LIFE MODEL FOR ROLLING CONTACT, APPLIED TO THE OPTIMIZATION OF A
TRIPODE CONSTANT VELOCITY JOINT
By
Eduardo Raul Mondragon-Parra

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILSOPHY
Mechanical Engineering
2010

ABSTRACT
LIFE MODEL FOR ROLLING CONTACT, APPLIED TO THE OPTIMIZATION OF A
TRIPODE CONSTANT VELOCITY JOINT
By
Eduardo R. Mondragon-Parra
A Constant Velocity Joint (CVJ) is a mechanical device capable of transmitting
torque and motion between two rotating shafts that are misaligned. Most front wheel
drive vehicles have torque and motion transmitted to the driven wheels through a set of
inboard CVJ’s (transmission side) and a set of outboard CVJ’s (wheel side). One of the
most common types of inboard joints is the Tripode Joint, which is composed of a
housing, a spider and three sets of roller assemblies that connect the spider to the
housing and allow for stroking motion and rotation of the joint at an angle. Wear fatiguelife in a Tripode Joint is defined by how many rotation cycles a Joint can complete
before experiencing spalling or flaking of the internal surfaces subject to contact forces.
Similarities and differences between a Tripode Joint and roller bearings are discussed.
Internal contacts are treated as non-Hertzian and the state of stress is determined
based on the kinematics of the joint, the geometry of the interacting components and
the way internal forces are distributed when a torque is applied.

A rolling contact wear fatigue-life model for Tripode Joints, that relates the critical
shear stress in the contacts to the number of cycles a joint can perform before spalling,
is proposed and compared to actual test data. Residual stresses were considered to
obtain the stress-life exponent of the Tripode equation.

Optimization of the geometric form of the contact surfaces in the spider is
performed, aimed to minimize the peak contact load and to minimize the load
differences between adjacent needle rollers. Reduction of the peak contact load allows
an increment in durability of the Tripode Joint. The optimized profile in the contact
surfaces of the spider is compared against traditional geometric forms, such as perfect
circular and elliptical. The proposed model has a direct application in the auto-industry.

Copyright by
EDUARDO R. MONDRAGON-PARRA
2010

Para Carolina, mi hija amada.
(To Carolina, my beloved daughter)

“… Of making many books there is no end, and much study wearies the body. Now all
has been heard; here is the conclusion of the matter: Fear God and keep his
commandments, for this is the whole duty of man.”
Ecclesiastes 12:12-13

v

ACKNOWLEDGMENTS

Conducting the research that led to this document was a unique experience. I
would like to thank the persons that contributed to this work in different manners and at
different stages.

To Dr. Patrick Kwon, my advisor, who risked to take me as his off-campus student
and guided me through the entire research. The time he dedicated reviewing my work,
his wisdom and valuable feedback made this possible. To my committee members Dr.
Ronald Averill, Dr. Dashin Liu and Dr. K. N. Subramanian, for their patience, for serving
on my committee and for their teaching inside and outside of the classroom.

To Mr. Jerry Eaton, Mr. William Skvarla and Mr. Robin Milavec, for their support,
encouragement and for granting me the flexibility and resources to combine work in
industry with my studies. To Dr. Keith Kozlowski, who led me by example when I started
to work in industry, encouraged me and planted the seeds of curiosity about contact
stress and fatigue-life.

To my parents, sister and friends, who made me laugh and laughed with me,
fished with me, brewed beer with me, allowed me to share their family time and prayed
with me and for me to help me be on track and keep things in perspective.

To JESUS, thank you for being my LORD and my HIDING PLACE.

vi

TABLE OF CONTENTS

LIST OF TABLES…………………………………………………………………………….…ix
LIST OF FIGURES……………………………………………………………………………..xi
LIST OF SYMBOLS………………………………………………………………………….xviii
CHAPTER 1
INTRODUCTION………………………………………………………………………………..1
1.1 Background……………….…………………………………………………………..1
1.2 Fixed center constant velocity joints……….……………………………………….3
1.3 Stroking constant velocity joints………………………………….…………………5
1.4 Rolling bearings and tripode joints……………….…………………………………6
1.5 Benefits of a life model………………………….………………………………….10
1.6 Joint size and packaging………………………………………………….………..10
1.7 Project description………………….……………………………………………….11
CHAPTER 2
KINEMATIC ANALYSIS OF A STROKING TRIPODE JOINT……………………………12
2.1 Introduction…………….…………………………………………………………….12
2.2 Assumptions………….…………………………………………………………......14
2.3 Definitions……………………………….…………………………………………...14
2.4 Spherical roller positions……………………………………….…………………..18
2.5 Numerical examples of spherical roller positions and displacements………...23
2.6 Needle roller positions and displacements……………………………….………31
2.7 Critical joint angle…………………………………………………………………...47
2.8 Critical joint angle, when ball tilting is neglected and ψ = ϕ…………….………48
CHAPTER 3
QUASI-STATIC ANALYSIS OF A STROKING TRIPODE JOINT………………………..56
3.1 Introduction……………………………………….………………………………….56
3.2 Normal forces at spider trunnions………………………………………………....57
3.3 Normal forces at ball bores………………………………………….……………..59
3.4 Numerical examples and GAF experimental results…………………………....64
CHAPTER 4
LOAD DISTRIBUTION ON A TRUNNION……………………………………………….…71
4.1 Introduction……………………………………………………………………….….71
4.2 Normal load distribution on needles (axial direction)………………………..…..72
4.3 Normal load distribution on needles (radial direction)……………………….….92
4.4 Numerical examples of load distribution on trunnions…………………………..99
4.5 Making trunnion load distribution a function of joint angle……………….……105

vii

CHAPTER 5
STATE OF STRESS AT THE ROLLER-TRUNNION CONTACT……………………….109
5.1 Introduction………………………………………………………………………...109
5.2 Hertzian formulation for line (cylindrical) contacts…………………………..…110
5.3 Non-Hertzian formulation for line (cylindrical) contacts…………………..……114
5.4 Non-Hertzian formulation for contact of rollers with crowning……….……….122
5.5 Sub-surface stress distribution……………………………………………..…….135
CHAPTER 6
FATIGUE-LIFE THEORIES…………………………………………………………………146
6.1 Introduction………………………………….……………………………………..146
6.2 Weibull equation…………………………………………………………………...146
6.3 Lundberg-Palmgren equation…………………………………………….………148
6.4 Ioannides-Harris equation……………………………………………………..….149
6.5 Zaretsky equation………………………………………………………………....150
6.6 Tripode equation……………………………………………………………….….151
6.7 Testing and prototypes to validate the Tripode model………………………...153
6.8 Load and stress conditions per test schedule……………………………….…157
6.9 Test results………………………………………………………………………....178
CHAPTER 7
TRUNNION FORM OPTIMIZATION……………………………………………………….191
7.1 Introduction……………………………………………….………………………..191
7.2 Elliptical vs. circular trunnion………………………….………………………….191
7.3 Optimized vs. elliptical trunnion………………………….………………………194
CHAPTER 8
CONCLUSIONS…………………………………………………………………………...…200
APPENDICES……………………………………………………………………………...…206
REFERENCES…………………………………………………………………….…………214

