1、90 FTM 3Simulation of Meshing, Transmission Errorsand Bearing Contact forSingle-Enveloping Worm-Gear Drivesby: Faydor L. Litvin and Vadim Kin, University of Illinois at Chicagoi iAmerican Gear Manufacturers AssociationlUlTECHNICAL PAPERSimulation of Meshing, Transmission Errors and Bearing Contact f
2、orSingle-Enveloping Worm-Gear DrivesFaydor L. Litvin and Vadim Kin,University of Ill:nois at ChicagoThe Statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:The authors hav
3、e developed a computerized method for simulation of meshing and bearing contact(TCA) for single-enveloping worm-gear drives. The developed computer programs enable to determinethe transmission errors and the shift of bearing contact that are caused by worm and gear misalignment.An important theorem
4、is proven for determination of the transfer point on the theoretical line of contactwhere the path of point contact starts for a misaligned worm-gear drive.A method of assembly for compensation of misalignment is proposed.Copyright 1990American Gear Manufacturers Association1500 King Street, Suite 2
5、01Alexandria, Virginia, 223 i4October, 1990ISBN: 1-55589-555-7Simulation of Meshing, Transmission Errors andBearing Contact for Single-Enveloping Worm.Gear DrivesbyFaydor L. Litvin (l), Vadim Kin (2)University of Illinois at Chicago(1) Professor, ASME Ptember, (2) Research FellowNomenclature misalig
6、nment itself plus due to compensatinggear rotation to restore contact.C - center distance. _ - angle of rotation of the ith gear during_i (iffif,h,s,q) - principal direction unit meshing.vectors. _i - surface of the ith gear (1-worm, 2-worm-m2 - gear ratio gear)._j_ number of teeth on the ith gear.
7、Ki (iffif,h,s,q) principal curvatures._1 - normal to the jth surface represented in k b - lead angle on the base cylinder of thecoordinate system Si involute worm.Pd - worm-gear diameterai pitchqi (i=1 . n) - worm-gear drive parameters 1. Introductionwhich, when deviate from the nominal values,cause
8、 misalignment.hi - position vector to a point on the ith Worm-gear drives have been a subject ofsurface, intensive research. Geometry of worm-gear drivesrb_- base cylinder radius of the involute worm. with ground worm surfaces and worm surfaces_ - displacement of the gear contact point generated by
9、blades have been developed byassociated with misalignment the actual (Litvin, 1968). (Winter et al, 1981) and (0ctrue,displacement due to misalignment plus the 1988) have concentrated their efforts on thedisplacement due to compensating gear turn determination of contact stresses and fatigue of_r(i)
10、to restore contact, worm-gear drives that mesh Jn line contact. Thetr - displacement of contact point in motion investigation performed by (Bagel, 1987) showed,.,with ith gear due to rotation, that the real contact stresses for the case of_/ - displacement of contact point in motion point contact ar
11、e much higher (up to six times),.,with ith gear due to misalignment, than for the case of iine contact._=/ - displacement of contact point in motionover the surface of the ith gear. Efforts have been made to develop modified(ul,Si) curvilinear coordinates of the ith methods for generation of worm-ge
12、ar drives based12)surface.on worm-hob mismatch (see (Janninck, 1988;v. - relative velocity of the point on the Colbourne, 1989). However, we consider thatsurface of the worm with respect to that on investigation of modified worm-gear drives mustthe surface of the worm-gear, be based on computerized
13、methods of simulation of- pressure angle, their meshing and contact - the Tooth Contacty - twist angle. Analysis _CA) programs. It is well known that_c - worm generating tool angle of rotation.such an approach applied for spiral bevel and_i - angle of rotation of the ith gear during hypoid gears (Gl
14、eason Works, 1970; Litvin and_ generation. Gutman, 1981) is a significant achievement that- change of orientation of the unit normal could improve substantially the technology andassociated with misaiignment - due to the quality of the gears. A similar approach to1the investigation of worm-gear driv
15、es is on the The authors of this paper have developed anagenda and its importance for the theory and algorithm for computerized simulation of meshingpractice of worm-gear drives is without doubts, and bearing contact that is applicable to alltypes of worm-gear drives - with involute worm,The TCA pro
16、gram for worm-gear drives must have Flender drives, etc. The developed algorithm hasspecific features taking into account the been applied to TCA for misaligned involute worm-following: gear drives.(i) Theoretically, the worm and the worm-gear The developed approach to computerizedtooth surfaces are
17、 in line contact if the hob is simulation of meshing and bearing contact ofidentlcal to the worm and there is no worm-gear drives covers the following topics=misalignment. However, the line contact of worm-gear drives is unstable and due to misaligment (l) Determination of worm surfaceturns into poi
