A Theoretical and Experimental Investigation on Bending Strength and Fatigue Life of Spiral Bevel and Hypoid Gears

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Mohammad A. Hotait

Graduate Program in Mechanical Engineering

The Ohio State University

2011

Dissertation Committee:

Professor Ahmet Kahraman, Advisor

Professor Gary L. Kinzel

Professor Dennis A. Guenther

Professor Anthony F. Luscher

Mohammad A. Hotait

2011

ABSTRACT

The tooth bending strength characteristics of spiral bevel and hypoid gears are investigated in this study both experimentally and theoretically, focusing specifically on the impact of gear alignment errors. On the experimental side, a new experimental set-up is developed for operating a hypoid gear pair under typical load conditions in the presence of tightly-controlled magnitudes of gear misalignments. The test set-up allows application of all four types of misalignments, namely the shaft offset error (V), the horizontal pinion position error

(H), the horizontal gear position error (G) and the shaft angle error (γ). An example face-hobbed hypoid gear pair from an automotive axle unit is instrumented with a set of strain gauges mounted at various root locations of multiple teeth and incorporated with digital signal acquisition and analysis system for collection and analysis of strain signals simultaneously. A number of tests covering typical ranges of misalignments and input torque under both drive and coast conditions are performed to quantify the influence of misalignments on the root stress distributions along the face width.

On the theoretical side, the computational model developed in earlier by Kolivand and

Kahraman [31] is expanded to generate the root surfaces of spiral bevel and hypoid gears cut by using either face-milling or face-hobbing processes. A new formulation is proposed to define the gear blank and a numerically efficient cutting simulation methodology is developed to compute the root surfaces from the machine settings, the cutter geometry and the basic design parameters,

including both Formate and Generate motions. The generated surfaces are used to define

customized finite element models of N-tooth segments of the pinion and the gears via an

automated mesh generator. Toot contact loads predicted by a previous load distribution model of

Ref. [31] is converted to nodal forces based on the same shape function used to interpolate for

nodal displacements. A skyline solver is used to compute the nodal displacements and the

resultant stresses at the Gauss points. An extrapolation matrix based on the least-square error formulation is applied to compute the stresses at the root surfaces. Predicted gear root stresses are shown to compare well with the measurements, including not only the extreme stress values but also the stress time histories. Through the same comparisons, the model is also shown to capture the impact of misalignments on the root stress distributions reasonably well.

At the end, a multiaxial, crack nucleation fatigue model of tooth bending is proposed; the model accounts for the multiaxial and non-proportional nature of the stress states predicted.

Fatigue lives predicted by the proposed model are compared to those estimated by using a

conventional uniaxial failure criterion to show that the predicted multiaxial fatigue lives are

significantly lower. The fatigue model is also used to quantify the influence of the misalignments

as well as certain key cutting tool parameters on the bending fatigue life of the hypoid gear pair.

iii

DEDICATION

This dissertation is dedicated to my dear family.

ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Ahmet Kahraman for the opportunity, guidance, and support throughout my graduate studies and research at The Ohio State University. It has been a privilege working with Prof. Kahraman. I am grateful to Prof. Gary Kinzel, Prof. Dennis

Guenther, and Prof. Anthony Luscher for serving on my dissertation committee.

Special thanks to Dr. Mohsen Kolivand for all his help and support that paved the way for my research in spiral bevel and hypoid gears area. I am thankful to Mr. David Talbot for sharing his finite element solver, to Mr. Sam Shon for helping in the experiments, and to Mr.

Jonny Harianto for providing the software support of HAP and LDP. I would like also to extend my thanks to Mr. Prashant Sondkar for the healthy discussions and conversations related to gears and other aspects of life. To all my lab mates, thank you for the support and friendship throughout my work at the GearLab.

To my grandmother, thank you for all the love and prayers. I miss you. To my brothers, thank you for the moral support and encouragement. I hope that I showed the good example of the eldest brother. To my mom and dad, thank you for all the time and effort you devoted in making me the person I am now. I am very blessed to have both of you in my life. Thank you for all the trust, advice, support, and encouragement. Without you this work would not have been possible. Last, and certainly not least, I would like to thank my wife, Reem, for her patience and belief in me throughout this journey. Thank you for making it easier.

VITA

April 28, 1982 Born in Jeddah, Saudi Arabia The son of Adel Hotait & Nabila Houmani

June, 2000 Lebanese Baccalaureate in Mathematics with Distinction Rafic Hariri High School, Saida Lebanon

October, 2004 Bachelor of Science in Mechanical Engineering with High Distinction American University of Beirut, Lebanon

2005-2006 University Fellowship The Ohio State University, Columbus, OH

June 2007 Masters of Science in Mechanical Engineering The Ohio State University, Columbus, OH

Summer 2008 Gear Engineering Intern American Axle & Manufacturing, Detroit, MI

2006-2011 Research Assistant GearLab, The Ohio State University, Columbus, OH

PUBLICATIONS

1. Hotait, M. and Kahraman, A., 2011, “An Investigation of Root Stresses of Hypoid Gears with

Misalignments,” Journal of Mechanical Design, in review.

2. Hotait, M. and Kahraman, A., 2008, “Experiments on Root Stresses of Helical Gears with

Lead Crown and Misalignments,” ASME Journal of Mechanical Design, v130, i7, p5.

FIELDS OF STUDY

Major Field: Mechanical Engineering vi

TABLE OF CONTENTS

ABSTRACT ...... ii

DEDICATION ...... iv

ACKNOWLEDGMENTS ...... v

VITA ...... vi

PUBLICATIONS ...... vi

FIELDS OF STUDY...... vi

TABLE OF CONTENTS ...... vii

LIST OF TABLES ...... xi

LIST OF FIGURES ...... xii

NOMENCLATURE ...... xx

TOOTH NOMENCLATURE ...... xxv

CHAPTER 1

1.1 Background and Motivation ...... 1

1.2 Literature Review...... 4

1.2.1 Gear Root Stresses Prediction Models ...... 4

1.2.2 Influence of Misalignments on Gear Bending Stresses ...... 7

vii

1.2.3 Gear Bending Fatigue Models ...... 8

1.3 Scope and Objectives ...... 11

1.4 Dissertation Outline ...... 13

CHAPTER 2

2.1 Introduction ...... 14

2.2 Test Machine ...... 15

2.3 Application of Misalignements ...... 21

2.4 Gear Specimens and Strain Gauging ...... 30

2.5 Instrumentation and Data Acquisition System ...... 35

2.6 Test Procedure and Data Analysis ...... 38

2.6.1 Test Matrix ...... 38

2.6.2 Data Analysis ...... 40

2.7 Experimental Results ...... 44

2.7.1 Experimental Results for a Gear Pair with No Misalignments ...... 44

2.7.2 Effect of Misalignments on the Measured Root Strains ...... 57

2.8 Summary and Conclusions ...... 71

CHAPTER 3

3.1 Introduction ...... 77

viii

3.2 Definition of the Hypoid Tooth Root Surfaces ...... 79

3.2.1 Kinematics of Face-Milling and Face-Hobbing Processes ...... 81

3.2.2 Blade Geometry and Relative Motions ...... 84

3.2.3 Root Surface as Envelope of Family of Surfaces ...... 89

3.3 Finite Element Formulation ...... 91

3.3.1 Element Type and Stiffness Matrix ...... 93

3.3.2 Automated Mesh Generator ...... 97

3.3.3 Boundary Conditions and Application of Contact Loads ...... 102

3.3.4 Finite Element Solution ...... 109

3.4 Numerical Results ...... 112

3.4.1 Optimal Number of Teeth Included in the Model ...... 112

3.4.2 An Example Analysis ...... 115

3.5 Model Validation and Comparisons to Experiments ...... 124

3.6 Summary ...... 130

CHAPTER 4

4.1 Introduction ...... 135

4.2 A Uniaxial Tooth Bending Fatigue Model ...... 138

4.3 A Multiaxial Tooth Bending Fatigue Model for Hypoid Gears ...... 139

4.3.1 Characteristic Plane Based Multiaxial Fatigue Criterion ...... 141

4.3.2 Implementation of the Characteristic Plane Criterion ...... 143

4.4 An Example Tooth Bending Fatigue Life Simulation ...... 147

4.5 Impact of Misalignments on Tooth Bending Fatigue Lives of Hypoid Gears ...... 152

4.6 Influence of Blade Edge Radius on on Tooth Bending Fatigue Life of an Hypoid Gear .... 161

4.7 Summary ...... 169

CHAPTER 5

5.1 Summary ...... 170

5.2 Conclusions and Contributions ...... 172

5.3 Recommendations for Future Work ...... 173

LIST OF REFERENCES ...... 176

LIST OF TABLES

Table 2.1: The basic design parameters for the tested gear pair used in this study...... 33

Table 2.2: The test matrix executed in the experimental study...... 39

Table 3.1: The basic design parameters of the example FM gear pair used in this study...... 116

Table 4.1: Example misalignment configurations used in the tooth bending fatigue parametric

study...... 153

Table 4.2: The worst case life conditions for the effect of misalignment combinations...... 163

LIST OF FIGURES

Figure 2.1: Hypoid gear test machine...... 16

Figure 2.2: Top view schematic layout of the hypoid test machine showing its main components

[67]...... 18

Figure 2.3: Mounting details for the hypoid gear pair used in this study [67]...... 20

Figure 2.4: Definition of misalignments on the gear pair...... 22

Figure 2.5: Definition of misalignments on the test machine...... 24

Figure 2.6: (a) Calibrated V blocks, (b) calibrated H blocks, and (c) calibrated G blocks used to

introduce misalignment...... 25

Figure 2.7: The V error set-up [67]...... 27

Figure 2.8: The H error set-up [67]...... 28

Figure 2.9: The G error set-up [67]...... 29

Figure 2.10: The γ error set-up [67]...... 31

Figure 2.11: Hypoid test gear pair used in this study...... 32

Figure 2.12: (a) Strain gauges installed on the tested gear, and (b) a schematic showing the

labeling of the strain gauges...... 34

Figure 2.13: Data acquisition set-up...... 36

Figure 2.14: An example of all 15 measured strain time histories for the baseline no

misalignment condition at T = 400 Nm...... 41

Figure 2.15: A four-gear-rotation segment of the strain time history recorded by gauge C1 for the

= baseline no misalignment condition at T 400 Nm...... 42

xii

Figure 2.16: Cycle-to-cycle variation of the peak-to-peak values of the measured strain of gauge

= C1 for the baseline no misalignment condition at T 400 Nm...... 43 Figure 2.17: (a) Strain time histories measured by the five gauges on tooth #2 for the baseline no

misalignment condition at T = 400 Nm in the drive side loading condition; (b)

zoomed view of one loading cycle...... 45

Figure 2.18: (a) Strain time histories measured by the five gauges on tooth #2 for the baseline no

misalignment condition at T = 400 Nm in the coast side loading condition; (b)

zoomed view of one loading cycle...... 47

Figure 2.19: Strain time histories of gauge B on tooth #1 for the baseline no misalignment

= condition with the contributions of the rim deflections highlighted. T 400 Nm. . 48 Figure 2.20: Measured strain time histories at the same face width location on the three

= consecutive teeth for the baseline no misalignment condition at T 400 Nm for gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh...... 49

Figure 2.21: Strain time histories of gauge C on tooth #2 for the baseline no misalignment

condition at different torque values...... 51

Figure 2.22: Strain time histories of gauge At on tooth #2 for the baseline no misalignment

condition at different torque values...... 52

Figure 2.23: Strain time histories of gauge B on tooth #2 for the baseline no misalignment

condition at different torque values...... 53

Figure 2.24: Strain time histories of gauge D on tooth #2 for the baseline no misalignment

condition at different torque values...... 54

Figure 2.25: Strain time histories of gauge Eh on tooth #2 for the baseline no misalignment

condition at different torque values...... 55

xiii

Figure 2.26: (a) Maximum and (b) minimum strain experienced by gauge C for the baseline no-

error condition as a function of input torque...... 56

Figure 2.27: Theoretical directions of the contact pattern shifts along the (a) drive side and (b)

coast side gear flanks for a right-hand gear due to different positive misalignments.

...... 58

Figure 2.28: Effect of the V error on the measured strain time histories for the baseline no

misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e)

Eh...... 59

Figure 2.29: Effect of the H error on the measured strain time histories for the baseline no

misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e)

Eh...... 60

Figure 2.30: Effect of the G error on the strain time histories for the baseline no misalignment

condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh...... 61

Figure 2.31: Effect of the γ error on the strain time histories for the baseline no misalignment

condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh...... 62

Figure 2.32: Effect of the V error on the (a) peak-to-peak and (b) normalized peak-to-peak root

strain distributions at T = 400 Nm on tooth # 2 for the drive side...... 64

Figure 2.33: Unloaded contact patterns for the baseline no misalignment condition...... 65

Figure 2.34: Unloaded contact patterns for (a) V = +0.2 mm and (b) V = -0.2 mm...... 66

Figure 2.35: Effect of the V error on the (a) peak-to-peak and (b) normalized peak-to-peak root

strain distributions at T = 400 Nm for the coast side...... 68

Figure 2.36: Effect of the H error on the (a) peak-to-peak and (b) normalized peak-to-peak root

strain distributions at T = 400 Nm for the drive side...... 69

Figure 2.37: Unloaded contact patterns for (a) H = +0.2 mm and (b) H = -0.2 mm...... 70

xiv

Figure 2.38: Effect of the G error on the (a) peak-to-peak and (b) normalized peak-to-peak root

strain distributions at T = 400 Nm for the drive side...... 72

Figure 2.39: Unloaded contact patterns for (a) G = +0.3 mm and (b) G = -0.05 mm...... 73

Figure 2.40: Effect of the γ error on the (a) peak-to-peak and (b) normalized peak-to-peak root

strain distributions at T = 400 N-m for the drive side...... 74

Figure 2.41: Unloaded contact patterns for (a) γ = +0.2o and (b) γ = -0.1o...... 75

Figure 3.1 Overall computational methodology used for prediction of root stresses...... 78

Figure 3.2: Cradle-based hypoid generator [31]...... 80

Figure 3.3: (a) Face-milling and (b) face-hobbing cutting processes...... 82

Figure 3.4: (a) Position of the blade on cutter head, (b) 3D geometry of the blade, and (c) the

shape of the cutting edge [31]...... 85

where the rotation matrices Mx and Mz are defined as follows: ...... 86

Figure 3.5: Relative motions and the corresponding transformation matrices from the cutter to the

blank...... 87

Figure 3.6: Two-dimensional gear blank region defined for the purpose of surface generation. .. 90

Figure 3.7: (a) Pinion concave side surface, and (b) mating gear convex side surface...... 92

Figure 3.8: 20-noded element used by the finite element model...... 94

Figure 3.9: (a) Coordinate transformation and (b) control points on each gear slice used for the

mesh generation...... 99

Figure 3.10: (a) The mesh template and (b) the meshed slices as seen in the Cartesian

Coordinates...... 101

Figure 3.11: Six-tooth (a) gear and (b) pinion segment examples used in the FE model...... 103

Figure 3.12: Fixed boundary regions of the FE gear segment...... 104

Figure 3.13: The load intensity representation on a contact line along a tooth surface...... 106

Figure 3.14: The projection of a three-dimensional element into the pre-defined two-dimensional

blank...... 107

Figure 3.15: (a) Load intensities along contact lines [31], (b) the corresponding nodal loads on the

gear, and (c) the corresponding nodal loads on the pinion...... 108

Figure 3.16: Maximum principal stresses as a function of the Nmesh at different virtual gauges

placed along the face width at (a) an arbitrary profile position and (b) the profile

position corresponding to the maximum principal stress experienced by the gear pair

throughout one loading cycle...... 114

Figure 3.17: Contact pressure contour plots of the example FM gear pair at T = 300 Nm under no

misalignment condition for (a) the drive and (b) the coast sides...... 117

Figure 3.18: Maximum principal stress contour plots of the gear root at different mesh instances

for the drive side of the example FM gear pair at T = 300 Nm under no

misalignment condition...... 118

Figure 3.19: Stress contour plots of the gear root at different mesh instances for the coast side of

the example FM gear pair at T = 300 Nm under zero misalignment condition...... 120

Figure 3.20: Components of the stress at the location of maximum principal stress at the gear root

for the drive side of example FM gear pair at T = 300 Nm with no misalignment.121

Figure 3.21: Maximum and minimum principal stress time histories along with the corresponding

virtual gauge readings at the location of maximum principal stress at the gear root

for the drive side of the example FM gear pair at T = 300 Nm with no misalignment.

...... 122

Figure 3. 22: Maximum principal stress contour plots of the pinion root at different mesh

instances for the drive side of the example FM gear pair at T = 300 Nm with no

misalignment ...... 123

xvi

Figure 3.23: Maximum and minimum principal stress time histories along with the corresponding

virtual gauge readings at the location of maximum principal stress at the pinion root

for the drive side of example FM gear pair at T = 300 Nm under zero misalignment

condition...... 125

Figure 3 24: (a) Predicted and (b) measured stress time histories of gauges At, B, C, D, and Eh for

tooth #1 at T = 300 Nm and no misalignment condition...... 126

Figure 3.25: Predicted effect of (a) the V error and (b) the G error on the peak-to-peak gear

principal stress distributions at T = 300 Nm for the drive side...... 128

Figure 3.26: Comparison of the measured and predicted peak-to-peak stress distributions along

the face width at T = 300 Nm for (a) V= − 0.2 mm, (b) V= − 0.1 mm, (c) V0= ,

(d) V= 0.1 mm, and (e) V= 0.2 mm...... 129

Figure 3.27: Comparison of the measured and predicted peak-to-peak stress distributions along

the face width at T = 300 Nm for (a) H= − 0.2 mm, (b) H= − 0.1 mm, (c) H0= ,

(d) H= 0.1 mm, and (e) H= 0.2 mm...... 131

Figure 3.28: Comparison of the measured and predicted peak-to-peak stress distributions along

the face width at T = 300 Nm for (a) G= − 0.05 mm, (b) G0= , (c) G= 0.1 mm,

(d) G= 0.2 mm, and (e) G= 0.3 mm...... 132

Figure 4.1: The two phases of fatigue failure as a function of the applied stress amplitude...... 137

Figure 4.2: Predicted stress tensor time histories at three different gear root locations for

T =1000 Nm...... 140

Figure 4.3: Definition of the Euler transformations...... 145

Figure 4.4: Definition of the fracture plane and the characteristic plane...... 146

Figure 4.5: Computational methodology of the proposed multiaxial tooth bending fatigue model.

