1、10FTM06AGMA Technical PaperFinite Element Analysisof High Contact RatioGearBy M. Rameshkumar, G.Venkatesan and P. Sivakumar,DRDO, Ministry of DefenceFinite Element Analysis of High Contact Ratio GearM. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of DefenceThe statements and opinions
2、contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractModern day vehicles demand higher load carrying capacity with less installed volume and weight. The gearsused in the vehicles should also have l
3、esser noise and vibration. Even though helical gears will meet therequirement, they are prone for additional axial thrust problem. High contact ratio (HCR) is one such gearingconcept used for achieving high load carrying capacity with less volume and weight. Contact ratio greater than2.0 in HCR gear
4、ing results in lower bending and contact stresses. Previously published literature deal withstudies on various parameters affecting performance of HCR gears but a comparison of HCR and normalcontact ratio (NCR) gears with same module and center distance has not been carried out so for. This paperdea
5、ls with finite element analysis of HCR, NCR gears with same module, center distance and the comparisonof bending, contact stress for both HCR, NCR gears. A two dimensional deformable body contact model ofHCR and NCR gears is analyzed in ANSYS software. ANSYS Parametric Designlanguage (APDL) is usedf
6、orstudying the bending and contact stress variation on complete mesh cycle of the gear pair for identical loadconditions. The study involves design, modeling, meshing and post processing of HCR and NCR gears usingsingle window modeling concept to avoid contact convergence and related numerical probl
7、ems.Copyright 2010American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October 2010ISBN: 978-1-55589-981-33Finite Element Analysis of High Contact Ratio GearM. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of DefenceIntroductionA majority of
8、 the heavily loaded transmissions usedin military applications use gears with a contact ratioless than 2.0. The contact ratios of these transmis-sions are in the range of 1.3 to 1.8. So, the numberof teeth in engagement at any instant is either one ortwo. Many gear designs use increased pressureangl
9、e for increasing the load carrying capacity ofgears with fixed module and center distance, but thecontact ratio decreases. Tooth dynamic loads andnoise increase due to decreased pressure angle.Hence increasing the load carrying capacity ofgears for the above conditions can be done by thedesign of ge
10、ars with a contact ratio greater than 2.0.High contact ratio gears having a contact ratiogreater than 2.0 have load sharing between two orthree teeth during engagement and less load pertooth 1. The high contact ratio (HCR) gearsguarantees that a minimum of two teeth alwaysshare the load. The variati
11、on of gear mesh stiffnessfor HCR gears is less than the normal contact ratio(NCR) gears; the transmission error for HCR gearsis minimum compared to NCR gears.The literature survey indicated that HCR gearingwas designed 2 and successfully used in heli-copter transmissions 3, to improve power toweight
12、 ratio of the gear trains. This study deals withestimation and comparison of tooth root bendingstress and contact stress over the path of contactfor high contact ratio gears 1, 4 and normal contactratio gears designed with identical module, centerdistance, gear ratio and face width using FiniteEleme
13、nt Analysis. In order to overcome the numer-ical and convergence difficulties 5 involved a newsingle window modeling approach 6 using ANSYSParametric Design Language (APDL). The contactstress and bending stress are compared and plottedfor identical load conditions.Design for HCR gear pairContact rat
14、io of a gear pair is defined as the averagenumber of teeth in contact during the course of en-gagement. The contact ratio of the gear pair playsan important role in increasing the load carrying ca-pacity of gears.The contact ratio (CR) for any gear pair is given byequation 1.+r2+ a2 r22cos2r1+ r2sin
15、 m cosCR =r1+ a2 r21cos2 m cos(1)wherer1,r2are the operating pitch radius of the pinionand gear, is the operating pressure angle;m is the module and a is the addendum(based on the operating pitch radius)which is equal to one module for standardgears.High contact ratio can be achieved by different wa
16、ysnamely:S Increasing the number of teeth;S Lowering the pressure angle;S Increasing the addendum factor.Figure 1, Figure 2 and Figure 3 show the variationsof contact ratio with respect to above parameters.In order to achieve a contact ratio more than 2.0 fora gear pair with identical module, center
17、 distance,gear ratio and pressure angle the addendum factorof the gears pair is increased from a standard 1.0 mto 1.25 m. The entire tooth parameters of a HCRgear pair are calculated using a Matlab code andtabulated in Table 1.4Figure 1. Contact ratio versus number ofteethFigure 2. Contact ratio ver
