Computerized Design Of Advanced Straight And Skew Bevel Gears Produced By Precision Forging

Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

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Comput. Methods Appl. Mech. Engrg.
journal homepage: www.elsevier.com/locate/cma

Computerized design of advanced straight and skew bevel gears produced
by precision forging
Alfonso Fuentes a,⇑, Jose L. Iserte b, Ignacio Gonzalez-Perez a, Francisco T. Sanchez-Marin b
a
b

Department of Mechanical Engineering, Polytechnic University of Cartagena (UPCT), Spain
Department of Mechanical Engineering and Construction, Universitat Jaume I, Castellon, Spain

a r t i c l e

i n f o

Article history:
Received 28 January 2011
Received in revised form 25 March 2011
Accepted 4 April 2011
Available online 13 April 2011
Keywords:
Bevel gears
Straight bevel
Skew bevel
Forging
TCA

a b s t r a c t
The computerized design of advanced straight and skew bevel gears produced by precision forging is proposed. Modifications of the tooth surfaces of one of the members of the gear set are proposed in order to
localize the bearing contact and predesign a favorable function of transmission errors. The proposed
modifications of the tooth surfaces will be computed by using a modified imaginary crown-gear and
applied in manufacturing through the use of the proper die geometry. The geometry of the die is obtained
for each member of the gear set from their theoretical geometry obtained considering its generation by
the corresponding imaginary crown-gear. Two types of surface modification, whole and partial crowning,
are investigated in order to get the more effective way of surface modification of skew and straight bevel
gears. A favorable function of transmission errors is predesigned to allow low levels of noise and vibration
of the gear drive. Numerical examples of design of both skew and straight bevel gear drives are included
to illustrate the advantages of the proposed geometry.
Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction
Bevel gears are used to transmit power between intersected
axes and are mainly used for automobile differentials. These gears
are cut or forged from conical blanks and connect shaft axes generally at 90° although designs for different shaft angles can be also
provided.
One of the most extended cutting technology for manufacturing
straight bevel gears is the coniflexÒ method (coniflex is a registered
trademark of The Gleason Works, Rochester, USA). This technology
takes advantage of the Phoenix free form flexibility and reduces
setup time to a minimum [1]. Coniflex straight bevel gears are
cut with a circular cutter with a circumferential blade arrangement. Nowadays, in the aim to look for more economical ways of
manufacturing bevel gears, cutting technologies might be replaced
for forming technologies [2–4].
The forging process of gear manufacturing was developed during the 1950s decade for manufacturing bevel gears for automobile
differentials, being stimulated by the lack of available gear cutting
equipment at that time [5]. The obtained precision was sufficient

for the automobiles of that period. The development of the technology for electric discharge machining of dies for precision forging
has allowed the manufacturing and application of forged bevel
gears to be extended during the recent years. It is based on the
⇑ Corresponding author.
E-mail address: (A. Fuentes).
0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.cma.2011.04.006

use of electrical discharged to remove material from the workpiece
of the die. Because the material removal is done point by point,
surface modifications can be applied to the forging die and therefore applied to the tooth surfaces of the manufactured gear. Following this idea, the reference geometry for the bevel gears is
obtained computationally in order to get the die geometry that will
achieve such geometry for the gears.
Among the different methods of forging, precision forging offers
the possibility of obtaining high quality parts, complex geometries
and good mechanical and technological properties [3]. It allows a
better material utilization in comparison to cutting, a reduction
of the costs of cutting because of shorter cycle times and new possibilities concerning the tooth surface geometry of the forged
gears. Precision forging contributes as well to fulfill the demand
of the production of highly loaded gears because of the fiber orientation which is favorable for carrying high oscillating loads [3].
In this paper, the computerized design of straight and skew bevel gears with localized bearing contact is proposed, partially based
on the ideas proposed by Professor Litvin et al. [6]. Modifications of
the tooth surfaces of one of the members of the gear set are proposed in order to localize the bearing contact and predesign a
favorable function of transmission errors. The proposed modifications of the tooth surfaces will be computed by using a modified
imaginary crown-gear and applied in manufacturing through the
use of the proper die geometry. The geometry of the die is obtained
for each member of the gear set from their theoretical geometry
obtained considering its generation by the corresponding

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imaginary crown-gear. Two types of surface modification, whole
and partial crowning, are investigated in order to get the more
effective way of surface modification of skew and straight bevel
gears.

According to typical design practice, the face width of a bevel
gear is generally chosen as one third of the outer pitch cone
distance,

Fw %
2. Basic design of a bevel gear transmission
The basic design parameters of a skew bevel gear transmission
(considering the straight bevel gear transmission as a particular
case of the mentioned above) are the module, m; the number of
teeth of pinion and gear, N1 and N2, respectively; the shaft angle,
R; the skew angle, b; and the pressure angle ad.
The gear ratio for bevel gears is given, as for other types of gears,
by

m12 ¼

x1 N 2
¼
;
x2 N 1



x2 N 1
;
m21 ¼
¼
x1 N 2

ð1Þ

where x1 and x2 are the angular velocities of pinion and gear,
respectively.
The pitch surfaces for bevel gears are cones. The larger end of
the pitch cone corresponds to the pitch diameter of the bevel gear.
Given the module and the number of teeth of pinion and gear, their
pitch radii are determined by

r p1 ¼

mN1
;
2

r p2 ¼

mN2
:
2

ð2Þ

Eq. (2) can be used for straight and skew bevel gears. Notice that the
skew angle is not considered when determining the pitch radii for
skew bevel gears.
The pitch angles of pinion and gear for any given shaft angle R
are determined by

c1 ¼ arctan




sin R
;
cos R þ m12

c2 ¼ arctan




sin R
:
cos R þ m21

ð3Þ

As shown in Fig. 1, the pitch cones are contained in a sphere of
radius Ro, the outer pitch cone distance, determined by

