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Tribology International 40 (2007) 613619Modeling of helical gear contact with tooth deflectionJuha Hedlund?, Arto LehtovaaraTampere University of Technology, Machine Design, P.O. Box 589, 33101 Tampere, FinlandAvailable online 4 January 2006AbstractThe majority of gear tribological studies are made on spur gears. However, helical gears are generally used in industry, and theircontact behavior deserves more attention to establish a realistic base for detailed friction, wear and life studies. This study focuses on themodeling of helical gear contact with tooth deflection. A calculation model for helical gear contact analysis is introduced. Helical gearsurface profiles are constructed from gear tool geometry by simulating the hobbing process. The model uses three-dimensional finiteelements for the calculation of tooth deflection including tooth bending, shearing and tooth foundation flexibility. The model combinescontact analysis with structural analysis to avoid large meshes. Tooth foundation flexibility was found to have an essential role in contactload sharing between the meshing teeth, whereas contact flexibility plays only a minor role. The capability of different local contactcalculation methods was also studied.r 2005 Elsevier Ltd. All rights reserved.Keywords: Helical gear; Contact; Deformation; Load distribution; Modeling1. IntroductionHelical gears are generally used in industry and theircontact behavior deserves more attention to establish arealistic base for the detailed study of gear friction, wearand life. The gear contact stresses derived from toothcontact forces and geometry are very important fordetermining gear pitting, i.e. life performance. Toothcontact forces along the line of action depend essentiallyon load sharing between meshing teeth, and, therefore,a realistic analysis of helical gear contact also requiresinformation on structural deformations, such as toothdeflection.The majority of gear contact analyses within tribologicalstudies are made on spur gears. Typically, teeth contactthrough the line of action is modeled as a constantlychanging roller contact, whose radius, speed and load areapproximated from ideal involute gear geometry in thegiven operating conditions. Gear contact ratio 12 andequal load distribution in the case of two teeth in contact(half of the single-tooth load) are often assumed. Deforma-tions are calculated according to the Hertz line contacttheory, otherwise assuming rigid tooth behavior. Somestudies are made by slicing the helical gear to a series ofspur gears and treating these slices as spur gears 1.Finite-element-basedcalculationmodelsarewidelyacceptedforcalculatingstructuraldeformationsandstresses in spur and helical gears in the case of concentratedloads. In gear transmission and dynamic analyses, typicaldeformation studies have used the two-dimensional (2D)finite element method with spur gears 24. Coy and Chao5 studied finite element grid size dimensions to cover theHertzian contact. Du et al. 6 and Arafa et al. 7 laterenhanced contact modeling as a part of structural analysisby using gap elements for the calculation of spur geardeformation.Only a few helical gear contact studies, which includestructural deformations of the gear, have been performedwith 3D finite elements 8,9. This is mainly because in thiscase FEM contact modeling is computationally expensiveand time-consuming due to the small grid size which isnecessary on the gear flank surface. Different methods havebeen introduced to overcome this problem. Vedmar 10separated structural and contact analysis by combining theARTICLE IN PRESS/locate/triboint0301-679X/$-see front matter r 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.triboint.2005.11.004?Corresponding author. Tel.: +358331154442.E-mail addresses: juha.hedlundtut.fi(J. Hedlund),arto.lehtovaaratut.fi(A. Lehtovaara).finite element method and the Weber & Banaschekdeformation formula to study the contact behavior ofinvolute helical gears. Brauer 11 utilized the globallocalFE approach, which combines local dense and globalcoarse element meshes, applying it to calculate transmis-sion error in anti-backlash conical involute gears. Pimsarnand Kazerounian 12 avoided the use of contact elementsby introducing a fast pseudo-interference method for thecalculation of mesh stiffness in the case of a spur gear pair.The calculation was done with 2D plane elements. Thismethod was based on an elastic foundation model.In this paper, a parameterized calculation model forhelical gear contact analysis is introduced. The model isbased on 3D finite elements. The model combines contactanalysis and structural analysis to avoid large meshes. Theeffect of tooth foundation and Hertzian deformation onload sharing between the meshing teeth is analyzed with thedeveloped model. The capability of pseudo-interference-based method utilizing 3D finite elements in