1、90 FTM 1Contact Stresses in Gear Teethby: John R. Colbourne, University of AlbertaAmerican Gear Manufacturers AssociationII Ill! IlllITECHNICAL PAPERContact Stresses in Gear TeethJohn R. Colbourne,University of AlbertaThe Statements and opinions contained herein are those of the author and should no
2、t be construed as anofficial action or opinion of the American Gear Manufacturers Association.ABSTRACT:It is shown that neither Hertzs line contact theory nor his point contact theory are entirely adequate forthe accurate calculation of contact stresses in gear teeth. A numerical procedure is descri
3、bed, which canbe used to find the contact stress in cases where the relative curvatures in the contact region are notconstant.Copyright 1990American Gear Manufacturers Association1500 King Street, Suite 201Alexandria, Virginia, 22314October, 1990ISBN: 1-55589-553-0CONTACT STRESSES IN GEAR TEETHJ.R.
4、Colbourne,Department of Mechanical Engineering,University of Alberta,Edmonton, Alberta T6G2G8, Canada.Introduction actual contact areas encountered in gearingoften differ from the elliptical shapesWith the use of mismatch in worm and predicted by conventional point contactbevel gears, and crowning i
5、n helical gears, theory.there is no longer line contact between theteeth, and it is evident that the contact In this paper, we will use examplesstress cannot be found accurately by Hertzs found in worm gearing, but the methodline contact theory. His point contact described for calculating the contac
6、t stresstheory is also sometimes unsuitable, because is equally applicable to bevel gears. It maythe assumption of constant curvature within also be useful for helical gears withthe contact area is not realistic, and the crowning, where the real stress distributionis probably similar to line contact
7、 near the00050 middle of the teeth, and to point contactr neartheends.0000_ _ The Need For a New Contact Stress TheoryA traditional line contact diagram fori a worm and worm gear is shown in Figure I.This is the contact diagram we obtain whenthere is no mismatch, so that the gear isI - | oo0o _ exac
8、tly conjugate with the worm. The threeI I lines shown in the diagram represent theooo5o contact lines on three adjacent teeth. ToFigure I. obtain the tooth forces and theGear pair Number I. Contact Lines. corresponding contact stresses, we assumethat the load intensity is constant along the gear pai
9、r of Figure I, when there is someeach contact line. From the specified mismatch between the worm and the gear. Thetorque, we calculate the total tooth force, worm is thread-milled with five threads, andWe then divide the total tooth force by the has a lead angle of 21.31 degrees. Thetotal length of
10、the contact lines, to obtain mismatch is obtained by modifying thethe load intensity, and to find the profile of the hob that is used to cut theindividual tooth forces we multiply the load gear, by making its lead slightly shorterintensity by the lengths of the individual than that of the worm, and
11、by swivelling thecontact lines. The contact stress at each direction of its axis when the gear is cut,point is found by Hertzs .line contact so that it is not exactly perpendicular totheory, using the relative curvature in the the gear axis. The contact diagram shows thedirection perpendicular to th
12、e contact line. contours of constant separation between theworm thread, and the envelope of the gearThe assumption of constant load tooth positions relative to the worm. Inintensity is questionable, even when the other words, it shows how closely the gearworm and the gear are exactly conjugate, and
13、teeth approach the worm thread during thethe contact lines extend to the edges of the entire meshing cycle, assuming that both areteeth. If the load intensity is constant rigid and that the angular velocity ratio isalong each contact line, it must be exactly constant. The two curves in theindependen
14、t of the relative curvature, which diagram show the contours where thedoes vary along the contact lines, and this separation is 0.0005 inches and 0.0010independence seems unlikely. However, in the inches. If we had included the curve wherecase where there is some mismatch the contact the separation
15、is 0.00025 inches, this wouldareas do not reach the edges of the teeth, correspond approximately to the area thatand it is quite certain that the load would be coloured by the marking compound inintensity is not uniform. Instead of line a bearing test.contact we have point contact, though thecontact
16、 areas may be very elongated, and we we now consider the gear pair in thetherefore have to consider the use of Hertzs same position as in Figure I, and wepoint contact theory, calculate the magnitude of the separationbetween the teeth in each of the three toothFigure 2 shows the contact diagram for
17、pairs. Figure 3 shows the contours where the0 separation between the teeth is 0.0010I iiii _ _ inches. It is obvious that we have point_-_ _ 0 contact instead of line contact, which iswhat we would expect when we introduce themismatch. It also appears that any stress. “_ “-“-_ _. calculation based o
