Date post: | 07-Apr-2015 |
Category: |
Documents |
Upload: | ashish-raj |
View: | 586 times |
Download: | 6 times |
DEVELOPMENT OF TOOTH CONTACT AND MECHANICAL EFFICIENCY
MODELS FOR FACE-MILLED AND FACE-HOBBED
HYPOID AND SPIRAL BEVEL GEARS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Mohsen Kolivand, B.S., M.S.
*****
The Ohio State University
2009
Dissertation Committee:
Professor Ahmet Kahraman, Advisor
Professor Donald R. Houser
Professor Gary L. Kinzel
Professor Henry H. Busby
Approved by
________________________________
Advisor
Graduate Program in Mechanical Engineering
© Copyright by
Mohsen Kolivand
2009
ii
ABSTRACT
A computationally efficient load distribution model is proposed for both face-
milled and face-hobbed hypoid gears produced by Formate and generate processes.
Tooth surfaces are defined directly from the cutter parameters and machine settings. A
novel methodology based on the ease-off topography is used to determine the unloaded
contact patterns. The proposed ease-off methodology finds the instantaneous contact
curves through a surface of roll angles, allowing an accurate unloaded tooth contact
analysis in a robust and accurate manner. Rayleigh-Ritz based shell models of teeth of
the gear and pinion are developed to define the tooth compliances due to bending and
shear effects efficiently in a semi-analytical manner. Base rotation and contact
deformation effects are also included in the compliance formulations. With this, loaded
contact patterns and transmission error of both face-milled and face-hobbed spiral bevel
and hypoid gears are computed by enforcing the compatibility and equilibrium conditions
of the gear mesh. The proposed model requires significantly less computational effort
than finite elements (FE) based models, making its use possible for extensive parameter
sensitivity and design optimization studies. Comparisons to the predictions of a FE
hypoid gear contact model are also provided to demonstrate the accuracy of the model
under various load and misalignment conditions.
iii
The proposed ease-off formulation is generalized next to include various types of
tooth surface deviations in the tooth contact analysis. These deviations are grouped in
two categories. The proposed ease-off based method is shown to be capable of modeling
both global deviations due to common manufacturing errors and heat treat distortions and
local deviations due to surface wear.
The proposed loaded contact model is combined at the end with a friction model
based on a mixed elastohydrodynamic lubrication model to predict the load dependent
(mechanical) power losses and efficiency of the hypoid gear pairs. The velocity, radius
of curvature and load information predicted by the contact model is input to the friction
model to determine the distribution of the friction coefficient along the contact surfaces.
At the end, the variations of predicted mechanical efficiency with geometry, surface and
lubricant parameters are quantified.
iv
Dedicated to my mother
v
ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my advisor, Prof. Ahmet
Kahraman, for this great research opportunity, his guidance throughout my research and
his effort in reviewing this dissertation. I would also like to express my appreciation to
Prof. Donald R. Houser, Prof. Gary L. Kinzel and Prof. Henry H. Busby for their patience
and effort in being a part of my dissertation committee. Also, thanks to Dr. Sandeep
Vijayakar who kindly permitted me to use CALYX package.
I would like to thank the sponsors of the Gear Power Transmission Research
Laboratory for their financial support throughout my study.
My sincere thanks go to Prof. Hermann J. Stadtfeld from The Gleason Works who
spent valuable time teaching me fundamental concepts of bevel gear design and
manufacturing, and also for having attended my dissertation defense.
I would also like to thank Jonny Harianto, Samuel Shon and all my lab mates for
their help and friendship throughout my study at OSU and beyond.
Finally, I deeply appreciate the love and trust shown toward me by my parents,
my grandparents, my uncles and aunts, my sisters and brother and my fiancée for all of
their support and encouragement.
vi
VITA
Oct. 19, 1976 ……………….…….. Born – Tehran, Iran
Sep. 1995 – Sep. 1999 ……………. B.S. in Mechanical Engineering, Solid Mechanics, Tehran University, Tehran, Iran
Sep. 1999 –Feb. 2002 ……………. M.S. in Mechanical Engineering, Applied Design, Tehran University, Tehran, Iran
Nov. 1999 –Apr. 2002 ……………. Design Engineer, Tarh Negasht Co., Tehran, Iran
Apr. 2002 –Aug. 2005 ……………. Design Engineer, TAM Co., Tehran, Iran
Jan. 2006 – present ……………… M.S. in Mechanical Engineering (Sep. 2008) Graduate Research Associate Gear and Power Transmission Research Laboratory Department of Mechanical Engineering The Ohio State University Columbus, Ohio
PUBLICATIONS
1. Kolivand M. and Kahraman A., “A Load Distribution Model for Hypoid Gears Using
Ease-off Topography and Shell Theory,” Journal of Mechanism and Machine Theory,
2009.
FIELDS OF STUDY
Major Field: Mechanical Engineering
vii
TABLE OF CONTENTS
Page
Abstract………………………………………………………………………………. ii
Dedication……………………………………………………………………………. iv
Acknowledgments………………………………………………………………….... v
Vita……………………………………………………………………………….….. vi
List of Tables..………………………………………………………………...…….. x
List of Figures...……………………………………………………………...……… xi
Nomenclature………………………………………………………………………... xvi
Chapters:
1 Introduction ……………………………………………………………………... 1
1.1 Motivation and background ………………………………………………… 1
1.2 Literature Review ………………………………………………………….... 4
1.3 Scope and Objectives ...………………...…………………………………… 14
1.4 Overall Modeling Methodology...……………...……………………………. 17
1.5 Dissertation Outline………………………………………………………….. 19
References of Chapter 1………………………………………………………….. 20
2 Definition of Face-milled and Face-hobbed Hypoid Gear Geometry and
Unloaded Tooth Contact Analysis ……………………………………..........…... 28
2.1 Introduction …………………….……………….…………………………… 28
2.2 Definition of Tooth Surface Geometry ……………………………………... 30
viii
2.2.1 Kinematics ………………………………………………………...… 32
2.2.2 Cutting Tool Geometry and the Relative Motion……………………. 36
2.2.3 Equation of Meshing…………………………………………………. 40
2.2.4 Principal Curvatures and Principal Directions…………………......... 42
2.3 Unloaded Tooth Contact Analysis ………………...………………………… 44
2.3.1 The Conventional Method of UTCA.……………….……………….. 47
2.3.2 Ease-off Based Method of UTCA …………………………………… 52
2.3.2.1 Construction of Ease-off and the Surface of Roll Angle………… 53
2.3.2.2 Contact Pattern and Transmission Error…………………………. 57
2.4 An Example Hypoid Unloaded Tooth Contact Analysis ……………………. 59
References for Chapter 2………………………………………………………. 65
3 Shell Based Hypoid Tooth Compliance Model and Loaded Tooth Contact
Analysis …............................................................................................................. 68
3.1 Introduction …………………………………………………………………. 68
3.2 Tooth Compliance Model …………………………………………………... 70
3.3 Loaded Tooth Contact Analysis ….…………………………………………. 81
3.4 An Example Hypoid Tooth Contact Analysis ………………………………. 84
References for Chapter 3………………………………………………………… 92
4 Loaded Tooth Contact Analysis of Hypoid Gears with Local and Global Surface
Deviations…………………………………..….………………………………… 94
4.1 Introduction …………………………………………………………………. 94
4.2 Construction of the Theoretical Ease-off Topography …..………………….. 98
4.3 Updating Ease-off Topography for Manufacturing Errors and Surface Wear 102
4.4 Unloaded and Loaded Tooth Contact Analyses ………………….…………. 107
4.5 Example Analyses ………………………………………………………..…. 111
4.5.1 A Face-milled Hypoid Gear Pair with Local Surface Deviations……. 111
ix
4.5.2 A Face-hobbed Hypoid Gear Pair with Global Deviations………….. 120
References for Chapter 4………………………………………………………... 130
5 Predictions of Mechanical Power losses of Hypoid Gear Pairs………………….. 133
5.1 Introduction ……………………………………………………………….…. 133
5.2 Hypoid Gear Mechanical Power Loss Model …………………………….…. 137
5.2.1 Definition of the Sliding and Rolling Velocities…………………….. 140
5.2.2 Friction Coefficient Model…………………………………………... 143
5.2.3 Derivation of a Friction Coefficient Formula………………………... 145
5.2.4 Computation of the Mechanical Power Loss of the Hypoid Gear Pair 151
5.3 Numerical Example ……………………………………..…………………... 152
5.4 Conclusion…………………………………………………………………… 167
References for Chapter 5………………………………………………………… 168
6 Conclusions and Recommendations for Future Work…...………………………. 172
6.1 Thesis Summary ……………………………………………………………... 172
6.2 Conclusion and Contributions……………………………………………….. 175
6.3 Recommendations for Future Work …………………………………………. 177
Bibliography …………………………..………………...…………………………... 178
x
LIST OF TABLES
Table Page
2.1 Basic drive side geometry and working parameters of the example hypoid
gear pair…………………………………………………………………..….. 60
3.1 The loaded transmission error predictions of the proposed model; 0G
mm and 0 for all cases.………………………….…………………..… 88
4.1 Basic drive side geometry and working parameters of the example hypoid
gear pair.…………………………………………………….……………….. 112
4.2 The transmission error amplitudes of theoretical and deviated surfaces…….. 118
4.3 Basic drive side geometry and working parameters of the example hypoid
gear pair.……………………………………………………………………... 121
4.4 The transmission error amplitudes of theoretical and deviated surfaces…..… 128
5.1 Parametric design for the development of the friction coefficient formula….. 147
5.2 Basic parameters of the 75W90 gear oil used in this study.…………….….... 148
5.3 Values of the coefficients in Eq. 5.11…………………..……………………. 150
5.4 Basic drive side geometry and working parameters of the examples hypoid
gear pairs……………………………………………………………………... 153
xi
LIST OF FIGURES
Figure Page
1.1 A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize
passenger cars and SUV’s.…………………………………………………… 2
1.2 A sample hypoid gear pair with a shaft angle and a shaft off-set ad .….... 4
1.3 Different gear types based on shaft arrangements…………………………… 6
1.4 Flowchart of overall hypoid gear loaded tooth contact analysis methodology 18
2.1 (a) Face-milling and (b) face-hobbing cutting processes…………………….. 31
2.2 Cradle based hypoid generator parameters…………………………………... 33
2.3 (a) Cutter head, (b) blade and (c) cutting edge geometry……………………. 37
2.4 Generation process…………………………………………………………… 41
2.5 Curvature computation procedure…………………………………………… 43
2.6 General case of approximating gear surfaces as two contacting ellipsoids to
orient instantaneous contact line……………………………………………... 49
2.7 Construction of the ease-off, action and Q surfaces…………………………. 55
2.8 Unloaded TCA computation procedure: (a) gear projection plane, ease-off
and Q surfaces, and (b) instantaneous contact curve, contact line and
unloaded transmission error…………………………………………………. 58
2.9 Unloaded transmission error of the example gear pair with misalignments
0.15 mm, 0.12 mm, 0, 0E P G ………………………….. 61
xii
2.10 Unloaded contact pattern of the example gear pair for three adjacent tooth
pairs 1i , i and 1i (i-1), (i) and (i+1) with
0.15 mm, 0.12 mm, 0E P G and 0 ……………………… 63
2.11 Unloaded contact pattern of the example gear pair (a) at nominal position
with 0E P G , (b) at toe with 0.08E mm, 0.10P
mm and 0G , (c) at heel with 0.15E mm, 0.10P mm
and 0G and (d) at toe with 0.05E mm, 0P G , and
4 min…………………………………………………………………. 64
3.1 Basic dimensions of a hypoid tooth used in the compliance formulation…… 72
3.2 Flowchart of the compliance computation.…………………………….…….. 79
3.3 Potential contact line discretization.……………………………...………….. 80
3.4 The comparison of the shell model deformation to FEM.…….……………... 82
3.5 Static equilibrium between torque applied on gear axis and torque produced
by the force of all contacting segments.………...………………………….... 85
3.6 Loaded transmission error of the example gear pair with 0.15,E
0.12, P 0G and 0 at (a) 50pT Nm, (b) 250pT Nm, and (c)
500pT Nm.…………………………………………………………............ 87
3.7 Comparison of loaded contact patterns predicted by the proposed model to
an FE model [3.11] for (a) 50 NmpT , 0.15E mm, 0.12P mm, (b)
250 NmpT , 0.15E mm, 0.12P mm, (c) 500 NmpT ,
0.15E mm, 0.12P mm, (d) 50 NmpT , 0.08E mm,
0.05P mm, and (e) 50 NmpT , 0.26E mm, 0.13P mm ( all at
0, 0G ).……………………………………………………….....… 89
4.1 Construction of the ease-off, action and Q surfaces………............................ 99
xiii
4.2 Graphical demonstration of the procedure to update ease-off surface for
surface deviations.………………………...…………………………………. 105
4.3 Graphical demonstration of the procedure to compute unloaded TCA; (a)
gear projection plane, ease-off and Q surfaces, and (b) instantaneous
contact curve, contact line and unloaded transmission error………………... 108
4.4 Theoretical contact curves of an example hypoid gear pair.………………… 109
4.5 Example local deviation surfaces for the gear and pinion tooth surfaces..….. 113
4.6 Ease-off update for the example deviation of Fig. 5. (a) Three-dimensional
view of the projection plane, and , , Q and Q surfaces, and contour
plots of (b) , (c) , and (d) the change of ease-off topography..………… 114
4.7 Predicted unloaded tooth contact pattern for separation value of 6 μm … 116
4.8 Transmission error (UTE) curves for theoretical and deviated surfaces at (a)
unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm. 117
4.9 Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)
theoretical and (b) deviated surfaces………………………………………… 119
4.10 Example global deviation surfaces measured by CMM for the gear and
pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured
deviation, (c) pinion deviation distribution in tooth active region and (d)
gear deviation distribution in tooth active region……………………………. 122
4.11 Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical ease-
off topography, (b) updated ease-off topography only with pinion deviation,
(c) updated ease-off topography only with gear deviation, and (d) updated
ease-off topography with both pinion and gear deviations…………………... 124
4.12 Predicted unloaded tooth contact pattern for separation value of 6 μm … 125
xiv
4.13 Transmission error curves for theoretical and deviated surfaces; (a)
unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm 127
4.14 Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)
theoretical and (b) deviated surfaces………………………………………… 129
5.1 Flowchart of overall hypoid gear efficiency computation
methodology.………………………………………………………………... 138
5.2 Sliding and rolling velocities and their projection in tangential plane along
and normal to the contact line direction.……………………………………... 141
5.3 Ease-off topography of (a) Design A with / 0.07a ad D and (b) Design B
with / 0.14a ad D …………………………………………………………... 154
5.4 Maximum contact pressure distribution of (a) Design A with / 0.07a ad D
and (b) Design B with / 0.14a ad D for 500 NmpT …………………… 155
5.5 Rolling velocity distribution of (a) Design A with / 0.07a ad D and (b)
Design B with / 0.14a ad D at 1500 rpmp …………………………… 157
5.6 Sliding velocity distribution of (a) Design A with / 0.07a ad D and (b)
Design B with / 0.14a ad D at 1500 rpmp …………………………… 158
5.7 Slide-to-roll ratio distribution of (a) Design A with / 0.07a ad D and (b)
Design B with / 0.14a ad D at 1500 rpmp …………………………... 159
5.8 Equivalent radius of curvature distribution of (a) Design A with
/ 0.07a ad D and (b) Design B with / 0.14a ad D ……………………….. 160
5.9 distribution of (a) Design A with / 0.07a ad D and (b) Design B with
/ 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT
and 1 2 0.8 mS S ……………………………………………………….... 161
xv
5.10 Friction coefficient distribution of (a) Design A with / 0.07a ad D and
(b) Design B with / 0.14a ad D at 1500 rpmp , 500 NmpT ,
90 CoilT and 1 2 0.8 mS S …………………………………………... 162
5.11 Power loss and efficiency of Design A (a1, b1) and Design B (a2,b2) at
90 CoilT and 1 2 0.8 mS S ……………………………………….…. 163
5.12 Efficiency of (a) Design A with / 0.07a ad D and (b) design B with
/ 0.14a ad D for different surface finish and oil temperatures at
1500 rpmp and 500 NmpT …………………………..……………… 166
xvi
NOMENCLATURE
ga Gear axis vector
pa Pinion axis vector
C Total compliance matrix
ad Pinion offset
aD Gear pitch diameter
1 2,e e Principal directions
E Overall efficiency
bE Blank offset
eqE Equivalent module of elasticity
kf Load per unit length of segment at time step k
F Force vector
h Film thickness
fh Tip of blade to reference point
toeh Tooth height at toe
heelh Tooth height at heel
ui Normal to the mid-surface of the shell
Ti Tilt angle
js Swivel angle
1 2,k k Normal curvatures
1 2,K K Principal curvatures
Segment index
gm Number of surface grid in lengthwise direction
xvii
ctbM Machine center to back
n Normal to the family of cutter surface
cln Number of potential contact lines at each time step
gn Number of surface grids in profile direction
n Total number of contact segments
sn Total number of time steps per pinion pitch
cN Total number of segment on all potential contact lines/curves
gN Number of teeth of gear
pN Number of teeth of pinion
tN Number of blade groups
PE Potential energy
q Roll angle
aq Pinion pitch
Q Surface of roll angle
r Gear ratio
cr Cutter radius
R Position vector of a point on circular cylindrical shell
eqR Equivalent Hertzian curvature
gR Vector of the distances of each segment to the gear axis
ij Ease-off value of point ij
Ease-off surface
s Distance of an arbitrary point to reference point on the blade edge
rS Radial setting
S Initial separation vector
xviii
SE Strain energy
SR Sliding to rolling velocity ratio
eqS Equivalent surface roughness
t Instantaneous potential contact line direction
t Perpendicular to the instantaneous potential contact line direction
toet Tooth thicknesses at toe-root
heelt Tooth thicknesses at heel-root
T Torque
oilT Oil temperature
( )gtu Gear surface velocity in t direction
( )gtu Gear surface velocity in t direction
( )ptu Pinion surface velocity in t direction
( )ptu Pinion surface velocity in t direction
UTE Unloaded transmission error
cV Velocity of the point being cut seen from cradle axis
ijv Total surface velocity vector
rV Rolling velocity in t direction
sV Sliding velocity in t direction
soV Overall sliding velocity
wV Velocity of the point being cut from work axis
W Transverse deflection
ijw Surface velocity vector along common normal
WF Work done by the external force
BX Sliding base
xix
Y Vector of slack variable
Variable of curvilinear cylindrical coordinate system (in tooth lengthwise
direction)
b Blade angle
pv Pressure viscosity coefficient
Variable of curvilinear cylindrical coordinate system (in tooth profile
direction)
x Shear rotation in profile direction
m Machine root angle
mn Shear strain
Paint thickness (separation)
b Blade offset angle
E Pinion offset error
G Gear mounting distance error
P Pinion mounting distance error
Shaft angle error
m Normal strain
k Rolling power loss of segment at time step k
Effective viscosity
0 Ambient viscosity
Ratio of the smooth condition minimum film thickness to the RMS of
surface roughness
k Friction coefficient of segment at time step k
Thermal correction factor
Shear stress
xx
c Cradle angle
g Blank phase angle
t Cutter phase angle
( )n Polynomial of order n for shape function in lengthwise direction
( )m x Polynomial of order m for shape function in profile direction
c Angular speed of the cradle axis
g Angular speed of the blank axis
t Angular speed of the cutter axis
p Pinion speed
Tangential plane
Superscript:
( ) Real or updated
( ) Theoretical
ˆ( ) Conjugate
( ) Interpolated / Extrapolated
p Pinion
g Gear
a Action
1
CHAPTER 1
INTRODUCTION
1.1. Motivation and background
Hypoid gears are widely used in many power trains to transfer power between
two non-intersecting crossed axes. Their most common and highest-volume applications
can be found in front and rear axles of rear-wheel-drive or all-wheel-drive vehicles [1.2].
Figure 1.1 shows a sample of hypoid gear application for the rear axle. A rear axle has
three primary functions: (i) transmit power from the drive train axis to the wheel axle,
that is usually perpendicular to the drive train axis with an offset, (ii) provide the
capability to the vehicle to turn corners without any slippage at its wheels through its
differential, and (ii) provide the final stage of speed reduction (torque increase) that is
typically of the order of three to four.
“Hypoid gears are the most general form of gearing and their solution has been long in coming. There is no form of gearing where so many guesses have been made, a few of them right, plenty of them wrong and some without consequences”
Ernest Wildhaber [1.1]
2
Figure 1.1: A cut-away of an ‘auxiliary’ axle (Rear Drive Module) used in midsize
passenger cars and SUV’s (Courtesy of American Axle & Manufacturing Inc.).
Hypoid pinion
Hypoid gear
Output
Input
3
A pair of hypoid gears is commonly used to deliver this third final drive function.
In the arrangement shown in Figure 1.1, the smaller of the hypoid gears, called the
pinion, is at the end of the drive shaft and is in mesh with the larger hypoid gear (called
the gear).
Hypoid gears can be considered as one of the most general cases of gearing based
on their geometry, such that other gear types can be obtained from it by assigning certain
values to some of the geometric parameters [1.1,1.3-1.5]. The main function of the
hypoid gear pair in a rear axle is to transmit power between two axes that are at a shaft
angle (usually90 ) [1.2,1.6] and at a certain amount of shaft off-set ad as shown in
Figure 1.2. A higher level of power transmission through such a kinematic configuration
is possible through use of a hypoid gear pair, which can provide a better balance amongst
all primary design requirements such as strength, noise and power density. The trade-off
between these performance characteristics while satisfying the kinematic constraints
results in the hypoid tooth form that is rather complex geometrically.
Figure 1.3 shows a schematic of different types of gearing, based on shaft
arrangements. The shaft offset, being the main difference between spiral bevel and
hypoid gears, provides several advantages to hypoid gears including larger pinion size,
smaller pinion tooth counts, higher contact ratio, and higher contact fatigue strength. On
the negative side, hypoid gears experience higher sliding velocities, resulting in higher
power losses due to excessive sliding friction. Increasing the pinion size without
4
Figure 1.2: A sample hypoid gear pair with a shaft angle and a shaft off-set ad .
ad
5
shaft offset (spiral bevel) increases the size of the final drive significantly, while the
hypoid pinion can be made larger due to shaft off-set to increase the strength of the gear
pair while minimizing the overall size of the gear pair.
Any attempt to improve the functional attributes of a hypoid gear pair in terms of
its strength, quality, noise and power efficiency requires an optimization of its design
either by fine-tuning its key parameters that have traditionally been chosen based on
certain empirical knowledge or by making use of additional motion capabilities provided
by new-generation hypoid gear cutting machines that allow application of many kinds of
surface modifications [1.7, 1.8]. Hypoid gear design procedures were developed within a
small number of hypoid gear cutting machine tool and cutting tool manufacturers and
practical and theoretical details of hypoid development are still propriety to these
companies [1.9].