...... 148

xvii

Figure 4.6: (a) The contact pressure distribution, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm with no misalignment...... 150

Figure 4.7: Comparison of the crack initiation lives from the uniaxial and multiaxial models along

(a) the root profile in the middle of the face width and (b) the face width in the

middle of the root profile. T =1000 Nm with no misalignment...... 151

Figure 4.8: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm with V0= , H= 0.1 mm, G= 0.1 mm, and

γ=0 ...... 154

Figure 4.9: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm with V= − 0.1 mm, H0= , G= 0.15 mm, and

γ=0 ...... 155

Figure 4.10: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm with V= − 0.1 mm, H= − 0.05 mm, G0= , and

γ=−0.1o ...... 156

Figure 4.11: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm with V= 0.1 mm, H0= , G= − 0.1 mm, and

γ=0 ...... 157

xviii

Figure 4.12: Combined influence of (a) the V and H errors and (b) the V and G errors on the

predicted crack initiation bending fatigue lives of the example FM gear set at

T = 800 Nm...... 159

Figure 4.13: Combined influence of (a) the V and γ errors and (b) the H and G errors on the

predicted crack initiation bending fatigue lives of the example FM gear set at

T = 800 Nm...... 160

Figure 4.14: Combined influence of (a) the H and γ errors and (b) the G and γ errors on the

predicted crack initiation bending fatigue lives of the example FM gear set at

T = 800 Nm...... 162

Figure 4.15: (a) Cutting blade shape with different edge radii and (b) a zoom-in view of the edge

part...... 165

Figure 4.16: Predicted (a) loaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm for ρ=t 0.5 mm...... 166

Figure 4.17: Predicted (a) loaded contact pattern, (b) the root stress distribution, (c) the uniaxial

fatigue life distribution, and (d) the multiaxial fatigue life distribution of the

example FM gear at T =1000 Nm for ρ=t 1.70 mm...... 167

Figure 4.18: Normalized life versus ρt of the example FM gear at T =1000 Nm...... 168

xix

NOMENCLATURE

αb Blade angle

(,,)αβγuuu Euler angles

B Strain displacement matrix c Blade parameter (design)

D Material properties matrix d Blade parameter (design)

δb Blade offset angle

Eb Blank offset

E Modulus of elasticity

ε Strain

F Nodal force

Φ Shape function

φt Cutter phase angle

φc Cradle rotation angle

φw Work rotation angle

G The G (gear axial) error value

γ The γ (shaft angle) error value

γm Root angle

H The H (pinion axial) error value

xx i Face width discretized point it Tilt angle j Profile width discretized point js Swivel angle

J Jacobian matrix k Stiffness matrix

K Global stiffness matrix

λ Surface parameter (angle)

M Transformation matrix

Mctb Machine center to back

µb Hook angle

Nt Number of blade groups

Nw Number of teeth on the work (pinion or gear)

Nh Total number of nodes on slice

Nmesh Number of teeth included in the mesh

Nn Total number of nodes of the finite element mesh of gear segment

N f Number of cycles to fatigue failure (finite number)

ν Poison’s ratio

ω Angular velocity

P Load intensity vector q Nodal displacement vector qc Cradle angle (or cradle angle change, setting parameter) r Position vector xxi

rc Cutter radius

Rtw Number of blades to number of teeth ratio

Ra Ratio of roll

ρt Blade edge radius

(,)RZ Projected blank coordinate system

s Surface parameter

Sr Radial Setting

σ Normal stress

σ Extrapolated stress

σ f Bending fracture strength (usually fully reversible)

σ Fatigue limit for a fully reversible bending loading R f =−1

t Time

T Input (applied) torque

τ Shear stress

τ f Torsional fracture strength (usually fully reversible)

τ Fatigue limit for a fully reversible torsional loading R f =−1

τb Rake angle

u Displacement vector

V Volume

V The V (offset) error value

(,,)υθφ “Hypoidal” coordinate system

xxii

X B Sliding Base

(,xyz ,) Cartesian coordinate system

(,ξηρ ,) Natural coordinate system

Subscript a Amplitude b Blade c Cradle in Node jn Node ig Gauss point ir Root location kn Node l Load m Mean max Maximum value min Minimum value n Normal to the plane t Tool w Work

1 Maximum principal direction

3 Minimum principal direction

xxiii

Superscript

cp Characteristic plane

e Element

fp Fracture plane

h Mesh slice

u Euler angle position

xxiv

TOOTH NOMENCLATURE

Convex Side

Topland Concave Side (Top) Start of Active Profile (S.A.P.) Base

Heel

Toe Bottom

Root Curve (R.C.)

xxv

CHAPTER 1

INTRODUCTION

1.1 Background and Motivation

Spiral bevel and hypoid type cross-axis gears are highly engineered machine elements

that are critical components of many power train systems. They find their most common and

high-volume applications in automotive drive trains, heavy-duty and off-highway vehicle

transmissions as well as rotorcraft gearboxes and industrial power transmission systems. They

provide perhaps the only reliable way of transmitting power between two intersecting or non- intersecting shafts at a right angle from each other. Spiral bevel gears are used when the rotation axes of the shafts intersect while hypoid gears are designed for applications with non-intersecting shaft axes with a certain shaft offset. With their geometric and manufacturing complexity, hypoid gears can be considered to be the most general case of gearing with the other types of gears can be derived from the hypoid geometry by restricting specific degrees of freedom in the manufacturing surface generation process.

There are two main cutting methods for spiral bevel and hypoid gears, named as the face-milling (FM) and the face-hobbing (FH) processes. The preference of each depends mainly on the industrial application and the desired finishing process. The common finishing method for

the FM process is grinding, while lapping (an accelerated surface wear process) is employed for the FH process. In the U.S. automotive industry, for instance, the FH process has been adopted as the common cutting method for the last decade [1]. The advantage of the FH cutting in this case

is the continuous indexing, which is more suitable for high-volume applications, given its shorter

manufacturing time and lower cost.

Design of spiral bevel and hypoid gears, like their parallel-axis counterparts, is dictated

by various system-level requirements established for the power transmission system. These

requirements are set to achieve increased levels of durability and reliability, and power density

(power-to-weight ratio) while reducing cost, noise levels as well as power losses of the drive

train. Gear design concepts that can deliver functional attributes to meet these multiple, and often

conflicting, requirements cannot be achieved by using conventional design formulae. Advanced

computational models are often required to arrive at designs satisfying these stated requirements.

Among these functional gear requirements, reliability of the gear pair is especially

critical. Requirements such as noise and efficiency are rather irrelevant if the gear set fails to

endure the duty cycles it is designed for. Spiral bevel and hypoid gears failure modes are

numerous with several mechanisms leading to them. The most common gear failures are induced

by material fatigue, temperature, impact or wear [2]. Surface wear and temperature-induced

failures such as scuffing are associated with the lubrication of the gear contact as well as the

ability of the lubricant in removing the heat generated at the contact interfaces. Gear fatigue

failures can occur in two distinct forms. One type is the contact fatigue failure, resulting in

deterioration of the contacting tooth surfaces due to micro-pitting or spalling. This type of failure

is dependent on a number of lubrication, load, speed and surface related parameters. Other

common type of fatigue failure observed in gears is the tooth bending fatigue failure. This failure

2 is more catastrophic than the contact failures as a broken tooth halts the operation of the drive train immediately. Such failures are directly related to the stress state along the root or fillet regions of the gear teeth. The cyclic nature of these stresses, combined with the overall tooth shapes that result in a stress concentration zone are responsible for this failure mode.

Many parameters have been identified to impact tooth bending fatigue lives, including geometry of the gear teeth at their root region, the load distribution along the teeth, surface finish, material properties, residual stresses, and operating conditions (speed, alignment, dynamic loads)

[3]. Two main categories are distinguished here. The first category includes factors that directly alter the root stresses and consequently the bending strength while the second group influences these stresses indirectly by changing the load distributions along the gear teeth and the resulting contact patterns. The cutter blade edge profile that determines the root fillet shape is recognized as a direct contributor to the state of stresses seen at the hypoid gear root [1]. An appropriate example of the indirect effects is gear misalignments. They are defined as the deviations of the gears forming the pair from their ideal positions. This can be caused by many reasons such as errors associated with the manufacturing of the housing and shafts, and deflections under load.

There are significant qualitative differences in the state of root stresses of different types of gears. While spur gears are subject to loading conditions that cause a primarily uniaxial stress state, other types of gearing such as spiral bevel and hypoid gears are subject to more complex loading schemes and geometries to experience multiaxial stress states. This study focuses on investigation of bending durability of spiral bevel and hypoid gears. Development of tools to predict the root stresses and bending fatigue lives of these types of gears is the main emphasis of this dissertation research as well as the validation of these tools through tightly-controlled laboratory experiments. Incorporation of different design and manufacturing factors, such as

3 misalignments, cutter blade profiles, and cutting machine settings into the models is also another motivation of this research.

1.2 Literature Review

1.2.1 Gear Root Stresses Prediction Models

A first step towards predicting bending fatigue life of a gear component is to determine the stress state at the tooth root. An accurate prediction of the stresses in the fillet region is a relatively complex task involving many factors and parameters. In their relatively comprehensive literature review of the published models to predict root stresses, Kawalec et al [4] grouped these models in two main categories based on their methodologies:

Semi-analytical models: As in Lewis formula, these models rely on a simplified parameterized set of factors that are calculated by empirical means.

Numerical models: These methods use a computational approach based on a model of the gear teeth developed by using discretization methods such as the boundary element method

(BEM) and the finite element method (FEM).

Regardless of the methodologies, most of the published work focused on the stresses of the parallel-axis gears such as spur and helical gears [5-15]. Some of these studies considered a single tooth segment or a gear segment consisting of up to three teeth, held at its cut boundaries via artificial constraints. A finite elements (FE) model of this segment developed by using a conventional finite elements package would typically be loaded by an external force at a certain tooth location to predict the resultant stresses in the root region. These models had two main shortcomings. One had to do with definition of the boundary conditions to emulate an actual

whole gear. The other had to do with the way the tooth (or in some cases, multiple teeth) was

loaded. As the actual gear mesh load distribution along the multiple contact zones formed

between sets of teeth at the gear mesh were not known, the simplified point forces applied in

these models did not reflect the actual loading conditions. More sophisticated computational

models that can handle two gears in mesh with all teeth and all contact zones have been proposed

in recent years to avoid these difficulties. These models, such as the quasi-prismatic FE models

of Vijayakar [13, 14], can predict the contact stresses, tooth and gear rim deflections, as well as

the root stresses of a gear pair simultaneously, at the expense of significant computational

demand.

Another group of studies on spur and helical gears employed simplified compliance

matrix formulations derived by using simplified tooth shapes such as a tapered plate or coarse FE

models and simplex algorithms to predict the load distribution [19]. With the load distribution

available, separate BEM or FEM formulations were then used to predict the root strain with or

without the rim deflections [18]. These models were shown to be significantly faster than the

deformable-body models such as those of Vijayakar [13, 14], while they correlate reasonably well with experiments [20].

Studies on prediction of the bending stresses of spiral bevel and hypoid gears are rather sparse. Majority of these studies were specific to a problem under investigation due to the complexity of the geometry and the difficulty of computing the load distributions. Wilcox [21] used the flexibility matrix method in conjunction with FEM to calculate gear tooth stresses of spiral bevel and hypoid gears. Handschuh and Bibel [22] used a commercial FE package to perform a contact analysis of a multi-mesh spiral bevel gear system. The model predictions were suggested to follow the general trends as the experiments while some discrepancies were found in

5 the peak stress amplitudes. Vijayakar [13, 14] used a hybrid finite element and semi-analytical model to simulate the contact of spiral bevel and hypoid gears. This model was used later by

Piazza and Vimercati [23] to compare the root stresses with published experimental data by

Handschuh [22] for a FH, spiral bevel gear set. The comparisons showed good agreement in the root area with some inconsistencies in the fillet region, which were attributed to the potential difference between the modeled and actual geometries. Simon [24] developed a computational model for stress analysis of hypoid gears. In this model, instead of using a FE model directly, equations based on a regression analysis and interpolation functions were proposed to compute tooth compliance. Argyris et al [25] presented a computerized approach for design, synthesis and stress analysis of spiral bevel gear set. They introduced an automated finite element solver for contact simulations and stress analysis including bending at the root. This study, however, was limited to FM spiral bevel gear sets and no details were provided on the mesh generator and finite element formulations. In addition, their model was not validated. Vecchiato [26] applied the boundary element method to a FH gear sets and performed a full body contact analysis to compute contact and bending stresses. No validation was presented for either the load distribution or the bending stress predictions. Finally, Litvin et al [27] proposed a methodology for the design, simulation of meshing and contact, and stress analysis of optimized spiral bevel gear drives. They used a commercial FEM package to predict the root stresses while providing no comparisons to experiments.

Detailed mathematical models for defining the root surfaces of spiral bevel and hypoid root surfaces are very limited. Some studies [25, 28] focused on the FM spiral bevel gear sets without extending the work to FH or hypoid gear sets. As for the FH gears, besides the work of

Vecchiato [26], Fan [29] presented a computerized “universal” model for simulating the face- hobbing processes. However, no details on how to handle the discrepancies between the actual 6 and nominal tooth heights as well as the undercutting were provided, which are both very common in FH gear set. Vimercati [30] provided more details on these phenomena in his mathematical model for FH hypoid surfaces. In a recent study, Kolivand and Kahraman [31] followed the methodology used by Conry and Seireg [19] for spur and helical gears to develop a load distribution model of spiral bevel and hypoid gears. In this study, a computationally efficient load distribution model was proposed for both FM and FH hypoid gears with tooth surface defined directly from the cutter and machine setting parameters. Rayleigh-Ritz based shell models of teeth of the gear and pinion were used to determine the tooth compliances due to the bending and shear effects in a very computationally efficient way in comparison to the FE method. Base rotation and contact deformation effects were also included in their compliance formulations. This model was shown to agree well with the FE models of Vijayakar [13, 14] while it was at least two orders of magnitude faster. Root geometries were left out of the model of Kolivand and Kahraman [31] as the main focus was on the loaded contact pattern, the contact stresses, the motion transmission error and the mechanical power losses.

1.2.2 Influence of Misalignments on Gear Bending Stresses

Sensitivity of the performance of a gear pair to the alignment errors is a very critical topic. The contact patterns, the load distributions as well as the resultant root stresses are all altered by gear misalignments, as this was demonstrated both experimentally and theoretically by

Hotait and Kahraman [20] for helical gears. Impact of misalignments on the contact patterns and load distributions of spiral bevel and hypoid gears is even more critical [32]. For this reason, several of the previous FE based models [13, 14, 24] as well as the model of Kolivand and

Kahraman [31] included misalignments to investigate the resultant shift of contact zone along the

tooth surfaces, while the impact of misalignments on root stresses of spiral bevel or hypoid gears

and on the resultant fatigue lives is yet to be investigated.

1.2.3 Gear Bending Fatigue Models

Bending fatigue failures at the tooth root take place in two phases: crack initiation (or

nucleation) and crack propagation. Provided that the gear set is designed to have relatively low root stress levels, most of the fatigue life of the tooth is spent in the crack initiation phase. At such low stress levels, the material response remains within the elastic region, and hence, the stress-life (S-N) approach is deemed proper in most of the gear bending fatigue experimental investigations [33-36], in which the gear material performance is characterized by an S-N curve.

Oda et al [33] investigated the position of fatigue crack initiation and the direction of crack propagation for helical gears under different loading conditions. Tobe et al [34] generated several

S-N curves to study the effect of surface treatment and residual stresses on bending fatigue lives of carburized spur gears. Similar fatigue tests were performed by Masuyama et al [36] to explore the effect of surface defects and inclusions on the fatigue life of spur gears. An optimized tooth shape was proposed in this work to improve bending durability of such gears.

Another group of studies [37-40] incorporated cumulative damage theories into high- cycle or S-N based gear bending fatigue lives. The first example is the work of Anno et al [37] where the Miner’s rule was applied to tooth bending life data collected using a single tooth bending procedure. They concluded that Miner’s rule is clearly not valid. Singh [38] used both

Miner’s Linear Damage Rule (LDR) and Manson’s Double Linear Damage Rule (DLDR) in an experimental study of bending fatigue life of spur gears. He concluded that DLDR performed well in predicting the service life while LDR failed to compare with the experiments. Similar

problems with LDR were reported in the experimental studies by Hanummana [39] and Oda [40].

Ligata [41] recently employed DLDR in a stress-life bending fatigue model for planetary gears.

The variations of time histories due to unequal planet loads, rim deflections and dynamic effects

were also included in his model.

On the other hand, several studies adapted the strain-life approach for assessing the

fatigue lives of gears. Among them, Glodez et al [42] employed the strain-life based Coffin-

Manson relation to model the crack initiation phase of spur gears while using the linear elastic

fracture mechanics (LEFM) approach to predict the propagation life. A similar approach was

used by Kramberger et al [43] to estimate the bending fatigue lives of thin rimmed spur gears.

Podrug et al [44] used the critical plane approach proposed by Socie and Bannatine [45] to model

the crack initiation phase and applied a numerical method using the Paris equation for modeling the propagation stage. They considered the effect of moving the loads along the profile rather than fixing it at the highest point of single tooth contact. Their results were compared to

experiments to show good correlation. Ural et al [46] presented a study on prediction of the crack

shape and fatigue life for a spiral bevel pinion using a parallel finite element method. A

commercial code (FRANC3D) was used in conjunction with a FE based contact analysis model

for simulating the 3D crack growth. Moving contact loads were considered in this analysis and

predictions were reported to compare well with experiments.

Regardless of the approach adopted, stress-life or strain-life, the fatigue models listed

above considered a uniaxial stress state at the root. This assumption is valid, to a certain extent,

for spur gears subjected to no shaft misalignments. In this case, the loading is nearly two-

dimensional and the principal stress does not change direction. However, in the cases of other

types of gears such as helical, spiral bevel and hypoid, three-dimensional loading conditions and

9 the complex shape of gear teeth result in a multiaxial stress state in the fillet region. Use of a uniaxial fatigue criterion for such cases is not appropriate.

In general, predicting the fatigue life of structural components under multiaxial loads has been one of the challenging tasks in the fatigue field. Numerous high-cycle fatigue criteria have been proposed to include the influence of multiaxial conditions [47-64]. These criteria can be classified in three main groups: the critical plane approach, the stress invariants approach, and the element volume approach. The most common models are based on the critical plane approach where fatigue failure is assumed to take place along a specific plane on which both normal and shear components contribute to the fatigue failure [57]. As the critical plane approach requires the fracture plane to be defined first, it is compatible with crack propagation analyses using fracture mechanics. A comprehensive review of various critical plane criteria was published by

Karolczuk and Macha [58] to suggest that none of these models [49-64] has been accepted as a universal criterion for assessing fatigue failure of components under multiaxial loading conditions. I t is also noted that some of these models considered non-proportional loading [47,

49, 51, 52, 55, 56] while others were just limited to cases where the loading is proportional [48].

Models that use a multiaxial fatigue criterion for gear problems are few. Borgianni et al

[57] used Vidal’s [50] multiaxial fatigue criterion to predict fatigue life at the root of face gears.

A commercial FE package was used in this study to predict the stresses in the fillet region. The

Vidal’s criterion is based on non-proportional loading where the principal stress direction does not change in time. Davoli et al [65] applied two different multiaxial criteria (the Sines criterion and the Dan Van criterion) to contact fatigue assessment of spur gears with the effects of residual stresses and variation of fatigue material properties with depth included. Recently, Li [66] used the characteristic plane approach proposed by Liu and Mahadevan [52, 54] along with two other

10 critical plane approaches of Matake [48] and McDiarmid [49] to model the contact fatigue lives of rollers and gears. The predictions were compared to experimental data to show that the characteristic plane approach was more accurate in terms of prediction of crack initiation locations and number of life cycles. None of the above studies were extended to the bending fatigue life problem of spiral bevel and hypoid gears.

1.3 Scope and Objectives

As evident from the above review of the literature, while a significant amount of work was dedicated to tooth bending fatigue failures of parallel-axis gears, cross-axis gears are still in the shadow in terms of bending fatigue modeling. Root stress prediction models for hypoid gears are quite sparse and the experimental data available is patchy, failing to provide a sufficient validation of the prediction models. Detailed mathematical representations of the root surfaces of hypoid gears are also very limited. The impact of gear misalignments on root stresses and bending fatigue lives is not described quantitatively. Application of multiaxial fatigue criteria to spiral bevel and hypoid gears also requires further attention. Accordingly, this study aims at bridging some of these gaps through detailed theoretical and experimental investigations of root stresses of hypoid gears and through development of a multiaxial bending fatigue life modeling methodology. In line with these goals, the following can be listed as specific objectives of this study:

(i) Develop an experimental methodology to measure the quasi-static root strains of hypoid

gear pairs under tightly-controlled loading and mounting error conditions. Execute an

extensive test matrix that includes a range of loads representative of the real applications

and misalignment values for all different types of errors.

(ii) Extend the computerized mathematical model developed in [31] to generate the root

surfaces of both FM and FH spiral bevel and hypoid gear pairs.

(iii) Develop a computationally efficient and generic methodology to predict the root stresses

of spiral bevel and hypoid gears under quasi-static conditions. Validate the model by

using the data collected in (i).

(iv) Propose a tooth bending fatigue life model for spiral bevel and hypoid gears that includes

the multiaxial and non-proportional loading conditions inherent in these types of gears.

(v) Using the models, perform parameter sensitivity studies to determine the effect of

misalignments and other relevant manufacturing parameters such as the blade edge radius

on the root strains and bending lives of hypoid and spiral bevel gears.

It is noted here that spiral bevel and hypoid gears share a common set of manufacturing processes and geometric formulations. The only main difference is that hypoid gear axes do not intersect, but they are at a certain shaft offset. Therefore, a spiral bevel gear is nothing but a hypoid gear with a zero shaft offset. With this in mind, the rest of this thesis refers to hypoid gears only while all of the models apply to spiral bevel gears as well.

Both modeling and experimental tasks proposed above will be performed under low- speed (quasi-static) conditions as the changes in tooth forces due to dynamic behavior are beyond the scope of this study. While the methodology provided in this study can apply any cross-axis hypoid or spiral bevel gearing, the experiments and the parametric studies will be kept limited to right-angle applications that dominate the automotive axle systems.

1.4 Dissertation Outline

In Chapter 2, the experimental setup devised for the measurement of root strains of

hypoid gear pairs will be presented. The methods used to apply intentional alignment errors will

be described and the details of the test gear specimens and the instrumentation will be provided.

A test matrix that includes different types and amounts of misalignments will be defined and the measurements for some of these test conditions will be presented within a range of input torque for under both drive and coast operation conditions.

