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공 학 박 사 학 논 문
CAD of Forged Involute Bevel Gear
2009년 8월
산 대 학 대 학 원
기 계 계 공 학 과
박 문 우
공 학 박 사 학 논 문
CAD of Forged Involute Bevel Gear
지 도 수 박 노 길
2009년 8월
산 대 학 대 학 원
기 계 계 공 학 과
박 문 우
박문우 공학 박사 학 논문 함
2009 년 7 월 3
원 시 복 ( )
원 정 ( )
원 배 원 병 ( )
원 조 용 주 ( )
원 박 노 길 ( )
Contents
Chapter 1. Introduction ....................................................................... - 1 -
1.1 Motivation ......................................................................................- 1 -
1.2 Literature Survey ..........................................................................- 4 -
1.3 Objective and Scopes .....................................................................- 7 -
1.4 Thesis Outline ................................................................................- 9 -
Chapter 2. Automotive Differential Bevel Gears .............................. - 10 -
2.1 Automotive Differentials: Their Function and Architecture ...... - 10 -
2.2 Manufacturing Differential Gears .............................................. - 14 -
2.2.1 Cutting Methods for Straight Bevel Gears for Differentials ....... - 14 -
2.2.2 Forging for Straight Bevel Gears .............................................. - 16 -
2.3 Problems: Differential Bevel Gear Tooth-Form and Forging
Processes ................................................................................................. - 21 -
Chapter 3. Spherical Involute Tooth Profile ..................................... - 23 -
3.1 Definition of the Spherical Involute Function ............................. - 23 -
3.2 Spherical Involute Tooth Profile ................................................. - 27 -
3.3 Coordinate System of Spherical Involute Tooth-Profile Curve..- 31 -
Chapter 4. CAD for Differential Bevel Gears ................................... - 33 -
4.1 CAD procedures .......................................................................... - 33 -
4.2 Specifying Parameters and Coordinate Systems ........................ - 37 -
4.3 Modeling the Spherical Involute Tooth-Form............................. - 41 -
4.3.1 Defining the Addendum and Dedendum Parameters.................. - 43 -
4.3.2 Generating Spherical Involute Curve ........................................ - 45 -
4.3.3 Defining the Tooth-Thickness Factors ....................................... - 47 -
4.3.4 Defining the Face and Root Parameters .................................... - 51 -
4.4 Tooth-Form Modification ............................................................ - 52 -
4.4.1 Geometrical Considerations ...................................................... - 53 -
4.4.2 Defining the Range of Modification .......................................... - 55 -
4.4.3 Profile Modification ................................................................. - 58 -
4.4.4 Crowning ................................................................................. - 62 -
4.5 Contact Pattern Simulation ......................................................... - 64 -
4.5.1 Simulation Algorithm ............................................................... - 64 -
4.5.2 Simulation Examples ................................................................ - 67 -
4.5.3 Simulation for the Shaft Angle Deviations ................................ - 73 -
4.6 Design Program that uses CATIA-VBA...................................... - 74 -
Chapter 5. Verification of the CAD Program.................................... - 77 -
5.1 Design Parameters ....................................................................... - 77 -
5.2 Design Results .............................................................................. - 79 -
5.3 Manufacturing ............................................................................. - 82 -
5.4 Contact Pattern Test .................................................................... - 82 -
Chapter 6 Conclusion ........................................................................ - 86 -
References .......................................................................................... - 88 -
Appendix A ......................................................................................... - 93 -
Nomenclature
B ; backlash
C*0 (x*
0, y*0, z*
0) ; space fixed coordinate system
C* (x*, y*, z*) ; body fixed coordinate system
Dp ; diameter of pitch circle
ı∗ ⃗ , ∗ ⃗ , k∗ ⃗ ; directional vectors of each coordinate system C*
m ; gear module
n ⃗ ; direction vector at point A
p ; pitch
t ; tooth thickness at an arbitrary point
tb ; tooth thickness of base circle
tp ; tooth thickness of pitch circle
RP ; radius of pitch circle
Z ; number of teeth
α ; tool pressure angle
αg ; gear pressure angle
β ; azimuthal angle
γ ; cone angle at an arbitrary point
γa ; addendum-cone angle
γb ; base-cone angle
γp ; pitch-cone angle
θg ; rotating angle of gear
ξ ; cone distance
φ ; azimuthal angle
ψ ; spherical-involute function
ψp ; spherical-involute function of pitch point
List of Figures
Figure 2.1 A kinematic diagram of a bevel gear differential. ......................... - 12 -
Figure 2.2 The architectures of automotive differentials. ............................... - 13 -
Figure 2.3 Coniflex tool of Gleason. ............................................................. - 15 -
Figure 2.4 Revacycle tool of Gleason............................................................ - 15 -
Figure 2.5 Flowchart for design and manufacturing of the differential bevel gear
forging die. ............................................................................................ - 18 -
Figure 2.6 Electrodes for differential bevel gears. ......................................... - 19 -
Figure 2.7 Machining the forging die through EDM. ..................................... - 20 -
Figure 3.1 A spherical involute curve. ........................................................... - 24 -
Figure 3.2 Basic parameters of a spherical involute bevel gear. ..................... - 29 -
Figure 3.3 Teeth profiles of a spherical involute bevel gear. .......................... - 30 -
Figure 3.4 A spherical involute tooth profile and its coordinate system. ......... - 32 -
Figure 4.1 Flowchart for the manufacture of forging dies with integrated CAD
…………………………………………………………………………..- 35 -
Figure 4.2 Flowchart of the modeling process for differential bevel gears through
the integrated CAD program. ................................................................. - 36 -
Figure 4.3 Coordinate System of a pair of spherical involute bevel gears. ..... - 39 -
Figure 4.4 Basic parameters of a pair of spherical involute bevel gears.......... - 40 -
Figure 4.5 Flowchart for modeling a spherical involute tooth-form. .............. - 42 -
Figure 4.6 Schematics of the addendum and dedendum for a spherical involute
curve. .................................................................................................... - 44 -
Figure 4.7 A spherical involute curve for differential bevel gears. ................. - 46 -
Figure 4.8 Tooth-thickness definitions. ......................................................... - 48 -
Figure 4.9 Tooth profile for the whole gear. .................................................. - 50 -
Figure 4.10 Flowchart of tooth-form modification. ....................................... - 54 -
Figure 4.11 Range of tooth-form modification. ............................................. - 57 -
Figure 4.12 The concept of tooth-profile modification. ................................. - 61 -
Figure 4.13 The crowning concept. ............................................................... - 63 -
Figure 4.14 A rolling action (envelope) of tooth flanks. ................................. - 65 -
Figure 4.15 Contact pattern between the pinion and the side gear. ................. - 66 -
Figure 4.16 Contact pattern for a bevel gear set (side gear). .......................... - 69 -
Figure 4.17 Contact pattern for a bevel gear set (pinion). .............................. - 70 -
Figure 4.18 Contact pattern simulations of the tooth flanks ........................... - 71 -
Figure 4.19 Contact pattern simulation of spiral bevel gear ........................... - 72 -
Figure 4.20 Contact pattern simulations for shaft angle deviations ................ - 73 -
Figure 4.21 Integrated design process by CATIA with VBA .......................... - 75 -
Figure 4.22 Input window of the CATIA-VBA ............................................. - 76 -
Figure 5.1 3D-CAD model of the differential bevel gears.............................. - 79 -
Figure 5.2 Simulation for CNC data transform .............................................. - 80 -
Figure 5.3 Simulation of the tooth contact pattern ......................................... - 81 -
Figure 5.4 Bevel gear contact test machine (Gleason Works)......................... - 83 -
Figure 5.5 Prototype contact test ................................................................... - 84 -
Figure 5.6 Machined bevel gears for forging ................................................. - 85 -
Figure 5.7 Contact Patterns ........................................................................... - 85 -
List of Tables
Table 4.1 Parameter definitions of the tooth modifications
Table 4.2 Parameters to simulate contact pattern
Table 4.3 Parameters of the differential bevel gears for forging
Table 4.4 Parameters of the tooth modifications
Table 5.1 Parameters of the differential bevel gears for forging
Table 5.2 Parameters of the tooth modifications
- 1 -
Chapter 1
Introduction
1.1 Motivation
Bevel gears are machine elements that are commonly used for transmitting
power and motion between intersecting shafts. These are conical with tapered teeth
that are larger at the periphery and smaller at the center. They are useful when the
direction of a shaft's rotation needs to be changed. The use of bevel gears with
various numbers of teeth can change the speed of rotation. Bevel gears are usually
mounted on a shaft and are offset from each other by 90°; however, they can be
designed to work at other angles as well [1]-[3].
