1、07FTM10The Gear Dynamic Factor, Historicaland Modern Perspectivesby: Dr. D.R. Houser and D. Talbot, The Ohio State UniversityTECHNICAL PAPERAmerican Gear Manufacturers AssociationThe Gear Dynamic Factor, Historical and ModernPerspectivesDr. Donald R. Houser and David Talbot, The Ohio State Universit
2、yThe 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 dynamic factor has been included in gear design and rating formulas since the 1930s. Its originalformulation wa
3、s based on an assessment of entering tooth impacts, but in modern gear design procedures,where tip relief and lead modifications are common, these impacts may be virtually eliminated. With thiselimination,onefindsthatgeardynamicsaremainlyexcitedbysteadystatephenomenasuchastransmissionerror, friction
4、 and axial shuttling of the mesh force. This paper first provides a historical progression of thedynamic factor equations that are based on impact theory and then discusses when this methodology isappropriate.Asophisticateddynamicfiniteelementsimulationofagearpairwithsignificantspacingerrorsisused t
5、o describe the entering impacts ofa spur gear pair.Anew steady state dynamic modeling approach isused to demonstrate the effects of the different excitations as well as manufacturing deviations on thepredicted dynamic loads.Copyright 2007American Gear Manufacturers Association500 Montgomery Street,
6、Suite 350Alexandria, Virginia, 22314October, 2007ISBN: 978-1-55589-914-11The Gear Dynamic Factor, Historical and Modern PerspectivesDr. Donald R. Houser and David Talbot, The Ohio State UniversityBackground:Impact dynamicsDynamic tooth loads have been the subject of nu-merousinvestigationsoverthepas
7、tcentury. Fisher1providesa wonderfulsummary ofthe initialstud-iesondynamicfactors(15referencesprior to1950)and Walker 2 provides a very cogent discussionand calculation of dynamic factors in which he alsointroduces the concept of tip relief in order to mini-mizeimpacteffects.HouserandSeireg3,4update
8、references through the 60s and Bradley 5 pres-ents a discussion of dynamic factor models andsome of the more recent concepts used in theASME/ANSI standards6.Literally all of the early dynamic factor equationswere based on the fact thatthere willbe anenteringimpact when the gear teeth first come into
9、 contact.This impact occurs whenever the sum of the meshdeflection plus the sum of the two worst case spac-ing errors of the respective mating gear teeth isgreater than the sum of the tip relief of the drivenmember and the root relief of the driving member(forhelicalgears,theamplitudeofleadcrownoren
10、drelief should be added to the tip and root relief val-ues). This impact, which occurs well in advance ofthe usual entering contact point (point D versuspoint E) ascharacterized inthe Fig.1(this figurewillbecleanedupinthefinal paperversion), createsanimpact force that is proportional to the differen
11、ce inthe instantaneous velocities of the respective con-tact points (tip of driven member and a positionabove the SAP start of active profile of the drivingmember).Earlyanalysessimulatedthisimpactwitha linear cam 7,8 that drives the simple single de-gree of freedom torsional model. This model hasoft
12、en been used to simulate gear dynamics 5.The original entering impact equations that weredeveloped did not consider accuracy, however, asaccuracies in gears improved, the equations wereadjusted so that they provided lower values of dy-namic factors. Some of the equations that havebeen developed are
13、given below.DF1 = 1 +600Walker 1,2DF2 = 1 +78Ross 7DF3 = 1 +50Gear Handbook 8Figure 1. Gear geometry needed to predict the entering impact of gears with spacing errors2DF4 = 1 +78AGMA 9DF5 = 1 +CBAGMA 6where B and C are constants that are related to theaccuracy of the gear set. The AGMA equation has
14、upper pitch line velocity limits that increase with in-creasing quality.DF6 = 1 +KeeCTgSeireg and Houser 4where K is a constant related to the gear geometryand type, eeis the effective spacing error, C is thecenter distance and Tgis the driven gear torque.This equation is appropriate up to ahigh spe
15、edlimitthat occurs when the duration of the impact forcebecomes much shorter than the natural period ofthe main torsional natural frequency of the geardrive (to be briefly discussed later).The DF6 equation was developed from tooth straindata taken from operating spur and helical gearsthathadintentio
16、nally largespacing errorsmanufac-tured in them. A typical tooth strain plot for a 0.006inchspacingerrorisshowninFigure2,andFigure3summarizes the peak strains for several operatingloads and speeds 3. The last tooth strain pulsehas a contact duration that would be considerednormal for this spur gear p
17、air, and the contact dura-tion of the first pair is cutshort, andthe middletoothpair has an elongated tooth strain pulse. Also, oneobserves that there is an instantaneous rise in thetooth strain that is caused by the entering impact.The peak strain at this particular condition is ex-tremelyhighwithF
