D of Forged Involute Bevel

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CAD of Forged Involute Bevel Gear

2009 8

2009 7 3

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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

, , 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

; tool pressure angle

g ; gear pressure angle

; azimuthal angle

; cone angle at an arbitrary point

a ; addendum-cone angle

b ; base-cone angle

p ; pitch-cone angle

g ; rotating angle of gear

; cone distance

; azimuthal angle

; spherical-involute function

p ; 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



- 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-

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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 Tredgolds 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

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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

Tredgolds 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 Klingelnbergs 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

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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 (R) is split by the

differential gears into the left and right axle-speeds (S1 and S2). This kinematical

relationship [27] is given as follows:

= (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

(s1)Right Axle

(s2)

Final Drive Pinion

Final Drive

Ring Gear (R)

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.






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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

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.

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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, , 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 , b, , , and are as follows.

cos = cos b

cos (3-1)

sin =

(3-2)

cos =

(3-3)

In the above, , b, and are angular parameters that represent COA, COB, and

BOA, respectively. The azimuthal angle, , is the angle between the arcs, BC and

AC. Eliminating the angle, , from Eqs. (3-1) and (3-2), we obtain:

sin = cos sin . (3-4)

The spherical involute function refers to the angle, , between the arcs, AC and

CD, in Figure 3.1. Since the lengths of BD and AB are identical,

- 26 -

= ( + ) sin . (3-5)

Expressing Eq. (3-5) in terms of , we get:

=

. (3-6)

In Eq. (3-6), the spherical involute function can be written as a function of and

the base-cone angle, b.

=

tan

cos

. (3-7)

Since b 0 if , Eq. (3-7) implies that asymptotically converges to

tan . From this, it can be confirmed that a spherical involute tooth-profile is

a generalized configuration of a cylindrical involute tooth profile. can be

expressed in terms of the cone angle, , instead of .

By applying Eq. (3-4) and rearranging terms, we obtain:

=

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, g.

In a standard gear, the gear pressure angle and the rack (or tool) pressure angle

are identical. The relationship between , b, and the pitch cone angle, p (angle

COP), in Figure 3.2 is obtained from Eq. (3-4).

sin = cos sin (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

+ 2 2 . (3-11)

- 28 -

In Eq. (3-11), R = Asin and R = Asin .

is obtained from either Eq. (3-7) or (3-8), and p is obtained by substituting

= into Eq. (3-7) or =

into Eq. (3-8).

=

tan

cos

(3-12)

The tooth thickness, tb, of the base circle is t when = in Eq. (3-11), at

which point = 0; then, tb becomes:

t = R

+ 2 . (3-13)

Where R = A sin .

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 a at m = 0.1, m = 3.0, = 20, Z = 20 and = 10~70.

The tooth-tip circle radius Rt becomes sin + , which is reduced as b

increases. When b reaches its maximum of 70, Rt is smaller than the pitch-circle

radius RP, tb increases in proportion to b. 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 = n (3-15)

n = sin sin + sin cos + cos k (3-16)

and

= (3-17)

In the above, and 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 gears angular motion, g,

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.






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 ,

p, and , 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 ().

(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 p1 and p2,

respectively, from the vector, OP. The rotating angles, p1 and p2, 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, 1 and 2, 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 p and b. The base cone angle, b, is calculated via Eq. (3-18) and

the basic parameters of , p, and for defining the coordinate system are obtained

from the gear parameters, m, , (shaft angle), Z1, and Z2. The relational

equations are as follows.

= tan

(4-1)

= (4-2)

=

=

(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

1

2

- 40 -

Figure 4.4 Basic parameters of a pair of spherical involute bevel gears.

AP

O

C

Base circle

Pitch circle

Contact normal

p2

y2

x10, y2

0

x1

x2z10, z1, x2

0

y10, z2

0, z2y1

b2

p1b1

- 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 (a1 and d1) of the pinion are

defined as per the following equations.

a = 1.17m .

(4-4)

d = 1.788m a (4-5)

= tan

(4-6)

= 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 = n (4-8)


= (4-10)

= + (4-11)

For the above equations, t1 and p1 are obtained through Eqs. (3-8) and (3-12),

respectively.

Once the angular variables, such as t1 and t1, are determined, the spherical

involute curve is defined from D to Pt1 and is also expressed by the following

vector, OA.