There are several different types of bevel gear, including straight, spiral, and
hypoid bevel gears. Straight bevel gears are the simplest ones that have a straight-
tooth geometry. Depending on the machining equipment, straight bevel gears are of
two types: Gleason and standard. The majority of such gears are of the Gleason
type with Coniflex and Revacycle tooth-forms that are manufactured on the
precision machine tools of Gleason Works [4]. Spiral and hypoid bevel gears have
teeth that are curved and oblique. They are also different tooth-forms in light of the
methods for generating the corresponding gear-tooth surfaces. Among these
methods are the Gleason method, the Klingelnberg system, and the Oerlikon
system [5]-[6].
Straight bevel gears are used in complex, agricultural and construction
equipment, special automotive applications, etc. Spiral and hypoid bevel gears also
are important components of automotive, rotorcraft, and marine drive-train systems
[7]. In particular, the most salient application of straight bevel gears is in bevel-
- 2 -
gear differentials. These bevel gears have generated or milled tooth-forms that are
machined by either the Coniflex or the Revacycle tools of Gleason Works [14].
Cutting methods that employ Gleason tools are mainly used to manufacture the
bevel gears.
Although these cutting methods can guarantee the stability of production, their
associated technologies entail low productivity, low utilization of materials, and
high capital expenditure on equipment. Therefore, in the case of automotive
applications, a recent approach for manufacturing differential bevel gears for mass
production is a precision cold forging method [8]. The forging method for
manufacturing differential bevel gears for automobiles comprises the following
processes.
(1) Cutting a master gear to make an electrode. The straight bevel gear that has
the same tooth-form geometry as the one that will be forged is manufactured
on a Gleason machine with Revacycle methods [9]. The manufactured master
gear has to undergo a contact test for ensuring the meshing of gears.
(2) 3D-CAD modeling for an electrode. The 3D-CAD model is constructed by
measuring the master gear on a 3D coordinate measuring machine (CMM). It
is also necessary to revise the 3D-CAD model for the bevel gear geometry to
be properly and easily forged [26].
(3) Machining an electrode. The electrode is machined on a CNC machine tool
based on the 3D-CAD model [9].
(4) Machining a precision forging die. The forging die is manufactured using
electric discharge machining (EDM) with the machined electrode [10].
(5) Forming.
The above processes, viz., (1) through to (5), make it difficult and complicated to
- 3 -
achieve high productivity and low cost of production for the forged bevel gears.
The difficulty and complexity of bevel-gear forging are mainly caused by processes
(1) through to (3) for making the bevel-gear electrode [9]. In addition, a trial-and-
error method is needed to ensure a suitable area of contact because it is difficult to
predict the area (or pattern) of contact between the straight bevel gears for which
the teeth are milled by the Revacycle tools of Gleason Works [11]. Furthermore,
there are major disadvantages of master gears; these are listed below.
- The bevel gear pairs with the Octoid tooth form (the contact path is similar to
the figure of ‘8’ [13]) that are machined on Gleason machines cannot be
perfectly conjugated, as a result of which kinematical errors occur and cause
variations in the angular velocities. This has been identified as the primary
cause of problems that are related to vibration and noise [32]-[37].
- The bevel gear tooth-form does not enable interchange. Therefore, one bevel
wheel works only with its complementary wheel and no other [34]-[37].
Therefore, it is necessary to improve both the kinematical performance (with
conjugated action and interchangeability) of automotive differential gears and the
productivity of the forging processes.
- 4 -
1.2 Literature Survey
The researches related to this thesis can be classified as follows;
(1) Manufacturing method of bevel gears
(2) Modeling of bevel gear tooth surface
(3) Straight bevel gear forging for differential gears
(4) Tooth contact analysis of the bevel gears
In practice, almost all straight bevel gears are cut with the standard, Coniflex and
Revacycle tools of Gleason. The standard and Coniflex tooth-forms are generated
with straight-edged tools that simulate an imaginary crown gear [1]-[4]. The
Revacycle tooth-form is milled with a circular broach-style form cutter for the
high-volume manufacture of straight bevel differential gearing [14]. Face hobbing
and milling are employed for cutting spiral and hypoid gears. The face-hobbing
process is of two types, non-generated (Formate) and generated methods, which are
applied for the generation of the tooth surfaces of gears [4]. With regard to other
means of manufacturing straight bevel gears, Ozel et al. [15] proposed a method of
using end mills in CNC milling through the CNC data that are generated from a
solid model that is based on Tredgold’s method of approximation [16].
The surface of a bevel-gear tooth can be mathematically modeled through a
spherical involute surface, which becomes the idealized tooth-form. Tsai and Chin
[18] proposed a mathematical model of the tooth surface for straight and spiral
bevel gears based on a spherical involute surface. The model was defined in terms
of a trace on the tangent plane that rolls over the base cone but the edge of the
tangent plane was regarded as a right line; hence, such a model cannot apply to an
exact tooth profile. Al-Daccak et al. [19] described an exact spherical involute
tooth-form and compared it with the model of Tsai and Chin. The solid models of
- 5 -
straight and spherical bevel gears were obtained by the means of simple sweeping
techniques. Shunmugam et al. [20] also discussed the normal deviation between the
idealized surface (exact spherical involute surface) and the model of Tsai and Chin
for straight bevel gears. Park and Chung [32]-[37] studied the kinematical
relationship of a spherical involute bevel gear set; the experimental verification
was carried out through CAD/CAM.
There is a method for modeling the bevel-gear tooth surface through the
simulation of the envelope of the locus of the tool surface. It is applicable to
existing manufacturing machines, such as those of Gleason Works and
Klingelnberg GmbH. Litvin et al. [21] proposed a computer-aided generation of
spiral bevel gears with an improved geometry in order to investigate the influence
of misalignments on both transmission errors and the shift of the bearing contact.
Ichino et al. [22] proposed a method for cutting straight bevel gears using a quasi-
complementary crown rack with a planar tool surface, instead of the usual crown
rack.
Studies of bevel-gear forging are also summarized. Benedict [30] proposed the
technique of the computer integrated manufacturing that enables the design and
manufacture of forge tooling for net-shaped bevel gears. Lee and Park [8] proposed
an application of CAD/CAM to straight bevel gears with crowning for modeling
and machining; it was verified through the contact pattern test. This research was
carried out to achieve the minimization of trial-and-error and the reproducibility of
differential gear forging. However, the tooth-form that is modeled is based on
Tredgold’s method for approximating a spherical tooth profile to a plane involute
profile; therefore, the gear cannot be correctly meshed and serious kinematical
errors arise. Kawasaki and Shinma [9] developed a system for designing and
manufacturing straight bevel gears for which the tooth-form is generated through a
quasi-complementary crown gear. In this study, a CAD/CAM method for the direct
- 6 -
milling forging die was also proposed to improve the productivity and lower the
cost of manufacture. Shih et al. [17] developed a program for generating both
standard and modified tooth-forms with spherical involutes of straight bevel gears.
Furthermore, it is advantageous to directly manufacture the forging die through a
CNC machine. Lelkes et al. [23] reported the contact analysis of uniformly-high
teeth of epicyclic spiral bevel gears that stemmed from Klingelnberg’s Cyclo-
Palloid System. The analysis was based on the simultaneous generation of gear
surfaces and contact simulation.
- 7 -
1.3 Objective and Scopes
This thesis focuses on the development of an integrated CAD program to design
a forging die (or an electrode for die EDM) for automotive differential bevel gears
through a spherical involute tooth-form. The integrated CAD program is able to
improve both the kinematical performance and the productivity of forging
processes because it is based on a spherical involute tooth-form and has the
following advantages.
- There is excellent kinematical performance through conjugated action, gear
interchangeability, and the good adjustment of assembly errors.
- The tooth form is mathematically formulated so that it is easily understood and
suitable for the construction of the CAD model. It enables the integration of
forging processes and is appropriate for the forged bevel gears.
Therefore, the objective of this thesis is to develop an integrated CAD program
for the forging die for overcoming the demerits and disadvantages that arise from
the use of a spherical involute tooth-form. Consequently, this thesis includes all the
CAD activities for integrating the design and manufacture of an electrode for
precision bevel gear forging. Hence, the scope of this thesis is determined as
follows.
(1) A spherical involute tooth profile is mathematically formulated and
implemented through a CAD program.
(2) The CAD program is developed for integrating the forging process.
(3) Through the CAD program, bevel gear sets with a spherical involute tooth-
form are modeled for automotive differentials.
(4) To quantitatively adjust the area of contact between gears, a lead-modification
(crowning) and profile-modification method is proposed for differential bevel
- 8 -
gears. It is also incorporated in the CAD program.
(5) Contact patterns are simulated under light load conditions.
(6) Prototypes of electrodes of two bevel gear sets are modeled and manufactured.
(7) The actual contact patterns of the prototype are compared with the simulated
patterns.
