METHOD FOR CUTTING STRAIGHT BEVEL GEARS USING QUASI-COMPLEMENTARY CROWN GEARS

METHOD FOR CUTTING STRAIGHT BEVEL GEARS
USING QUASI-COMPLEMENTARY CROWN GEARS

Koreaki Ichino
Nagaoka Gear Works Co., Ltd
Nagaoka, Japan

Hisashi Tamura and Kazumasa Kawasaki
Department of Mechanical and Production Engineering
Niigata University
Niigata, Japan

ABSTRACT

In the manufacture of straight bevel gears, there is usually a need for some
trial-and-error process by which a good tooth bearing is obtained. In this
paper, a new straight bevel gear cutting method in which trial-and-error process
is not necessary is proposed. The method can be applied at once without the
necessity of generator alteration. Furthermore, each gear and pinion is
generated by a newly introducedgquasi-complementary crown gearh. Gears made
on an experimental basis showed the expected good tooth bearing.

1. INTRODUCTION

The tooth surface of a straight bevel gear is a simple surface as compared
with that of a spiral bevel gear; nevertheless, trial-and-error process is still
employed to obtain a good tooth bearing in the straight bevel gear manufacturing.
The authors think the trial-and-error process actually covers the defect of gear
cutting method. The gear cutting method based on the kinematic theory employs
the pitch cone of the generated gear and the pitch plane of the complementary
crown gear which are arranged to roll with each other.
The straight bevel gear cutting method has been developed by Gleason Works
[1, 2]. However, the method using the generator such as the Gleason No. 14
straight bevel gear generator, employs the members which are not arranged to
roll. As a result, the real tooth surfaces generated in machining are slightly
different from the theoretical surfaces and the tooth bearing is unsuitable for
practical use when those surfaces are meshed. The unsuitable tooth bearing can
be corrected by the trial-and-error process of the skilled worker.
In this paper, a new method for cutting straight bevel gears is proposed.
The method employs the pitch cone of the generated gear and the pitch cone of a
newly introducedgquasi-complementary crown gearhwhich are arranged to roll
with each other and keeps the kinematical strictness of gear tooth generation.
The pitch cone angle of the quasi-complementary crown gear is designated strictly
as 90ß minus the root angle of the gear to be cut. The tool surface of the
quasi-complementary crown gear is the plane with pressure angle ƒ¿. The plane
tool surface produces the modified tooth surface with depthwise tooth taper.
Besides, angle ƒ¿ determines the position of tooth bearing on the generated gear
tooth surface.
In the previous papers [3, 4], a method using a quasi- complementary crown
gear in spiral bevel gear cutting was proposed. The quasi-complementary crown
gear can be realized by the Gleason No. 116 spiral bevel gear generator and can
generate only the modified tooth surface with depthwise tooth taper. However,
the position of tooth bearing was controlled by another technique.
The proposed method in this paper can be used in combination with the usual
crowning process in pinion machining. The process of this method is the same as
the usual process except that the ratio of roll and the pressure angle of the
tool are theoretically determined taking the gear generation method and the
position of tooth bearing into consideration respectively, as stated above.
This method can be applied at once without the necessity of any alteration of
the usual generator.

2. QUASI-COMPLEMENTARY CROWN GEAR

In general, the complementary crown gear which generates the straight bevel
gear has a depthwise tooth taper. Therefore, it is necessary for the tool which
simulates the crown gear tooth to move tooth-depthwise, namely, perpendicularly
to the cradle plane [5] theoretically. This tool movement is in need of the
mechanism of tool inclination to cope with the various root angles of the gear
to be cut and complicates the machine, so that the workpiece is set to the

Fig. 1 Relations between three gears

Fig. 2 Clearance between tooth surfaces

machine with more inclination angle than its pitch cone angle in practice to get
around the machine complexity. More inclination angle of the workpiece is the
root angle ƒÆf of the gear to be generated. However, this additional inclination
of workpiece breaks the kinematical strictness of gear tooth generation.
In other words, the arrangement that the pitch cone of the workpiece and the
pitch plane of the complementary crown gear roll on each other is broken by this
workpiece inclination. The broken arrangement is strictly restored by the
quasi-complementary crown gear as follows:
First, we call the complementary crown gear the crown gear for short. Next,
we imagine that the crown gear with the plane tooth surface xc generates both
the workpiece and gear-I with a large pitch cone angle at the same time as shown
in Fig. 1. The pitch plane of the crown gear lies between the workpiece and
gear-I, keeps in contact with each pitch cone, and rolls on each cone [6].
The pitch cone angle of the workpiece is ƒÂ0 and the large pitch cone angle of
gear-I is ( 90ß-ƒÆf ). The ratio of roll between the crown gear and workpiece
is 1/sinƒÂ0 from the condition of roll. The ratio of roll between the crown gear
and gear-I is 1/cosƒÆf . Therefore, the ratio of roll i between the gear-I and
workpiece is:

i = cosƒÆf / sinƒÂ0 (1)