Ro ¼

r p1
r p2
¼
:
sin c1 sin c2

Ro
:
3

ð6Þ

3. Geometry of the imaginary generating crown-gear
The proposed geometry for straight and skew bevel gears is
achieved by considering an imaginary generating crown-gear as
the theoretical generating tool. The generating surfaces of the
imaginary crown-gear will be modified to apply the required surface modifications to the to-be-generated bevel gear.
The number of teeth of the theoretical crown gear, Ncg, is given
by

Ncg ¼

2Ro
;
m

ð7Þ

where Ro is the outer pitch cone distance. The number of teeth of
the theoretical crown gear can be a decimal number.
Fig. 2 shows the applied coordinate systems for the theoretical
generation of a straight or skew bevel gear by an imaginary
crown-gear. The crown gear is rotated around axis ycg and the
being-generated bevel gear is rotated around axis zi. Rotations of
the being-generated bevel gear (straight or skew) and the imaginary crown-gear are related by

wi ¼ wcg

Ncg
;
Ni

ði ¼ 1; 2Þ;

ð8Þ

where wi and Ni are the angle of rotation and number of teeth of the
pinion (i = 1) or the gear (i = 2), respectively, during their theoretical
generation, and wcg is the corresponding angle of rotation of the
generating crown-gear.
Two types of surface modifications, whole and partial crowning,
will be investigated in order to get the more effective way to modify a forged bevel gear. Fig. 3 shows a bevel gear tooth surface divided in nine zones wherein partial crowning (Fig. 3(a)) is applied,
or in four zones wherein conventional or whole parabolic crowning

ð4Þ

The face and root angles of the pinion and gear tooth surfaces, cF
and cR, will be determined by

cF 1;2 ¼ c1;2 þ

ha m
;
Ro

cR1;2 ¼ c1;2 À

hf m
:
Ro

ð5Þ

Here, ha and hf are the addendum and dedendum coefficients,
usually chosen to be 1.0 and 1.25, respectively.

Fig. 1. Pitch cones of bevel gears.

Fig. 2. Applied coordinate systems for theoretical generation of a straight bevel
gear by an imaginary crown-gear.


A. Fuentes et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

2365

Fig. 3. Areas of profile and longitudinal crowning to be applied to straight and skew bevel gears; (a) partial crowning and (b) whole crowning.

(Fig. 3(b)) is applied. With respect to Fig. 3(a), representing the
application of partial crowning:
(i) Zone 5 is an area of the bevel gear tooth surface where profile and longitudinal crowning are not applied.
(ii) Zones 1, 3, 7, and 9 are areas of crowning in profile and longitudinal directions.
(iii) Zones 2 and 8 are areas of crowning only in profile direction.
(iv) Zones 4 and 6 are areas of crowning only in longitudinal
direction.
When whole crowning of the gear tooth surface is applied, only
four areas exist provided with crowning in longitudinal and profile
directions (Fig. 3(b)). Those zones correspond to zones 1, 3, 7 and 9
in Fig. 3(a), because areas 2, 4, 5, 6, and 8 (Fig. 3(a)) do not exist
when whole crowning is applied.
In order to achieve the surface modifications described above, a
modified imaginary generating crown-gear will be applied for
computerized generation of the geometry of the bevel gear.
3.1. Geometry of the reference blade profile
The geometry of the imaginary generating crown-gear is based
on the geometry of a reference blade profile (Fig. 4). Both sides of

the blade profile will be defined in coordinate system Sc, fixed to
the blade, with its origin Oc placed on the middle of the segment
Oa Ob , with axis xc directed along the pitch line and the axis yc directed towards the addendum height of the reference blade. Auxiliary
coordinate systems Sa and Sb (see Fig. 4), with origins in Oa and Ob,
are rigidly connected to the blade profiles that will define the driving and coast sides of the theoretical crown gear, respectively, and
having their origins on the intersection of the pitch line with the
respective blade profiles. The axes ya and yb of coordinate systems
Sa and Sb are directed along the reference straight profile of the
blade towards the addendum height of the blade.
The profile of the blade is represented in coordinate systems Sa
and Sb (see Fig. 4) for left and right sides as