Chapter 3 is devoted to the development of the root stresses prediction model. The methodology developed to define the root surfaces of both face-milled and face-hobbed hypoid gears will be presented first through simulation of the cutting process defined by the blank dimensions, cutter geometry, and key machine settings. A finite element model of gear segments with an automatic mesh generator will be developed. An FM hypoid gear pair will be simulated as an example analysis. At the end, experimental results from Chapter 2 will be used to validate the root stress predictions of the proposed computational model.

In Chapter 4, a new gear bending fatigue life model based on a characteristic plane

multiaxial fatigue criterion will be proposed for hypoid gears. Effect of blade edge radius as well

as alignment errors on the root stresses and the resultant fatigue lives will be discussed.

Chapter 5 provides an extended summary of this study. It lists the major conclusions in

terms of modeling approach and accuracy and experimental techniques as well as the accuracy of

the predictions. A list of recommendations for future work is also included in this chapter in an

attempt to guide future work on this topic.

CHAPTER 2

EXPERIMENTAL INVESTIGATION OF ROOT STRAINS OF HYPOID GEAR PAIRS WITH MISALIGNMENTS

2.1 Introduction

A novel generalized experimental methodology was developed in this study for

quantifying the sensitivity of root strains of hypoid gears to input loads and gear mounting errors.

A new concept to introduce different types of misalignment errors in an accurate and repeatable

manner was developed and implemented. These misalignments include the position errors (V and

H) of the pinion along the horizontal and vertical directions, the position error (G) of the gear

along its axis as well as the angle error (γ) between the two gear axes. The experimental set-up

consists of a test machine, a strain-gauged hypoid gear pair, and a data acquisition system. The

test machine consists of a pinion spindle that is adjustable in V and H direction, a gear spindle

that is adjustable in the G direction, a high-torque and low-speed DC motor, and a load unit formed by a speed increaser belt drive and a precision magnetic particle brake. In addition, the pinion spindle can be rotated about the vertical axis passing through its cone apex to induce desired amounts of shaft error γ. An example face-hobbed hypoid gear pair from an automotive axle application was instrumented via a set of strain gauges positioned at the roots along the faces of multiple consecutive teeth for the purpose of this study. Instrumentation and data analysis 14

system to measure the root strains for a hypoid gear pair with or without the position errors were

developed and incorporated within the test machine. A test matrix that includes a range of pinion

torque values between 20 and 500 Nm and ranges of the V, H, G and γ errors was executed. The

test matrix included tests using both drive and coast sides of the gear pair. The main purpose of

this experimental study was to measure the influence of torque and misalignments on distribution

of root strains along the gear face direction. The measured data was processed to determine the

variation of the peak-to-peak root strains recorded with misalignments under various loads

applied on both drive and coast sides.

In this chapter, details about the hypoid test machine with its main components, as well as its capabilities are described first. The methods used to introduce different alignment errors are defined. The instrumented face-hobbed hypoid gear pair is described, followed by the method devised for the acquisition and post-processing of the data. Examples of measured root strains are presented to show the effect of different types of misalignments on the root strain distributions at roots of hypoid gear pairs. In addition, the corresponding unloaded contact patterns obtained under the same misalignment conditions are shown to better describe the measured behavior observed in terms of root strains.

2.2 Test Machine

A “four-axis” test machine shown in Fig. 2.1 was designed and procured for this study.

The same test machine was used jointly for a companion study for investigating the influence of

misalignments on the motion transmission error of hypoid gears [67]. While some of the details of the test set-up were described by Makam [67], a complete description will be provided here as well, focusing specifically on the root strain aspects.

Figure 2.1: Hypoid gear test machine.

The top-view layout of the machine depicted in Fig. 2.2 shows its main components.

They include two high-precision and rigidly-supported spindles that hold the pinion and the gear

in their desired error positions. The input (pinion) side spindle was connected to a torque motor

through a universal joint and a torque-meter. The drive motor was capable of producing high

torque values under very low speed conditions such that the measurement of the root strains

under quasi-static (no dynamic effects) and realistic torque conditions can be done effectively.

As the gear ratios of interest were in the order of 4:1 to represent a typical automotive axle speed

reduction ratio, any reasonable input torque level (say, up to 500 Nm) would produce very large

torque levels (up to 2000 Nm) at the output side, that would require a very large-capacity brake.

In the arrangement shown in Fig. 2.1 and 2.2, the load was applied on the output side (gear spindle) by a magnetic particle brake via a speed-increaser belt drive so that the higher torque values close to the real applications can be accommodated with a reasonable size brake. A precision torque sensor on the input side of the drive train measured the instantaneous input torque applied by the torque motor. The input (pinion) speed was measured from the signal produced by the input-side encoder while the output-side encoder shown in Fig. 2.1 and 2.2 was not needed for this study, but used for the measurement of the transmission error by Makam [67].

As the objective of this study was to measure the root strains under high loads, a DC torque-motor (Sierra Magnedyne, 491-10) was used here. This motor had seven pairs of brushes with 28 poles and 253. The magnetic particle brake (Placid Industries, PFB-400) utilized in this set-up was capable of applying loads up to 400 Nm. With the 1:5 ratio beslt drive speed increaser between the brake and the output spindle, this represented a maximum output spindle torque value of 2000 Nm that was sufficient for this application. This brake had a cycle-to-cycle accuracy of ±2% that was rated at the maximum load.

Belt Drive

Bearings Bearings

Coupling Coupling

Optical Encoder Magnetic Particle Brake Torque Motor 18

Output Spindle

Gear Flange

Gear

Pinion Pinion Input Optical Coupling Universal Torque Cartridge Spindle Encoder Joint Sensor

Figure 2.2: Top view schematic layout of the hypoid test machine showing its main components [67].

The torque and speed throughout the tests were controlled by a PLC controller designed

specifically for this set-up. A torque-meter (Lebow, 1228-10K) was used to measure the torque applied to the pinion and provided a feedback to the PLC unit. The sensor had a working range of

1130 Nm. The angular speed of the input spindle was measured by a high-precision optical encoder (Heidenhain RON88). While an encoder with a much lower resolution could also be sufficient in this study, the companion transmission error investigation of Makam [67] required such precise encoders at both sides.

As it will be described later, the example test hypoid gear pair was borrowed from an actual passenger vehicle axle unit. Hence, it was critical to mount both the pinion and the gear in the same way as they were held within the axle. Figure 2.3 shows a detailed cross-sectional view of the gear pair mounting employed in this study. Here, the gear is attached to the output spindle by a flange in exactly the same way as the gear was supported in the actual axle differential. The flange was custom-made to accommodate a multi-channel, end-of-shaft type slip ring used in acquiring the strain signal as will be discussed later in this chapter. The output spindle supported the ring gear only on one side through this flange. In actual application, the output shaft was supported by a pair of tapered roller bearings on both sides of the ring gear.

Therefore, the arrangement shown in Fig. 2.3 to hold the ring gear was a compromise, requiring that the output spindle to be as rigid as possible to represent the actual intended application. On the other side, a custom designed cartridge held the pinion with its tapered roller bearings in the exact same way as the actual axle application. This cartridge allowed the pinion to be pre-loaded axially as it is the case for the real application. In this arrangement, both the pinion and the gear were supported rather rigidly, ensuring minimal support deflections. With this, the misalignments were largely caused by the intentional errors induced through the mechanisms described later in this chapter. Likewise, in order to avoid any support deflections, the test

Pinion Pinion End of the Production End of the Gear Cartridge Cartridge Output Gear Pinion Bearings Input Spindle Flange Holder Spindle 20

Figure 2.3: Mounting details for the hypoid gear pair used in this study [67].

machine employed massive pedestals to hold the pinion and the gear in precise user-defined

positions. In addition, the overall test rig was supported by a thick steel base to prevent any

fixture deflections under loaded conditions.

2.3 Application of Misalignements

As mentioned in Section 2.1, the main purpose of this chapter is to investigate the

sensitivity of gear mounting errors (often called gear misalignments in the gearing community)

on the root strains of hypoid gear pair at different root locations in the face width direction. A

simple and repeatable method of imposing misalignments at user-defined levels was crucial to the

success of this study. Here, the components of the test machine were designed to move into pre-

defined fixed misalignment positions to shift the contact zone to different locations along the

tooth surfaces.

Four different of position errors are sufficient to specify the relative position of the pinion

and gear in mesh completely. These errors, as shown on the gear pair in Fig. 2.4, are referred to

as the V (or E), H (or P), G (or R), and γ (or A) errors. Using the notation of Fig. 2.4, they are defined as follows:

(i) The V Error: This error represents the deviation of the shaft offset from its nominal

value. The shaft offset is what differentiates the hypoid from the spiral bevel gear sets.

The offset allows a larger pinion size, smaller pinion tooth count, higher contact ratio,

and higher contact fatigue strength. However, the bigger the offset, the higher the sliding

and the less efficient gear pair [32]. An increase of the offset from the design value due to

manufacturing errors, mounting errors, or deflections is represented by a positive V error.

This error is also referred to in the literature as the E error.

+γ

Figure 2.4: Definition of misalignments on the gear pair.

(ii) The H Error: This error denotes the deviation of the pinion nominal position with

respect to the gear along the pinion axis. A positive H error is movement of the pinion

away from the gear axis as shown in Fig. 2.4. This error is also referred to in the

literature as the P error.

(iii) The G Error: As shown in Fig. 2.4, this error represents the deviation of the gear from its

nominal position along its axis of rotation. A positive G error results in increasing the

distance between the pinion and the gear, and hence, a higher backlash value. This error

is also called the R error.

(iv) The γ Error: Also referred to as the A error, this error represents the deviation of the

angle between the two shafts axes as shown in Fig. 2.4. A positive γ error represents a

shaft angle larger than its nominal value.

Figure 2.5 shows how these four position errors are defined on the hypoid gear test

machine used in this study. Here, predefined amounts of these intentional errors were applied through sets of calibrated gauge blocks, as depicted in Fig. 2.6. Different blocks were placed between the movable components and the foundation, depending on the desired type and amount of the error. By varying the thickness of these blocks, the V, H and G errors of different amounts were obtained. Similarly, the table holding the pinion spindle was pivoted to the base plate at the cone center of the gear pair. This allowed the pinion axis to be rotated to other angles about the perfect 90 position to introduce certain values of shaft error γ , also illustrated in Fig. 2.5. The adjustment of the vertical position of the pinion relative to the gear (i.e. shaft offset error) required that the pinion spindle height to be altered. By clamping V gauge blocks of

G H γ gauge V

G gauge H gauge V gauge

Figure 2.5: Definition of misalignments on the test machine.

(a) (b) (c) 25

Figure 2.6: (a) Calibrated V blocks, (b) calibrated H blocks, and (c) calibrated G blocks used to introduce misalignment.

different thicknesses between the table of the pinion spindle and the base of the machine at three different leg locations, the V error represented by the thickness of the gauge blocks would be implemented as shown in Fig. 2.7. A hydraulic system lifted the table of the input spindle once the three locking bolts are loosened. After placing the desired set of V gauges, the table was lowered and locking bolts were tightened. In addition to a set of no-error V blocks, other calibrated gauge blocks at different thicknesses as shown in Fig. 2.6(a) were also procured, such that discrete error values of V=−− 0.2, 0.1, 0, 0.1 and 0.2 mm can be achieved incrementally.

The pinion spindle shown in Fig. 2.8 was allowed to slide horizontally about the rotation axis of the pinion, as depicted by the arrow placed on the pinion spindle. With the set of input spindle bolts clamping the spindle to its table loosened and the locking bolt shown in the figure removed, an ACME thread jack type positioning device was used to move the table away from the rigid stop mounted on the base. A calibrated H block of a desired thickness from Fig. 2.6(b) was then placed between the rigid stop and the pinion spindle assembly. The locking bolt was tightened as well as the set of spindle bolts to fix the pinion spindle (hence the pinion) in this horizontal position. Here, one H block of a specific thickness represented the nominal position of the pinion at H0= . By producing other blocks with thicknesses at ±0.1 and ±0.2 from this reference thickness, discrete H error values of H=−− 0.2, 0.1, 0, 0.1 and 0.2 mm were achieved.

Figure 2.9 illustrates the way the position of the gear was varied in the horizontal

direction to induce a G error of a given value in a manner similar to the H error. The table holding

the output spindle with the gear was allowed to slide in the direction of the rotation axis of the

gear, as marked by an arrow in Fig. 2.9. By loosening the set of bolts that fix the table on the base

and removing the locking bold on the side, the table was moved away from the rigid stop

mounted on the base. A G block of certain thickness from Fig. 2.6(c) was used to position the 26

V 27

Locking Rigid Base of the Calibrated Calibrated Bolt Leg Input Spindle V blocks V blocks

Figure 2.7: The V error set-up [67].

H 28

Positioning Rigid Locking Calibrated Input Spindle Locking Rigid Calibrated Device Stop Bolt H block Bolts Bolt Stop H block

Figure 2.8: The H error set-up [67].

G 29

Calibrated Rigid Stop Locking Positioning Output Rigid Stop Calibrated Locking G block Bolt Device Spindle Bolts G block Bolt

Figure 2.9: The G error set-up [67].

gear horizontally such that G0= . The other G blocks in this figure corresponded to the discrete

error values of G= − 0.05, 0.1, 0.2 and 0.3 mm.

Finally, the adjustment to deviate the angle between the two shafts from the nominal 90

position ( γ=0 ) required the pinion assembly to be rotated about its apex relative to the gear (or the machine base). The entire pinion spindle support structure consisting of the pinion spindle, the pinion spindle table and the three-legged machine base was pivoted at the point of projection

of the pinion cone apex onto the base as shown in Fig. 2.10. Once the bolts fixing this assembly

onto the base of the machine loosened, this assembly holding the pinion could be rotated freely

about its pivot. By matching a particular pair of dowel pinholes and placing the pin in that

location, a certain γ value was obtained. Five such dowel pin holes are seen in Fig. 2.10 for

γ=−0.2 , − 0.1  , 0 , 0.1  and 0.2 , corresponding to the shaft angles of 89.8 , 89.9 90 90.1

and 90.2 , respectively.

2.4 Gear Specimens and Strain Gauging

As mentioned previously, a mid-size, 10-41 tooth face-hobbed hypoid gear pair from an

automotive axle application was used in this study. Figure 2.11 shows the gear pair with a slip

ring attached to the gear flange. The basic design parameters of the gear pair are shown in Table

2.1. The pinion is generated and the gear is Formate cut. The nominal shaft offset is 35 mm with

the nominal angle between shafts held at 90o. In order to investigate the impact of misalignments

on the root strain amplitudes, the test gear was instrumented with a set of root strain gauges as

illustrated in Fig. 2.12(a). Referring to Fig. 2.12(b), the roots of three

Dowel Pin 31

Locking Bolt

Calibrated Error Holes

Figure 2.10: The γ error set-up [67].

Gear Slip Ring Pinion

Figure 2.11: Hypoid test gear pair used in this study.

Table 2.1: The basic design parameters for the tested gear pair used in this study.

Parameter Pinion Gear

Number of teeth 10 41

Hand of Spiral Left Right

Mean spiral angle (deg) 52 27.35

Shaft angle (deg) 90

Shaft offset (mm) 35

Outer cone distance (mm) 93.34 102.17

Generation type Generate Formate ®

Depthwise tooth taper FH

(a)

(b)

D C B At 3 2 1 Root Center Line

Figure 2.12: (a) Strain gauges installed on the tested gear, and (b) a schematic showing the labeling of the strain gauges.

consecutive teeth (called tooth #1, #2 ad #3) were strain gauged to (i) capture the instantaneous

tooth-to-tooth load distribution as the contact ratio of the gear set is about three, and (ii) check for the tooth-to-tooth repeatability of the strain signals. Minor variations in the location of the

mounted gauges along the root, especially in the profile direction, were quite likely, resulting in

potential variations in the measured strain signals. Having three consecutive teeth gauged

allowed one to check for consistency of the gauge locations. It also allowed some averaging of

the signals from the three teeth to be done, if desired. Finally, it provided certain amount of

redundancy in the event of some of the gauges being damaged during the tests. As shown in Fig.

2.12(b), the root of each tooth was instrumented with 5 gauges along the face direction from heel

to toe. Each gauge was mounted along the root center such that the selected gauges (Micro

Measurements, model EA-06-031EC-350) with 0.787 mm by 0.787 mm dimensions did not

interfere with the teeth of the pinion even under a zero-backlash (tight-mesh) condition. A given

set of gauges mounted along a given root at a spacing of 5 mm were named as Ah, B, C, D, and

Et as shown in Fig. 2.12(b) with “h” and “t” denoting heel and toe side gauges, respectively.

With three teeth instrumented, a total of 15 gauges were used in the experiments (Ah1, B1, C1,

D1, Et1, Ah2, …, C3, D3 and Et3).

2.5 Instrumentation and Data Acquisition System

A data acquisition system was set-up to capture the strain signals from all of the gauges

simultaneously. A 36-channel slip ring (Michigan Scientific, model SR36M) was installed on the

gear flange as shown previously in Fig. 2.11. This setup allowed reading the output of all 15

strain gauges concurrently and transferring these signals to the non-rotating frame where the rest

of the data acquisition system was installed, as illustrated in Fig.2.13. The strain gauge outputs

SR36M Slip Ring NI SCXI-1100 Chassis Michigan Scientific

NI SCXI-1520 NI SCXI-1314 Module Terminal Block

Strain Gauge Wheatstone Bridge 15 Strain Input Module (Quarter Bridge Configuration) Gauge Signals

DAQ PC

LabVIEW Program

NI-6052E DAQ Card

Figure 2.13: Data acquisition set-up.

were connected to a front-mounting terminal block (NI SCXI-1314) for the universal strain gauge input module (NI SCXI-1520). Each terminal block included a total of 8 quarter-bridge configured circuitry for a 350 Ω strain gauge resistor. A total of two terminal blocks were used here to be able to collect the data from all 15 strain gauges simultaneously.

The excitation voltage for each of the channels was provided by the SCXI-1520 input module. Here, each of the 350 Ω strain gauges instrumented on the test gear was excited by 10 volts and a gain factor of 100 applied to the output in order to increase the measurement resolution and improve on the signal-to-noise ratios. In order to capture the strain signals simultaneously, the module was operated in the multiplexing mode at a rate of 333,000 samples/sec. Finally, the output of each of the channels was multiplexed into the NI-6052E data acquisition card installed on a PC.

Here, the analog-to-digital converted data was read into a custom written LabVIEW virtual instrument program. The program provided the interface as well as control of the sampling conditions. Although the set-up was capable of sampling at higher rates, up to 333 kHz, the strains were sampled at 1 kHz since the input (pinion) speed was limited to 15 rpm for all the test conditions, corresponding to a mesh frequency of only 2.5 Hz. A software based low-pass filter of 40 Hz was applied to the output signals to eliminate any high-frequency noises. The hardware was configured for continuous sampling and the block size was set to 5000 samples.

Null compensation was applied to the strain signals when the gauges were far from the mesh and the gear was unloaded. This insured the elimination of any residual strain voltages in the system.

2.6 Test Procedure and Data Analysis

At the beginning of each test, the required type and amount of alignment error was

applied as discussed in Sect. 2.3. First, the gear was decoupled from the load and the gear pair

was rotated to a position where the instrumented teeth were away from the gear mesh interface.

Next, the LabVIEW program was initiated and the output signals were recorded for the purpose

of offsetting the null. The gear was then connected to the load and the desired torque and speed

were applied. After the gear pair reached its steady-state (constant speed and constant torque)

conditions, the LabVIEW program was triggered to start collecting the data for five complete

gear rotations.

2.6.1 Test Matrix

The test matrix used in this study is given in Table 2.2. It included tests with both drive

and coast sides loaded up to an input (pinion) torque level of T = 500 Nm. In addition to tests under no error conditions (i.e. VHG= = =γ= 0), the test matrix included tests with the other

error conditions of V= ± 0.1 and ±0.2 mm (with HG= =γ= 0), H= ± 0.1 and ±0.2 mm (with

VG= =γ= 0), G= − 0.05 , 0.1, 0.2, and 0.3 mm (with VH= =γ= 0), and γ=±0.1 and ±0.2

(with VHG0= = = ). With this, a total of 34 error conditions listed in Table 2.2 were considered. For the no error condition, a 20 Nm input torque increment for loads between 0 and

100 Nm and a 50 Nm increment for loads from 100 to 500 Nm were used. For the other misalignment conditions, an increment of 100 Nm was considered throughout. Consequently, the

database collected consisted of 186 individual tests.

Table 2.2: The test matrix executed in the experimental study.

