ROTARY DRESSING ROLLER AND METHOD AND APPARATUS FOR DRESSING CUP-SHAPED GRINDING WHEELS

DRESSEUR ROTATIF ET METHODE ET APPAREIL DESTINES A DRESSER LES MEULES CUPULAIRES

English Abstract

37
Abstract of the Disclosure
A rotary dressing roller of the generating type and a
method and apparatus for dressing a substantially cup-
shaped grinding wheel for longitudinally curved tooth
gears. The rotary dressing roller (40) is provided with a
base portion (42) which is rotatably mounted to the
dressing motor (38) of a dressing apparatus (15). The
dressing apparatus (15) is used to dress a grinding wheel
(50). At the other end of base portion (42) there is
provided a working portion (44) having an outer concave
surface (45), an inner convex surface (46) and an outer
cutting rim portion (47). The outer concave surface (45)
and inner convex surface (46) of working portion (44) are
shaped so as to allow dressing of the inside working
surface (52) and outside working surface (54) of grinding
wheel (50) by rotation of the rotary dressing roller (40)
in a single direction.

Claims

22
The embodiments of the invention in which an
exclusive property or privilege is claimed are defined
as follows:
1. A rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears
comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and said inner convex surface;
wherein said outer cutting rim portion has an inner
cutting surface and an outer cutting surface, said inner
cutting surface having an effective radius of curvature at
a first point of contact with the grinding wheel at a
maximum working depth of the grinding wheel less than a
normal radius of curvature of the inside working surface of
the grinding wheel at the first point of contact with said
dressing roller at the maximum working depth, and said
outer cutting surface having an effective radius of
curvature at a second point of contact with the grinding
wheel at the working depth of the grinding wheel greater
than a normal radius of curvature of the outside working
surface of the grinding wheel at the second point of
contact with said dressing roller at the maximum working
depth.
2. A rotary dressing roller according to claim 1
wherein said effective radius of curvature of said inner
cutting surface is no greater than about 80 per cent of the
normal radius of curvature of the inside working surface of
the grinding wheel at the first point of contact with said
dressing roller at the maximum working depth.

23
3. A rotary dressing roller according to claim 2
wherein said effective radius of curvature of said outer
cutting surface is equal to or greater than 1.25 times the
normal radius of curvature of the outside working surface of the grinding wheel at the
second point of contact with said dressing roller at the
maximum working depth.
4. A rotary dressing roller for dressing
inside and outside working surfaces of a substantially cup-
shaped grinding wheel for longitudinally curved tooth gears
comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and inner convex surface;
wherein said outer cutting rim portion has an inner
cutting surface and an outer cutting surface, said outer
cutting rim having a configuration in accordance with the
following relationship:
Po - Rwo ? Rwi - Pi
wherein: Pi is an effective radius of curvature of
said inner cutting surface of said rotary dressing roller
at a first point of contact with the grinding wheel at a
maximum working depth of the grinding wheel; Po is an
effective radius of curvature of said outer cutting surface
of said rotary dressing roller at a second point of contact
with the grinding wheel at the maximum working depth of the
grinding wheel; Rwi is a normal radius of curvature of the
inside working surface of the grinding wheel at the first
point of contact with said dressing roller at the maximum

24
working depth; and, Rwo is a normal radius of curvature of
the outside working surface of the grinding wheel at the
second point of contact with said dressing roller at the
maximum working depth.
5. A rotary dressing roller according to claim 4
wherein said outer concave surface of said dressing roller
together with the outside working surface of the grinding
wheel define a first clearance angle and said inner convex
surface of said dressing roller together with the inside
working surface of the grinding wheel define a second
clearance angle, said first and second clearance angles
being in a range of 2° to 6°.
6. A rotary dressing roller according to claim 4
wherein said outer cutting rim portion has a substantially
circular cross sectional profile at least in the vicinity
of the respective points of contact between said dressing
roller and the grinding wheel at the maximum working depth.
7. A rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears, the
inside and outside working surfaces of the grinding wheel
being inclined at respective pressure angles with respect
to an axis of rotation of the grinding wheel comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and inner convex surface;

wherein said dressing roller has a diameter Dx which
meets the following relationship:
Dx > 2 Rwa x TAN (?i)
wherein: Rwa is a radius of the grinding wheel at a
maximum ? working depth on the inside working surface of
the grinding wheel; and TAN (?i) is a tangent function of
the pressure angle of the inside working surface of the
grinding wheel at the maximum working depth.
8. A rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears, the
inside and outside working surfaces of the grinding wheel
being inclined at respective pressure angles with respect
to an axis of rotation of the grinding wheel comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and inner convex surface;
wherein said inner convex surface is inclined to a
line perpendicular to said axis of rotation of said dressing
roller through angle .gamma. i in accordance with the following
relationship:
.gamma.i = ?i + .beta. - .alpha.i
wherein: ?i is the pressure angle of the inside
working surface of the grinding wheel; .beta. is an angle of
inclination of said axis of rotation of said dressing
roller with respect to a line perpendicular to the axis of
rotation of the grinding wheel; and .alpha.i; is a clearance angle

26
between the inside working surface of said grinding wheel
and said inner convex surface of said dressing roller.
9. A rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears, the
inside and outside working surfaces of the grinding wheel
being inclined at respective pressure angles with respect
to an axis of rotation of the grinding wheel comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and inner convex surface;
wherein said outer concave surface is inclined to a
line perpendicular to said axis of rotation of said
dressing roller through angle .gamma.o in accordance with the
following relationship:
.gamma.o = .beta. + .alpha.o - ?o
wherein: .beta. is an angle of inclination of said axis of
rotation of said dressing roller with respect to a line
perpendicular to the axis of rotation of the grinding
wheel; .alpha. is a clearance angle between the outside working
surface of the grinding wheel and said outer concave
surface of said dressing roller; and ?o is the pressure
angle of the outside working surface of the grinding wheel.
10. A rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears, the

