1、11FTM11AGMA Technical PaperMarine Reversing MainGear Rating FactorVersus Number ofLoading Reversals andShrink Fit StressBy E.W. Jones, S. Ismonov andS.R. Daniewicz, Mississippi StateUniversityMarine Reversing Main Gear Rating Factor Versus Number ofLoading Reversals and Shrink Fit StressE. William J
2、ones, Shakhrukh Ismonov and Steven R. Daniewicz, Mississippi StateUniversityThe 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.AbstractThe marine vessel reversing main gear to
3、oth is subjected to three different loading cycles:- Ahead travel with load pulsing from zero to 100% of full power;- Astern travel with load pulsing from zero to about minus 66% of full power;- Reversal of direction with load changing from 100% of full power to about minus 66% of full power.Thenumb
4、erofrepetitionsofthesethreedifferentloadingcyclesvarieswiththevesseldutycycleandlife. Thepublishedvaluesofallowabledesignstressforteetharebasedonpulsingloads,whichmustbe modifiedforthisthirdloadingcycle. Thetoothmayalsobesubjectedtomeanstressduetoshrinkfittingofthegearontoahub.This paper evaluates t
5、he derating factor for marine reversing main gear tooth allowable bending stressusing the Goodman fatigue line and Miners equation as a function of the average number of changes invessel direction per hour, shrink fitting stress values, and different materials based on the AGMA valuesfor allowable s
6、tress and life factor.Copyright 2011American Gear Manufacturers Association1001 N. Fairfax Street, 5thFloorAlexandria, Virginia 22314October 2011ISBN: 978-1-61481-010-03 11FTM11Marine Reversing Main Gear Rating FactorVersus Number of Loading Reversals and Shrink Fit StressE. William Jones, Shakhrukh
7、 Ismonov and Steven R. Daniewicz,Mississippi State UniversityIntroductionTheobjectiveofthiswork,whichwasinitiatedbytheAmericanGearManufacturersAssociationMarineUnitsCommittee, is to adjust the allowable tooth stress derating factor for the marine vessels reversing main geartomatchthevesselsdutycycle
8、andshrinkfittingstresses. Thetoothflankofamarinevesselsreversingmaingear experiences three different types of load cycles as illustrated in Figure 1 and described below.- Pulsing, tensile bending stress cycles occur during ahead travel, which vary from zero to smax.- Pulsing, compressive bending str
9、ess cycles occur during astern travel, which vary from zero top smax. The value of p is the ratio: astern torque divided by ahead torque.- Onehalfofareversingbendingfatiguecycleoccurseachtimethedirectionofthevesselisreversed. Thestress varies between smaxand p smax.Thefatiguelifeofthereversingmainge
10、artoothisafunctionofthenumberofrepetitionsofthesethreeloadingcycles, which vary with the vessels duty cycle, and the design life. The fatigue life may also be affected bystresses induced into the tooth flank by shrink fitting the gear to the shaft.Current design practice is similar to treatment of a
11、n idler gear tooth, which has the allowable bending stressnumber reduced by a derating factor to compensate for the reduction in fatigue strength when the stress ischanged from uni-directional pulsing to fully reversing bending stress. For an idler gear tooth, a deratingfactor of 0.7 is given by the
12、 American Bureau of Shipping Rules 1, MAAG Gear Book 6 and ANSI/AGMA2001-D04 2. A derating factor for the reversing main gear tooth allowable bending stress number of 0.9 isspecifiedbytheAmericanBureauofShippingRulesandInternationalAssociationofClassificationSocieties,LTD12toaccountfortheinfluenceof
13、thereversingtoothstressandthestressatthetoothduetoshrinkfittingof the gear onto the shaft.Figure 1. Tooth bending stress during ahead travel, reversal of direction, astern travel, andreversal back to ahead travel4 11FTM11This study estimates the deratingfactor forthe allowablebending stressnumber as