- 9 -
1.4 Thesis Outline
The rest of this thesis is presented in five chapters. Chapter 2 introduces the
theoretical background of an automotive differential and its technology. In addition
the manufacturing technology of the differential gears is presented. Chapter 3
introduces the spherical involute surface and its function as defined in a
mathematical procedure. The geometry of the spherical involute curve is described
in this chapter as well as its basic parameters used to define the spherical involute
tooth profile in the fixed coordinate system. Also included in Chapter 3 is the
kinematical relationship of the spherical involute bevel gears. Chapter 4 focuses on
the design considerations in the manufacture of bevel gears for automotive
differentials by forging. First, the definitions of the tooth addendum and dedendum
values and tooth thickness to avoid tooth interference between the tip and root are
presented. Later, details of the geometrical considerations are described. Also
included in Chapter 4 is the tooth modification to be properly applied to bevel
gears for automotive differentials. Chapter 4 also focuses on the modeling of the
spherical involute bevel gears by using CAD S/W and its significant modeling
parameters, which are summarized in this chapter. Chapter 5 applies actual bevel
gear models (side gear and pinion) for automotive differentials to the CAD
modeling. Chapter 6 summarizes the conclusion from this study and states the
contributions of this study. Recommendations for future work are also listed in
Chapter 6.
- 10 -
Chapter 2
Automotive Differential Bevel Gears
2.1 Automotive Differentials: Their Function and
Architecture
The primary function of an automotive differential is to allow two-wheel drive
vehicles to rotate at different speeds in order to avoid tire slippage on the road. If
there is a rigid connection between the two driving wheels, slippage will appear
when driving on straight as well as curved roads because the diameters of the two
tires cannot be perfectly equal.
Figure 2.1 shows a kinematic diagram of an automotive differential that is
mounted on the axle in the case of a vehicle with either rear wheel drive (RWD) or
four-wheel drive (4WD) transmission. The final-drive speed (ωR) is split by the
differential gears into the left and right axle-speeds (ωS1 and ωS2). This kinematical
relationship [27] is given as follows:
= ω (2-1)
This relationship shows that the numbers of teeth of the differential pinions and
side gears do not affect the differential. The numbers of teeth could be determined
by the criteria of mechanical design and manufacturing.
The most common architecture of differentials is presented in Figure 2.2. It is
mainly composed of a final drive pinion, a ring gear, two differential-pinions, two
side-gears, left/right axles, and a differential cage. The two differential-pinions (at
- 11 -
least) are mounted on the shafts (or studs) that are attached to the differential cage.
The two side-gears, which are meshed with the differential pinions, drive the left
and right axles, respectively. The differential cage is attached to the final drive ring
gear (commonly, a hypoid gear). The power flow through the differential is
summarized as follows.
(1) The final drive pinion rotates the ring gear.
(2) The ring gear rotates the differential cage.
(3) The differential pinions, as they rotate with the cage, force the pinion gears
against the side gears, which, in turn, rotate the axles.
As a matter of pure kinematics, a single differential-pinion is enough but to
reduce the forces that act on the teeth of the pinion, two pinions are usually used.
Moreover, in the case of a single pinion, the contact force on the tooth induces
moments on the points of mounting between the left and right axles and the
differential cage.
Other architectures are sometimes used to increase the number of differential
pinions. It is thus possible to have three or four pinions for a differential; most
cases employ four pinions. Under this option, it is possible to transmit a higher
torque in the same space but more parts and additional costs are entailed. In the
case of the four pinions, its exploded view is shown in Figure 2.2.
- 12 -
Figure 2.1 A kinematic diagram of a bevel gear differential.
Left Axle
(ωs1)Right Axle
(ωs2)
Final Drive Pinion
Final Drive
Ring Gear (ωR)
Differential Cage
Differential Pinion
Differential Side Gear
- 13 -
(a) Two pinion type
(b) Four pinion type
Figure 2.2 The architectures of automotive differentials.
Left Axle
Right Axle
Final Drive Pinion
Final Drive Ring Gear
Differential Cage
Final Drive Shaft
Differential Pinion
Differential Side Gear
- 14 -
2.2 Manufacturing Differential Gears
The differential pinion and side gears, which are core components of
differentials, are generally straight bevel gear pairs with a shaft angle of 90°. These
bevel gears can be manufactured by cutting or forging.
2.2.1 Cutting Methods for Straight Bevel Gears for Differentials
The cutting process for a straight bevel gear is of two types, generated and
milled methods; these are applied to the cutting of the tooth-form of the gear. The
generated and milled teeth are commonly cut on Gleason machines through
Coniflex and Revacycle tools [28]. The Coniflex tool (Figure 2.3) allows a simple
plunge-and-roll for completing the cycle and eliminates the typical need for
additional motions, such as grinding and finishing. The Coniflex tool is used to cut
teeth-forms that have a crowning (sometimes, it is called a localized lengthwise
tooth bearing). These Coniflex-type straight bevel gears provide good control of
tooth contact through the crowning. The crowned tooth permits the minor
adjustment of gears in assembly and allows for some displacement due to
deflection under operating loads without the concentration of the load on the end of
the tooth.
In other cases, the Revacycle tool (Figure 2.4) is used for the high-volume
manufacture of straight bevel differential gearing. Through the Revacycle tool, the
teeth are completed in one revolution of the circular broach-type cutter, which has
three types of blade (roughening, semi-finishing, and finishing).
- 15 -
Figure 2.3 Coniflex tool of Gleason.
Figure 2.4 Revacycle tool of Gleason.
- 16 -
2.2.2 Forging for Straight Bevel Gears
Although the cutting methods through the use of Gleason tools can guarantee the
stability of production, the associated technology entails low productivity, low
utilization of materials, and high capital expenditure on equipment.
Today, there is a trend towards net-shaped forged differential gears. Furthermore,
in the 28 models studied, 18 were forged and 10 were machined (i.e., 64% were
forged) [29]. The net-shaped process minimizes waste through precision dies and
often eliminates machining; however, the processes are quite expensive in terms of
tooling and the capital expenditure that is required [30]. The forging processes for
designing and manufacturing differential bevel gears for automobiles are shown in
Figure 2.5; they are described below.
(1) Master gear machining: A pair of straight bevel gears that have the same
tooth-forms as the forged gears are machined on a Gleason machine with
Coiflex or Revacycle tools.
(2) Contact pattern test of the master gear set: The contact patterns of a
machined pair of bevel gears are verified to ensure the meshing of gears. The
contact patterns are obtained when the gear set is run under a light load in a
rolling test machine. If the contact patterns are not around the middle portion
of the tooth flank, the master gear set should be machined again.
(3) Measuring the tooth flanks of the master gear set: The tooth flank surface
data are obtained by measuring the gear set through a 3D coordinate
measuring machine (CMM).
(4) Constructing 3D-CAD models for electrodes: 3D-CAD models are
constructed for electrodes based on the tooth-surface data that are measured
through CMM. It is also necessary to revise the 3D-CAD models so that the
- 17 -
bevel-gear geometry can be properly and easily forged [9], [26].
(5) Machining an electrode: The electrode is machined on a CNC machine tool
that is based on the 3D-CAD model [9]. Figure 2.6 shows the machining
process and electrodes that are machined for automotive differential gears.
(6) Contact pattern test of electrodes of the gear set: The contact patterns of
electrodes of the gear set are verified for ensuring the meshing of gears.
These contact patterns are also obtained on a rolling test machine.
(7) Machining a precision forging die: The forging die is manufactured using
electric discharge machining (EDM) with the machined electrode [10].
Figure 2.7 shows the machining of the forging die through EDM.
- 18 -
Figure 2.5 Flowchart for design and manufacturing of the differential bevel gear
forging die.
Completed Die
OK
Start
Master Gear Machining
Measuring the Tooth Flank of the Master Gear
Constructing a 3D-CAD model for an Electrode
Contact Pattern Test of the
Master Gear
Machining the Electrode
Contact Pattern Test of the Electrode
Machining the Forging Die by EDM
Failure
OK
Failure
- 19 -
(a) Process for cutting electrodes through a CNC machine
(b) Machined electrodes
Figure 2.6 Electrodes for differential bevel gears.
- 20 -
Figure 2.7 Machining the forging die through EDM.
- 21 -
2.3 Problems: Differential Bevel Gear Tooth-Form and
Forging Processes
Almost all differential bevel gears, including those that are machined and forged,
have the Octoid tooth-form; they have several disadvantages, which are listed
below.
(1) They are not interchangeable, i.e., the bevel gears operate as inseparable pairs.
One bevel wheel is designed to work with its complementary wheel and no
other.
(2) They cannot be perfectly conjugated, as a result of which kinematical errors
occur and cause variations in the angular velocities. This has been identified
as the primary cause of problems that are related to vibration and noise.
(3) There is poor adjustment of assembly errors.
(4) The tooth-form is not mathematically formulated and is distinct from a
spherical involute tooth-form; this is a limitation in practice.
The above disadvantages are caused by the kinematical characteristics of the
Octoid tooth-form, the tooth of which is practically generated through a straight-
edged crown tooth [31]. In addition, the processes of bevel gear forging make it
difficult and complicated to achieve high productivity and low cost of production
for the forged bevel gears. These demerits of bevel-gear forging are listed as
follows.
