[1]United States Patent Office 31438,279 3,438,279 PITCH POINT ACTION GEARING William S. Rouverol, 2120 Haste St., Berkeley, Calif. 94704 Filed June 7, 1967, Ser. No. 644,173 Int. Cl. F16h 55108 U.S. Cl. 74-462 10 Claims ABSTRACT OF THE DISCLOSURE A system of helical or spiral bevel gearing that by violating the "law of gearing" restricts the tooth action entirely to the pitch point. The greatly reduced sliding velocity between the teeth of mating gears permits them to be made of "dry bearing" or other m6ldable materialls without exceeding rated PV values, and a multiplicity Of contact points moving transversely along the pitch line insures continuity of action. This invention relates to the profile and shape of gear teeth. The gearing system herein disclosed in intended to be especially adapted to gears molded from dry bearing materials, but its advantages are such that it will find many applications in gears made of a wide variety of materials, including elastomers, plastics, die casting alloys, sintered metals and hardened steel, as well as c(>mbinations of these materials. Designing gears of reasona:ble power capacity and wear life which can be made of moldable materials has generally been considered an impossibility. The problem has been that the involute system of gearing, because it has proven to be most satisfactory for cut gearing, must necessarily be the optimum system for molded gearing. This is an incorrect assumption, and the present invention is intended to disclose a new system of gearing that is considerably more appropriate for molded gearing than the involute system. As a class, moldable materials have certain properties which differ from those of the harder materials from which gears are customarily machined (steel, cast iron, bronze). One of these properties, allowable compressive stress, is considerE@bly lower than that for the hard materials, and it has generally been assumed that because of this the capacity of molded gearing could never be very great. This assumption is also incorrect, because moldable materials are generally softer (have a lower modulus of elasticity) than cut gear materials, so that tooth surface loads tend to be distributed over much larger areas. In addition, local deformations are much greater, so that considerably wider gear faces may be used without the inequalities of loading which occur in hard gears as a result of slight axial misalignment. Between the larger surface areas of contact and the wider gear faces possible with moldable materials, gears can readily be designed that will accommodate tooth loads substantially as great as those of the best heat treated steei. The real problem of making satisfartory molded geari.ng 'is, therefore, not one of limited tooth load, but is rather a problem of low resistance to abrasion and low thermal conductivity. These problems have long been recognized in the design of molded bearings, where they have been taken into consideration by means of what is called the PV factor. This quantity, which is the product of bearing pressure in pounds per square inch and sliding velocity in feet per minute, reflects both the rate of wear in a rubbing contact and, in extreme service, the power converted to frictiona@l heat per unit projected area of bearing surface which must be dissipated if the bearing is not to be destroyed by overheating. PV factors for molded bearings vary considerably, from Patented Apr. 15, 1969 2 a few hundred to as high as thirty or forty thousand, depending on the materials, lubrication, ambient temperatures, lining thickness, required wear life, etc. A typical value, for molybdenum disulfied filled nylon, unlubricated, would be a@bout 8,000. The example of a dry bearing material with a PV of 8,000 is instructive to consider in connection with the unsuitability of the involute system ifor molded gearing. This particular material has a compressive strength of 10 20,000 p.s@i., and ff this is to be fully utilized, and yet the allowable PV is not to be exceeded, the sliding velocity for the teeth should not exceed 8,000/20,000 or 0.4 foot per minute. I l@f involute gears are designed for minimum surface 15 stress in the tooth contact area, they must be designed with the minimum number of teeth that can be mated without tooth interference. This means that engagement starts substantially at the point where the line of action is tangent to the base circle of the driving gear and con20 tinues to the point where the addendum circle of the driving gear intersects the line of action. This produces an angle of action for 20' stub teeth of about 36', or '/lo revolution, and a sliding -velocity V,, between mating teeth which starts at 25 v@=V(R@+1) Si, 5 R2 where V is the pitch line velocity, R, and R2 the pitch radii of the gears, and 0 the pressure angle. From this 30 maximum value at the point of en.