《AGMA-93FTM1-1993.pdf》由会员分享，可在线阅读，更多相关《AGMA-93FTM1-1993.pdf（12页珍藏版）》请在三一文库上搜索。
1、93FTM1 Undercuttingin Worms and Worm-Gears by: John R. Colbourne Universityof Alberta,Edmonton,Alberta, Canada AmericanGearManufacturersAssociation I TECHNICALPAPER Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing,
2、 Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,- Undercuttingin Worms and Worm-Gears John R. Colbourne Universityof Alberta,Edmonton,Alberta,Canada Thestatementsand opinionscontainedherein arethoseof theauthorandshould notbeconslru
3、edas anofficialactionor opinion of the American GearManufacturers Association. Copyright 1993 AmericanGear Manufacturers Association 1500King Street, Suite201 Alexandria, Virginia,22314 October,1993 ISBN:1-55589-594-8 Copyright American Gear Manufacturers Association Provided by IHS under license wi
4、th AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,- UNDERCUTTINGIN WORMS AND WORM-GEARS John R. Colbourne Department of Mechanical Engineering University- of Alberta Edmonton,Albert
5、a,Canada T6G 2G8 Abstractuse this method the hob surface must be described by analytical equations.In the case of the thread-milled An equation is developed, which can be used toworm,the threadsurface shapeis generallyfound ensure that there is no undercuttingin a worm.Fornumerically.And in nearly a
6、ll practical cases, the hob worm-gears,the possibilityof undercuttingdepends onprofile is modified from that of the worm, so analytical manyvariables,and nosimplecriterionhas beenequations are unlikely to exist.It was therefore not found. Procedures are therefore described for checkingfoundpossiblet
7、o developanysimpleequationto whether there is undercutting,and also whether therepredictthe occurenceof non-conjugatecontactor are other potentialproblems,such as interferenceorundercutting,but proceduresare described for checking non-conjugatecontact,any particulargear-pairdesign. It is shown that
8、there is a maximum value for the gear face-width,beyond Introductionwhich non-conjugate contact will occur at the exit end of the gear tooth.Additionaltopics discussedin the In the design of worms and worm-gears, it haspaper are the possibility of interferenceat the worm been commonpracticefor many
9、years to use largerthread fillet, and of non-conjugatecontact at the worm pressure angles when the lead angle is large, and onethread tip caused by very small pressure angles.And of the reasons for this practice is to avoid undercuttinglastly, the paper describes a methodfor determining in the worm.
10、In this paper, a study is made of thewhether there will be undercuttingin the teeth of the conditionswhen undercuttingmay occur, and it isgear. shown that these depend not only on the pressure angle and the lead angle, but also on the number of threads.A number of exampleshave been considered, An eq
11、uation is derived which can be used to ensuresome of conventional designs in which the pitch point that there is no undercutting,coincides with the mean point, and some of recess- action designs.Itcan beconcludedfrom these The paper then examines the possibility of non-examples that non-conjugatecon
12、tact and undercutting conjugate contact between the worm and the gear, andare very much more likely to occur in recess-action of undercuttingin the gear.Previous researchers1-3designs.In addition, it was found that in some recess- haveshown that undercuttingoccurs ff the contactaction gears, even th
13、ough there is no undercutting, the lines on the generating surface form an envelope, anddepth of the fillet may be greaterthan that of the the surface extends beyond the envelope.However, toactive profile, and the fillet may be cut so deeply that Copyright American Gear Manufacturers Association Pro
14、vided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,- the tooth is considerablyweakened. So,althoughthereX = ( Row- Yw)(2) are some significant advantages
15、 to recess action, it istan 0r important that the designs be checked for the possible problems discussed in this paper.Y = Row + Rpg - Yw(3) = _ X 2 +y2(4) Rg Conjugate Action where Ywis the coordinate of worm point A, Rpw and The meshing of a worm and its conjugate gearR_o are the pitch circle radi
16、i, and R is the radius of has been fully analysed by Buckingham4,5.Wett/-_“ conjugate point on the gear.When the shape of regard the worm as a rack, since an axial advance ofthe worm thread is specified, the axial coordinate zwof the worm along its axis is kinematicallyequivalent topoint A is known.