(1) The difficulty and complexity of bevel-gear forging are mainly caused by
processes for making the bevel-gear electrode. (Refer Subsection 2.2.)
(2) A trial-and-error method is needed to ensure a suitable area of contact because
- 22 -
it is difficult to predict the area (or pattern) of contact between the straight
bevel gears for which the teeth are milled by the Revacycle tools of Gleason
Works.
To improve both the kinematical performance of the Octoid tooth-form and the
productivity of the forging processes, a spherical involute tooth-form can be a
particularly good solution. The main advantages are as follows.
- There is excellent kinematical performance through conjugated action, gear
interchangeability, and the good adjustment of assembly errors.
- The tooth form is mathematically formulated so that it is easily understood and
suitable for the construction of the CAD model. It enables the integration of
forging processes and is appropriate for the forged bevel gears.
- 23 -
Chapter 3
Spherical Involute Tooth Profile
3.1 Definition of the Spherical Involute Function
The involute function is a means of mathematically defining the tooth-profile
curve of a spur gear. Whereas a tooth-profile curve generally involves complex
mathematical procedures, the involute function facilitates the manipulation of the
curve. Since it is a widely known fact that the bevel gear system is a generalized
configuration of the parallel-axis gear system, it is believed that the geometrical
procedures for the tooth-profile curve in a complex bevel gear system can be
simplified by deriving the spherical involute function.
This is a generalized form of the cylindrical involute function. Unless specified
otherwise, in this thesis, a bevel gear configuration will be expressed on a
transverse sphere. A transverse sphere refers to a reference sphere with a radius of ξ
and two axes that intersect at the center of the sphere. When the transverse sphere
converges to a cylindrical gear, it becomes a transverse plane that is perpendicular
to the gear axis. Figure 3.1 displays a spherical involute curve.
- 24 -
Figure 3.1 A spherical involute curve.
- 25 -
The edge of the base cone is referred to as the base circle, the center of which is
represented by point C on the transverse sphere. The trajectory of point A at the tip
of arc AB is the spherical involute curve, DA. The tooth-profile angle, φ, at an
arbitrary point, A, on the curve is defined by the angle between the tangent to the
curve and the arc, AC. The curve, DA, and the arc, AB, are always perpendicular.
Therefore, BC and AB are perpendicular and the spherical triangle, ABC, is a right
triangle. The relationships between the angles and the sides of a spherical triangle
are different from those of a plane triangle (refer Appendix 1). In Figure 3.1, the
relationships between γ, γb, χ, φ, and β are as follows.
cos γ = cos γb
cos χ (3-1)
sinφ =
(3-2)
cos β =
(3-3)
In the above, γ, γb, and χ are angular parameters that represent COA, COB, and
BOA, respectively. The azimuthal angle, β, is the angle between the arcs, BC and
AC. Eliminating the angle, χ, from Eqs. (3-1) and (3-2), we obtain:
sin γ = cosφ sin γ. (3-4)
The spherical involute function refers to the angle, ψ, between the arcs, AC and
CD, in Figure 3.1. Since the lengths of BD and AB are identical,
- 26 -
χ = (β + ψ) sinγ . (3-5)
Expressing Eq. (3-5) in terms of ψ, we get:
ψ =
− β. (3-6)
In Eq. (3-6), the spherical involute function can be written as a function of φ and
the base-cone angle, γb.
ψ =
tan
− cos
. (3-7)
Since γb → 0 if ξ → ∞, Eq. (3-7) implies that asymptotically converges to
tanφ − φ. From this, it can be confirmed that a spherical involute tooth-profile is
a generalized configuration of a cylindrical involute tooth profile. ψ can be
expressed in terms of the cone angle, γ, instead of φ.
By applying Eq. (3-4) and rearranging terms, we obtain:
ψ =
cos
− cos
. (3-8)
- 27 -
3.2 Spherical Involute Tooth Profile
Figure 3.2 depicts the basic parameters of a spherical involute gear. The gear's
pitch circle is used as a reference for defining the tooth profile. A pitch point, P, is
the intersection of a pitch circle and the center line of the axis. The common
normal of the tooth-contact point passes through the pitch point. When the tooth-
profile curve passes through a pitch point, the angle of the tooth-profile curve at
that point is referred to as the gear pressure angle, αg.
In a standard gear, the gear pressure angle and the rack (or tool) pressure angle α
are identical. The relationship between α, γb, and the pitch cone angle, γp (angle
COP), in Figure 3.2 is obtained from Eq. (3-4).
sin γ = cos α sin γ (3-9)
The relationship between the tooth thickness on the pitch circle, the space width,
and the backlash, B, is:
t =
. (3-10)
Accordingly, the tooth thickness, t, at an arbitrary point, A, on the tooth-profile
curve is:
t = R
+ 2ψ − 2ψ . (3-11)
- 28 -
In Eq. (3-11), R = Asin γ and R = Asinγ .
ψ is obtained from either Eq. (3-7) or (3-8), and ψp is obtained by substituting
φ = α into Eq. (3-7) or γ = γ
into Eq. (3-8).
ψ =
tan
− cos
(3-12)
The tooth thickness, tb, of the base circle is t when γ = γ in Eq. (3-11), at
which point ψ = 0; then, tb becomes:
t = R
+ 2ψ . (3-13)
Where R = A sinγ .
The module, m, determines the tooth size. The bevel gear's module is defined in
the same way as that of a spur gear. If the diameter of the pitch circle is Dp and the
number of teeth is Z,
m =
. (3-14)
- 29 -
Figure 3.2 Basic parameters of a spherical involute bevel gear.
- 30 -
Spherical involute tooth profiles for different base-cone angles in a single
module are depicted in Figure 3.3, which displays tooth-profile curves with
addendum angle γa at m ξ⁄ = 0.1, m = 3.0, α = 20°, Z = 20 and γ = 10~70°.
The tooth-tip circle radius Rt becomes ξ sin γ + γ
, which is reduced as γb
increases. When γb reaches its maximum of 70°, Rt is smaller than the pitch-circle
radius RP, tb increases in proportion to γb. Since a bevel gear has a thick tb, it can be
learned that it has greater strength than a spur gear.
Figure 3.3 Teeth profiles of a spherical involute bevel gear.
- 31 -
3.3 Coordinate System of a Spherical Involute Tooth-
Profile Curve
A spherical involute tooth-profile curve can be expressed with reference to the
fixed coordinate system, Cg, of a gear. In Cg, zg is the rotational axis of the gear and
the yg axis is directed toward the intersection, Pr, of the tooth-profile curve and the
tooth pitch circle, as shown in Figure 3.4.
Therefore,
OA ⃗ = ξn ⃗ (3-15)
n ⃗ = sin γ sin η ı ⃗ + sin γ cos η ⃗ + cos γ k ⃗ (3-16)
and
η = ψ −ψ (3-17)
In the above, η and γ are the angles of CgA and CgPr, respectively. The rigid body
motion of the gear can be defined by the relationship between the spatially-fixed
reference coordinate system of the gear, Cg0, and Cg. Cg is constructed by
rotationally displacing Cg0 in the positive direction by the gear’s angular motion, θg,
with zg0 as the reference axis.
- 32 -
Figure 3.4 A spherical involute tooth profile and its coordinate system.
- 33 -
Chapter 4
CAD for Differential Bevel Gears
4.1 CAD procedures
The existing processes for designing and manufacturing a forging die for
differential bevel gears are shown in Figure 2.5. The existing processes mainly
consist of master gear machining, a contact pattern test, CMM, construction of the
CAD model of the electrode from CMM data, electrode machining, and EDM die
machining. These processes are based on trial-and-error; hence, the design and
characteristics of contact are difficult to modify and improve upon. In addition, the
processes for designing and obtaining the electrode are varied and complicated.
Moreover, forged differential bevel gears have the Octoid tooth form, which has
several disadvantages as follows.
(1) They are not interchangeable, i.e., the bevel gears operate as inseparable pairs.
One bevel wheel is designed to work with its complementary wheel and no
other.
(2) They cannot be perfectly conjugated, as a result of which kinematical errors
occur and cause variations in the angular velocities. This has been identified
as the primary cause of problems that are related to vibration and noise.
(3) There is poor adjustment of assembly errors.
(4) The tooth-form is not mathematically formulated and is distinct from a
spherical involute tooth-form; this is a limitation in practice.
Therefore, an integrated CAD process that uses a spherical involute tooth profile
- 34 -
is required and it overcomes these disadvantages. The main advantages of a
spherical involute tooth profile are as follows.
- There is excellent kinematical performance through conjugated action, gear
interchangeability, and the good adjustment of assembly errors.
- The tooth form is mathematically formulated so that it is easily understood and
suitable for the construction of the CAD model. It enables the integration of
forging processes and is appropriate for the forged bevel gears.