-agement, which amounts to about two thirds of the pitch line velocity if the gears have equal diameters and 20' teeth, the sliding velocity reduces almost linearly to zero at the pitch point, then increases to almost the same as the s ing 35 value at the end of the contact angle. Hence the most serious condition is for the flanks and tips of the mating teeth, and this is the condition which limits the utility of the gears. With regard to actual values of sliding velocity, it is 40 probably reasonable to consider an average service for power transmission gearing to be perhaps a 6-inch gear attached to a 1725 r.p.m. induction motor (or a 3-inch gear attached to a 3450 r.p.m. motor). This gives a pitch line velocity of about 2700 feet per minute, and a maxi45 mum sliding velocity at the initiation of tooth contact of about 1800 feet per minute. When this value is compared to the 0.4 foot per minute that would be recommerided on the basis of the PV for mdlded nylon gears, it becomes evident why such gears have been of necessity 50 restricted to "light service." If the sliding velocity of 1800 ifeet per minute is to be taken into consideration, the compressive load on the teeth should be reduced to 8000/1800 or about 4.4 p.s.i., instead of 20,000 ps.i. Some modifications in the above evaluation should 55 be made to take account of the fact that the ctitical c6mbination of load and sliding velocity is only imposed on a particular point on a tooth for a few degrees of each 360' revolution. This would lead to longer service lives than would be obtained from continuously operated 60 bearings of equivalent PV loading, or, altematively somewhat higher PV values could be employed in gear design to give corresponding service lives. However these corrections are nowhere near sufficient to make up for the fact that for rotational speeds normally encountered in 65 most power transmission applications, involute tooth shapes impose too high sliding velocities to permit the use of molded gearing in any circumstances except those which impose tooth loads that are only a fraction of the tooth surface compressive strength. 70 The object of the present invention is therefore to provide a new tooth form that will cause the sliding velocity between mating teeth to be as small as possible
[2]@,438 79 3 relative to the pitch line velocity -and hdncd to enable these teeth to be made of moldable material on which can be imposed the fuil value of the allowable compressive strength for this material without exceeding its allowable PV value. Further objects of the invention are to provide a form of gearing adapted to be made of dry bearing materials, so that in addition to low molding costs relative to the machining expense of cut gearing, there will also be economies in the elimination of an oil tight housing, oil seals, etc. Finally, it is an objective of the invention to provide g,-ating that is efficient, quiet and durable. The means to achieve ihese ari@ other objects and advanta,-es of the invention will be evident from ihe drawings as explained in the specification that follows: FIG. I is an enlarged partial section of' a pair of mating gears taken perpendicularly to the pitch line and showing mating tooth profiles embodyin., the invention. FIG. 2 is a plan view of a herringbone g6air embodying the invention. FIG. 3 is an enlarged partial section of FIG. 2 showing tooth profiles in a plane perpendicular t6 th6 tooth length direction. . FIG. 4 is a greatly enlarged section of a tooth embody@ in,a the invention, taken perpendicularly to the tooth direction (i.e., in the "normal plane"). In detail, and referring to FIG. 1, the method of confinin.- the tooth action to the pitch line is disclosed. In this figure, a driving gear generally designated 1 en.-ages a driven gear generally designated 2 at the pitch point P. Gear 1 is centered at Cl, has pitch radius RI, pitch circle 5-5', and typical tooth profile 3. Gear 2 is centered at C2, has pitch radius R2, pitch circle 6-6' and typical tooth profile 4. Other parts of the two gears, such as hubs, webs, rims, keyways, etc., are standard and are ornitted in the interest of clarity. In this system of gearing the tooth contact is purposely confined @to the pitch point P, which may be defined as the point of intersection of the pitch line with the plane of the figure, the pitch line in turn being defined as the line of tangency of the pitch surfaces of the mating gears. These surfaces will be cylinders in the case of helical and herringbone gears and cones in the case of spiral bevel gears. It will be evident that if the gears are to transmit uniform continuous motion and still only have contact at the pitch line, the teeth will have to be helical or herringbone, or in the case of bevel gearing, spiral. The steepness of the helix or spiral will in addition have to be sufficiently great so that two o-r more teeth are always in -contact (i.e., the circular pitch must be less than the tooth advance across the face of the gear). The method of confining the gear tooth action to the region exactly at the pitch point is to deliberately violate the "law of gearing." This law states that if inating gears are to transmit a uniform velocity, the teeth must be shaped in such a way that