2

6
6
ra;b ðuÞ ¼ 6
6
4

Æapf ðu À u0 Þ2
u
0

3

7
7
7:
7
5

ð9Þ

1
Here, u is the blade profile parameter, apf is the parabola coefficient
for profile crowning, and u0 is the value of parameter u at the tangency point of the parabolic profile with the corresponding ya or yb
axis. The upper and lower signs of apf correspond to representation
of profile geometry in coordinate systems Sa and Sb for the left and
right sides, respectively.
The following conditions are established in order to apply profile crowning by considering three parts for the active part of the

reference blade profile:
 If u > u0t , then apf ¼ apft and u0 ¼ u0t (area A of zones 1, 2, and 3
in Fig. 3(a)).
 If u 6 u0t and u P u0b , then apf = 0 and u0 = 0 (area B of zones 4,
5, and 6 in Fig. 3(a)).
 If u < u0b , then apf ¼ apfb and u0 ¼ u0b (area C of zones 7, 8, and 9
in Fig. 3(a)).
Parameters ðapft ; u0t Þ, and ðapfb ; u0b Þ control the crowning and position of areas A and C, respectively, for profile crowning. By considering u0t ¼ u0b ¼ 0 and apft ¼ apfb we can take into account a
conventional parabolic profile for the reference blade profile. Similarly, by considering apft ¼ apfb ¼ 0 we can take into account a conventional straight profile for the reference blade profile.
Blade profiles corresponding to the left and right sides, are represented in coordinate system Sc as

Fig. 4. Reference blade profile definition.

rc ðuÞ ¼ Mca;b ra;b ðuÞ;

ð10Þ


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where

Mca;b

2

cos ad
6 Ç sin a

6
d
¼6
4
0
0

Æ sin ad
cos ad
0
0

0
0
1
0

Ç pm
4

0
0
1

3.2. Geometry of the imaginary generating crown-gear

3
7
7
7:

5

ð11Þ

Here, ad represents the pressure angle of the reference blade profile,
and the upper and lower signs correspond to the left and right blade
profiles.
By considering Eqs. (9)–(11), the reference blade profiles are
represented in coordinate system Sc as

2

6
6
rc ðuÞ ¼ 6
6
4

Æapf ðu À u0 Þ2 cos ad Æ u sin ad Ç p4m
ðiÞ

Àapf ðu À u0 Þ2 sin ad þ u cos ad
0
1
ðiÞ

3

7
7

7:
7
5

ð12Þ

As mentioned above, the upper and lower signs correspond to the
left and right blade profiles, respectively.

The following ideas are applied for definition of the geometry of
the generating crown-gear:
 The reference blade profile is developed over the outer sphere
defined by the pitch cones of the to-be-generated pinion and
gear, i.e., each point M of the reference blade profile has its corresponding point M0 on the sphere with radius Ro, the outer
pitch cone distance, as shown in Fig. 1.
 The pitch plane of the generating crown-gear is defined by the
pitch line of the reference blade profile and the center of the
sphere.
 The geometry of the imaginary generating crown-gear will be
obtained in coordinate system Scg, with origin in the center of
the outer sphere and axis zcg containing the origin Oc of the
reference blade profile coordinate system Sc, and axes xcg and
ycg parallel to axes xc and yc of the reference blade profile,
respectively (Fig. 5).

Fig. 5. Towards determination of the geometry of the imaginary generating crown-gear.

Fig. 6. For determination of the geometry of: (a) an imaginary straight generating crown-gear and (b) an imaginary skew generating crown-gear.

A. Fuentes et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

2367

 For definition of an imaginary straight crown-gear generating a
straight bevel gear, any given point M0 of the reference blade
profile over the outer sphere is projected towards the origin
Oh of coordinate system Sh, where Oh coincides with the center
of the outer sphere (Fig. 6(a)), defining lines of the generating
surface of a non-modified straight crown-gear.
 For definition of a skew imaginary crown-gear generating a
skew bevel gear, the projection point Oh, origin of coordinate
system Sh, for any given point M0 , is not the center of the outer
sphere but the tangent point with a circle defined on the pitch
plane of the generating crown-gear as shown in Fig. 6(b), whose
radius Rb is given by

Rb ¼ Ro sinðbÞ;

ð13Þ

where b is the skew angle of the bevel gear. The skew angle b is
considered positive for a right-hand skew bevel gear (as shown
in Fig. 6(b)) and negative for a left-hand skew bevel gear.
Fig. 7. Towards application of longitudinal crowning.

 For any given point M of the reference blade profile with coorðMÞ
ðMÞ
dinates xc and yc in coordinate system Sc (see Fig. 5), the corresponding point M0 on the outer sphere is defined considering
that: (i) point A0 on the outer sphere is obtained considering

_
that it is in the pitch plane and (ii) the length of arc Oc A0 is equal
ðMÞ
to jxc j. Point M0 on
_ the outer sphere is obtained knowing that
the length of arc A0 M 0 measured over the great circle defined by
ðMÞ
a plane normal to the pitch plane, is equal to jyc j.
 An auxiliary coordinate system Sh is defined for description of
the geometry of the imaginary crown-gear for each point M0
of the reference blade profile over the outer sphere. Coordinate
system Sh has the origin Oh in the center of the outer sphere for
a crown-gear generating a straight bevel gear or as mentioned
below for generation of skew bevel gears (see Fig. 6). Axis yh
is parallel to axis yc of the reference blade profile, and axis zh
is contained in the pitch plane of the crown gear with direction
of the projection of vector Oh M 0 on the pitch plane (Fig. 5).

