CA1211306A - Method of generating involute tooth forms with a milling cutter - Google Patents

Abstract

METHOD OF GENERATING INVOLUTE
TOOTH FORMS WITH A MILLING CUTTER
ABSTRACT OF THE DISCLOSURE
An involute external tooth profile is cut into the periphery of a gear blank by positioning a rotating milling cutter with its cutting path perpendicular to the plane of action of the desired base surface of revolution within the gear blank so as to penetrate the plane of action from the side thereof opposite to the base surface and with a predetermined line of intersection which generates the involute profile as the base surface rolls upon the plane of action. The rolling action causes the generating line to traverse the blank between its addendum surface and a depth sufficient to provide the desired length of active profile.

Description

METHOD OF GENE~ATING INVOI,UTE
TOOT~ FORMS WITH A MIL~ING CUTT~R
.

My invention is a method of milling or otherwise machining the involute profiles of the teeth of conical (i.e., bevel) or oylindrical gears, with teeth either axially straight, or helical, or curved.
My method is a generalized method in ceveral senses. Firstly, it generate= tooth profiles individually, rather than cutting teeth as such, and thus allows the gear designer great freedom in the design of gears. Secondly, the method i~ generalized in the sense that~ being independent of specific tooth formc E~ se, it can be used to machine all formc of external involute ~ears having true conjugate action, the method treating the familiar spur and helical gearC
~erely as a limiting case of the more general case o conical gears.
BACKG~OUND OF THE INV~NTION
The several methods now widely used commercially for cutting gear~ from gear blanks, i.e., hobbing, shaping, and milling with rotary form cutters, are all based upon the use of a dedicated tool which will cut only teeth of a single form and cize.
Hobbing is a continuous process in which the involute profiles of the teeth of cylindrical gearC are generated by the rotation of a series of helically arrayed cutters whose individual cut~ing e~ges sweep a conical path. In this arrangemen~, the cutting speed o~
the tool and the generating movement of the tool with respect to the gear blank are interdependent, and the gear teeth are generated incrementally about the entire periphery of the gear blank as the cutter is slowly fed axially of the rotatin~ gear blank.

In disc shaping, or gear shaping, the reciprocatiny cutter itself i5 in the form of an involute gear~ and both the shaping cutter and the gear blank are incrementally indexed by rotating both with S the same pitch-surface advance before each cutting stroke after the cutter has entered the gear blank radially to the desired cutting depth.
With the rack shaping method, i.e., where the reciprocating tool assumes the form of a rack to be 10 meshed with the gear to be formed, the involute toDth profile is also generated by incremental indexing rotation of the gear blank with concurrent tangential index of the rack equal to the pitch circle index of the gear blank before each stroke of the cutter after the 15 rack has entered the gear blank ra~ially to the necessary depth. The process differs from disc or gear shaping in that the length of the rack cutter is iimi~ed by practical considerations, an~ require tooth-indexing of the gear blank relative to the cutter.
Both disc and rack shaping are intermittent processes as the tool in each case cuts only on the forward stroke and is idle on the return.
Milling with rotary form cutters, i.e., an axial or helical cut with cutting edges shaped to the 25 involute profile to be left upon the gear tee~h, likewise re~uires tooth indexing of the gear blank relativ2 to the cutter. In some instances, slot milling with an ordinary cutter is employed as a preli~inary roughing operation to be followed by a finishing 30 operation with a rotary cutt~r having the correct involute form9 or by hobbing or rack shaping.
Compared with each other~ these prevailing metho~s have their advantages and disadvanta~es. For example, the c~tting paths swept by hobs, ~eing conical s~rfaces of revolution~ leave helical scallops on the profile of the tooth. The resulting surface o~ the profile may be undesirably erose if the axial feed of I the hob d~rin~ or between s~ccecsive passes is not ; limited, resulting in relatively slow production due 3 either to the limited axial feed required for the sake 5 of accepta~le finish, or to the s~bsequenk sha~ing or grinding operation which may be required to achieve it.
The primary common disadvantaye, ho~lever, of all of these cl~tting methods is their reliance upon the ; concept of basic racks having standardized tooth and 10 tooth-space proportions and pressure angles. The hob, the disc shaper, the rack shaper, and the rotary form cutter embody a sing]e tooth form dic~ated by one of the basic racks. A different tool is therefore needed for each variation in diametral pitch, circular pitch, and 15 metric module; for each variation in pressure angle; for each variation in depth p~oportions, whether of full depth or cne of the stub tooth variations; for each variation of root-fillet radius; and~ finally, for each variation of function in the production sequence, i.e., 20 roughing, pre-grind, pre-~have, or finishing.
In addition, different tools are required in some system~ to adapt them ~or helical gears~ and even for the hand of the helix, righ~ or leftO
Bevel gears~ whether straight toothed or 25 helical, require still different machines and tooling sy~tems~
Moreover~ as the decign of gear teeth is to some degree the compromise o~ conflicting criteriat the relatively complex calculations involved in resolving 30 them~ combined with the cost of tool inventory for gear~cutting syste~ premised on the rack form, has led to the development of standard data for ~tandardized gears which has put gear design into fairly rigid con~inement.

,~

OBJECTIV~S AN~ BRIEF DESCRIPTION OF T~E INVENTION

The general aim of this inven~ion, accordingly, is to free the design and manufacture of gears froM the restraints :imposed by rack-based cutting syste~Ds by pro~id-ing an improved method of cutting gears which recogni~es no fundamental difference between cyl:indrical and conical (bevel) gears, and in which the involute profile of both straight and helical. gear teeth is generatecl by a rotary milling cutter having a plane face. The cutter may therefore be of very simple construction, may use indexable inserts, and the same cutter may be employed to cut tooth forms of a number of sizes, pitches, depths, and pressure angles, or which are asymmetric, or modified in profile by under-cutting or tip-relieving, or modified axially by crowning or tapering. Moreover, the method of the invention can, if desired, combine roughing and finishing into a single operati.on, cut multiple tooth fianks simultaneously in the same gear blank, and cut gears of large diameter and extended face width.
While the method is explained in the following specification in its application to the machining of the gear teeth of right circular conical and cylindrical gears, which dominate the field of gearing, it is not so limited, but may be used, for example, to machine the teeth of meshable gears with axes askew in different planes, or of conical or cylindrical gears of variable curvature, i.e., whose directrices are ellipses, spirals, etc.
Broadly speaking, the present invention provides the manufacture of a gear, the method of machining a gear blank to produce a tooth profile which-is involute from an imaginary base surface of revolution within the gear blank, the surface having a straight~line generatrix and having an imaginary plane of action tangent to the surface the method comprising the steps of rotating a cutter havi~g a plurality of cutting edges uniformly spaced about the entire periphery of the cutter sweeping a cutting path in the form of a surface of revolution about the axis of the cutter so that lcm/l~

- 4a -the plurality of cutting edges are distributed substantially uniformly about the common surface of revolution which they define, the cutting-path surface comprising a plunge-cuttlng rim portion of cutting thickness not exceeding the de~sired tooth space at the tooth root and a contiguous tooth-profile cutting portion, positioning the rotating cutter on the side of the plane of action opposite to che base surface with the rim portion penetrating the plane of action and with the tooth-profile cutting portion intersecting the plane of action along a predetermined generating line and with the tooth-profile cutting portion perpendicular to the plane of action at least at the center of the generating line, in-dependently controlling the rotation of the cutter, and effecting a relative feeding movement of the gear blank and rotating cutter independently of the rotation of the cutter while maintaining the aforesaid position of the cutter rel.ative to the plane of action, the feeding movement being such as: to cause a relative rolling motion between the base surface and the plane of acti.on without slippage; to cause the generating line at all points therealong to maintain a controlled angularity with respect to the instantaneous direction of its movement relative to the line of tangency of the base surface with the plane of action during the rolling motion; and to cause the rotating cutter to pene-trate the gear blank and the generating line to transverse the gear blank between its addendum surface and a depth at which the desired active tooth profile is achieved at the center of the generating llne.

THE DRAWINGS
The accomplishment of these objectives will become I apparent from the following detailed description made in conjunction with the accompanying drawings, in which:
FIGURE 1 is a perspective view of twin-head, 8-axis milling machine designed to perform the method of lcm/ ~

L; ~

this invention on both cylindrical and conical gears;
FIGU~E 2 is a fragmentary side elevational view of a modification of the machine of FIGURE 1 to convert the same to 9-axis operation, FIGURE 3 is a fragmentary elevatiQnal view of the back side of a plane milling cutter designed for practici.ng the method of the i.nvention;
FIGURE 4 is a fragmentary sectional view taken along the line 4-4 of FIGURE 3;
~IGUR~ 5 is a diagramma~ic perspec~ive view of the generation of the involute tooth form in a conical gear blank by a plane c~tter such as that of FIGU~S 3 and 4, showing the positions of the cutter relative to the plane of action an~ to the gear blank for generatinq 15 either straight or helical teeth;
FIGURE 6 is a diagrammatic view of nature r similar to FIGURE 5 illustrating the generatiGn of the involute tooth profile in a cylindrical gear by a plane cutter, with similar indication of the cutter placement 20 relative to the plane of action and to the sear blank for straight and helical teeth;
FIGVRES 7 and 8 are fragmentary diagrammatic views showing schematically the progression of the cutter and the gear blank while generating the involute tooth profile with a plane cutter on a machine such as that of FIGVRE 1 and FIGURE 27 FIGURE 9 is a fragmentary elevational view oE
the face of a cylindrical spur gear positioned~as it would be in the machine of FIGURE 1, showing the plane 30 cutter of FIGVRE 4 in schematic form and diagramming its path of movement in generating the involute tooth profile of a sear of ~ubstan~ial face width;
FIGURE 10 is a fragmentary cross-sectional view taken on the line 10-10 of FIGURE 9;
FIGURES ll(a) and ll~b~ are views ~imilar to FIGUR~S 7 and 8 showing in schematic form the simultaneous milling of two tooth fla~ks on a ~win-head machine such as that of FIGURE 1 and FIGU~E 2;
FIC.URE 12 is a diagram of the relative placement of the proEile qenerating lines of ~wo 5 complementary plane cutters for the simul~anec7us genera~ion of two tooth profiles, ~tralght or helical, for a cy]indrical gear;
FIGURE 13 is a similar diagram of the placement of complernentary plane ~u~ters in the circular plane of 10 action of a bèvel gear for the simultaneous generation of two tooth profiles on the machine of FIGURE 1 modified as in F~GUR~ 2 for 9-axis operation;
FIGURES ~4ta-d), 15(a-d), 16(a-d), and 17(a-d), re~pectively, are diagrammatic illu~trations of plane, 15 conical, and cylindrical cutters and ~heir placement relative to the plane of action of the gear for the practice of the invention, and also showing diagrammatically the complementary forms of cutter of each type for milling opposite tooth flanks;
FIGURES 18(a-~) illustrzte forms of unmodi~ied straight and helical involute tooth profiles cut with the plane milling cutter of FIGURE 14;
FIGURES 19(a-f~ inclusive illustrate the curved tooth forms cut by the complementary milling cutters of 25 any of FIGURES 15 through 17;
~ IGURE 20 illustrates the asymmetric or "but~ess" teeth which the method of the invention is capable of cutting;
FIGUPE 21 i~ a cross-section of an involute 30 tooth with its engaging tips relieved in an outline involut~ to a smaller base circle, FIGURE 22 is a cross-sectio7lal view o an involute tooth undercut in anticipatiol7 of a subseauent milling or grinæing operation after hardening;
FI~URE 23 is a fragmer)tary plan view of an axially crowned involute tooth made by the method of the invention;
FIGURE 24 is a ~imilar fragmentary vie~ of meshing o~lrved gear teeth of different c~rvature produced by the method of the invention;
FIGURE 25 is a similar fragmentary and diaqrammatic view of a pair of mesning gears with their teeth tapered op~ositely for backlash control;
FIGURE 26 is similar to FIGURE 1 but illustrates the cutting of a bevel gear with only one of 10 two machine heads, ~sing the method of the invention in one of its modifications;
FIGURE 27 is a perspective view of the gear blank of FIGURE 26 at an intermediate stage of generation; and FIGU~ES 28.1 to 28.8 inclu~ive are diagrams illustrating the analytic geometry of involute generation of conical gears, including generation on the machine shown in FIGURE 26, utilizing five of the eight axes.
FIGU~ES 29.1 to 29~4, inclusive, are diagrams illustrating the applicability Df the method of the invention to the generation of the tooth profiles of hyperboloidal gear~.
EQUIPMENT FOR PR~CTICE OF THE INVENTION
A machine 30 especially adapted for the utilization of the method of the invenk;on to cut very large gears~ and particularly for cutting two opposing tooth profiles at the same time; is shown in FIGVRE 1.
It includes a rotary work table 32 for supporting the 30 gear blank 34 for rotation about a vertical axis on a carriage 36 which is movable horizontally on covered ways 38 and ~ositioned therealong by a ball screw 40 turned by a servo motor 42 at one end of the ways. The work table is driven from its underside by gear 35 connections (not shown) to a pair of oppo~ing servo motors 44 mounted on the carriage sides~

.. .