In general, two different basic cutting methods are used to generate hypoid gears,
namely face-milling (FM) or single indexing, and face-hobbing (FH) or continuous
indexing, which have their own advantages over each other. The FM process that was
the primary hypoid cutting method for decades has been taken over by the FH process in
automotive axle applications, mostly due to its productivity advantages caused by
continuous indexing [1,2, 1.10-1.12]. However, it is safe to state that the technology
level and design understanding of the FH process is almost a decade behind the face
milling process [1.13]. One reason for this is that newer machining methods
6
Figure 1.3: Different gear types based on shaft arrangements.
Worm gears
High reduction hypoid gears (HRH) or Spiroid
Hypoid gears, Face gear
Bevel gears (Straight, Spiral and Zerol), Face gear
Parallel axis gears (Spur, Helical and Herringbone)
Sli
ding
Pin
ion
size
Eff
icie
ncy
7
such as grinding are applicable to FM process, while there is still no such alternative
method or machinery developed for face-hobbing method.
Having high gear ratios in hypoid gears in automotive applications causes the gear
to have usually 3 to 4 times the number of teeth of the pinion, which justifies designing
the gear surface as simple as possible to increase production efficiency and minimize
manufacturing time. One typical cost-effective cutting method, called Formate®, is much
faster than the Generating methods. In Formate®, only a few degrees of freedom of
motions are allowed between the cutter and the gear blank (compared to the Generate
cutting method). Therefore, most of the surface modifications are applied to pinion tooth
surfaces, rather than the gear tooth surfaces.
Quality of a hypoid gear pair is defined by a number of performance
characteristics including its contact pattern, the motion transmission error (TE),
efficiency and sensitivity to misalignments. The geometric accuracy of a single gear has
limited significance here as the geometry of the mating gear and the assembly errors can
change the performance characteristics drastically. These performance characteristics
have been quantified either by using FE-based hypoid gear load distribution models or by
experimental means, both of which are very time-consuming and expensive. Due to their
significant computational burden, FE-based hypoid gear contact models are not suitable
for design and parameter and misalignment sensitivity studies. The aim of this study is to
develop computationally efficient, semi-analytical loaded tooth contact models for both
8
FM and FH hypoid gears with or without misalignments. The main motivation for this
dissertation research is to develop formulations to analytically describe the hypoid gear
surfaces and employ them in a gear contact mechanics formulation to predict unloaded
and loaded contact characteristics as well as functional metrics such as the transmission
error and mechanical efficiency.
1.2. Literature Review
In his writings, Aristotle (about 330 BC), made mention of gears and their
commonality. The earliest recognized relic of ancient time gearing is the south pointing
chariot with pinned gears used by the Chinese in about 2600 BC [1.14]. According to the
historical perspective provided by Litvin [1.15], theoretical development of gears as we
know today starts with Euler (1781) who proposed the concept of an involute curve
(1781), followed by others such as Willis (1841), Olivier ( 1842) and Gochman (1886)
who developed basic ideas of conjugacy and the foundations of modern gear geometry.
As for spiral bevel gears, Monneret (1899) filed the first patent for spiral bevel
generating method. About two decades later in 1910, Böttcher was issued a series of
patents that addressed both face hobbing and face milling methods [1.2]. Wildhaber’s
earlier papers and patents formed the basis for many of today’s hypoid gear geometry and
generation approaches [1.1, 1.3]. Wildhaber pointed out the significance of using
principal curvatures and directions in establishing hypoid gear geometry [1.16, 1.17].
9
Baxter’s later formulas, based on vector notation, helped condense the formulations to
facilitate the use of computers for definition of surfaces and tooth contact analysis (TCA)
[1.18]. He also developed one of the first unloaded tooth contact model for simulation of
mismatched surfaces of gears generated by Gleason type machines and studied the effects
of various misalignments on contact pattern [1.19], which was later expanded by
Coleman [1.20, 1.21]. Krenzer published a series of formulations for unloaded tooth
contact analysis (UTCA) of spiral bevel and hypoid gears [1.22]. These formulations
were useful for the gears manufactured by a class of machinery but were quite difficult to
adapt since the logic behind his formulae were not given. Nearly two decades later, he
proposed a loaded tooth contact analysis model without providing details of the geometry
and the contact analysis. This model used the simplified cantilever beam formula of
Westinghouse to estimate the compliance of the tooth [1.23]. Within the same time
frame, Litvin and Gutman [1.24-1.27] published a series of papers on synthesis and
analysis of FM hypoid gears. They calculated machine settings based on predetermined
contact characteristics at a mean point. They determined the contact points, the
instantaneous contact length and direction by conventionally using the surface principal
curvatures and directions. They used a conventional approach to find the contact points,
the instantaneous contact directions and the instantaneous contact lengths utilizing
surface principal curvatures and directions [1.24-1.27]. The effort of computing
optimized machine settings for limited cutting methods such as spiral bevel gears cut by
face-milling method was later continued by Litvin and Fuentes [1.28]. As these studies
10
focused on calculating surface coordinate of FM spiral bevel and hypoid gears, there are
very few studies on geometry of FH hypoid gears [1.29-1.33].
Among the few published studies on calculating and “optimizing” TCA, Stadtfeld
[1.2] appears to be the only investigator who used the ease-off approach. He provided a
consistent definition for ease-off as well as a procedure to calculate TCA from ease-off,
and calculated instantaneous contact between two surfaces as a line that maintains its
orientation over the tooth area [1.2]. Moreover, he utilized ease-off to optimize UTCA of
both FM and FH gears by applying various kinds of modifications through machine
settings and cutter geometry [1.34]. However, he did not provide a detailed procedure on
how to determine orientation of the instantaneous contact curves. Meanwhile, Fan [1.35]
focused on how to calculate surface coordinates and normal vectors for FH spiral bevel
and hypoid gears cut by using the generation method. The solution to the set of equations
that determines contact points where the collinearity condition for the normal vectors of
two mating surfaces is satisfied is typically subject to various numerical instabilities. Fan
[1.35] used the conventional approach for UTCA in conjunction with the Euler-
Rodrigues’ formula to avoid these stability issues. He also used minimization of the
separation between the tooth surfaces to determine the direction and length of the
instantaneous contact lines [1.8]. Later Vogel et al [1.36, 1.37] proposed an alternate
approach to compute both tooth surfaces and the UTCA by using Singularity Theory.
They considered the generated tooth flank as a first-order singularity of the particular
11
function that models the generating process. They also used numerical differentiation to
investigate the sensitivity of tooth contact to machine settings [1.36]. Simon also used
the conventional system of five scalar nonlinear equation and six unknowns to find
contact points on both surfaces. He calculated contact lines orientation by minimizing the
separation function between two contact surfaces and applied his method for a FM
Gleason type gear pair with a generated pinion and a Formate® gear [1.38].
Published studies on modeling of hypoid tooth contact under loaded conditions
are quite sparse. Simon [1.39-1.41] used a FE model to calculate deflection and
displacement under load from which interpolation functions were obtained to estimate
stresses and deflections via regression analysis of FE results. Gosselin et al [1.42]
developed a loaded tooth contact analysis (LTCA) model for spiral bevel gears by using
tooth compliances obtained by curve-fitting to the FE deformation results of a single
pinion and gear tooth pair. Wilcox et al [1.43] also developed a FE-based model to
calculate the spiral bevel and hypoid gear tooth compliances by using a three-dimensional
model of a tooth including base deformations, which was later employed by Fan and
Wilcox [1.44] to perform LTCA analyses. Vimercati and Piazza [1.45] also calculated
FH gear pair surfaces and incorporated them with a commercially available finite
elements (FE) package [1.46] to calculate both TCA and LTCA. This particular hypoid
FE package that employs FE away from the contact zone and a semi-analytical contact
formulation at the contact zone [1.47] is perhaps the most advanced hypoid LTCA model
12
available to accurately simulate a hypoid gear contact. The major drawback of these FE-
based models is that they require a considerable amount of computation time, making
them more of an analysis tool. Their use for design tasks such as parameter and assembly
variation sensitivity studies is not very practical.
Beside the FE method, the Boundary Element (BE) method was also used in
several studies for performing LTCA. For instance, Sugyarto [1.48] sliced gear and
pinion teeth into a number of sections and considered each section as an independent
plate from its neighboring sections, and applied a two dimensional BEM formulation to
each slice to compute bending and shear deflections. Liu [1.49] applied same compliance
methodology to face gears [1.49] with a correction intended to couple each slice with
adjacent slices using Borner’s coefficient [1.50], which was originally proposed for
parallel axis gear. Vecchiato [1.51] used a three-dimensional BE approach for loaded
tooth contact predictions of FH hypoid gears. As for unloaded contact analysis, he used
conventional approach [1.8,1.11] and studied misalignment effects.
Besides these computational models, some semi-analytical models were also
proposed for determining tooth compliance of parallel-axis gears through elasticity-based
deformation solutions. Adding linear thickness variation along the profile to the
originally proposed plate solution [1.52], Yakubek [1.53] used the Rayleigh-Ritz Energy
Method to calculate the approximate deflection of a tapered plate for estimating the
compliance of spur and helical gears. Bending deformations of a tooth were considered
13
as a sum of shape functions that satisfy clamped-free and free-free boundary conditions,
and the unknown coefficients of the shape functions were determined by minimizing the
potential energy. Later, Yau [1.54, 1.55] expanded this compliance model to add shear
deformations to the energy function and found more realistic deformation for spur and
helical gears and Stegemiller [1.56, 1.57] used the FE package ANSYS to propose an
approximate interpolation based formula to compensate for base rotation and base
translation. All of these analytical compliance methods are valid for tooth having
constant height along face width and either constant or linearly varying thickness along
its profile, which is not the case for hypoid gears. Vaidyanathan [1.58-1.60] proposed an
analytical compliance model for a tooth with linearly varying thickness in the profile and
lengthwise directions as well as linearly varying tooth height along the face width. His
Rayleigh-Ritz based formulation used polynomial shape functions and was applied to
both sector and shell geometries. The sector model represents straight bevel gear
geometry closely while the shell model is sufficiently close to a spiral bevel gear tooth in
terms of its geometry.
The challenges mentioned above in terms of performing a loaded hypoid tooth
contact analysis in a practical and computationally efficient way have been the major
road block to the development of other models to study other functional behavior of
hypoid gears. One such behavior is the efficiency of the hypoid gear pair. In addition to
the analytical surface geometries and surface velocities, an accurate description of the
14
contact load distribution is required at many rotational increments (of pinion angle) to
predict the distribution of the friction coefficient and the resultant mechanical power
losses. While this hypoid efficiency methodology has been demonstrated by Xu and
Kahraman [1.61-1.63] by using the FE-based loaded tooth contact model of Vijayakar
[1.64], the amount of computations required was reported to be very significant for it to
be used extensively as a design tool. Likewise, other hypoid gear models for simulation
of surface wear [1.65] and finishing processes such as lapping have also been hampered
by the difficulties in obtaining load distribution.
1.3. Scope and Objectives
Computation of the contact pressure distributions and the relative surface
velocities forms the basis for predicting the required functional parameters of the hypoid
gear pair, including the transmission error, contact stresses, root bending stresses, fatigue
life and mechanical power losses. It is evident from the review of the literature that a
model to compute the load distribution accurately and efficiently without resorting to
computationally demanding FE methods does not exist. This is mainly due to three
primary reasons:
(i) A detailed description of general and reliable formulation to define the geometry
of FH and FM hypoid tooth surfaces from cutter parameters, machine motions
and settings is not available in the literature.
15
(ii) Conventional methods of matching the tooth surfaces (bringing them to contact)
present major numerical difficulties for UTCA.
(iii) There is no published model available for hypoid gear LTCA based on semi-
analytical tooth compliance formulations.
Accordingly, the main objective of this study is to develop FH and FM hypoid
gear contact models that address these issues. This study performs the following specific
tasks to achieve this objective:
(i) Development of a methodology that simulates the FM and FH processes to define
surface geometries of hypoid gears including the coordinates, normal vectors and
radii of curvatures.
(ii) Development of a novel formulation for unloaded tooth contact analysis by using
the ease-off topography, surface of action and roll angle surface to predict
unloaded transmission error and unloaded contact pattern in addition to potential
contact lines/curves to be used for loaded tooth contact analysis.
(iii) Development of a semi-analytical tooth compliance model tailored for both FH
and FM hypoid and spiral bevel gears.
16
(iv) Development of a LTCA model for FH and FM hypoid gears to predict pressure
distribution and loaded transmission error with or without misalignments of
various types.
Actual manufactured surfaces of gears include inevitable machining errors, heat
treatment distortions and lapping surface changes as globally distributed deviations on
tooth surfaces, which affect contact patterns and transmission error significantly.
Moreover, wear or lapping simulations (as accumulated wear) changes surface geometry
usually in a very local fashion that conventional tooth contact analysis approaches are not
capable of capturing. A novel ease-off based approach will also be developed to modify
ease-off topography of the theoretically generated tooth surfaces to account for both
global deviations due to the manufacturing process and local surface deviations due to
factors such as wear and lapping process.
A second objective of this dissertation is to develop a capability to predict load
dependent (mechanical) power losses of hypoid gear pairs. For this purpose, the
proposed loaded tooth contact model will be combined with a new friction model
according to the methodology proposed by Xu and Kahraman [1.62, 1.63] to predict
mechanical power losses and gear pair efficiency including all relevant contact, surface,
and lubricant parameters as well as the operating conditions. This hypoid gear efficiency
model will be used to investigate the impact of basic design parameters, and surface and
17
lubricant conditions, on mechanical power losses of hypoid gear pairs and to arrive at
guidelines on how to reduce such losses.
1.4. Overall Modeling Methodology
The overall methodology used to develop the hypoid load distribution model is
illustrated in the flowchart of Figure 1.4. Gear blank dimensions, cutter geometry,
machine settings, assembly dimensions and misalignments, torque and speed are all
included as input parameters for the load distribution model. These parameters are
commonly put together by hypoid gear manufacturers in a standard form that is called a
special analysis file. The pinion and gear cutter surfaces are first constructed and used to
define the extended pinion and gear surfaces (including surface coordinates, normals and
curvatures) by applying fundamental equation of meshing between a gear blank and its
respective cutter surfaces. These extended tooth surfaces are then trimmed in 3D space
so that they are contained by the blanks, and transformed to a global coordinate system
where any misalignments in the directions of shaft offset (ΔE), pinion axis (ΔP), gear axis
(ΔG) as well as the shaft angle error (ΔΣ) can be applied. Next, ease-off and surface of
roll angle are constructed and an UTCA model is developed by bringing the tooth
surfaces together and an unloaded contact pattern is defined by choosing a separation
tolerance between the tooth surfaces.
Figure 1.4: Flowchart of overall hypoid gear loaded tooth contact analysis methodology.
18
FH Method(Generate / Formate)
Extended Pinion and Gear surfaces and Curvatures (FM)
FM Method(Generate / Formate)
Surfaces in Global Coordinates System
(Misalignments)
Extended Pinion and Gear Surfaces and Curvatures (FH)
Global coordinates transformation Global coordinates transformation-Under Development
Unloaded TCA
Loaded Tooth Contact Analysis
Required Data for Efficiency Analysis
Contact Pressure Distribution
Loaded Transmission Error
Reading Design File.HAP or .SPA file
Tooth Compliances
Ease-off Construction
Equation of Meshing
Cutter axis
Root (clamped edge)
Tip (free edge)
Toe-BaseHeel-Base
Toe-height
Heel-height
Toe (free edge)
Heel (free edge)
Heel Toe
Root
Top
RPPLPP CPP
1M
2MContact lines
1M 2M
TE (μ rad)
100
Maximum UTE
Pitch Pinion phase angle (deg.)
Adjacent tooth pairs
Equation of Meshing
c
Cutter
Generating gear
Extended
epicycloids trace
t
IB OB tC
(b)
GC
Blade
FH Method
Generating gear
Cutter
Circular
t
IB
OB
(a)
GC
Fixed
FM Method
-20 -15 -10 -5 0 5 10 15 20
X (mm) Root
-4
-2
0
2
Y (
mm
) T
oe
(MPa)
758674590506421337253169
840
Updating Ease-off by Surface Deviations
Elastohydrodynamic Friction Coefficient Model
Hypoid Gear Mechanical Power Loss and Efficiency
19
Next the tooth compliance matrices comprising bending, shear, Hertzian and base
rotation deflections are computed. Finally, a set of equilibrium and compatibility
conditions are defined and solved simultaneously to compute the load distribution and the
loaded transmission error of the hypoid gear pair. Moreover, all required information for
efficiency and lapping simulations are computed.
1.5. Dissertation Outline
In Chapter 2, the hypoid gear tooth surfaces will be defined through simulation of
the face-milling and face-hobbing processes with all relevant cutter and machine related
parameters included. A new formulation of unloaded tooth contact analysis based on the
principles of ease-off and a newly introduced surface of roll angle will be proposed as
well.
In Chapter 3, a semi-analytical tooth compliance model will be employed and a
loaded tooth contact model will be described. A novel approach will be introduced in
Chapter 4 to compute loaded tooth contacts of gear surfaces that have deviations from
their theoretically intended surfaces either in local or global fashion.
Chapter 5 proposes a model to predict the mechanical efficiency of hypoid gear
pairs. This mode combines the developed computationally efficient contact model and
the mixed elastohydrodynamic lubrication (EHL) based friction model of Li and
20
Kahraman [1.66] to predict gear mesh power losses and mechanical efficiency. A
summary, major conclusions and contributions of this research to the state-of-the-art as
well as a list of recommendation for future work will be included in Chapter 6.
References of Chapter 1
[1.1] Wildhaber, E., 1946, Basic Relationship of Hypoid Gears, McGraw-Hill.
[1.2] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute
of Technology.
[1.3] Stewart, A. A., and Wildhaber, E., 1926, "Design, Production and Application of
the Hypoid Rear-Axle Gear." J. SAE, 18, pp. 575-580.
[1.4] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,
Elsevier Science B. V.
[1.5] Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A
Concurrent Engineering Approach, John Wiley & Sons Inc.
[1.6] Stadtfeld, H., J., 1995, Gleason Advanced Bevel Gear Technology, The Gleason
Works.
[1.7] Coleman, W., 1963, Design of Bevel Gears, The Gleason Works.
[1.8] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear
Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.
Des., 129(1), pp. 31-37.
21
[1.9] Dooner, D. B., 2002, "On the Three Laws of Gearing." ASME J. Mech. Des., 124,
pp. 733-744.
[1.10] Krenzer, T. J., 2007, The Bevel Gears, http://www.lulu.com/content/1243519.
[1.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd
ed.), Cambridge University Press, Cambridge.
[1.12] Krenzer, T. J., 1990, "Face Milling or Face Hobbing." AGMA, Technical Paper
No. 90FTM13.
[1.13] Stadtfeld, H. J. (2000). "The Basics of Gleason Face Hobbing." The Gleason
Works.
[1.14] Dudley, D. W., 1969, The Evolution of the Gear Art, American Gear
Manufacturers Association, Washington, D. C.
[1.15] Litvin, F. L. (2000). "Development of Gear Technology and Theory of Gearing."
NASA RP1406.
[1.16] Wildhaber, E., 1956, "Surface Curvature." Product Engineering, pp. 184-191.
[1.17] Dyson, A., 1969, A General Theory of the Kinematics and Geometry of Gears in
Three Dimensions, Clarendon Press, Oxford.
[1.18] Baxter, M. L., 1964, "An Application of Kinematics and Vector Analysis to the
Design of a Bevel Gear Grinder." ASME Mechanism Conference, Lafayette, IN.
[1.19] Baxter, M. L., and Spear, G. M. "Effects of Misalignment on Tooth Action of
Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.
22
[1.20] Coleman, W. "Analysis of Mounting Deflections on Bevel and Hypoid Gears."
SAE 750152.
[1.21] Coleman, W. "Effect of Mounting Displacements on Bevel and Hypoid Gear
Tooth Strength." SAE 750151.
[1.22] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason
Works.
[1.23] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears
under Load." SAE Earthmoving Industry Conference, Peoria, IL.
[1.24] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for
hypoid gear-drives of formate and helixform; Part I-Calculation for machine
setting for member gear manufacture of the formate and helixform hypoid gears."
ASME J. Mech. Des., 103, pp. 83-88.
[1.25] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for
hypoid gear-drives of formate and helixform; Part II-Machine setting calculations
for the pinions of formate and helixform gears." ASME J. Mech. Des., 103, pp.
89-101.
[1.26] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for
hypoid gear-drives of formate and helixform; Part III-Analysis and optimal
synthesis methods for mismatched gearing and its application for hypoid gears of
formate and helixform." ASME J. Mech. Des., 103, pp. 102-113.
[1.27] Litvin, F. L., and Gutman, Y., 1981, "A Method of Local Synthesis of Gears
Grounded on the Connections Between the Principal and Geodetic of Surfaces."
ASME J. Mech. Des., 103, pp. 114-125.
23
[1.28] Litvin, F. L., Fuentes, A., Fan, Q., and Handschuh, R. F., 2002, "Computerized
design, simulation of meshing, and contact and stress analysis of face-milled
formate generated spiral bevel gears." J. Mechanism and Machine Theory, 37(5),
pp. 441-459.
[1.29] Fong, Z. H., and Tsay, C.-B., 1991, "A Mathematical Model for the Tooth
Geometry of Circular-Cut Spiral Bevel Gears." ASME J. Mech. Des., 113, pp.
174-181.
[1.30] Fong, Z. H., 2000, "Mathematical Model of Universal Hypoid Generator with
Supplemental Kinematic Flank Correction Motions." ASME J. Mech. Des.,
122(1), pp. 136-142.
[1.31] Tsai, Y. C., and Chin, P. C., 1987, "Surface Geometry of Straight and Spiral
Bevel Gears." J. Mechanism, Transmission and Automation in Design, 109, pp.
443-449.
[1.32] Fong, Z. H., and Tsay, C.-B., 1991, "A Study on the Tooth Geometry and Cutting
Machine Mechanisms of Spiral Bevel Gears." ASME J. Mech. Des., 113, pp. 346-
351.
[1.33] Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, "Determination of
Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel
Gears." J. Mechanism, Transmission and Automation in Design, 110, pp. 495-
500.
[1.34] Stadtfeld, H. J., and Gaiser, U., 2000, "The Ultimate Motion Graph." ASME J.
Mech. Des., 122(3), pp. 317-322.
24
[1.35] Fan, Q., 2006, "Computerized Modeling and Simulation of Spiral Bevel and
Hypoid Gears Manufactured by Gleason Face Hobbing Process." ASME J. Mech.
Des., 128(6), pp. 1315-1327.
[1.36] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for
Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany.
[1.37] Vogel, O., Griewank, A., and Bär, G., 2002, "Direct gear tooth contact analysis
for hypoid bevel gears." Computer Methods in Applied Mechanics and
Engineering, 191(36), pp. 3965-3982.
[1.38] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME
International Power Transmission and Gearing Conference ASME, 88, pp. 789-
798.
[1.39] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,
122(44), pp. 529-535.
[1.40] Simon, V., 2000, "FEM stress analysis in hypoid gears." J. Mechanism and
Machine Theory, 35(9), pp. 1197-1220.
