1、99FTM18Effects of Wear on the Meshing Contactof Worm Gearingby: D.R. Houser, X. Su, M. Vaishya, The Ohio State Universityand S.M. Vijayakar, Advanced Numerical SolutionsvAmerican Gear Manufacturers Association- TECHNICAL PAPEREffects of Wear on the Meshing Contact of Worm GearingDonald R. Houser, Xi
2、aogen Su, Manish Vaishya, The Ohio State University andSandeep M. Vijayakar, Advanced Numerical SolutionsThe statements and opinions contained herein are those of the author and should not be construed asan official action oropinion of the American Gear Manufacturers Association.AbstractIn this pape
3、r, a combination of loaded tests, coordinate measurements, surface reverse engineering and a special finiteelement method is employed to study the effect of break-in wear on meshing contact between the mating surfaces ofworm gearing parts. Three different wheel tooth geometries are investigated in t
4、his paper: the as-cut geometry cut by afully oversized hob, the conjugate tooth geometry generated by the mating worm thread and the broken-in wheel toothsurface. The broken-in wheel tooth is measured with a coordinate measurement machine and reverse engineered. Thecontact stresses between the worm
5、thread and these three different wheel tooth geometries are studied with the CAPPSoftware (Contact Analysis Program Package 13). Based on the obtained contact pressure values, the effects of wear oncontact stresses are investigated. It is found that the three maximum contact pressures differ signifi
6、cantly. The contactpatterns, the load sharing and the transmission errors are also discussed.Copyright 1999American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1999ISBN: 1-55589-756-8AvEFFECTS OF WEAR ON THE MESHING CONTACT OF WORM GEARINGDonald. R. H
7、ouser1(Professor)Xiaogen Su (Research Associate)Manish Vaishya (Research Associate)Department of Mechanical EngineeringThe Ohio State University206 W 18th Ave., Columbus, OH 43210, USASandeep M. VijayakarAdvanced Numerical Solutions3554 Mark Twain Court, Hilliard OH 43026, USALIn the literature, the
8、 as-cut geometry of the1. INTRODUCTION wheel is often used by researchers on kinematicalaspects 1, 2, 3, 4 while the conjugate geometry isIn this paper, a combination of loaded tests, used both on kinematical 5, 6, 7, 8 and on load andcoordinate measurements, surface reverse stress calculations 5, 9
9、, 10. Because neither ofengineering and a special finite element method is these two extreme cases represents the actualused to investigate the effect of wear on meshing geometry of the wheel, it is not appropriate to use- contact of worm gearing. The hob used to cut the either of these two surfaces
10、 to do contact stressworm wheel can be viewed as a replica of the mating calculation. It is not a surprise that the contactworm. In practice, the hob cannot be made exactly stresses found by Narayan 11 and Narayan et althe same as the worm and it differs from the worm 12 based on the as-cut geometry
11、 are significantlynot only in having a larger outside diameter to higher than those estimated based on the conjugateproduce clearance between the worm tip and the geometry 10. In this paper, a third geometry iswheel root area, but also with a larger pitch diameter introduced:to allow re-sharpening.
12、The difference between thehob diameter and the worm diameter is called the Broken-in geometry: Due to considerable wear onhob oversize. After each re-sharpening, the oversize the wheel teeth during the break-in period of a wormdecreases and the hob has to be replaced prior to gearing mesh, the wheel
13、 tooth topographythe oversize reaching zero. Two extreme cases of experiences significant change during this stage. Atthe wheel teeth are cut by two extreme dimensions the end of the break-in period, the contact area isofthe hob: increasedand the contactpressuresare reducedtoa level where the lubric
14、ant film is strong enough toAs-cut geometry: defined as the shape of the wheel separate the two mating surfaces. From then on, thetooth cut by a new hob with full oversize. The as-cut wear rate slows down and the wheel tooth geometrygeometry of the wheel is determined by the design becomes stable. T
15、his shape of the wheel teeth isgeometry of the hob and the corresponding hobbing referred as the broken-in geometry in this study.setup. A worm gear drive test rig was built tointroduce break-in wear. The wheel tooth surfaceConjugate geometry: defined as the shape of the after the break-in period wa
16、s studied through reversewheel tooth generated by the mating worm (viewed engineering, wherein best fitting is used to define theas a hob with zero oversize). The conjugate measured broken-in tooth surface. For the abovegeometry is determined by the design geometry of three geometries, the contact s
17、tresses are analyzedthe worm and the corresponding meshing setup, with the CAPP Software (Contact Analysis ProgramPackage 13). Finally, the effect of cumulative wearA on contactcharacteristicsis investigated.v Corresponding author. Fax(614) 292-31632. CAPP PROGRAM AND TEETH four wheel teeth used by