A point P(u, h) on the imaginary generating crown-gear tooth
surface (Fig. 7) is defined by profile parameter u of the blade (that
defines the reference point M on the reference blade profile and
corresponding point M0 on the outer sphere) and its longitudinal
direction parameter h, measured from Oh on the projection line
Oh M 0 (Fig. 6).
For any given point M0 defined by profile parameter u of the reference blade profile, angles ab and aa can be determined (Fig. 5).
Angle ab defines point M0 in coordinate system Sh. Then, by considering angle aa and skew angle b, point M0 might be determined in
coordinate system Scg (Fig. 6). Angles ab and aa are given, for a nonmodified imaginary crown-gear, by (see Fig. 5):
_

A0 M 0 yc ðuÞ

ab ðuÞ ¼
¼
;
Ro
Ro

ð14Þ

_

aa ðuÞ ¼

Oc A0 xc ðuÞ
¼
:
Ro
Ro

Fig. 8. Coordinate systems applied for bevel gear generation by an imaginary crown-gear.

ð15Þ


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A. Fuentes et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

Fig. 9. Coordinate systems applied for TCA of bevel gears.

Longitudinal crowning is applied to the generating surfaces of

the imaginary crown-gear by modifying angle aa with Daa, determined by

Daa ðhÞ ¼ Æ

ald ðh À h0 Þ2
:
h

ð16Þ

Here, ald is the parabola coefficient for longitudinal crowning, h is
the longitudinal parameter, defined as mentioned above, and h0 is
the value of parameter h where modifications of the generating surface start.
By choosing appropriately different values for h0 and ald for the
toe and heel areas of the crown-gear generating tooth surface, partial longitudinal crowning can be applied, as shown in Fig. 7. The
upper sign in Eq. (16) is applied for generation of the driving side
of the bevel gear (left side) and the lower sign is applied for generation of the coast side of the bevel gear (right side). The modified
angle aa will be denoted as aÃa and is given by
Ã
a ðu; hÞ

a

xc ðuÞ ald ðh À h0 Þ2
¼ aa ðuÞ þ Daa ðhÞ ¼
Æ
:
Ro
h

ð17Þ

The following conditions have to be observed in Eq. (17) in order
to provide longitudinal partial crowning to the surfaces of the
imaginary generating crown-gear (Fig. 7). Three areas will be
considered:
 If h < h0t , then ald ¼ aldt and h0 ¼ h0t (area D of zones 1, 4, and 7
in Fig. 3(a)).

 If h P h0t and h 6 h0h , then ald = 0 (area E of zones 2, 5, and 8 in
Fig. 3(a)).
 If h > h0h , then ald ¼ aldh and h0 ¼ h0h (area F of zones 3, 6, and 9
in Fig. 3(a)).
Parameters ðaldt ; h0t Þ and ðaldh ; h0h Þ control the crowning and position of areas D and F, respectively, for longitudinal crowning. By
considering h0t ¼ h0h ¼ Ro À F w =2 and aldt ¼ aldh we can take into
account a conventional longitudinal parabolic crowned surface
for the imaginary crown-gear. Similarly, by considering
aldt ¼ aldh ¼ 0 we can take into account a non-modified surface in
longitudinal direction for the imaginary generating crown-gear.
According to the ideas described above, a point P(u, h) is given in
coordinate system Sh by (see Fig. 5)

3
0
6 h sin a ðuÞ 7
b
7
6
rh ðu; hÞ ¼ 6
7:

4 h cos ab ðuÞ 5
2

ð18Þ

1
Considering coordinate transformation from Sh to Scg as shown
in Fig. 6(b), the generating surfaces of an imaginary skew crown
gear are given by

rcg ðu; hÞ ¼ Mcgh ðaÃa ðu; hÞÞrh ðu; hÞ;
where

ð19Þ


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A. Fuentes et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

Fig. 10. Errors of alignments: (a) axial displacement of the pinion DA1, (b) axial displacement of the gear DA2, (c) change of the shaft angle DR and (d) shortest distance
between axes DE.

2

cosðb À aÃa Þ 0 À sinðb À aÃa Þ Rb cosðb À aÃa Þ

6
0
6

Mcgh ðaÃa Þ ¼ 6
6
4 sinðb À aÃa Þ
0

1

0

0

cosðb À aÃa Þ

0

0

3

@ aÃa dxc =du
;
¼
Ro
@u
@ ab dyc =du
;
¼
Ro
@u

7
0
7
7:
à 7
Rb sinðb À aa Þ 5
1

@a
2ald ðh À h0 Þh À ald ðh À h0 Þ
;
¼Æ
@h
h2
@ ab
¼ 0:
@h

Considering Eqs. (18)–(20), Eq. (19) can be represented by

Rb cosðb À aÃa Þ À h cosðab Þ sinðb À aÃa Þ

3

Step 3.

6
7
6
7

h sinðab Þ
7
rcg ðu; hÞ ¼ 6
6 R sinðb À aÃ Þ þ h cosða Þ cosðb À aÃ Þ 7:
4 b
b
a
a 5
1

ð21Þ

The derivative of the reference blade profile (Eq. (12))
with respect to the profile parameter is obtained as

2 dxc 3
du

2

Æ2apf ðu À u0 Þ cos ad Æ sin ad

3

drc 6 dy 7 6
7
¼ 4 c 5 ¼ 4 À2apf ðu À u0 Þ sin ad þ cos ad 5:
du
du
0

0

Step 2.

