1、12FTM17AGMA Technical PaperDynamic Analysis of aCycloidal GearboxUsing Finite ElementMethodBy S.V. Thube and T.R. Bobak,Sumitomo Drive TechnologiesDynamic Analysis of a Cycloidal Gearbox Using FiniteElement MethodSandeep V. Thube and Todd R. Bobak, Sumitomo Drive TechnologiesThe statements and opini
2、ons contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractSpeed reducers incorporating cycloidal technology as their primary reduction mechanism have always beenactive topics of research given their
3、 unique trochoidal tooth profile. A cycloidal reducer is recognized for itsstrengthandmainlystudiedforrotationalperformanceimprovement. Nowadays,thisstudycanbeperformedbydigitalprototyping,whichhasbecomeavaluabletoolforsimulatingexactscenarioswithoutexperimentingon actual model.This paper discusses
4、the stress distribution, modeled in a dynamic simulation environment, on the rotatingparts of Cycloidal reducer. A three dimensional finite element model is developed using Algor FEAcommercialcodetosimulatethecombinedeffectofexternalloadinganddynamicaswellasinertialforcesonone-cycloid disc system. T
5、his model utilizes surface-to-surface contact to define interaction betweenrotating parts of the reducer assembly. The results are analyzed for the variation in stress and deformationwith respect to time for a certain simulation period. This study gives an insight of internal load sharing ofrotating
6、 parts and their capability of carrying shock loads.Copyright 2012American Gear Manufacturers Association1001 N. Fairfax Street, Suite 500Alexandria, Virginia 22314October 2012ISBN: 978-1-61481-048-33 12FTM17Dynamic Analysis of a Cycloidal Gearbox Using Finite Element MethodSandeep V. Thube and Todd
7、 R. Bobak, Sumitomo Drive TechnologiesIntroductionThe cycloidal style of speed reducer is commonly used in many industrial power transmission applications.This type of mechanism, known for its high torque density and extreme shock load capacity, incorporates aunique reduction mechanism which is diff
8、erent from that of the more commonly understood involutegearing.To recognize the technical benefits of the Cycloidal reduction mechanism, one needs to understand theforces, load distribution and contact stresses associated with the reduction components within the mechan-ism. This type of study is al
9、so essential in designoptimization processesto improvethe overallperformanceof the reducer 1-6.This study can be facilitated through an example of one tooth difference type cycloidal reducer with lowreduction ratio. For simplicity, let us consider one disc reducer. The main rotating components of su
10、chreducermechanismareshowninanexplodedviewinFigure 1. Here,acycloidaldiscwitheightholesrotateson an eccentric bearing (cam). A rotation of the input shaft mounted eccentric cam generates swaying androtationalcomponentsofmotioninthesystem. Theswayingmotionisafunctionoftheamountofeccentricityin the ec
11、centric cam. The larger the eccentricity, the lower the reduction ratio. The large amount of eccentri-city results in making bigger hole design on the cycloidal disc to accommodate low speed shaft rollers asshown by equation (1).D = d+ 2 e(1)whereD is the diameter of the disc holes,d is low speed sh
12、afts roller diameter, ande is the eccentricity.Consequently, this bigger hole design reduces more material from the cycloidal disc and as a result, the discundergoesgreater stresses. Theanalysis ofload distributionin dynamicconditions givesaccess toexaminesuch stresses and their effects on the rotat
13、ing parts.Thereareseveralstudiesperformedonstaticanddynamicproblemsofthegears. Numerical,analytical,andexperimental approaches are used to investigate different gear profiles for load distribution, contact stressand dynamic effects. Such work helps togather vitalinformation todefine designguidelines
14、 andfind waystooptimize the torque transmission. However, very few publications are available addressing the sameanalyses on the cycloidal tooth profiled gears.Figure 1. An exploded view of one tooth difference, one disc type of cycloidal reducer4 12FTM17Thepurposeofthispaperistoinvestigatetheloadan
15、dstressdistributiononthecycloiddiscunderdynamicaswellasinertialeffectusing3-dimensionalfiniteelementanalysis. Indynamicconditions,theloadsareappliedas a function of time. Authors of this paper previously worked on the rigid body dynamic simulation of thecycloidal reducer to calculate motion related
16、dynamic forces of the reducer 7. Those forces were thenapplied to the rotating parts in a static FEA environment. The results, however, are limited by the rigid-bodyassumption which cannot deduce material-based contact stiffness and consequently accurate contactstresses. This limitation is overcome
17、in the current study by performing analysis in a dynamic FEAenvironment.To the authors knowledge, the following studies have been done on this topic.Malhotra and Parameswaran proposed a theoretical method to calculate contact force distribution on thecycloidal disc 8. The analysis is based on one di