contactcalculations is also studied.2. Geometry modelNumerical approach is used to create helical geargeometry by simulating the hobbing process. This methodis based on a wide set of numerical calculation points andtheir synchronizing, which allows deviations from involutegeometry. Two movable coordinate systems are used,which are rigidly connected to gear (x1,y1) and rack(x2,y2), as shown in Fig. 1. Coordinate transformationmatrix M21consists of a rotation matrix and a translationmatrix. An inversed transformation matrix gives the gearprofile in a stationary coordinate system. The rotationmatrix is given asMg1cosf?sinf00sinfcosf00001000012666437775(1)and the translation matrix in the case of a helical gear asM2g100rp1f010?rp1 h1001000012666437775,(2)M21 M2gMg1,(3)ARTICLE IN PRESSNomenclatureacontact ellipse radiibcontact ellipse radiibggear widthdcontact deformation at center of contactEelasticity modulusE0reduced modulus of elasticity 2=E0121 ?v21=E1 1 ? v22=E2?etijdifference between contact surface profilesFccalculated total contact forceFijforce at surface node i,jh1distance from pitch linei,jindicates grid nodesKijstiffness value at grid point (i,j)Keqijreduced total stiffnessLeffective length of the line of actionMg1rotation matrixM2gtranslation matrixM21transformation matrixM12transformation matrixmnnormal modulepcontact pressurep0maximum contact pressurer1x,r1yprincipal radii of solid 1r2x,r2yprincipal radii of solid 2rp1pitch radiussinitial separation of surfacesTpinion torqueucontact normal deformationWapplied total forcex,x0coordinatesy,y0coordinateszdepth coordinatez1,z2number of teethbhelix angledcomposite rigid-body movementegtotal contact ratiozijforced nodal displacementnpoisson ratioxcoordinate along the line of actionrradiusfrotation angleSubscripts1refers to body 12refers to body 2caseapplied test caseJ. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619614Transformation matrix M12describes tool rack rollingwithout sliding around the pitch circle of the gear, as in thecase of hobbing. The gear teeth profile can be created fromthe set of curves shown in Fig. 2. The gear teeth profile isextruded along to a helicoid curve to 3D surface geometry.Contact line is determined by searching the minimumdistance between two undeformed mating surfaces. Finally,the tooth pair surface geometry and the contact line arenumerically obtained.3. Finite element modelEight-node trilinear hexahedral elements are exploited inthe FEM model. The developed model has its own mesher,solver and postprocessor. MatlabTMis used as a program-ming tool.The surface geometry data obtained from the geometrymodelisutilizedinthefiniteelementmodel.Theparameterized mesher creates 3D solid elements from thesurface data and adds another parameterized mesh belowthe tooth, which covers the tooth foundation. Afterstiffness and mass matrix integration, these two meshesare summed up in an assembly process.The stiffness matrix and mass matrix assembly processesare performed by freedom-pointer-technique using anelement freedom table. Contact load values are shared toequivalent nodal loads. Finally, the model gives nodalstiffness, displacement and stress values as output. Thetotal stiffness values of gear mesh can be calculated fromthese nodal displacement values.4. Contact formulationGear contact geometry is described with two non-conformal smooth surfaces, as shown in Fig. 3.Under the action of normal load W, the two bodies aredeformed and approach each other by distance d. Withinthe real contact, elastic deformation u(x) added with initialARTICLE IN PRESSFig. 2. A tool rack rolling around the pitch circle, the set of curves.Fig. 1. Rotating and translating coordinate systems 13.M12 M?121cosfsinf0sinfrp1 sinfh1? cosfrp1f?sinfcosf0cosfrp1 cosfh1 sinfrp1f00cosf2 sinf20000cosf2 sinf226666643777775.4Fig. 3. Schematic view of undeformed smooth surface profiles in contact.J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619615separation s(x) must be equal to the rigid-body movementd and outside the real contact greater than d as follows:ux sx d;px40,(5)ux sx4d;px 0.(6)In addition, the resulting pressure distribution mustsatisfy the force balance in normal direction with totalforce W applied on the contacting bodies. It follows:Z1?1Z1?1px;ydxdy W.(7)For 3D elastic contact problems the Boussinesq for-mulation can be used. The basic equation for surfacepressuredeformationofsurfacesinz-directionu(x)between semi-infinite solids is 14ux 2pE0Z1?1Z1?1px0;y0dx0dy0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix ? x02 y ? y02q.