DEVELOPMENT OF TOOTH CONTACT AND MECHANICAL EFFICIENCY MODELS FOR FACE-MILLED AND FACE-HOBBED HYPOID AND SPIRAL BEVEL GEARS

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

By Mohsen Kolivand, B.S., M.S. ***** The Ohio State University 2009

Dissertation Committee: Approved by Professor Ahmet Kahraman, Advisor Professor Donald R. Houser ________________________________ Professor Gary L. Kinzel Advisor Professor Henry H. Busby Graduate Program in Mechanical Engineering

© Copyright by Mohsen Kolivand 2009

ABSTRACT

A computationally efficient load distribution model is proposed for both facemilled and face-hobbed hypoid gears produced by Formate and generate processes. Tooth surfaces are defined directly from the cutter parameters and machine settings. A novel methodology based on the ease-off topography is used to determine the unloaded contact patterns. The proposed ease-off methodology finds the instantaneous contact curves through a surface of roll angles, allowing an accurate unloaded tooth contact analysis in a robust and accurate manner. Rayleigh-Ritz based shell models of teeth of the gear and pinion are developed to define the tooth compliances due to bending and shear effects efficiently in a semi-analytical manner.

Base rotation and contact

deformation effects are also included in the compliance formulations. With this, loaded contact patterns and transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears are computed by enforcing the compatibility and equilibrium conditions of the gear mesh. The proposed model requires significantly less computational effort than finite elements (FE) based models, making its use possible for extensive parameter sensitivity and design optimization studies. Comparisons to the predictions of a FE hypoid gear contact model are also provided to demonstrate the accuracy of the model under various load and misalignment conditions. ii

The proposed ease-off formulation is generalized next to include various types of tooth surface deviations in the tooth contact analysis. These deviations are grouped in two categories. The proposed ease-off based method is shown to be capable of modeling both global deviations due to common manufacturing errors and heat treat distortions and local deviations due to surface wear. The proposed loaded contact model is combined at the end with a friction model based on a mixed elastohydrodynamic lubrication model to predict the load dependent (mechanical) power losses and efficiency of the hypoid gear pairs. The velocity, radius of curvature and load information predicted by the contact model is input to the friction model to determine the distribution of the friction coefficient along the contact surfaces. At the end, the variations of predicted mechanical efficiency with geometry, surface and lubricant parameters are quantified.

iii

Dedicated to my mother

iv

ACKNOWLEDGMENTS

I would like to express my sincere gratitude to my advisor, Prof. Ahmet Kahraman, for this great research opportunity, his guidance throughout my research and his effort in reviewing this dissertation. I would also like to express my appreciation to Prof. Donald R. Houser, Prof. Gary L. Kinzel and Prof. Henry H. Busby for their patience and effort in being a part of my dissertation committee. Also, thanks to Dr. Sandeep Vijayakar who kindly permitted me to use CALYX package. I would like to thank the sponsors of the Gear Power Transmission Research Laboratory for their financial support throughout my study. My sincere thanks go to Prof. Hermann J. Stadtfeld from The Gleason Works who spent valuable time teaching me fundamental concepts of bevel gear design and manufacturing, and also for having attended my dissertation defense. I would also like to thank Jonny Harianto, Samuel Shon and all my lab mates for their help and friendship throughout my study at OSU and beyond. Finally, I deeply appreciate the love and trust shown toward me by my parents, my grandparents, my uncles and aunts, my sisters and brother and my fiancée for all of their support and encouragement.

v

VITA Oct. 19, 1976 ……………….……..

Born – Tehran, Iran

Sep. 1995 – Sep. 1999 ……………. B.S. in Mechanical Engineering, Solid Mechanics, Tehran University, Tehran, Iran Sep. 1999 –Feb. 2002 …………….

M.S. in Mechanical Engineering, Applied Design, Tehran University, Tehran, Iran

Nov. 1999 –Apr. 2002 ……………. Design Engineer, Tarh Negasht Co., Tehran, Iran Apr. 2002 –Aug. 2005 ……………. Design Engineer, TAM Co., Tehran, Iran Jan. 2006 – present ………………

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

PUBLICATIONS 1. Kolivand M. and Kahraman A., “A Load Distribution Model for Hypoid Gears Using Ease-off Topography and Shell Theory,” Journal of Mechanism and Machine Theory, 2009.

FIELDS OF STUDY Major Field: Mechanical Engineering vi

TABLE OF CONTENTS

Page Abstract……………………………………………………………………………….

ii

Dedication…………………………………………………………………………….

iv

Acknowledgments…………………………………………………………………....

v

Vita……………………………………………………………………………….…..

vi

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

x

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

xi

Nomenclature………………………………………………………………………...

xvi

Chapters: 1 Introduction ……………………………………………………………………...

1

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

1

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

4

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

14

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

17

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

19

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

20

2

Definition of Face-milled and Face-hobbed Hypoid Gear Geometry and Unloaded Tooth Contact Analysis ……………………………………..........…...

28

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

28

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

30

vii

3

4

2.2.1

Kinematics ………………………………………………………...…

32

2.2.2

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

36

2.2.3

Equation of Meshing………………………………………………….

40

2.2.4

Principal Curvatures and Principal Directions………………….........

42

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

44

2.3.1

The Conventional Method of UTCA.……………….………………..

47

2.3.2

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

52

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

53

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

57

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

59

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

65

Shell Based Hypoid Tooth Compliance Model and Loaded Tooth Contact Analysis ….............................................................................................................

68

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

68

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

70

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

81

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

84

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

92

Loaded Tooth Contact Analysis of Hypoid Gears with Local and Global Surface Deviations…………………………………..….…………………………………

94

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

94

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

98

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

102

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

107

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

111

4.5.1

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

4.5.2

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

References for Chapter 4………………………………………………………... 130 5

Predictions of Mechanical Power losses of Hypoid Gear Pairs………………….. 133 5.1 Introduction ……………………………………………………………….…. 133 5.2 Hypoid Gear Mechanical Power Loss Model …………………………….…. 137 5.2.1

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

5.2.2

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

5.2.3

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

5.2.4

Computation of the Mechanical Power Loss of the Hypoid Gear Pair

151

5.3 Numerical Example ……………………………………..…………………... 152 5.4 Conclusion…………………………………………………………………… 167 References for Chapter 5………………………………………………………… 168 6

Conclusions and Recommendations for Future Work…...………………………. 172 6.1 Thesis Summary ……………………………………………………………... 172 6.2 Conclusion and Contributions………………………………………………..

175

6.3 Recommendations for Future Work …………………………………………. 177 Bibliography …………………………..………………...…………………………... 178

ix

LIST OF TABLES

Table 2.1

Page Basic drive side geometry and working parameters of the example hypoid gear pair…………………………………………………………………..…..

3.1

The loaded transmission error predictions of the proposed model; G  0 mm and   0 for all cases.………………………….…………………..…

4.1

60

88

Basic drive side geometry and working parameters of the example hypoid gear pair.…………………………………………………….……………….. 112

4.2

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

4.3

Basic drive side geometry and working parameters of the example hypoid gear pair.……………………………………………………………………... 121

4.4

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

5.1

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

5.2

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

5.3

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

5.4

Basic drive side geometry and working parameters of the examples hypoid gear pairs……………………………………………………………………... 153

x

LIST OF FIGURES

Figure 1.1

Page A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize passenger cars and SUV’s.……………………………………………………

2

1.2

A sample hypoid gear pair with a shaft angle  and a shaft off-set d a .…....

4

1.3

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

6

1.4

Flowchart of overall hypoid gear loaded tooth contact analysis methodology

18

2.1

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

31

2.2

Cradle based hypoid generator parameters…………………………………...

33

2.3

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

37

2.4

Generation process……………………………………………………………

41

2.5

Curvature computation procedure……………………………………………

43

2.6

General case of approximating gear surfaces as two contacting ellipsoids to orient instantaneous contact line……………………………………………...

49

2.7

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

55

2.8

Unloaded TCA computation procedure: (a) gear projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded transmission error………………………………………………….

2.9

58

Unloaded transmission error of the example gear pair with misalignments

E  0.15 mm, P  0.12 mm, G  0,   0 …………………………..

xi

61

2.10

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

i 1

,

i

and

i 1

(i-1),

(i)

and

(i+1)

with

E  0.15 mm, P  0.12 mm, G  0 and   0 ……………………… 2.11

63

Unloaded contact pattern of the example gear pair (a) at nominal position with E  P  G    0 , (b) at toe with E  0.08 mm, P  0.10 mm and G    0 , (c) at heel with E  0.15 mm, P  0.10 mm and G    0 and (d) at toe with

E  0.05 mm, P  G  0 , and

  4 min………………………………………………………………….

64

3.1

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

72

3.2

Flowchart of the compliance computation.…………………………….……..

79

3.3

Potential contact line discretization.……………………………...…………..

80

3.4

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

82

3.5

Static equilibrium between torque applied on gear axis and torque produced by the force of all contacting segments.………...…………………………....

3.6

85

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

P  0.12, G  0 and   0 at (a) Tp  50 Nm, (b) Tp  250 Nm, and (c) Tp  500 Nm.…………………………………………………………............ 3.7

87

Comparison of loaded contact patterns predicted by the proposed model to an FE model [3.11] for (a) Tp  50 Nm , E  0.15 mm, P  0.12 mm, (b)

Tp  250 Nm , E  0.15 mm,

E  0.15 mm,

P  0.12 mm,

P  0.12 mm,

(d)

(c)

Tp  50 Nm ,

Tp  500 Nm , E  0.08 mm,

P  0.05 mm, and (e) Tp  50 Nm , E  0.26 mm, P  0.13 mm ( all at

4.1

G  0,   0 ).……………………………………………………….....…

89

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

99

xii

4.2

Graphical demonstration of the procedure to update ease-off surface for surface deviations.………………………...…………………………………. 105

4.3

Graphical demonstration of the procedure to compute unloaded TCA; (a) gear projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded transmission error………………...

108

4.4

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

4.5

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

4.6

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

113

view of the projection plane, and  ,  , Q and Q surfaces, and contour plots of (b)  , (c)  , and (d) the change of ease-off topography..………… 114 4.7

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

4.8

Transmission error (UTE) curves for theoretical and deviated surfaces at (a) unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm. 117

4.9

Predicted contact pressure distribution for a pinion toque of 200 Nm for (a) theoretical and (b) deviated surfaces………………………………………… 119

4.10

Example global deviation surfaces measured by CMM for the gear and pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured deviation, (c) pinion deviation distribution in tooth active region and (d) gear deviation distribution in tooth active region……………………………. 122

4.11

Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical easeoff topography, (b) updated ease-off topography only with pinion deviation, (c) updated ease-off topography only with gear deviation, and (d) updated ease-off topography with both pinion and gear deviations…………………... 124

4.12

Predicted unloaded tooth contact pattern for separation value of   6 μm … 125 xiii

4.13

Transmission error curves for theoretical and deviated surfaces; (a) unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm

4.14

127

Predicted contact pressure distribution for a pinion toque of 200 Nm for (a) theoretical and (b) deviated surfaces………………………………………… 129

5.1

Flowchart

of

overall

hypoid

gear

efficiency

computation

methodology.………………………………………………………………... 5.2

138

Sliding and rolling velocities and their projection in tangential plane along and normal to the contact line direction.……………………………………... 141

5.3

Ease-off topography of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 …………………………………………………………... 154

5.4

Maximum contact pressure distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 for T p  500 Nm …………………… 155

5.5

Rolling velocity distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm …………………………… 157

5.6

Sliding velocity distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm …………………………… 158

5.7

Slide-to-roll ratio distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm …………………………... 159

5.8

Equivalent radius of curvature distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 ……………………….. 160

5.9

 distribution of (a) Design A with d a / Da  0.07 and (b) Design B with

d a / Da  0.14

at

 p  1500 rpm ,

T p  500 Nm ,

Toil  90 C

and S1  S 2  0.8 m ……………………………………………………….... 161

xiv

5.10

Friction coefficient  distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm , T p  500 Nm ,

Toil  90 C and S1  S2  0.8 m …………………………………………... 162 5.11

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

Toil  90 C and S1  S2  0.8 m ……………………………………….…. 163 5.12

Efficiency of (a) Design A with d a / Da  0.07 and (b) design B with d a / Da  0.14

for different surface finish and oil temperatures at

p

  1500 rpm and T p  500 Nm …………………………..……………… 166

xv

NOMENCLATURE

ag

Gear axis vector

ap

Pinion axis vector

C

Total compliance matrix

da

Pinion offset

Da

Gear pitch diameter

e1 , e2

Principal directions

E

Overall efficiency

Eb

Blank offset

Eeq

Equivalent module of elasticity

f k

Load per unit length of segment  at time step k

F

Force vector

h

Film thickness

hf

Tip of blade to reference point

htoe

Tooth height at toe

hheel

Tooth height at heel

iu

Normal to the mid-surface of the shell

iT

Tilt angle

js

Swivel angle

k1 , k2

Normal curvatures

K1 , K 2

Principal curvatures



Segment index

mg

Number of surface grid in lengthwise direction xvi

M ctb

Machine center to back

n

Normal to the family of cutter surface

ncl

Number of potential contact lines at each time step

ng

Number of surface grids in profile direction

n

Total number of contact segments

ns

Total number of time steps per pinion pitch

Nc

Total number of segment on all potential contact lines/curves

Ng

Number of teeth of gear

Np

Number of teeth of pinion

Nt

Number of blade groups

PE

Potential energy

q

Roll angle

qa

Pinion pitch

Q

Surface of roll angle

r

Gear ratio

rc

Cutter radius

R

Position vector of a point on circular cylindrical shell

Req

Equivalent Hertzian curvature

Rg

Vector of the distances of each segment to the gear axis

ij

Ease-off value of point ij



Ease-off surface

s

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

Sr

Radial setting

S

Initial separation vector xvii

SE

Strain energy

SR

Sliding to rolling velocity ratio

Seq

Equivalent surface roughness

t

Instantaneous potential contact line direction

t

Perpendicular to the instantaneous potential contact line direction

ttoe

Tooth thicknesses at toe-root

theel

Tooth thicknesses at heel-root

T

Torque

Toil

Oil temperature

(u g )t

Gear surface velocity in t direction

(u g )t 

Gear surface velocity in t direction

(u p )t

Pinion surface velocity in t direction

(u p )t 

Pinion surface velocity in t direction

UTE

Unloaded transmission error

Vc

Velocity of the point being cut seen from cradle axis

vij

Total surface velocity vector

Vr

Rolling velocity in t direction

Vs

Sliding velocity in t direction

Vso

Overall sliding velocity

Vw

Velocity of the point being cut from work axis

W

Transverse deflection

w ij

Surface velocity vector along common normal

WF

Work done by the external force

XB

Sliding base xviii

Y

Vector of slack variable



Variable of curvilinear cylindrical coordinate system (in tooth lengthwise direction)

b

Blade angle

 pv

Pressure viscosity coefficient



Variable of curvilinear cylindrical coordinate system (in tooth profile direction)

x

Shear rotation in profile direction

m

Machine root angle

 mn

Shear strain



Paint thickness (separation)

b

Blade offset angle

E

Pinion offset error

G

Gear mounting distance error

P

Pinion mounting distance error



Shaft angle error

m

Normal strain

k

Rolling power loss of segment  at time step k



Effective viscosity

0

Ambient viscosity



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

 k

Friction coefficient of segment  at time step k



Thermal correction factor



Shear stress xix

c

Cradle angle

g

Blank phase angle

t

Cutter phase angle

 n ()

Polynomial of order n for shape function in lengthwise direction

 m ( x)

Polynomial of order m for shape function in profile direction

c

Angular speed of the cradle axis

g

Angular speed of the blank axis

t

Angular speed of the cutter axis

p

Pinion speed



Tangential plane

Superscript: ( )

Real or updated

( )

Theoretical

(ˆ)

Conjugate

( )

Interpolated / Extrapolated

p

Pinion

g

Gear

a

Action

xx

CHAPTER 1

INTRODUCTION

“Hypoid gears are the most general form of

gearing and their solution has been long in coming. There is no form of gearing where so many guesses have been made, a few of them right, plenty of them wrong and some without consequences” Ernest Wildhaber [1.1] 1.1. Motivation and background Hypoid gears are widely used in many power trains to transfer power between two non-intersecting crossed axes. Their most common and highest-volume applications can be found in front and rear axles of rear-wheel-drive or all-wheel-drive vehicles [1.2]. Figure 1.1 shows a sample of hypoid gear application for the rear axle. A rear axle has three primary functions: (i) transmit power from the drive train axis to the wheel axle, that is usually perpendicular to the drive train axis with an offset, (ii) provide the capability to the vehicle to turn corners without any slippage at its wheels through its differential, and (ii) provide the final stage of speed reduction (torque increase) that is typically of the order of three to four.

1

Hypoid gear Input

Output

Hypoid pinion

Figure 1.1: A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize passenger cars and SUV’s (Courtesy of American Axle & Manufacturing Inc.).

2

A pair of hypoid gears is commonly used to deliver this third final drive function. In the arrangement shown in Figure 1.1, the smaller of the hypoid gears, called the pinion, is at the end of the drive shaft and is in mesh with the larger hypoid gear (called the gear). Hypoid gears can be considered as one of the most general cases of gearing based on their geometry, such that other gear types can be obtained from it by assigning certain values to some of the geometric parameters [1.1,1.3-1.5]. The main function of the hypoid gear pair in a rear axle is to transmit power between two axes that are at a shaft angle  (usually 90 ) [1.2,1.6] and at a certain amount of shaft off-set d a as shown in Figure 1.2. A higher level of power transmission through such a kinematic configuration is possible through use of a hypoid gear pair, which can provide a better balance amongst all primary design requirements such as strength, noise and power density. The trade-off between these performance characteristics while satisfying the kinematic constraints results in the hypoid tooth form that is rather complex geometrically. Figure 1.3 shows a schematic of different types of gearing, based on shaft arrangements. The shaft offset, being the main difference between spiral bevel and hypoid gears, provides several advantages to hypoid gears including larger pinion size, smaller pinion tooth counts, higher contact ratio, and higher contact fatigue strength. On the negative side, hypoid gears experience higher sliding velocities, resulting in higher power losses due to excessive sliding friction. Increasing the pinion size without 3

 

da 

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

4

shaft offset (spiral bevel) increases the size of the final drive significantly, while the hypoid pinion can be made larger due to shaft off-set to increase the strength of the gear pair while minimizing the overall size of the gear pair. Any attempt to improve the functional attributes of a hypoid gear pair in terms of its strength, quality, noise and power efficiency requires an optimization of its design either by fine-tuning its key parameters that have traditionally been chosen based on certain empirical knowledge or by making use of additional motion capabilities provided by new-generation hypoid gear cutting machines that allow application of many kinds of surface modifications [1.7, 1.8]. Hypoid gear design procedures were developed within a small number of hypoid gear cutting machine tool and cutting tool manufacturers and practical and theoretical details of hypoid development are still propriety to these companies [1.9]. In general, two different basic cutting methods are used to generate hypoid gears, namely face-milling (FM) or single indexing, and face-hobbing (FH) or continuous indexing, which have their own advantages over each other. The FM process that was the primary hypoid cutting method for decades has been taken over by the FH process in automotive axle applications, mostly due to its productivity advantages caused by continuous indexing [1,2, 1.10-1.12]. However, it is safe to state that the technology level and design understanding of the FH process is almost a decade behind the face milling process [1.13].

One reason for this is that newer machining methods 5

Worm gears

 

High reduction hypoid gears (HRH) or Spiroid

Pinion size

Sliding

Efficiency

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

Parallel axis gears (Spur, Helical and Herringbone)

Figure 1.3: Different gear types based on shaft arrangements.

6

such as grinding are applicable to FM process, while there is still no such alternative method or machinery developed for face-hobbing method. Having high gear ratios in hypoid gears in automotive applications causes the gear to have usually 3 to 4 times the number of teeth of the pinion, which justifies designing the gear surface as simple as possible to increase production efficiency and minimize manufacturing time. One typical cost-effective cutting method, called Formate®, is much faster than the Generating methods. In Formate®, only a few degrees of freedom of motions are allowed between the cutter and the gear blank (compared to the Generate cutting method). Therefore, most of the surface modifications are applied to pinion tooth surfaces, rather than the gear tooth surfaces. Quality of a hypoid gear pair is defined by a number of performance characteristics including its contact pattern, the motion transmission error (TE), efficiency and sensitivity to misalignments. The geometric accuracy of a single gear has limited significance here as the geometry of the mating gear and the assembly errors can change the performance characteristics drastically. These performance characteristics have been quantified either by using FE-based hypoid gear load distribution models or by experimental means, both of which are very time-consuming and expensive. Due to their significant computational burden, FE-based hypoid gear contact models are not suitable for design and parameter and misalignment sensitivity studies. The aim of this study is to develop computationally efficient, semi-analytical loaded tooth contact models for both 7

FM and FH hypoid gears with or without misalignments. The main motivation for this dissertation research is to develop formulations to analytically describe the hypoid gear surfaces and employ them in a gear contact mechanics formulation to predict unloaded and loaded contact characteristics as well as functional metrics such as the transmission error and mechanical efficiency.