OA = n (4-12)


= (4-14)

In the above, the concerned parameters satisfy the inequality, +

. As before, 1 and p1 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

a1

d1Pt1

Pr1

Tip circle

Root circle

t1P

b1

B

t1

z10, z1

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 (t1 and t2), as follows.

t =

(a a ) tan + 0.05m (4-15)

t = m t (4-16)

t = t

( )

(4-17)

t = t

( )

(4-18)

= tan

(4-19)

= tan

(4-20)

- 48 -

Figure 4.8 Tooth-thickness definitions.

P

O

Base circle

Pitch circle

t2

[Pinion]

C

t1

[Side Gear]

t1

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, 1, which

is the angular thickness of the pinion tooth at any point on the calculated spherical

involute curve, is found from t1 and P1. Then, 1 can be calculated as follows:

= 2( ) (4-21)

Through the foregoing value of 1, the coordinates for the mirror-image of the

spherical involute curve are given by:

OA = n (4-22)


= + (4-24)

In the above, the concerned parameters satisfy the inequality, +

; 1 and p1 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

t1

a1

d1

Pt1

Pr1

Tip circle

Root circle

1

1

PP

1

- 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 (m) It determines the extend t of tip relief.

Amount of Root Relief (m) It determines the extend of root relief.

Amount of Crowning (m) 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 ca (the angular extent of tip relief), cd (the angular

extent of root relief), ca (the angular range of tip relief), and cd (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 (ca and cd).

(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 (ca and cd). Therefore, Pt-r and

Pr-r are obtained in the gear-fixed coordinate system as follows.

OP = x + y + z k (4-25)

x = [sin sin cos +sin cos sin ] (4-26)

y = [sin sin sin +sin cos cos ] (4-27)

z = cos (4-28)

- 59 -

= (4-29)

= + (4-30)

t and p, which feature in the above equations, are obtained through Eqs. (3-8)

and (3-12), respectively.

OP = x + y + z k (4-31)

x = [sin sin cos +sin cos sin ] (4-32)

y = [sin sin sin +sin cos cos ] (4-33)

z = cos (4-34)

= (4-35)

As before, p 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 ca and

cd; these are defined in the gear-fixed coordinate system as follows.

OP = n (4-36)


= (4-38)

= + (4-39)

- 60 -

trr and p, which feature in the above equations, are obtained through Eqs. (3-8)

and (3-12), respectively.

OP = n (4-40)


= (4-42)

= + + (4-43)

As before, trr and p, 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

cacd

Pr

ca

Not modified tooth surface

Pr-r

Pt-r

Pt

Pt-r-r

Pr-r-r

cd

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 c-heel (angular

extent of heel crowning), c-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 (c-heel and c-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, c-heel and c-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

c-toe

Not modified tooth surface

Ctoe

Tip-Root Relief and Crowned Tooth

Amount of crown

Pcrown-wheel

Pcrown-toe

c-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 100m.

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 (m) 100 100

Amount of Root Relief (m) 100 100

Amount of Crowning (m) 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.

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Table 4.3 Parameters of differential bevel gears for forging.


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 3532 5426

FACE ANGLE 4829 6140

ROOT ANGLE 2820 4131

Table 4.4 Parameters of tooth modifications.


Amount of Lead Contact (%) 40~80% 40~80%


Positions of Contact Area center, toe and heel center, toe and heel



Amount of Crown (m) 100 100

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(a) Central contact

(b) Toe contact

(c) Heel contact

Figure 4.16 Contact pattern for a bevel gear set (side gear).

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(a) Central contact

(b) Toe contact

(c) Heel contact

Figure 4.17 Contact pattern for a bevel gear set (pinion).

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(a) Central contact

(b) Toe contact

(c) Heel contact

Figure 4.18 Contact pattern simulations of the tooth flanks

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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 100m. The contact pattern

simulation is shown below.

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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 3532 5426

FACE ANGLE 4829 6140

ROOT ANGLE 2820 4131



Amount of Lead Contact (%) 50% 50%


Center of Contact center center



Amount of Crown (m) 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 90and 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.

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Figure 5.4 Bevel gear contact test machine (Gleason Works)

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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.

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(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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D of Forged Involute Bevel

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