Title of Invention

METHOD FOR DEFINING THE TOOTHING GEOMETRIES OF A GEAR PAIR COMPRISING TWO GEARS WITH INTERSECTING AXES

Abstract A method for establishing the gearing geometries of a gear pairing of two gears with intersecting axes comprises the following steps: predetermining a first gearing geometry of a virtual first gear; calculating the geometry of the gearing of a virtual second gear, this geometry resulting during conjugated creation from a roll-off process of a tooth (1) of the predetermined geometry of the virtual first gear, whereby the roll-off process of the tooth (1) of the virtual first gear, this roll-off process underlying the calculation, is ended in the symmetrical position thereof in the tooth gap (3) of the second virtual tooth; calculating the tooth face geometry of the teeth of the first gear, this tooth face geometry resulting during the reverse conjugated creation, while underlying a full roll-off process of a tooth of the virtual second gear with the first; establishing the final geometry of the gearing of the first gear according to the tooth face geometry of the gearing of the first gear, this tooth face geometry having been calculated in the previous step, and; establishing the final geometry of the gearing of the second gear according to the previously calculated tooth face geometry of the gearing of the virtual second gearing.
Full Text WO 2006/082038 PCT/EP2006/000874
1
Method for defining the toothing geometries
of a gear pair comprising two gears
with intersecting axes
The present invention relates to a method for defining the
toothing geometries of a gear pair comprising two gears with
intersecting axes, especially a gear pair comprising crown gear
and pinion. The present invention also relates to a gear pair
comprising crown gear and pinion with toothings of mutually
conjugate form.
In the prior art, there has been widely used, to define the
geometries of the toothings of a gear pair comprising crown gear
and pinion, a method based on numerical design, wherein the exact;
geometry of one of the partners of the gear pair is predosiqnated
and a complete cycle of rolling of the pinion over the crown gear
is simulated, whereby the geometry of the toothing of the other
partner of the gear pair can be defined. This method is known as
"conjugate generation". A good overview of the related prior art
can be found in the book entitled "Development of Gear Technology
and Theory of Gearing." by Faydor L. Litvin, NASA (National
Aeronautics and Space Administration) from 1997. A gear pair
comprising crown gear and pinion or two bevel-toothed gears and
defined in this way can then be generated by means of" known chip-
removing and/or forming manufacturing methods from the knowledge
of the exact toothing geometries of both gears. For this purpose,
however, the toothing geometry of one of the partners of the gear

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pair must always be predesignated exactly.
According to the known prior art, cylindrical geometry of the
pinion toothing, wherein the tooth flanks of the pinion teeth
have a linear shape over the axial width of the teeth, is usually
assumed in the case of a gear pair comprising crown gear and
pinion.
To define advantageous toothing geometries of a gear pair
comprising crown gear and pinion, however, there has also boon
proposed already for the pinion teeth a more complex geometry, in
which the standard cylindrical toothing geometry is abandoned in
order to reduce the undercut that would otherwise develop at. the
tooth flanks of the crown-gear teeth. For example, it has been
proposed for this purpose that the tooth-flank load-bearing
capacity of the crown-gear teeth be improved by providing the
pinion teeth with a pressure-angle and/or profile-displacement
variation over the axial width thereof. In this connection, an
undercut of the crown-gear teeth can also be further reduced
advantageously with a tip shortening that increases over the
axial width of the pinion teeth. Here also the tooth flanks of
the pinion teeth have a geometry that is always characterized by
a linear shape in axial direction.
The method of conjugate generation has already proved to be less
than optimal for defining the aforesaid more complex toothing
geometries, since the exact geometry of one of the partners of
the gear pair must already be known for this purpose in each

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case. Thus optimization and definition of the toothing geometry
frequently necessitate numerous computing runs, in order to
approximate the desired geometric characteristics by varying the
most diverse toothing parameters of a gear pair.
It is therefore the object of the present invention to provide a
method for defining the toothing geometries of a gear pair
comprising two gears with intersecting axes, especially a gear
pair comprising crown gear and pinion, wherein optimal toothing
geometries for crown gear and pinion can be defined with low
computational effort to achieve the greatest; possible tooth-flank
load-bearing capacity of the crown-gear teeth and the highest
possible operating strength of the toothing on the whole. Another
object is to provide a gear pair comprising crown gear and
pinion, wherein the toothing geometries of mutually conjugate
form are advantageously superior to the prior art in terms of
operating strength of the gear pair.
This object is achieved for a gear pair comprising two gears with
beveled toothing by a method according to independent claims 1 or
2. Although the inventive method - subject to certain secondary
conditions - is suitable for arbitrary gear pairs with
intersecting axes, the method and its advantages will be
explained hereinafter on the basis of the embodiment according to
claim 3, according to which the gear pair is one comprising crown
gear and pinion having axes that intersect at an angle of 90°.