1.2. Literature Review In his writings, Aristotle (about 330 BC), made mention of gears and their commonality. The earliest recognized relic of ancient time gearing is the south pointing chariot with pinned gears used by the Chinese in about 2600 BC [1.14]. According to the historical perspective provided by Litvin [1.15], theoretical development of gears as we know today starts with Euler (1781) who proposed the concept of an involute curve (1781), followed by others such as Willis (1841), Olivier ( 1842) and Gochman (1886) who developed basic ideas of conjugacy and the foundations of modern gear geometry. As for spiral bevel gears, Monneret (1899) filed the first patent for spiral bevel generating method. About two decades later in 1910, Böttcher was issued a series of patents that addressed both face hobbing and face milling methods [1.2].

Wildhaber’s

earlier papers and patents formed the basis for many of today’s hypoid gear geometry and generation approaches [1.1, 1.3].

Wildhaber pointed out the significance of using

principal curvatures and directions in establishing hypoid gear geometry [1.16, 1.17]. 8

Baxter’s later formulas, based on vector notation, helped condense the formulations to facilitate the use of computers for definition of surfaces and tooth contact analysis (TCA) [1.18]. He also developed one of the first unloaded tooth contact model for simulation of mismatched surfaces of gears generated by Gleason type machines and studied the effects of various misalignments on contact pattern [1.19], which was later expanded by Coleman [1.20, 1.21]. Krenzer published a series of formulations for unloaded tooth contact analysis (UTCA) of spiral bevel and hypoid gears [1.22]. These formulations were useful for the gears manufactured by a class of machinery but were quite difficult to adapt since the logic behind his formulae were not given. Nearly two decades later, he proposed a loaded tooth contact analysis model without providing details of the geometry and the contact analysis. This model used the simplified cantilever beam formula of Westinghouse to estimate the compliance of the tooth [1.23]. Within the same time frame, Litvin and Gutman [1.24-1.27] published a series of papers on synthesis and analysis of FM hypoid gears. They calculated machine settings based on predetermined contact characteristics at a mean point. They determined the contact points, the instantaneous contact length and direction by conventionally using the surface principal curvatures and directions. They used a conventional approach to find the contact points, the instantaneous contact directions and the instantaneous contact lengths utilizing surface principal curvatures and directions [1.24-1.27].

The effort of computing

optimized machine settings for limited cutting methods such as spiral bevel gears cut by face-milling method was later continued by Litvin and Fuentes [1.28]. As these studies 9

focused on calculating surface coordinate of FM spiral bevel and hypoid gears, there are very few studies on geometry of FH hypoid gears [1.29-1.33]. Among the few published studies on calculating and “optimizing” TCA, Stadtfeld [1.2] appears to be the only investigator who used the ease-off approach. He provided a consistent definition for ease-off as well as a procedure to calculate TCA from ease-off, and calculated instantaneous contact between two surfaces as a line that maintains its orientation over the tooth area [1.2]. Moreover, he utilized ease-off to optimize UTCA of both FM and FH gears by applying various kinds of modifications through machine settings and cutter geometry [1.34]. However, he did not provide a detailed procedure on how to determine orientation of the instantaneous contact curves. Meanwhile, Fan [1.35] focused on how to calculate surface coordinates and normal vectors for FH spiral bevel and hypoid gears cut by using the generation method. The solution to the set of equations that determines contact points where the collinearity condition for the normal vectors of two mating surfaces is satisfied is typically subject to various numerical instabilities. Fan [1.35] used the conventional approach for UTCA in conjunction with the EulerRodrigues’ formula to avoid these stability issues. He also used minimization of the separation between the tooth surfaces to determine the direction and length of the instantaneous contact lines [1.8]. Later Vogel et al [1.36, 1.37] proposed an alternate approach to compute both tooth surfaces and the UTCA by using Singularity Theory. They considered the generated tooth flank as a first-order singularity of the particular

10

function that models the generating process. They also used numerical differentiation to investigate the sensitivity of tooth contact to machine settings [1.36]. Simon also used the conventional system of five scalar nonlinear equation and six unknowns to find contact points on both surfaces. He calculated contact lines orientation by minimizing the separation function between two contact surfaces and applied his method for a FM Gleason type gear pair with a generated pinion and a Formate® gear [1.38]. Published studies on modeling of hypoid tooth contact under loaded conditions are quite sparse.

Simon [1.39-1.41] used a FE model to calculate deflection and

displacement under load from which interpolation functions were obtained to estimate stresses and deflections via regression analysis of FE results. Gosselin et al [1.42] developed a loaded tooth contact analysis (LTCA) model for spiral bevel gears by using tooth compliances obtained by curve-fitting to the FE deformation results of a single pinion and gear tooth pair. Wilcox et al [1.43] also developed a FE-based model to calculate the spiral bevel and hypoid gear tooth compliances by using a three-dimensional model of a tooth including base deformations, which was later employed by Fan and Wilcox [1.44] to perform LTCA analyses. Vimercati and Piazza [1.45] also calculated FH gear pair surfaces and incorporated them with a commercially available finite elements (FE) package [1.46] to calculate both TCA and LTCA. This particular hypoid FE package that employs FE away from the contact zone and a semi-analytical contact formulation at the contact zone [1.47] is perhaps the most advanced hypoid LTCA model

11

available to accurately simulate a hypoid gear contact. The major drawback of these FEbased models is that they require a considerable amount of computation time, making them more of an analysis tool. Their use for design tasks such as parameter and assembly variation sensitivity studies is not very practical. Beside the FE method, the Boundary Element (BE) method was also used in several studies for performing LTCA. For instance, Sugyarto [1.48] sliced gear and pinion teeth into a number of sections and considered each section as an independent plate from its neighboring sections, and applied a two dimensional BEM formulation to each slice to compute bending and shear deflections. Liu [1.49] applied same compliance methodology to face gears [1.49] with a correction intended to couple each slice with adjacent slices using Borner’s coefficient [1.50], which was originally proposed for parallel axis gear. Vecchiato [1.51] used a three-dimensional BE approach for loaded tooth contact predictions of FH hypoid gears. As for unloaded contact analysis, he used conventional approach [1.8,1.11] and studied misalignment effects. Besides these computational models, some semi-analytical models were also proposed for determining tooth compliance of parallel-axis gears through elasticity-based deformation solutions.

Adding linear thickness variation along the profile to the

originally proposed plate solution [1.52], Yakubek [1.53] used the Rayleigh-Ritz Energy Method to calculate the approximate deflection of a tapered plate for estimating the compliance of spur and helical gears. Bending deformations of a tooth were considered 12

as a sum of shape functions that satisfy clamped-free and free-free boundary conditions, and the unknown coefficients of the shape functions were determined by minimizing the potential energy. Later, Yau [1.54, 1.55] expanded this compliance model to add shear deformations to the energy function and found more realistic deformation for spur and helical gears and Stegemiller [1.56, 1.57] used the FE package ANSYS to propose an approximate interpolation based formula to compensate for base rotation and base translation.

All of these analytical compliance methods are valid for tooth having

constant height along face width and either constant or linearly varying thickness along its profile, which is not the case for hypoid gears. Vaidyanathan [1.58-1.60] proposed an analytical compliance model for a tooth with linearly varying thickness in the profile and lengthwise directions as well as linearly varying tooth height along the face width. His Rayleigh-Ritz based formulation used polynomial shape functions and was applied to both sector and shell geometries. The sector model represents straight bevel gear geometry closely while the shell model is sufficiently close to a spiral bevel gear tooth in terms of its geometry. The challenges mentioned above in terms of performing a loaded hypoid tooth contact analysis in a practical and computationally efficient way have been the major road block to the development of other models to study other functional behavior of hypoid gears. One such behavior is the efficiency of the hypoid gear pair. In addition to the analytical surface geometries and surface velocities, an accurate description of the

13

contact load distribution is required at many rotational increments (of pinion angle) to predict the distribution of the friction coefficient and the resultant mechanical power losses. While this hypoid efficiency methodology has been demonstrated by Xu and Kahraman [1.61-1.63] by using the FE-based loaded tooth contact model of Vijayakar [1.64], the amount of computations required was reported to be very significant for it to be used extensively as a design tool. Likewise, other hypoid gear models for simulation of surface wear [1.65] and finishing processes such as lapping have also been hampered by the difficulties in obtaining load distribution.

1.3. Scope and Objectives Computation of the contact pressure distributions and the relative surface velocities forms the basis for predicting the required functional parameters of the hypoid gear pair, including the transmission error, contact stresses, root bending stresses, fatigue life and mechanical power losses. It is evident from the review of the literature that a model to compute the load distribution accurately and efficiently without resorting to computationally demanding FE methods does not exist. This is mainly due to three primary reasons: (i)

A detailed description of general and reliable formulation to define the geometry of FH and FM hypoid tooth surfaces from cutter parameters, machine motions and settings is not available in the literature. 14

(ii)

Conventional methods of matching the tooth surfaces (bringing them to contact) present major numerical difficulties for UTCA.

(iii) There is no published model available for hypoid gear LTCA based on semianalytical tooth compliance formulations. Accordingly, the main objective of this study is to develop FH and FM hypoid gear contact models that address these issues. This study performs the following specific tasks to achieve this objective: (i)

Development of a methodology that simulates the FM and FH processes to define surface geometries of hypoid gears including the coordinates, normal vectors and radii of curvatures.

(ii)

Development of a novel formulation for unloaded tooth contact analysis by using the ease-off topography, surface of action and roll angle surface to predict unloaded transmission error and unloaded contact pattern in addition to potential contact lines/curves to be used for loaded tooth contact analysis.

(iii) Development of a semi-analytical tooth compliance model tailored for both FH and FM hypoid and spiral bevel gears.

15

(iv) Development of a LTCA model for FH and FM hypoid gears to predict pressure distribution and loaded transmission error with or without misalignments of various types. Actual manufactured surfaces of gears include inevitable machining errors, heat treatment distortions and lapping surface changes as globally distributed deviations on tooth surfaces, which affect contact patterns and transmission error significantly. Moreover, wear or lapping simulations (as accumulated wear) changes surface geometry usually in a very local fashion that conventional tooth contact analysis approaches are not capable of capturing. A novel ease-off based approach will also be developed to modify ease-off topography of the theoretically generated tooth surfaces to account for both global deviations due to the manufacturing process and local surface deviations due to factors such as wear and lapping process. A second objective of this dissertation is to develop a capability to predict load dependent (mechanical) power losses of hypoid gear pairs.

For this purpose, the

proposed loaded tooth contact model will be combined with a new friction model according to the methodology proposed by Xu and Kahraman [1.62, 1.63] to predict mechanical power losses and gear pair efficiency including all relevant contact, surface, and lubricant parameters as well as the operating conditions. This hypoid gear efficiency model will be used to investigate the impact of basic design parameters, and surface and

16

lubricant conditions, on mechanical power losses of hypoid gear pairs and to arrive at guidelines on how to reduce such losses.

1.4. Overall Modeling Methodology The overall methodology used to develop the hypoid load distribution model is illustrated in the flowchart of Figure 1.4. Gear blank dimensions, cutter geometry, machine settings, assembly dimensions and misalignments, torque and speed are all included as input parameters for the load distribution model. These parameters are commonly put together by hypoid gear manufacturers in a standard form that is called a special analysis file. The pinion and gear cutter surfaces are first constructed and used to define the extended pinion and gear surfaces (including surface coordinates, normals and curvatures) by applying fundamental equation of meshing between a gear blank and its respective cutter surfaces. These extended tooth surfaces are then trimmed in 3D space so that they are contained by the blanks, and transformed to a global coordinate system where any misalignments in the directions of shaft offset (ΔE), pinion axis (ΔP), gear axis (ΔG) as well as the shaft angle error (ΔΣ) can be applied. Next, ease-off and surface of roll angle are constructed and an UTCA model is developed by bringing the tooth surfaces together and an unloaded contact pattern is defined by choosing a separation tolerance between the tooth surfaces.

17

 

  (a)   

Generating gear 

FM Method (Generate / Formate)

Circular  CG

OB 

Reading Design File .HAP or .SPA file

(b)

Generating gear 

FH Method (Generate / Formate)

Extended  epicycloids trace

CG

t

t  

Blade

IB 

c

Fixed 

OB

IB 

Ct

Cutter

Cutter 

Equation of Meshing

FM Method

Equation of Meshing

Extended Pinion and Gear surfaces and Curvatures (FM)

FH Method

Extended Pinion and Gear Surfaces and Curvatures (FH) Surfaces in Global Coordinates System (Misalignments)

Global coordinates transformation

Global coordinates transformation-Under Development

18 Ease-off Construction

Updating Ease-off by Surface Deviations

Tooth Compliances

Unloaded TCA   Maximum  

Top M1

   Toe

LPP M2

Pitch

UTE

M1

CPP Root

Contact lines

Pinion phase angle (deg.)

 

M2

Tip (free edge)

Root (clamped edge)

100 TE (μ rad)

Adjacent tooth pairs

Toe-Base

Required Data for Efficiency Analysis

Y (mm) Toe

(MPa)

2 0 -2 -4 -20

-15

-10

-5 0 5 X (mm) Root

10

15

20

Contact Pressure Distribution

Loaded Transmission Error

Heel-height

Toe-height

Loaded Tooth Contact Analysis

758 674 590 506 421 337 253 169 84 0

Heel (free edge)

Toe (free edge)

RPP Heel

Cutter axis

Heel-Base

Hypoid Gear Mechanical Power Loss and Efficiency

Elastohydrodynamic Friction Coefficient Model

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

Next the tooth compliance matrices comprising bending, shear, Hertzian and base rotation deflections are computed.

Finally, a set of equilibrium and compatibility

conditions are defined and solved simultaneously to compute the load distribution and the loaded transmission error of the hypoid gear pair. Moreover, all required information for efficiency and lapping simulations are computed.

1.5. Dissertation Outline In Chapter 2, the hypoid gear tooth surfaces will be defined through simulation of the face-milling and face-hobbing processes with all relevant cutter and machine related parameters included. A new formulation of unloaded tooth contact analysis based on the principles of ease-off and a newly introduced surface of roll angle will be proposed as well. In Chapter 3, a semi-analytical tooth compliance model will be employed and a loaded tooth contact model will be described. A novel approach will be introduced in Chapter 4 to compute loaded tooth contacts of gear surfaces that have deviations from their theoretically intended surfaces either in local or global fashion. Chapter 5 proposes a model to predict the mechanical efficiency of hypoid gear pairs. This mode combines the developed computationally efficient contact model and the mixed elastohydrodynamic lubrication (EHL) based friction model of Li and 19

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

A

summary, major conclusions and contributions of this research to the state-of-the-art as well as a list of recommendation for future work will be included in Chapter 6.

References of Chapter 1 [1.1]

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

[1.2]

Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute of Technology.

[1.3]

Stewart, A. A., and Wildhaber, E., 1926, "Design, Production and Application of the Hypoid Rear-Axle Gear." J. SAE, 18, pp. 575-580.

[1.4]

Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears, Elsevier Science B. V.

[1.5]

Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A Concurrent Engineering Approach, John Wiley & Sons Inc.

[1.6]

Stadtfeld, H., J., 1995, Gleason Advanced Bevel Gear Technology, The Gleason Works.

[1.7]

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

[1.8]

Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech. Des., 129(1), pp. 31-37. 20

[1.9]

Dooner, D. B., 2002, "On the Three Laws of Gearing." ASME J. Mech. Des., 124, pp. 733-744.

[1.10] Krenzer, T. J., 2007, The Bevel Gears, http://www.lulu.com/content/1243519. [1.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd ed.), Cambridge University Press, Cambridge. [1.12] Krenzer, T. J., 1990, "Face Milling or Face Hobbing." AGMA, Technical Paper No. 90FTM13. [1.13] Stadtfeld, H. J. (2000). "The Basics of Gleason Face Hobbing." The Gleason Works. [1.14] Dudley, D. W., 1969, The Evolution of the Gear Art, American Gear Manufacturers Association, Washington, D. C. [1.15] Litvin, F. L. (2000). "Development of Gear Technology and Theory of Gearing." NASA RP1406. [1.16] Wildhaber, E., 1956, "Surface Curvature." Product Engineering, pp. 184-191. [1.17] Dyson, A., 1969, A General Theory of the Kinematics and Geometry of Gears in Three Dimensions, Clarendon Press, Oxford. [1.18] Baxter, M. L., 1964, "An Application of Kinematics and Vector Analysis to the Design of a Bevel Gear Grinder." ASME Mechanism Conference, Lafayette, IN. [1.19] Baxter, M. L., and Spear, G. M. "Effects of Misalignment on Tooth Action of Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.

21

[1.20] Coleman, W. "Analysis of Mounting Deflections on Bevel and Hypoid Gears." SAE 750152. [1.21] Coleman, W. "Effect of Mounting Displacements on Bevel and Hypoid Gear Tooth Strength." SAE 750151. [1.22] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason Works. [1.23] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears under Load." SAE Earthmoving Industry Conference, Peoria, IL. [1.24] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for hypoid gear-drives of formate and helixform; Part I-Calculation for machine setting for member gear manufacture of the formate and helixform hypoid gears." ASME J. Mech. Des., 103, pp. 83-88. [1.25] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for hypoid gear-drives of formate and helixform; Part II-Machine setting calculations for the pinions of formate and helixform gears." ASME J. Mech. Des., 103, pp. 89-101. [1.26] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for hypoid gear-drives of formate and helixform; Part III-Analysis and optimal synthesis methods for mismatched gearing and its application for hypoid gears of formate and helixform." ASME J. Mech. Des., 103, pp. 102-113. [1.27] Litvin, F. L., and Gutman, Y., 1981, "A Method of Local Synthesis of Gears Grounded on the Connections Between the Principal and Geodetic of Surfaces." ASME J. Mech. Des., 103, pp. 114-125.

22

[1.28] Litvin, F. L., Fuentes, A., Fan, Q., and Handschuh, R. F., 2002, "Computerized design, simulation of meshing, and contact and stress analysis of face-milled formate generated spiral bevel gears." J. Mechanism and Machine Theory, 37(5), pp. 441-459. [1.29] Fong, Z. H., and Tsay, C.-B., 1991, "A Mathematical Model for the Tooth Geometry of Circular-Cut Spiral Bevel Gears." ASME J. Mech. Des., 113, pp. 174-181. [1.30] Fong, Z. H., 2000, "Mathematical Model of Universal Hypoid Generator with Supplemental Kinematic Flank Correction Motions." ASME J. Mech. Des., 122(1), pp. 136-142. [1.31] Tsai, Y. C., and Chin, P. C., 1987, "Surface Geometry of Straight and Spiral Bevel Gears." J. Mechanism, Transmission and Automation in Design, 109, pp. 443-449. [1.32] Fong, Z. H., and Tsay, C.-B., 1991, "A Study on the Tooth Geometry and Cutting Machine Mechanisms of Spiral Bevel Gears." ASME J. Mech. Des., 113, pp. 346351. [1.33] Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, "Determination of Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel Gears." J. Mechanism, Transmission and Automation in Design, 110, pp. 495500. [1.34] Stadtfeld, H. J., and Gaiser, U., 2000, "The Ultimate Motion Graph." ASME J. Mech. Des., 122(3), pp. 317-322.