[1.41] Simon, V., 2001, "Optimal Machine Tool Setting for Hypoid Gears Improving
Load Distribution." ASME J. Mech. Des., 123(4), pp. 577-582.
[1.42] Gosselin, C., Cloutier, L., and Nguyen, Q. D., 1995, "A general formulation for
the calculation of the load sharing and transmission error under load of spiral
bevel and hypoid gears." J. Mechanism and Machine Theory, 30(3), pp. 433-450.
[1.43] Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite
Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth."
AGMA, Technical Paper No. 97FTM05.
25
[1.44] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis
(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,
Technical Paper No. 05FTM08.
[1.45] Vimercati, M., and Piazza, A., 2005, "Computerized Design of Face Hobbed
Hypoid Gears: Tooth Surfaces Generation, Contact Analysis and Stress
Calculation." AGMA, Technical Paper No. 05FTM05.
[1.46] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced
Numerical Solution Inc., Hilliard, Ohio.
[1.47] Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element
Solution for a Three-Dimensional Contact Problem." International J. for
Numerical Methods in Engineering, 31, pp. 525-545.
[1.48] Sugyarto, E., 2002, "The Kinematic Study, Geometry Generation, and Load
Distribution Analysis of Spiral Bevel and Hypoid Gears," M.Sc. Thesis, The Ohio
State University, Columbus, Ohio.
[1.49] Liu, F., 2004, "Face Gear Design and Compliance Analysis," M.Sc. Thesis, The
Ohio State University, Columbus, Ohio.
[1.50] Borner, J., Kurz, N., and Joachim, F. (2002). "Effective Analysis of Gears with
the Program LVR (Stiffness Method)."
[1.51] Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth
Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University
of Illinois at Chicago.
[1.52] Timoshenko, S. P., and Woinowsky-Krieger, S., 1959, Theory of Plates and
Shells, McGraw-Hill Book Company Inc.
26
[1.53] Yakubek, D., Busby, H. R., and Houser, D. R., 1985, "Three-Dimensional
Deflection Analysis of Gear Teeth Using Both Finite Element Analysis and a
Tapered Plate Approximation." AGMA, Technical Paper No. 85FTM4.
[1.54] Yau, H., 1987, "Analysis of Shear Effect on Gear Tooth Deflections Using the
Rayleigh-Ritz Energy Method," M.Sc. Thesis, The Ohio State University,
Columbus, Ohio.
[1.55] Yau, H., Busby, H. R., and Houser, D. R., 1994, "A Rayleigh-Ritz Approach to
Modeling Bending and Shear Deflections of Gear Teeth." J. of Computers &
Structures, 50(5), pp. 705-713.
[1.56] Stegmiller, M. E., 1986, "The Effects of Base Flexibility on Thick Beams and
Plates Used in Gear Tooth Deflection Models," M.Sc. Thesis, The Ohio State
University, Columbus, Ohio.
[1.57] Stegmiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of
the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192.
[1.58] Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded
Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The
Ohio State University, Columbus, Ohio.
[1.59] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz
Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears
Using Shell Theory." AGMA, Technical Paper No. 93FTM03.
[1.60] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical
Approach to the Static Analysis of an Annular Sector Mindlin Plate with
Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp.
255-266.
27
[1.61] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction
Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University,
Columbus, Ohio.
[1.62] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of
Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part
K: J. Multi-body Dynamics, 221(3), pp. 387-400.
[1.63] Xu, H., Kahraman, A., and Houser, D. R., 2006, "A Model to Predict Friction
Losses of Hypoid Gears." AGMA, Technical Paper No. 0FTM06.
[1.64] Vijayakar, S. M., 2003, Calyx User Manual, Advanced Numerical Solution Inc.,
Hilliard, Ohio.
[1.65] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear
Pairs." In press, Wear.
[1.66] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric
Integrated Control Volume Discretization." Tribology International, Hiroshima,
Japan.
28
CHAPTER 2
DEFINITION OF FACE-MILLED AND FACE-HOBBED HYPOID GEAR
GEOMETRY AND UNLOADED TOOTH CONTACT ANALYSIS
2.1. Introduction
Unlike most types of gears that have closed-form equations defining their
geometry, the geometry of hypoid gears can only be computed by solving implicit
equations governed by the manufacturing process, including its machine settings and
cutter specifications. Besides the gear blank dimensions and basic geometry
requirements, a set of performance or functionality related requirements must be met.
Among them, the contact pattern (location, size, shape) on the gear tooth surfaces and the
motion transmission error amplitude of the gear pair are two of the most common ones
checked routinely in the design of hypoid gear pairs. The contact pattern and the
transmission error are both determined via a contact analysis of the pinion and gear tooth
surfaces under a very small amount of load.
29
In general, contact of hypoid gear surfaces is single-mismatched. This is the most
general case of point contact condition between two surfaces [2.1]. The purpose of the
unloaded tooth contact analysis (UTCA) is to determine a contact point path (CPP) on
each surface in addition to area on each surface in the neighborhood of each
instantaneous contact point that falls in a specified separation distance (usually 6.3
micron of separation distance is commonly used in hypoid gear industry) [2.2]. In
addition, UTCA results in the function of motion transmission error between two gear
axes that is viewed as a key metric used to estimate the noise/level of the hypoid gear pair
in operation [2.3, 2.4].
In this chapter, as the first basic step in the analysis of hypoid gears, the geometry
of both face-milled (FM) and face-hobbed (FH) hypoid gear pairs produced by using both
Formate® and Generate cutting methods will be computed. This will be done by
simulating individual cutting processes. Basic machine tool settings, cutter geometry
parameters and gear blank dimensions will form the input for this computation. Next, a
novel method based on the ease-off topography will used to determine the unloaded
contact patterns. The proposed ease-off based methodology finds the instantaneous
contact curve through a surface of roll angles, allowing an unloaded tooth contact
analysis in a robust and accurate manner.
30
2.2. Definition of Tooth Surface Geometry
The concept of the generating gear is a key to the basic understanding of hypoid
gears because this hypothetical gear can be treated as cutting tool for both the pinion and
the gear [2.5]. In a FM cutter head, blades are arranged around the cutter head axis on an
equal radius for inside and outside blades (IB and OB, respectively) to form a conical
shape due to cutter axis rotation. The inside blades cut convex side of a tooth slot while
the outside blades cut concave side of the same slot, as shown in Figure 2.1(a). Face-
hobbing cutter heads (such as PENTAC® or TRI-AC®) like the one shown in Figure
2.1(b) roll while cutting such that each set of IB-OB blades (called blade group) will pass
through a different tooth slot. The cutting process can be considered as rolling of two
gears together, except the teeth of one of the gears are replaced by blade group of the
cutter head. By rolling the cutter head and the gear blank together while advancing the
cutter head into the blank, the gear is cut by the continuous indexing method. While the
axis of the generating gear for FM process is fixed, it is located on the center of a circle
tC for FH process that rolls on the generating gear circle GC , as shown in Figure 2.1(b).
Therefore, the edges of a blade in FH process traces extended epicycloids since they
usually lie on a radius that is larger than the radius of rolling circle tC .
31
Figure 2.1: (a) Face-milling and (b) face-hobbing cutting processes.
c
t
(a)
Cutter
Generating gear
Extended epicycloids trace
IBOB
tC
(b)
GC
Blade group
Generating gear
Cutter
Circular arc trace
t
IB
OB
GC
Fixed
32
2.2.1. Kinematics
Earlier cradle-based hypoid generators were designed to provide the required
degrees of freedom and relative motions through machine settings to accommodate the
cutting process of the gear and pinion blanks by means of the cutter blades. Figure 2.2
shows a typical cradle-based hypoid generator with machine settings and relative motions
defined as the cutter phase angle t , the tilt angle Ti , the swivel angle js , the radial
setting rS , the cradle angle c , the sliding base BX , the machine root angle m ,
machine center to back ctbM , the blank offset bE , the blank phase angle g , angular
speed of the cutter axis t , angular speed of the cradle axis c , angular speed of the
blank axis g , and the cradle angle change q. Newer-generation hypoid gear machine
tools still use settings in the form of older cradle-based machines. While the new cutting
machines are not set as the old mechanical machines, their working principles are still the
same such that they generate the same gear surface with an added capability of
controlling higher-order surface geometrical parameters as well. While most of the
machine settings are typically fixed (kept constant), a number of them are defined as a
polynomial function of q. These parameters that are dependent on q are called higher-
order motions such as the modified roll (ratio of roll change), the helical motion (sliding
base change) and the vertical motion (blank offset change).
33
Figure 2.2: Cradle based hypoid generator parameters.
c
bE
ctbM m
rS
js
Ti
t g
BX
34
In the FM process, the teeth are cut individually by blade edges that rotate fast
about the cutter axis. The cutting edges of the blades form a conical surface and their
envelope seen from a coordinate attached to the blank is the gear or pinion surface. In the
Formate® case for the FM process, the blank is fixed, the cutter advances towards the
gear blank while rotating, in the process replicating its surface on the blank. Then, the
cutter head retreats, the blank is rotated by one tooth spacing, and the same cutting
process is repeated as shown in Figure 2.1(a). Meanwhile, the face-hobbing process
using Formate® requires the blank and cutter head to be rolled together according to a
kinematic relationship defined as
Rog t
tgt g
NR
N
(2.1)
where t is cutter head angular velocity, Rog is the rolling portion (also called
indexing) of angular velocity of the gear blank, and tN and gN are numbers of blade
groups and gear teeth being cut. Typically, edges of the blades in FH process do not
intersect with cutter head axis so that the cutting surface is a hyperboloid of revolution.
In the generate process for both FM and FH processes, there is an additional
relative rotation between the gear blank and the cradle axis in Figure 2.1(b). The ratio of
roll is given as the ratio of relative rotations, i.e.
35
,RgFM
ac
(2.2a)
GegFH
ac
R
(2.2b)
where FMaR and FH
aR are the ratios of rolls for the FM and FH processes (they are zero
for Formate® process), respectively, c is the angular velocity of the cradle axis, g is
the angular velocity of the blank axis during face-milling and Geg is the generating
portion of blank angular velocity for the FH process. As a result, the total blank angular
velocities for FM and for FH are given, respectively:
,FMg g (2.3a)
FH Ro Geg g g (2.3b)
Corresponding gear blank rotation angles FMg and FH
g for FM and FH processes are
given respectively as:
,FM FMg aR q (2.4a)
( )FH FHg a tg tR q R . (2.4b)
36
Here, the relative motion between the cutter and the gear blank for the FM process can be
treated as a special case of the FH process. Therefore, the formulation for the FH process
will be explained here in detail while its differences from the FM process will also be
specified, as required.
2.2.2. Cutting Tool Geometry and the Relative Motion
Figure 2.3 shows a typical FH blade with its geometry along its cutting edge is
defined by the blade angle b , the rake angle , the hook angle , the blade offset angle
b , the cutter radius cr , and the distance from the tip of blade to reference point fh .
The cutting edge is divided into four different sections as the edge (or tip radius), toprem,
profile and flankrem that are all shown in Figure 2.3(c). The edge and flankrem are
usually circular arcs while toprem is usually a straight line at a slight angle from the
profile section. Most of the cutting is done by the profile section of the blade that is
usually a straight line or a circular arc. For a typical FM cutter, 0b .
Referring to Figure 2.3, an arbitrary point A on cutting edge is at position ( )sr r
relative to the local coordinate system bX fixed to the cutter head (with its origin at
reference point M) where s is the distance of point A to point M along the blade edge.
With this, the unit tangent vector is s t r , and if the cutting edge is a line, it can be
reduced to
37
Figure 2.3: (a) Cutter head, (b) blade and (c) cutting edge geometry.
31
18
Flankrem
b
bcr
Mtx
ty
,b tx x
bz
M
b
M
tz
z
bx Profile
Toprem
cr
by fh
y
,bz z
,x x
y
,x x bx
Inside Blade
n
A
tr
s
Reference Plane
Edge
(a)
(b) (c)
38
[ sin 0 cos 1]Tb b t (2.5)
where the superscript T denotes a matrix transpose. Position and unit tangent vectors, tr
and tt , in the coordinate system tX whose z axis coincides with the cutter axis and xy
plane passes through reference point M as shown in Figure 2.3(a) are given as
( ) ( ) ( )t z b x z t = M M M t , (2.6a)
t c i tr sr = t t (2.6b)
where
[cos sin 0 1]Ti b b t = (2.6c)
and ( )k M is a rotation matrix facilitating a rotation angle about axis k
( [ , , ])k x y z such that
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
( ) ,x
M (2.6d)
cos 0 sin 0
0 1 0 0
sin 0 cos 0
0 0 0 1
( ) ,y
M (2.6e)
39
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
( )z
M . (2.6f)
Position vector of point A is transformed to the coordinate system gX fixed to the
blank as [2.6]:
(
( ) ( ) ( )
( ) ( ) ( )
b ctb m B B c
r
g x g E M y m X X z c
z S j z x T z t tq js i
r M M M M M M
M M M M M ) r (2.7a)
where the transformation matrices are defined as
1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,r
r
S j
S
M (2.7b)
1 0 0 0
0 1 0 0
0 0 1
0 0 0 1
,B c
BX X
M (2.7c)
1 0 0
0 1 0 0
0 0 1 0
0 0 0 1
,m B
ctb
X
M
M (2.7d)
40
1 0 0 0
0 1 0
0 0 1 0
0 0 0 1
.b ctb
bE M
E
M (2.7e)
Variables influencing gr , with the exception of s and t , are either fixed or dependent
on q (in case of higher order motion).
2.2.3. Equation of Meshing
Equation (2.7a) represents a family of surfaces in a coordinate system fixed to the
blank whose envelope is the generated surface on the blank. The envelope of gr with
three independent variables s, t and q is given mathematically as [2.7-2.9]:
0g g g
ts q
r r r. (2.8)
This equation is mathematically equivalent to the fundamental equation of meshing,
( ) 0c w n V V , which states for each point to lie on the envelope surface that the
normal vector n to the family of the cutter surfaces should be perpendicular to relative
velocity between the blank (w) and the cutter (c) as shown in Figure 2.4.
41
Figure 2.4: Generation process.
Blank axis
Cradle axis
Cutter head
Blank
n
c
wVVc
V Vc w
V Vc w
w
42
Using Eq. (2.8), each point on the generated surface can be found by solving a
system of two implicit nonlinear equations for a pair of unknown parameters that can be
chosen as ( , )ts , ( , )s q or ( , )tq .
2.2.4. Principal Curvatures and Principal Directions
As shown in Figure 2.5 for any point 0 0 0( , )ts P defined by independent surface
curvilinear variables s and t on the gear surface, the unit normal to the surface is given
as
0
g g
t
g g
t
s
s
r r
nr r
. (2.9)
Moving from point 0 0 0( , )ts P to 1 0 0( , )ts ds P by only infinitesimally changing one
of the surface variables s, the change of unit normal to the surface is defined as [2.10]
1 11 1( , ) t tt
dk
d s
n
t v . (2.10)
Here
1 01
1 0
P P
tP P
, (2.11a)
43
Figure 2.5: Curvature computation procedure.
0n1n
0P 1P
2P
2n
1t
2t
1v
1C
2C
44
0 11
0 1
n t
vn t
(2.11b)
and ( , )ts defines the distance from 0P to 1P along the gear surface (along curve 1C )
as a function of s and t (here, 0 constantt t and only s varies). 1t
k is the normal
curvature in the 1t direction and 1t
is the geodesic torsion in the direction of 1v . Here
0n , 1t and 1v form a Frenet trihedron. Following the same procedure, but this time
moving from 0 0 0( , )ts P to 2 0 0( , )t ts d P by infinitesimally changing t , the normal
curvatures 2t
k and geodesic torsion 2t
in 2t and 2v directions are found according to
2 22 2( , ) t tt
dk
d s
n
t v (2.12)
with
2 02
2 0
P P
tP P
, (2.13a)
0 22
0 2
n t
vn t
(2.13b)
and ( , )ts defines the distance from 0P to 2P along the gear surface (along curve 2C
in Figure 2.5) as a function of s and t (here 0s s is constant and only t varies). With
45
1tk ,
2tk ,
1t and
2t in hand, Euler equation [2.11, 2.12] is applied to compute the
principle directions ( 1e and 2e ) and the principal curvatures ( 1K and 2K ).
Having the principal curvatures and directions of every possible contact point
(points on the projection plane) on the pinion and gear, principal directions of the
difference surface are defined as directions in which two contacting surfaces have the
extremes of the relative curvatures [2.11-2.13]. The curvatures pR and gR of pinion
and gear surfaces in the direction of maximum relative curvatures are used to find the
equivalent curvature ( )p g p geqR R R R R that will be required later to determine
Hertzian deflections in loaded TCA [2.14].
2.3. Unloaded Tooth Contact Analysis
In unloaded tooth contact analysis (UTCA), the goal is to calculate:
(i) the contact point path (CPP) on each of the gear surfaces in addition to the zone on
each surface in the neighborhood of each instantaneous contact point that is as close
as the specified separation distance [2.15, 2.16], and
(ii) the function of transmission error between two gear axes.
Two different approaches were used in the past for performing UTCA of hypoid gears
with mismatched surfaces. In the conventional method, tooth surfaces are treated as two
46
arbitrary surfaces, rotating about the pinion and gear axes. The contact point path (CPP)
on each surface is computed by satisfying two contact conditions. The first condition is
the coincidence of position vector tips of the points on the gear and pinion surfaces in
three dimensional space. The second condition is the collinearity of the normal vectors
of the both of the surfaces at the contact point.
The second method of performing UTCA is based on the ease-off procedure. The
current literature lacks a clear and accurate mathematical definition of ease-off as well as
its construction including the instantaneous contact lines/curves [2.17]. Ease-off has
often been defined in the literature as the change in pinion surface with the application of
modifications. While these changes directly reshape ease-off, they do not constitute the
ease-off itself.
Finding the location and orientation of potential instantaneous contact lines is one
major step in the UTCA. In general, the instantaneous contact shape in the projection
plane (plane that includes the gear axes) is slightly curved as opposed to commonly used
approximate straight lines, which is the contact shape in action surface. The
instantaneous contact line directions are conventionally found based on principal
curvatures and directions of contacting surfaces of gear members at contacting points as
the direction of minimal relative normal curvature between contacting surfaces.
47
In the next two sections, a brief overview of the conventional approach will be
provided, followed by a detailed ease-off based UTCA formulation developed in this
thesis.
2.3.1. The Conventional Method of UTCA
The position vector 1 1( , )p r and normal vector 1 1( , )p n of any point on the
surface of the pinion can be defined by two independent curvilinear local surface
variables 1 and 1 . Similarly, the position and normal vectors of any point on the
surface of the gear are given as 2 2( , )g r and 2 2( , )g n where 2 and 2 are the
independent curvilinear local variables of the gear surface. Pinion surface coordinate
1 1( , )p r and normal 1 1( , )p n are rotated about the pinion axis pa as much as an
angle p while the gear surface coordinate 2 2( , )g r and normal 2 2( , )g n are
rotated about the gear axis ga by an angle g to satisfy the two contact conditions
defined below:
1 1 2 2( ) ( , ) ( ) ( , )p p g gz z M r M r OE , (2.14a)
1 1 2 2( ) ( , ) ( ) ( , )p p g gz z M n M n . (2.14b)
48
Here, OE is the offset vector that connects the origins of the pinion and the gear. Eq.
(2.14) constitutes a system of five nonlinear equations (since p gn n ) and six
unknowns 1 , 1 , p and 2 , 2 , g . Defining the value of one of these six
parameters (usually p ) as an input parameter, a set of five nonlinear equations defined
by Eq. (2.14) can be solved for the remaining five unknowns. However, due to the high
level of conformity of the pinion and gear surfaces in the vicinity near the contact point,
the solution of this system of equations is subject to several numerical instabilities. In
addition, the solution is very sensitive to the initial guesses. Provided these numerical
difficulties can be overcame, the solution of Eq. (2.14) yields the coordinates of contact
point path (CPP) on the pinion and gear surfaces as well as the angular position of the
gear as a function of the pinion angle.
With the conventional method, at each point of the contact point path (CPP), a
direction in which separation between two surfaces (here pinion and gear surfaces) is
minimum [2.11, 2.12, 2.18, 2.19] is assumed to be potential contact line. For this
purpose, the two contacting surfaces are approximated as two contacting ellipsoids with
an instantaneous point contact M as shown in Figure 2.6, The principal curvatures of
both pinion ( 1pK and 2
pK ) and gear ( 1gK and 2
gK ) surfaces respectively and the
corresponding principal directions ( 1pe , 2
pe , 1ge and 2
ge ) are all required to find
direction in which relative normal curvature between ellipsoids is minimal. This direction
49
Figure 2.6: General case of approximating gear surfaces as two contacting ellipsoids to
orient instantaneous contact line.
1e p
p
1eg
eg2
n
M
T 1pk
(a)
u
v
2e p
2pk
1gk
2gk
g
10
u v
2L
1L
(b)
50
u, as shown in Figure 2.6, having an unknown angle from 1pe in the tangential plane
T is the major direction of instantaneous contact ellipse. Utilizing the Euler equation
[2.11, 2.12], the pinion normal curvature along any arbitrary direction u with an angle
from 1pe in the tangential plane T is
2 21 2cos ( ) sin ( )p pp
uk k k (2.15a)
while gear normal curvature along same direction is
2 21 2cos ( ) sin ( )g gg
uk k k . (2.15b)
Hence, the relative curvature along u is given as
2 2 2 21 2 1 2cos ( ) sin ( ) cos ( ) sin ( )g g p ppg
uk k k k k . (2.16)
The value of angle that minimizes pguk is found by
0pg
udk
d
(2.17)
such that
1
12
1 sin(2 )tan
2 cos(2 )pgK
, (2.18a)
51
1 212
1 2
p ppg
g g
k kK
k k
. (2.18b)
The length of the unloaded instantaneous contact line is defined as 1 8 pguL k
where is the unloaded separation distance [2.12]. In the conventional method, it has
been assumed that instantaneous contact lines spread equally on both sides of
instantaneous contact points M along assuming contact direction u since approximating
contacting surfaces with local ellipsoids results in the same relative curvatures on both
sides of contact points. In other words, the conventional method fails to include the
variation of the curvature along a given contact line, resulting in contact line length
estimations that might be erroneous.
Simon [2.20] found the direction of minimum separation by minimizing a
function that defines separation between contacting surfaces in the direction of the
normal to the gear surface at each contact point. Although his approach does not acquire
principal curvatures and directions information on each contact point, it still bears the
some level of computational complexity and inefficiency since such minimization of the
separation function requires the solution of a system of seven nonlinear equations and
seven unknowns. He later mentioned that the instantaneous contact form is a curve
rather than the generally assumed line [2.21]. Fan [2.8] found instantaneous contact line
direction and length without using second order information by finding minimum
52
separation direction. He constructed a cylinder centered at an instantaneous contact point
with an axis that is collinear with the common normal then searched for a direction on the
tangent plane in which the separation between the contacting surfaces is minimal. With
this, another search was performed to find the distance required to move on the both sides
of the minimum separation direction in order to reach the predetermined separation value
. He eliminated the need for the second order surface information and computed more
realistic contact line length estimating different lengths on both sides of contact point
along contact line. However, this method still assumed an instantaneous contact line, as
opposed to more realistic curved shape.