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According to the invention, there is first predesignated a first
toothing geometry of a "virtual" pinion, which is intended to
serve as the starting basis for the subsequent steps of
calculating and defining the toothings. Starting from this
predesignated toothing geometry of the virtual pinion, there is
calculated the toothing geometry of a "virtual" crown gear that
results from a cycle of rolling of the virtual pinion in
conjugate generation. In a departure from the prior art method of
conjugate generation, only a "half" rolling cycle of the virtual
pinion is used as basis instead of a complete rolling cycle,
which introduced an undercut of the tooth flanks of the crown
gear. For this purpose, the calculation of the tooth-flank
geometry of the virtual crown gear resulting from the rolling
cycle is ended at the point at which the tooth of the virtual
pinion used as reference for calculating the tooth flanks during
the rolling cycle is in the symmetric position in the tooth gap
of the virtual crown gear. Because of the early termination of
conjugate generation, there is calculated in this way the
geometry of a tooth flank of a tooth of a virtual crown gear on
which the predesignated virtual pinion tooth can roll halfway.
Thereby the calculated tooth flank is mathematically represented
as the envelope of the virtual pinion tooth after it has rolled
halfway. The corresponding mating flank is then generated either
by reflecting the half-rolled, undercut-free tooth flank or by
assuming a further half rolling cycle in opposite running
direction. The two tooth flanks of each tooth of the virtual
crown gear then have mutually symmetric geometry - starting from

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a toothing geometry of the virtual pinion that is symmetrically
predesignated for both running directions. It is expressly
pointed out that the predesignated virtual pinion cannot roll
completely on the calculated virtual crown gear.
Starting from the toothing geometry of the virtual crown gear, in
which, by virtue of the underlying calculation procedure using
only half of a rolling cycle, the tooth flanks do not have any
undercut, the geometry of both tooth flanks of a pinion tooth
satisfying the toothing law is then calculated by means of
reverse conjugate generation. In the process, it is possible; that
excessive tapering of the (separately) calculated tooth flanks of
the pinion may occur in the region of the pinion toothing located
at the radially inner end - relative to the revo.Luti.on of.' the
pinion on the crown gear, meaning that the two tooth flanks of: a
pinion tooth intersect. In this case the region of: the two tooth
flanks located above the line of intersection - relative to the
height of the tooth - is technically meaningless; the resulting
excessive tapering is avoided with appropriate shortening of the
tooth flanks, resulting in a corresponding shortening of the
tooth tip. The pinion-toothing geometry obtained in this way is
then defined as the final geometry of the pinion toothing. In
extremely advantageous manner, a pinion tooth created in this way
therefore has a tip shortening at the radially inner end relative
to its revolution on the crown gear; in contrast to the pinion
toothings known heretofore, whose tooth flanks have axiaU.y
linear shape, the tooth flanks of the pinion tooth having the
geometry defined with the inventive method have a shape that is

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not linear over the entire axial width but instead is curved.
According to the first variant of the method according to claim
1, the toothing geometry of the virtual crown gear can then be
defined as the final geometry of the crown-gear toothing. The
pinion, whose geometry has already been finally defined, was
; indeed obtained directly from reverse conjugate generation, and
so, according to the invention, two mutually conjugate gears have
been defined.
Alternatively, as the last step of the inventive method according
to claim 2, and starting from the pinion whose geometry was
finally defined on the basis of a complete pinion rolling cycle,
there can be calculated, by means of conjugate generation, the
geometry of a crown-gear toothing; this is then defined as the
final geometry of the crown-gear toothing. This last step is used
as it were to check the previous calculation steps, since the
resulting and defined geometry of the crown-gear toothing must
correspond substantially to that of the previously calculated
virtual crown gear.
The net effect according to the invention is therefore that the
toothing geometry of a gear pair comprising crown gear and pinion
has been defined in a manner that completely avoids an undercut
of the tooth flanks of the crown gear. The tooth flanks of the
crown gear teeth, whose geometry has been optimally defined by
the method, have high load-bearing capacity. The pinion teeth -
and therefore the gear pair as a whole - have been advantageously
designed for the greatest possible operating strength, especially