27
inside and outside working surfaces of the grinding wheel
being inclined at respective pressure angles with respect
to an axis of rotation of the grinding wheel comprising:
a base portion for mounting said rotary dressing
roller for rotation about an axis of said dressing roller;
and
a working portion formed at one end of said base
portion having an outer concave surface, an inner convex
surface and an outer cutting rim portion connecting said
outer concave surface and inner convex surface;
wherein a difference angle .DELTA..gamma. is defined between
respective inclination angles of said inner convex surface
and said outer concave surface with respect to a line
perpendicular to said axis of rotation of said dressing
roller in accordance with the following relationship:
.DELTA..gamma. = (?; + ?o) - 2 .alpha.
wherein: ?; and ?o are the respective pressure angles
of the inside and outside working surfaces of the grinding
wheel, and .alpha. is the average of two clearance angles, one
clearance angle being defined between the inside working
surface of the grinding wheel and said inner convex surface
of said dressing roller and the other of said two clearance
angles being defined between the outside working surface of
the grinding wheel and said outer concave surface of said
dressing roller.
11. A rotary dressing roller according to claim 1
wherein said outer cutting rim portion has a substantially
circular cross sectional profile at least in the vicinity
of the respective points of contact between said dressing
roller and the grinding wheel at the maximum working depth.

28
12. A rotary dressing roller according to claim 1
wherein said rotary dressing roller further comprises a
neck portion between said base portion and said working
portion for providing clearance between said dressing
roller base portion and the grinding wheel during use.
13. A rotary dressing roller according to claim 11
wherein said cutting rim portion comprises a layer of
diamond grit.
14. A rotary dressing roller according to claim 1
wherein said outer concave surface of said dressing roller
together with the outside working surface of the grinding
wheel define a first clearance angle and said inner convex
surface of said dressing roller together with the inside
workings surface of the grinding wheel define a second
clearance angle, said first and second clearance angles
being in a range of 2° to 6°.
15. A rotary dressing roller according to claim 1
wherein the grinding wheel is a straddle grinding wheel
having concentric inner and outer rims, the inner rim
including the outside working surface, the outer rim
including the inside working surface.
16. A rotary dressing roller according to claim 10
wherein said dressing roller has a diameter Dx which meets
the following relationship:
Dx > 2 Rwa x TAN (?i)

29
wherein: Rwa is a radius of the grinding wheel at the
maximum working depth on the inside working surface of the
grinding wheel and TAN (?i) is a tangent function of the
pressure angle of the inside working surface of the
grinding wheel at the maximum working depth.
17. A rotary dressing roller according to claim 1
wherein the inside and outside working surfaces of the
grinding wheel are inclined at respective pressure angles
with respect to an axis of rotation of the grinding wheel
and a difference angle .DELTA..gamma. is defined between respective
inclination angles of said inner convex surface and said
outer concave surface with respect to a line perpendicular
to said axis of rotation of said dressing roller in
accordance with the following relationship:
.DELTA..gamma. = (?i + ?o) - 2 .alpha.
wherein: ?i and ?o are the respective pressure angles
of the inside and outside working surfaces of the grinding
wheel; and ~ is the average of two clearance angles, one
clearance angle being defined between the inside working
surface of said grinding wheel and said inner convex
surface of said dressing roller and the other of said two
clearance angles being defined between the outside working
surface of the grinding wheel and said outer concave
surface of said dressing roller during use.
18. A rotary dressing roller according to claim 17
wherein said dressing roller has a diameter Dx which meets
the following relationship:
Dx > 2 Rwa x TAN (oi)

wherein: Rwa is a radius of the grinding wheel at the
maximum working depth on the inside working surface of the
grinding wheel and TAN (?i) is a tangent function of the
pressure angle of the inside surface of the grinding wheel
at the maximum working depth.
19. A dressing apparatus for dressing cup shaped
grinding wheels comprising:
a rotary dressing roller for dressing inside and
outside working surfaces of a substantially cup-shaped
grinding wheel, the inside and outside working surfaces of
the grinding wheel being inclined at respective pressure
angles with respect to an axis of rotation of the grinding
wheel; said dressing roller having an axis of rotation, an
inner cutting surface, an outer cutting surface and a
circular contour rim connecting said inner and outer
cutting surfaces;
means for rotating the grinding wheel about its axis;
means for rotating said dressing roller about its
axis;
means for moving said dressing roller with respect to
the grinding wheel to effect dressing of the working
surfaces of the grinding wheel; and
said dressing roller axis being angularly oriented
through angle .beta. with respect to a plane perpendicular to
the axis of rotation of the grinding wheel in a range
between .beta.min and .beta.max according to the following
relationships:
.beta.max = Arcsin <IMG> + ?o
and
.beta.min = Arcsin <IMG> - ?i

31
wherein: Rwi is a normal radius of curvature of the
inside working surface of the grinding wheel at a working
depth of the grinding wheel; Rwo is a normal radius of
curvature of the outside working surface of the grinding
wheel at a working depth of the grinding wheel; R is a
radius of said circular contour rim of said dressing
roller; ?i and ?o are the respective pressure angles of the
inside and outside working surfaces of the grinding wheel;
and Dx is the diameter of said dressing roller.
20. A dressing apparatus according to claim 19
wherein the diameter Dx of said dressing roller is defined
such that the angle .beta.max is greater than .beta.min.
21. A dressing apparatus according to claim 20
wherein the diameter Dx of said dressing roller which meets
the following relationship:
Dx > 2 Rwa x TAN (?i)
wherein: Rwa is a radius of the grinding wheel at the
maximum working depth on the inside working surface of the
grinding wheel and TAN (?i) is the tangent function of the
pressure angle of the inside working surface of the
grinding wheel at the maximum working depth.
22. The dressing apparatus of claim 20 wherein said
means for moving said dressing roller with respect to the
grinding wheel to effect dressing of the working surfaces
of the grinding wheel provides for a range of relative
travel T of said dressing roller with respect to the
grinding wheel perpendicular to the axis of the grinding
wheel.