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[1.35] Fan, Q., 2006, "Computerized Modeling and Simulation of Spiral Bevel and Hypoid Gears Manufactured by Gleason Face Hobbing Process." ASME J. Mech. Des., 128(6), pp. 1315-1327. [1.36] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany. [1.37] Vogel, O., Griewank, A., and Bär, G., 2002, "Direct gear tooth contact analysis for hypoid bevel gears." Computer Methods in Applied Mechanics and Engineering, 191(36), pp. 3965-3982. [1.38] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME International Power Transmission and Gearing Conference ASME, 88, pp. 789798. [1.39] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des., 122(44), pp. 529-535. [1.40] Simon, V., 2000, "FEM stress analysis in hypoid gears." J. Mechanism and Machine Theory, 35(9), pp. 1197-1220. [1.41] Simon, V., 2001, "Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution." ASME J. Mech. Des., 123(4), pp. 577-582. [1.42] Gosselin, C., Cloutier, L., and Nguyen, Q. D., 1995, "A general formulation for the calculation of the load sharing and transmission error under load of spiral bevel and hypoid gears." J. Mechanism and Machine Theory, 30(3), pp. 433-450. [1.43] Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth." AGMA, Technical Paper No. 97FTM05. 24

[1.44] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA, Technical Paper No. 05FTM08. [1.45] Vimercati, M., and Piazza, A., 2005, "Computerized Design of Face Hobbed Hypoid Gears: Tooth Surfaces Generation, Contact Analysis and Stress Calculation." AGMA, Technical Paper No. 05FTM05. [1.46] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced Numerical Solution Inc., Hilliard, Ohio. [1.47] Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem." International J. for Numerical Methods in Engineering, 31, pp. 525-545. [1.48] Sugyarto, E., 2002, "The Kinematic Study, Geometry Generation, and Load Distribution Analysis of Spiral Bevel and Hypoid Gears," M.Sc. Thesis, The Ohio State University, Columbus, Ohio. [1.49] Liu, F., 2004, "Face Gear Design and Compliance Analysis," M.Sc. Thesis, The Ohio State University, Columbus, Ohio. [1.50] Borner, J., Kurz, N., and Joachim, F. (2002). "Effective Analysis of Gears with the Program LVR (Stiffness Method)." [1.51] Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University of Illinois at Chicago. [1.52] Timoshenko, S. P., and Woinowsky-Krieger, S., 1959, Theory of Plates and Shells, McGraw-Hill Book Company Inc. 25

[1.53] Yakubek, D., Busby, H. R., and Houser, D. R., 1985, "Three-Dimensional Deflection Analysis of Gear Teeth Using Both Finite Element Analysis and a Tapered Plate Approximation." AGMA, Technical Paper No. 85FTM4. [1.54] Yau, H., 1987, "Analysis of Shear Effect on Gear Tooth Deflections Using the Rayleigh-Ritz Energy Method," M.Sc. Thesis, The Ohio State University, Columbus, Ohio. [1.55] Yau, H., Busby, H. R., and Houser, D. R., 1994, "A Rayleigh-Ritz Approach to Modeling Bending and Shear Deflections of Gear Teeth." J. of Computers & Structures, 50(5), pp. 705-713. [1.56] Stegmiller, M. E., 1986, "The Effects of Base Flexibility on Thick Beams and Plates Used in Gear Tooth Deflection Models," M.Sc. Thesis, The Ohio State University, Columbus, Ohio. [1.57] Stegmiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192. [1.58] Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. [1.59] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears Using Shell Theory." AGMA, Technical Paper No. 93FTM03. [1.60] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical Approach to the Static Analysis of an Annular Sector Mindlin Plate with Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp. 255-266. 26

[1.61] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. [1.62] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part K: J. Multi-body Dynamics, 221(3), pp. 387-400. [1.63] Xu, H., Kahraman, A., and Houser, D. R., 2006, "A Model to Predict Friction Losses of Hypoid Gears." AGMA, Technical Paper No. 0FTM06. [1.64] Vijayakar, S. M., 2003, Calyx User Manual, Advanced Numerical Solution Inc., Hilliard, Ohio. [1.65] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear Pairs." In press, Wear. [1.66] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric Integrated Control Volume Discretization." Tribology International, Hiroshima, Japan.

27

CHAPTER 2

DEFINITION OF FACE-MILLED AND FACE-HOBBED HYPOID GEAR GEOMETRY AND UNLOADED TOOTH CONTACT ANALYSIS

2.1. Introduction Unlike most types of gears that have closed-form equations defining their geometry, the geometry of hypoid gears can only be computed by solving implicit equations governed by the manufacturing process, including its machine settings and cutter specifications. Besides the gear blank dimensions and basic geometry requirements, a set of performance or functionality related requirements must be met. Among them, the contact pattern (location, size, shape) on the gear tooth surfaces and the motion transmission error amplitude of the gear pair are two of the most common ones checked routinely in the design of hypoid gear pairs. The contact pattern and the transmission error are both determined via a contact analysis of the pinion and gear tooth surfaces under a very small amount of load.

28

In general, contact of hypoid gear surfaces is single-mismatched. This is the most general case of point contact condition between two surfaces [2.1]. The purpose of the unloaded tooth contact analysis (UTCA) is to determine a contact point path (CPP) on each surface in addition to area on each surface in the neighborhood of each instantaneous contact point that falls in a specified separation distance (usually 6.3 micron of separation distance is commonly used in hypoid gear industry) [2.2]. In addition, UTCA results in the function of motion transmission error between two gear axes that is viewed as a key metric used to estimate the noise/level of the hypoid gear pair in operation [2.3, 2.4]. In this chapter, as the first basic step in the analysis of hypoid gears, the geometry of both face-milled (FM) and face-hobbed (FH) hypoid gear pairs produced by using both Formate® and Generate cutting methods will be computed.

This will be done by

simulating individual cutting processes. Basic machine tool settings, cutter geometry parameters and gear blank dimensions will form the input for this computation. Next, a novel method based on the ease-off topography will used to determine the unloaded contact patterns.

The proposed ease-off based methodology finds the instantaneous

contact curve through a surface of roll angles, allowing an unloaded tooth contact analysis in a robust and accurate manner.

29

2.2. Definition of Tooth Surface Geometry The concept of the generating gear is a key to the basic understanding of hypoid gears because this hypothetical gear can be treated as cutting tool for both the pinion and the gear [2.5]. In a FM cutter head, blades are arranged around the cutter head axis on an equal radius for inside and outside blades (IB and OB, respectively) to form a conical shape due to cutter axis rotation. The inside blades cut convex side of a tooth slot while the outside blades cut concave side of the same slot, as shown in Figure 2.1(a). Facehobbing cutter heads (such as PENTAC® or TRI-AC®) like the one shown in Figure 2.1(b) roll while cutting such that each set of IB-OB blades (called blade group) will pass through a different tooth slot. The cutting process can be considered as rolling of two gears together, except the teeth of one of the gears are replaced by blade group of the cutter head. By rolling the cutter head and the gear blank together while advancing the cutter head into the blank, the gear is cut by the continuous indexing method. While the axis of the generating gear for FM process is fixed, it is located on the center of a circle Ct for FH process that rolls on the generating gear circle CG , as shown in Figure 2.1(b). Therefore, the edges of a blade in FH process traces extended epicycloids since they usually lie on a radius that is larger than the radius of rolling circle Ct .

30

(a)

Generating gear

 

Circular arc trace

CG OB

t IB Fixed

Cutter (b)

Generating gear Extended epicycloids trace

CG

t

Blade group

c IB

Ct

OB Cutter

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

2.2.1. Kinematics Earlier cradle-based hypoid generators were designed to provide the required degrees of freedom and relative motions through machine settings to accommodate the cutting process of the gear and pinion blanks by means of the cutter blades. Figure 2.2 shows a typical cradle-based hypoid generator with machine settings and relative motions defined as the cutter phase angle t , the tilt angle iT , the swivel angle js , the radial setting Sr , the cradle angle

c , the sliding base X B , the machine root angle  m ,

machine center to back M ctb , the blank offset Eb , the blank phase angle  g , angular speed of the cutter axis t , angular speed of the cradle axis c , angular speed of the blank axis  g , and the cradle angle change q. Newer-generation hypoid gear machine tools still use settings in the form of older cradle-based machines. While the new cutting machines are not set as the old mechanical machines, their working principles are still the same such that they generate the same gear surface with an added capability of controlling higher-order surface geometrical parameters as well. While most of the machine settings are typically fixed (kept constant), a number of them are defined as a polynomial function of q. These parameters that are dependent on q are called higherorder motions such as the modified roll (ratio of roll change), the helical motion (sliding base change) and the vertical motion (blank offset change).

32

 

c Sr js iT

Eb

t

m

g

M ctb

XB

Figure 2.2: Cradle based hypoid generator parameters.

33

In the FM process, the teeth are cut individually by blade edges that rotate fast about the cutter axis. The cutting edges of the blades form a conical surface and their envelope seen from a coordinate attached to the blank is the gear or pinion surface. In the Formate® case for the FM process, the blank is fixed, the cutter advances towards the gear blank while rotating, in the process replicating its surface on the blank. Then, the cutter head retreats, the blank is rotated by one tooth spacing, and the same cutting process is repeated as shown in Figure 2.1(a).

Meanwhile, the face-hobbing process

using Formate® requires the blank and cutter head to be rolled together according to a kinematic relationship defined as

Rtg 

where

Ro g t



Nt Ng

(2.1)

t is cutter head angular velocity, Ro g is the rolling portion (also called

indexing) of angular velocity of the gear blank, and Nt and N g are numbers of blade groups and gear teeth being cut. Typically, edges of the blades in FH process do not intersect with cutter head axis so that the cutting surface is a hyperboloid of revolution. In the generate process for both FM and FH processes, there is an additional relative rotation between the gear blank and the cradle axis in Figure 2.1(b). The ratio of roll is given as the ratio of relative rotations, i.e.

34

RaFM 

RaFH



g c

(2.2a)

,

Ge g

(2.2b)

c

where RaFM and RaFH are the ratios of rolls for the FM and FH processes (they are zero for Formate® process), respectively, c is the angular velocity of the cradle axis, g is the angular velocity of the blank axis during face-milling and Ge g is the generating portion of blank angular velocity for the FH process. As a result, the total blank angular velocities for FM and for FH are given, respectively: FM g  g ,

(2.3a)

Ro Ge FH g  g  g

(2.3b)

and  FH for FM and FH processes are Corresponding gear blank rotation angles  FM g g given respectively as: FM  FM g   Ra q,

(2.4a)

FH  FH g  ( Ra q  Rtg t ) .

(2.4b)

35

Here, the relative motion between the cutter and the gear blank for the FM process can be treated as a special case of the FH process. Therefore, the formulation for the FH process will be explained here in detail while its differences from the FM process will also be specified, as required.

2.2.2. Cutting Tool Geometry and the Relative Motion Figure 2.3 shows a typical FH blade with its geometry along its cutting edge is defined by the blade angle b , the rake angle  , the hook angle  , the blade offset angle

b , the cutter radius rc , and the distance from the tip of blade to reference point h f . The cutting edge is divided into four different sections as the edge (or tip radius), toprem,

profile and flankrem that are all shown in Figure 2.3(c). The edge and flankrem are usually circular arcs while toprem is usually a straight line at a slight angle from the profile section. Most of the cutting is done by the profile section of the blade that is For a typical FM cutter,     b  0 .

usually a straight line or a circular arc.

Referring to Figure 2.3, an arbitrary point A on cutting edge is at position r  r ( s) relative to the local coordinate system Xb fixed to the cutter head (with its origin at reference point M) where s is the distance of point A to point M along the blade edge. With this, the unit tangent vector is t  r s , and if the cutting edge is a line, it can be reduced to

36

(a)

yt Reference Plane

x , x

b

rc

xt zb , z z (b)

M



Inside Blade

31

(c)

b

Flankrem r

Profile

xb

x , x

xb

A t s

xb , xt

M

 yb

M y

y

n

b

hf

Toprem

18

Edge

rc

zb

zt

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

37

t  [ sin b

0  cos b 1]T

(2.5)

where the superscript T denotes a matrix transpose. Position and unit tangent vectors, rt and t t , in the coordinate system Xt whose z axis coincides with the cutter axis and xy plane passes through reference point M as shown in Figure 2.3(a) are given as t t = M z (  b ) M x () M z () t ,

(2.6a)

rt = rc t i  s t t

(2.6b)

where t i = [cos b

sin b

0 1]T

(2.6c)

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

axis k

(k  [ x, y, z ]) such that 0 0 0 1  0 cos  sin  0  , M x ( )    0  sin  cos  0   0 0 0 1 

(2.6d)

 cos   0 M y ( )    sin   0

(2.6e)

0  sin  0  1

0

0

0

cos 

0

0

0

,

1 

38

 cos  sin  0 0    sin  cos  0 0   . M z ( )   0 1 0  0  0 0 0 1 

(2.6f)

Position vector of point A is transformed to the coordinate system X g fixed to the blank as [2.6]: rg  M x ( g ) M Eb M ctb M y (  m ) M  m X B M X B c M z (c )  M z (q ) M Sr j M z (  js ) M x (iT ) M z ( t   ) rt

(2.7a)



where the transformation matrices are defined as 1 0 M Sr j   0  0

0 0 Sr  1 0 0 , 0 1 0 0 0

1 0 M X B c   0  0 1 0 M m X B   0  0

0 0 1 0

(2.7b)

1 

0  0 

,

(2.7c)

0 1 XB  0 0 1 

0 0  M ctb  1 0 0 1

0 0

0 0

1

 ,  

(2.7d)

39

M Eb M ctb

1 0  0  0

0 1 0 Eb  . 0 1 0 0 0

0 0

(2.7e)

1 

Variables influencing rg , with the exception of s and t , are either fixed or dependent on q (in case of higher order motion).

2.2.3. Equation of Meshing Equation (2.7a) represents a family of surfaces in a coordinate system fixed to the blank whose envelope is the generated surface on the blank. The envelope of rg with three independent variables s, t and q is given mathematically as [2.7-2.9]:  rg rg  rg   0.    s t  q

(2.8)

This equation is mathematically equivalent to the fundamental equation of meshing, n  (Vc  Vw )  0 , which states for each point to lie on the envelope surface that the normal vector n to the family of the cutter surfaces should be perpendicular to relative velocity between the blank (w) and the cutter (c) as shown in Figure 2.4.

40

Vc  Vw

 

Vc

Vw

n Cradle axis Vc  Vw Cutter head

c

w

Blank

Blank axis

Figure 2.4: Generation process.

41

Using Eq. (2.8), each point on the generated surface can be found by solving a system of two implicit nonlinear equations for a pair of unknown parameters that can be chosen as ( s, t ) , ( s, q) or (q, t ) .

2.2.4. Principal Curvatures and Principal Directions

As shown in Figure 2.5 for any point P0 ( s0 , t 0 ) defined by independent surface curvilinear variables s and t on the gear surface, the unit normal to the surface is given as rg n0 

s rg s

 

rg t . rg

(2.9)

t

Moving from point P0 ( s0 , t 0 ) to P1 ( s0  ds, t 0 ) by only infinitesimally changing one of the surface variables s, the change of unit normal to the surface is defined as [2.10] dn   kt1 t1  t1 v1 . d ( s, t )

(2.10)

Here P P t1  1 0 , P1  P0

(2.11a) 42

n2

 

n0

n1

C2

P2

C1

t2 P1

t1

P0 v1

Figure 2.5: Curvature computation procedure.

43

v1 

n0  t1 n0  t1

(2.11b)

and ( s, t ) defines the distance from P0 to P1 along the gear surface (along curve C1 ) as a function of s and t (here, t  t 0  constant and only s varies). kt1 is the normal curvature in the t1 direction and t1 is the geodesic torsion in the direction of v1 . Here n0 , t1 and v1 form a Frenet trihedron.

Following the same procedure, but this time

moving from P0 ( s0 , t 0 ) to P2 ( s0 , t 0  d t ) by infinitesimally changing t , the normal curvatures kt2 and geodesic torsion t2 in t 2 and v 2 directions are found according to

dn  kt2 t 2  t2 v 2 d ( s, t )

(2.12)

with

t2 

P2  P0 , P2  P0

(2.13a)

v2 

n0  t 2 n0  t 2

(2.13b)

and ( s, t ) defines the distance from P0 to P2 along the gear surface (along curve C2 in Figure 2.5) as a function of s and t (here s  s0 is constant and only t varies). With

44

kt1 , kt2 , t1 and t2 in hand, Euler equation [2.11, 2.12] is applied to compute the

principle directions ( e1 and e2 ) and the principal curvatures ( K1 and K 2 ). Having the principal curvatures and directions of every possible contact point (points on the projection plane) on the pinion and gear, principal directions of the difference surface are defined as directions in which two contacting surfaces have the extremes of the relative curvatures [2.11-2.13]. The curvatures R p and R g of pinion and gear surfaces in the direction of maximum relative curvatures are used to find the equivalent curvature Req  R p R g ( R p  R g ) that will be required later to determine Hertzian deflections in loaded TCA [2.14].

2.3. Unloaded Tooth Contact Analysis In unloaded tooth contact analysis (UTCA), the goal is to calculate: (i) the contact point path (CPP) on each of the gear surfaces in addition to the zone on each surface in the neighborhood of each instantaneous contact point that is as close as the specified separation distance  [2.15, 2.16], and (ii) the function of transmission error between two gear axes. Two different approaches were used in the past for performing UTCA of hypoid gears with mismatched surfaces. In the conventional method, tooth surfaces are treated as two 45

arbitrary surfaces, rotating about the pinion and gear axes. The contact point path (CPP) on each surface is computed by satisfying two contact conditions. The first condition is the coincidence of position vector tips of the points on the gear and pinion surfaces in three dimensional space. The second condition is the collinearity of the normal vectors of the both of the surfaces at the contact point. The second method of performing UTCA is based on the ease-off procedure. The current literature lacks a clear and accurate mathematical definition of ease-off as well as its construction including the instantaneous contact lines/curves [2.17]. Ease-off has often been defined in the literature as the change in pinion surface with the application of modifications. While these changes directly reshape ease-off, they do not constitute the ease-off itself. Finding the location and orientation of potential instantaneous contact lines is one major step in the UTCA. In general, the instantaneous contact shape in the projection plane (plane that includes the gear axes) is slightly curved as opposed to commonly used approximate straight lines, which is the contact shape in action surface.

The

instantaneous contact line directions are conventionally found based on principal curvatures and directions of contacting surfaces of gear members at contacting points as the direction of minimal relative normal curvature between contacting surfaces.

46

In the next two sections, a brief overview of the conventional approach will be provided, followed by a detailed ease-off based UTCA formulation developed in this thesis.

2.3.1. The Conventional Method of UTCA The position vector r p (1, 1 ) and normal vector n p (1, 1 ) of any point on the surface of the pinion can be defined by two independent curvilinear local surface variables 1 and 1 . Similarly, the position and normal vectors of any point on the surface of the gear are given as r g ( 2 , 2 ) and n g ( 2 , 2 ) where  2 and 2 are the independent curvilinear local variables of the gear surface. Pinion surface coordinate

r p (1, 1 ) and normal n p (1, 1 ) are rotated about the pinion axis a p as much as an angle  p while the gear surface coordinate r g ( 2 , 2 ) and normal n g ( 2 , 2 ) are rotated about the gear axis a g by an angle  g to satisfy the two contact conditions defined below: M z ( p ) r p (1, 1 )  M z ( g ) r g ( 2 , 2 )  OE ,

(2.14a)

M z ( p ) n p (1, 1 )  M z ( g ) n g ( 2 , 2 ) .

(2.14b)

47

Here, OE is the offset vector that connects the origins of the pinion and the gear. Eq. (2.14) constitutes a system of five nonlinear equations (since n p  n g ) and six unknowns 1 , 1 ,  p and  2 , 2 ,  g .

Defining the value of one of these six

parameters (usually  p ) as an input parameter, a set of five nonlinear equations defined by Eq. (2.14) can be solved for the remaining five unknowns. However, due to the high level of conformity of the pinion and gear surfaces in the vicinity near the contact point, the solution of this system of equations is subject to several numerical instabilities. In addition, the solution is very sensitive to the initial guesses.

Provided these numerical

difficulties can be overcame, the solution of Eq. (2.14) yields the coordinates of contact point path (CPP) on the pinion and gear surfaces as well as the angular position of the gear as a function of the pinion angle. With the conventional method, at each point of the contact point path (CPP), a direction in which separation between two surfaces (here pinion and gear surfaces) is minimum [2.11, 2.12, 2.18, 2.19] is assumed to be potential contact line. For this purpose, the two contacting surfaces are approximated as two contacting ellipsoids with an instantaneous point contact M as shown in Figure 2.6, The principal curvatures of both pinion ( K1p and K 2p ) and gear ( K1g and K 2g ) surfaces respectively and the corresponding principal directions ( e1p , e 2p , e1g and e2g )

are all required to find

direction in which relative normal curvature between ellipsoids is minimal. This direction 48

 



e 2p

(a)

e 2g

g

u

n



e1p

v

T



k2g

k1p e1g

M

k2p

k1g p  

u

v

(b)

1L20 L1

Figure 2.6: General case of approximating gear surfaces as two contacting ellipsoids to orient instantaneous contact line.