The underframing of the covered ~able carriage ways 3~ is joined to a cross frame 46 of triangular cross-section whose sloping side facing the table carriage is similarly provided ~ith covered ways 4B upon 5 which two machine-head columns ~0 and 52 are mou~lted for movement transversely of the table carriage ways, each column being positioned along its ~upporting wa~ by a separate ball screw, not shown, similar to that which positions the work table carriage, and each screw is turned by its own servo motor~ of which only the motor 54 for the near column S0 of FIGURE 1 is show~.
Vertically movable on each of the machine head columns on rails or wayE 56 thereon are machine-head slides 58 positioned at the desired height by ball screws 60 each driven by a servo motor 61. ~ach slide 58 in turn carries on its fron~ face a cutter head 62 with self con~ained variable speed spindle drive motor 63. The cutt~r head is pivotable a~ a body through at least a li~ited arc about an axis perpendicular to ~he front face of the slide~ i.e. 9 extending horizontally forwardly as seen in FIGVRE 1~ parallel to the ways 38 on which the work table carriage is movable. The cutter head i s pivoted about that axis by a servo motor (not shown) housed within the slide 58 and driving a pinion engaged without backlash with a gear segment integral with the cutter head. The pivot axis of each cutter head 62 is preferably positioned to lie in the cutting plane of its cutter 64 at its center of rotatiQn, i.e., to pass through the spindle axisO
3~ The cutter heads 62 are constructed as mirror images so that the plane cutters 64 mounted on their respective spindles face each other and can be placed in reasonably close ~uxtaposition for simultaneou~ly cutting the same gear blank. The machine 3~ is so constructed that the axes of the spindles o~ both cutter heads 62 are always positioned in a common vertical plane, notwithstanding that each cutter head is ~eparately rotatable upon it5 ~wn vertical:Ly movable ~lide 5~, each machine head slide i 6 separately vertically movable on its own ways 56, and the columns 50 are separately movable horizontally on their common ways 48.
The two cutters 64, in facing relation to one another when mounted in their respective spindles, are of similar con~truction but of opposite "hand" so that their operative segments can be rotated dGwnwardly through the gear blank to deposit their chips below~
The arrangemerlt of FIGUXE 1 will thus be seen to embody ~ive axes of relative movement between each c~tter 64 and the gear blank 34, namely, the three axes 15 of mutually perpendicular linear movement represented by the movement of the work table carriage 36 on its ways 38, the transverse movement of each cutting head column 50 and 52 on the ways 4B, and the vertical movement of each cutter head slide 58, and twc axes of relative rotational ~ovement represented by the vertical rotational axis of the work table 32 and the horizontal axes of rotation of the individual cutter heads 62 on the slides 58. As the work table translation and rotation enter into each 5~axis relationship between 25 cutter and work piece, the machine actually embodie~
only eight, rather than ten, axes o movement. Movement ~long or about each of those axe~ is controlled by a separate servo motor or motors under the overall control of a computerized numerical control, not shown.
As each cutter head 62 incorporates its own spindle drive motor 63, and all movement on or alony the five axes of motio~ of the gear blank relative to each cutter are independently powered an~ controlled, the cutting speeds of the cutters 64 may be chosen at will for optimum cutting performance and finish as they are independent o the involute generating movement.

It will be apparent from later explanation o~
the cutting method of the invention that the method can be carried o~t on a 5-axis machine of greater or lecser proportion9 and to a limited extent, i.e., f~r 5 cylindrical spur gears only~ on a 4-axis machine, typically one in which only the gear blank is rotatable.
FIGURE 2 shows schematically a modification of the machine of FIGURE l in which the work table carriage 36' is constructed in two parts, namely a cradle 66 lO which is tiltable toward the aforementioned common vertical plane of rotation of the pindle axes, and a supporting frame 680 positionable, like the carriage 36 of FIGURE l, along the way~ 38'. The cradle 66 carries the rotary work table 32' and its servo motors 44'. The 15 tilt axis of the cradle is defined by tran~versely extending trunnions 70 journalled in bearings in the supporting frame 68, while the cradle 66 itself is variably tiltable and maintained in tilted position by a mo~or-driven pinion 72 and segmental gear 74 secured to 20 the cradle. The placement of the cradle trunnions well forwardly of the center of gravity o~ the cradle and gear blank assures the maintenance of a substantial moment opposing that of the cutting forces to assure the rigid support of the gear blank being machined.
The preferred form of plane cutter 64 shown in FIGURES 3 and 4 includes several sets of indexable cutting inserts 76 spaced angularly about both faces of the cutter body, and each po~itioned in a pocket 78 l~cated to place the cutting edge of the insert adjacent the trailing side of a gener~ally radial chip channel 80~ The inserts 76 are preferably of the so-called "on edge" type similar to the inse ts of Figure 9 of Erkfrit~ U.S. Patent 3,708~0430 The forward face 82 of the cutter body lies in 35 a plane perpendicular to the axls of rotation of the cuttert and the rear face 84 of the c~tter body is conical~ In both facec of the cutter body~ the inserts 76 are spaced radially alongside successive chip channels so that their respective outting edges sweep overlapping paths to generate a continu~us cutting path, 5 conical on the back side of the cutter, and an annular plane on the front side of the cutter perpendicular to the axis of rotation.
Referring to FIGt1~E 4/ it will be seen that the radially outermost inserts of both the front and rear lO faces of the cutter sweep paths which overlap at their ti.ps to form a plunge-cutting rim 86 which is relieved by a sli~ht radius at the tip of each radially outermost cutter insert, the radius 88 at the corner of the outermost ~utting in~ert of the front face 82 being 15 later refle~ted in the gear as the root fillet of the tooth.
As indicated, the form of cutter ~hown in FI~URES 3 and 4 is the preferred form of plane cutter, because the buttress cross section of the outer s~pporting rim of the c~tter body is sturdy~ and sufiiciently large to permit the use of indexable and replaceable inserts, while the angle incl~ded bet.ween the two cutting faces, preferably about 25 to 30~ is not sufficiently great to interfere with the facing 25 profile of the adjacent tooth.
As will later be seen, the plunge-cutting rim 86 and conical back face 84 of the cutter perform the rough cutting and incur the greater wearO wher~as the plane front face 82 removes considerably less metal in 30 ~orming the involute profile~ in what is thus essentially a finishing operation.
~ here gear tooth size is too small to permit the use of a cutter with indexable inserts~ the cutting inserts are bra~ed onto the cutter body~ and sharpened~
35 as required~ in the conventional way.

IJ~

GENEXATIMG THE INVOLUTE TOOTH PROFII,~
The basic principle of the genera~ion of a ~ooth profile which is involute from a right ~ur~ace of revolution is illustrated diagrammatic311y in FIGUR~S 5 5 and 6 for the predominant cases of the right circular cone and right circular cylinder. Both diagrams al50 serve to illustrate the method of the invention for cutting involute gear teeth with a plane cutter.
a Conical Generation~ The General Case, Figure 5 _ .
FIGURE 5 illustrates in broken line a cone 90, referred to in gear terminology as "the base cone", resting upon a plane g2, called the "plane of action~', upon which it may roll in a circular path about an axis 15 94 perpendicular to the plane of action at the apex of the cone.
~ he radius of ~he circular path is the same as the cone distance of the ~ase cone, i.eO~ the length of its generating element, which is alfio the length of the 20 line of tangency of the base cone with the plane of action. It may be convenient9 therefore, to think and speak of the plane of action g2 as circular for it is only the circular portion of that plane, i.e~/ the circular locus of the line of tangency of the rolling 25 base cone, with which we are concernedO
Insofar as the gear blank itself is concerned~
we may further focus ~ttention on the intercept of the base cone by the conical gear blank 96~ i.e9 ~ the fru.~trum of the base cone be~ween its base circle and 30 the lesser circle which defines the opposite and parallel face of the ba~e cone intercepted by the gear blankO That frustrum o the base cone rolls in a circular path delineated by ~he circle 98 which circumscribes the plane of action and by an inner 35 concentric circle 100.
As the base cone rolls over the plane of action, any point such as point 102 in the plane traces a path 104 away from the su{face of the base cone to which it was momentarily tangent. The path 104 is involute to the surface of the cone at the location of 5 the separation of the po.int 102 in the plane of actior-therefrom, as though from a circle of equal rolling radius, namely a circle perpendicular to the plane of action having a radius measured perpendicularly to the plane of action, from the given point of tangency, to 10 the axis of the cone. That radi~s is termed a "tran~verse radius" and is e~al to the cone radius to the point 102 when it was tangent ~o the cone divided by the cosine of the cone angle. In FIC.URE 5, the base circle radius of the base cone is labelled R and the 15 transverse radius at the base of the base cone is labelled RT~
Conversely, if the rolling movement of the base cone on the plane of action is stopped and then reversed, the given point 102 in the plane of action 20 retraces the same involute path 104 back to the surface of the base cone ~0 as the latter rolls back to its starting location, i.e., again to include the point 102 within the line of tangency of the ba~e coneO
It may be appreciated that inasmuch as the rGlling base cone 90 is constantly changing direction as it roll~ upon the plane o action, the trace 104 of the given point is a 3-dimensional curve in space, as distin~uished from the usual concept of a plane curve involute from or t~ a circle in a plane.
I instead of the single point 102 consi~ered above, all of the points in the plane of action which 1 ie in a line 106 in the plane are considered simultaneously~ the path traced by that line in its movement away from the rolling base cone is a surface 35 108 which is the envelope o:E the individual 3-dimensiona~ involute traces of all point6 on the given 3 ~` ~

1~-line 106 in the plane. The line 106 in the plane of action may accordingly be called the "generating line'l.
If the generating line i5 straight, and if it is disposed radially in the plane of action~ viz., the 5 line 106 of FIGURE 5, it will coincide with an instan~aneo~s line of tangency of the rolling base cone with the plane of actionO The development of the involute traces of all points on the generating line 106 accordingly proceeds simultaneously as a surface which 10 for convenience is here termed an "involute surface".
~11 straight lines in the resulting surface pass through the apex of the ba~e cone.
If the generating line is straight but skewed from radial alignment in the plane of action~ e.g.~ the 15 generating line 110 of FIGURE 5, the incremental involute traces of successive points along the line are generated pr~gressively, exactly as though the ~enerating line 110 were a conical helix being unwrapped from the surface of the base cone. Similarly, if the 20 generating line i~ curved, the progression of the generation of the involute traces by the individual points on the generating line i5 governed by the slope of the curve. In all cases, the resulting surface is the envelope of the individual invo~ute traces of all 25 points on the generating line, and for conveniencet that surface is referred to as ~involute" from or to the base coneO
If, then, the plane of action were re~uced to the two broken lines ga and 100 considered as rails, and 30 the base cone 90 were imagined to extend axially beyond the rails and to have a concentric outer ~ru~to-conical layer between the circular rails~ and if the generating lines 106 and 110 were though~ to be taut wires stretched between the circular rails and capable of 35 cutting the concentric outer layer (gear blank ~6), and if the base ~one 90 were visualized as rol~ing from left to ri~ht to the po~ition shown, the wire 106 wo~ld c~t an involwte surface 107 (a left-hand tooth profile) through the outer layer from its periphery ~addendum) down to the base cone 90. As the base cone continued to 5 roll, the same wire 105 wo~ld cut another and complementary involute surface 108' (a right-hand profile) in the outer layer as t.he beginning p~rtion of the extended involute sheet 10~ shown in FIGU~E 5D The net effect wo~ld be to carve a cusp like groove into the 10 frusto-conical outer layer, the walls of which are complementary invol~te surfaces.
Similarly, the wire 110 would first cut the helical involute surface 111 down to the surface of the ba~e cone 90 and then the complementary opposing surface 15 112' as the beginning of the more extended involute sheet 112.
This, in effect, is how the involute profile~
of gear teeth are generated by the method of the invention, althou~h with the facing involute profiles 20 107 and 108', or 111 and 112'~ machined separate~y and separated peripherally to provide the desired tooth thickness/ and by effecting a relative rolling movement of the base cone with respect to the plane of action.
As it is inherent to the geometry of involute 25 yeneration that each elemental involute curve ~f what has here been termed an involute surface is perpendicular to the plane of action at it6 intersection with that plane, the imaginary taut wires 106 and 110 of the foregoing illustration are replaceable with the 30 rotary cutter 6~ whose cut~ing edges sweep a circular cutting plane perpendic~lar to the plane of action. The cutter axis 116 is positioned below and parallel to the plane 3f action, with an arc segment of the cutting plane protruding through the p l ane of action so that the 35 chord 118 along which the cuttin~ plane intersects the plane of act.on is substituted for the taut- wire cutter i.eO, becomes the generating line.
Still referring to FIGU~E 5, the ro~ary plane cutter 6~ is illustrated at its point of maximum penetration, i.e., with the rim of ~he cutter arc which 5 i 5 above the plane of action plunged to the maximum desired depth in the gear blank 96, viz., to a depth coincident with the desired root or dedend~lm depth of the gear tooth. The near face of the c~tter 64, a~
shown diagrammatically in FIGURE 5, is the active 10 generating face, and the chordal intersec~ion of that face with the plane of action is the generating line 118. As illu~trated, the cutter p7ane is disposed radially of the plane of action, i.e., transverse to the rolling path of the base cone.
To cut successive profiles 114 of straight teeth in a conical or bevel gear, as illustrated in FIGURE 5, the cutter 64 is re-positioned successively along the circular path with its generating line 118 moved successively to positions separated from each 20 other by the plane-of-~ction angular eq~ivalent of the transverse base pitch angle of the gear ~
To cut a helical tooth profile of either right-or left-hand helix, the cutter plane of the cutter 64 is turned cn an axi~ perpendicular to the plane of action 25 so tha~ its resulting generating lines llB' and llB'' are askew from radial in the plane of action, ~t will J also be apparent that when the cutter is so turned~ its new generating lines 118' and 118~' (being straight lines in the illustrated case) would make different 30 ang]es with the line of tangency of the base cone at eve~y point of their sequential intersection; making a ~arger angle at the lesser rolling radius than at the greater. ~lthough the helix angle is accor~ingly variable, notwithstanding the fixed position of the 35 generating line in the plane of action~ the position of the generating line may nevertheless be speGified by spec.ifying the radius of a circle 120 abou~ the axis 94 by the plane of action to which the genera~ing lines 118' and 118'' are tangent, a circle which may be termed the nbase helix base circlen.
When generating ~he tooth profiles of a bevel gear by the method of the invention using the modified machine of FIGU~E 2, the axis of the bevel gear blank 96 and of the i.maginary base cone included wi~hin it is tilted toward the vertical plane of the spindle axes 10 until an element of the base cone is vertical, i.e., parallel to the plane of the spindle axes7 The carriage 36~ is then advanced toward the operating zone of the rotatin~ ~tters 64 until the imaginary plane of action, tangent to the vertical element of the imaginary base 15 cone, is penetrated by each rotating eutter to ~he desired depth~
r The principles of involute generation illustrated in FIGURE 5 are applied by effectin~ a relati_ rolling motion between the imayinary base cone 20 and the imaginary tangent and ve~tical plane of action, tha~ is, by rotating the gear blank about its own axis and rotating the imaginary plane of action about its axis at an angular velocity ratio cuch as to synthesize the rolling motion of the base cone on the plane of 25 action without slippage. What this amounts to is swinging the cutter~ in a vertical plane as though fixed in the rotatiny plane of action, until the generating line of the cutter has traversed the gear blan~ between the outer or addendu~ surface thereof and a depth of 30 penetration ~uch as to have ~enerated the involute profile down to the desire~ depth, at least to the so called Wstart of active profilen, i.e.~ the point on the profile at which contact by the meshing gear tooth ceases~
In the machine of FIGURE 2, the swing of the cutters 64 on any radius is accomplished by the simultaneo~ls translation of the c~tters horizontally and vertically in the transverse vertical plane of their axes, and by the pivoting of the cutter heads to maintain the constant angularity of their ~enerating 5 lines with the plane-of-action radii through their centers.
The same relative generating movement can also be accomplished for the genera~ion of one conical tooth profile at a time using the machine of FIGURE 1, as shown in FIGURE 26, i.e.~ with a 5-axis relationship between the cutter 64 and the gear blank 164, as will be explained later herein.
b. Cvlindrical Generation, The Special Case Although external cylindrical gears may 15 predominate numerically, the generation of involute tooth profiles of cylindrical gears is essentially one of two limits of the general or conical case~ of which the crown gear, or circular rack, is the other.
The cylindrical case is approached as a limit 20 when the apex angle of the base cone becomes s~aller and smaller~ and its circular path of rolling movement upon the plane of action commensurately larger 9 until the apex angle is zers, the base surface cylindrical, and the path of rolling movement of the base surface is a straight path. The opposite limit, the crown ~ear, is approached as the apex angle of the base cone becomes larger and larger until the surface of the base cone merges into the plane of action~
Cylindrical generation of involute tooth 30 profiles is illustrated diagrammatically in FIGURE 6, comparably with the illustration of conical generation in FIGURE 5D
The base cylinder 120, shown in broken lines, rests upon the plane of action 124 I!pon which lt may 35 roll in the ~traight path defir-ed by the broken lines 126~ As the base cylinder rolls over the plane, any 3~

