1、93FTM1Undercutting in Wormsand Worm-Gearsby: John R. ColbourneUniversity of Alberta, Edmonton, Alberta, CanadaAmerican Gear Manufacturers AssociationITECHNICAL PAPERUndercutting in Worms and Worm-GearsJohn R. ColbourneUniversity of Alberta, Edmonton, Alberta, CanadaThestatementsand opinionscontained
2、herein arethose of theauthor andshould notbe conslrued as an officialactionoropinion of the American GearManufacturers Association.Copyright 1993AmericanGear Manufacturers Association1500King Street, Suite 201Alexandria, Virginia,22314October, 1993ISBN: 1-55589-594-8UNDERCUTTING IN WORMS AND WORM-GE
3、ARSJohn R. ColbourneDepartment of Mechanical EngineeringUniversity- of AlbertaEdmonton, Alberta, Canada T6G 2G8Abstract use this method the hob surface must be described byanalytical equations. In the case of the thread-milledAn equation is developed, which can be used to worm, the thread surface sh
4、ape is generally foundensure that there is no undercutting in a worm. For numerically. And in nearly all practical cases, the hobworm-gears, the possibility of undercutting depends on profile is modified from that of the worm, so analyticalmany variables, and no simple criterion has been equations a
5、re unlikely to exist. It was therefore notfound. Procedures are therefore described for checking found possible to develop any simple equation towhether there is undercutting, and also whether there predict the occurence of non-conjugate contact orare other potential problems, such as interference o
6、r undercutting, but procedures are described for checkingnon-conjugatecontact, any particular gear-pairdesign. It is shown that thereis a maximum value for the gear face-width, beyondIntroduction which non-conjugate contact will occur at the exit endof the gear tooth. Additional topics discussed in
7、theIn the design of worms and worm-gears, it has paper are the possibility of interference at the wormbeen common practice for many years to use larger thread fillet, and of non-conjugate contact at the wormpressure angles when the lead angle is large, and one thread tip caused by very small pressur
8、e angles. Andof the reasons for this practice is to avoid undercutting lastly, the paper describes a method for determiningin the worm. In this paper, a study is made of the whether there will be undercutting in the teeth of theconditions when undercutting may occur, and it is gear.shown that these
9、depend not only on the pressure angleand the lead angle, but also on the number of threads. A number of examples have been considered,An equation is derived which can be used to ensure some of conventional designs in which the pitch pointthat there is no undercutting, coincides with the mean point,
10、and some of recess-action designs. It can be concluded from theseThe paper then examines the possibility of non- examples that non-conjugate contact and undercuttingconjugate contact between the worm and the gear, and are very much more likely to occur in recess-actionof undercutting in the gear. Pr
11、evious researchers 1-3 designs. In addition, it was found that in some recess-have shown that undercutting occurs ff the contact action gears, even though there is no undercutting, thelines on the generating surface form an envelope, and depth of the fillet may be greater than that of thethe surface
12、 extends beyond the envelope. However, to active profile, and the fillet may be cut so deeply thatthe tooth is considerably weakened. So, althoughthere X = ( Row- Yw) (2)are some significant advantages to recess action, it is tan 0rimportant that the designs be checked for the possibleproblems discu
13、ssed in this paper. Y = Row + Rpg - Yw (3)= _ X 2 + y 2 (4)RgConjugate Actionwhere Ywis the coordinate of worm point A, Rpw andThe meshing of a worm and its conjugate gear R_o are the pitch circle radii, and R is the radius ofhas been fully analysed by Buckingham 4,5. We tt/-_“conjugate point on the
14、 gear. When the shape ofregard the worm as a rack, since an axial advance of the worm thread is specified, the axial coordinate zw ofthe worm along its axis is kinematically equivalent to point A is known. The axial translation of the worma rotation. The directions of the xw and Yw axes, and the cor
15、responding rotation of the gear can then beshown in Figure 1, therefore remain fixed as the worm determined, and hence the polar coordinate 0g of thetranslates in the zw direction. The plane Xw=0 in the gear tooth point can be found.worm will be called the central axial plane, and anyparallel plane
16、at Xw=COnStant will be called an offsetplane. We calculate the thread profile in an offset Cw _,/Wrmaxissection, and this profile, regarded as a rack, is used to yw/ ,/generate the conjugate tooth profile in the gear _ i % Rpwtransverse section zg=-x w. At a typical point A of awrm Prffie the angle
17、_r between the pr6file tangent Sand 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) Rpgtan_r = (1)tanwhere 0, 0t and _t are the polar coordinate, thetransverse pressure angle and the helix angle at point
18、 YA.X “- C9j_-Worm Figure 2. Path of contact in a typical offset section. wzw ( xw Yw Yw Undercutting in the WormZY It is clear from Equation (2) that the coordinatek X approaches infinity as 0r tends toward zero. AtX /_ %g Zg _ Cg points where Or is negative, Equation (2) is no longervalid, and con
19、jugate action is impossible. The locus ofGear axis points at which 0r is equal to zero is known as thelimit of conjugate action. Buckingham has shown 4Figure 1. Coordinate axes. that for involute helicoid worms the limit of conjugateaction lies along the two horizontal tangents to thebase circle, as
