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Tribology International 40 (2007) 613619 Modeling of helical gear contact with tooth defl ection Juha Hedlund?, Arto Lehtovaara Tampere University of Technology, Machine Design, P.O. Box 589, 33101 Tampere, Finland Available online 4 January 2006 Abstract The majority of gear tribological studies are made on spur gears. However, helical gears are generally used in industry, and their contact behavior deserves more attention to establish a realistic base for detailed friction, wear and life studies. This study focuses on the modeling of helical gear contact with tooth defl ection. A calculation model for helical gear contact analysis is introduced. Helical gear surface profi les are constructed from gear tool geometry by simulating the hobbing process. The model uses three-dimensional fi nite elements for the calculation of tooth defl ection including tooth bending, shearing and tooth foundation fl exibility. The model combines contact analysis with structural analysis to avoid large meshes. Tooth foundation fl exibility was found to have an essential role in contact load sharing between the meshing teeth, whereas contact fl exibility plays only a minor role. The capability of different local contact calculation methods was also studied. r 2005 Elsevier Ltd. All rights reserved. Keywords: Helical gear; Contact; Deformation; Load distribution; Modeling 1. Introduction Helical gears are generally used in industry and their contact behavior deserves more attention to establish a realistic base for the detailed study of gear friction, wear and life. The gear contact stresses derived from tooth contact forces and geometry are very important for determining gear pitting, i.e. life performance. Tooth contact forces along the line of action depend essentially on load sharing between meshing teeth, and, therefore, a realistic analysis of helical gear contact also requires information on structural deformations, such as tooth defl ection. The majority of gear contact analyses within tribological studies are made on spur gears. Typically, teeth contact through the line of action is modeled as a constantly changing roller contact, whose radius, speed and load are approximated from ideal involute gear geometry in the given operating conditions. Gear contact ratio 12 and equal 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 contact theory, otherwise assuming rigid tooth behavior. Some studies are made by slicing the helical gear to a series of spur gears and treating these slices as spur gears 1. Finite-element-basedcalculationmodelsarewidely acceptedforcalculatingstructuraldeformationsand stresses in spur and helical gears in the case of concentrated loads. In gear transmission and dynamic analyses, typical deformation studies have used the two-dimensional (2D) fi nite element method with spur gears 24. Coy and Chao 5 studied fi nite element grid size dimensions to cover the Hertzian contact. Du et al. 6 and Arafa et al. 7 later enhanced contact modeling as a part of structural analysis by using gap elements for the calculation of spur gear deformation. Only a few helical gear contact studies, which include structural deformations of the gear, have been performed with 3D fi nite elements 8,9. This is mainly because in this case FEM contact modeling is computationally expensive and time-consuming due to the small grid size which is necessary on the gear fl ank surface. Different methods have been introduced to overcome this problem. Vedmar 10 separated structural and contact analysis by combining the ARTICLE IN PRESS 0301-679X/$-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). fi nite element method and the Weber px40,(5) ux sx4d;px 0.(6) In addition, the resulting pressure distribution must satisfy the force balance in normal direction with total force W applied on the contacting bodies. It follows: Z 1 ?1 Z 1 ?1 px;ydxdy W.(7) For 3D elastic contact problems the Boussinesq for- mulation can be used. The basic equation for surface pressuredeformationofsurfacesinz-directionu(x) between semi-infi nite solids is 14 ux 2 pE0 Z 1 ?1 Z 1 ?1 px0;y0dx0dy0 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi x ? x02 y ? y02 q.(8) Contact problems with arbitrary undeformed surface profi les need to be solved by numerical methods. The numerical solution process is typically iterative, because pressure distribution and real contact area distribution are unknown, whereas total load, material properties and initial contact geometry are known parameters. The well-known Hertz solution to the contact problem is based on cases, where the undeformed geometry of contacting solids can be represented in general terms by two ellipsoids. The solution requires the calculation of ellipticity parameter and complete elliptic integrals. A simplifi ed solution of the classical Hertz theory of elliptical contact solution is presented in 15. This calculation method is non-iterative and fast. The solution includes an elastic half space assumption. 