point 128 in the plane which is momentarily tang~nt t~
the base cylinder traces an involute pa~h 130 away from the surface of the cylinder as the rolliny movement proceeds, and retraces the same path if the rolling 5 movement is reversed in direction. The involute curve 130 is, howevex, a plane ~urve because the cylinder rolls in a straight path rather than in the circular path of the general or conical case.
A s~raight generating line 132 in the plane of 10 action disp~sed ~ransversely of the path of the rolling base cylinder 120 parallel with the line of ~angency of the cylinder with the plane of action will accordingly first cut the involute surface 133 in the concentric outer layer 121 and then ~he complementary involute ~urface 134' therein as the be~inning of the more extended involute sheet 134 representing the envelope of the involute traces of all points in the generating line~ Beca~se the base surface is a cylinder~ all straight lines of the involute surfaces are parallel to 20 the axis of the cylinder and ~o its line of tangency wi'ch 'che plane of actionO
If the generating line in the plane of action be disposed transversely of the cylinder path but askew from parallelism with the line of tangency of the base 25 cylinder with the plane, e.g. ~ the line 136~ the development of ~he complementary involute surfaces 137 and 138' in the outer layer 121 as the base cylinder rolls proceeds pro~re~sively rather than ~imultaneously. The progressive trace of the generating 30 line upon the surface of the~base cylinder 120, i.eO, the line ~t the bottom of the cusp-like groove defined by the surfaces 137 and 138', i5 a helix having a base helix angle measured by the angular divergence of the generating line 136 from parallelism with t~e line o tangencyO
FIGURE 6 similarly shows the plane cutter 64 with its axis 115 disposed below the plane of action 124 upon which the base cylinder 120 is presumed ~o roll, with an arc segment protruding upwardly thro~gh the plane of action to penetrate the gear blank to the S desired depth ei~her at the commencement or ~t the ~ermination sf the profile generating movement, as will later be explained. The intercection of the plane face of the cutter 64 with the plane of activn prsvides the generating line 118, as in the conical case~ With the 10 direction o~ rolling movemen~ as depicted in the diagrammatic illustration of FIGURE 6, the tooth profile 114 engaged with the forward plane face of ~he cutter is ~hown at the completion of generation ready for withdrawal of the cutter~
Just as in the conical case, succe~sive tooth profiles 11~ are generated by indexing the cutter 64 along the rolling path of the base cylinder by one "transverse base pitch", i.e., the circumference of the base cylinder divided by ~he number of teeth of the gear.
~or the cutting of a helical cylindrical gear, the c~tter plane, while maintained perpendicular to ~he plane of action; is turned one way or the other to the desired helix angle away from parallelism with the line of tangency o the base cylinder, to cut helical teeth 25 of ~ither left- or right-hand helix~
In the machine o~ FIGU~E 1, the principles of involute generation illustrated in FIGURE 6 are applied by effecting a relative rolling motion between the imayinary base cylinder of the cylindrical gear blank 34 30 which is disposed horizontally for rotation about a vertical axis, and an imaginary vertical plane o~ action which is penetrated by the two plane cutters Ç4 in the manner described in connection with the diagrammatic showing o ~IG~RE Ç. This relationship is accomplished 35 by advanci.ng the carriage 36 toward the operating zone of the rotating cutters 64 until the imaginary plane of , action tangent to the inlaginary base cylinder i5 penetrated by each rotating cut:ter to ~he desired depth.
cO The Mechanics of Generation _ The progressive generation of the involute 5 tooth profile of a gear using a rotary plane cutter o~
the preferred type shown in FIGURES 3 and 4 is illustrated diagrammatically in FIGURES 7 and B. These diagrammatic illustrations may be taken as the "rolled out" development of the back cone of a straight-toothed 10 bevel gear into the plane of the drawing, i.e., the A ~`ans.1~,^5e plane, and equally as the cross-section of a straight-tooth cylindrical gear.
Initially assuming operation with only one cutter, that cutter may be positioned as shown in FIGURE
15 7 by the broken line trace of the cutter 64, i.e., at a far left position clear of the addendum surface of the gear blank. When the gear blank is ro~ated counter-clockwi~e as seen in ~IGURE 7, the cutter is advanced simultaneously from left to right at a lineal 2U speed equal ~o the peripheral speed of the imaginary base surface, thereby to effect a relative rolling movement of the base surface from right to left upon the plane of action. In that processl i.e., as the cutter plane 82 advances toward the 7ine of tan~ency of the 25 base surface with the plane of action, the face of the cutter perpendicular to the plane of action generates the involute tooth profile 114a, as is shown incrementally by the ~uccession of views of FIGURE 7.
FIGURE 8 illustrates how the same generating 30 movement can be accomplished in reverse, i e., by first plunging the cutter 64 to the desired depth, and then by rotating the gear blank in the opposite direction and ~imilarly reversing the feeding direction of the cutter to retrace the same involute path~
In either feeding direction~ the greater part of the material removed to create the tooth space i~

removed by the cutting edges of the plunging ri~ 86 and by the cutting edges on the conical back face 84 of th~
cutter. The front plane face 82 of the cutter, in contrast, removes only the relatively small amount of 5 material in the open, wedge-shaped space between the cutter and the involute surface 114a generated (far right~hand view of FIGURE 7~, enabling the cutter to produce a good finish on the too~h surface while the plunging rim and the back face of the rigid cu~ter are 10 roughing out metal to form the tooth space.
The limit of the movement of the perpendicular cutter plane toward the line o tangency is a matter of choice. If the plane face 82 of the cutter in FIGURES
7, 8, and 10 were traversed to coincide with the line of 15 tangency of the base surface, the involute profile would be generated fully to the base surface, i.e., the cutter plane 82 would be on the center line of the gear blank.
Any further rvlling motion would cause the cutter rim to undercut the tooth profile inwardly of the base surface, 20 but this may be done deliberately where an undercut tooth is desired.
Actually~ as gear ~esigners will understand~ it is not necessary to generate the involute profile to a depth greater than the so-called ~start of active 25 profile", which is the point ~n the profile at which contact is made with the tip of the tooth of the meshing gear~ An additional margin of involute profile below the designed "start of active profile" may be desirable to allow for center distance tolerances, but further 30 involute ~eneration is unnecessary.
Inasmuch as the cutter segment which extends through the plane of action produces the clearance space for the tip of the meshing tooth, the depth of penetration of the plane of action by a given cutter 3~ determines the maximum depth of penetration of the gear blank by the cu~ter. It also determines the length ~f ~ 3 w23-the generating line of any give~ cutter, and thus the number of passes neces~ary to cut a tooth profile of given face width.
That is to say, where, contrary to the 5 diagrammatic illustration of FIGURES 5 and 6, the chordal generating line 118 is not long eno~gh to generate a tooth profile of the desired face width in a single generating pass, the cutter may be moved laterally of the rolling path of the hase surface for a 10 second and a~ many further yenera~ing passes as are neces~ary to extend the tooth profile to the entire face width of the gear. As shown ln ~I5URE 9 for the cylin~rical case/ this is preferably done continuously, i.e., with a contin~ous component of axial feeding 15 movement of the cutter to translate its generating line 118 in extension of itself while generating the profile by first rolling the base cylinder in one direction ~FIGURE 7) until the generating line traverses the gear blank from the addendum surface to the de~ired depth of 20 active tooth profile, and then reversing the rolling - movement ~FIGURE 8~ to withdraw the cutter from the blank along the same path projected. This sequence is repeated while the cutter is fed axially of the gear blank, the re~ulting co~posite movement tracing a 25 zig-zag path across the face o the gear (FIGURE 9)O
When the tooth profile generation is completed across the face width of ~he gear blank in multiple zig-æag passes as shown in FIGURE 9y the bottom of the tooth space is characterized by a series of cusps and 3~ scallops, which may be removed in a concluding axial slotting pass of the cutter, combined with suitable rotation of the gear blank in the case of helical gears to provide the required root helix.
The opposite ~ooth profile, shown in broken 35 line in FIGURE 10, is cut by the same method, with the plane cutter~ preferably a complementary cutter of - ~2~31~

opposite hand, facing in the oæposit~ direo~ion and traversing a feeding path disposed symmetrically (for symmetric teeth) on the opposite side of the line of tangency of the base cylinder with the plane of action.
Whether the opposite tooth profiles are cut subsequen~ly or simultaneously as later explained, the gear blank i~ indexed by one pitch angle for the cutting of each ~uccessive profile, and the cutting planes of the opposing cutters are separated by a distance which 10 is the sum of the transverse base thickness of the tovth, i,e., tooth thickness measured along ~he involute intercept by the base surface, and any convenient integer multiple of the transverse base pitch.
TWO-CUTTER GENERATION
It will be appreciated from the foregoing explanation that multiple cutters may be employed to operate upon the same gear blank if they are disposed at appropriate interYals along the rolling path of the base surface on the same plane of action so as ~o intercept 20 the gear blank requen~ially as its imaginary base surface rolls upQn that plane of action. Moreover~ when a relative rolling m~vement of the base surface upon a plane of action is effected, as in the machine of FIGU~
1 for cylindrical gears, or as modified by FIGURE 2 for 25 bevel gears9 by a rotation o the gear blank in place about its own axis and a transverse movement of the generating lines of the cutters tangentially of the base surface of the gear blank, it would similarly be possible to so position multiple cutters for transverse 30 feeding movement~ each with i~s own plarle of action tangent to the base surface, and inter.seeting the plane or planes of the other cuttersO
Practically~ however, multiple cutters are employed in pairs in the same plane of action with their 35 cutting planes facing each other~ iOe~ ~ in opposite directions of rotation o the gear blank~