______Error [mm or degrees] Loading Direction V H G γ ______Drive 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 0 0 0 0 -0.05 0 0 0 0.1 0 0 0 0.2 0 0 0 0.3 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 ______Coast 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 0 0 0 0 -0.05 0 0 0 0.1 0 0 0 0.2 0 0 0 0.3 0 0 0 0 -0.2 0 0 0 -0.1 0 0 0 0.1 0 0 0 0.2 ______

2.6.2 Data Analysis

Figure 2.14 shows an example recording of all the 15 strain measurements at an input

torque value of 400 Nm for the baseline case of no errors. In this example, the concave flank of

the pinion is the driving side. Here, data for one loading cycle (one complete gear revolution)

segment is shown for illustration purposes. The data from the first strain loading cycle was used

as a control signal and the other four were stored for further analysis. Figure 2.15 shows the

remaining four loading cycles recorded for gauge C1 at the same testing conditions as Fig. 2.14.

In addition to the actual strain signals, the maximum, minimum and peak-to-peak values of each

strain signal were also quantified to be used as metrics in determining the impact of

misalignments. Averaging was feasible because of the minimal cycle-to-cycle variability

examined throughout all the tests performed. Figure 2.16 depicts the cycle-to-cycle fluctuation

for the example gauge C1 at 5 different loads (100-500 N-m). As shown, the variability of the peak-to-peak strain from the average at each load is less than 5% in the worst case condition that is noticed to be at 100 N-m.

The following section of this chapter will present some of the raw and processed test results in order to empirically describe the behavior of hypoid root strains under different misalignment and loading conditions. Some of these measurements data will also be used in

Chapter 3 for validating the computational model devised to predict root stresses of face-hobbed and face-milled hypoid and spiral bevel gears.

1500

1000

500

-500 Strain

µ -1000

-1500

-2000

-2500

-3000 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.14: An example of all 15 measured strain time histories for the baseline no misalignment condition at T = 400 Nm.

1500

1000

500

-500 Strain µ -1000

-1500

-2000

-2500 0 1 2 3 4 Ouput Rotations

Figure 2.15: A four-gear-rotation segment of the strain time history recorded by gauge C1 for the = baseline no misalignment condition at T 400 Nm.

800 Cycle 1 700 Cycle 2 Cycle 3 600 Cycle 4

500

400 Strain µ 300

200

100

0 100 200 300 400 500 Torque [N-m]

Figure 2.16: Cycle-to-cycle variation of the peak-to-peak values of the measured strain of gauge = C1 for the baseline no misalignment condition at T 400 Nm.

2.7 Experimental Results

In this section, representative results of the experimental study performed according to

the test matrix of Table 2.2 are presented. Different outputs based on the actual measured strain

time histories and averaged maximum and minimum strain values are used to demonstrate the combined influence of load and alignment errors on the root strain values and distributions along the face width of a hypoid gear pair. First, the results for the baseline condition with no misalignments are presented. Next, the results for the cases of different errors (V, H, G, and γ) of different magnitudes are presented.

2.7.1 Experimental Results for a Gear Pair with No Misalignments

For the case of VHG= = =γ= 0, Fig. 2.17 shows the strain time histories for gauges

At, B, C, D, and Eh of tooth #2 at an input torque of 400 Nm. The gear is loaded in the drive direction that means the concave sides of the pinion and the convex sides of the gear are in contact. The plot in Fig. 2.17(a) shows one loading cycle or 41 mesh cycles (corresponding to one complete gear rotations as the number of the teeth on the gear is 41) while Fig. 2.17(b) is a zoomed in view of the mesh cycles when the gauged teeth are in the gear meshing zone. As expected, the gauges experience loads in a shifted sequence at different levels suggesting a non- uniform stress distribution along the root in the face width direction. Gauge Eh experiences the strain first, indicating that the contact is initiated at the heel of the gear teeth and it moves from the heel to the toe. The compressive part of the strain signal for all the gauges is larger than the tensile part, indicating that the gauges were mounted away from the critical section of maximum tensile stresses. Gauge D reads the highest tensile stress in this case while gauge C registers the 44

(a) (b) 1500 D 1000 Eh

500 At

0 B

-500 Strain µ -1000

-1500 C 45

-2000

-2500 0 5 10 15 20 25 30 35 41 18 19 20 21 22 23 Mesh Cycles Mesh Cycles

Figure 2.17: (a) Strain time histories measured by the five gauges on tooth #2 for the baseline no misalignment condition at T = 400 Nm in the drive side loading condition; (b) zoomed view of one loading cycle.

highest compressive stress value. Gauges near the center of the tooth read higher stress values

than those near the toe (At) and heel (Eh). Also, gauge D is loaded for more than two mesh cycles in line with the theoretical contact ratio value of this example hypoid gear pair.

A similar behavior is seen in Fig. 2.18 for the same test condition, but for the coast side loading (i.e. the concave side of the gear tooth flanks are loaded). Obviously, in this case, the gear tooth starts meshing at its toe and the contact moves from its toe to its heel. As a result, gauge At experiences strain first rather than Eh. Also noted from Fig. 2.18 is that the tensile part of any strain signal is larger since the concave side of the gear is loaded in this case.

Going back to Fig. 2.17(a), the gauges are seen to experience some low levels strains when their teeth are away from mesh zone. These lower levels of strains are mainly caused by the deflections in the gear blank. Since the gear blank thickness is smallest at the toe, the rim deflections, and hence the strains away from the meshing zone, are largest at this end, as depicted by strain traces of gauge At. Figure 2.19 shows the strain time history for gauge B on tooth #1 for the same conditions in Fig. 2.17. Here, markers “a” and “b” mark the entry and exit points of tooth to and from the mesh zone. Between these two positions, tooth #1 has a contact with a tooth from the pinion. The strain level right before the position “a” and after the position “b” are connected by a dashed line in this figure to illustrate the tooth bending and rim deflection components of the resultant strain signal.

The tooth-to-tooth variation of the strain signal is demonstrated in Fig. 2.20 for all the

= five gauges along the tooth root at T 400 Nm. For instance, in Fig. 2.20(a), gauges At on all three teeth measure about the same levels of tensile stresses while compressive stresses show a

240 µstrain spread amongst the three gauges. Similar differences are evident in Fig. 2.20(b-e) for

(a) (b) 2500 C 2000 Eh 1500

1000

500

Strain At µ 0 B -500 D

47 -1000

-1500

-2000 0 5 10 15 20 25 30 35 41 15 16 17 18 19 20 Mesh Cycles Mesh Cycles

Figure 2.18: (a) Strain time histories measured by the five gauges on tooth #2 for the baseline no misalignment condition at T = 400 Nm in the coast side loading condition; (b) zoomed view of one loading cycle.

1000 a b

500

-500 Strain µ

-1000

-1500

-2000 15 20 25 Mesh Cycles

Figure 2.19: Strain time histories of gauge B on tooth #1 for the baseline no misalignment = condition with the contributions of the rim deflections highlighted. T 400 Nm.

2000 (a) (b) 1000

0 Strain µ -1000

-2000

-3000 2000 (c) (d) 1000

0 Strain µ -1000

-2000

-3000 0 5 10 15 20 25 2000 Mesh Cycles (e) 1000

0 Strain µ -1000

-2000

-3000 0 5 10 15 20 25 Mesh Cycles

Figure 2.20: Measured strain time histories at the same face width location on the three = consecutive teeth for the baseline no misalignment condition at T 400 Nm for gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh. 49

the other four gauge groups as well. While the signals from each tooth are very similar

qualitatively indicating that their locations along the face width are quite consistent, differences in

the maximum and minimum values point to some variations of the gauge positions in the profile direction. As the strain gradient is very steep along the tooth root fillet, any slight deviations of the actual position of the gauges from an intended root location should be expected to cause sizable chances of the measured strains. This demonstrates the difficulties in strain gauging gears in general, as it was pointed to by others such as Ref. [68, 69].

Figure 2.21 shows the effect of the amount of load applied on the strain trace of gauge C for the baseline misalignment condition. The shape of the signal, as demonstrated, does not change with the torque value, while the maximum and minimum values of the measured strain remain proportional to the torque transmitted. It is observed that these signals that the gauged tooth extends its time in mesh with increased torque. For instance, the tooth is in the meshing

= = zone for 2.4 mesh cycles at T 100 Nm, while it has 3.3 mesh cycles in contact when T 500 Nm. Likewise, the strain values when the gauged tooth is outside the meshing zone are also

increased proportionally with T, indicating load dependence of rim deflections.

Figures 2.22 to 2.25 show the same behavior for the other gauges At, B, D, and Eh,

respectively. The same kind of a dependence on T is evident in all these figures. It is also noted

that the rim deflection related strain component contributes the most to the tow side of the tooth

root (Fig. 2.22 for gauge At), whose effect diminished in locations towards the heel of the gear.

Figure 2.26 plots the maximum and minimum values of the root strain measured by gauge C as a

function of T to further demonstrate this linear relationship.

1500

1000

500

-500 Strain

µ -1000

-1500 T = 100 Nm -2000 T = 200 Nm T = 300 Nm -2500 T = 400 Nm T = 500 Nm -3000 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.21: Strain time histories of gauge C on tooth #2 for the baseline no misalignment condition at different torque values.

600

400

200

-200 Strain

µ -400

-600 T = 100 Nm -800 T = 200 Nm T = 300 Nm -1000 T = 400 Nm T = 500 Nm -1200 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.22: Strain time histories of gauge At on tooth #2 for the baseline no misalignment condition at different torque values.

1000

500

-500 Strain µ

-1000 T = 100 Nm T = 200 Nm -1500 T = 300 Nm T = 400 Nm T = 500 Nm -2000 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.23: Strain time histories of gauge B on tooth #2 for the baseline no misalignment condition at different torque values.

1500

1000

500

-500 Strain

µ -1000

-1500 T = 100 Nm -2000 T = 200 Nm T = 300 Nm -2500 T = 400 Nm T = 500 Nm -3000 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.24: Strain time histories of gauge D on tooth #2 for the baseline no misalignment condition at different torque values.

1500

1000

500

0 Strain µ -500

-1000 T = 100 Nm T = 200 Nm T = 300 Nm -1500 T = 400 Nm T = 500 Nm -2000 0 5 10 15 20 25 30 35 41 Mesh Cycles

Figure 2.25: Strain time histories of gauge Eh on tooth #2 for the baseline no misalignment condition at different torque values.

180 (a) 160

140

120

100 Strain µ 80

20 0 (b) -100

-200

-300 Strain

µ -400

-500

-600

-700 0 50 100 150 200 250 300 350 400 450 500 Torque [Nm]

Figure 2.26: (a) Maximum and (b) minimum strain experienced by gauge C for the baseline no- error condition as a function of input torque.

2.7.2 Effect of Misalignments on the Measured Root Strains

One of the main design objectives for a hypoid gear pair is to account for the gear

performance to misalignments. The direct effect of misalignments (V, H, G or γ) on the gear pair in mesh is evidenced in the measured contact pattern position on the gear and pinion flanks.

Figures 2.27(a) and (b) show qualitative schematics of the expected theoretical shifts of the contact pattern on the gear flank with application of each type of the alignment error for a right- hand gear loaded on the drive and coast flanks, respectively. On the drive side, a positive V error should move the contact towards the toe tip while a positive H and G shifts it to the tip heel.

Also, a positive γ error result in a contact pattern biased towards the tip as shown in the schematic of Fig. 2.27(a). Such shifts of contact pattern along the tooth surface should be critical to the distribution of the tooth bending stresses along its root fillet. In this section, the effect of misalignments on the contact patterns and the resultant root stresses will be investigated.

Figure 2.28 shows the sensitivity of strain time histories for the 5 gauges (At, B, C, D, and Eh) along tooth #2 to the V error at an input torque value of 400 Nm. The two extreme cases

= ± = of the V error ( V 0.2 mm) together with the baseline condition of V0 mm are shown in each of these plots. While the qualitative shape and duration of each gauge signal remains the

same for different V values, the amplitudes are altered especially for end gauges At and Eh. For

gauge At, both the absolute maximum and minimum strain values are increased significantly by

applying a positive V error. Meanwhile, a negative V error causes the same effect to the signal

from gauge Eh. A similar behavior is observed for the other gauges in a more subdued manner,

suggesting that root stresses in the middle of the tooth are less sensitive to the V error.

Similar plots are shown in Figures 2.29 to 2.31 demonstrate the influence of the other

error types. For the H error in Fig. 2.29, the impact on all gauges along the gear face width 57

(a) Drive Side Top γ + +H +V +G

Toe Heel

Root

(b) Coast Side Top γ +G +H + +V

Toe Heel

Root

Figure 2.27: Theoretical directions of the contact pattern shifts along the (a) drive side and (b) coast side gear flanks for a right-hand gear due to different positive misalignments.

2000 (a) (b) 1000

0 Strain µ -1000

-2000

-3000 2000 (c) (d) 1000

0 Strain µ -1000

-2000

-3000 0 1 2 3 4 5 2000 Mesh Cycles (e) 1000

0 V = -0.2 mm

Strain V = 0 µ -1000 V = 0.2 mm

-2000

-3000 0 1 2 3 4 5 Mesh Cycles

Figure 2.28: Effect of the V error on the measured strain time histories for the baseline no misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh. 59

2000 (a) (b) 1000

0 Strain µ -1000

-2000

-3000 2000 (c) (d) 1000

0 Strain µ -1000

-2000

-3000 0 1 2 3 4 5 2000 Mesh Cycles (e) 1000

0 H = -0.2 mm

Strain H = 0 µ -1000 H = 0.2 mm

-2000

-3000 0 1 2 3 4 5 Mesh Cycles

Figure 2.29: Effect of the H error on the measured strain time histories for the baseline no misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh.

2000 (a) (b) 1000

0 Strain µ -1000

-2000

-3000 2000 (c) (d) 1000

0 Strain µ -1000

-2000

-3000 0 1 2 3 4 5 2000 Mesh Cycles (e) 1000

0 G = -0.05 mm

Strain G = 0 µ -1000 G = 0.3 mm

-2000

-3000 0 1 2 3 4 5 Mesh Cycles

Figure 2.30: Effect of the G error on the strain time histories for the baseline no misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh. 61

2000 (a) (b) 1000

0 Strain µ -1000 γ = -0.1 mm -2000 γ = 0 γ = 0.2 mm -3000

2000 (c) (d) 1000

0 Strain µ -1000

-2000

-3000 0 1 2 3 4 5 0 1 2 3 4 5 Mesh Cycles Mesh Cycles

Figure 2.31: Effect of the γ error on the strain time histories for the baseline no misalignment condition at T = 400 Nm of gauges (a) At, (b) B, (c) C, (d) D, and (e) Eh.

appears to be the same with the positive H error, causing slight increases in the peak-to-peak

values of the root strains. Likewise, the measured influences of the G and γ errors are shown in

Figures 2.30 and 2.31, respectively. A positive G error is shown to cause a similar variation to the strain traces resulted from a negative V error and vice versa. The shaft angle error also results in considerable variation in the strain amplitudes while keeping the basic shape the same. Here, the time (mesh cycles) at which the gauges register their peak values changes significantly with the amount of the γ error. Note that the results of γ−= 0.2 are not included in Fig. 2.31 since this error condition resulted in a tight mesh (zero backlash) condition with contact in both flanks of gear teeth.

In order to better quantify the variations of the root strains with different alignment errors, the peak-to-peak (p-p) values of measured strains along with the normalized distributions are considered next for the same gear pair within the same ranges of the V, H, G and γ errors.

Figure 2.32(a) shows the p-p root strain distributions along the face width of the gear for the five different V errors at 400 Nm. Here, the p-p stress distribution curve for the baseline case of

VHG= = =γ= 0 is rotated clockwise with positive V indicating increased stress levels on the

toe side while the heel side experiences less stress. The opposite is true with a negative V error

when the heel gauges read higher strains and the toe experiences lower strains. In order to better

understand the root cause of this rotation of the root strain distribution, the actual contact patterns

in Fig. 2.33 for the case of no error are compared to those in Fig. 2.34 with the V errors applied.

Figure 2.33 represents the desired unloaded contact pattern, in which the tooth bearing is in the

middle of the face width for both the drive and coast sides. Application of a positive error of

V= 0.2 mm in Fig. 2.34(a) results in a cross bearing contact pattern where one side of the tooth

carries the load on one end while the other side is being loaded at the opposite end. In this case, 63

4000 (a)

3500

3000

2500 Strain µ

2000 V = -0.2 mm V = -0.1 mm 1500 V = 0 V = 0.1 mm V = 0.2 mm 1000

1.4 (b)

1.3

1.2

1.1 Strain µ 1

0.9

0.8

0.7 At B C D Eh

Figure 2.32: Effect of the V error on the (a) peak-to-peak and (b) normalized peak-to-peak root strain distributions at T = 400 Nm on tooth # 2 for the drive side.

No Error

Desired contact pattern

Figure 2.33: Unloaded contact patterns for the baseline no misalignment condition.

(a)

V = +0.2 mm

Cross bearing contact pattern

(b)

V = -0.2 mm

Cross bearing contact pattern

Figure 2.34: Unloaded contact patterns for (a) V = +0.2 mm and (b) V = -0.2 mm.

the drive side has a toe bearing to describe the increase of the p-p strains at gauges At and B.

Again, the opposite effect is observed in Fig. 2.34(b) for V= − 0.2 mm when the contact moves to the heel on the drive side (gear convex) and, consequently, higher strains recorded for gauge

Eh. The normalized p-p strains (normalized by the corresponding no-error values) shown in Fig.

2.33(b) further quantify the effect of the offset error. For instance, gauge At reads about 40% more p-p strain when V= 0.2 mm compared to V0= while the p-p strain on gauge Eh is reduced about 15%. Likewise, an error of V= − 0.2 mm causes a 30% reduction of the p-p strain

of gauge At at the expense of a 15% increase at the location of gauge Eh. Note here that the At and Eh gauges are positioned 5 mm away from the toe and heel ends and thus the effect of the V error should be even more drastic at the edges, with some edge contact that must definitely to be avoided.

The effect of the V error on p-p root strains during the coast side operation is shown in

Fig. 2.35. In this case, the positive V error shifts the contact to the heel side and causes counter- clockwise rotation of the root strain distributions from the baseline condition. This complies with the contact patterns on the coast side (Fig. 2.34). Finally, while a positive offset mostly alters the p-p strain of gauge At on the drive side, a negative V error is seen to cause the p-p strain of gauge

B to be highest and about 38% more than the no-error condition.

The similar plots are produced in Fig. 2.36 for the H error. Here, a positive H error is

seen to increase strain amplitudes along the entire tooth root while the opposite happens for the

negative H errors. This is mainly due to the fact that a positive H error moves the contact zone

towards the tip with a skew towards the heel side, resulting in a larger bending moment. The

corresponding unloaded contact patterns for H= ± 0.2 mm H errors shown in Fig. 2.37 confirm

this. While gauge C in the middle of the tooth showed minimum sensitivity to V error,

4500 (a)

4000

3500

3000 Strain µ 2500

2000 V = -0.2 mm V = -0.1 mm V = 0 1500 V = 0.1 mm V = 0.2 mm 1000

1.4 (b)

1.3

1.2

1.1 Strain µ 1

0.9

0.8

0.7 At B C D Eh

Figure 2.35: Effect of the V error on the (a) peak-to-peak and (b) normalized peak-to-peak root strain distributions at T = 400 Nm for the coast side.

4000 (a)

3500

3000

2500 Strain µ

2000 H = -0.2 mm H = -0.1 mm 1500 H = 0 H = 0.1 mm H = 0.2 mm 1000

1.2 (b)

1.15

1.1 Strain µ 1.05

0.95 At B C D Eh

Figure 2.36: Effect of the H error on the (a) peak-to-peak and (b) normalized peak-to-peak root strain distributions at T = 400 Nm for the drive side.

(a)

H = +0.2 mm

High bearing contact pattern (b)

H = -0.2 mm

Low bearing contact pattern

Figure 2.37: Unloaded contact patterns for (a) H = +0.2 mm and (b) H = -0.2 mm.

it is not the case here increases as high as 18% are shown in Fig. 2.36(b) solely due to the H error.

The influence of G error on the root strains distribution is somewhat similar to that of the

H error with a biased shift towards the heel as seen in the p-p strain distributions of Fig. 2.38(a).

This bias causes the distribution to rotate in the counter-clockwise direction from the baseline case with increasing G error values. While gauge C exhibits very little change, the remaining four gauges register certain sensitivity to the G error. For instance, gauge D has p-p stresses up to

22% higher than the no error case as depicted in Fig. 2.38(b). The contact patterns in Fig. 2.39 for G= − 0.05 and 0.3 mm confirm these conclusions. It is rather obvious in Fig. 2.39(a) that the

contact zone is at the heel and close to the topland when G= 0.3 mm.