viii

LIST OF TABLES

Table 3.1

Properties of CVJ greases. (*) stands for “Extreme Pressure”………...67

Table 4.1

Load distribution (N) at different proportions of yield torque………….100

Table 5.1

Convergence of maximum pressure and deflection…………………...119

Table 5.2

Convergence of maximum pressure and deflection…………………...121

Table 5.3

Convergence of maximum pressure and deflection…………………...126

Table 5.4

Convergence of maximum pressure and deflection………………...…129

Table 6.1

Summary of wear tests conditions……………………………………....155

Table 6.2

Needle-trunnion load as a function of joint angle at 272Nm………….163

Table 6.3

Needle-trunnion load as a function of applied torque………………….163

Table 6.4

Inputs to calculate non-Hertzian contacts…………...………………….165

Table 6.5

Key characteristics of non-Hertzian contacts……….………………….167

Table 6.6

Key characteristics of Hertzian contacts………………………...……...168

Table 6.7

Sub-surface Critical shear stresses at the center of the contact……..169

Table 6.8

Sub-surface Critical shear stresses at the point of
maximum pressure………………………………………………………..170

Table 6.9

Ratio of yield shear strength to critical shear stress…………………...172

Table 6.10

Elastic stress and residual stress components at z = 0.047mm……..177

Table 6.11

Elastic stress and residual stress components
at z = 0.042mm…………………………………………………………….179

Table 6.12

Test results summary……………………………………………………..181

Table 6.13

Inputs to best-fit life equation…………………………………………….185

Table 6.14

Life prediction summary…………………………………………………..186

ix

Table 6.15

Life prediction summary, residual stresses included………………..…188

Table 7.1

Load distribution, circular vs. elliptical trunnion (et = 1.003)………….193

Table 7.2

Life comparison, circular vs. elliptical trunnion at 816Nm and 7°…….194

Table 7.3

Load distribution, circular vs. elliptical vs. optimized trunnion………..198

Table 7.4

Life comparison, different trunnion forms at 816Nm and 7°………….199

Table B.1

Residual stresses……………………………………………………….…208

Table C.1

Chemical composition………………………………………………….…209

Table C.2

Hardness…………………………………………………………………...210

x

LIST OF FIGURES

Figure 1.1a

Steering motion (schematic aerial view)……………..………………….…2

Figure 1.1b

Suspension motion (schematic front view)…………...……………………2

Figure 1.2

Universal Joint……….…………………………………………………….…3

Figure 1.3

Double Hooke Joint………….………………………………………..……..4

Figure 1.4

Fixed Center CVJ…………….…………………………………..…………..4

Figure 1.5

CGJ (left) and DOJ (right)…….…………………………………...……..….5

Figure 1.6

Tripode Joint……….………………………………………………….………6

Figure 1.7

Axial motion and support in Roller Bearing and Tripode Joint……….….7

Figure 1.8

Outer member support in Roller Bearing and Tripode Joint…………..…8

Figure 1.9

Variation of load distribution in a tripode joint…………….…………….…8

Figure 1.10

Inner member shape in Roller Bearing and Tripode Joint…………….…9

Figure 2.1

Exploded view of tripode joint………………………………………….….12

Figure 2.2

Housing Reference Frame…………………………………………………16

Figure 2.3

Spider Reference Frame…………………………………………………...16

Figure 2.4

Fixed and mobile coordinate frames……………………………………...17

Figure 2.5

Coordinates and orientation of spider shaft……………………………...21

Figure 2.6

Fluctuation of joint angle in a revolution……………………………….…24

Figure 2.7

Deviation from nominal angle in a revolution…………………………….25

Figure 2.8

Orbiting path of spider on spider plane……………………………….…..26

Figure 2.9

Fluctuation of eccentricity within a revolution……………………………27

Figure 2.10

Difference between input and output angular displacements……….…27

xi

Figure 2.11

Orbiting path of spider on housing plane…………………………….…..28

Figure 2.12

Variations of the axial displacement of the spider center………………29

Figure 2.13

Displacement of the spherical roller center along ball bore axis………30

Figure 2.14

Radial position of the spherical roller center, from spider center……...30

Figure 2.15

Radial displacement of the spherical roller center,
along trunnion axis………………………………………………………….31

Figure 2.16

Trunnion, local cylindrical coordinate system……………………………32

Figure 2.17

Secondary-rotating coordinate systems………………………………….33

Figure 2.18

Angle δ, per Eq. (2.30)………………………………………………….….36

Figure 2.19

Angle δ, per Eq. (2.31)……………………………………………….…….36

Figure 2.20

Partial rolling of ball due to tilting…………………………………….……37

Figure 2.21

Actual arch displaced at ball outer diameter……………………………..40

Figure 2.22

Difference in arch displacement without and with ball tilting effect……40

Figure 2.23

Graphical representation of angle ε.......................................................41

Figure 2.24

Angle ε, per expression (2.39)……………………………………….……42

Figure 2.25

Angle ε', per expression (2.40)…………………………………………....43

Figure 2.26

Angle η, per expression (2.41)…………………………………………....44

Figure 2.27

Angular displacements and positions in trunnion-rollers set…………...45

Figure 2.28

Angular position of needle j = 1, per expression (2.44)………………...46

Figure 2.29

Deviation from nominal angle in a revolution at θ o = 23°, 6°………….48

Figure 2.30

Maximum proportional deviation from nominal angle vs. θ o …………..49

Figure 2.31

Difference in arch displacement without and with ball tilting, at 6°……51

Figure 2.32

Relative error of ignoring ball tilting, at 6°………………………………..51

xii

Figure 2.33

Relative error of arch-length displacement, ignoring tilting, vs. θ o …...52

Figure 2.34

Needle roller displacement, complete vs. simplified solution………..…54

Figure 3.1

Forces on spider plane……………………………………………………..60

Figure 3.2

Rotating coordinate systems and joint orientation when ψ i = 0..……..62

Figure 3.3

Graphical representation of angle ε and forces q i and pi ………….....62

Figure 3.4

Magnitude of p1 with (solid line) and without joint angle variation…....64

Figure 3.5

Magnitude of q1 with (solid line) and without joint angle variation...…..66

Figure 3.6

Tangential force w1, at 600Nm, 7.5°, grease A…………………...…….68

Figure 3.7

Generated axial force, at 600Nm, 7.5°, grease A……………………….68

Figure 3.8

Generated axial force, at 300N, experimental vs. theoretical values….69

Figure 3.9

Generated axial force, at 600N, experimental vs. theoretical values….70

Figure 4.1

Case 1: Spherical roller fully supported, no offset…………………...….73

Figure 4.2a

Case 2: Positive offset (Λ)………………………………………………....74

Figure 4.2b

Case 3: Negative offset (–Λ)……………………………………………....74

Figure 4.3

Clearance between needle, roller-retainer and spacer ring……………76

Figure 4.4a

Needle axial displacement at 6° joint angle……………………………...77

Figure 4.4b

Needle axial displacement at 23° joint angle…………………………….78

Figure 4.5

Offset between ball equator and needles symmetry plane…………….79

Figure 4.6

Effective contact length between needle roller and ball, at 23°………..81

Figure 4.7

Load per unit length at upper end of needle, at 19.5°…………………..90

Figure 4.8

Load per unit length at lower end of needle, at 19.5°…………………..91

Figure 4.9

Spider Geometry……………………………………………………………92

xiii

Figure 4.10

Cross section of spherical roller……………………………………….…..95

Figure 4.11

Schematic view of loads…………………………………………………...96

Figure 4.12

Load distribution at 25% and 50% of yield torque……………………..101

Figure 4.13

Load distribution at 50% of yield torque, perfect vs. lobed trunnion…102

Figure 4.14

Load distribution at 50% of yield torque,
circular vs. elliptical trunnion……………………………………………..104