18、sus pressure angleFigure 3. Contact ratio versus addendumfactorGeneration of gear pair modelSpur gear geometryThe profile of an involute spur gear tooth is com-prised of two curves. The working portion is the in-volute and the fillet portion is the trochoid. Thetrochoid tooth fillet as generated by
19、a rack cutter ismodeled exactly using the procedure suggested byBuckingham 7. An “APDL” computer languagecode in ANSYS was developed for generating anexact tooth profile with a trochoidal fillet. The troch-oidal fillet form is generated from the dedendumcircle up to the limiting circle, where it mee
20、ts the in-volute profile at the common point of tangency andthe involute profile extends up to the addendumcircle.Table 1. Gear parametersSI No. Parameters NCR HCR1. Profile Involute Involute2. DIN accuracy class 7 73. Module, m 2.5 mm 2.5 mm4. Number of teeth ingear, Z150 505. Number of teeth inpin
21、ion, Z247 477. Profile correction ingear, X10.1552 0.15528. Profile correction inpinion, X20.152 0.15210. Center distance, Cd122 mm 122 mm11. Reduction ratio, Gr1.06383 1.0638312. Addendum factor,Ya1.0 1.2513. Contact ratio, CR 1.6860 2.0626614. Face width, F 18 mm 18 mmFigure 4 shows generation of
22、a trochoidal fillet bythe tip of the basic rack with a = 0 (sharp cutter).Coordinates of the trochoid are calculated using theAPDL program using equation 2 through 8 as notedin 7. The type of Trochoidal fillet changes withparameters of the cutter like type of cutter (pinion orrack), edge radius (a),
23、 addendum (b), pressureangle and profile correction required in the gear.t= tan1R2t (R b)2(R b)R2t (R b)2R(2)tan t=RR b2 R2tRR2tR b22(3)5Figure 4. Trochoid profile generationThe co-ordinates of the actual fillet are determinedfrom:Rf= R2t+ A2 2 ARtsint(4)f= t+ cos1Rt A sintRf (5)xi= Rfsin f(6)yi= Rf
24、cos f(7)f= t+ f(8)whereA is the cutter tip edge radius;b is the distance between the pitch line ofthe cutter and the center of the outeredge;R is the operating pitch circle radius;Rt,Rfis the radius vector of the trochoid and rootfillet; is the angle between the radius vector andtangent to trochoid;
25、tis the angle between the radius vector andcenterline of trochoid; is the angle between the center of the geartooth and the center of the trochoid;fis the vectorial angle of fillet in reference toselected center line;fis the original vectorial angle of fillet;xiis the abscissa of the profile co-ordi
26、nate;yiis the ordinate of the profile co-ordinate.Finite element analysis of NCR and HCRgear pairA two dimensional deformable body symmetric con-tact model of HCR and NCR gear pairs wasmodeled using a ANSYS APDL looping programand a quasi static Finite element analysis of the gearpair was carried ou
27、t. The various parameters suchas load sharing ratio, bending stress, and contactstress are estimated over the path of contact forboth NCR and HCR gearing.Assumptions for finite element models andmeshingA list of the assumptions adopted in the presentwork is given below:S The gear material is assumed
28、 to be homogen-eous, isotropic and elastic according to Hookeslaw, and the material properties required for theanalysis are Youngs modulus of elasticity(2.1e5 N/mm2) and Poissons ratio (0.3).S The load distribution along the face width isassumed uniform and plane strain method isadopted. 8S The effe
29、ct of case hardness, case depth and theoil film thickness is neglected.S The surface asperities and waviness is neg-lected.S Root fillet curves are assumed to be circular 9,here assumed actual trochoid.S Sliding friction between the mating gear teeth isneglected, since its effect on deflection is sm
30、all.10S All the manufacturing errors and geometricalerrors are neglected.Finite element model of gear and meshgenerationBoth the NCR and HCR gears are kept in contact bypositioning at the stipulated center distance (122mm) with respect to the global coordinate systemand only the plane area models ar
31、e used for theFEA. Quadratic two dimensional (2D) plane 183higher order elements as shown in Figure 5 areused with plane a strain option 8. The element isdefined by eight nodes having two degrees of free-dom at each node which are nodal translations in xand y directions. For easy convergence of a co
32、ntactsolution, the finite element models are meshed with6a very fine mesh i.e., 0.01mm element edge lengthwhere the tooth will experience contact. The finiteelement mesh of NCR and HCR gear pairs areshown in Figure 6 and Figure 7, respectively.Figure 5. Plane 183 quad elementFigure 6. Finite element
33、 meshed model ofNCR gear pairFigure 7. Finite element meshed model ofHCR gear pairLoading and boundary conditionsIn order to study the load sharing, identical bendingstress and contact stress loads were applied forboth NCR and HCR gear pairs. A maximum of 373Nm is applied at all nodes lying on the c