14、a functionof theaveragenumberofvesselreversalsperhourofoperation,thenumberofstresscyclesduringaheadtravel,thenumberofstresscyclesduringasterntravel,andtheshrinkfitstressinducedatthetoothflank. Thisanalysisisbasedon the Goodman fatigue line, Miners Rule, allowable stresses per AGMA, and Life Factors
15、for bending perAGMA.Material strengthValuesoftheallowablebendingstressnumber,sat,fordifferentgearsteelsunderpulsingloads arepublishedin ANSI/AGMA 2001-D04 based on 99% reliability for 10,000,000 cycles. A lifefactor, YN,adjusts theseval-ues for different numbers of load cycles. Gear fatigue strength
16、 is determined experimentally under pulsing,i.e. onedirectionalloadingofa“non-running”geartooth10. Thestressconcentrationfactorisincorporatedas a stress increasing term via the geometry J-factor. The materials fatigue ratio, M, is the ratio of fullyreversing load fatigue strength, Sf, to ultimate te
17、nsile strength, Su. Values of Mvary between 0.35 and 0.6 forSuvalues less than 200,000 psi and tend to be constant for larger values of Su9.(1)M =SfSuwhereM material fatigue ratio;Sfis fatigue strength;Suis ultimate strength, psi.Goodman fatigue lineThe Goodman fatigue line defines the relationship
18、between alternating stress, sa, mean stress, sm, fatiguestrength,Sf,andultimatestrength,Su. TheGoodmanfatiguelinerepresentsthefatiguefailureofhighstrengthsteelsquitewell,butductilematerialsfollowtheGerberlinemoreclosely.3 Basedonthefactthatcompress-ive mean stresses at long lives are beneficial, the
19、 modified Goodman equation canbe convenientlyextrapol-ated to the compressive mean stress region as shown in Figure 2. 9 Theequation forthe Goodmanfatigueline, which lies between the yield lines, is shown in Figure 2.(2)saSf+smSu= 1The load ratio is defined as:R =sminsmaxThe alternating and mean str
20、esses are:(3)sa=smax smin2(4)sm=smax+ smin2wheresais alternating stress, psi;smis mean stress;R is ratio of minimum to maximum stress values;sminis minimum stress, psi;smaxis maximum stress, psi.Combining the above equations gives the Goodman equation as follows:(5)smax(1 R)2 Sf + smax(1 + R)2 Su =
21、15 11FTM11Figure 2. Fatigue and yielding criteria for constant life as a function of alternating and meanstresses, modified Goodman diagram.Idler gear tooth derating factorThe idler gear teeth are subjected to fully reversing bending stresses. The published allowable gear toothbending stresses, whic
22、h are based pulsing loads, are derated by a factor to produce equivalent fatiguestrength for a fully reversing bending cycle. The following example showing how this derating factor may beobtained for the idler will indicate part of the theory used for the development of the derating factor for there
23、versing main gear tooth. The bending stress in the idler tooth varies from Ki satto Ki sat.Thederating factor, Ki, must be multiplied times the AGMA published allowable stress number, sat, to obtain theallowable stress number for an idler tooth. Therefore, the maximum and minimum stresses in the idl
24、er toothare:(6)smax= Kisat(7)smin= KisatwhereKiderating factor for satif all tooth load cycles are fully reversing;satis published allowable stress number.For fully reversing loading of the idler teeth, the endurance strength is Sfand R is minus one. Substitution ofEquations 6 and 7 into Equation 5,
25、 Goodmans equation, gives:(8)KisatSf+0Su= 1So, the “allowable bending stress number” as published by AGMA is equal to the endurance strength underfully reversing bending load, Sf, divided by a derating factor, Ki.(9)sat=SfKiWritetheGoodmanequation relatingthe pulsingstresses ofthe geartooth fatiguet