Figure 4.15

Example of initial wear in a trunnion…………………………………….105

Figure 4.16

Position of roller within load profile………………………………………106

Figure 4.17

Roller position with respect to symmetry plane of loading profile……107

Figure 4.18

Load profile at joint angle of 6±………………………………………..…108

Figure 5.1

Ideal cylindrical contact…………………………………………………...111

Figure 5.2

Subsurface stresses along the plane of symmetry of the contact……113

Figure 5.3

Two bodies in contact……………………………………………………..115

Figure 5.4

Pressure distribution in a sphere-plane contact……………………..…119

Figure 5.5

Pressure distribution in a cylinder contact………………………………122

Figure 5.6

Needle roller dimensions…………………………………………………123

Figure 5.7

Crowning of 10.81mm needle roller……………………………………..124

Figure 5.8a

Pressure distribution of needle roller with crowning…………………...125

Figure 5.8b

Lateral view of pressure distribution of needle roller with crowning….125

Figure 5.9

View of needle skewing…………………………………………………..126

Figure 5.10a

Pressure distribution of needle roller with crowning and skewing……127

Figure 5.10b

Lateral view of pressure distribution of needle roller
with crowning and skewing……………………………………………….128

Figure 5.11

Undercut on trunnion surface………………………………………….…129

xiv

Figure 5.12a

Pressure distribution of skewed needle roller
with undercut on trunnion………………………………………………..130

Figure 5.12b

Lateral view of pressure distribution of skewed needle roller
with undercut on trunnion………………………………………………..130

Figure 5.13

Case of asymmetrical load, positive offset……………………………..131

Figure 5.14a

Pressure distribution of needle roller with crowning and a load
offset Λi of –1mm, no skewing…………………………………………133

Figure 5.14b

Lateral view of pressure distribution of needle roller
with crowning and a load offset Λi of –1mm, no skewing……………134

Figure 5.15a

Pressure distribution of needle roller with crowning and a load
offset Λi of –1mm, skewing of 4º……………………………………..…134

Figure 5.15b

Lateral view of pressure distribution of needle roller
with crowning and a load offset Λi of –1mm, skewing of 4º ….....…..135

Figure 5.16a

Stress distribution along the depth of the contact, σ x component…..141

Figure 5.16b

Stress distribution along the depth of the contact, σ y component…..141

Figure 5.16c

Stress distribution along the depth of the contact, σ z component…..142

Figure 5.16d

Stress distribution along the depth of the contact, τ max ……………...142

Figure 5.17a

Stress distribution along the depth of the contact,
σ x component represented with a solid line,
σ y component represented by a dashed line………………………….144

Figure 5.17b

Stress distribution along the depth of the contact, σ z component…..144

Figure 5.17c

Stress distribution along the depth of the contact, τ max ……………...145

Figure 6.1

CVJ wear test bench………………………………………………..........154

Figure 6.2

Example of roundness check…………………………………………….156

Figure 6.3

Needle displacement……………………………………………………...158

Figure 6.4

Load offset………………………………………………………………….159

xv

Figure 6.5a

Variation of load on ball bore, housing plane at 272Nm……………....160

Figure 6.5b

Loads on ball bore, housing plane……………………………………....161

Figure 6.6

Plane formed by housing axis and spider axis,
zero plane for rotation…………………………………………………….161

Figure 6.7a

Virtual load of needle to trunnion at spider plane, at 272Nm…………162

Figure 6.7b

Virtual load of needle to trunnion at spider plane………………………162

Figure 6.8

Pressure distribution, input torque of 272Nm…………………………..166

Figure 6.9

Pressure distribution, input torque of 466Nm…………………………..166

Figure 6.10

Pressure distribution, input torque of 816Nm…………………………..166

Figure 6.11

Max Shear and Octahedral shear at center of contact………………..168

Figure 6.12

Max Shear and Octahedral shear at point of maximum pressure……169

Figure 6.13

Max Shear and Octahedral shear at point of maximum pressure……174

Figure 6.14

Shear stress τ zxr at a depth of 0.047mm………………………………178

Figure 6.15

Parts A-1 and A-2………………………………………………………….181

Figure 6.16

Parts B-3 and B-1………………………………………………………….182

Figure 6.17

Part C-2…………………………………………………………………….182

Figure 6.18a

Part B-2 (units are microns)……………………………..……………….183

Figure 6.18b

Part B-2 (units are microns)…………………………………………..….184

Figure 6.19

Actual versus theoretical life……………………………………………..186

Figure 6.20

Residual stresses in Spider C-4…………………………………………187

Figure 6.21

Actual versus theoretical life after superposing residual stresses……188

Figure 6.22

Hardness gradient in trunnion……………………………………………190

Figure 7.1

Load distribution on a trunnion…………………………………………..195

Figure 7.2

Pareto Front………………………………………………………………..197

xvi

Figure A.1

Surface texture before testing……………………………………………206

Figure A.2

Surface texture after testing……………………………………………...206

Figure D.1

Roundness with filter 0-15UPR…………………………………………..209

Figure D.2

Roundness with filter 0-50UPR…………………………………………..210

Figure D.3

Roundness with filter 0-150UPR…………………………………………211

Figure D.4

Roundness with filter 0-500UPR…………………………………………212

xvii

LIST OF SYMBOLS

BCD

=

ball circle diameter

a

=

bcd / 2, radial distance between housing axis and ball bore axis

=

Orientation matrix of spider reference frame X iS - YiS - Z iS , with
respect to housing reference frame X iH - YiH - Z iH

b

=

spider shaft length, from spider center to spherical joint center

co

=

initial stroke position of spider center with respect to housing plane

d

=

needle roller diameter

h, g, k

=

coordinates of spherical joint center (end of spider axle bar)

e

=

eccentricity of spider center to housing axis

ex , ey

=

coordinates of spider center on the frame X * - Y * - Z *

fi

=

radial displacements of ball center, measured along trunnions

mi

=

spherical roller (ball) inner diameter

mo

=

spherical roller (ball) outer diameter

n

=

number of needle rollers per trunnion

ri

=

radial distance between spider axis and ball bore centerlines,
measured along trunnions

si

=

arch length displaced on ball outer diameter due to ball travel
along ball bore

t

=

trunnion diameter

X-Y-Z

=

fixed reference frame

i

A

X ' -Y ' - Z ' =

mobile spider reference frame

xviii

X * - Y * - Z * = mobile spider offset reference frame
x, y, z

=

coordinates of spider center with respect to housing reference frame

X i - Yi - Z i =

coordinates of spherical roller centers with respect to fixed
reference frame X-Y-Z