2

ð26Þ

ð27Þ

3
ð28Þ

The surface of the imaginary crown-gear is given by Eq.
(19). The normal will therefore be obtained by

ð22Þ

Here, the upper and lower signs correspond to the left and
right blade profiles, respectively.
The derivatives to the angles aÃa and ab with respect to
surface parameters u and h are also needed. We recall that
the mentioned angles are defined by Eqs. (17) and (14),
respectively. Their derivatives are given by

0

@rh 6
7
¼ 4 sin ab 5:

@h
cos ab
Step 4.

ð25Þ

The position vector of a point P(u, h) of the imaginary
crown-gear in coordinate system Sh is given by Eq. (18).
Its derivatives with respect to surface parameters u and
h, used below for determination of the normal to the
imaginary crown-gear generating surfaces, are given by

2
3
0
@rh 6
dab 7
¼ 4 h cos ab du 5;
@u
ab
Àh sin ab ddu

By considering b = 0 in Eq. (21), and therefore considering
Rb = 0, the generating surfaces of an imaginary straight crown-gear
are obtained in coordinate system Scg.
For determination of the equation of meshing, the normal to the
generating surfaces, represented by Eq. (21), is determined by the
following steps:

Step 1.

ð24Þ
2

Ã
a

ð20Þ

2

ð23Þ

Ncg ðu; hÞ ¼



@rcg @rcg
Â
:
@u
@h

ð29Þ

Derivatives @rcg/@u and @rcg/@h are given by

@rcg @Mcgh
@rh
¼

rh þ Mcgh
;
@u
@u
@u
@rcg @Mcgh
@rh
¼
rh þ Mcgh
:
@h
@h
@h

ð30Þ
ð31Þ


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A. Fuentes et al. / Comput. Methods Appl. Mech. Engrg. 200 (2011) 2363–2377

Table 1
Details of coordinate system transformation from S2 to S1.
Matrix

Transformation

Magnitude

Axis

Ml2
Mml
Mnm
Mfn
M1f

Rotation CCW
Rotation CCW
Rotation CCW
Translation
Rotation CCW

/2

z2  zl
zl  zm
xm  xn
N/A
zf  z1

p
R + DR
[ÀDE, DA2, ÀDA1]T
/1

CCW = Counterclockwise; CW = Clockwise; N/A = Not applicable.

Table 2

Main design parameters of two forged bevel gear drives.

Module, m (mm)
Number of pinion teeth, N1
Number of gear teeth, N2
Shaft angle, R (deg)
Skew angle, b (deg)
Pressure angle, an (deg)

Drive A (straight)

Drive B (skew)

4
25
36
90°
0°
25°

4
25
36
90°
10°
25°

Here, derivatives @rh/@u and @rh/@h are obtained by Eqs. (27) and
(28), respectively. Derivatives @Mcgh/@u and @Mcgh/@h can be obtained by derivation of Eq. (20), considering that angle aÃa ¼ aÃa ðu; hÞ.
4. Geometry of straight and skew bevel gears

The proposed new geometry of straight and skew bevel gears is
obtained by considering their computerized generation by an
imaginary crown-gear, whose geometry has been described in
the previous section. A modified crown-gear will be used for the
theoretical generation of the pinion whereas a non-modified
crown-gear is used for generation of the gear.
Fig. 8 shows the coordinate systems applied for the theoretical
generation of a bevel gear (straight or skew) by a crown-gear, and
complements those coordinate systems illustrated in Fig. 2.
Coordinate systems Scg and Si are rigidly connected to the generating imaginary crown-gear and the being generated bevel gear
(i = 1 for the pinion and i = 2 for the gear), respectively. Coordinate
systems Sj, Sk, and Sl are auxiliary coordinate systems. Angle ci is
the pitch angle of the being-generated gear (Eq. (3)).
We recall that the imaginary crown-gear generating tooth surfaces are given by Eq. (21). The bevel gear tooth surfaces are determined as the envelope of the family of generating crown-gear

Table 3
Studied cases of design with design characteristics for tooth surface modifications of
the pinion member of the gear set according to Section 3.
Case 1 (Non-modified)

Case 2 (Whole crowned)

À1

À1

Case 3 (Partial crowned)

apft ¼ 0:0 mm

apft ¼ 0:0004 mm

apftop ¼ 0:001 mmÀ1

apfb ¼ 0:0 mmÀ1
u0t = 0.0 mm
u0b = 0.0 mm
h0t = 73.0584 mm
h0h = 73.0584 mm
aldt ¼ 0:0 mmÀ1
aldh ¼ 0:0 mmÀ1

apfb ¼ 0:0004 mmÀ1
u0t = 0.0 mm
u0b = 0.0 mm
h0t = 73.0584 mm
h0h = 73.0584 mm
aldt ¼ 0:0001 mmÀ1
aldh ¼ 0:0001 mmÀ1

apfbottom ¼ 0:0004 mmÀ1
u0t = 0.5517 mm
u0b = À1.1034 mm
h0t = 64.2984 mm
h0h = 81.8184 mm
aldtoe ¼ 0:001 mmÀ1
aldheel ¼ 0:001 mmÀ1

Table 4
Studied misaligned conditions.

