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1、07FTM17 Simulation Model for the Emulation of the Dynamic Behavior of Bevel Gears by: A. Gacka, C. Brecher and T. Schrder, RWTH Aachen University TECHNICAL PAPER American Gear Manufacturers Association Simulation Model for the Emulation of the Dynamic Behavior of Bevel Gears Adam Gacka, Christian Br
2、echer and Tobias Schrder, Laboratory for Machine Tools and Production Engineering (WZL), RWTH Aachen University The statements and opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract S
3、tarting point: Today the impact of bevel gear deviations on the noise excitation behaviour can only be examined insufficiently under varying working conditions such as different rotational speed and torque. The vibration excitation of bevel gears resulting from the tooth contact is primarily determi
4、ned by the contact conditions and the stiffness properties of the gears. By the use of a detailed tooth contact analysis the geometry based gear properties can be developed and provided for a dynamical analysis of the tooth mesh. Researchobjective: Amodelhasbeendevelopedforthesimulationofthedynamicb
5、ehaviourofbevelgears. Withtheaidofaload-freetoothcontactanalysisthegeometry-basedpartofthepathexcitationisdetermined at first. With a tooth contact analysis under load the path excitation caused by deflections can be calculated. Thegeometrybasedpartofthepathexcitationandacharacteristicsurfaceoftheex
6、citationvaluesiscreated andprovidedfordynamicsimulation.Thebehaviourofthemodelhasbeenverifiedwithasetofhypoidgears by describing the tooth contact force depending on the rotary speed. Thisdynamicmodelisabletoconsidereverydeviationofthemicro- andmacrogeometryfromtheidealflank topography, i.e. waves a
7、nd/or groves in the surface structure in combination with two and three dimensional flank deviations like profile deviations, helix deviations and twists. It is also possible toconsider theinfluence of friction and the contact impact caused by load and/or manufacturing. errors with a test rig to ver
8、ify the calculations. Results: Theresultofthestudyistheinvestigationofthecomplexinfluenceofsurfacestructureswhichresult from manufacturing processes, manufacturing deviations and flank corrections on the noise excitation of bevel gears. The resulting noise excitation can be rated in form of the exci
9、tation level with the aid of this dynamic model. Copyright 2007 American Gear Manufacturers Association 500 Montgomery Street, Suite 350 Alexandria, Virginia, 22314 October, 2007 ISBN: 978-1-55589-921-9 1 Simulation Model for the Emulation of the Dynamic Behavior of Bevel Gears Adam Gacka, Christian
10、 Brecher, and Tobias Schrder, RWTH Aachen University Introduction Due to their high efficiency, bevel gear sets are widely used in industry to transmit torque, generate high rotational speeds and change rotation direc- tions. Mostly studies on bevel gear dynamics are based on experiments or simple f
11、ormulations con- sideringonlythetorsionalvibrations. Inthispapera precise dynamic simulation model of bevel gear pairs is developed. The method combines a finite element based tooth contact approach with a multi body dynamic simulation to provide a more accu- rateandcomprehensiveanalysisoftheexcitat
12、ionin the tooth mesh. The tooth mesh is modelled as a spring-damper set. The spring-damper set is able to consider all six degrees of freedom. The varying characteristic mesh forces of the spring and the damper are calculated using a detailed, finite ele- ment based tooth contact analysis. The force
13、s are multidimensional functions of the rotational trans- mission error under load, the pinion displacement, the rolling position and if necessary additionally the sliding speed.The considered excitations in the tooth mesh are transmission errors due to profile deviations, the changing stiffness of
14、the meshing teethasthenumberofteethincontactchangesand thevaryingslidingfriction. Finally,abasictransmis- sion with a bevel gear pair is analyzed for different operating speeds. Excitation mechanisms in the tooth mesh Themainsource forthe dynamicexcitation ofbevel gears is the tooth mesh 1. In the t
15、ooth contact dif- ferent excitation mechanisms are combined to a complexresultingexcitation. Thethreemostsignifi- cant terms are the unloaded transmission error, the changingstiffnessofthemeshingteethandthepre- mature tooth contact, Figure 1. The load-free transmission error is caused by gear deviat
16、ions which lead to a path excitation 3. The most important gear deviations influencing the excitation behaviour under load-free operating condition are pitch, flank and profile variations 4, 5, 6. But path excitations can also be caused by devations of 3rd order, as e.g., generated cut deviationsand
17、evenof4thorder,ase.g.,roughness describing tooth flank surface structures 7.All deviations lead to changing operating conditions, i.e., changing rotational speeds and different load carrying torques. In numerous practical tests it is shown that for operation conditions of low specific loads deviatio