X i ' - Yi ' -0 =

coordinates of spherical roller centers with respect to spider
reference frame, X ' - Y ' - Z '

X i * - Yi * -0 =

coordinates of spherical roller centers with respect to spider
offset reference frame, X * - Y * - Z *

X*

=

X '+e x

Y*

=

Y '+e y

X iH - YiH - Z iH = Housing, secondary mobile reference frame
X iS - YiS - Z iS = Spider, secondary mobile reference frame
XS , XH
i
i

=

Vectors that define orientation of spider and housing rays

α, β, θ

=

direction angles of spider axis, with respect to fixed reference
frame

κ i , λi , µ i =

direction angles of spider rays, with respect to housing mobile
reference frame, X iH - YiH - Z iH

U i - Φ i - Vi =

trunnion local reference frame, cylindrical coordinate system

θ

=

joint angle, angle between spider and housing axes

θo

=

joint angle, angle between spider and housing axes, when ϕ = 0

θ cr

=

critical joint angle

θ Λ
cr

=

critical joint angle to assumption of linear load distribution

xix

φ ij

=

Needle roller angular position, with respect to trunnion local
reference frame U i - Φ i -Vi

δi

=

angle between spider ray and housing plane X iH - YiH , measured
on plane formed by vectors X S and Z H
i
i

εi

=

angle between vectors YiS and YiH

ηi

=

angular displacement of contact point between spherical roller
and ball bore, with respect to trunnion local reference frame

ϕ

=

angular displacement of housing shaft (input angle)

ψ

=

angular displacement of spider (output angle)

ψ1

=

ψ 2 − 2π / 3 = ψ 3 + 2π / 3

ϕ1

=

ϕ 2 − 2π / 3 = ϕ 3 + 2π / 3

i

=

Trunnion index, 1, 2, 3

j

=

Needle roller index, 1, 2, 3, … , n

ξ

=

Angular displacement of O*O’ with respect to X *

pij

=

concentrated normal force from needle j to trunnion i

pi

=

force normal to trunnion axis, on spider plane

qi

=

force normal to ball bore axis, on housing plane

wi

=

tangential force at ball bore, parallel to ball bore axis

tϕ

=

input torque

tψ

=

output torque

GAF

=

generated axial force, measured on the housing axis

µb

=

sliding friction coefficient between ball and ball bore

xx

µr

=

rolling friction coefficient between ball and ball bore

ct

=

effective contact length of needle on trunnion

cb

=

effective contact length of needle on ball

c bu

=

effective upper contact length of needle on ball

c bd

=

effective lower contact length of needle on ball

Λi

=

distance between plane passing thru equator of spherical roller
and transversal symmetry plane of needle rollers, offset

Λo

=

initial offset, when joint angle is zero

Θi

=

axial displacement of needle along trunnion

Θo

=

clearance between needle, roller retainer and ring spacer

c buo

=

initial effective upper contact length of needle on ball

c bdo

=

initial effective lower contact length of needle on ball

ptmij

=

mean load per unit of length on trunnion

pbmij

=

mean load per unit of length on ball

ptuij

=

load per unit of length on trunnion at point TU

ptdij

=

load per unit of length on trunnion at point TD

pbuij

=

load per unit of length on ball at point BU

pbdij

=

load per unit of length on ball at point BD

espider

=

height of spider shoulder from spider axis

c spacer

=

thickness of spacer

xxi

bneedle

=

total needle length

Rb

=

radius to the centroid of the radial cross section of the ball

hb

=

distance from the centroid of the cross section of the ball to the
extreme fiber on the concave side

c bw

=

ball height (lateral face to lateral face distance)

αb

=

semi-arch of ball width

y bA

=

radius to intersection point of circular and rectangular segments
of the radial cross section of the ball

y b1

=

radius to centroid of circular segment

y b2

=

radius to centroid of rectangular segment

ab1

=

area of circular segment of ball cross section

ab 2

=

area of rectangular segment of ball cross section

ab

=

area of ball radial cross section

I b1x

=

second moment of area of circular segment of ball cross section

I b2 x

=

second moment of area of rectangular segment of ball cross section

I bxx

=

second moment of area of ball radial cross section

φu

=

angular position on inner diameter of ball when φ i 1 = 0

φv

=

angular position of needle roller v when φ i 1 = 0

u

=

auxiliary needle spacing index, 2, 3, 4, … , n

v

=

auxiliary needle roller index, 2, 3, 4, … , n

pCuv

=

influence coefficient for ring (ball) deflection due to a roller load

qCu

=

influence coefficient for ring (ball) deflection due applied load q i

xxii

δb

=

radial displacement of ball center with respect to trunnion center

δu

=

relative radial approach of ball to trunnion at angle φu

υu

=

combined radial deformation of ball, for needle roller at angle φu

Ωu

=

radial deviation of actual trunnion surface with respect to trunnion
basic circle at angle φu

ru

=

radial distance to trunnion surface at angle φu

EY

=

modulus of elasticity of steel = 207GPa

k pt

=

material constant for contact between needle and trunnion

k pb

=

material constant for contact between needle and ball

kp

=

combined material constant for needle contacts

pv

=

radial load transmitted by needle roller v, when φ i 1 = 0

hw

=

order of dominant harmonica causing distortion on trunnion

Ωo

=

profile or roundness tolerance of trunnion

nr

=

constant for elastic properties roller bearings = 10 / 9

=

radial deflection of ball at angle φu due to a load pv

q ∆u

=

radial deflection of ball at angle φu due to a load q i

Γo

=

constant that takes its value based on position of first roller

Γ

=

upper limit to index v

Γv

=

0.5 for φv = 0. Otherwise, Γv = 1

χ ij

=

angle between the plane passing thru the center of trunnion i and

pv

∆u

the center of roller j, and plane X ' - Y '

xxiii

Ψ

=

roller to trunnion continuous load profile function

a

=

semi-width of rectangular contact element (for section 5 only)

b

=

semi-length of rectangular contact element (for section 5 only)

bh

=

semi-width of line contact area, perpendicular to contact line

lh

=

length of line contact area, parallel to contact line, Hertz

σh

=

surface pressure in a cylindrical contact, Hertz

σ h max

=

maximum surface pressure in a cylindrical contact, Hertz

σ hx

=

stress component in the x-direction in the region of contact, Hertz

σ hy

=

stress component in the y-direction in the region of contact, Hertz

σ hz

=

stress component in the z-direction in the region of contact, Hertz

τh

=

maximum principal shear stress in the region of contact, Hertz

τ h max

=

peak value of the maximum principal shear stress, Hertz

σ x ,σ y ,σ z

=

normal stress components in the region of contact, numeric

τ xy ,τ yz ,τ xz

=

shear stress components in the region of contact, numeric

σ Ι ,σ ΙΙ ,σ ΙΙΙ

=

stress invariants

σ p1, σ p2 , σ p3

=

principal stresses in the region of contact, numeric

σ p1 = σ p max

=

max principal stress in the region of contact, numeric

σ p3 = σ p min

=

min principal stress in the region of contact, numeric

τ max

=

maximum principal shear stress in the region of contact, numeric

EY

=

modulus of elasticity, for steel = 207GPa

νs

=

Poisson’s ratio, for steel = 0.3

xxiv

κ

=

material constant

X-Y-Z

=

reference frame at the contact region (for section 5 only)

x, y, z

=

coordinates of a point in the contact region (for section 5 only)