the common no-rmal to the contact surface must intersect the line of centers at a fixed point (the pitch point), which point divides the line of centers in inverse proportion to the velocity ratio. In the system of gearing herein disclosed, the contact point is at the pitch point, and as the mating surfaces move beyond the pitch point their common normal intersects the line of centers at a point between the pitch point and the center of the driving gear. Since this intersection point divides the line of centers in inverse proportion to the velocity ratio, its movement toward the driving gear would require a Teduction in speed of the driven gear. ]But because of the helical (or spiral) nature of the teeth this reduction in speed does not occur. Instead, the contact point moves along the pitch line, and the tooth surfaces @that would have allowed the reduction in speed if they haci remained in contact, simply separate. The essential conditions for ensuring this violation 4 of the "law of gearing" are illustrated iti' FIG. 1. In this figure the common norinal to the contact surfaces at the pitch point P is the line Q,-Q2, which makes an angle 95 (the pressure angle) with the pitch plane 7-7' (a plane, shown in edge view in this figure, which is tangent to the pitch surfaces of the mating gears 1, 2). In this system the optimum tooth profile is a circular arc, from the standpoint of allowable surface stress, ease of manufacture, freedom from interference and en-- 10 sured violation of the "law of gearing." Numerous other curves will also serve, such as elliptical arcs, but whatever profile curves are employed, the r@quirement with regard to radius of curvature at the pitch point are the sanie. It will be evident that when mating profiles are 1.5 in contact, the common normal to the contacting surfaces must contain the centers of curvaturelof both profiles. In FIG. I these centers are T, and T2, respectively. It will be noted that the radii of curvature i-I and i-2 are substantially less than QIP and Q2P, resp@ctively, Q, and 20 Q2 being the points at which perpendiculars to.the common normal contain the respective gear centers C, and C2; they also correspond to the points where the "line of action" for involute teeth would be tangent to the involute base circle. The reason why T, and T2 must be closer to P than 25 Q, and Q2 iS that if these centers of, curvature coi@cided with points Q, and Q2, then a small rotation of the two gears 1, 2 would cau@e these centers to move to two new positions which would still lie substantially on the line QI-Q2 so that the line joining them would still pass 30 through the pitch point P, and the "law of -earing" would not have been violated and the tooth profiles would maintain contact beyond the pitch point, much as they do in inv6lute gears which have the same local radii ofcurvatlire Q,P and Q2P at the pitch point. 35 However, if the centers of curvature Tl, T2 are in between Q, and Q2, as shown in FIG. 1, then a small rotation of the gears 1, 2 will move them to positions Tl', T2' which are both displaced from the QI-Q2 line on the side closest to the driving gear 1. A line joining these new 40 centers of curvature Tl' and T2' intersects the line of centers Cl-C2 at point P', which is substantially displaced from P, also on the side of the driving gear 1. Based on a contact point at P, the speed N2 of the driven gear 2 is N, x CIPIC2,P, where N, is the speed of the driv45 ing gear 1. If the contact point were at P', on the other hand, the speed of the driven gear 2 would be N, x C,P'IC2,P', which is slightly less. Since the speed of the driven gear 2 50 cannot be restrained by the portions of the tooth profiles centered at Tl' and T2', these portions must be moving away from each other. The line Tl', T2' must therefore be longer than the line TI, T2, and the difference must appear in the form of profile separation. An analogous. 55 argument can -be made to show that the profiles must be separated before they reach the pitch point P, and allso that if the radii of curvature r, and i-2 are greater than PQI and PQ2, the profiles will interfere rather than separate. 60 In mathematical terms, the conditions for the violation of the "law of gearing" described above are that the relative radius of curvature r must satisfy the equation: K sin 65 r= IR1+,IR2 (2) the relative radius of curvature being: 70 r = 'Irl +'lr2 (3) and K being the "separation rate factor" which must besubstantially less than unity. It may be noted that there is nothin.- in Equation 3 that 75 prevents one radius of curvature i-I or r2 from being
