DUPLEX SPREAD BLADE METHOD FOR CUTTING HYPOID GEARS
WITH MODIFIED TOOTH SURFACE
Kazumasa Kawasaki and Hisashi Tamura
Department of Mechanical and Production Engineering
Niigata University
Niigata, Japan
ABSTRACT
In this paper, a duplex spread blade method for cutting hypoid gears with modified tooth
surface is proposed. The duplex spread blade method provides a rapid and economical
manufacturing method because both the ring gear and pinion are cut by a spread blade method.
In the proposed method, the non-generated ring gear is manufactured with cutting edge that
is altered from the usual straight line to a circular arc with a large radius of curvature
and the circular arc cutting edge produces a modified tooth surface. The pinion is
generated by a cutter with straight cutting edges as usual. The main procedure of this
method is the determination of the cutter specifications and machine settings.
The proposed method was validated by gear manufacture.
1. INTRODUCTION
The widespread use of hypoid gears for power transmission originated with the automobile
rear axles. These gears are now used in many applications such as transportation,
earth-moving equipment, and construction equipment [1]. For hypoid gears, there is a need
for smoother and quieter running gears. At the same time, a rapid and economical
manufacturing method is essential to the industries in their products.
For many years, Gleason Works has provided the machinery and equipment for manufacture of
hypoid gears [2-4]. The cutter specifications and machine settings are generally calculated
using the Calculating Instructions developed by Gleason Works. However, the authors believe
that the formulae of Gleason method are rather complicated. Until now, the geometry and
characteristics of the gears have been investigated. Baxter [5] investigated the effect of
various types of misalignment on tooth contact analysis in bevel and hypoid gears. Krenzer
[6] investigated the effect of the cutter radius in spiral bevel and hypoid tooth contact
behavior. Litvin et al. [7, 8] proposed local synthesis of Formate and Helixform hypoid
gears for conjugate gear tooth surfaces and investigated the transmission errors and shift
of bearing contact for hypoid gear drive. Sugimoto et al. [9] investigated the effect of
the tooth contact and alignment error of the gears under loaded condition.
It is well known that the tooth surfaces of the gears are generally modified to obtain a
better tooth bearing and to absorb the transmission errors caused by the misalignment.
The authors [10] presented Formate gear cutting method of hypoid gears with modified tooth
surface. In the method, the convex and concave sides of hypoid pinion tooth were cut
separately, namely, the pinion was manufactured by a single side method. However, the
production cost of the gears was increased substantially.
In this paper, a duplex spread blade method for cutting hypoid gears with modified tooth
surface is proposed instead of the single side method. The duplex spread blade method is
that both the ring gear (gear for short) and pinion are cut by the spread blade cutter, that
is, both sides of a tooth space are finished simultaneously by each cutter blade [2].
In the proposed method, the non-generated gear is manufactured with cutting edge that is
altered from the usual straight line to a circular arc with a large radius of curvature and
the pinion is generated by a cutter with straight cutting edges as usual. The circular arc
cutting edges enable the duplex spread blade method to work theoretically and produce a
modified tooth surface. The main procedure of this method is the determination of the
cutter specifications and machine settings that satisfy the duplex spread blade conditions
taking into account of the position of tooth bearing. The production cost of the gears is
decreased by the duplex spread blade method.
Fig. 1 Face-mill cutter in ring gear cutting
2. TOOL SURFACE
The cutting edge of the face-mill type of cutter in ring gear cutting is a circular arc
as shown in Fig. 1. The cutting edge turns about the cutter axis and produces a surface of
revolution close to a conical surface. The surface is called a pseudo-conical surface in
this paper. Oc-xcyczc is the coordinate system attached to the cutter. Origin Oc is the
cutter center. zc is the cutter axis. R is the cutter radius. W is the point width.
r and r' are the radii of curvature of circular arc and the values influence the level and
shape of transmission errors of the gears. 1' and 2 are the blade angles of outside and
inside cutting edges respectively. S' is the shim thickness of the outside cutter blade and
can be altered to adjust the distance between the outside and inside cutter blades. y0, z0
and y0', z0' are the coordinates of the center of the curvature. The circular arc of the
inside cutting edge passes through two points A(0, R-W/2, 0) and B(0, R-W/2-Rtan2/8, -R/8)
in plane xc=0. Therefore, the following equations are derived:
y0 = R - W/2 - R tan2 / 16 - 2cos22 - (R / 16)2
(1)
z0 = 2sin22 - (R tan2 / 16)2 - R / 16
From Fig. 1, the following equation yields:
tan0 = z0 / (R - W/2 - y0) (2)
, ' are the parameters representing the curved line and u, u' are the parameters represent-
ing the angle of revolution, that is, any point on the pseudo-conical surface is defined by a
combination (u, ) or (u',').