α ij

=

displacement of roller j with respect to trunnion i, caused by deformation

v 1,v 2

=

displacements of points in or near the contact area,
parallel to the direction of the applied force to the contact

z1,z2

=

initial separation of points in or near the contact area,
parallel to the direction of the applied force to the contact

zcw

=

crowning, deviation from nominal cylinder along its axis

zsk

=

equivalent crowning cause by needle skewing

zξ

=

equivalent crowning due to asymmetrical load distribution

zt

=

total equivalent crowning

rsk

=

radial clearance caused by needle skewing

sn

=

uniform pressure on element n

n, m

=

dummy variables, 1, 2, 3, … , r

r

=

number of rectangles in the mesh used to calculate contact area

fmn

=

influence coefficients

ζ

=

skewing angle of needle roller with respect to trunnion

ξ ij

=

slope of equivalent contact deformation on needle j of trunnion i
due to slope of asymmetric load distribution

ε

=

Weibull slope or Weibull modulus

ϑ

=

Exponent, experimentally obtained

c

=

critical shear-stress life exponent

xxv

σw

=

fracture strength of a specimen

σ wβ

=

characteristic fracture strength of a specimen

S

=

specimen survival to fracture strength σ w

V

=

stressed volume in a specimen

N

=

number of cycles to fracture

L

=

life of a component, in cycles

τ cr

=

critical shear stress

τo

=

maximum octahedral shear stress

τu

=

Fatigue limit

Ci

=

constants, for i = 1, 2, 3, …

zcr

=

depth of critical shear stress

nh

=

life exponent for Hertzian contacts

l eff

=

effective, non-Hertzian contact length

φ eff

=

effective angular displacement of a needle roller within a
revolution at a given joint angle

τ eff

=

effective shear stress (varies depending on the
static failure theory used)

YS

=

uni-axial yield strength

TS

=

uni-axial tensile strength

Hv

=

Vickers Hardness

HRc

=

Rockwell-C Hardness

SF

=

Ratio of shear yield strength to critical shear stress

xxvi

1. INTRODUCTION

1.1 Background
This work focuses on the wear life prediction of the Tripode (TP) type Constant
Velocity Joint (CVJ). Internally, the TP CVJ is composed of rolling elements which
function similar to those of roller bearings. Rolling bearing technology, as we know it
today, began to develop in the nineteenth century. In 1881, H. Hertz published his
contact stress analysis beginning the analytical study of bearings. The first paper
discussing life bearing prediction was published in 1924 by A. Palmgren [1]. Bearing
technology, contact stress and life prediction theories continued to develop in the
second half of the twentieth century and are still developing.

A Constant Velocity Joint (CVJ) is a mechanical device capable of transmitting
torque and motion between two rotating shafts that are not aligned. One of the first
applications of constant velocity joints in the U.S. auto industry was in the 1960’s, in the
halfshafts of front wheel drive vehicles [2]. A halfshaft is an assembly of two constant
velocity joints and an axle bar. Its purpose is to transmit motion and torque from the
transmission to the powered wheels of a vehicle. One of the most common halfshaft
configurations includes a fixed center constant velocity joint and a stroking constant
velocity joint. Such configuration allows the halfshaft to articulate and change its length
as a result of steering and suspension motions in a vehicle, as shown schematically in
Figures 1.1a and 1.1b

1

Figure 1.1a. Steering motion (schematic aerial view)

Figure 1.1b. Suspension motion (schematic front view)

2

Even though constant velocity joints have been around for almost one hundred
years little has been done to develop specific analytical models to predict its wear life.
It’s the purpose of this work to develop a wear life model for a specific type of constant
velocity joint, the Tripode joint.

1.2 Fixed center constant velocity joints
The first predecessor of the constant velocity joint was the Universal Joint or
Cardan Joint, named after Geronimo Cardano who was the first to describe it in the mid
sixteenth century [3]. The Universal joint allows transmission of torque and motion
between two shafts that are not aligned but with variation of torque and speed. A
schematic view of the Universal Joint is shown in Figure 1.2.

Figure 1.2. Universal Joint

Almost 100 years later Robert Hooke had the idea of combining two universal
joints to eliminate the non-uniformity, which was the first mechanism with true constant
velocity properties [3].

3

Figure 1.3. Double Hooke Joint

One of the first ball-type constant velocity joints, as we know it today, was
patented by Alfred Rzeppa in 1934 [4]. This joint is composed on an inner race member,
an outer race member, balls connecting the two races and a cage to keep the balls in
the constant velocity plane. The constant velocity transmission is achieved by keeping
the balls on a bisecting plane, which is the same principle that is used to transmit
constant velocity using bevel gears.

Figure 1.4. Fixed Center CVJ

4

1.3 Stroking constant velocity joints
There are several types of stroking or plunging constant velocity joints. They can
be classified in ball-type joints and tripode-type joints. The first type uses the same
principle of the bisecting plane to transmit constant velocity. The two most common
types of ball-type stroking joints are the cross groove joint (CGJ) and the double offset
joint (DOJ) [2].

Figure 1.5. CGJ (left) and DOJ (right)

The tripode-type joints are not perfect constant velocity joints but they approximate
constant velocity at the typical operating angles in a vehicle, and can be considered a
constant velocity joint for practical purposes. Section 2 of this work explains in detail the
kinematics of the tripode joint and explains why it is a quasi-constant velocity joint.

5

Figure 1.6. Tripode Joint

1.4 Rolling Bearings and Tripode Joints
A tripode joint is composed of a spider assembly and a housing. The spider
assembly is composed of a spider and three sets of rollers that rotate and slide along
the trunnions of the spider. Some of the characteristics that make the roller sets in a
tripode joint different from a typical roller bearing are the following:

6

Figure 1.7. Axial motion and support in Roller Bearing and Tripode Joint

1. In most roller bearing applications the race members remain axially static or
quasi-static with respect to themselves. In a tripode joint the outer member of the
roller set, named roller or ball, moves axially with respect to the inner member,
named trunnion. This is shown schematically in Figure 1.7.

2. In most roller bearing applications the needle rollers connecting the races are
fully supported. In a tripode joint the needle rollers are not fully supported all the
time. As shown in Figure 1.7.