18、ns have the main influence on the excitation in the tooth mesh and on the noise. Figure 1. Excitation mechanisms 2 2 The second excitation mechanism in the tooth contact results from changing stiffness of the meshing teeth as the number of teeth in contact changes. This mechanism named as parametric
19、 excitationcausesadditionalvibrations foroperation conditions under load.The domination of the parametric excitation increases by higher load carrying torques 8. A further excitation mechanism in the tooth mesh can be traced back to the premature tooth contact, 9.Under loaded operating conditions th
20、e teeth and the gear body are deformed.This leads to disturbed kinematic contact conditions in the tooth mesh. Consequently, the first tooth contact occurs earlier than under undisturbed conditions and the driving tooth flank penetrates theoretical the driven tooth flank. Out of this results a tooth
21、 impact and a force impulse is generated combined with an additional sound stimulation. Fundamental procedure At first, themanufactured flanktopography mustbe determined in consideration of the manufacturing process, and further of the prescribed quality re- spectively the prescribed flank modificat
22、ions. The flank topography canbe determinedwith ageneral- lykeptapproachforthedescriptionofmachinekine- matics of bevel gear cutting machines 10. The methodsimulatesthemanufacturingprocessbyfol- lowing the penetration of the single tool cutting bladesthroughthematerial. Thetoolgeometryand the chosen
23、 machining parameters can be conside- red. Based on the geometries of the pinion and the gear resulting from the manufacturing process FE models are generated in the next step. To reach an exact reproduction of gear properties theuseofahigh-gradecomputationalapproach,as the Finite Element Analysis (
24、FEA), is necessary. But the formulation of the contact conditions is problematic in the Finite Element Analysis, since it is a non-linear problem. FEA is suitable for solving the stiffness of gears but the modelling of the tooth contact is a very difficult task. Hence an FE based tooth contact analy
25、sis is used.This FE based approach uses the structure properties of an FE modelandcombinesthemwithasimple mathematical spring model to examine the tooth contact conditions 11, Figure 2. Figure 2. Mathematical spring model 3 Thestiffnessesofthegearsarecharacterizedbyso called influence numbers. An in
26、fluence number ij determines the displacement of a point j according to a unit force F = 1 N on point i.The influence numbers are calculated by the Finite Element Anal- ysis.After the influence numbers are known to consider the stiffness behavior of the gears the tooth mesh can be examined. The FE b
27、ased tooth contact analysis determines the path excitation caused by flank deviations and the parametric excitation caused by elastical deformations. Both typesofdeviationsfromthe idealinvolute flanklead to rotational transmission errors.Deformations caused by load carrying torques create a rotation
28、al transmission error under load TEuL. For the tooth contact the reason of the deviation regarding to the rotational transmission error isnt relevant. There- fore the components resulting from a load free, purely geometrical deviation and those whichresult from deformations can be combined to a tota
29、l rotational transmission error TEtotal. The rotational transmissionerrorunderloadspecifiesthe parametrical excitation and the load free rotational transmissionerrorTELfspecifiesthepath excitation. A dynamical simulation model of the tooth mesh is combinedwiththequasi-staticFEbasedtoothcon- tactanal
30、ysis. Inthisprocessthetoothcontactanal- ysis examines the load free rotational transmission errorTELfasafunctionoftherollingpositionp. And the rotational transmission error under load TEuLis afunctionoftherollingpositionpandthesixcompo- nents of the load carrying torque Fx, Fy, Fz, Mx, My andMz. The
31、resultingcharacteristicdiagramforthe torqueMzaroundz-axisasafunctionofrollingposi- tion p and the rotational transmission error under load TEuLis shown in Figure 3. The tooth contact of a bevel gear set is a three dimensional problem.The load carrying torque causes pinion displacements and changing
32、tooth contacts.A precise dynamic simulation has to consider the changing tooth contact under load caused by pinion displacement relative to the gear. According to the current state of the tooth contact analysisunderload,thetoothspringforces mustbe functions of following parameters:the rolling positi
33、on p, the rotational transmission error under load TEuLand the position of the pinion relative to the gear, given by H, V, G and .The multi- dimensional characteristic diagrams are available in form of tables. Figure 3. Characteristic tooth spring diagram 4 Intheliterature,theslidingfrictionbetweenm