2.3.2. Ease-off Based Method of UTCA
This study proposes a novel surface of roll angle and utilizes it to orient
instantaneous contact lines/curves without using principal curvatures and directions. This
method requires significantly less computational effort since (i) it does not result in a set
of nonlinear algebraic equations that must be solved numerically, and (ii) it only requires
the coordinates and the normal vector of one contacting surface and the spatial
orientation of the axes of both gears. The instantaneous contact curves are defined
between two conjugate surfaces, namely a given surface and the conjugate surface to the
reference surface with respect to the given spatial orientation of the axes of the gears.
Since the instantaneous contact line orientation is extremely insensitive to local surface
53
changes as it will be demonstrated later, the validity of using a conjugate surfaces instead
of the actual surface is well justified.
2.3.2.1. Construction of Ease-off and the Surface of Roll Angle
Ease-off will be defined in this study as the deviation of real gear surface from
the conjugate of its real mating pinion surface. It can also be defined as the deviation of
real pinion surface from the conjugate of its real mating gear. These definitions are
respectively called the gear-based ease-off and pinion-based ease-off [2.5]. The
conventional method of UTCA seeks a contact between two arbitrary surfaces, failing to
benefit from the fact that the designed gear and pinion surfaces are indeed close to the
corresponding conjugate surfaces. Closeness of the actual and the corresponding
conjugate surfaces enables the use of the ease-off concept. The proposed ease-off
approach for UTCA has several advantages such as
(i) providing an overview of the contact pattern and transmission error as well as
interference between pinion and gear teeth especially at the edges,
(ii) providing more accurate instantaneous contact curves instead of commonly used
approximate contact lines,
(iii) eliminating the need to compute the curvature in order to estimate length and
direction of the contact lines/curves, and
54
(iv) avoiding the need for initial guesses required by the conventional method to
locate first contact point (some initial guesses might end up divergent solutions).
In this study, the ease-off surface is constructed directly from the relationships between
the continuous cutter surfaces. Therefore, any surface fits to pinion and gear surfaces is
not needed.
The first step in constructing ease-off is to specify an area in the gear projection
plane with the possibility of contact between pinion and gear, as shown in Figure 2.7.
Such an area is the projection of a volume bounded by the faces and front and back cones
of the pinion and gear into the gear projection plane. The projection plane shown in
Figure 2.7 is a gear-based projection plane since its ease-off is defined on gear
tooth area. Using the gear machine settings and Eq. (2.8), the real gear surface
coordinates (shown in Figure 2.7) are computed for every point of the projection plane as
gijr where [1, ]gi m and [1, ]gj n are the indices in lengthwise and profile directions
of the surface point with gm and gn as number of surface grids respective directions.
Then, each point of the gear projection plane is transformed to the pinion coordinate
system to construct the projection plane for the pinion. Next, the real pinion surface
points and unit normals to the surface are computed from Eq. (2.8) for every point of the
pinion projection plane as pijr and p
ijn . Both real pinion and gear surface coordinates and
unit normal vectors are transformed into the global coordinate system where
55
Figure 2.7: Construction of the ease-off, action and Q surfaces.
Conjugate of
pinion, r̂gij
Real pinion
surface, r pij
Action
surface, raij
Q Surface
a p
a g
Gear projection plane
Ease-off surface, ij
Real gear
surface, rgij
G
P
E
56
misalignments E , P , G and shown in Figure 2.7 are applied as well. Having
pijr , p
ijn , the pinion and gear axis vectors pa and ga , and the gear ratio R , the action
surface position vector ijar is found from conjugacy equation in 3D space as [2.5]:
ijij ij ij( ) ( )p p pp g aR a r n a r n . (2.19)
The same steps are repeated for all points on the real pinion surface to define the
action surface completely. As seen in Figure 2.7, the action surface for a hypoid gear
pair, while quite flat, is not a plane. Any point and its unit normal vector on the real
pinion surface are rotated around the pinion axis in order to satisfy Eq. (2.19). The angle
of rotation ijp of each point that satisfies Eq. (2.19) is then plotted on the projection
plane to construct the pinion roll angle surface Q . The angle ij ij=g p R corresponds to
the amount of rotation from the surface of action to reach the conjugate of pinion surface.
Therefore,
ijij ij( ) g g az r M r . (2.20)
If this conjugate surface of the pinion were to match perfectly with the real gear
surface at any point, then a perfect meshing condition with zero unloaded transmission
error would exist. The difference between these two surfaces (conjugate of pinion and
57
real gear surfaces) in projection plane domain is defined as ease-off surface where the
differences between these two surface at each grid is ij .
2.3.2.2. Contact Pattern and Transmission Error
Any value , of any point on the Q surface is the pinion roll angle. As shown in
Figure 2.8 for a specific pinion roll angle i , intersection of the plane iz and the
Q surface defines x and y coordinates of all points on the projection plane that have the
same roll angle, stating theoretically that they lie on the same contact line/curve. Since
Q is not a plane, this intersection for hypoid gears is usually a curve rather than a
straight line as assumed by most studies. The instantaneous contact curve ( )iC shown
in Figure 2.8 is determined by projecting the intersection curve first on projection plane
and then projecting this projected curve once more on the ease-off surface. The
minimum distance from ( )iC to the projection plane at point ( )iH is the
instantaneous unloaded transmission error ( )iTE . Moreover, moving in both directions
from point ( )iH along ( )iC within a preset separation distance , gives the unloaded
contact line length ( )iL . Repeating this procedure for every pinion roll angle
increment, unloaded transmission error curve ( )TE and the unloaded tooth contact
pattern are computed. Here the contact curves are between real pinion and conjugate
gear.
58
Figure 2.8: Unloaded TCA computation procedure: (a) gear projection plane, ease-off
and Q surfaces, and (b) instantaneous contact curve, contact line and unloaded
transmission error.
Ease-off surface
Projection plane
Q surface
iz
iQ
Contact curve projection
Instantaneous contact curve, ( )iC
z
y
(a)
A
x
A
( )iC z
( )iTE
( )iL
( )iH
(b)
x
59
Replacing the conjugate gear with the real one practically does not change the contact
line orientation and shape, since the effect of microscopic changes of pinion and gear
surfaces on Q surface is negligible.
2.4. An Example Hypoid Unloaded Tooth Contact Analysis
An example hypoid gear pair whose basic parameters are listed in Table 2.1 for
the drive-side contact (concave side of pinion and convex side of gear) is considered to
demonstrate the capabilities of the proposed hypoid gear geometry computation and
unloaded tooth contact pattern models. This is a FM gear set representative of
automotive rear axle gear sets.
The predicted unloaded transmission error (UTE) curves computed by the model
for three adjacent tooth pairs 1i , i and 1i are shown in Figure 2.9. Here UTE is
plotted against the mesh cycles (pinion roll angle) where each of the individual UTE
curves corresponds to a single tooth pair in mesh. Individual curves for two adjacent
tooth pairs are one mesh cycle apart. At the intersection point of the two adjacent UTE
curves ( 1M or 2M in Figure 2.9), transition from one tooth pair to adjacent tooth pair
occurs. The transmission error value of the intersection point is the maximum UTE,
which is attempted to be minimized for unloaded tooth contact pattern optimization
procedures. The corresponding predicted unloaded tooth contact pattern is shown in
60
__________________________________________________________________
Parameter Pinion Gear
__________________________________________________________________
Number of teeth 11 41
Hand of Spiral Left Right
Mean spiral angle (deg) 40.5 28.5
Shaft angle (deg) 90
Shaft offset (mm) 20
Outer cone distance (mm) 115.0 111.0
Generation type Generate Formate
Cutting method FM
__________________________________________________________________
Table 2.1: Basic drive side geometry and working parameters of the example hypoid gear
pair.
61
Figure 2.9: Unloaded transmission error of the example gear pair with misalignments
0.15 mm, 0.12 mm, 0, 0E P G .
1
15
3
30
2
TE rad]
Mesh cycles
1i
i 1i
1M 2M P-P UTE
62
Figure 2.10. The curve marked as CPP is the locus of all instantaneous contact
points between gear and pinion tooth surfaces. On both sides of CPP are right point path
(RPP) and left point path (LPP), which are as far from CPP as it is required to reach
chosen separation value of 0.006 mm between ease-off topography and projection
plane shown in Figure 2.8. The CPP, RPP and LPP curves are computed for a single
tooth pair in contact while in case of multiple teeth in contact these curves are partially
active (usually middle part of the curves are active) since the adjacent pairs will take over
the motion. In Figure 2.10, unloaded contact pattern of the tooth of interest is bounded
by the instantaneous contact lines at 1M and 2M and the RPP and LPP curves.
Using the same example gear pair, the influence of misalignments effects on
unloaded contact patterns are illustrated next. Figure 11(a) shows unloaded contact
pattern of the drive side at a nominal position where 0E P G .
Misalignments of 0.08E mm, 0.10P mm and 0G move the unloaded
contact pattern to toe as shown in Figure 11(b) while misalignments 0.15E mm,
0.10P mm and 0G move the contact pattern to heel as shown in Figure
11(c). Similarly, Figure 10(d) shows another contact pattern near toe for the gear pair
with misalignments of 0.05E mm, 0P G , and 4 min. Comparison of
Figures 11(b) and 11(d) indicates that similar shifts in the contact patterns can be caused
by different sets of misalignments.
63
Figure 2.10: Unloaded contact pattern of the example gear pair for three adjacent tooth
pairs 1i , i and 1i (i-1), (i) and (i+1) with 0.15 mm, 0.12 mm, 0E P G
and 0 .
Tip
RootToe
Heel
RPP
Contact lines
LPP CPP
2M
1M
64
Figure 2.11: Unloaded contact pattern of the example gear pair (a) at nominal position
with 0E P G , (b) at toe with 0.08E mm, 0.10P mm and
0G , (c) at heel with 0.15E mm, 0.10P mm and 0G and
(d) at toe with 0.05E mm, 0P G , and 4 min.
(a)
(b)
(c)
(d)
65
References for Chapter 2
[2.1] Baxter, M. L., and Spear, G. M., 1961, "Effects of Misalignment on Tooth Action
of Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.
[2.2] Krenzer, T. J., 1981, Understanding Tooth Contact Analysis, The Gleason Works.
[2.3] Smith, R. E., 1984, "What Single Flank Measurement Can Do For You." AGMA,
Technical Paper No. 84FTM2.
[2.4] Smith, R. E., 1987, "The Relationship of Measured Gear Noise to Measured Gear
Transmission Errors." AGMA, Technical Paper No. 87FTM6.
[2.5] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute
of Technology.
[2.6] Zhang, Y., and Wu, Z., 2007, "Geometry of Tooth Profile and Fillet of Face-
Hobbed Spiral Bevel Gears." IDETC/CIE 2007, Las Vegas, Nevada, USA.
[2.7] Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A
Concurrent Engineering Approach, John Wiley & Sons Inc.
[2.8] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear
Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.
Des., 129(1), pp. 31-37.
[2.9] Vecchiato, D., 2005, "Design and Simulation of Face-Hobbed Gears and Tooth
Contact Analysis by Boundary Element Method," Ph.D. Dissertation, University
of Illinois at Chicago.
66
[2.10] Wu, D., and Luo, J., 1992, A Geometric Theory of Conjugate Tooth Surfaces,
World Scientific, River Edge, NJ.
[2.11] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd
ed.), Cambridge University Press, Cambridge.
[2.12] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,
Elsevier Science B. V.
[2.13] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears
under Load." Earthmoving Industry Conference, Peoria, IL.
[2.14] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their
Load Carrying Capacity (Part I)." D.S.I.R., London.
[2.15] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason
Works.
[2.16] Shtipelman, B. A., 1979, Design and manufacture of hypoid gears, John Wiley &
Sons, Inc.
[2.17] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for
Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany.
[2.18] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.
[2.19] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis
(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,
Technical Paper No. 05FTM08.
67
[2.20] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME
International Power Transmission and Gearing Conference ASME, 88, pp. 789-
798.
[2.21] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,
122(44), pp. 529-535.
68
CHAPTER 3
SHELL BASED HYPOID TOOTH COMPLIANCE MODEL AND LOADED
TOOTH CONTACT ANALYSIS
3.1. Introduction
In the previous chapter, an ease-off topography defined by the theoretical tooth
surface was developed and used to determine the unloaded contact characteristics of a
hypoid gear pair, including the unloaded single and multiple tooth contact patterns and
the unloaded transmission error. This chapter builds on this formulation to predict the
same under loaded contact conditions.
Published studies on the modeling of tooth contact of hypoid gears under loaded
conditions are quite sparse. They can basically be divided into two major groups:
Computational models that use Finite Element (FE) or Boundary Element (BE)
formulations, and analytical models. As an example of models from the first group,
69
Wilcox et al [3.1] developed a FE-based model to calculate spiral bevel and hypoid gear
tooth compliance using 3D model of tooth including base deformations, which was later
employed by Fan and Wilcox [3.2] to develop a loaded tooth contact analysis (LTCA).
Vijayakar [3.3] developed another FE based hypoid LTCA package. This model employs
a hybrid approach with FE away from the contact zone and a semi-analytical contact
formulation at the contact zone. This model is perhaps the most advanced hypoid LTCA
model available today to simulate the loaded contacts of a hypoid gear pair accurately.
The major drawback of these computational models however is that they require a
considerable amount of computation time, which makes them more of an analysis tool.
Their use for design tasks such as parameter and assembly variation sensitivity studies is
not very practical for the same reason.
Besides these computational models, some semi-analytical models were also
proposed for determining tooth compliance of parallel-axis gears trough elasticity-based
deformation solutions. A detailed literature review of such studies was provided in
Chapter 1. All of these analytical compliance models were valid for a tooth having
constant height along its face width and either constant or linearly varying thickness
along its profile, which is not the case for hypoid gears. Vaidyanathan [3.4-3.6] proposed
an analytical compliance model for a tooth with linearly varying thickness in the profile
and lengthwise directions as well as linearly varying tooth height along the face width.
His Rayleigh-Ritz based formulation used polynomial shape functions and was applied to
both sector and shell geometries. The sector model represents straight bevel gear
70
geometry closely while the shell model can be deemed sufficiently close to that of spiral
bevel gear.
In this chapter, a Rayleigh-Ritz based shell model similar to the one proposed by
Vaidyanathan [3.4-3.6] will be applied to face-hobbed and face-milled hypoid pinion and
gear teeth to define the tooth compliances due to bending and shear effects efficiently in a
semi-analytical manner. Base rotation and contact deformation effects will also be
included in this compliance formulation. With this, loaded contact patterns and
transmission error of both face-milled and face-hobbed spiral bevel and hypoid gears will
be predicted by enforcing the compatibility and equilibrium conditions associated with
the load distribution at the gear mesh.
3.2. Tooth Compliance Model
According to methodology outlined in the flowchart of Figure 1.4, the last step
before a LTCA can be performed is determining the tooth compliances of both contacting
members. The compliance of a tooth is defined as the amount of deflection at any
contact point due to a unit load applied at various points on the same tooth surface [3.5].
The compliance of a gear tooth must include tooth bending deflections, shear and
Hertzian deformations as well as the base rotation, since each might contribute to tooth
deflections significantly. As the computational efficiency of the model is a major
consideration, a semi-analytical shell model [3.5] is employed here instead of a FE
71
model. The model considers a hypoid gear tooth as part of a shell with a linearly varying
thickness and height, as parameterized in Figure 3.1. The tooth thicknesses toet , heelt
and tipt at the toe-root, heel-root and tooth tip locations are calculated and a linear, two-
variable function is fit to these calculated thicknesses to obtain a thickness function as
( ) ( ) ( )
tip toe heel toe ctoe
x t t t t r h xt t
h f h (3.1a)
where h is the height of the tooth defined as
[ ]heel toetoe c
h hh h r
f
. (3.1b)
Here, f is the face width, cr is the cutter radius, toeh and heel heelh are tooth height
values at the toe and the heel of the tooth, is the angle between any contact points on
the tooth to toe measured from cutter center, and x is the tooth height at that specific
contact point as defined in Figure 3.1.
In cylindrical coordinates with independent variables , and z, the position
vector of a point on the circular cylindrical shell is defined as [3.5]:
( , , ) ( , ) R r iuz z . (3.2a)
Here iu is the normal to the mid-surface of the shell, defined as
72
Figure 3.1: Basic dimensions of a hypoid tooth used in the compliance formulation.
6
heelh
Root (clamped)
Heel (free)
toeh
Tip (free)
Toe (free)
heelt
toet
tipt
x
Cutter axis
73
sinu
r ri , 1cos
r r
r r. (3.2b,c)
The normal and shear strains, m and mn , are given as ( , [1,3]m n , m n )
[3.5]
3
1
1
2
m k m
mkm m km k
U U g
gg g, (3.3a)
1( ) ( )
m nmn m n
m nm n m n
U Ug g
g g g g (3.3b)
where
21 α=[ (1+ )]g A z R , (3.3c)
22 =[ (1+ )]g A z R , (3.3d)
3=1g (3.3e)
and m,nU is displacement component.
In cylindrical coordinates, normal and shear strain relations are derived as [3.5]:
74
1 1,
(1 )
U V A Wz A AB R
R
(3.4a)
1 1,
(1 )
V U B Wz B AB R
R
(3.4b)
,zW
z
(3.4c)
(1 )(1 )
[ ] [ ],(1 ) (1 ) (1 ) (1 )
zz BARR U V
z z z zB A A B
R R R R
(3.4d)
1(1 ) [ ],
(1 ) (1 )
z
W z UA
z zR zA AR R
(3.4e)
1(1 ) [ ].
(1 ) (1 )
z
W z VB
z zR zB BR R
(3.4f)
For a cylindrical shell, R , cR r , x , , 1A and cB r with and
as the independent curvilinear coordinates along and perpendicular to tooth root line,
respectively. Employing a displacement assumption based on the Mindlin type shear
75
theory [3.7] that assumes a constant shear strain throughout the thickness [3.5], Eq. (3.3)
is reduces to:
2
2,
xx
Wz
x x (3.5a)
2
2
1,
( ) ( )
c c c
z W W
r r z r z (3.5a)
2( ) (2 ),
( ) ( )
c c xx
c c c c
z r z z r z W z
r x r r z x r z (3.5c)
(1 ) , zc
z
r (3.5d)
. zx x (3.5e)
The transverse deflection W and shear rotations, x and , are obtained by
setting the first variation of the potential energy to zero by using the Rayleigh-Ritz
method. The potential energy (PE) for a conservative system is the difference between
the strain energy SE and the work done by the external force WF, i.e. PE SE WF .
Here, SE and WF due to an external force p of a deformed shell surface are [3.5]
2 22
2 2 212
( ) ( 2 )2(1 )
(1 )( ) ,
c x xx z
x xz z
ESE r z
dzd dx
(3.6a)
76
c
x
WF pWr dxd . (3.6b)
where E and are the modulus of elasticity and the Poisson’s ratio for the gear material.
Setting the first variation of the potential energy to zero one obtains
2
12
( ) ( ( )(1 )
(1 )( ) 0.
c x x x xx z
x x xz xz z z cx
EPE r z
dzd dx p Wr dxd
(3.7)
The transverse deflection and shear rotations are written as linear combinations of
finite number of polynomials with unknown coefficients as
( ) ( ) mn m nm n
W A x , (3.8a)
( ) ( ) x mn m nm n
B x , (3.8b)
( ) ( ) mn m nm n
C x , (3.8c)
all of which must satisfy the following boundary conditions of a shell-shaped tooth
shown in Figure 3.1:
0( , ) 0
xW x ,
0
( , )
x
W x
x
, (3.9a,b)
77
0( , ) 0x xx , 0
( , ) 0x
x . (3.9c,d)
The polynomial functions ( )m x and ( ) n must satisfy all essential boundary
conditions, in addition to being continuous, linearly independent and complete. Equation
(3.8) is substituted in Eq. (3.5) that is needed to evaluate normal and shear strains defined
in Eq. (3.6a) to compute strain energy. Computations of SE and WF are done
numerically using Gauss-Quadrature method to yield the linear set of equations:
11 12 13 1
21 22 23 2
31 32 33 3
K K K FA
K K K B F
CK K K F
(3.10)
where the sub-matrices ( , [1,3])s sm ns sm n K are determined based on volume integral
of material properties and assumed trial functions, and ( [1,3])F smsm forms the force
vector that depends on type (point, line, etc) and location of the applied load.
Computation of the tooth compliance is done by solving Eq. (3.10) numerically for
coefficients mnA , mnB and mnC . Polynomial functions for pinned-free and clamped-free
boundary conditions are assumed as ( ) ( ) mm x x a and 1( ) ( ) m
m x x a ,
respectively, where a is the tooth height. The free-free condition for ( ) n is
represented by the polynomial 1( ) ( )nn
, where is the subtended angle by
78
circular segment of the shell of a length equal to the face width of the gear [3.5], which is
approximately equal to the ratio of the tooth length to the cutter radius.
Figure 3.2 shows a flowchart of the tooth compliance computation methodology.
While the task of computing unknown coefficients of shape functions is time consuming,
it is done only once and is valid for all mesh positions. Suppose that total contact
lines/curves as is shown (for one contact line) in Figure 3.3 are divided into cN segments
(and each segment has its own local load, which is yet to be computed), then the total
compliance matrix of a pinion or gear tooth is written as
11 1
1
.
. . .
.
Cc
c c c
N
N N N
w w
w w
(3.11)
where ( , [1, ])s si j s s cw i j N is the deflection at segment is due to the load applied at
segment js. With this, the total deflection at segment is due to all of the applied discrete
loads (load vector) is 1 cs s ss
Ni i jjW w .
Closed-form formula of Weber [3.8] was used here for computing the Hertzian
deformations while the base rotation and base translation effects on total tooth
compliance were introduced by using an approximate interpolation method similar to the
one developed for helical gears by Stegemiller [3.9].
79
Figure 3.2: Flowchart of the compliance computation.
Construct compliance matrix for pinion and gear and calculate
total compliance matrix
Use polynomial shape functions with unknown coefficients for
deformation that satisfy boundary conditions
Fit a linear function to tooth thickness and height
Tooth thickness and height calculation
Calculate strain energy based on shape function
Choose number of mode shapes IFF: Free-Free
ICF: Clamped-Free
Blank dimensions and machine settings
Potential contact lines from Unloaded TCA calculation
Minimize potential energy (strain energy + work) using
Rayleigh-Ritz method
Calculate unknown coefficients of polynomial shape functions
80
Figure 3.3: Potential contact line discretization.
