DEVELOPMENT OF TOOTH CONTACT AND MECHANICAL EFFICIENCY

MODELS FOR FACE-MILLED AND FACE-HOBBED

HYPOID AND SPIRAL BEVEL GEARS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

By

Mohsen Kolivand, B.S., M.S.

*****

The Ohio State University

2009

Dissertation Committee:

Professor Ahmet Kahraman, Advisor

Professor Donald R. Houser

Professor Gary L. Kinzel

Professor Henry H. Busby

Approved by

________________________________

Advisor

Graduate Program in Mechanical Engineering

© Copyright by

Mohsen Kolivand

2009

ii

ABSTRACT

A computationally efficient load distribution model is proposed for both face-

milled and face-hobbed hypoid gears produced by Formate and generate processes.

Tooth surfaces are defined directly from the cutter parameters and machine settings. A

novel methodology based on the ease-off topography is used to determine the unloaded

contact patterns. The proposed ease-off methodology finds the instantaneous contact

curves through a surface of roll angles, allowing an accurate unloaded tooth contact

analysis in a robust and accurate manner. Rayleigh-Ritz based shell models of teeth of

the gear and pinion are developed to define the tooth compliances due to bending and

shear effects efficiently in a semi-analytical manner. Base rotation and contact

deformation effects are also included in the compliance formulations. With this, loaded

contact patterns and transmission error of both face-milled and face-hobbed spiral bevel

and hypoid gears are computed by enforcing the compatibility and equilibrium conditions

of the gear mesh. The proposed model requires significantly less computational effort

than finite elements (FE) based models, making its use possible for extensive parameter

sensitivity and design optimization studies. Comparisons to the predictions of a FE

hypoid gear contact model are also provided to demonstrate the accuracy of the model

under various load and misalignment conditions.

iii

The proposed ease-off formulation is generalized next to include various types of

tooth surface deviations in the tooth contact analysis. These deviations are grouped in

two categories. The proposed ease-off based method is shown to be capable of modeling

both global deviations due to common manufacturing errors and heat treat distortions and

local deviations due to surface wear.

The proposed loaded contact model is combined at the end with a friction model

based on a mixed elastohydrodynamic lubrication model to predict the load dependent

(mechanical) power losses and efficiency of the hypoid gear pairs. The velocity, radius

of curvature and load information predicted by the contact model is input to the friction

model to determine the distribution of the friction coefficient along the contact surfaces.

At the end, the variations of predicted mechanical efficiency with geometry, surface and

lubricant parameters are quantified.

iv

Dedicated to my mother

v

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Prof. Ahmet

Kahraman, for this great research opportunity, his guidance throughout my research and

his effort in reviewing this dissertation. I would also like to express my appreciation to

Prof. Donald R. Houser, Prof. Gary L. Kinzel and Prof. Henry H. Busby for their patience

and effort in being a part of my dissertation committee. Also, thanks to Dr. Sandeep

Vijayakar who kindly permitted me to use CALYX package.

I would like to thank the sponsors of the Gear Power Transmission Research

Laboratory for their financial support throughout my study.

My sincere thanks go to Prof. Hermann J. Stadtfeld from The Gleason Works who

spent valuable time teaching me fundamental concepts of bevel gear design and

manufacturing, and also for having attended my dissertation defense.

I would also like to thank Jonny Harianto, Samuel Shon and all my lab mates for

their help and friendship throughout my study at OSU and beyond.

Finally, I deeply appreciate the love and trust shown toward me by my parents,

my grandparents, my uncles and aunts, my sisters and brother and my fiancée for all of

their support and encouragement.

vi

VITA

Oct. 19, 1976 ……………….…….. Born – Tehran, Iran

Sep. 1995 – Sep. 1999 ……………. B.S. in Mechanical Engineering, Solid Mechanics, Tehran University, Tehran, Iran

Sep. 1999 –Feb. 2002 ……………. M.S. in Mechanical Engineering, Applied Design, Tehran University, Tehran, Iran

Nov. 1999 –Apr. 2002 ……………. Design Engineer, Tarh Negasht Co., Tehran, Iran

Apr. 2002 –Aug. 2005 ……………. Design Engineer, TAM Co., Tehran, Iran

Jan. 2006 – present ……………… M.S. in Mechanical Engineering (Sep. 2008) Graduate Research Associate Gear and Power Transmission Research Laboratory Department of Mechanical Engineering The Ohio State University Columbus, Ohio

PUBLICATIONS

1. Kolivand M. and Kahraman A., “A Load Distribution Model for Hypoid Gears Using

Ease-off Topography and Shell Theory,” Journal of Mechanism and Machine Theory,

2009.

FIELDS OF STUDY

Major Field: Mechanical Engineering

vii

TABLE OF CONTENTS

Page

Abstract………………………………………………………………………………. ii

Dedication……………………………………………………………………………. iv

Acknowledgments………………………………………………………………….... v

Vita……………………………………………………………………………….….. vi

List of Tables..………………………………………………………………...…….. x

List of Figures...……………………………………………………………...……… xi

Nomenclature………………………………………………………………………... xvi

Chapters:

1 Introduction ……………………………………………………………………... 1

1.1 Motivation and background ………………………………………………… 1

1.2 Literature Review ………………………………………………………….... 4

1.3 Scope and Objectives ...………………...…………………………………… 14

1.4 Overall Modeling Methodology...……………...……………………………. 17

1.5 Dissertation Outline………………………………………………………….. 19

References of Chapter 1………………………………………………………….. 20

2 Definition of Face-milled and Face-hobbed Hypoid Gear Geometry and

Unloaded Tooth Contact Analysis ……………………………………..........…... 28

2.1 Introduction …………………….……………….…………………………… 28

2.2 Definition of Tooth Surface Geometry ……………………………………... 30

viii

2.2.1 Kinematics ………………………………………………………...… 32

2.2.2 Cutting Tool Geometry and the Relative Motion……………………. 36

2.2.3 Equation of Meshing…………………………………………………. 40

2.2.4 Principal Curvatures and Principal Directions…………………......... 42

2.3 Unloaded Tooth Contact Analysis ………………...………………………… 44

2.3.1 The Conventional Method of UTCA.……………….……………….. 47

2.3.2 Ease-off Based Method of UTCA …………………………………… 52

2.3.2.1 Construction of Ease-off and the Surface of Roll Angle………… 53

2.3.2.2 Contact Pattern and Transmission Error…………………………. 57

2.4 An Example Hypoid Unloaded Tooth Contact Analysis ……………………. 59

References for Chapter 2………………………………………………………. 65

3 Shell Based Hypoid Tooth Compliance Model and Loaded Tooth Contact

Analysis …............................................................................................................. 68

3.1 Introduction …………………………………………………………………. 68

3.2 Tooth Compliance Model …………………………………………………... 70

3.3 Loaded Tooth Contact Analysis ….…………………………………………. 81

3.4 An Example Hypoid Tooth Contact Analysis ………………………………. 84

References for Chapter 3………………………………………………………… 92

4 Loaded Tooth Contact Analysis of Hypoid Gears with Local and Global Surface

Deviations…………………………………..….………………………………… 94

4.1 Introduction …………………………………………………………………. 94

4.2 Construction of the Theoretical Ease-off Topography …..………………….. 98

4.3 Updating Ease-off Topography for Manufacturing Errors and Surface Wear 102

4.4 Unloaded and Loaded Tooth Contact Analyses ………………….…………. 107

4.5 Example Analyses ………………………………………………………..…. 111

4.5.1 A Face-milled Hypoid Gear Pair with Local Surface Deviations……. 111

ix

4.5.2 A Face-hobbed Hypoid Gear Pair with Global Deviations………….. 120

References for Chapter 4………………………………………………………... 130

5 Predictions of Mechanical Power losses of Hypoid Gear Pairs………………….. 133

5.1 Introduction ……………………………………………………………….…. 133

5.2 Hypoid Gear Mechanical Power Loss Model …………………………….…. 137

5.2.1 Definition of the Sliding and Rolling Velocities…………………….. 140

5.2.2 Friction Coefficient Model…………………………………………... 143

5.2.3 Derivation of a Friction Coefficient Formula………………………... 145

5.2.4 Computation of the Mechanical Power Loss of the Hypoid Gear Pair 151

5.3 Numerical Example ……………………………………..…………………... 152

5.4 Conclusion…………………………………………………………………… 167

References for Chapter 5………………………………………………………… 168

6 Conclusions and Recommendations for Future Work…...………………………. 172

6.1 Thesis Summary ……………………………………………………………... 172

6.2 Conclusion and Contributions……………………………………………….. 175

6.3 Recommendations for Future Work …………………………………………. 177

Bibliography …………………………..………………...…………………………... 178

x

LIST OF TABLES

Table Page

2.1 Basic drive side geometry and working parameters of the example hypoid

gear pair…………………………………………………………………..….. 60

3.1 The loaded transmission error predictions of the proposed model; 0G

mm and 0 for all cases.………………………….…………………..… 88

4.1 Basic drive side geometry and working parameters of the example hypoid

gear pair.…………………………………………………….……………….. 112

4.2 The transmission error amplitudes of theoretical and deviated surfaces…….. 118

4.3 Basic drive side geometry and working parameters of the example hypoid

gear pair.……………………………………………………………………... 121

4.4 The transmission error amplitudes of theoretical and deviated surfaces…..… 128

5.1 Parametric design for the development of the friction coefficient formula….. 147

5.2 Basic parameters of the 75W90 gear oil used in this study.…………….….... 148

5.3 Values of the coefficients in Eq. 5.11…………………..……………………. 150

5.4 Basic drive side geometry and working parameters of the examples hypoid

gear pairs……………………………………………………………………... 153

xi

LIST OF FIGURES

Figure Page

1.1 A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize

passenger cars and SUV’s.…………………………………………………… 2

1.2 A sample hypoid gear pair with a shaft angle and a shaft off-set ad .….... 4

1.3 Different gear types based on shaft arrangements…………………………… 6

1.4 Flowchart of overall hypoid gear loaded tooth contact analysis methodology 18

2.1 (a) Face-milling and (b) face-hobbing cutting processes…………………….. 31

2.2 Cradle based hypoid generator parameters…………………………………... 33

2.3 (a) Cutter head, (b) blade and (c) cutting edge geometry……………………. 37

2.4 Generation process…………………………………………………………… 41

2.5 Curvature computation procedure…………………………………………… 43

2.6 General case of approximating gear surfaces as two contacting ellipsoids to

orient instantaneous contact line……………………………………………... 49

2.7 Construction of the ease-off, action and Q surfaces…………………………. 55

2.8 Unloaded TCA computation procedure: (a) gear projection plane, ease-off

and Q surfaces, and (b) instantaneous contact curve, contact line and

unloaded transmission error…………………………………………………. 58

2.9 Unloaded transmission error of the example gear pair with misalignments

0.15 mm, 0.12 mm, 0, 0E P G ………………………….. 61

xii

2.10 Unloaded contact pattern of the example gear pair for three adjacent tooth

pairs 1i , i and 1i (i-1), (i) and (i+1) with

0.15 mm, 0.12 mm, 0E P G and 0 ……………………… 63

2.11 Unloaded contact pattern of the example gear pair (a) at nominal position

with 0E P G , (b) at toe with 0.08E mm, 0.10P

mm and 0G , (c) at heel with 0.15E mm, 0.10P mm

and 0G and (d) at toe with 0.05E mm, 0P G , and

4 min…………………………………………………………………. 64

3.1 Basic dimensions of a hypoid tooth used in the compliance formulation…… 72

3.2 Flowchart of the compliance computation.…………………………….…….. 79

3.3 Potential contact line discretization.……………………………...………….. 80

3.4 The comparison of the shell model deformation to FEM.…….……………... 82

3.5 Static equilibrium between torque applied on gear axis and torque produced

by the force of all contacting segments.………...………………………….... 85

3.6 Loaded transmission error of the example gear pair with 0.15,E

0.12, P 0G and 0 at (a) 50pT Nm, (b) 250pT Nm, and (c)

500pT Nm.…………………………………………………………............ 87

3.7 Comparison of loaded contact patterns predicted by the proposed model to

an FE model [3.11] for (a) 50 NmpT , 0.15E mm, 0.12P mm, (b)

250 NmpT , 0.15E mm, 0.12P mm, (c) 500 NmpT ,

0.15E mm, 0.12P mm, (d) 50 NmpT , 0.08E mm,

0.05P mm, and (e) 50 NmpT , 0.26E mm, 0.13P mm ( all at

0, 0G ).……………………………………………………….....… 89

4.1 Construction of the ease-off, action and Q surfaces………............................ 99

xiii

4.2 Graphical demonstration of the procedure to update ease-off surface for

surface deviations.………………………...…………………………………. 105

4.3 Graphical demonstration of the procedure to compute unloaded TCA; (a)

gear projection plane, ease-off and Q surfaces, and (b) instantaneous

contact curve, contact line and unloaded transmission error………………... 108

4.4 Theoretical contact curves of an example hypoid gear pair.………………… 109

4.5 Example local deviation surfaces for the gear and pinion tooth surfaces..….. 113

4.6 Ease-off update for the example deviation of Fig. 5. (a) Three-dimensional

view of the projection plane, and , , Q and Q surfaces, and contour

plots of (b) , (c) , and (d) the change of ease-off topography..………… 114

4.7 Predicted unloaded tooth contact pattern for separation value of 6 μm … 116

4.8 Transmission error (UTE) curves for theoretical and deviated surfaces at (a)

unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm. 117

4.9 Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)

theoretical and (b) deviated surfaces………………………………………… 119

4.10 Example global deviation surfaces measured by CMM for the gear and

pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured

deviation, (c) pinion deviation distribution in tooth active region and (d)

gear deviation distribution in tooth active region……………………………. 122

4.11 Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical ease-

off topography, (b) updated ease-off topography only with pinion deviation,

(c) updated ease-off topography only with gear deviation, and (d) updated

ease-off topography with both pinion and gear deviations…………………... 124

4.12 Predicted unloaded tooth contact pattern for separation value of 6 μm … 125

xiv

4.13 Transmission error curves for theoretical and deviated surfaces; (a)

unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm 127

4.14 Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)

theoretical and (b) deviated surfaces………………………………………… 129

5.1 Flowchart of overall hypoid gear efficiency computation

methodology.………………………………………………………………... 138

5.2 Sliding and rolling velocities and their projection in tangential plane along

and normal to the contact line direction.……………………………………... 141

5.3 Ease-off topography of (a) Design A with / 0.07a ad D and (b) Design B

with / 0.14a ad D …………………………………………………………... 154

5.4 Maximum contact pressure distribution of (a) Design A with / 0.07a ad D

and (b) Design B with / 0.14a ad D for 500 NmpT …………………… 155

5.5 Rolling velocity distribution of (a) Design A with / 0.07a ad D and (b)

Design B with / 0.14a ad D at 1500 rpmp …………………………… 157

5.6 Sliding velocity distribution of (a) Design A with / 0.07a ad D and (b)

Design B with / 0.14a ad D at 1500 rpmp …………………………… 158

5.7 Slide-to-roll ratio distribution of (a) Design A with / 0.07a ad D and (b)

Design B with / 0.14a ad D at 1500 rpmp …………………………... 159

5.8 Equivalent radius of curvature distribution of (a) Design A with

/ 0.07a ad D and (b) Design B with / 0.14a ad D ……………………….. 160

5.9 distribution of (a) Design A with / 0.07a ad D and (b) Design B with

/ 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT

and 1 2 0.8 mS S ……………………………………………………….... 161

xv

5.10 Friction coefficient distribution of (a) Design A with / 0.07a ad D and

(b) Design B with / 0.14a ad D at 1500 rpmp , 500 NmpT ,

90 CoilT and 1 2 0.8 mS S …………………………………………... 162

5.11 Power loss and efficiency of Design A (a1, b1) and Design B (a2,b2) at

90 CoilT and 1 2 0.8 mS S ……………………………………….…. 163

5.12 Efficiency of (a) Design A with / 0.07a ad D and (b) design B with

/ 0.14a ad D for different surface finish and oil temperatures at

1500 rpmp and 500 NmpT …………………………..……………… 166

xvi

NOMENCLATURE

ga Gear axis vector

pa Pinion axis vector

C Total compliance matrix

ad Pinion offset

aD Gear pitch diameter

1 2,e e Principal directions

E Overall efficiency

bE Blank offset

eqE Equivalent module of elasticity

kf Load per unit length of segment at time step k

F Force vector

h Film thickness

fh Tip of blade to reference point

toeh Tooth height at toe

heelh Tooth height at heel

ui Normal to the mid-surface of the shell

Ti Tilt angle

js Swivel angle

1 2,k k Normal curvatures

1 2,K K Principal curvatures

Segment index

gm Number of surface grid in lengthwise direction

xvii

ctbM Machine center to back

n Normal to the family of cutter surface

cln Number of potential contact lines at each time step

gn Number of surface grids in profile direction

n Total number of contact segments

sn Total number of time steps per pinion pitch

cN Total number of segment on all potential contact lines/curves

gN Number of teeth of gear

pN Number of teeth of pinion

tN Number of blade groups

PE Potential energy

q Roll angle

aq Pinion pitch

Q Surface of roll angle

r Gear ratio

cr Cutter radius

R Position vector of a point on circular cylindrical shell

eqR Equivalent Hertzian curvature

gR Vector of the distances of each segment to the gear axis

ij Ease-off value of point ij

Ease-off surface

s Distance of an arbitrary point to reference point on the blade edge

rS Radial setting

S Initial separation vector

xviii

SE Strain energy

SR Sliding to rolling velocity ratio

eqS Equivalent surface roughness

t Instantaneous potential contact line direction

t Perpendicular to the instantaneous potential contact line direction

toet Tooth thicknesses at toe-root

heelt Tooth thicknesses at heel-root

T Torque

oilT Oil temperature

( )gtu Gear surface velocity in t direction

( )gtu Gear surface velocity in t direction

( )ptu Pinion surface velocity in t direction

( )ptu Pinion surface velocity in t direction

UTE Unloaded transmission error

cV Velocity of the point being cut seen from cradle axis

ijv Total surface velocity vector

rV Rolling velocity in t direction

sV Sliding velocity in t direction

soV Overall sliding velocity

wV Velocity of the point being cut from work axis

W Transverse deflection

ijw Surface velocity vector along common normal

WF Work done by the external force

BX Sliding base

xix

Y Vector of slack variable

Variable of curvilinear cylindrical coordinate system (in tooth lengthwise

direction)

b Blade angle

pv Pressure viscosity coefficient

Variable of curvilinear cylindrical coordinate system (in tooth profile

direction)

x Shear rotation in profile direction

m Machine root angle

mn Shear strain

Paint thickness (separation)

b Blade offset angle

E Pinion offset error

G Gear mounting distance error

P Pinion mounting distance error

Shaft angle error

m Normal strain

k Rolling power loss of segment at time step k

Effective viscosity

0 Ambient viscosity

Ratio of the smooth condition minimum film thickness to the RMS of

surface roughness

k Friction coefficient of segment at time step k

Thermal correction factor

Shear stress

xx

c Cradle angle

g Blank phase angle

t Cutter phase angle

( )n Polynomial of order n for shape function in lengthwise direction

( )m x Polynomial of order m for shape function in profile direction

c Angular speed of the cradle axis

g Angular speed of the blank axis

t Angular speed of the cutter axis

p Pinion speed

Tangential plane

Superscript:

( ) Real or updated

( ) Theoretical

ˆ( ) Conjugate

( ) Interpolated / Extrapolated

p Pinion

g Gear

a Action

1

CHAPTER 1

INTRODUCTION

1.1. Motivation and background

Hypoid gears are widely used in many power trains to transfer power between

two non-intersecting crossed axes. Their most common and highest-volume applications

can be found in front and rear axles of rear-wheel-drive or all-wheel-drive vehicles [1.2].

Figure 1.1 shows a sample of hypoid gear application for the rear axle. A rear axle has

three primary functions: (i) transmit power from the drive train axis to the wheel axle,

that is usually perpendicular to the drive train axis with an offset, (ii) provide the

capability to the vehicle to turn corners without any slippage at its wheels through its

differential, and (ii) provide the final stage of speed reduction (torque increase) that is

typically of the order of three to four.

“Hypoid gears are the most general form of gearing and their solution has been long in coming. There is no form of gearing where so many guesses have been made, a few of them right, plenty of them wrong and some without consequences”

Ernest Wildhaber [1.1]

2

Figure 1.1: A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize

passenger cars and SUV’s (Courtesy of American Axle & Manufacturing Inc.).

Hypoid pinion

Hypoid gear

Output

Input

3

A pair of hypoid gears is commonly used to deliver this third final drive function.

In the arrangement shown in Figure 1.1, the smaller of the hypoid gears, called the

pinion, is at the end of the drive shaft and is in mesh with the larger hypoid gear (called

the gear).

Hypoid gears can be considered as one of the most general cases of gearing based

on their geometry, such that other gear types can be obtained from it by assigning certain

values to some of the geometric parameters [1.1,1.3-1.5]. The main function of the

hypoid gear pair in a rear axle is to transmit power between two axes that are at a shaft

angle (usually90 ) [1.2,1.6] and at a certain amount of shaft off-set ad as shown in

Figure 1.2. A higher level of power transmission through such a kinematic configuration

is possible through use of a hypoid gear pair, which can provide a better balance amongst

all primary design requirements such as strength, noise and power density. The trade-off

between these performance characteristics while satisfying the kinematic constraints

results in the hypoid tooth form that is rather complex geometrically.