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by virtue of the fact that their tooth flanks are curved in axial
direction.
Obviously the method is not limited merely to defining the
geometry of a gear pair comprising crown gear and pinion. To the
contrary, as already mentioned hereinabove, it can be applied for
arbitrary gear pairs with intersecting axes, if an undercut that:
may develop on one of the two gears is to be avoided. In
particular, the method' can therefore also be applied to gear
pairs in which the gear axes intersect at an angle different from
90°. Even in such cases the inventive method permits rapid and
optimal definition of the toothing geometries that are
advantageous for the purpose.
In a further preferred configuration of the inventive method, the
pinion-toothing geometry to be predesignated has a constant tooth
height as well as a variation of the pressure angle and/or of the
profile displacement over the axial width of the teeth. Such a
pinion-toothing geometry, which is already advantageous and
deviates from the cylindrical tooth form that has been standard
heretofore, is therefore further optimized in extremely
advantageous manner. The tip shortening developing on the pinion
is advantageously limited to the minimum extent necessary to
avoid an undercut.
Particularly advantageously, the method is applied to a gear pair
comprising crown gear and pinion with an axis intersection angle
of 90°, wherein there is no axis offset between crown gear and
pinion and the pinion has straight toothing. A gear pair

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exhibiting the aforesaid characteristics is preferably suitable
precisely as a component of a differential gear mechanism
subjected to high loads, in which case the gear pair needs
advantageously optimal characteristics in terms of its operating
strength. However, this should not be construed as a limitation,
since the present method is also applicable in particular to
crown-gear/pinion pairs with helical toothing.
Finally, the present invention also relates to a gear pa.i r
comprising crown gear and pinion with toothings of mutually
conjugate form, characterized in that the pinion teeth have, over
their width, a tip shortening that increases radially inward
relative to the revolution of the pinion on the crown gear, while
the tooth flanks of the pinion teeth are curved over their width.
This gear pair is therefore characterized by those features that
appear in the inventive method; it is distinguished from the
already known prior art in extremely advantageous manner, by the
fact that the pinion teeth have an increasing tip shortening over
their axial width as well as a curved shape of their tooth
flanks.
Starting from the curved shape of the tooth flanks of the pinion
teeth over the axial tooth width and from the tip shortening
found at the radially inner end relative to the revolution of the
pinion on the crown gear, a gear pair optimized in terms of high
operating strength is obtained by conjugate generation of the
crown-gear toothing. According to an advantageous embodiment of
the inventive gear pair, it can then be further provided that the

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tooth flanks of the crown gear bear load over their full surface
and thus do not have any undercut. The advantageous effects of
the unnecessary undercut on the tooth-flank load-bearing capacity
of the crown-gear teeth are evident.
A practical example of the inventive method and of the .inventive
gear pair comprising crown gear and pinion will be explained in
more detail hereinafter on the basis of the drawing, wherein
Figs, la, lc, ld each show a perspective view of a
schematic diagram of the geometry of a
pinion tooth that is obtained by execution
of a practical example of the inventive
method,
Figs, lb, le each show a perspective view of: a
schematic diagram of the geometry of a
crown-gear tooth gap that is obtained by
execution of the practical example of the
inventive method,
Fig. 2 shows a diagram of a pinion tooth of a
practical example of the inventive gear
pair comprising crown gear and pinion in
several cross-sectional diagrams, and

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Fig. 3 shows a diagram of a crown-gear tooth of a
practical example of the inventive gear
pair comprising crown gear and pinion in
several cross-sectional diagrams.
Fig. la shows a schematic perspective diagram of a tooth 1 of the
virtual pinion, whose geometry is predesignated in connection
with execution of a practical example of the inventive method.
The two tooth flanks 2 of illustrated virtual pinion tooth ] have
mutually symmetric form, and can have, for example, a decreasing
profile displacement and an increasing pressure ang.le over their
axial width. The limit conditions known for a pinion/crown-gcar
toothing, such as the taper limit and the interference Limit, are
advantageously met in this way.
Fig. lb) then shows a schematic perspective diagram of a tooth
gap 3 obtained between two adjacent teeth of the virtual crown
gear by simulation of a half rolling cycle of the virtual pinion
according to Fig. la. The two tooth flanks 4 bounding tooth gap 3
of the crown-gear toothing have mutually symmetric form and do
not have any undercut, since the rolling cycle on which the
calculation is based was ended when tooth 1 of the virtual, pinion
reached its symmetry position in- tooth gap 3 of the toothing of
the virtual crown gear.
Fig. lc then shows a schematic view of the tooth flanks 6 of a
pinion tooth 5, which flanks were obtained by reverse conjugate