.1
2 3
.
cN
1cN
cFN
1cFN
1F 2F
3F
81
In order to validate the proposed tooth compliance computation procedure, a tooth
of a face-milled Formate hypoid gear was modeled by using a commercially available
finite elements package (ANSYS). As an example, a 500 N load was applied in the
middle of the lengthwise direction and the middle of the profile, and the tooth deflections
predicted by ANSYS and the proposed semi-analytical shell model along the middle of
the top-land of the tooth were compared as shown in Figure 3.4. It is observed in this
figure that increasing the number of mode shapes in Eq. (3.8) (IFF mode shapes for the
free-free boundary conditions and ICF mode shapes for the clamped-free boundary
conditions) improves shell model predictions, converging the predicted deflections to
those from ANSYS. However, it also increases the computational time require as shown
in this figure as well. Vaidyanathan conducts an extensive comparison of his developed
shell model with ANSYS for various loading conditions and number of mode shapes and
proved accuracy of the shell model [3.6].
3.3. Loaded Tooth Contact Analysis
The number of tooth pairs in contact depends on the gear contact ratio, roll angle
of the pinion (or gear) and amount of applied torque. Under unloaded conditions, a
hypoid gear pair having a contact ratio greater than one has always at least one tooth pair
in contact. Once the load is applied, this number increases due to the deflection of the
contacting teeth. In the loaded tooth contact model, all the tooth pairs that are likely to
Figure 3.4: The comparison of the shell model deformation to FEM.
82
0.5
1
1.5
2
0
2.5
5 10 15 20 25 30 35 40 45
3
-0.50
Def
lect
ion
of f
ree
edge
(m
icro
n)
Location along face width direction (mm)
5, 3, 8 s ICF IFF t
5, 5, 15 s ICF IFF t
5, 9, 50 s ICF IFF t
5, 15, 135 s ICF IFF t
FEM Shell model
83
geometrically share the torque must be taken into consideration with their respective
separation distances. Potential contact lines/curves of all contacting tooth pairs are
computed and discretized into a finite number of segments. The length, separation,
surface curvatures of both members along each line segment are computed and used as
input for the LTCA model.
Conditions of compatibility and equilibrium must be satisfied simultaneously in
the load distribution model [3.10]. According to the compatibility condition, in order for
the contact to occur along each of the contact lines/curves, the sum of total elastic
deformation of two contacting teeth CF and the initial separation vector S must be
greater than or equal to the rigid body rotation Θ gR , i.e.
Θ g CF S R (3.12)
where F is force vector, C is the total compliance matrix that is the sum of the pinion
and gear tooth compliance matrices pC and gC (Eq. (3.11)), and the Hertzian
compliance matrix hC , and gR is the vector that contains the distances of each segment
to the gear axis. Eq. (3.11) can be written in form of an equality constraint by
introducing slack variable Y as
Θ g CF R Y = S . (3.13)
84
Since two bodies must be at contact for any force on a given segment si to exist, either
0si F for 0
siY or 0
siF for 0
siY ( [1, ])s ci N .
Meanwhile, the equilibrium condition assures that the total moment caused by
forces acting on all contacting segments about the gear axis as shown in Figure 3.5 must
be equal to external torque gT applied on the gear axis:
T g gTF R (3.14)
where superscript T denotes matrix transpose. The load distribution and loaded
transmission error are computed by solving compatibility and equilibrium equations
simultaneously.
3.4. An Example Hypoid Tooth Contact Analysis
The same face-milled example hypoid gear pair used for the UTCA whose basic
parameters are listed in Table 2.1 for the drive-side contact (concave side of pinion and
convex side of gear) is considered to demonstrate the capabilities of the proposed hypoid
gear load distribution model. This is a FM gear set representative of an automotive rear
axle gear sets.
85
Figure 3.5: Static equilibrium between torque applied on gear axis and torque produced
by the force of all contacting segments.
1F
2F
3F
1gR 2
gR 3gR
gT
ga
86
Loaded transmission error (LTE) of the example gear pair predicted by the
proposed model are shown in Figure 3.6 at three different pinion torque values of
50,pT 250 and 500 Nm for a set of fixed misalignment values of 0.15 E
mm, 0.12 P mm, 0G mm and 0 . These LTE time histories indicate that the
shape and the average value of LTE change with pT , as expected. Table 3.1 lists the
peak-to-peak value (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-mean-
square (RMS) value of LTE corresponding to the cases of Figure 3.6. The LTE functions
in each pT level are dictated primarily by the first harmonic order. Modest increases in
p-p, 1st harmonic and the root-mean-square values of LTE are observed with increasing
pT .
Figures 3.7(a-c) show the pressure distributions predicted by the proposed model
for the same cases of Figure 3.6. It is seen in Figure 3.7(a) that the contact is localized at
the center of the tooth when pT is low (50 Nm) with no edge loading. An increase in
pT causes the contact pattern to spread, in the process exhibiting edge loading at the tip
and root regions as it is evident from Figures 3.7(b) and (c). Figures 3.7(a-c) also show
the loaded contact patterns (maximum contact pressure distributions) predicted by a FE-
based hypoid contact model [3.11] for the same cases are in good agreement. It is
worthwhile to mention here that each simulation with the proposed model required 45
seconds of CPU time (about 25 seconds for compliance matrix computations and 1
second per roll angle) on a 3.0 GHz PC while the same
87
Figure 3.6: Loaded gear transmission error of the example gear pair with
0.15 mm, 0.12 mm, 0E P G and 0 at (a) 50pT Nm, (b) 250pT
Nm, and (c) 500pT Nm.
(c) 500 Nmp T =
Mesh cycles
TE [rad]
90
100
110
120
130
270
280
290
300
310
580
590
600
610
620
0.0 0.5 1.0 1.5 2.0
(b) 250 Nmp T =
(a) 50 Nmp T =
88
_________________________________________________________________
Loaded Transmission Error [μrad]
pT Errors ______________________________________
[Nm] [mm] p-p 1st 2nd 3rd RMS _________________________________________________________________
50 0.15, 0.12 E P 12.8 6.3 0.6 0.1 6.3
250 0.15, 0.12 E P 15.9 7.9 0.3 0.4 7.9
500 0.15, 0.12 E P 21.9 9.9 2.3 0.5 10.1
_________________________________________________________________
50 0.08, 0.05 E P 12.2 5.5 0.7 0.6 5.6
50 0.26, 0.13 E P 12.5 6.3 0.4 0.1 6.3
_________________________________________________________________
Table 3.1: The loaded transmission error predictions of the proposed model; 0G
mm and 0 for all cases.
Figure 3.7: Comparison of loaded contact patterns predicted by the proposed model to an FE model [3.11] for (a)
50 NmpT , 0.15E mm, 0.12P mm, (b) 250 NmpT , 0.15E mm, 0.12P mm, (c) 500 NmpT ,
0.15E mm, 0.12P mm, (d) 50 NmpT , 0.08E mm, 0.05P mm, and (e) 50 NmpT , 0.26E mm, 0.13P mm ( all at 0, 0G ).
(a) 50 Nmp T = , 0.15 mmE = , 0.12 mmP =
(b) 250 Nmp T = , 0.15 mmE = , 0.12 mmP =
(c) 500 Nmp T = , 0.15 mmE = , 0.12 mmP =
Proposed Model FE Model [3.11]
89
Continued
Figure 3.7 continued
Proposed Model FE Model [3.11]
(d) 50 Nmp T = , 0.08 mmE = , 0.05 mmP =
(e) 50 Nmp T = , 0.26 mmE = , 0.13 mmP =
90
Continued
91
analysis using the FE model took about 15 minutes using the same computer. This
highlights the main advantage of this proposed model as a design tool, even if it might
not be as accurate as the full FE model [3.11].
Next, the same gear pair is simulated by using the proposed model and the FE
model [3.11] at 50pT Nm for two other misalignment conditions. Here, two of the
errors are kept constant at 0G mm and 0 , and the other two errors E and
P are varied. In Figure 3.7(d), error values of 0.08 E mm and 0.05 P mm
cause the predicted loaded contact pattern to move towards toe and root, compared to
Figure 3.7(a). Meanwhile, the loaded contact for 0.26 E mm and 0.13 P mm
moves the contact in the opposite direction towards the heel. In the process, the
maximum contact pressure is reduced since there is larger area in the heel that carries the
same load. In addition, equivalent radii of curvature are larger at heel than toe, which
directly decreases maximum Hertzian pressure from Weber equation [3.8]. The FE
simulations of the same error combinations shown in the same figures are again in good
agreement with the predictions of the proposed model. This suggests that the sensitivity
of the hypoid gear contact to gear errors is captured sufficiently by this model. Finally
the LTE parameters listed in Table 3.1 for these two cases reveal slight reduction in LTE
amplitudes compared to the first case of Figure 3.7(a), suggesting that a good contact
pattern does not necessarily mean lower LTE.
92
References for Chapter 3
[3.1] Wilcox, L. E., Chimner, T. D., and Nowell, G. C., 1997, "Improved Finite
Element Model for Calculating Stresses in Bevel and Hypoid Gear Teeth."
AGMA, Technical Paper No. 97FTM05.
[3.2] Fan, Q., and Wilcox, L., 2005, "New Developments in Tooth Contact Analysis
(TCA) and Loaded TCA for Spiral Bevel and Hypoid Gear Drives." AGMA,
Technical Paper No. 05FTM08.
[3.3] Vijayakar, S. M., 1991, "A Combined Surface Integral and Finite Element
Solution for a Three-Dimensional Contact Problem." International J. for
Numerical Methods in Engineering, 31, pp. 525-545.
[3.4] Vaidyanathan, S., 1993, "Application of Plate and Shell Models in the Loaded
Tooth Contact Analysis of Bevel and Hypoid Gears," Ph.D. Dissertation, The
Ohio State University, Columbus, Ohio.
[3.5] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1993, "A Rayleigh-Ritz
Approach to Determine Compliance and Root Stresses in Spiral Bevel Gears
Using Shell Theory." AGMA, Technical Paper No. 93FTM03.
[3.6] Vaidyanathan, S., Houser, D. R., and Busby, H. R., 1994, "A Numerical
Approach to the Static Analysis of an Annular Sector Mindlin Plate with
Applications to Bevel Gear Design." J. of Computers & Structures, 51(3), pp.
255-266.
[3.7] Mindlin, R. D., 1951, "Influence of Rotary Inertia and Shear on Flexural Motions
of Isotropic Elastic Plates. ." J. of Applied Mechanics, 18, pp. 31-38.
93
[3.8] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their
Load Carrying Capacity (Part I)." D.S.I.R., London.
[3.9] Stegemiller, M. E., and Houser, D. R., 1993, "A Three Dimensional Analysis of
the Base Flexibility of Gear Teeth." ASME J. Mech. Des., 115(1), pp. 186-192.
[3.10] Conry, T. F., and Seireg, A., 1972, "A Mathematical Programming Technique for
the Evaluation of Load Distribution and Optimal Modification for Gear Systems."
ASME J. of Industrial Engineering.
[3.11] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced
Numerical Solution Inc., Hilliard, Ohio.
94
CHAPTER 4
LOADED TOOTH CONTACT ANALYSIS OF HYPOID GEARS WITH
LOCAL AND GLOBAL SURFACE DEVIATIONS
4.1. Introduction
Hypoid gears used in various highest-volume applications such as automotive
axles are subject to various manufacturing errors and heat treatment distortions that
deviate the actual (real) tooth contact surfaces from the intended (theoretical) ones. Such
errors impact the quality of a hypoid gear pair, defined by a number of performance
indicators including its contact pattern, the motion transmission error (TE), efficiency as
well as its sensitivity to misalignments. These deviations represented by these
manufacturing errors typically follow patterns that shift, rotate or twist the surfaces
relative to the theoretical ones. Therefore, they can be characterized as global deviations.
Other more local deviations occur during the life span of hypoid gears in the form of
95
surface wear. Since the surface wear depths are proportional to the contact pressure and
the sliding distance, deviations due to surface wear are rather local and cannot be
captured by using the surface fitting methods developed to approximate the global
deviations. The motivation of this chapter is to develop a unified, ease-off based
methodology that allows loaded and unloaded tooth contact analysis of hypoid gears
having both global and local deviations.
Tooth contact analysis has usually been performed by considering the theoretical
pinion and gear surfaces defined by simulation of the hypoid cutting processes. The
analysis results presented in Chapters 2 and 3 also considered such theoretical tooth
surface errors with no deviations.. There are only a few published studies on hypoid
gear tooth contact analysis using the real surfaces. In such an analysis, Gosselin [4.1]
proposed an approach to compute tooth contact of real spiral bevel gear surfaces. He
interpolated measured surfaces with rational functions to predict their unloaded contact
pattern and transmission error. Since pinion and gear normal vectors of low-mismatch
(high-conformity) surfaces, a wide span of potential contact line around contact point can
be identified and the condition of collinearity for normal vectors is subjects to numerical
stability issues. In order to simplify the task of locating the contact point, Gosselin [4.1]
computed the difference between pinion and gear normal vectors at several points along
the lengthwise direction and estimated a location where the difference between the
normal vector is zero.
96
Zhang et al [4.2] proposed an approach to analyze unloaded tooth contact of real
hypoid gears based on a generalization of the work of Kin [4.3, 4.4] on spur gears. The
real pinion and gear tooth surfaces were divided into two vectorial surfaces of theoretical
and deviation surfaces. Separating theoretical and deviation surfaces, finding the
theoretical surfaces through cutting simulation, and applying interpolation only to the
deviation surfaces made his approach simpler and more accurate. Zhang [4.2] defined
the deviation surface in the normal direction of the theoretical surface by comparing the
theoretical and the real (measured) surfaces, and fit a bicubic surface to it. The normal to
the real surfaces (sum of the theoretical and interpolated deviation surfaces) were
computed by taking the derivative of the real surfaces. Having continuous functions for
the surfaces and normal of the pinion and the gear, he employed conventional system of
five nonlinear equations and five unknowns used in many other studies [4.5-4.7] to
simulate unloaded tooth contact of real hypoid surfaces.
Gosselin proposed a method called “surface matching” that attempts to define
changes to the machine settings that define the theoretical surfaces so that real gear
surfaces can be computed approximately from the cutting simulation [4.8, 4.9]. This
method found machine settings that generate a theoretical surface close to (but not
identical to) the real surface and the difference of the two surfaces was defined as
“residual error surfaces” for the pinion and the gear. He computed transmission error and
the contact pattern of the generated surfaces with this new set of machine settings. Then,
he used the residual error surface to modify predicted transmission error and contact
pattern without providing the details of this process [4.10]. This method is suitable in
97
capturing the effect of global errors such as the pressure angle error, the spiral angle
error, and lengthwise or profile crowning errors for the cases when residual errors are
rather small, while the same cannot be said for localized deviations such as surface wear
and global deviations with large residual errors.
In Chapter 2, it was mentioned that the unloaded tooth contact analysis (UTCA)
of hypoid gears with mismatched surfaces were performed using two fundamentally
different methods. The first method that was used widely defines the tooth surfaces as
two arbitrary surfaces, rotating about pinion and gear axes [4.7, 4.11-4.13]. In this
method, the contact point path (CPP) on each surface was determined by satisfying two
contact conditions: (i) coincidence of position vector tips of the points on the gear and
pinion surfaces and (ii) collinearity of the normals of the both of the surfaces. The
second method that was proposed in Chapter 2 was based on the ease-off topography. A
detailed formulation for construction of ease-off and determining the instantaneous
contact curves from surface of roll angle was provided. In Chapter 3, UTCA results were
combined with a semi-analytical compliance model based on shell theory to predict the
loaded contact patters and loaded TE. Using the ease-off approach for TCA of real gear
surfaces instead of conventional approach was shown to increase the computational
efficiency of TCA since surface interpolations for measured pinion and gear surfaces as
well as the solution of the system of five governing nonlinear equations are not needed.
In this chapter, ease-off is defined the same way as Chapter 2 as the deviation of
real gear surface from the conjugate of its real mating pinion surface. The formulation
98
that will be proposed to handle local and global deviations is based on the premise that all
surface deviations, both local and global, can be handled through modifications of the
ease-off topography. Having machine settings and blank dimensions, surface coordinates
and normal vectors of both pinion and gear will be defined using the methodology
proposed in Chapter 2. These theoretical surfaces will be used to establish the theoretical
ease-off topography and a theoretical surface of roll angle. As the main contribution of
this study, a procedure will be proposed to update the ease-off topography by taking into
account pinion and gear surface deviations. The updated ease-off and roll angle surfaces
will be used to determine the unladed tooth contact characteristics of the gear pair. These
UTCA results will be combined with the semi-analytical LTCA methodology of Chapter
3 to predict the loaded contact patterns and the transmission error of hypoid gears having
local and global surface deviations.
4.2. Construction of the Theoretical Ease-off Topography
The first step in defining the theoretical ease-off surface is specifying an area in
the gear projection plane with the possibility of contact between pinion and gear, as
illustrated in Figure 4.1. This area represents the projection of a volume bounded by the
faces and front and back cones of the pinion and gear into the gear projection plane. The
projection plane shown in Figure 4.1 is a gear-based projection plane since its ease-off
is defined on the gear tooth area. Using the gear machine settings and applying the
equation of meshing, the theoretical gear surface coordinates of a point of the projection
99
Figure 4.1: Construction of the ease-off, action and Q surfaces.
Gear projection plane
ij
Q paga
gijr
ˆ gijr
pijr
aijr
100
plane shown in Figure 4.1 are computed as gijr where [1, ]i m and [1, ]j n are the
indices in lengthwise and profile directions of the surface point with m and n as number
of surface grids in respective directions, and the superscript g denotes gear surface.
Then, gijr is transformed to the pinion coordinate system to construct the projection plane
for the pinion.
Next, the theoretical pinion surface point pijr and its unit normal to the surface
pijn are computed again through machine settings and applying the equation of meshing
[4.1]. Both theoretical pinion and gear surface coordinates and unit normal vectors are
transformed into the global coordinate system. Having pijr , p
ijn , the pinion and gear
axis vectors pa and ga , and the gear ratio R, the position vector aijr of the corresponding
point on the action surface is found from the conjugacy equation in 3D space as [4.14]:
( ) ( )p p pp g aijij ij ijR a r n a r n , [1, ]i m , [1, ]j n . (4.1)
The same procedure is repeated for all points on the real pinion surface to define the
action surface completely. As observed from Figure 4.1, the action surface for a hypoid
gear pair, while quite flat, is not a plane. Here, any point and its unit normal vector on
the theoretical pinion surface are rotated around the pinion axis in order to satisfy Eq.
(4.1). The angle of rotation pijq of each point ij required to satisfy Eq. (4.1) is then
101
plotted on the gear projection plane to construct the theoretical pinion roll angle surface
Q .
Here, only the relative values of pijq (i.e. the shape of the Q surface) are of
interest. Therefore, for computational simplicity and graphical demonstration purposes,
the Q surface is shifted in the direction normal to the projection plane to bring it to
contact with the projection plane such that at least one grid point has zero pijq value.
Mathematically, this shift is equivalent to rigidly rotating pinion tooth surface around the
pinion axis, which has no effect on the pinion surface. The angle ˆ =g pij ijq q R
corresponds to the amount of rotation required to travel from the surface of action to the
conjugate of theoretical pinion surface. Therefore, the position vector of the conjugate
of the theoretical pinion surface is found as
ˆ ˆ( ) g g az ijij ijqr M r , [1, ]i m , [1, ]j n , (4.2a)
where the rotation matrix about the z axis at an angle ˆ gijq is
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
ˆ ˆ
ˆ ˆˆ( )
g gij ij
g ggij ijz ij
q q
q qq
M . (4.2b)
102
If this conjugate surface of the pinion were to match perfectly with the real gear
surface at any point, then a perfect meshing condition with zero unloaded transmission
error would exist. For simplicity, the conjugate of pinion will be called here the
conjugate gear. The difference between the conjugate gear and the theoretical gear is
defined as theoretical ease-off topography . In general, the conjugate gear and the
theoretical gear are located at different angular positions with respect to the gear axis. In
order to compare these two surfaces, conjugate gear surface is rotated around gear axis
ga by an angle such that a grid point of conjugate gear surface touches the
corresponding grid point on the theoretical gear surface. In this position, the radial
distances ij between the grid points ij on these surfaces define the theoretical ease-off
surface . The and Q surfaces were used in Chapters 2 and 3 for both unloaded and
loaded tooth contact analyses.
4.3. Updating Ease-off Topography for Manufacturing Errors and Surface Wear
Deviations of the pinion and gear surfaces a grid point ij from their respective
theoretical surfaces are defined as pij and g
ij ( [1, ]i m , [1, ]j n ) as shown in Figure
4.2. Normal vectors to both pinion and gear surfaces are considered in inward direction.
Measured and/or worn pinion and gear surfaces are written as ( [1, ]i m , [1, ]j n ):
p p p pij ij ij ij r r n , (4.3a)
103
g g g gij ij ij ij r r n . (4.3b)
Here, it is assumed that the normal vectors of the real and theoretical surfaces are the
same, since both surfaces are practically very close to each other [4.4].
Any point of the theoretical ease-off surface ij can be updated by using
deviations on pinion and gear surfaces. The goal here is to update theoretical ease-off
surface directly from surface deviations rather that updating original pinion and gear
surfaces and conducting whole tooth contact procedure between new surfaces as it has
been done in Ref. [4.2].
Assume that an ease-off value ij of a point on the theoretical ease-off surface
is computed based on a corresponding theoretical surface vectors
, ,Tp p p p
ij ij ij ijx y z r , pijn and , ,
Tg g g gij ij ij ijx y z r . At the same grid point ij, the
pinion roll angle is pijq , and hence, the corresponding roll angle of the gear surface
(conjugate to the theoretical pinion surface) is g pij ijq q R and the distance of the same
grid points on the pinion and gear surfaces to their own rotation axes ( pa and ga
respectively) are
2 2( ) ( )p p pij ij ijL x y , (4.4a)
104
2 2( ) ( )g g gij ij ijL x y (4.4b)
as shown in Figure 4.2. With this, the changes in the pinion and gear roll angles due to
the pinion and gear surface errors pij and g
ij are defined as
( ),
h h hij ij ijh
ij hij
qL
u n = ,h p g. (4.5a)
Here piju and g
iju are the unit normal vectors in the radial (circular) direction of the
pinion and gear axes pa and ga , respectively, as shown in Figure 4.2. They are
defined as:
ppijp
ij ppij
a ru
a r,
ggijg
ij ggij
a ru
a r (4.5b,c)
Hence, the updated pinion roll angle taking the pinion deviation into account is
=p p pij ij ijq q q , (4.6)
which can be used to find locations and directions of contact curves corresponding to the
deviated tooth surfaces.
105
Figure 4.2: Graphical demonstration of the procedure to update ease-off surface for
surface deviations.
Pinion projection plane
Gear projection plane
pa
ga
gijr
pijrg
iju
pijup
ijn
gijn
pijL
gijL
gij
pij
Q
106
Change to the theoretical ease-off surface can be described in two components.
The first component is related to the pinion surface deviation that is formulated as
p gp gij ij ijq L (4.7)
where =gp pij ijq q R is the gear roll angle change due to the pinion surface deviation.