18、sc cycloidal reducer with one tooth difference. It was as-sumed that only half number of the low speed shaft and housing (outer)rollers participatein torquetransmis-sion at any instance. They also discuss the effect of various design parameters on forces and contactstressesusingHertzianformula. Thew
19、orkfurtheraddressestheoreticalefficiencyofthereducerconsideringfriction. BlancheandYanginvestigatedbacklashandtorquerippleofthecycloidaldriveusingamathematicalmodel which considers machining tolerances 9. The above mentioned analytical (conventional) methods,however, cannot predict precise contact l
20、oading because of their inherent assumptions and simplifications.In the last two decades, there has been ongoing research performed using finite element based numericalmethods. Gamez et al. mentioned contact stress analysisin trochoidalgear pumpapplication 10. Hisworkgives a comparison between analy
21、tical model and FEM analysis ofcontact stress. Photoelasticitytechniqueis also elaborated to evaluate the stress. For finite element analysis, his work mainly focuses on two dimen-sionalquasi-staticmodeloftrochoidaltoothprofiledgearpump. Lietal. usedvarioustools,includingAbaqusfiniteelementsoftware,
22、toobservecontactstressesinpinion-gearsystembyvaryingpressureangle11. Theworkalsoaddressesstressesinplanetarygearboxesbyvaryinggaptolerance. Thefiniteelementmodelwasdeveloped using two dimensional, four node bilinear element for nonlinear sliding contact solution.Barone et al. investigated load shari
23、ng and stresses of face gears building a three dimensional FEA modelinAnsys software to simulate the effects of shaft misalignment and tooth profile modification. The results areshownbyplottinggraphsofloadsharing,contactpressure,contactstress,andcontactpathagainstrotationalangle 12. This model imple
24、ments penalty and augmented Lagrangian methods for surface to surfacecontact.Geometry of cycloidal reducerOne tooth difference Cycloidal reducer can reduce the input speed up to 87:1 in a single stage. The gearboxCycloCNH609-15, which is a horizontal, foot mounted, concentric shaft speed reducer of
25、ratio 15:1, is se-lected for the simulation modeling and analysis 13. The gearbox is chosen from the low reduction ratio cat-egory offered by the manufacturer. As discussed in Introduction section, the low reduction ratio results ingreater induced stresses in thecycloidal disc on account of reduced
26、material content of the disc. This modelwouldbeobservedforsuchstressesinthecycloidaldisc. Thereducerdesigniscomprisedofonediscmech-anism along with a counterweight (as shown in Figure 1), acting as a substitute to the second disc which isapparent in high torque transmitting reducers for dynamic bala
27、nce and load sharing. This mechanism givesan advantage to focus on the contact loading behavior of only one cycloidal disc.Theepi-trochoidaltoothprofileofthecycloidaldisc isa vitalfeature inthe reducermechanism. Theoretically,fortheonetoothdifferencereducer,alldiscteethorlobesontheprofileremainincon
28、tactwithsubsequentringgearhousingrollers(outerrollers)andhalfoftherollersareconsideredparticipatingintorquetransmissionatany instance 8. However, the manufacturing errors and clearances for lubrication etc. impede theapplicationofalltooth-rollercontacts. Thetrochoidalprofileinthismodelisdevelopedfro
29、mthemanufacturingdrawing which takes tooth modifications and tolerances into account.5 12FTM17Modeling descriptionAlgor Graphics User Interface (GUI) and its finite element code are used to build and simulate the dynamicbehavior of the cycloidal reducer. The torque transmitting components shown in F
30、igure 2are importedfromthe CAD model to Algor GUI. For more contact simplification, eccentric rollers of the cam are excluded andthecorrespondinggapisfilledbyraisingthediameterofthecam. ThedynamicFEAmodelisbuiltcreating3Dsolidmeshofbrickelements. Abrickelementcomprisesof8nodeswithonlytranslationalde
31、greesoffreedomin3Dspace. Thiselement typeis preferredin thestudy sincethe solidmesh enginegenerates moreconsist-ent brick elements than any other type, like tetrahedral. Several attempts are made to establish a suitablemesh density which would capture exact geometry of the parts. For instance, the t
32、rochoidal profile on thecycloidaldiscperipheryhastobedefinedpreciselyinordertoobtaincorrectmatingoftheassembledrotatingparts.“Surface-to-surface” contact is established between two interacting parts; namely, eccentric bearing (cam)and the disc, the cycloidal disc and outer rollers, and lastly, the c
33、ycloidal disc and low speed rollers.Aconstantnodalprescribedrotationalmotionof20or30revolutionspersecond,assumedtobegeneratedbymotor, is applied to the center point of the cam hole. In the absence of high speed shaft, beam joints arecreatedtotransferprescribedmotiontotheeccentriccamasshowninFigure 3