FIGURES ll(a) and (b) ill~strate ~he simple case of two plane cutters 64a and 64b disposed in facing relation ~o as t~ cut the opposite flanks of different teeth of a gear at the same time in a single transverse 5 feeding pass. A5 shown in FIGURE ll(b), this requires that the cutting planes 82a and 82b be spaced apart at the proper distance, which is to say that they must be spaeed a distance whi.ch is e~ual to an integer multiple of the transverse base pitch plus the ~ra~sverse 10 thickness of the gear ~th measured along .its intercept by the base surface~ This distance has been ~ermed a ~transverse base tangen~n.
To generate a pair o opposing tooth profiles, such a~s those of FIGURE ll(b)~ completely in any given 15 gearl the integer multiple of the transverse base pitch, and thus the value of the transverse base tangent, is chosen so that when one cutter is at its innermost position, e~g., right-hand cut~er of FIGURE ll(b), the other cut~er is clear of the opposite profile 1143~
20 While time and space considerations make it preferable that the transverse feeding movement of the two cutters be held to a minimum, the integer multiple may be increased arbit~arily beyond the minimum multiple, i~, for example, the distance between the cutter faces has a 25 minimum limit which requires a larger multiple.
In FIGURE ll(a), the gear blank has been advanced toward the cutter~ so a~ to have plunged the left-hand cutter into the gear blank to the desired depth while the right-hand cutter stands clear of the 30 work piece. As the gear blank is rotated clockwise from the position of FIGURE ll(a) to the position of FIGURE
ll(b~, both cutters are simultaneously fed transversely from right to left at the base-surface peripheral speed unt~l the let-hand cutter 64ar ~tanding clear of the 35 periphery of the gear blank in FIGURE ll(b~ has generated the involute profile 114a while the right-hand cut~er 64b has generated a similar, but complemen~ary, involute profile 114b of the tooth two pitches removed.
It will be understood, m~reover, that as the feeding direction of the cutters is reversed for cutting 5 a tooth of face wi.dth exceeding the length of the generating lines, the direction of rotation of the gear blank is also reversed, as explained in connection with FIGURES 7 and 8~
In roughing out the bulk of the tooth space, 10 the back side oF each cutter leaves a conical surface opposing the involute surface generated by it~ plane front face, or a series of conical surfaces if the cut~ers are simultaneously fed axially of the gear face during the generating movement, as in FIGURE 9. Such 15 oppo~ing surface at its maximum penetration is indicated by the straight dotted lines 140 in FIGURE ll(b), which simil~rly indicate the relatively fimall amount of remaining ~ooth-space material to be cleaned out by the left-hand cutter when the cutters are ~ubsequently withdrawn and the gear blank indexed through one circular pitch for the subsequent ~enerating operationO
FIGURES 12 and 13 illustrate diagrammatically in plan views of the planes of ac~ion the positions and spacing of the generating lines of the complementary cutters of FIGURES ll(a) and ll(b) for the c~lindrical and conical or bevel gear cases, respectively, and for face widths which exceed the lengths of the generating lines of the two complementary plane cutters. Inasmuch as the cutter heads of the machine of FIG~RE 1 are pivotable on axes through the centers of their cutting planes, the generating lines of the cutters positioned for straight teeth and for helical teeth oF either hand are shown as intersecting in a common point ~or both the cylindrical case of FIGURE 12 and the conical case of FIGURE 13~ Those common points are spaced apart in the cylindrical case by the minimum transverse base tangentD

viz., as illustrated in FIG~RE 11 by two ~ransverse base pitches pl~3s one tran~verse base tooth thickness, and are spaced apart in the bevel gear case of FIGU~E 13 by a transverse base tangent angle in the circular plane of 5 action which is the eg~3ivalent of an integral n~mber of angular pitches plus the angular thickness of one tooth~
_L ERNATIVE FORMS OF_CUTTER FO~ CURVED GENERATING L rNES
FIGURES 15 to 17 inclusive illustrate collectively and diagrammatically several forms of lO cutters suited to the production of axially curve~ teeth by the method of the invention. They are there displayed for convenient orientation by reference to the general purpose plane ~utter 64 of FIGURES 3 and 4 which is shown diagrammatically in FIGURE 140 lS FIGURES 14~c) and ~d) illustrate diagrammatically a pair of plane cutters 64a and 64b ~hich are complementary in the sense that their cutting edges are disposed to cut in opposit2 directions of ro~ation. This is ~he preferred arrangement for 23 simul~aneous cutting by two cutters, as in FIGURE l, where, as earlier explained, the engaged segments of both cutters cut downwardly through the gear blankl and the feed v the cutters axial~y of the gear blank is upward so as to clear the chips downwardly through the

2~ tooth space already cut. FIGURE 14(a~ shows in solid li~e the segment of the cutter 64 which protrudes through the plane of action, whereas FIGU~E 14~b) shows the same cutter endwise in elevation and in relation to the plane of action~
3Q FIGURE 15 illustrates a conical cutter 142 which intersects the plane of action in a hyperbolic generating arc 144, the cone angle being ~uite large, while FIGURE 15(b~ shows the rotational axis of the cutter angled with respect to the plane of action to the 35 extent necessary to position the conical cutting path perpendicular to the plane of action at the center D~

the generating line 14~.
It will b~ recalled from the foreg~ing explanation of the geometry of involute ger)eration that the i.nvolute curve req~ires that the cutter be 5 perpendicular to the plane of action in order to be tangent to an involute from the base s~rface.
Therefore, when employing a cutter of the kind illustrated in FIGURE lS, a true involute lS generated only At the center of the generating line 144, i.e., by 10 that element of the conical surface which is perpendicular t~ the plane of ac~ion. On e.ither side of that perpendicular, the cone elements deviate progressively from the perpendicular and the curve which they generate i6 accordingly modified from truly lS involute but tolerable if kept within the limits of in~olute distortion normally to be expected from tooth deflection under load.
Cutters which are complementary in form are required, the cutter 142a oE FI~URE 15(c) having its 20 involu~e-generating cu~ting edges on the inside of the cone to cut the axially CQnveX profiles of the teeth, while the cutter 142b of FIGURE 15(d) has its generating cutting edges Oli the outside of a cone preferably of identica~ dimension to cut the axially concave opposing 25 sides vf the ~eeth~
FIGVRE 16 illustrates a conical cutter 146 having a much reduced cone angle so as to intercept the plane of action with the elliptical generating arc 14B
seen in FIGURE 16 ~a) ~ FIGVRE 16 (b~ illustrates that the 30 element of the cor,ica~ cutting surface in the center of the elliptical arc 148 of FIGURE 16(a) is perpendic~lar to the plane of action~ with progressive deviation from perpendicularity along the generating line in both directions rom centerO The complementary forms are shown in FIGURES 16~c~ and (d~, the cutter 146a o~
FIGU~E 16(c3 having ;ts generating cutting edges on the

3 L; ~7 _~9_ in~ide of the cDne, and the cutter 146b of FIGURE (d) having its generating cutting edges on the Gutside.
In similar fashion, FIGURES 17(a) and 17(b) illustrate the circular arc generating line lS2 of a 5 cylindrical cutter 150 intercepted by the indicated face width of the gear, and penetrating the plane of action with its rotational axls perpendicular to the plane of action. FIGURF. 17(c) illustrates such ~ cutter 150a with its generating cutting edges on the inside of the 10 cylinder for generating the convex invol~te surfaces of axially circular teeth, while F~GURE 17(d) illustrates the complemen~ary cutt2r 150b with its involute-generating cutting edges on the outside, to sweep a cylindrical generating path for cutting the concave 15 profiles.
When machini~g curved teeth, it is necessary that the radii of curvature at the midpoint of the generating arcs 144, 148, and 152 o cutters 142a, 1~6a, and 150a not be greater than tho~e produced by the 20 complementary cutters 142b, 146b, and 150b, respectively. Failure to observe this requirement will re~ult in teeth which will touch only at their ends, Inasmuch as the cutter axes, i.e., spindle axes, are parallel to the plane of action in the machine 25 form of FIGURE 1 and in its modification of FIGURE 2y it will be apparent that modification of the cutter heads 62 o the machine, or of their mountings, would be necessary for the use of conical or cylindrica~ cutters to place the spindle axes in the required relation to 30 the plane of action ill~strated in FIGU~ES 15(b), 16(b), and 17(b), one candidate being the nutatable-spindle machine head of my colleague, ~eith Goode9 illustrated B in U.S~ Patent ~3700~ ~i' J~"~,~"~ 23, ~q~3~
The manifolc9 tooth forms of which the plane 35 cutter 64 is capable when used in the method of the invention are shown in FIGURE 18D FIGURE 18(a~

~ 30-illustrates a straight spur gear, while FIGURES 18(b) and (c) ill~strate helical cylindrical gears of right-and left-hand helix respectively. FIGUR~ 18 (d) shows a bevel spur gear, and FIGURES 18(e) and (f) ill~strat2 5 respectively helical bevel gears of left~ and right-hand helix~
The diagrammatic illustrations of FIGURE l9 may be taken to illustrate the tooth forms of which the conical and cylindrical cutters of FIGU~ES 15, 16, and lO 17 are capable. FIGURE l9 (a) illustrates a cylindrical gear in which the curved teeth are symmetrical about the mid-plane of the blank, i.e.~ formed by any of the complementary cutters of FIGURES 15 through 17 with ~he cutter axis in the mid~plane of the gear blank~ FIGURES
15 19(b) and ~c) represent curved helical teeth ~uch as would be generated by turning either of the conical cutters 142 or 146 about the axis of its element perpendicular ~o the plane of action, i.eO, rotating the generating lines clockwise or counter-clockwise as 20 viewed i.n FIGURES 15(a) and 16~a). The same diagrammatic illustrations of FIGURES 19 ~b) and (c) may also be taken to repre~ent curved helical teeth generated by the complementary cylindrical cutters of FIGURE~ 17(c~ and (d), with the rotational axes of the 25 cutters positioned below the mid-plane of the gear blank of FIGURE l9~b~ and above the mid-plane of the blank o~
FIGURE l9(c).
Similarly, the ourved teeth of FIGURE l9~d) are those generated by the complementary cutters of either 30 of FIGURES 15, 16l or 17, with their generating lines tangent to a radiu~ of the circular plane of action at the centers o~ the generating lines, whereas the spiral bevel gears ~hown diagrammatically as left~hand and right~hand spirals in FIGURES l9(e) and (f~ are 35 generated by the complementary conical cutters of either FIGURES 15 or 16 turned, as earlier indicated, about , their respective elements perpendicular to the plane of action 50 ~hat they are askew from ~angency ~v a plane of action radius at their centers, but instead are tangent at their centers to lines 118' and 118'', 5 respectively, of FIGURE 5 which ~re tangent to the base helix base circle 120.
The spiral bevel gears of FIGVRES 19(e) and (f) may also be cut by the complementary cylindrical cutters of FIGURE 17 by rotating their axes in circular pa~hs in 10 the plane of action at a radius such that the intercept of the circular generating arc is tangent to the tangents 118' and 118'' to the base helix base circle 120 ~FIGURE 5~ radially midway ketween the circles 98 and 100 which delineate the rolling path of the gear 15 blank's intercept of the base cone.
MODIFIED TOOTH FORMS
It can be appreciated from the foregoing explanation of involute profile generation by ~he method of the invention that a change of the transverse feediny 20 speed of the cutter rela~ive to the rotational speed of the base surface in effect changes ~he radius of the imaginary base cylinder or the cone angle of the imaginary base cone within the gear blank from which the generated tooth proile is involute. If the transverse 25 feed velocity of the cutters is increased, or if the rotation of the gear blank is slowed~ the efect is to increase the radius or cone angle of the base surface to which the tooth profile ~hereafter generated is involute, which is to say that for a given pitch 3C surfacey the pressure angle, i.e.~ the angle at which a plane tangent to two meshing gear s at their pitch points intersects the plane of action, is reduced.
Conversely, if the transverse feed rate of the cutters is reduced, or the rotation of the gear blank speeded up, the effect is to generate a tooth profile involute from a base cylinder of smaller radius~ or from . .