17、The axial translationof the worm a rotation.The directionsof the xw and Yw axes,and the corresponding rotation of the gear can then be shown in Figure 1, therefore remain fixed as the wormdetermined,and hence the polar coordinate0g of the translates in the zw direction.The plane Xw=0 in thegear toot
18、h point can be found. worm will be called the central axial plane, and any parallelplane at Xw=COnStant will be called an offset plane.We calculatethe thread profile in an offsetCw_,/Wrm axis section, and this profile, regardedas a rack, is used toyw/,/ generatetheconjugatetoothprofileinthe gear_i %
19、Rpw transversesection zg=-x w. At a typical point A of a wrm Prffie the angle _r between the pr6file tangentS and the vertical will be called the rack pressure angle, and its value is given 6 by the following expression,Path of contact ( tan 0t sin O - cos O)Rpg tan _r=(1) tan where0, 0t and _t are
20、the polarcoordinate,the transverse pressure angle and the helix angle at point Y A. X “-C9 j_-Worm Figure 2. Path of contact in a typical offset section. wzw( xw YwYw Undercuttingin the Worm Z YIt is clear from Equation (2) that the coordinate k X approachesinfinity as 0r tends towardzero.At X/_%gZg
21、_Cgpoints where Oris negative, Equation (2) is no longer valid, and conjugate action is impossible.The locus of Gear axispoints at which 0r is equal to zero is known as the limit of conjugate action.Buckingham has shown 4 Figure 1. Coordinate axes.that for involute helicoid worms the limit of conjug
22、ate action lies along the two horizontaltangentsto the base circle, as shown in Figure 3, and that for other types of worm the locus is asymptoticto the same The gear tooth is in contact with the wormhorizontal lines. profile at point A when the profile normal at A passes through the pitch point P,
23、as shown in Figure 2. TheFigure 3 is drawn looking in the positivezw positionin space where the contact occurs, i.e. thedirection (See Figure 1).We are assuminga right- coordinates of a point on the path of contact, can thenhanded thread, and in order to drive the gear clockwise be read from Figure
24、2,in Figure 1, the worm in Figure 3 must rotate counter- 2 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license fro
25、m IHS -,-,- F-WormIt is not made clear by Buckingham exactly what problemoccurs if larger lead angles are used, or how he arrived at these particularlimiting values.A similar set of maximumlead angle values is given in AGMA /L_Screw Helicid341.02 7, and these values are Iisted in Table2. Again, no e
26、xplanation is presented, and it is interesting thatthevaluesareconsiderablyhigherthan _-_/“-InvoluteHelicoidBuckinghams. High PA -/“_-11/“-LowPAIt has been suggested that larger lead angles than EntryendI,IExitendthose recomendedin Tables 1 or 2 may lead to Ixx-Gearundercutting in the worm. However,
27、 the author ofthe t 1 presentpaperbelievesthesituationismore complicated,and that the possibilityof undercutting Figure 3. Limit of conjugate action,depends both on the lead and pressure angles, and on the number of threads.To simplify the analysisof undercutting,we will consider a worm which is gro
28、und by a straight-sidedgrinding wheel of infinitediameter. clockwise.Theleft-handside of thediagramisThis is the limiting case of both the involute helicoid therefore the entry end, where the pressure angle_)risworm and the thread-milled worm, when the grinding larger, and the right-handside is the
29、exit end, where _rwheel diameter increases to infinity. is smaller.It can be seen from the diagram that contactwithpointsof the wormat the limitof/_ conjugacyis only a danger near the exit end of the/Grinding wheel gear tooth. MaximumLead Angles(Buckingham) (_n_maxVh 14.515.95 2024.23A 2536.87/q _m
30、Buckingham5 has giventablesof suitable valuesfor wormdesigns.Referringto one set ofFigure 4. Worm and grinding wheel. values, he states:“Table 1 stops when the value of the lead angle approaches16 degrees, because with theThe worm and the grinding wheel are shown in 14.5 degree form, effectivecontac
31、t conditions do notFigure 4.The worm has Nw threads, a lead Lw, and exist beyond that point.“Table 1 in this paper givesan axial pitch Pzw equal to (Lw/Nw).We choose a the largest lead angles proposed by Buckingham,forradius Rmw at the mean thread height, and the lead the three most commonly used no
32、rmal pressure angles,angle _mat this radius is given by Table 2tan _Lw-(5) 2 _zRmw MaximumLead Angles (AGMA 341.02) The grinding wheel is generally set at this lead angle, as shown in Figure 4, and the traversing velocity vh is _n_mmaxchosen so that the grinding wheel advances a distance 14.517Lwfor