Figure 4.1 shows the flowchart for the integrated process of bevel gear design
through forging. Several existing processes (in Figure 2.5) are eliminated; the
entire process is integrated into one process of “3D-CAD by using a spherical
involute tooth profile.” This one process has several steps that are shown in Figure
4.2. The 3D-CAD modeling is carried out according to the flowchart in Figure 4.2.
The modeling processes for differential bevel gears with a spherical involute tooth
form are developed for ensuring these specific conditions. The modeling processes
are summarized in Figure 4.2.
In this thesis, all the results, such as the solid models, are directly used for
machining the electrode or forging die because all the CAD activities are carried
out on CATIA V5 and its VBA programming interface.
- 35 -
Figure 4.1 Flowchart for the manufacture of forging dies with integrated CAD.
Completed Die
OK
Start
3D-CAD modeling of Differential Bevel Gears by
the Use of aSphericalInvolute Tooth Profile
Machining the Electrode
Contact Pattern Test of the Electrode
Machining Forging Die by EDM
Failure
- 36 -
Figure 4.2 Flowchart of the modeling process for differential bevel gears through
the integrated CAD program.
Start
Preliminary Step: Specifying Parameters and Coordinate
Systems
Modeling a Spherical InvoluteTooth-Form
Tooth-Form Modification
End
Contact Pattern
Simulation Need Modification
OK
- 37 -
4.2 Preliminary Step: Specifying Parameters and
Coordinate Systems
The first step in the flowchart is concerned with gear parameters for defining the
spherical involute bevel-gear geometry. These parameters are referred as the
module, pressure angle, shaft angle, number of teeth, addendum, dedendum,
backlash, etc. Furthermore, these parameters are customized for Gleason machines
that are used for making various differential gears.
However, in order to model the differential bevel gears through the spherical
involute tooth profile, these parameters are converted into the basic parameters of α,
γp, and ξ, which are used to define the coordinate system. This procedure is
described in the flowchart and the coordinate system is shown in Figure 4.3. The
procedure for defining the coordinate system is described below.
(1) Determine an origin.
(2) Define the vector OP (a line from the origin to the pitch point) by using the
cone-distance parameter (ξ).
(3) Define the spatially-fixed reference coordinate systems (C10 and C2
0).
- Place the coordinate axes of z10 and z2
0, which are rotated by γp1 and γp2,
respectively, from the vector, OP. The rotating angles, γp1 and γp2, are the
pitch-cone angles of the pinion and side gear. Both the vector, OP, and the
coordinate axes of z10 and z2
0 are on the same plane.
- Define the coordinate axes of x10 and y2
0 as being normal to the plane
between z10 and z2
0.
- Lastly, define the coordinate axes of y10 and x2
0.
(4) Define the gear-fixed coordinate systems (C1 and C2).
- 38 -
- Rotate the spatially-fixed coordinate systems of the pinion and side gear
around z10 and z2
0 by the angles, θ1 and θ2, respectively.
In the gear-fixed coordinate systems, the point, O, is the apex of the pitch and
base cones. The pitch and base circles are also determined by the pitch and base
cone angles of γp and γb. The base cone angle, γb, is calculated via Eq. (3-18) and
the basic parameters of α, γp, and ξ for defining the coordinate system are obtained
from the gear parameters, m, α, Σ(shaft angle), Z1, and Z2. The relational
equations are as follows.
γ = tan
(4-1)
γ = Σ − γ (4-2)
ξ =
=
(4-3)
In addition, the necessary geometries for the bases of the spherical involute
bevel gears are defined in terms of the basic parameters. Figure 4.4 shows these
basic geometries, such as the pitch point, pitch circles, base circles, and contact
normal (disc of action).
- 39 -
Figure 4.3 Coordinate System of a pair of spherical involute bevel gears.
AP
O
C
x2z10, z1, x2
0
Base circle
Pitch circle
Contact normal
[Pinion]
[Side Gear]
y2
x10, y2
0
x1
y10, z2
0, z2 y1
θ1
θ2
- 40 -
Figure 4.4 Basic parameters of a pair of spherical involute bevel gears.
AP
O
C
Base circle
Pitch circle
Contact normal
α
ξ
γp2
y2
x10, y2
0
x1
x2z10, z1, x2
0
y10, z2
0, z2y1
γb2
γp1γb1
- 41 -
4.3 Modeling the Spherical Involute Tooth-Form
Figure 4.5 shows the procedures for modeling the spherical involute tooth-form
for a differential bevel gear set. For modeling the spherical involute tooth-form,
these procedures are performed on the basis of the gear-fixed coordinate systems
(C1 and C2) that were defined in Subsection 4.2. The main parameters in this
procedure are the addendum, dedendum, tooth thickness, face-width, and the face
and root angles. These parameters are defined as angles or ratios.
For automotive differential bevel gears, the tooth addendum and dedendum are
specially modified to avoid tooth-tip interference. In other words, the addendum of
a pinion is longer but the addendum of a side gear is shorter by the addendum
modification. The long-and-short addendum system for side gears and pinions is
used to reduce the undercut of the pinion and to more nearly equalize the strength
and durability of the side gear and pinion. For the application of the addendum and
dedendum to the spherical involute tooth form, these two parameters are defined in
terms of angles. In addition, the tooth thickness has to be properly resized because
this also improves the strength and durability characteristics of the teeth. For
various reasons, the tooth shapes of the bevel gear have to be properly modified
and these modifications have to be modeled.
- 42 -
Figure 4.5 Flowchart for modeling a spherical involute tooth-form.
Start
Defining Addendum and Dedendum Parameters
Generating Spherical InvoluteCurve
Generating Whole Tooth Profile
Defining Tooth Tip and Root
End
Defining Tooth Thickness
Sweeping Tooth Profile
- 43 -
4.3.1 Defining the Addendum and Dedendum Parameters
Figure 4.6 shows the schematics of the addendum and dedendum for the
spherical involute curve of the pinion. The spherical involute curve is trimmed
from the base circle to the addendum line (the tip circle) and will apply to the long-
addendum pinion. However, if the spherical involute curve has a shorter length
from the pitch point to the base line, its curve has to be extended to the dedendum
line (the root circle) by extrapolation. In this manner, the tooth profile is
determined. The addendum and dedendum angles (θa1 and θd1) of the pinion are
defined as per the following equations.
a = 1.17m − .
(4-4)
d = 1.788m − a (4-5)
θ = tan
(4-6)
θ = tan
(4-7)
These equations should be used only for differential gears. For the side gear, the
tooth profile can be similarly obtained.
- 44 -
Figure 4.6 Schematics of the addendum and dedendum for a spherical involute
curve.
P
O
Base circle
Pitch circleAddendum angle
[Pinion]
Spherical Involute Curve
C
dedendumangle
Extrapolated
Addendumdedendum
- 45 -
4.3.2 Generating Spherical Involute Curve
The spherical involute curve for the pinion, which is shown in Figure 4.7, is
defined in the gear-fixed coordinate system that is defined in Subsection 4.2. The
spherical involute curve is generated from the base-circle point (D) to the tip-circle
point (Pt1), which is determined through the following vector, OPt1.
OP ⃗ = ξn ⃗ (4-8)
n ⃗ = sinγ sin η ı ⃗ + sin γ cos η ⃗ + cos γ k ⃗ (4-9)
η = ψ − ψ (4-10)
γ = γ + θ (4-11)
For the above equations, ψt1 and ψp1 are obtained through Eqs. (3-8) and (3-12),
respectively.
Once the angular variables, such as ηt1 and γt1, are determined, the spherical
involute curve is defined from D to Pt1 and is also expressed by the following
vector, OA.
OA ⃗ = ξn ⃗ (4-12)
n ⃗ = sin γ sin η ı ⃗ + sinγ cos η ⃗ + cos γ k ⃗ (4-13)
η = ψ −ψ (4-14)
In the above, the concerned parameters satisfy the inequality, γ ≤ γ ≤ γ +
θ . As before, ψ1 and ψp1 are obtained via Eqs. (3-8) and (3-12), respectively.
- 46 -
Figure 4.7 A spherical involute curve for differential bevel gears.
P
O
Base circle
Pitch circle
[Pinion]
C
θa1
θd1Pt1
Pr1
Tip circle
Root circle
ψt1
ψP
γb1
B
γt1
z10, z1
ξ
D
Spherical Involute
- 47 -
4.3.3 Defining the Tooth-Thickness Factors
The tooth-thickness factor of a differential bevel gear is a very distinct
characteristic from that of a standard bevel gear; hence, it is applicable only to
differential bevel gears. The long-and-short addendum system can reduce the
strength and durability of the pinion because the tooth form of the pinion is
relatively thinner than that of the side gear. In the case of the differential bevel gear,
the tooth-thickness factors are specified to improve the strength and durability in
line with the knowhow of experienced manufacturers such as Gleason Works.