49

u, as shown in Figure 2.6, having an unknown angle  from e1p in the tangential plane

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

[2.11, 2.12], the pinion normal curvature along any arbitrary direction u with an angle  from e1p in the tangential plane T is kup  k1p cos 2 ( )  k2p sin 2 ()

(2.15a)

while gear normal curvature along same direction is kug  k1g cos 2 (  )  k2g sin 2 (  ) .

(2.15b)

Hence, the relative curvature along u is given as kupg  k1g cos 2 (  )  k2g sin 2 (  )  k1p cos 2 ()  k2p sin 2 () .

(2.16)

The value of angle  that minimizes kupg is found by dkupg 0 d

(2.17)

such that



  1 sin(2) , tan 1  pg 2  K12  cos(2) 

(2.18a)

50

p p pg k1  k2 K12  k1g  k2g

.

(2.18b)

The length of the unloaded instantaneous contact line is defined as L1  8 kupg where  is the unloaded separation distance [2.12]. In the conventional method, it has been assumed that instantaneous contact lines spread equally on both sides of instantaneous contact points M along assuming contact direction u since approximating contacting surfaces with local ellipsoids results in the same relative curvatures on both sides of contact points. In other words, the conventional method fails to include the variation of the curvature along a given contact line, resulting in contact line length estimations that might be erroneous. Simon [2.20] found the direction of minimum separation by minimizing a function that defines separation between contacting surfaces in the direction of the normal to the gear surface at each contact point. Although his approach does not acquire principal curvatures and directions information on each contact point, it still bears the some level of computational complexity and inefficiency since such minimization of the separation function requires the solution of a system of seven nonlinear equations and seven unknowns.

He later mentioned that the instantaneous contact form is a curve

rather than the generally assumed line [2.21]. Fan [2.8] found instantaneous contact line direction and length without using second order information by finding minimum

51

separation direction. He constructed a cylinder centered at an instantaneous contact point with an axis that is collinear with the common normal then searched for a direction on the tangent plane in which the separation between the contacting surfaces is minimal. With this, another search was performed to find the distance required to move on the both sides of the minimum separation direction in order to reach the predetermined separation value  . He eliminated the need for the second order surface information and computed more

realistic contact line length estimating different lengths on both sides of contact point along contact line. However, this method still assumed an instantaneous contact line, as opposed to more realistic curved shape.

2.3.2. Ease-off Based Method of UTCA This study proposes a novel surface of roll angle and utilizes it to orient instantaneous contact lines/curves without using principal curvatures and directions. This method requires significantly less computational effort since (i) it does not result in a set of nonlinear algebraic equations that must be solved numerically, and (ii) it only requires the coordinates and the normal vector of one contacting surface and the spatial orientation of the axes of both gears. The instantaneous contact curves are defined between two conjugate surfaces, namely a given surface and the conjugate surface to the reference surface with respect to the given spatial orientation of the axes of the gears. Since the instantaneous contact line orientation is extremely insensitive to local surface 52

changes as it will be demonstrated later, the validity of using a conjugate surfaces instead of the actual surface is well justified.

2.3.2.1. Construction of Ease-off and the Surface of Roll Angle Ease-off will be defined in this study as the deviation of real gear surface from the conjugate of its real mating pinion surface. It can also be defined as the deviation of real pinion surface from the conjugate of its real mating gear. These definitions are respectively called the gear-based ease-off and pinion-based ease-off [2.5].

The

conventional method of UTCA seeks a contact between two arbitrary surfaces, failing to benefit from the fact that the designed gear and pinion surfaces are indeed close to the corresponding conjugate surfaces.

Closeness of the actual and the corresponding

conjugate surfaces enables the use of the ease-off concept.

The proposed ease-off

approach for UTCA has several advantages such as (i)

providing an overview of the contact pattern and transmission error as well as interference between pinion and gear teeth especially at the edges,

(ii)

providing more accurate instantaneous contact curves instead of commonly used approximate contact lines,

(iii) eliminating the need to compute the curvature in order to estimate length and direction of the contact lines/curves, and 53

(iv) avoiding the need for initial guesses required by the conventional method to locate first contact point (some initial guesses might end up divergent solutions). In this study, the ease-off surface is constructed directly from the relationships between the continuous cutter surfaces. Therefore, any surface fits to pinion and gear surfaces is not needed. The first step in constructing ease-off is to specify an area in the gear projection plane with the possibility of contact between pinion and gear, as shown in Figure 2.7. Such an area is the projection of a volume bounded by the faces and front and back cones of the pinion and gear into the gear projection plane. The projection plane shown in Figure 2.7 is a gear-based projection plane since its ease-off is defined on gear tooth area.

Using the gear machine settings and Eq. (2.8), the real gear surface

coordinates (shown in Figure 2.7) are computed for every point of the projection plane as

rijg where i  [1, mg ] and j  [1, ng ] are the indices in lengthwise and profile directions of the surface point with mg and n g as number of surface grids respective directions. Then, each point of the gear projection plane is transformed to the pinion coordinate system to construct the projection plane for the pinion. Next, the real pinion surface points and unit normals to the surface are computed from Eq. (2.8) for every point of the pinion projection plane as rijp and nijp . Both real pinion and gear surface coordinates and unit normal vectors are transformed into the global coordinate system where 54

 

P

Real gear surface, rijg Gear projection plane E

Ease-off surface, ij  



G

ap

ag

Q Surface

Real pinion surface, rijp

Action surface, rija Conjugate of pinion, rˆijg

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

misalignments E , P , G and  shown in Figure 2.7 are applied as well. Having

rijp , nijp , the pinion and gear axis vectors a p and a g , and the gear ratio R , the action surface position vector rija is found from conjugacy equation in 3D space as [2.5]: R (a p  rijp )  nijp  (a g  rija )  nijp .

(2.19)

The same steps are repeated for all points on the real pinion surface to define the action surface completely. As seen in Figure 2.7, the action surface for a hypoid gear pair, while quite flat, is not a plane. Any point and its unit normal vector on the real pinion surface are rotated around the pinion axis in order to satisfy Eq. (2.19). The angle of rotation ijp of each point that satisfies Eq. (2.19) is then plotted on the projection plane to construct the pinion roll angle surface Q . The angle ijg = ijp R corresponds to the amount of rotation from the surface of action to reach the conjugate of pinion surface. Therefore, rijg  M z (ijg ) rija .

(2.20)

If this conjugate surface of the pinion were to match perfectly with the real gear surface at any point, then a perfect meshing condition with zero unloaded transmission error would exist. The difference between these two surfaces (conjugate of pinion and

56

real gear surfaces) in projection plane domain is defined as ease-off surface  where the differences between these two surface at each grid is ij .

2.3.2.2. Contact Pattern and Transmission Error Any value , of any point on the Q surface is the pinion roll angle. As shown in Figure 2.8 for a specific pinion roll angle i , intersection of the plane z  i and the

Q surface defines x and y coordinates of all points on the projection plane that have the same roll angle, stating theoretically that they lie on the same contact line/curve. Since

Q is not a plane, this intersection for hypoid gears is usually a curve rather than a straight line as assumed by most studies. The instantaneous contact curve C (i ) shown in Figure 2.8 is determined by projecting the intersection curve first on projection plane and then projecting this projected curve once more on the ease-off surface.

The

minimum distance from C (i ) to the projection plane at point H (i ) is the instantaneous unloaded transmission error TE (i ) . Moreover, moving in both directions from point H (i ) along C (i ) within a preset separation distance  , gives the unloaded contact line length L(i ) .

Repeating this procedure for every pinion roll angle 

increment, unloaded transmission error curve TE () and the unloaded tooth contact pattern are computed. Here the contact curves are between real pinion and conjugate gear. 57

Ease-off surface

z

  Instantaneous contact curve, C (i )

(a)

Projection plane A

Contact curve projection

y z  i x

Q  i

Q surface

A (b)

C (i )

z L(i ) H (i )

 x

TE (i )

Figure 2.8: Unloaded TCA computation procedure: (a) gear projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded transmission error. 58

Replacing the conjugate gear with the real one practically does not change the contact line orientation and shape, since the effect of microscopic changes of pinion and gear surfaces on Q surface is negligible.

2.4. An Example Hypoid Unloaded Tooth Contact Analysis An example hypoid gear pair whose basic parameters are listed in Table 2.1 for the drive-side contact (concave side of pinion and convex side of gear) is considered to demonstrate the capabilities of the proposed hypoid gear geometry computation and unloaded tooth contact pattern models.

This is a FM gear set representative of

automotive rear axle gear sets. The predicted unloaded transmission error (UTE) curves computed by the model for three adjacent tooth pairs i  1 , i and i  1 are shown in Figure 2.9. Here UTE is plotted against the mesh cycles (pinion roll angle) where each of the individual UTE curves corresponds to a single tooth pair in mesh. Individual curves for two adjacent tooth pairs are one mesh cycle apart. At the intersection point of the two adjacent UTE curves ( M1 or M 2 in Figure 2.9), transition from one tooth pair to adjacent tooth pair occurs. The transmission error value of the intersection point is the maximum UTE, which is attempted to be minimized for unloaded tooth contact pattern optimization procedures. The corresponding predicted unloaded tooth contact pattern is shown in 59

__________________________________________________________________ Parameter

Pinion

Gear

__________________________________________________________________ Number of teeth

11

41

Hand of Spiral

Left

Right

Mean spiral angle (deg)

40.5

28.5

Shaft angle (deg)

90

Shaft offset (mm)

20

Outer cone distance (mm) Generation type Cutting method

115.0

111.0

Generate

Formate FM

__________________________________________________________________

Table 2.1: Basic drive side geometry and working parameters of the example hypoid gear pair.

60

  TE rad]

30

i 1

i

15

i 1

M1

M2

1

2

P-P UTE 3

Mesh cycles

Figure 2.9: Unloaded transmission error of the example gear pair with misalignments

E  0.15 mm, P  0.12 mm, G  0,   0 .

61

Figure 2.10. The curve marked as CPP is the locus of all instantaneous contact points between gear and pinion tooth surfaces. On both sides of CPP are right point path (RPP) and left point path (LPP), which are as far from CPP as it is required to reach chosen separation value of   0.006 mm between ease-off topography and projection plane shown in Figure 2.8. The CPP, RPP and LPP curves are computed for a single tooth pair in contact while in case of multiple teeth in contact these curves are partially active (usually middle part of the curves are active) since the adjacent pairs will take over the motion. In Figure 2.10, unloaded contact pattern of the tooth of interest is bounded by the instantaneous contact lines at M1 and M 2 and the RPP and LPP curves. Using the same example gear pair, the influence of misalignments effects on unloaded contact patterns are illustrated next. Figure 11(a) shows unloaded contact pattern of the drive side at a nominal position where E  P  G    0 . Misalignments of E  0.08 mm, P  0.10 mm and G    0 move the unloaded contact pattern to toe as shown in Figure 11(b) while misalignments E  0.15 mm, P  0.10 mm and G    0 move the contact pattern to heel as shown in Figure

11(c). Similarly, Figure 10(d) shows another contact pattern near toe for the gear pair with misalignments of E  0.05 mm, P  G  0 , and   4 min. Comparison of Figures 11(b) and 11(d) indicates that similar shifts in the contact patterns can be caused by different sets of misalignments.

62

Tip

 

M1

LPP

CPP RPP

Toe

Contact lines

M2

Root Heel

Figure 2.10: Unloaded contact pattern of the example gear pair for three adjacent tooth pairs i  1 , i and i  1 (i-1), (i) and (i+1) with E  0.15 mm, P  0.12 mm, G  0 and   0 .

63

(a)

(b)

(c)

(d)

Figure 2.11: Unloaded contact pattern of the example gear pair (a) at nominal position with E  P  G    0 , (b) at toe with E  0.08 mm, P  0.10 mm and G    0 , (c) at heel with E  0.15 mm, P  0.10 mm and G    0 and

(d) at toe with E  0.05 mm, P  G  0 , and   4 min.

64

References for Chapter 2 [2.1]

Baxter, M. L., and Spear, G. M., 1961, "Effects of Misalignment on Tooth Action of Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.

[2.2]

Krenzer, T. J., 1981, Understanding Tooth Contact Analysis, The Gleason Works.

[2.3]

Smith, R. E., 1984, "What Single Flank Measurement Can Do For You." AGMA, Technical Paper No. 84FTM2.

[2.4]

Smith, R. E., 1987, "The Relationship of Measured Gear Noise to Measured Gear Transmission Errors." AGMA, Technical Paper No. 87FTM6.

[2.5]

Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute of Technology.

[2.6]

Zhang, Y., and Wu, Z., 2007, "Geometry of Tooth Profile and Fillet of FaceHobbed Spiral Bevel Gears." IDETC/CIE 2007, Las Vegas, Nevada, USA.

[2.7]

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

Concurrent Engineering Approach, John Wiley & Sons Inc. [2.8]

Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.

Des., 129(1), pp. 31-37. [2.9]

Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University of Illinois at Chicago.

65

[2.10] Wu, D., and Luo, J., 1992, A Geometric Theory of Conjugate Tooth Surfaces, World Scientific, River Edge, NJ. [2.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd

ed.), Cambridge University Press, Cambridge. [2.12] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears, Elsevier Science B. V. [2.13] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears under Load." Earthmoving Industry Conference, Peoria, IL. [2.14] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their Load Carrying Capacity (Part I)." D.S.I.R., London. [2.15] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason Works. [2.16] Shtipelman, B. A., 1979, Design and manufacture of hypoid gears, John Wiley & Sons, Inc. [2.17] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany. [2.18] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212. [2.19] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA, Technical Paper No. 05FTM08.

66

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

International Power Transmission and Gearing Conference ASME, 88, pp. 789798. [2.21] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,

122(44), pp. 529-535.

67

CHAPTER 3

SHELL BASED HYPOID TOOTH COMPLIANCE MODEL AND LOADED TOOTH CONTACT ANALYSIS

3.1. Introduction In the previous chapter, an ease-off topography defined by the theoretical tooth surface was developed and used to determine the unloaded contact characteristics of a hypoid gear pair, including the unloaded single and multiple tooth contact patterns and the unloaded transmission error. This chapter builds on this formulation to predict the same under loaded contact conditions. Published studies on the modeling of tooth contact of hypoid gears under loaded conditions are quite sparse.

They can basically be divided into two major groups:

Computational models that use Finite Element (FE) or Boundary Element (BE) formulations, and analytical models. As an example of models from the first group,

68

Wilcox et al [3.1] developed a FE-based model to calculate spiral bevel and hypoid gear tooth compliance using 3D model of tooth including base deformations, which was later employed by Fan and Wilcox [3.2] to develop a loaded tooth contact analysis (LTCA). Vijayakar [3.3] developed another FE based hypoid LTCA package. This model employs a hybrid approach with FE away from the contact zone and a semi-analytical contact formulation at the contact zone. This model is perhaps the most advanced hypoid LTCA model available today to simulate the loaded contacts of a hypoid gear pair accurately. The major drawback of these computational models however is that they require a considerable amount of computation time, which makes them more of an analysis tool. Their use for design tasks such as parameter and assembly variation sensitivity studies is not very practical for the same reason. Besides these computational models, some semi-analytical models were also proposed for determining tooth compliance of parallel-axis gears trough elasticity-based deformation solutions. A detailed literature review of such studies was provided in Chapter 1. All of these analytical compliance models were valid for a tooth having constant height along its face width and either constant or linearly varying thickness along its profile, which is not the case for hypoid gears. Vaidyanathan [3.4-3.6] proposed an analytical compliance model for a tooth with linearly varying thickness in the profile and lengthwise directions as well as linearly varying tooth height along the face width. His Rayleigh-Ritz based formulation used polynomial shape functions and was applied to both sector and shell geometries. The sector model represents straight bevel gear 69

geometry closely while the shell model can be deemed sufficiently close to that of spiral bevel gear. In this chapter, a Rayleigh-Ritz based shell model similar to the one proposed by Vaidyanathan [3.4-3.6] will be applied to face-hobbed and face-milled hypoid pinion and gear teeth to define the tooth compliances due to bending and shear effects efficiently in a semi-analytical manner.

Base rotation and contact deformation effects will also be

included in this compliance formulation.

With this, loaded contact patterns and

transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears will be predicted by enforcing the compatibility and equilibrium conditions associated with the load distribution at the gear mesh.

3.2. Tooth Compliance Model According to methodology outlined in the flowchart of Figure 1.4, the last step before a LTCA can be performed is determining the tooth compliances of both contacting members.

The compliance of a tooth is defined as the amount of deflection at any

contact point due to a unit load applied at various points on the same tooth surface [3.5]. The compliance of a gear tooth must include tooth bending deflections, shear and Hertzian deformations as well as the base rotation, since each might contribute to tooth deflections significantly.

As the computational efficiency of the model is a major

consideration, a semi-analytical shell model [3.5] is employed here instead of a FE

70

model. The model considers a hypoid gear tooth as part of a shell with a linearly varying thickness and height, as parameterized in Figure 3.1.

The tooth thicknesses ttoe , theel

and ttip at the toe-root, heel-root and tooth tip locations are calculated and a linear, twovariable function is fit to these calculated thicknesses to obtain a thickness function as

t  ttoe 

x (ttip  ttoe ) h



(theel  ttoe )  rc (h  x) fh

(3.1a)

where h is the height of the tooth defined as

h  htoe  [

hheel  htoe ] rc . f

(3.1b)

Here, f is the face width, rc is the cutter radius, htoe and heel hheel are tooth height values at the toe and the heel of the tooth,  is the angle between any contact points on the tooth to toe measured from cutter center, and x is the tooth height at that specific contact point as defined in Figure 3.1. In cylindrical coordinates with independent variables  ,  and z, the position vector of a point on the circular cylindrical shell is defined as [3.5]: R (, , z )  r (, )  z i u .

(3.2a)

Here i u is the normal to the mid-surface of the shell, defined as

71

Heel (free)

 

Tip (free) hheel

ttip

htoe

x



theel Root (clamped)

6

ttoe Toe (free)

Cutter axis

Figure 3.1: Basic dimensions of a hypoid tooth used in the compliance formulation.

72

iu 

r  r sin 

,

 r  r   cos 1   r r

 . 

(3.2b,c)

The normal and shear strains,  m and  mn , are given as ( m, n  [1,3] , m  n ) [3.5]

m 

  m

 mn 

 Um   g m

 1   2 g m

 U k g m  ,  g k 1  k  k  3



 U U    ( m )  gn ( n )  gm  n g m g n   m gm g n  1

(3.3a)

(3.3b)

where g1 =[A(1+ z Rα )]2 ,

(3.3c)

g 2 =[A(1+ z R )]2 ,

(3.3d)

g3 =1

(3.3e)

and U m,n is displacement component. In cylindrical coordinates, normal and shear strain relations are derived as [3.5]:

73

 1 U V A W  ,   z  A  AB  R  (1  ) R 1

 

 

 1 V U B W    ,  z (1  )  B  AB  R  R 1

(3.4a)

(3.4b)

W , z

(3.4c)

z z B(1  ) ) R  R  U V  [ ] [ ], z z z  z B(1  )  A(1  ) A(1  ) B (1  ) R R R R

(3.4d)

z 

A(1 

 

 z 

 z 

W z  U  A(1  ) [ ], z  z  R z  A(1  ) A(1  ) R R 1

W z  V  B(1  ) [ ]. z R z B(1  z ) B(1  )  R R 1

(3.4e)

(3.4f)

For a cylindrical shell, R   , R  rc ,   x ,    , A  1 and B  rc with  and  as the independent curvilinear coordinates along and perpendicular to tooth root line, respectively. Employing a displacement assumption based on the Mindlin type shear

74

theory [3.7] that assumes a constant shear strain throughout the thickness [3.5], Eq. (3.3) is reduces to:   x  2W  x  z   2 , x   x

 

z rc

 x 

(3.5a)

  1  2W  W ,    2   (rc  z )   (rc  z )

z (rc  z )  z (2rc  z )  2W z  x   , x rc (rc  z ) x (rc  z )  rc

 z  (1 

z ) , rc

(3.5a)

(3.5c)

(3.5d)

 zx   x .