18、CAPP are shown in Figs. 2MODELING and 3, respectively. The four teeth are numbered astooth 1, 2, 3 and 4 from top to bottom for both theCAPP is a Contact Analysis Program Package worm and the gear. Figure 4 shows the grid of cellsdeveloped by Vijayakar 13. CAPP uses finite quasi- generated by CAPP i
19、n potential contact areas afterprismatic elements to model the gear teeth 14. Its the two gear models are assembled at one meshingapproach is to combine the finite element model with position (only the wheel is shown in this figure).an elastic half-space model 15. The finite elementmodel is used to
20、obtain gross deformations of thegear away from the contact zone, while the elastichalf-space model is used to obtain relativedeformations within the contact zone. The finiteelement deformations are then matched with the ,o_,local deformations in a least square sense. In orderto use the elastic half-
21、space formulae for the localdeformation, the contact zone is discretized into acontact grid consisting of many small cells. Thecontact pressure is assumed to be constant over ,oo_2each individual cell, and the local deformations dueto the load on a cell is obtained by numericallyintegrating the Bous
22、inesq solution over that cell.CAPP reads in the tooth geometry through a =_3finite element mesh file for each gear 13. A meshgenerator is needed for each type of gear in order togenerate the mesh file. Two mesh generators werewritten, one for the worm tooth and the other for the =wheel tooth. The ex
23、plicit wheel tooth equationsderived in 16 are used to calculate the wheel teeth.The finitequasi-prismaticelementsusedin CAPPrequire the modeled gear teeth to be smooth alongthe face width directions. The top boundary of thewheel tooth forms two sharp corners at positions Band C (Fig. 1) when the thr
24、oat edge intersects the Fig.2. Finite element model of four worm teeth.top edge. While CAPP is capable of modeling thiskind of discontinuity by dividing the tooth into threeportions 11, this practice is less efficient incomputation. A different approach was attempted inwhich the whole top edge was f
25、it by a fourth orderpolynomial (shown in Fig. 1). This model providesunlimitedorder of continuitvalona the too edae.B tp_edq.e c _II _olh2J t.ooth top boundary= WhOle top boundary smoothed using 41h order polynomial =s./s._ s.tFig 1. Modelingof topedge of a wheeltooth.For the studyof wear effects,a
26、four-threadedZK type of wormgear drivewas used. Fourpairsofteeth were modeledfor the finite element analysis,The finite element models of four wormteeth and Fig.3. Finiteelementmodelof fourwheel teeth.2Table 1. Parameters of the worm ,4,1,parameter value units: worm profile type ZKI ,oo_, meshing ce
27、nter distance 88.9 mmnumberofwormthreads 4Iwormthreadlead 41.891 mmi wormoutsidediameter 50.990 mmL wormpitchdiameter 44.456 mm,oo_2 half angle of worm grinding 22.5 degwheelnumberof gearteeth 40gear blank face width 31.75 mm_3 gear blank outside diameter 146.812 mmgear pitch diameter 133.344 mmdiam
28、etric hob oversize 1.143 mmhob lead 41.798 mm,o_, gear profile tip relief 0.0076 mmgear profile root relief 0.0279 mmTable2.Wormgearboxspecifications.Fig. 4. Grid of cells in potential contact areas, parameter valueratio 10:1An output torque of 236 N-m (2089 in-lbf) and input shaft speed 1750 rpma c
29、onstant friction coefficient of 0.05 are used in the rated output torque 189 Nm (1671 in-lbf)stress calculations. This coefficient varies across the rated horsepower 5gear tooth surface as shown by Hohn and- Steingrover 17.3. WEAR TEST AND COORDINATEMEASUREMENT OF TOOTH WEARWear Test Rig: In order t
30、o investigatethe effectofbreak-in wear of the wheel teeth on the contactcharacteristics,a worm gear drivetest rigwas buitat the Gear Dynamics and Gear Noise ResearchLaboratory of the Qhio State University. A four-threaded,ZK type of worm gear drivewas usedforthe tests. The geometry parameters of the
31、 gear pairhave been listed in Table 1. The specifications of thegearbox are listed in Table 2.The test rig is shown in Fig. 5. The input shaft Fig. 5. Worm gear drive test rig usedfor wear tests.was driven by a DC dynamometer and the outputshaft was loaded by a pneumatically actuated,water-cooled, f
32、riction brake. The speed of the input Wear Measurements: A 125 mm x 75 mm (5“ x 3“)shaft was monitored by an optical tachometer and hole was cut at the bottom of the gearbox. This holeallows measurement of the tooth surface withoutthe torque of the output shaft was measured by aslip ring torque tran
33、sducer. Also monitored were the disassembling the gear pair (see Fig. 6). Thelubricant temperature and the room temperature. It surfaces of four teeth were chosen for measurement,was found that the output torque fluctuated by about such that these four teeth mesh with the samethread of the worm. As
