the procedure of AGMA 927-A01 2, whichuses the
9、 factor KH. Both factors are similar defined as the maximum load per unit face width divided by themean load per unit face width as one can see in equations 3 and 4.(2)H1,2= ZBDH0KAKvKHKH(3)KH=FbbmaxFmb(4)KH=FgnpeakFgnavewhereH1,2is contact stress, N/mm2;ZBDis single pair tooth contact factors for t
10、he pinion, for the wheel;KAis application factor;Kvis dynamic factor;KHis face load factor (contact stress);KHis transverse load factor (contact stress).Fbis nominal transverse load in plane of action, N;Fmis mean transverse tangential load at the reference or operating circle, N;KHis load distribut
11、ion factor;Fgis total load in the plane of action, N;n is number of discrete points on the face width.Motivation and objectiveWith the usage of helical gears the imprecise nature of the definition emerges. The line of contact is nothorizontaloverthefacewidthanymore;moreoveritisinclinedandrunsdiagona
12、lacrosstheflank,asshownin5 11FTM09Figure 2. Thewholeproblem has changedfrom atwodimensionalintoathree dimensionalproblem andthedefinitions in equations 3 and 4 are not really clear. The new situation leads to a lot of new questions: Is theloadper unit facewidthdefinedfor oneinstantaneous lineof cont
13、act?Whichof theinfinitemeshingpositionsandthereforeresultinginstantaneous lines of contact betweenthestart ofactiveprofile(SAP) andtheendoftheactiveprofile(EAP)havetobetakenintoaccount?Howtohandletheaxialoverlapsandvariablelengthofthe contact line between SAP and EAP? Actually ISO 6336, DIN3990 andA
14、GMA 927-A01use asimplifica-tioninorder togivethedesigners ahandy tool. Thecalculation of the loaddistribution is made withconstantmeshstiffness inevery meshingpositionandsotheinfluenceof thevaryinginstantaneous lines of contact isnot apparent anymore.Manydefinitionshavebeenmadewithintheyears. Eachof
15、 thesesuggestions hasits ownperspectivetothisspecific problem depending on the available metrological technology and calculation possibilities.State of the artDescriptions of load distribution measurements for helical gears were given in the 1970s 6. There weresuggestionstomeasurestraingaugesignalsf
16、romrootfilletsandtocomparethemaximumvalueofthepeakswith the mean value. Signals from root fillets are shown inFigure 3. It is also describedthat dueto thehelixanglethepeaks willappear sequentialby at. However, thereflectionofvaryinginstantaneouslines ofcon-tact and the non proportionality of root fi
17、llet stresses to flank loads was not done. Nevertheless, it is still theeasiest way to determine a load distribution without calculation of the flank loads and it is used very often.Figure 2. Load distribution on a helical gearFigure 3. Root fillet strain gauge signals in a helical gear6 11FTM09Wink
18、elmann 10 described numerical methods to calculate resulting loads on teeth of a helical planetarygearbox duetodeviations anddynamics. AKHvaluewas definedfor eachinstantaneous contactline. Thecalculated maximum of the specific load was compared to the maximum specific load of the ideal, i.e.,deviati
19、on and dynamic free, teeth. Equation 5 shows this relation.(5)KH= maxFbb maxFbb max 0whereIndex 0 is ideal and dynamic free.WiththeintentiontodetermineavaluewhichwasusableinthecalculationsaccordingtotheDIN3990, awaytochoosetherightinstantaneous contactlinewasshowninhis work10. Hedecidedthat theinsta
20、ntaneouscontactlinewiththemaximumofthefactorsKHandKHvwasrightone. KHandKHvdescribetheinfluenceofloadsharinganddynamicstothecontactstressonthegears. FurthermoreKHforthiscontactlinehadtobedivided by the squared helix angle factor Zdefined in DIN 3990 to achieve the KHused in DIN 3990(equation 6).(6)KH
21、=KHmaxKHKHvZ2whereKHis load sharing factor calculated by contact stress comparison;KHvis dynamic factor calculated by contact stress comparison.ThedefinitionoftheloadanddynamicfactorsandofthehelixanglefactorZaccordingtoDIN3990areshownin equations 7 and 8.(7)KHKHv= max Hi(n 0)max H02Hiis max value of
22、 the contact stress on the contact line, N/mm2;H0is(8)Z= cos ,N/mm2; is helix angle, degrees.Newresearchresults9haveledtoanewdefinitionofthehelixanglefactorZintheISO6336. Actuallyithaschanged the value to its opposite as described in equation 9.(9)Z=1cosThis changeof definitionmeans that comparedtot