(8)Contact problems with arbitrary undeformed surfaceprofiles need to be solved by numerical methods. Thenumerical solution process is typically iterative, becausepressure distribution and real contact area distribution areunknown, whereas total load, material properties andinitial contact geometry are known parameters.The well-known Hertz solution to the contact problem isbased on cases, where the undeformed geometry ofcontacting solids can be represented in general terms bytwo ellipsoids. The solution requires the calculation ofellipticity parameter and complete elliptic integrals. Asimplified solution of the classical Hertz theory of ellipticalcontact solution is presented in 15. This calculationmethod is non-iterative and fast. The solution includes anelastic half space assumption.4.1. FEM-based contact modelInitially, geometrical overlap between the contact bodiesis chosen to produce a calculation domain greater than thefinal contact area. The loading vector acting in thecalculation domain is a combination of nodal displace-ments (overlap) and zero loads, producing non-homoge-nous FEM boundary conditions. This approach is appliedto calculate force distribution over the domain. Forcedistribution Fij, which is located at the surface, is used forthe calculation of contact stiffness values Kijin every node(i,j) as follows:KijFijBij.(9)Reduced stiffness values in the grid between the contactbodies are determined as follows:KeqijKij1Kij2Kij1 Kij2.(10)After the reduced contact stiffness is established and theinitial separation of the undeformed surfaces is known,contactparametersarecalculatedbyapproachingthe contact bodies gradually until load balance is achieved(Eq. (11). The condition of the contact is checked at eachiteration step at the nodal points, i.e. parameter etijisupdated.Total contact force is determined asFcXiXjKeqijetijif etij40; etij 0.(11)The FEM-based contact model takes into accountstructure boundaries, i.e. no half space assumption isneeded.5. Result and discussion5.1. Contact model test caseThe FEM-based contact model was tested againstsimplified Hertzs formulas 15 in the case of circularand elliptical contact. The test case dimensions and loadconditions are shown in Table 1 and the results in Table 2.Both surfaces have the same material properties. Theelliptical test case was chosen to evaluate the crowned spurgear contact. Mesh size was limited to 4500 elements percontact body in the test case calculation. One surfacecalculation domain consists of 900 nodal points. Thedeformed contact surface and calculation grid are shown inthe case of circular contact (Fig. 4).The results show that the FEM-based contact modelgives reasonable approximation of contact parameterstaking into account the fairly coarse grid size. The minorsemi-axisoftheellipseespeciallysuffersfromgriddimensions. Mesh size and shape have a certain effect onthe results unless the mesh size is fine enough. It is obviousthat the accuracy of results will decrease as the ellipticityratio of the contact increases.ARTICLE IN PRESSTable 2Comparison resultsModela (mm)b (mm)d (mm)p0(GPa)Hertz/circular0.370.3718.13.56FEM/circular0.40.419.13.11Hertz/elliptical0.274.1320.72.11FEM/elliptical0.44.020.22.36Table 1Test case specificationsCaseCircularEllipticalE (GPa)206n0.3r1x(mm)12.312.3r2x(mm)18.618.6r1y(mm)12.31000r2y(mm)18.61000W (N)10005000J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 6136196165.2. Helical gear contact caseIn helical gear mesh, contact is more complex than in thecase of spur gears. The contact area has a real 3D nature,which makes the contact curvature change along thecontact line and the line of action. Also, load sharingbetween the gear teeth is complicated partly because thetotal force is often shared between three teeth pairs. Therealistic force acting on a single tooth at any location alongthe line of action is the basic parameter in tribologicalcontact studies.Force distribution between meshing teeth pairs wasstudied in the developed model. Four different test caseswere established where the model allows:(1) tooth and tooth foundation deformations with rigidcontact;(2) tooth and tooth foundation deformations with contactdeformation;(3) tooth deformation with rigid foundation and rigidcontact;(4) tooth and contact deformation with rigid foundation.The stiffness vector for a tooth pair consists of 30calculation points along the line of action. Two thousandnine hundred and twenty-six elements per tooth weresolved at every calculation point. The stiffness vector of asingle gear pair was copied with offset to represent otherteeth in contact. The total mesh stiffness vector of a gearpair was obtained by summing up these single stiffnessvectors. Finally, displacement along the line of action wascalculated and the contact force of a single tooth wassolved.Contact stiffness along the line of action was calculatedon the Hertz line contact formula by using the forcedistribution from test case 1. The values of tooth flankradius are calculated over the contact line and estimated asa mean value. The overall reduced stiffness was obtainedby the iterative method, as in Eq. (10).Helical gear contact was studied with the FZG test rigrelated gear data shown in Table 3. In this example, totalcontact ratio was over 2, which means that there are alwaysat least two teeth pairs in contact. The test case elementmesh and the mating tooth surface for pinion are shown inFig. 