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generation and exhibit excessive taper 7. This technically
meaningless crossover of tooth flanks 6 is avoided by shortening
tooth flanks 6 in the region of excessive taper 7.
Fig. Id shows the resulting pinion tooth 8, whose geometry .is
defined as the final geometry of the pinion toothing. Tip
shortening 10 is clearly evident. The resulting tooth flanks 9
have been shortened relative to tooth flanks 6 in the region of
excessive taper 7. Starting from this pinion tooth, the geometry
of a crown-gear toothing is then calculated by means of conjugate
generation. A tooth gap 11 of this toothing is schematicaily
illustrated in Fig. le, together with the two tooth flanks 12 of
two adjacent crown-gear teeth bounding the tooth gap. Finally,
this crown-gear toothing geometry, which substantially coincides
with that of Fig. 1b, is defined as the final geometry.
Figs. 2 and 3 then show diagrams of a pinion and crown-gear tooth
respectively of a practical example of an inventive gear pair,
whose toothing geometries were defined according to the method
explained hereinabove.
In the diagram at top center, Fig. 2 shows a side view of a tooth
8 of the pinion, formed as a spur gear, of an inventive gear
pair. Tip shortening 10 resulting from execution of the inventive
method can be clearly recognized on pinion tooth 8.

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To the left and below the side view of tooth 8 there is further
illustrated an overhead view of tooth 8 along arrow L of the side
view. Thereunder there are then presented the diagrams of
sections A-A, B-B, C-C, D-D and E-E according to the section
planes indicated in the side view. The shaded area in sections A-
A to E-E represents the part of tooth 8 located in the respective
section plane. Over their axial width, which is oriented from
left to right in the respective sections, the tooth flanks of
pinion tooth 8 are not straight but instead are curved, as is
evident in particular from the clearly recognizable (convex)
bulge in the region of the tooth tip in sections A-A and B-B.
Furthermore, it can be seen from the overhead view of the tooth
(view L) that the tooth flanks are also slightly curved in the
region of the tooth root, whose shape is evident from the upper
and lower bounding lines of view L, although in this case the
tooth flanks have the form not of a bulge but rather of a slight
(concave) indentation.
To the right and below the side view of tooth 8 there is further
illustrated a view of tooth 8 according to arrow K of the side
view. Tip shortening 10 can be clearly recognized here also.
Thereunder there are presented diagrams of sections F-F to J-J
through tooth 8, wherein the part of tooth 8 located in the
respective section plane is again highlighted by shading.
In the diagram at top center, Fig. 3 shows a side view of a tooth
13 of the crown gear of an inventive gear pair, wherein the

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toothing has a form conjugate to that of the pinion toothing
according to Fig. 2.
To the left and below the side view of tooth 13 there is further
illustrated an overhead view of tooth 13 along arrow L of the
side view. Thereunder there are then presented the diagrams of
sections A-A, B-B, C-C, D-D and E-E according to the section
planes indicated in the side view. The shaded area in sections A-
A to E-E represents the part of tooth 13 located in the
respective section plane.
To the right and below the side view of tooth 13 there is further
illustrated a view of tooth 13 according to arrow K of the side
view. Thereunder there are presented diagrams of sections F-F to
J-J through tooth 13, wherein the part of tooth 13 located in the
respective section plane is again highlighted by shading. In
extremely advantageous manner, the tooth flanks of crown-gear
tooth 13 do not have any undercut.

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Claims
1. A method for defining the toothing geometries of a gear pair-
comprising two gears with intersecting axes, wherein the
method includes the following steps:
predesignating a first toothing geometry of a virtual
first gear;
calculating the toothing geometry of a virtual second
gear of the gear pair that results in conjugate
generation from a cycle of rolling of a tooth (1)
having the predesignated geometry of the virtual first
gear, wherein the cycle of rolling of the tooth (1) of
the virtual first gear on which the calculation is
based is ended in its symmetric position in the tooth
gap (3) of the second virtual gear;
calculating the tooth-flank geometry of the teeth of
the first gear that result from reverse conjugate
generation on the basis of a complete cycle of rolling
of a tooth of the virtual second gear with the first
gear, wherein any resulting excessive taper (7) is
avoided by reduction of the tooth flanks in this
region;
defining the final geometry of the toothing of the
first gear in a manner corresponding to the tooth-flank
geometry of the toothing of the first gear calculated
in the previous step;
defining the final geometry of the toothing of the
second gear in a manner corresponding to the previously
calculated tooth-flank geometry of the toothing of the
virtual second gear.