33、eachrotationoftheworm, 2030Lwco 2545Vh =(6) 2_ 3 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,
34、- _-Rackcutter Nw Ptr tBy combining Equations (5,7 and8),wecanshow tr _hthat the pitch circle radius Ro is identical with the -meancircle radius Rmw of the worm.Fromthe conventional theory of involute gears, we know that 1there is no undercuttingif point H on the rack toothprofile where the straight
35、 section ends cuts the path of _-mcontact between the pitch point P and the interference point E.The distance between point E and the rack Figure 5. Worm and rack cutter,pitchlineis Rp sin2_t, where(_tis the transverse pressureangle.Most worms are designedwitha where 03 radians/secis the angular vel
36、ocityof thededendum of 1.157m, where the module m is defined worm.as (pzw/_). Allowing for a circular rounding at the tip of the grinding wheel, the distance between point H In Figure 5 the grinding wheel of Figure 4 isand the rack pitch line is then about 1.05m.The conditionfor no undercuttingis th
37、erefore replaced by a rack cutter moving to the left at the same velocity viaas the grinding wheel. The shape of theRpsin2_t 1.05m(9) wormthreadscutby therackcutterwouldbe thesame as thosegroundbya grindingwheelof infiniteWe express Ro in terms of the module, and(_tin terms diameter.We now replace t
38、he horizontal velocity vl_ofOn andX m, toobtaintheconditionforno of the rack cutter by a vertical velocity Vr, equal toundercutting in its final form, (vJtan_m)“ The two rack motions are kinematically equivalent,but with the verticalrack velocitywe cantan2_n2.1 _,(10) regard the worm as a convention
39、al helical gear meshedtan _m ( sin2_m+ tan2(_n )Nw with a rack, whose pitch Ptr in the vertical directionis givenbyIt is clearthatthemaximum valueofthelead Pzwangle _m dependson the numberof threadsNw,as Ptr = _(7)well as on the normalpressureangle On“Maximum tan Lmvalues of _hnare given in Table 3
40、for worms with 1 to 12 threads.The table also gives minimumvalues for The pitch circle of the worm when it meshes as the diameter quotient q, which is defined as the mean a gear with a conventionalrack is shown in Figure 6. Its radius Rp is given 8 by the followingexpression, Tv _f_Pathof contact s,
41、iil, i,oh, e Figure 7.Worm thread axial section. Figure 6. Meshing diagram for worm and rack cutter.Nw = 3,_n = 14“5,Xm= 20“557- 4 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007
42、 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,- circle diameterdivided by the module.Comparison with Equation (5) shows that the diameter quotient is directlyrelated to the lead angle. 2Rmw q = ._(11) m tanXm = Nw(12) q Table 3 _nNw_mmaxqmin 14.5114.133.97 219.575
43、.62 323.246.99Figure 8. Worm thread normal section. 426.088.17Nw = 3,_n = 14“5,Lm = 20“557“ 528.459.23 630.5010.19 732.3111.07Thevaluesin Table3 werecalculatedassuming 833.9411.89a grindingwheel of infinitediameter. A real grinding 935.4312.65wheel would be contained entirely inside an infinite- 103
44、6.8013.37diameter grinding wheel.Hence, if a worm is 1138.0714.04designedso thatthereis no undercuttingfroman 1239.2714.68infinite-diametergrinding wheel,then therewill be no undercuttingfromtherealgrindingwheel.Most 20116.493.38wormsare designed with diameterquotientsof 7 or 223.454.61higher,so it
45、is clearfrom Table3 that undercuttingof 328.105.62the worm is not often a problem, at least when the 431.706.48numberof threadsis not greaterthan5. 534.677.23 637.227.90 739.468.50NumericalExample 841.479.05 943.299.55In order to considersome specificcases, the 1044.9610.01profiles were calculated f
46、or a number of worms with 1146.5010.44the following specifications,all lengths beinggiven in 1247.9210.83inches.Axial pitch pzw=0.5;Outer diameter Dow=l.592;MeandiameterDmw=l.2732,givinga 25118.203.04diameter quotient q=8.0;Axial thread thickness 226.464.02tzmw=0.245;Grinding wheel meandiameter 331.
47、974.81D=160The examplesusedin thispaper arefor mEw“ “ 436.195.47involutehelicoidworms,butthe methodsapplyequally 539.656.03to thread-milledor screw helicoidworms. When the 642.606.52wormhas 1, 2, 3, 4 or 5 threads,the lead anglesare 745.176.96respectively7.125, 14.037,20.557,26.566and 32.006 847.467
48、.34degrees. 949.517.69 1051.377.99Figures7 and 8 show the axial sectionand 1153.068.27normal section profiles of a 3-threadworm with a 1254.628.52normalpressure angle of 14.5degrees. This example was chosen becausethe lead angle of 20.557 degrees 5 Copyright American Gear Manufacturers Association Provided by IHS under license with AGMA Licensee=IHS Employees/1111111001, User=Wing, Bernie Not for Resale, 04/18/2007 11:59:45 MDTNo reproduction or networking permitted without license from IHS -,-,- is below the maximum value given in