Finally, Fig. 2.40 shows the influence of the shaft angle error on the p-p root strain distributions. Similar to the H error, a positive γ error increase the p-p strain levels at all gauge locations as it moves the contact pattern towards the topland. This high bearing tooth contact is

evident in Fig. 2.41(a) for γ = +0.2ο where tooth bearing is shifted towards the heel in addition to

placing it high on the tooth flank. Quantitatively, the middle gauges (B, C, and D) seem to be

affected more by the γ error than the toe and heel gauges. Here, gauge C experiences an increase

of more than 20% when the error is varied from γ = −0.1ο to 0.2ο as seen in Fig.2.40(b).

2.8 Summary and Conclusions

In this chapter, a novel the experimental methodology was developed to measure the root strains of hypoid gears under tightly-controlled alignment errors condition within a wide range of input torque. The methodology was applied to an automotive axle hypoid gear pair to quantify the influence of all four basic types of misalignments on root stresses. The gear was instrumented with 15 strain gauges distributed along the roots of three consecutive teeth with 5 gauges on each. 71

4000 (a)

3500

3000

2500 Strain µ

2000 G = -0.05 mm G = 0 1500 G = 0.1 mm G = 0.2 mm G = 0.3 mm 1000

1.2 (b) 1.15

1.1

1.05

1 Strain µ 0.95

0.9

0.85

0.8 At B C D Eh

Figure 2.38: Effect of the G error on the (a) peak-to-peak and (b) normalized peak-to-peak root strain distributions at T = 400 Nm for the drive side.

(a)

G = +0.3 mm

Cross bearing contact pattern

(b)

G = -0.05 mm

Close to desired contact bearing (acceptable)

Figure 2.39: Unloaded contact patterns for (a) G = +0.3 mm and (b) G = -0.05 mm.

4000 (a)

3500

3000

2500 Strain µ

2000 γ = -0.1o γ = 0 1500 γ = 0.1o γ = 0.2o 1000

1.2 (b)

1.15

1.1

1.05 Strain µ 1

0.95

0.9

0.85 At B C D Eh

Figure 2.40: Effect of the γ error on the (a) peak-to-peak and (b) normalized peak-to-peak root strain distributions at T = 400 N-m for the drive side.

(a)

γ=+0.2o

Cross bearing and high contact pattern

(b)

γ=−0.1o

Low bearing contact pattern

Figure 2.41: Unloaded contact patterns for (a) γ = +0.2o and (b) γ = -0.1o.

A data acquisition system was devised to capture the strain signals of all the gauges simultaneously. A test matrix consisting of a range of torque values, various levels of misalignment amounts for each error type (V, H, G, andγ) and both the drive and coast sides was executed. Experimental results were presented in different forms to investigate the combined influence of the load and the alignment errors on strain time histories and peak-to-peak values along the tooth root width.

Based on the results of this study, the proposed experimental methodology as well as the data collection system can be deemed to be reasonably accurate and reliable. The measured root strains and the corresponding contact patterns under various error conditions have shown to vary considerably, indicating that they must be included in the design of hypoid gears. The measured strain time histories also point to certain amount of rim deflections that must be accounted for in modeling root stresses. The distribution of root strains along the face width follows the same pattern as the change of contact pattern position with different misalignments.

In view of the literature review presented in Chapter 1, this set of root measurement can be considered to form the most extensive database to date including the wide ranges of errors as well as realistic load levels. These measurements will be used in the next chapter to validate the predictions of a new hypoid gear root stress model.

CHAPTER 3

A MODEL TO PREDICT ROOT STRESSES OF SPIRAL BEVEL AND HYPOID GEARS

3.1 Introduction

A finite element based model is proposed in this chapter for prediction of root stresses of

spiral bevel and hypoid gears. The main tasks involved in the computational methodology are

summarized in the flowchart of Fig. 3.1. Due to the complexity of both the load distribution and

the geometry of hypoid gears, the resultant stress states at the root are multiaxial, requiring an

accurate representation of the root surfaces of the gears forming the pair. Hence, the first task in

Fig. 3.1 is the generation (i.e. three-dimensional definition) of the root fillet surfaces. The basic gear design parameters, the cutting machine settings, and the cutter geometry are input to the model to define the gear root and fillet regions through a simulation of the actual cutting process.

With the root geometries defined, the exact tooth thicknesses based on the actual geometry and a recently developed hypoid load distribution model [31] are used to predict the load intensities along the contact lines through an entire gear mesh cycle. An automated mesh generator and a customized finite element formulation are developed next to obtain a deformable-body model of

N-tooth segments of the pinion and the gear with appropriate displacement boundary

Cutter Geometry Machine Settings Basic Gear Parameters

Tooth Hypoid Load Thickness Root Surface Distribution Model Definition [31]

Flank Surfaces Automated Mesh Generator

Load Intensities along Contact Lines Definition of Nodal Forces

Stress and Deflection Predictions

Figure 3.1 Overall computational methodology used for prediction of root stresses.

conditions. The distributed tooth loads predicted by a contact model are applied to the FE models

of the gear segments to predict deflections of gear and pinion teeth as well as the multiaxial root stresses.

3.2 Definition of the Hypoid Tooth Root Surfaces

Unlike other common types of gears, hypoid gear geometry cannot be defined explicitly in closed-form. It must rather be defined by solving a set of implicit equations representing the cutting process that includes the basic gear dimensions, machine setting parameter as well as the cutter geometry parameters. This is accomplished by simulating the cutting process, either face- milling (FM) or face-hobbing (FH), starting from the cutter blade geometry and then applying several motion transformations prescribed by the machine settings on the cradle-based hypoid generator shown in Fig. 3.2. The figure shows the conventional cutting machine with its key setting parameters. More contemporary CNC machines use the same machine settings as well.

These machines settings and the relative motions shown in Fig 3.2 are the cutter phase ( φt ), the tilt angle (it ), the swivel angle ( js ), the radial setting ( Sr ), the cradle rotation ( φc ), the sliding

base ( X B ), the root angle ( γm ), the machine center to back ( Mctb ), the blank offset ( Eb ), and the blank or work-piece phase angle ( φw ). These machine setting parameters are either fixed or

allowed to change as a function of the cradle angle ( qc ). Changing the machine settings with the

cradle angle allows certain higher order motions such as the modified roll, the helical motion

(sliding base change), and the vertical motion (blank offset change) [31].

φc

Sr j i Eb q φw φt

Mctb γm

Figure 3.2: Cradle-based hypoid generator [31].

3.2.1 Kinematics of Face-Milling and Face-Hobbing Processes

In both FM and FH cutting processes, two motion types, namely Formate® and

Generate, are used. In the Formate case, the tooth surface of the gear is basically a copy of the

generating surface that is defined by the relative motion between the cutter head and work piece

blank (pinion or gear). For the FM process, the cutter axis is fixed and the edges of the blades cut

through the gear blank as they rotate around this axis. The gear blank is held at rest while the

cutter advances towards the work piece, cutting one tooth slot at a time (single indexing). The

generating surface for the FM process is a conical one with circular arc traces along the

lengthwise direction of the tooth as illustrated in Fig. 3.3(a). For the FH process, in order to

allow for the continuous indexing, the kinematics of the cutter head and work piece blank are

rotate according to the relation

(1) ωwtN = = Rtw (3.1) ωtwN

ω(1) ω where w is the component of the work piece angular velocity related to the cutter rotation, t

is the cutter head angular velocity, Nt is the number of blade groups, and Nw is the number of teeth on the gear blank. This action can be considered as rolling of two gears with pitch circles

(called roll) and base circle, as illustrated in Fig. 3.3(b). Here, the resultant generated surfaces are extended epicycloid traces along the tooth face direction.

In the Generate case, after the cutter reaches the required tooth depth, a generating roll action is applied by which the cradle and the work piece are rotated synchronously according to the ratio of roll Ra

(a) Work Blank (b) Work-piece Rotation Circular Arcs Extended Epicycloids

Inside Blade Outside Blade Inside Blade 82

Fixed Outside Blade

Cutter Rotation Base Circle Roll Circle Cutter Rotation

Face-Milling Face-Hobbing

Figure 3.3: (a) Face-milling and (b) face-hobbing cutting processes.

(2) ωw = Ra (3.2) ωc

ω(2) where w is the component of the work piece angular velocity related to the cradle rotation

ω (generating action) and c is the cradle angular velocity. Unlike the Formate case, the tooth surface is the envelope of the family of surfaces resulted from the generating action.

A generic mathematical representation of the blank angular speed that can be applied to both the FM and FH processes with Formate or Generate motions is given by

(1) (2) ωww =ω +ω w (3.3)

(1) where ω=w 0 for the FM process. Similarly, a general equation can be defined for the blank rotation as [31]

φw =−()RR a φ+ c tw φ t (3.4)

= = where Ra 0 for the Formate case and Rtw 0 for the FM cutting process.

Finally, in terms of simulating the cutting process, the FM process can be considered as a special case of the FH process, as the conditions for the FM process can be obtained from the FH process by setting certain parameter (the rake, hook, and offset angles, and other certain blade parameters) to zero. With this observation, the FH process will be used here to illustrate the methodology applied to define root surfaces required for the FE model.

3.2.2 Blade Geometry and Relative Motions

A typical FH blade with its cutting edge geometry is shown in Fig. 3.4. Here, the key

cutter parameters are the blade angle ( αb ), the rake angle ( τb ), the hook angle ( µb ), the blade offset angle ( δb ), the cutter radius ( rc ), the tip radius ( ρt ), and the tip of blade to the reference point ( hf ). In general, the cutting edge is divided into four sections: flankrem, profile, toprem, and edge. The flankrem and edge sections are usually circular arcs while the shapes of the toprem and the profile sections can be any type of curve. The main function of the toprem and the flankrem sections is to relieve the tooth root and tip in order to avoid interferences.

Meanwhile, the edge section is responsible for generating the root areas, and hence, is the primary focus of this work.

The position vector of an arbitrary point A on the cutting edge (Fig. 3.4(c)) in the fillet region relative to a coordinate system Xb attached to the blade is given as

T rb (λ ) =[cd −ρ sin λ 0 +ρ cos λ] . (3.5)

The position vector of the same point A can be defined on the coordinate system Xt attached to the cutter as

rt()λ= Mr tb b () λ. (3.6a)

Here, Mtb is a transformation matrix defined by

cos(−δb ) sin( −δ b ) 0r cb cosδ  −sin( −δ ) cos( −δ ) 0r sin δ M=b b cbMM()µτ () (3.6b) tb 0 010 x b z b  0 0 01 84

(a) yt Reference Plane

xb xxτµ,

xt rc δb M zzb , τ zµ Inside Blade 31 (b) (c)

Flankrem

Profile xb rb xxbt, xxτµ,

h yb Toprem f y µ yτ Edge A

λ yt rc c

Figure 3.4: (a) Position of the blade on cutter head, (b) 3D geometry of the blade, and (c) the shape of the cutting edge [31].

where the rotation matrices Mx and Mz are defined as follows:

1 0 00  0 cosϕϕ sin 0 M ()ϕ=, (3.6c) x 0−ϕ sin cos ϕ 0  0 0 01

cosϕϕ sin 0 0  −ϕsin cos ϕ 0 0 M (ϕ=) . (3.6d) z 0 0 10  0 0 01

Through several transformations, each representing certain relative motions between the cutter and blank as listed in Fig. 3.5, the position vector rt ()λ defined in Eq. (3.6a) is defined in a coordinate system Xw fixed to the blank (work piece) as [31]

rw(,λφ t , φ c ) = M wo MM op pr M rs M sm M mc MMMMMr cr ce ej ji it t ()λ (3.7a) where the transformation matrices are defined as

MMit= z(2 −π −φ t ), (3.7b)

MMji= x()i t , (3.7c)

MMej= z()j s , (3.7d)

100Sr  010 0 M = , (3.7e) ce 001 0  000 1

Cutter Rotation Mit

Tilt M ji

Swivel Mej

Radial Setting Mce

Cradle Angle Mcr M rt ()λ Cradle Rotation mc

Sliding Base Msm

Blank Offset Mop

Machine Center to Back M pr

Machine Root Angle Mrs

Work-piece Rotation Mwo

Figure 3.5: Relative motions and the corresponding transformation matrices from the cutter to the blank.

MMcr= z()q c , (3.7f)

MMmc= z() φ c , (3.7g)

100 0  010 0 =  Msm , (3.7h) 001−X B  000 1

cosγγmm 0 sin 0  0 100 −γ =  M ym() , (3.7i) −γsinmm 0 cos γ 0  0 001

100−Mctb  010 0 M = , pr 001 0 (3.7j)  000 1

100 0  010E M = b , op 001 0 (3.7k)  000 1

MMwo= x() φ w (3.7l)

3.2.3 Root Surface as Envelope of Family of Surfaces

The position vector in Eq. (3.7a) represents a family of surfaces, the envelope of which

represents the actual root surface of the gear (or pinion). Analytically, this envelope is defined by

the equation of meshing that enforces the conjugacy condition between the surfaces

∂∂∂rrrwww × ⋅=0 . (3.8) ∂λ ∂φtc ∂φ

Due to the complexity of the governing equations for the tool geometry and cutting simulations, a

closed-form solution to these equations is not possible. They must be solved implicitly for a

desired set of points to define the tooth root surfaces. A two-dimensional (2D) gear blank region,

shown in Fig. 3.6, bounded by the gear blank geometry in the (,)RZ coordinates, is devised for

this purpose. This area is discretized into mn× grid points. For each grid point (,ij ), the

λ φ φ surface parameters , t , and c generate rwwww(,xyz ,) such that

 =22 += Rw xyR w w(, ij )  (3.9)  zZw= (, ij )

λ φ φ Here, there are only two unknowns and t since c is solved for implicitly by applying the

conjugacy equation (3.8). A nonlinear solver based on the Broyden’s Method [70] is employed

here, in which the Jacobian is computed numerically for the first iteration only using a finite

difference approximation. It is noted here that the equation of meshing is not required for the

Formate case since the actual root surface is simply a copy of the generated surface of the cutter

as mentioned previously.

Gear Blank

Gear Axis

View A

Face Cone R Pitch Cone Root Cone 2D Blank

(,ij )

Actual Root Curve

View A Z

Figure 3.6: Two-dimensional gear blank region defined for the purpose of surface generation.

As mentioned earlier, the same task was taken on by Kolivand and Kahraman [31] as well. The proposed formulation differs from Ref. [31] in the way the root boundary line is defined. Kolivand and Kahraman [31] defined the root line approximately as a straight line, as determined by the root angle (root cone) of the gear blank. This was a reasonable approximation as long as the focus is limited to the load distribution and contact stresses only. However, in general, this root line is a curve, not a straight line, as shown in Fig. 3.6. Defining this curve requires not only the blank dimensions, but also the machine settings and the cutter geometry.

φ φφ First, the range of t required to generate the flank surface is computed as [,tt,min ,max ]. Then, equally spaced discrete points that fall within this range are used to generate the corresponding surface points at λ=0 . These surface points are then projected back to the 2D blank and a quadratic curve is fit into their (,)RZ coordinates. The 2D blank is redefined based on the new root line and the tooth root surface is generated following the methodology discussed earlier.

Tooth surfaces including the root and fillet regions are obtained as illustrated in Fig. 3.7 for the concave side of a pinion and the convex side of the mating gear teeth including the root regions.

3.3 Finite Element Formulation

Customized finite element models of the pinion and gear segments will be developed here to compute the stresses at the root and fillet regions of spiral bevel and hypoid gears in a computationally efficient way. The model will be generic in terms of incorporating any spiral bevel or hypoid design that is cut using the FH or FM processes with Formate or Generate motions. An automatic mesh generator will be devised by mapping the actual surfaces onto a pre-defined mesh template. This will allow the model to be used with little prior FE

(a)

(b)

Figure 3.7: (a) Pinion concave side surface, and (b) mating gear convex side surface.

knowledge.

According to Fig. 3.1, this model relies on the contact model of Kolivand and Kahraman

[31] in terms the prediction of the load intensities along the contact lines. A method is employed

to transform these loads to finite element nodal forces. The displacement vectors and stress

tensors are then computed at each mesh position for different loading and misalignment

conditions.

3.3.1 Element Type and Stiffness Matrix

In the finite element model, 20-noded isoparametric quadratic elements [71], shown in

Fig. 3.8, with shear modeling ability are used. This type of an element is capable of modeling the curved boundaries such as the root surfaces. Figure 3.8 defines the element in two different coordinate systems, the original Cartesian system defined by (,xyz ,) and a natural coordinate system, which is a normalized local system, attached to the element and defined by (,ξηρ ,).

Each element in the Cartesian coordinate system is mapped onto a rectangular parallelepiped in

the natural coordinates [72]. Both the displacement vector u and geometric coordinates r inside

the element are interpolated by using the same element interpolation functions as

20 uq= Φ , (3.10a) ∑ iinn in =1

20 rx= Φ (3.10b) ∑ iinn in =1

Transformed Element in Actual Element in Natural Coordinates Cartesian Coordinates η

η ρ 8 15 7 ρ

20 19 ξ 11 4 3 14 16 x 12 13 10 ξ 5 6 17 z 18

1 9 2 y

Figure 3.8: 20-noded element used by the finite element model.

where q and x are the nodal displacements and nodal coordinates of node i (i ∈[1,20] ) in in n n

and Φ are the interpolation functions (or shape functions) defined in the natural coordinate in system by

1  (1+ξ0 )(1 +η 0 )(1 +ρ 00 )( ξ +η 0 +ρ 0 −2),in = 1 − 8, 8 1  (1−ξ2 )(1 +η )(1 +ρ ) and η= 0, i =9,11,13,15, 4 00 n Φ= (3.10c) in 1  (1+ξ )(1 −η2 )(1 +ρ ), i =10,12,14,16, 4 00 n  1  (1+ξ )(1 +η )(1 −ρ2 ), i =17,18,19,20. 4 00 n

e e The stiffness matrix k for an element of volume V is derived from the strain energy as

kee= ∫ BT DB dV (3.11a) V e

where B is the strain displacement matrix defined as

ΦΦ1, xx0 0 ... 20, 0 0  0 ΦΦ1, yy0 ... 0 20, 0 0 0ΦΦ ... 0 0 1, zz20, B= [ BB1 2... B 20 ] (3.11b) ΦΦ1,yx 1, 0 ... Φ20,y Φ 20, x0  0 ΦΦ1,zy 1, ... 0 ΦΦ20,z 20, y  Φ1, z0 ΦΦ1, xz ... 20, 0 Φ20, x

∂Φi where Φ=n ( v ∈ [xyz , , ]) . In Eq. (3.11a), D is the material properties matrix in ,v ∂v

ddd122000  d212 dd 000 ddd221000 D =  (3.11c) 000d3 00 0000d 0 3 00000d3 where

 E(1−ν ) d1 =  (1+ν )(1 − 2 ν )   Eν d2 = (3.11d)  (1+ν )(1 − 2 ν )   E d3 =  2(1+ν ) with E and ν as the modulus of elasticity and Poisson’s ratio for the gear material.

The strain displacement matrix as defined in Eq. (3.11b) requires the partial derivatives of the shape functions with respect to the Cartesian coordinates. The Jacobian matrix J is defined as

∂∂∂xyz  ∂ξ ∂ξ ∂ξ ΦΦ 1,ξξ... 20, xyz111 ∂∂∂xyz J = =ΦΦ1,ηη... 20,  (3.12) ∂η ∂η ∂η  ΦΦ... xyz ∂∂∂xyz1,ρρ 20, 20 20 20  ∂ρ ∂ρ ∂ρ such that

ΦΦ ξ ixnn,,  i −1 Φ=Φiy,,J  iη . (3.13) nn  ΦΦ iznn,,  iρ

With the determinant of the Jacobian matrix J in hand, the element stiffness matrix is computed

11 1 ke = ∫∫∫BT DB J dddξηρ. ρ=−η=−ξ=−111

(3.14)

In this study, the above integral equation is evaluated numerically using the 3x3x3 Gauss

Quadrature Scheme. Here, the integral is reduced to the sum of the function evaluated at 27

Gauss Points. In order to reduce computational effort, ke is computed nodewise. For a pair of nodes i and j , the stiffness sub-matrix k is obtained as [71] n n ijnn

T kij= B DB j. (3.15) nn in n

The sub-matrices are then placed in the global stiffness matrix K based on the nodal connectivity obtained by the mesh generator. It is noted here according to the Maxwell’s Reciprocity Theorem that kk= , which reduces the computations significantly. in j n ji nn

3.3.2 Automated Mesh Generator

The purpose here is to generate the finite element mesh automatically for any hypoid or

spiral bevel gear pair design using the actual tooth profile and root surfaces. Such a mesh 97

generator would eliminate the need for a commercial FE package as well as reducing the burden

on the user. The first step is to define the tooth from the generated surfaces. One of the

generated surfaces, concave or convex, is rotated until the required tooth thickness is reached.