For a spherical involute bevel gear, the tooth geometry is quite different from the
Octoid tooth-form of Gleason bevel gears but the difference in the tooth size is
slight; hence, the tooth-thickness factors proposed by Gleason Works are applied to
this study.
Figure 4.8 shows definitions of the tooth thickness for a spherical involute bevel
gear; these are expressed as the circular tooth thickness (t1 and t2), chordal
thickness (tc1 and tc2), and angular thickness (θt1 and θt2), as follows.
t =
− (a − a ) tanα + 0.05m (4-15)
t = πm− t (4-16)
t = t −
( ) −
(4-17)
t = t −
( ) −
(4-18)
θ = tan
(4-19)
θ = tan
(4-20)
- 48 -
Figure 4.8 Tooth-thickness definitions.
P
O
Base circle
Pitch circle
θt2
[Pinion]
C
t1
[Side Gear]
θt1
t2
- 49 -
Figure 4.9 shows the whole tooth profile with a mirrored curve. For modeling
the whole bevel gear tooth, a mirror-image of the spherical involute curve is
obtained for a tooth about the tooth center line. To achieve this, the angle, θ1, which
is the angular thickness of the pinion tooth at any point on the calculated spherical
involute curve, is found from θt1 and ψP1. Then, θ1 can be calculated as follows:
θ = θ − 2(ψ − ψ ) (4-21)
Through the foregoing value of θ1, the coordinates for the mirror-image of the
spherical involute curve are given by:
OA ⃗ = ξn ⃗ (4-22)
n ⃗ = sin γ sin η ı ⃗ + sinγ cos η ⃗ + cos γ k ⃗ (4-23)
η = ψ −ψ + θ (4-24)
In the above, the concerned parameters satisfy the inequality, γ ≤ γ ≤ γ +
θ ; ψ1 and ψp1 are obtained via Eqs. (3-8) and (3-12), respectively.
In the case of the side gear, the coordinates for the mirror-image are similarly
calculated.
- 50 -
Figure 4.9 Tooth profile for the whole gear.
P
O
Base circle
Pitch circle
[Pinion]
C
θt1
θa1
θd1
Pt1
Pr1
Tip circle
Root circle
θ1
ψ1
ψP
ψP
γ1
- 51 -
4.3.4 Defining the Face and Root Parameters
In the case of standard or commonly used bevel gears, simple geometric
parameters, such as the addendum, dedendum, tooth thickness, and backlash, are
defined to specify the tooth dimensions. However, differential bevel gears are
usually manufactured through forging; hence, it is necessary to consider the
suitability of geometries for forging. Therefore, in practice, the face and root
parameters are also defined for modifying the tooth.
Figure 4.7 shows the tooth-form in light of the face and root parameters; the
tooth tip and bottom are sloped more steeply.
- 52 -
4.4 Tooth-Form Modification
Tooth-form modification is generally known as tooth flank grinding through
very thin cuts of the gear tooth face. The methods of modification are mainly
divided into two categories: profile and lead modifications. Profile modification is
commonly applied to reduce the tooth noise that arises from tooth deformations
under conditions of loading. Lead modification [38], which is sometimes referred
as crowning (the ground shape of the tooth is similar to that of a crown; hence, it is
also called crowning), is applied to position the area of contact at the middle
without concentrating the load on the end of the tooth [1]-[2]. In other words, the
common purposes of gear tooth-form modification are: localizing the tooth contact
area; improving upon the tooth strength; reducing noise; and reducing transmission
errors [39].
In the case of straight bevel gear tooth-form modification, the modification has
to be applied for enabling some adjustment of the gears in assembly and for
compensating loads without concentrating loads on the ends of teeth. Straight bevel
gears with a generated tooth-form (via Coniflex tools) are widely used for various
purposes. Crowning is directly achieved by the tools. In addition, straight bevel
gears with a milled tooth-form (via Revacycle tools) enable manual tooth-form
modification [28]. In this thesis, a method of tooth-form modification, which is
effectively applied to the CAD model of a spherical involute bevel gear, is
proposed.
- 53 -
4.4.1 Geometrical Considerations
For differential bevel gears that are machined or forged, the tooth form is based
on generation or milling; further, some modification of the tooth form is entailed.
Therefore, differential bevel gears with a spherical involute tooth-form must have
similar tooth-form modifications.
For modeling tooth-form modifications on a spherical involute tooth flank, the
tooth-flank model must be obtained, as shown in Figure 4.11. For modeling the
tooth modification, the parameters for modeling are also required and modeling
procedures are formulated for adjusting the modified tooth flank.
The parameters to be applied for modification are as shown in Figure 4.11 and
the flowchart of tooth-form modification is shown in Figure 4.10. Table 4.1 lists
the definitions of the parameters for modifying the tooth flank. These parameters
are customized to differential bevel gears.
- 54 -
Figure 4.10 Flowchart of tooth-form modification.
Start
Preparing a Completely Modeled Tooth Flank
Input Parameters of Tooth Modification
Defining the Modification Range
Profile Modification(root and tip relief)
Lead Modification (Crowning)
End
- 55 -
4.4.2 Defining the Range of Modification
Almost all differential bevel gears have modified tooth flanks but it is difficult to
know the exact range of modification as well as the extent of modification, such as
the tip and root reliefs. However, these ranges have to be parametrically defined for
modeling tooth-form modification. Therefore, the range is defined as the center-
point dimensions of the contact area and modifiable flank dimensions. (Refer to
Figure 4.11.) These dimensions of the range of modification are the basis for the
next steps. The procedures for defining the range are as follows.
(1) Prepare the modifiable tooth as determined by the basic geometries.
(2) Specify the position of the contact area. The position that is specified in this
step will be the origin of tooth-form modification.
(3) Specify the modifiable flank dimensions.
- 56 -
Table 4.1 Definitions of the parameter for tooth modification.
Parameters Description
Amount of Lead Contact (%) It determines the lead crowning ranges.
Amount of Profile Contact (%)
It determines the ranges of profile modification.
Position of Contact Area
It is determined by the coordinates (Distance1, Distance2) in Figure 4.11. In the case of central contact, these are automatically calculated given that the contact is always at the center of the tooth flank.
Amount of Tip Relief (μm) It determines the extend t of tip relief.
Amount of Root Relief (μm) It determines the extend of root relief.
Amount of Crowning (μm) It determines the extend of crowning.
- 57 -
Figure 4.11 Range of tooth-form modification.
Tip relief
Root relief
Center point of contact area
Amount of lead contact
Distance 1
Distance 2
Amount of profile contact
- 58 -
4.4.3 Profile Modification
The range of the tooth flank to be modified is first defined. The sectors of the
spherical involute tooth profile are also determined. These sectors are shown in
Figure 4.11. The sectors near the tip and root will be eliminated and the middle
sector remains a pure spherical involute. The profile-modification concept and the
relevant parameters are shown in Figure 4.12. The applied parameters are defined
as angular values, such as θca (the angular extent of tip relief), θcd (the angular
extent of root relief), Φca (the angular range of tip relief), and Φcd (the angular
range of root relief). These parameters are determined from the values in Table 4.1.
To define the profile modification, several important points are required, such as
the tip-relief point (Pt-r), root-relief point (Pr-r), tip-relief range point (Pt-r-r), and
root-relief range point (Pr-r-r).
The modification procedure is as follows.
(1) Confirm the sectors defined in Subsection 4.3.2.
(2) Specify extents of the modification (Ca and Cd) and calculate these in angular
terms (θca and θcd).
(3) Create offset points (Pt-r and Pr-r) that are rotated from the tip and base points
(Pt and Pb) as the angular amounts of modification (θca and θcd). Therefore, Pt-r and
Pr-r are obtained in the gear-fixed coordinate system as follows.
OP ⃗ = x ı ⃗ + y ⃗ + z k ⃗ (4-25)
x = ξ[sinγ sinη cos θ +sinγ cos η sinθ ] (4-26)
y = ξ[−sinγ sin η sin θ +sin γ cos η cos θ ] (4-27)
z = ξ cos γ (4-28)
- 59 -
η = ψ −ψ (4-29)
γ = γ + θ (4-30)
ψt and ψp, which feature in the above equations, are obtained through Eqs. (3-8)
and (3-12), respectively.
OP ⃗ = x ı ⃗ + y ⃗ + z k ⃗ (4-31)
x = ξ[sin γ sin η cos θ +sin γ cos η sin θ ] (4-32)
y = ξ[−sin γ sin η sin θ +sin γ cos η cos θ ] (4-33)
z = ξ cos γ (4-34)
η = −ψ (4-35)
As before, ψp is obtained via Eq. (3-12).
(4) Specify the tip and root relief range points (Pt-r-r and Pr-r-r) between the tip and
base points (Pt and Pb). The points can be obtained by using the angles of φca and
φcd; these are defined in the gear-fixed coordinate system as follows.