These three gears are conjugate gears, so that gear-I can generate the same
gear with pitch cone angle ƒÂ0 that the crown gear generates. The gear-I is a
straight bevel gear with the tooth surface

Fig. 3 Quasi-complementary crown gear and workpiece

x‡T of octoid profile. The tooth surface x‡T is very close to a plane because
the pitch cone angle of gear-I is close to 90ß. Figure 2 shows the surface x‡T
is in contact with the plane tooth surface xc of the crown gear at point Q.
There is a gap between x‡T and xc with the exception of point Q. The gap is very
small in the neighborhood of point Q. Therefore, the real plane tool surface x
substitutes for x‡T as the tooth surface of gear-I. A part of the substituted
tool plane x in the neighborhood of point Q can generate a conjugate tooth
surface in the workpiece. The gap between the tooth surfaces away from point Q
is not so large that a part of the tool plane x away from point Q can generate a
modified tooth surface. In this study, the gear-I with tool plane x is called
the quasi-complementary crown gear, namely, the quasi-crown gear for short, which
generates the tooth surface of the straight bevel gear with modified profile and
depthwise tooth taper.
The characteristics of the quasi-crown gear is as follows:
1. The pitch cone angle is ( 90ß-ƒÆf ).
2. The ratio of roll is cosƒÆf / sinƒÂ0.
3. The tool surface is a plane as usual.
4. The generated tooth profile becomes the modified one.
5. The pressure angle of the quasi-crown gear is theoretically determined
relating to the position of tooth bearing as state in next section.
Figure 3 shows the relation between the quasi-crown gear and workpiece. In the
machining of workpiece, the cradle axis is the quasi-crown gear axis, the set
angle of the workpiece axis is angle (ƒÂ0 - ƒÆf ), and the straight sided tool is
employed as usual. However, the setting of the ratio of roll i is only different
from the usual setting.
From above, it is clear that the quasi-crown gear is realized easily by the
employment of an ordinary gear generator such as the Gleason No. 14 straight
bevel gear generator which is the most fundamental machine, without any
alteration.

Fig. 4 Engagement of teeth in transverse plane

Fig. 5 Pressure angle ƒ¿ of quasi-complementary crown gear

3. TOOTH BEARING AND PRESSURE ANGLE

In the proposed method, a pair of quasi-crown gears is employed instead of a
crown gear except the case of miter gear generation. One is for gear generation
and the other is for pinion generation. However, the crown gear is a fundamental
gear in this method and is very useful for the analysis of the relationship
between the position of tooth bearing and the pressure angle of each quasi-crown
gear.
Figure 4 shows the crown gear tooth, the pinion tooth, and the gear tooth each
engage at point Q on the common transverse plane of the bevel gear. The
kinematical meaning of point Q is the same as that of Fig. 2. That is, the
parts of both the pinion tooth surface and gear tooth surface which are away from
point Q are the modified non-conjugate tooth surfaces causing a small
transmission error, and the parts of both the tooth surfaces in the neighborhood
of point Q are the conjugate surfaces, so that the tooth bearing always appears
about point Q when a pair of generated gears are meshed. Thus, the position of
point Q on the crown gear tooth surface represents the position of tooth bearing
on the meshed tooth surfaces, and besides, relates the tool pressure angle ƒ¿ of
each quasi-crown gear as shown

Fig. 6 Coordinate systems of each gear

in Fig. 5.
The tooth surface xc of the crown gear which rotates about its axis by angle
ƒÓ is used as the plane tool surface x of the quasi-crown gear which rotates
about its axis by angle ƒÕ. The pressure angle ƒ¿ of tool surface x is unknown.
ƒ¿ is determined as follows:
Referring to Fig. 4, we can introduce the following equation according to the
condition of meshing of spur gear kinematics:

H = - L sinƒ¿ccosƒ¿c (2)

where L is the moving length of tooth surface xc in the transverse plane, ƒ¿c is
the pressure angle of the crown gear and H (positive) is the height of point Q
from pitch plane S toward addendum. H (negative) is toward dedendum.
The length L can be expressed using the mean cone distance Rm and angle ƒÓ:

