1、93FTM2A- Topological Tolerancing ofWorm-Gear Tooth surfacesby: Vadim KinM Litvin, 1989a; Kin, sufficient for the study of gear tooth surface tolerance1990), a different form of this Equation is preferred, and bands.we will use it later in this paper to prove a theoreticalresult. This form is represe
2、nted asApplication: ZA Worm-Gear Drivesn_- v_ = 0 (3a)In this Section we consider an application of thewhere n_ is the unit normal to the worm surface and v_ is theory developed above to a particular case of worm-gearthe relative velocity of the point on the worm surface with geometry. ZA is one of
3、the geometries in wide use today.respect to that of the point on the worm-gear surface. The other such types are ZI, ZN, ZK and ZC. While theinformation below is only pertinent to ZA geometry, theThe preference that form (3a) has enjoyed over above theory is general enough to be applied to anythe ye
4、ars is due to the fact that its ufdization results in worm-gear.much less complicated expressions. This is not, however,a strong factor in our approach, since we make extensive The worm surface in a ZA-type worm gear driveuse of a symbolic algebra package to derive the necessary is an Archimedes hel
5、icoid, i.e. a helicoid with a straightexpressions symbolically, or to approximate them line in the axial section. The vector equation of the wormnumerically. The disadvantage of form (3a) is that the surface (equation (1) in this case becomesvector of relative velocity v_ has to be derived from firs
6、tprinciples. While we have completed this task previously,this results in a lesser degree of automation of thesolution, and is thus potentially more prone to errors. “ucos2,p cos0)ucosXp sinO_r, (u,0) = (8)Gear Tooth Surface Deviations p0-1sin _,pJWe now turn our attention to the problem ofrepresent
7、ing the deviations of the worm-gear toothsurface resuking from the utilization of a worse-than-ideal where _,pis the lead angle at the pitch diameter, p-rptar_phob. The surface of such a hob is represented by is worm lead per radian, and rp is the worm pitch radius.No derivations are performed from
8、this point on.r_*(u,0)=rw(u,0)+fiw(u,0 ) (5) The tolerance bands can now be obtained by a directapplication of (6), which is soNed by a symbolic-where _(u,0) represents the worm thread surface error numerical method outlined in (Kin, 1993).function. In line with equation (4) above, the worm-geartoot
9、h error function _ then be represented asExamples 1992). One way of overcomingthe problemis tosupplement 3-D graphics by axial or cross-sections of theThe parameters of the worm-gear drive used in tolerance map. One of the cross-sections of the abovethe examples are summarized in Table 1. map is sho
10、wn in Fig. 5.We first apply the system of equations (4) toconstruct the tooth surface for one of the flanks. Thisflank and the gear root surface are shown in Fig. 2. Thesame tooth surface is shown in greater detail in Fig. 3.Center Distance, in 8.135GearRatio 90:1Worm Pitch Diameter, in 2Pitch Lead
11、Angle, deg 8.5328Numberof Starts 1Normal Pressure Angle, deg 20Fig. 3Table1 Worm-GearToothSurface:DetailWorm-Gear Drive Parameters0.$o.x-,.-._Fig. 2 ,., -,._Worm-GearToothandRoot Surface “We then apply the system (6) to compute the Fig.4three-dimensional tolerance “band“ for the worm-gear Three-Dime
12、nsional Worm-Gear Toothsurface (Fig. 4). The band shows the maximum surface Surface Tolerance Mapdeviations that can be generated by a worm whose proneis within the+0.001“ tolerance band in the axial section. Such worm o.surface in the case of ZA geometry is represented by a.000s _, UCOS_,psin 0 / +
13、/_ COS_,psin0- -/ _rw(uO)= p0-usinXp/ 5sinZp Fig. 51 ) _, 0 Tolerance Map Cross-Section(9)It should be noted that the cross-section shown inHere the first member of the vector sum is recognized as Fig. 5 is not constant throughout the map in Fig. 4. Thethe ideal worm surface (8), and the second - th
14、e worm section depicted is merely an example, and in order toerror surface corresponding to the rectangtflar band. fully visualize the error surface, many such cross-sectionshave to be constructed.Tolerance maps like that of Fig. 4 are not alwayseasy to visualize without the benefits of a sophistica