4.1. FEM-based contact model Initially, geometrical overlap between the contact bodies is chosen to produce a calculation domain greater than the fi nal contact area. The loading vector acting in the calculation domain is a combination of nodal displace- ments (overlap) and zero loads, producing non-homoge- nous FEM boundary conditions. This approach is applied to calculate force distribution over the domain. Force distribution Fij, which is located at the surface, is used for the calculation of contact stiffness values Kijin every node (i,j) as follows: Kij Fij Bij .(9) Reduced stiffness values in the grid between the contact bodies are determined as follows: Keqij Kij1Kij2 Kij1 Kij2 .(10) After the reduced contact stiffness is established and the initial separation of the undeformed surfaces is known, contactparametersarecalculatedbyapproaching the contact bodies gradually until load balance is achieved (Eq. (11). The condition of the contact is checked at each iteration step at the nodal points, i.e. parameter etijis updated. Total contact force is determined as Fc X i X j Keqijetijif etij40; etij 0.(11) The FEM-based contact model takes into account structure boundaries, i.e. no half space assumption is needed. 5. Result and discussion 5.1. Contact model test case The FEM-based contact model was tested against simplifi ed Hertzs formulas 15 in the case of circular and elliptical contact. The test case dimensions and load conditions are shown in Table 1 and the results in Table 2. Both surfaces have the same material properties. The elliptical test case was chosen to evaluate the crowned spur gear contact. Mesh size was limited to 4500 elements per contact body in the test case calculation. One surface calculation domain consists of 900 nodal points. The deformed contact surface and calculation grid are shown in the case of circular contact (Fig. 4). The results show that the FEM-based contact model gives reasonable approximation of contact parameters taking into account the fairly coarse grid size. The minor semi-axisoftheellipseespeciallysuffersfromgrid dimensions. Mesh size and shape have a certain effect on the results unless the mesh size is fi ne enough. It is obvious that the accuracy of results will decrease as the ellipticity ratio of the contact increases. ARTICLE IN PRESS Table 2 Comparison results Modela (mm)b (mm)d (mm)p0(GPa) Hertz/circular0.370.3718.13.56 FEM/circular0.40.419.13.11 Hertz/elliptical0.274.1320.72.11 FEM/elliptical0.44.020.22.36 Table 1 Test case specifi cations CaseCircularElliptical E (GPa)206 n0.3 r1x(mm)12.312.3 r2x(mm)18.618.6 r1y(mm)12.31000 r2y(mm)18.61000 W (N)10005000 J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619616 5.2. Helical gear contact case In helical gear mesh, contact is more complex than in the case of spur gears. The contact area has a real 3D nature, which makes the contact curvature change along the contact line and the line of action. Also, load sharing between the gear teeth is complicated partly because the total force is often shared between three teeth pairs. The realistic force acting on a single tooth at any location along the line of action is the basic parameter in tribological contact studies. Force distribution between meshing teeth pairs was studied in the developed model. Four different test cases were established where the model allows: (1) tooth and tooth foundation deformations with rigid contact; (2) tooth and tooth foundation deformations with contact deformation; (3) tooth deformation with rigid foundation and rigid contact; (4) tooth and contact deformation with rigid foundation. The stiffness vector for a tooth pair consists of 30 calculation points along the line of action. Two thousand nine hundred and twenty-six elements per tooth were solved at every calculation point. The stiffness vector of a single gear pair was copied with offset to represent other teeth in contact. The total mesh stiffness vector of a gear pair was obtained by summing up these single stiffness vectors. Finally, displacement along the line of action was calculated and the contact force of a single tooth was solved. Contact stiffness along the line of action was calculated on the Hertz line contact formula by using the force distribution from test case 1. The values of tooth fl ank radius are calculated over the contact line and estimated as a mean value. The overall reduced stiffness was obtained by the iterative method, as in Eq. (10). Helical gear contact was studied with the FZG test rig related gear data shown in Table 3. In this example, total contact ratio was over 2, which means that there are always at least two teeth pairs in contact. The test case element mesh and the mating tooth surface for pinion are shown in Fig. 5. The element mesh of the gear is equal. The calculated force distribution curves of the test case are shown in Fig. 6. The line of action is described with non-dimensionalparameterc x/L, where Listhe effective length of the line of action. Fig. 6 shows that the general trend in force distribution remains in the different