26、est tothe fatiguestrengthof the material, Sf:(10)sa=smax2=sat26 11FTM11(11)sm=smax2=sat2(12)sat2 Sf+sat2 Su= 1(13)sat=112 Sf+12 SuEquate equations for satabove to obtain:(14)SfKi=112 Sf+12 SuTherefore, the value of Kiis:(15)Ki=12+Sf2 SuSubstitute the fatigue ratio, per equation 1:(16)Ki=1 + M2The ab
27、ove equation is the same as given by Det Norske Veritas 10 for idlers, which have R equal to minusone. Values of the idler gear tooth derating factor, Ki, for different values of fatigue ratio, M, are given inTable 1. Note that when M equals 0.40, the value of Kiis 0.70 as specified in the literatur
28、e.Fatigue strength for fully reversing load on toothThe fatigue strength for a fully reversing load on a gear tooth as a function of number of fatigue cycles, N,isSf(N). Sf(N) varies along the Goodman fatigueline asa functionof thenumber ofcycles applied,N,whichisrepresented by the AGMA stress cycle
29、 factor, YN.2(17)Sf(N) = KiYNsatwhereYNstress cycle factor for bending strength.Forafatigueratio,M,of0.4,casecarburizedsteel,andsatof65,000psithevaluesofSf(N)aregiveninTable 2for 1,000, 3,000,000 and 1010cycles. These numbers of cycles are located at the ends of two straight lineportions on the log-
30、log plot of the YNversus cycles plot in ANSI/AGMA 2001-D04. 2Table 1. Idler gear tooth derating factor, Ki, for different values of material fatigue ratio, MFatigue ratio, M Idler tooth derating factor, Ki0.30 0.650.40 0.700.50 0.75Table 2. Fully reversing, bending tooth strength for case carburized
31、 AGMA Grade II steel fordifferent numbers of fatigue cycles.Number of cycles, N YNFatigue strength, Sf(N),for Ki= 0.70, psi1,000 2.70 Sf(103) = 123,0003,000,000 1.04 Sf(3 106) = 47,30010,000,000,000 0.90 Sf(1010) = 40,9507 11FTM11The fatigue strength, Sfa,when the number of cycles, N, is between 1,0
32、00 and 3,000,000 cycles may beobtained from the equation of a line:(18)logSfa= logSf103+log Sf3106Sf103log3106103log (N) log103The fatigue strength, Sfb, when the number of cycles, N, is between 3,000,000 and 10,000,000,000 cyclesmay be obtained from the equation of a line:(19)logSfb= logSf3 106+log
33、Sf1010Sf3106log10103106logN3 106whereSfaequivalent stress when 1,000 Na 3,000,000 cycles;Sfbequivalent stress when 3,000,000 Nb 10,000,000,000 cycles;Na number of life cycles at equivalent stress Sfa,if1,000 Na 3,000,000 cyclesNb number of life cycles at equivalent stress Sfb, if 3,000,000 Nb 10,000
34、,000,000 cyclesShrink fitting induced stresses at toothStresses over a section of a gear due to shrink fitting of the gear onto a shaft are developed for two differentgear designs, a solid gear and a fabricated gear. The specifications for the two gears are given in Table 3.Figure 3 shows the two ge
35、ars, Figure 4 and Figure 5 shows sections cut from half of each gear.Figure 3. Fabricated gear and solid gear8 11FTM11Table 3. Data for two gears analyzed in this study for shrink fitting induced stresses in tooth areaSolid gear Double helicalfabricated gearNormal diametral pitch 3.175 (8 module) 4
36、(6.35 module)Pitch diameter 41.874” 31.243”Normal pressure angle 20 degrees 20 degreesNumber of teeth 129 113Helix angle 14 degrees 25.285 degreesInterference on diameter/bore 0.004”/7.000” 0.005”/8.500”Figure 4. Sectors cut from half of each gearFigure 5. Finite element model of segment of half of
37、double helical fabricated gear9 11FTM11The ANSYS finite element code is used to evaluate stresses and deflections of these gears due to shrink fitbetween the shaft and gear. The academic version of ANSYS has a restriction on number of elements andnodes, which limits the size of part and the mesh den
