1、91 FTM 16AVContact Analysis of Gears Using a CombinedFinite Element and Surface Integral Methodby: S. M. Vijayakar, Advanced Numerical Solutions andD. R. Houser, Ohio State University,LAmerican Gear Manufacturers AssociationII ITECHNICAL PAPERContact Analysis of Gears Using a Combined Finite Element
2、 and SurfaceIntegral MethodS. M. Vijayakar, Advanced Numerical Solutions andD. R. Houser, Ohio State UniversityThe Statementsandopinionscontained hereinarethoseof theauthorandshouldnotbe construed asan official action oropinion of the American Gear Manufacturers Association.ABSTRACT:A new method is
3、described for the solutionof the contactproblemingears. Themethod usesa combination of the finiteelement method and a surfaceintegral form of the Bousinesq and Cermtisolutions. Numerical examples are presentedof contacting hypoid gears, helical gears andcrossed axis helical gears.Copyright 1991Ameri
4、can Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1991ISBN: 1-55589-614-6Contact Analysis of Gears using a Combined FiniteElement and Surface Integral MethodSandeep M. VijayakarAdvanced Numerical Solutions2085 Pine Grove Lane, Columbus OH 43232,andDonal
5、d R. HouserProfessor, Dept. of Mech. Eng. The Ohio StateUniversityColumbus OH 43210INTRODUCTIONResearch in the mid and late eighties showed thatThe complete and accurate solution of the contact the gear contact problem was not unsurmountable, butproblem of three-dimensional gears has been, for the r
6、equired an approach that combined the strengths ofpast several decades, one of the more sought after, the finite element method with those of otheralbeit elusive solutions in the engineering community, techniques such as boundary element and surfaceEven the arrival of finite element techniques on th
7、e integral methods. Concepts from mathematicalscene in the mid seventies failed to produce the programming could be used to advantage in solvingsolution to any but the most simple gear contact the contact equations. An innovative approach towardsproblems, the formulation of the finite elements thems
8、elvescould go a long way towards solving the meshThe reasons for this are manyfold. When gears are generation and geometric accuracy problems. With thebrought in contact, the width of the contact zone is idea of incorporating the best of these and othertypically an order of magnitude smaller than th
9、e other technologies in mind, development of what is nowdimensions of the gears. This gives rise to the need for CAPP (Contact Analysis Program Package) was beguna very highly refined finite element mesh near the four years ago. It has evolved into a powerful collectioncontact zone. But given the fa
10、ct that the contact zone of computer programs that provide the gear designermoves over the surface of the gear, one would need a with an insight into the state of stress in gears that hasvery highly refined mesh all over the contacting thus far never been possible. Some of the features thatsurface.
11、Finite element models refined to this extent CAPP supports are: friction, sub-surface stresscannot be accommodated on even the largest of todays calculation, stress contours, transmission error, contactcomputers. Compounding this difficulty is the fact that pressure distributions and load distributi
12、on calculation.the contact conditions are very sensitive to thegeometry of the contacting surfaces. General purpose Figures 1 to 5 show examples of gear sets for whichfinite element models cannot provide the required this process has been successfully used.level of geometric accuracy. Finally, the d
13、ifficulties ofgenerating an optimal three-dimensional mesh that CONTACT ANALYSIScan accurately model the stress gradients in the criticalregions while minimizing the number of degrees of In earlier studies Vijayakar 1988,1989; Bathe 1985,freedom of the model have kept the finite element Chowdhury 19
14、86 of contact modeling, a pure finitemethod from being widely used to solve the complete element approach was used to obtain compliance termsgear contact problem, relating traction at one location of a body to the normaldisplacement at another location on the contacting1AvFigure 4- Contact analysis
15、of a 90 crossed axis externalhelical gear set.Figure 1- Contact analysis of helical gears.Figure 5- Contact an_YSciSlga9_tcrossed axis externalgear body. It became apparent that in order to obtainsufficient resolution in the contact area, the size of theFigure 2-Contact analysis of hypoid gears, fin
16、ite element model would have to be inordinatelylarge. A finite element mesh that is locally refinedaround the contact region cannot be used when thecontact zone travels over the surfaces of the two bodies.Other researchers working in the tribology area deMul 1985, Seabra 1987, Lubrecht 1987 have obt
17、ainedcompliance relationships in surface integral form byintegrating the Greens function for a point load on thesurface of a half space (the Bousinesq solution) over theareas of individual cells demarcated on the contactzone. This method works well as long as the extent ofthe contacting bodies is mu
