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

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 g

Computerized design of advanced straight and skew bevel gears produced

by precision forging

Alfonso Fuentesa,⇑, Jose L Iserteb, Ignacio Gonzalez-Pereza, Francisco T Sanchez-Marinb

a

Department of Mechanical Engineering, Polytechnic University of Cartagena (UPCT), Spain

b

Department of Mechanical Engineering and Construction, Universitat Jaume I, Castellon, Spain

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 pro-posed 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

gener-ally 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 coniflexÒmethod (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

arrange-ment 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

dur-ing the 1950s decade for manufacturdur-ing 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

tech-nology 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

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 there-fore applied to the tooth surfaces of the manufactured gear Fol-lowing 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 pos-sibilities 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 orien-tation which is favorable for carrying high oscillating loads[3]

In this paper, the computerized design of straight and skew

bev-el 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 pro-posed in order to localize the bearing contact and predesign a favorable function of transmission errors The proposed modifica-tions 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

0045-7825/$ - see front matter Ó 2011 Elsevier B.V All rights reserved.

⇑ Corresponding author.

E-mail address: alfonso.fuentes@upct.es (A Fuentes).

Contents lists available atScienceDirect Comput Methods Appl Mech Engrg.

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c m a

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

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, N1and N2, respectively; the shaft angle,

R; the skew angle, b; and the pressure anglead

The gear ratio for bevel gears is given, as for other types of gears,

by

x2¼N2

x1¼N1

where x1 andx2 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

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 angleR

are determined by

As shown in Fig 1, the pitch cones are contained in a sphere of

radius Ro, the outer pitch cone distance, determined by

rp 2

The face and root angles of the pinion and gear tooth surfaces,cF

andcR, will be determined by

cF1;2¼c1;2þham

Ro

Ro

Here, ha and hf are the addendum and dedendum coefficients,

usually chosen to be 1.0 and 1.25, respectively

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

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 sur-face modifications to the to-be-generated bevel gear

The number of teeth of the theoretical crown gear, Ncg, is given by

where Rois the outer pitch cone distance The number of teeth of the theoretical crown gear can be a decimal number

Fig 2shows 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 imagi-nary crown-gear are related by

Ncg

Ni

wherewiand Niare the angle of rotation and number of teeth of the pinion (i = 1) or the gear (i = 2), respectively, during their theoretical generation, andwcgis 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 mod-ify a forged bevel gear.Fig 3shows a bevel gear tooth surface di-vided in nine zones wherein partial crowning (Fig 3(a)) is applied,

or in four zones wherein conventional or whole parabolic crowning

Fig 2 Applied coordinate systems for theoretical generation of a straight bevel

(Fig 3(b)) is applied With respect toFig 3(a), representing the

application of partial crowning:

(i) Zone 5 is an area of the bevel gear tooth surface where

pro-file and longitudinal crowning are not applied

(ii) Zones 1, 3, 7, and 9 are areas of crowning in profile and

lon-gitudinal 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

inFig 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 Ocplaced on the middle of the segment

OaOb, with axis xcdirected along the pitch line and the axis yc direc-ted towards the addendum height of the reference blade Auxiliary coordinate systems Saand Sb(seeFig 4), with origins in Oaand Ob, are rigidly connected to the blade profiles that will define the driv-ing 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 yaand ybof coordinate systems

Saand Sbare 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(seeFig 4) for left and right sides as

u 0 1

2 6 6 4

3 7 7

Here, u is the blade profile parameter, apfis the parabola coefficient for profile crowning, and u0is the value of parameter u at the tan-gency point of the parabolic profile with the corresponding yaor yb

axis The upper and lower signs of apfcorrespond to representation

of profile geometry in coordinate systems Saand Sbfor the left and right sides, respectively

The following conditions are established in order to apply pro-file crowning by considering three parts for the active part of the reference blade profile:

 If u > u0 t, then apf¼ apf tand u0¼ u0 t(area A of zones 1, 2, and 3

inFig 3(a))

 If u 6 u0 t and u P u0b, then apf= 0 and u0= 0 (area B of zones 4,

5, and 6 inFig 3(a))

 If u < u0 b, then apf ¼ apf band u0¼ u0 b(area C of zones 7, 8, and 9

inFig 3(a))

Parameters ðapf t;u0 tÞ, and ðapf b;u0 bÞ control the crowning and po-sition of areas A and C, respectively, for profile crowning By con-sidering u0 t ¼ u0b¼ 0 and apf t¼ apfb we can take into account a conventional parabolic profile for the reference blade profile Sim-ilarly, by considering apft¼ apf b¼ 0 we can take into account a con-ventional straight profile for the reference blade profile

Blade profiles corresponding to the left and right sides, are rep-resented in coordinate system Scas

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

4

2

6

6

3 7

Here,adrepresents 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 Scas

4

0 1

2

6

6

4

3 7 7

As mentioned above, the upper and lower signs correspond to the

left and right blade profiles, respectively

3.2 Geometry of the imaginary generating crown-gear 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 cor-responding point M0 on the sphere with radius Ro, the outer pitch cone distance, as shown inFig 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 xcgand

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.