(3.5e)

The transverse deflection W and shear rotations,  x and  , are obtained by setting the first variation of the potential energy to zero by using the Rayleigh-Ritz method. The potential energy (PE) for a conservative system is the difference between the strain energy SE and the work done by the external force WF, i.e. PE  SE  WF . Here, SE and WF due to an external force p of a deformed shell surface are [3.5]

SE 

E 2

   (rc  z ) ( x

2(1   ) x  z

2

 2  2 x  )

(3.6a)

 1 (1  )(  x2   xz 2   z 2 )  dzd dx, 2  75

WF    pWrc dxd  .

(3.6b)

x

where E and  are the modulus of elasticity and the Poisson’s ratio for the gear material. Setting the first variation of the potential energy to zero one obtains

PE 

E

   (rc  z ) ( x  x    ( x    x )

(1   2 ) x  z

(3.7)

 12 (1  )(  x x   xz  xz   z  z )  dzd dx    pWrc dxd   0. x

The transverse deflection and shear rotations are written as linear combinations of finite number of polynomials with unknown coefficients as W    Amn  m ( x) n () ,

(3.8a)

 x    Bmn  m ( x) n () ,

(3.8b)

    Cmn  m ( x) n () ,

(3.8c)

m n

m n

m n

all of which must satisfy the following boundary conditions of a shell-shaped tooth shown in Figure 3.1:

W ( x, ) x 0  0 ,

W ( x, ) , x x 0

(3.9a,b)

76

 x ( x, ) x  0  0 ,

 ( x, ) x 0  0 .

(3.9c,d)

The polynomial functions  m ( x) and  n () must satisfy all essential boundary conditions, in addition to being continuous, linearly independent and complete. Equation (3.8) is substituted in Eq. (3.5) that is needed to evaluate normal and shear strains defined in Eq. (3.6a) to compute strain energy.

Computations of SE and WF are done

numerically using Gauss-Quadrature method to yield the linear set of equations:  K11 K12   K 21 K 22  31 K 32  K

K13   A   F1       K 23   B   F 2      K 33   C   F3    

(3.10)

where the sub-matrices K ms ns (ms , ns  [1,3]) are determined based on volume integral

of material properties and assumed trial functions, and F ms (ms  [1,3]) forms the force vector that depends on type (point, line, etc) and location of the applied load. Computation of the tooth compliance is done by solving Eq. (3.10) numerically for coefficients Amn , Bmn and Cmn . Polynomial functions for pinned-free and clamped-free boundary conditions are assumed as

 m ( x)  ( x a)m

respectively, where a is the tooth height.

and

 m ( x)  ( x a) m1 ,

The free-free condition for  n () is

represented by the polynomial  n ()  (  ) n1 , where  is the subtended angle by

77

circular segment of the shell of a length equal to the face width of the gear [3.5], which is approximately equal to the ratio of the tooth length to the cutter radius. Figure 3.2 shows a flowchart of the tooth compliance computation methodology. While the task of computing unknown coefficients of shape functions is time consuming, it is done only once and is valid for all mesh positions. Suppose that total contact lines/curves as is shown (for one contact line) in Figure 3.3 are divided into N c segments (and each segment has its own local load, which is yet to be computed), then the total compliance matrix of a pinion or gear tooth is written as  w11  C . w  1N c

wNc 1   . .  . wNc Nc  .

(3.11)

where wis js (is , js  [1, N c ]) is the deflection at segment is due to the load applied at segment js. With this, the total deflection at segment is due to all of the applied discrete loads (load vector) is Wis   Nj c1 wis js . s

Closed-form formula of Weber [3.8] was used here for computing the Hertzian deformations while the base rotation and base translation

effects on total tooth

compliance were introduced by using an approximate interpolation method similar to the one developed for helical gears by Stegemiller [3.9].

78

Choose number of mode shapes IFF: Free-Free ICF: Clamped-Free

Blank dimensions and machine settings

Calculate strain energy based on shape function

Tooth thickness and height calculation

Minimize potential energy (strain energy + work) using Rayleigh-Ritz method

Fit a linear function to tooth thickness and height

Calculate unknown coefficients of polynomial shape functions

Use polynomial shape functions with unknown coefficients for deformation that satisfy boundary conditions

Construct compliance matrix for pinion and gear and calculate total compliance matrix

Potential contact lines from Unloaded TCA calculation

Figure 3.2: Flowchart of the compliance computation.

79

F1

F2

F3

. 1

2

3

FN 1 c

.

FN c

Nc  1 Nc

Figure 3.3: Potential contact line discretization.

80

In order to validate the proposed tooth compliance computation procedure, a tooth of a face-milled Formate hypoid gear was modeled by using a commercially available finite elements package (ANSYS). As an example, a 500 N load was applied in the middle of the lengthwise direction and the middle of the profile, and the tooth deflections predicted by ANSYS and the proposed semi-analytical shell model along the middle of the top-land of the tooth were compared as shown in Figure 3.4. It is observed in this figure that increasing the number of mode shapes in Eq. (3.8) (IFF mode shapes for the free-free boundary conditions and ICF mode shapes for the clamped-free boundary conditions) improves shell model predictions, converging the predicted deflections to those from ANSYS. However, it also increases the computational time require as shown in this figure as well. Vaidyanathan conducts an extensive comparison of his developed shell model with ANSYS for various loading conditions and number of mode shapes and proved accuracy of the shell model [3.6].

3.3. Loaded Tooth Contact Analysis The number of tooth pairs in contact depends on the gear contact ratio, roll angle of the pinion (or gear) and amount of applied torque. Under unloaded conditions, a hypoid gear pair having a contact ratio greater than one has always at least one tooth pair in contact. Once the load is applied, this number increases due to the deflection of the contacting teeth. In the loaded tooth contact model, all the tooth pairs that are likely to 81

3 Shell model

FEM

82

Deflection of free edge (micron)

2.5

ICF  5, IFF  3, t  8 s ICF  5, IFF  5, t  15 s

ICF  5, IFF  9, t  50 s

2

ICF  5, IFF  15, t  135 s 1.5

1

0.5

0

-0.5 0

5

10

15

20

25

30

Location along face width direction (mm) Figure 3.4: The comparison of the shell model deformation to FEM.

35

40

45

geometrically share the torque must be taken into consideration with their respective separation distances.

Potential contact lines/curves of all contacting tooth pairs are

computed and discretized into a finite number of segments. The length, separation, surface curvatures of both members along each line segment are computed and used as input for the LTCA model. Conditions of compatibility and equilibrium must be satisfied simultaneously in the load distribution model [3.10]. According to the compatibility condition, in order for the contact to occur along each of the contact lines/curves, the sum of total elastic deformation of two contacting teeth C F and the initial separation vector S must be greater than or equal to the rigid body rotation ΘR g , i.e. C F  S  ΘR g

(3.12)

where F is force vector, C is the total compliance matrix that is the sum of the pinion and gear tooth compliance matrices C p and C g (Eq. (3.11)), and the Hertzian compliance matrix Ch , and R g is the vector that contains the distances of each segment to the gear axis.

Eq. (3.11) can be written in form of an equality constraint by

introducing slack variable Y as  C F  ΘR g  Y = S .

(3.13)

83

Since two bodies must be at contact for any force on a given segment is to exist, either Fis  0 for Yis  0 or Fis  0 for Yis  0 (is  [1, N c ]) .

Meanwhile, the equilibrium condition assures that the total moment caused by forces acting on all contacting segments about the gear axis as shown in Figure 3.5 must be equal to external torque T g applied on the gear axis: FT R g  T g

(3.14)

where superscript T denotes matrix transpose.

The load distribution and loaded

transmission error are computed by solving compatibility and equilibrium equations simultaneously.

3.4. An Example Hypoid Tooth Contact Analysis

The same face-milled example hypoid gear pair used for the UTCA whose basic parameters are listed in Table 2.1 for the drive-side contact (concave side of pinion and convex side of gear) is considered to demonstrate the capabilities of the proposed hypoid gear load distribution model. This is a FM gear set representative of an automotive rear axle gear sets.

84

Tg

ag

F3

F2

R3g R2g

F1

R1g

Figure 3.5: Static equilibrium between torque applied on gear axis and torque produced by the force of all contacting segments.

85

Loaded transmission error (LTE) of the example gear pair predicted by the proposed model are shown in Figure 3.6 at three different pinion torque values of T p  50, 250 and 500 Nm for a set of fixed misalignment values of E  0.15

mm, P  0.12 mm, G  0 mm and   0 . These LTE time histories indicate that the shape and the average value of LTE change with T p , as expected. Table 3.1 lists the peak-to-peak value (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-meansquare (RMS) value of LTE corresponding to the cases of Figure 3.6. The LTE functions in each T p level are dictated primarily by the first harmonic order. Modest increases in p-p, 1st harmonic and the root-mean-square values of LTE are observed with increasing T p.

Figures 3.7(a-c) show the pressure distributions predicted by the proposed model for the same cases of Figure 3.6. It is seen in Figure 3.7(a) that the contact is localized at the center of the tooth when T p is low (50 Nm) with no edge loading. An increase in T p causes the contact pattern to spread, in the process exhibiting edge loading at the tip

and root regions as it is evident from Figures 3.7(b) and (c). Figures 3.7(a-c) also show the loaded contact patterns (maximum contact pressure distributions) predicted by a FEbased hypoid contact model [3.11] for the same cases are in good agreement. It is worthwhile to mention here that each simulation with the proposed model required 45 seconds of CPU time (about 25 seconds for compliance matrix computations and 1 second

per

roll

angle)

on

a

3.0 86

GHz

PC

while

the

same

(a) T p = 50 Nm

130 120 110 100 90

(b) T p = 250 Nm

310 300

TE [rad]

290 280 270

(c) T p = 500 Nm

620 610 600 590 580 0.0

0.5

1.0

Mesh cycles

1.5

2.0

Figure 3.6: Loaded gear transmission error of the example gear pair with E  0.15 mm, P  0.12 mm, G  0 and   0 at (a) T p  50 Nm, (b) T p  250 Nm, and (c) T p  500 Nm.

87

_________________________________________________________________

T

p

Loaded Transmission Error [μrad] ______________________________________

Errors

[Nm] [mm] p-p 1st 2nd 3rd RMS _________________________________________________________________ 50 E  0.15, P  0.12

12.8

6.3

0.6

0.1

6.3

250 E  0.15, P  0.12

15.9

7.9

0.3

0.4

7.9

500 E  0.15, P  0.12

21.9

9.9

2.3

0.5

10.1

_________________________________________________________________ 50 E  0.08, P  0.05

12.2

5.5

0.7

0.6

5.6

50 E  0.26, P  0.13

12.5

6.3

0.4

0.1

6.3

_________________________________________________________________

Table 3.1: The loaded transmission error predictions of the proposed model; G  0 mm and   0 for all cases.

88

Proposed Model

(a) T p = 50 Nm , E = 0.15 mm , P = 0.12 mm

FE Model [3.11]

(b) T p = 250 Nm , E = 0.15 mm , P = 0.12 mm  

89

(c) T p = 500 Nm , E = 0.15 mm , P = 0.12 mm  

Continued Figure 3.7: Comparison of loaded contact patterns predicted by the proposed model to an FE model [3.11] for (a) T  50 Nm , E  0.15 mm, P  0.12 mm, (b) T p  250 Nm , E  0.15 mm, P  0.12 mm, (c) T p  500 Nm , p

E  0.15 mm, P  0.12 mm, (d) T p  50 Nm , E  0.08 mm, P  0.05 mm, and (e) T p  50 Nm , E  0.26 mm, P  0.13 mm ( all at G  0,   0 ).

Continued Proposed Model

(d) T p = 50 Nm , E = 0.08 mm , P = 0.05 mm

 

(e) T p = 50 Nm , E = 0.26 mm , P = 0.13 mm

  90

Figure 3.7 continued

FE Model [3.11]

analysis using the FE model took about 15 minutes using the same computer. This highlights the main advantage of this proposed model as a design tool, even if it might not be as accurate as the full FE model [3.11]. Next, the same gear pair is simulated by using the proposed model and the FE model [3.11] at T p  50 Nm for two other misalignment conditions. Here, two of the errors are kept constant at G  0 mm and   0 , and the other two errors E and P are varied. In Figure 3.7(d), error values of E  0.08 mm and P  0.05 mm cause the predicted loaded contact pattern to move towards toe and root, compared to Figure 3.7(a). Meanwhile, the loaded contact for E  0.26 mm and P  0.13 mm moves the contact in the opposite direction towards the heel.

In the process, the

maximum contact pressure is reduced since there is larger area in the heel that carries the same load. In addition, equivalent radii of curvature are larger at heel than toe, which directly decreases maximum Hertzian pressure from Weber equation [3.8]. The FE simulations of the same error combinations shown in the same figures are again in good agreement with the predictions of the proposed model. This suggests that the sensitivity of the hypoid gear contact to gear errors is captured sufficiently by this model. Finally the LTE parameters listed in Table 3.1 for these two cases reveal slight reduction in LTE amplitudes compared to the first case of Figure 3.7(a), suggesting that a good contact pattern does not necessarily mean lower LTE.

91

References for Chapter 3

[3.1]

Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth." AGMA, Technical Paper No. 97FTM05.

[3.2]

Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis (TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA, Technical Paper No. 05FTM08.

[3.3]

Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem." International J. for Numerical Methods in Engineering, 31, pp. 525-545.

[3.4]

Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The Ohio State University, Columbus, Ohio.

[3.5]

Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears Using Shell Theory." AGMA, Technical Paper No. 93FTM03.

[3.6]

Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical Approach to the Static Analysis of an Annular Sector Mindlin Plate with Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp. 255-266.

[3.7]

Mindlin, R. D., 1951, "Influence of Rotary Inertia and Shear on Flexural Motions of Isotropic Elastic Plates. ." J. of Applied Mechanics, 18, pp. 31-38.

92

[3.8]

Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their Load Carrying Capacity (Part I)." D.S.I.R., London.

[3.9]

Stegemiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192.

[3.10] Conry, T. F., and Seireg, A., 1972, "A Mathematical Programming Technique for the Evaluation of Load Distribution and Optimal Modification for Gear Systems." ASME J. of Industrial Engineering. [3.11] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced Numerical Solution Inc., Hilliard, Ohio.

93

CHAPTER 4

LOADED TOOTH CONTACT ANALYSIS OF HYPOID GEARS WITH LOCAL AND GLOBAL SURFACE DEVIATIONS

4.1. Introduction Hypoid gears used in various highest-volume applications such as automotive axles are subject to various manufacturing errors and heat treatment distortions that deviate the actual (real) tooth contact surfaces from the intended (theoretical) ones. Such errors impact the quality of a hypoid gear pair, defined by a number of performance indicators including its contact pattern, the motion transmission error (TE), efficiency as well as its sensitivity to misalignments.

These deviations represented by these

manufacturing errors typically follow patterns that shift, rotate or twist the surfaces relative to the theoretical ones. Therefore, they can be characterized as global deviations. Other more local deviations occur during the life span of hypoid gears in the form of 94

surface wear. Since the surface wear depths are proportional to the contact pressure and the sliding distance, deviations due to surface wear are rather local and cannot be captured by using the surface fitting methods developed to approximate the global deviations.

The motivation of this chapter is to develop a unified, ease-off based

methodology that allows loaded and unloaded tooth contact analysis of hypoid gears having both global and local deviations. Tooth contact analysis has usually been performed by considering the theoretical pinion and gear surfaces defined by simulation of the hypoid cutting processes. The analysis results presented in Chapters 2 and 3 also considered such theoretical tooth surface errors with no deviations..

There are only a few published studies on hypoid

gear tooth contact analysis using the real surfaces. In such an analysis, Gosselin [4.1] proposed an approach to compute tooth contact of real spiral bevel gear surfaces. He interpolated measured surfaces with rational functions to predict their unloaded contact pattern and transmission error. Since pinion and gear normal vectors of low-mismatch (high-conformity) surfaces, a wide span of potential contact line around contact point can be identified and the condition of collinearity for normal vectors is subjects to numerical stability issues. In order to simplify the task of locating the contact point, Gosselin [4.1] computed the difference between pinion and gear normal vectors at several points along the lengthwise direction and estimated a location where the difference between the normal vector is zero.

95

Zhang et al [4.2] proposed an approach to analyze unloaded tooth contact of real hypoid gears based on a generalization of the work of Kin [4.3, 4.4] on spur gears. The real pinion and gear tooth surfaces were divided into two vectorial surfaces of theoretical and deviation surfaces. Separating theoretical and deviation surfaces, finding the theoretical surfaces through cutting simulation, and applying interpolation only to the deviation surfaces made his approach simpler and more accurate. Zhang [4.2] defined the deviation surface in the normal direction of the theoretical surface by comparing the theoretical and the real (measured) surfaces, and fit a bicubic surface to it. The normal to the real surfaces (sum of the theoretical and interpolated deviation surfaces) were computed by taking the derivative of the real surfaces. Having continuous functions for the surfaces and normal of the pinion and the gear, he employed conventional system of five nonlinear equations and five unknowns used in many other studies [4.5-4.7] to simulate unloaded tooth contact of real hypoid surfaces. Gosselin proposed a method called “surface matching” that attempts to define changes to the machine settings that define the theoretical surfaces so that real gear surfaces can be computed approximately from the cutting simulation [4.8, 4.9]. This method found machine settings that generate a theoretical surface close to (but not identical to) the real surface and the difference of the two surfaces was defined as “residual error surfaces” for the pinion and the gear. He computed transmission error and the contact pattern of the generated surfaces with this new set of machine settings. Then, he used the residual error surface to modify predicted transmission error and contact pattern without providing the details of this process [4.10]. 96

This method is suitable in

capturing the effect of global errors such as the pressure angle error, the spiral angle error, and lengthwise or profile crowning errors for the cases when residual errors are rather small, while the same cannot be said for localized deviations such as surface wear and global deviations with large residual errors. In Chapter 2, it was mentioned that the unloaded tooth contact analysis (UTCA) of hypoid gears with mismatched surfaces were performed using two fundamentally different methods. The first method that was used widely defines the tooth surfaces as two arbitrary surfaces, rotating about pinion and gear axes [4.7, 4.11-4.13]. In this method, the contact point path (CPP) on each surface was determined by satisfying two contact conditions: (i) coincidence of position vector tips of the points on the gear and pinion surfaces and (ii) collinearity of the normals of the both of the surfaces. The second method that was proposed in Chapter 2 was based on the ease-off topography. A detailed formulation for construction of ease-off and determining the instantaneous contact curves from surface of roll angle was provided. In Chapter 3, UTCA results were combined with a semi-analytical compliance model based on shell theory to predict the loaded contact patters and loaded TE. Using the ease-off approach for TCA of real gear surfaces instead of conventional approach was shown to increase the computational efficiency of TCA since surface interpolations for measured pinion and gear surfaces as well as the solution of the system of five governing nonlinear equations are not needed. In this chapter, ease-off is defined the same way as Chapter 2 as the deviation of real gear surface from the conjugate of its real mating pinion surface. The formulation 97

that will be proposed to handle local and global deviations is based on the premise that all surface deviations, both local and global, can be handled through modifications of the ease-off topography. Having machine settings and blank dimensions, surface coordinates and normal vectors of both pinion and gear will be defined using the methodology proposed in Chapter 2. These theoretical surfaces will be used to establish the theoretical ease-off topography and a theoretical surface of roll angle. As the main contribution of this study, a procedure will be proposed to update the ease-off topography by taking into account pinion and gear surface deviations. The updated ease-off and roll angle surfaces will be used to determine the unladed tooth contact characteristics of the gear pair. These UTCA results will be combined with the semi-analytical LTCA methodology of Chapter 3 to predict the loaded contact patterns and the transmission error of hypoid gears having local and global surface deviations.

4.2. Construction of the Theoretical Ease-off Topography The first step in defining the theoretical ease-off surface is specifying an area in the gear projection plane with the possibility of contact between pinion and gear, as illustrated in Figure 4.1. This area represents the projection of a volume bounded by the faces and front and back cones of the pinion and gear into the gear projection plane. The projection plane shown in Figure 4.1 is a gear-based projection plane since its ease-off is defined on the gear tooth area. Using the gear machine settings and applying the equation of meshing, the theoretical gear surface coordinates of a point of the projection

98

 

rijg

Gear projection plane ij

ap

Q

ag

 

rija

rijp

rˆijg

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

plane shown in Figure 4.1 are computed as rijg where i  [1, m] and j  [1, n] are the indices in lengthwise and profile directions of the surface point with m and n as number

of surface grids in respective directions, and the superscript g denotes gear surface. Then, rijg is transformed to the pinion coordinate system to construct the projection plane for the pinion. Next, the theoretical pinion surface point rijp and its unit normal to the surface nijp are computed again through machine settings and applying the equation of meshing

[4.1]. Both theoretical pinion and gear surface coordinates and unit normal vectors are transformed into the global coordinate system. Having rijp , nijp , the pinion and gear axis vectors a p and a g , and the gear ratio R, the position vector rija of the corresponding point on the action surface is found from the conjugacy equation in 3D space as [4.14]:

R (a p  rijp )  nijp  (a g  rija )  nijp , i  [1, m] ,

j  [1, n] .