Condition a

Condition b

Condition c

Condition d

DA1 = 0.0 mm
DA2 = 0.0 mm
DE = 0.0 mm
DR = 0.0 deg

DA1 = 0.0 mm
DA2 = 0.0 mm
DE = 0.05 mm
DR = 0.0 deg

DA1 = 0.0 mm
DA2 = 0.0 mm
DE = 0.0 mm
D R = 0.5 deg

DA1 = 0.1 mm
DA2 = 0.0 mm
DE = 0.0 mm
DR = 0.0 deg

tooth surfaces in coordinate system Si, fixed to the pinion (i = 1)
or fixed to the gear (i = 2), and represented as (Fig. 8)

ri ðu; h; wi Þ ¼ Mik ðwi ÞMkj Mjcg ðwcg ðwi ÞÞrcg ðu; hÞ:

ð32Þ

Here,

2

cos wcg
6 0
6
Mjcg ðwcg ðwi ÞÞ ¼ 6
4 sin wcg
0
2

1
0
6 0 cos c
6
i
Mkj ¼ 6
4 0 À sin ci

0
sin ci

0

0

0
2

cos wi
6 À sin w
6
i
Mik ðwi Þ ¼ 6
4
0
0

0 À sin wcg
1

0

0

cos wcg

0

0

cos c

3

07
7
7;
05

ð33Þ

1

3
0
07
7
7;
05

ð34Þ

1

0

3
0 0
0 07
7
7:
1 05

0

0 1

sin wi
cos wi

0

Fig. 11. (a) Contact pattern and (b) function of transmission errors for case A1a (straight(A) non-modified(1) aligned(a) bevel gear drive).

ð35Þ


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Fig. 12. Contact patterns for: (a) case A1b, (b) case A1c, (c) case A1d and (d) functions of transmission errors for previous cases of design.

Angles wcg and wi are the angles of rotation of the imaginary
generating crown-gear and the being-generated bevel gear, related
by

wcg ðwi Þ ¼

Ni
w:
Ncg i

ð36Þ

@ri @Mik ðwi Þ
¼
Mkj Mjcg ðwcg Þrcg ðu; hÞ
@wi
@wi
@Mjcg ðwcg Þ @wcg
rcg ðu; hÞ:
þ Mik ðwi ÞMkj
@wi
@wcg
Here,

2
The derivation of the bevel gear tooth surfaces is based on the
simultaneous consideration of Eq. (32) and the equation of
meshing,

À sin wi

@Mik ðwi Þ 6
6 À cos wi
¼6
4
@wi
0

cos wi

ð37Þ

À sin wcg
@Mjcg ðwcg Þ 6
0
6
¼6
4 cos wcg
@wcg

Eq. (37) is represented in differential geometry [7] as



@ri @ri
@ri
¼ 0:
Â
Á
@u @h
@wi

0

ð38Þ

0
0

0 0

0 À cos wcg
0

0

0

À sin wcg

0

0

ð41Þ

0

3

07
7
7;
05

ð43Þ

Using Eqs. (39) and (40), the equation of meshing is represented

ð39Þ

where Ncg(u, h) represents the normal to the imaginary generating
crown-gear surface represented in coordinate system Scg (Eq.
(29)), and matrices L are 3 Â 3 matrices, which may be obtained
by eliminating the last row and the last column of the corresponding matrices M (Eqs. (33)–(35)). Derivative @ri/@ wi in Eq. (38) is represented as

ð42Þ

0

@wcg
Ni
¼
:
@wi
Ncg

Here,



@ri @ri
¼ Ni ðu; hÞ ¼ Lik ðwi ÞLkj Ljcg ðwcg ðwi ÞÞNcg ðu; hÞ;
Â
@u @h

2

3

0 07
7
7;
0 05

À sin wi

0
ficg ðu; h; wi Þ ¼ 0:

0 0

as



@Mik ðwi Þ
Mkj Mjcg ðwcg Þ
@wi
#
@Mjcg ðwcg Þ @wcg
rcg ðu; hÞ ¼ 0:
þ Mik ðwi ÞMkj
@wcg
@wi

ficg ðu; h; wi Þ ¼ Ni ðu; hÞ Á

ð44Þ

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Fig. 13. (a) Contact pattern and (b) function of transmission errors for case A2a (straight(A) whole-crowned(2) aligned(a) bevel gear drive).

Fig. 14. (a) Contact pattern and (b) function of transmission errors for case A3a (straight(A) partial-crowned(3) aligned(a) bevel gear drive).

Simultaneous consideration of Eqs. (32) and (44) allows determination of the geometry of a straight or skew bevel gear with
modified geometry to be manufacturing by forging.
5. Computerized simulation of meshing and contact
A new general purpose algorithm for tooth contact analysis
(TCA) of gear drives has been developed and applied for tooth contact analysis of forged straight and skew bevel gears. It is based on
a numerical method that takes into account the positional study of
the surfaces and minimization of the distances until contact is
achieved. A virtual marking compound thickness of 0.0065 mm
has been used for determination of the contact patterns for all
cases. This algorithm for tooth contact analysis does not depend
on the precondition that the surfaces are in point contact or the
solution of any system of nonlinear equations as the existing approaches, and can be applied for tooth contact analysis of gear
drives in point, lineal or edge contact as it will be shown below.
Alternative algorithms that can be used for tooth contact analysis
are found in [7–9]. All TCA analyses are conducted under rigidbody assumptions so that no elastic tooth deformation due to actual loading is considered when TCA results are shown.
5.1. Applied coordinate systems
Fig. 9 represents the applied coordinate systems for tooth contact analysis (TCA) of straight and skew bevel gears.

