1、07FTM17Simulation Model for the Emulation of theDynamic Behavior of Bevel Gearsby: A. Gacka, C. Brecher and T. Schrder,RWTH Aachen UniversityTECHNICAL PAPERAmerican Gear Manufacturers AssociationSimulation Model for the Emulation of the DynamicBehavior of Bevel GearsAdamGacka,ChristianBrecherandTobi
2、asSchrder,LaboratoryforMachineTools and Production Engineering (WZL), RWTH Aachen UniversityThe 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.AbstractStarting point: Today th
3、e impact of bevel gear deviations on the noise excitation behaviour can only beexamined insufficiently under varying working conditions such as different rotational speed and torque. Thevibration excitation of bevel gears resulting from the tooth contact is primarily determined by the contactconditi
4、ons and the stiffness properties of the gears. By the use of a detailed tooth contact analysis thegeometry based gear properties can be developed and provided for a dynamical analysis ofthe tooth mesh.Researchobjective: Amodelhasbeendevelopedforthesimulationofthedynamicbehaviourofbevelgears.Withthea
5、idofaload-freetoothcontactanalysisthegeometry-basedpartofthepathexcitationisdeterminedat first. With a tooth contact analysis under load the path excitation caused by deflections can be calculated.Thegeometrybasedpartofthepathexcitationandacharacteristicsurfaceoftheexcitationvaluesiscreatedandprovid
6、edfordynamicsimulation.Thebehaviourofthemodelhasbeenverifiedwithasetofhypoidgearsby describing the tooth contact force depending on the rotary speed.Thisdynamicmodelisabletoconsidereverydeviationofthemicro- andmacrogeometryfromtheidealflanktopography, i.e. waves and/or groves in the surface structur
7、e in combination with two and three dimensionalflankdeviationslikeprofiledeviations,helixdeviationsandtwists.Itisalsopossibletoconsidertheinfluenceof friction and the contact impact caused by load and/or manufacturing. errors with a test rig to verify thecalculations.Results: Theresultofthestudyisth
8、einvestigationofthecomplexinfluenceofsurfacestructureswhichresultfrom manufacturing processes, manufacturing deviations and flank corrections on the noise excitation ofbevel gears. The resulting noise excitation can be rated in form of the excitation level with the aid of thisdynamic model.Copyright
9、 2007American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2007ISBN: 978-1-55589-921-91Simulation Model for the Emulation of the Dynamic Behavior of Bevel GearsAdam Gacka, Christian Brecher, and Tobias Schrder, RWTH Aachen UniversityIntroductionDu
10、e to their high efficiency, bevel gear sets arewidely used in industry to transmit torque, generatehigh rotational speeds and change rotation direc-tions. Mostly studies on bevel gear dynamics arebased on experiments or simple formulations con-sideringonlythetorsionalvibrations. Inthispaperaprecise
11、dynamic simulation model of bevel gearpairs is developed. The method combines a finiteelement based tooth contact approach with a multibody dynamic simulation to provide a more accu-rateandcomprehensiveanalysisoftheexcitationinthe tooth mesh. The tooth mesh is modelled as aspring-damper set. The spr
12、ing-damper set is ableto consider all six degrees of freedom. The varyingcharacteristic mesh forces of the spring and thedamper are calculated using a detailed, finite ele-ment based tooth contact analysis. The forces aremultidimensional functions of the rotational trans-mission error under load, th
13、e pinion displacement,the rolling position and if necessary additionally thesliding speed. The considered excitations in thetooth mesh are transmission errors due to profiledeviations, the changing stiffness of the meshingteethasthenumberofteethincontactchangesandthevaryingslidingfriction. Finally,a
14、basictransmis-sion with a bevel gear pair is analyzed for differentoperating speeds.Excitation mechanisms in the tooth meshThemainsource forthe dynamicexcitation ofbevelgears is the tooth mesh 1. In the tooth contact dif-ferent excitation mechanisms are combined to acomplexresultingexcitation. Theth
15、reemostsignifi-cant terms are the unloaded transmission error, thechangingstiffnessofthemeshingteethandthepre-mature tooth contact, Figure 1.The load-free transmission error is caused by geardeviations which lead to a path excitation 3. Themost important gear deviations influencing theexcitation beh
16、aviour under load-free operatingcondition are pitch, flank and profile variations 4,5, 6. But path excitations can also be caused bydevations of 3rd order, as e.g., generated cutdeviationsandevenof4thorder,ase.g.,roughnessdescribing tooth flank surface structures 7. Alldeviations lead to changing op
17、erating conditions,i.e., changing rotational speeds and different loadcarrying torques. In numerous practical tests it isshown that for operation conditions of low specificloads deviations have the main influence on theexcitation in the tooth mesh and on the noise.Figure 1. Excitation mechanisms 22T