The second component is due to the gear surface deviation that is given as
g g gij ij ijq L (4.8)
Therefore, the total change to the ease-off topography is the sum of its two components
p gij ij ij . (4.9)
With this, the new ease-off surface is found as
ij ij ij . (4.10)
This updated ease-off surface and the corresponding updated surface of roll
angle Q as defined by Eq. (4.6) are used for unloaded and loaded tooth contact analyses
according to the methodology proposed in Chapters 2 and 3 As in Q , the updated
surface of roll angle Q is also shifted to touch the projection plane since the absolute
values pijq do not have any effect on unloaded and loaded tooth contact analyses.
107
4.4. Unloaded and Loaded Tooth Contact Analyses
In Figure 4.3, surfaces and Q are constructed on opposite sides of projection
plane to provide an insight into how UTCA is conducted. For any value q on surface Q ,
a corresponding instantaneous contact curve can be defined. As shown in Figure 4.3 for
a specific pinion roll angle kq , the intersection of the plane kz q and the surface Q
defines the x and y coordinates of all points on the projection plane that have same roll
angle, stating theoretically that they lie on the same contact curve. Since Q is not a
plane, this intersection for hypoid gears is usually a curve rather than a straight line as
assumed in most of the previous studies.
Figure 4.4 shows the theoretical contact curves of the drive and coast sides of a
sample hypoid gear pair on the projection plane for different pinion angles as shown on
the contact curves. The instantaneous contact curve ( )kC q shown in Figure 4.3 is
obtained by first projecting this intersection curve on projection plane and then projecting
this projected curve on the ease-off surface . The minimum distance from ( )kC q to
the projection plane at point H is instantaneous unloaded transmission error ( )kTE q .
Moreover, moving in both directions from point H along the curve ( )kC q within a preset
separation distance yields the unloaded contact line length ( )kU q . Repeating this
procedure for every pinion roll angle increment kq q ( [1, ]lk N where lN is total
number of contact curves considered), unloaded transmission error curve ( )kTE q and the
unloaded tooth contact pattern are computed.
108
Figure 4.3: Graphical demonstration of the procedure to compute unloaded TCA; (a) gear
projection plane, ease-off and Q surfaces, and (b) instantaneous contact curve, contact
line and unloaded transmission error.
Contact curve projection
kq q
kz q
Gear projection plane
z
x
Q
( )kC q
( )kU q
H
z
x
( )kTE q
109
Figure 4.4: Theoretical contact curves of an example hypoid gear pair.
10° 25° 40° 55° 70° 85°
Root
Toe
(a) Drive Side
10° 25° 40° 55° 70°
85°
Root
Toe
(b) Coast Side
110
It is noted here that the contact curves are defined between the real pinion
surface and conjugate of the real pinion surface (conjugate gear surface), instead of using
the real gear surface. Replacing the real gear surface with the conjugate one causes a
very little change to the orientation and shape of the contact curve, since the effect of
micro-geometry deviations has a negligible influence on the Q surface, i.e. Q and Q
are practically identical.
Number of tooth pairs in contact depends on the gear contact ratio, the roll angle
of the pinion (or gear) and the amount of torque applied. Under unloaded conditions, a
hypoid gear pair having a contact ratio greater than one has always at least one tooth pair
in contact. Once the load is applied, this number increases due to deflection of the
contacting teeth. In the LTCA model of Chapter 3, all the tooth pairs that are likely to
share the torque geometrically are taken into consideration with their respective
separation distances. Potential contact curves of all contacting tooth pairs are computed
and discretized into a finite number of segments ( cN ). The length of the separation at
each segment along each contact curve are computed and used as input for the LTCA
model. With the theoretical ease-off topography replaced by the modified ease-off
topography corresponding to real surfaces with deviations, the formulations of Chapters 2
and 3 can be applied to predict the unloaded and loaded tooth contact conditions,
respectively.
111
4.5. Example Analyses
4.5.1. A Face-milled Hypoid Gear Pair with Local Surface Deviations
A face-milled hypoid gear set with local deviations whose basic parameters are
listed in Table 4.1 for the drive-side contact (concave side of pinion and convex side of
gear) will considered for an example loaded contact analysis of surfaces with local
deviations. This gear set is representative of an automotive rear axle gear set. An
example case of local deviations from theoretical surfaces is shown in Figure 4.5 for both
pinion deviation pij and gear deviation g
ij . This case represents worn tooth surfaces
predicted by a hypoid gear wear model [4.15] from a companion study. In Figure 4.5, the
“Root” line refers to the lower limit of active contact region and it is not actual gear root
line.
Following the proposed ease-off update approach, gear projection plane, the
theoretical and updated ease-off surfaces, and , and the theoretical and updated roll
angle surfaces, Q and Q , are computed. Figure 4.6(a) shows these surfaces in relation
to each other while the theoretical ease-off topography, updated ease-off and the amount
of ease-off change computed from Eq. (4.9) are shown in Fig. 4.6(b) to (d) in contour plot
format, respectively. As shown here, the maximum changes take place in the vicinity of
the locations where pinion and gear deviations are maximum, according to Figure 4.5 and
the rest of the projection plane does not exhibit any considerable ease-off change.
112
_____________________________________________________________
Parameter Pinion Gear _____________________________________________________________
Number of teeth 11 41 Hand of Spiral Left Right Mean spiral angle (deg) 40.5 28.5 Shaft angle (deg) 90 Shaft offset (mm) 20 Outer cone distance (mm) 115 111 Generation type Generate Formate Cutting method FM ___________________________________________________________________________
Table 4.1: Basic drive side geometry and working parameters of the example hypoid gear
pair.
113
Figure 4.5: Example local deviation surfaces for the gear and pinion tooth surfaces.
Toe
Toe
Root
Root
pij
gij
μm
0246810
Pinion deviation
Root
Toe
012345
μmGear deviation
Root
Toe
114
Figure 4.6: Ease-off update for the example deviation of Fig. 5. (a) Three-dimensional
view of the projection plane, and , , Q and Q surfaces, and contour plots of (b) ,
(c) , and (d) the change of ease-off topography.
Gear projection plane
(a)
Q , Q
0
20
40
60 μm
(c)
Toe
Root
1 2 3 4
5 6
μm (c)
Toe
Root
0
20
40
60μm
(b)
Toe
Root
115
The corresponding predicted unloaded tooth contact patterns are shown in Figure
4.7 for a maximum separation value of 6 μm for both cases of (a) theoretical and (b)
deviated tooth surfaces, indicating that the unloaded contact patterns are influenced by
the local deviation as well. It is clear from this figure that the length of the contact lines
are elongated for UTCA of deviated surfaces since ease-off in the contact region is
flattened.
The predicted unloaded transmission error (UTE) curves are shown in Figure
4.8(a) for the theoretical and deviated surfaces. The corresponding peak-to-peak
amplitude (p-p), first three Fourier harmonics (1st, 2nd, 3rd) and the root-mean-square
(RMS) value of these curves are listed in Table 4.2(a) to show that all components of
UTE are influenced by the local deviation introduced. Both the root-mean-square (RMS)
and p-p amplitudes are reduced with deviated surfaces.
Next, LTCA is performed for the theoretical and the locally deviated surfaces as
before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded
transmission errors (LTE) at this torque value are shown in Figure 4.8(b) for the
theoretical and deviated surfaces. It is noted here that both curves are identical for certain
mesh positions where areas of the local deviation are not in contact while they differ
significantly in certain mesh positions. Table 4.2(b) lists the same LTE amplitudes for
theoretical and deviated surfaces to show that such local deviations also impact the LTE.
Finally, predicted contact pressure distributions are shown in Figures 4.9(a) and (b) for
the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it is
116
Figure 4.7: Predicted unloaded tooth contact pattern for separation value of 6 μm .
Toe
Root
(a) Theoretical surfaces
Toe
Root
(b) Deviated surfaces
117
Figure 4.8: Transmission error (UTE) curves for theoretical and deviated surfaces at (a)
unloaded conditions and (b) loaded conditions at a pinion torque of 200 Nm.
0.5 1.0 1.5 2.0 0
5
10
15
20
25
0 Mesh cycles
Deviated surfaces
0
5
10
15
20
25 (a) Unloaded TE [µrad]
Theoretical surfaces
Mesh cycles
Deviated surfaces
225 0.5 1.0 1.5 2.0 0
230
235
240
245
250
(b) Loaded TE [µrad]
Theoretical surfaces
245
250
255
260
265
270
118
(a) Unloaded Transmission Error in [µrad]
__________________________________________________________ p-p 1st 2nd 3rd RMS
__________________________________________________________ Theoretical surfaces 21.1 8.3 3.7 2.0 9.3 Deviated surfaces 15.0 4.3 4.3 1.4 6.3
__________________________________________________________
(b) Loaded Transmission Error at 200 Nm in [µrad] __________________________________________________________
p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 14.2 7.3 0.3 0.5 7.4 Deviated surfaces 10.2 4.5 1.4 0.4 4.7
__________________________________________________________
Table 4.2: The transmission error amplitudes of theoretical and deviated surfaces.
119
Figure 4.9: Predicted contact pressure distribution for a pinion toque of 200 Nm
for (a) theoretical and (b) deviated surfaces.
0
400
800
1200
Toe
Root
(a) Theoretical surfacesMpa
0
400
800
1200
Toe
Root
(b) Deviated surfacesMpa
120
observed that the edge loading condition experienced by the theoretical surfaces on the
gear root (pinion tip) is reduced significantly for the surfaces with the local deviations
considered since ease-off topography shown in Figure 4.6(d) is considerably flattened
due to surface deviation in the same region.
4.5.2. A Face-hobbed Hypoid Gear Pair with Global Deviations
A face-hobbed hypoid gear set with basic parameters listed in Table 4.3 for its
drive-side contact (concave side of pinion and convex side of gear) is considered as
example for loaded contact analysis of surfaces with global deviations. This gear pair is
also representative of an automotive rear axle gear sets. The measured pinion and gear
deviation surfaces ( pij and g
ij ) from their theoretical geometry after heat treatment and
the lapping process are shown in Figures 4.10(a) and (b), along with their respective
interpolated surfaces on active surface domain shown as pij and g
ij . Figures 4.10(c) and
(d) are the contour plots of the same, showing a maximum 50 µm of error for pinion in
heel-top region and 25 µm for gear in toe-root. The source of deviation here could be
due to manufacturing errors, heat treatment distortion and surface wear caused by lapping
process. In the most common measurement procedure used by the axle manufacturers,
measured coordinates of points on a 5x9 grid are compared to the corresponding
theoretical surface coordinates to determine the measured surface deviations in the
direction normal to the surface. In Figure 4.10, these deviations are shifted for
demonstration purposes so that all are in the positive side.
121
_____________________________________________________________
Parameter Pinion Gear
_____________________________________________________________
Number of teeth 12 41
Hand of Spiral Left Right
Mean spiral angle (deg) 50.0 24.0
Shaft angle (deg) 90
Shaft offset (mm) 45.0
Outer cone distance (mm) 105 130
Generation type Generate Formate
Cutting method FH
_____________________________________________________________
Table 4.3: Basic drive side geometry and working parameters of the example hypoid gear
pair.
122
Figure 4.10: Example global deviation surfaces measured by CMM for the gear and
pinion tooth surfaces, (a) pinion measured deviation, (b) gear measured deviation, (c)
pinion deviation distribution in tooth active region and (d) gear deviation distribution in
tooth active region.
Toe
Root
Measured gear deviation µm (d)
0
5 10
15
20 25
Toe
Root
Measured pinion deviation µm (c)
0
10
20
30
40
(a)
Root
Toe pij
pij
(b)
Root
Toe
gij
gij
123
It should be noted here that a simple weighted average is used to interpolate (or
extrapolate) the deviations pij to p
ij and gij to g
ij at a point ij that is not one of the 45
measurement points. No interpolation is used to estimate surface coordinate and normal
vectors as is the case in previous studies, hence the required accuracy and complexity of
this interpolation is by no means comparable to the case that interpolation is required for
surface coordinates and normal vectors estimation.
Following the proposed method described in the previous section , the theoretical
and updated ease-off surfaces are computed. Figures 4.11(a-d) respectively show (a)
theoretical ease-off topography, (b) updated ease-off topography by only considering
pinion surface deviations, (c) updated ease-off topography by only considering gear
surface deviations and (d) updated ease-off topography by considering both pinion and
gear surface deviations. Figure 4.11(a) shows a localized well defined ease-off
topography as a result of interaction between theoretical pinion and gear surfaces.
Although pinion and gear deviation effects on ease-off, when applied alone, are
considerable as shown in Figure 4.11(b,c), the combination of these deviations alleviates
their adverse effect of alone, resulting in the ease-off topography shown in Figure
4.11(d).
The corresponding predicted unloaded tooth contact patterns of this design are
shown in Figure 4.12(a,b) for a maximum separation value of 6 μm for both cases of
theoretical and deviated tooth surfaces, respectively, indicating that the unloaded contact
patterns are influenced by the deviations given in Figure 4.10 as well. With the deviation
124
Figure 4.11: Ease-off update for the example deviation of Fig. 4.10. (a) Theoretical ease-
off topography, (b) updated ease-off topography only with pinion deviation, (c) updated
ease-off topography only with gear deviation, and (d) updated ease-off topography with
both pinion and gear deviations.
Toe
Root
µm(a)
0
40
120
80
Toe
Root
µm(b)
0
40
120
80
Toe
Root
µm(c)
0
40
120
80
Toe
Root
µm(d)
2
4
6
8
0
40
120
80
125
Figure 4.12: Predicted unloaded tooth contact pattern for separation value of 6 μm .
Toe
Root
(b) Deviated surfaces
Toe
Root
(a) Theoretical surfaces
126
included, unloaded contact pattern slightly shifted toward heel and becomes narrower as
it approaches gear tip.
The predicted unloaded transmission error (UTE) curves are shown in Figure
4.13(a) for theoretical and deviated surfaces against mesh cycles. The corresponding
peak-to-peak amplitude, first three Fourier harmonics and the RMS value of these curves
are listed in Table 4.4(a) to show that the all components of UTE are influenced by the
local deviation introduced. The RMS, peak-to-peak and 1st harmonic amplitudes are
almost doubled with deviations included.
Next, LTCA is performed for the theoretical and the globally deviated surfaces as
before. A pinion torque of 200 Nm was applied in this analysis. The predicted loaded
transmission errors (LTE) at this torque value are shown in Figure 4.13(b) for the
theoretical and globally deviated surfaces. Table 4.4(b) lists the same LTE amplitudes
for theoretical and deviated surfaces to show that such local deviations also impact the
LTE. Finally, predicted contact pressure distribution is also shown in Figure 4.14(a) and
(b) for the theoretical and the deviated surfaces at 200 Nm pinion torque value. Here, it
is observed that the contact pressure distributions for the theoretical and deviated surfaces
are rather close since pinion and gear surface deviations tend to compensate each other
in this particular example set of deviation (Figure 4.10).
127
Figure 4.13: Transmission error curves for theoretical and deviated surfaces; (a) unloaded
conditions and (b) loaded conditions at a pinion torque of 200 Nm.
Theoretical surfaces
0
20
40
60
Deviated surfaces
0
20
40
60
0 0.5 1.0 1.5 2.0
(a) Unloaded TE [µrad]
(b) Loaded TE [µrad]
Theoretical surfaces
305
315
325
335
0 0.5 1.0 1.5 2.0 340
350
360
370
Mesh cycles
Deviated surfaces
128
(a) Unloaded Transmission Error [µrad] __________________________________________________________
p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 36.9 15.1 4.0 1.8 15.7
Deviated surfaces 61.5 28.9 2.1 1.5 29.0
__________________________________________________________
(b) Loaded Transmission Error at 200 Nm [µrad] __________________________________________________________
p-p 1st 2nd 3rd RMS __________________________________________________________ Theoretical surfaces 23.9 12.1 0.5 0.7 12.1
Deviated surfaces 29.8 15.8 2.8 2.3 16.2
_______________________________________________________
Table 4.4: The transmission error amplitudes of theoretical and deviated surfaces.
129
Figure 4.14: Predicted contact pressure distribution for a pinion toque of 200 Nm for (a)
theoretical and (b) deviated surfaces.
Toe
Root
Mpa(a) Theoretical surfaces
0
200
400
600
800
Toe
Root
Mpa(b) Deviated surfaces
0
200
400
600
800
130
4.6. Summary
In this chapter, an accurate and practical method based on ease-off topography
was proposed to perform loaded and unloaded tooth contact analysis of spiral bevel and
hypoid gears having both types of local and global deviations. Manufacturing errors
causing global errors and localized surface deviations were considered to update the
theoretical ease-off to form a new ease-off surface that was used to perform a loaded
tooth contact analysis. Two numerical examples of (i) face-milled hypoid gear set with
local deviations and (ii) face-hobbed hypoid gear set with global deviations measured by
CMM were presented to demonstrate the effectiveness of the proposed methodology as
well as quantifying the effect of such deviations on load distribution and the unloaded
and loaded motion transmission error.
References for Chapter 4
[4.1] Gosselin, C., et al. 1991, "Tooth Contact Analysis of High Conformity Spiral
Bevel Gears." Proceedings of JSME Int. Conf. on Motion and Power
Transmission, Hiroshima, Japan.
[4.2] Zhang, Y., Litvin, F. L., Maryuama, N., Takeda, R., and Sugimoto, M., 1994,
"Computerized Analysis of Meshing and Contact of Gear Real Tooth Surfaces."
116, pp. 677-682.
[4.3] Kin, V., 1992, "Tooth Contact Analysis Based on Inspection." Proceedings of 3rd
World Congress on Gearing, Paris, France.
131
[4.4] Kin, V., 1992, "Computerized Analysis of Gear Meshing Based on Coordinate
Measurement Data." ASME Int. Power Transmission and Gearing Conference,
Scottsdale, AZ.
[4.5] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.
[4.6] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd
ed.), Cambridge University Press, Cambridge.
[4.7] Fan, Q., 2007, "Enhanced Algorithms of Contact Simulation for Hypoid Gear
Drives Produced by Face-Milling and Face-Hobbing Processes." ASME J. Mech.
Des., 129(1), pp. 31-37.
[4.8] Gosselin, C., Nonaka, T., Shiono, Y., Kubo, A., and Tatsuno, T., 1998,
"Identification of the Machine Settings of Real Hypoid Gear Tooth Surfaces."
ASME J. Mech. Des., 120(3), pp. 429-440.
[4.9] Gosselin, C., Jiang, Q., Jenski, K., and Masseth, J., 2005, "Hypoid Gear Lapping
Wear Coefficient and Simulation." AGMA, Technical Paper No. 05FTM09.
[4.10] Gosselin, C., Guertin, T., Remond, D., and Jean, Y., 2000, "Simulation and
Experimental Measurement of the Transmission Error of Real Hypoid Gears
Under Load." ASME J. Mech. Des., 122(1), pp. 109-122.
[4.11] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,
Elsevier Science B. V.
[4.12] Litvin, F. L., and Gutman, Y., 1981, "Methods of synthesis and analysis for
hypoid gear-drives of formate and helixform; Part III-Analysis and optimal
synthesis methods for mismatched gearing and its application for hypoid gears of
formate and helixform." ASME J. Mech. Des., 103, pp. 102-113.
132
[4.13] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME
International Power Transmission and Gearing Conference ASME, San Diego.
[4.14] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute
of Technology.
[4.15] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear
Pairs." In press, Wear, pp.
133
CHAPTER 5
PREDICTION OF MECHANICAL POWER LOSSES OF
HYPOID GEAR PAIRS
5.1. Introduction
Prediction of power losses of automotive drive trains is becoming increasingly
critical to power train designers. This is mainly because the government regulations in
regards to fuel economy and carbon emissions are becoming more stringent. Forecasted
increases in oil prices also add to the motivation to predict and reduce power losses of
drive trains. In rear-wheel drive vehicles, the rear axle-differential unit is one of the
major sources of power losses. The axle efficiency values can be typically as low as 90
to 95% [5.1]. Considering that rear-wheel drive vehicles comprise a significant share of
the global passenger vehicle market, any sizable improvements to the axle efficiency can
have a significant positive impact on environment and energy consumption.
Axle power losses can be divided into two groups. One group of losses is
independent of the torque transmitted. These load independent (spin) power losses are
134
due to viscous bearing losses (including the losses due to pre-load) and gear windage and
oil churning losses [5.2, 5.3]. Such losses are outside the scope of this research. The
other group of losses are induced by friction at bearing and hypoid gear pair locations
under load. These power losses are called load-dependent (mechanical) losses. Focusing
on the hypoid gear pair, mechanical power losses are associated with the relative sliding
and the lubricated contact conditions along the tooth contacts. The shaft off-set, being
the main difference between spiral bevel and hypoid gears, causes increased relative
sliding in hypoid gears and the power losses associated with friction [5.4]. The
motivation of this chapter is to develop a mechanical power loss model of face-milled
and face-hobbed hypoid gears.
Modeling mechanical losses of a gear pair involves (i) computation of surface
geometry parameters and velocities, and the normal load at each contact point from a
tooth contact analysis model as the one proposed in Chapters 2 and 3, (ii) a friction model
to determine the coefficient of friction at each contact points (iii) computing the surface
traction from the distributions of the friction coefficient and normal force, and (iv)
determining the friction torque and the resultant power loss. Published gear efficiency
models differ mostly in the way they determine the friction coefficient. The first group of
models used a constant friction coefficient μ [5.5-5.7] in computing the power losses.
Recognizing the fact that μ is dependent on various contact parameters, including rolling
velocity, sliding to roll ratio, radii of curvature of the contacting surfaces and normal
135
load, all of which vary as gears roll, experiment based µ empirical formulae [5.8-5.11]
were employed by another group of studies [5.8, 5.12-5.16]. However, the applicability
of these models was limited to narrow ranges of the operating temperatures, speed, load,
and surface roughness conditions represented by the empirical formula. The third group
of models predicted the friction coefficient using the elastohydrodynamic lubrication
(EHL) theory [5.17-5.22]. This approach, while physics-based and potentially more
accurate, requires a significant computational effort as several hundreds of EHL analyses
are required to predict the mechanical losses of a gear pair. In order to avoid this
difficulty, Xu et al. [5.23] proposed a methodology to derive a gear contact friction
formula up-front by using the EHL model of Cioc et al [5.24]. Using this EHL model,
they conducted a large parameter study, covering wide ranges of contact and surface
parameters as well as operating conditions representative of gears. The predicted surface
traction data was reduced into a single formula by using linear regression technique.
All of the models cited above were limited to spur or helical gears. Efficiency
models for hypoid gears are very sparse. Approximating the hypoid gear power loss as
the sum of losses from the corresponding spiral bevel and worm gears, Buckingham
[5.25] recommended a power loss equation. Coleman [5.1] proposed a simple closed-
form formula to estimate bevel and hypoid gears efficiency. This heuristic formula used
a constant friction coefficient of 0.05 at every contact point and was a function of the
136
normal load, pressure angle, and pinion and gear mean spiral angles. Simon [5.26]
applied a smooth surface EHL model to simulate hypoid gear lubrication.