5.2. Errors of alignment
The errors of alignment considered are: (i) DA1 – the axial displacement of the pinion (Fig. 10(a)), (ii) DA2 – the axial displacement of the gear (Fig. 10(b)), (iii) DR – the change of the shaft
angle R (Fig. 10(c)), and (iv) DE – the shortest distance between

axes of the pinion and the gear when these axes are not intersected
but crossed (Fig. 10(d)). The mentioned errors of alignment can
also be observed in Fig. 9.
Coordinate systems S1 and S2 are movable coordinate systems
rigidly connected to the pinion and gear, respectively. Angles /1
and /2 are the angles of rotation of the pinion and the gear, respectively. Table 1 shows details of coordinate transformation from S2
to S1.
Transformation Mml is needed if pinion and gear have been generated following the same coordinate transformations, so that one
of the members of the gear drive have to be rotated an angle p to
face corresponding surfaces for tooth contact analysis.
5.3. Discussion of obtained results
Two bevel gear drives manufactured by forging, one straight
and the other skew, with main design parameters shown in Table 2
will be considered for tooth contact analysis. Three cases of design
for each gear drive, with design characteristics shown in Table 3,
will be considered. Those design characteristics correspond to the
tooth surface modifications of the pinion member of the gear set.


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Fig. 15. Contact patterns for: (a) case A2b, (b) case A2c, (c) case A2d and (d) functions of transmission errors for previous cases of design.

The wheel member will be considered non-modified for all three
cases of design. Then, four misaligned conditions, shown in Table 4,
will be investigated in order to check up the sensitivity of the contact pattern to the appearance of errors of alignment and the obtained function of transmission errors. Condition a corresponds
to the aligned gear drive. Only results of tooth contact analysis concerning the driving side of the bevel gear transmission will be
shown below.

Each investigated gear drive, case of design and misalignment
condition will be denoted by three letters. As an example, case
B2a will correspond to a skew gear drive, with whole crowned surfaces, and aligned conditions (see Tables 2–4).
Fig. 11 shows the contact pattern and the obtained function of
transmission errors for case A1a corresponding to a straight nonmodified and aligned bevel gear drive. The bevel gear drive, under
aligned conditions, has no transmission errors, and the contact pattern covers the whole surface of the teeth.
Fig. 12 shows the contact patterns for cases A1b (12(a)), A1c
(12(b)), and A1d (12(c)). Fig. (12(d)) shows the obtained functions
of transmission errors for previous cases of design. The shortest
distance between axes DE (misaligned condition b) and the change
of shaft angle DR (misaligned condition c) do not cause high transmission errors for the non-modified bevel gears. However, function
of transmission errors is very sensitive to the axial displacement of
the pinion (misaligned condition d) and the axial displacement of
the gear (not shown in this paper), having in these cases lineal
functions of transmission errors that are the source of high levels
of noise and vibration of the gear drive.
In order to absorb the lineal functions of transmission errors
caused by errors of alignment in general and the axial displacements of pinion or gear in particular, Designs 2 and 3 (see Table 3)

are proposed with whole-crowned and partial-crowned bevel gear
tooth surfaces, respectively. Figs. 13 and 14 shows the contact patterns and the predesigned functions of transmission errors for
cases A2a and A3a corresponding to a straight whole-crowned
and aligned bevel gear drive (Fig. 13) and to a straight partialcrowned and aligned bevel gear drive (Fig. 14). Parabolic functions
of transmission errors have been predesigned with levels of
8.5 arcsec for whole-crowned surfaces (Design 2) and 5.5 arcsec
for partial-crowned surfaces (Design 3). Design 3 (partial-crowning) allows the contact pattern to cover a larger area of the bevel
gear contacting surfaces, by creating an area of no modification
of the tooth surfaces. The main advantage of this geometry is that
the lower the misalignment is, the bigger the contact pattern is obtained, allowing contact stresses to be reduced.
Fig. 15 shows the contact patterns for cases A2b (15(a)), A2c

(15(b)), and A2d (15(c)). Fig. (15(d)) shows the obtained functions
of transmission errors for previous cases of design. For cases of design A2b and A2c, the contact pattern is kept inside de contacting
surfaces although for case of design A2d, it is shifted towards the
top edge of the wheel, and might cause high contact stresses. All
functions of transmission errors are obtained with parabolic shape,
absorbing efficiently the lineal functions of transmission errors
caused by errors of alignment for non-modified bevel gear tooth
surfaces.
Fig. 16 shows the contact patterns for cases A3b (16(a)), A3c
(16(b)), and A3d (16(c)). Fig. (16(d)) shows the obtained functions
of transmission errors for previous cases of design. For cases of design A3b and A3c, the contact patterns are localized inside the contacting surfaces, avoiding edge contacts, although for case of
design A3d the contact pattern is slightly shifted towards the top
edge of the pinion. The lineal function of transmission errors


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Fig. 16. Contact patterns for: (a) case A3b, (b) case A3c, (c) case A3d and (d) functions of transmission errors for previous cases of design.

