10FTM06

AGMA Technical Paper

Finite Element Analysisof High Contact RatioGearBy M. Rameshkumar, G.Venkatesan and P. Sivakumar,DRDO, Ministry of Defence

Finite Element Analysis of High Contact Ratio Gear

M. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of Defence

[The 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.]

AbstractModern day vehicles demand higher load carrying capacity with less installed volume and weight. The gearsused in the vehicles should also have lesser 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 gearing 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 paperdeals 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 Design language (APDL) is used forstudying 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 problems.

Copyright 2010

American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314

October 2010

ISBN: 978--1--55589--981--3

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Finite Element Analysis of High Contact Ratio Gear

M. Rameshkumar, G. Venkatesan and P. Sivakumar, DRDO, Ministry of Defence

Introduction

A majority of 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 pressureangle 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 gears 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 variation 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 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 FiniteElement 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 arecompared andplottedfor identical load conditions.

Design for HCR gear pair

Contact ratio 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+ a

2− r2

2 cos2α − r1+ r2 sinα

π m cosα

CR=r1+ a

2− r2

1 cos2απ m cosα

(1)

where

r1, r2 are the operating pitch radius of the pinionand gear,

α 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 waysnamely:

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 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.

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Figure 1. Contact ratio versus number ofteeth

Figure 2. Contact ratio versus pressure angle

Figure 3. Contact ratio versus addendumfactor

Generation of gear pair model

Spur gear geometry

The 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 a rack cutter is

modeled 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 meets the in-volute profile at the common point of tangency andthe involute profile extends up to the addendumcircle.

Table 1. Gear parameters

SI No. Parameters NCR HCR1. Profile Involute Involute

2. DIN accuracy class 7 7

3. Module, m 2.5 mm 2.5 mm

4. Number of teeth ingear, Z1

50 50

5. Number of teeth inpinion, Z2

47 47

7. Profile correction ingear, X1

0.1552 0.1552

8. Profile correction inpinion, X2

0.152 0.152

10. Center distance, Cd 122 mm 122 mm

11. Reduction ratio, Gr 1.06383 1.06383

12. Addendum factor,Ya

1.0 1.25

13. Contact ratio, CR 1.6860 2.06266

14. Face width, F 18 mm 18 mm

Figure 4 shows generation of 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), addendum (b), pressureangle and profile correction required in the gear.

θt= tan−1⎪⎡⎣

R2t− (R− b)2

(R− b) ⎪⎤⎦−⎪⎡⎣

R2t− (R− b)2

R ⎪⎤⎦(2)

tan ψt=R R− b2

− R2t

R R2t − R− b2

2 (3)

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Figure 4. Trochoid profile generation

The co-ordinates of the actual fillet are determinedfrom:

Rf= R2t + A2− 2 A Rt sinψt (4)

θf= θt+ cos−1 Rt− A sinψtRf (5)

x i= Rf sin θ′′f (6)

y i= Rf cos θ′′f (7)

θ′′f = δt+ θf (8)

where

A is the cutter tip edge radius;b is the distance between the pitch line of

the cutter and the center of the outeredge;

R is the operating pitch circle radius;Rt , Rf is the radius vector of the trochoid and root

fillet;

ψ is the angle between the radius vector andtangent to trochoid;

θt is the angle between the radius vector andcenterline of trochoid;

δ is the angle between the center of the geartooth and the center of the trochoid;

θf is the vectorial angle of fillet in reference to

selected center line;θf is the original vectorial angle of fillet;xi is the abscissa of the profile co-ordinate;yi is the ordinate of the profile co-ordinate.

Finite element analysis of NCR and HCRgear pair

A 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 thegearpair was carried out. 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 andmeshing

A list of the assumptions adopted in the presentwork is given below:

S The gear material is assumed to be homogen-eous, isotropic and elastic according to Hooke’slaw, and the material properties required for theanalysis are Young’s modulus of elasticity(2.1e5 N/mm2) and Poisson’s ratio (0.3).