Figure 1.3 shows a schematic of different types of gearing, based on shaft

arrangements. The shaft offset, being the main difference between spiral bevel and

hypoid gears, provides several advantages to hypoid gears including larger pinion size,

smaller pinion tooth counts, higher contact ratio, and higher contact fatigue strength. On

the negative side, hypoid gears experience higher sliding velocities, resulting in higher

power losses due to excessive sliding friction. Increasing the pinion size without

4

Figure 1.2: A sample hypoid gear pair with a shaft angle and a shaft off-set ad .

 

ad

5

shaft offset (spiral bevel) increases the size of the final drive significantly, while the

hypoid pinion can be made larger due to shaft off-set to increase the strength of the gear

pair while minimizing the overall size of the gear pair.

Any attempt to improve the functional attributes of a hypoid gear pair in terms of

its strength, quality, noise and power efficiency requires an optimization of its design

either by fine-tuning its key parameters that have traditionally been chosen based on

certain empirical knowledge or by making use of additional motion capabilities provided

by new-generation hypoid gear cutting machines that allow application of many kinds of

surface modifications [1.7, 1.8]. Hypoid gear design procedures were developed within a

small number of hypoid gear cutting machine tool and cutting tool manufacturers and

practical and theoretical details of hypoid development are still propriety to these

companies [1.9].

In general, two different basic cutting methods are used to generate hypoid gears,

namely face-milling (FM) or single indexing, and face-hobbing (FH) or continuous

indexing, which have their own advantages over each other. The FM process that was

the primary hypoid cutting method for decades has been taken over by the FH process in

automotive axle applications, mostly due to its productivity advantages caused by

continuous indexing [1,2, 1.10-1.12]. However, it is safe to state that the technology

level and design understanding of the FH process is almost a decade behind the face

milling process [1.13]. One reason for this is that newer machining methods

6

Figure 1.3: Different gear types based on shaft arrangements.

 Worm gears

High reduction hypoid gears (HRH) or Spiroid

Hypoid gears, Face gear

Bevel gears (Straight, Spiral and Zerol), Face gear

Parallel axis gears (Spur, Helical and Herringbone)

Sli

ding

Pin

ion

size

Eff

icie

ncy

7

such as grinding are applicable to FM process, while there is still no such alternative

method or machinery developed for face-hobbing method.

Having high gear ratios in hypoid gears in automotive applications causes the gear

to have usually 3 to 4 times the number of teeth of the pinion, which justifies designing

the gear surface as simple as possible to increase production efficiency and minimize

manufacturing time. One typical cost-effective cutting method, called Formate®, is much

faster than the Generating methods. In Formate®, only a few degrees of freedom of

motions are allowed between the cutter and the gear blank (compared to the Generate

cutting method). Therefore, most of the surface modifications are applied to pinion tooth

surfaces, rather than the gear tooth surfaces.

Quality of a hypoid gear pair is defined by a number of performance

characteristics including its contact pattern, the motion transmission error (TE),

efficiency and sensitivity to misalignments. The geometric accuracy of a single gear has

limited significance here as the geometry of the mating gear and the assembly errors can

change the performance characteristics drastically. These performance characteristics

have been quantified either by using FE-based hypoid gear load distribution models or by

experimental means, both of which are very time-consuming and expensive. Due to their

significant computational burden, FE-based hypoid gear contact models are not suitable

for design and parameter and misalignment sensitivity studies. The aim of this study is to

develop computationally efficient, semi-analytical loaded tooth contact models for both

8

FM and FH hypoid gears with or without misalignments. The main motivation for this

dissertation research is to develop formulations to analytically describe the hypoid gear

surfaces and employ them in a gear contact mechanics formulation to predict unloaded

and loaded contact characteristics as well as functional metrics such as the transmission

error and mechanical efficiency.

1.2. Literature Review

In his writings, Aristotle (about 330 BC), made mention of gears and their

commonality. The earliest recognized relic of ancient time gearing is the south pointing

chariot with pinned gears used by the Chinese in about 2600 BC [1.14]. According to the

historical perspective provided by Litvin [1.15], theoretical development of gears as we

know today starts with Euler (1781) who proposed the concept of an involute curve

(1781), followed by others such as Willis (1841), Olivier ( 1842) and Gochman (1886)

who developed basic ideas of conjugacy and the foundations of modern gear geometry.

As for spiral bevel gears, Monneret (1899) filed the first patent for spiral bevel

generating method. About two decades later in 1910, Böttcher was issued a series of

patents that addressed both face hobbing and face milling methods [1.2]. Wildhaber’s

earlier papers and patents formed the basis for many of today’s hypoid gear geometry and

generation approaches [1.1, 1.3]. Wildhaber pointed out the significance of using

principal curvatures and directions in establishing hypoid gear geometry [1.16, 1.17].

9

Baxter’s later formulas, based on vector notation, helped condense the formulations to

facilitate the use of computers for definition of surfaces and tooth contact analysis (TCA)

[1.18]. He also developed one of the first unloaded tooth contact model for simulation of

mismatched surfaces of gears generated by Gleason type machines and studied the effects

of various misalignments on contact pattern [1.19], which was later expanded by

Coleman [1.20, 1.21]. Krenzer published a series of formulations for unloaded tooth

contact analysis (UTCA) of spiral bevel and hypoid gears [1.22]. These formulations

were useful for the gears manufactured by a class of machinery but were quite difficult to

adapt since the logic behind his formulae were not given. Nearly two decades later, he

proposed a loaded tooth contact analysis model without providing details of the geometry

and the contact analysis. This model used the simplified cantilever beam formula of

Westinghouse to estimate the compliance of the tooth [1.23]. Within the same time

frame, Litvin and Gutman [1.24-1.27] published a series of papers on synthesis and

analysis of FM hypoid gears. They calculated machine settings based on predetermined

contact characteristics at a mean point. They determined the contact points, the

instantaneous contact length and direction by conventionally using the surface principal

curvatures and directions. They used a conventional approach to find the contact points,

the instantaneous contact directions and the instantaneous contact lengths utilizing

surface principal curvatures and directions [1.24-1.27]. The effort of computing

optimized machine settings for limited cutting methods such as spiral bevel gears cut by

face-milling method was later continued by Litvin and Fuentes [1.28]. As these studies

10

focused on calculating surface coordinate of FM spiral bevel and hypoid gears, there are

very few studies on geometry of FH hypoid gears [1.29-1.33].

Among the few published studies on calculating and “optimizing” TCA, Stadtfeld

[1.2] appears to be the only investigator who used the ease-off approach. He provided a

consistent definition for ease-off as well as a procedure to calculate TCA from ease-off,

and calculated instantaneous contact between two surfaces as a line that maintains its

orientation over the tooth area [1.2]. Moreover, he utilized ease-off to optimize UTCA of

both FM and FH gears by applying various kinds of modifications through machine

settings and cutter geometry [1.34]. However, he did not provide a detailed procedure on

how to determine orientation of the instantaneous contact curves. Meanwhile, Fan [1.35]

focused on how to calculate surface coordinates and normal vectors for FH spiral bevel

and hypoid gears cut by using the generation method. The solution to the set of equations

that determines contact points where the collinearity condition for the normal vectors of

two mating surfaces is satisfied is typically subject to various numerical instabilities. Fan

[1.35] used the conventional approach for UTCA in conjunction with the Euler-

Rodrigues’ formula to avoid these stability issues. He also used minimization of the

separation between the tooth surfaces to determine the direction and length of the

instantaneous contact lines [1.8]. Later Vogel et al [1.36, 1.37] proposed an alternate

approach to compute both tooth surfaces and the UTCA by using Singularity Theory.

They considered the generated tooth flank as a first-order singularity of the particular

11

function that models the generating process. They also used numerical differentiation to

investigate the sensitivity of tooth contact to machine settings [1.36]. Simon also used

the conventional system of five scalar nonlinear equation and six unknowns to find

contact points on both surfaces. He calculated contact lines orientation by minimizing the

separation function between two contact surfaces and applied his method for a FM

Gleason type gear pair with a generated pinion and a Formate® gear [1.38].

Published studies on modeling of hypoid tooth contact under loaded conditions

are quite sparse. Simon [1.39-1.41] used a FE model to calculate deflection and

displacement under load from which interpolation functions were obtained to estimate

stresses and deflections via regression analysis of FE results. Gosselin et al [1.42]

developed a loaded tooth contact analysis (LTCA) model for spiral bevel gears by using

tooth compliances obtained by curve-fitting to the FE deformation results of a single

pinion and gear tooth pair. Wilcox et al [1.43] also developed a FE-based model to

calculate the spiral bevel and hypoid gear tooth compliances by using a three-dimensional

model of a tooth including base deformations, which was later employed by Fan and

Wilcox [1.44] to perform LTCA analyses. Vimercati and Piazza [1.45] also calculated

FH gear pair surfaces and incorporated them with a commercially available finite

elements (FE) package [1.46] to calculate both TCA and LTCA. This particular hypoid

FE package that employs FE away from the contact zone and a semi-analytical contact

formulation at the contact zone [1.47] is perhaps the most advanced hypoid LTCA model

12

available to accurately simulate a hypoid gear contact. The major drawback of these FE-

based models is that they require a considerable amount of computation time, making

them more of an analysis tool. Their use for design tasks such as parameter and assembly

variation sensitivity studies is not very practical.

Beside the FE method, the Boundary Element (BE) method was also used in

several studies for performing LTCA. For instance, Sugyarto [1.48] sliced gear and

pinion teeth into a number of sections and considered each section as an independent

plate from its neighboring sections, and applied a two dimensional BEM formulation to

each slice to compute bending and shear deflections. Liu [1.49] applied same compliance

methodology to face gears [1.49] with a correction intended to couple each slice with

adjacent slices using Borner’s coefficient [1.50], which was originally proposed for

parallel axis gear. Vecchiato [1.51] used a three-dimensional BE approach for loaded

tooth contact predictions of FH hypoid gears. As for unloaded contact analysis, he used

conventional approach [1.8,1.11] and studied misalignment effects.

Besides these computational models, some semi-analytical models were also

proposed for determining tooth compliance of parallel-axis gears through elasticity-based

deformation solutions. Adding linear thickness variation along the profile to the

originally proposed plate solution [1.52], Yakubek [1.53] used the Rayleigh-Ritz Energy

Method to calculate the approximate deflection of a tapered plate for estimating the

compliance of spur and helical gears. Bending deformations of a tooth were considered

13

as a sum of shape functions that satisfy clamped-free and free-free boundary conditions,

and the unknown coefficients of the shape functions were determined by minimizing the

potential energy. Later, Yau [1.54, 1.55] expanded this compliance model to add shear

deformations to the energy function and found more realistic deformation for spur and

helical gears and Stegemiller [1.56, 1.57] used the FE package ANSYS to propose an

approximate interpolation based formula to compensate for base rotation and base

translation. All of these analytical compliance methods are valid for tooth having

constant height along face width and either constant or linearly varying thickness along

its profile, which is not the case for hypoid gears. Vaidyanathan [1.58-1.60] proposed an

analytical compliance model for a tooth with linearly varying thickness in the profile and

lengthwise directions as well as linearly varying tooth height along the face width. His

Rayleigh-Ritz based formulation used polynomial shape functions and was applied to

both sector and shell geometries. The sector model represents straight bevel gear

geometry closely while the shell model is sufficiently close to a spiral bevel gear tooth in

terms of its geometry.

The challenges mentioned above in terms of performing a loaded hypoid tooth

contact analysis in a practical and computationally efficient way have been the major

road block to the development of other models to study other functional behavior of

hypoid gears. One such behavior is the efficiency of the hypoid gear pair. In addition to

the analytical surface geometries and surface velocities, an accurate description of the

14

contact load distribution is required at many rotational increments (of pinion angle) to

predict the distribution of the friction coefficient and the resultant mechanical power

losses. While this hypoid efficiency methodology has been demonstrated by Xu and

Kahraman [1.61-1.63] by using the FE-based loaded tooth contact model of Vijayakar

[1.64], the amount of computations required was reported to be very significant for it to

be used extensively as a design tool. Likewise, other hypoid gear models for simulation

of surface wear [1.65] and finishing processes such as lapping have also been hampered

by the difficulties in obtaining load distribution.

1.3. Scope and Objectives

Computation of the contact pressure distributions and the relative surface

velocities forms the basis for predicting the required functional parameters of the hypoid

gear pair, including the transmission error, contact stresses, root bending stresses, fatigue

life and mechanical power losses. It is evident from the review of the literature that a

model to compute the load distribution accurately and efficiently without resorting to

computationally demanding FE methods does not exist. This is mainly due to three

primary reasons:

(i) A detailed description of general and reliable formulation to define the geometry

of FH and FM hypoid tooth surfaces from cutter parameters, machine motions

and settings is not available in the literature.

15

(ii) Conventional methods of matching the tooth surfaces (bringing them to contact)

present major numerical difficulties for UTCA.

(iii) There is no published model available for hypoid gear LTCA based on semi-

analytical tooth compliance formulations.

Accordingly, the main objective of this study is to develop FH and FM hypoid

gear contact models that address these issues. This study performs the following specific

tasks to achieve this objective:

(i) Development of a methodology that simulates the FM and FH processes to define

surface geometries of hypoid gears including the coordinates, normal vectors and

radii of curvatures.

(ii) Development of a novel formulation for unloaded tooth contact analysis by using

the ease-off topography, surface of action and roll angle surface to predict

unloaded transmission error and unloaded contact pattern in addition to potential

contact lines/curves to be used for loaded tooth contact analysis.

(iii) Development of a semi-analytical tooth compliance model tailored for both FH

and FM hypoid and spiral bevel gears.

16

(iv) Development of a LTCA model for FH and FM hypoid gears to predict pressure

distribution and loaded transmission error with or without misalignments of

various types.

Actual manufactured surfaces of gears include inevitable machining errors, heat

treatment distortions and lapping surface changes as globally distributed deviations on

tooth surfaces, which affect contact patterns and transmission error significantly.

Moreover, wear or lapping simulations (as accumulated wear) changes surface geometry

usually in a very local fashion that conventional tooth contact analysis approaches are not

capable of capturing. A novel ease-off based approach will also be developed to modify

ease-off topography of the theoretically generated tooth surfaces to account for both

global deviations due to the manufacturing process and local surface deviations due to

factors such as wear and lapping process.

A second objective of this dissertation is to develop a capability to predict load

dependent (mechanical) power losses of hypoid gear pairs. For this purpose, the

proposed loaded tooth contact model will be combined with a new friction model

according to the methodology proposed by Xu and Kahraman [1.62, 1.63] to predict

mechanical power losses and gear pair efficiency including all relevant contact, surface,

and lubricant parameters as well as the operating conditions. This hypoid gear efficiency

model will be used to investigate the impact of basic design parameters, and surface and

17

lubricant conditions, on mechanical power losses of hypoid gear pairs and to arrive at

guidelines on how to reduce such losses.

1.4. Overall Modeling Methodology

The overall methodology used to develop the hypoid load distribution model is

illustrated in the flowchart of Figure 1.4. Gear blank dimensions, cutter geometry,

machine settings, assembly dimensions and misalignments, torque and speed are all

included as input parameters for the load distribution model. These parameters are

commonly put together by hypoid gear manufacturers in a standard form that is called a

special analysis file. The pinion and gear cutter surfaces are first constructed and used to

define the extended pinion and gear surfaces (including surface coordinates, normals and

curvatures) by applying fundamental equation of meshing between a gear blank and its

respective cutter surfaces. These extended tooth surfaces are then trimmed in 3D space

so that they are contained by the blanks, and transformed to a global coordinate system

where any misalignments in the directions of shaft offset (ΔE), pinion axis (ΔP), gear axis

(ΔG) as well as the shaft angle error (ΔΣ) can be applied. Next, ease-off and surface of

roll angle are constructed and an UTCA model is developed by bringing the tooth

surfaces together and an unloaded contact pattern is defined by choosing a separation

tolerance between the tooth surfaces.

Figure 1.4: Flowchart of overall hypoid gear loaded tooth contact analysis methodology.

18

FH Method(Generate / Formate)

Extended Pinion and Gear surfaces and Curvatures (FM)

FM Method(Generate / Formate)

Surfaces in Global Coordinates System

(Misalignments)

Extended Pinion and Gear Surfaces and Curvatures (FH)

Global coordinates transformation Global coordinates transformation-Under Development

Unloaded TCA

Loaded Tooth Contact Analysis

Required Data for Efficiency Analysis

Contact Pressure Distribution

Loaded Transmission Error

Reading Design File.HAP or .SPA file

Tooth Compliances

Ease-off Construction

Equation of Meshing

 

Cutter axis

Root (clamped edge)

Tip (free edge)

Toe-BaseHeel-Base

Toe-height

Heel-height

Toe (free edge)

Heel (free edge)

   

Heel Toe

Root

Top

RPPLPP CPP

1M

2MContact lines

   

1M 2M

TE (μ rad)

100

Maximum UTE

Pitch Pinion phase angle (deg.)

Adjacent tooth pairs

Equation of Meshing

 

c

Cutter

Generating gear 

Extended 

epicycloids trace

t

IB OB tC

(b)

GC

Blade

FH Method

 

  Generating gear 

Cutter 

Circular 

t  

IB 

OB 

(a) 

GC

Fixed 

FM Method

-20 -15 -10 -5 0 5 10 15 20

X (mm) Root

-4

-2

0

2

Y (

mm

) T

oe

(MPa)

758674590506421337253169

840

Updating Ease-off by Surface Deviations

Elastohydrodynamic Friction Coefficient Model

Hypoid Gear Mechanical Power Loss and Efficiency

19

Next the tooth compliance matrices comprising bending, shear, Hertzian and base

rotation deflections are computed. Finally, a set of equilibrium and compatibility

conditions are defined and solved simultaneously to compute the load distribution and the

loaded transmission error of the hypoid gear pair. Moreover, all required information for

efficiency and lapping simulations are computed.

1.5. Dissertation Outline

In Chapter 2, the hypoid gear tooth surfaces will be defined through simulation of

the face-milling and face-hobbing processes with all relevant cutter and machine related

parameters included. A new formulation of unloaded tooth contact analysis based on the

principles of ease-off and a newly introduced surface of roll angle will be proposed as

well.

In Chapter 3, a semi-analytical tooth compliance model will be employed and a

loaded tooth contact model will be described. A novel approach will be introduced in

Chapter 4 to compute loaded tooth contacts of gear surfaces that have deviations from

their theoretically intended surfaces either in local or global fashion.

Chapter 5 proposes a model to predict the mechanical efficiency of hypoid gear

pairs. This mode combines the developed computationally efficient contact model and

the mixed elastohydrodynamic lubrication (EHL) based friction model of Li and

20

Kahraman [1.66] to predict gear mesh power losses and mechanical efficiency. A

summary, major conclusions and contributions of this research to the state-of-the-art as

well as a list of recommendation for future work will be included in Chapter 6.

References of Chapter 1

[1.1] Wildhaber, E., 1946, Basic Relationship of Hypoid Gears, McGraw-Hill.

[1.2] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute

of Technology.

[1.3] Stewart, A. A., and Wildhaber, E., 1926, "Design, Production and Application of

the Hypoid Rear-Axle Gear." J. SAE, 18, pp. 575-580.

[1.4] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,

Elsevier Science B. V.

[1.5] Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A

Concurrent Engineering Approach, John Wiley & Sons Inc.

[1.6] Stadtfeld, H., J., 1995, Gleason Advanced Bevel Gear Technology, The Gleason

Works.

[1.7] Coleman, W., 1963, Design of Bevel Gears, The Gleason Works.

[1.8] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear

Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.

Des., 129(1), pp. 31-37.

21

[1.9] Dooner, D. B., 2002, "On the Three Laws of Gearing." ASME J. Mech. Des., 124,

pp. 733-744.

[1.10] Krenzer, T. J., 2007, The Bevel Gears, http://www.lulu.com/content/1243519.

[1.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd

ed.), Cambridge University Press, Cambridge.

[1.12] Krenzer, T. J., 1990, "Face Milling or Face Hobbing." AGMA, Technical Paper

No. 90FTM13.

[1.13] Stadtfeld, H. J. (2000). "The Basics of Gleason Face Hobbing." The Gleason

Works.

[1.14] Dudley, D. W., 1969, The Evolution of the Gear Art, American Gear

Manufacturers Association, Washington, D. C.

[1.15] Litvin, F. L. (2000). "Development of Gear Technology and Theory of Gearing."

NASA RP1406.

[1.16] Wildhaber, E., 1956, "Surface Curvature." Product Engineering, pp. 184-191.

[1.17] Dyson, A., 1969, A General Theory of the Kinematics and Geometry of Gears in

Three Dimensions, Clarendon Press, Oxford.

[1.18] Baxter, M. L., 1964, "An Application of Kinematics and Vector Analysis to the

Design of a Bevel Gear Grinder." ASME Mechanism Conference, Lafayette, IN.

[1.19] Baxter, M. L., and Spear, G. M. "Effects of Misalignment on Tooth Action of

Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.

22

[1.20] Coleman, W. "Analysis of Mounting Deflections on Bevel and Hypoid Gears."

SAE 750152.

[1.21] Coleman, W. "Effect of Mounting Displacements on Bevel and Hypoid Gear

Tooth Strength." SAE 750151.

[1.22] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason

Works.

[1.23] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears

under Load." SAE Earthmoving Industry Conference, Peoria, IL.