L = RmƒÓ (3)

From Eqs. (2) and (3), the rotation angle ƒÓ of the crown gear is:

ƒÓ = - H / (Rm sinƒ¿ccosƒ¿c ) (4)

Three coordinate systems shown in Fig. 6 are introduced to analyze the
relationship between xc and x. O-XYZ is fixed in space, O-xcyczc is attached to
the crown gear and O-xyz is attached to the quasi-crown gear. O-xcyczc and O-xyz
rotate by angles ƒÓ and ƒÕ about axis zc and axis z respectively. When both ƒÓ
and ƒÕ are zero, axes xc and x agree with axis X, and yc agrees with axis Y which
is the pitch surface generator [6]. In Y-Z plane, axis z is inclined by angle
ƒÆf which is the root angle of the gear to be cut.
In each coordinate system, nc represents the unit surface normal of xc which is
the simulated plane of the crown gear tooth surface as shown in Fig. 7. nc is
changed in O-XYZ only by angle ƒÓ expressed as a function of H, Rm, and ƒ¿c. n
represents the unit surface normal of x which is the real tool plane of the quasi
-crown gear as shown in Fig. 8. In O-XYZ, the surface xc attached to O-xcyczc
coincides with

Fig. 7 Tooth surface of complementary crown gear

Fig. 8 Tooth surface of quasi-complementary crown gear

the surface x attached to O-xyz, namely, nc coincides with n. Therefore, the
following equation is obtained:

A(-ƒÆf ) C(ƒÕ) n(ƒ¿) = C(ƒÓ) nc (5)

where A and C are the coordinate transformation matrices and are as follows:

1 0 0
A(ƒÆf) = m 0 cosƒÆf -sinƒÆf n
0 sinƒÆf cosƒÆf
(6)
cosƒÕ -sinƒÕ 0
C(ƒÕ) = m sinƒÕ cosƒÕ 0 n
0 0 1

Two unknowns ƒ¿ and ƒÕ are determined as the solutions of Eq. (5). The solutions
are as follows:

ƒ¿ = sin-1 (sinƒÓcosƒ¿c sinƒÆf + sinƒ¿c cosƒÆf)
sinƒÓcosƒ¿c cosƒÆf - sinƒ¿c sinƒÆf (7)
ƒÕ = tan-1m \\\\\\\\\\\\\\\\\\\ n
cosƒÓcosƒ¿c

According to Eqs. (4) and (7), the pressure angle ƒ¿ of the quasi-crown gear is
determined directly by H which is the position of tooth bearing.
The pressure angle ƒ¿' of mating quasi-crown gear is as follows:

ƒ¿' = sin-1(-sinƒÓcosƒ¿c sinƒÆf' + sinƒ¿c cosƒÆf') (8)

Table 1 Dimensions of straight bevel gears

Table 2 Dimensions of quasi-complementary crown gears

where ƒÆf' is the root angle of mating gear.
Now, the crown gear is not actually employed in this method, so that the
pressure angle ƒ¿c of the crown gear can be omitted from Eqs. (7) and (8),
replacing sinƒÓ with ƒÓ. As a result, the following equation is introduced:

aƒÔ2 + bƒÔ2 + c = 0 (9)

where ƒÔ = sinƒ¿', a = - sinƒÆf cosƒÆf , b = sin(ƒÆf -ƒÆf')sinƒ¿,
(10)
c = sinƒÆf 'cosƒÆf' sin2ƒ¿ + H sin2(ƒÆf +ƒÆf') / Rm

The pressure angle ƒ¿' is determined from Eq. (9) when ƒÆf, ƒÆf', H, Rm and an
arbitrary angle ƒ¿ are given. That is, the position of tooth bearing is
determined by the designation of ƒ¿' corresponding to ƒ¿.
In case of miter gear, for example, ƒÆf = ƒÆf' and H = 0 yield, so that the
result of ƒ¿' = ƒ¿ is obtained. This means that, according to this method, the
tooth bearing appears on the pitch cone (H = 0) without fail when a pair of gears
is generated using the same straight sided tool with an arbitrary pressure angle.