23、heoldresults, whichimpliedaworseloaddistributionwithincreasing helix angle, the new results just state the opposite.WithhisexactmodelPlaczek7couldcalculateloaddistributionsonthewholeflankforanymeshingposition.Havingtheinformationoftheloadonevery pointof theflank hemadeasuggestiontoreplace KHwithoneo
24、fthefactors describedintheequations 10and11. Inorder toderivethese, additionalfactors(seeequation12and13) havetobedeterminedbefore. ThefactorkFcomparesthemaximumspecific loadontheflank tothespecificloadatthebasecircle. Thefactorkpcomparesthemaximumcontactstressontheflanktothecontactstress at thepitc
25、hcircle. Nowtheequivalentfactors toKHcomparesthefactorscalculatedoncefor anidealgear andoncefor agear withdeviations. Ontheonehand, if thedeviations leadtoincreasingloads thenthe7 11FTM09factor rises aboveone. Ontheother handthe factor can get smaller thanone whenthe deviations decreasetheload. Thes
26、edecreasingdeviations canbeappliedtotheflanksonobviousby meansof leadmodificationfor example.(10)KF=kF deviationkFideal(11)Kp=kp deviationkp ideal(12)kF=FmaxbiFbtb(13)kp=pmaxpc2kFis ratio of maximum force and reference value;kPis ratio of maximum contact stress and pC;Fmaxis maximum value of load di
27、stribution, N;biis distance between calculation nodes, mm;Fbtis nominal transverse load (base tangent plane), N;pmaxis maximum value of local contact stress, N/mm2;pCis contact stress at pitch diameter, N/mm2.Placzekdidnotderiveanequivalenttoaloaddistribution;infacthewascomparingtheeffectof deviatio
28、nsonmaximum loads on a flank.Wikidal 11 also stated in his work, that for helical gears the differentiation in a transverse load distributionfactor KHand a face load distribution factor KHis impossible. Just like Placzek, he suggested to replaceboth factors in the load carrying capacity calculation
29、by one factor. His proposal was KH1/2.(14)KHa,12= pHeq 12pH0eq 122KHais summary of KHand KH;pHeqis equivalent contact stress of a deviation afflicted gear pair, N/mm2;pH0eqis equivalent contact stress of an ideal gear pair, N/mm2;The proposal in equation 14 compares damage equivalent contact stresse
30、s of ideal gears and gears withdeviations. Thedecisionforaninstantaneouscontactlineissolvedbydivisionoftheinfinitemeshingpositioninto a discrete number of positions. For each position the maximum damage equivalent contact stress iscalculatedfor theidealgear. Ineachpositionthemaximum valueof thewhole
31、contactlineispickedout. Inasecondstepthemaximumvalueofallthesecontactstresses ischosenasthereferencevalue. For thismax-imumvaluearecalculationofthecontactstressismadeconsideringmodificationsanddeviations. Theresult-ingcontactstressis thecomparedonetothereferencevalue. Thisreflectionmakesit possiblet
32、oincludethepositive effect of modifications within this factor, because with the right modification the damage equivalentcontact stresses can be lower than the contact stress of idealgears. This canlead toa factor which is lowerthan one.Meeusen 5 defined KH3Das a three dimensional value, which shall
33、 cover face and transverse loaddistributioninonevalue. Inordertoachieveathreedimensionalloaddistributiontheloadinonesectionoftheface width is averaged in height direction, as shown in equation 15. As a next step the maximum value iscompared to a mean value of the averages (equation 16).8 11FTM09(15)
34、Fbb3D=nni=nfbb3Di(16)KH3D=Fbb3D maxFbb3D meanwhere(Fb/b)3Dis tooth force at position along tooth width, N/mm;n is number of calculation point along path of contact;is contact ratio;fbis local tooth force on tooth flank, N;KH 3Dis face load factor for contact stress for a 3D load distribution;(Fb/b)3
35、Dmaxis maximum tooth force along tooth width, N/mm;(Fb/b)3Dmeanis mean tooth force along tooth width, N/mm.All these calculations lead to a reflection of the flank at every position. Furthermore they show how moderncalculationmethods canoptimizegears andtheloaddistributionoverthewholeflank. Neverthe