[1.24] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for

hypoid gear-drives of formate and helixform; Part I-Calculation for machine

setting for member gear manufacture of the formate and helixform hypoid gears."

ASME J. Mech. Des., 103, pp. 83-88.

[1.25] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for

hypoid gear-drives of formate and helixform; Part II-Machine setting calculations

for the pinions of formate and helixform gears." ASME J. Mech. Des., 103, pp.

89-101.

[1.26] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for

hypoid gear-drives of formate and helixform; Part III-Analysis and optimal

synthesis methods for mismatched gearing and its application for hypoid gears of

formate and helixform." ASME J. Mech. Des., 103, pp. 102-113.

[1.27] Litvin, F. L., and Gutman, Y., 1981, "A Method of Local Synthesis of Gears

Grounded on the Connections Between the Principal and Geodetic of Surfaces."

ASME J. Mech. Des., 103, pp. 114-125.

23

[1.28] Litvin, F. L., Fuentes, A., Fan, Q., and Handschuh, R. F., 2002, "Computerized

design, simulation of meshing, and contact and stress analysis of face-milled

formate generated spiral bevel gears." J. Mechanism and Machine Theory, 37(5),

pp. 441-459.

[1.29] Fong, Z. H., and Tsay, C.-B., 1991, "A Mathematical Model for the Tooth

Geometry of Circular-Cut Spiral Bevel Gears." ASME J. Mech. Des., 113, pp.

174-181.

[1.30] Fong, Z. H., 2000, "Mathematical Model of Universal Hypoid Generator with

Supplemental Kinematic Flank Correction Motions." ASME J. Mech. Des.,

122(1), pp. 136-142.

[1.31] Tsai, Y. C., and Chin, P. C., 1987, "Surface Geometry of Straight and Spiral

Bevel Gears." J. Mechanism, Transmission and Automation in Design, 109, pp.

443-449.

[1.32] Fong, Z. H., and Tsay, C.-B., 1991, "A Study on the Tooth Geometry and Cutting

Machine Mechanisms of Spiral Bevel Gears." ASME J. Mech. Des., 113, pp. 346-

351.

[1.33] Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, "Determination of

Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel

Gears." J. Mechanism, Transmission and Automation in Design, 110, pp. 495-

500.

[1.34] Stadtfeld, H. J., and Gaiser, U., 2000, "The Ultimate Motion Graph." ASME J.

Mech. Des., 122(3), pp. 317-322.

24

[1.35] Fan, Q., 2006, "Computerized Modeling and Simulation of Spiral Bevel and

Hypoid Gears Manufactured by Gleason Face Hobbing Process." ASME J. Mech.

Des., 128(6), pp. 1315-1327.

[1.36] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for

Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany.

[1.37] Vogel, O., Griewank, A., and Bär, G., 2002, "Direct gear tooth contact analysis

for hypoid bevel gears." Computer Methods in Applied Mechanics and

Engineering, 191(36), pp. 3965-3982.

[1.38] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME

International Power Transmission and Gearing Conference ASME, 88, pp. 789-

798.

[1.39] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,

122(44), pp. 529-535.

[1.40] Simon, V., 2000, "FEM stress analysis in hypoid gears." J. Mechanism and

Machine Theory, 35(9), pp. 1197-1220.

[1.41] Simon, V., 2001, "Optimal Machine Tool Setting for Hypoid Gears Improving

Load Distribution." ASME J. Mech. Des., 123(4), pp. 577-582.

[1.42] Gosselin, C., Cloutier, L., and Nguyen, Q. D., 1995, "A general formulation for

the calculation of the load sharing and transmission error under load of spiral

bevel and hypoid gears." J. Mechanism and Machine Theory, 30(3), pp. 433-450.

[1.43] Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite

Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth."

AGMA, Technical Paper No. 97FTM05.

25

[1.44] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis

(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,

Technical Paper No. 05FTM08.

[1.45] Vimercati, M., and Piazza, A., 2005, "Computerized Design of Face Hobbed

Hypoid Gears: Tooth Surfaces Generation, Contact Analysis and Stress

Calculation." AGMA, Technical Paper No. 05FTM05.

[1.46] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced

Numerical Solution Inc., Hilliard, Ohio.

[1.47] Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element

Solution for a Three-Dimensional Contact Problem." International J. for

Numerical Methods in Engineering, 31, pp. 525-545.

[1.48] Sugyarto, E., 2002, "The Kinematic Study, Geometry Generation, and Load

Distribution Analysis of Spiral Bevel and Hypoid Gears," M.Sc. Thesis, The Ohio

State University, Columbus, Ohio.

[1.49] Liu, F., 2004, "Face Gear Design and Compliance Analysis," M.Sc. Thesis, The

Ohio State University, Columbus, Ohio.

[1.50] Borner, J., Kurz, N., and Joachim, F. (2002). "Effective Analysis of Gears with

the Program LVR (Stiffness Method)."

[1.51] Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth

Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University

of Illinois at Chicago.

[1.52] Timoshenko, S. P., and Woinowsky-Krieger, S., 1959, Theory of Plates and

Shells, McGraw-Hill Book Company Inc.

26

[1.53] Yakubek, D., Busby, H. R., and Houser, D. R., 1985, "Three-Dimensional

Deflection Analysis of Gear Teeth Using Both Finite Element Analysis and a

Tapered Plate Approximation." AGMA, Technical Paper No. 85FTM4.

[1.54] Yau, H., 1987, "Analysis of Shear Effect on Gear Tooth Deflections Using the

Rayleigh-Ritz Energy Method," M.Sc. Thesis, The Ohio State University,

Columbus, Ohio.

[1.55] Yau, H., Busby, H. R., and Houser, D. R., 1994, "A Rayleigh-Ritz Approach to

Modeling Bending and Shear Deflections of Gear Teeth." J. of Computers &

Structures, 50(5), pp. 705-713.

[1.56] Stegmiller, M. E., 1986, "The Effects of Base Flexibility on Thick Beams and

Plates Used in Gear Tooth Deflection Models," M.Sc. Thesis, The Ohio State

University, Columbus, Ohio.

[1.57] Stegmiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of

the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192.

[1.58] Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded

Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The

Ohio State University, Columbus, Ohio.

[1.59] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz

Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears

Using Shell Theory." AGMA, Technical Paper No. 93FTM03.

[1.60] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical

Approach to the Static Analysis of an Annular Sector Mindlin Plate with

Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp.

255-266.

27

[1.61] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction

Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University,

Columbus, Ohio.

[1.62] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of

Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part

K: J. Multi-body Dynamics, 221(3), pp. 387-400.

[1.63] Xu, H., Kahraman, A., and Houser, D. R., 2006, "A Model to Predict Friction

Losses of Hypoid Gears." AGMA, Technical Paper No. 0FTM06.

[1.64] Vijayakar, S. M., 2003, Calyx User Manual, Advanced Numerical Solution Inc.,

Hilliard, Ohio.

[1.65] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear

Pairs." In press, Wear.

[1.66] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric

Integrated Control Volume Discretization." Tribology International, Hiroshima,

Japan.

28

CHAPTER 2

DEFINITION OF FACE-MILLED AND FACE-HOBBED HYPOID GEAR

GEOMETRY AND UNLOADED TOOTH CONTACT ANALYSIS

2.1. Introduction

Unlike most types of gears that have closed-form equations defining their

geometry, the geometry of hypoid gears can only be computed by solving implicit

equations governed by the manufacturing process, including its machine settings and

cutter specifications. Besides the gear blank dimensions and basic geometry

requirements, a set of performance or functionality related requirements must be met.

Among them, the contact pattern (location, size, shape) on the gear tooth surfaces and the

motion transmission error amplitude of the gear pair are two of the most common ones

checked routinely in the design of hypoid gear pairs. The contact pattern and the

transmission error are both determined via a contact analysis of the pinion and gear tooth

surfaces under a very small amount of load.

29

In general, contact of hypoid gear surfaces is single-mismatched. This is the most

general case of point contact condition between two surfaces [2.1]. The purpose of the

unloaded tooth contact analysis (UTCA) is to determine a contact point path (CPP) on

each surface in addition to area on each surface in the neighborhood of each

instantaneous contact point that falls in a specified separation distance (usually 6.3

micron of separation distance is commonly used in hypoid gear industry) [2.2]. In

addition, UTCA results in the function of motion transmission error between two gear

axes that is viewed as a key metric used to estimate the noise/level of the hypoid gear pair

in operation [2.3, 2.4].

In this chapter, as the first basic step in the analysis of hypoid gears, the geometry

of both face-milled (FM) and face-hobbed (FH) hypoid gear pairs produced by using both

Formate® and Generate cutting methods will be computed. This will be done by

simulating individual cutting processes. Basic machine tool settings, cutter geometry

parameters and gear blank dimensions will form the input for this computation. Next, a

novel method based on the ease-off topography will used to determine the unloaded

contact patterns. The proposed ease-off based methodology finds the instantaneous

contact curve through a surface of roll angles, allowing an unloaded tooth contact

analysis in a robust and accurate manner.

30

2.2. Definition of Tooth Surface Geometry

The concept of the generating gear is a key to the basic understanding of hypoid

gears because this hypothetical gear can be treated as cutting tool for both the pinion and

the gear [2.5]. In a FM cutter head, blades are arranged around the cutter head axis on an

equal radius for inside and outside blades (IB and OB, respectively) to form a conical

shape due to cutter axis rotation. The inside blades cut convex side of a tooth slot while

the outside blades cut concave side of the same slot, as shown in Figure 2.1(a). Face-

hobbing cutter heads (such as PENTAC® or TRI-AC®) like the one shown in Figure

2.1(b) roll while cutting such that each set of IB-OB blades (called blade group) will pass

through a different tooth slot. The cutting process can be considered as rolling of two

gears together, except the teeth of one of the gears are replaced by blade group of the

cutter head. By rolling the cutter head and the gear blank together while advancing the

cutter head into the blank, the gear is cut by the continuous indexing method. While the

axis of the generating gear for FM process is fixed, it is located on the center of a circle

tC for FH process that rolls on the generating gear circle GC , as shown in Figure 2.1(b).

Therefore, the edges of a blade in FH process traces extended epicycloids since they

usually lie on a radius that is larger than the radius of rolling circle tC .

31

Figure 2.1: (a) Face-milling and (b) face-hobbing cutting processes.

  

c

t

(a)

Cutter

Generating gear

Extended epicycloids trace

IBOB

tC

(b)

GC

Blade group

Generating gear

Cutter

Circular arc trace

t

IB

OB

GC

Fixed

32

2.2.1. Kinematics

Earlier cradle-based hypoid generators were designed to provide the required

degrees of freedom and relative motions through machine settings to accommodate the

cutting process of the gear and pinion blanks by means of the cutter blades. Figure 2.2

shows a typical cradle-based hypoid generator with machine settings and relative motions

defined as the cutter phase angle t , the tilt angle Ti , the swivel angle js , the radial

setting rS , the cradle angle c , the sliding base BX , the machine root angle m ,

machine center to back ctbM , the blank offset bE , the blank phase angle g , angular

speed of the cutter axis t , angular speed of the cradle axis c , angular speed of the

blank axis g , and the cradle angle change q. Newer-generation hypoid gear machine

tools still use settings in the form of older cradle-based machines. While the new cutting

machines are not set as the old mechanical machines, their working principles are still the

same such that they generate the same gear surface with an added capability of

controlling higher-order surface geometrical parameters as well. While most of the

machine settings are typically fixed (kept constant), a number of them are defined as a

polynomial function of q. These parameters that are dependent on q are called higher-

order motions such as the modified roll (ratio of roll change), the helical motion (sliding

base change) and the vertical motion (blank offset change).

33

Figure 2.2: Cradle based hypoid generator parameters.

 

c

bE

ctbM m

rS

js

Ti

t g

BX

34

In the FM process, the teeth are cut individually by blade edges that rotate fast

about the cutter axis. The cutting edges of the blades form a conical surface and their

envelope seen from a coordinate attached to the blank is the gear or pinion surface. In the

Formate® case for the FM process, the blank is fixed, the cutter advances towards the

gear blank while rotating, in the process replicating its surface on the blank. Then, the

cutter head retreats, the blank is rotated by one tooth spacing, and the same cutting

process is repeated as shown in Figure 2.1(a). Meanwhile, the face-hobbing process

using Formate® requires the blank and cutter head to be rolled together according to a

kinematic relationship defined as

Rog t

tgt g

NR

N

(2.1)

where t is cutter head angular velocity, Rog is the rolling portion (also called

indexing) of angular velocity of the gear blank, and tN and gN are numbers of blade

groups and gear teeth being cut. Typically, edges of the blades in FH process do not

intersect with cutter head axis so that the cutting surface is a hyperboloid of revolution.

In the generate process for both FM and FH processes, there is an additional

relative rotation between the gear blank and the cradle axis in Figure 2.1(b). The ratio of

roll is given as the ratio of relative rotations, i.e.

35

,RgFM

ac

(2.2a)

GegFH

ac

R

(2.2b)

where FMaR and FH

aR are the ratios of rolls for the FM and FH processes (they are zero

for Formate® process), respectively, c is the angular velocity of the cradle axis, g is

the angular velocity of the blank axis during face-milling and Geg is the generating

portion of blank angular velocity for the FH process. As a result, the total blank angular

velocities for FM and for FH are given, respectively:

,FMg g (2.3a)

FH Ro Geg g g (2.3b)

Corresponding gear blank rotation angles FMg and FH

g for FM and FH processes are

given respectively as:

,FM FMg aR q (2.4a)

( )FH FHg a tg tR q R . (2.4b)

36

Here, the relative motion between the cutter and the gear blank for the FM process can be

treated as a special case of the FH process. Therefore, the formulation for the FH process

will be explained here in detail while its differences from the FM process will also be

specified, as required.

2.2.2. Cutting Tool Geometry and the Relative Motion

Figure 2.3 shows a typical FH blade with its geometry along its cutting edge is

defined by the blade angle b , the rake angle , the hook angle , the blade offset angle

b , the cutter radius cr , and the distance from the tip of blade to reference point fh .

The cutting edge is divided into four different sections as the edge (or tip radius), toprem,

profile and flankrem that are all shown in Figure 2.3(c). The edge and flankrem are

usually circular arcs while toprem is usually a straight line at a slight angle from the

profile section. Most of the cutting is done by the profile section of the blade that is

usually a straight line or a circular arc. For a typical FM cutter, 0b .

Referring to Figure 2.3, an arbitrary point A on cutting edge is at position ( )sr r

relative to the local coordinate system bX fixed to the cutter head (with its origin at

reference point M) where s is the distance of point A to point M along the blade edge.

With this, the unit tangent vector is s t r , and if the cutting edge is a line, it can be

reduced to

37

Figure 2.3: (a) Cutter head, (b) blade and (c) cutting edge geometry.

31

18

Flankrem

b

bcr

Mtx

ty

,b tx x

bz

M

b

M

tz

z

bx Profile

Toprem

cr

by fh

y

,bz z

,x x

y

,x x bx

Inside Blade

n

A

tr

s

Reference Plane

Edge

(a)

(b) (c)

38

[ sin 0 cos 1]Tb b t (2.5)

where the superscript T denotes a matrix transpose. Position and unit tangent vectors, tr

and tt , in the coordinate system tX whose z axis coincides with the cutter axis and xy

plane passes through reference point M as shown in Figure 2.3(a) are given as

( ) ( ) ( )t z b x z t = M M M t , (2.6a)

t c i tr sr = t t (2.6b)

where

[cos sin 0 1]Ti b b t = (2.6c)

and ( )k M is a rotation matrix facilitating a rotation angle about axis k

( [ , , ])k x y z such that

1 0 0 0

0 cos sin 0

0 sin cos 0

0 0 0 1

( ) ,x

M (2.6d)

cos 0 sin 0

0 1 0 0

sin 0 cos 0

0 0 0 1

( ) ,y

M (2.6e)

39

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

( )z

M . (2.6f)

Position vector of point A is transformed to the coordinate system gX fixed to the

blank as [2.6]:

(

( ) ( ) ( )

( ) ( ) ( )

b ctb m B B c

r

g x g E M y m X X z c

z S j z x T z t tq js i

r M M M M M M

M M M M M ) r (2.7a)

where the transformation matrices are defined as

1 0 0

0 1 0 0

0 0 1 0

0 0 0 1

,r

r

S j

S

M (2.7b)

1 0 0 0

0 1 0 0

0 0 1

0 0 0 1

,B c

BX X

M (2.7c)

1 0 0

0 1 0 0

0 0 1 0

0 0 0 1

,m B

ctb

X

M

M (2.7d)

40

1 0 0 0

0 1 0

0 0 1 0

0 0 0 1

.b ctb

bE M

E

M (2.7e)

Variables influencing gr , with the exception of s and t , are either fixed or dependent

on q (in case of higher order motion).

2.2.3. Equation of Meshing

Equation (2.7a) represents a family of surfaces in a coordinate system fixed to the

blank whose envelope is the generated surface on the blank. The envelope of gr with

three independent variables s, t and q is given mathematically as [2.7-2.9]:

0g g g

ts q

r r r. (2.8)

This equation is mathematically equivalent to the fundamental equation of meshing,

( ) 0c w n V V , which states for each point to lie on the envelope surface that the

normal vector n to the family of the cutter surfaces should be perpendicular to relative

velocity between the blank (w) and the cutter (c) as shown in Figure 2.4.

41

Figure 2.4: Generation process.

 

Blank axis

Cradle axis

Cutter head

Blank

n

c

wVVc

V Vc w

V Vc w

w

42

Using Eq. (2.8), each point on the generated surface can be found by solving a

system of two implicit nonlinear equations for a pair of unknown parameters that can be

chosen as ( , )ts , ( , )s q or ( , )tq .

2.2.4. Principal Curvatures and Principal Directions

As shown in Figure 2.5 for any point 0 0 0( , )ts P defined by independent surface

curvilinear variables s and t on the gear surface, the unit normal to the surface is given

as

0

g g

t

g g

t

s

s

r r

nr r

. (2.9)

Moving from point 0 0 0( , )ts P to 1 0 0( , )ts ds P by only infinitesimally changing one

of the surface variables s, the change of unit normal to the surface is defined as [2.10]

1 11 1( , ) t tt

dk

d s

n

t v . (2.10)

Here

1 01

1 0

P P

tP P

, (2.11a)

43

Figure 2.5: Curvature computation procedure.

 

0n1n

0P 1P

2P

2n

1t

2t

1v

1C

2C

44

0 11

0 1

n t

vn t

(2.11b)

and ( , )ts defines the distance from 0P to 1P along the gear surface (along curve 1C )

as a function of s and t (here, 0 constantt t and only s varies). 1t

k is the normal

curvature in the 1t direction and 1t

is the geodesic torsion in the direction of 1v . Here

0n , 1t and 1v form a Frenet trihedron. Following the same procedure, but this time

moving from 0 0 0( , )ts P to 2 0 0( , )t ts d P by infinitesimally changing t , the normal

curvatures 2t

k and geodesic torsion 2t

in 2t and 2v directions are found according to

2 22 2( , ) t tt

dk

d s

n

t v (2.12)

with

2 02

2 0

P P

tP P

, (2.13a)

0 22

0 2

n t

vn t

(2.13b)

and ( , )ts defines the distance from 0P to 2P along the gear surface (along curve 2C

in Figure 2.5) as a function of s and t (here 0s s is constant and only t varies). With

45

1tk ,

2tk ,

1t and

2t in hand, Euler equation [2.11, 2.12] is applied to compute the

principle directions ( 1e and 2e ) and the principal curvatures ( 1K and 2K ).

Having the principal curvatures and directions of every possible contact point

(points on the projection plane) on the pinion and gear, principal directions of the

difference surface are defined as directions in which two contacting surfaces have the

extremes of the relative curvatures [2.11-2.13]. The curvatures pR and gR of pinion

and gear surfaces in the direction of maximum relative curvatures are used to find the

equivalent curvature ( )p g p geqR R R R R that will be required later to determine

Hertzian deflections in loaded TCA [2.14].

2.3. Unloaded Tooth Contact Analysis

In unloaded tooth contact analysis (UTCA), the goal is to calculate:

(i) the contact point path (CPP) on each of the gear surfaces in addition to the zone on

each surface in the neighborhood of each instantaneous contact point that is as close

as the specified separation distance [2.15, 2.16], and

(ii) the function of transmission error between two gear axes.

Two different approaches were used in the past for performing UTCA of hypoid gears

with mismatched surfaces. In the conventional method, tooth surfaces are treated as two

46

arbitrary surfaces, rotating about the pinion and gear axes. The contact point path (CPP)

on each surface is computed by satisfying two contact conditions. The first condition is

the coincidence of position vector tips of the points on the gear and pinion surfaces in

three dimensional space. The second condition is the collinearity of the normal vectors

of the both of the surfaces at the contact point.

The second method of performing UTCA is based on the ease-off procedure. The

current literature lacks a clear and accurate mathematical definition of ease-off as well as

its construction including the instantaneous contact lines/curves [2.17]. Ease-off has

often been defined in the literature as the change in pinion surface with the application of

modifications. While these changes directly reshape ease-off, they do not constitute the

ease-off itself.

Finding the location and orientation of potential instantaneous contact lines is one

major step in the UTCA. In general, the instantaneous contact shape in the projection

plane (plane that includes the gear axes) is slightly curved as opposed to commonly used

approximate straight lines, which is the contact shape in action surface. The

instantaneous contact line directions are conventionally found based on principal

curvatures and directions of contacting surfaces of gear members at contacting points as

the direction of minimal relative normal curvature between contacting surfaces.