The model proposed recently by Xu and Kahraman [5.27] extended their helical
gear efficiency model to hypoid gears. They used a commercially available FE-based
hypoid gear contact model CALYX [5.28] to determine all required contact load and
geometry parameters including curvatures. Employing set of equations developed by
Litvin [5.29] primarily to describe relationships between curvatures of mating surfaces,
they computed sliding and rolling velocities at each contact point along and perpendicular
to the contact line. While this model [5.30] was physics-based and included most of the
key surface, lubricant, geometry and operating parameters, its FE load distribution
computation required significant computational time, making it impractical for design
and parameter sensitivity studies. It relied on the same FE model for its geometry and
curvature information as well. More importantly, the EHL model [5.24] it employed to
derive the friction coefficient formula was not designed for simulating mixed type of
lubrication condition. Therefore, the fidelity of the model Xu and Kahraman [5.30] was
limited to contact conditions with no or limited asperity interactions. However, in most
automotive hypoid gear applications, mixed EHL conditions characterized by excessive
metal-to-metal contacts of the asperity peaks occur commonly. Recently, Li and
Kahraman developed transient mixed or boundary EHL models for line [5.31] and point
[5.32] contacts that can handle any lubricated gear contact conditions ranging from
137
almost dry to full-film EHL. Li et al [5.33] demonstrated the effectiveness of their line
EHL model by applying them to the methodology of Xu et al. [5.23] to predict helical
gear efficiency.
The hypoid gear efficiency model that will be developed in this chapter improves
the methodology of Xu and Kahraman [5.32] by (i) employing the gear geometry defined
in Chapter 2 to derive contact curvature and surface velocity values at each contact point
instead of relying on any particular commercial FE package, (ii) employing the loaded
tooth contact model of proposed in Chapter 3 for computation of the normal load for
minimizing the computational time required for this task, and (iii) by incorporating a new
µ formula derived by using the mixed EHL model of Li and Kahraman [5.32] such that
any degree of asperity interactions can be modeled.
5.2. Hypoid Gear Mechanical Power Loss Model
Figure 5.1 shows the flowchart of the methodology used to compute the power
losses of hypoid gear pairs in the proposed model. It combines the developed ease-off
based hypoid gear contact model of Chapters 2 and 3 and the EHL-based friction
coefficient model of Li and Kahraman [5.33] to compute mechanical efficiency. Blank
dimensions, machine settings, cutter geometry, misalignments, load and speed are input
data for the gear contact model.
138
Figure 5.1: Flowchart of overall hypoid gear efficiency computation methodology.
Instantaneous Mechanical Efficiency
, , ,eq r sR V V f
,
No
Blank dimensions, Machine settings, Cutter geometry,
Misalignments, Load and Speed
Hypoid Gear Contact Model
Friction Coefficient Model
Lubricant property, Temperature and Surface
roughness
sk n
Overall Mechanical Efficiency
Yes
139
At each time step k with pinion roll angle kq q k q ( [1, ]sk n ), there are
multiple potential contact lines between gear and pinion surfaces of adjacent tooth pairs
(depending on the contact ratio and the pinion roll angle). Here, sn is total number of
time steps per one pinion pitch 2a pq N , pN is the number of teeth of the pinion and
a sq q n is pinion roll angle increment. At any incremental position k, there are cln
(integer) number of potential contact lines, each divided into cpn number of contact
segments. With this, a total of cl cpn n n number of potential contact segments are
defined at each increment k . Each line segment k ( [1, ]sk n and [1, ]n ) has a
length kL and carries a constant load per unit length kf . The contact at the same
segment has an equivalent radius of curvature ( )keqR between contacting surfaces as well
as constant sliding velocity ( )ksV and rolling velocity ( )k
rV . Here, ( )krV and ( )k
eqR are
computed in a direction perpendicular to the potential contact line they belong to.
At each line segment k , contacting surfaces are approximated as two cylinders in
combined rolling and sliding. Friction at each segment has two components of sliding and
rolling. The friction due to sliding is because of relative sliding between contacting
surfaces and friction coefficient k is defined as the ratio of the tangential force
produced due to sliding to the normal force applied between contacting surfaces.
140
The friction caused by rolling is because of resistance of contacting surfaces
against rolling over each other [5.23] and empirical rolling friction coefficient of
Goksem [5.34] is used to predict friction force ( )krF . In addition to the lubricant
properties and the surface roughness amplitude S , other contact parameters at each
contact segment, namely kL , kf , ( )ksV , ( )k
rV and ( )keqR must be defined to determine
the friction coefficient k and the rolling loss ( ) ( )k k kr rF V . Distributions of k
and k are used to computed the instantaneous mechanical power loss kP at increment k
due to sliding and rolling that are averaged over k to compute the average gear mesh
power loss as 1sn k
kP P .
5.2.1. Definition of the Sliding and Rolling Velocities
The load kf at each line segment k are computed using the hypoid gear contact
model proposed in Chapter 2 and 3. The computation of the rolling and sliding
velocities, ( )ksV and ( )k
rV , require a kinematic analysis beyond what is provided in
Chapter 2. As shown in Figure 5.2(a) for any point on the ease-off surface with position
vector ijM , the surface velocities of the pinion and gear are defined as:
( )p pp pij ij v a r ( )g gg g
ij ij v a r ( [1, ]i m , [1, ]j n ) (5.1)
141
Figure 5.2: Sliding and rolling velocities and their projection in tangential plane along and normal to the contact line direction.
a p
a g
ad
ijgr
ijpr
gijv
ijpvijn
t
t piju
ijgu
ijM
(a)
(b)
ijpu
ijgu
( ) , ( )p gt tij iju u
( )ptij u
( )gtij u
tt
142
where p gR . The surface velocity hijv of gear h ( ,h p g ) at this contact point has
two components: hijw in the common normal direction ijn and h
iju in the tangential plane
. Noting
p gij ijw w , (5.2)
( ) ( ) 0p g p gij ijij ij ij ij v v n u u n , (5.3)
it can be stated that hiju ( ,h p g ) lies in the tangential plane and has two components,
one component ( )hij tu along the instantaneous potential contact line direction t and
another component ( )hij tu perpendicular to t in direction t as shown in Figure 5.2(a).
With this, sliding velocities at the same contact point along t and t are given,
respectively, as
( ) ( ) ( )p gsij t t tij ijv u u , (5.4a)
( ) ( ) ( )p gsij t t tij ijv u u . (5.4b)
Hence, the total sliding velocity at the same point is
2 2( ) ( )s s s
ij ij t ij tv v v . (5.5)
143
The rolling velocity along t is defined as
( ) ( )( )
2
p gt tij ijr
ij t
u uv
. (5.6)
Sliding ( )ksV and rolling ( )k
rV velocities at the contact segment k ( [1, ]sk n
and [1, ]n ) are found through weighted averaging of the values at four corners of
quadrilateral grid cell on the ease-off surface that contains the middle point of this
segment. These two velocities will be computed for each contact segment and will be
used to determine the local friction coefficient.
5.2.2. Friction Coefficient Model
The surface shear traction consists of the viscous shear within the fluid regions
and the asperity traction in the regions of metal-to-metal contact. Considering a one-
dimensional flow, the sliding viscous shear stress acting on the contact segment k is
defined as
( )( , )
( , )
ksV
x th x t
(5.7)
144
where ( , )h x t is the instantaneous film thickness distribution and is the effective
viscosity. Parameter x defines the coordinate of a point within the contact in the direction
of sliding. Within the asperity contact regions, the shear stress is defined as
( , ) ( , )dx t p x t (5.8)
where ( , )p x t is the instantaneous pressure distribution and d is the dry contact friction
coefficient. With these, the average friction coefficient at a contact line segment k
( [1, ]sk n and [1, ]n ) is given as
1
( , )1
e
ts
x
nNxk
knt
x t dx
N f
(5.9)
here sx and ex are the start and end points of lubricated contact in the direction of
sliding, tN is the number of time steps at which the lubrication analysis performed.
The rolling traction formula of Goksem [5.34] is used here to find the rolling
traction in its corrected form for thermal effects [5.35]
0.658 0.01264.318( ) ( )( )
keqk
rpv
GU W RF
. (5.10)
145
Here ( )k keq eqW f E R , 0( ) ( )k k
r eq eqU V E R , pv eqG E , pv is the
pressure-viscosity coefficient for the lubricant used, is thermal correction factor [5.31],
eqE equivalent module of elasticity of two contacting bodies and 0 is viscosity at
ambient conditions. With this, rolling power loss at each contact segment k is
( ) ( )k k kr rF V .
5.2.3. Derivation of a Friction Coefficient Formula
In Eq. (5.7) to (5.9), the distribution of the surface shear ( ( , )x t ), normal pressure
( ( , )p x t ) and film thickness ( ( , )h x t ) of every contact segment k must be predicted by
using a mixed elastohydrodynamic lubrication (MEHL) model. Considering that there
are n contact line segments at each rotational increment k and there are a total of sn
increments, a total of sn n number of MEHL analyses are required to find the
distribution of the friction coefficient and the resultant gear mesh power loss. This would
require several hours of CPU time, hampering the usefulness of the model.
In accordance with the methodology first proposed by Xu et al [5.23], an upfront
detailed parametric study of MEHL conditions of hypoid gear contacts will be performed
here by including all key contact parameters. These parameters and their selected ranges
146
and incremental values are listed in Table 5.1. These ranges of Hertzian pressure hp
(0.5 to 2.5 GPa), radius of curvature eqR (5 to 40 mm), rV (1 to 20 m/s), slide-to-roll
ratio SR (0 to 1), and viscosity of a typical axle fluid (75W90) within a temperature
range (25 to 100 C ) cover most of the contact conditions present in automotive hypoid
gear pairs. In addition, a typical measured roughness profile from a gear surface with
0.5μmqR is considered and different RMS roughness profiles with different
amplitudes are obtained from this baseline profile by multiplying it by a constant. The
mechanical properties of 75W90 gear oil are listed in Table 5.2 is used.
A total of 31,500 contact conditions as a result of all combinations of the
parameter values listed in Table 5.1 were analyzed by using the model of Li et al [5.31]
and a regression analysis to the µ values predicted for each or these conditions to obtain
the following friction coefficient formula:
For 1 :
4 5
8 96 7 10
ln ln20 1 2 3
ln ln
exp
,
c
c
a G a S
a a Sa G a a Heq
a H a SR a G a SR
P S
(5.11a)
147
_____________________________________________________________
Lubricant 75W90 gear oil
_____________________________________________________________
Hertzian Pressure hp (GPa) 0.5, 1, 1.5, 2, 2.5
Equivalent Radius of Curvature Req (mm) 5, 20, 40
Rolling Velocity Vr (m/s) 1, 5, 10, 15, 20
Slide to Roll Ratio SR 0.025, 0.05, 0.1, 0.25, 0.5, 0.75, 1
Inlet Lubricant Temperature oilT (°C) 25, 50, 75, 100
Surface 1 roughness RMS 1qR (µm) 0.1, 0.35, 0.6, 0.85, 1
Surface 2 roughness RMS 2qR (µm) 0.1, 0.35, 0.6, 0.85, 1
_____________________________________________________________
Table 5.1: Parametric design for the development of the friction coefficient formula.
148
________________________________________________________________________
Temperature Pressure-Viscosity Coefficient Dynamic Viscosity Density
oilT (°C) 1 (GPa-1) 0 (Pa.s) 0 (kg/m3)
________________________________________________________________________
25 18.0 0.1626 844.30
50 13.9 0.0499 829.30
75 11.4 0.0208 814.30
100 9.7 0.0106 799.30
________________________________________________________________________
Table 5.2: Basic parameters of the 75W90 gear oil used in this study.
149
For 1 3 :
2 3 4 5
7 8 9 106
ln ln ln0 1
ln ln
exp ( )
,c
b U b G b H b H
b b H b H b Sb H
b b SR SR
U G H
(5.11b)
For 3 :
3 4 5 6 7
8 9 10 11
ln ln ln ln0 1 2
ln
exp ( )
.
cc G c H c H c c S
c G c H c G c SReq
c c G c SR U SR
U H S
(5.11c)
Here is the lambda ratio (ratio of the smooth condition minimum film thickness to the
RMS surface roughness value). These formulae are dependent on a number of
dimensionless parameters: , SR , U , G , h eqH p E , two roughness parameters
c c eqS S R and eq eq eqS S R (1 2
2 2c q qS R R and
1 2 1 2( )eq q q q qS R R R R ). Here,
coefficients 0a to 10a , 0b to 10b and 0c to 11c are constants representative of the
lubricant considered. For the 75W90 gear oil, these parameters are listed in Table 5.3.
150
___________________________________________________________________
i ia ib ic
___________________________________________________________________
1 -0.62538 16.1512 5.0435
2 -51.411 -0.60156 -0.00576307
3 -0.0371532 -0.0466305 248228480
4 2.06770 -0.348239 -0.396002
5 -0.031750 -0.358514 -0.405254
6 -0.046276 22.568 35.618
7 -7.754e-5 -11.2295 -0.109382
8 1.18821 -6.49095 -0.112364
9 0.170432 -1.31986 -0.00016056
10 -0.136892 -0.881279 -0.0690238
11 -7.8882 -4398.1 -0.00038810
12 0.052821
___________________________________________________________________
Table 5.3: Values of the coefficients in Eq. 5.11.
151
5.2.4. Computation of the Mechanical Power Loss of the Hypoid Gear Pair
With the sliding friction coefficient ( k ) and rolling loss ( k ) computed in the
previous section for every contact segment k , the mechanical power loss due to the
contact of k ( [1, ]sk n , [1, ]n ) is computed as
( )k k k k k ksP f L V , (5.12)
With this, the instantaneous gear pair power loss at a given rotational increment k
becomes
1
lnk kP P
(5.13a)
and the average mechanical power loss of the hypoid gear pair is found as
1
1 snk
ks
P Pn
. (5.13b)
Having input pinion torque pT and speed p and power loss kP loss at each rotational
increment k , the instantaneous mechanical gear pair efficiency is defined as
1k
kp p
PE
T
, (5.14a)
152
and overall the average gear mesh efficiency is estimated as
1p p
PE
T
. (5.14b)
5.3. Numerical Example
Two sets of face-hobbed hypoid gear designs defined in Table 5.4 are borrowed
from automotive applications to study the influence of shaft offset along with working
conditions that affect on hypoid gear efficiency. Designs A and B have two levels of
offset ratios of 0.07a ad D and 0.14, respectively, while the other geometry and
operating parameters (such as number of teeth, shaft angle and gear pitch diameter aD )
of the two designs are kept the same or very close to each other to isolate the offset
influences from those of the other parameters ( ad is pinion shaft offset). For this
purpose, similar ease-off topographies as shown in Figure 5.3 are developed for the two
gear sets. Despite all geometrical differences of two designs, they have similar contact
pressure distribution as shown in Figure 5.4 for pinion torque of 500pT Nm. This is
mainly due to matching ease-off topographies of two design sets through machine setting
changes.
153
_____________________________________________________________________
Set A Set B
Parameter Pinion Gear Pinion Gear
_____________________________________________________________________
Number of teeth 12 41 12 41
Hand of Spiral Left Right Left Right
Mean spiral angle (deg) 40.0 31.3 47.0 29.6
Shaft angle (deg) 90 90
Shaft offset (mm) 15.0 30.0
Gear pitch diameter (mm) 220.0 220.0
Generation type Generate Formate Generate Formate
Cutting Method FH FH
_____________________________________________________________________
Table 5.4: Basic drive side geometry and working parameters of the examples hypoid
gear pairs.
154
Figure 5.3: Ease-off topography of (a) Design A with / 0.07a ad D and (b) Design B
with / 0.14a ad D .
0
5
(a)
Toe Root
µm
0
50
100
(b)
Toe Root
µm
0
50
100
155
Figure 5.4: Maximum contact pressure distribution of (a) Design A with / 0.07a ad D
and (b) Design B with / 0.14a ad D for 500 NmpT .
Mpa
200
400
600
800
1000(b)
Toe Root
200
400
600
800
1000
Mpa (a)
Toe Root
156
At 1500p rpm, rV and sV of the two gear sets are compared in Figures 5.5
and 5.6, respectively, for these two designs. It is seen that design B with higher offset
ratio ( 0.14a ad D ) has higher rV and sV compared to design A. Likewise, the slide-
to-roll ratio (SR) values over the tooth face of gear set B is higher than that of gear set A,
as shown in Figure 5.7. Additionally, the eqR and distributions shown in Figures 5.8
and 5.9 reveal slight differences in eqR values observed between the two gear designs,
the values for the gear set B are larger than those for the design set A. With all the
parameters required by Eq. (5.11) computed, the distributions along the tooth faces can
be determined, as shown in Figure 5.10. Although gear set B has higher ratio
compared to design A, its higher SR levels causes higher values, resulting in higher
sliding friction force.
Next, the influences of operating and surface conditions, including load pT ,
speed p , oil temperature oilT and surface roughness S , on mechanical power loss P
and mechanical efficiency E of these gear pairs are quantified. In Figures 5.11(a1, a2)
and (b1, b2), variation of P and E with p and pT are shown. It is observed in Figure
5.11(a2, b2) that the E decreases slightly with increasing p when the speed is low.
This trend is reversed at higher speed ranges as the slope between E and p becomes
positive. Similar conclusions can be drawn for the influence of pT .
157
Figure 5.5: Rolling velocity distribution of (a) Design A with / 0.07a ad D and (b)
Design B with / 0.14a ad D at 1500 rpmp .
(a)
Toe Root
1.0 1.5 2.00.5
2.5 3.0 3.5
(b)
Toe Root
1.0
2.52.0
1.5
3.53.0 4.54.0
158
Figure 5.6: Sliding velocity distribution of (a) Design A with / 0.07a ad D
and (b) Design B with / 0.14a ad D at 1500 rpmp .
(a)
Toe Root
1.0 1.5
2.0
(b)
Toe Root
1.5 2.02.5
3.0
159
Figure 5.7: Slide-to-roll ratio distribution of (a) Design A with / 0.07a ad D and (b)
Design B with / 0.14a ad D at 1500 rpmp .
0
0
0
0
0
0
0
0
0.10.20.30.40.50.60.70.8(a)
Toe Root
0.10.20.30.40.50.60.70.8(b)
Toe Root
160
Figure 5.8: Equivalent radius of curvature distribution of (a) Design A with
/ 0.07a ad D and (b) Design B with / 0.14a ad D .
(a)
Toe Root
35 40 4550
(b)
Toe Root
35 40 45
50 55
161
Figure 5.9: distribution of (a) Design A with / 0.07a ad D and (b) Design B with
/ 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT and 1 2 0.8 mS S .
0.10
0.15
0.20
0.25
0.30(a)
Toe Root
0
0
0
0
0.10
0.15
0.20
0.25
0.30(b)
Toe Root
162
Figure 5.10: Friction coefficient distribution of (a) Design A with / 0.07a ad D and
(b) Design B with / 0.14a ad D at 1500 rpmp , 500 NmpT , 90 CoilT and
1 2 0.8 mS S .
0.01
0.02
0.03
0.04
0.05
0.06 (b)
Toe Root
0.01
0.02
0.03
0.04
0.05
0.06(a)
Toe Root
163
Figure 5.11: Power loss and efficiency of Design A (a1, a2) and Design B (b1, b2) at
90 CoilT and 1 2 0.8 mS S .
1
10
100
1,000
10,000
0 500 1000 1500 2000 2500 3000
100 Nm 500 Nm 1000 Nm
p
(a1)
[w]P
1
10
100
1,000
10,000
0 500 1000 1500 2000 2500 3000
100 Nm 500 Nm 1000 Nm
(b1)
[w]P
p Continued
164
Figure 5.11 continued
96.0
96.5
97.0
97.5
98.0
98.5
99.0
99.5
0 500 1000 1500 2000 2500 3000
100 Nm 500 Nm 1000 Nm
[%]E
p
(a2)
96.0
96.5
97.0
97.5
98.0
98.5
99.0
99.5
0 500 1000 1500 2000 2500 3000
100 Nm 500 Nm 1000 Nm
(b2)
[%]E
p
Continued
165
In the lower speed range, where the power loss is relatively small, higher pT
results in higher efficiency since the increase in power loss due to the heavier load
applied is smaller than the corresponding input power increase, while the opposite is true
in the medium and high speed ranges. Between the two designs, gear set A with a
smaller shaft off-set is consistently more efficient (on average, gear set A has about 1.5%
higher efficiency than gear set B).
Figure 5.12 illustrates the effects of surface finish as well as the lubricant
temperature on the gear mesh efficiency. It is found that reduction in surface roughness
amplitude effectively increases the mechanical efficiency of both gear pairs. As for the
operating temperature, within the low roughness range, where a substantial amount of
contact area is separated by the hydrodynamic fluid film and the viscous shear dominates,
an increase in temperature will reduces the sliding friction and the power loss P
accordingly through the reduction in lubricant viscosity. In the medium and high
roughness ranges, when the contact zone might experience severe asperity contacts,
lower lubricant viscosity at high temperature results in thinner fluid film and larger
boundary friction force, reducing the gear mesh efficiency.
166
Figure 5.12: Efficiency of (a) Design A with / 0.07a ad D and (b) design B with
/ 0.14a ad D for different surface finish and oil temperatures at 1500 rpmp and
500 NmpT .
96.0
96.5
97.0
97.5
98.0
98.5
99.0
99.5
0.1 0.3 0.5 0.7 0.9 1.1 1.3
50° C 75° C 100° C
[%]Ef
S
oilT
(b)
96.0
96.5
97.0
97.5
98.0
98.5
99.0
99.5
0.1 0.3 0.5 0.7 0.9 1.1 1.3
50° C 75° C 100° C
[%]Ef
S
(a)
oilT
167
5.4. Conclusion
In this chapter, a new spiral bevel and hypoid gear mechanical efficiency model is
proposed for both face-milling and face-hobbing cutting methods. The proposed
efficiency model combines the computationally efficient contact model developed in
chapters 2 and 3 with an EHL based friction coefficient model developed by Li et al
[5.33] to estimate sliding friction loss and employed a conventionally developed
formulation for rolling loss. The developed model improved the methodology of Xu and
Kahraman [5.30] by (i) employing the gear geometry defined in Chapter 2 to derive
contact curvature and surface velocity values at each contact point instead of relying on
any particular commercial FE package, (ii) employing the loaded tooth contact model
proposed in Chapter 3 for computation of the normal load for minimizing the
computational time required for this task, and (iii) by incorporating a new µ formula
derived by using the mixed EHL model of Li et al [5.33] such that any degree of asperity
interactions can be modeled.
Limited numerical results show that the shaft off-set is critical to the efficiency of
the gear set as lower off-sets resulting in significant increases in mechanical efficiency.
Likewise, reduction in surface roughness was also shown to reduce power losses of
hypoid gear pairs.
168
References for Chapter 5
[5.1] Coleman, W., 1975, "Computing efficiency for bevel and hypoid gears." Machine
Design, 47, pp. 64-65.
[5.2] Seetharaman, S., and Kahraman, A., 2009, "Load-Independent Spin Power Losses
of a Spur Gear Pair: Model Formulation." ASME J. of Tribology, 131(2), pp.
022201.
[5.3] Seetharaman, S., Kahraman, A., Moorhead, M. D., and Petry-Johnson, T. T.,
2009, "Oil Churning Power Losses of a Gear Pair: Experiments and Model
Validation." ASME J. of Tribology, 131(2), pp. 022202.