38、sity for the model. Therefore, the model of the gear istaken as a symmetric segment from half of the gear. In order to simplify the problem, the helix angle waschanged to zero for the model. See Figure 6 through Figure 9.Figure 6. Solid gear von Mises stress distributionFigure 7. Fabricated gear von
39、 Mises stress distribution10 11FTM11Figure 8. Radial deflection of segment of half of solid gearFigure 9. Radial deflection of segment of half of double helical fabricated gearThe von Mises stresses are shown in the stress plots as they represent the stresses at any point withoutregardtodirectionoft
40、hestresstensors. ThevonMisesstressisindependentofthechangesindirectionofthestressatthetoothfillet,sinceitisafunctionofthedistortionenergyatapoint. ThevonMisesstress, svm,maybe evaluated using the three principal stresses, s1, s2, and s3.(20)svm= s1 s22+s2 s32+s3 s120.5wheresvmis von Mises stress, ps
41、i;s1is most positive principal stress, psi;11 11FTM11s2is intermediate value of principal stress, psi;s3is least positive principal stress, psi.These results show small stresses at the tooth flank due to shrink fitting of the gear onto the shaft. Cast andfabricated gear blanks may experience non-uni
42、form elastic distortion of the rim and non-uniform toothstresses at the bottom of the root due to the position and geometry of cutouts and ribs.Stresses due to the shrink fitting of the gear onto the shaft are evaluated by the finite element method. ThemaximumvonMisesstressesintherootoftheteethdueto
43、shrinkfittingofthegearontotheshaftareshowninFigure 10 and Figure 11, and are given in Table 4. These stresses include the influence of stress concentra-tion produced by changes in geometry. The maximum stresses occur in the roots of the teeth, where thestress is increased by the theoretical stress c
44、oncentration factor, Kt, 5. In the AGMA process of gear toothdesign,thefatiguestressconcentrationfactor,KfisappliedaspartoftheJ-Factorwhencalculatingthestressnumber per AGMA 918-A93 4. These shrink fitting induced stressesdecrease rapidlyas theyrise fromtherootanduptheflankofthetooth. Thefabricatedg
45、earshowshigherrootstressesandawidervariationacrossthegearfaceduetothevariationinelasticsupport. Inthisstudy,theinfluenceofshrinkfittingstress,ssf,onthederating factor are included.Table 4. Variation in von Mises stress across the root of teeth due to shrink fitting of gear ontoshaftSolid gear root s
46、tresses, psi Fabricated gear root stresses828 - 977 3627 - 7832Figure 10. Maximum von Mises stresses due to shrink fitting occur at root of tooth of fabricatedgear and peak at rib. Stresses include the effects of stress concentration due to tooth geometry12 11FTM11Figure 11. Maximum von Mises stress
47、es due to shrink fitting occur at root of tooth of solid gearand decrease at un-buttressed end. Stresses include the effects of stress concentration due totooth geometryMarine reversing main gear tooth derating factor when all cycles are reversal of travelThe reversing main gear teeth are subjected
48、to three different stress cycles. This section considers all lifecyclestobeduetovesselreversaltoestablishalowerlimitforthederatingfactor. Thenextsectionevaluatesthe derating factor for a combination of the three different stress cycles. The cycledue toreversing theves-sels travel direction may have
49、a relatively large number of cycles for applications like fire fighting tugs, shipdocking tugs, etc. A derating factor, Krev, is determined for the tooth stress cycling between smaxand- p smax,wherepistheratioofasterntorquetoaheadtorque. Therelationshipsbetweenmaximumstress,smax, minimum stress, smin, the derating factor, Krev, and the allowable stress number, sat, for the reversingcycle are:(21)smax= Krevsat(22)smin= pKrevsatwherep is ratio of a