The circular arc of the outside cutting edge passes through two points A'(0, R+W/2+S', 0)
and B'(0, R+W/2+S'+Rtan1'/8, -R/8) in plane xc=0. y0', z0', and 0' can be derived in the
same manner as y0, z0, and 0.
The cutting edges of the cutter in pinion cutting are usual straight lines and are
represented by parameters v and v'. Then, the straight line cutting edges turn about the
cutter axis and produce the conical surfaces.
Each pseudo-conical and conical surface produces a pair of gears which meshes with each
other at a point at every instant with negligible small non-conjugate.
When the inside and outside cutting edges turn about the cutter axis zc, the loci of the
edges are the surfaces of revolution and are expressed as:
Xgc(ug,g)
-r cosg sin ug - y0 sin ug
= r cosg cos ug + y0 cos ug
-r sing + z0
Xgc'(ug',g')
r' cosg' sin ug' - y0' sin ug'
= -r' cosg' cos ug' + y0' cos ug'
-r' sing' + z0'
Xpc(up, vp;1p , Sp) (3)
-vp sin1p sin up - (R + W/2 +Sp) sin up
= vp sin1p cos up + (R + W/2 +Sp) cos up
vp cos1p
Xpc'(up', vp';2p')
vp'sin2p 'sin up' - (R - W/2) sin up'
= -vp'sin2p' cos up' + (R - W/2) cos up'
vp'cos2p'
g = g + 0, g' = g' + 0'
In Eq. (3),"'"is related to both the concave side of the gear tooth surface and convex side
of the pinion tooth surface. The subscripts"g","p", and"c"indicate that each is related to
the gear, pinion, and cutter respectively. Then, the unit normals of the surfaces expressed
as Xgc, Xgc', Xpc, Xpc' are Ngc(ug,g), Ngc'(ug',g'), Npc(up;1p), Npc'(up';2p')
respectively. 1p, 2p', and Sp are unknowns at this stage and are determined from the
duplex conditions.
3. DESIGN POINT
The hypoid gears and its coordinate system are shown in Fig. 2. The gear axis coincides
with axis z and the pinion axis is parallel to axis y. The common perpendicular of the gear
and pinion axes is axis x. Offset distance e is measured along axis x. The gear and pinion
are in mesh with each other in O-xyz and pinion cutting is also considered in O-xyz.
In hypoid gear design, the direction of the tooth trace of the gears is made to coincide
with that of the relative velocity between the pinion and gear at a design point, and the
tooth bearing is made to appear around the point.
Fig. 2 Hypoid gears and its coordinate system
In this method, however, the design point is treated only as a reference point to determine
the direction of the tooth trace of the gear, but not the appearance point of tooth bearing.
After the design point P is determined, the pinion root cone is established so that the cone
passes through point P in order to prevent an irregular tooth surface from appearing on the
pinion tooth flank [10]. The gear face cone is determined using the pinion root cone and
top clearance. At first, the gear face cone whose cone angle is gi in case of top clearance
equal zero is considered. When the top clearance is ensured, the gear face cone anglegi is
reduced and the reduced angle is denoted by gf.
Gear ratio i, offset distance e, and gear radius Rg are generally given at first. In this
method, pinion radius Rp and gear face cone angle gi in addition to i, e, Rg are also given
as design conditions in order to determine design point P. The coordinates of design point
P in O-xyz are (X0, Y0, Z0) and denoted by the position vector X0. From Fig. 3, we obtain
the following equations:
Rg2 = X02 + Y02
(4)
Rp2 = (e - X0)2 + Z02
The gear face cone is tangent to the relative velocity W between the pinion and gear at
point P. The unit normal vector Ng of the gear face cone is perpendicular to W [11]:
Ng(X0, Y0) W(X0, Y0, Z0) = 0 (5)
where
X0 cosgi/Rg
Ng(X0, Y0) = Y0 cosgi/Rg
singi
W(X0, Y0, Z0) = Vp - Vg (6)
iZ0 -Y0
Vp = 0 , Vg = X0
i(e - X0) 0
and Vp is the velocity vector of the pinion and Vg is the velocity vector of the gear.
From Eqs. (4) and (5), (X0, Y0, Z0), namely, X0 is determined.