18、he second excitation mechanism in the toothcontact results from changing stiffness of themeshing teeth as the number of teeth in contactchanges. This mechanism named as parametricexcitationcausesadditionalvibrations foroperationconditions under load. The domination of theparametric excitation increa
19、ses by higher loadcarrying torques 8.A further excitation mechanism in the tooth meshcan be traced back to the premature tooth contact,9. Under loaded operating conditions the teethand the gear body are deformed. This leads todisturbed kinematic contact conditions in the toothmesh. Consequently, the
20、 first tooth contact occursearlier than under undisturbed conditions and thedriving tooth flank penetrates theoretical the driventooth flank. Out of this results a tooth impact and aforce impulse is generated combined with anadditional sound stimulation.Fundamental procedureAt first, themanufactured
21、 flanktopography mustbedetermined in consideration of the manufacturingprocess, and further of the prescribed quality re-spectively the prescribed flank modifications. Theflank topography canbe determinedwith ageneral-lykeptapproachforthedescriptionofmachinekine-matics of bevel gear cutting machines
22、 10. Themethodsimulatesthemanufacturingprocessbyfol-lowing the penetration of the single tool cuttingbladesthroughthematerial. Thetoolgeometryandthe chosen machining parameters can be conside-red. Based on the geometries of the pinion and thegear resulting from the manufacturing process FEmodels are
23、 generated in the next step.To reach an exact reproduction of gear propertiestheuseofahigh-gradecomputationalapproach,asthe Finite Element Analysis (FEA), is necessary.But the formulation of the contact conditions isproblematic in the Finite Element Analysis, since itis a non-linear problem. FEA is
24、suitable for solvingthe stiffness of gears but the modelling of the toothcontact is a very difficult task. Hence an FE basedtooth contact analysis is used. This FE basedapproach uses the structure properties of an FEmodel and combines them with a simplemathematical spring model to examine the toothc
25、ontact conditions 11, Figure 2.Figure 2. Mathematical spring model3Thestiffnessesofthegearsarecharacterizedbysocalled influence numbers. An influence number ijdetermines the displacement of a point j accordingto a unit force F = 1 N on point i. The influencenumbers are calculated by the Finite Eleme
26、nt Anal-ysis. After the influence numbers are known toconsider the stiffness behavior of the gears thetooth mesh can be examined. The FE based toothcontact analysis determines the path excitationcaused by flank deviations and the parametricexcitation caused by elastical deformations. Bothtypesofdevi
27、ationsfromthe idealinvolute flankleadto rotational transmission errors. Deformationscaused by load carrying torques create a rotationaltransmission error under load TEuL. For the toothcontact the reason of the deviation regarding to therotational transmission error isnt relevant. There-fore the comp
28、onents resulting from a load free,purely geometrical deviation and those whichresultfrom deformations can be combined to a totalrotational transmission error TEtotal. The rotationaltransmission error under load specifies theparametrical excitation and the load free rotationaltransmission error TELfs
29、pecifies the pathexcitation.A dynamical simulation model of the tooth mesh iscombinedwiththequasi-staticFEbasedtoothcon-tactanalysis. Inthisprocessthetoothcontactanal-ysis examines the load free rotational transmissionerrorTELfasafunctionoftherollingpositionp. Andthe rotational transmission error un
30、der load TEuLisafunctionoftherollingpositionpandthesixcompo-nents of the load carrying torque Fx,Fy,Fz,Mx,MyandMz. TheresultingcharacteristicdiagramforthetorqueMzaroundz-axisasafunctionofrollingposi-tion p and the rotational transmission error underload TEuLis shown in Figure 3.The tooth contact of
31、a bevel gear set is a threedimensional problem. The load carrying torquecauses pinion displacements and changing toothcontacts. A precise dynamic simulation has toconsider the changing tooth contact under loadcaused by pinion displacement relative to the gear.According to the current state of the to
32、oth contactanalysisunderload,thetoothspringforces mustbefunctions of following parameters: the rollingposition p, the rotational transmission error underload TEuLand the position of the pinion relative tothe gear, given by H, V, G and . The multi-dimensional characteristic diagrams are availablein f
33、orm of tables.Figure 3. Characteristic tooth spring diagram4Intheliterature,theslidingfrictionbetweenmeshingtooth flanks is known as a significant mechanism inbevel gear systems 12. Hence for the damping asliding friction formulation is considered in the dy-namical simulation model. According to the