This thickness is either given by the basic gear parameters or measured physically and reported at

a given cone distance and addendum location.

Taking advantage of the “cone” that distinguishes hypoid and spiral bevel gears

geometry, a coordinate transformation is applied to the tooth surfaces and the mesh is defined in a

specially devised “hypoidal” coordinate system (,,)υθφ as shown in Fig. 3.9(a). The

transformation is performed such that

zZ=c −υcos θ, (3.16a)

xR=cos φ , (3.16b)

yR=sin φ , (3.16c)

R =υθsin . (3.16d)

Here, the same concept of the 2D blank defined earlier for the surface generation is used for

meshing of a tooth as well. The 2D blank (or the tooth) is sliced into m slices in the face width

θ direction. Each slice hm∈[1, ] is defined by fixing (or Zc ) as shown in Fig. 3.9(a). Mesh control points on each slice are then computed as the basis of the required mesh. Figure 3.9(b)

h h shows the control points defined for each slice. Points CX and CV ( jn∈[1, ] ) are the actual j j surface points on both the concave and convex curves generated by letting ih= in Eq. (3.9).

Any point TP that belongs to the topland is computed such that

(a) (b) R Topland Slice h φ TP CVn Projected slice h CXn

CVj CXj Concave

Convex CV1

CX1 Base

99 Concave

Base Convex θ u

Z BV c BM Z BX Bottom

Figure 3.9: (a) Coordinate transformation and (b) control points on each gear slice used for the mesh generation.

υh TP h rTP= θ TP (3.17)  φh TP

hh h hh h h hh where υ=υ=υTP CVn CXn , θ=θ=θTP CVn CXn and φTP ∈φ[,] CVn φ CXn . Control points BV and

BX in Fig. 3.9(b) are the intersection points between the bottom curve and the base concave and base convex lines. These two points have the same R and Z coordinates, defined by the base cone

hh angle parameter. The third coordinate for point BV is φ=φBV and for point BX is CV1

hh φ=φBX . CX1

Any point BM that belongs to the bottom is defined by

υh BM h rBM= θ BM (3.18)  φh BM

h hhh hh h hh where υBM =υ=υ BX BV , θBM =θ=θ BX BV and φBM ∈φ[,] BV φ BX .

After defining all the control points, the coordinates (,,)υθφkkkfor each node kN∈[1,h ]

( Nh is the total number of nodes on each tooth slice h) are computed based on the mesh template

shown in Fig. 3.10(a). The resultant meshed slices, as seen in Fig. 3.10(b), are then connected to

form the mesh of an entire tooth. A gear segment formed by Nmesh teeth is meshed by cloning one tooth and rotating by a base pitch. Here, the nodal connectivity and nodal coordinates are stored for the solution phase.

100

(a) (b) slice h+1 mesh slice h mesh

-22

-24

z -26 -28

-30

-32 101 -34

-75

-70 -5

-65 y -10 x

-60 -15

Figure 3.10: (a) The mesh template and (b) the meshed slices as seen in the Cartesian Coordinates.

Figures 3.11(a) and (b) show examples of FE meshes of 6-tooth segments of a hypoid

gear and a mating hypoid pinion, respectively. Given the nodal connectivity, the global stiffness

matrix K is constructed and stored in skyline form in order to increase the computational efficiency [18]. The most costly computational task here is the factorization of the global stiffness matrix. The K matrix is factorized into its upper and lower triangular matrices using the

LU Decomposition scheme so that the forward elimination and the backward substitution can be

applied in the solution phase [72]. The K matrix is independent of the loading and misalignment conditions. Therefore, the same factorized K matrix can be stored to be used for any loading and

misalignment condition, allowing an efficient way of performing sensitivity analyses.

3.3.3 Boundary Conditions and Application of Contact Loads

A gear (or pinion) segment formed by Nmesh teeth is included in the finite element

simulation. Here, certain boundary conditions representative of the actual constraints must be

imposed. The nodes of the bottom section of the gear segment including the cut edges are fixed

in all directions, as shown in Fig. 3.12. While applying such boundary conditions results in a

structure with reduced compliance since it prevents most of the rim deflections, it is a good

engineering compromise as long as the root stresses are the main focus. Any attempt to include

the rim deflections would result in modeling the entire gear and pinion bodies at the expense of

significant computational burden.

The hypoid load distribution model of Kolivand and Kahraman [31] is used to predict the

load intensities along the contact lines during one complete mesh cycle of a hypoid gear pair

under given torque and alignment conditions. The load intensity is represented by a concentrated

102

(a) (b) 103

Figure 3.11: Six-tooth (a) gear and (b) pinion segment examples used in the FE model.

Fixed Boundary Conditions

Figure 3.12: Fixed boundary regions of the FE gear segment.

104

load vector Pl at location rl in a direction normal to the loaded tooth surface as shown in Fig.

3.13. The procedure of translating the load intensity distribution to the nodal forces first requires one to specify the loaded tooth surface is concave or convex. Next, the corresponding loaded element is detected by using a simple search method. Since the loads can only be applied to the surface nodes of the FE model, this search is reduced to a 2D one by projecting the element surfaces back to the plane where the blank was first defined as shown in Fig. 3.14, in the (,)RZ process saving considerable computational time. The corresponding nodal location (,ξηρ ,)l of

the applied force in the natural coordinate system is determined by using inverse mapping. Here

again, two parameters are unknown since only one surface of the element is concerned. The

equivalent nodal forces vector applied at is given as

Nl FP= Φ(, ξηρ ,) , inn∑ i ll (3.19) l=1

where in ∈[1, 8] are the nodes along the contact surface of the loaded element and Nl is the total

number of concentrated loads on the same element. The load vector is constructed consequently

F1  F2 F =  (3.20)   F Nn

where Nn is the total number of nodes of the FE model. Figure 3.15(a) shows the load

intensities along contact lines as computed by the hypoid load distribution model [31] and Fig.

3.15(b,c) show the corresponding translated nodal forces acting on both the gear and pinion

105

Contact Line

rl Pl

Figure 3.13: The load intensity representation on a contact line along a tooth surface.

106

Projected Element in 3D Element in 2D

Loaded face

Figure 3.14: The projection of a three-dimensional element into the pre-defined two-dimensional blank.

107

(a)

(b)

(c)

Figure 3.15: (a) Load intensities along contact lines [31], (b) the corresponding nodal loads on the gear, and (c) the corresponding nodal loads on the pinion.

108 surfaces. The gear pair in this example is simulated in the drive direction and consequently the gear convex and pinion concave surfaces are loaded.

3.3.4 Finite Element Solution

Given the decomposed global stiffness matrix K and the force vector F , the solution is carried on by computing the nodal displacements represented by the nodal displacement vector

Q using the linear matrix equation at a given mesh instant

KQ= F . (3.21)

The elements along the roots of the gear or pinion teeth are then identified for further strain and stress computations. Since numerical integration is performed in the computation of the element stiffness matrix ke , the best sampling points for strain and consequently stress calculations are the Gauss points. The strain vector ε at a given gauss point is given by

εxx  ε yy  εzz = =Bq(, ξηρ ,)e (3.22a) εig iiiggg γxy  γ yz  γxz

∈ e where ig [1, 27] and q is the displacement vector of the element nodes given by

109

q1  q2 qe = . (3.22b)   q20

The stress vector σ at the same point is determined as

σxx  σyy  σzz = =D ε (3.23) σiigg τxy  τyz  τxz where matrix D was defined earlier in Eq. (3.11c,d).

The highest stresses occur at the surface boundaries of the root elements. Therefore, it is sufficient to stresses along the surface elements, provided that the residual stresses are negligible.

As applying Eq. (3.22) and (3.23) at the nodal points is likely to result in inaccurate estimates of the stresses [73], a quadratic least-square extrapolation scheme is used here to compute the stresses at the root nodes. In this scheme, stresses at the nodes are extrapolated from the stresses computed at the 27 Gauss points of each root element. The formulation for the extrapolation matrix is presented here for any stress component of interest and the same matrix can be incorporated for other stress components the same way. The difference between the actual stress

σ and the extrapolated stress σ (i.e. the error in the extrapolated stress) at any point (,ξηρiii ,) is given by

Εiii=σ −σ . (3.24)

110

At the Gauss points, Eq. (3.24) is written as

Ε =σ −σ . (3.25) iiiggg

The extrapolated stress at a Gauss points is interpolated by using the shape functions from the

nodal stresses. Mathematically, this is expressed as

20 σ= Φξηρσ(, ,) . (3.26) ig∑ ii ng ii gg i n in =1

With this, the least-square extrapolation requires one to minimize the following functional

27 2 χ = σ −Φ( ξ , η , ρ ) σ , i ∈ [1,20] . (3.27a) in∑  i g ii ng ii gg i n n ig =1

∂χi The equation n = 0 yields the minimum value of this functional (ik,∈ [1,20] ): ∂σ nn in

27 σ−Φξηρσ(, ,)  −Φξηρ (, ,) (3.27b) ∑  ig kiii ng gg k n iiii ng gg ig =1 or

27 27 ΦξηρΦ(,,)(,,)= ξηρσ Φξηρσ (,,) . (3.27c) ∑∑iiiing gg kiii ng gg k n iiiii ng gg g iigg=11=

Equation (3.27c) is written in matrix form as

AGσσ = (3.28a)

111

where σ is the vector of 20 unknown nodal stresses, σ is vector of 27 known Gauss point stresses,

27 A = ΦξηρΦ(, ,) (, ξηρ ,) (3.28b) ∑ iiiing gg kiii ng gg i =1 g

and

Φξηρ1(, 1 1 ,) 1 Φξηρ 1 ( 2 , 2 , 2 ) Φξ1 ( 27 , η 27 , ρ 27 )  Φξηρ2(, 1 1 ,) 1 Φξηρ 2 ( 2 , 2 , 2 ) Φξ2 ( 27 , η 27 , ρ 27 ) G = . (3.28c)     ΦξηρΦξηρ20(, 1 1 ,) 1 20 ( 2 , 2 , 2 ) Φξηρ 20 ( 27 , 27 , 27 )

Finally, the extrapolated stresses at the nodes are computed by

−1 σσ = AG. (3.29)

3.4 Numerical Results

3.4.1 Optimal Number of Teeth Included in the Model

The FE model proposed in the previous section considered segments of the gear and

pinion. The number of teeth Nmesh forming a gear or pinion segment was introduced as a user-

defined parameter. Before any extensive analyses can be performed, certain guidelines must be

established on the value of Nmesh . Hypoid gears have a relatively high contact ratio, i.e. under

any torque values, three or more teeth are typically in contact at any given instant. Hence, a

minimum of four teeth should be included in the simulations to define the segments. With the

rigid boundary conditions imposed at the cut ends, additional teeth might be required to minimize

any adverse effects of boundary conditions on the predicted deflections and stresses. In this 112

section, a set of example analyses are performed with different Nmesh values to determine its optimum value.

The hypoid gear pair used in the experiments of Chapter 2 with basic parameters given in

Table 2.1 is used for this purpose. The stresses are predicted at the face width locations where the strain gauges were mounted (five gauges, named 1 to 5, placed along the root in the face direction at 5 mm apart starting at 5 mm from the toe end). A pinion load of 300 Nm is applied and no misalignments are introduced. The predictions from the variations of the model with

Nmesh = 4,5,6 and 7 are compared in Fig. 3.16(a,b). In Fig. 3.16(a), the maximum principal stress σ1 predicted at an arbitrary fixed radius along the root fillet using different Nmesh are

σ compared. Likewise, Fig. 3.16(b) compares the same at a radius where 1 is maximum. From this example, up to 13% difference in stresses is observed between Nmesh = 4 and Nmesh = 5

while there are less than 6% difference between Nmesh = 5 and Nmesh = 6 . Beyond Nmesh = 6 ,

predictions converge as less than 1% difference is observed between the predictions with

Nmesh = 6 and Nmesh = 7 .

As a compromise between the computationally time required and the accuracy of the

= predictions, FE models with Nmesh 6 is deemed to be sufficient in this application. While

this could be used for gear pairs of sizes similar to this example system, type of sensitivity study

presented in Fig. 3.16 might be necessary for gear pair of other sizes and designs.

113

(a) 300 4 teeth 5 teeth 250 6 teeth 7 teeth 200

150 [MPa] 1 σ 100

0 (b) 700

600

500

400 [MPa] 1 300 σ

200

100

0 Gauge 1 Gauge 2 Gauge 3 Gauge 4 Gauge 5

Figure 3.16: Maximum principal stresses as a function of the Nmesh at different virtual gauges placed along the face width at (a) an arbitrary profile position and (b) the profile position corresponding to the maximum principal stress experienced by the gear pair throughout one loading cycle.

114

3.4.2 An Example Analysis

A face-milled gear pair with the basic design parameters given in Table 3.1 is analyzed here at a pinion torque value of T = 300 Nm and with no misalignments. Finite element models

= with Nmesh 6 is used in this simulation. A similar analysis will be performed in the next

section for the experimental FH gear set of Chapter 2. These two analyses are intended to

demonstrate the ability of the proposed model in handling both FH and FM gear sets. Figures

3.17(a,b) show contour plots of the maximum contact pressure distributions on the drive and

coast sides of the gear pair as predicted by the load distribution model of Kolivand and Kahraman

[31]. The horizontal and vertical axes in these figures are normalized with respect to the tooth

dimensions along the face width and profile (flank) directions, respectively, such that a tooth

surface can be mapped to a rectangular domain bounded by the toe and heel in the tooth

longitudinal direction and start of active profile (S.A.P.) and topland in the profile direction.

Figure 3.17 shows that the predicted maximum contact pressure distribution, and hence, the

approximate contact pattern location for both the drive and coast side conditions. Larger contact stresses at locations near the tip and S.A.P. are also evident, suggesting a certain amount of edge loading. These contact patterns imply that the locations of the maximum root stresses should shift towards the toe end as well.

Figure 3.18 shows the contour plots of the predicted instantaneous gear tooth root maximum principal stresses at various mesh instances for the load distribution provided on the drive side. Here, again and for plotting purposes, the root area is mapped onto a rectangular region bounded by the toe and heel in the face width direction and root center (R.C.) and the start of active profile (S.A.P.) in the profile direction. Six contour plots are shown for the convex side

115

Table 3.1: The basic design parameters of the example FM gear pair used in this study.

Parameter Pinion Gear

Number of teeth 11 41

Hand of Spiral Left Right

Mean spiral angle (deg) 40.1291 27.9828

Shaft angle (deg) 90

Shaft offset (mm) 19.0500

Outer cone distance (mm) 108.6592 112.4064

Generation type Generate Formate ®

Depthwise tooth taper FM

116

(a) 1.5 Topland [

1 GPa ]

0.5

R.C. 0 (b) 1.5 Topland [

1 GPa ]

0.5

R.C. 0 Toe Heel

Figure 3.17: Contact pressure contour plots of the example FM gear pair at T = 300 Nm under no misalignment condition for (a) the drive and (b) the coast sides.

117

Step 1 of 32 500 S.A.P.

450 R.C.

Step 9 of 32 S.A.P. 400

R.C. 350 Step 13 of 32 S.A.P. 300 [ MPa

R.C. ]

Step 18 of 32 250 S.A.P.

200 R.C. Step 25 of 32 S.A.P. 150

R.C. 100

Step 32 of 32

S.A.P. 50

R.C. 0 Toe Heel

Figure 3.18: Maximum principal stress contour plots of the gear root at different mesh instances for the drive side of the example FM gear pair at T = 300 Nm under no misalignment condition.

118

of the gear tooth that experiences the maximum load through the entire mesh cycles. Here, as the

instantaneous contact line moves from the heel to the toe, the maximum principal stress location

at the root follows it, reaching its absolute maximum (about 500 MPa) at the gear mesh increment

of 25 (one mesh cycle is represented here by 32 incremental positions. The same behavior is

observed in Fig. 3.19 along the coast side with peak principle stress value now reaching 550 MPa.

The plot in Fig. 3.20 shows the time histories for the six stress components from a virtual gauge,

named gauge 1, placed at the location of the maximum principal stress recorded for the drive side

at R = 81.6 mm and Z = 32.6 mm. In addition to the normal stresses ( σxx , σyy and σzz ), the shear stresses and in specific τ contributes significantly to the overall stress state at the root of yz

the gear. The corresponding maximum and minimum principal stress time histories ( σ1 and σ3 )

are plotted in Fig. 3.21. Also shown on the same plot are the stress values one would measure

with an actual gauge if a strain gauge were to be physically mounted on the gear at the same

specified location. This gauge would experience load for about three mesh cycles in line with the

theoretical contact ratio for this gear pair. As anticipated, the gauge would undergo a compressive

cycle prior to the tensile loading when the previous tooth is under load.

Maximum principle root stress contour plots similar to those in Fig. 3.19 are shown in

Fig. 3.22 for the pinion for the same mesh increments. As for the gear case, the maximum value

of the first principal stress occurs at the 25th mesh increment at a location closer to the toe than

the heel. This peak is around 500 MPa that indicate that this design balances the stresses between

the gear and the pinion well in terms of bending strength. It is also noted that the difference in

the root stress contour shapes between the pinion and gear is a direct result of the difference in

their root geometries. The time histories of both the first and third principal stresses at the

119

Step 1 of 32 550 S.A.P.

500 R.C.

Step 9 of 32 S.A.P. 450

R.C. 400

Step 13 of 32 S.A.P. 350 [ MPa

300

R.C. ]

Step 18 of 32 S.A.P. 250

R.C. 200 Step 25 of 32 S.A.P. 150

R.C. 100 Step 32 of 32

S.A.P. 50

R.C. 0 Toe Heel

Figure 3.19: Stress contour plots of the gear root at different mesh instances for the coast side of the example FM gear pair at T = 300 Nm under zero misalignment condition.

120

400 σ 350 xx τ xy 300 τ xz 250 σ yy 200 τ yz σ 150 zz 100 Stress [MPa] 50

-50

-100 0 1 2 3 4 5 6 Mesh Cycles

Figure 3.20: Components of the stress at the location of maximum principal stress at the gear root for the drive side of example FM gear pair at T = 300 Nm with no misalignment.

121

600 σ 1 500 σ 3 Virtual Gauge 400

300

200 Stress [MPa]

100

-100 0 1 2 3 4 5 6 Mesh Cycles

Figure 3.21: Maximum and minimum principal stress time histories along with the corresponding virtual gauge readings at the location of maximum principal stress at the gear root for the drive side of the example FM gear pair at T = 300 Nm with no misalignment.

122

Step 1 of 32 550 S.A.P.

500 R.C.

Step 9 of 32 S.A.P. 450

R.C. 400

Step 13 of 32 S.A.P. 350 [ MPa

300

R.C. ]

Step 18 of 32 S.A.P. 250

R.C. 200 Step 25 of 32 S.A.P. 150

R.C. 100 Step 32 of 32

S.A.P. 50

R.C. 0 Toe Heel

Figure 3. 22: Maximum principal stress contour plots of the pinion root at different mesh instances for the drive side of the example FM gear pair at T = 300 Nm with no misalignment .

123

location where the maximum value of the pinion principal stresses is observed are plotted in Fig.

3.23. Again, the predicted readings of the virtual gauge are also shown in the same figure. This figure shows both the compressive and tensile parts of the stress signal as the pinion goes through mesh. The compressive component here follows the tensile as the pinion rotates in the opposite direction of the gear.

3.5 Model Validation and Comparisons to Experiments

The FH hypoid gear set used in the strain gauge experiments of Chapter 2 is modeled in this section under the same loading and misalignment conditions. The basic parameters of this

= gear pair were listed in Table 2.1. Finite element models with Nmesh 6 are used in this simulation as well. Only the gear root stress predictions at the locations of the strain gauges are presented as the main goal of this section is to provide a direct comparison between the measured and predicted root strains.

Figure 3.24(a) shows the predicted stress time histories at the five gauge locations shown in Figure 2.12 during a drive side operation at T = 300 Nm and the baseline no misalignment

condition. The corresponding measurements are shown in Fig. 3.24(b). It is seen that the stress

traces are in reasonable agreement both qualitatively and quantitatively. The predicted stresses

reach their peak values at around the same mesh positions as measurements. gauge D records the

highest stress while gauge C records the minimum among all the gauges across the face width.