Fig. 3 Design point P
Since X0 is determined, the position vector Eg=(0, 0, zg) of apex Og of the gear face cone,
the position vector Ep=(e, yp, 0) of apex Op of the pinion root cone, and the pinion root
cone angle pr are determined (see Fig. 3). Furthermore, spiral angle g and p of the
gear and pinion are determined at point P respectively. Considering that the tooth height
hk and top clearance b are given, the pinion face cone angle pf, the gear face cone
angle gf, and the gear root cone angle gr are also determined.
4. RING GEAR CUTTING
The ring gear is the non-generated spiral bevel gear, so that the surface of each side of
gear tooth is just a copy of the tool surface. Therefore, the surfaces of the convex and
concave sides of the gear tooth are also expressed as Xgc and Xgc' respectively. The method
for cutting the gear is shown in Fig. 4(a). Om-xmymzm is the coordinate system attached to
the gear cutting machine. Om is the machine center, axis xm, axis ym, and axis zm coincide
with so-called the V-axis, the H-axis, and the cradle axis respectively. The cutter axis zc
does not tilt.
We transform point P in O-xyz into Om-xmymzm as shown in Fig. 4(b) and the transformed
point is denoted by Pm. The way of transformation of point P is as follows: First, point P
is turned about axis z by angle until P approaches the y-z plane. is represented by:
= tan-1(X0/Y0) + (1/4) (2/n) (7)
where n is the number of gear teeth. Next, after moving the gear blank along axis z until
Og reaches O, the gear blank is made a half turn about axis y, and is rotated about axis x
by angle -=(gr - /2) radian. The state after this procedure is shown in Fig. 4(a).
The position vector x0 of point Pm in Om-xmymzm is represented by:
x0 = A() B() {C() X0 - Eg} (8)
where A, B, C are the matrices of the coordinate transformation relating relating to the
rotation about x, y, and z axis respectively [11]:
Fig. 4 Method of ring gear cutting
1 0 0
A() = 0 cos -sin
0 sin cos
cos 0 sin
B() = 0 1 0 (9)
-sin 0 cos
cos -sin 0
C() = sin cos 0
0 0 1
Since the relative velocity W determines the direction of the tooth trace of the gear, the
direction of W is necessary when the gear is cut. Therefore, W in O-xyz is also transformed
into Om-xmymzm and the transformed W is denoted by w:
w = A() B() C() W (10)
The convex side of the gear tooth surface is cut by the tool surface Xgc whose center Oc
is positioned at Dg =(Vg, Hg, 0)T in Om-xmymzm. Dg is determined so that the tool surface
Xgc is tangent to w at point Pm in Om-xmymzm:
Ngc(ug,g) w = 0
(11)
Xgc(ug,g) + Dg - x0 = 0
Equations (11) are four scalar equations with four unknowns Vg, Hg, and ug, g. Therefore,
Vg, Hg and ug, g are determined, namely, the machine settings and parameters representing
the point Pm on the pseudo-conical surface are determined. In the duplex spread blade
hypoid gear cutting method, the concave side of the gear tooth surface is also cut
simultaneously under these machine settings.
Fig. 5 Sketch showing the cutter used for pinion generation
and the ring gear
5. PINION CUTTING
If the pinion is generated by the tool surface which coincides with the gear tooth
surface, the pinion tooth surface becomes conjugate with the gear tooth surface. However,
the tool surface and gear tooth surface become mismatched because of depthwise tooth taper.
In this study, hypoid gears with depthwise tooth taper are regarded as non-conjugate gears
which are close to conjugate gears, so that pinion cutting is considered as follows:
The pseudo-conical surface of the convex side of the gear tooth is replaced by the conical
tool surface Xpc. The pseudo-conical surface of the concave side of the gear tooth is also
replaced by the conical tool surface Xpc'. In this case, tool surface Xpc is determined so
that Xpc is in contact with the surface Xgc of the convex side of the gear tooth at the
designated point Qm. Therefore, the pinion tooth surface meshes conjugably with the gear
tooth surface only at point Qm. That is, the tooth bearing appears around point Qm.
The pinion tooth surface meshes with non-conjugate motion with the gear tooth surface except
at point Qm. Tool surface Xpc' is determined in the same manner as Xpc. The concave and
convex sides of the pinion tooth surfaces are generated by Xpc and Xpc' by a spread blade
method.