18、nt contact. (2) Determination of worm-gear surface(3) Determination of instantaneous lines of(ii) To localize the bearing contact and fight contact for the theoretical conditions ofwith the influence of mlsalignment, it has been meshing.proposed: (a) to oversize the hob in comparison (4) Transition
19、from line to point contact forwith the operating worm, (b) to use a hob with an a misaligned worm-gear drive.increased number of threads and to change (5) Determination of the real contact pointrespectively the crossing angle for cutting, (c) path, transmission errors, and the bearingto change the l
20、ead angle on the hob and the contact as a set of instantaneous contactcrossing angle for cutting, and (d) to modify the ellipses.profile of the hob. To the authors knowledge,the above-mentioned modifications are based onthe experience of manufacture and have not been 2. Representation of Worm Surfac
21、einvestigated by a TCA program.There are various types of worm surfaces thatA TCA program for worm-gear drives must cover have found application in the industry (Litvintwo parts: () Determination of the theoretical 1968, 1989): () Archimedes worm surface that islines of contact and the curvatures of
22、 the worma ruled undeveloped surface. Such a surface isand the worm-gear surfaces that enable one to generated by a straight line L that performscompare the conditions of lubrication and load screw motion with respect to the worm axis andcapacity for various types of worm-gear drives; intersects the
23、 worm axis in this motion. Line L(2) Investigation of the influence of is the shape of the blade that is used formodifications mentioned above as well as manufacturing of the worm. The cross-section ofmisalignment. Such an investigation must cover the worm is the Archimedes spiral. The termnot only
24、the determination of real bearing “undeveloped“ means that a small part of the wormcontact but also the transmission errors, surface cannot be developed on a plane becausethe normas to the worm surface along theA difficult topic of the TCA program is the generating line are not collinear.determinati
25、on of the transfer point where the (2) Convolute worm surface that is also a ruledtheoretical line contact is changed for point undeveloped surface. The worm surface iscontact. The real bearing contact in the case of generated by a blade that is installedpoint contact will be found as a set of conta
26、ct perpendicular to the helix on the pitch cylinder.ellipses. The cross-section of the worm is an extendedinvolute.The authors hope that the approach developed (3) Screw involute surface that is a ruledin this paper will become the basis for a TCA developed surface that can be generated (ground)prog
27、ram that will cover all the above-mentioned by a plane.topics for various types of worm-gear drives. It (4) Concave-convex worm surface that is generatedwill also give the opportunity to determine the by a grinding wheel whose axial section is adeviations of real worm and worm-gear surfaces by circu
28、lar arc (proposed by Flender). The wormcoordinate measurements and minimize those surface is the envelope of the family of surfacesdeviations by correction of machlne-tool of the grinding wheel. A particular case of thesettings. However, the scope of this paper is concave-convex worm surface represe
29、nts a set oflimited to the discussions of the influence of circular arcs (proposed by Litvln, see (Litvin,mlsalignment but does not cover the investigation 1989a).of surface modifications. Hopefully, this will be (5) Klingelnbergs type of worm surface that isa subject of future research, generated b
30、y a peripheral cutter with conicalsurface. The worm surface is the envelope of theThe development of TCA programs is a powerful family of cone surfaces that is generated intool that will open the way to the solution of reference frame rigidly connected to the worm.significant technological problems:
31、 (i)ocalizatlon of bearing contact, (ii) predesign In accordance with the above-mentioned methodsof a parabolic function of transmission errors of worm generation the worm surface can bethat will reduce the level of vibrations (Litvin represented:et al, 1989b), (iii) generation of modified worm- (1)
32、 In two-parametric form for Archimedes,gear drives, (iv) determination of contact convolute, involute and the above-mentionedstresses for worm-gear drives, etc. modification of the concave-convex worm_1 _1 following equations=_I = _I (u,9), NFI) _ x _ 0 (I)N_I)- . _12) = f(u,e,) = 0 (5)Here (u,8) ar
33、e the cyTyilinear coordinates(surface parameters) and N-“ is the worm surfacenormal. Equations(ii) In three-parameteric form for a regularconcave-convex form and Klingeinbergs type of _2 = _2 (u,8,), f(u,8,) = 0 (6)wormrepresent the worm-gear surface in three-N(C) (cl) parameteric form, but with rel