47

In the next two sections, a brief overview of the conventional approach will be

provided, followed by a detailed ease-off based UTCA formulation developed in this

thesis.

2.3.1. The Conventional Method of UTCA

The position vector 1 1( , )p r and normal vector 1 1( , )p n of any point on the

surface of the pinion can be defined by two independent curvilinear local surface

variables 1 and 1 . Similarly, the position and normal vectors of any point on the

surface of the gear are given as 2 2( , )g r and 2 2( , )g n where 2 and 2 are the

independent curvilinear local variables of the gear surface. Pinion surface coordinate

1 1( , )p r and normal 1 1( , )p n are rotated about the pinion axis pa as much as an

angle p while the gear surface coordinate 2 2( , )g r and normal 2 2( , )g n are

rotated about the gear axis ga by an angle g to satisfy the two contact conditions

defined below:

1 1 2 2( ) ( , ) ( ) ( , )p p g gz z M r M r OE , (2.14a)

1 1 2 2( ) ( , ) ( ) ( , )p p g gz z M n M n . (2.14b)

48

Here, OE is the offset vector that connects the origins of the pinion and the gear. Eq.

(2.14) constitutes a system of five nonlinear equations (since p gn n ) and six

unknowns 1 , 1 , p and 2 , 2 , g . Defining the value of one of these six

parameters (usually p ) as an input parameter, a set of five nonlinear equations defined

by Eq. (2.14) can be solved for the remaining five unknowns. However, due to the high

level of conformity of the pinion and gear surfaces in the vicinity near the contact point,

the solution of this system of equations is subject to several numerical instabilities. In

addition, the solution is very sensitive to the initial guesses. Provided these numerical

difficulties can be overcame, the solution of Eq. (2.14) yields the coordinates of contact

point path (CPP) on the pinion and gear surfaces as well as the angular position of the

gear as a function of the pinion angle.

With the conventional method, at each point of the contact point path (CPP), a

direction in which separation between two surfaces (here pinion and gear surfaces) is

minimum [2.11, 2.12, 2.18, 2.19] is assumed to be potential contact line. For this

purpose, the two contacting surfaces are approximated as two contacting ellipsoids with

an instantaneous point contact M as shown in Figure 2.6, The principal curvatures of

both pinion ( 1pK and 2

pK ) and gear ( 1gK and 2

gK ) surfaces respectively and the

corresponding principal directions ( 1pe , 2

pe , 1ge and 2

ge ) are all required to find

direction in which relative normal curvature between ellipsoids is minimal. This direction

49

Figure 2.6: General case of approximating gear surfaces as two contacting ellipsoids to

orient instantaneous contact line.

 

1e p

p

1eg

eg2

n

M

T 1pk

(a)

u

v

2e p

2pk

1gk

2gk

g

 

10

u v

2L

1L

(b)

50

u, as shown in Figure 2.6, having an unknown angle from 1pe in the tangential plane

T is the major direction of instantaneous contact ellipse. Utilizing the Euler equation

[2.11, 2.12], the pinion normal curvature along any arbitrary direction u with an angle

from 1pe in the tangential plane T is

2 21 2cos ( ) sin ( )p pp

uk k k (2.15a)

while gear normal curvature along same direction is

2 21 2cos ( ) sin ( )g gg

uk k k . (2.15b)

Hence, the relative curvature along u is given as

2 2 2 21 2 1 2cos ( ) sin ( ) cos ( ) sin ( )g g p ppg

uk k k k k . (2.16)

The value of angle that minimizes pguk is found by

0pg

udk

d

(2.17)

such that

1

12

1 sin(2 )tan

2 cos(2 )pgK

, (2.18a)

51

1 212

1 2

p ppg

g g

k kK

k k

. (2.18b)

The length of the unloaded instantaneous contact line is defined as 1 8 pguL k

where is the unloaded separation distance [2.12]. In the conventional method, it has

been assumed that instantaneous contact lines spread equally on both sides of

instantaneous contact points M along assuming contact direction u since approximating

contacting surfaces with local ellipsoids results in the same relative curvatures on both

sides of contact points. In other words, the conventional method fails to include the

variation of the curvature along a given contact line, resulting in contact line length

estimations that might be erroneous.

Simon [2.20] found the direction of minimum separation by minimizing a

function that defines separation between contacting surfaces in the direction of the

normal to the gear surface at each contact point. Although his approach does not acquire

principal curvatures and directions information on each contact point, it still bears the

some level of computational complexity and inefficiency since such minimization of the

separation function requires the solution of a system of seven nonlinear equations and

seven unknowns. He later mentioned that the instantaneous contact form is a curve

rather than the generally assumed line [2.21]. Fan [2.8] found instantaneous contact line

direction and length without using second order information by finding minimum

52

separation direction. He constructed a cylinder centered at an instantaneous contact point

with an axis that is collinear with the common normal then searched for a direction on the

tangent plane in which the separation between the contacting surfaces is minimal. With

this, another search was performed to find the distance required to move on the both sides

of the minimum separation direction in order to reach the predetermined separation value

. He eliminated the need for the second order surface information and computed more

realistic contact line length estimating different lengths on both sides of contact point

along contact line. However, this method still assumed an instantaneous contact line, as

opposed to more realistic curved shape.

2.3.2. Ease-off Based Method of UTCA

This study proposes a novel surface of roll angle and utilizes it to orient

instantaneous contact lines/curves without using principal curvatures and directions. This

method requires significantly less computational effort since (i) it does not result in a set

of nonlinear algebraic equations that must be solved numerically, and (ii) it only requires

the coordinates and the normal vector of one contacting surface and the spatial

orientation of the axes of both gears. The instantaneous contact curves are defined

between two conjugate surfaces, namely a given surface and the conjugate surface to the

reference surface with respect to the given spatial orientation of the axes of the gears.

Since the instantaneous contact line orientation is extremely insensitive to local surface

53

changes as it will be demonstrated later, the validity of using a conjugate surfaces instead

of the actual surface is well justified.

2.3.2.1. Construction of Ease-off and the Surface of Roll Angle

Ease-off will be defined in this study as the deviation of real gear surface from

the conjugate of its real mating pinion surface. It can also be defined as the deviation of

real pinion surface from the conjugate of its real mating gear. These definitions are

respectively called the gear-based ease-off and pinion-based ease-off [2.5]. The

conventional method of UTCA seeks a contact between two arbitrary surfaces, failing to

benefit from the fact that the designed gear and pinion surfaces are indeed close to the

corresponding conjugate surfaces. Closeness of the actual and the corresponding

conjugate surfaces enables the use of the ease-off concept. The proposed ease-off

approach for UTCA has several advantages such as

(i) providing an overview of the contact pattern and transmission error as well as

interference between pinion and gear teeth especially at the edges,

(ii) providing more accurate instantaneous contact curves instead of commonly used

approximate contact lines,

(iii) eliminating the need to compute the curvature in order to estimate length and

direction of the contact lines/curves, and

54

(iv) avoiding the need for initial guesses required by the conventional method to

locate first contact point (some initial guesses might end up divergent solutions).

In this study, the ease-off surface is constructed directly from the relationships between

the continuous cutter surfaces. Therefore, any surface fits to pinion and gear surfaces is

not needed.

The first step in constructing ease-off is to specify an area in the gear projection

plane with the possibility of contact between pinion and gear, as shown in Figure 2.7.

Such an area is the projection of a volume bounded by the faces and front and back cones

of the pinion and gear into the gear projection plane. The projection plane shown in

Figure 2.7 is a gear-based projection plane since its ease-off is defined on gear

tooth area. Using the gear machine settings and Eq. (2.8), the real gear surface

coordinates (shown in Figure 2.7) are computed for every point of the projection plane as

gijr where [1, ]gi m and [1, ]gj n are the indices in lengthwise and profile directions

of the surface point with gm and gn as number of surface grids respective directions.

Then, each point of the gear projection plane is transformed to the pinion coordinate

system to construct the projection plane for the pinion. Next, the real pinion surface

points and unit normals to the surface are computed from Eq. (2.8) for every point of the

pinion projection plane as pijr and p

ijn . Both real pinion and gear surface coordinates and

unit normal vectors are transformed into the global coordinate system where

55

Figure 2.7: Construction of the ease-off, action and Q surfaces.

 

Conjugate of

pinion, r̂gij

Real pinion

surface, r pij

Action

surface, raij

 

Q Surface

a p

a g

Gear projection plane

Ease-off surface, ij

Real gear

surface, rgij

G

P

E

56

misalignments E , P , G and shown in Figure 2.7 are applied as well. Having

pijr , p

ijn , the pinion and gear axis vectors pa and ga , and the gear ratio R , the action

surface position vector ijar is found from conjugacy equation in 3D space as [2.5]:

ijij ij ij( ) ( )p p pp g aR a r n a r n . (2.19)

The same steps are repeated for all points on the real pinion surface to define the

action surface completely. As seen in Figure 2.7, the action surface for a hypoid gear

pair, while quite flat, is not a plane. Any point and its unit normal vector on the real

pinion surface are rotated around the pinion axis in order to satisfy Eq. (2.19). The angle

of rotation ijp of each point that satisfies Eq. (2.19) is then plotted on the projection

plane to construct the pinion roll angle surface Q . The angle ij ij=g p R corresponds to

the amount of rotation from the surface of action to reach the conjugate of pinion surface.

Therefore,

ijij ij( ) g g az r M r . (2.20)

If this conjugate surface of the pinion were to match perfectly with the real gear

surface at any point, then a perfect meshing condition with zero unloaded transmission

error would exist. The difference between these two surfaces (conjugate of pinion and

57

real gear surfaces) in projection plane domain is defined as ease-off surface where the

differences between these two surface at each grid is ij .

2.3.2.2. Contact Pattern and Transmission Error

Any value , of any point on the Q surface is the pinion roll angle. As shown in

Figure 2.8 for a specific pinion roll angle i , intersection of the plane iz and the

Q surface defines x and y coordinates of all points on the projection plane that have the

same roll angle, stating theoretically that they lie on the same contact line/curve. Since

Q is not a plane, this intersection for hypoid gears is usually a curve rather than a

straight line as assumed by most studies. The instantaneous contact curve ( )iC shown

in Figure 2.8 is determined by projecting the intersection curve first on projection plane

and then projecting this projected curve once more on the ease-off surface. The

minimum distance from ( )iC to the projection plane at point ( )iH is the

instantaneous unloaded transmission error ( )iTE . Moreover, moving in both directions

from point ( )iH along ( )iC within a preset separation distance , gives the unloaded

contact line length ( )iL . Repeating this procedure for every pinion roll angle

increment, unloaded transmission error curve ( )TE and the unloaded tooth contact

pattern are computed. Here the contact curves are between real pinion and conjugate

gear.

58

Figure 2.8: Unloaded TCA computation procedure: (a) gear projection plane, ease-off

and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded

transmission error.

 Ease-off surface

Projection plane

Q surface

iz

iQ

Contact curve projection

Instantaneous contact curve, ( )iC

z

y

(a)

A

x

A

( )iC z

( )iTE

( )iL

( )iH

(b)

x

59

Replacing the conjugate gear with the real one practically does not change the contact

line orientation and shape, since the effect of microscopic changes of pinion and gear

surfaces on Q surface is negligible.

2.4. An Example Hypoid Unloaded Tooth Contact Analysis

An example hypoid gear pair whose basic parameters are listed in Table 2.1 for

the drive-side contact (concave side of pinion and convex side of gear) is considered to

demonstrate the capabilities of the proposed hypoid gear geometry computation and

unloaded tooth contact pattern models. This is a FM gear set representative of

automotive rear axle gear sets.

The predicted unloaded transmission error (UTE) curves computed by the model

for three adjacent tooth pairs 1i , i and 1i are shown in Figure 2.9. Here UTE is

plotted against the mesh cycles (pinion roll angle) where each of the individual UTE

curves corresponds to a single tooth pair in mesh. Individual curves for two adjacent

tooth pairs are one mesh cycle apart. At the intersection point of the two adjacent UTE

curves ( 1M or 2M in Figure 2.9), transition from one tooth pair to adjacent tooth pair

occurs. The transmission error value of the intersection point is the maximum UTE,

which is attempted to be minimized for unloaded tooth contact pattern optimization

procedures. The corresponding predicted unloaded tooth contact pattern is shown in

60

__________________________________________________________________

Parameter Pinion Gear

__________________________________________________________________

Number of teeth 11 41

Hand of Spiral Left Right

Mean spiral angle (deg) 40.5 28.5

Shaft angle (deg) 90

Shaft offset (mm) 20

Outer cone distance (mm) 115.0 111.0

Generation type Generate Formate

Cutting method FM

__________________________________________________________________

Table 2.1: Basic drive side geometry and working parameters of the example hypoid gear

pair.

61

Figure 2.9: Unloaded transmission error of the example gear pair with misalignments

0.15 mm, 0.12 mm, 0, 0E P G .

 

1

15

3

30

2

TE rad]

Mesh cycles

1i

i 1i

1M 2M P-P UTE

62

Figure 2.10. The curve marked as CPP is the locus of all instantaneous contact

points between gear and pinion tooth surfaces. On both sides of CPP are right point path

(RPP) and left point path (LPP), which are as far from CPP as it is required to reach

chosen separation value of 0.006 mm between ease-off topography and projection

plane shown in Figure 2.8. The CPP, RPP and LPP curves are computed for a single

tooth pair in contact while in case of multiple teeth in contact these curves are partially

active (usually middle part of the curves are active) since the adjacent pairs will take over

the motion. In Figure 2.10, unloaded contact pattern of the tooth of interest is bounded

by the instantaneous contact lines at 1M and 2M and the RPP and LPP curves.

Using the same example gear pair, the influence of misalignments effects on

unloaded contact patterns are illustrated next. Figure 11(a) shows unloaded contact

pattern of the drive side at a nominal position where 0E P G .

Misalignments of 0.08E mm, 0.10P mm and 0G move the unloaded

contact pattern to toe as shown in Figure 11(b) while misalignments 0.15E mm,

0.10P mm and 0G move the contact pattern to heel as shown in Figure

11(c). Similarly, Figure 10(d) shows another contact pattern near toe for the gear pair

with misalignments of 0.05E mm, 0P G , and 4 min. Comparison of

Figures 11(b) and 11(d) indicates that similar shifts in the contact patterns can be caused

by different sets of misalignments.

63

Figure 2.10: Unloaded contact pattern of the example gear pair for three adjacent tooth

pairs 1i , i and 1i (i-1), (i) and (i+1) with 0.15 mm, 0.12 mm, 0E P G

and 0 .

  Tip

RootToe

Heel

RPP

Contact lines

LPP CPP

2M

1M

64

Figure 2.11: Unloaded contact pattern of the example gear pair (a) at nominal position

with 0E P G , (b) at toe with 0.08E mm, 0.10P mm and

0G , (c) at heel with 0.15E mm, 0.10P mm and 0G and

(d) at toe with 0.05E mm, 0P G , and 4 min.

(a)

(b)

(c)

(d)

65

References for Chapter 2

[2.1] Baxter, M. L., and Spear, G. M., 1961, "Effects of Misalignment on Tooth Action

of Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.

[2.2] Krenzer, T. J., 1981, Understanding Tooth Contact Analysis, The Gleason Works.

[2.3] Smith, R. E., 1984, "What Single Flank Measurement Can Do For You." AGMA,

Technical Paper No. 84FTM2.

[2.4] Smith, R. E., 1987, "The Relationship of Measured Gear Noise to Measured Gear

Transmission Errors." AGMA, Technical Paper No. 87FTM6.

[2.5] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute

of Technology.

[2.6] Zhang, Y., and Wu, Z., 2007, "Geometry of Tooth Profile and Fillet of Face-

Hobbed Spiral Bevel Gears." IDETC/CIE 2007, Las Vegas, Nevada, USA.

[2.7] Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A

Concurrent Engineering Approach, John Wiley & Sons Inc.

[2.8] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear

Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.

Des., 129(1), pp. 31-37.

[2.9] Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth

Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University

of Illinois at Chicago.

66

[2.10] Wu, D., and Luo, J., 1992, A Geometric Theory of Conjugate Tooth Surfaces,

World Scientific, River Edge, NJ.

[2.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd

ed.), Cambridge University Press, Cambridge.

[2.12] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,

Elsevier Science B. V.

[2.13] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears

under Load." Earthmoving Industry Conference, Peoria, IL.

[2.14] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their

Load Carrying Capacity (Part I)." D.S.I.R., London.

[2.15] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason

Works.

[2.16] Shtipelman, B. A., 1979, Design and manufacture of hypoid gears, John Wiley &

Sons, Inc.

[2.17] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for

Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany.

[2.18] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.

[2.19] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis

(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,

Technical Paper No. 05FTM08.

67

[2.20] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME

International Power Transmission and Gearing Conference ASME, 88, pp. 789-

798.

[2.21] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,

122(44), pp. 529-535.

68

CHAPTER 3

SHELL BASED HYPOID TOOTH COMPLIANCE MODEL AND LOADED

TOOTH CONTACT ANALYSIS

3.1. Introduction

In the previous chapter, an ease-off topography defined by the theoretical tooth

surface was developed and used to determine the unloaded contact characteristics of a

hypoid gear pair, including the unloaded single and multiple tooth contact patterns and

the unloaded transmission error. This chapter builds on this formulation to predict the

same under loaded contact conditions.

Published studies on the modeling of tooth contact of hypoid gears under loaded

conditions are quite sparse. They can basically be divided into two major groups:

Computational models that use Finite Element (FE) or Boundary Element (BE)

formulations, and analytical models. As an example of models from the first group,

69

Wilcox et al [3.1] developed a FE-based model to calculate spiral bevel and hypoid gear

tooth compliance using 3D model of tooth including base deformations, which was later

employed by Fan and Wilcox [3.2] to develop a loaded tooth contact analysis (LTCA).

Vijayakar [3.3] developed another FE based hypoid LTCA package. This model employs

a hybrid approach with FE away from the contact zone and a semi-analytical contact

formulation at the contact zone. This model is perhaps the most advanced hypoid LTCA

model available today to simulate the loaded contacts of a hypoid gear pair accurately.

The major drawback of these computational models however is that they require a

considerable amount of computation time, which makes them more of an analysis tool.

Their use for design tasks such as parameter and assembly variation sensitivity studies is

not very practical for the same reason.

Besides these computational models, some semi-analytical models were also

proposed for determining tooth compliance of parallel-axis gears trough elasticity-based

deformation solutions. A detailed literature review of such studies was provided in

Chapter 1. All of these analytical compliance models were valid for a tooth having

constant height along its face width and either constant or linearly varying thickness

along its profile, which is not the case for hypoid gears. Vaidyanathan [3.4-3.6] proposed

an analytical compliance model for a tooth with linearly varying thickness in the profile

and lengthwise directions as well as linearly varying tooth height along the face width.

His Rayleigh-Ritz based formulation used polynomial shape functions and was applied to

both sector and shell geometries. The sector model represents straight bevel gear

70

geometry closely while the shell model can be deemed sufficiently close to that of spiral

bevel gear.

In this chapter, a Rayleigh-Ritz based shell model similar to the one proposed by

Vaidyanathan [3.4-3.6] will be applied to face-hobbed and face-milled hypoid pinion and

gear teeth to define the tooth compliances due to bending and shear effects efficiently in a

semi-analytical manner. Base rotation and contact deformation effects will also be

included in this compliance formulation. With this, loaded contact patterns and

transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears will

be predicted by enforcing the compatibility and equilibrium conditions associated with

the load distribution at the gear mesh.

3.2. Tooth Compliance Model

According to methodology outlined in the flowchart of Figure 1.4, the last step

before a LTCA can be performed is determining the tooth compliances of both contacting

members. The compliance of a tooth is defined as the amount of deflection at any

contact point due to a unit load applied at various points on the same tooth surface [3.5].

The compliance of a gear tooth must include tooth bending deflections, shear and

Hertzian deformations as well as the base rotation, since each might contribute to tooth

deflections significantly. As the computational efficiency of the model is a major

consideration, a semi-analytical shell model [3.5] is employed here instead of a FE

71

model. The model considers a hypoid gear tooth as part of a shell with a linearly varying

thickness and height, as parameterized in Figure 3.1. The tooth thicknesses toet , heelt

and tipt at the toe-root, heel-root and tooth tip locations are calculated and a linear, two-

variable function is fit to these calculated thicknesses to obtain a thickness function as

( ) ( ) ( )

tip toe heel toe ctoe

x t t t t r h xt t

h f h (3.1a)

where h is the height of the tooth defined as

[ ]heel toetoe c

h hh h r

f

. (3.1b)

Here, f is the face width, cr is the cutter radius, toeh and heel heelh are tooth height

values at the toe and the heel of the tooth, is the angle between any contact points on

the tooth to toe measured from cutter center, and x is the tooth height at that specific

contact point as defined in Figure 3.1.

In cylindrical coordinates with independent variables , and z, the position

vector of a point on the circular cylindrical shell is defined as [3.5]:

( , , ) ( , ) R r iuz z . (3.2a)

Here iu is the normal to the mid-surface of the shell, defined as

72

Figure 3.1: Basic dimensions of a hypoid tooth used in the compliance formulation.