15、ted The tolerance map can also be constructed for thesymbolic algebra system that allows to color and render case when K-bands are used for worm profile tolerancing.surfaces and to rotate 3-D graphics at will (Wolfram,For our example, we choose a symmetrical K-band with Kin, V., 1993a, Symbolic and
16、Numerical Solutions in0.0015“ height in the middle and 0.002“ height at the Theory of Gearing, Proc. 3rd National Conference ontails. A cross-section of the corresponding worm-gear Applied Mechanisms and Robotics, Cincinnati, Ohio.tolerance map is shown in Fig. 6, and the map itself- inFig. 7. Kin,
17、V., 1993b, Computerized Analysis of GearMeshing Based on Coordinate Measurement Data,ASME I. Mechanical Design, in Press.ConclusionLitvin,F.L., Rahman, P., and Goldrich, R.N., 1982,A method has been presented to determine the Mathematical Models for Synthesis and Optimization ofworm-gear tooth surfa
18、ce three-dimensional tolerance Spiral Bevel Gear Tooth Surfaces, NASA Contractormaps from the worm (or hob) tolerance bands. When Report 3553, NASA Lewis Research Center,applied to the worm profile tolerances specified in a Cleveland, Ohio.standard, the procedure yields “equivalent“ worm-gearsurface
19、 topography tolerances. This equivalence means Litvin, F.L., 1989a, Theory of Gearing, NASAthat the same amount of geometry mismatch between the Reference Publication 1212, NASA Lewis Researchworm thread and the worm-gear tooth surface is achieved Center, Cleveland, Ohio.when either (i) a worm of gi
20、ven precision is used with anideal worm-gear, or (ii) an ideal worm is used with the Litvin, F.L., Zhang, Y., Kieffer, I. and Handschuh,worm-gear of precision “equivalent“ to that of the non- R.F., 1989b, Identification and Minimization ofideal worm. Deviations of Real Gear Tooth Surfaces, Proc. 15t
21、hASME Design Aurora. Conf., Montreal._ Litvin, F.L., and Kin, V., 1990, Simulation of_il - Meshing, Bearing Contact and Transmission Errors for. Single-Enveloping Worm-Gear Drives, AGMA paper,: _. 90FTM3, Am. Gear Manufacturers Assoc., Alexandria,VA./_._1-_ S. Wolfram, 1992, Mathematica: A System fo
22、r DoingMathema_cs by Computer, Second Edition, Addison-Fig. 6 Wesley.Tolerance Map Cross-Section:Worm K-BandAppendix A. Worm-Gear Error SurfaceLet the tooth surface of an ideal worm-gear berepresented byr_=M_r_ (A1)N w -v_ = 0where rw and rg are “then worm and gear surface positionvectors, respectiv
23、ely, M_ is the coordinateFig. 7 transformation matrix, N_ is the worm surface normal,Worm-Gear Tolerance Map: and vw is the relative velocity of the point on the wormWorm K-Band surface with respect to the point on the gear surface. Itcan be shown (Litvin, 1989a) that the relative velocity canbe rep
24、resented asReferencesvw =%wrw+Rxeg (A2)Kin, V., 1990, Computerized Simulation of Meshingand Bearing Contact in Single-Enveloping Worm-GearDrives, Ph.D. Thesis, The University of Illinois atChicago.where _ and mg are the worm and the gear angularvelocities, respectively,m_-_f_ w is the relative angul
25、ar velocity, and R is thevector OwOs (Fig. 1). The second of (AI) is thenrepresented asNw “(ow, xr w +Rxog)= 0 (A3)Let us without loss of generality assume that the surfaceerrors are measured along the normal to the surface. Thesurface of.the worm with deviations is thenr;=r_+_nw (A4)where n_ is the
26、 surface unit normal and8=8(u,e) is the error function. Let us also assume that thesurface errors are small enough for the unit normal to notchange its orientation significantly because of them. Theassumption is a very reasonable one and has been usedsuccessfully in (Litvin et al, 1989b) and (Kin, 1
27、993b).The equation (A3) for the case of the gear tooth surfaceproduced by the imperfect worm then becomesN w-os_ x(r w+Snw) +R xog = 0 (A5)But sinceNw o w 5nw= 0because the vectors nw and N_ are collinear, equation(A5) becomes identical to (A3) - the Equation of Meshingfor the ideal worm-gear drive.