3. In most roller bearing applications the outer race is fully supported. In a tripode
joint the outer member of the roller set is supported at only one point. This is
shown schematically in Figure 1.8.

7

Figure 1.8. Outer member support in Roller Bearing and Tripode Joint

4. In most roller bearing applications possible misalignment is constant, making
easier to predict load distributions. In a tripode, equivalent misalignment in the
roller set continuously changes within a revolution and is a function of the joint
angle, making it necessary to predict instantaneous load distributions. This is
shown schematically in Figure 1.9 and will be covered in detail in Section 4.

Figure 1.9. Variation of load distribution in a tripode joint

8

5. In most roller bearing applications the race members are round. In a tripode joint
the trunnion may have non-round shapes, which changes the load distribution in
the needle rollers. A circular and a non-circular shape are shown schematically in
Figure 1.10. The non-circular shape has been exaggerated for visualization
purposes.

Figure 1.10. Inner member shape in Roller Bearing and Tripode Joint

For the reasons formerly mentioned, traditional approaches to estimate the life of the
roller set in a tripode joint, such as the ISO-ANSI/AFBMA standards, can not be
employed. A new specific analytical model, with experimental verification, to predict the
life of tripode joints is necessary.

9

1.5 Benefits of a life model
Traditionally, a semi-empiric approach has been used by some CVJ manufacturers
to estimate the life of a tripode joint, resulting in joints that may be over-designed or
under-designed. By knowing the kinematics of the joint, the internal load conditions, the
generated stress fields in the contacts and its relationship to wear life, optimized joints
can be developed, resulting in packaging, mass and cost reductions.

Another benefit of developing a valid analytic model is that in a family of products
only one size needs to be tested and validated. Larger and smaller sizes can be
developed with a bigger level of confidence, reducing the development time and cost of
the project.

1.6 Joint Size and Packaging
The size of a tripode joint is typically defined by the torque capacity of the largest
axle bar that can be plugged into the spider. “Yield Torque” is a common term used by
CVJ manufacturers and refers to the maximum torque an axle bar or splined interface
can transmit before having plastic deformation. The joint used for the experimental
portion of this work is a “21-size” joint. The yield torque of the largest axle bar that can
be used in combination with the joint is around 2100Nm.

The “packaging” of the joint is another common term used by CVJ manufacturers
and is used to refer to the maximum swing diameter of the joint.

10

1.7 Project description
The sequence of steps taken in this project is as follows:
1. Description of the architecture of the tripode joint
2. Study of the kinematics of the tripode joint
3. Study of the forces at the internal contacts of the tripode joint
4. Calculation of the stress field resulting from the forces at the contacts
5. Use of the state of stress to predict wear life using fatigue theories
6. Experimentation, analysis and correlation of results
7. Optimization of the roller assemblies to maximize life and/or minimize
packaging.

The experimental portion of this work was conducted at an industrial facility that
has specialized test machines for halfshafts, proprietary to the CVJ manufacturer
sponsoring this work. Details of the testing performed such as speed, torque, angle and
test time, and inspection results of tested parts will be discussed in Section 7.

11

2. KINEMATIC ANALYSIS OF A STROKING TRIPODE JOINT

2.1 Introduction
The tripode joint has been around for almost a hundred years [2]. However,
many investigations have not been published. Most of them are related to motion
analysis, kinematic structure and properties [5-10], dynamic effects such as Generated
Axial Force (GAF) [11-14] and general design guidelines [2, 3, 15-16]. The purpose of
this section is not to propose a totally new kinematic approach to analyze the tripode
joint but to use, improve and expand what is currently available, knowing its limitations
and related assumptions, in order to define internal displacements, which will be
subsequently used to predict internal forces and the state of stress at the contacts
within the joint.

Figure 2.1. Exploded view of tripode joint

12

This work treats the tripode joint as a single universal joint. Therefore, the joint
angle and the respective positions of the input and output shafts are chosen arbitrarily.
As shown in Figure 2.1, the joint is composed of:

•

A housing (1) with three straight parallel and equally spaced ball bores (grooves
or slots).

•

A spider (2) whose three trunnions (legs) lie on the same plane, are equally
spaced and converge to the same point.

•

Three spherical rollers (3) which can rotate and slide on the trunnions thanks to
the interaction with the needle rollers (4). This interaction can be simplified to a
cylindrical pair for purposes of kinematic analysis. The interaction between
spherical balls and housing ball bores can be simplified to a sphere-groove
kinematic pair.

•

A set of rings (6) and roller retainers (5) whose main function is to keep all
components in place. Their kinematic and dynamic effects can be neglected
since they are not mobile parts and its mass is very small compared to the mass
of other components formerly mentioned.

•

Optional spacer rings (7) which can be used to eliminate edge contact on the
needle rollers at the base of the trunnion. Its kinematic and dynamic effects can
also be neglected.

13

2.2 Assumptions
Individual components are considered as rigid bodies. Geometrical errors (i.e.
spacing errors) and clearances are neglected during the kinematic and quasi-static
analysis since they are very small compared to the displacements and joint dimensions.

2.3 Definitions

The reference frames used in this section are shown in Figure 2.2, Figure 2.3
and Figure 2.4. The housing is typically connected to the transmission output in a
vehicle and its only motion is rotational about its axis. Thus, a fixed reference frame XY-Z is chosen on the housing side, where plane X-Y is the housing plane and Z is the
housing axis. The spider rotates together with the housing and at the same time it can
displace and change its orientation with respect to the housing due to suspension and
steering motion in a vehicle. Thus, two moving frames are chosen on the spider side.
The first one, denoted by X ' - Y ' - Z ' has its origin at the spider center O’ and Z ' is the
spider axis. Plane X ' - Y ' is the spider plane. Y ' is oriented parallel to the common
perpendicular of the Z and Z ' axes. The second moving frame, X * - Y * - Z * , has its
axes, X * , Y * and Z * parallel to axes X ' , Y ' and Z ' , respectively. Origin O* is the
intersection of the spider plane and the housing axis. X * and Y * lie on the spider
plane. The following terms are defined:

Housing plane: An arbitrary plane normal to the housing axis, Z (Figure 2.2).

14

BCD: Ball Circle Diameter, the diameter of the circle on the housing plane passing
through the intersections of ball bore centerlines and housing plane (Figure 2.2). This is
a main joint parameter and will be constantly referenced.
BCR: Ball Circle Radius, defined in this work as variable a. It is half of the BCD
Ball bore plane: A plane that passes through the housing axis and a ball bore centerline
or ball bore axis (Figure 2.2).
Housing rays: Equally spaced (each 120° apart), represent the intersection between
housing plane and ball bore planes. The three rays intersect at the housing axis (Figure
2.2).
Spider rays: The centerline or axis of each trunnion (leg) of the spider, equally spaced
(each 120° apart) (Figure 2.3).
Spider center: The common intersection of the three spider rays (Figure 2.3).
Spider plane: The plane that passes through the three spider rays (Figure 2.3).
Spider axis: Axis perpendicular to the spider plane passing through the spider center
(Figure 2.3).
Spherical roller (ball) center: The center point of each of the three spherical rollers
(Figure 2.3).