4. AMOUNT OF PROFILE MODIFICATION

As stated in section 2, there is a gap between the theoretical tooth

Fig. 9 Backlash and kickout

surface of gear-I with octoid profile and the tooth surface of the quasi-crown
gear (see Fig. 2). The amount of gap becomes that of the profile modification
because a small gap directly turns to the generated tooth surface. Furthermore,
the amounts become essentially the definite amounts according to the root angle
ƒÆf of the gear to be cut. The amount of gap at arbitrary point on x is obtained
from the following vector equation:

x(u, v) + t n(ƒ¿) = x‡T(U, ƒ³) (11)

where U, ƒ³, and u, v are the parameters to express the surface of x‡T and x
respectively, t is the amount of gap or the amount of profile modification, n is
the unit surface normal of plane x, and ƒ¿ is the known constant as stated above.
The formulation and analysis of the generated tooth surface such as worm gear
surface, hypoid gear surface, spiral bevel gear surface, and straight bevel gear
surface x‡T in this study, have been developed by many researchers [7-18] using
the tool geometry, machine tool kinematics, differential geometry, theory of
conjugate surface and technique of coordinate transformation matrix.

5. EXAMPLE OF MANUFACTURING STRAIGHT BEVEL GEARS

The dimensions of the gears manufactured as an example are shown in Table 1.
These dimensions are designated according to Gleason method. The root angle of
the gear is very large angle 8ß51' and dedendum is twice the length of addendum.
If the position of tooth bearing is designated as a point 1 mm apart from the
pitch cone toward dedendum, the tooth bearing will appear at the middle of the
working depth of the gears. Therefore, the gear manufacturing in two cases is
carried out. One is the tooth bearing on the pitch cone (H = 0 mm) and the other
is H = -1 mm on the gear tooth (H = +1 mm on the pinion tooth). The
corresponding quasi-crown gear dimensions are shown in Table 2.
The practical inspections of bevel gears are the check of the position of tooth
bearing by the eye-measurement and the check of commonly calledgkickouthwhich
is a difference between the backlashes measured in two cases shown in Fig. 9.
It is desirable that the amount of kickout is zero.
In case of H = 0 mm, the kickout is zero and tooth bearing appears

Fig. 10 Amounts of profile modification [ƒÊm ]
(without crowning)

apart from the middle of tooth depth as arranged previously. In caseof H = -1 mm
on the gear tooth (H = +1 mm on the pinion tooth), the kickout is undesirably
large value 0.07 mm, but the position of tooth bearing is the middle of tooth
depth as arranged.
The amount of kickout of the gears generated by this method can be calculated
from the amount of profile modification. The amount of modification t is
calculated in case of the manufactured gears and the results are shown in Fig.10.
The points A, A', B, and B' in Fig. 10 nearly correspond to the points in Fig. 9
respectively. The profile modification causes the additional backlash and two
times the amount of modification becomes the amount of kickout. Therefore, in
case of H = 0 mm, (A‡C + B ‡E = ) 10ƒÊm of additional backlash minus (A'‡C + B'‡E
= ) 10ƒÊm is zero kickout which coincides with experimental result. In case of
H = -1 mm on gear tooth and H = +1 mm on pinion, (A 14 + B 24 = ) 38ƒÊm minus (A'
0 + B' 0 = ) 0ƒÊm is 38ƒÊm which is nearly one half of 0.07 mm of experimental
result. There are good agreements in two cases.
In case of H = -1 mm of both gear and pinion, (A 0 + B 24 = ) 24ƒÊm minus
(A' 14 + B' 0 = ) 14ƒÊm is 10ƒÊm, so that twice 10ƒÊm will be the kickout.
A pair of gears with H = -1 mm was manufactured, and the position of tooth
bearing and the kickout were

Fig. 11 Tooth bearing on gear tooth surface
(Gear H = -1 mm, Pinion H = -1 mm)

checked. Figure 11 shows the tooth bearing which is very close to the previous
arrangement. The measured kickout is about 0.015 mm which is slightly smaller
than the simulation result.

6. CONCLUSION

In this paper, a new method for cutting straight bevel gears was proposed.
In this method, the gear and pinion are generated by a correspondingly
quasi-complementary crown gear which is a newly introduced tool gear instead of
a usual complementary crown gear. The proposed method is a simple one and can be
applied at once without any alteration of a usual gear generator or the necessity
of trial-and-error process in manufacturing. This method is based on the
kinematics of gear generation. Therefore, the ratio of roll, the pressure angle
of the tool surface which is a quasi-complementary crown gear tooth surface, the
machine setting of workpiece, and the amount of tooth profile modification can be
theoretically determined. The usefulness of the proposed method was confirmed by
some experiments of gear manufacturing.

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