S The load distribution along the face width isassumed uniform and plane strain method isadopted. [8]

S The effect 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 small.[10]

S All the manufacturing errors and geometricalerrors are neglected.

Finite element model of gear and meshgeneration

Both 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 are 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 contactsolution, the finite element models are meshed with

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a 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 element

Figure 6. Finite element meshed model ofNCR gear pair

Figure 7. Finite element meshed model ofHCR gear pair

Loading and boundary conditions

In 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 circumfer-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 created at the involuteportion of the gear pair in contact.

Solution and post processing

Estimation of percentage load sharing

Each 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 gear 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, loadsharing ratioandthe contact stress are obtained for all the 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 maximum 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 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 double tooth contact) which isabove the pitch circle and only 20% load is sharedby the tip of the teeth during the course ofengagement.

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Load sharing comparison NCR and HCR

Load sharing ratio in terms of percentage loadshared from root to tip of a particular tooth for theNCR gearing 47/50 is shown in 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 8.5 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 observed 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. Again 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 8o to 8.5 in view of single to two tooth contact

and the load sharing ratio decreases gradually to-wards tip. This phenomenon 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 single 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 contact 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 minimum 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% and minimum percentage ofload shared is 42.5%.

Figure 8. Load sharing of 47/50 teeth

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Figure 9. Load sharing of 467/50 teeth HCR gear

Bending stress comparison of NCR andHCR gear pair

The 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 ratio.However, the stress level is more near the tip regionin view of the cantilever effect and the tooth shape.The bending stress at the root is about 175.7 MPaand it increases up to 500 MPa at a 6 rotation anglecorresponding to that of single tooth contact. Thestress varies from 500 MPa to 580 MPa throughoutsingle tooth contact when the load sharing ratio isconstant i.e. 100%. The variation is again due tothe cantilever effect on the tooth. The bendingstress again reduces from 580 MPa to 445 MPa atthe tip. However, the bending stress is more nearthe tip compared to the root as explained above.The maximum bending stress occurs at the HPSTCpoint and the minimum bending stress occurs at theroot.

The variation of bending stress from root to tip ofany tooth of the 47 tooth HCR gear which is in con-tact with the 50 tooth gear is shown in Figure 11.The load is shared by two teeth in the region BC and

DE, while load is shared by three teeth in the regionAB, CD and EF. 20% load sharing occurs at the tipand as the load increases, the stress also increasesand the maximum bending stress of 478 MPa isobserved at the first lowest point of double tooth(FLPDTC) contact. As the load increases, in thethree teeth contact zone CD, the bending stressvariation is identical to the load sharing behavior.Further, in the two tooth and three tooth regions, DEand EF, respectively, the bending stress decreasesas the load decreases. From Figure 10 andFigure 11, it is noted that the maximum bendingstress occurs at the highest point of single toothcontact (HPSTC) for the NCR gear and it occurs atthe FLPDTC for the HCR gear. The maximum toothroot bending stress is 18% less in the HCR gearcompared to the NCR gear.

Li [12] has found that the increase in addendum in-creases the number of teeth in contact therebydecreases the tooth contact stress and the rootbending stress. But this increase also makes thetooth depth long and allows the tooth to bedeformed easily. Therefore, though, the incrementin addendum can reduce the tooth contact stress,there is no guarantee that this increment can cer-tainly reduce the tooth root bending stress. Toothroot stress is increased if the addendum becomes

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longer and the number of contact teeth has nochange. Tooth root stress can also be reducedwhen the number of teeth in contact is increasedthrough increasing the addendum. But there is noguarantee that this increment of the number of teeth

can certainly reduce the root stress. This is be-cause the increase of addendum also makes thewhole depth of the teeth longer so that a largermoment occurred at the tooth root.