20、 shown in Figure 3, and that for othertypes of worm the locus is asymptotic to the sameThe gear tooth is in contact with the worm horizontal lines.profile at point A when the profile normal at A passesthrough the pitch point P, as shown in Figure 2. The Figure 3 is drawn looking in the positive zwpo
21、sition in space where the contact occurs, i.e. the direction (See Figure 1). We are assuming a right-coordinates of a point on the path of contact, can then handed thread, and in order to drive the gear clockwisebe read from Figure 2, in Figure 1, the worm in Figure 3 must rotate counter-2F-Worm It
22、is not made clear by Buckingham exactly whatproblem occurs if larger lead angles are used, or howhe arrived at these particular limiting values. A similarset of maximum lead angle values is given in AGMA/ L_ Screw Helicid 341.02 7, and these values are Iisted in Table 2.Again, no explanation is pres
23、ented, and it is interestingthat the values are considerably higher than_-_/“-Involute Helicoid Buckinghams.High PA -/ “_ -11/ “-Low PA It has been suggested that larger lead angles thanEntryend I ,I Exitend those recomendedin Tables 1 or 2 may lead toIxx-Gear undercutting in the worm. However, the
24、author of thet 1present paper believes the situation is morecomplicated, and that the possibility of undercuttingFigure 3. Limit of conjugate action, depends both on the lead and pressure angles, and onthe number of threads. To simplify the analysis ofundercutting, we will consider a worm which is g
25、roundby a straight-sided grinding wheel of infinite diameter.clockwise. The left-hand side of the diagram is This is the limiting case of both the involute helicoidtherefore the entry end, where the pressure angle _)r is worm and the thread-milled worm, when the grindinglarger, and the right-hand si
26、de is the exit end, where _r wheel diameter increases to infinity.is smaller. It can be seen from the diagram thatcontact with points of the worm at the limit of /_conjugacy is only a danger near the exit end of the / Grinding wheelgear tooth.Maximum Lead Angles (Buckingham)(_n _ max Vh14.5 15.9520
27、24.23 A25 36.87 /q_mBuckingham 5 has given tables of suitablevalues for worm designs. Referring to one set of Figure 4. Worm and grinding wheel.values, he states: “Table 1 stops when the value of thelead angle approaches 16 degrees, because with the The worm and the grinding wheel are shown in14.5 d
28、egree form, effective contact conditions do not Figure 4. The worm has Nw threads, a lead Lw, andexist beyond that point.“ Table 1 in this paper gives an axial pitch Pzw equal to (Lw/Nw). We choose athe largest lead angles proposed by Buckingham, for radius Rmw at the mean thread height, and the lea
29、dthe three most commonly used normal pressure angles, angle _m at this radius is given byTable 2 tan _ Lw- (5)2 _zRmwMaximum Lead 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 _mmax chosen so that the grinding wh
30、eel advances a distance14.5 17 Lwfor eachrotationof the worm,20 30 Lw co25 45 Vh = (6)2_3_-Rack cutter Nw Ptrt By combiningEquations(5, 7 and 8), we can showtr_h that the pitch circle radius Ro is identical with the- mean circle radius Rmw of the worm. From theconventional theory of involute gears,
31、we know that1 there is no undercuttingif point H on the rack toothprofile where the straight section ends cuts the path of_-m contact between the pitch point P and the interferencepoint E. The distance between point E and the rackFigure 5. Worm and rack cutter, pitch line is Rp sin2_t, where (_t is
32、the transversepressure angle. Most worms are designed with awhere 03 radians/sec is the angular velocity of the dedendum of 1.157m, where the module m is definedworm. as (pzw/_). Allowing for a circular rounding at the tipof the grinding wheel, the distance between point HIn Figure 5 the grinding wh
33、eel of Figure 4 is and the rack pitch line is then about 1.05m. Thecondition for no undercutting is thereforereplaced by a rack cutter moving to the left at the samevelocity via as the grinding wheel. The shape of the Rp sin2_t 1.05m (9)wormthreadscut by the rack cutterwouldbe the sameas those groun
34、d by a grinding wheel of infinite We express Ro in terms of the module, and (_t in termsdiameter. We now replace the horizontal velocity vl_ of On and Xm, to obtain the condition for noof the rack cutter by a vertical velocity Vr, equal to undercutting in its final form,(vJtan _m)“ The two rack moti
35、ons are kinematicallyequivalent,but with the verticalrack velocity we can tan2_n 2.1_, (10)regard the worm as a conventional helical gear meshed tan _m ( sin2_m + tan2(_n ) Nwwith a rack, whose pitch Ptr in the vertical direction isgivenby It is clearthatthemaximumvalueof the leadPzw angle _m depend
36、son the number of threads Nw, asPtr = _ (7) well as on the normalpressure angle On“ Maximumtan Lm values of _hnare given in Table 3 for worms with 1 to12 threads. The table also gives minimum values forThe pitch circle of the worm when it meshes asthe diameter quotient q, which is defined as the mea
37、na gear with a conventional rack is shown in Figure 6.Its radius Rp is given 8 by the following expression,Tv_f_ _Path of contacts,iil, i,oh, eFigure 7. Worm thread axial section.Figure 6. Meshing diagram for worm and rack cutter. Nw = 3, _n = 14“5, Xm= 20“557-4circle diameter divided by the module.