Fig. 17. (a) Contact pattern and (b) function of transmission errors for case B1a (skew(B) non-modified(1) aligned(a) bevel gear drive).

caused by the axial displacement of the pinion (misaligned condition c) is not completely absorbed, so that the partial profile
crowning is not working properly for this geometry.
The skew bevel gear transmission with main design parameters
shown in Table 2 is also designed using parameters shown in
Table 3 for three different cases of design. Fig. 17 shows the contact
pattern and the obtained function of transmission errors for case
B1a corresponding to a skew non-modified and aligned bevel gear

drive. The skew bevel gear drive with the proposed geometry, under aligned conditions, has no transmission errors, and the contact

pattern covers the whole surface of the teeth as shown in
Fig. 17(a).
Fig. 18 shows the contact patterns for cases B1b (18(a)), B1c
(18(b)), and B1d (18(c)). Fig. (18(d)) shows the obtained functions
of transmission errors for previous cases of design. All misaligned
conditions (from b to d) cause lineal functions of transmission errors. The skew bevel gear drive is very sensitive to the change of
shaft angle DR (misaligned condition c) and the axial displacement
of pinion (misaligned condition d) and the axial displacement of
the gear (not shown in this paper).


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Fig. 18. Contact patterns for: (a) case B1b, (b) case B1c, (c) case B1d and (d) functions of transmission errors for previous cases of design.

Fig. 19. (a) Contact pattern and (b) function of transmission errors for case B2a (skew(B) whole-crowned(2) aligned(a) bevel gear drive).

In order to absorb those lineal functions of transmission errors
caused by errors of alignment for the skew bevel gear drive, Designs 2 and 3 (see Table 3) are proposed also for this transmission,
with whole-crowned and partial-crowned bevel gear tooth surfaces, respectively. Figs. 19 and 20 shows the contact patterns
and the predesigned functions of transmission errors for cases
B2a and B3a corresponding to a skew whole-crowned and aligned
bevel gear drive (Fig. 19) and to a skew partial-crowned and
aligned bevel gear drive (Fig. 20). A parabolic function of transmission errors with maximum level of 7 arcsec has been predesigned
for the whole-crowned skew bevel gear drive (Design 2). However,
for the case of partial-crowned skew bevel gear drive (Design 3), a

function of transmission error of 2 arcsec is obtained taking advantage of an area of non-modified tooth surface due to partial crowning. Again, Design 3 (partial-crowning) allows the contact pattern
to cover a larger area of the bevel gear contacting surfaces.
Fig. 21 shows the contact patterns for cases B2b (21(a)), B2c
(21(b)), and B2d (21(c)). Fig. (21(d)) shows the obtained functions
of transmission errors for previous cases of design. Although for
cases of design B2b and B2c the contact pattern is localized inside
the contacting surfaces, avoiding undesirable edge contacts, when
an axial displacement of the pinion occurs, the contact pattern is
shifted towards the edge of the gear as shown in Fig. 21(c). All
functions of transmission errors are obtained with parabolic shape,


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Fig. 20. (a) Contact pattern and (b) function of transmission errors for case B3a (skew(B) partial-crowned(3) aligned(a) bevel gear drive).

Fig. 21. Contact patterns for: (a) case B2b, (b) case B2c, (c) case B2d and (d) functions of transmission errors for previous cases of design.

absorbing efficiently the lineal functions of transmission errors
caused by errors of alignment for non-modified bevel gear tooth
surfaces.
Fig. 22 shows the contact patterns for cases B3b (22(a)), B3c
(22(b)), and B3d (22(c)). Fig. (22(d)) shows the obtained functions
of transmission errors for previous cases of design. For this design,
the contact patterns are also localized inside the contacting surfaces, avoiding edge contacts, and the predesigned function of
transmission errors is able to absorb the lineal functions of transmission errors caused by errors of alignment.

6. Conclusions
The performed research work allows the following conclusions
to be drawn:
1. The computerized design of advanced straight and skew bevel
gears produced by precision forging has been developed. The
developed approach takes into account modified tooth surfaces
for the pinion member of the gear set in order to localize the
bearing contact by the use of the proper die geometry.


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Fig. 22. Contact patterns for: (a) case B3b, (b) case B3c, (c) case B3d and (d) functions of transmission errors for previous cases of design.

2. The geometry of the die for the pinion is computed considering
its theoretical generation by a modified imaginary crown-gear
so that the applied surface modifications are applied directly
to the pinion when it is precision forged. The wheel member
of the gear set remains non-modified.
3. Two types of surface modification, whole and partial crowning,
are investigated in order to get the more effective way of surface modification of skew and straight bevel gears. Whole
crowning has been probed to be the most effective way of modification of straight and skew bevel gear tooth surfaces, keeping
the contact localized, and yielding low levels of parabolic functions of transmission errors.
Acknowledgments
The authors express their deep gratitude to the Spanish Ministry of Science and Innovation (MICINN) for the financial support of
research projects Ref. DPI2010-20388-C02-01 (financed jointly by
FEDER) and DPI2010-20388-C02-02.

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