Figure 10. Bending stress of 47 teeth NCR gear

Figure 11. Bending stress of 47 teeth HCR gear

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Contact stress comparison of NCR andHCR gear pair

In this study, the contact stress at the meshes wascalculated based on the tooth load distributed on aunit contact area of the tooth surface. Tooth contactstress was analyzed using the Hertz formula but it isnot precise enough [13]. Tooth contact stresses areoften calculated with the Hertz formula when thetooth loads are known. Since the Hertz formula wasdeduced from two symmetric elastic cylinders, it isnot precise enough for contact stress calculations ofgear teeth, a cantilever structure with involute pro-files. Especially at engagement positions such astooth tip and root contacts, the Hertz formula doesnot give the correct contact stress values [6]. Themaximum load applied at the FLPDTC, where twoteeth are in contact, is only 57% and hence the max-imum tooth contact stress is lesser by 19% in theHCR gear compared to the NCR gear. The reduc-tion in the contact stress is due to the increase of theaddendum factor and the number of contact teeth,resulting in a lesser load.

Tooth loading condition is a critical parameterespecially in HCR gear applications. The toothcontact stress which is critical to wear, pitting andoperating temperature was studied by Cornell andWestervelt [14]. Wang and Howard [6] carried outan analysis of HCR spur gears with tooth profilemodification. They concluded that the contactstress was significantly reduced compared to theunmodified gears. The contact stresses tended to

be smooth with only small stress irregularities at therelief starting point.

The variation of contact stress on the 47 tooth NCRgear as shown in Figure 12 resembles its loadsharing behavior. It varies from 820.3 MPa at theroot to 1200 MPa at the start of single tooth contact,corresponding to 6 degrees, and the stress ismaintained constant till 8 degrees and furtherdecreases to 811 MPa at the tip, similar to the loadsharing pattern.

The contact stress pattern of the 47 tooth HCR gearis shown in Figure 13. As in the case of the NCRgear, its variation in contact stress of the HCR gearresembles its load sharing behavior. The contactstress increases from 770.5 MPa at the root to 988MPa at the start of FLPDTC. The contact stress re-duces in the double tooth contact region initially andfurther the contact stress increases and reaches themaximum value of 988 MPa at the start of FLPDTCequivalent to rotation angle of 10.5 degrees. There-after the contact stress decreases with respect tothe rotation angle in both the two tooth and threetooth region. Wang and Howard (2008) carried outresearch on the contact zone of the NCR and theHCR gears with profile modification [6]. For the tiprelieved HCR gears, the width of single, double andtriple contact zone will change dramatically as theload increases. The HCR gears will have singletooth contact under light loads. Single contact zonein the mesh cycle can quickly disappear when theload is increased to become a triple contact zonethat will expand in zone width as the load is furtherincreased.

Figure 12. Contact stress of 47 teeth NCR gear

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Figure 13. Contact stress of 47 teeth HCR gear

Conclusions

Quasi static finite element analysis was carried outfor NCR and HCR gears with fixed module, centerdistance and gear ratio. Here the increased contactratio is obtained by increasing the addendum factorfrom 1.0 to 1.25 m. Hence a contact ratio of morethan 2.0 was achieved for the same number ofteeth.

Two dimensional deformable body contact modelsfor both HCR gear and NCR gears were created us-ing the ANSYS-APDL loop program. Various para-meters such as load sharing ratio, bending stressand contact stress were evaluated and comparedover the path of contact.

The maximum percentage of load sharing occurs atthe HPSTC point in the case of a NCR gear and theFLPDTC point in the case of a HCR gear pair. Atany point of time, the HCR gear tooth experiences amaximum of 57% of the total load against 100% inthe NCR gear pair. At the tip of the tooth, a HCRgear shares 20% of the total load against 40% in aNCR gear pair. The maximum bending stress for aHCR gear is 18% less and contact stress is 19%less as compared to a NCR gear for the specificcase of same module and fixed center distance.

Hence the load carrying capacity of the HCR gear is18% more than the NCR gear designed for thesame weight and volume for fixed module and samecenter distance gear pairs.

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