15

Figure 2.2. Housing Reference Frame

Figure 2.3. Spider Reference Frame

16

The eccentricity e shown in Figure 2.4 is defined as the distance from O* to O’.
The rotation of the spider shaft (output) is defined as ψ and the rotation of the housing
shaft (input) is defined as ϕ. The joint angle θ is defined as the angle between the spider
and housing axes. Due to the small variations between input and output angular
displacements, which are inherent to the kinematic structure of the joint, the tripode joint
is in fact a non-constant velocity joint.

ϕ −ψ ≠ 0

(2.1)

Figure 2.4. Fixed and mobile coordinate frames

17

Neither ϕ nor ψ are constants, the derivatives of Eq. (2.1) with respect to time
clearly indicate that the variations between input and output displacements translate into
the variations between input and output angular velocities. Such variation has little
influence in the displacements of the spherical rollers during normal operation
conditions in most commercial applications (typically, joint angle is below 25deg). This
problem has been treated in detail by Akbil and Lee [7-8] and by Mariot and K’Nevez [9]
and will be discussed in the following subsections. Both investigations lead to the same
equations of motion.

2.4 Spherical roller positions
When the tripode joint is angulated an eccentricity e of the spider center with
respect to the housing axis is generated since the pivoting point is not the center of the
spider (Figure 2.3 and Figure 2.4). The magnitude of such eccentricity, O*O’, is [2, 5, 7,
9].

e=

a (1 − cos θ )
2 cos θ

(2.2)

This eccentricity when projected on the X* and Y* axes has the components

e x = e cos(π − ξ )

(2.3)

18

e y = e sin(π − ξ )

(2.4)

where ξ is the angular displacement of O*O’ with respect to X * . Because of the
symmetry of the tripode joint, the orientation of O*O’ with respect to X * will repeat
three times per revolution. The expression relating rotation of the spider and angular
displacement of the eccentricity vector O*O’ is [4],

ξ = 3ψ − π

(2.5)

Then, Eqs. (2.3) and (2.4) can be rewritten as a function of spider rotation

ex = −

a (1 − cos θ )
cos 3ψ
2 cos θ

(2.6)

ey = −

a (1 − cos θ )
sin 3ψ
2 cos θ

(2.7)

When the joint rotates, the eccentricity previously described causes an orbiting
motion of the spider. Position of the spider center with respect to the fixed reference
frame is [7],

x = e(sin 3ψ sin(ϕ − ψ ) − cosθ cos 3ψ cos(ϕ − ψ ))

(2.8)

y = e(− sin 3ψ cos(ϕ − ψ ) − cosθ cos 3ψ sin(ϕ − ψ ))

(2.9)

19

and the position of the center of the spherical rollers with respect to the fixed reference
frame is given by [7]

Z i = (a / 2)(2 cosψ i + (1 − cos θ )cos 3ψ ) tanθ + z

(2.10)

z = b(cos θ o − cos θ ) + c o

(2.11)

for i = 1, 2, 3, where “z” is the position of O’ with respect to the fixed reference frame in
the Z-direction, c o is the initial stroke position of the spider (axial position) and θ o is the
joint angle when ϕ = 0. These initial conditions correspond to the stroke-angle position
of the joint, a function of the suspension and steering motions in an automobile. It is
assumed in this work that the tripode joint will be coupled to a fixed-center type constant
velocity (C/V) joint, through the spider shaft. A fixed-center C/V joint can be simplified as
a spherical joint. The case when two tripode joints are coupled has been studied by by
Mariot and K’Nevez [9] and will not be considered here.

20

Figure 2.5. Coordinates and orientation of spider shaft

The position of the end of the shaft that is coupled to the spherical joint can be
identified as point P(h,g,k). The length of spider shaft or distance from the spider center
to the spherical joint center is defined as distance b. At the same time the orientation of
the shaft can be defined by three direction angles, α, β, θ, with respect to the fixed
coordinate system X-Y-Z (Figure 2.5). Then, the direction cosines of Z ' (spider axis) are

cos α = (h − x ) / b

(2.12)

21

cos β = (g − y ) / b

(2.13)

cosθ = (k − z ) / b

(2.14)

If g, h and b are treated as independent variables, k may be obtained by simple
trigonometry. Eqs. (2.12), (2.13) and (2.14) can be combined with Eqs. (2.8) and (2.9),
which is simplified to get

h 2 + g 2 = (b sinθ + e cos θ cos 3ψ )2 + (e sin 3ψ )2

(2.15)

Then, Eq. (2.2) can be substituted into Eq. (2.15) and solved numerically for θ. An
important expression that relates the input and output angular displacements, derived
by Akbil and Lee [7-8], is

tan(ϕ − ψ ) = cos β / cos α

(2.16)

Eq. (2.16) can be solved for ϕ, numerically, by substituting Eqs. (2.12), (2.13),
(2.8) and (2.9) and using the value of θ obtained from solving Eq. (2.15). The radial
position of the ball centers, with respect to the spider center is given by [4, 6]

ri =

a  (1 + cos θ ) 2(1 − cos θ )

+
cos 2ψ i  = a + e(1 + 2 cos 2ψ i )

2  cos θ
cos θ


for i = 1, 2, 3

22

(2.17)

Therefore, the displacement of the spherical roller center is

fi = e(1+ 2 cos 2ψ i )

(2.18)

for i = 1, 2, 3

2.5 Numerical examples of spherical roller positions and displacements
It is interesting to note that Eqs. (2.17) and (2.18) indicate that spherical roller
positions depend only on joint angle (orientation of spider shaft), BCD and rotational
displacement of the joint. The reader is reminded that variable a is defined as half of the
BCD. A computer program was written to perform calculations. The code was debugged
by verifying the outputs with the numerical results of the examples published by Akbil
and Lee [8]. Graphics and numerical values in this and subsequent sections were
obtained using the characteristics of the actual tripode joint design that will be used to
do the experimental portion of this work, unless otherwise indicated. The joint
characteristics are:

Nominal BCD

=

40.8mm

Shaft length

=

300mm

Maximum functional angle

=

23°

Trunnion diameter

=

20.97mm

Needle roller diameter

=

1.9975mm

Number of needles (n)

=

36

23

Spherical roller outer diameter =

34.942mm

Figure 2.6 describes the fluctuation of joint angle in a revolution, when the joint is
operating at a nominal joint angle θ o of 23°, and was obtained solving Eq. (2.15). The
coordinates of the center of the spherical joint were obtained using Eqs. (2.19) and
(2.20).

h = −b sinθ o −

a
(1 − cosθ o )
2

(2.19)

k = b2 − g 2 − h2

(2.20)

q @degD

23.30
23.25
23.20
23.15
23.10
23.05
1

2

3

4

5

6

Figure 2.6. Fluctuation of joint angle in a revolution

24

y @radD

Dq @degD

0.30
0.25
0.20
0.15
0.10
0.05
1

2

3

4

5

6

y @radD

Figure 2.7. Deviation from nominal angle in a revolution

Figure 2.7 was obtained from Eq. (2.21), which represents the angle variation with
respect to the nominal joint angle.