34、 currentstate of the tooth contact analysis under load, thetooth damping forcesmust befunctions offollowingparameters: the rolling position p, the rotationaltransmission error under load TEuL, the slidingspeed vGand the position of the pinion relative tothe gear, given by H, V, G and . The multidime
35、n-sional characteristic diagrams are available in formof tables. Since only one pitch is generated in thetooth contact analysis, the multidimensional char-acteristic diagrams for the tooth spring/dampingforcesincludedataonlyforonepitchaswell. There-fore the current rolling position has to be standar
36、d-ized to a rolling position in one gear pitch before itcan be considered in the mechanical model of thetoothmesh. Inthelaststepamulti-bodysimulationmodelwithrigidbodiesofabevel geartransmissionhas been developed, which uses the model of thetooth mesh. The fundamental procedure is de-scribed in Figu
37、re 4.Mechanical model of the tooth meshIn this section the basic steps of the mechanicalmodel are described. The characteristic diagramdata has to be generated as tables and the tablescanthereforebeinterpolatedby thesimulationsys-tem. First, the angular positions of the gearGandthe pinion Pare neede
38、d as input data of the toothmesh model, Figure 5. With the two angular posi-tions under considering the transmission ratio i thetotal rotational transmission error TEtotalcan be de-termined. Second,theangularpositionofthepinionPhas to be standardised to one pitch before it canbe considered together
39、with the pinion positiondescribed by V, G, H and in the evaluation of theloadfree rotational transmission error diagram.Third, the result of the evaluation is thestandardised loadfree rotational transmission errorTELf. The difference of the total rotationaltransmission error TEtotaland loadfree rota
40、tionaltransmission error TELfdeterimes the rotationaltransmission error under load TEuL. Fourth, theevaluationofthecharacteristictoothspringdiagramhas to be done with the following parameters: PTEuL,V,G,Hand. Fifth, the difference of theangular speeds of the gear .Gand the pinion .Punder considerati
41、on the transmission ratio i can beused together with P,TEuL,V,G,Hand toevaluate the characteristic tooth damping diagram.Finally, the evaluated forces and torquesMD=(Mx,My,Mz)D,Mts=(Mx,My,Mz)ts,Fts=(Fx,Fy,Fz)ts,FD=(Fx,Fy,Fz)Dofthecharacteristic spring/damper diagrams are thedynamical response of the
42、 tooth mesh system.Figure 4. Fundamental procedure5Figure 5. Mechanical model of the tooth meshA very simplified multi-body simulation model withrigid bodies of a bevel gear transmission has beendeveloped in MSC.ADAMS, Figure 6. It consistsonlyofafewcomponents. Thesearethebevelgearset, three shafts
43、(one driving and two driven), adifferential gear housing and several specificelement-types of the multi-body mechanics, e.g.,to model additionally couplings or to reduce thedegrees of freedom. The gear and the pinion of thebevel gear set are connected with the developedmodel of the tooth mesh in for
44、m of a dynamicallyloadable library (e.g., .dll on Microsoft Windows).All elements of the multi-body simulation are rigidbodies and the only dynamically active componentis the tooth mesh. This simplified approach issuitable for an evaluation of the model behaviour.Figure 6. MBS-Model with rigid bodie
45、s of a bevel gear transmission in MSC.ADAMS 136Analysis of the mechanical modelInfluence of the rotation speedIn Figure 7 the resulting transmission error for twodifferent rotation speeds can be seen. In the leftpartofthepicturethecomputedtransmissionerrorsfor a speed of 500 min- 1and the right part
46、 the com-puted transmission errors for a imaginary speed of50000 min- 1can be seen. The driving torque wasset to a constant rate of Mdriven= 1200 Nm. For asimplification of themodel thedamping wasset toaconstant value. For both speeds the gears are aswell excited by the path excitation TELfdue to th
47、egeometrical conditions, as by the time variant stiff-ness under load.At slow rotation speeds the resulting tooth springdeformationovertimeofthedeviation freegear atastandard force is the reciprocal value of the toothmesh stiffness and does therefore correspond tothe loaded transmission error TEuL.
48、At increasingrotation speeds dynamical effects begin to over-weight this behavior. This leads to the resultingtransmission error at the undercritical speed of500 min- 1. In the overcritical range of the rotationspeed (n = 50000 min- 1) the gears are not able tofollow the excitation due to the varyin
49、g stiffness be-causeoftheirmassinertia. Therefore nooscillationis anymore superimposed to the rotation of thegears. The resulting transmission error is aconstant value at these speeds and the additionalloads for the tooth mesh can be calculated with theamplitude of the load free and the loaded transmis-sion error.IntheupperpartofFigure8thecoursesof thetoothmeshtorqueatarotationspeedof500 min- 1andof50000min- 1aredisplayed. Also thecorrespondingFourier analysis are shown. The driving torque issetagainto Mdriven= 1200Nm