36、less, aslongas there is no local calculation method established for the local ratingof thepitting resistanceof gears theseresults can be used for an internal design review but not for calculations according to the establishedstandards and not for certifications.ApproachThispapersuggeststogetbacktoth
37、erootsoftheloaddistributionfactoroverthefacewidthandtoprovidetheonedimensionalcalculationmethodwiththeinputneeded. Thesuggestionistocomparethelocalloadsoverthe face width with each other, even though these loads will not act on one instantaneous line of contact forhelical gears. The idea is to have
38、an exact value of the load at the criticalpoints accordingto thecalculationmethods derived from the torque applied. As described before these points with the worst combination ofcontact stress and negative sliding are point B for the driver and point D for the driven gear.Theapproachis tohavetheload
39、distributiononthegear flankas aninput, seeFigure 4. This distributioncanbederivedfromcalculationmethodsshownin5orfromcalculationprograms. ThenextstepistofiltertheforKHrelevantloadsoutoftheselinesofcontact. Sooutofeveryinstantaneouslineofcontacttheloadthatlieson the desired diameter is selected. In F
40、igure 5 the selection for the diameters of the single engagementpointsoutoflinesinFigure 4areshown. Itcanbeseenthatnowagainaloaddistributionalongthefacewidthis given, even though the lines of contact are inclined in Figure 4.Figure 4. Several instantaneous lines of contact with contact forces for a
41、helical gear9 11FTM09Figure 5. Filtered loads out of instantaneous lines of contact for determination of KHAnalogtoequation3, the KHis definedas themaximum loaddividedby themeanloadalongthefacewidth.But now the differentiation of the reflected diameter is taken into account. Equation 17 shows the ne
42、wdefinition of the load distribution value, and equation 18 shows the selection of the load on the correctdiameter.(17)KH BCD=Fselbmax BCDFselbmean BCD(18)Fselb =1nni=nfbbBCDwhereKH B/C/Dis face load factor (contact stress) at diameters according to B/C/D;Fselis force on instantaneous line of contac
43、t at diameters according to B/C/D, N.DiscussionIn order to compare the different interpretations of the load distribution value, a simple helical gear set isselectedandmodelledinacalculation program for determinationof theflank loads 8. The dataof theusedgear set is shown in Table 1.The reference ge
44、ar set has no modifications. From this basic gear set variations of face width, load andmodifications are derived. These variants of gear set and modifications are listed in Table 2.Table 1. Gear data of the used gear setTerm Symbol Units Pinion WheelNumber of teeth z 14 59Profile shift x 0.4713 0.4
45、161Module mnmm 8Normal angle 20Transverse pressure angle t 20.344Working transverse pressure angle wt 23.471Helix angle - 11 11Center distance a mm 304.068Reference diameter d mm 114.1 480.83Pitch diameter dwmm 116.63 491.51Mesh stiffness cN/mm/mm 16.7Single stiffness csthN/mm/mm 18.4410 11FTM09Tabl
46、e 2. Variants of the investigated gear setVariant Term SymbolUnits Set 1 Set 2 Set 3Pinion Wheel Pinion Wheel Pinion WheelTransmitted torque T Nm 6000 25286 6000 25286 2000 8429Face width b mm 127 160 71Transverse contact ratio 1.445 1.445 1.445Overlap ratio 0.964 1.177 0.501Total contact ratio 2.40
47、9 2.622 1.946Var 0Angle correction Cmm 0 0 0 0 0 0Crowning Ccmm 0 0 0 0 0 0Var 1Angle correction Cmm 0 25 0 17 0 100Crowning Ccmm 0 0 0 0 0 0Var 2Angle correction Cmm 0 - 25 0 - 17 0 - 8Crowning Ccmm 0 18 0 25 0 2Var 3Angle correction Cmm 0 - 25 0 - 17 0 - 8Crowning Ccmm 0 18 0 25 0 2Linear tip reli
48、ef Camm 30 10 20 5 30 10Var 4Angle correction Cmm 0 - 5Crowning Ccmm 0 11Linear tip relief Camm 40 20Var 5 Generated tip relief Cgwmm 50 100Variant 1includes awrongleadmodificationwhichleads toan expectedworse loaddistribution comparedtoVariant 0. For gear set 3 the deviation is even so high that pa
49、rts of the flank are not carrying any load at all.Variant 2 is optimized for an optimum load distribution calculated according to the standards with constantmeshstiffnessineverymeshingposition. Variant3isaddingatiprelieftothegearsetinordertodecreasetheload in the regions with high sliding amounts and to avoid mesh interference.The difference betw