34、ated parametersc _c = fc(Uc,ec,c) = 0The angles of rotation of the worm and the gear_l = _l(UcSc_c ) = Mlc _c (2) are related in the process of genration anddesignatei_.the generalized parameter of motion.z iHere= Vector _ s represented in Sl and designatesthe relative (sliding) velocity of the worm
35、 with_c = _c(Uc,Sc ) respect to the worm-gear.is the vector equation that represents the 4. Representaton of Lines of Contact.generating surface - the surface of theperipheral cutter (grinding wheel) that generates The ines of contact on the worm surface arethe worm and determined by the equationsBc
36、 BCc 1 = C1(u,e), f (u,e,) = o (7)-(N6c)= - x_uc _8c where the value of _ is fixed for each contactline. Similarly, the lines of contact on theis the normal to the generating surface of the worm-gear surface are determined by the equationstool that, LS represented in coordinate system Sc“Vector _jcl
37、) represents the sliding velocity in _2(u,e,2) = M21(2) _i(u,8), f(u,8,_) = 0 (8)relative motion (it is a screw motion) withrespect to the being generated worm. Designation where again the value of is fixed for each_c indicates the motion parameter in the contact line. The lines of contact in the fi
38、xedgenerating process. Matrix M1c describes the coordinate system are determined by the equationscoordinate transformation in transition from Scto SI. _f(u,e,) = Mf1(l) _l(U8) f(u,8,) = O (9)The worm surface _I and the generating surface _c where Mfl describes the coordinatecontact each other aong a
39、 line and the normal to transformation from Sl to Sf. The totality of thethe worm surface coincides with that to the contact lines in Sf represents the so-calledgenerating surface at each point of that line. surface of action.This normal can be represented by 5. Principal Directions and Curvatures o
40、n_1 = L 1c _c (3) Worm.Gear Surface. Instantaneous ContactEllipse.where Llc is obtained from _lc by eliminatingthe fourth row and column. The determination of the principal curvaturesand directions on the worm-gear surface is aIt is obvious that the representation of the difficult problem due to the
41、 extreme complexityworm surface by of the worm-gear surface. We recall that theseequations represent the surface in three-_l = _(Uc,Sc,_c), fc(Uc,ec,_c ) = 0 (4) parametric form. The solution to this problem canbe substantially simplified using the approachis equivalent to the direct representation
42、of the proposed in (Litvin, 1989a) that enables toworm surface in two-parameteric form express the principal curvatures and directionsof the envelope in terms of those of the_1 = _1 (u,e) generating surface (see Appendix A). Knowing theprincipal curvatures and directions of the twoHenceforth, for th
43、e purpose of simpification of contacting surfaces, it becomes possible tonotation we wil consider that the worm surfaces determine the dimensions and orientation of theof al types are represented by equations (1). instantaneous contact ellipse (Appendix B). Thedimensions of the contact ellipse are3.
44、 Representation of Worm.Gear Surface proportional to the elastic approach of thecontact surfaces.The worm-gear surface _2 is generated by a hobO whose surface is identica to worm surface Zl“ 6. Klgorithm for Simulation of Meshing andThe meshing of the worm with the being generated Beariilg Contact f
45、or a MisaUgned Worm-Gearworm-gear is simulated in the process of Drive,generation. The worm-gear surface _2 isrepresented as an envelope of the family of The misalignment of a worm-gear drive can besurfaces _l that is generated in S2 by the caused by a change of center distance, the twistangle betwe
46、en the axes of the worm and the worm- D(fl,f2,f3,f4,fs,f 6)gear, the axial displacement of the worm-gear, 46 = (16)and similar reasons. The development of the D(Ul,el,U2,e2,_, _)algorithm for simulation of meshing and bearingcontact can be divided into the folowing stages: Jacobian 46 differs from 0
47、 if the values ofparameters _ differ from the nominal (like inStep 1: Derivation of equations of tangency of the case a misaigned worm-gear drive).contacting surfaces. Consider that equations (15) are satisfied by theConsider that the equations of the worm surface set of parameters pO=(lo=u _,81,u2,
48、82,o o o o,_o) and_l and the worm-gear surface _2 are represented _6_0 at pO. Then, equations (15) can be solved inin S1 and S2, respectively, as follows: (i) _1 the neighborhood of pO by functionsequations areuI(1) 81(_) u2(i) 82(;), _(T), 2(;) (17)_l = _l (u1,el), _I = _I (Ul,el) (lO)Due to the co
49、ntinuity of tangency of the(ii) _2 equations are contacting surfaces, functions (17) can bedetermined for a new set of parameters P and_2 = C2(u2e2ql . qn ), system (15) can be solved for a whole range of_1_2 = _2(u2e2_ql . qn ), Step 3: Determination of the contact point pathf(u2,82,q I . qn ) = 0 (11) and transmission errors.The contact point path on _1 is represented bywhere parameters qj (j=l,2, n) represent theassembly parameters or the machine tool settings _l = _l(ul,el), Ul = Ul(i), el =