[3]3)438,279 5 negative. This would simply mean that the tooth profile of negative radius of curvature was concave instead of convex, and had its center of curvature between T, and Ql, or T 2 and Q2) - Unlike the Wildhaber -Novikov system, however (U.S. Patent No. 1,601,750; Prodtict Engineering, Sept. 17, 1962, p. 91), there is no special advantage in mating concave and convex teeth. In the gearing hereindisclosed, the maximum relative radius of curvature r that can be used withoutinterference and with rapid profile separation on either side of the pitch point P is the same for both convexconcave and convex-convex teeth ' namely the value indicated in Equation 2. Since the Hertzian contact stress is based on relative radius of curvature r, it follows that the only effect of using concaveconvex profiles in the present invention is a loss in interchangeability without any compensating gain in allowable tooth load. With regard to the Wildhaber-Novikov system, it miy be noted that while it allows an increase in tooth load relative to the involute system in the case of lubricated cut gearing, it is basically even less suitable for unlubricated molded gearing than the involute system. This is because the Wildhaber-Novikov system places the centers of curvature of the mating concave-convex profiles substantially at the pitch point, so that the contact surfaces niust be considerably removed from the pitch point and consequently lie in a region of relatively rapid sliding. e average sliding velocity for a typical Wildhaber-Novikov gear is about half the pitch line velocity, as compared to one third for the involute system. in contrast to these average sliding velocities of thirty to fifty percent of the pitch line velocity for typical involute and Wildhabor-Noviko-v gears respectively, the average sliding velocity for gears embodying the herein disclosed invention is of the order of one percent of the pitch line velocity, and the increase in capacity based on PV values for molded dry bearing materials will be commensurate. That there is any sliding velocity at all in the subject gearing system is due to the fact that the moduli of elasticity of moldable materials is such th@,t the tooth contact extends over a relatively large area, onl,y the center of which has zero velocity by virtue of its being exactly at the pitch point. The edges of the contact area may be a degree or so of rotation in front of or beyond the pitch point, depending on the magnittide c)f the helix or spiral angle. But these fringe areas that are subjected to minor sliding are more lightly loaded than the center of the contact area, in any case. It should also be noted, in connection with Equations 2 and 3, that R, or R2 MaY 90 tO infinity in the case of a rack, and r, or r2 may also go to infinity if one of any pair of mating gears has flat teeth; the equations are still valid for these conditions, however. With regard to the other drawings, FIG. 2 is a plan view of gear I of FIG. 1, showing a herringbone construction. It will be noted that the face length is quite wide relative to the gear diameter, since there are no serious ali,-nment problems in this system. Also, since unit costs are not greatly increased by using herringbone teeth, and the face is amply wide, advantage may be taken of the herringbone construction to eliminate external axial thrust. FIG. 3 shows the front and back tooth profiles in a sectional plane of FIG. 2 normal to the tooth directions. As in conventional helical gearing, the pressure angle On is related to that in the plane of rotation by the standard equation tan On=tan 0 cos iP (4) where ip is the helix angle. Similarly, the radius of curvature in the normal plane r,,, relates to that in the plane of rotation r, as follows: COS2 1/2 r,,,=rl COI12 @ +sin2 For a pair of mating gears, the relative radius of curvature r, will be: r.=r COS2 @ (cos2 q5+sin2 q5)3/2 COS2 (6) 5 where r is the relative radius of curvature in the plane of rotation as given by Equation 3. iIn practice, the most useful pressure and helix angles are such (20 to 40') that the argument of Equation 6 is not far from unity, 10 and since the teeth in this system tend to be quite fine, profiles cut to an exact circular arc in the normal plane will be substantially circular arcs in the plane of rotation, and vice versa. FIG. 4, showing an enlarged section of a tooth 3 in 15 the normal plane, indicates the pressure distribution over the slant height of the tooth. Unlike the involute and epicyclic gear systems in which mating teeth slide over one another, the teeth in this system are subjected to almost no sliding and hence the tooth load should be 20 distributed over substantially the full slant height of the tooih. Except for teeth involving elastomer materials, the height of the contact area 2b is usually less than 1/1(; of an inch, and the circular pitch correspondingly small. For example, a gear such as that shown in FIG. 2 made 25 of nylon filled with molybdenum disulfide wou d ave the following specifications: Pitch diameter ---------------------------- 3" Length ----------------------------------- 3" Pressure angle ------------------------ deg-- 30 30 Helix angle ------------------------------- 32'30' D iametral pitch -- ------------------------- 6 0 N umber of teeth -- ------------------------ 1 80 N umber of contact points across full face of gear ----------------------------------- 36 35 Maximum surface stress -------------- p.s.i-- 20,000 Power capacity at 3450 - r.p.m. and PV of 8000 ------------------------- HP-- Approx. 