38、 Comparisonwith Equation (5) shows that the diameter quotient isdirectly related to the lead angle.2Rmwq = ._ (11)mtanX m = Nw (12)qTable 3_n Nw _mmax qmin14.5 1 14.13 3.972 19.57 5.623 23.24 6.99 Figure 8. Worm thread normal section.4 26.08 8.17 N w = 3, _n = 14“5, Lm = 20“557“5 28.45 9.236 30.50 1
39、0.197 32.31 11.07 The values in Table 3 were calculatedassuming8 33.94 11.89 a grindingwheel of infinite diameter. A real grinding9 35.43 12.65 wheel would be contained entirely inside an infinite-10 36.80 13.37 diameter grinding wheel. Hence, if a worm is11 38.07 14.04 designed so that there is no
40、undercutting from an12 39.27 14.68 infinite-diameter grinding wheel, then there will be noundercutting from the real grinding wheel. Most20 1 16.49 3.38 worms are designed with diameterquotientsof 7 or2 23.45 4.61 higher, so it is clear from Table 3 that undercuttingof3 28.10 5.62 the worm is not of
41、ten a problem, at least when the4 31.70 6.48 numberof threadsis not greaterthan 5.5 34.67 7.236 37.22 7.907 39.46 8.50 NumericalExample8 41.47 9.059 43.29 9.55 In order to consider some specificcases, the10 44.96 10.01 profiles were calculated for a number of worms with11 46.50 10.44 the following s
42、pecifications, all lengths being given in12 47.92 10.83 inches. Axial pitch pzw=0.5; Outer diameterDow=l.592; Mean diameter Dmw=l.2732, giving a25 1 18.20 3.04 diameter quotient q=8.0; Axial thread thickness2 26.46 4.02 tzmw=0.245; Grinding wheel mean diameter3 31.97 4.81 D =160 The examplesused in
43、this paper are formEw “ “4 36.19 5.47 involute helicoid worms, but the methods applyequally5 39.65 6.03 to thread-milled or screw helicoid worms. When the6 42.60 6.52 worm has 1, 2, 3, 4 or 5 threads,the lead angles are7 45.17 6.96 respectively 7.125, 14.037,20.557, 26.566 and 32.0068 47.46 7.34 deg
44、rees.9 49.51 7.6910 51.37 7.99 Figures 7 and 8 show the axial section and11 53.06 8.27 normal section profiles of a 3-thread worm with a12 54.62 8.52 normal pressure angle of 14.5 degrees. This examplewas chosen because the lead angle of 20.557 degrees5is below the maximum value given in Table 3, bu
45、t wellabove the values given in Tables 1 and 2. There is notrace of undercuttingin eitherprofile,and, in the _ _ _ _authors opinion, this worm would mesh satisfactorilywith a suitable conjugate gear.Maximum Gear Face-WidthWhile it has been shown that larger lead anglesthan those in Tables 1 and 2 do
46、 not necessarily causeundercutting of the worm, there are other problems that Figure 10. Contact area diagram.may occur. In particular, the area of conjugate contact Nw = 1, (_n= 20, Ng = 50.may be limited at the exit end of the gear tooth, andthe face-width should be chosen so that the tooth doesno
47、t extend beyond this point. Nw=l and _n=20. In this example, the gear has 50teeth, and the center distance is 4.615, which meansThe path of contact is shown in Figure 9, with that the pitch point coincides with the mean point. Thethe end points labelled Tg and Tw. When the worm is gear outer diamete
48、r is 8.44, its throat radius is 0.478,driving, the initial contact point is To, and the final and its face-width is 1.1, which is 0.864 times thecontact point is T Point T is the point where the mean diameter. In Figure 10, and in all the contactW“ gpath of contact is intersected by the tip circle o
49、f the area diagrams shown in this paper, the axes are plottedgear The radius R,- of this circle is equal to the outer through the pitch point.“ lgradius Ro., except in the throat part of the gear toothThe other end Tw of the path of contact corresponds to In each contact area diagram, the upper edge ofcontact at the tip of the worm thread, and lies at the the contact area coincides with the tip of the gearintersection of the path of contact and the horizontal tooth. The lowest curve in the disgram is a circularline Ywmaxbelow the wor