(4.1)

The same procedure is repeated for all points on the real pinion surface to define the action surface completely. As observed from Figure 4.1, the action surface for a hypoid gear pair, while quite flat, is not a plane. Here, any point and its unit normal vector on the theoretical pinion surface are rotated around the pinion axis in order to satisfy Eq. (4.1). The angle of rotation qijp of each point ij required to satisfy Eq. (4.1) is then 100

plotted on the gear projection plane to construct the theoretical pinion roll angle surface Q.

Here, only the relative values of qijp (i.e. the shape of the Q surface) are of interest. Therefore, for computational simplicity and graphical demonstration purposes, the Q surface is shifted in the direction normal to the projection plane to bring it to contact with the projection plane such that at least one grid point has zero qijp value. Mathematically, this shift is equivalent to rigidly rotating pinion tooth surface around the pinion axis, which has no effect on the pinion surface.

The angle qˆijg = qijp R

corresponds to the amount of rotation required to travel from the surface of action to the conjugate of theoretical pinion surface. Therefore, the position vector of the conjugate of the theoretical pinion surface is found as

rˆijg  M z (qˆijg ) rija , i  [1, m] ,

j  [1, n] ,

(4.2a)

where the rotation matrix about the z axis at an angle qˆijg is

 cos qˆ g ij  g  M z (qˆijg )    sin qˆij  0  0 

g

sin qˆij

g cos qˆij

0 0

0 0

 0 0 .  1 0  0 1

101

(4.2b)

If this conjugate surface of the pinion were to match perfectly with the real gear surface at any point, then a perfect meshing condition with zero unloaded transmission error would exist.

For simplicity, the conjugate of pinion will be called here the

conjugate gear. The difference between the conjugate gear and the theoretical gear is defined as theoretical ease-off topography  . In general, the conjugate gear and the theoretical gear are located at different angular positions with respect to the gear axis. In order to compare these two surfaces, conjugate gear surface is rotated around gear axis

a g by an angle  such that a grid point of conjugate gear surface touches the corresponding grid point on the theoretical gear surface. In this position, the radial distances ij between the grid points ij on these surfaces define the theoretical ease-off surface  . The  and Q surfaces were used in Chapters 2 and 3 for both unloaded and loaded tooth contact analyses.

4.3. Updating Ease-off Topography for Manufacturing Errors and Surface Wear Deviations of the pinion and gear surfaces a grid point ij from their respective theoretical surfaces are defined as ijp and ijg ( i  [1, m] , j  [1, n] ) as shown in Figure 4.2. Normal vectors to both pinion and gear surfaces are considered in inward direction. Measured and/or worn pinion and gear surfaces are written as ( i  [1, m] , j  [1, n] ):

rijp  rijp  ijp nijp ,

(4.3a) 102

rijg  rijg  ijg nijg .

(4.3b)

Here, it is assumed that the normal vectors of the real and theoretical surfaces are the same, since both surfaces are practically very close to each other [4.4]. Any point of the theoretical ease-off surface ij can be updated by using deviations on pinion and gear surfaces. The goal here is to update theoretical ease-off surface  directly from surface deviations rather that updating original pinion and gear surfaces and conducting whole tooth contact procedure between new surfaces as it has been done in Ref. [4.2]. Assume that an ease-off value ij of a point on the theoretical ease-off surface



is

computed

based

on

a

corresponding

T

theoretical

surface

vectors

T

rijp   xijp , yijp , zijp  , nijp and rijg   xijg , yijg , zijg  . At the same grid point ij, the    

pinion roll angle is qijp , and hence, the corresponding roll angle of the gear surface (conjugate to the theoretical pinion surface) is qijg  qijp R and the distance of the same grid points on the pinion and gear surfaces to their own rotation axes ( a p and a g respectively) are

Lijp  ( xijp ) 2  ( yijp ) 2 ,

(4.4a)

103

Lijg  ( xijg )2  ( yijg ) 2

(4.4b)

as shown in Figure 4.2. With this, the changes in the pinion and gear roll angles due to the pinion and gear surface errors ijp and ijg are defined as

qijh



ijh (uijh  nijh ) Lhij

, h =p, g.

(4.5a)

Here uijp and uijg are the unit normal vectors in the radial (circular) direction of the pinion and gear axes a p and a g , respectively, as shown in Figure 4.2.

They are

defined as:

uijp



a p  rijp a

p

 rijp

,

uijg



a g  rijg a

g

 rijg

(4.5b,c)

Hence, the updated pinion roll angle taking the pinion deviation into account is

qijp =qijp  qijp ,

(4.6)

which can be used to find locations and directions of contact curves corresponding to the deviated tooth surfaces.

104

Pinion projection plane

p nijp uij

ijp Lijp

Gear projection plane ijg

Q

nijg

uijg

rijg

rijp

ap

Lijg

ag

Figure 4.2: Graphical demonstration of the procedure to update ease-off surface for surface deviations.

105

Change to the theoretical ease-off surface  can be described in two components. The first component is related to the pinion surface deviation that is formulated as

ijp  qijgp Lijg

(4.7)

where qijgp = qijp R is the gear roll angle change due to the pinion surface deviation. The second component is due to the gear surface deviation that is given as

ijg  qijg Lijg

(4.8)

Therefore, the total change to the ease-off topography is the sum of its two components

ij  ijp  ijg .

(4.9)

With this, the new ease-off surface is found as

ij  ij  ij .

(4.10)

This updated ease-off surface  and the corresponding updated surface of roll angle Q as defined by Eq. (4.6) are used for unloaded and loaded tooth contact analyses according to the methodology proposed in Chapters 2 and 3

As in Q , the updated

surface of roll angle Q is also shifted to touch the projection plane since the absolute values qijp do not have any effect on unloaded and loaded tooth contact analyses. 106

4.4. Unloaded and Loaded Tooth Contact Analyses In Figure 4.3, surfaces  and Q are constructed on opposite sides of projection plane to provide an insight into how UTCA is conducted. For any value q on surface Q , a corresponding instantaneous contact curve can be defined. As shown in Figure 4.3 for a specific pinion roll angle qk , the intersection of the plane z   qk and the surface Q defines the x and y coordinates of all points on the projection plane that have same roll angle, stating theoretically that they lie on the same contact curve. Since Q is not a plane, this intersection for hypoid gears is usually a curve rather than a straight line as assumed in most of the previous studies. Figure 4.4 shows the theoretical contact curves of the drive and coast sides of a sample hypoid gear pair on the projection plane for different pinion angles as shown on the contact curves. The instantaneous contact curve C (qk ) shown in Figure 4.3 is obtained by first projecting this intersection curve on projection plane and then projecting this projected curve on the ease-off surface  . The minimum distance from C (qk ) to the projection plane at point H is instantaneous unloaded transmission error TE (qk ) . Moreover, moving in both directions from point H along the curve C (qk ) within a preset separation distance  yields the unloaded contact line length U (qk ) . Repeating this procedure for every pinion roll angle increment q  qk ( k  [1, Nl ] where Nl is total number of contact curves considered), unloaded transmission error curve TE (qk ) and the unloaded tooth contact pattern are computed. 107

z

 



C (qk )

Gear projection plane

Contact curve projection

z   qk x

q  qk

Q

z U (qk ) 

H x

TE (qk )

Figure 4.3: Graphical demonstration of the procedure to compute unloaded TCA; (a) gear projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded transmission error.

108

(a) Drive Side

Toe

10°

25°

40°

85°

70°

55°

Root

(b) Coast Side 10° Toe

25°

40°

55°

70° 85°

Root

Figure 4.4: Theoretical contact curves of an example hypoid gear pair.

109

It is noted here that the contact curves are defined between the real pinion surface and conjugate of the real pinion surface (conjugate gear surface), instead of using the real gear surface.

Replacing the real gear surface with the conjugate one causes a

very little change to the orientation and shape of the contact curve, since the effect of micro-geometry deviations has a negligible influence on the Q surface, i.e. Q and Q are practically identical. Number of tooth pairs in contact depends on the gear contact ratio, the roll angle of the pinion (or gear) and the amount of torque applied. Under unloaded conditions, a hypoid gear pair having a contact ratio greater than one has always at least one tooth pair in contact. Once the load is applied, this number increases due to deflection of the contacting teeth. In the LTCA model of Chapter 3, all the tooth pairs that are likely to share the torque geometrically are taken into consideration with their respective separation distances. Potential contact curves of all contacting tooth pairs are computed and discretized into a finite number of segments ( N c ). The length of the separation at each segment along each contact curve are computed and used as input for the LTCA model.

With the theoretical ease-off topography replaced by the modified ease-off

topography corresponding to real surfaces with deviations, the formulations of Chapters 2 and 3 can be applied to predict the unloaded and loaded tooth contact conditions, respectively.

110

4.5. Example Analyses 4.5.1. A Face-milled Hypoid Gear Pair with Local Surface Deviations A face-milled hypoid gear set with local deviations whose basic parameters are listed in Table 4.1 for the drive-side contact (concave side of pinion and convex side of gear) will considered for an example loaded contact analysis of surfaces with local deviations. This gear set is representative of an automotive rear axle gear set. An example case of local deviations from theoretical surfaces is shown in Figure 4.5 for both pinion deviation ijp and gear deviation ijg . This case represents worn tooth surfaces predicted by a hypoid gear wear model [4.15] from a companion study. In Figure 4.5, the “Root” line refers to the lower limit of active contact region and it is not actual gear root line. Following the proposed ease-off update approach, gear projection plane, the theoretical and updated ease-off surfaces,  and  , and the theoretical and updated roll angle surfaces, Q and Q , are computed. Figure 4.6(a) shows these surfaces in relation to each other while the theoretical ease-off topography, updated ease-off and the amount of ease-off change computed from Eq. (4.9) are shown in Fig. 4.6(b) to (d) in contour plot format, respectively. As shown here, the maximum changes take place in the vicinity of the locations where pinion and gear deviations are maximum, according to Figure 4.5 and the rest of the projection plane does not exhibit any considerable ease-off change.

111

_____________________________________________________________ Parameter Pinion Gear _____________________________________________________________ Number of teeth Hand of Spiral Mean spiral angle (deg) Shaft angle (deg) Shaft offset (mm) Outer cone distance (mm) Generation type Cutting method

11 Left 40.5 115 Generate

90 20 FM

41 Right 28.5 111 Formate

___________________________________________________________________________

Table 4.1: Basic drive side geometry and working parameters of the example hypoid gear pair.

112

ijp

Toe Root

ijg

Root

Pinion deviation

Toe

μm

Toe

Root Gear deviation Toe

Root

μm

10 8 6 4 2 0 5 4 3 2 1 0

Figure 4.5: Example local deviation surfaces for the gear and pinion tooth surfaces.

113

 

 

Gear projection plane

(a)

(b)

Q,Q

μm

60 40

Toe

20 Root

0

μm

(c)

60 40

Toe

20 Root

0

μm

(c) Toe

Root

6 5 4 3 2 1

Figure 4.6: Ease-off update for the example deviation of Fig. 5. (a) Three-dimensional view of the projection plane, and  ,  , Q and Q surfaces, and contour plots of (b)  , (c)  , and (d) the change of ease-off topography. 114

The corresponding predicted unloaded tooth contact patterns are shown in Figure 4.7 for a maximum separation value of   6 μm for both cases of (a) theoretical and (b) deviated tooth surfaces, indicating that the unloaded contact patterns are influenced by the local deviation as well. It is clear from this figure that the length of the contact lines are elongated for UTCA of deviated surfaces since ease-off in the contact region is flattened. The predicted unloaded transmission error (UTE) curves are shown in Figure 4.8(a) for the theoretical and deviated surfaces.

The corresponding peak-to-peak

amplitude (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-mean-square (RMS) value of these curves are listed in Table 4.2(a) to show that all components of UTE are influenced by the local deviation introduced. Both the root-mean-square (RMS) and p-p amplitudes are reduced with deviated surfaces. Next, LTCA is performed for the theoretical and the locally deviated surfaces as before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded transmission errors (LTE) at this torque value are shown in Figure 4.8(b) for the theoretical and deviated surfaces. It is noted here that both curves are identical for certain mesh positions where areas of the local deviation are not in contact while they differ significantly in certain mesh positions. Table 4.2(b) lists the same LTE amplitudes for theoretical and deviated surfaces to show that such local deviations also impact the LTE. Finally, predicted contact pressure distributions are shown in Figures 4.9(a) and (b) for the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it is 115

(a)

Theoretical surfaces

Toe

Root (b)

Deviated surfaces

Toe

Root

Figure 4.7: Predicted unloaded tooth contact pattern for separation value of   6 μm .

116

25 20

(a) Unloaded TE [µrad] Theoretical surfaces

15 10 5 0 25 20

Deviated surfaces

15 10 5 0

0

0.5

1.0 Mesh cycles

1.5

2.0

(b) Loaded TE [µrad] 270 Theoretical surfaces 265 260 255 250 245 250

Deviated surfaces

245 240 235 230 225

0

0.5

1.0 Mesh cycles

1.5

2.0

Figure 4.8: Transmission error (UTE) curves for theoretical and deviated surfaces at (a) unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm. 117

(a) Unloaded Transmission Error in [µrad] __________________________________________________________ p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 21.1 8.3 3.7 2.0 9.3 Deviated surfaces 15.0 4.3 4.3 1.4 6.3 __________________________________________________________ (b) Loaded Transmission Error at 200 Nm in [µrad] __________________________________________________________ p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 14.2 7.3 0.3 0.5 7.4 Deviated surfaces 10.2 4.5 1.4 0.4 4.7 __________________________________________________________

Table 4.2: The transmission error amplitudes of theoretical and deviated surfaces.

118

(a)

Theoretical surfaces

Mpa

1200 800

Toe

400 Root (b)

Deviated surfaces

0 Mpa

1200 800

Toe

400 Root

0

Figure 4.9: Predicted contact pressure distribution for a pinion toque of 200 Nm for (a) theoretical and (b) deviated surfaces.

119

observed that the edge loading condition experienced by the theoretical surfaces on the gear root (pinion tip) is reduced significantly for the surfaces with the local deviations considered since ease-off topography shown in Figure 4.6(d) is considerably flattened due to surface deviation in the same region.

4.5.2. A Face-hobbed Hypoid Gear Pair with Global Deviations A face-hobbed hypoid gear set with basic parameters listed in Table 4.3 for its drive-side contact (concave side of pinion and convex side of gear) is considered as example for loaded contact analysis of surfaces with global deviations. This gear pair is also representative of an automotive rear axle gear sets. The measured pinion and gear deviation surfaces ( ijp and ijg ) from their theoretical geometry after heat treatment and the lapping process are shown in Figures 4.10(a) and (b), along with their respective interpolated surfaces on active surface domain shown as  ijp and  ijg . Figures 4.10(c) and (d) are the contour plots of the same, showing a maximum 50 µm of error for pinion in heel-top region and 25 µm for gear in toe-root. The source of deviation here could be due to manufacturing errors, heat treatment distortion and surface wear caused by lapping process. In the most common measurement procedure used by the axle manufacturers, measured coordinates of points on a 5x9 grid are compared to the corresponding theoretical surface coordinates to determine the measured surface deviations in the direction normal to the surface. In Figure 4.10, these deviations are shifted for demonstration purposes so that all are in the positive side. 120

_____________________________________________________________ Parameter

Pinion

Gear

_____________________________________________________________ Number of teeth

12

41

Hand of Spiral

Left

Right

Mean spiral angle (deg)

50.0

24.0

Shaft angle (deg)

90

Shaft offset (mm)

45.0

Outer cone distance (mm) Generation type Cutting method

105

130

Generate

Formate FH

_____________________________________________________________

Table 4.3: Basic drive side geometry and working parameters of the example hypoid gear pair.

121

(a)

 

ijp

Toe

 

 ijp

(b)

Root

Toe ijg  ijg

Root

(c)

Measured pinion deviation

µm 40 30

Toe

20 10 0

Root (d)

Measured gear deviation

Toe

µm

25 20 15 10 5 0

Root Figure 4.10: Example global deviation surfaces measured by CMM for the gear and pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured deviation, (c) pinion deviation distribution in tooth active region and (d) gear deviation distribution in tooth active region. 122

It should be noted here that a simple weighted average is used to interpolate (or extrapolate) the deviations  ijp to ijp and  ijg to ijg at a point ij that is not one of the 45 measurement points. No interpolation is used to estimate surface coordinate and normal vectors as is the case in previous studies, hence the required accuracy and complexity of this interpolation is by no means comparable to the case that interpolation is required for surface coordinates and normal vectors estimation. Following the proposed method described in the previous section , the theoretical and updated ease-off surfaces are computed. Figures 4.11(a-d) respectively show (a) theoretical ease-off topography, (b) updated ease-off topography by only considering pinion surface deviations, (c) updated ease-off topography by only considering gear surface deviations and (d) updated ease-off topography by considering both pinion and gear surface deviations.

Figure 4.11(a) shows a localized well defined ease-off

topography as a result of interaction between theoretical pinion and gear surfaces. Although pinion and gear deviation effects on ease-off, when applied alone, are considerable as shown in Figure 4.11(b,c), the combination of these deviations alleviates their adverse effect of alone, resulting in the ease-off topography shown in Figure 4.11(d). The corresponding predicted unloaded tooth contact patterns of this design are shown in Figure 4.12(a,b) for a maximum separation value of   6 μm for both cases of theoretical and deviated tooth surfaces, respectively, indicating that the unloaded contact patterns are influenced by the deviations given in Figure 4.10 as well. With the deviation 123

(a)

µm 120

Toe

80 40 0

Root (b)

µm 120

Toe

80 40 0

Root (c)

µm 120

Toe

80 40 0

Root (d)

µm 120

Toe

8

80

6 4

40

2

Root

0

Figure 4.11: Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical easeoff topography, (b) updated ease-off topography only with pinion deviation, (c) updated ease-off topography only with gear deviation, and (d) updated ease-off topography with both pinion and gear deviations. 124

(a)

Theoretical surfaces

Toe

Root (b)

Deviated surfaces

Toe

Root

Figure 4.12: Predicted unloaded tooth contact pattern for separation value of   6 μm .

125

included, unloaded contact pattern slightly shifted toward heel and becomes narrower as it approaches gear tip. The predicted unloaded transmission error (UTE) curves are shown in Figure 4.13(a) for theoretical and deviated surfaces against mesh cycles. The corresponding peak-to-peak amplitude, first three Fourier harmonics and the RMS value of these curves are listed in Table 4.4(a) to show that the all components of UTE are influenced by the local deviation introduced. The RMS, peak-to-peak and 1st harmonic amplitudes are almost doubled with deviations included. Next, LTCA is performed for the theoretical and the globally deviated surfaces as before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded transmission errors (LTE) at this torque value are shown in Figure 4.13(b) for the theoretical and globally deviated surfaces. Table 4.4(b) lists the same LTE amplitudes for theoretical and deviated surfaces to show that such local deviations also impact the LTE. Finally, predicted contact pressure distribution is also shown in Figure 4.14(a) and (b) for the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it is observed that the contact pressure distributions for the theoretical and deviated surfaces are rather close since pinion and gear surface deviations tend to compensate each other in this particular example set of deviation (Figure 4.10).

126

(a) Unloaded TE [µrad] Theoretical surfaces

60 40 20 0

Deviated surfaces

60 40 20 0

0

0.5

1.0

1.5

2.0

(b) Loaded TE [µrad] Theoretical surfaces

335 325 315 305 370

Deviated surfaces

360 350 340

0

0.5

1.0 2.0 1.5 Mesh cycles Figure 4.13: Transmission error curves for theoretical and deviated surfaces; (a) unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm. 127

(a) Unloaded Transmission Error [µrad] __________________________________________________________ p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 36.9 15.1 4.0 1.8 15.7 Deviated surfaces 61.5 28.9 2.1 1.5 29.0 __________________________________________________________ (b) Loaded Transmission Error at 200 Nm [µrad] __________________________________________________________ p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 23.9 12.1 0.5 0.7 12.1 Deviated surfaces 29.8 15.8 2.8 2.3 16.2 _______________________________________________________

Table 4.4: The transmission error amplitudes of theoretical and deviated surfaces.