18、ch larger than thedimensions of the contact zone, and the contact zone isfar enough from the other surface boundaries so thatthe two contacting bodies may be treated as elastichalfspaces. These conditions are, however, not satisfiedby gearsThe approach that is described here is based on theassumptio
19、n that beyond a certain distance from thecontact zone, the finite element model predictsdeformations well. The elastic half space model isFigure 3- Contact analysis of worm gears, accurate in predicting relative displacements of pointsJ=-6near the contact zone. Under these assumptions, it is A possi
20、ble to make predictions of surface displacements _ 0that make use of the advantages of both, the finitev element method, as well as the surface integralapproach.This method is related to asymptotic matchingmethods that are commonly used to solve singularperturbation problems. Schwartz and Harper Sch
21、wartz1971 have used such an asymptotic matching methodcylinders pressed against an elastic cylinder in planestrain.In order to combine the surface integral solutionwith the finite element solution, a reference ormatching interface embedded in the contacting body is Figure 6- Computational grid in th
22、e contact zone of theused. This matching surface is far enough removed gears.from the principal point of contact so that the finiteelement prediction of displacements along this surfaceis accurate enough. At the same time, it is close enough force applied at the location p which is on the surface of
23、to the principal point of contact so that the effect of the the gear. The superscripts (si) and (fe) on a term willfinite extent of the body does not significantly affect the mean that the term has been calculated using surfacerelative displacements of points on this surface with integral formulae a
24、nd a finite element model,respect to points in the region of contact, respectively. Subscripts 1 and 2 will denote gearsnumber 1 and 2, respectively. When this subscript isContact analysis is carried out in several steps. The omitted in an equation, the equation will befirst step is to lay out a gri
25、d at each contact zone. Then understood to apply to both the gearscross compliance terms between the various grid pointsare calculated using a combination of a surface integral Let u(p;q) = -u(p;q).n be the inward normalform of the Bousinesq and Cerruti solutions and the component of the displacemen
26、t vector u(p;q), where nfinite element model of the contacting bodies. Finally is the outward unit normal vector at the point p.load distributions and rigid body movements arecalculated using an algorithm based on the Simplex The displacement u(rij;r) of a field point r due to amethod Vijayakar 1988
27、. load at the surface grid point r can be expressed as:llIn order to discretize the contact pressuredistribution that is applied on the surfaces of two u(rij;r) = (u(rij;r)- u(rij;q ) )+ u(rij;q )contacting gear teeth, a computational grid is set up.Figure 6 shows such a computational grid that has
28、been where q is some location in the interior of the body,set up in the contact zone of the gears. The entire face sufficiently removed from the surface (Figure 7). If thewidth of one of the gears (gear 1) which is mapped onto first two terms are evaluated using the surface integral_:_ -1,+1 is divi
29、ded into 2N +1 slices. N is a user formulae and the third term is computed from theselectable quantity. The thickness of each slice in the _ finite element model, then we obtain the displacementparameter space is A_ = 2/(2N+1). For each slice j=-N to estimate:+N, a cross section of gear 1 is taken a
30、t the middle of theslice, and a point is located on this slice that approaches u(rij;r)(q ) = (u(Si)(rij;r) _ u(Si)(rij;q) ) + u(fe)(rij;q)the surface of the mating gear (gear 2) the closest. Thisselection is carried out using the undeformed geometry.If the separation between the two gears at this c
31、losest The term in parentheses is the deflection of r withpoint is larger than a user selectable separation respect to the reference point q. This relativetolerance, then the entire gear slice is eliminated from component is better estimated by a local deformationfurther consideration. Otherwise, a
32、set of grid cells field based on the Bousinesq and Cerruti half spaceidentified by the grid cell location indices (i,j), i = - M solutions than by the finite element model. The grossto M and the position vectors r is set up centered deformation of the body due to the fact that it is not a1j half spa
33、ce will not significantly affect this term. On thearound this closest point of slice, j. The number M is contrary, the remaining term u(fe)(rij;q) is notuser selectable. The dimension of the grid cells in theprofile direction As is also user selectable, significantly affected by local stresses at th