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2. A method for defining the toothing geometries of a gear pair
comprising two gears with intersecting axes, wherein the
method includes the following steps:
predesignating a first toothing geometry of a virtual
first gear;
calculating the toothing geometry of a virtual second
gear of the gear pair that results in conjugate
generation from a cycle of rolling of a tooth (1)
having the predesignated geometry of the virtual first
gear, wherein the cycle of rolling of the tooth (1) of
the virtual first gear on which the calculation is
based is ended in its symmetric position in the tooth
gap (3) of the second virtual gea re;
calculating the tooth-flank geometry of the teeth of
the first gear that result from reverse conjugate
generation on the basis of a complete cycle of roil ing
of a tooth of the virtual second gear with the fi rst
gear, wherein any resulting excessive taper (7) is
avoided by reduction of the tooth flanks in this
region;
defining the final geometry of the toothing of the
first gear in a manner corresponding to the tooth-flank
geometry of the toothing of the first gear calculated
in the previous step;
calculating the tooth-flank geometry of the teeth of
the second gear that results by conjugate generation
from a complete cycle of rolling of the first gear
having the defined geometry;
defining the final geometry of. the toothing of the

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second gear in a manner corresponding to the tooth-
flank geometry of the toothing of the second gear
calculated in the previous step.
3. A method according to claim 1 or claim 2, characterized in
that the first gear is a pinion and the second gear is a
crown gear, wherein the axes intersect at an angle of 90°.
4. A method according to claim 3, characterized :i n that; the
predesignated toothing geometry of the virtual pinion has a
constant tooth height as well as a variation of the pressure
angle and/or of the profile displacement over the axial,
width of the teeth.
5. A gear pair comprising crown gear and pinion, wherein t;he
toothings of crown gear and pinion have mutually conjugate
form and the tooth flanks (9) of the pinion teeth are curved
over their width, characterized in that the pinion teeth (8)
have, over their width, a tip shortening (10) that increases
radially inward relative to the revolution of the pinion on
the crown gear.
6. A gear pair comprising crown gear and pinion according to
claim 4, characterized in that the tooth flanks of the crown
gear (12) are fully load-bearing, and an undercut is
completely avoided.

A method for establishing the gearing geometries of a gear pairing of two gears with intersecting axes comprises the
following steps: predetermining a first gearing geometry of a virtual first gear; calculating the geometry of the gearing of a virtual
second gear, this geometry resulting during conjugated creation from a roll-off process of a tooth (1) of the predetermined geometry
of the virtual first gear, whereby the roll-off process of the tooth (1) of the virtual first gear, this roll-off process underlying the
calculation, is ended in the symmetrical position thereof in the tooth gap (3) of the second virtual tooth; calculating the tooth face
geometry of the teeth of the first gear, this tooth face geometry resulting during the reverse conjugated creation, while underlying
a full roll-off process of a tooth of the virtual second gear with the first; establishing the final geometry of the gearing of the first
gear according to the tooth face geometry of the gearing of the first gear, this tooth face geometry having been calculated in the
previous step, and; establishing the final geometry of the gearing of the second gear according to the previously calculated tooth face
geometry of the gearing of the virtual second gearing.

Documents:

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Patent Number 269296
Indian Patent Application Number 2250/KOLNP/2007
PG Journal Number 42/2015
Publication Date 16-Oct-2015
Grant Date 14-Oct-2015
Date of Filing 19-Jun-2007
Name of Patentee SONA BLW PRAZISIONSSCHMIEDE GMBH
Applicant Address FRANKFURTER RING 227, 80807 MUNCHEN
Inventors:
# Inventor's Name Inventor's Address
1 GUTMANN, PETER GRIMMEISENSTR. 11, 91927 MUNCHEN
PCT International Classification Number F16H 55/08
PCT International Application Number PCT/EP06/000874
PCT International Filing date 2006-02-01
PCT Conventions:
# PCT Application Number Date of Convention Priority Country
1 102005005169.3 2005-02-02 Germany