model test cases. The area between the sharp edges near the middle represents a situation where three teeth pairs carry the total load. In the case of helical gears, this transition from two to three teeth and vice versa occurs quite smoothly. Contrary to spur gears, single tooth force is high, when all three teeth are in contact. This is because fl exibility is lower at the tooth tip corners than in the middle and root area. The different test cases produce clear differences in contact force behavior. The fl exibility of tooth foundation has the most crucial effect on the distribution of contact force along the line of action. Contact fl exibility has less impact, but interestingly, it shifts the force distribution curve slightly to the right in certain areas. This is because the combined contact radius is asymmetric over the pitch point. The load distribution was observed to be sensitive to stiffness properties at the start and end points of the line of action. One contact point (c ?0.236), shown in Fig. 6, was chosen for a closer study. This contact point was estimated ARTICLE IN PRESS Fig. 5. Element mesh of pinion tooth used in the test case. Fig. 4. Deformed contact surface. Table 3 Test case gear data mn(mm)2.75 bg(mm)20 b (deg)12 z126 z239 eg2.084 T (Nm)143 J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619617 with the Hertzian elliptical contact formula 16 and the FEM-based contact model. This contact point represents the situation where two teeth pairs carry the total load. The chosen contact situation was calculated with two different forces corresponding to the calculated test cases 2 and 4. The load sharing differs depending on the modeling of the tooth foundation. The radius of surface profi les was approximated with the circum circle method and these values are shown in Table 4. Some crowning was included in the contact line direction. At the studied contact point, different load sharing between cases 2 and 4 has only a minor effect on contact parameters (Table 5). However, force difference between the different test cases is greater at some other contact points, as shown in Fig. 6. Especially in the beginning of gearpairengagement,forcedifferencebetweenthe different test cases may be remarkable. In the studied helical gear contact, the estimated ellipticity ratio becomes very high. This is the case even when the studied gear was rather narrow. This is the main reason why the calculated FEM results are less accurate than in the earlier contact model test case. How accurate the assumption of elliptical contact is in helical gear contact is not studied here. Future studies will determine the fi nal capability of the used contact models. However, the FEM-based contact model has potential especially in the calculation of edge contacts, i.e. in cases which are not fully covered by analytical formulas. 6. Conclusions A calculation model for the analysis of helical gear contact is introduced. Helical gear surface profi les are constructed from gear tool geometry by simulating the hobbing process. This procedure allows deviations from ideal involute geometry. The gear pair contact line is numerically defi ned direct from the gear surface geometry. The model uses 3D fi nite elements for the calculation of tooth defl ection including tooth bending, shearing and tooth foundation fl exibility. The model combines contact analysis and structural analysis to avoid large meshes. The fl exibility of tooth foundation was found to have an essential role in contact load sharing between the meshing teeth, whereas contact fl exibility plays only a minor role. This indicates that reasonable distribution of tooth contact force along the line of action may be generated by using fl exible teeth and fl exible tooth foundation, but allowing rigid contact. ARTICLE IN PRESS Fig. 6. Single tooth contact force along the line of action. Table 4 Contact specifi cations E (GPa)220 n0.3 r1x(mm)8.7 r2x(mm)23.2 r1y(mm)7000 r2y(mm)7000 Table 5 Calculation results CaseCase 4Case 2 HertzFEMHertzFEM Wcase(N)15701723.5 a (mm)5.74.55.84.5 b (mm)0.120.20.120.2 d (mm)5.64.946.15.27 p0(GPa)1.151.281.181.37 J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619618 The FEM-based contact model gives a reasonable approximation of contact parameters when the mesh size is fi ne enough. Contact shapes, such as in helical gears, require small element size, i.e. a large number of elements to avoid element dimensional distortion. However, the FEM-based contact model has potential in calculating edge contacts. References 1 Flodin A. Simulation of mild wear in helical gears. Wear 2000; 241:1238. 2 Wallace DB, Seireg A. Computer simulation of dynamic stress, deformation, and fracture of gear teeth. J Eng Ind 1973;95:110815. 3 Wilcox L, Coleman W. Application of fi nite elements to the analysis of gear tooth stresses. J Eng Ind 1973;95:113948. 4 Wang KL, Cheng HS. A numerical solution to the dynamic load, fi lm thickness, and surface temperatures in spur gears, part I analysis. J Mech Des 1981;103:17787. 5 Coy JJ, Chao CH-C.