 

6

heelh

Root (clamped)

Heel (free)

toeh

Tip (free)

Toe (free)

heelt

toet

tipt

x

Cutter axis

73

sinu

r ri , 1cos

r r

r r. (3.2b,c)

The normal and shear strains, m and mn , are given as ( , [1,3]m n , m n )

[3.5]

3

1

1

2

m k m

mkm m km k

U U g

gg g, (3.3a)

1( ) ( )

m nmn m n

m nm n m n

U Ug g

g g g g (3.3b)

where

21 α=[ (1+ )]g A z R , (3.3c)

22 =[ (1+ )]g A z R , (3.3d)

3=1g (3.3e)

and m,nU is displacement component.

In cylindrical coordinates, normal and shear strain relations are derived as [3.5]:

74

1 1,

(1 )

U V A Wz A AB R

R

(3.4a)

1 1,

(1 )

V U B Wz B AB R

R

(3.4b)

,zW

z

(3.4c)

(1 )(1 )

[ ] [ ],(1 ) (1 ) (1 ) (1 )

zz BARR U V

z z z zB A A B

R R R R

(3.4d)

1(1 ) [ ],

(1 ) (1 )

z

W z UA

z zR zA AR R

(3.4e)

1(1 ) [ ].

(1 ) (1 )

z

W z VB

z zR zB BR R

(3.4f)

For a cylindrical shell, R , cR r , x , , 1A and cB r with and

as the independent curvilinear coordinates along and perpendicular to tooth root line,

respectively. Employing a displacement assumption based on the Mindlin type shear

75

theory [3.7] that assumes a constant shear strain throughout the thickness [3.5], Eq. (3.3)

is reduces to:

2

2,

xx

Wz

x x (3.5a)

2

2

1,

( ) ( )

c c c

z W W

r r z r z (3.5a)

2( ) (2 ),

( ) ( )

c c xx

c c c c

z r z z r z W z

r x r r z x r z (3.5c)

(1 ) , zc

z

r (3.5d)

. zx x (3.5e)

The transverse deflection W and shear rotations, x and , are obtained by

setting the first variation of the potential energy to zero by using the Rayleigh-Ritz

method. The potential energy (PE) for a conservative system is the difference between

the strain energy SE and the work done by the external force WF, i.e. PE SE WF .

Here, SE and WF due to an external force p of a deformed shell surface are [3.5]

2 22

2 2 212

( ) ( 2 )2(1 )

(1 )( ) ,

c x xx z

x xz z

ESE r z

dzd dx

(3.6a)

76

c

x

WF pWr dxd . (3.6b)

where E and are the modulus of elasticity and the Poisson’s ratio for the gear material.

Setting the first variation of the potential energy to zero one obtains

2

12

( ) ( ( )(1 )

(1 )( ) 0.

c x x x xx z

x x xz xz z z cx

EPE r z

dzd dx p Wr dxd

(3.7)

The transverse deflection and shear rotations are written as linear combinations of

finite number of polynomials with unknown coefficients as

( ) ( ) mn m nm n

W A x , (3.8a)

( ) ( ) x mn m nm n

B x , (3.8b)

( ) ( ) mn m nm n

C x , (3.8c)

all of which must satisfy the following boundary conditions of a shell-shaped tooth

shown in Figure 3.1:

0( , ) 0

xW x ,

0

( , )

x

W x

x

, (3.9a,b)

77

0( , ) 0x xx , 0

( , ) 0x

x . (3.9c,d)

The polynomial functions ( )m x and ( ) n must satisfy all essential boundary

conditions, in addition to being continuous, linearly independent and complete. Equation

(3.8) is substituted in Eq. (3.5) that is needed to evaluate normal and shear strains defined

in Eq. (3.6a) to compute strain energy. Computations of SE and WF are done

numerically using Gauss-Quadrature method to yield the linear set of equations:

11 12 13 1

21 22 23 2

31 32 33 3

K K K FA

K K K B F

CK K K F

(3.10)

where the sub-matrices ( , [1,3])s sm ns sm n K are determined based on volume integral

of material properties and assumed trial functions, and ( [1,3])F smsm forms the force

vector that depends on type (point, line, etc) and location of the applied load.

Computation of the tooth compliance is done by solving Eq. (3.10) numerically for

coefficients mnA , mnB and mnC . Polynomial functions for pinned-free and clamped-free

boundary conditions are assumed as ( ) ( ) mm x x a and 1( ) ( ) m

m x x a ,

respectively, where a is the tooth height. The free-free condition for ( ) n is

represented by the polynomial 1( ) ( )nn

, where is the subtended angle by

78

circular segment of the shell of a length equal to the face width of the gear [3.5], which is

approximately equal to the ratio of the tooth length to the cutter radius.

Figure 3.2 shows a flowchart of the tooth compliance computation methodology.

While the task of computing unknown coefficients of shape functions is time consuming,

it is done only once and is valid for all mesh positions. Suppose that total contact

lines/curves as is shown (for one contact line) in Figure 3.3 are divided into cN segments

(and each segment has its own local load, which is yet to be computed), then the total

compliance matrix of a pinion or gear tooth is written as

11 1

1

.

. . .

.

Cc

c c c

N

N N N

w w

w w

(3.11)

where ( , [1, ])s si j s s cw i j N is the deflection at segment is due to the load applied at

segment js. With this, the total deflection at segment is due to all of the applied discrete

loads (load vector) is 1 cs s ss

Ni i jjW w .

Closed-form formula of Weber [3.8] was used here for computing the Hertzian

deformations while the base rotation and base translation effects on total tooth

compliance were introduced by using an approximate interpolation method similar to the

one developed for helical gears by Stegemiller [3.9].

79

Figure 3.2: Flowchart of the compliance computation.

Construct compliance matrix for pinion and gear and calculate

total compliance matrix

Use polynomial shape functions with unknown coefficients for

deformation that satisfy boundary conditions

Fit a linear function to tooth thickness and height

Tooth thickness and height calculation

Calculate strain energy based on shape function

Choose number of mode shapes IFF: Free-Free

ICF: Clamped-Free

Blank dimensions and machine settings

Potential contact lines from Unloaded TCA calculation

Minimize potential energy (strain energy + work) using

Rayleigh-Ritz method

Calculate unknown coefficients of polynomial shape functions

80

Figure 3.3: Potential contact line discretization.

.1

2 3

.

cN

1cN

cFN

1cFN

1F 2F

3F

81

In order to validate the proposed tooth compliance computation procedure, a tooth

of a face-milled Formate hypoid gear was modeled by using a commercially available

finite elements package (ANSYS). As an example, a 500 N load was applied in the

middle of the lengthwise direction and the middle of the profile, and the tooth deflections

predicted by ANSYS and the proposed semi-analytical shell model along the middle of

the top-land of the tooth were compared as shown in Figure 3.4. It is observed in this

figure that increasing the number of mode shapes in Eq. (3.8) (IFF mode shapes for the

free-free boundary conditions and ICF mode shapes for the clamped-free boundary

conditions) improves shell model predictions, converging the predicted deflections to

those from ANSYS. However, it also increases the computational time require as shown

in this figure as well. Vaidyanathan conducts an extensive comparison of his developed

shell model with ANSYS for various loading conditions and number of mode shapes and

proved accuracy of the shell model [3.6].

3.3. Loaded Tooth Contact Analysis

The number of tooth pairs in contact depends on the gear contact ratio, roll angle

of the pinion (or gear) and amount of applied torque. Under unloaded conditions, a

hypoid gear pair having a contact ratio greater than one has always at least one tooth pair

in contact. Once the load is applied, this number increases due to the deflection of the

contacting teeth. In the loaded tooth contact model, all the tooth pairs that are likely to

Figure 3.4: The comparison of the shell model deformation to FEM.

82

0.5

1

1.5

2

0

2.5

5 10 15 20 25 30 35 40 45

3

-0.50

Def

lect

ion

of f

ree

edge

(m

icro

n)

Location along face width direction (mm)

5, 3, 8 s ICF IFF t

5, 5, 15 s ICF IFF t

5, 9, 50 s ICF IFF t

5, 15, 135 s ICF IFF t

FEM Shell model

83

geometrically share the torque must be taken into consideration with their respective

separation distances. Potential contact lines/curves of all contacting tooth pairs are

computed and discretized into a finite number of segments. The length, separation,

surface curvatures of both members along each line segment are computed and used as

input for the LTCA model.

Conditions of compatibility and equilibrium must be satisfied simultaneously in

the load distribution model [3.10]. According to the compatibility condition, in order for

the contact to occur along each of the contact lines/curves, the sum of total elastic

deformation of two contacting teeth CF and the initial separation vector S must be

greater than or equal to the rigid body rotation Θ gR , i.e.

Θ g CF S R (3.12)

where F is force vector, C is the total compliance matrix that is the sum of the pinion

and gear tooth compliance matrices pC and gC (Eq. (3.11)), and the Hertzian

compliance matrix hC , and gR is the vector that contains the distances of each segment

to the gear axis. Eq. (3.11) can be written in form of an equality constraint by

introducing slack variable Y as

Θ g CF R Y = S . (3.13)

84

Since two bodies must be at contact for any force on a given segment si to exist, either

0si F for 0

siY or 0

siF for 0

siY ( [1, ])s ci N .

Meanwhile, the equilibrium condition assures that the total moment caused by

forces acting on all contacting segments about the gear axis as shown in Figure 3.5 must

be equal to external torque gT applied on the gear axis:

T g gTF R (3.14)

where superscript T denotes matrix transpose. The load distribution and loaded

transmission error are computed by solving compatibility and equilibrium equations

simultaneously.

3.4. An Example Hypoid Tooth Contact Analysis

The same face-milled example hypoid gear pair used for the UTCA whose basic

parameters are listed in Table 2.1 for the drive-side contact (concave side of pinion and

convex side of gear) is considered to demonstrate the capabilities of the proposed hypoid

gear load distribution model. This is a FM gear set representative of an automotive rear

axle gear sets.

85

Figure 3.5: Static equilibrium between torque applied on gear axis and torque produced

by the force of all contacting segments.

1F

2F

3F

1gR 2

gR 3gR

gT

ga

86

Loaded transmission error (LTE) of the example gear pair predicted by the

proposed model are shown in Figure 3.6 at three different pinion torque values of

50,pT 250 and 500 Nm for a set of fixed misalignment values of 0.15 E

mm, 0.12 P mm, 0G mm and 0 . These LTE time histories indicate that the

shape and the average value of LTE change with pT , as expected. Table 3.1 lists the

peak-to-peak value (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-mean-

square (RMS) value of LTE corresponding to the cases of Figure 3.6. The LTE functions

in each pT level are dictated primarily by the first harmonic order. Modest increases in

p-p, 1st harmonic and the root-mean-square values of LTE are observed with increasing

pT .

Figures 3.7(a-c) show the pressure distributions predicted by the proposed model

for the same cases of Figure 3.6. It is seen in Figure 3.7(a) that the contact is localized at

the center of the tooth when pT is low (50 Nm) with no edge loading. An increase in

pT causes the contact pattern to spread, in the process exhibiting edge loading at the tip

and root regions as it is evident from Figures 3.7(b) and (c). Figures 3.7(a-c) also show

the loaded contact patterns (maximum contact pressure distributions) predicted by a FE-

based hypoid contact model [3.11] for the same cases are in good agreement. It is

worthwhile to mention here that each simulation with the proposed model required 45

seconds of CPU time (about 25 seconds for compliance matrix computations and 1

second per roll angle) on a 3.0 GHz PC while the same

87

Figure 3.6: Loaded gear transmission error of the example gear pair with

0.15 mm, 0.12 mm, 0E P G and 0 at (a) 50pT Nm, (b) 250pT

Nm, and (c) 500pT Nm.

(c) 500 Nmp T =

Mesh cycles

TE [rad]

90

100

110

120

130

270

280

290

300

310

580

590

600

610

620

0.0 0.5 1.0 1.5 2.0

(b) 250 Nmp T =

(a) 50 Nmp T =

88

_________________________________________________________________

Loaded Transmission Error [μrad]

pT Errors ______________________________________

[Nm] [mm] p-p 1st 2nd 3rd RMS _________________________________________________________________

50 0.15, 0.12 E P 12.8 6.3 0.6 0.1 6.3

250 0.15, 0.12 E P 15.9 7.9 0.3 0.4 7.9

500 0.15, 0.12 E P 21.9 9.9 2.3 0.5 10.1

_________________________________________________________________

50 0.08, 0.05 E P 12.2 5.5 0.7 0.6 5.6

50 0.26, 0.13 E P 12.5 6.3 0.4 0.1 6.3

_________________________________________________________________

Table 3.1: The loaded transmission error predictions of the proposed model; 0G

mm and 0 for all cases.

Figure 3.7: Comparison of loaded contact patterns predicted by the proposed model to an FE model [3.11] for (a)

50 NmpT , 0.15E mm, 0.12P mm, (b) 250 NmpT , 0.15E mm, 0.12P mm, (c) 500 NmpT ,

0.15E mm, 0.12P mm, (d) 50 NmpT , 0.08E mm, 0.05P mm, and (e) 50 NmpT , 0.26E mm, 0.13P mm ( all at 0, 0G ).

(a) 50 Nmp T = , 0.15 mmE = , 0.12 mmP =

(b) 250 Nmp T = , 0.15 mmE = , 0.12 mmP =

(c) 500 Nmp T = , 0.15 mmE = , 0.12 mmP =

 

 

Proposed Model FE Model [3.11]

89

Continued

Figure 3.7 continued

Proposed Model FE Model [3.11]

 

 

(d) 50 Nmp T = , 0.08 mmE = , 0.05 mmP =

(e) 50 Nmp T = , 0.26 mmE = , 0.13 mmP =

90

Continued

91

analysis using the FE model took about 15 minutes using the same computer. This

highlights the main advantage of this proposed model as a design tool, even if it might

not be as accurate as the full FE model [3.11].

Next, the same gear pair is simulated by using the proposed model and the FE

model [3.11] at 50pT Nm for two other misalignment conditions. Here, two of the

errors are kept constant at 0G mm and 0 , and the other two errors E and

P are varied. In Figure 3.7(d), error values of 0.08 E mm and 0.05 P mm

cause the predicted loaded contact pattern to move towards toe and root, compared to

Figure 3.7(a). Meanwhile, the loaded contact for 0.26 E mm and 0.13 P mm

moves the contact in the opposite direction towards the heel. In the process, the

maximum contact pressure is reduced since there is larger area in the heel that carries the

same load. In addition, equivalent radii of curvature are larger at heel than toe, which

directly decreases maximum Hertzian pressure from Weber equation [3.8]. The FE

simulations of the same error combinations shown in the same figures are again in good

agreement with the predictions of the proposed model. This suggests that the sensitivity

of the hypoid gear contact to gear errors is captured sufficiently by this model. Finally

the LTE parameters listed in Table 3.1 for these two cases reveal slight reduction in LTE

amplitudes compared to the first case of Figure 3.7(a), suggesting that a good contact

pattern does not necessarily mean lower LTE.

92

References for Chapter 3

[3.1] Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite

Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth."

AGMA, Technical Paper No. 97FTM05.

[3.2] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis

(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,

Technical Paper No. 05FTM08.

[3.3] Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element

Solution for a Three-Dimensional Contact Problem." International J. for

Numerical Methods in Engineering, 31, pp. 525-545.

[3.4] Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded

Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The

Ohio State University, Columbus, Ohio.

[3.5] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz

Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears

Using Shell Theory." AGMA, Technical Paper No. 93FTM03.

[3.6] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical

Approach to the Static Analysis of an Annular Sector Mindlin Plate with

Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp.

255-266.

[3.7] Mindlin, R. D., 1951, "Influence of Rotary Inertia and Shear on Flexural Motions

of Isotropic Elastic Plates. ." J. of Applied Mechanics, 18, pp. 31-38.

93

[3.8] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their

Load Carrying Capacity (Part I)." D.S.I.R., London.

[3.9] Stegemiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of

the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192.

[3.10] Conry, T. F., and Seireg, A., 1972, "A Mathematical Programming Technique for

the Evaluation of Load Distribution and Optimal Modification for Gear Systems."

ASME J. of Industrial Engineering.

[3.11] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced

Numerical Solution Inc., Hilliard, Ohio.

94

CHAPTER 4

LOADED TOOTH CONTACT ANALYSIS OF HYPOID GEARS WITH

LOCAL AND GLOBAL SURFACE DEVIATIONS

4.1. Introduction

Hypoid gears used in various highest-volume applications such as automotive

axles are subject to various manufacturing errors and heat treatment distortions that

deviate the actual (real) tooth contact surfaces from the intended (theoretical) ones. Such

errors impact the quality of a hypoid gear pair, defined by a number of performance

indicators including its contact pattern, the motion transmission error (TE), efficiency as

well as its sensitivity to misalignments. These deviations represented by these

manufacturing errors typically follow patterns that shift, rotate or twist the surfaces

relative to the theoretical ones. Therefore, they can be characterized as global deviations.

Other more local deviations occur during the life span of hypoid gears in the form of

95

surface wear. Since the surface wear depths are proportional to the contact pressure and

the sliding distance, deviations due to surface wear are rather local and cannot be

captured by using the surface fitting methods developed to approximate the global

deviations. The motivation of this chapter is to develop a unified, ease-off based

methodology that allows loaded and unloaded tooth contact analysis of hypoid gears

having both global and local deviations.

Tooth contact analysis has usually been performed by considering the theoretical

pinion and gear surfaces defined by simulation of the hypoid cutting processes. The

analysis results presented in Chapters 2 and 3 also considered such theoretical tooth

surface errors with no deviations.. There are only a few published studies on hypoid

gear tooth contact analysis using the real surfaces. In such an analysis, Gosselin [4.1]

proposed an approach to compute tooth contact of real spiral bevel gear surfaces. He

interpolated measured surfaces with rational functions to predict their unloaded contact

pattern and transmission error. Since pinion and gear normal vectors of low-mismatch

(high-conformity) surfaces, a wide span of potential contact line around contact point can

be identified and the condition of collinearity for normal vectors is subjects to numerical

stability issues. In order to simplify the task of locating the contact point, Gosselin [4.1]

computed the difference between pinion and gear normal vectors at several points along

the lengthwise direction and estimated a location where the difference between the

normal vector is zero.

96

Zhang et al [4.2] proposed an approach to analyze unloaded tooth contact of real

hypoid gears based on a generalization of the work of Kin [4.3, 4.4] on spur gears. The

real pinion and gear tooth surfaces were divided into two vectorial surfaces of theoretical

and deviation surfaces. Separating theoretical and deviation surfaces, finding the

theoretical surfaces through cutting simulation, and applying interpolation only to the

deviation surfaces made his approach simpler and more accurate. Zhang [4.2] defined

the deviation surface in the normal direction of the theoretical surface by comparing the

theoretical and the real (measured) surfaces, and fit a bicubic surface to it. The normal to

the real surfaces (sum of the theoretical and interpolated deviation surfaces) were

computed by taking the derivative of the real surfaces. Having continuous functions for

the surfaces and normal of the pinion and the gear, he employed conventional system of

five nonlinear equations and five unknowns used in many other studies [4.5-4.7] to

simulate unloaded tooth contact of real hypoid surfaces.

Gosselin proposed a method called “surface matching” that attempts to define

changes to the machine settings that define the theoretical surfaces so that real gear

surfaces can be computed approximately from the cutting simulation [4.8, 4.9]. This

method found machine settings that generate a theoretical surface close to (but not

identical to) the real surface and the difference of the two surfaces was defined as

“residual error surfaces” for the pinion and the gear. He computed transmission error and

the contact pattern of the generated surfaces with this new set of machine settings. Then,

he used the residual error surface to modify predicted transmission error and contact

pattern without providing the details of this process [4.10]. This method is suitable in

97

capturing the effect of global errors such as the pressure angle error, the spiral angle

error, and lengthwise or profile crowning errors for the cases when residual errors are

rather small, while the same cannot be said for localized deviations such as surface wear

and global deviations with large residual errors.

In Chapter 2, it was mentioned that the unloaded tooth contact analysis (UTCA)

of hypoid gears with mismatched surfaces were performed using two fundamentally

different methods. The first method that was used widely defines the tooth surfaces as

two arbitrary surfaces, rotating about pinion and gear axes [4.7, 4.11-4.13]. In this

method, the contact point path (CPP) on each surface was determined by satisfying two

contact conditions: (i) coincidence of position vector tips of the points on the gear and

pinion surfaces and (ii) collinearity of the normals of the both of the surfaces. The

second method that was proposed in Chapter 2 was based on the ease-off topography. A

detailed formulation for construction of ease-off and determining the instantaneous

contact curves from surface of roll angle was provided. In Chapter 3, UTCA results were

combined with a semi-analytical compliance model based on shell theory to predict the

loaded contact patters and loaded TE. Using the ease-off approach for TCA of real gear

surfaces instead of conventional approach was shown to increase the computational

efficiency of TCA since surface interpolations for measured pinion and gear surfaces as

well as the solution of the system of five governing nonlinear equations are not needed.

In this chapter, ease-off is defined the same way as Chapter 2 as the deviation of

real gear surface from the conjugate of its real mating pinion surface. The formulation

98

that will be proposed to handle local and global deviations is based on the premise that all

surface deviations, both local and global, can be handled through modifications of the

ease-off topography. Having machine settings and blank dimensions, surface coordinates

and normal vectors of both pinion and gear will be defined using the methodology

proposed in Chapter 2. These theoretical surfaces will be used to establish the theoretical

ease-off topography and a theoretical surface of roll angle. As the main contribution of

this study, a procedure will be proposed to update the ease-off topography by taking into

account pinion and gear surface deviations. The updated ease-off and roll angle surfaces

will be used to determine the unladed tooth contact characteristics of the gear pair. These

UTCA results will be combined with the semi-analytical LTCA methodology of Chapter

3 to predict the loaded contact patterns and the transmission error of hypoid gears having

local and global surface deviations.