34、. Thesebeamsaredefinedasrigidelementswithnomassdensity. Similarly,asdescribedinFigure 4,rigidbeamelementsarealso usedtodefine the relative motion between the low speed shaftrollers. The outputshaft loadis assignedby placinganodalmomentwhichactsatthecenterofthebeamelementlinkage,intheoppositedirectio
35、noftherotationoflowspeedshaftrollers. Theouterrollersarefixedinalldegreesoffreedomsattheirinnerdiameters. Thetotaltime duration of the simulation is set for 360 rotation of the eccentric cam. This time duration is divided intosmall time increment steps to capture simulation history at every 2 rotati
36、on.Figure 2. CAD imported components6 12FTM17Figure 3. Rigid beam members (in red color) at the center of eccentric camFigure 4. Linkage for low speed shaft rollers to depict low speed shaftA total Lagrangian formulation method is adopted to determine the equilibrium of strains at each timeincrement
37、. By this adoption, therefore, the model becomes geometrically non-linear where the stiffness andthegeometryareupdatedateachstep. IfthesimulationresultisanimatedinGUI,themovementsofthepartswould be seen because of the geometric updating. AISI E52100 material properties are assigned to theelements of
38、 all parts. The material defined in this study is elastic, homogenous and isotropic.The model is built with certain assumptions considering limitations of the finite element code and thesimplification adopted in this study. It is assumed that the sliding motion is absent between the contactingsurfac
39、es. So,thetorqueistransmittedfromoneparttoanotherbyperfectrollingcontact. Themodelingdoesnot take into account the friction between rotating parts, manufacturing errors, wear, thermal and vibrationeffects.7 12FTM17SimulationFiniteelementcommercialcodeusessurface-to-surfacecontactalgorithmtoanalyzeth
40、econtactforcesgen-erated by the interaction between parts. The contact algorithm involves an iterative process where the solu-tionisderivedfromdisplacement-basedconvergence. Thesolutionisreachediterativelyateachdefinedtimeincrement step by solving non-linear equations using full Newton-Raphson integ
41、ration method 14.ThisiterativeprocessispresentedgraphicallyinFigure 5wherethethickbluecoloredlineshowsatheoreticalstiffnesscurve. Here,threeiterationsareperformedtocalculatestiffnesscurvesk1,k2andk3. Theforce,F,isconsidered as an appropriate solution for displacement, D, when an increment d at the l
42、ast iteration, toobtain curve k3, is within the defined tolerance. To obtain precise approximation of the stiffness, one candefineverysmalltolerance;however,itwouldincreasethenumberofiterationsandultimatelythetotaltimetocomplete the simulation.Thecontactbetweentwosurfacesisidentifiedbyapplyingcontac
43、tstiffnessbetweennodesofthosesurfaces.Thiscontactstiffnessisapplied,whichimpliesthatthecontacthasoccurred,atthemomentwhenthedefinedcontact tolerance is greater than the distance between the surfaces. For too small contact stiffness, the sur-face nodes are unable to experience the reaction force from
44、 the nodes of the othercontacting surface,whichresults in surface penetration. If the contact stiffness is too high, the iterative time step may be reduced to asmall value causing a convergence problem which makes the model unstable. The contact stiffness iscalculated by the software from the proper
45、ties of material of the parts using the following equation:K =fsEL2S(2)whereK is contact stiffness,fsis the scaling parameter,E is the Youngs Modulus of the material,L is the contact surface area, andS is the volume of the element.Figure 5. Pictorial representation of stiffness curve iterations8 12F
46、TM17In dynamic analysis, stresses are estimated considering the material behavior, motion and gravity of theparts. Equation 3 of the simulation in matrix form relies on the combination of Hooks law and Newtonssecond law.Ma+ Cv+ Kd = f(3)whereM is the mass,a is the acceleration vector,C is the dampen
47、ing constant,v is the velocity vector,K is stiffness,d is the displacement vector due to force, andf is the force vector.The involvement of the acceleration term allows simulating impact motion of the cycloidal disc on the outerrollers.Results and discussionUnliketheinvolutetoothprofilegearingwheret
48、hecontactloadisconcentratedonlyatasmallareaofthegearpair,thecycloidalreduceriswellknownfordiffusingtheloadoveritscomponents. Thisloaddistributionabilityof thecycloidal gear mechanism forms a high shock load absorption capacity. Further discussionin thissec-tion investigates contact load and stress d
49、istribution in dynamic conditions. The analysis is performed at fullload 134.47 kN-mm moment which acts on the low speed shaft in the direction opposite to the rotation ofoutput shaft. The eccentric cam rotates one revolution under the prescribed displacement of 1200 rpm.Figure 6 presents the Von Mises stress contour of the disc, housing rollers and low speed shaft rollers at 54from X-axis at time instance 0.005 sec. After analyzing simulation data history collected at 180 time in-stances, it is inferred that the cycloidal disc shares the