a base cone of smaller cone angle, that is, effectively to increase the pressure angle of the gear tooth.
As a result, the ~ethod of ~he invention, whether employing cutters having straight or curved 5 generating lines, can produce gear teeth of different pressure angle on opposite profiles of ~he same tooth, or indeed in different sections of the same tooth profile~
The former is shown in FIGURE 20, which 10 illustrates the design layout of a pair o~ cylindrical gears of "b~ttress~ or asymmetrical form with a pitch pressure angle of 15 on the meshing tooth profiles 154 and 156 of the meshed gears when driving in the preferred direction indicated by the arrow, and a 15 pressure angle of 25 in the opposite direction~ It is understood by gear designers ~hat smaller pressure angles are desirable for the increased tooth-contact ratio which they allow, but undesirable from the standpoi~t tha~, in symmetrical teeth, the lower 20 pressure angle results in a narrower base width of the too~h with resulting weakening of the toothO For those applications in which the gear rotation is predominately unidirec~ional~ the advantage of the low press~re angle, without the disadvantage of the weakened tooth, is 2S obtainable with an asymmetric tooth as indicated in FIGURE 20, where the predominant working pressure angle of 15 at the indicated base pitch readily allows a contact ratio of 2, i.e., with two teeth of ea~ch gear always in meshing engage~ent7 but with teeth whose bendîng strength is enhanced by their asymmetric profile~
Asymmetric teeth are readily achieved in the practice of the method of this invention9 as earlier explained~ by changing the transverse eed rate o the cutters relative to the rotational speed of the gear blank when cutting the opposite profiles. Moeeover, opposing tooth flanks of different pressure angle may be ClJt simultaneously by feeding the two cutters at different r~es, ~aking d~e care tv adjust the feed stroke of the faster traversing cutter to terminate its generating movement simultaneously with that of the 5 slower cutter when the faster moving cl~tte~ is generating from the addendum surface inwardly.
A similar speed-charlge technique may be employed to relieve the tips of gear teeth when required to prevent interference as teeth meshu Referring to 10 FIG[JRE ~1, it may be seen that the outermost portions 158 o~ the tooth profiles near the tip of the tooth, h~ving greater curvature, are involute from a surface of revolution of lesser radius ~or cone angle), i~e., tha~
the transverse feeding speed of the cutter was reduced 15 for the generation of the tip of the gear, and speeded up for the generation of the balance of the profile~
Inasmuch as the cutter plane is maintained perpendicular to the plane of action irrespectiYe of the feeding speed, it will be understood that the two invclute 20 portions vf the tooth profile intersect at a common cylinder or cone at their point of mexger so that the load is transferred smoothly ~s the contact line of the tooth with the tooth of its meshing gear moves from one involute portion of the tooth profile to the other.
2~ However, the transition from one plane of action to another effects a change of the length of the generating line which must be taken into account, particularly where the face width of the gear requires multiple generating passes.
~o FIGURE 22 shows diagrammatically yet another form o~ pro~ile modification which in other methods requires special cutters, eOg.~ a protuberance hob, b~t which in my method uses the standard cutters~ This modification is useful when heat treatment and subsequent machining are contemplated, e.gO, either further milling by the method o~ the invention, or by grinding. Post-heat treatment milling with available ~utting materials is feasible at gear-blank hardnesses up to 62 Rockwell C. The tooth 160 shown in cross-section in FIGURE 22 has an underc~t 163 which is 5 produced by a suitable relative positioning of the cutter and the gear when finishing the root with an axial slotting pass as earlier described~ The undercut is typically sufficient to allow for the removal of a few thousandths inches of material af~er heat treatment, 10 and so that the ~rinding wheel or skive finishing tool does not interere with the fillet radius. The final tooth profile is generated from ~he same base surface so that the final involute profile indicated by the dotted lines 162 of FIGURE 22 will have an involute profile 15 parallel to that shown in solid line above the undercut~
A few of ~he feasible axial tooth modifications of which the method of the invention is capable are illustrated in FIGUXE5 23 to 25 inclusive.
In FIGURE 23, a straight or helical cylindrical 2~ tooth is slightly crowned on both profile~ for the sake of localizing the contact pressure with meshiny teeth which may be straight or similarly crowned, and for insuring that shaft axis misalignment from perfect parallelism doe~ no~ result in point contact at the 25 axial edge of the teeth of either gear~ Where a tooth of extended ~ace width is generated by repeated feedin~
passes of a plane cutter with translation of the cutter endwise to extend the generating line, a gradual rotatiGn of the cutter plane whilst maintaining its 30 perpendicularity to the plane of action will produce the crowned profile~
FIGURE 24 illustrates the curved teeth produced by the conical or cylindrica~ cutters of FIGURES 1.5 to 17 inclusiv2, where cutters of s7ightly di~ferent 35 curvature are used respectively for the two ~eshiny gears for the same ultimate purpose served by the crowning of straight teeth, namely, localization of the contact of meshing teeth theoretically to a point, but in actual fact to a more extended area of contact resulting fr~m the resilient ~eformation of the tooth 5 materials under load. Such curved teeth also are use~ul for applications where perfec~ shaft alignmen~ cannot be achieved by design or in practice, as indeed are me~hing curved teeth of circular outline of the same radius (not shown).
FIGU~E 25 illustrates diagrammatically meshing teeth which are tapered axiallyt the teeth of one of the meshing gears being shown in broken line for contrast.
It will be understood that either or both of the tooth profiles has a slight modification of ~he base helix 15 angle, the purpose of the reversely tapered teeth being the achievement of backlash con~rvl by relative axial movement of the two meshing gears.
Not shown is the meshing of crowned with uncrowned teeth, nor of circular teeth of ~he same curvature, nor of other combinat ons of meshable tooth - form~ within the capability of the versatile method of the invention.
FIVE-AXIS GENERATION OF BEVEL GEA~5 The foregoing explanation of specific application of the method of the invention to the milling of bevel gears by the method of the invention has been limited to the 9~axis relationship of gear blank and individual cutters illustrated by way of example in the FIGURE 2 modification of the machine of 30 FIGURE 1~ i~e., a set-up in which two cutters may be employed simultaneously to cut opposing tooth flanks at the same time because the plane of action ls vertical~
i~e,., parallel to the plane of ~he ax2s of the twt cutters 64.
It is also possiblel however, to mill any of the tooth forms of FIGURES 18(d) through tf~ inclusive by single-cutter generation on a 5-axis machine, or on fiv~ axes of the B-axis machine illustrated in FIGURE 1, which is to say, using the two ~eparate plane cu~ters 64 of the machine of FIGURE 1 singly to ~enerate all of the 5 le~t-hand flanks and then (or alternately) to generate the right-hand flanks.
FIGURE 26 illustrates the machine of FIGURE 1 cutting-~ bevel spur teeth in gear blank 164 with only the cutter and machine head of column 5~, the near 10 column in FIGURE 26, the cutter and machine head of the far column 52 being idle while its counterpart is active, and YiCe versa.. FIGURE 27 shows a closer perspective view of the teeth of the gear blank of FIGURE 26 at an intermediate point of the process of 15 generation by only the first cutter, i.e., showing one profile 166 with involute form and root finishing completed, while the ot~er wall 158 of the groove, subsequently to become the facing involute profile of the adjacent tooth/ is straight and inclined from the 20 base cone surface at an angle complementary to the cone angle of the back side of the cutter, For simplificationt the scalloped effect of the pre-generation surface o the opposing ~lank 168 is omitted from FIGURE 27~
The milling of the involute surface on a 5-axis or 8-axis machine where, as in FIGURE 26~ ~he plane o action tangent to the imaginary base cone of the gear blank is tilted away from the vertical plane of the spindle axes, using cutters whose axes are confined to 30 but pivotable in that vertical plane, i~ better explained by reference to the diagrams of FIGURES 5 and 28.
From the simplistic if not ideal diagrammatic illustration of FIGURE 5, it may be appreciated that the 35 rolling movement of the base cone upon the circular plane of action may be effected as a relatlYe rolling moYement in several ways.
One of these, already discussed in connection with the machine modificati~n of FIGURE 2, i.e.7 with the base cone vertically tangent to a vertical plane of 5 action, is to rotate the base cone about its own axis while simultaneously rotating the circular plane of action, with the cutters positioned therein, about ~he fixed axis of the circular plane of action at a speed such that there is no slippage between the base cone and 10 the plane of action at their fixed and vertical line of tangency. Inasmuch as the plane of action is imaginary, this amounts to swinging the ro~ating cutters about the apex of the imaginary base cone as though they were rotating with the plane of action.
The same would equally be true of the machine of FIGURE 1, i~e.~ ~ith the base-c~ne axis vertical, if the cutter head mountings were modified for rotation of the cutter axes in planes parallel to the plane of action and or controlled linear mobility of the cutters 20 independently on all three rectilinear axes, 50 as to maintain the cutters constantly perpendicular to the resulting inclined plane of action with suhstantially constant penetration thereof throughout the generating swing of the cutters, while the ~ear blank also rotates 25 to maintain the relative rolling action.
Not yet suggested, but also pos~ible, is the reverse of the simplistic arrangement of FIGURE 5, namely, with the base cone 90 fixed non-rotatably in space and with the circular plane of action rolling on 30 the base coneO In such a rol7ing movement, the axis 94 oE the circular plane of action would nutate about the axls of the base cone at the apex of the cone, and the entire circular plane of action would nutate as he plane rolls upon the surface of the cone The generation of bevel teeth on the unmodified 8-axis machine of FIGURE 1, as illustrated by FIGURE 26, 3t ~

results, in fact, from the creation of the relative rolling movement between the base cone and the plane of action by a combination of aspects of all of the foregoing, viz., by an absolute rotation of the base 5 cone on its own axis and by an absolute nutative rolling motion of the clrcular plane oE action on the base cone.
These absolute motions are the sum or resul~ant of the relative movement between the base cone and the plane of action as a mutual rotation about their lO respective axes in non-slipping .rolling contact at a given line of tangency, together with a simultaneous rotation of that system~ as a whole, about the axis of the base cone while the axis of the plane cutter is rotated in its confining vertical plane to maintain ~he 15 cutter face constantly perpendicular to the plane of action, and the cutter axis is simultaneously translated with respect to all three orthogonal axes to maintain the penetration of the plane of action by the cutter, and its placement therein, ar the plane nutates 2~ a. Mathematical Developmen~
Because the generating movement of the cutter relative to the gear blank is a movement of the cutter toward or away from the line of tangency of the base surface with the plane of action, it is convenient first 25 to derive the mathema~ical relatiunships for t3 e general (conical) case within the framework of the right-handed orthogonal system i~lustrated in FIGURE 2~, wherein the system origin ls centered at the apex of the base cone of the gear blank with the vertical or Z-axis coinciding 30 wit.h the ax~s of the base cone, with the XZ plane passing through the axis of the base cone and its line of tangency to the plane of action~ the tangent being assumedly fixed in the XZ plane by the re~ative rotational velocities of the base cone and plane of 35 aGtion a~out their respective axes.
Within this system, the required roll angle of -3g-the base cone ~gear blank) about its axis, the roll ~nyle of the plane of action (cutter1 about its axis, and the coordinates of the center of ~he c~t~er, are determined for single cutter generation without regard 5 to the machine constraint of the c~tter axis to a vertical pl.~ne, i.e., as though the cutter axis were freely rotatable in a plane parallel to the plane of action a~ the latter rotates.
The two roll angles and the coordinates are 10 determined as funotions of the _ransverse ~essure an~le of the base cone and (assuming multiple-pass generation) of the cone distance to the center of the generating line as two independently variable parameters, and with respect t~ certain constant parameters fixed b~ the 15 characteristics of the gear to be cut~ namely, the base ~one angley the radiu~ of the base helix base circle (120 in FIGVRE 5), the initial distance between the plane of action and the axis of ~he cutter spindle, and the angle between the plane of action and the plane in 20 which the cutter zxis is confined.
Then, because the cutter axis is confined in a ~ertical plane tran~verse to the axis of movement of the machine carriage, those mathem~tical relationships are adapted tv the real system, i e.~ with respect to a 25 coordinate system in which the plane of rotation of the cutter axis is parallel to the YZ plane of the machine.
This amounts to rotating the original coordinate system about its Z-axis through a variable angle ~ until the new Y-axi~, Y', is parallel, and the new X-axis, Xl, is 30 perpendicular, to the vertical plane of the cutter axis. That angle~ o, like the other variables, ls derived as a function of the in~ependent variables, namely~ the transverse pressure angle and the cone distance to the cent~r of the cutter plane, or more 35 exactly~ the radial distance from the center of the plane of action to the center of the generating line.

~ J

~40-FIGURE 28.1 sh~ws the base cone, pl~ne of action, cutter, and cutter axis pro jected ~o the XZ
plane, FIGURE 28.2 shows the same projected to the XY
plane; FIGURE 28.3 the ~ame projected to the YZ plane;
5 FIGURE 2~.4 the same projected to the plane of action;
FIGURE 28.5 the same projected to the cutter plane;
PIGURE 2806 the same pxojected to the Y'Z plane and FIGURE 28.7 is a pro]ection of the same to the transverse plane, i.e., a plane perpendicular to ~he 10 line of tangency. The latter projection is "rolled out"~ i.e~, the back oone of the base cone is developed in the plane of the drawing as a transverse base circle, i~eO, a full circl~ of radius egual to the back cone distance, and the arcuate traversing path of the cutter 15 relative to the line of tangency during ~he relative rolling movement of the base cone and plane of action i s developed as the equivalent linear distance d in the plane of the drawing from the point projection of the line of tangency in FIGURE 28~7 to the projection P of 20 the center C of the cutter plane.
The ~ransverse pressure angle selected as the basic independently variabl2 parameter is the angle ~T
in FIGURE 28.7~ It is the angle in the transverse plane between the transverse radius of the base cone to the 25 line of tangency~ and a transverse radius to the intersection of the involute tooth profile 114 with the plane of action 92 (FIG~RE 530 The minimum value of the transverse pressure angle ~T is determined by the designated start-of-30 active profile and would have a value o ~ero if theinvolute profi~e were to be generated right down to the surface of the base cone. The maximum value of ~T i~
determined by the addendum surface o the gear blank and is that value which is necessaxy to assure that, on the 35 generating pass of the cutter away rom the line of tangency9 the trai1ing end o the generating line has cleared the addendu~ s~rface.
The simultaneo~s angular positions of the ha~e cone about its own axis and of the plane of action (i.e., the c~tter) about the axis of the plane of 5 action, are derived from the transverse pressure angle.
From FIGURE 28.7, it will he seen that the transverse roll angle, ~T~ of the ba~e cone to generate the involute 114 to the extent there ~hown, if measured in radians from the given line of tangency, 10 equals the tangent of the transverse pressure angleO
~ T = tan~T (1) That is to say, as the transverse arc subtended by the angle ~T is equal in length to the developed distance d of the center P of the generating line from the line 15 of tangency, dividing both by the base cone transverse radius RBT demonstratec that 5T in radians equal tan ~T.
From FIGVRE 2~.1 and 28~7, i~ may be appreciated that the developed distance d of the center 20 of the generating line from the line of tangency, considered respectively as the arc of the transverse base circle of the base cone, as the arc of the actual base circle of the base cone9 and as the arc of the circular plane of action, subtends different but related 25 angles in the transverse plane, in the plane of rotation of the base cone about its own axis, and in the plane of action, and that those angles are inversely proportional to the base cone transverse radius (back cone diEtance~
RBT, the base circle radius R, and the base cone 30 distance A. Therefore, as the cone angle of the base cone is r, the transverse roll angle of the ba~e cone is ~T ~ T R~cosr ~2) the ro~l angle P of the base cone (gear blank) about its own axis is P = R (3) ~j ~