32
23. The dressing apparatus of claim 22 wherein said
travel T required to move said dressing roller with respect
to the grinding wheel for generating both the inside and
outside working surfaces of the grinding wheel in a
direction perpendicular to the grinding wheel axis is
limited according to the following relationship:
T < WT + DX X COS (?)
wherein: WT is the thickness of the grinding wheel at
the working depth; Dx is a diameter of said dressing
roller; and COS (?) is a cosine function of an average
pressure angle of the respective pressure angles of the
inside and outside working surfaces of the grinding wheel.
24. The dressing apparatus of claim 22 wherein said
means for moving said dressing roller with respect to the
grinding wheel to effect dressing of the working surfaces
of the grinding wheel provides for a range of relative
travel of said dressing roller with respect to the grinding
wheel in a direction along the axis of the grinding wheel.
25. A method of dressing inside and outside working
surfaces of a substantially cup-shaped grinding wheel for
longitudinally curved tooth gears with a rotary dressing
roller, the inside and outside working surfaces of the
grinding wheel being inclined at respective pressure angles
with respect to an axis of rotation of the grinding wheel,
said rotary dressing roller having a base portion for
mounting the rotary dressing roller for rotation about an
axis of said dressing roller and a working portion formed
at one end of said base portion including an outer concave
surface, an inner convex surface and an outer cutting rim
portion connecting said outer concave surface and said

33
inner convex surface; wherein said outer cutting rim
portion has an inner cutting surface and an outer cutting
surface, said inner cutting surface having an effective
radius of curvature at a working depth of the grinding
wheel less than a normal radius of curvature of the inside
working surface of the grinding wheel at a point of contact
with said dressing roller at the working depth, and said
outer cutting surface having an effective radius of
curvature at the working depth of the grinding wheel
greater than a normal radius of curvature of the outside
working surface of the grinding wheel at a point of contact
with said dressing roller at the working depth, comprising
the steps of:
rotating the grinding wheel and said dressing roller
about their respective axes in the same respective
directions while dressing both the inside and outside
working surfaces of the grinding wheel, and
moving said dressing roller with respect to the
grinding wheel for generating both the inside and outside
working surfaces of the grinding wheel.
26. A method of dressing a cup-shaped grinding wheel
according to claim 25 including the step of:
orienting said axis of rotation of said dressing
roller with respect to a line perpendicular to the axis of
rotation of the grinding wheel through a fixed angle .beta.
while dressing at least one of the inside and outside
working surfaces of the grinding wheel.
27. A method of dressing a cup-shaped dressing wheel
of claim 26 wherein said rotary dressing roller remains
substantially in contact with the grinding wheel and said
axis of rotation of said dressing roller remains oriented
with respect to the line perpendicular to the axis of

34
rotation of the grinding wheel through the fixed angle .beta.
while dressing both the inside and outside working surfaces
of the grinding wheel.
28. A method of dressing a cup-shaped grinding wheel
according to claim 27 further comprising the step of:
initially contacting the grinding wheel with said
rotary dressing roller below the maximum cutting depth of
one of the working surfaces of the grinding wheel.
29. A method of dressing a cup-shaped grinding wheel
according to claim 25 wherein travel T required to move
said dressing roller with respect to the grinding wheel for
generating both the inside and outside working surfaces of
the grinding wheel in a direction perpendicular to the
grinding wheel axis is limited according to the following
relationship.
T < WT + DX X COS (?)
wherein: WT is the thickness of the grinding wheel at
the maximum working depth; Dx is the diameter of said
dressing roller; and COS (?) is a cosine function of an
average pressure angle of the respective pressure angles of
the inside and outside working surfaces of the grinding
wheel.
30. A method of dressing a cup-shaped grinding wheel
according to claim 29 wherein travel TAB required to move
said dressing roller from a point of contact A with one of
the working surfaces of the grinding wheel at the working
depth to a point of contact B with the other of the working
surfaces of the grinding wheel at the working depth in a

direction perpendicular to the grinding wheel axis is
limited according to the following relationship:
TAB < WT + 2 R
wherein: R is the radius of said outer cutting rim
portion of said dressing roller.
31. A method of dressing inside and outside working
surfaces of a substantially cup-shaped grinding wheel
according to claim 25 wherein said dressing roller axis is
angularly oriented through angle .beta. with respect to a plane
perpendicular to the axis of rotation of the grinding wheel
in a range between .beta.min and .beta.max according to the following
relationships.
.beta.max = Arcsin <IMG> + ?o
and
.beta.min = Arcsin <IMG> - ?i
wherein: Rwi is a normal radius of curvature of the
inside working surface of the grinding wheel at a working
depth of the grinding wheel; Rwo is a normal radius of
curvature of the outside working surface of the grinding
wheel at a working depth of the grinding wheel; R is a
radius of said circular contour rim of said dressing
roller; ?i and ?o are the respective pressure angles of the
inside and outside working surfaces of the grinding wheel;
and Dx is the diameter of said dressing roller.

36
32. A method of dressing inside and outside working
surfaces of a substantially cup-shaped grinding wheel
according to claim 31 wherein the diameter Dx of said
dressing roller is defined such that the angle .beta.max is
greater than .beta.min.

Description

~3~0
ROTARY DRESSING ROLLER AND METHOD AND APPARATUS
FOR DRESSING CUP-SHAPED GRINDING WHEELS
The present invention is directed to a rotary
5 generating dressing roller and a method and apparatus for
dressing a cup-shaped grinding wheel for producing
longitudinally curved tooth gears.
In the manufacture of curved tooth gears a cup-shaped
10 grinding wheel is used. When the grinding wheel becomes
worn to the point wherein it no longer is suitable for
producing precise tooth surfaces, the grinding wheel is
dressed (renewed) to its original cutting condition. The
dressing operation may be accomplished by providing a
15 diamond cutting point which removes the worn portion of the
grinding wheel. However, a single cutting point is subject
to wear relatively quickly thereby affecting the sharpness
and precision of the grinding wheel.
Rotary dressing rollers may also be used to dress
grinding wheels. Rotary dressing rollers may be of the
"form" or "generating" type. In a form-type rotary
dressing roller, the outer configuration of the rotary
dressing roller is shaped so as to conform to a working
25 surface of the grinding wheel to dress the entire profile
of one side of the grinding wheel. This type of dressing
roller is more expensive than a generating-type rotary
dressing roller and is limited in use to a single grinding
wheel profile configuration. Additionally, this type of
A 30 roller is quite sensitive to localized ~wear at any point
along its dressing surface.
A rotary dressing roller of the generating type
contacts the grinding wheel at a single point and can be
35 used to generate any desired configuration. For example, a
known generating-type dressing roller is oriented with its
axis aligned with a working surface on one side (e.g.,