5. The element mesh of the gear is equal.The calculated force distribution curves of the test caseare shown in Fig. 6. The line of action is described withnon-dimensionalparameterc x/L, where Listheeffective length of the line of action.Fig. 6 shows that the general trend in force distributionremains in the different model test cases. The area betweenthe sharp edges near the middle represents a situationwhere three teeth pairs carry the total load. In the case ofhelical gears, this transition from two to three teeth andvice versa occurs quite smoothly. Contrary to spur gears,single tooth force is high, when all three teeth are incontact. This is because flexibility is lower at the tooth tipcorners than in the middle and root area.The different test cases produce clear differences incontact force behavior. The flexibility of tooth foundationhas the most crucial effect on the distribution of contactforce along the line of action. Contact flexibility has lessimpact, but interestingly, it shifts the force distributioncurve slightly to the right in certain areas. This is becausethe combined contact radius is asymmetric over the pitchpoint. The load distribution was observed to be sensitive tostiffness properties at the start and end points of the line ofaction.One contact point (c ?0.236), shown in Fig. 6, waschosen for a closer study. This contact point was estimatedARTICLE IN PRESSFig. 5. Element mesh of pinion tooth used in the test case.Fig. 4. Deformed contact surface.Table 3Test case gear datamn(mm)2.75bg(mm)20b (deg)12z126z239eg2.084T (Nm)143J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619617with the Hertzian elliptical contact formula 16 and theFEM-based contact model. This contact point representsthe situation where two teeth pairs carry the total load. Thechosen contact situation was calculated with two differentforces corresponding to the calculated test cases 2 and 4.The load sharing differs depending on the modeling of thetooth foundation. The radius of surface profiles wasapproximated with the circum circle method and thesevalues are shown in Table 4. Some crowning was includedin the contact line direction.At the studied contact point, different load sharingbetween cases 2 and 4 has only a minor effect on contactparameters (Table 5). However, force difference betweenthe different test cases is greater at some other contactpoints, as shown in Fig. 6. Especially in the beginning ofgearpairengagement,forcedifferencebetweenthedifferent test cases may be remarkable.In the studied helical gear contact, the estimatedellipticity ratio becomes very high. This is the case evenwhen the studied gear was rather narrow. This is the mainreason why the calculated FEM results are less accuratethan in the earlier contact model test case. How accuratethe assumption of elliptical contact is in helical gearcontact is not studied here. Future studies will determinethe final capability of the used contact models. However,the FEM-based contact model has potential especially inthe calculation of edge contacts, i.e. in cases which are notfully covered by analytical formulas.6. ConclusionsA calculation model for the analysis of helical gearcontact is introduced. Helical gear surface profiles areconstructed from gear tool geometry by simulating thehobbing process. This procedure allows deviations fromideal involute geometry. The gear pair contact line isnumerically defined direct from the gear surface geometry.The model uses 3D finite elements for the calculation oftooth deflection including tooth bending, shearing andtooth foundation flexibility. The model combines contactanalysis and structural analysis to avoid large meshes.The flexibility of tooth foundation was found to have anessential role in contact load sharing between the meshingteeth, whereas contact flexibility plays only a minor role.This indicates that reasonable distribution of tooth contactforce along the line of action may be generated by usingflexible teeth and flexible tooth foundation, but allowingrigid contact.ARTICLE IN PRESSFig. 6. Single tooth contact force along the line of action.Table 4Contact specificationsE (GPa)220n0.3r1x(mm)8.7r2x(mm)23.2r1y(mm)7000r2y(mm)7000Table 5Calculation resultsCaseCase 4Case 2HertzFEMHertzFEMWcase(N)15701723.5a (mm)4.5b (mm)20.2d (mm)5.64.946.15.27p0(GPa)1.151.281.181.37J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619618The FEM-based contact model gives a reasonableapproximation of contact parameters when the mesh sizeis fine enough. Contact shapes, such as in helical gears,require small element size, i.e. a large number of elementsto avoid element dimensional distortion. However, theFEM-based contact model has potential in calculating edgecontacts.Refer
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