18、igure3showingthevalues ofthepeak strains relative to the static strains. Also,notethat following the impact, there is an oscillation at afrequency of about 3 kHz. This frequency corre-sponds to the main mesh torsional natural frequen-cy. Each of the three torque traces that are showncorrespondtooneo
19、fthetoothstraintraces. Torquerepeatabilitycanbeobservedfromthesetracesandone also sees torque oscillations occurring at amuch lower frequency.Figure 2. Strain histories for 106 tooth spurgear teeth in region of positive 0.006 in. error(individual traces do not have the same gain)(taken from Ref. 3)F
20、igure 3. Tooth strain data for central gage oftooth with positive 0.006 in error (106 toothspur gear) (taken from Ref. 3)Steady state dynamicsIf one applies adequate tip relief, the entering im-pactsduetospacingerrorsandprofileerrorscanbecompletely eliminated. However, that does notmean that dynamic
21、 loads are eliminated. Excita-3tionssuch astransmission error,time varyingmeshstiffness change, friction, and force axial shuttlingmoments 12 can still excite thegear dynamic sys-temandhenceresult indynamic toothloading. Thestrongmeshresonanceeffectthatisshowninmuchexperimentaldata13-16isduetosteady
22、stateex-citations and not due tothe impacteffects thatwerepreviously discussed.The ISO Standard 17 interestingly has portions ofitsdynamicfactorthatconsidertheimpactphenom-enon and other portions that relate to steady stateresponses. Since these phenomena are totally dif-ferent and their peak forces
23、 may occur at differentangular positions in the mesh cycle, it is debatablewhether theymay simplybe addedas isdone intheISO Standard.Basic gear dynamics modelingThe authors gear reference database containsover 1000 papers on the subject of gear dynamics,with many of them being reviewed in the review
24、 pa-per by zguven and Houser 18. A goal of this pa-per is to look at some of the more recent modelingtechniques that are appropriate for predicting thegear dynamic factor. This section will start with thepresentation of a simple lumped parameter modelandwillthenproceedtolookata coupleof moreso-phist
25、icated techniques for first dealing with the en-tering impact problem of spur gears and anothermethod for dealing with the steady state analysis ofhelical gears.The dynamic factor has a fairly simple definition(shownbelow),yethassomesubtleinterpretations.DF =Fs+ FiFswhere Fsis the normal load carrie
26、d by the mesh atvery slowspeeds andFiisthe incrementalincreaseintoothnormalloadthatoccurswhenthegearpairsspeed is increased.In this definition, the timing of the load is not impor-tant,sooneneednotbeconcernedwithwhethertheangularpositionofthepeakloadisatthesameposi-tion as the peak stress. Likewise,
27、 the strain gagereadings of Figure 3 are really dynamic strains andnotdynamicloadssooneneedstobecarefulinhowthe strain values are interpreted.Perhapsa betterdefinition wouldbe todeal withdy-namic stresses as opposed to dynamic loads.Houser and Harianto created such definitions andusedthemintheirpred
28、ictionsofdynamicfactorsforspur 19 and helical gears 20. However, the sim-plicityofusingthedynamicloadisthemainreasonitwill be used in the analyses discussed here.Simple dynamic modelFigure 4 shows a relatively simple spring mass dy-namic model of a drive train. The inertias andmassesofthegearwheels,
29、areI2,I3,m2andm3,re-spectively, and the end inertias, I1and I4are thedriverandtheloadinertias,respectively. Thespringbetween the two gears depicts the stiffness of thegear teeth and the cam signifies a transmission er-ror displacementexcitation. The springs,k2andk3,are simplifications of the bearing
30、s supporting thegears, with each stiffness including not only thebearingstiffnessalongthelineofaction,butalsotheshaft and housing deflections. The purely torsionalsprings connecting the gears to the drive and loadmotors, respectively, are kt1and kt4. By making thedrive and load inertias zero and mak
31、ing the bearingstiffnessesinfinite,wegetthemodelofFigure5,thesimplest possible dynamics model for a gear pair.This simplified model will have one natural frequen-cy that is often referred to as the main gear meshtorsional natural frequency and may be calculatedas:fn=12 kmr22I2+r23I3 ,Hz.where r1and
32、r2are the respective base radii of thetwo gears.Figure 4. 6DOF dynamic model of gear train4Figure 5. SDOF dynamic model of gear pairThis frequency is referred to in ISO Standards andis the frequency of oscillation shown in Figure 2 forthe tooth pair having the impact. It is also the fre-quency that
33、is excited by transmission error andotherexcitationstogivesignificantdynamicfactors.The accuracy of the equation is certainly related totheaccuracywithwhichthemeshstiffnessandiner-tias are calculated. The mesh stiffness is affectednot only by the tooth bending and shear, but alsobythe Hertzian conta