~33~
inside) of the cup-shaped grinding wheel and is moved in a
direction parallel to the working surface for generating
the required profile. For dressing the working surface on
the other side (e.g., outside) of the cup-shaped grinding
5 wheel, the dressing roller is pivoted to a new orientation
with its axis aligned with the other working surface and is
rotated in the opposite direction relative to the direction
of rotation of the grinding wheel. It is generally
understood in the art that substantially the same relative
10 motion between the cutting surface of the dressing roller
and working surfaces of the grinding wheel is required to
dress the inside and outside working surfaces to
substantially equal sharpness. This requirement is met
using the known generating-type dressing roller by rotating
15 the roller in opposite directions relative to the
rotational direction of the grinding wheel while dressing
the inside and outside working surfaces of the grinding
wheel, respectively. However, rotating the roller in one
direction while dressing one of the working surfaces causes
20 the cutting surface, usually in the form of bonded diamond
grit, on the rotary dressing roller to form a wear pattern.
When the roller is rotated in the opposite direction to
dress the other side, the initial wear pattern tends to
cause the abrasive grit to break free from the roller
25 thereby reducing the useful life of the roller and the
overall quality of the dressing operation. Known
generating-type dressing rollers also tend to wear rapidly
because most of the dressing stock of the grinding wheel is
removed by a very limited zone of contact on the dressing
30 roller.
Dressing mechanisms for rotating and transporting the
known generating-type dressing roller of the prior art are
undesirably complex and difficult to control during use to
35 obtain the required accuracy. For example, the mechanisms
are required to accurately pivot the axis of the dressing

13133~0
roller between two angular orientations for separately
dressing working surfaces on opposite sides of the cup-
shaped grinding wheel. A third orientation may be required
for dressing the end surface of the grinding wheel. The
5 dressing mechanisms are also req~ired to provide an
extended range of travel for the dressing roller in a
direction substantially perpendicular to the grinding wheel
axis. This extended range of travel is required to
position diametrically opposite points on the circumference
10 of the roller which dress the inside and outside working
surfaces of the grinding wheel respectively.
Further, the known generating-type dressing roller is
not appropriate for dressing "straddle-type" cup-shaped
15 grinding wheels. In the straddle-type grinding wheel, the
inside and outside wor~ing surfaces are mounted facing one
another on separate concentric annular supports. It would
not be possible to dress either working surface of the
grinding wheel with the Xnown generating-type roller
20 because of interference between the roller and the working
surface opposite the wor~ing surface of the grinding wheel
to be dressed.
The present invention is directed to a rotary dressing
25 roller of the generating type and to a method and apparatus
for dressing a substantially cup-shaped grinding wheel for
longitudinally curved tooth gears which minimize or
overcome the problems of the prior art.
SUMMARY OF THE INVENTION
In one aspect of the present invention there is
provided a rotary generating dressing roller for dressing a
substantially cup-shaped grinding wheel for longitudinally
35 curved tooth gears. The roller has a base portion for
mounting the rotary dressing roller to a rotatable drive

131~3~0
spindle. A working portion is provided at the other end of
the base portion having an outer concave surface, an inner
convex surface and an outer cutting rim portion connecting
the outer concave surface and inner convex surface.
In another aspect of the present invention there is
provided a method of dressing a substantially cup-shaped
grinding wheel for longitudinally curved tooth gears having
inside and outside working surfaces. The substantially
10 cup-shaped grinding wheel is dressed by rotating the rotary
generating dressing roller in the same direction for
dressing both the inside and outside working surfaces of
the grinding wheel.
In yet another aspect of the present invention there
is provided a dressing apparatus for dressing a
substantially cup-shaped grinding wheel having 2 rotary
generating dressing roller and means for moving the rotary
generating dressing roller with respect to the
20 substantially cup-shaped grinding wheel. The rotary
generating dressing roller has a configuration which allows
dressing both working surfaces of the grinding wheel by
rotating the dressing roller in a single direction.
In yet still another aspect of the present invention
there is provided an apparatus for making longitudinally
curved tooth gears having a rotatable substantially cup-
shaped grinding wheel for making curved tooth gears, means
for rotating the grinding wheel about its axis of rotation
30 and means for dressing the grinding wheel. The means for
dressing the grinding wheel includes a rotary generating
dressing roller. The roller has a base portion and a
working portion having an outer concave surface and an
inner convex surface and an outer cutting rim portion
35 connecting the concave surface and the inner convex
surface. The outer cutting rim portion has an inner

33~0
s
cutting surface and an outer cutting surface. Means are
provided for moving the rotary dressing roller in a manner
so as to dress the cup-shaped grinding wheel.
DETAILED DESCRIPTION OF THE DRAWINGS
Figure 1 is a fragmentary perspective view of an
apparatus having a grinding wheel for making longitudinally
curved tooth gears and an apparatus for dressing the
10 grinding wheel;
Figure 2 is a perspective view of a rotary generating
dressing roller made in accordance with the present
invention;
Figure 3 is a cross sectional view of the rotary
generating dressing roller of figure 2 taken along lines 3-
3;
Figure 4 is an enlarged side elevational view of the
rotary dressing roller of figure 3 contacting the inside
working surface of the grinding wheel for curved tooth
gears;
Figure 5 is an enlarged side elevational view of the
rotary dressing roller of figure 3 contacting the outside
working surface of the grinding wheel for curved tooth
gearS;
Figure 6 a and b are enlarged fragmentary cross
sectional views of figures 4 and 5, respectively,
illustrating the outer peripheral cutting surface of the
dressing roller as it contacts the grinding wheel;
Figure 7 is a diagrammatical representation of the
geometric properties of a dressing roller made in