4.2. Construction of the Theoretical Ease-off Topography

The first step in defining the theoretical ease-off surface is specifying an area in

the gear projection plane with the possibility of contact between pinion and gear, as

illustrated in Figure 4.1. This area represents the projection of a volume bounded by the

faces and front and back cones of the pinion and gear into the gear projection plane. The

projection plane shown in Figure 4.1 is a gear-based projection plane since its ease-off

is defined on the gear tooth area. Using the gear machine settings and applying the

equation of meshing, the theoretical gear surface coordinates of a point of the projection

99

Figure 4.1: Construction of the ease-off, action and Q surfaces.

 

Gear projection plane

ij

Q paga

gijr

 

ˆ gijr

pijr

aijr

100

plane shown in Figure 4.1 are computed as gijr where [1, ]i m and [1, ]j n are the

indices in lengthwise and profile directions of the surface point with m and n as number

of surface grids in respective directions, and the superscript g denotes gear surface.

Then, gijr is transformed to the pinion coordinate system to construct the projection plane

for the pinion.

Next, the theoretical pinion surface point pijr and its unit normal to the surface

pijn are computed again through machine settings and applying the equation of meshing

[4.1]. Both theoretical pinion and gear surface coordinates and unit normal vectors are

transformed into the global coordinate system. Having pijr , p

ijn , the pinion and gear

axis vectors pa and ga , and the gear ratio R, the position vector aijr of the corresponding

point on the action surface is found from the conjugacy equation in 3D space as [4.14]:

( ) ( )p p pp g aijij ij ijR a r n a r n , [1, ]i m , [1, ]j n . (4.1)

The same procedure is repeated for all points on the real pinion surface to define the

action surface completely. As observed from Figure 4.1, the action surface for a hypoid

gear pair, while quite flat, is not a plane. Here, any point and its unit normal vector on

the theoretical pinion surface are rotated around the pinion axis in order to satisfy Eq.

(4.1). The angle of rotation pijq of each point ij required to satisfy Eq. (4.1) is then

101

plotted on the gear projection plane to construct the theoretical pinion roll angle surface

Q .

Here, only the relative values of pijq (i.e. the shape of the Q surface) are of

interest. Therefore, for computational simplicity and graphical demonstration purposes,

the Q surface is shifted in the direction normal to the projection plane to bring it to

contact with the projection plane such that at least one grid point has zero pijq value.

Mathematically, this shift is equivalent to rigidly rotating pinion tooth surface around the

pinion axis, which has no effect on the pinion surface. The angle ˆ =g pij ijq q R

corresponds to the amount of rotation required to travel from the surface of action to the

conjugate of theoretical pinion surface. Therefore, the position vector of the conjugate

of the theoretical pinion surface is found as

ˆ ˆ( ) g g az ijij ijqr M r , [1, ]i m , [1, ]j n , (4.2a)

where the rotation matrix about the z axis at an angle ˆ gijq is

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

ˆ ˆ

ˆ ˆˆ( )

g gij ij

g ggij ijz ij

q q

q qq

M . (4.2b)

102

If this conjugate surface of the pinion were to match perfectly with the real gear

surface at any point, then a perfect meshing condition with zero unloaded transmission

error would exist. For simplicity, the conjugate of pinion will be called here the

conjugate gear. The difference between the conjugate gear and the theoretical gear is

defined as theoretical ease-off topography . In general, the conjugate gear and the

theoretical gear are located at different angular positions with respect to the gear axis. In

order to compare these two surfaces, conjugate gear surface is rotated around gear axis

ga by an angle such that a grid point of conjugate gear surface touches the

corresponding grid point on the theoretical gear surface. In this position, the radial

distances ij between the grid points ij on these surfaces define the theoretical ease-off

surface . The and Q surfaces were used in Chapters 2 and 3 for both unloaded and

loaded tooth contact analyses.

4.3. Updating Ease-off Topography for Manufacturing Errors and Surface Wear

Deviations of the pinion and gear surfaces a grid point ij from their respective

theoretical surfaces are defined as pij and g

ij ( [1, ]i m , [1, ]j n ) as shown in Figure

4.2. Normal vectors to both pinion and gear surfaces are considered in inward direction.

Measured and/or worn pinion and gear surfaces are written as ( [1, ]i m , [1, ]j n ):

p p p pij ij ij ij r r n , (4.3a)

103

g g g gij ij ij ij r r n . (4.3b)

Here, it is assumed that the normal vectors of the real and theoretical surfaces are the

same, since both surfaces are practically very close to each other [4.4].

Any point of the theoretical ease-off surface ij can be updated by using

deviations on pinion and gear surfaces. The goal here is to update theoretical ease-off

surface directly from surface deviations rather that updating original pinion and gear

surfaces and conducting whole tooth contact procedure between new surfaces as it has

been done in Ref. [4.2].

Assume that an ease-off value ij of a point on the theoretical ease-off surface

is computed based on a corresponding theoretical surface vectors

, ,Tp p p p

ij ij ij ijx y z r , pijn and , ,

Tg g g gij ij ij ijx y z r . At the same grid point ij, the

pinion roll angle is pijq , and hence, the corresponding roll angle of the gear surface

(conjugate to the theoretical pinion surface) is g pij ijq q R and the distance of the same

grid points on the pinion and gear surfaces to their own rotation axes ( pa and ga

respectively) are

2 2( ) ( )p p pij ij ijL x y , (4.4a)

104

2 2( ) ( )g g gij ij ijL x y (4.4b)

as shown in Figure 4.2. With this, the changes in the pinion and gear roll angles due to

the pinion and gear surface errors pij and g

ij are defined as

( ),

h h hij ij ijh

ij hij

qL

u n = ,h p g. (4.5a)

Here piju and g

iju are the unit normal vectors in the radial (circular) direction of the

pinion and gear axes pa and ga , respectively, as shown in Figure 4.2. They are

defined as:

ppijp

ij ppij

a ru

a r,

ggijg

ij ggij

a ru

a r (4.5b,c)

Hence, the updated pinion roll angle taking the pinion deviation into account is

=p p pij ij ijq q q , (4.6)

which can be used to find locations and directions of contact curves corresponding to the

deviated tooth surfaces.

105

Figure 4.2: Graphical demonstration of the procedure to update ease-off surface for

surface deviations.

Pinion projection plane

Gear projection plane

pa

ga

gijr

pijrg

iju

pijup

ijn

gijn

pijL

gijL

gij

pij

Q

106

Change to the theoretical ease-off surface can be described in two components.

The first component is related to the pinion surface deviation that is formulated as

p gp gij ij ijq L (4.7)

where =gp pij ijq q R is the gear roll angle change due to the pinion surface deviation.

The second component is due to the gear surface deviation that is given as

g g gij ij ijq L (4.8)

Therefore, the total change to the ease-off topography is the sum of its two components

p gij ij ij . (4.9)

With this, the new ease-off surface is found as

ij ij ij . (4.10)

This updated ease-off surface and the corresponding updated surface of roll

angle Q as defined by Eq. (4.6) are used for unloaded and loaded tooth contact analyses

according to the methodology proposed in Chapters 2 and 3 As in Q , the updated

surface of roll angle Q is also shifted to touch the projection plane since the absolute

values pijq do not have any effect on unloaded and loaded tooth contact analyses.

107

4.4. Unloaded and Loaded Tooth Contact Analyses

In Figure 4.3, surfaces and Q are constructed on opposite sides of projection

plane to provide an insight into how UTCA is conducted. For any value q on surface Q ,

a corresponding instantaneous contact curve can be defined. As shown in Figure 4.3 for

a specific pinion roll angle kq , the intersection of the plane kz q and the surface Q

defines the x and y coordinates of all points on the projection plane that have same roll

angle, stating theoretically that they lie on the same contact curve. Since Q is not a

plane, this intersection for hypoid gears is usually a curve rather than a straight line as

assumed in most of the previous studies.

Figure 4.4 shows the theoretical contact curves of the drive and coast sides of a

sample hypoid gear pair on the projection plane for different pinion angles as shown on

the contact curves. The instantaneous contact curve ( )kC q shown in Figure 4.3 is

obtained by first projecting this intersection curve on projection plane and then projecting

this projected curve on the ease-off surface . The minimum distance from ( )kC q to

the projection plane at point H is instantaneous unloaded transmission error ( )kTE q .

Moreover, moving in both directions from point H along the curve ( )kC q within a preset

separation distance yields the unloaded contact line length ( )kU q . Repeating this

procedure for every pinion roll angle increment kq q ( [1, ]lk N where lN is total

number of contact curves considered), unloaded transmission error curve ( )kTE q and the

unloaded tooth contact pattern are computed.

108

Figure 4.3: Graphical demonstration of the procedure to compute unloaded TCA; (a) gear

projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact

line and unloaded transmission error.

 

Contact curve projection

kq q

kz q

Gear projection plane

z

x

Q

( )kC q

( )kU q

H

z

x

( )kTE q

109

Figure 4.4: Theoretical contact curves of an example hypoid gear pair.

10° 25° 40° 55° 70° 85°

Root

Toe

(a) Drive Side

10° 25° 40° 55° 70°

85°

Root

Toe

(b) Coast Side

110

It is noted here that the contact curves are defined between the real pinion

surface and conjugate of the real pinion surface (conjugate gear surface), instead of using

the real gear surface. Replacing the real gear surface with the conjugate one causes a

very little change to the orientation and shape of the contact curve, since the effect of

micro-geometry deviations has a negligible influence on the Q surface, i.e. Q and Q

are practically identical.

Number of tooth pairs in contact depends on the gear contact ratio, the roll angle

of the pinion (or gear) and the amount of torque applied. Under unloaded conditions, a

hypoid gear pair having a contact ratio greater than one has always at least one tooth pair

in contact. Once the load is applied, this number increases due to deflection of the

contacting teeth. In the LTCA model of Chapter 3, all the tooth pairs that are likely to

share the torque geometrically are taken into consideration with their respective

separation distances. Potential contact curves of all contacting tooth pairs are computed

and discretized into a finite number of segments ( cN ). The length of the separation at

each segment along each contact curve are computed and used as input for the LTCA

model. With the theoretical ease-off topography replaced by the modified ease-off

topography corresponding to real surfaces with deviations, the formulations of Chapters 2

and 3 can be applied to predict the unloaded and loaded tooth contact conditions,

respectively.

111

4.5. Example Analyses

4.5.1. A Face-milled Hypoid Gear Pair with Local Surface Deviations

A face-milled hypoid gear set with local deviations whose basic parameters are

listed in Table 4.1 for the drive-side contact (concave side of pinion and convex side of

gear) will considered for an example loaded contact analysis of surfaces with local

deviations. This gear set is representative of an automotive rear axle gear set. An

example case of local deviations from theoretical surfaces is shown in Figure 4.5 for both

pinion deviation pij and gear deviation g

ij . This case represents worn tooth surfaces

predicted by a hypoid gear wear model [4.15] from a companion study. In Figure 4.5, the

“Root” line refers to the lower limit of active contact region and it is not actual gear root

line.

Following the proposed ease-off update approach, gear projection plane, the

theoretical and updated ease-off surfaces, and , and the theoretical and updated roll

angle surfaces, Q and Q , are computed. Figure 4.6(a) shows these surfaces in relation

to each other while the theoretical ease-off topography, updated ease-off and the amount

of ease-off change computed from Eq. (4.9) are shown in Fig. 4.6(b) to (d) in contour plot

format, respectively. As shown here, the maximum changes take place in the vicinity of

the locations where pinion and gear deviations are maximum, according to Figure 4.5 and

the rest of the projection plane does not exhibit any considerable ease-off change.

112

_____________________________________________________________

Parameter Pinion Gear _____________________________________________________________

Number of teeth 11 41 Hand of Spiral Left Right Mean spiral angle (deg) 40.5 28.5 Shaft angle (deg) 90 Shaft offset (mm) 20 Outer cone distance (mm) 115 111 Generation type Generate Formate Cutting method FM ___________________________________________________________________________

Table 4.1: Basic drive side geometry and working parameters of the example hypoid gear

pair.

113

Figure 4.5: Example local deviation surfaces for the gear and pinion tooth surfaces.

Toe

Toe

Root

Root

pij

gij

μm

0246810

Pinion deviation

Root

Toe

012345

μmGear deviation

Root

Toe

114

Figure 4.6: Ease-off update for the example deviation of Fig. 5. (a) Three-dimensional

view of the projection plane, and , , Q and Q surfaces, and contour plots of (b) ,

(c) , and (d) the change of ease-off topography.

 

Gear projection plane

(a)

Q , Q

0

20

40

60 μm

(c)

Toe

Root

1 2 3 4

5 6

μm (c)

Toe

Root

0

20

40

60μm

(b)

Toe

Root

115

The corresponding predicted unloaded tooth contact patterns are shown in Figure

4.7 for a maximum separation value of 6 μm for both cases of (a) theoretical and (b)

deviated tooth surfaces, indicating that the unloaded contact patterns are influenced by

the local deviation as well. It is clear from this figure that the length of the contact lines

are elongated for UTCA of deviated surfaces since ease-off in the contact region is

flattened.

The predicted unloaded transmission error (UTE) curves are shown in Figure

4.8(a) for the theoretical and deviated surfaces. The corresponding peak-to-peak

amplitude (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-mean-square

(RMS) value of these curves are listed in Table 4.2(a) to show that all components of

UTE are influenced by the local deviation introduced. Both the root-mean-square (RMS)

and p-p amplitudes are reduced with deviated surfaces.

Next, LTCA is performed for the theoretical and the locally deviated surfaces as

before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded

transmission errors (LTE) at this torque value are shown in Figure 4.8(b) for the

theoretical and deviated surfaces. It is noted here that both curves are identical for certain

mesh positions where areas of the local deviation are not in contact while they differ

significantly in certain mesh positions. Table 4.2(b) lists the same LTE amplitudes for

theoretical and deviated surfaces to show that such local deviations also impact the LTE.

Finally, predicted contact pressure distributions are shown in Figures 4.9(a) and (b) for

the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it is

116

Figure 4.7: Predicted unloaded tooth contact pattern for separation value of 6 μm .

Toe

Root

(a) Theoretical surfaces

Toe

Root

(b) Deviated surfaces

117

Figure 4.8: Transmission error (UTE) curves for theoretical and deviated surfaces at (a)

unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm.

0.5 1.0 1.5 2.0 0

5

10

15

20

25

0 Mesh cycles

Deviated surfaces

0

5

10

15

20

25 (a) Unloaded TE [µrad]

Theoretical surfaces

Mesh cycles

Deviated surfaces

225 0.5 1.0 1.5 2.0 0

230

235

240

245

250

(b) Loaded TE [µrad]

Theoretical surfaces

245

250

255

260

265

270

118

(a) Unloaded Transmission Error in [µrad]

__________________________________________________________ p-p 1st 2nd 3rd RMS

__________________________________________________________ Theoretical surfaces 21.1 8.3 3.7 2.0 9.3 Deviated surfaces 15.0 4.3 4.3 1.4 6.3

__________________________________________________________

(b) Loaded Transmission Error at 200 Nm in [µrad] __________________________________________________________

p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 14.2 7.3 0.3 0.5 7.4 Deviated surfaces 10.2 4.5 1.4 0.4 4.7

__________________________________________________________

Table 4.2: The transmission error amplitudes of theoretical and deviated surfaces.

119

Figure 4.9: Predicted contact pressure distribution for a pinion toque of 200 Nm

for (a) theoretical and (b) deviated surfaces.

0

400

800

1200

Toe

Root

(a) Theoretical surfacesMpa

0

400

800

1200

Toe

Root

(b) Deviated surfacesMpa

120

observed that the edge loading condition experienced by the theoretical surfaces on the

gear root (pinion tip) is reduced significantly for the surfaces with the local deviations

considered since ease-off topography shown in Figure 4.6(d) is considerably flattened

due to surface deviation in the same region.

4.5.2. A Face-hobbed Hypoid Gear Pair with Global Deviations

A face-hobbed hypoid gear set with basic parameters listed in Table 4.3 for its

drive-side contact (concave side of pinion and convex side of gear) is considered as

example for loaded contact analysis of surfaces with global deviations. This gear pair is

also representative of an automotive rear axle gear sets. The measured pinion and gear

deviation surfaces ( pij and g

ij ) from their theoretical geometry after heat treatment and

the lapping process are shown in Figures 4.10(a) and (b), along with their respective

interpolated surfaces on active surface domain shown as pij and g

ij . Figures 4.10(c) and

(d) are the contour plots of the same, showing a maximum 50 µm of error for pinion in

heel-top region and 25 µm for gear in toe-root. The source of deviation here could be

due to manufacturing errors, heat treatment distortion and surface wear caused by lapping

process. In the most common measurement procedure used by the axle manufacturers,

measured coordinates of points on a 5x9 grid are compared to the corresponding

theoretical surface coordinates to determine the measured surface deviations in the

direction normal to the surface. In Figure 4.10, these deviations are shifted for

demonstration purposes so that all are in the positive side.

121

_____________________________________________________________

Parameter Pinion Gear

_____________________________________________________________

Number of teeth 12 41

Hand of Spiral Left Right

Mean spiral angle (deg) 50.0 24.0

Shaft angle (deg) 90

Shaft offset (mm) 45.0

Outer cone distance (mm) 105 130

Generation type Generate Formate

Cutting method FH

_____________________________________________________________

Table 4.3: Basic drive side geometry and working parameters of the example hypoid gear

pair.

122

Figure 4.10: Example global deviation surfaces measured by CMM for the gear and

pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured deviation, (c)

pinion deviation distribution in tooth active region and (d) gear deviation distribution in

tooth active region.

Toe

Root

Measured gear deviation µm (d)

0

5 10

15

20 25

Toe

Root

Measured pinion deviation µm (c)

0

10

20

30

40

 (a)

Root

Toe pij

pij 

(b)

Root

Toe

gij

gij

123

It should be noted here that a simple weighted average is used to interpolate (or

extrapolate) the deviations pij to p

ij and gij to g

ij at a point ij that is not one of the 45

measurement points. No interpolation is used to estimate surface coordinate and normal

vectors as is the case in previous studies, hence the required accuracy and complexity of

this interpolation is by no means comparable to the case that interpolation is required for

surface coordinates and normal vectors estimation.

Following the proposed method described in the previous section , the theoretical

and updated ease-off surfaces are computed. Figures 4.11(a-d) respectively show (a)

theoretical ease-off topography, (b) updated ease-off topography by only considering

pinion surface deviations, (c) updated ease-off topography by only considering gear

surface deviations and (d) updated ease-off topography by considering both pinion and

gear surface deviations. Figure 4.11(a) shows a localized well defined ease-off

topography as a result of interaction between theoretical pinion and gear surfaces.

Although pinion and gear deviation effects on ease-off, when applied alone, are

considerable as shown in Figure 4.11(b,c), the combination of these deviations alleviates

their adverse effect of alone, resulting in the ease-off topography shown in Figure

4.11(d).

The corresponding predicted unloaded tooth contact patterns of this design are

shown in Figure 4.12(a,b) for a maximum separation value of 6 μm for both cases of

theoretical and deviated tooth surfaces, respectively, indicating that the unloaded contact

patterns are influenced by the deviations given in Figure 4.10 as well. With the deviation

124

Figure 4.11: Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical ease-

off topography, (b) updated ease-off topography only with pinion deviation, (c) updated

ease-off topography only with gear deviation, and (d) updated ease-off topography with

both pinion and gear deviations.

Toe

Root

µm(a)

0

40

120

80

Toe

Root

µm(b)

0

40

120

80

Toe

Root

µm(c)

0

40

120

80

Toe

Root

µm(d)

2

4

6

8

0

40

120

80

125

Figure 4.12: Predicted unloaded tooth contact pattern for separation value of 6 μm .

Toe

Root

(b) Deviated surfaces

Toe

Root

(a) Theoretical surfaces

126

included, unloaded contact pattern slightly shifted toward heel and becomes narrower as

it approaches gear tip.

The predicted unloaded transmission error (UTE) curves are shown in Figure

4.13(a) for theoretical and deviated surfaces against mesh cycles. The corresponding

peak-to-peak amplitude, first three Fourier harmonics and the RMS value of these curves

are listed in Table 4.4(a) to show that the all components of UTE are influenced by the

local deviation introduced. The RMS, peak-to-peak and 1st harmonic amplitudes are

almost doubled with deviations included.

Next, LTCA is performed for the theoretical and the globally deviated surfaces as

before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded

transmission errors (LTE) at this torque value are shown in Figure 4.13(b) for the

theoretical and globally deviated surfaces. Table 4.4(b) lists the same LTE amplitudes

for theoretical and deviated surfaces to show that such local deviations also impact the

LTE. Finally, predicted contact pressure distribution is also shown in Figure 4.14(a) and

(b) for the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it

is observed that the contact pressure distributions for the theoretical and deviated surfaces

are rather close since pinion and gear surface deviations tend to compensate each other

in this particular example set of deviation (Figure 4.10).