and the roll angle of the plane of acti~n (cutter) ab~ut the roll axis of the plane is d d B = A ~ ~/sinr Dividing equation (3) by t2), a~d (~) ~y (3) p ~ ~T/cosr (5) ~ = Psinr (6) and the ratio of the angular velocity of the plane of action (cutter) about its axis to the angular velocity of the base cone (gear blank) about its axis for 10 relative rolling movement without slipping is B/P - sinr (7) While the values ascribed to R and A in the foregoing explanation were those at the base circle of the base cone, the derived relationships are valid at 15 any selected lesser value of the cone distance A, which has the constant relation to ~ corresponding radius R of the cone expressed in equation~ (4)~ namely R - Asinr (8) As will be~t be appreciated from FIGURE 28.4, ~o the cutter 64 i~ positioned perpendicular to the plane of action 92 and with its axis parallel thereto (FIGURE
28.1). It is shown there and in all views of FIGURE 28 with the center P of its generating line 118 at the edge of the p~ane of action 92 and with the generating line, ~5 extended in the plane of act.ion, tangent to a base helix base circle 120 of radi~s R~, resulting in a helix angle ~ with a plane-of-action radius to the center P of the generating line 118 in the illustrated case at the perimeter o~ the plane of ac~tion~ From ~IGURE 28.4, 30 ~ = sin~l(~ /A~ (91 Thu~, the angle of the projection of the cutter axis to the plane of action (FI~URE 28.4)~ measured with respect to the Y~axi~, becomes ~
The cutter axis projected at the X'~ plane makes 35 the angle G with the Z-axis ~FIGURE 28.1)~ an angle . . . . . . . ~;

s ~` ;

equal in value to the cone angle r. Projected to the YZ
plane (FIGURE 2B~3 and FIGURE 280~) 9 the cutter a~is makes the anyle ~ relative to the Y-axis. Then, from tan~ = a CosG (10) tan(3~) ~ b (11) wherefore, ~ - tan~l(tan(B~)cosG) (12) Projected to the XY plane (F1GURE 2~.2), the cutter axis makes the angle ~ relative to the Y-axis. 0 Again, from FI~URE 28.3~ tana = a slnG (13) wherefore, a = tan~l(tarl(B+~)sinG) (14) The coordinates of point C, the center of the cutter~ in the XYZ system (FXGURES 28~1 and 2B.4), are:
XC = A cos~sinr + HcosG (15) YC = A sln~ (16) ZC ~ -A cosBcosr ~ HsinG (17) where A is the base cone distance, R is the radius of the base cone at such distance and H is the distance of 20 the cutter axis from the plane of action~
From the foregoing equations, it will be seen that all concurrent instantaneou~ values of the angle of rotation P of the gear blank about its axis~ the angie of rotation ~ of the cutter about the axis of the plane 25 of rotation9 the helix angle ~, the angles made by the projections of the cutter axis in all three orthogonal planes, and the coordinates of the center C of the cuttex, are all ultimately expressed in terms of the two independently variable parame~ter~ ~T and A~ and the 30 constants r7 G, RE~, and ~lo The value o A, namely, the radial location in the plane of action of the center P of the chordal increment 118 of the generating line, may be varied in several ways to generate a tooth proflle o~ facs~ width 35 exceeding the radia~ pro]ection of that chcSrdal ;3IJ~;

-4~-increment in the plane of action~ FIGURE 28~4 shows a shaded segment of the circular plane of action which representE as an area o~ the plane the overall generating sweep fvr a gear of face width exceeding the

5 l.enyth of the chordal ~enerator of the cutter.
I~ is obvious that if the radial projection ~f the chordal generating line 118 3f the c~tter in the plane of action exceeds the face width of the gear, A
may be held constant because the tooth profile will be 10 generated in a single traverse~
Where that projection is less than ~he face width of the gear, the value of A may be varied in steps between generating traverses so as to sweep the generatiny path in contiguous or overlapping circular 15 bands, or the value of A may be varied slowly and continuously to traverse radially of the plane of action while the cutter is swung by the rotation of the plane of action ahout its own radius. If the zig-zag pattern analogous to FIGURE 9 is to be followed by including a 20 continuous radial feed, the total radial traverse of one generating pass may not exceed one-half the length of the radial projection of the chordal increment 118, with th~ result that most, if not all~ of the face width of the tooth profile is traversed twice by the generating 25 line. In that circumstance, A may be varied as a function of ~T such that no ungenerated areas are left on the tooth profile~
b. Mathematical Relationships Modified For 5-Axis Generation _ As shown in FIGU~E 28~2, a secondary coordinate system consistent with the five of the eight axes of the actual machine of FIGURE 26 may be established by rotating the X-axis and Y-axis about the ~ axis until the Y~axis i5 parallel to the projection of the rotational axis of the cutter to the XY pl.ane~ which is to say, par~llel to the transverse plane of the machine in which the spindle axes are conf1ned~
So rotated, the X'-axis and Y'~a~is make the variable angle c with their original positions, and the roll angle of the plane of action and of the base cone 5 (gear blank) D ~easured oriyinally from the line of tangency in the XZ plane, must be referred to the new X'-axis.
Thus, the roll angle p of the gear blank abo~t its axis becomes p-a with respect to the X'-axis, and p-o = tan~T/cosr - tan l(tan(tan~Ttanr -~
sin~l(RH/A))si.nG) (18) In the vertical plane of the machine which contains the spindle axis and is parallel to the Y'Z
plane (FIGURE 28.6), the angle of the cutter axis 15 relative to the Y'-axis, ~', may be determined from FIGURE 28.8, as follo~s:
a/sin(~+~) (19) sin' = sin(~+~)cosG (20) and substituting the values of ~ and ~ using eguations 20 (13, (5), (6~o and (9), ~' = sin-l(sin(tan~Ttanr ~ sin 1(~ /A))cosG) (21 The coordinates of the center of the cutter, adjusted to the X'Y'Z system are similarly recalculated~ From FIGURE 23.2 it will be seen ~hat 25xc' = xccosa + ycsina (22) From FIGURE 28.2~ it will also be seen that Yc = Yccos~ ~ xCsinu (23) Lastly ~C ZC (24) It will be seen that all concurrent coordinate 30 values and values of the angles of rotation p-a of the gear blank and of the spindle axis ~' are expressed ultimately in terms of the transverse pressure angle ~T
and the radial distance A to the center o~ the incremental generating line in the plane o aetion as 35 indepetldent variable,s., J

3~

The res~lt is that the 8-axis machine o FIGURE
26 cuts the bevel tooth profile by a genera~ive movement which amounts to the relative rDlling motion of ~he base cone and circular plane of action at a given line of 5 tangency which itself is rotated about the base cone axis to enable the plane cutter to maintain its perpendicularity to, and its position in, the nutating plane of action by rotation of its axis in its fixed vertical plane and by concurrent relative linear 10 movements of the cutter and gear blank.
It will be appreciated from FIGU~E 28,7 and from earlier discussion that simultaneou~ generation of opposing tooth profiles implies different simultaneous values and signs of the transverse pressure angles ~T
15 of opposing profiles at the same time, resulting in different simultaneous values of the X' coordinates of the two cutters. As this is not. possible in the machine of FIGURE 1 or FIGURE 26, that machine is limited to single cutter generation of conical gear s one tooth 20 profile at a time.
cO Mathematical Conditions For Two-Cutter Conical Generation .
In the modified machine of FI~URE 2, having the additional rotational axis represented by the tiltable 25 table~ the ang~e G IFIGURE 28.1~ is reduced to zero by the tilting of the ta~le 32 by an amount equal to the cone angle r to position the plane of act70n 92 vertical and parall~l to the common plane of the cutter axes.
Inasmuch as the angle G was equal to r, G = r - r - n (25) with the result th~t sinG - 0 ~anG = 0 and cosG = 1 35 Substituting these values in Equations (12), (:l4), and (2~), ' J

a = O (26) = B~ (27) (28) and p-~ = p-0 = p (29) 5With a equal to zero, i~e., no longer a function of the transverse press~re angle 4T~ the same gear blank rotation P satisfies both cutters whose axes must lie in the same plane, namely the condition provided by the machine of FIGURE 2.
10do Mathematical Demonstration That Cylindrical Generation Is a Limit of the General Method .
The following analysis shows that the generation of cylindrical gears is simply a particular limiting c~se of the generation of helical bevel gears.
15For the cylindrical case, the spindle axis angle ~ in the Y'Z plane must be set at the base helix angle, i.e., a' = ~ (30) the spindle axis angle ~ in the X'Y' plane must be zero, 20 i.e~, a = 0 (31) the gear blank rotation angle must equal the transver~e roll angle, i~e., T ( 3 ~ ) ~5 the x' coordinate o~ the cutter center must eq~al the base circle radius dimension pl~5 the distance of the cutter axis from the plane of action, i~e~l xc' = R ~ H ~ (33) and the y~ coordinate of the cutter center must be 3~ Yc' = ~ tan~T (34) ~ o demonstrate, as the cone angle of a cylinde is zero~
sinG = sinr - 0 (35) tanG = tanr - 0 (36) 35and cosG - cosr - 1 (37) Accordinyly, the value of the ang~e of --~8-rotation P of the base surface Igear blank) taken from Equation (5) and (1), narnely P ~- tan~T/coSr is reduced to S P = tan~T (38) The roll angle ~ of the plane of acti~n (the swing of the cutter), taken from Equations (6), (5) and (1) as ~ = tan~tanr 10 is red~ced to ~ = o.
The angle ~' of the c~tter axis projected to the Y'Z plane, using Equations (20), (37), and (39), is ~' = sin~l(sin(~+~)cosG~
- sin~l(sin(O+~)xl) = sin~l(sin~) = ~ Q.E.D.
The foregoing will be recognized as Equation (30).
The angle o of the cutter axis projected to the XY plane~ from Equations (14)~ (39~ ~ and (35), is a = tan~l (tarl ( ~8+y, ) sinG~
= ~an~l(tan(O~)xO) a = O Q.E.D. (40) which will be recognized as Equation (31).
The angle of rotati~n of the gear blank, p-a, 25 from Equations (38) and (40), and ~1~, is p-a ~ tan~T ~
T ~.E.D.
which will be recognized as Eq~ati3n (32)~
The x' coordinate o~ the center C of the 30 cut~ex7 from Equatlons (22) and (40)~ is Xc xccoso + ycsina = xC x 1 + YC x O
= xc (from Eq. lS) - A cos,~sinr + HcosG
35 (from Eq. 8) = R cos~ + HcosG (41) (from Eqsv 37 & 39) = R x 1 ~ H x 1 -4g-xc' = R + H Q.E.D.
which is Eq~ation (33~O
From ~quations (2) and (4~
RBT~T A~ (42) 5 and from Eq~ation (6), it is apparent that as r approaches ~ero~ so also does ~ and therefore the sine of ~. Thus, in the li~it, sin~ = ~. (43) The y' coordinate of the cutter cen~er C, from Equation 10 ~23) is Yc -xcsina ~ ycCos~
(from Eq. 31) - ~XC x + Yc x 1 YC
(fxom Eq. 16) = A sinB
15 (from Eq. 43) = A~
(from Eq. 42) ~T T
(from ~q. 2) ~ (R/cosr)6T
(from Eq. 37) = (R/l)~T
= RaT
20 ~from Eq. 1) Ycl = R tan~T Q.E.D.
which is Equation (34).
HYPER~OLOIDAL GEARS
_ The generation o~ involute tooth profiles of gear teeth has been considered earlier herein on the 25 usual basis of meshing gears whose axes lie in the fiame plane, intersectln~ in the conical case, and par~llel in the cylindr~cal case. The method of the invention is, however, applicable equally to the manufacture of gear pairs designed for the direct connection of shafts 30 having non-parallel~ non-intçrsecting axes9 i.e., gears whose pitch surfaces are essentially the frustra of single-sheet hyperboloids of revolution tangent along a shared generatrix.
Two such pitch-surfaces hyperboloids 171 and 172 are shown in orthographic projection in FI~URE5 29.1~ 29~2t and 29.3.

, .