3~0
accordance with the present invention as it contacts the
inside surface of the grinding wheel as taken along a plane
passing through the axis of rotation of the rotary grinding
wheel and rotary dressing roller;
Figure 8 is a diagrammatical representation of the
geometric properties of a dressing roller made in
accordance with the precent invention as it contacts the
outside surface of the grinding wheel as taken along a
10 plane passing through the a~is of rotation of the rotary
grinding wheel and rotary dressing roller;
Figure 9 is an enlaxged side elevational view of the
rotary dressing roller of figure 3 contacting a straddle
lS type grinding wheel;
Figure 10 is a side elevational view of a modified
form of a dressing roller made in accordance with the
present invention;
Figure 11 is a side elevational view of another
modified embodiment of a dressing roller made in accordance
with the present invention; and
Figure 12 is yet another modified form of a dressing
roller made in accordance with the present invention.
DETAILED DESCRIPTION OF THE INVENTION
Referring to figure 1 there is illustrated a
fragmentary perspective view of an apparatus 10 used to
make longitudinally curved tooth gears. The apparatus 10
comprises a support column 12 which supports a spindle
housing 14 in which is mounted a substantially cup-shaped
35 grinding wheel 50. Apparatus 10 is provided wit~ an
appropriate drive means (~e~ shown) to rotate grinding

3~0
wheel 50 about its axis. The particular means used to
rotate wheel 50 and move spindle housing 14 may be of any
conventional design presently used in the art. A dressing
apparatus 15 pivotable about the grinding wheel axis is
5 mounted to supporting column 12 within guideway 17 by a
pair of mounting brackets 18. Secured to brackets 18 is a
support frame 24 having a pair of guide rods 22 and a
rotatable ball drive screw 26 secured thereto. Mounted to
frame 24 is a ball screw drive motor 28 which engages a
10 gear box 30 which turns ball drive screw 26. A support
bracket 20 is slidably mounted to guide rods 22 and is
caused to slide along guide rods 22 by a drive nut (not
shown) secured to bracket 20 which is threadedly engaged to
ball drive screw 26. Mounted to support bracket 20 is a
"Y" axis drive motor 21 which moves slide 23 along the Y
axis as illustrated by the arrow labelled "Y". Mounted to
the top of slide 23 is an "X" axis drive motor 25 which
moves slide 27 in the X axis direction as illustrated by
arrow labelled by "X". A mounting plate 29 is securely
20 mounted to the top of slide 27. A dressing motor 38 is
mounted to plate 29 and has a dressing roller 40 rotatably
mounted thereto. Bellows 31 are placed around guide rods
22 to protect them from the dust and grit in the
environment.
The dressing apparatus as illustrated in figure 1 is
in the fully retracted position. When it is desired to
dress grinding wheel 50 the motor 28 is activated to turn
ball screw 26 in the appropriate direction to move the
30 support bracket 20 supporting Y axis and X axis drive
motors 21, 25 toward grinding wheel 50. The motor 28 is
used to obtain the approximate position of dressing roller
for the dressing operation. The X axis and Y axis drive
motors 21, 25 are operated so as to move dressing roller 40
35 in the appropriate path for dressing grinding wheel 50.
Typically, the operation of drive motors 21, 25 are

13133~0
controlled by a microprocessor or other control means as is
commonly done in the prior art. By appropriately
controlling the operation of motors 21, 25 any desired
configuration may be generated by dressing roller 40. In
5 the particular embodiment illustrated the means for
determining the positions of the slides 23, 27 is not
illustrated as this may take any conventional form as is
presently used as is well known to those skilled in the
art.
Referring to figures 2 and 3 the rotary dressing
roller 40 comprises a base port~on 42 which is mounted to
A the rotatable drive spindle (~shown) on dressing motor
38 in any conventional manner. In the particular
15 embodiment illustrated, dressing roller 40 is provided with
an axially centrally located opening 41 for receiving a
retaining bolt ~ shown) which is secured to the drive
spindle of dressing motor 38. The rotary dressing roller
40 has a working portion 44 disposed at the other end of
20 the base portion 42. In the particular embodiment
illustrated, roller 40 is also provided with a necked
portion 43 connecting base portion 42 and working portion
44 for providing clearance between the roller 40 and
grinding wheel 50 during the dressing operation. The
25 rotary dressing roller 40 is rotatable about central axis
XD. Although the illustrated dressing roller 40 is shown
with integral base portion 42 and necked portion 43,
working portion 44 may be supported for rotation by other
types of base portions which may be arranged to mount
30 working portion 44 to a drive spindle and provide
appropriate clearance for not interfering with the grinding
wheel during use. In addition, the illustrated base and
neck portions of dressing roller 40 could be incorporated
into an arbor for supporting the dressing roller, whereas
35 the base portion of the dressing roller may comprise only a
mounting surface for the arbor. The working portion 44

1;~133~0
9 ~ 6Q(~ JC~ f~C-~t,
comprises an,outer concave surface ~tand a connecting
outer cutting rim portion 47. The outer cutting rim
portion 47 comprises a relatively thin layer of diamond
grit or other abrasive materials normally used for dressing
5 grinding wheels. The outer cutting rim portion 47 has an
inner cutting surface 48 and an outer cutting surface 49.
Referring to figure 4, there is illustrated a cross
sectional view of the rotary dressing roller 40 in its
10 position when it first contacts inside working surface 52
of grinding wheel 50. The axis XD of rolle~rL40 is
positioned at an angle ~ with respect to a ~*ee 51 which
is perpendicular to the axis of rotation XW of the grinding
wheel 50 ~see figure 7).
Referring to figure 5, there is illustrated the rotary
dressing roller 40 as it initially contacts the outside
working surface 54 of ~rinding wheel 50. Rotary dressing
roller 40 is disposed at an angle ~ with respect to a plane
20 perpendicular to the axis XW of grinding wheel 50 (see
figure 8).
Referring to figure 6a there is illustrated an
enlarsed fragmentary cross sectional view of the outer
25 cutting rim portion 47 of figure 4 after it has initially
contacted the inside working surface 52 at its working
depth. Outer cutting rim portion 47 has a substantially
circular outer configuration having a radius R which
provides a surface of revolution for dressing grinding
30 wheel 50. In the particular embodiment illustrated, radius
R is about .254 cm (.1 inch), however, the radius may be
any desired value and is limited only by the structural
integrity required of the roller 40. The outer rim portion
47~preferably follows this circular contour at least to the
35 c ~ normal point CN. For the purposes of this
invention the contact normal point CN is that point wherein