127

Figure 4.13: Transmission error curves for theoretical and deviated surfaces; (a) unloaded

conditions and (b) loaded conditions at a pinion torque of 200 Nm.

Theoretical surfaces

0

20

40

60

Deviated surfaces

0

20

40

60

0 0.5 1.0 1.5 2.0

(a) Unloaded TE [µrad]

(b) Loaded TE [µrad]

Theoretical surfaces

305

315

325

335

0 0.5 1.0 1.5 2.0 340

350

360

370

Mesh cycles

Deviated surfaces

128

(a) Unloaded Transmission Error [µrad] __________________________________________________________

p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 36.9 15.1 4.0 1.8 15.7

Deviated surfaces 61.5 28.9 2.1 1.5 29.0

__________________________________________________________

(b) Loaded Transmission Error at 200 Nm [µrad] __________________________________________________________

p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 23.9 12.1 0.5 0.7 12.1

Deviated surfaces 29.8 15.8 2.8 2.3 16.2

_______________________________________________________

Table 4.4: The transmission error amplitudes of theoretical and deviated surfaces.

129

Figure 4.14: Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)

theoretical and (b) deviated surfaces.

Toe

Root

Mpa(a) Theoretical surfaces

0

200

400

600

800

Toe

Root

Mpa(b) Deviated surfaces

0

200

400

600

800

130

4.6. Summary

In this chapter, an accurate and practical method based on ease-off topography

was proposed to perform loaded and unloaded tooth contact analysis of spiral bevel and

hypoid gears having both types of local and global deviations. Manufacturing errors

causing global errors and localized surface deviations were considered to update the

theoretical ease-off to form a new ease-off surface that was used to perform a loaded

tooth contact analysis. Two numerical examples of (i) face-milled hypoid gear set with

local deviations and (ii) face-hobbed hypoid gear set with global deviations measured by

CMM were presented to demonstrate the effectiveness of the proposed methodology as

well as quantifying the effect of such deviations on load distribution and the unloaded

and loaded motion transmission error.

References for Chapter 4

[4.1] Gosselin, C., et al. 1991, "Tooth Contact Analysis of High Conformity Spiral

Bevel Gears." Proceedings of JSME Int. Conf. on Motion and Power

Transmission, Hiroshima, Japan.

[4.2] Zhang, Y., Litvin, F. L., Maryuama, N., Takeda, R., and Sugimoto, M., 1994,

"Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces."

116, pp. 677-682.

[4.3] Kin, V., 1992, "Tooth Contact Analysis Based on Inspection." Proceedings of 3rd

World Congress on Gearing, Paris, France.

131

[4.4] Kin, V., 1992, "Computerized Analysis of Gear Meshing Based on Coordinate

Measurement Data." ASME Int. Power Transmission and Gearing Conference,

Scottsdale, AZ.

[4.5] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.

[4.6] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd

ed.), Cambridge University Press, Cambridge.

[4.7] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear

Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.

Des., 129(1), pp. 31-37.

[4.8] Gosselin, C., Nonaka, T., Shiono, Y., Kubo, A., and Tatsuno, T., 1998,

"Identification of the Machine Settings of Real Hypoid Gear Tooth Surfaces."

ASME J. Mech. Des., 120(3), pp. 429-440.

[4.9] Gosselin, C., Jiang, Q., Jenski, K., and Masseth, J., 2005, "Hypoid Gear Lapping

Wear Coefficient and Simulation." AGMA, Technical Paper No. 05FTM09.

[4.10] Gosselin, C., Guertin, T., Remond, D., and Jean, Y., 2000, "Simulation and

Experimental Measurement of the Transmission Error of Real Hypoid Gears

Under Load." ASME J. Mech. Des., 122(1), pp. 109-122.

[4.11] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,

Elsevier Science B. V.

[4.12] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for

hypoid gear-drives of formate and helixform; Part III-Analysis and optimal

synthesis methods for mismatched gearing and its application for hypoid gears of

formate and helixform." ASME J. Mech. Des., 103, pp. 102-113.

132

[4.13] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME

International Power Transmission and Gearing Conference ASME, San Diego.

[4.14] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute

of Technology.

[4.15] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear

Pairs." In press, Wear, pp.

133

 

 

CHAPTER 5

 

PREDICTION OF MECHANICAL POWER LOSSES OF

HYPOID GEAR PAIRS

 

5.1. Introduction

Prediction of power losses of automotive drive trains is becoming increasingly

critical to power train designers. This is mainly because the government regulations in

regards to fuel economy and carbon emissions are becoming more stringent. Forecasted

increases in oil prices also add to the motivation to predict and reduce power losses of

drive trains. In rear-wheel drive vehicles, the rear axle-differential unit is one of the

major sources of power losses. The axle efficiency values can be typically as low as 90

to 95% [5.1]. Considering that rear-wheel drive vehicles comprise a significant share of

the global passenger vehicle market, any sizable improvements to the axle efficiency can

have a significant positive impact on environment and energy consumption.

Axle power losses can be divided into two groups. One group of losses is

independent of the torque transmitted. These load independent (spin) power losses are

134

 

due to viscous bearing losses (including the losses due to pre-load) and gear windage and

oil churning losses [5.2, 5.3]. Such losses are outside the scope of this research. The

other group of losses are induced by friction at bearing and hypoid gear pair locations

under load. These power losses are called load-dependent (mechanical) losses. Focusing

on the hypoid gear pair, mechanical power losses are associated with the relative sliding

and the lubricated contact conditions along the tooth contacts. The shaft off-set, being

the main difference between spiral bevel and hypoid gears, causes increased relative

sliding in hypoid gears and the power losses associated with friction [5.4]. The

motivation of this chapter is to develop a mechanical power loss model of face-milled

and face-hobbed hypoid gears.

Modeling mechanical losses of a gear pair involves (i) computation of surface

geometry parameters and velocities, and the normal load at each contact point from a

tooth contact analysis model as the one proposed in Chapters 2 and 3, (ii) a friction model

to determine the coefficient of friction at each contact points (iii) computing the surface

traction from the distributions of the friction coefficient and normal force, and (iv)

determining the friction torque and the resultant power loss. Published gear efficiency

models differ mostly in the way they determine the friction coefficient. The first group of

models used a constant friction coefficient μ [5.5-5.7] in computing the power losses.

Recognizing the fact that μ is dependent on various contact parameters, including rolling

velocity, sliding to roll ratio, radii of curvature of the contacting surfaces and normal

135

 

load, all of which vary as gears roll, experiment based µ empirical formulae [5.8-5.11]

were employed by another group of studies [5.8, 5.12-5.16]. However, the applicability

of these models was limited to narrow ranges of the operating temperatures, speed, load,

and surface roughness conditions represented by the empirical formula. The third group

of models predicted the friction coefficient using the elastohydrodynamic lubrication

(EHL) theory [5.17-5.22]. This approach, while physics-based and potentially more

accurate, requires a significant computational effort as several hundreds of EHL analyses

are required to predict the mechanical losses of a gear pair. In order to avoid this

difficulty, Xu et al. [5.23] proposed a methodology to derive a gear contact friction

formula up-front by using the EHL model of Cioc et al [5.24]. Using this EHL model,

they conducted a large parameter study, covering wide ranges of contact and surface

parameters as well as operating conditions representative of gears. The predicted surface

traction data was reduced into a single formula by using linear regression technique.

All of the models cited above were limited to spur or helical gears. Efficiency

models for hypoid gears are very sparse. Approximating the hypoid gear power loss as

the sum of losses from the corresponding spiral bevel and worm gears, Buckingham

[5.25] recommended a power loss equation. Coleman [5.1] proposed a simple closed-

form formula to estimate bevel and hypoid gears efficiency. This heuristic formula used

a constant friction coefficient of 0.05 at every contact point and was a function of the

136

 

normal load, pressure angle, and pinion and gear mean spiral angles. Simon [5.26]

applied a smooth surface EHL model to simulate hypoid gear lubrication.

The model proposed recently by Xu and Kahraman [5.27] extended their helical

gear efficiency model to hypoid gears. They used a commercially available FE-based

hypoid gear contact model CALYX [5.28] to determine all required contact load and

geometry parameters including curvatures. Employing set of equations developed by

Litvin [5.29] primarily to describe relationships between curvatures of mating surfaces,

they computed sliding and rolling velocities at each contact point along and perpendicular

to the contact line. While this model [5.30] was physics-based and included most of the

key surface, lubricant, geometry and operating parameters, its FE load distribution

computation required significant computational time, making it impractical for design

and parameter sensitivity studies. It relied on the same FE model for its geometry and

curvature information as well. More importantly, the EHL model [5.24] it employed to

derive the friction coefficient formula was not designed for simulating mixed type of

lubrication condition. Therefore, the fidelity of the model Xu and Kahraman [5.30] was

limited to contact conditions with no or limited asperity interactions. However, in most

automotive hypoid gear applications, mixed EHL conditions characterized by excessive

metal-to-metal contacts of the asperity peaks occur commonly. Recently, Li and

Kahraman developed transient mixed or boundary EHL models for line [5.31] and point

[5.32] contacts that can handle any lubricated gear contact conditions ranging from

137

 

almost dry to full-film EHL. Li et al [5.33] demonstrated the effectiveness of their line

EHL model by applying them to the methodology of Xu et al. [5.23] to predict helical

gear efficiency.

The hypoid gear efficiency model that will be developed in this chapter improves

the methodology of Xu and Kahraman [5.32] by (i) employing the gear geometry defined

in Chapter 2 to derive contact curvature and surface velocity values at each contact point

instead of relying on any particular commercial FE package, (ii) employing the loaded

tooth contact model of proposed in Chapter 3 for computation of the normal load for

minimizing the computational time required for this task, and (iii) by incorporating a new

µ formula derived by using the mixed EHL model of Li and Kahraman [5.32] such that

any degree of asperity interactions can be modeled.

 

5.2. Hypoid Gear Mechanical Power Loss Model

Figure 5.1 shows the flowchart of the methodology used to compute the power

losses of hypoid gear pairs in the proposed model. It combines the developed ease-off

based hypoid gear contact model of Chapters 2 and 3 and the EHL-based friction

coefficient model of Li and Kahraman [5.33] to compute mechanical efficiency. Blank

dimensions, machine settings, cutter geometry, misalignments, load and speed are input

data for the gear contact model.

138

 

Figure 5.1: Flowchart of overall hypoid gear efficiency computation methodology.

Instantaneous Mechanical Efficiency

, , ,eq r sR V V f

,

No

Blank dimensions, Machine settings, Cutter geometry,

Misalignments, Load and Speed

Hypoid Gear Contact Model

Friction Coefficient Model

Lubricant property, Temperature and Surface

roughness

sk n  

Overall Mechanical Efficiency

Yes

139

 

At each time step k with pinion roll angle kq q k q ( [1, ]sk n ), there are

multiple potential contact lines between gear and pinion surfaces of adjacent tooth pairs

(depending on the contact ratio and the pinion roll angle). Here, sn is total number of

time steps per one pinion pitch 2a pq N , pN is the number of teeth of the pinion and

a sq q n is pinion roll angle increment. At any incremental position k, there are cln

(integer) number of potential contact lines, each divided into cpn number of contact

segments. With this, a total of cl cpn n n number of potential contact segments are

defined at each increment k . Each line segment k ( [1, ]sk n and [1, ]n ) has a

length kL and carries a constant load per unit length kf . The contact at the same

segment has an equivalent radius of curvature ( )keqR between contacting surfaces as well

as constant sliding velocity ( )ksV and rolling velocity ( )k

rV . Here, ( )krV and ( )k

eqR are

computed in a direction perpendicular to the potential contact line they belong to.

At each line segment k , contacting surfaces are approximated as two cylinders in

combined rolling and sliding. Friction at each segment has two components of sliding and

rolling. The friction due to sliding is because of relative sliding between contacting

surfaces and friction coefficient k is defined as the ratio of the tangential force

produced due to sliding to the normal force applied between contacting surfaces.

140

 

The friction caused by rolling is because of resistance of contacting surfaces

against rolling over each other [5.23] and empirical rolling friction coefficient of

Goksem [5.34] is used to predict friction force ( )krF . In addition to the lubricant

properties and the surface roughness amplitude S , other contact parameters at each

contact segment, namely kL , kf , ( )ksV , ( )k

rV and ( )keqR must be defined to determine

the friction coefficient k and the rolling loss ( ) ( )k k kr rF V . Distributions of k

and k are used to computed the instantaneous mechanical power loss kP at increment k

due to sliding and rolling that are averaged over k to compute the average gear mesh

power loss as 1sn k

kP P .

 

5.2.1. Definition of the Sliding and Rolling Velocities

The load kf at each line segment k are computed using the hypoid gear contact

model proposed in Chapter 2 and 3. The computation of the rolling and sliding

velocities, ( )ksV and ( )k

rV , require a kinematic analysis beyond what is provided in

Chapter 2. As shown in Figure 5.2(a) for any point on the ease-off surface with position

vector ijM , the surface velocities of the pinion and gear are defined as:

( )p pp pij ij v a r ( )g gg g

ij ij v a r ( [1, ]i m , [1, ]j n ) (5.1)

141

 

 

 

Figure 5.2: Sliding and rolling velocities and their projection in tangential plane along and normal to the contact line direction.

 

a p

a g

ad

ijgr

ijpr

gijv

ijpvijn

t

t piju

ijgu

 

ijM

(a)

(b)

 

ijpu

ijgu

( ) , ( )p gt tij iju u

( )ptij u

( )gtij u

tt

142

 

where p gR . The surface velocity hijv of gear h ( ,h p g ) at this contact point has

two components: hijw in the common normal direction ijn and h

iju in the tangential plane

. Noting

p gij ijw w , (5.2)

( ) ( ) 0p g p gij ijij ij ij ij v v n u u n , (5.3)

it can be stated that hiju ( ,h p g ) lies in the tangential plane and has two components,

one component ( )hij tu along the instantaneous potential contact line direction t and

another component ( )hij tu perpendicular to t in direction t as shown in Figure 5.2(a).

With this, sliding velocities at the same contact point along t and t are given,

respectively, as

( ) ( ) ( )p gsij t t tij ijv u u , (5.4a)

( ) ( ) ( )p gsij t t tij ijv u u . (5.4b)

Hence, the total sliding velocity at the same point is

2 2( ) ( )s s s

ij ij t ij tv v v . (5.5)

143

 

The rolling velocity along t is defined as

( ) ( )( )

2

p gt tij ijr

ij t

u uv

. (5.6)

Sliding ( )ksV and rolling ( )k

rV velocities at the contact segment k ( [1, ]sk n

and [1, ]n ) are found through weighted averaging of the values at four corners of

quadrilateral grid cell on the ease-off surface that contains the middle point of this

segment. These two velocities will be computed for each contact segment and will be

used to determine the local friction coefficient.

 

5.2.2. Friction Coefficient Model

The surface shear traction consists of the viscous shear within the fluid regions

and the asperity traction in the regions of metal-to-metal contact. Considering a one-

dimensional flow, the sliding viscous shear stress acting on the contact segment k is

defined as

( )( , )

( , )

ksV

x th x t

(5.7)

144

 

where ( , )h x t is the instantaneous film thickness distribution and is the effective

viscosity. Parameter x defines the coordinate of a point within the contact in the direction

of sliding. Within the asperity contact regions, the shear stress is defined as

( , ) ( , )dx t p x t (5.8)

where ( , )p x t is the instantaneous pressure distribution and d is the dry contact friction

coefficient. With these, the average friction coefficient at a contact line segment k

( [1, ]sk n and [1, ]n ) is given as

1

( , )1

e

ts

x

nNxk

knt

x t dx

N f

(5.9)

here sx and ex are the start and end points of lubricated contact in the direction of

sliding, tN is the number of time steps at which the lubrication analysis performed.

The rolling traction formula of Goksem [5.34] is used here to find the rolling

traction in its corrected form for thermal effects [5.35]

0.658 0.01264.318( ) ( )( )

keqk

rpv

GU W RF

. (5.10)

145

 

Here ( )k keq eqW f E R , 0( ) ( )k k

r eq eqU V E R , pv eqG E , pv is the

pressure-viscosity coefficient for the lubricant used, is thermal correction factor [5.31],

eqE equivalent module of elasticity of two contacting bodies and 0 is viscosity at

ambient conditions. With this, rolling power loss at each contact segment k is

( ) ( )k k kr rF V .

 

5.2.3. Derivation of a Friction Coefficient Formula

In Eq. (5.7) to (5.9), the distribution of the surface shear ( ( , )x t ), normal pressure

( ( , )p x t ) and film thickness ( ( , )h x t ) of every contact segment k must be predicted by

using a mixed elastohydrodynamic lubrication (MEHL) model. Considering that there

are n contact line segments at each rotational increment k and there are a total of sn

increments, a total of sn n number of MEHL analyses are required to find the

distribution of the friction coefficient and the resultant gear mesh power loss. This would

require several hours of CPU time, hampering the usefulness of the model.

In accordance with the methodology first proposed by Xu et al [5.23], an upfront

detailed parametric study of MEHL conditions of hypoid gear contacts will be performed

here by including all key contact parameters. These parameters and their selected ranges

146

 

and incremental values are listed in Table 5.1. These ranges of Hertzian pressure hp

(0.5 to 2.5 GPa), radius of curvature eqR (5 to 40 mm), rV (1 to 20 m/s), slide-to-roll

ratio SR (0 to 1), and viscosity of a typical axle fluid (75W90) within a temperature

range (25 to 100 C ) cover most of the contact conditions present in automotive hypoid

gear pairs. In addition, a typical measured roughness profile from a gear surface with

0.5μmqR is considered and different RMS roughness profiles with different

amplitudes are obtained from this baseline profile by multiplying it by a constant. The

mechanical properties of 75W90 gear oil are listed in Table 5.2 is used.

A total of 31,500 contact conditions as a result of all combinations of the

parameter values listed in Table 5.1 were analyzed by using the model of Li et al [5.31]

and a regression analysis to the µ values predicted for each or these conditions to obtain

the following friction coefficient formula:

For 1 :

4 5

8 96 7 10

ln ln20 1 2 3

ln ln

exp

,

c

c

a G a S

a a Sa G a a Heq

a H a SR a G a SR

P S

(5.11a)

147

 

_____________________________________________________________

Lubricant 75W90 gear oil

_____________________________________________________________

Hertzian Pressure hp (GPa) 0.5, 1, 1.5, 2, 2.5

Equivalent Radius of Curvature Req (mm) 5, 20, 40

Rolling Velocity Vr (m/s) 1, 5, 10, 15, 20

Slide to Roll Ratio SR 0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1

Inlet Lubricant Temperature oilT (°C) 25, 50, 75, 100

Surface 1 roughness RMS 1qR (µm) 0.1, 0.35, 0.6, 0.85, 1

Surface 2 roughness RMS 2qR (µm) 0.1, 0.35, 0.6, 0.85, 1

_____________________________________________________________

Table 5.1: Parametric design for the development of the friction coefficient formula.

148

 

________________________________________________________________________

Temperature Pressure-Viscosity Coefficient Dynamic Viscosity Density

oilT (°C) 1 (GPa-1) 0 (Pa.s) 0 (kg/m3)

________________________________________________________________________

25 18.0 0.1626 844.30

50 13.9 0.0499 829.30

75 11.4 0.0208 814.30

100 9.7 0.0106 799.30

________________________________________________________________________

Table 5.2: Basic parameters of the 75W90 gear oil used in this study.

149

 

For 1 3 :

2 3 4 5

7 8 9 106

ln ln ln0 1

ln ln

exp ( )

,c

b U b G b H b H

b b H b H b Sb H

b b SR SR

U G H

(5.11b)

For 3 :

3 4 5 6 7

8 9 10 11

ln ln ln ln0 1 2

ln

exp ( )

.

cc G c H c H c c S

c G c H c G c SReq

c c G c SR U SR

U H S

(5.11c)

Here is the lambda ratio (ratio of the smooth condition minimum film thickness to the

RMS surface roughness value). These formulae are dependent on a number of

dimensionless parameters: , SR , U , G , h eqH p E , two roughness parameters

c c eqS S R and eq eq eqS S R (1 2

2 2c q qS R R and

1 2 1 2( )eq q q q qS R R R R ). Here,

coefficients 0a to 10a , 0b to 10b and 0c to 11c are constants representative of the

lubricant considered. For the 75W90 gear oil, these parameters are listed in Table 5.3.

150

 

___________________________________________________________________

i ia ib ic

___________________________________________________________________

1 -0.62538 16.1512 5.0435

2 -51.411 -0.60156 -0.00576307

3 -0.0371532 -0.0466305 248228480

4 2.06770 -0.348239 -0.396002

5 -0.031750 -0.358514 -0.405254

6 -0.046276 22.568 35.618

7 -7.754e-5 -11.2295 -0.109382

8 1.18821 -6.49095 -0.112364

9 0.170432 -1.31986 -0.00016056

10 -0.136892 -0.881279 -0.0690238

11 -7.8882 -4398.1 -0.00038810

12 0.052821

___________________________________________________________________

 

Table 5.3: Values of the coefficients in Eq. 5.11.