OP ⃗ = ξn ⃗ (4-36)
n ⃗ = sinγ sinη ı ⃗ + sin γ cos η ⃗ + cos γ k ⃗ (4-37)
η = ψ −ψ (4-38)
γ = γ + θ −φ (4-39)
- 60 -
ψtrr and ψp, which feature in the above equations, are obtained through Eqs. (3-8)
and (3-12), respectively.
OP ⃗ = ξn ⃗ (4-40)
n ⃗ = sinγ sin η ı ⃗ + sin γ cos η ⃗ + cos γ k ⃗ (4-41)
η = ψ −ψ (4-42)
γ = γ + θ +φ (4-43)
As before, ψtrr and ψp, which feature in the above equations, are obtained through
Eqs. (3-8) and (3-12), respectively.
(5) Connect the points of Pt-r, Pt-r-r, Pr-r-r, Pr-r, and Pr by a B-spline. Therefore, the
modified tooth-profile can be obtained as follows.
C(t) = ∑ P N (t)
(4-44)
In the above, Pi are the specified position vectors and
N (t) =
N (t) +
N (t) (4-45)
N (t) =
1 for t ≤ t < t 0 for otherwise
(4-46)
- 61 -
Figure 4.12 The concept of tooth-profile modification.
Base circle
Pitch circle
C
Tip circle
Root circle
Not modified
cd
Modified tooth profile
ca
θca
θcd
Pr
Φca
Not modified tooth surface
Pr-r
Pt-r
Pt
Pt-r-r
Pr-r-r
Φcd
OPt-r
Pb
OPr-r
- 62 -
4.4.4 Crowning
The sectors in the lead direction are determined by the procedure of Subsection
4.3.2 and are shown in Figure 4.11. The sectors near the toe and heel will be
eliminated and the middle sector remains. The crowning concept and its relevant
parameters are shown in Figure 4.13. The parameters are defined as θc-heel (angular
extent of heel crowning), θc-toe (angular extent of toe crowning), Cheel (crown length
of heel crowning), and Ctoe (crown length of toe crowning). These parameters are
determined from the values of Table 4.1. To define the crowning, several important
points, such as the crowning points between the heel and toe (Pcrown-r and Pcrown-t),
are needed.
The modification procedure is as follows.
(1) Confirm the sectors defined in Subsection 4.3.2.
(2) Specify the extent of crowning for the heel and toe and calculate this in
angular terms (θc-heel and θc-toe).
(3) To determine the crowned tooth, rotate the modified profile (in Subsection
4.4.3) around the z-axis of the gear-fixed coordinate system with respect to the two
angles, θc-heel and θc-toe.
(4) Define the crowning points between the heel and toe (Pcrown-r and Pcrown-t);
these are scaled points from the tip and root points and are therefore calculated as
follows.
OP ⃗ =
OP ⃗ (4-47)
OP ⃗ =
OP ⃗ (4-48)
(5) Join the offset points and the toe and heel points by the B-spline.
- 63 -
Figure 4.13 The crowning concept.
Base circle
O
Tip circle
Root circle
Tip and root relieved tooth profile
Θc-toe
Not modified tooth surface
Ctoe
Tip-Root Relief and Crowned Tooth
Amount of crown
Pcrown-wheel
Pcrown-toe
Θc-wheel
Cwheel
Pr-r
Pt-r
Pr
F
- 64 -
4.5 Contact Pattern Simulation
4.5.1 Simulation Algorithm
The tooth contact pattern refers to the contact trace that is obtained by rolling
between the tooth flanks of the bevel gears on a low-load condition. A prediction of
the tooth contact pattern between the pinion and side-gear tooth flanks for which
the tooth-forms have been modified is required for adjusting the contact area and
avoiding load concentration. This is commonly undertaken for differential bevel
gears. To predict the tooth contact pattern, the rolling action (envelope) of the side
gear around the pinion is simulated. Figure 4.14 shows this rolling action. The
predicted contact pattern is shown in Figure 4.15. This example shows that the
rolling areas of the side-gear flank are in good contact with the pinion flank.
The example parameters of the tooth modification are listed in Table 4.2. This
modification makes the tooth contact to be 65% around the center area; the extent
of relief and crowning is 100μm.
Table 4.2 Parameters for simulating the contact pattern.
Parameters Pinion Side Differential
Amount of Lead Contact (%) 65% 65%
Amount of Profile Contact (%) 65% 65%
Position of Contact Area center center
Amount of Tip Relief (μm) 100 100
Amount of Root Relief (μm) 100 100
Amount of Crowning (μm) 100 100
Figure 4.15 shows the contact pattern, which indicates that the contact is good
and that the modifications are suitable.
- 65 -
Figure 4.14 A rolling action (envelope) of tooth flanks.
Rolling trace of side gear flank
Pinion flank
- 66 -
Figure 4.15 Contact pattern between the pinion and the side gear.
- 67 -
4.5.2 Simulation Examples
Examples of simulations of the tooth contact pattern are shown in this subsection
for demonstrating the ability to adjust and customize the contact patterns via the
proposed profile modification and crowning methods. The differences among the
central, toe and heel contact are shown in Figure 4.16 through to Figure 4.18.
These contact patterns are simulated by using the parameters of profile
modification and crowning listed in Tables 4.3 and 4.4.
In the case of the central contact ((a) of Figure 4.16, Figure 4.17 and Figure
4.18), the areas of contact are located in the middle of the tooth surface (the areas
of contact are shown as red-colored areas on the tooth flanks of the pinion and side
gear). The central contact shows that the tooth flank is properly modified. The toe
and heel contacts ((b) and (c) of Figure 4.16, Figure 4.17 and Figure 4.18) are also
simulated to show the contact is shifted toward one of the outer (heel) or inner (toe)
portions of the tooth surface by changing the parameter as necessary to effect such
a shifting of the contact position.
By using the developed program for simulating the tooth contact patterns of
bevel gear sets, the contact patterns are properly and easily adjusted and
customized. It is very practical to minimize a trial-and-error and to correct the tooth
contact pattern in the manufacturing processes because the program enables to
predict the proper contact pattern. The program is also used to predict the contact
pattern of the spiral bevel gears and the example is shown in the Figure 4.19. The
tooth contact area is properly simulated in the middle of the tooth surface.
- 68 -
Table 4.3 Parameters of differential bevel gears for forging.
Parameters Pinion Side Differential
NO. TEETH 10 EA 14 EA
MODULE 5.107
PRESSURE ANGLE 22.5˚
AXIS ANGLE 90˚
WHOLE DEPTH 10.32
ADDENDUM 5.63 3.60
DEDENDUM 4.69 6.72
PITCH ANGLE 35˚32’ 54˚26’
FACE ANGLE 48˚29’ 61˚40’
ROOT ANGLE 28˚20’ 41˚31’
Table 4.4 Parameters of tooth modifications.
Parameters Pinion Side Differential
Amount of Lead Contact (%) 40~80% 40~80%
Amount of Profile Contact (%) 80% 80%
Positions of Contact Area center, toe and heel center, toe and heel
Amount of Tip Relief (μm) 100 100
Amount of Root Relief (μm) 100 100
Amount of Crown (μm) 100 100
- 69 -
(a) Central contact
(b) Toe contact
(c) Heel contact
Figure 4.16 Contact pattern for a bevel gear set (side gear).
- 70 -
(a) Central contact
(b) Toe contact
(c) Heel contact
Figure 4.17 Contact pattern for a bevel gear set (pinion).
- 71 -
(a) Central contact
(b) Toe contact
(c) Heel contact
Figure 4.18 Contact pattern simulations of the tooth flanks
- 72 -
Figure 4.19 Contact pattern simulation of spiral bevel gear
- 73 -
4.5.3 Simulation for the Shaft Angle Deviations
Common bevel gears are designed to be mounted on shafts which offset from
each other by 90degs; the designed shaft angle is 90degs and the gear set is
machined on the condition that the shaft angle is determined as 90degs. However,
the common bevel gears have some machining errors or assembling errors and
these errors make the tooth contact poor. On the other hands, spherical involute
bevel gears have better tooth contact and it is a unique merit.
The simulation is carried out to verify the contact merits of the spherical involute
tooth form by using various shaft angles from 85degs to 95degs. The tooth contact
simulation results for the shaft angle deviations are shown in Figure 4.20 and the
simulation shows that the each tooth contacts are equally made but its profile
direction position is moved to the root or top. Therefore, although the spherical
involute bevel gears have some machining errors or assembling errors, they can be
smoothly contacted.
Figure 4.20 Contact pattern simulations for shaft angle deviations
Shaft angle: 91deg 92deg 93deg 94deg 95deg
Shaft angle: 89deg 88deg 87deg 86deg 85deg90deg
- 74 -
4.6 Design Program that uses CATIA-VBA
For better productivity, the complete, integrated design process is implemented
in the CATA-VBA module. This is summarized in Figure 4.21. All the procedures
from inputting parameters through to complete solid modeling are programmed in
CATIA through VBA. Figure 4.22 shows the input windows of the program, which
has mainly two categories, one for spherical involute tooth profiles and the other
for tooth-form modifications. The parameters that have been defined in Subsection
4.2 are used in the program.
