Note: Descriptions are shown in the official language in which they were submitted.
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.
-10-
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
1 1 -
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~
-12-
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
-13-
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
-15-
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
-16-
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
-18-
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~
-19-
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
-20-
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
,
-21-
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~
-24-
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~
-25-
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
-26-
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
-27-
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~
28-
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
,
-31-
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
. .
32-
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
-33-
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
-35-
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
-36-
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 ~
-38-
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
-41-
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 ~
-42-
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 ~` ;
-43-
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
FIGURE 29
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
-45-
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~
-46-
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
-A7-
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.
, .
-50-
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
-51-
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
,
-52-
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
-
--55--
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
-56-
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 ~ !
-57-
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~.
.