The elevational view of FIGU~E 29.2 is the projectior of the hyperboloids to a plane perpe~dicu1ar to the mutual perpend.ic~lar 174 to the two hyperboloid axes 176 and l78r which iE acoordi~gly pro-jected in 5 FIGURE 29.~ as the point P defined by the in~ersec~ion of the two axes. In FIGURE 29.2, the line of tangency 180, i.eO, the common generatrix, is accordingly projected at full length, and the angle ~ between the hyperboloid axes, and the angles Gl and G2 between 10 each of them respectively and ~he projected line of tangency 180 of the two hyperboloidal curfaces, are proje~ted at their maximum valuesO
In the end view of FIGU~E 29.3, the axes 176 and 178 of the two hyperboloids are projected as 15 parallel lines which may al o be taken as the projections of the par~llel planes containing the two r skewed axes 176 and 178, those planes being spaced apart at a distance C by the mutual perpendicular 174 to the two axes. Because the line of tangency 180 is depicted 20 horizontally in FIGURE 29.2, i~ projects as a mere point - on the mutual perpendicular 174 in FIG~RE 29.3, dividing the mutual perpendicular into two segments Xl and X2, which are proportional to the projected angles Gl and G2O
From FIGURES 29.1 and 29.2 it may be appreciated that the meshing engagement of hyperboloidal gears9 whose pitch-surface tangent 180 ic a~ke~ from both axes of rotation, is accordingly ~ rolling action combined with a relative lateral ~liding motion along 30 the tangent line 180 of the ~pitch surfaces~ as distinguished from the imple rolling motion of the pitch surfaces of gears having coplanar axes.
Ina~much as a single-sheet hyperboloid may be regarded as the envelope of two symmetrical series of 35 coaxial cones of decrea~ing and increacing cone angle who~e common axis i5 the locus of th2ir apices, and which merge in a cylinder at the least radius of the hyperboloid, an essentially hyperboloidal gear connection between non-parallel, non-intersecting shafts may be made in ~wo ways. Xf the connectior be made at 5 locations along the tangent pitch surfaces axially remote from the mutual perpendic~lar 174 to the axes 176 and l78 of the shafts, i~e~0 where axially li~ited frustra of the hyperboloidal pitch surfaces are essentially conical, the connection can be made by lO conical c~ears commonly referred to as "hypoid" gears.
If the connection between the shafts is made at the least distance between the axes, i.e., so as to include their c~mmon perpendicular 174 in the two meshing gears, the connection can be made with two cylindrical gears 15 whose pitch surfaces are the essentially cylindrical "waist" portions of the two hyperboloids. ~ch gears are commonly reerred to as cross-helical or "~kew"
geaxs.
As a point of departure from which to explain 20 the generation of conjugate tooth profiles of hyperboloidal beve~ gears, it will be well to recall that in the ordinary bevel gear case, i.e., where the shaft axes intersect, the base cones of the two gears are tangent to opposite sides of the same circular plane 25 o action with their apices coincident at its center, and with the meshing tooth profiles of the two gears generated as though by the same generating line in that plane of action. Moreover 7 in the usual case of symmetrical teethO the plane of action of the opposite 30 tooth profiles of both gears, intersecting the first plane of action on the pitch line of the gears, is also tangent to the same two base cones.
In applying the method of the invention to the generation of hyperboloidal gears of the general conical 35 or "hypoid" case, the meshing tooth profiles are similarly generated from two base cone~ which are , respectively tangent to the opposite ~ides of a common plane of action, but the apices of the cone6 do not coincide. Accordingly, each base cone has a separate circular path in the common plane of action. Conjugate 5 action of s~ch gears, i.e., a constant ratio of ang~lar velocities, is nevertheless obtained by using the same generating line for the meshing profiles of the two gears, cr, more specifically, in acknowledgment of the separate generation of the meshir,g profiles, by assuring 10 that the generating lines of the meshing profiles of the two gears coincide throu~hout the zone of action of the two profiles in the overlap of the separate circular paths of their base cones in the common plane of action.
As it will further be apparent that it is not 15 po~sible for two base cones with non-coincident apices to be tangent simultaneously to the opposite sides of two different planes of action, the opposite me~hing profiles of the two hypoid gears are generated from a second pair of base cones each respectively coaxial with 20 the first base cone of its gear, but having a different apex, a different cone angle, and typically, but not necessarily, a diFferent base circle.
These criteria for the generation of hyper-boloidal bevel gears are developed graphically in 25 ~IGURES 29.2, 29.3, and 29.4 for the general case where the two gears include the transverse hyperboloidal sections through the point Q on the common tangent 180.
Pitch cones 181 and 182 are tangent respectively to the hyperboloidal surfaces 171 and 172 at the indicated 30 transverse sections 184 and 186 which are accordingly bounded by pitch circles 185 and 187. The pitch cones lBl and 182 are thus in contact at the point Q on the line of tangency lB0 of the hyperboloidal pitch sur-faces. In FIGURE 2903, the pitch circles 185 and 187 of 35 the pitch cones project as ellipses, and the pitch plane 18B, i.e., the plane mutually tangent to the pitch cones along the contact l.ine, projects as a straight line.
To develop the geometry for ~,ne set of rneshing profiles, a plane 190 is passed through ~he pitch line 180 at the desired transverse pressure angle 9TL to 5 the pitch plane 188 to constitute the plane of action for the left tooth profiles. Lesser concentric circles 191 and 192 in the bases of the pitch cones 181 and 182 tanqent to the plane of action 190 are the base circles of the two corresponding base cones 201 and 202, wherea~
10 the intersection.~ of the plane of action 190 with the respective axes 176 and 178 determine the apices 198 and 20D of the two ba~e cones 201 and 202 tangent to that plane of action. In the illustrated case, a transverse press~re angle of 20 determines the configuration of 15 the base cone 201 for the larger gear and the base cone 202 for the smaller.
The selection of a desired transverse press~re angle ~ for the engagement of the opposite or right profiles o~ the teeth of the two gears determine~ t.he 20 location of the opposite plane of action 204, which, by the same procedure, results in the definition of a second pair of base cones 206 and 208 for the opposite tooth profiles~ Thus~ the second base cone for each gear has a different apex and a different cone angle, 25 and, if the transverse pressure angle ~T~ is different from that for the left profiles, will also have a different base circle.
In FIGURE 2~.4, the circular paths 211 and 212 of the base circles 191 and 192 of the base cone~ 201 3a and 202 for the left-hand tooth profiles of both gear~
are developed by projection from FIGURES 29~3 and 29.2.
The points of tangency Sl and S~ of the base cone base circles with the left~hand pl.ane o action 190 are projected to the transverse sectior)s 184 35 and 186 in FIGURE 29.2, and the projections of the lines of tangency 194 and 196 of each of those base cone~ to i, ` ) -5~-the left-hand plane of action 190 are drawn in FIGU~E
29.2 to determine the point R oE their common intersection with pitch line 180 extended.
Then in FIGURE 29.4, the pitch line 180 is 5 projected perpendic~larly from the plane of action 190 in FIGURE 29.3, a convenien~ point is selected for the point Rp the intersections of the plane of action 190 with the projections of the axes 176 and 17B in FIGURE
29~3 are similarly projected, and the distances of the 10 base cone apices 198 and 200 from the poin~ R, projected to the pitch ]ine 180, are transferred from FIGURE 29.2 to FIGURE 29.4 to determine the locations of the points 198 and 200 by projection from the line 180 to intersect with the prViectionc 3f ~he points 198 and 200 from 15 FIGURE 29.3. The apices 198 and 200 of the base cones 201 and 202, as thus located in FIGU~E 2904, are the centers of the cîrcular planes of action for each of the base cones, or, more precisely~ the respective circular paths of relakive rolling movement of the base cones 201 20 and 202 upon the opposite sides of their common plane of action 1~0.
Lines drawn in FIGURE 2g>4 from the projected apices 198 and 200 of the base cones to the point R are therefore the lines of tan~ency 194 and 196 of each base 25 cone with the lef~-hand plane of action 190, and thus also the projections of the two axes 176 and 178 to that plane of actionn From the points 198 and 200 in FIGURE 29~4, arcs struck at the respective cone distances of the base 30 cones 201 and 202 are the circular paths of the base circles 131 and 192 of those cones in the common plane of action. A face width Wl, selected within limits and with the circular path 2il approximately centered therein, may be taken to determine, by its intersection~
35 with the pitch line lQ0, a suitable face width W2 for the meshing gear~ The overlap of those ann~7ar hands in -the space between the lines of tangency 194 and 196 of the two base cones in FIGURE 29.4, determines ~he ol~ter limits of the zone of action 300 of the ~reshing left-hand tooth profiles in their plane of action 190, 5 the sctual size and shape thereof within those limitE
being governed ~sually by the intersection of the addendum surface with the plane of action It is within this limited ~one of action that the tooth profiles are engaged, and within this zone of 10 action that the generating lines for the meshing profiles of both gears must therefore coincide for conjugate action.
If, for example, the gear designer ~hould decide that the ~eeth generated from the base cone 201 15 should be axial, the generating line in the plane of action would remain aligned radially with respect to the center 198 in its ~ovement through the zone of action, whereas it would be concomitantly necessary to pivot the cutter, i.e., the generating line~ for generation ~rom 20 the base cone 202 a~ the generating line swung about the center 200 while traversing the zone of action in order to provide tooth contact along the identical line in the zone of action.
If, for example, it were desired that the line 25 of contact of the meshing tooth profiles be parallel to the pitch line 18~, it would be necessary to pivot the generating lines of both gears as they swing about the centexs 198 and 200 in order to maintain the coincidence of the two generating lines during their separate 3Q traversee of the zone of action.
As indicated earlier, the pivoting of the straight generating line provided ~y the plane c~tter of FIGURE5 1 t~ 6 is provided by the pivoting of the cutter head S2 on an axis which lies in the plane of the cutter 35 64 and intersects the rotational axis of the cutter.
Locating the base cones 206 and 20æ in the same manller by the selection of an appropriate transverse pressure angle ~TR to determirle the right-hand plane of action 204, the necessary geometry is eskablished ~or generating the right-hand tooth profiles. In this way~
5 the opposing flanks of a single hyper~oloidal be-~el gear are generated from base cones which are coaxial but which have different apices and different cone angles.
In the cylindrical case, i.e~, where the gear connection between two such non-parallel~
10 non-intersecting shafts is made at the least distance between them, there is no common or shared plane of action as in ~he skewed conical case because the cylinders interiorly tangent to the hyperboloids at their least radius have no apices. Rather, each 15 cylindrical gear has its own plane of action, which is tangent to its base cylinder, passes through the point of cont3ct (pitch point) of the two pitch cylinders, and intersects the plane o action of the other in a straight line which passes through the pitch point and ~0 is the locus of the point contact of the active profiles of the engaged teeth of the two gears.
Such gears can be generated like any other helical cylindrical gear in the manner explained earlier herein~
CONCLUSION
The method of the invention, whether practiced in limited scope to generate cylindrical gears on a 4-axis machine, or more fully to generate cylindrical and bevel gears on five-y six-, eight-, or nine-axis 30 machlnes, produces gears at a much greater rate than is possible by the exi6ting procedures referred to at the beginning of this specification, due to the complete independence of the cutting speed of the cutter from its generating movement and the ability to remove metal at a 35 very substantial rate without detrimental effect upon the finish of the tooth profiles produced~ The time I ~ !

advantage over the prevailing hobbing proced~re i5 very s~bstantial and the finish of the tooth pro~iles produced is far superior to the scalloped finish left by the hob.
The superiority of the method here disclosed over all known machining procedures, over and above its speed, i6 its versatility in the production of gears of many kinds, size6, and design specifications with cutters of relatively few sizes, which, as illustrated 10 herein in the case of plane cu~ters, can perform the rough and finish machining in a single operation.
The features of the invention believed new and patentable are set forth in the following clai~.

.

Claims (40)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. In the manufacture of a gear, the method of machining a gear blank to produce a tooth profile which is involute from an imaginary base surface of revolution within said gear blank, said surface having a straight-line generatrix and having an imaginary plane of action tangent to said surface said method comprising the steps of rotating a cutter having a plurality of cutting edges uniformly spaced about the entire periphery of the cutter sweeping a cutting path in the form of a surface of revolution about the axis of the cutter so that said plur-ality of cutting edges are distributed substantially uni-formly about the common surface of revolution which they define, said cutting-path surface comprising a plunge-cutting rim portion of cutting thickness not exceeding the desired tooth space at the tooth root and a contiguous tooth-profile cutting portion, positioning the rotating cutter on the side of said plane of action opposite to said base surface with said rim portion penetrating said plane of action and with said tooth-profile cutting portion intersecting said plane of action along a predetermined generating line and with said tooth-profile cutting portion perpendicular to said plane of action at least at the center of said generating line, independently controlling the rotation of the cutter, and effecting a relative feeding movement of said gear blank and rotating cutter independently of the rotation of the cutter while maintaining the aforesaid position of the cutter relative to said plane of action) said feeding movement being such as:
to cause a relative rolling motion between said base surface and said plane of action without slippage;
to cause said generating line at all points therealong to maintain a controlled angularity with respect to the instantaneous direction of its movement relative to the line of tangency of said base surface with said plane of action during said rolling motion; and to cause said rotating cutter to penetrate said gear blank and said generating line to traverse said gear blank between its addendum surface and a depth at which the desired active tooth profile is achieved at the center of said generating line.

2. The method of Claim 1 wherein the tooth-profile cutting portion of the cutting path of the cutter is a circular plane, the rim portion thereof extends axially of the cutter at the periphery thereof, and the generating line is a straight line.

3. The method of Claim 1 wherein the tooth-profile cutting portion of the cutting path of the cutter is a cone, and the generating line is a conic section arc to the center of which an element of the cone is perpendicular, the rim portion being turned outwardly for an axially convex tooth flank and turned inwardly for cutting an axially concave tooth flank.

4. The method of Claim 1 wherein the tooth-profile cutting portion of the cutting path of the cutter is a cylinder with its axis perpendicular to the plane of action and the generating line is a circular arc, the rim portion of the cutting path being turned outwardly of the cylinder for cutting an axially convex tooth flank and inwardly for cutting an axially concave tooth flank.

5. The method of Claim 1 wherein said controlled angularity of the generating line remains constant throughout its said traverse.