160 Separation rate factor ------------------- --- 0.5 40 The especially unusual features of this type of gear may be noted: the large number of teetb. (180), the large number of separate contact areas in the three inch length of face (36), and the exceptionally high power capacity for unlubricated molded plastic gearing (160 HP). 45 It should also be noted in connection with FIG. 4 that an accurate determination of the minor semiaxis b of the contact ellipse may be calculated from the general Hertz equations for pressure between bodies with curvature in two directions (e.f., Theory of Elasticity, S. 50 Timoshenko, McGraw-Hill, 1934, pp. 344-7). In thesecalculations the lengthwise radii of curvature rL of the teeth in the normal section may be obtained from the equation: RI, 2 55 rLI, 2- sin2 V, sin (7) Approximate values of the semiaxis b, however, may be obtained from the expression: 60 b= 1.825Sr@ + El Y2) (8) where S is maximum allowable surface stress, rl is from Equations Z, 3, and 6, and El and E2 are the moduli of elasticity for the mating gears. 65 Equation 8 is useful primarily for selecting the pitch for a particular set of mating gears. Optimum capacity in this system is obtained not by using the minimum number of teeth that can be mated without interference, as in the involute system, but by using the maximum 70 number of teeth that can be employed without the minor axis of the contact area ellipse (2b) exceeding the slant height of the teeth. This insures that the teeth will have the minimum possible circular pitch, which in turn maximizes the number of contact points for any given face (@o-s2@ 0) (5) 75 width and helix angle. This general bbjective of mini-
[4]3)438)279 7 mizing the circular pitch to obtain a large number of contact points will also be served by making the working profile extend over more or less equal arcs on either side of the pitch point, as shown in FIG. 4. 'fhis ftirther optiri'lizes capacity by locating the maximum surface stress 5 S at the point where the sliding velocity is zero. While' the tooth system h--rein disclosed may be used for cut gearin@ in applications requirin.- higher - efficiencies than are obtainable with involute gearing (i.e., gears inclined tb overheat due to brief intensive use, as in rac- 10 ing vehicles), the primary objective of the invention is to devise a type of gear adapted to be molded. In this context, as well as in the claims, a "moldable material" is intended to encompass materials capable of beina fbrmed against a finished die or mold surface with no - separate 15 finishing operation'involvin- scraping, cutting or - grinding 6f material from the tooth surface, @s for example by inoldin.-, plastic da@tin- extruding or die casting. A "dry beariti.-' material" is one intended' to be used in - unlubricat@d j6urnal @b6arings; These asually carry a PV rating 20 and at present include such materials as tetr afluoroethylene or graphite- nylon, tetrafluoroethylene or - phenolic filled with mol@bd,num disulfide, glass fiber, lead, or graphite; -sintered materials such as bronze, aluminum, iron, or nylon filled with oil, lead, tetrafl uoroethylene, 25 molybdenum disulfide or -raphite or combinations thereof; and hard materials such as steel or die cast - aluminum which bave a low friction surface of vapor deposited dry film or baked tetrafluoroethylene. From the above explanation it will be evident that 0( numerous advantages accrue from the -earing herein disclosed. The use of tooth profiles specifically designed to violate the "law of gcaring" enables the action of the gearing to be entirely confined to the re.-ion of the pitch line so that slidin.- velocities between mating teeth are - substan- 35 tially zero. The effect of this is to reduce the frictional losses to the point where dry bearing materials may be used for the teeth as well as a variety of soft rnoldable materials of low thermal conductivity. Gears made of 0 4 these materials will be exceptionally economical. In addition to being economical, gears embodyin.- the present invention are highly efficient, extremely - durable, quiet and capable of transmitting, at least at speeds of 1725 r.p.m. and above, substantially as mtich power as 4,5 the finest cut gearin.-. In some high speed applications where the operating conditions are closely controlled, gears may be designed embodying the invention in such a way that their radial growth or cold flow due to - centrifugal effect will substantially offset tooth wear. 50 Further advantages of the gears embodying the system herein disclosed are that they niay be made interchangeable with no loss of capacity and are relatively insensitive to axial misalignment, molding distortion or changes in center distance. 55 The specific description given above of the preferred form of the invention should not be taken as restricti:ve as it will be apparent that various modifications in design may be resorted to by those skilled in the art without departing from the scope of the following