34、a result, a close similarity of+10% of its mean value. The fluctuation was at a the contact pattern and the wear pattern wasrate of once per revolution of the output shaft due to observed amongst these teeth pairs. Each tooth_. irregularity in the friction brake, flank was discretized into a 21 x 77
35、 grid of points-._ and coordinate measurement data were taken at3each grid point. The axis of the output shaft was the whole tooth surface as shown in Fig. 7. Then theused as the reference for the measurement, tooth surface was measured and treated as thebroken-in surface for contact stress calculat
36、ion. The_V torquewas increased in order to accelerate wear.4. REVERSE ENGINEERING OF BROKEN-INWHEEL TEETHWe use the conjugate wheel tooth shape as areference. The measured broken-in tooth iscompared to the reference shape and the differencebetween these two surfaces is modeled. Therefore,the broken-
37、in tooth is reverse engineered throughthe conjugate tooth shape plus a difference function.The following steps are taken for the reverseengineering of the broken-in tooth.1. The worm thread is measured and reverseengineered with the method described in 18. It isFig. 6. CMM measurement of tooth topog
38、raphy, found that the wear of the worm thread surfaceintroduced during the break-in process wasThe gearbox was tested for 200,000 cycles at negligible.50% of the rated torque, followed by 2,000,000 2. The measured broken-in wheel tooth iscycles at 100% of the rated torque and 15,000,000 compared wit
39、h a tooth shape which is conjugate tocycles at 125% of the rated torque. After the total of the reverse engineered worm thread. The17.2 million cycles, the contact area had spread over topographical differences between these two toothshapes are plotted in Fig. 8. Large differences areobserved along
40、the top edge and in two areas closeto the root, Area A and Area B. It is observed thatArea A is beyond the meshing contact and Area B isa recess. Both areas have not participated in themeshing process. The large negative differencesfound along the top edge might have originated fromthe measurement p
41、rocess. The target points mightbe too close to the top edge and the measurementprobe might have missed the tooth flank andtouched the top edge of the tooth, which would thenproduce a false touch and the negative difference.Points that are out of the meshing area or recordeddue to false measurements
42、are excluded in theconstruction of the difference function.“68_wfacez_(r.m) -15Fig.7. Contact patterns obtained by paintingthewheel teeth. Upper chart: after the total of Fig.8. Topographical plot of normal differences17,200,O00cycles;lowerchart:after the first between the measured broken-in wheel7,
43、200,000 cycles, tooth and the conjugate tooth geometry.45. EFFECT OF WEAR ON MESHING3. A two-parameter, fifth order polynomial (with CONTACT21 degrees of freedom) is designed to model thedifferences between the broken-in tooth and the During one gear meshing cycle, the worm rotates byconjugate tooth
44、. The 21 coefficients of the 90. The load distribution and the contact pressurespolynomial were found through a best fit between the are calculated at 10 discrete meshing positionsmeasured topographical differences and the throughout one gear meshing cycle, 10 apart frommodeled difference function.
45、The obtained shape ofone another. The 10th meshing position of the wormthe difference function is plotted in Fig. 9. The fitting is one gear meshing cycle ahead from the 1sterror between the reverse engineered tooth and the meshing position and the results at the 10th meshingmeasured tooth (or fitti
46、ng deviation between the position should repeat those at the 1st meshingmodeled topographical differences i normalized contact areas (or potential contact areas),o “ between the four pairs of teeth. The four blocks, from-_ _x _ left to right, correspond to tooth pair 4, 3, 2 and 1.4, , _._ The botto
47、m edge of the block corresponds to the left. “_“-o_-_-_,try edge of the wheel tooth and the top edge_,_ corresponds to the right edge. The width of thef_c_z_,_,) -“ histogram, which corresponds to the width of thecontact area often designated as 2b in the HertzianFig. 10. Fitting deviation plot betw
48、een the modeled formulae, has only relative physical meaning. Thistopographical differences and the measured should be examined by comparing the areas undertopographical differences, contact in Figs. 12 and 13, i.e. the areas covered bynon-zero contact pressures). It is interesting to notethat the c
49、onjugate case has fairly even load_,= distributionbut alsohasedgecontactat the endofeachcontactregion.Edgecontactis notevidentforthe as-cut case and there is no edge contact for the5broken-in case. It is also apparent that the broken-in summarized in Table 4. Note that there is a verytooth has much different contact from the conjugate small contact area at the top edge of wheel tooth 1case and the pressure distribution is very different for the as-cut case and the highest contact pressurefrom the as-cut case and the conjugate