Through this program, the conventional design steps, including the machining
and measuring processes for the master gear and 3D-CAD of the master gear
model for an electrode, are reduced to one step. In addition, this program can
simulate the contact pattern of the gear flanks; hence, it can easily improve the
contact performance. Furthermore, because it has a spherical involute tooth profile,
the kinematical performance is improved.
- 75 -
Figure 4.21 Integrated design process by CATIA with VBA
2. Establish Gear Coordinate system
3 . Generating Spherical Involute Curve
4. Determine Tooth Form 5. Completed Bevel Gears
1. Input Parameters
Tip relief
Root relief
Center point of
contact area
Amount of
lead contact
Distance 1
Distance 2
Amount of profile contact
5. Tooth Form Modification and its Contact Analysis
- 76 -
Figure 4.22 Input window of the CATIA-VBA
- 77 -
Chapter 5
Verification of the CAD Program
5.1 Design Parameters
For applying this integrated design program (refer Chapter 4), a conceptual
model of a pair of automotive differential gears for forging is considered and its
parameters are obtained. The differential model is mounted on the transaxle of a
commercial car of Hyundai Motors.
The design parameters, which are input to the program, are listed in Table 5.1.
The tooth-modification parameters are listed in Table 5.2. These modifications
make the tooth contact to be 50% in the lead length but 100% in the profile
direction. The extents of relief and crowning are 100μm. The contact pattern
simulation is shown below.
- 78 -
Table 5.1 Parameters of the differential bevel gears for forging
Parameters Pinion Side Differential
NO. TEETH 10 EA 14 EA
MODULE 5.107
PRESSURE ANGLE 22.5˚
AXIS ANGLE 90˚
WHOLE DEPTH 10.32
ADDENDUM 5.63 3.60
DEDENDUM 4.69 6.72
PITCH ANGLE 35˚32’ 54˚26’
FACE ANGLE 48˚29’ 61˚40’
ROOT ANGLE 28˚20’ 41˚31’
Table 5.2 Parameters of the tooth modifications
Parameters Pinion Side Differential
Amount of Lead Contact (%) 50% 50%
Amount of Profile Contact (%) 100% 100%
Center of Contact center center
Amount of Tip Relief (μm) 100 100
Amount of Root Relief (μm) 100 100
Amount of Crown (μm) 100 100
- 79 -
5.2 Design Results
The main results of the design are a 3D-CAD model of the differential bevel
gears and the contact pattern. The 3D-CAD model can be converted to CAM data
for machine electrodes by PowerMill. The contact pattern will be analyzed through
the contact test for the electrodes.
(1) A 3D-CAD model is constructed for the electrode.
Figure 5.1 3D-CAD model of the differential bevel gears
- 80 -
(2) CNC CAM data simulations are undertaken through PowerMill.
Figure 5.2 Simulation for CNC data transform
(a) Rough cutting
(b) 2nd semi-rough cutting
(c) Finishing
- 81 -
(3) Tooth Contact Pattern Simulations: The contact pattern is shown in the
middle area of the tooth flank.
Figure 5.3 Simulation of the tooth contact pattern
- 82 -
5.3 Manufacturing
The 3D-CAD model has to be converted to CAM data through CAM S/W, such
as PowerMill, for manufacturing electrodes. Based on these CAM data, the
electrodes that are modeled by the integrated design program can be manufactured.
In this thesis, prototypes of the electrodes were machined by a machining center
with a cutting speed of 42,000 RPM but the materials were replaced by common
alloy steel (AISI4140).
5.4 Contact Pattern Test
To verify the 3D-CAD model of the differential bevel gears that were designed
by the integrated program, prototypes of the differential bevel gears were machined
and these were tested on a bevel gear contact test machine, as shown in Figure 5.4.
A pair of the prototype gears was installed on the contact test machine to ensure
that the shaft angle was exactly 90°and that the shafts intersected. Figure 5.5 shows
the contact test scene of the prototype gears. As shown in Figure 5.6, a compound
was pasted on the surface of the bevel gears for carrying out the contact test on the
test-bench.
Figure 5.7 shows the results of the contact test; the contact areas have been
properly located on the surfaces of the bevel gears. These show that crowning has
been reasonably applied on the gear surface and that the spherical involute profiles
are perfectly modeled.
- 83 -
Figure 5.4 Bevel gear contact test machine (Gleason Works)
- 84 -
Figure 5.5 Prototype contact test
- 85 -
Figure 5.6 Machined bevel gears for forging
Figure 5.7 Contact Patterns
- 86 -
Chapter 6
Conclusion
In this thesis, an integrated CAD program was developed for an automotive
differential bevel gear set by the use of a spherical involute tooth form. The CAD
program integrated the processes for the modeling and contact-pattern analysis of
electrodes of a forged differential gear set. Therefore, the prototype of the
electrodes was directly machined, which reduced the extent of trial-and-error. In
addition, because the tooth form was replaced with a spherical involute tooth form,
the kinematical performance was improved.
To verify the tooth forms and gear meshes of the modeled electrodes for a bevel
gear set, prototypes were developed of electrodes for one gear set that had localized
tooth-form modifications. The results of the rolling test of the gear were compared
with the calculated results. The tooth flank surface was measured on a CMM.
The results of this thesis are listed as follows.
(1) A spherical involute tooth profile was mathematically formulated and
implemented in a CAD program.
(2) A CAD program was developed for integrating the forging process.
(3) By the use of the integrated CAD program, bevel gear sets with spherical
involute tooth forms were modeled for automotive differentials.
(4) To quantitatively adjust the contact area between the gears, lead modification
(crowning) and profile modification methods were proposed for differential bevel
gears.
(5) The contact patterns were simulated under conditions of light loads.
- 87 -
(6) A prototype of a bevel gear set for an electrode was modeled and
manufactured.
(7) The actual contact patterns of the prototype were compared with the
simulated results.
- 88 -
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- 93 -
Appendix A
A.1 Geometry of the Spherical Triangle
A line connecting two points on the surface of a reference sphere is an arc. The
relationships in the triangle formed by connecting three points on a sphere surface
will be different from those of a plane triangle. Figure A.1 displays triangle
ΔA1B2C3 on the surface of a sphere with center O and radius ξ. If we denote the
tangent lines that contact arcs A1B2 and A1C3 at apex A1 as A1B1 and A1C1,
respectively, the following relationship holds for plane triangle ΔA1B1C1.
(B C ) = (A B )
+ (A C ) − 2(A B ) (A C )cosα (A-1)
In triangle ΔOB1C1,
(B C ) = (OB )
+ (OC ) − 2(OB )(OC )cosA (A-2)
and in ΔOA1B1 and ΔOA1C1
A B = OA tanΓ (A-3)
OB = OA /cosΓ (A-4)
A C = OA tanB (A-5)
OC = OA /cosB (A-6)
Substituting Equations (A-2) ~ (A-6) into Equation (A-1) and rearranging, we
obtain,
- 94 -
cosA = cosBcosΓ + sinBsinΓcosα (A-7)
A.1 Spherical triangle and its relations
Likewise, if we apply similar processes to plane triangles ΔA2B2C2 and ΔA3B3C3,
we obtain the following relationships:
cosB = cosΓcosA + sinΓsinAcosβ (A-8)
cosΓ = cosAΓcosB + sinAsinBcosγ (A-9)
If ΔA1B2C3 is a right spherical triangle with α= 90deg from Equation (A-7)
cosA = cosBcosΓ (A-10)
Eliminating Β or Α from Equations (A-8) and (A-10) and rearranging,
- 95 -
tanΓ = tanAcosβ (A-11)
tanB = tanβcosγ (A-12)
Eliminating Γ from Equations (A-9) and (A-10), we obtain
tanB = tanAcosγ (A-13)
- 96 -
CAD of Forged Involute Bevel Gear
산 학 학원 계 계공학과
박 문 우
본 논문에 는 동차 차동 치 베벨 어 전극 전한
벌루트 치 하여 계하 다. 본 베벨 어 실제
상 차에 적 는 차동 치 베벨 어 제원 그 적 하
벌루트 치 학적 특 그 갖도 3D 러(CATIA)
에 해 만들어졌다. 그리고 베벨 어 에 크라우닝 적 하 고
치물림 시물 하 다. 본 베벨 어 검증 하여 5
니싱 터 제품 가공하 고 물림 시험 실시하여 시물 결과
비 하 다. 그 결과 제품 시물 과 치함 었다.
라 본 논문 통하여 벌루트 치 갖는 베벨 어 실
보여졌고 존 리슨 치 베벨 어 체 가능함 었다.
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