128

(a)

Theoretical surfaces

Mpa 800 600

Toe

400 200 Root (b)

Deviated surfaces

0

Mpa 800 600

Toe

400 200 Root

0

Figure 4.14: Predicted contact pressure distribution for a pinion toque of 200 Nm for (a) theoretical and (b) deviated surfaces.

129

4.6. Summary In this chapter, an accurate and practical method based on ease-off topography was proposed to perform loaded and unloaded tooth contact analysis of spiral bevel and hypoid gears having both types of local and global deviations. Manufacturing errors causing global errors and localized surface deviations were considered to update the theoretical ease-off to form a new ease-off surface that was used to perform a loaded tooth contact analysis. Two numerical examples of (i) face-milled hypoid gear set with local deviations and (ii) face-hobbed hypoid gear set with global deviations measured by CMM were presented to demonstrate the effectiveness of the proposed methodology as well as quantifying the effect of such deviations on load distribution and the unloaded and loaded motion transmission error.

References for Chapter 4 [4.1]

Gosselin, C., et al. 1991, "Tooth Contact Analysis of High Conformity Spiral Bevel Gears." Proceedings of JSME Int. Conf. on Motion and Power Transmission, Hiroshima, Japan.

[4.2]

Zhang, Y., Litvin, F. L., Maryuama, N., Takeda, R., and Sugimoto, M., 1994, "Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces."

116, pp. 677-682. [4.3]

Kin, V., 1992, "Tooth Contact Analysis Based on Inspection." Proceedings of 3rd World Congress on Gearing, Paris, France. 130

[4.4]

Kin, V., 1992, "Computerized Analysis of Gear Meshing Based on Coordinate Measurement Data." ASME Int. Power Transmission and Gearing Conference, Scottsdale, AZ.

[4.5]

Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.

[4.6]

Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd ed.), Cambridge University Press, Cambridge.

[4.7]

Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech. Des., 129(1), pp. 31-37.

[4.8]

Gosselin, C., Nonaka, T., Shiono, Y., Kubo, A., and Tatsuno, T., 1998, "Identification of the Machine Settings of Real Hypoid Gear Tooth Surfaces." ASME J. Mech. Des., 120(3), pp. 429-440.

[4.9]

Gosselin, C., Jiang, Q., Jenski, K., and Masseth, J., 2005, "Hypoid Gear Lapping Wear Coefficient and Simulation." AGMA, Technical Paper No. 05FTM09.

[4.10] Gosselin, C., Guertin, T., Remond, D., and Jean, Y., 2000, "Simulation and Experimental Measurement of the Transmission Error of Real Hypoid Gears Under Load." ASME J. Mech. Des., 122(1), pp. 109-122. [4.11] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears, Elsevier Science B. V. [4.12] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for hypoid gear-drives of formate and helixform; Part III-Analysis and optimal synthesis methods for mismatched gearing and its application for hypoid gears of formate and helixform." ASME J. Mech. Des., 103, pp. 102-113.

131

[4.13] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME International Power Transmission and Gearing Conference ASME, San Diego. [4.14] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute of Technology. [4.15] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear Pairs." In press, Wear, pp.

132

 

CHAPTER 5  

PREDICTION OF MECHANICAL POWER LOSSES OF HYPOID GEAR PAIRS  

5.1. Introduction Prediction of power losses of automotive drive trains is becoming increasingly critical to power train designers. This is mainly because the government regulations in regards to fuel economy and carbon emissions are becoming more stringent. Forecasted increases in oil prices also add to the motivation to predict and reduce power losses of drive trains.

In rear-wheel drive vehicles, the rear axle-differential unit is one of the

major sources of power losses. The axle efficiency values can be typically as low as 90 to 95% [5.1]. Considering that rear-wheel drive vehicles comprise a significant share of the global passenger vehicle market, any sizable improvements to the axle efficiency can have a significant positive impact on environment and energy consumption. Axle power losses can be divided into two groups. One group of losses is independent of the torque transmitted. These load independent (spin) power losses are 133  

due to viscous bearing losses (including the losses due to pre-load) and gear windage and oil churning losses [5.2, 5.3]. Such losses are outside the scope of this research. The other group of losses are induced by friction at bearing and hypoid gear pair locations under load. These power losses are called load-dependent (mechanical) losses. Focusing on the hypoid gear pair, mechanical power losses are associated with the relative sliding and the lubricated contact conditions along the tooth contacts. The shaft off-set, being the main difference between spiral bevel and hypoid gears, causes increased relative sliding in hypoid gears and the power losses associated with friction [5.4].

The

motivation of this chapter is to develop a mechanical power loss model of face-milled and face-hobbed hypoid gears. Modeling mechanical losses of a gear pair involves (i) computation of surface geometry parameters and velocities, and the normal load at each contact point from a tooth contact analysis model as the one proposed in Chapters 2 and 3, (ii) a friction model to determine the coefficient of friction at each contact points (iii) computing the surface traction from the distributions of the friction coefficient and normal force, and (iv) determining the friction torque and the resultant power loss. Published gear efficiency models differ mostly in the way they determine the friction coefficient. The first group of models used a constant friction coefficient μ [5.5-5.7] in computing the power losses. Recognizing the fact that μ is dependent on various contact parameters, including rolling velocity, sliding to roll ratio, radii of curvature of the contacting surfaces and normal 134  

load, all of which vary as gears roll, experiment based µ empirical formulae [5.8-5.11] were employed by another group of studies [5.8, 5.12-5.16]. However, the applicability of these models was limited to narrow ranges of the operating temperatures, speed, load, and surface roughness conditions represented by the empirical formula. The third group of models predicted the friction coefficient using the elastohydrodynamic lubrication (EHL) theory [5.17-5.22]. This approach, while physics-based and potentially more accurate, requires a significant computational effort as several hundreds of EHL analyses are required to predict the mechanical losses of a gear pair. In order to avoid this difficulty, Xu et al. [5.23] proposed a methodology to derive a gear contact friction formula up-front by using the EHL model of Cioc et al [5.24]. Using this EHL model, they conducted a large parameter study, covering wide ranges of contact and surface parameters as well as operating conditions representative of gears. The predicted surface traction data was reduced into a single formula by using linear regression technique. All of the models cited above were limited to spur or helical gears. Efficiency models for hypoid gears are very sparse. Approximating the hypoid gear power loss as the sum of losses from the corresponding spiral bevel and worm gears, Buckingham [5.25] recommended a power loss equation. Coleman [5.1] proposed a simple closedform formula to estimate bevel and hypoid gears efficiency. This heuristic formula used a constant friction coefficient of   0.05 at every contact point and was a function of the

135  

normal load, pressure angle, and pinion and gear mean spiral angles. Simon [5.26] applied a smooth surface EHL model to simulate hypoid gear lubrication. The model proposed recently by Xu and Kahraman [5.27] extended their helical gear efficiency model to hypoid gears. They used a commercially available FE-based hypoid gear contact model CALYX [5.28] to determine all required contact load and geometry parameters including curvatures. Employing set of equations developed by Litvin [5.29] primarily to describe relationships between curvatures of mating surfaces, they computed sliding and rolling velocities at each contact point along and perpendicular to the contact line. While this model [5.30] was physics-based and included most of the key surface, lubricant, geometry and operating parameters, its FE load distribution computation required significant computational time, making it impractical for design and parameter sensitivity studies. It relied on the same FE model for its geometry and curvature information as well. More importantly, the EHL model [5.24] it employed to derive the friction coefficient formula was not designed for simulating mixed type of lubrication condition. Therefore, the fidelity of the model Xu and Kahraman [5.30] was limited to contact conditions with no or limited asperity interactions. However, in most automotive hypoid gear applications, mixed EHL conditions characterized by excessive metal-to-metal contacts of the asperity peaks occur commonly.

Recently, Li and

Kahraman developed transient mixed or boundary EHL models for line [5.31] and point [5.32] contacts that can handle any lubricated gear contact conditions ranging from 136  

almost dry to full-film EHL. Li et al [5.33] demonstrated the effectiveness of their line EHL model by applying them to the methodology of Xu et al. [5.23] to predict helical gear efficiency. The hypoid gear efficiency model that will be developed in this chapter improves the methodology of Xu and Kahraman [5.32] by (i) employing the gear geometry defined in Chapter 2 to derive contact curvature and surface velocity values at each contact point instead of relying on any particular commercial FE package, (ii) employing the loaded tooth contact model of proposed in Chapter 3 for computation of the normal load for minimizing the computational time required for this task, and (iii) by incorporating a new µ formula derived by using the mixed EHL model of Li and Kahraman [5.32] such that any degree of asperity interactions can be modeled.  

5.2. Hypoid Gear Mechanical Power Loss Model Figure 5.1 shows the flowchart of the methodology used to compute the power losses of hypoid gear pairs in the proposed model. It combines the developed ease-off based hypoid gear contact model of Chapters 2 and 3 and the EHL-based friction coefficient model of Li and Kahraman [5.33] to compute mechanical efficiency. Blank dimensions, machine settings, cutter geometry, misalignments, load and speed are input data for the gear contact model. 137  

Blank dimensions, Machine settings, Cutter geometry, Misalignments, Load and Speed

Lubricant property, Temperature and Surface roughness

Req , Vr ,Vs , f Hypoid Gear Contact Model

, 

Friction Coefficient Model

Instantaneous Mechanical Efficiency

Yes

k  ns   No

Overall Mechanical Efficiency

Figure 5.1: Flowchart of overall hypoid gear efficiency computation methodology.

138  

At each time step k with pinion roll angle q  q k  k q ( k  [1, ns ] ), there are multiple potential contact lines between gear and pinion surfaces of adjacent tooth pairs (depending on the contact ratio and the pinion roll angle). Here, ns is total number of time steps per one pinion pitch qa  2 N p , N p is the number of teeth of the pinion and q  qa ns is pinion roll angle increment. At any incremental position k, there are ncl

(integer) number of potential contact lines, each divided into ncp number of contact segments. With this, a total of n  ncl ncp number of potential contact segments are defined at each increment k . Each line segment k ( k  [1, ns ] and   [1, n ] ) has a length Lk

and carries a constant load per unit length fk . The contact at the same

segment has an equivalent radius of curvature ( Req )k between contacting surfaces as well as constant sliding velocity (Vs )k and rolling velocity (Vr )k . Here, (Vr )k and ( Req )k are computed in a direction perpendicular to the potential contact line they belong to. At each line segment k , contacting surfaces are approximated as two cylinders in combined rolling and sliding. Friction at each segment has two components of sliding and rolling. The friction due to sliding is because of relative sliding between contacting surfaces and friction coefficient k is defined as the ratio of the tangential force produced due to sliding to the normal force applied between contacting surfaces. 139  

The friction caused by rolling is because of resistance of contacting surfaces against rolling over each other [5.23] and empirical rolling friction coefficient of Goksem [5.34] is used to predict friction force ( Fr )k . In addition to the lubricant properties and the surface roughness amplitude S , other contact parameters at each contact segment, namely Lk , fk , (Vs )k , (Vr )k and ( Req )k must be defined to determine the friction coefficient k and the rolling loss k  ( Fr )k  (Vr )k . Distributions of k and k are used to computed the instantaneous mechanical power loss P k at increment k due to sliding and rolling that are averaged over k to compute the average gear mesh power loss as P   nk s1 P k .  

5.2.1. Definition of the Sliding and Rolling Velocities The load fk at each line segment k are computed using the hypoid gear contact model proposed in Chapter 2 and 3.

The computation of the rolling and sliding

velocities, (Vs )k and (Vr )k , require a kinematic analysis beyond what is provided in Chapter 2. As shown in Figure 5.2(a) for any point on the ease-off surface with position vector M ij , the surface velocities of the pinion and gear are defined as:

v ijp   p (a p  rijp ) v ijg  g (a g  rijg )

( i  [1, m] , 140

 

j  [1, n] )

(5.1)

  t (a)

nij

t

uijp

v ijp

u gij

v ijg

M ij

ap

rijp

 

rijg da

ag

(b)

  t

(uijp )t 

t

u ijp

(uijg )t 

u gij 

(uijp )t , (uijg )t

   

Figure 5.2: Sliding and rolling velocities and their projection in tangential plane along and normal to the contact line direction.

141  

where  p  R g . The surface velocity v ijh of gear h ( h  p, g ) at this contact point has two components: w ijh in the common normal direction nij and uijh in the tangential plane  . Noting

w ijp  w ijg ,

(5.2)

( v ijp  v ijg )  nij  (uijp  uijg )  nij  0 ,

(5.3)

it can be stated that uijh ( h  p, g ) lies in the tangential plane and has two components, one component (uijh )t along the instantaneous potential contact line direction t and another component (uijh )t perpendicular to t in direction t  as shown in Figure 5.2(a). With this, sliding velocities at the same contact point along t and t  are given, respectively, as (vijs )t  (uijp )t  (uijg )t ,

(5.4a)

(vijs )t  (uijp )t  (uijg )t  .

(5.4b)

Hence, the total sliding velocity at the same point is

2

2

vijs  (vijs )t   (vijs )t  .    

(5.5) 142

 

The rolling velocity along t  is defined as

(vijr )t 

(uijp )t  (uijg )t 2

.

(5.6)

Sliding (Vs )k and rolling (Vr )k velocities at the contact segment k ( k  [1, ns ] and   [1, n ] ) are found through weighted averaging of the values at four corners of quadrilateral grid cell on the ease-off surface that contains the middle point of this segment. These two velocities will be computed for each contact segment and will be used to determine the local friction coefficient.  

5.2.2. Friction Coefficient Model The surface shear traction consists of the viscous shear within the fluid regions and the asperity traction in the regions of metal-to-metal contact. Considering a onedimensional flow, the sliding viscous shear stress acting on the contact segment k is defined as

( x, t )  

(Vs )k h ( x, t )

(5.7)

143  

where h( x, t ) is the instantaneous film thickness distribution and  is the effective viscosity. Parameter x defines the coordinate of a point within the contact in the direction of sliding. Within the asperity contact regions, the shear stress is defined as ( x , t )   d p ( x , t )

(5.8)

where p( x, t ) is the instantaneous pressure distribution and  d is the dry contact friction coefficient. With these, the average friction coefficient at a contact line segment k ( k  [1, ns ] and   [1, n ] ) is given as xe

1 k  Nt

Nt



n 1

 ( x, tn )dx

xs

(5.9)

f k

here xs and xe are the start and end points of lubricated contact in the direction of sliding, N t is the number of time steps at which the lubrication analysis performed. The rolling traction formula of Goksem [5.34] is used here to find the rolling traction in its corrected form for thermal effects [5.35]

( Fr )k



4.318(GU )0.658W 0.0126 ( Req )k   pv

144  

.

(5.10)

Here W  fk  Eeq ( Req )k  , U  0 (Vr )k  Eeq ( Req )k  , G   pv Eeq ,  pv    

is the

pressure-viscosity coefficient for the lubricant used,  is thermal correction factor [5.31],

Eeq equivalent module of elasticity of two contacting bodies and 0 is viscosity at ambient conditions. With this, rolling power loss at each contact segment k is

k  ( Fr )k  (Vr )k .  

5.2.3. Derivation of a Friction Coefficient Formula

In Eq. (5.7) to (5.9), the distribution of the surface shear ( ( x, t ) ), normal pressure ( p( x, t ) ) and film thickness ( h( x, t ) ) of every contact segment k must be predicted by using a mixed elastohydrodynamic lubrication (MEHL) model. Considering that there are n contact line segments at each rotational increment k and there are a total of ns increments, a total of n ns number of MEHL analyses are required to find the distribution of the friction coefficient and the resultant gear mesh power loss. This would require several hours of CPU time, hampering the usefulness of the model. In accordance with the methodology first proposed by Xu et al [5.23], an upfront detailed parametric study of MEHL conditions of hypoid gear contacts will be performed here by including all key contact parameters. These parameters and their selected ranges 145  

and incremental values are listed in Table 5.1.

These ranges of Hertzian pressure ph

(0.5 to 2.5 GPa), radius of curvature Req (5 to 40 mm), Vr (1 to 20 m/s), slide-to-roll ratio SR (0 to 1), and viscosity of a typical axle fluid (75W90) within a temperature range (25 to 100 C ) cover most of the contact conditions present in automotive hypoid gear pairs.

In addition, a typical measured roughness profile from a gear surface with

Rq  0.5 μm is considered and different RMS roughness profiles with different amplitudes are obtained from this baseline profile by multiplying it by a constant. The mechanical properties of 75W90 gear oil are listed in Table 5.2 is used. A total of 31,500 contact conditions as a result of all combinations of the parameter values listed in Table 5.1 were analyzed by using the model of Li et al [5.31] and a regression analysis to the µ values predicted for each or these conditions to obtain the following friction coefficient formula: For   1 :



  exp a0  H  a1  SR   a2G   a3 2

  SR a lnG a ln S  4

5

c

 a G  a   a ln   a9 ln Sc  S  a10 H  , P 6 7  8 eq

146  

(5.11a)

_____________________________________________________________ Lubricant

75W90 gear oil

_____________________________________________________________ Hertzian Pressure ph (GPa)

0.5, 1, 1.5, 2, 2.5

Equivalent Radius of Curvature Req (mm) Rolling Velocity Vr (m/s)

5, 20, 40 1, 5, 10, 15, 20

Slide to Roll Ratio SR

0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1

Inlet Lubricant Temperature Toil (°C)

25, 50, 75, 100

Surface 1 roughness RMS Rq1 (µm)

0.1, 0.35, 0.6, 0.85, 1

Surface 2 roughness RMS Rq2 (µm)

0.1, 0.35, 0.6, 0.85, 1

_____________________________________________________________

Table 5.1: Parametric design for the development of the friction coefficient formula.

147  

________________________________________________________________________ Temperature

Pressure-Viscosity Coefficient

Dynamic Viscosity

Density

Toil (°C)

1 (GPa-1)

0 (Pa.s)

0 (kg/m3)

________________________________________________________________________ 25

18.0

0.1626

844.30

50

13.9

0.0499

829.30

75

11.4

0.0208

814.30

100

9.7

0.0106

799.30

________________________________________________________________________

Table 5.2: Basic parameters of the 75W90 gear oil used in this study.

148  

For 1    3 : b2 ln U b3 ln  G b4 ln  H b5 H  

  exp b0  b1  SR   ( SR) 

b H  b b ln H  H b9 ln H   b10Sc  , U 6 G 7 8

(5.11b)

For   3 : c3 ln  G  c4 ln  H  c5 H  c6 ln    c7 ln  Sc   

  exp c0  c1G  c2  SR  U  ( SR) 

c8G  c9 ln  H    c10G  H S

U

eq

c11 SR   .

(5.11c) Here  is the lambda ratio (ratio of the smooth condition minimum film thickness to the RMS surface roughness value).

These formulae are dependent on a number of

dimensionless parameters:  , SR , U , G , H  ph Eeq , two roughness parameters

Sc  Sc Req and Seq  Seq Req ( Sc  Rq21  Rq22 and Seq  Rq1 Rq2 ( Rq1  Rq2 ) ). Here, coefficients a0 to a10 , b0 to b10 and c0 to c11 are constants representative of the lubricant considered. For the 75W90 gear oil, these parameters are listed in Table 5.3.

149  

___________________________________________________________________ ai

i

bi

ci

___________________________________________________________________ 1

-0.62538

16.1512

5.0435

2

-51.411

-0.60156

-0.00576307

3

-0.0371532

-0.0466305

248228480

4

2.06770

-0.348239

-0.396002

5

-0.031750

-0.358514

-0.405254

6

-0.046276

22.568

35.618

7

-7.754e-5

-11.2295

-0.109382

8

1.18821

-6.49095

-0.112364

9

0.170432

-1.31986

-0.00016056

10

-0.136892

-0.881279

-0.0690238

11

-7.8882

-4398.1

-0.00038810

12

0.052821

___________________________________________________________________  

Table 5.3: Values of the coefficients in Eq. 5.11.

150  

5.2.4. Computation of the Mechanical Power Loss of the Hypoid Gear Pair With the sliding friction coefficient ( k ) and rolling loss ( k ) computed in the previous section for every contact segment k , the mechanical power loss due to the contact of k ( k  [1, ns ] ,   [1, n ] ) is computed as

Pk  fk Lk k (Vs )k  k ,

(5.12)

With this, the instantaneous gear pair power loss at a given rotational increment k becomes nl

P k   Pk

(5.13a)

1

and the average mechanical power loss of the hypoid gear pair is found as

P

1 ns

ns

 Pk .