Point Qm is designated as a point on the line of intersection of the gear tooth surface
Xgc(ug,g) and right circular cone whose half angle is g0, and is a point Rg0 apart from
the gear axis as shown in Fig. 5. Considering that g0 and Rg0 are given, ug and g are
determined. Therefore, the position vector xm0 and unit normal Ngc of surface Xgc at point
Qm are determined.
The state of the gear tooth surfaces being cut by tool surfaces Xgc and Xgc' is shown in
Fig. 6(a). Point Qm' is on the gear tooth surface being cut by Xgc'. However, tool
surface Xpc' is determined so that
Fig. 6 Duplex spread blade pinion cutting
Xpc' is in contact with the gear tooth surface at point Qm" as shown in Fig. 6(b).
In other words, point Qm" changes by one pitch in comparison with point Qm'. Therefore,
after the position vector xm0' of point Qm' and the unit normal Ngc' of surface Xgc' are
determined in the same manner as point Qm, point Qm' is changed by one pitch, namely, Qm'
is transformed into Qm". The position vector xm0" of Qm" and the unit normal Ngc" of the
surface are represented by:
xm0" = A() C() A-1() xm0' (12)
Ngc" = A() C() A-1() Ngc'
where (radians) is represented by:
= 2/ n + C0 / Rg0 (13)
and C0 is the backlash.
The cutter axis zc is inclined by the angle which is the rotation angle about axis xm
as shown in Fig. 5 so that the crest of the cutter generates the pinion tooth bottom land
securing clearance b. The axis zc can be also inclined by a very small angle (unknown)
about axis ym after inclination of . This process increases the degree of freedom of the
cutter settings.
Using this additional freedom, both the tool surfaces Xpc and Xpc' whose center position
at D=(V, H, Z )T are determined so that Xpc and Xpc' are in contact with the surfaces of the
convex and concave sides of the gear tooth at point Qm and Qm" respectively:
B() A() Npc(up ;1p) - Ngc = 0
B() A() Xpc(up, vp ;1p, Sp) + D = xm0
B) A() Npc'(up' ;2p') - Ngc" = 0
(14)
B() A() Xpc'(up', vp' ;2p') + D = xm0"
vp cos1p = lg0 sin (gf - g0 + b / lg0)
lg0 = Rg0 / sing0
where lg0 is the length of OmQm. Equations (14) represents a system of twelve scalar
equations with 12 unknowns V, H, Z, up, vp, 1p, Sp, up', vp', 2p', , and lg0.
The solutions of these equations for the unknowns provide the cutter specifications and
machine settings. The unit normal vector toward the cutter axis is as follows:
a = B() A() ( 0, 0, -1)T (15)
D and a are transformed into O-xyz and are denoted by Dp and ap respectively. Both the
concave and convex sides of the pinion tooth surfaces are generated simultaneously under
these machine settings.
In this method, there is a clearance between the tool surface Xpc in pinion cutting and
the gear tooth surface Xgc as shown in Fig. 6(b). The amount of the clearance can be
regarded as that of the tooth surface modification because it is very small. Moreover, the
transmission errors depend on the amounts.
6. NUMERICAL EXAMPLE
A numerical example is presented based on the method described above. The basic data of
hypoid gears are shown in Table 1 and the cutter specifications used in this method are
shown in Table 2. The calculated results are shown in Table 3.
Since the gear tooth surface is a copy of each tool surface, the shim thickness Sg' in
Table 2 is determined considering that the gear tooth thickness is given. In this study,
the gear tooth thickness is determined so that the ratios of the tooth and space widths on
the circle of radius Rg0, namely, ratios of QmQm" and QmQm' in Fig. 6(a), are 0.47 to
0.53. The reason for making a difference is that the generated pinion tooth becomes thinner
than the gear tooth.
Since the position vectors and unit normals of the surfaces coincide with each other at
point Qm (Qm") as expressed in Eq. (14), mating tooth surfaces of the gear and pinion are in
point contact. However, the interference between them occasionally occurs. The interference
here means that a part of the two surfaces penetrates into mating material side. It is
sufficient to investigate the presence of the interference between the tool surface in pinion
cutting and the gear tooth surface because the tool surface which replaces the gear tooth
surface generates the pinion tooth surface. The relative total curvature K which is defined
by Yokota [12] is calculated for the surfaces in point contact. There is no interference
when K is positive. If K is negative, it can be changed into positive by shortening r and r'
which are the radii of curvature of circular cutting edges shown in Fig. 1. The calculated
results of K with the variation of r and r' are shown in Table 4. We consider that the
maximum value of r when the interference does not occur is slightly smaller than 400 mm in
case of drive side. In this study, however, r of 200 mm is selected in order to obtain gears
insensitive to misalignment. r' of 100 mm is selected in case of coast side for the same
purpose.