34、ircumfer-ence of the inner hub diameter of the pinion gear (47teeth) and the pinion is arrested in the radial direc-tion with respect to the local coordinate system.The gear of the 47-50 teeth gear pair (ie) 50 teethgear is fully constrained in all directions. A symmet-ric contact element 11 was cre
35、ated at the involuteportion of the gear pair in contact.Solution and post processingEstimation of percentage load sharingEach gear is rotated as a rigid body according to thegear ratio. The solution is repeated for both NCRand HCR gears rotated with same amount ofangular increment according to the g
36、ear ratio.Approximately 30-50 angular increments with 0.5degree steps are used for this analysis and the ana-lysis is carried out with the help of the customizedAPDL (ANSYS Parametric Design Language) loop-ing program 11. Root stress, load sharing ratio andthe contact stress are obtained for all the
37、 gear meshpositions. The nodal forces at each node of thecontact element are captured from the ANSYS postprocessing for each individual gear tooth. By thismethodology, the percentages of load sharingbetween teeth are estimated for both the gear pairsthroughout the path of contact.Accordingly, the ma
38、ximum percentage of loadshared by the individual teeth for the NCR and HCRgears are estimated for the entire path of contact.The individual tooth loads have been determined bycomparing the total normal load to the sum of thenormal loads contributed by each pair of contactequally.It is observed from
39、the above FEA that for the NCRgearing the maximum load of 100% is taken by thesingle tooth at the HPSTC point (Highest point ofsingle tooth contact) and at the tip of teeth only 40%load is shared. However, for the HCR gearing, themaximum load of 57% load is taken at the FLPDTC(First lowest point of
40、double tooth contact) which isabove the pitch circle and only 20% load is sharedby the tip of the teeth during the course ofengagement.7Load sharing comparison NCR and HCRLoad sharing ratio in terms of percentage loadshared from root to tip of a particular tooth for theNCR gearing 47/50 is shown in
41、Figure 8. The loadsharing ratio is plotted with respect to rotation anglefor a single tooth of the 47 gear tooth from root to tipwhich corresponds to a rotation angle of 0.5deg(root) and 13.5 deg (tip). It can be seen from thegraph that the double tooth contact band is from 0.5deg to 5.5 and from 8.
42、5 deg to 13.5 deg, the maxim-um and minimum percentage of load shared inthese double tooth contact bands are 59% and 40%respectively. The single tooth contact band startsfrom 5.5 deg and ends at 8.5 deg. In the entirerange of the single tooth contact, 100% load is takenby the single tooth. It is obs
43、erved that the rate of in-crease of percentage load sharing from 0.5 deg to5.5 deg is gradual from 40% to 59%, whereas from arotation angle of 5.5 deg to 6.5 deg, the rate of in-crease of percentage load sharing is drastic ie from59% to 100% during the change over from doubletooth to single tooth. A
44、gain two tooth contact startsat 8.5 and gradually the load sharing reduced to40% at the tip, corresponding to 13.5. It can alsobe observed that the rate of decrease is very rapidfrom 8oto 8.5 in view of single to two tooth contactand the load sharing ratio decreases gradually to-wards tip. This phen
45、omenon is common to anytooth of the 47 tooth gear which is in contact.However, the tooth of the mating gear which comesin to contact carries the balance of the load in boththe cases i.e., 47/ 50 teeth gears.Figure 9 shows the load sharing ratio in terms ofpercentage of load from root to tip for a si
46、ngle toothof the HCR gearing (47/50 teeth). The load sharingratio is plotted with respect to rotation angle for asingle tooth of the 47 tooth gear from root to tipwhich corresponds to rotation angle of 0.5 deg (root)and after 18 deg (tip) period. It can be seen from theplot that the triple tooth con
47、tact band is from 0.5 degto 3 deg, 7 deg to 10.5 deg and from 15.5 deg to 18deg along the path of contact. On a single tooth, thetriple tooth contact occurs three times and duringthis period the maximum percentage of load sharedby any single tooth during the three tooth contactband is 46% and minimu
48、m percentage load is 20%.Similarly, the double tooth contact is from 3 deg to 7deg and from 10.5 deg to 15.5 deg, along the path ofcontact. On a single tooth, the double tooth contactoccurs twice and the maximum percentage of loadshared by any single tooth during the double toothcontact band is 57%
49、and minimum percentage ofload shared is 42.5%.Figure 8. Load sharing of 47/50 teeth8Figure 9. Load sharing of 467/50 teeth HCR gearBending stress comparison of NCR andHCR gear pairThe variation of bending stress from root to tip ofany tooth of the 47 tooth NCR gear which is in con-tact with the 50 tooth gear is shown in Figure 10. Itcan be seen from the figure that the rate of increaseand decrease of bending stress during two teethcontact is the same as the load sharing