34、eshing tooth flanks is known as a significant mechanism in bevel gear systems 12. Hence for the damping a sliding friction formulation is considered in the dy- namical simulation model. According to the current state of the tooth contact analysis under load, the tooth damping forcesmust befunctions
35、offollowing parameters:the rolling position p, the rotational transmission error under load TEuL, the sliding speed vGand the position of the pinion relative to the gear, given by H, V, G and . The multidimen- sional characteristic diagrams are available in form of tables. Since only one pitch is ge
36、nerated in the tooth contact analysis, the multidimensional char- acteristic diagrams for the tooth spring/damping forcesincludedataonlyforonepitchaswell. There- fore the current rolling position has to be standard- ized to a rolling position in one gear pitch before it can be considered in the mech
37、anical model of the toothmesh. Inthelaststepamulti-bodysimulation modelwithrigidbodiesofabevel geartransmission has been developed, which uses the model of the tooth mesh.The fundamental procedure is de- scribed in Figure 4. Mechanical model of the tooth mesh In this section the basic steps of the m
38、echanical model are described. The characteristic diagram data has to be generated as tables and the tables can therefore be interpolated by the simulation sys- tem. First, the angular positions of the gear Gand the pinion Pare needed as input data of the tooth mesh model, Figure 5. With the two ang
39、ular posi- tions under considering the transmission ratio i the total rotational transmission error TEtotalcan be de- termined. Second,theangularpositionofthepinion Phas to be standardised to one pitch before it can be considered together with the pinion position described by V, G, H and in the eval
40、uation of the loadfree rotational transmission error diagram. Third,theresultoftheevaluationisthe standardised loadfree rotational transmission error TELf.Thedifferenceof thetotal rotational transmission error TEtotaland loadfree rotational transmission error TELfdeterimes the rotational transmissio
41、n error under load TEuL.Fourth, the evaluationofthecharacteristictoothspringdiagram has to be done with the following parameters: P TEuL, V, G, H and . Fifth, the difference of the angular speeds of the gear . G and the pinion . P under consideration the transmission ratio i can be used together wit
42、h P, TEuL, V, G, H and to evaluate the characteristic tooth damping diagram. Finally,theevaluatedforcesandtorques MD= (Mx, My, Mz)D,Mts= (Mx, My, Mz)ts, Fts= (Fx, Fy, Fz)ts,FD= (Fx, Fy, Fz)Dofthe characteristic spring/damper diagrams are the dynamical response of the tooth mesh system. Figure 4. Fun
43、damental procedure 5 Figure 5. Mechanical model of the tooth mesh A very simplified multi-body simulation model with rigid bodies of a bevel gear transmission has been developed in MSC.ADAMS, Figure 6. It consists onlyofafewcomponents. Thesearethebevelgear set, three shafts (one driving and two driv
44、en), a differential gear housing and several specific element-types of the multi-body mechanics, e.g., to model additionally couplings or to reduce the degrees of freedom. The gear and the pinion of the bevel gear set are connected with the developed model of the tooth mesh in form of a dynamically
45、loadable library (e.g., .dll on Microsoft Windows). All elements of the multi-body simulation are rigid bodies and the only dynamically active component is the tooth mesh.This simplified approach is suitable for an evaluation of the model behaviour. Figure 6. MBS- -Model with rigid bodies of a bevel
46、 gear transmission in MSC.ADAMS 13 6 Analysis of the mechanical model Influence of the rotation speed In Figure 7 the resulting transmission error for two different rotation speeds can be seen. In the left partofthepicturethecomputedtransmissionerrors for a speed of 500 min- -1and the right part the
47、 com- puted transmission errors for a imaginary speed of 50000 min- -1can be seen. The driving torque was set to a constant rate of Mdriven= 1200 Nm. For a simplification of themodel thedamping wasset toa constant value. For both speeds the gears are as well excited by the path excitation TELfdue to
48、 the geometrical conditions, as by the time variant stiff- ness under load. At slow rotation speeds the resulting tooth spring deformationovertimeofthedeviation freegear ata standard force is the reciprocal value of the tooth mesh stiffness and does therefore correspond to the loaded transmission er
49、ror TEuL. At increasing rotation speeds dynamical effects begin to over- weight this behavior.This leads to the resulting transmission error at the undercritical speed of 500 min- -1. In the overcritical range of the rotation speed (n = 50000 min- -1) the gears are not able to follow the excitation due to the varying stiffness be- causeoftheirmassinertia. Therefore nooscillation is anymore superimposed to the rotation of the gears.The resulting transmission error is a constant value at these speeds and the additional load
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