Some differences between the predictions and experiments are observed in Fig. 3.24

especially for the gauge locations At and B close to the toe of the gear tooth. The maximum and

124

600 σ 500 1 σ 3 400 Virtual Gauge

300

200

100 Stress [MPa] 0

-100

-200

-300 0 1 2 3 4 5 6 Mesh Cycles

Figure 3.23: Maximum and minimum principal stress time histories along with the corresponding virtual gauge readings at the location of maximum principal stress at the pinion root for the drive side of example FM gear pair at T = 300 Nm under zero misalignment condition.

125

(a) (b) 200 D D B Eh B 100 Eh At At 0

-100

Stress [MPa] -200 126

C -300 C

-400 0 1 2 3 4 5 6 0 1 2 3 4 5 6 Mesh Cycles Mesh Cycles

Figure 3 24: (a) Predicted and (b) measured stress time histories of gauges At, B, C, D, and Eh for tooth #1 at T = 300 Nm and no misalignment condition.

minimum values at these locations are slightly larger in the predictions. There might be various

potential reasons for this. First of all, the location of the gauge is very critical to the resultant

stresses. While an effort was made to mount the gauges in precise locations, certain amount of

position errors is expected to potentially cause such differences. Also noted is that the stresses go

to near zero values in the predictions when a gauged tooth is outside the meshing zone while it

experiences certain amount of strain in measurements. This indicates that there are some gear rim

(blank) deflections that are not captured by the model. Since the gear blank gets thinner at the

toe, gauges towards that end are more affected by these deflections.

In order to assess the sensitivity of the model to misalignments, plots of peak-to-peak stresses for gauges placed at locations of At, B, C, D, and Eh are given for different misalignment values. The same misalignment amounts applied in the experiments are simulated here. Figure

3.25 shows predictions of the model for the V and G error conditions. Here, the predicted influence of the V error shown in Fig. 3.25(a) is very similar to the measured sensitivity presented in Fig. 2.32(a). Likewise the predicted sensitivities to the G error in Figure 3.25(b) are close agreement with measurements shown in Fig. 2.38(a). In both predictions and the measurements, introducing a positive V error resulted in a shift of the load distribution towards the tow while a positive G error caused this distribution to shift in the opposite direction from toe to heel. Similar agreements were obtained with the H and γ errors as well.

Besides the qualitative comparisons made between Fig. 3.25(a) and 2.32(a) as well as between Fig, 3.25(b) and 2.38(b), a direct comparison of predicted and measured peak-to-peak root stress values will be made next. Figure 3.26 compares the predictions to measurements at different V error values. In Fig 3.26(c) for V0= , the model and the measurements are in almost

127

(a) 700

600

500

400

300 Stress [MPa]

200 V = -0.2 mm V = -0.1 mm V = 0 100 V = 0.1 mm V = 0.2 mm 0 (b) 700

600

500

400

300 Stress [MPa]

200 G = -0.05 mm G = 0 G = 0.1 100 G = 0.2 mm G = 0.3 mm 0 At B C D Eh Gauges

Figure 3.25: Predicted effect of (a) the V error and (b) the G error on the peak-to-peak gear principal stress distributions at T = 300 Nm for the drive side.

128

700 (a) (b) 600

500 400

300

Stress [MPa] 200

100

0 700 (c) (d) 600

500

400

300

Stress [MPa] 200

100

0 At B C D Eh 700 Gauges (e) 600

500

400 Measurements 300 Predictions Stress [MPa] 200

100

0 At B C D Eh Gauges

Figure 3.26: Comparison of the measured and predicted peak-to-peak stress distributions along the face width at T = 300 Nm for (a) V= − 0.2 mm, (b) V= − 0.1 mm, (c) V0= , (d) V= 0.1 mm, and (e) V= 0.2 mm. 129 perfect agreement with the maximum deviation less than 4% at gauge B location. As the magnitude of the V increase, the model tends to deviate more from the measurements, indicating that model is more sensitive to the V error that the actual measurements. This is anticipated since the model is less compliant than the actual structure, given the rigid boundary conditions enforced.

Similar behavior is seen for the other three types of errors: H, G, and γ. The similar plots are shown for these errors in Figures 3.27 to 3.29. As stated in Chapter 2, this FH design is less sensitive to the H error than the V error. This is also depicted in the predictions as they compare better to measurements for all H error values as shown in Fig. 3.27. Still, the model tends to predict higher p-p stresses at higher H error values. Comparisons for the G error exhibit an opposite behavior than the one observed for the V error. The model predicts the p-p stresses variations more accurately for lower levels of the G error. As the contact zone shifts more severely towards one of the ends of the tooth, the predicted p-p stresses of at the locations of the gauges at these ends deviate from measurements the most. Finally, the comparisons for the case of different γ errors are shown in Fig. 3.29. Again, a reasonably good agreement is observed for all γ values except for γ=0.2 where the model is more sensitive to this error value. In summary, based to the comparisons presented above under various error conditions, it can be stated that the model is suitable to quantify the root stresses of hypoid gears with or without misalignments.

3.6 Summary

In this chapter, a root stress prediction model for spiral bevel and hypoid gears was proposed. Details of the definition of root geometries of face-milled and face-hobbed hypoid

130

700 (a) (b) 600

500 400

300

Stress [MPa] 200

100

0 700 (c) (d) 600

500

400

300

Stress [MPa] 200

100

0 At B C D Eh 700 Gauges (e) 600

500

400 Measurements 300 Predictions Stress [MPa] 200

100

0 At B C D Eh Gauges

Figure 3.27: Comparison of the measured and predicted peak-to-peak stress distributions along the face width at T = 300 Nm for (a) H= − 0.2 mm, (b) H= − 0.1 mm, (c) H0= , (d) H= 0.1 mm, and (e) H= 0.2 mm. 131

700 (a) (b) 600

500 400

300

Stress [MPa] 200

100

0 700 (c) (d) 600

500

400

300

Stress [MPa] 200

100

0 At B C D Eh 700 Gauges (e) 600

500

400 Measurements 300 Predictions Stress [MPa] 200

100

0 At B C D Eh Gauges

Figure 3.28: Comparison of the measured and predicted peak-to-peak stress distributions along the face width at T = 300 Nm for (a) G= − 0.05 mm, (b) G0= , (c) G= 0.1 mm, (d) G= 0.2 mm, and (e) G= 0.3 mm. 132

700 (a) (b) 600

500

400

300

Stress [MPa] 200 Measurements 100 Predictions 0

700 (c) (d) 600

500

400

300

Stress [MPa] 200

100

0 At B C D Eh At B C D Eh Gauges Gauges

Figure 3.29: Comparison of the measured and predicted peak-to-peak stress distributions along the face width, at T = 300 Nm for (a) γ=−0.1 , (b) γ=0 , (c) γ=0.1 and (d) γ=0.2 .

133

gears were presented. An automated mesh template was devised to construct finite element

models of segments of the pinion and the gear. The steps of the finite element solution were

outlined. An example analysis of a FM gear was presented to show time-varying and multiaxial nature of the root stresses. At the end, the model was compared to the experimental measurements of a FH gear pair presented Chapter 2 to show that the proposed model is reasonably accurate in predicting the root stresses of gear pairs having various misalignment condition.

134

CHAPTER 4

A MULTIAXIALTOOTH BENDING FATIGUE MODEL FOR SPIRAL BEVEL AND HYPOID GEARS

4.1 Introduction

The predicted and measured stress (or strain) time histories indicate that any point on a tooth root surface is subject to cyclic stress variations that have both tensile and compressive components. Each time the tooth passes through the gear mesh, these compression and tension cycles are repeated to cause fatigue damage leading to tooth breakage that represents a catastrophic failure of the gearbox [74]. The entire process of fatigue failure involves two stages, namely crack nucleation (initiation) and crack propagation. The process starts first with nucleation of micro cracks usually from surface discontinuities such as inclusions, second-phase particles, corrosion pits, grain boundaries, twin boundaries, pores, voids, and slip bands [75].

Under repetitive loading, these micro cracks grow on planes of maximum shear until they become visible to the naked eye, which are then called macro cracks. If the fluctuating stresses are sufficiently high, these macro cracks continue to grow in the plane of maximum tensile stress.

This mode is called the tensile mode or Stage II.

The type of fatigue depends on the level of stresses. If the fluctuating stresses are relatively low, the material will spend most of its life in the crack initiation phase and this is what

135

is referred to high-cycle fatigue. On the other hand, if the strains are high enough to cause large

plastic deformations then a larger portion of metal life is spent in the propagation or crack growth

phase. Figure 4.1 describes the difference between high-cycle and low-cycle fatigue phenomena.

Here, at a given applied stress amplitude σa , the fatigue life is partitioned between certain

number of cycles for crack initiation, labeled N fi, , and another number of cycles to fracture

σ g marked by N fp, . For higher a values, N fp, constitutes a large portion of the total life while

σ N fi, dominate the total life at lower a values. The latter condition is more representative of the loading conditions on gears, making the prediction of crack initiation life the main focus [38]. In other words, tooth bending failures of gears is generally high-cycle fatigue failures. Such failures occur in the root region, far from the contact zone. The failure starts with crack nucleation on the surface of the root at the loaded side where it experiences the maximum tensile stress [75]. The crack then propagates to the point of zero-stress down into the root where it changes direction to proceeds to the free stress point on the other side of the root.

To date, uniaxial gear bending fatigue models using the Stress-Life (S-N) approach have been used for gear bending fatigue problems. The advantage of these models is that they are simple to apply. As illustrated in Fig. 3.20, root regions of spiral bevel and hypoid gears are subject to more complex multiaxial stress states where use of such uniaxial models might be flawed. In this chapter, first, a conventional uniaxial crack nucleation model will be applied to an example hypoid gear pair. Then, a multiaxial fatigue model based on Liu’s characteristic plane criterion [54] will be proposed to assess the accuracy of the uniaxial model. Finally, the multiaxial bending fatigue model will be used to perform various parametric studies of the effect of misalignments and the blade edge radius on the bending fatigue life of an example hypoid gear pair. 136

Fatigue Crack Nucleation

σ a Fatigue Crack Growth

g σ a Final Fracture

N g g fp, N fi,

N f

Figure 4.1: The two phases of fatigue failure as a function of the applied stress amplitude.

137

4.2 A Uniaxial Tooth Bending Fatigue Model

The uniaxial high-cycle bending fatigue models for crack initiation life predictions

consider the principal stresses at the root. These stresses are indeed equivalent to the strain gauge

readings from the measurements of Chapter 2 as demonstrated in Figures 3.21 and 3.23 since the

direction of the maximum principal stress coincides with the normal to the fillet region. In

practice, the conventional way of predicting the fatigue life of a gear pair starts with finding the

critical location on the root. This is the location where the first principal normal stress exhibits a

maximum value throughout the loading cycle. Given the location, the first and third principal

cr cr cr stress time histories, namely σσ13(tt ) and ( ) are used to determine the alternating ( σa ) and

cr the mean ( σm ) values of the stress as

1 σ=cr max( σcr ) − min( σ cr ) , (4.1a) a 2 13

1 σ=cr max( σcr ) + min( σ cr ) . (4.1b) m 2 13

In order to account for the mean stress, an empirical relation such as the Goodman’s relationship is often used. This criterion states that failure occurs when the following relation is satisfied:

σσcr cr am+≥1 (4.2) σσ Nuf

138

where σ and σ are the fatigue and the ultimate strength values of the gear material. Here, the N f u

fatigue strength σN is determined empirically for a specific material using a fully reversed f

bending test procedure that constitutes the empirical S-N curve. It represents the number of cycles N required for the material to fail for a given alternating normal stress σ . It has f N f

often been assumed that there is a certain stress value, called endurance limit, below which the

material will not fail through cyclic loading. More conservative approaches assume no

endurance limit and a relationship between the σN and N f . The well-known Basquin’s f

equation is used for the finite life calculations:

σ=σ()N bσ (4.3) Nf ff

σ where f and bσ are material constants that are determined empirically. Equations (4.2) and

(4.3) are applied to determine the predicted number of crack initiation life cycles N for a given f

cr cr set of σa and σm values.

4.3 A Multiaxial Tooth Bending Fatigue Model for Hypoid Gears

As stated in Chapter 1, there are a large number of multiaxial fatigue criteria in the literature, none of which has been adapted as a widely accepted criterion. Studies that applied a multiaxial fatigue criterion to gear fatigue problems are also a few [41, 76]. No published model is available for predicting the tooth bending fatigue lives of hypoid gears.

The complexity of the geometry and loading conditions in hypoid gears dictate a multiaxial and non-proportional stress state at the fillet and root regions. Figure 4.2 shows plots

139

1200 S.A.P.

1000 [

(81.7,30.8) MPa 800 ] (94.9,37.7) 600 (72.9,28.9) 400 R.C. 200

Toe Heel

600 600 600

400 400 400

200 200 200 140

0 0 0

-200 σ -200 σ -200 σ xx xx xx τ τ τ xy xy xy Stress [MPa] Stress [MPa] -400 Stress [MPa] -400 -400 τ τ τ xz xz xz -600 σ -600 σ -600 σ yy yy yy τ τ τ -800 yz -800 yz -800 yz σ σ σ zz zz zz -1000 -1000 -1000 0 2 4 6 8 0 2 4 6 8 0 2 4 6 8 Mesh Cycles Mesh Cycles Mesh Cycles

Figure 4.2: Predicted stress tensor time histories at three different gear root locations for T =1000 Nm.

of all six components (normal and shear) of stress time histories at three different (,)RZ

locations of (72.9,28.9) , (81.7,30.8) , and (94.9,37.7) mm. All these arbitrary root locations experience multiaxial stresses. In addition to having normal and shear stress components contributing to the overall stress state at the root, the direction of the maximum principal stress changes throughout the mesh cycle such that the loading is non-proportional.

4.3.1 Characteristic Plane Based Multiaxial Fatigue Criterion

In the characteristic plane approach proposed by Liu et al [54] and applied earlier to roller [76] and gear [66] contact problems successfully, will be used here to predict the crack nucleation tooth bending fatigue lives of hypoid gear pairs. The characteristic plane criterion includes multiaxial and non-proportional stress states. Two different planes are distinguished here, (i) the fracture plane (fp) and (ii) the characteristic plane (cp). The fracture plane is assumed to be the plane that experiences maximum normal stress amplitude while the characteristic plane is a material-dependent plane where the hydrostatic stress amplitude is zero.

The fatigue damage will be evaluated here on this characteristic plane. The angle αcp that

defines the orientation of the characteristic plane from the fracture plane is a material dependent

parameter, defined as [54]

11 4− 4( − 3)(5 −− 4s2 ) − 2 22f cp 1 −1 ssff α=cos  (4.4a) 2 1 2 2(5−− 4s f ) s2 f

141

where s is the ratio of fatigue limit for a fully reversible shear (torsion) loading to that for a f

fully reversed bending (axial) loading and given by

τR =−1 s = f . (4.4b) f σ R f =−1

Note that 13≤≤s f 1 for typical gear materials. The damage criterion defined on this plane

[54] is

2 2 cp σ  1 σ N f σcp 1 +η xx, m + () τcp +τ cp 2 =σ (4.5a) βxx, a N ff στ  xy,, a xz a N f NNff  

cp cp where σxx, a and σxx, m are the predicted alternating amplitude and mean value of the normal stresses in the direction normal to the characteristic plane. Note that these stresses are given in a local coordinate system defined based on the orientation of the characteristic plane, as will be described later in this chapter. Meanwhile, τcp and τcp are the amplitudes of the shear xy, a xz, a

components on the characteristic plane. The other variables in Eq. (4.5a) are both material

dependent parameters defined as [54]

β=s22cos (2 αcp ) + sin 2 (2 αcp ) ff (4.5b)

σN −3 f  τσ 31NNff η=+ for > 1 (4.5c) N f 44 − τ 31 N f  

142

Here τ is the Basquin’s parameter for modeling the finite-life behavior of a material under N f pure shear loading. Here, the S-N plot for the shear loading can be represented by

τ=τ()N bτ (4.6) Nf ff

where, as in the pure bending case, τ and b are empirical material constants. f τ

4.3.2 Implementation of the Characteristic Plane Criterion

The complete stress states of every surface point along the root region of a hypoid gear

pair were predicted by using the model proposed in Chapter 3. For this purpose, the root surface

× of one tooth were discretized into ()mnrr grid points. The stress tensor time history σ()t at

each grid point (,ijrr ), imrr∈[1, ] and jnrr∈[1, ] is given as

σττ()ttt () () xx xy xz σ()t=τστxy () t yy () tt yz () (4.7)  ττσxz()ttt yz () zz ()

where t is time (or mesh cycles in this case).

Given the stress tensor for the grid point (,ijrr ), a search method to determine the

orientation of the plane of maximum normal stress amplitude is devised first. Here, the Euler

Transformation is employed in order to cover all the possible directions in the search. Using this

transformation, the new coordinate system (,xyzuuu ,) is defined by three unique rotation angles

143

αβuu, , and γu , as illustrated in Fig. 4.3. The stress tensor about this coordinate frame is given

u σσ()tt=Mu ( αβγ uuu , , ) () (4.8a)

where Mu is the transformation matrix that is defined by

cosγγuu sin 0 1 0 0  cosααu sin u 0     Mu(αβγ uuu , , ) =− sin γ u cos γ u 0 0 cosβu sin β u  − sin α u cos α u 0  . (4.8b) 0 0 1 0−β sin cos β  0 0 1  uu  

The search spans all planes within the ranges of the three angles α∈u [0, π / 2],β∈u [0, π / 2] ,

u and γ∈u [0, π / 2] . At each orientation, the stress state σ ()t is used to determined the normal

u amplitude σ1,a . The fracture plane is specified then as the orientation at which the following

relation is satisfied:

uu σ=xx, a max( σ1,a ) . (4.9)

fp fp′′ fp The coordinate system X fp′ = (,xy , z ) at this position, as shown in Fig. 4.4, is defined by

fp fp′ fp fp the angles αu , βu and γu . This system is then rotated about the x axis of this new coordinate system until the amplitude of the shear component on the plane in the y-direction,

τ fp' τ=fp′′ βfp fp β∈πfp' xy, a is maximized, i.e. max(xy, a ) maxMu (0, u ,0)σ (t ) with u [0, / 2]. The Euler

fp fp fp angles that determine the new coordinate system X fp = (,xyz ,) attached to the fracture

fp fp fp plane are αu , βu and γu . The stress tensor is written with respect to X fp as

144

yu zu

βu

y x γ u αu u x

N u

Figure 4.3: Definition of the Euler transformations.

145

fp x Characteristic Plane ycp xcp zzcp, fp z fp' fp αcp y Fracture Plane

y fp'

x z

Figure 4.4: Definition of the fracture plane and the characteristic plane.

146

fp fp fp fp σσ()tt=Muu ( αβγ , u , u ) (). (4.10)

In order to obtain the characteristic plane, the coordinate system X is rotated about z fp , which fp the direction of the minimum in-plane shear amplitude, by the material dependent angle αcp

defined in Eq. (4.4a). Here, the characteristic plane is governed by a new coordinate system

X = (,xyzcp cp ,) cp where zcp coincides z fp . The stress tensor with respect to X is cp cp

cp cp fp σσ(tt )=Muu ( α ,0,0) ( ). (4.11)

Finally, the fatigue equation given in Eq. (4.5a) is applied and the number of cycles N f

predicted for crack initiation at (,ijrr ) is computed. The overall computational methodology is summarized in the flow chart of Fig. 4.5. The same methodology is applied to all the grid points along the gear root surface to generate crack initiation life map.

4.4 An Example Tooth Bending Fatigue Life Simulation

First, the crack initiation tooth bending fatigue life of the gear of the example face-milled hypoid gear set of Table 3.1 is simulated by using the uniaxial procedure outlined in Section 4.2 at a pinion torque value of T =1000 Nm and no misalignments. The maximum principal gear root stress contour plot was shown earlier in Fig. 4.2(a). Here, the peak stress is around 1200 MPa and occurs in the middle of the tooth at (RZ , )= (83.5, 33.4) mm. Using Eq. (4.1), the alternating

σ=cr amplitude and mean values at this critical point are calculated to be a 673.8 MPa and

cr σ=m 471.8 MPa . Using the material properties and the pure bending S-N curve given by Li and

Kahraman [76] for a representative gear steel, and applying Eq. (4.2) and (4.3), the number 147

Discretize Root (mnrr× ) Surface grid

Hypoid Root Stresses Prediction Model

new (ijrr , ) imrr∈[1, ] π jnrr∈[1, ]  α∈u 0, 2 new αβγ , , ( uuu) Apply Euler π β∈u 0, Transformation 2 π γ∈u 0, u 2 σ()tt= Mσu ()

No uu σ=xx, a max( σ1,a ) ?

Yes αβγfp,, fp fp fp fp ( uuu) Find βτu / xy is max. σ fp ()t

cp cp cp αβγuuu,, cp cp ( ) Apply α=αu σcp ()t

Material Apply Fatigue Properties Criterion

N f

Figure 4.5: Computational methodology of the proposed multiaxial tooth bending fatigue model.