18、n line contact will be4 _ 0000s,. extremely inaccurate, particularly in the_ %_ j_/ case of the upper tooth pair, where thepossible contact area is now much shorterthan the line in Figure I.encountered when we try to use Hertzs point0theory. First, the theory was developed toFigure 2. calculate the
19、contact stress when theGear Pair Number I. Contact Diagram. contact force is known. In our case, we knowvO 0050C- ooooo wi_ _ii EquivalentSpringSystem.qthree springs have initial gaps, and the _-_00000 _ relation between force and compression for_ooo5o each spring is non-linear, as in all contactFig
20、ure 3. problems.Gear Pair Number I. Separation Contours.The second diffulty we find in usingHertzs point contact theory is that Hertzthe total contact force, but we do not know dealt only with contacting bodies which havehow it is shared between the tooth pairs, constant curvatures in the contact re
21、gion.This problem is made more difficult by the This condition implies that the bodyfact that there is initially no contact at surfaces near the initial point of contactthe upper and lower tooth pairs, and that can be represented by ellipsoids, and thatthis contact only occurs when there is some con
22、tours of constant separation are thenflexibity in the middle tooth pair. The concentric ellipses. The contours inminimum separations in the direction of the Figure 3 are almost elliptical, so it appearsworm axis between the three tooth pairs were that Hertzs condition may be satisfied forfound to be
23、 0.00033, 0.00003 and 0.00083 this gear pair. However, Figure 5 shows twoinches. These were calculated assuming the separation contours for the upper toothteeth are rigid, and that the angular pair, the inner one at 0.0010 inches and thevelocity ratio is constant. To understand outer one at 0.0020 i
24、nches. Under light loadthe effect of these separations we will the contact will look like Hertzian pointconsider the gear as fixed. First, the worm contact, within the elliptical contour, butcan make a small rotation, corresponding to as the load is increased the contact areaan axial displacement of
25、 0.00003 inches,O0050before there is any contact whatsoever. Thisrotation represents the transmission error -“_. F_I- o oooounder no load, for the current position ofthe gear pair. Once contact is obtained atthe central tooth pair the worm may continue I _-_to rotate, due to the contact stressindent
26、ation and the bending flexibility ofthe teeth. As the rotation continues, the _upper and lower tooth pairs will eventually _come into contact, and the rotation willcease when the three contact forces are in I _00000 _equilibrium with the applied torque. The problem of finding the individual tooth pa
27、ir oo05ocontact forces can be represented by the Figure 5.spring system shown in Figure 4. Two of the Gear Pair Number I. Separation Contours.-0,0050r iO0050e-seso i -_ ,-_ ccoc I: o.oolo“ 1 0.0005; . “,; I _ _, _- - . .0.0020“ /,9-eBB. v?_CCC - _-_s0050 ooosoFigure 6. Figure 8.Gear Pair Number 2. C
28、ontact Diagram. Gear Pair Number 2. Separation Contours.will spread to the right, and now looks much which passes close to the throat, such as themore like the line contact that we rejected one shown in Figure 7, is in fact far fromearlier, making contact near its center, while theremay be contact t
29、aking place near the ends.The conclusion that Hertzs point The real separation contours, forcontact theory is sometimes unsuitable for separations of 0.0010 and 0.0020 inches, aregearing problems gains more support, if we shown in Figure 8. Under light load there areconsider the gear pair whose cont
30、act diagram two separate contact areas, and these willis shown in Figure 6. The worm is again join up when the load is increased, to formthread-milled with five threads, and in this an hour-glass shaped contact area. It iscase the lead angle is 25.72 degrees. The evident that this particular tooth p
31、air doesupper diagram in Figure 6 shows the not come close to satisfying Hertzsseparation along the tip of the gear tooth, condition of constant curvature.and it can be seen that there is considerableseparation in the region near the throat. What is needed, for the solution of theThis means that any
32、 traditional contact line contact problem in gearing, is a method forcalculating the contact stress and the00o50 amount of indentation, when the shapes of the. ,.-_ _-_ - _osO_._ _. _ _ contacting bodies are entirely arbitrary.The second part of this paper will describesuch a method., - _ _ Beforepr
33、oceedingwith thedescription,it is worthwhileto explainwhy the shapesof. /-_ the gear teeth in Figures 7 and 8 may appearrather odd, particularly along the upperedges. If we neglect the effect of theL _ mismatch, the contact between a worm and itsiiii gear can be analysed as a number of rack andL0 pi