34、e surface. Thisis because q is chosen to be far enough beneath theLet u(p;q) denote the displacement vector at the surface. This term is therefore best computed using alocation q on a gear due to a unit normal compressive finite element model of the body. The value u(rij;r)(q)OJ o 0.4 of the finite
35、element were created semi-automatically.Only a sector containing three teeth of each gear was2o 0_1 0_2 053 o._ o.55 0.5_ 0_ 0._ o.69 0_ modelled, with each tooth being identical. The gear_p_c_c_,_ (gear no. 1) and the pinion (gear no. 2, the smaller gear)were then oriented in space as per the assem
36、blyFigure 14: Variation of sub-surface shear stress with drawings, and the analysis was carried out for eachindividual time step. Figure 2 shows the six tooth geardepth under the point of maximum contact pressure, and pinion model. Sectoral symmetry is used togenerate stiffness matrices from the sti
37、ffness matrix ofmanufacture these gears have many kinematic settings, one tooth. For this particular gear set, a three toothThe settings are chosen such that the contact zone model suffices because at the most two teeth contact at aremains in the center of the tooth surfaces as the gears time. Figur
38、e 15 shows the surfaces of the three toothroll against each other. A heuristic procedure is gear. Figures 16, 17 and 18 show the contact patternavailable to select the settings, but in practice, these (which is the locus of the contact zone as the gears rollsettings have to be selected after a tedio
39、us iterative against each other), for a gear torque of 240, 480 and 960process involving cutting and testing actual gears. Even in-lbs, respectively. Figures 19 and 21 show views of theso, it is very difficult to predict the actual contact contact zone with contact pressure contours on the gearstres
40、ses, fatigue life, kinematic errors and other design for two particular angular positions. Figures 20 and 22criteria, especially when not installed in ideal show magnified views of the contact zone for these twoconditions. The contact stresses are so sensitive to the positions. They show contours of
41、 normal contactactual surface profile that conventional 3-D contact pressures on the surfaces. Computational grids of 11x25analysis is not feasible, cells were used on these surfaces to obtain the pressuredistributions. Finally, the position of the pinion wasA sample 90 hypoid gear set from the rear
42、 axle of a perturbed slightly from the design location, and thecommercial vehicle was selected. The gear ratio of this Figures 23 (a) to (d) show the contact patterns that wereset was 41:1I and the axial offset was 1.5 inches. The obtained. When compared to the contact pattern for thegear surfaces h
43、ad been experimentally shown to be ideal unperturbed position in Figure 16, it shows that the bestfor this particular gear ratio and axial offset. In other contact pattern does indeed occur at the designedwords, the contact zone was found to remain in the position, lending credence to the notion tha
44、t ancentral portion of the gear teeth in the operational analysis of the kind described in this paper has thepotential to be used in the design process itself.vFigure 16: The locus of the contact zone at a gear torqueof 240 in-lbs. Figure 19: Contact pressure contours for position 1.Figure 17: The l
45、ocus of the contact zone at a gear torqueof 480 in-lbs. Figure 20: Contact pressure contours for position 1.normal stress calculated at various sections in a pair ofcontacting helical gear teeth. Figure 25 shows contourcurves of maximum principal normal stress drawnalong the surface of the gear toot
46、h, and Figure 26 showsa contour surface of maximum principal normal stresswithin a gear tooth. It is also possible to draw contourcurves and surfaces for the minimum principal normalstress and the Von Mises octahedral shear stress. Figure27 is an example of an arrow diagram that can be usedto show b
47、oth the magnitude as well as direction of theprincipal normal stresses. Stresses are depicted byarrows pointing in the principal directions. Tensile_ stresses are depicted by outward pointing arrows andcompressive stresses are depicted by inward pointingarrows. The length of an arrow is proportional
48、 to theFigure 18: The locus of the contact zone at a gear torqueof 960 in-lbs, magnitude of the principal stress.CONCLUSIONSExamples of other postprocessing featuresUsing a combination of finite element and surfaceA variety of post-processing options are available for integral methods seems to be, i
49、n the authors opinion,the display of the state of stress in contacting gears, the most practical method of modelling stiffnessFigure 24 shows contour curves of maximum principal behaviour of contacting bodies. Used along with an7_|rt _ in AvFigure 21: Contact pressure contours for position 2.Figure 23(b): The effect of an Y translation on thecontact patternFigure 22: Contact pressure contours for position 2.Figure 23(c): The effect of an Z translation on thecontact patternefficient algorithm for solving contact equations, onecan predict contact stress distributions