Table 1 Basic data of hypoid gears
Table 2 Cutter specifications (mm)
The level and shape of transmission errors have an immediate connection with the relative
total curvature K which depends on the values of r and r'. The transmission errors under
unloaded condition are calculated [7, 8]. The results are illustrated in Fig. 7 and Fig. 8
for the drive side and coast side respectively. With the negative K, the functions of
transmission errors for cycles of meshing are discontinuous ones. This means that the next
pair of the meshing teeth is not in contact when one pair of teeth finishes their meshing.
Therefore, the discontinuous functions increase noise and vibration. With the positive K,
the functions of transmission errors become parabolic ones. The parabolic functions enable
the gears to absorb the errors caused by misalignment. Using r of 200 mm and r' of 100 mm,
it is possible to absorb the respective error of about }0.05 mm in misalignment which is
the displacement of the pinion along x, y, and z axis shown in Fig. 2.
With the duplex spread blade method, the machine settings may be done only two times,
namely, once is in gear cutting and once more is in pinion cutting. 1`4 in Table 3 are
the angles which determine the cutter machine settings of Gleason No. 116 hypoid generator.
They are the setting angles called tilt, swivel, eccentric, and cradle angles respectively
and can be calculated from Dp and ap. The calculation method was presented by Litvin [13].
Table 3 Calculated results of hypoid gear design (mm)
Table 4 Relative total curvature K ( 10-7/mm2)
7. ACTUAL GEAR CUTTING
To confirm the validity of this method, a pair of hypoid gears according to Table 1`3 was
manufactured. Then, the tooth surfaces of the ring gear and pinion were measured using a
coordinate measuring machine and the machine setting errors in gear cutting were detected
separately. The detection of the machine setting errors is performed using the tooth
surface measurement data of both the convex and concave sides.
Table 5 shows the detected results of the machine setting errors in ring gear cutting.
These results are obtained using a method for inspection developed by the authors [14].
t in Table 5 means the average deviation of the real tooth surface from the estimated
theoretical tooth surface and it is defined as the length along the circumference. When the
value of t is small, the estimated tooth surface fits the measured data well. Therefore,
t also means the accuracy of fit.
The coordinates of the cutter center Vg and Hg are transformed into the radial setting Rsg
( = Vg2 + Hg2) and base cradle angle Asg { = tan-1(Hg/Vg)}. Zg is the z coordinate of the
cutter center Oc and the design value of Zg is zero. Lg is the mounting distance and Lxg is
the distance along axis xm in Om-xmymzm with respect to the workpiece setting. From Table 5,
we find that each machine setting error is very small.
Fig. 7 Calculated transmission errors of drive side and coast side
The detected results of the machine setting errors in pinion cutting are shown in Table 6.
The subscript"p"is used instead of "g". The machine setting errors in pinion cutting are
also small. However, the smallest value of in Table 6 is 11.6m and is more than two
times as large as that of the ring gear. The reason is that the real cutter blade angle
2p' is larger than the design value by about 33 min. This is confirmed by measurement of
a copy of tool surface Xpc'
The gear and pinion were meshed with each other under light loaded condition. Figure 9(a)
shows the photograph of mating gears and Fig. 9(b) shows the sketched result of tooth bearing
on the gear tooth surface of convex side. In Fig. 9(b), the black point is the designated
center point of tooth bearing. The trial-and-error gear cutting is not carried out to
develop the desirable tooth bearing. The tooth bearing is acceptable although it appears
near the tooth top land.
Table 5 Detected results of errors in ring gear cutting
Table 6 Detected results of errors in pinion cutting
Fig. 9 Tooth bearing of mating gears
8. CONCLUSION
A duplex spread blade method for cutting hypoid gears with modified tooth surface was
proposed. In the proposed method, each inside and outside cutting edge of the cutter in
ring gear cutting is altered from the usual straight line to a circular arc with a large
radius of curvature. The circular cutting edges produce a proper tooth surface modification
for the duplex spread blade method. The duplex spread blade method is suitable for mass
production because each machine setting in ring gear and pinion cutting is done only one
time. A numerical example was presented and a pair of hypoid gears was manufactured
according to the proposed method.
ACKNOWLEDGMENTS
The authors are grateful for the support in gear cutting from the Suzuki Motor Co., Ltd
and Nagaoka Gear Works Co., Ltd.
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