 For any given point M of the reference blade profile with

coor-dinates xðMÞ

c and yðMÞ

c in coordinate system Sc(seeFig 5), the cor-responding point M0on 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_A0is equal

to jxðMÞ

c j Point M0on the outer sphere is obtained knowing that

the length of arc A0_M0measured over the great circle defined by

a plane normal to the pitch plane, is equal to jyðMÞ

c j

 An auxiliary coordinate system Shis 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 Shhas the origin Ohin 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 (seeFig 6) Axis yh

is parallel to axis ycof the reference blade profile, and axis zh

is contained in the pitch plane of the crown gear with direction

of the projection of vector OhM0on the pitch plane (Fig 5)

 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

Ohof coordinate system Sh, where Ohcoincides 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 inFig 6(b), whose radius Rbis given by

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

inFig 6(b)) and negative for a left-hand skew bevel gear

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 M0on the outer sphere) and its longitudinal direction parameter h, measured from Oh on the projection line

OhM0(Fig 6)

For any given point M0defined by profile parameter u of the ref-erence blade profile, anglesabandaacan be determined (Fig 5) Angleabdefines point M0in coordinate system Sh Then, by consid-ering angleaaand skew angle b, point M0might be determined in coordinate system Scg(Fig 6) Anglesabandaaare given, for a non-modified imaginary crown-gear, by (seeFig 5):

0

_

Ro

0 _

Ro

Fig 7 Towards application of longitudinal crowning.

Longitudinal crowning is applied to the generating surfaces of

the imaginary crown-gear by modifying angleaawithDaa,

deter-mined by

2

Here, aldis the parabola coefficient for longitudinal crowning, h is

the longitudinal parameter, defined as mentioned above, and h0is

the value of parameter h where modifications of the generating

sur-face start

By choosing appropriately different values for h0and aldfor the

toe and heel areas of the crown-gear generating tooth surface,

par-tial longitudinal crowning can be applied, as shown inFig 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

gener-ation of the coast side of the bevel gear (right side) The modified

angleaawill be denoted asaand is given by

Ro

2

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 < h0 t, then ald¼ ald tand h0¼ h0 t (area D of zones 1, 4, and 7

inFig 3(a))

 If h P h0 t and h 6 h0h, then ald= 0 (area E of zones 2, 5, and 8 in

Fig 3(a))

 If h > h0 h, then ald¼ ald h and h0¼ h0 h(area F of zones 3, 6, and 9

inFig 3(a))

Parameters ðald t;h0 tÞ and ðaldh;h0 hÞ control the crowning and po-sition of areas D and F, respectively, for longitudinal crowning By considering h0 t ¼ h0h ¼ Ro Fw=2 and aldt¼ ald h we can take into account a conventional longitudinal parabolic crowned surface for the imaginary crown-gear Similarly, by considering

aldt¼ ald h¼ 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 Shby (seeFig 5)

0

1

2 6 6

3 7

Considering coordinate transformation from Shto Scgas shown

inFig 6(b), the generating surfaces of an imaginary skew crown gear are given by

where

Fig 9 Coordinate systems applied for TCA of bevel gears.

McghðaÞ ¼

2

6

6

4

3 7 7

5: ð20Þ

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

hsinðabÞ

1

2

6

6

4

3 7 7

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 The derivative of the reference blade profile (Eq (12))

with respect to the profile parameter is obtained as

drc

dx c

du

dy c

du

0

2

6

3 7

5 ¼

0

2 6

3 7

Here, the upper and lower signs correspond to the left and

right blade profiles, respectively

Step 2 The derivatives to the anglesa and abwith 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

@a

Ro

@ab

Ro

@a

2

@ab

Step 3 The position vector of a point P(u, h) of the imaginary

crown-gear in coordinate system Shis 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

0

du

du

2 6

3 7

0

2 6

3 7

Step 4 The surface of the imaginary crown-gear is given by Eq

(19) The normal will therefore be obtained by

@h

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

@rh

@rh

Fig 10 Errors of alignments: (a) axial displacement of the pinion DA 1 , (b) axial displacement of the gear DA 2 , (c) change of the shaft angle DRand (d) shortest distance between axes DE.