6. The method of Claim 5 wherein the relative feeding movement of the gear blank and rotating cutter is accomplished by rotating the gear blank about the axis of its base surface and by simultaneously moving the cutter so as to cause every point on said generating line to maintain an instantaneous component of velocity in said instantaneous direction equal to the instantaneous peripheral velocity of the base surface at the projection of each such point to said line of tangency along the path of said relative rolling motion.

7. The method of Claim 6 applied to the simultaneous machining of the opposite flanks of two spaced teeth of the same gear blank, comprising the complementary use of two such cutters disposed with the tooth-profile cutting portions of their respective cutting paths in facing relation and with their rotational axes making the same angles with a normal to the plane of action, and being separated by a transverse base tangent measurement, said relative feeding movements of the cutters occurring in unison and causing the generating line of one cutter to traverse the gear blank from the addendum surface to said depth while the other cutter traverses oppositely.

8. The method of Claim 7 applied to the machining of cylindrical gears and further comprising the use of a pair of such cutters having planar tooth-profile cutting portions in facing parallel relation at a transverse base tangent between their planes and with their generating lines substantially of equal length and substantially coincident projection to the line of tangency of the base cylinder to the plane of action.

9. The method of Claim 7 applied to the machining of bevel gears which further comprises employing a pair of such cutters having planar tooth-profile cutting portions facing toward one another with their generating lines at the centers thereof intersecting equal radii of the circular plane of action at equal angles and with said radii spaced apart by a transverse base tangent angle.

10. The method of Claim 5 employed to machine gears having a tooth width greater than the length of said generating line, wherein the tooth-profile cutting portion of the cutting path of the cutter is a circular plane, the rim portion thereof extends generally axially of the cutter at the periphery thereof and the generating line is a straight line, the additional steps of translating the cutter along an extension of said generating line to displace said generating line endwise a distance not greater than the length of said line for successive traverses of said gear blank by said generating line, and repeating said relative feeding movement and displacing the generating line endwise as aforesaid as frequently as may be necessary to extend the generated tooth-profile to the desired width.

11. The method of Claim 10 wherein the endwise displacement of said generating line and the relative feeding movement of the cutter and gear blank occur simultaneously, and wherein said successive traverses of the gear blank by the generating line occur in opposite directions.

12. The method of Claim 6 employed to machine cylindrical gears wherein the gear blank is mounted for rotation about the axis of the base cylinder and the cutter axis is rotatable in a plane parallel to the axis of the base cylinder, the cutter axis also has two degrees of rectilinear motion in said cutter-axis plane, said cutter-axis plane is movable relative to the gear blank axis toward and away from the same, and said relative rolling action of the base cylinder and the plane of action is accomplished by rotating the gear blank on its own axis while simultaneously moving the cutter axis linearly in the cutter-axis plane so as to move said generating line with a component of motion perpendicular to the line of tangency of the base cylinder to said plane of action in the direction and with the velocity of the peripheral movement of said base cylinder at said line of tangency.

13. The method of Claim 12 adapted for the simultaneous machining of two opposite tooth profiles of the same gear blank by the simultaneous employment of a second cutter positioned and movable and moved in the manner specified for the single cutter of Claim 12, with the axes of both cutters in the same cutter-axis plane, with the tooth-profile cutting portions of said two cutters parallel and facing each other at a transverse base tangent distance and with their generating lines of substantially equal length and projection to said line of tangency.

14. The method of Claim 1 employed to machine the tooth profiles of bevel gears on a 5-axis machine wherein the gear blank is mounted for rotation about its own axis, the cutter axis is movable linearly along three mutually perpendicular axes which define planes respectively parallel and perpendicular to the gear blank axis, and pivotable about an axis perpendicular to one of said planes parallel to the axis of the gear blank, and said relative rolling motion of the base cone and the circular plane of action is accomplished in part by rotation of the gear blank on its own axis and in part by the nutation of the plane of action about the axis of the gear blank, said nutation of the plane of action being effected by pivoting the axis of the cutter while translating the same along said mutually perpendicular axes in order to maintain the perpendicularity of said tooth profile cutting portion to, and its penetration of, the plane of action, and to maintain said angularity of the generating line.

15. The method of Claim 14 in which the tooth-profile cutting portion of the cutting path of the cutter is a circular plane, the rim portion thereof extends generally axially of the cutter at the periphery of said circular plane, the generating line is a straight line, said controlled angularity is constant, and the axis of the cutter lies in a plane parallel to the axis of the gear blank, the pivot axis of the cutter axis lies in said circular plane, and wherein the simultan-eous values of a) the angular displacement of the gear blank, b) the angular displacement of the cutter axis, and c) the three rectinlinear coordinates of the center of the circular cutting plane, are specified respectively, with respect to said five axes, as follows:

(a) tan .PHI.T/cos .GAMMA.- tan -1(tan(tan .PHI.T tan .GAMMA.+sin -1(RH/A))sin G) (b) sin -1(sin(tan .PHI.T tan .GAMMA.+ sin -1(RH/A))cos G) (cx)(A cos .beta.sin .GAMMA. + H cos G)cos .sigma. +A sin .beta. sin .sigma.
(cy)A sin .beta.cos .sigma. - (A cos .beta. sin .GAMMA. +H cos G)sin .sigma.
(cz)- A cos .beta. cos .GAMMA. +H sin G
wherein .PHI.T is the instantaneous transverse pressure angle .GAMMA. is the cone angle of the base cone RH is the radius of the base helix base circle .DELTA. is the base cone distance G is equal to the cone angle .GAMMA. of the base cone .beta. is tan .PHI.T tan .GAMMA.
H is the distance of the cutter axis from the plane of action .sigma. is tan -1(tan(.beta.+.omega.)sin G) .omega. is sin -1(RH/A) and wherein the values of .PHI.T and A are independently variable.

16. The method of Claim 6 applied to the machin-ing of an external bevel gear, wherein the rotating cutter is movable linearly relative to the gear blank along three mutually perpendicular axes two of which determine a reference plane, the axis of the base cone is tilted to place an element of said base cone parallel to said reference plane and tangent to said plane of action, the path of said relative rolling motion is a circular path in the plane of action centered on the inter-section of the base cone axis with the plane of action, and said instantaneous direction of any point on the generating line is perpendicular to a radius in said plane of action from said intersection to such point.

17. The method of Claim 16 wherein the tooth-profile cutting portion of the cutting path is a circular plane perpendicular to the plane of action, the axis of the cutter is parallel to the reference plane and rotated about an axis perpendicular to said reference plane and translated parallel to said reference plane to achieve said instantan-eous velocity in said circular path in the plane of action

18. The method of Claim 16 wherein the tooth-profile cutting portion of said cutting path is a cone and the generating line is a conic section arc to the center of which an element of the conical cutting path is perpendicular, said arc spanning the face width of the gear blank, and the axis of the cutter pivots about an axis perpendicular to the reference plane and translates parallel to said reference plane to move said generating line in said circular path as the gear blank rotates.

19. The method of Claim 16 wherein the tooth-profile cutting portion of said cutting path is a cylinder whose axis is perpendicular to said reference plane, the generating line is a circular arc spanning the face width of the gear blank, and, as said gear blank rotates, said cutter axis is translated parallel to said reference plane in a circular path to move the generating line in said circular path of relative rolling motion.

20. The method of Claim 7 applied to the machining of an external bevel gear, wherein the two rotating cutters are each movable linearly relative to the gear blank along three mutually perpendicular axes and two of said three axes of linear movement of each cutter determine a common reference plane, the axis of the base cone is tilted to place an element thereof parallel to said reference plane and tangent to said plane of action, the path of said relative rolling motion is a circular path in the plane of action centered on the intersection of the base cone axis with the plane of action, said instantaneous direction of any point on the generating line of either cutter is perpendicular to a radius in said plane of action from said intersection to such point, and the generating lines of said cutters at the centers thereof intersect equal radii of the circular plane of action at equal angles and said radii are spaced apart by a transverse base tangent angle.

21. The method of Claim 20 wherein the tooth-profile cutting portion of the cutting path of each cutter is a circular plane perpendicular to the plane of action, and the axis of each cutter is parallel to said reference plane, pivots about an axis perpendicular to said reference plane, and translates parallel to said reference plane to achieve said instantaneous velocity in said circular path in the plane of action.

22. The method of Claim 20 wherein the tooth-profile cutting portion of the cutting path of each cutter is a cone and the generating line of each is a conic section arc to the center of which an element of the cone is perpendicular, each said arc spans the face width of the gear blank, said generating lines at the centers thereof intersect equal radii of the circular plane of action at equal angles and with said radii spaced apart by a transverse base surface angle, and the axis of each cutter is rotated about an axis perpendicular to the reference plane and translated paallel to said reference plane to move each generating line in said circular path of relative rolling motion as the gear blank rotates.

23. The method of Claim 20 wherein the tooth-profile cutting portion of the cutting path of each cutter is a cylinder whose axis is perpendicular to said reference plane and whose generating line is a circular arc spanning the face width of the gear blank, and the axis of each cutter translates in a circular path parallel to said reference plane to move the generating line of each cutter in said circular path of relative rolling motion as the gear blank rotates.

24. The method of Claim 6 utilized to produce buttress teeth on cylindrical gears, wherein the machining of the opposite profiles of the gear teeth is effected from concentric base cylinders of different radii.

25. The method of Claim 12 utilized to produce buttress teeth on cylindrical gears wherein the cutter axis is moved linearly in the cutter-axis plane so that said component of motion of its generating line has a velocity equal to the peripheral velocity of a base cylinder of one radius for one tooth profile and of a base cylinder of different radius for the opposite profile.

26. The method of Claim 8 utilized to produce buttress teeth, wherein the relative feeding movements of the two cutters proceed simultaneously with different values of said component velocities equal respectively to the peripheral velocities of base cylinders of the different radii necessary for involute profiles of the different desired pressure angles on opposite profiles of the same tooth.

27. The method of Claim 5 utilized to produce buttress teeth on bevel gears wherein the machining of the opposite profiles of the gear teeth is effected from concentric base cones of coincident apices and different apex angles.

28. The method of Claim 14 utilized to produce buttress teeth on bevel gears wherein the machining of the opposite profiles of the gear teeth is effected from concentric base cones of coincident apices and different apex angle 3

29. The method of Claim 16 utilized to produce buttress teeth on bevel gears wherein the machining of the opposite profiles of the gear teeth is effected from concentric base cones of coincident apices and different apex angles.

30. The method of Claim 1 wherein the relative feeding movement of the gear blank and rotating cutter is accomplished by rotating the gear blank about the axis of its base surface and by simultaneously moving the cutter so as to cause at least one point on said generating line to maintain an instantaneous component of velocity in said instantaneous direction equal to the instantaneous peripheral velocity of the base surface at the projection of said one point to said line of tangency along the path of said relative rolling movement, and wherein any change of said controlled angularity is achieved by pivoting said generating line in the plane of action about said one point.

31. The method of Claim 30 wherein said one point is located at the center of said generating line.

32. The method of Claim 30 applied to machine meshing conical gears with non-intersecting axes wherein the contacting profiles of the teeth of the two gears are generated from a pair of base cones having non-coincident apices in a common plane of action tangent to both cones, the circular paths of said pair of base cones in said common plane of action overlap, the zone of action of the contacting profiles occurs within said overlap, and said angularity of the generating line of each of the contacting tooth profiles is controlled so as to cause said generating line to coincide with the generating line of the contacting tooth profile of the meshing gear as each generating line traverses the zone of action.

33. The method of Claim 32 wherein the opposite contacting profiles of the teeth of the meshing gears are generated as specified in Claim 32 from a second pair of base cones tangent with non-coincident apices to a second plane of action, the two base cones of each gear being coaxial and having different apices.

34. The method of Claim 30 applied to machine one of a pair of meshing conical gears with non-intersecting axes wherein the opposite profiles of the teeth are generated from different base cones which are coaxial but have non-coincident apices.

35. The method of Claim 2 wherein the cutting path surface includes a conical back portion extending from said rim portion toward the cutter axis in diverging spaced relation to said tooth-profile cutting potion, said rim and back portions of the cutting path surface serving to remove metal from the gear blank between adjacent teeth and the tooth-profile cutting portion serving to generate the tooth profile.

36. The method of Claim 11 wherein the cutting path surface includes a back portion extending from said rim portion toward the cutter axis in spaced relation to said tooth-profile cutting portion, said rim and back portions serving to remove metal from the blank between adjacent teeth during alternate traverses.

37. The method of Claim 13 wherein the cutting path surface of each cutter includes a back portion extending from said rim portion toward the cutter axis in spaced relation to said tooth-profile cutting portion, said rim and back portions serving to remove metal from the blank between adjacent teeth during alternate traverses.

38. The method of Claim 8 wherein the cutting path surface of each cutter includes a back portion extending from said rim portion toward the cutter axis in spaced relation to said tooth-profile cutting portion, said rim and back portions serving to remove metal from the blank between adjacent teeth during alternate traverses.

39. The method of Claim 9 wherein the cutting path surface of each cutter includes a back portion extending from said rim portion toward the cutter axis in spaced relation to said tooth-profile cutting portion, said rim and back portions serving to remove metal from the blank between adjacent teeth during alternate traverses.

40. The method of Claim 21 wherein the cutting path surface of each cutter includes a back portion extending from said rim portion toward the cutter axis in spaced relation to said tooth-profile cutting portion, said rim and back portions serving to remove metal from the blank between adjacent teeth during alternate traverses.