148

of cycles to failure at the critical location is predicted to be 16.8 million cycles. Repeating the

same procedure to all other root surface grid points, the contour plot the uniaxial fatigue life

shown in Fig. 4.6(c) is obtained.

Next, the same example application will be simulated by using the characteristic plane

based multiaxial fatigue criterion defined in Section 4.3. Here, the multiaxial model predicts only

1.14 million cycles at the critical location, nearly 15 times less than the value predicted by the

uniaxial model. The critical location predicted using the multiaxial approach is at

(RZ , )= (83.1, 33.3) mm that is slightly different than the critical location defined above by using the uniaxial scheme. The predicted multiaxial bending fatigue life distribution along the root region is shown in Fig. 4.6(d). Comparing the life contour maps for the uniaxial and multiaxial models, it is observed that the life values at ant given location are at least an order of magnitude lower for the multiaxial model. For more direct comparison, the variations of the predicted number of cycles to failure from both approaches are plotted along two different directions.

Figure 4.7(a) compares the predictions of the models along the root profile at the middle of the face width. Here, the trends are very similar while the curve for the multiaxial model is shifted down. A similar behavior is seen in Fig. 4.7(b) for the life traces along a trajectory in the longitudinal (face with) direction at the middle of the root profile.

This example analysis shows that the multiaxial and non-proportional nature of the root stresses of hypoid and spiral bevel gears is critical to the predicted life values. They cannot be overlooked for the sake of the simplicity of the uniaxial model if one seeks accurate quantitative predictions of tooth bending life. For this reason, the parametric studies presented in the next section will consider only the multiaxial model for quantifying the influence of misalignments

149

(a) Topland 2.5

2 ]

1.5 GPa [

0.5 R.C. 0 (b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.6: (a) The contact pressure distribution, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm with no misalignment.

150

(a) 9.5 Uniaxial 9 Multiaxial

8.5

f 8 N 10 7.5 Log

6.5

6 0 0.2 0.4 0.6 0.8 1 Normalized Root Profile

(b) 20 Uniaxial 18 Multiaxial

f 14 N 10 12 Log

6 0 0.2 0.4 0.6 0.8 1 Normalized Face Width

Figure 4.7: Comparison of the crack initiation lives from the uniaxial and multiaxial models along (a) the root profile in the middle of the face width and (b) the face width in the middle of the root profile. T =1000 Nm with no misalignment.

151

and cutter geometry, while the uniaxial life contours will still be included for completeness

purposes.

4.5 Impact of Misalignments on Tooth Bending Fatigue Lives of Hypoid Gears

In this section, different amounts of misalignments are applied to the same FM hypoid

gear pair used in the previous section and the resultant changes from the baseline (no

misalignment) life values of Fig. 4.6 are quantified at T =1000 Nm. Table 4.1 summarizes four example misalignment conditions considered here. The first of these four error conditions with

V0= , H= 0.1 mm, G= 0.1 mm and γ=0 shifts the contact zone to a high bearing position in

the middle of the tooth as shown in the simulated unloaded contact pattern of Fig. 4.8(a). The

resultant maximum principal root stresses as well as the predicted fatigue lives using uniaxial and

multiaxial approaches are given in Fig. 4.11(b), (c), and (d), respectively. The location with the

minimum life is at (RZ , )= (86.5, 35.1) mm and the minimum life is 0.7 million cycles that is

about 40% less the zero misalignment case.

Figure 4.9 shows the predictions for the second misalignment case of V= − 0.1 mm,

H0= , G= 0.15 mm and γ=0 . With the contact shifted towards root and heel, the minimum life at the critical point of (RZ , )= (91.8, 36.7) is 0.8 million cycles that is about 30% less the

zero misalignment case. The results for the other two misalignment conditions are presented in

Figures 4.10 and 4.11. In Fig. 4.10 with V= − 0.1 mm, H= − 0.05 mm, G0= and γ=0.1 ,

contact patterns and the resultant root stresses are such that the tooth bending fatigue life

improves about 6 times to 8.7 million cycles, suggesting that the influence misalignments on

152

Table 4.1: Example misalignment configurations used in the tooth bending fatigue parametric study.

Case V [mm] H [mm] G [mm] γ [deg.]

1 0 0.1 0.1 0

2 -0.1 0 0.15 0

3 -0.1 -0.05 0 -0.1

4 0.1 0 -0.1 0

153

(a) Topland

R.C.

(b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.8: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm with V0= , H= 0.1 mm, G= 0.1 mm, and γ=0 .

154

(a) Topland

R.C.

(b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.9: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm with V= − 0.1 mm, H0= , G= 0.15 mm, and γ=0 .

155

(a) Topland

R.C.

(b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.10: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm with V= − 0.1 mm, H= − 0.05 mm, G0= , and γ=−0.1o .

156

(a) Topland

R.C.

(b) 1.75 S.A.P. 1.5 1.25 ]

1 GPa [ 0.75 0.5 0.25 R.C. 0 (c) 20 S.A.P. )]

15 f N (

10 10

5 [Log R.C. 2 (d) 20 S.A.P. )] 15 f N (

10 10

5 [Log R.C. 2 Toe Heel

Figure 4.11: Predicted (a) unloaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm with V= 0.1 mm, H0= , G= − 0.1 mm, and γ=0 .

157

bending fatigue performance is not always negative. On the other hand, misalignment conditions

of V= 0.1 mm, H0= , G= − 0.1 mm and γ=0 has a detrimental effect on contact patters that is moved to an extreme toe-edge location of the tooth. As a result, the predicted bending fatigue life is about four orders of magnitude lower than the baseline case of no misalignment.

Figure 4.12(a) shows the combined influence of the V and the H errors within ranges of

±0.2 mm on the minimum bending fatigue lives of the same FM gear at T = 800 Nm with

G0=γ= . Here the minimum fatigue life N f values are normalized with respect to the corresponding no misalignment fatigue life value of N f ,no error . Here, it is seen that the influence

of the H error on N f diminishes for a negative value of the V error. Likewise, the sensitivity of the N f to the V error is somewhat reduced in the presence of a positive H error. Overall trend is

that a combination of a positive V error and a negative H error is the most harmful to the fatigue

life of the gear set. For instance, at V= 0.2 mm and H= − 0.2 mm, N f value is reduced by

almost 6 orders of magnitude in comparison to the baseline no misalignment case. A very similar

influence is observed in Fig. 4.12(b), which shows the combined influence of the V and G errors.

One observes here that the system becomes totally insensitive to the V error at G= 0.2 mm while the same is true for the G error when V= − 0.2 mm. In combination, a negative G and a positive

V is the most detrimental life condition.

Combined influences of error pairs V and γ, and H and G are shown next in Fig. 4.13(a) and (b), respectively, for the same gear set at T = 800 Nm. In Fig. 4.13(a), the behavior for the V

and γ errors is very similar to those observed in Fig. 4.12, where the γ error plays the same role the H and G errors played earlier. The worst case condition is at a positive V and a negative γ.

158

(a)

) 0

f,no error -2 /N f -4 (N 10 -6 Log -8 0.2 0.2 0 0 V [mm] H [mm] -0.2 -0.2 (b)

) 0

f,no error -2 /N f -4 (N 10 -6 Log -8 0.2 0.2 0 V [mm] 0 G [mm] -0.2 -0.2

Figure 4.12: Combined influence of (a) the V and H errors and (b) the V and G errors on the predicted crack initiation bending fatigue lives of the example FM gear set at T = 800 Nm.

159

(a)

2 ) 0 f,no error

/N -2 f (N

10 -4 Log -6 0.2 0.2 0 V [mm] 0 γ [deg.] -0.2 -0.2 (b)

) 0

f,no error -2 /N f -4 (N 10 -6 Log -8 0.2 0.2 0 G [mm] H [mm] 0 -0.2 -0.2

Figure 4.13: Combined influence of (a) the V and γ errors and (b) the H and G errors on the predicted crack initiation bending fatigue lives of the example FM gear set at T = 800 Nm.

160

In Fig. 4.13(b), N f values remain unchanged if either H or G error is on the positive side, with

the negative valued H and G representing the minimum values of N f .

Finally, the influences of the remaining two pairs of errors on N f are displayed in Fig.

4.14. In Fig. 4.14(a), the coupling between the H and γ errors is quite involved. For a negative γ

value, N f is reduced significantly as H is moved from its positive extreme to negative. This sensitivity to H is reversed when γ is positive. The same is true for the influence of γ on N f with a positive or negative H. The worst case is when both H and γ are negative (about 5 orders of

o magnitude lower N f at H= − 0.2 mm and γ=−0.2 ). The trend in Fig. 4.14(b) for the error pair

G and γ is also very similar with G taking the place of the H error.

In view of Fig. 4.8 to 4.14, it can be stated that the misalignments of a hypoid gear set has a significant influence on the tooth bending strength of hypoid gears set. As summarized in Table

4.2, the worst case conditions exist when the V error is positive while at the same time one or more of the other three errors (H, G, or γ) are negative.

4.6 Influence of Blade Edge Radius on on Tooth Bending Fatigue Life of an Hypoid Gear

The blade edge radius, dictates the root geometry (fillet radius) of a hypoid gear pair for both face-milled and face-hobbed designs [77]. Due to the difficulty in finishing the surface at the root of spiral bevel and hypoid gear pairs, the soft cut defines the final root shape almost exclusively, making the choice of the blade edge (as parameterized in Fig. 3.4) a very critical task in design of these types of gears. Traditionally, the largest possible edge radius is applied to the cutting blade such that it does not (i) eliminate the Toprem (if exists), (ii) interfere with the 161

(a)

2 ) 0 f,no error

/N -2 f (N

10 -4 Log -6 0.2 0.2 0 γ H [mm] 0 [deg.] -0.2 -0.2 (b)

) 0

f,no error -2 /N f -4 (N 10 -6 Log -8 0.2 0.2 0 0

G [mm] -0.2 -0.2 γ [deg.]

Figure 4.14: Combined influence of (a) the H and γ errors and (b) the G and γ errors on the predicted crack initiation bending fatigue lives of the example FM gear set at T = 800 Nm..

162

Table 4.2: The worst case life conditions for the effect of misalignment combinations.

Combination V [mm] H [mm] G [mm] γ [deg.]

V-H + −

V-G + −

V-γ + −

H-G − −

H-γ − −

G-γ − −

Overall + − − −

163

tip cone of the other member, or (iii) cause mutilation (cutting the unintended side of the surface).

Although it has been a common practice to include a larger edge radius, the effect of this radius on the bending fatigue of hypoid gears is not known quantitatively. The goal here is to investigate the influence of this parameter on the bending strength of a hypoid gear.

Using the same example FM gear pair, six different edge radii ( ρ=t 0.50 ,1.0, 1.25, 1.50,

1.65 and 1.70 mm) in addition to the given “baseline” radius of ρ=0.762 mm, are applied and t the corresponding root stresses and fatigue lives are predicted at T =1000 Nm. These cutting

edge geometries are shown in Fig. 4.15. Contact and root stress distributions, and the

corresponding fatigue life distributions for the smallest ( ρ=0.50 mm) and the largest ( ρ=1.70 t t mm) edge radii are presented in Figures 4.16 and 4.17, respectively. For ρ=0.50 mm, the t contact pressure distribution presented in Fig. 4.16 is very close to Fig. 4.6(a) for ρ=0.762 mm. t This is expected since the blade edge radius has nothing to do with the generation of contact surface areas. Yet, the maximum principal root stress is increased by about 2% for ρ=0.50 t mm, resulting in a 32% reduction in bending life (using the multiaxial model) as compared to the

case when ρ=0.762 mm. On the other hand, the results for ρ=1.70 mm in Fig. 4.17 show a t t reduction in the maximum principal root stress of about 8% compared to the baseline case of

ρ=0.762 mm. As a result, the bending fatigue life is increased 10 times. Figure 4.18 plots the t

variation of the bending fatigue life of the gear (normalized with respect to the life N f for

ρ=0.762 mm) as a function of ρ . It reveals one-order-of-magnitude spread of the tooth t t bending fatigue life as a direct result of ρt . It also demonstrates the ability of the proposed methodology in quantifying the influence of such key parameters on durability of the gear set.

164

(a) (b) 2.5 -1.2

2 -1.3

1.5 ρ=1.70 mm -1.4 t 1 1.65 mm -1.5 1.50 mm 0.5 -1.6 [mm]

1.25 mm t

z 0 -1.7 1.00 mm -0.5 0.762 mm -1.8 -1

165 0.50 mm -1.9 -1.5

-2 -2 72.5 73 73.5 74 74.5 75 75.5 76 76.5 74.5 75 75.5 76 x [mm] x [mm] t t

Figure 4.15: (a) Cutting blade shape with different edge radii and (b) a zoom-in view of the edge part.

(a) Topland 2.5

2 ]

1.5 GPa [

0.5 R.C. 0 (b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.16: Predicted (a) loaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm for ρ=t 0.5 mm.

166

(a) Topland 2.5

2 ]

1.5 GPa [

0.5 R.C. 0 (b) 1.25 S.A.P.

1 ] 0.75 GPa [ 0.5 0.25 R.C. 0 (c) 20 S.A.P. )] 15 f N ( 10 10 [Log R.C. 5.5 (d) 20 S.A.P. )] f

15 N ( 10 10 [Log R.C. 5.5 Toe Heel

Figure 4.17: Predicted (a) loaded contact pattern, (b) the root stress distribution, (c) the uniaxial fatigue life distribution, and (d) the multiaxial fatigue life distribution of the example FM gear at T =1000 Nm for ρ=t 1.70 mm.

167

NNff/ 7 6

0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 ρ [mm] ρ= t t 0.762

Figure 4.18: Normalized life versus ρt of the example FM gear at T =1000 Nm.

168

4.7 Summary

In this chapter, a multiaxial fatigue criterion using the characteristic plane approach was

applied to the hypoid gear problem to predict the location and the life of the crack initiation tooth

bending failure. A companion uniaxial model was also developed for the sole purpose of

demonstrating its inaccuracy, given the multi-axial and non-proportional characteristics of the root stress states. At the end, the multiaxial model was used to investigate the influence of misalignments and the blade edge radius on the tooth bending fatigue lives of hypoid gears.

Both were shown to have significant impact on the durability of the example hypoid gear.

169

CHAPTER 5

CONCLUSIONS

5.1 Summary

In this study, the tooth bending strength of spiral bevel and hypoid gears was investigated

both experimentally and theoretically, focusing specifically on the impact of gear alignment

errors. On the experimental side, a new experimental set-up was designed and developed for operating a hypoid gear pair under typical load conditions in the presence of tightly-controlled magnitudes of gear misalignments. The test set-up allowed application of all four types of misalignments, namely the shaft offset error (V), the horizontal pinion position error (H), the horizontal gear position error (G) and the shaft angle error (γ). An example face-hobbed hypoid gear pair from an automotive axle unit was instrumented with a set of miniature strain gauges mounted at various root locations of three consecutive teeth. A multi-channel, digital signal acquisition and analysis system was devised for collection and analyzing strain signals simultaneously. A test matrix that included a number of test within the ranges of misalignments and input torque under both drive and coast conditions was executed. The experimental results were used to quantify the influence of misalignments on the root stress distributions along the face width.

170

On the theoretical side, the computational model developed in earlier by Kolivand and

Kahraman [31] was expanded to generate the root surfaces of spiral bevel and hypoid gears cut by

using either face-milling or face-hobbing processes. A new formulation was proposed to define

the gear blank and a numerically efficient cutting simulation methodology was developed to

compute the root surfaces from the machine settings, the cutter geometry and the basic design

parameters, including both Formate and Generate motions. The generated surfaces were used to develop customized finite element models of the N-tooth segments of the pinion and the gears via an automated mesh generator. The finite element model employed 20-noded isoparametric elements. Toot contact loads predicted by the model of Kolivand and Kahraman [31] model were converted to nodal forces based on the same shape function used to interpolate for nodal displacements. A skyline solver was used to compute the nodal displacements and the resultant stresses at the Gauss points. An extrapolation matrix based on the least-square error formulation was applied to compute the stresses at the root surfaces. Predicted gear root stresses were compared to the measurements to demonstrate a good correlation, including not only the extreme stress values but also the stress time histories. Through the same comparisons, the model was also shown to capture the impact of misalignments on root stress distributions reasonably well.

With the predicted root stress information in hand, a multiaxial, crack nucleation fatigue model of tooth bending was proposed. The model employed Liu’s [52] characteristic plane approach and accounted for the multiaxial and non-proportional nature of the stress states of the spiral bevel and hypoid gear pairs. Fatigue lives predicted by the proposed model were compared to those estimated by using a conventional uniaxial failure criterion to show that the predicted multiaxial fatigue lives were consistently an order of magnitude lower that those predicted by the uniaxial model. The fatigue model was used to quantify the influence of the misalignments on the bending fatigue life of the gear pair. Certain combinations of misalignments were shown to be 171 detrimental to the bending fatigue strength of hypoid gears. At the end, the impact of edge blade radius on the predicted crack initiation bending lives was also demonstrated to be significant.

5.2 Conclusions and Contributions

This study resulted in extensive experimental and theoretical results to enhance the current level of understanding of the tooth bending strength of spiral bevel and hypoid gears.

Some of the main contributions of this study can be listed as follows:

• The experimental set-up proposed to induce misalignments in a tightly controlled and

repeatable manner was shown perform well to allow the study of the effect of

misalignments. The set-up is novel and can be considered as a major contribution to the

experimental hypoid gear literature. Equally important is the ability of the set-up to

operate a hypoid gear pair under realistic load levels at very low (quasi-static) speeds

such that the collected data is compatible with loaded tooth contact models of hypoid

gears.

• An experimental database generated in this study is extensive to be considered as the

most comprehensive set of hypoid gear experiments in the gear literature including

misalignments.

• The hypoid gear root stress prediction model proposed in this study can be considered as

a fast and accurate alternative to full-scale deformable body models, presenting itself as

an efficient design tool, rather than being a computational analysis tool. The formulation

proposed to define the hypoid gear root geometries is the first formulation of its kind as

all the loaded tooth contact models available in the literature fail to extend the analysis to

172

the root regions. It is generic in the sense it simulates both face-milled and face-hobbed

designs using Formate or Generate motions.

• The model compared reasonably well to experiments under even extreme misalignment

conditions, making it the first validated hypoid gear root stress prediction model.

• The predicted hypoid gear root stresses were shown to be multiaxial and non-

proportional, indicating clearly that a uniaxial failure criterion is inadequate. The crack

nucleation life prediction methodology proposed here is the first such model that uses a

multiaxial fatigue criterion.

• Influence of misalignments and the cutter geometry were both shown to be critical to the

bending fatigue lives of hypoid gear pairs. For any new gear pair design, similar

parametric studies can be performed by using this model to establish acceptable ranges of

the tolerances dictating allowable misalignment ranges as well as designing the cutter

blade optimally.

5.3 Recommendations for Future Work

The following are listed as potential future studies towards enhancing and complementing the work presented in this dissertation:

• The experimental methodology is such that measurements from face-milled hypoid gear

pairs as well as spiral bevel and other right-angle, cross-axis gear pairs can also be added

to the database generated.

173

• The mathematical model for computing the exact root surfaces of spiral bevel and hypoid

gears provides the root geometry information that is required to predict the gear mesh

pocketing (pumping) losses of spiral bevel and hypoid gears. Such losses were shown to

represent a significant portion of load independent (spin) power losses of spur gears [78,

79] where the spaces between the gears were straightforward to compute. In the case of

spiral bevel and hypoid gears, computation of the change of volume of the pockets

formed between the teeth of the mating gears is not a trivial matter, yet possible with the

aid of the root surface generation formulation of this study.

• The methodology used to predict root stresses is generic, allowing the resultant root

stresses due to any load distribution. This study was kept limited to low speed behavior

where dynamic effects are negligible. Provided that gear load distributions can be

computed under dynamic conditions, this model can be used in its current form to

compute the resultant dynamic root stresses.

• The multiaxial fatigue model proposed requires basic material data in the form of purely

torsional and purely axial stress-life data. Such data is not available in the literature for

even most common gear materials. Generation of such data is critical to the fidelity of

the quantitative life prediction values using this model.

• The multiaxial fatigue model can be extended to include crack propagation predictions as

the initial angle of the crack direction has already been specified by the computed

characteristic plane orientations.

• The model can be used to perform additional sensitivity analyses to assess the effect of

other system-level factors, machine settings, and design parameters on the bending

174 performance of spiral bevel and hypoid gears, in the process, generating general guidelines in regards to bending strength of such gears.

175

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