34、nions lying in parallel planes, each rackFigure 7. being a section through the worm thread in aGear Pair Number 2. Contact Lines. plane perpendicular to the gear axis.4Worm axis shape of the worm thread tip, but the upperedge (corresponding to points A and B inE _._Geartipcircle Figure 9) is no long
35、er the same shape as thegear tooth tip. The difference is obvious inFigures 7 and 8, but is not visible inP_,_/-Low_ PA. Figures I, 3 and 5, since this particularlygear pair has only recess action, and the/Hi_h/_L g PA u_ paths of contact would all start at point E in Figure 9.Figure 9.Paths of Cont
36、act.Hertzs Contact Stress TheoryFigure 9 shows the path of contact for twosuch racks, one with a high pressure angle Hertzs theory of point contact stressand the other with a low pressure angle. The is described in books on elasticity, such asupper end point of each path of contact is Timoshenko and
37、 Goodier I . We have twodetermined by the maximum radius in the gear bodies touching at a point, as in Figure 10,tooth, while the lower end point is and we set up an (x, y) coordinate system indetermined by the maximum depth below the the common tangent plane. The separation sworm axis of the partic
38、ular rack section we when the bodies are undeformed is a knownare dealing with. If we consider two function of x and y. Points A I and A2 of thesections, equidistant from the central two bodies lie on the common normal, remotesection and on opposite sides, then the from the contact point. We hold AI
39、 fixed, andmaximum depths of the two sections are equal, apply a force at A2, causing it to move aas shown in Figure 9. However, the end points distance 6 towards AI, as shown in Figure 11.C and D of the two paths of contact are at The quantity 6 is called the approach of thedifferent radii, measure
40、d from the gear two bodies, or the indentation.axis. Hence, if we plot the contact diagramon the gear teeth (i.e., we plot values of Under load, the point of contactthe gear coordinates Rg and zg for the becomes an area of contact, due to theoutline of the contact region), the top edge flexibility o
41、f the bodies. If we consider thewill follow the shape of the gear tooth tip, two bodies separated, as in Figure 12, thebut the lower edge will not be exactly the contact stress acts with equal magnitude andcircular shape of the worm thread tip. This in opposite directions on the two bodies. Thecan b
42、e seen in the contact diagrams in corresponding displacements w I and w2 mustFigures 2 and 6, which are plotted on the be such that the deformed shapes touch withingear teeth, as is customary for contact .A2diagrams.The remaining diagrams, Figures I, 3, _ /5, 7 and 8, show the contact lines or the _
43、 _./_s _ xseparation diagrams on a number of wormthreads, seen one behind the other. It _therefore seems more natural to plot these ona worm thread, so we calculate the outline ofthe contact region using the x and y “Atcoordinates in the worm. In this case the Figure 10.lower edge follows exactly th
44、e circular Initial Separation.IA2Figure 13._ EquivalentHalf-SpaceProblem._o dA (2)w(x,y) - = Pwhere p is the distance from the load elementIAI to the point (x, y) at which the displacementFigure 11. is being calculated, and C is the materialDeformed Bodies in Contact. flexibility, defined in terms o
45、f Youngsmodulus and Poissons ratio for the twothe contact area. bodies.l-v_ I-_w I + w2 = 6 - S(X, y) (I) C - E I + E-2- (3)For the purpose of the analysis, we can The displacement w in this equation isreplace the two bodies by the system shown in known from Equation (I). The stress o isFigure 13, w
46、hich is a half-space loaded by unknown, so the equation is an integralthe same stress distribution as the contact equation, but it is unusual in that thestress between the two bodies. The flexibity boundary of the contact region is alsoof the half-space is equal to the combined unknown. Hertz was ab
47、le to find solutions forflexibilities of the two bodies, and the cases in which the surface curvatures of thedisplacement w at the surface of the two bodies in the contact region arehalf-space is equal to the sum of the constant. The separation function then hasdisplacements of the two bodies. Withi
48、n the the form,loaded area, w must satisfy Equation (I),while outside the contact area w must be s(x, y) = Ax2 + By2 (4)greater than (6 - s).where A and B are constants. These are theTo relate the displacement to the cases discussed earlier, where the contoursapplied stress, Hertz used the integral
49、form of constant separation are elliptical, andof Boussinesqs equation, are therefore not always suitable forw2 _ gearingproblems.A More General MethodThe author of this paper has recentlydevelopeda numericalprocedurefor solvingthe contact problem 2, in cases where theseparation is completely arbitrary.We set up a grid of nodes on the surfaceFigure 12. of the half-space, with spacing c between theContact Stress Distribution. nodes in both the x and the y directions. We6apply a load to the surface at one