~33~
the tangent of the surface of the dressing roller coincides
with the tangent of the surface of grinding wheel 50. The
inner cutting surface 4~ of rim portion 47 extends at least
to point 58 prior to where the rotary dressing roller 40
5 initially contacts the grinding wheel 50 as the rotary
dressing roller 40 is brought along the grinding wheel 50.
.
When the rotary dressing roller 40 is first brought
into contact with the grinding wheel 50, the rotary
10 dressing roller 40 is positioned so that point of contact
of the roller 40 with the grinding wheel 50 occurs at a
oint A (see figure 6a). Point A being defined as the
point of maximum working depth of the inside working
surface 52 of grinding wheel 50. Therefore, the roller 40
15 first contacts grinding wheel 50 at a point which is just
beyond the region of the working surface used to make
longitudinal curved tooth gears. After the roller 40 has
been brought to its working depth, it is moved outward
toward the tip of the grinding wheel 50, as indicated by
20 arrow 55, dressing the inside working surface as it moves
along. The dash line 57 indicating the shape of inside
working surface 52 after the dressing operation. As the
rotary dressing roller 40 is brought along inside working
surface 52 the initial contact of inner cutting surface 48
25 with the inside working surface will occur at a zone Z, as
indicated by circle shown in dash lines, prior to the
contact normal point CN. Zone Z will take the brunt of
initial contact with grinding wheel 50 thereby causing the
greatest wear to occur prior to the contact normal point CN
30 which provides the final configuration to the grinding
wheel 50. Zone Z not only provides for removing a
substantial portion of the abrasive material on the
grinding wheel 50 which is ultimately removed by the
dressing roller 40 but also helps preserve the diamond grit
35 in the vicinity of the contact normal point CN. The inner
cutting surface 48 in zone Z tends to wear back into the

1~33~0
roller at an angle extending from the contact normal point
and is a function of the depth of feed of the dressing
roller 40. It can be appreciated that if the dressing
roller 40 was moved in the opposite direction to arrow 55
5 along the working profile of the grinding wheel 50, the
circular portion of outer cutting rim 47 would be first to
contact the grinding wheel 50 and would be required to
absorb most of the wear associated with such contact. This
would result in increased wear at the normal contact point
10 CN and would lead to inaccuracies in the dressed profile of
the grinding wheel 50. The inner cutting surface 48 takes
the form of underlying inner convex surface 46 of the
working portion of the dressing roller 40. The inner
convex surface 46 of the roller being configured so as to
15 provide a clearance angle ~j with inside working surface 52
of the grinding wheel 50 of at least about 2, preferably
. ` in the range of 2 to 6.
Figure 6b illustrates the contact of the outer cutting
20 rim portion 47 with the outside working surface 54 of
grinding wheel 50. The outer cutting surface 49 of rim
portion 47 is configured in the same manner as the inner
cutting surface 48. The outer cutting surface 49 extends
to a point 58 prior to where the roller 40 wherein first
25 contacts grinding wheel 50. The rim portion 47 continues
to follow a circular contour having a radius R, at least to
the contact normal point CN. The preceding zone Z portion
of cutting surface 49 functions in the same manner as zone
Z on inner cutting surface 48. The outer cutting surface
30 49 takes the form of the underlying outer concave surface
45. A clearance angle ~0 is provided between the outer
concave surface 45 of the dressing roller 40 and the
outside working surface 54 of grinding wheel 50 so as to
prevent any substantial contact there between. Preferably
35 clearance angle ~0 is at least 2 ~ ~ 61~6c~ (

13~33~0
Referring to figure 7 there is diagrammatically
illustrated the rotary dressing roller 40 as it contacts
the inside surface 52 of grinding wheel 50 at point A. The
grinding wheel 50 has an axis of rotation XW about which
5 the inside working surface 52 revolves. ~j indicates
inclination of the inside working surface 52 with respect
to a line parallel to the axis of rotation XW. ~k~ 0~
represents the angle of inclination of the outside working
surface 54 with respect to a line parallel to the axis of
10 rotation XW of the grinding wheel 50. ~; and ~0 are
commonly referred to in the art as the "pressure angle" of
the inside working surface and the "pressure angle" of the
outside working surface, respectively. From point A, on
the grinding wheel 50 there is a normal radius of curvature
15 of the inside working surface 52 identified as R~;. Rw; is
defined as the length of a line from point A, drawn normal
to the inside working surface 52 to where it crosses the
axis XW of grinding wheel 50. P; is the effective radius
of curvature of the inner cutting surface of the rotary
20 dressing roller 40 and is defined as the length of a line
from point A, drawn normal to the point of contact to where
it crosses the axis XD of the rotary dressing roller 40.
Likewise, as illustrated in figure 8, the outside working
surface 54 of grinding wheel 50 has a normal radius of
25 curvature Rwo at point B, point B being the maximum working
depth of outside working surface 54. The rotary dressing
roller 40 has an effective radius of curvature P0 defined
along the normal line from point B to axis XD of the
roller.
It may now be appreciated that in order to achieve
contact at point A without interference between the
dressing roller 40 and the grinding wheel 50, the effective
radius of curvature P; of the dressing roller 40 must be
35 less than normal radius of curvature R~j of the inside
surface 52 of the grinding wheel 50. Conversely, in order

~3~:~3~0
to achieve contact at point B without interference, the
effective radius of curvature P0 of the dressing roller
must be greater than the normal radius of curvature F~o of
the outside surface 54 of the grinding wheel 50. In order
5 to provide an optimum shape to roller 40 whereby the
greatest potential amount of cutting surface is provided on
the inside surface, while at the same time providing a
roller 40 that will fit within the substantially cup-shaped
grinding wheel S0, the dressing roller 40 is designed such
10 that the effective radius of curvature P; at point A is
equal to some constant K times the normal radius of
curvature (Pj = K x Rwj)~ Pj is generally no greater than
about 80 percent of the normal radius of curvature R~; of
A the inside working surface 52 [P; ~ .8 R~j]. The effective
15 radius of curvature P0 of outer cu~ling surface 49 of
dressing roller 40 at point B is preferably greater than or
equal to about 1.25 times the normal radius of curvature
[P0 2 1.25 ~0].
The effective radius of curvature Pj may be controlled
by modifying the diameter Dx of the dressing roller 40 or
by modifying the inclination of angle ~ of axis XD with
respect to a line perpendicular to the axis XD of grinding
wheel 50. For example, a decrease in diameter Dx or an
25 increase in inclination angle ~ results in a decrease in
effective radius of curvature Pj. It is preferred,
however, that diameter Dx be made as large as possible so
that the point of contact between the dressing roller 40
and grinding wheel 50 will be dressed by as many diamond
30 grits as possible positioned at the same radius about the
circumference (i.e., surface of revolution) of the dressing
roller 40. Accordingly, it is preferred to orient the
dressing roller 40 at as large an angle ~ as possible so
that the largest dressing roller 40 having a diameter Dx
35 may be used. However, the amount of permissible
inclination of angle ~ ~ be limited by geometric