Here, derivatives @rh/@u and @rh/@h are obtained by Eqs.(27) and

(28), respectively Derivatives @Mcgh/@u and @Mcgh/@h can be

ob-tained by derivation of Eq.(20), considering that anglea¼aðu; 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 8shows 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 inFig 2

Coordinate systems Scgand Siare rigidly connected to the

gen-erating 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 Slare auxiliary coordinate systems Angleciis

the pitch angle of the being-generated gear (Eq.(3))

We recall that the imaginary crown-gear generating tooth

sur-faces are given by Eq.(21) The bevel gear tooth surfaces are

deter-mined as the envelope of the family of generating crown-gear

tooth surfaces in coordinate system Si, fixed to the pinion (i = 1)

or fixed to the gear (i = 2), and represented as (Fig 8)

Here,

2 6 6

3 7

2 6 6

3 7

2 6 6

3 7

Table 1

Details of coordinate system transformation from S 2 to S 1

N/A

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

Table 2

Main design parameters of two forged bevel gear drives.

Drive A (straight) Drive B (skew)

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) Case 3 (Partial crowned)

a pf t ¼ 0:0 mm 1 a pf t ¼ 0:0004 mm 1 a pf top ¼ 0:001 mm 1

apfb¼ 0:0 mm 1 apfb¼ 0:0004 mm 1 apfbottom¼ 0:0004 mm 1

h 0t = 73.0584 mm h 0t = 73.0584 mm h 0t = 64.2984 mm

h 0h = 73.0584 mm h 0h = 73.0584 mm h 0h = 81.8184 mm

a ld t ¼ 0:0 mm 1

a ld t ¼ 0:0001 mm 1

a ld toe ¼ 0:001 mm 1

a ld h ¼ 0:0 mm 1 a ld h ¼ 0:0001 mm 1 a ld heel ¼ 0:001 mm 1

Table 4 Studied misaligned conditions.

DA 1 = 0.0 mm DA 1 = 0.0 mm DA 1 = 0.0 mm DA 1 = 0.1 mm

DA 2 = 0.0 mm DA 2 = 0.0 mm DA 2 = 0.0 mm DA 2 = 0.0 mm

Angles wcg and wi are the angles of rotation of the imaginary

generating crown-gear and the being-generated bevel gear, related

by

Ncg

The derivation of the bevel gear tooth surfaces is based on the

simultaneous consideration of Eq (32) and the equation of

meshing,

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

@ri

@ri

@h

i

Here,

@ri

@ri

@h

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  3 matrices, which may be obtained

by eliminating the last row and the last column of the

correspond-ing matrices M (Eqs.(33)–(35)) Derivative @ri/@wiin Eq.(38)is

rep-resented as

@ri

@wi

MkjMjcgðwcgÞrcgðu; hÞ

@wi

rcgðu; hÞ:

Here,

@wi

¼

2 6 6

3 7

¼

2 6 6

3 7

Ncg

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

ficgðu; h; wiÞ ¼ Ni



#

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.

Simultaneous consideration of Eqs.(32) and (44)allows

deter-mination 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

con-tact 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

ap-proaches, 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

rigid-body assumptions so that no elastic tooth deformation due to

ac-tual loading is considered when TCA results are shown

5.1 Applied coordinate systems

Fig 9represents the applied coordinate systems for tooth

con-tact analysis (TCA) of straight and skew bevel gears

5.2 Errors of alignment The errors of alignment considered are: (i)DA1– the axial dis-placement of the pinion (Fig 10(a)), (ii)DA2– the axial displace-ment of the gear (Fig 10(b)), (iii)DR– the change of the shaft angleR(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 inFig 9

Coordinate systems S1and S2are movable coordinate systems rigidly connected to the pinion and gear, respectively Angles /1 and /2are the angles of rotation of the pinion and the gear, respec-tively.Table 1shows details of coordinate transformation from S2

to S1 Transformation Mmlis needed if pinion and gear have been gen-erated following the same coordinate transformations, so that one

of the members of the gear drive have to be rotated an anglepto 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 inTable 2

will be considered for tooth contact analysis Three cases of design for each gear drive, with design characteristics shown inTable 3, will be considered Those design characteristics correspond to the tooth surface modifications of the pinion member of the gear set

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).