34、ct deflection and the rotation oftooth base. Load distribution also affects the stiff-ness, so one must evaluate the load distribution ofthe gear pair in order to obtain a reasonable esti-mate of the mesh stiffness. Also, the calculation ofthe inertias is not as simple as one might expect.For instan
35、ce, for the 6.2:1 ratio spur gear set foundinTable1,thenaturalfrequencyisdominatedbytheinertia of the small diameter pinion. Since the shaftsupporting the pinion is likely to be about the samediameter as the pinion, one must decide what widthisusedintheinertiacalculation. Finally,whenusingthe 6 degr
36、ee of freedom model of Figure4, thecou-pling between thetorsional andlateral motionsmayshift the main torsional natural frequency by asmuch as 20%.Table 1 provides pertinent information on the twogear pairs that are to be used as examples in thispaper. Each of the gear sets, when modeled withthe 6 d
37、egrees of freedom (DOF) model of Figure 4will have 5 natural frequencies that are shown inTable 2. The important mode with respect to dy-namic loading of the teeth is mode 5 while modes 3and 4 are also important when considering thedynamic forces that are transmitted through thebearings.Transient mo
38、del for capturing impactdynamicsIn order to capture the entering impacts using mod-ern modeling techniques, it is necessary to apply atime domain integration method to a model thatcombines the analysis of gear kinematics with thedynamics analysis. However, most available com-mercial methods are eith
39、er very good at the kine-maticanalysisorverygoodatthedynamicanalysis,but seldom do both well enough to capture the offline of action impacts that occur.Table 1. Information for gear pairs used in this studySpur gear pair(6DOF,FEM)Helical gear pair(FEMTooth numbers 17 106 25 31Normal module (mm) 3.14
40、1 2.780Pressure angle (deg) 21.98 22.21Helix angle (deg) 0 28.9Outside diameter (mm) 60.94 332.63 85.29 104.34Face width (mm) 76.2 76.2 31.75 31.75Root diameter (mm) 45.73 317.43 71.40 71.40Shaft diameter (mm) 50.8 69.9 38.1 38.1Shaft length (m) 1.02 0.61 1.15 0.76Bearing radial stiffnesses (N/m) 3.
41、80e86.81e83.50e83.50e8Mesh stiffness (N/m) 1.237e96.460e8Drive and load inertia (kg-m2) 1.514 0.508 3.789 0.508Shaft stiffnesses (N-m/rad) 2.99e43.04e5- - - -Pinion torque (N-m) 290 3405Table 2. Predicted natural frequencies (Hz) for the 17-106 spur gear pairMode 6DOF FEM Shaft Nature of the mode1 1
42、04.7 103.4 Driver/load torsional, in phase2 226.7 221.9 Driver/driven torsional, out of phase3 994.9 914.9 Gear bearing bounce4 2438 2400 Pinion bearing bounce5 6911 6093Main resonance coupled torsional/bearing bounceSDOF 5416 Pure torsional resonance calculationHowever, one such time domain method
43、that doesdothistypeofanalysishasbeendevelopedbyTam-minana et. al 21. This method uses a complete fi-niteelementmodelofthegearthatincludesdetailedmodels of each tooth and then performs a reason-ably extensive number crunching to perform thetime domain integrations. Due to the extensivecomputation req
44、uirements, the method has not yetbeen extended to three dimensions, so it may onlybe used for the analysis of spur gears.This model was set up for the spur gear pair whosetested results are provided in Figure 2 and whosegeometryisprovidedinTable1. Figure6presentsasnapshotofthedynamicstressesonthepar
45、tjustaf-ter the impact. One flaw of the dynamic load con-cept as compared with the use of dynamic stressesisportrayedinthisplot. Notethatthereisahighrootstress in the gear tooth, but a low root stress in thepinion tooth at this point of contact. Here the peakdynamic load occurs on the pinion at a lo
46、cationwhere stresses are low; hence, one could beapply-ing the appropriate dynamic effect to the gear, butan effect that is too large for the pinion. Figure 7showstheactualtimetracesthatarepredicted. Theprediction for the “entering” tooth pair shows thenormal period of contact. The period of contact
47、 forthe tooth before the error is much shortened, whichis similar to what happens in Figure 2. The toothwith the error has an impact response similar to thesame tooth in Figure 2 with tooth oscillations at themesh natural frequency. The dynamic stresses arenot nearly has high because the mean load u
48、sed inthesimulationwashigherthanthe valueused intheexperiment.More results using this modeling technique are notpresented since the authors feel that the steadystate dynamics that are discussed the remainder ofthis paper are more relevant to todays gears.Figure 6. Dynamic finite element stress plotF
49、igure 7. Root stresses of four consecutiveteeth of the 106 tooth spur gear with a largespacing error on the third toothFinite element model for capturing steadystate dynamicsPrediction of steady state gear dynamics usinglumped parameter models of the typeshown inFig-ures 4 and 5 may work well in certain circum-stances, but require some astute assumptions bythe modeler in order to give good results. In6addition, lumped parameter models have difficultycapturing certain characteristics of the shafting andb