1;~1;~3~0
14
constraints associated with dressing the outside working
surface of the grinding wheel S0.
Referring to figure 8, there is illustrated the
5 geometric relationships for determining the diameter and
orientation of the dressing roller 40 with respect to point
B of outside working surface 54 of the grinding wheel 50.
In order to achieve contact at point B, the effective
radius of curvature P0 of the roller 40 must be larger than
10 the normal radius of curvature F~o of the outside working
surface 54 of the grinding wheel 50 at point B. As angle
increases, the effective radius of curvature PO decreases
and the difference between radii of curvature PO and R~o is
also decreased. Thus, although an increase in angle
15 permits a desirable increase in roller diameter Dx for
dressing the inside working surface 52, large increases in
angle ~ may undesirably decrease the effective radius of
curvature PO and lead to interference between the dressing
roller and the outside working surface 54 of the grinding
20 wheel 50. It is preferred that the roller 40 be made in
accordance with the following relationship:
Po - F~o 2 R~; - p
wherein:
P0 is the effective radius of curvature of the outer
cutting surface of said rotary dressing roller at the
maximum working depth of the grinding wheel;
Pj is the effective radius of curvature of the inner
cutting surface of said rotary dressing roller at the
maximum working depth of the grinding wheel;
X~O is the normal radius of curvature of the outside
working surface of the grinding wheel at the point of
contact with the dressing roller at the maximum
working depth; and
Rwj is the normal radius of curvature of the inside
working surface of the grinding wheel at the point of
. ,~....
. .~,
' ' '

13133~0
contact with the dressing roller at the maximum
working depth.
In the preferred embodiment angle ~ is fixed for
5 dressing both sides of grinding wheel 50 and preferably
lies in a range between ~min and ~max as defined by the
following relationships:
( 5 Dx ~ R)
~max Arcsin ~ + ~ and
(-5 Dx ~ R)
~ 15 ~min = Arcsin -----___~
f~, ( . 80 R"; ~ R)
wherein:
R is the radius of the circular contour of rim portion
47 joining inner cutting surface 48 and outer cutting
surface 49;
Dx is the diameter of dressing roller 40;
~0 (pressure angle) is the angle of inclination of the
outside working surface 54 of grinding wheel S0; and
~pressure angle) is the angle of inclination of the
inside working surface of the grindinq wheel 50.
Although for economic reasons it is preferred that the
30 diameter Dx of the dressing roller does not exceed 12 . 7 cm
(5 inches), it is important that Dx be selected so that
~max is greater than or equal to ~min to define a preferred
range of values for ~, and it is otherwise preferred that
diameter Dx meets the following relationship:
Dx 22 R~a X TAN (~j)
wherein:
R~a is the radius of the grinding wheel 50 at point A
on inside working surface 52 (see figure 7), and

13~33~0
16
TAN (~j) is the tangent function of the pressure angle
of inside working surface 52 at point A.
A rotary dressing roller made in accordance with the
5 present invention has a configuration such that the outer
cutting rim 47 may be used to cut both the inside working
surface 52 and the outside working surface 54. It should
be appreciated that the particular cross sectional
configuration of roller 40 may vary greatly and is
10 dependent upon the orientation of roller 40 with the
grinding wheel 50 during the dressing operation for both
the inside working surface 52 and outside working surface
54. The outer concave surface 45 is recessed into roller
40 to allow dressing of the outside working surface 54.
15 Likewise inner convex surface 46 of roller 40 is shaped so
as not to interfere with the dressing operation of the
inside working surface 52 of wheel 50 and permit dressing
of the generating type. Accordingly, the outer concave
surface has a concave configuration as illustrated in
20 section and inner convex surface 46 has a convex surface as
illustrated in section. An example of a concave surface
for the purposes of this invention would be the inside
surface of a cone or sphere and an example of a convex
surface for the purposes of this invention would be the
25 outside surface of a sphere or cone. Preferably the inner
convex surface 46, as shown in figure 4, is inclined at an
angle with respect to a line perpendicular to the axis XD
of roller 40 in accordance with the following relationship:
Y i = ~i +
wherein:
y; is the inclination of inner convex surface 46 of
the dressing roller with respect to a line
perpendicular to the axis XD of the roller;
~j is the pressure angle of inside working surface 52,
~ is the inclination of roller axis XD; and

13~33~0
17
is the clearance angle between the inside working
surface 52 and inner convex surface 46 of the roller,
preferably in the range of 2 to 6.
Outer concave surface 45 of the roller is preferably
inclined with respect to the same perpendicular to axis XD
in accorda~nce with the following relationship:
A ~=~+~o~~o
wherein:
rO is the inclination of outer concave surface 45 of
the dressing roller with respect to a line
perpendicular to axis XD of the roller;
is the inclination of axis XD;
~0 is the clearance angle between outside working
surface 54 and outer concave surface 45 of the roller,
preferably in the range of 2 to 6; and
is the pressure angle of outside ~orking surface
54.
From the above equations forY ; and ~0 it follows that
the difference between the respective inclinations of inner
convex surface 46 and outer concave surface 45 of the
dressing roller may be determined from the following
relationship:
~ (~i + ~0) - 2
wherein:
~Y is the difference between the respective
inclinations of inner convex surface 46 and outer
concave surface 4S measured with respect to a
perpendicular to axis XD of the dressing roller~;
~; and ~0 are the respective pressure angles of the
inside (52) and outside (54) working surfaces of the
grinding wheel 50; and
~ is the average clearance angle between the
respective contacting surfaces of the dressing roller