151

 

5.2.4. Computation of the Mechanical Power Loss of the Hypoid Gear Pair

With the sliding friction coefficient ( k ) and rolling loss ( k ) computed in the

previous section for every contact segment k , the mechanical power loss due to the

contact of k ( [1, ]sk n , [1, ]n ) is computed as

( )k k k k k ksP f L V , (5.12)

With this, the instantaneous gear pair power loss at a given rotational increment k

becomes

1

lnk kP P

(5.13a)

and the average mechanical power loss of the hypoid gear pair is found as

1

1 snk

ks

P Pn

. (5.13b)

Having input pinion torque pT and speed p and power loss kP loss at each rotational

increment k , the instantaneous mechanical gear pair efficiency is defined as

1k

kp p

PE

T

, (5.14a)

152

 

and overall the average gear mesh efficiency is estimated as

1p p

PE

T

. (5.14b)

 

5.3. Numerical Example

Two sets of face-hobbed hypoid gear designs defined in Table 5.4 are borrowed

from automotive applications to study the influence of shaft offset along with working

conditions that affect on hypoid gear efficiency. Designs A and B have two levels of

offset ratios of 0.07a ad D and 0.14, respectively, while the other geometry and

operating parameters (such as number of teeth, shaft angle and gear pitch diameter aD )

of the two designs are kept the same or very close to each other to isolate the offset

influences from those of the other parameters ( ad is pinion shaft offset). For this

purpose, similar ease-off topographies as shown in Figure 5.3 are developed for the two

gear sets. Despite all geometrical differences of two designs, they have similar contact

pressure distribution as shown in Figure 5.4 for pinion torque of 500pT Nm. This is

mainly due to matching ease-off topographies of two design sets through machine setting

changes.

153

 

_____________________________________________________________________

Set A Set B

Parameter Pinion Gear Pinion Gear

_____________________________________________________________________

Number of teeth 12 41 12 41

Hand of Spiral Left Right Left Right

Mean spiral angle (deg) 40.0 31.3 47.0 29.6

Shaft angle (deg) 90 90

Shaft offset (mm) 15.0 30.0

Gear pitch diameter (mm) 220.0 220.0

Generation type Generate Formate Generate Formate

Cutting Method FH FH

_____________________________________________________________________

 

 

 

Table 5.4: Basic drive side geometry and working parameters of the examples hypoid

gear pairs.

154

 

Figure 5.3: Ease-off topography of (a) Design A with / 0.07a ad D and (b) Design B

with / 0.14a ad D .

0

5

(a)

Toe Root

µm 

0

50

100

(b)

Toe Root

µm 

0

50

100

155

 

Figure 5.4: Maximum contact pressure distribution of (a) Design A with / 0.07a ad D

and (b) Design B with / 0.14a ad D for 500 NmpT .

Mpa 

200

400

600

800

1000(b)

Toe Root

200

400

600

800

1000

Mpa  (a)

Toe Root

156

 

At 1500p rpm, rV and sV of the two gear sets are compared in Figures 5.5

and 5.6, respectively, for these two designs. It is seen that design B with higher offset

ratio ( 0.14a ad D ) has higher rV and sV compared to design A. Likewise, the slide-

to-roll ratio (SR) values over the tooth face of gear set B is higher than that of gear set A,

as shown in Figure 5.7. Additionally, the eqR and distributions shown in Figures 5.8

and 5.9 reveal slight differences in eqR values observed between the two gear designs,

the values for the gear set B are larger than those for the design set A. With all the

parameters required by Eq. (5.11) computed, the distributions along the tooth faces can

be determined, as shown in Figure 5.10. Although gear set B has higher ratio

compared to design A, its higher SR levels causes higher values, resulting in higher

sliding friction force.

Next, the influences of operating and surface conditions, including load pT ,

speed p , oil temperature oilT and surface roughness S , on mechanical power loss P

and mechanical efficiency E of these gear pairs are quantified. In Figures 5.11(a1, a2)

and (b1, b2), variation of P and E with p and pT are shown. It is observed in Figure

5.11(a2, b2) that the E decreases slightly with increasing p when the speed is low.

This trend is reversed at higher speed ranges as the slope between E and p becomes

positive. Similar conclusions can be drawn for the influence of pT .

157

 

Figure 5.5: Rolling velocity distribution of (a) Design A with / 0.07a ad D and (b)

Design B with / 0.14a ad D at 1500 rpmp .

 

(a)

Toe Root

1.0 1.5 2.00.5

2.5 3.0 3.5

(b)

Toe Root

1.0

2.52.0

1.5

3.53.0 4.54.0

158

 

 

 

 

Figure 5.6: Sliding velocity distribution of (a) Design A with / 0.07a ad D

and (b) Design B with / 0.14a ad D at 1500 rpmp .

(a)

Toe Root

1.0 1.5

2.0

(b)

Toe Root

1.5 2.02.5

3.0

159

 

Figure 5.7: Slide-to-roll ratio distribution of (a) Design A with / 0.07a ad D and (b)

Design B with / 0.14a ad D at 1500 rpmp .

0

0

0

0

0

0

0

0

0.10.20.30.40.50.60.70.8(a)

Toe Root

0.10.20.30.40.50.60.70.8(b)

Toe Root

160

 

Figure 5.8: Equivalent radius of curvature distribution of (a) Design A with

/ 0.07a ad D and (b) Design B with / 0.14a ad D .

(a)

Toe Root

35 40 4550

(b)

Toe Root

35 40 45

50 55

161

 

Figure 5.9: distribution of (a) Design A with / 0.07a ad D and (b) Design B with

/ 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT and 1 2 0.8 mS S .

0.10

0.15

0.20

0.25

0.30(a)

Toe Root

0

0

0

0

0.10

0.15

0.20

0.25

0.30(b)

Toe Root

162

 

Figure 5.10: Friction coefficient distribution of (a) Design A with / 0.07a ad D and

(b) Design B with / 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT and

1 2 0.8 mS S .

0.01

0.02

0.03

0.04

0.05

0.06 (b)

Toe Root

0.01

0.02

0.03

0.04

0.05

0.06(a)

Toe Root

163

 

Figure 5.11: Power loss and efficiency of Design A (a1, a2) and Design B (b1, b2) at

90 CoilT and 1 2 0.8 mS S .

1

10

100

1,000

10,000

0 500 1000 1500 2000 2500 3000

100 Nm 500 Nm 1000 Nm

p

(a1)

[w]P

1

10

100

1,000

10,000

0 500 1000 1500 2000 2500 3000

100 Nm 500 Nm 1000 Nm

(b1)

[w]P

p Continued

164

 

Figure 5.11 continued

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

0 500 1000 1500 2000 2500 3000

100 Nm 500 Nm 1000 Nm

[%]E

p

(a2)

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

0 500 1000 1500 2000 2500 3000

100 Nm 500 Nm 1000 Nm

(b2)

[%]E

p

Continued

165

 

In the lower speed range, where the power loss is relatively small, higher pT

results in higher efficiency since the increase in power loss due to the heavier load

applied is smaller than the corresponding input power increase, while the opposite is true

in the medium and high speed ranges. Between the two designs, gear set A with a

smaller shaft off-set is consistently more efficient (on average, gear set A has about 1.5%

higher efficiency than gear set B).

Figure 5.12 illustrates the effects of surface finish as well as the lubricant

temperature on the gear mesh efficiency. It is found that reduction in surface roughness

amplitude effectively increases the mechanical efficiency of both gear pairs. As for the

operating temperature, within the low roughness range, where a substantial amount of

contact area is separated by the hydrodynamic fluid film and the viscous shear dominates,

an increase in temperature will reduces the sliding friction and the power loss P

accordingly through the reduction in lubricant viscosity. In the medium and high

roughness ranges, when the contact zone might experience severe asperity contacts,

lower lubricant viscosity at high temperature results in thinner fluid film and larger

boundary friction force, reducing the gear mesh efficiency.

166

 

Figure 5.12: Efficiency of (a) Design A with / 0.07a ad D and (b) design B with

/ 0.14a ad D for different surface finish and oil temperatures at 1500 rpmp and

500 NmpT .

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

0.1 0.3 0.5 0.7 0.9 1.1 1.3

50° C 75° C 100° C

[%]Ef

S

oilT

(b)

96.0

96.5

97.0

97.5

98.0

98.5

99.0

99.5

0.1 0.3 0.5 0.7 0.9 1.1 1.3

50° C 75° C 100° C

[%]Ef

S

(a)

oilT

167

 

5.4. Conclusion

In this chapter, a new spiral bevel and hypoid gear mechanical efficiency model is

proposed for both face-milling and face-hobbing cutting methods. The proposed

efficiency model combines the computationally efficient contact model developed in

chapters 2 and 3 with an EHL based friction coefficient model developed by Li et al

[5.33] to estimate sliding friction loss and employed a conventionally developed

formulation for rolling loss. The developed model improved the methodology of Xu and

Kahraman [5.30] by (i) employing the gear geometry defined in Chapter 2 to derive

contact curvature and surface velocity values at each contact point instead of relying on

any particular commercial FE package, (ii) employing the loaded tooth contact model

proposed in Chapter 3 for computation of the normal load for minimizing the

computational time required for this task, and (iii) by incorporating a new µ formula

derived by using the mixed EHL model of Li et al [5.33] such that any degree of asperity

interactions can be modeled.

Limited numerical results show that the shaft off-set is critical to the efficiency of

the gear set as lower off-sets resulting in significant increases in mechanical efficiency.

Likewise, reduction in surface roughness was also shown to reduce power losses of

hypoid gear pairs.

 

168

 

References for Chapter 5

[5.1] Coleman, W., 1975, "Computing efficiency for bevel and hypoid gears." Machine

Design, 47, pp. 64-65.

[5.2] Seetharaman, S., and Kahraman, A., 2009, "Load-Independent Spin Power Losses

of a Spur Gear Pair: Model Formulation." ASME J. of Tribology, 131(2), pp.

022201.

[5.3] Seetharaman, S., Kahraman, A., Moorhead, M. D., and Petry-Johnson, T. T.,

2009, "Oil Churning Power Losses of a Gear Pair: Experiments and Model

Validation." ASME J. of Tribology, 131(2), pp. 022202.

[5.4] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute

of Technology.

[5.5] Denny, C. M., 1998, "Mesh Friction in Gearing." AGMA, Technical Paper No.

98FTM2.

[5.6] Pedrero, J. I., 1999, "Determination of The Efficiency of Cylindrical Gear Sets."

4th World Congress on Gearing and Power Transmission, Paris, France.

[5.7] Michlin, Y., and Myunster, V., 2002, "Determination of Power Losses in Gear

Transmissions with Rolling and Sliding Friction Incorporated." J. Mechanism and

Machine Theory, 37, pp. 167.

[5.8] Benedict, G. H., and Kelly, B. W., 1960, "Instantaneous Coefficients of Gear

Tooth Friction." ASLE.

[5.9] O’Donoghue, J. P., and Cameron, A., 1966, "Friction and Temperature in Rolling

Sliding Contacts." ASLE Transactions, 9, pp. 186-194.

169

 

[5.10] Drozdov, Y. N., and Gavrikov, Y. A., 1967, "Friction and Scoring Under The

Conditions of Simultaneous Rolling and Sliding of Bodies." Wear, pp. 291-302.

[5.11] Misharin, Y. A., 1958, "Influence of The Friction Condition on The Magnitude of

The Friction Coefficient in The Case of Rollers with Sliding." Int. Conference On

Gearing, London, UK.

[5.12] Heingartner, P., and Mba, D., 2003, "Determining Power Losses in The Helical

Gear Mesh; Case Study." Proceeding of DETC3, Chicago, Illinois, USA.

[5.13] Anderson, N. E., and Loewenthal, S. H., 1986, "Efficiency of Nonstandard and

High Contact Ratio Involute Spur Gears." J. Mechanisms, Transmissions and

Automation in Design, 108, pp. 119-126.

[5.14] Anderson, N. E., and Loewenthal, S. H., 1982, "Design of Spur Gears for

Improved Efficiency." J. Mechanical Design, 104, pp. 767-774.

[5.15] Barnes, J. P., 1997, "Non-Dimensional Characterization of Gear Geometry, Mesh

Loss and Windage." AGMA, Technical Paper No. 97FTM11.

[5.16] Vaishya, M., and Houser, D. R., 1999, "Modeling and Measurement of Sliding

Friction for Gear Analysis." AGMA, Technical Paper No. 99FTMS1.

[5.17] Martin, K. F., 1981, "The Efficiency of Involute Spur Gears." ASME J. Mech.

Des., 103, pp. 160-169.

[5.18] Dowson, D., and Higginson, G. R., 1964, "A Theory of Involute Gear

Lubrication." Institute of Petroleum, Gear Lubrication, Elsevier, London, UK.

170

 

[5.19] Simon, V., 1981, "Load Capacity and Efficiency of Spur Gears in Regard to

Thermo-End Lubrication." International Symposium on Gearing and Power

Transmissions, Tokyo, Japan.

[5.20] Simon, V., 2009, "Influence of machine tool setting parameters on EHD

lubrication in hypoid gears." J. Mechanism and Machine Theory, 44, pp. 923-

937.

[5.21] Larsson, R., 1997, "Transient non-Newtonian elastohydrodynamic lubrication

analysis of an involute spur gear." Wear, 207, pp. 67-73.

[5.22] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,

Elsevier Science B. V.

[5.23] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, "Prediction

of Mechanical Efficiency of Parallel-Axis Gear Pairs." ASME J. Mech. Des.,

129(1), pp. 58-68.

[5.24] Cioc, C., Cioc, S., Kahraman, A., and Keith, T., 2002, "A Non-Newtonian,

Thermal EHL Model of Contacts with Rough Surfaces." Tribology Transactions,

45, pp. 556-562

[5.25] Buckingham, E., 1949, Analytical Mechanics of Gears, McGraw-Hill.

[5.26] Simon, V., 1981, "Elastohydrodynamic Lubrication of Hypoid Gears." ASME J.

Mech. Des., 103, pp. 195-203.

[5.27] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction

Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University,

Columbus, Ohio.

171

 

[5.28] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced

Numerical Solution Inc., Hilliard, Ohio.

[5.29] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd

ed.), Cambridge University Press, Cambridge.

[5.30] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of

Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part

K: J. Multi-body Dynamics, 221(3), pp. 387-400.

[5.31] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric

Integrated Control Volume Discretization." Tribology International, Hiroshima,

Japan.

[5.32] Li, S., and Kahraman, A., 2009, "An Asymmetric Integrated Control Volume

Method for Transient Mixed Elastohydrodynamic Lubrication Analysis of

Heavily Loaded Point Contact Problems." Tribology International.

[5.33] Li, S., Vaidyanathan, A., Harianto, J., and Kahraman, A., 2009, "Influence od

Design Parameters and Micro-Geometry on Mechanical Power Losses of Helical

Gear Pairs." JSME International, Hiroshima, Japan.

[5.34] Goksem, P. G., and Hargreaves, R. A., 1978, "The effect of viscous shear heating

on both film thickness and rolling traction in an EHL line contact." J. Lubrication

Technology, 100, pp. 346–352.

[5.35] Wu, S., and Cheng, H., S., 1991, "A Friction Model of Partial-EHL Contacts and

its Application to Power Loss in Spur Gears." Tribology Transactions, 34(3), pp.

398-407.

172

 

CHAPTER 6

CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK

6.1. Thesis Summary

A computationally efficient load distribution model was proposed for both face-

milled and face-hobbed hypoid gears produced by Formate and generate processes.

Tooth surfaces were defined directly from the cutter parameters and machine settings.

This study defined and utilized a new surface of roll angle as an essential tool to simplify

the task of locating instantaneous contact lines of any general type of gearing in the

projection plane. First, the position vector and normal to one of the mating surfaces of

contacting members were computed, and the action surface and the surface of roll angle

were introduced by applying equation of meshing between any general axis arrangement.

Once the surface of roll angle was constructed, the instantaneous contact lines location

and orientation were computed through a novel approach inspired by analogy to parallel

axes gears. Gear surfaces were assumed conjugate only in computation of contact line

173

 

locations and orientation of the proposed approach. However it was shown that surface

of roll angle is very insensitive to any practical modifications of contacting surfaces, and

hence, real instantaneous contact lines practically remain identical to their conjugate

counterparts. For any other steps of contact analysis, theoretically generated surfaces

based on machine settings were employed.

Rayleigh-Ritz based shell models of teeth of the gear and pinion were developed

to define the tooth compliances due to bending and shear effects efficiently in a semi-

analytical manner. Base rotation and contact deformation effects were also included in

the compliance formulations. With this, loaded contact patterns and transmission error of

both face-milled and face-hobbed spiral bevel and hypoid gears were computed by

enforcing the compatibility and equilibrium conditions of the gear mesh. The proposed

model requires significantly less computational effort than finite elements (FE) based

models, making its use possible for extensive parameter sensitivity and design

optimization studies. Comparisons to the predictions of a FE hypoid gear contact model

were also provided to demonstrate the accuracy of the model under various load and

misalignment conditions.

Two applications of the proposed model were also introduced. First application

combined the proposed model with a newly introduced approach of modifying the ease-

off topography to investigate the effect of errors occurred in manufacturing and heat

treatment of gear surfaces or surface deviations due to wear or lapping. Manufacturing

errors typically cause real (measured) spiral bevel and hypoid gear surfaces to deviate

174

 

from the theoretical ones globally. Tooth surface wear patterns accumulated through the

life span of the gear set are typically local deviations that are aggravated especially in

case of edge contact conditions. An accurate and practical methodology based on the

developed ease-off topography approach was proposed in this study to perform loaded

tooth contact analysis of spiral bevel and hypoid gears having both types of local and

global deviations. Manufacturing errors and localized surface wear deviations were used

to update the theoretical ease-off and surface of roll angle to form a new ease-off surface

that was used to perform a loaded tooth contact analysis. Two numerical examples of

face-milled and face-hobbed hypoid gear sets with local and global deviated surfaces,

respectively, were analyzed to demonstrate the effectiveness of the proposed

methodology as well as quantifying the effect of such deviations on load distribution and

the loaded motion transmission error.

As another vital application of the proposed model, a hypoid gear mechanical

efficiency model was developed next for both face-milling and face-hobbing cutting

methods. The proposed efficiency model combined the computationally efficient contact

model and a mixed EHL based surface traction model to predict friction power losses.

The contact area, pressure distribution and rolling and sliding velocities were determined

employing the developed loaded tooth contact model. The EHL traction model

considered specific ranges of the key contact parameters, including Hertzian pressure,

contact radii, surface speeds, lubricant temperature and surface roughness amplitude of

hypoid type of gears, covering wide range of lubrication conditions from full film to

175

 

boundary regimes. The efficiency model for hypoid gear was applied to two face-hobbing

examples with similar overall dimensions, but different off-sets, to investigate the effects

of several working and design parameters including offset, load, speed, surface roughness

and lubricant temperature on mechanical efficiency.

6.2. Conclusion and Contributions

Computation of the contact pressure distributions is essential to every hypoid gear

analysis intended to predict required functional parameters of the hypoid gear pair,

including the transmission error, contact stresses, root bending stresses, fatigue life and

mechanical power losses. The hypoid gear literature lacked a model to compute the load

distribution accurately and efficiently without resorting to computationally demanding

FE methods. The main potential reasons for that was the absence of a general and

reliable formulation to define the geometry of FH and FM hypoid tooth surfaces from

cutter parameters, machine motions and settings. This void, combined with the

numerical difficulties in matching the tooth surfaces using the conventional methods and

lack of a semi-analytical tooth compliance formulation for hypoid gears, has hampered in

design, analysis and optimization of hypoid gears. This research study fills some of this

void.

The model proposed in this study to simulate the contacts of FM and FH hypoid

gear pairs under both unladed and loaded condition provides major enhancements to the

current state of hypoid analysis. Specifically:

176

 

(i) Methodology that simulates the FM and FH processes to define surface

geometries of hypoid gears including the coordinates, normal vectors and radii of

curvatures is accurate and computationally efficient. It does not employ

simplifying assumptions such as conjugacy of the tooth surfaces.

(ii) The method using ease-off topography to compute the unladed contact conditions

is novel. It is superior to the conventional method in various aspects, including its

numerical stability and computational efficiency. This method and its surface of

roll angle concept is also general such that it can be applied to the other gear

types.

(iii) Application of mounting errors and inclusion of global and local deviations are

rather straightforward with the proposed model while these have typically been

difficult or impossible tasks when the conventional methods were used.

(iv) The efficiency model that combines the loaded tooth contact model proposed in

this study with an accurate and computationally efficient friction model is

superior to any published hypoid gear efficiency model as it includes all key

geometry, surface, load and lubricant parameters as well as operating conditions.

Its ability to handle variety of lubrication conditions ranging from almost dry

contact to full film EHL makes this model applicable to wide ranges of hypoid

and spiral bevel gear applications from automotive and aerospace systems.

177

 

6.3. Recommendations for Future Work

The following items can be considered as the potential future studies to improve

or add to the model presented in this study.

(i) Extensive parameter studies can be performed to determine cost-effective ways of

improving hypoid gear efficiency and define efficiency guidelines to be used in

design of hypoid gears.

(ii) The loaded TCA formulation can be improved by enhancing the base rotation

formulation to account for all geometric complexities.

(iii) A shell model for non-circular cylinders can be developed to compute a more

accurate compliance matrix for a generated gear and the face-hobbed generation

method that have epicycloids tooth traces.

(iv) The proposed model lends itself to optimization studies to refine machine settings

for desired transmission error amplitudes and loaded tooth contact patterns.

(v) The way local deviation and mounting errors are included in the proposed model

is general such that this model can be used to predict surface wear as well as to

simulate the lapping process that is commonly used in manufacturing of FH

hypoid gears.

178

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