(5.13b)

k 1

Having input pinion torque T p and speed  p and power loss P k loss at each rotational increment k , the instantaneous mechanical gear pair efficiency is defined as

Ek  1

Pk T p p

,

(5.14a) 151

 

and overall the average gear mesh efficiency is estimated as

E  1

P T p p

.

(5.14b)

 

5.3. Numerical Example Two sets of face-hobbed hypoid gear designs defined in Table 5.4 are borrowed from automotive applications to study the influence of shaft offset along with working conditions that affect on hypoid gear efficiency. Designs A and B have two levels of offset ratios of d a Da  0.07 and 0.14, respectively, while the other geometry and operating parameters (such as number of teeth, shaft angle and gear pitch diameter Da ) of the two designs are kept the same or very close to each other to isolate the offset influences from those of the other parameters ( d a is pinion shaft offset). For this purpose, similar ease-off topographies as shown in Figure 5.3 are developed for the two gear sets. Despite all geometrical differences of two designs, they have similar contact pressure distribution as shown in Figure 5.4 for pinion torque of T p  500 Nm. This is mainly due to matching ease-off topographies of two design sets through machine setting changes.

152  

_____________________________________________________________________ Set A Parameter

Set B

Pinion

Gear

Pinion

Gear

_____________________________________________________________________ Number of teeth

12

41

12

41

Hand of Spiral

Left

Right

Left

Right

Mean spiral angle (deg)

40.0

31.3

47.0

29.6

Shaft angle (deg)

90

90

Shaft offset (mm)

15.0

30.0

Gear pitch diameter (mm)

220.0

220.0

Generation type Cutting Method

Generate

Formate

FH

Generate

Formate FH

_____________________________________________________________________      

Table 5.4: Basic drive side geometry and working parameters of the examples hypoid gear pairs.

153  

µm 

(a)

100

5 50

Toe

0

Root

(b)

µm 

100

50

Toe

0

Root

Figure 5.3: Ease-off topography of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 .

154  

Mpa 

(a)

1000 800 600 400 200 Toe

Root Mpa 

(b)

1000 800 600 400 200

Toe

Root

Figure 5.4: Maximum contact pressure distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 for T p  500 Nm .

155  

At  p  1500 rpm, Vr and Vs of the two gear sets are compared in Figures 5.5 and 5.6, respectively, for these two designs. It is seen that design B with higher offset ratio ( d a Da  0.14 ) has higher Vr and Vs compared to design A. Likewise, the slideto-roll ratio (SR) values over the tooth face of gear set B is higher than that of gear set A, as shown in Figure 5.7. Additionally, the Req and  distributions shown in Figures 5.8 and 5.9 reveal slight differences in Req values observed between the two gear designs, the  values for the gear set B are larger than those for the design set A. With all the parameters required by Eq. (5.11) computed, the  distributions along the tooth faces can be determined, as shown in Figure 5.10.

Although gear set B has higher  ratio

compared to design A, its higher SR levels causes higher  values, resulting in higher sliding friction force. Next, the influences of operating and surface conditions, including load T p , speed  p , oil temperature Toil and surface roughness S , on mechanical power loss P and mechanical efficiency E of these gear pairs are quantified. In Figures 5.11(a1, a2) and (b1, b2), variation of P and E with  p and T p are shown. It is observed in Figure 5.11(a2, b2) that the E decreases slightly with increasing  p when the speed is low. This trend is reversed at higher speed ranges as the slope between E and  p becomes positive. Similar conclusions can be drawn for the influence of T p . 156  

(a)

0.5

1.0

1.5

Toe

2.0

2.5

3.0

3.5

Root

(b)

1.0 1.5

2.0

2.5

3.0

Toe

3.5

4.0

4.5

Root

Figure 5.5: Rolling velocity distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm .  

157  

(a) 1.0

1.5 2.0

Toe

Root

(b) 1.5

2.0 2.5 3.0

Toe

Root

     

Figure 5.6: Sliding velocity distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm .

158  

(a)

Root

0.8 0.7 0 0.6 0 0.5 0 0.4 0 0.3 0 0.2 00.1

Root

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

0 0

Toe (b)

Toe

Figure 5.7: Slide-to-roll ratio distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm .

159  

(a) 35

40

Toe

45

50

Root

(b) 35 40

Toe

45

50

55

Root

Figure 5.8: Equivalent radius of curvature distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 .

160  

(a)

0.30 0.25 0.20 0.15

Toe

Root

(b)

0.10 0.30 0.25

0

0.20

0

0.15

0

Toe

Root

0.10

0

Figure 5.9:  distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm , T p  500 Nm , Toil  90 C and S1  S 2  0.8 m .

161  

(a)

0.06 0.05 0.04 0.03 0.02

Toe (b)

0.01

Root

0.06 0.05 0.04 0.03 0.02

Toe

0.01

Root

Figure 5.10: Friction coefficient  distribution of (a) Design A with d a / Da  0.07 and (b) Design B with d a / Da  0.14 at  p  1500 rpm , T p  500 Nm , Toil  90 C and S1  S 2  0.8 m .

162  

10,000

(a1)

1,000

P [w]

100

10 100 Nm

500 Nm

1000 Nm

1 0

500

1000

1500

2000

2500

3000

p

(b1)

10,000

1,000

P [w]

100

10 100 Nm

500 Nm

1000 Nm

1 0

500

1000

1500

2000

2500

3000

Continued p Figure 5.11: Power loss and efficiency of Design A (a1, a2) and Design B (b1, b2) at

Toil  90 C and S1  S 2  0.8 m . 163  

Continued

99.5

(a2)

99.0 98.5

E[%] 98.0 97.5 97.0 96.5

100 Nm

500 Nm

1000 Nm

96.0 0

500

1000

1500

2000

2500

3000

p

(b2)

99.5 99.0 98.5

E[%] 98.0 97.5 97.0 96.5

100 Nm

500 Nm

1000 Nm

96.0 0

500

1000

1500

2000 p

Figure 5.11 continued 164  

2500

3000

p In the lower speed range, where the power loss is relatively small, higher T

results in higher efficiency since the increase in power loss due to the heavier load applied is smaller than the corresponding input power increase, while the opposite is true in the medium and high speed ranges. Between the two designs, gear set A with a smaller shaft off-set is consistently more efficient (on average, gear set A has about 1.5% higher efficiency than gear set B). Figure 5.12 illustrates the effects of surface finish as well as the lubricant temperature on the gear mesh efficiency. It is found that reduction in surface roughness amplitude effectively increases the mechanical efficiency of both gear pairs. As for the operating temperature, within the low roughness range, where a substantial amount of contact area is separated by the hydrodynamic fluid film and the viscous shear dominates, an increase in temperature will reduces the sliding friction and the power loss P accordingly through the reduction in lubricant viscosity.

In the medium and high

roughness ranges, when the contact zone might experience severe asperity contacts, lower lubricant viscosity at high temperature results in thinner fluid film and larger boundary friction force, reducing the gear mesh efficiency.

165  

99.5

(a)

99.0 98.5 Ef [%]

98.0 97.5 97.0 Toil

96.5

50° C

75° C

100° C

96.0 0.1 99.5

0.3

0.5

0.7

0.9

1.1

1.3

0.9

1.1

1.3

S

(b)

99.0 98.5 Ef [%]

98.0 97.5 97.0 Toil

96.5

50° C

75° C

100° C

96.0 0.1

0.3

0.5

0.7 S

Figure 5.12: Efficiency of (a) Design A with d a / Da  0.07 and (b) design B with d a / Da  0.14 for different surface finish and oil temperatures at  p  1500 rpm and T p  500 Nm . 166  

5.4. Conclusion In this chapter, a new spiral bevel and hypoid gear mechanical efficiency model is proposed for both face-milling and face-hobbing cutting methods. The proposed efficiency model combines the computationally efficient contact model developed in chapters 2 and 3 with an EHL based friction coefficient model developed by Li et al [5.33] to estimate sliding friction loss and employed a conventionally developed formulation for rolling loss. The developed model improved the methodology of Xu and Kahraman [5.30] by (i) employing the gear geometry defined in Chapter 2 to derive contact curvature and surface velocity values at each contact point instead of relying on any particular commercial FE package, (ii) employing the loaded tooth contact model proposed in Chapter 3 for computation of the normal load for minimizing the computational time required for this task, and (iii) by incorporating a new µ formula derived by using the mixed EHL model of Li et al [5.33] such that any degree of asperity interactions can be modeled. Limited numerical results show that the shaft off-set is critical to the efficiency of the gear set as lower off-sets resulting in significant increases in mechanical efficiency. Likewise, reduction in surface roughness was also shown to reduce power losses of hypoid gear pairs.   167  

References for Chapter 5 [5.1]

Coleman, W., 1975, "Computing efficiency for bevel and hypoid gears." Machine Design, 47, pp. 64-65.

[5.2]

Seetharaman, S., and Kahraman, A., 2009, "Load-Independent Spin Power Losses of a Spur Gear Pair: Model Formulation." ASME J. of Tribology, 131(2), pp. 022201.

[5.3]

Seetharaman, S., Kahraman, A., Moorhead, M. D., and Petry-Johnson, T. T., 2009, "Oil Churning Power Losses of a Gear Pair: Experiments and Model Validation." ASME J. of Tribology, 131(2), pp. 022202.

[5.4]

Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute of Technology.

[5.5]

Denny, C. M., 1998, "Mesh Friction in Gearing." AGMA, Technical Paper No. 98FTM2.

[5.6]

Pedrero, J. I., 1999, "Determination of The Efficiency of Cylindrical Gear Sets." 4th World Congress on Gearing and Power Transmission, Paris, France.

[5.7]

Michlin, Y., and Myunster, V., 2002, "Determination of Power Losses in Gear Transmissions with Rolling and Sliding Friction Incorporated." J. Mechanism and Machine Theory, 37, pp. 167.

[5.8]

Benedict, G. H., and Kelly, B. W., 1960, "Instantaneous Coefficients of Gear Tooth Friction." ASLE.

[5.9]

O’Donoghue, J. P., and Cameron, A., 1966, "Friction and Temperature in Rolling Sliding Contacts." ASLE Transactions, 9, pp. 186-194. 168

 

[5.10] Drozdov, Y. N., and Gavrikov, Y. A., 1967, "Friction and Scoring Under The Conditions of Simultaneous Rolling and Sliding of Bodies." Wear, pp. 291-302. [5.11] Misharin, Y. A., 1958, "Influence of The Friction Condition on The Magnitude of The Friction Coefficient in The Case of Rollers with Sliding." Int. Conference On Gearing, London, UK. [5.12] Heingartner, P., and Mba, D., 2003, "Determining Power Losses in The Helical Gear Mesh; Case Study." Proceeding of DETC3, Chicago, Illinois, USA. [5.13] Anderson, N. E., and Loewenthal, S. H., 1986, "Efficiency of Nonstandard and High Contact Ratio Involute Spur Gears." J. Mechanisms, Transmissions and Automation in Design, 108, pp. 119-126. [5.14] Anderson, N. E., and Loewenthal, S. H., 1982, "Design of Spur Gears for Improved Efficiency." J. Mechanical Design, 104, pp. 767-774. [5.15] Barnes, J. P., 1997, "Non-Dimensional Characterization of Gear Geometry, Mesh Loss and Windage." AGMA, Technical Paper No. 97FTM11. [5.16] Vaishya, M., and Houser, D. R., 1999, "Modeling and Measurement of Sliding Friction for Gear Analysis." AGMA, Technical Paper No. 99FTMS1. [5.17] Martin, K. F., 1981, "The Efficiency of Involute Spur Gears." ASME J. Mech. Des., 103, pp. 160-169. [5.18] Dowson, D., and Higginson, G. R., 1964, "A Theory of Involute Gear Lubrication." Institute of Petroleum, Gear Lubrication, Elsevier, London, UK.

169  

[5.19] Simon, V., 1981, "Load Capacity and Efficiency of Spur Gears in Regard to Thermo-End Lubrication." International Symposium on Gearing and Power Transmissions, Tokyo, Japan. [5.20] Simon, V., 2009, "Influence of machine tool setting parameters on EHD lubrication in hypoid gears." J. Mechanism and Machine Theory, 44, pp. 923937. [5.21] Larsson, R., 1997, "Transient non-Newtonian elastohydrodynamic lubrication analysis of an involute spur gear." Wear, 207, pp. 67-73. [5.22] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears, Elsevier Science B. V. [5.23] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, "Prediction of Mechanical Efficiency of Parallel-Axis Gear Pairs." ASME J. Mech. Des., 129(1), pp. 58-68. [5.24] Cioc, C., Cioc, S., Kahraman, A., and Keith, T., 2002, "A Non-Newtonian, Thermal EHL Model of Contacts with Rough Surfaces." Tribology Transactions, 45, pp. 556-562 [5.25] Buckingham, E., 1949, Analytical Mechanics of Gears, McGraw-Hill. [5.26] Simon, V., 1981, "Elastohydrodynamic Lubrication of Hypoid Gears." ASME J. Mech. Des., 103, pp. 195-203. [5.27] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University, Columbus, Ohio. 170  

[5.28] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced Numerical Solution Inc., Hilliard, Ohio. [5.29] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd ed.), Cambridge University Press, Cambridge. [5.30] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part K: J. Multi-body Dynamics, 221(3), pp. 387-400. [5.31] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric Integrated Control Volume Discretization." Tribology International, Hiroshima, Japan. [5.32] Li, S., and Kahraman, A., 2009, "An Asymmetric Integrated Control Volume Method for Transient Mixed Elastohydrodynamic Lubrication Analysis of Heavily Loaded Point Contact Problems." Tribology International. [5.33] Li, S., Vaidyanathan, A., Harianto, J., and Kahraman, A., 2009, "Influence od Design Parameters and Micro-Geometry on Mechanical Power Losses of Helical Gear Pairs." JSME International, Hiroshima, Japan. [5.34] Goksem, P. G., and Hargreaves, R. A., 1978, "The effect of viscous shear heating on both film thickness and rolling traction in an EHL line contact." J. Lubrication Technology, 100, pp. 346–352. [5.35] Wu, S., and Cheng, H., S., 1991, "A Friction Model of Partial-EHL Contacts and its Application to Power Loss in Spur Gears." Tribology Transactions, 34(3), pp. 398-407.

171  

CHAPTER 6

CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK

6.1. Thesis Summary A computationally efficient load distribution model was proposed for both facemilled and face-hobbed hypoid gears produced by Formate and generate processes. Tooth surfaces were defined directly from the cutter parameters and machine settings. This study defined and utilized a new surface of roll angle as an essential tool to simplify the task of locating instantaneous contact lines of any general type of gearing in the projection plane. First, the position vector and normal to one of the mating surfaces of contacting members were computed, and the action surface and the surface of roll angle were introduced by applying equation of meshing between any general axis arrangement. Once the surface of roll angle was constructed, the instantaneous contact lines location and orientation were computed through a novel approach inspired by analogy to parallel axes gears. Gear surfaces were assumed conjugate only in computation of contact line 172

 

locations and orientation of the proposed approach. However it was shown that surface of roll angle is very insensitive to any practical modifications of contacting surfaces, and hence, real instantaneous contact lines practically remain identical to their conjugate counterparts. For any other steps of contact analysis, theoretically generated surfaces based on machine settings were employed. Rayleigh-Ritz based shell models of teeth of the gear and pinion were developed to define the tooth compliances due to bending and shear effects efficiently in a semianalytical manner. Base rotation and contact deformation effects were also included in the compliance formulations. With this, loaded contact patterns and transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears were computed by enforcing the compatibility and equilibrium conditions of the gear mesh. The proposed model requires significantly less computational effort than finite elements (FE) based models, making its use possible for extensive parameter sensitivity and design optimization studies. Comparisons to the predictions of a FE hypoid gear contact model were also provided to demonstrate the accuracy of the model under various load and misalignment conditions. Two applications of the proposed model were also introduced. First application combined the proposed model with a newly introduced approach of modifying the easeoff topography to investigate the effect of errors occurred in manufacturing and heat treatment of gear surfaces or surface deviations due to wear or lapping. Manufacturing errors typically cause real (measured) spiral bevel and hypoid gear surfaces to deviate 173

 

from the theoretical ones globally. Tooth surface wear patterns accumulated through the life span of the gear set are typically local deviations that are aggravated especially in case of edge contact conditions. An accurate and practical methodology based on the developed ease-off topography approach was proposed in this study to perform loaded tooth contact analysis of spiral bevel and hypoid gears having both types of local and global deviations. Manufacturing errors and localized surface wear deviations were used to update the theoretical ease-off and surface of roll angle to form a new ease-off surface that was used to perform a loaded tooth contact analysis. Two numerical examples of face-milled and face-hobbed hypoid gear sets with local and global deviated surfaces, respectively, were analyzed to demonstrate the effectiveness of the proposed methodology as well as quantifying the effect of such deviations on load distribution and the loaded motion transmission error. As another vital application of the proposed model, a hypoid gear mechanical efficiency model was developed next for both face-milling and face-hobbing cutting methods. The proposed efficiency model combined the computationally efficient contact model and a mixed EHL based surface traction model to predict friction power losses. The contact area, pressure distribution and rolling and sliding velocities were determined employing the developed loaded tooth contact model. The EHL traction model considered specific ranges of the key contact parameters, including Hertzian pressure, contact radii, surface speeds, lubricant temperature and surface roughness amplitude of hypoid type of gears, covering wide range of lubrication conditions from full film to 174

 

boundary regimes. The efficiency model for hypoid gear was applied to two face-hobbing examples with similar overall dimensions, but different off-sets, to investigate the effects of several working and design parameters including offset, load, speed, surface roughness and lubricant temperature on mechanical efficiency.

6.2. Conclusion and Contributions Computation of the contact pressure distributions is essential to every hypoid gear analysis intended to predict required functional parameters of the hypoid gear pair, including the transmission error, contact stresses, root bending stresses, fatigue life and mechanical power losses. The hypoid gear literature lacked a model to compute the load distribution accurately and efficiently without resorting to computationally demanding FE methods. The main potential reasons for that was the absence of a general and reliable formulation to define the geometry of FH and FM hypoid tooth surfaces from cutter parameters, machine motions and settings.

This void, combined with the

numerical difficulties in matching the tooth surfaces using the conventional methods and lack of a semi-analytical tooth compliance formulation for hypoid gears, has hampered in design, analysis and optimization of hypoid gears. This research study fills some of this void. The model proposed in this study to simulate the contacts of FM and FH hypoid gear pairs under both unladed and loaded condition provides major enhancements to the current state of hypoid analysis. Specifically: 175

 

(i)

Methodology that simulates the FM and FH processes to define surface geometries of hypoid gears including the coordinates, normal vectors and radii of curvatures is accurate and computationally efficient.

It does not employ

simplifying assumptions such as conjugacy of the tooth surfaces. (ii)

The method using ease-off topography to compute the unladed contact conditions is novel. It is superior to the conventional method in various aspects, including its numerical stability and computational efficiency. This method and its surface of roll angle concept is also general such that it can be applied to the other gear types.

(iii) Application of mounting errors and inclusion of global and local deviations are rather straightforward with the proposed model while these have typically been difficult or impossible tasks when the conventional methods were used. (iv) The efficiency model that combines the loaded tooth contact model proposed in this study with an accurate and computationally efficient friction model is superior to any published hypoid gear efficiency model as it includes all key geometry, surface, load and lubricant parameters as well as operating conditions. Its ability to handle variety of lubrication conditions ranging from almost dry contact to full film EHL makes this model applicable to wide ranges of hypoid and spiral bevel gear applications from automotive and aerospace systems.

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6.3. Recommendations for Future Work The following items can be considered as the potential future studies to improve or add to the model presented in this study. (i)

Extensive parameter studies can be performed to determine cost-effective ways of improving hypoid gear efficiency and define efficiency guidelines to be used in design of hypoid gears.

(ii)

The loaded TCA formulation can be improved by enhancing the base rotation formulation to account for all geometric complexities.

(iii) A shell model for non-circular cylinders can be developed to compute a more accurate compliance matrix for a generated gear and the face-hobbed generation method that have epicycloids tooth traces. (iv) The proposed model lends itself to optimization studies to refine machine settings for desired transmission error amplitudes and loaded tooth contact patterns. (v)

The way local deviation and mounting errors are included in the proposed model is general such that this model can be used to predict surface wear as well as to simulate the lapping process that is commonly used in manufacturing of FH hypoid gears.

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