13~33~0
18
and grinding wheel 50 during use, preferably in the
range of 2 to 6.
Additionally, surfaces 46 and 45 are spaced so that
5 the roller 4~ has sufficient structural strength to provide
rigid support during the dressing operation. The amount of
spacing between surfaces 46 and 45 is substantially
determined by radius R which is preferably about .254 cm
(.1 inch).
It may also be appreciated that since the opposite
(inside 52 and outside 54) working surfaces of the grinding
wheel 50 are espectivel dressed ~ opposite (
~i6R ~5~ S~ 6g ~ ~6~
A ~-- --
15 directions of relative rotation between the grinding wheel
50 and dressing roller are maintained during dressing
operations on both sides of the grinding wheel 50. Thus,
the rotary dressing roller may be rotated in a single
direction for dressing both the inside working surface 52
20 and the outside working surface 54 of grinding wheel 50.
A description of the operation of the dressing
apparatus 15 will now be discussed. Referring to figure 4,
dressing apparatus 15 is positioned such that rotational
25 axis XD of the rotary dressing roller 40 is oriented at an
angle ~ when it contacts the maximum working depth of the
grinding wheel 50. In the embodiment illustrated, grinding
wheel 5~ has substantially straight inside working surface
52 in cross section. Accordingly, once the rotary dressing
30 roller 40 is positioned to its cutting depth, rotary
dressing roller 40 is simply moved in a direction parallel
to inside working surface 52 of the grinding wheel 50
toward the outer end (see figure 6a). If inside working
surface 52 is something other than a straight line in cross
35 section, the contact normal point CN will move along the
surface 47 as re~uired by the surface of the grinding wheel

lg ~3133~0
50. The same procedure is provided for dressing outside
working surface 54 of grinding wheel 50. Since rotary
generating roller 40 is rotated in a single direction, the
entire profile of the grinding wheel 50 can be generated
5 without removing the roller 40 from contact with grinding
wheel 50. As the roller reaches the outer end of inside
working surface 52 it may continue around the end of
grinding wheel 50 on to outside working surface 54
traveling toward point B. However, it is preferred that
10 inside working surface 52 and outside working surface 54
each be dressed by initially contacting each surface beyond
its maximum working depth and dressed in a direction toward
the outer end.
A rotary generating dressing roller made in accordance
with the present invention need only be rotated in a single
direction for dressing both sides of the grinding wheel 50
thereby avoiding the problems of prior art. Additionally
because of the roller's configuration, the roller need only
20 be moved in a single plane thereby avoiding any unnecessary
pivoting of the dressing roller that may be required in a
single point cutting tool or rotary dressing roller of the
prior art. This enables use of a simpler mechanism for
moving dressing roller thereby improving efficiency and
25 accuracy of the dressing process.
It may also be appreciated that because the inner and
outer cutting surfaces for dressing opposite side working
surfaces of the grinding wheel 50 are closely spaced, the
30 required travel of the dressing roller in a direction
perpendicular to the grinding wheel 50 axis for dressing
both sides of the grinding wheel 50 is minimized. It is
contemplated that the amount of required travel be
significantly less than the travel required of known rotary
35 dressing rollers of the generating type according to the
following relationship:

~33~Q
T < WT + DX X CS (~)
wherein:
T is equal to the required travel of the dressing
roller in a direction perpendicular to the grinding
wheel 50 axis XD for dressing both the inside and
outside working surfaces of the grinding wheel 50;
WT is the thickness of the grinding wheel 50 at the
maximum working depth (i.e., the distance between
points A and B illustrated in figures 4 and 5);
DX is the diameter of the dressing roller; and
COS (~) is the cosine function of the average pressure
angle of the inside and outside working surfaces of
the grinding wheel 50.
Preferably, the amount of required travel of the
dressing roller 40 between dressing positions A and B on
opposite side surfaces of the grinding wheel 50 is
approximately equal to the following relationship:
TAB S WT + 2R
wherein:
TAa is equal to the required travel of the dressing
roller in a direction perpendicular to the grinding
wheel axis XD between points A and B at the maximum
working clepth of the grinding wheel 50; and
WT is the thickness of the grinding wheel 50 at the
maximum working depth; and R is the radius of the
cutting rim.
Referring to figure 9 there is illustrated a rotary
30 dressing roller 40 made in accordance with the present
invention used to dress a "straddle type" cup grinding
wheel 150. Straddle grinding wheel 150 has an inner
cutting rim 152 and an outer cutting rim 154. The inner
cutting rim 152 has an outside working surface 156 and
35 inside clearance surface 157. Outer cutting rim 154 has an
outside clearance surface 158 and an inside working surface

3~0
21
159 which faces surface 156. The inner cutting rim 152 is
spaced a relative short distance from outer cutting rim 154
for grinding opposite sides of a gear tooth. Accordingly,
it would not be possible to dress surfaces 156 or 159 of
5 grinding wheel 150 with a known generating type roller
because of the interference between the roller and the
working surface opposite the surface to be dressed. A
rotary dressing roller made in accordance with the present
invention has a configuration which allows orientation of
10 the roller which permits dressing of the outer surface 156
of inner cutting rim 152 and inside surface 159 of outer
cutting rim 154. Additionally, the amount of travel
perpendicular to the axis of grinding wheel 50 required by
a roller of the present invention is minimal.
Referring to figures 10, 11, 12 there are illustrated
various modified configuration of the rotary dressing
rollers made in accordance with present invention. The
rotary dressing roller illustrated in figure 10 is similar
20 to configuration of the illustrated in figures 3 and 4
except that the outer working portion is substantially
smaller in diameter. Figures 11 and 12 illustrate various
other configurations of the working portion of a dressing
roller made in accordance with the present in~ention.
25 Various other configurations may be used as required.
Various modifications and changes may be made without
departing from the scope of the present invention. For
example, but not by way of limitation, the inner convex
30 surface 46 and outer convex surface 45 may take other
shapes other than straight as illustrated.

Representative Drawing