[5.4] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute
of Technology.
[5.5] Denny, C. M., 1998, "Mesh Friction in Gearing." AGMA, Technical Paper No.
98FTM2.
[5.6] Pedrero, J. I., 1999, "Determination of The Efficiency of Cylindrical Gear Sets."
4th World Congress on Gearing and Power Transmission, Paris, France.
[5.7] Michlin, Y., and Myunster, V., 2002, "Determination of Power Losses in Gear
Transmissions with Rolling and Sliding Friction Incorporated." J. Mechanism and
Machine Theory, 37, pp. 167.
[5.8] Benedict, G. H., and Kelly, B. W., 1960, "Instantaneous Coefficients of Gear
Tooth Friction." ASLE.
[5.9] O’Donoghue, J. P., and Cameron, A., 1966, "Friction and Temperature in Rolling
Sliding Contacts." ASLE Transactions, 9, pp. 186-194.
169
[5.10] Drozdov, Y. N., and Gavrikov, Y. A., 1967, "Friction and Scoring Under The
Conditions of Simultaneous Rolling and Sliding of Bodies." Wear, pp. 291-302.
[5.11] Misharin, Y. A., 1958, "Influence of The Friction Condition on The Magnitude of
The Friction Coefficient in The Case of Rollers with Sliding." Int. Conference On
Gearing, London, UK.
[5.12] Heingartner, P., and Mba, D., 2003, "Determining Power Losses in The Helical
Gear Mesh; Case Study." Proceeding of DETC3, Chicago, Illinois, USA.
[5.13] Anderson, N. E., and Loewenthal, S. H., 1986, "Efficiency of Nonstandard and
High Contact Ratio Involute Spur Gears." J. Mechanisms, Transmissions and
Automation in Design, 108, pp. 119-126.
[5.14] Anderson, N. E., and Loewenthal, S. H., 1982, "Design of Spur Gears for
Improved Efficiency." J. Mechanical Design, 104, pp. 767-774.
[5.15] Barnes, J. P., 1997, "Non-Dimensional Characterization of Gear Geometry, Mesh
Loss and Windage." AGMA, Technical Paper No. 97FTM11.
[5.16] Vaishya, M., and Houser, D. R., 1999, "Modeling and Measurement of Sliding
Friction for Gear Analysis." AGMA, Technical Paper No. 99FTMS1.
[5.17] Martin, K. F., 1981, "The Efficiency of Involute Spur Gears." ASME J. Mech.
Des., 103, pp. 160-169.
[5.18] Dowson, D., and Higginson, G. R., 1964, "A Theory of Involute Gear
Lubrication." Institute of Petroleum, Gear Lubrication, Elsevier, London, UK.
170
[5.19] Simon, V., 1981, "Load Capacity and Efficiency of Spur Gears in Regard to
Thermo-End Lubrication." International Symposium on Gearing and Power
Transmissions, Tokyo, Japan.
[5.20] Simon, V., 2009, "Influence of machine tool setting parameters on EHD
lubrication in hypoid gears." J. Mechanism and Machine Theory, 44, pp. 923-
937.
[5.21] Larsson, R., 1997, "Transient non-Newtonian elastohydrodynamic lubrication
analysis of an involute spur gear." Wear, 207, pp. 67-73.
[5.22] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,
Elsevier Science B. V.
[5.23] Xu, H., Kahraman, A., Anderson, N. E., and Maddock, D. G., 2007, "Prediction
of Mechanical Efficiency of Parallel-Axis Gear Pairs." ASME J. Mech. Des.,
129(1), pp. 58-68.
[5.24] Cioc, C., Cioc, S., Kahraman, A., and Keith, T., 2002, "A Non-Newtonian,
Thermal EHL Model of Contacts with Rough Surfaces." Tribology Transactions,
45, pp. 556-562
[5.25] Buckingham, E., 1949, Analytical Mechanics of Gears, McGraw-Hill.
[5.26] Simon, V., 1981, "Elastohydrodynamic Lubrication of Hypoid Gears." ASME J.
Mech. Des., 103, pp. 195-203.
[5.27] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction
Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University,
Columbus, Ohio.
171
[5.28] Vijayakar, S., 2004, Calyx Hypoid Gear Model, User Manual, Advanced
Numerical Solution Inc., Hilliard, Ohio.
[5.29] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd
ed.), Cambridge University Press, Cambridge.
[5.30] Xu, H., and Kahraman, A., 2007, "Prediction of Friction-Related Power Losses of
Hypoid Gear Pairs." Proceedings of the Institution of Mechanical Engineers, Part
K: J. Multi-body Dynamics, 221(3), pp. 387-400.
[5.31] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric
Integrated Control Volume Discretization." Tribology International, Hiroshima,
Japan.
[5.32] Li, S., and Kahraman, A., 2009, "An Asymmetric Integrated Control Volume
Method for Transient Mixed Elastohydrodynamic Lubrication Analysis of
Heavily Loaded Point Contact Problems." Tribology International.
[5.33] Li, S., Vaidyanathan, A., Harianto, J., and Kahraman, A., 2009, "Influence od
Design Parameters and Micro-Geometry on Mechanical Power Losses of Helical
Gear Pairs." JSME International, Hiroshima, Japan.
[5.34] Goksem, P. G., and Hargreaves, R. A., 1978, "The effect of viscous shear heating
on both film thickness and rolling traction in an EHL line contact." J. Lubrication
Technology, 100, pp. 346–352.
[5.35] Wu, S., and Cheng, H., S., 1991, "A Friction Model of Partial-EHL Contacts and
its Application to Power Loss in Spur Gears." Tribology Transactions, 34(3), pp.
398-407.
172
CHAPTER 6
CONCLUSION AND RECOMMENDATIONS FOR FUTURE WORK
6.1. Thesis Summary
A computationally efficient load distribution model was proposed for both face-
milled and face-hobbed hypoid gears produced by Formate and generate processes.
Tooth surfaces were defined directly from the cutter parameters and machine settings.
This study defined and utilized a new surface of roll angle as an essential tool to simplify
the task of locating instantaneous contact lines of any general type of gearing in the
projection plane. First, the position vector and normal to one of the mating surfaces of
contacting members were computed, and the action surface and the surface of roll angle
were introduced by applying equation of meshing between any general axis arrangement.
Once the surface of roll angle was constructed, the instantaneous contact lines location
and orientation were computed through a novel approach inspired by analogy to parallel
axes gears. Gear surfaces were assumed conjugate only in computation of contact line
173
locations and orientation of the proposed approach. However it was shown that surface
of roll angle is very insensitive to any practical modifications of contacting surfaces, and
hence, real instantaneous contact lines practically remain identical to their conjugate
counterparts. For any other steps of contact analysis, theoretically generated surfaces
based on machine settings were employed.
Rayleigh-Ritz based shell models of teeth of the gear and pinion were developed
to define the tooth compliances due to bending and shear effects efficiently in a semi-
analytical manner. Base rotation and contact deformation effects were also included in
the compliance formulations. With this, loaded contact patterns and transmission error of
both face-milled and face-hobbed spiral bevel and hypoid gears were computed by
enforcing the compatibility and equilibrium conditions of the gear mesh. The proposed
model requires significantly less computational effort than finite elements (FE) based
models, making its use possible for extensive parameter sensitivity and design
optimization studies. Comparisons to the predictions of a FE hypoid gear contact model
were also provided to demonstrate the accuracy of the model under various load and
misalignment conditions.
Two applications of the proposed model were also introduced. First application
combined the proposed model with a newly introduced approach of modifying the ease-
off topography to investigate the effect of errors occurred in manufacturing and heat
treatment of gear surfaces or surface deviations due to wear or lapping. Manufacturing
errors typically cause real (measured) spiral bevel and hypoid gear surfaces to deviate
174
from the theoretical ones globally. Tooth surface wear patterns accumulated through the
life span of the gear set are typically local deviations that are aggravated especially in
case of edge contact conditions. An accurate and practical methodology based on the
developed ease-off topography approach was proposed in this study to perform loaded
tooth contact analysis of spiral bevel and hypoid gears having both types of local and
global deviations. Manufacturing errors and localized surface wear deviations were used
to update the theoretical ease-off and surface of roll angle to form a new ease-off surface
that was used to perform a loaded tooth contact analysis. Two numerical examples of
face-milled and face-hobbed hypoid gear sets with local and global deviated surfaces,
respectively, were analyzed to demonstrate the effectiveness of the proposed
methodology as well as quantifying the effect of such deviations on load distribution and
the loaded motion transmission error.
As another vital application of the proposed model, a hypoid gear mechanical
efficiency model was developed next for both face-milling and face-hobbing cutting
methods. The proposed efficiency model combined the computationally efficient contact
model and a mixed EHL based surface traction model to predict friction power losses.
The contact area, pressure distribution and rolling and sliding velocities were determined
employing the developed loaded tooth contact model. The EHL traction model
considered specific ranges of the key contact parameters, including Hertzian pressure,
contact radii, surface speeds, lubricant temperature and surface roughness amplitude of
hypoid type of gears, covering wide range of lubrication conditions from full film to
175
boundary regimes. The efficiency model for hypoid gear was applied to two face-hobbing
examples with similar overall dimensions, but different off-sets, to investigate the effects
of several working and design parameters including offset, load, speed, surface roughness
and lubricant temperature on mechanical efficiency.
6.2. Conclusion and Contributions
Computation of the contact pressure distributions is essential to every hypoid gear
analysis intended to predict required functional parameters of the hypoid gear pair,
including the transmission error, contact stresses, root bending stresses, fatigue life and
mechanical power losses. The hypoid gear literature lacked a model to compute the load
distribution accurately and efficiently without resorting to computationally demanding
FE methods. The main potential reasons for that was the absence of a general and
reliable formulation to define the geometry of FH and FM hypoid tooth surfaces from
cutter parameters, machine motions and settings. This void, combined with the
numerical difficulties in matching the tooth surfaces using the conventional methods and
lack of a semi-analytical tooth compliance formulation for hypoid gears, has hampered in
design, analysis and optimization of hypoid gears. This research study fills some of this
void.
The model proposed in this study to simulate the contacts of FM and FH hypoid
gear pairs under both unladed and loaded condition provides major enhancements to the
current state of hypoid analysis. Specifically:
176
(i) Methodology that simulates the FM and FH processes to define surface
geometries of hypoid gears including the coordinates, normal vectors and radii of
curvatures is accurate and computationally efficient. It does not employ
simplifying assumptions such as conjugacy of the tooth surfaces.
(ii) The method using ease-off topography to compute the unladed contact conditions
is novel. It is superior to the conventional method in various aspects, including its
numerical stability and computational efficiency. This method and its surface of
roll angle concept is also general such that it can be applied to the other gear
types.
(iii) Application of mounting errors and inclusion of global and local deviations are
rather straightforward with the proposed model while these have typically been
difficult or impossible tasks when the conventional methods were used.
(iv) The efficiency model that combines the loaded tooth contact model proposed in
this study with an accurate and computationally efficient friction model is
superior to any published hypoid gear efficiency model as it includes all key
geometry, surface, load and lubricant parameters as well as operating conditions.
Its ability to handle variety of lubrication conditions ranging from almost dry
contact to full film EHL makes this model applicable to wide ranges of hypoid
and spiral bevel gear applications from automotive and aerospace systems.
177
6.3. Recommendations for Future Work
The following items can be considered as the potential future studies to improve
or add to the model presented in this study.
(i) Extensive parameter studies can be performed to determine cost-effective ways of
improving hypoid gear efficiency and define efficiency guidelines to be used in
design of hypoid gears.
(ii) The loaded TCA formulation can be improved by enhancing the base rotation
formulation to account for all geometric complexities.
(iii) A shell model for non-circular cylinders can be developed to compute a more
accurate compliance matrix for a generated gear and the face-hobbed generation
method that have epicycloids tooth traces.
(iv) The proposed model lends itself to optimization studies to refine machine settings
for desired transmission error amplitudes and loaded tooth contact patterns.
(v) The way local deviation and mounting errors are included in the proposed model
is general such that this model can be used to predict surface wear as well as to
simulate the lapping process that is commonly used in manufacturing of FH
hypoid gears.
178
BIBLIOGRAPHY
[1] Anderson, N. E., and Loewenthal, S. H., 1982, "Design of Spur Gears for
Improved Efficiency." J. Mechanical Design, 104, pp. 767-774.
[2] 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.
[3] Barnes, J. P., 1997, "Non-Dimensional Characterization of Gear Geometry, Mesh
Loss and Windage." AGMA, Technical Paper No. 97FTM11.
[4] Baxter, M. L., 1964, "An Application of Kinematics and Vector Analysis to the
Design of a Bevel Gear Grinder." ASME Mechanism Conference, Lafayette, IN.
[5] Baxter, M. L., and Spear, G. M., 1961, "Effects of Misalignment on Tooth Action
of Bevel and Hypoid Gears." ASME Design Conference, Detroit, MI.
[6] Benedict, G. H., and Kelly, B. W., 1960, "Instantaneous Coefficients of Gear
Tooth Friction." ASLE.
[7] Borner, J., Kurz, N., and Joachim, F. (2002). "Effective Analysis of Gears with
the Program LVR (Stiffness Method)."
[8] Buckingham, E., 1949, Analytical Mechanics of Gears, McGraw-Hill.
[9] 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
[10] Coleman, W., 1963, Design of Bevel Gears, The Gleason Works.
179
[11] Coleman, W., 1975, "Analysis of Mounting Deflections on Bevel and Hypoid
Gears." SAE 750152.
[12] Coleman, W., 1975, "Computing efficiency for bevel and hypoid gears." Machine
Design, 47, pp. 64-65.
[13] Coleman, W., 1975, "Effect of Mounting Displacements on Bevel and Hypoid
Gear Tooth Strength." SAE 750151.
[14] Conry, T. F., and Seireg, A., 1973, "A Mathematical Programming Technique for
the Evaluation of Load Distribution and Optimal Modification for Gear Systems."
ASME J. of Industrial Engineering.
[15] Denny, C. M., 1998, "Mesh Friction in Gearing." AGMA, Technical Paper No.
98FTM2.
[16] Dooner, D. B., 2002, "On the Three Laws of Gearing." ASME J. Mech. Des., 124,
pp. 733-744.
[17] Dooner, D. B., and Seireg, A., 1995, The Kinematic Geometry of Gearing: A
Concurrent Engineering Approach, John Wiley & Sons Inc.
[18] 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.
[19] Dowson, D., and Higginson, G. R., 1964, "A Theory of Involute Gear
Lubrication." Institute of Petroleum, Gear Lubrication, Elsevier, London, UK.
[20] Dudley, D. W., 1969, The Evolution of the Gear Art, American Gear
Manufacturers Association, Washington, D. C.
180
[21] Dyson, A., 1969, A General Theory of the Kinematics and Geometry of Gears in
Three Dimensions, Clarendon Press, Oxford.
[22] 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.
[23] 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.
[24] 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.
[25] 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.
[26] 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.
[27] Fong, Z. H., and Tsay, C.-B., 1991, "A Study on the Tooth Geometry and Cutting
Machine Mechanisms of Spiral Bevel Gears." ASME J. Mech. Des., 113, pp. 346-
351.
[28] 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.
181
[29] 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.
[30] 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.
[31] 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.
[32] Gosselin, C., Jiang, Q., Jenski, K., and Masseth, J., 2005, "Hypoid Gear Lapping
Wear Coefficient and Simulation." AGMA, Technical Paper No. 05FTM09.
[33] 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.
[34] Heingartner, P., and Mba, D., 2003, "Determining Power Losses in The Helical
Gear Mesh; Case Study." Proceeding of DETC3, Chicago, Illinois, USA.
[5] Krenzer, T. J., 1965, TCA Formulas and Calculation procedures, The Gleason
Works.
[36] Krenzer, T. J., 1981, "Tooth Contact Analysis of Spiral Bevel and Hypoid Gears
under Load." SAE Earthmoving Industry Conference, Peoria, IL.
[37] Krenzer, T. J., 1981, Understanding Tooth Contact Analysis, The Gleason Works.
182
[38] Krenzer, T. J., 1990, "Face Milling or Face Hobbing." AGMA, Technical Paper
No. 90FTM13.
[39] Krenzer, T. J., 2007, The Bevel Gears, http://www.lulu.com/content/1243519.
[40] Kin, V., 1992, "Computerized Analysis of Gear Meshing Based on Coordinate
Measurement Data." ASME Int. Power Transmission and Gearing Conference,
Scottsdale, AZ.
[41] Kin, V., 1992, "Tooth Contact Analysis Based on Inspection." Proceedings of 3rd
World Congress on Gearing, Paris, France.
[42] Li, S., and Kahraman, A., 2009, "A Mixed EHL Model with Asymmetric
Integrated Control Volume Discretization." Tribology International, Hiroshima,
Japan.
[43] 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.
[44] 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.
[45] Litvin, F. L. (1989). "Theory of Gearing." NASA RP-1212.
[46] Litvin, F. L. (2000). "Development of Gear Technology and Theory of Gearing."
NASA RP1406.
[47] Litvin, F. L., and Fuentes, A., 2004, Gear Geometry and Applied Theory (2nd
ed.), Cambridge University Press, Cambridge.
183
[48] 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.
[49] 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.
[50] 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.
[51] 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.
[52] 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.
[53] Litvin, F. L., Zhang, Y., Lundy, M., and Heine, C., 1988, "Determination of
Settings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel
Gears." J. Mechanism, Transmission and Automation in Design, 110, pp. 495-
500.
184
[54] Liu, F., 2004, "Face Gear Design and Compliance Analysis," M.Sc. Thesis, The
Ohio State University, Columbus, Ohio.
[55] Martin, K. F., 1981, "The Efficiency of Involute Spur Gears." ASME J. Mech.
Des., 103, pp. 160-169.
[56] 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.
[57] 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.
[58] 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.
[59] O’Donoghue, J. P., and Cameron, A., 1966, "Friction and Temperature in Rolling
Sliding Contacts." ASLE Transactions, 9, pp. 186-194.
[60] Park, D., and Kahraman, A., 2008, "A Surface Wear Model for Hypoid Gear
Pairs." In press, Wear.
[61] Pedrero, J. I., 1999, "Determination of The Efficiency of Cylindrical Gear Sets."
4th World Congress on Gearing and Power Transmission, Paris, France.
185
[62] 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.
[63] 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.
[64] Shtipelman, B. A., 1979, Design and manufacture of hypoid gears, John Wiley &
Sons, Inc.
[65] Simon, V., 1981, "Elastohydrodynamic Lubrication of Hypoid Gears." ASME J.
Mech. Des., 103, pp. 195-203.
[66] 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.
[67] Simon, V., 1996, "Tooth Contact Analysis of Mismatched Hypoid Gears." ASME
International Power Transmission and Gearing Conference ASME, San Diego.
[68] Simon, V., 2000, "FEM stress analysis in hypoid gears." J. Mechanism and
Machine Theory, 35(9), pp. 1197-1220.
[69] Simon, V., 2000, "Load Distribution in Hypoid Gears." ASME J. Mech. Des.,
122(44), pp. 529-535.
[70] Simon, V., 2001, "Optimal Machine Tool Setting for Hypoid Gears Improving
Load Distribution." ASME J. Mech. Des., 123(4), pp. 577-582.
186
[71] Simon, V., 2009, "Influence of machine tool setting parameters on EHD
lubrication in hypoid gears." J. Mechanism and Machine Theory, 44, pp. 923-
937.
[72] Smith, R. E., 1984, "What Single Flank Measurement Can Do For You." AGMA,
Technical Paper No. 84FTM2.
[73] Smith, R. E., 1987, "The Relationship of Measured Gear Noise to Measured Gear
Transmission Errors." AGMA, Technical Paper No. 87FTM6.
[74] Stadtfeld, H., J., 1993, Handbook of Bevel and Hypoid Gears, Rochester Institute
of Technology.
[75] Stadtfeld, H., J., 1995, Gleason Advanced Bevel Gear Technology, The Gleason
Works.
[76] Stadtfeld, H. J. (2000). "The Basics of Gleason Face Hobbing." The Gleason
Works.
[77] Stadtfeld, H. J., and Gaiser, U., 2000, "The Ultimate Motion Graph." ASME J.
Mech. Des., 122(3), pp. 317-322.
[78] Stegemiller, 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.
[79] 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.
[80] Stewart, A. A., and Wildhaber, E., 1926, "Design, Production and Application of
the Hypoid Rear-Axle Gear." J. SAE, 18, pp. 575-580.
187
[81] 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.
[82] Timoshenko, S. P., and Woinowsky-Krieger, S., 1959, Theory of Plates and
Shells, McGraw-Hill Book Company Inc.
[83] 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.
[84] 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.
[85] 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.
[86] 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.
[87] Vaishya, M., and Houser, D. R., 1999, "Modeling and Measurement of Sliding
Friction for Gear Analysis." AGMA, Technical Paper No. 99FTMS1.
[88] 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.
188
[89] 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.
[90] Vijayakar, S. M., 2003, Calyx User Manual, Advanced Numerical Solution Inc.,
Hilliard, Ohio.
[91] Vijayakar, S. M., 2004, Calyx Hypoid Gear Model, User Manual, Advanced
Numerical Solution Inc., Hilliard, Ohio.
[92] 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.
[93] Vogel, O., 2006, "Gear-Tooth-Flank and Gear-Tooth-Contact Analysis for
Hypoid Gears," Ph.D. Dissertation, Technical University of Dresden, Germany.
[94] 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.
[95] Wang, X. C., and Ghosh, S. K., 1994, Advanced Theories of Hypoid Gears,
Elsevier Science B. V.
[96] Weber, C., 1949, "The Deformation of Loaded Gears and the Effect on Their
Load Carrying Capacity (Part I)." D.S.I.R., London.
[97] 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.
189
[98] Wildhaber, E., 1946, Basic Relationship of Hypoid Gears, McGraw-Hill.
[99] Wildhaber, E., 1956, "Surface Curvature." Product Engineering, pp. 184-191.
[100] Wu, D., and Luo, J., 1992, A Geometric Theory of Conjugate Tooth Surfaces,
World Scientific, River Edge, NJ.
[101] 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.
[102] Xu, H., 2005, "Development of a Generalized Mechanical Efficiency Prediction
Methodology for Gear Pairs," Ph.D. Dissertation, The Ohio State University,
Columbus, Ohio.
[103] 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.
[104] 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.
[105] Xu, H., Kahraman, A., and Houser, D. R., 2005 "A Model to Predict Friction
Losses of Hypoid Gears." AGMA, Technical Paper No. 0FTM06.
[106] 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.
190
[107] 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.
[108] Yau, H., Busby, H. R., and Houser, D. R., 1991, "A Rayleigh-Ritz Approach for
Modeling the Bending and Shear Deflections of Gear Teeth." JSME International
Conference on Motion and Powertransmission.
[109] 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.
[110] Zhang, Y., and Wu, Z., 2007, "Geometry of Tooth Profile and Fillet of Face-
Hobbed Spiral Bevel Gears." IDETC/CIE 2007, Las Vegas, Nevada, USA.