∆θ = θ − θ o

(2.21)

Figure 2.8 represents the spider orbiting path and was obtained by generating a
parametric plot of e x versus e y where ψ is the parameter. The dashed line represents
the orbiting path neglecting joint angle fluctuation (ψ = ϕ), while the solid line represents
the orbiting path including joint angle fluctuation ∆θ . Orbiting path on spider plane is a
circle.

25

Y*

0.5

- 0.5

0.5

X*

- 0.5

Figure 2.8. Orbiting path of spider on spider plane

Figure 2.9 was obtained from solving simultaneously Eqs. (2.2) and (2.15). If joint
angle fluctuation ∆θ was ignored the function plotted in Figure 2.9 would be a straight
horizontal line.

26

e @mmD
0.905
0.900
0.895
0.890
0.885
1

2

3

4

5

6

y @radD

Figure 2.9. Fluctuation of eccentricity within a revolution

j-y @degD

0.4
0.2
0.0
- 0.2
- 0.4

1

2

3

4

5

6

y @radD

Figure 2.10. Difference between input and output angular displacements

Figure 2.10 is obtained solving Eq. (2.16). It shows the difference between input
and output angular displacements caused by joint angle fluctuation ∆θ . From Figure

27

2.10 it can be observed that the joint behaves as a true C/V joint at six specific positions
within a revolution. Such positions are ψ = nπ /3, where n = 0, 1, 2, …, ¶.

Y

0.5

- 0.5

0.5

X

- 0.5

Figure 2.11. Orbiting path of spider on housing plane

Figure 2.11 represents the spider orbiting path on the housing plane. It was
obtained generating a parametric plot of x versus y where ψ is the parameter. The

28

dashed line represents the orbiting path neglecting joint angle fluctuation (ψ = ϕ), while
the solid line represents the orbiting path including joint angle fluctuation ∆θ . Orbiting
path on housing plane is an ellipse.

z @mmD
0.7
0.6
0.5
0.4
0.3
0.2
0.1
1

2

3

4

5

6

y @radD

Figure 2.12. Variations of the axial displacement of the spider center

Figure 2.12 describes the movement of the spider center along axis Z, when the
joint is operating at a nominal joint angle θ o of 23°. It was obtained solving Eq. (2.11).
Similarly, Figure 2.13 was obtained solving Eq. (2.10).

29

Z1 @mmD

5

1

2

3

4

5

6

y @radD

-5

Figure 2.13. Displacement of the spherical roller center along ball bore axis

r1 @mmD
23.0
22.5
22.0
21.5
21.0
20.5
20.0
1

2

3

4

5

6

y @radD

Figure 2.14. Radial position of the spherical roller center, from spider center

30

Figure 2.14 represents the radial position of ball center with respect to the spider
center. It was obtained solving Eq. (2.17). The position at zero joint angle would be half
of the BCD, a = 20.4mm.

f1 @mmD
2.5
2.0
1.5
1.0
0.5
0.0
- 0.5

1

2

3

4

5

6

y @radD

Figure 2.15. Radial displacement of the spherical roller center, along trunnion axis

Figure 2.15 represents the displacement of the ball center along the trunnion with
respect to its “zero” position, which is defined then ri = a. It was obtained solving Eq.
(2.18).

2.6 Needle roller positions and displacements
There is no published work about needle roller positions and displacements within
a tripot joint. Motion of spherical rollers (balls) causes rolling and sliding motion of

31

needle rollers on contacting trunnions. A cylindrical coordinate system U i - Φ i -Vi ,
whose origin is O’, can be defined for each trunnion, i = 1, 2, 3. Axis Vi is a spider ray
and Φ i is measured positive from the spider plane around Vi in a CCW direction, as
shown in Figure 2.16. Variable φ ij , also shown in Figure 2.16, defines the angular
position of each needle roller, labeled j = 1, 2, 3, … , n. It is assumed that needles are
uniformly spaced around the trunnion and clearance between them is neglected. On the
other hand, interference is not allowed by design.

Figure 2.16. Trunnion, local cylindrical coordinate system

32

A secondary housing reference frame, X iH - YiH - Z iH , and a secondary spider
reference frame, X iS - YiS - Z iS , which rotate together with the joint are also defined. Axis

Z iH is always parallel to axis Z and axis Z iS is always parallel to axis Z ' (Figure 2.17).

Figure 2.17. Secondary-rotating coordinate systems

To define the angular position of a needle on a trunnion, it is necessary to know
the amount of rolling of the spherical roller (ball) due to translation of its center along
ball bore axis. If the rolling axes of spherical rollers were parallel to housing rays, there
would be pure rolling motion, no sliding. On the other hand, if they were perpendicular
there would be pure sliding, no rolling at all. In a tripode joint, spider rays are the rolling
axes of spherical rollers and they are neither parallel, with the exception of few specific

33

positions, nor perpendicular to housing rays. Ball rolling is a function of trunnion axis
inclination, relative to the housing plane, measured on an auxiliary plane formed by
vectors X S and Z H since ball rolling can only occur around axis X S and ball
i
i
i
translation along ball bore can only occur on Z H direction. The orientation of the
i
trunnion axis X S is defined by the direction angles κ i , λi , µ i , measured from X H , YiH
i
i
and Z H , respectively. Trunnion axis inclination with respect to plane X iH - YiH is
i
identified by angle δ i and is measured on auxiliary plane X iS - Z iH , as previously
indicated.

To calculate the angles formerly mentioned the orientation matrix developed by
K’Nevez, Mariot and Diaby [10] (obtained using different reference frames) is modified
such that orientation can be described with respect to the reference frames used in this
work. Modification consisted of adding a π / 2 phase to the input and output angular
displacements. After some algebraic and trigonometric manipulation the matrix that
defines orientation of the spider reference frame X iS - YiS - Z iS with respect to the
housing reference frame X iH - YiH - Z iH can be written as,

i

 cos θ cos ϕ cosψ + sin ϕ sinψ

A =  cos θ sin ϕ cosψ − cos ϕ sinψ

− cosψ sinθ


cos θ cos ϕ sinψ − cosψ sin ϕ cos ϕ sinθ 

cos θ sin ϕ sinψ + cos ϕ cosψ sinθ sin ϕ 
− sinθ sinψ
cos θ 


where ψ = ψ i and ϕ = ϕ i ; for i = 1, 2, 3

(2.22)

34

To calculate the direction angles of spider rays the dot products between unit
vector X S and unit vectors X H , YiH , Z H , are obtained
i
i
i

cos θ cos ϕ i cosψ i + sin ϕ i sinψ i 
S
X i = cos θ cos ϕ i sinψ i − cosψ i sin ϕ i 




cos ϕ i sinθ



T

(2.23)

X H = [1 0 0]
i

(2.24)

YiH = [0 1 0]

(2.25)

Z H = [0 0 1]
i

(2.26)

cos κ i = cosθ cos ϕ i cosψ i + sin ϕ i sinψ i

(2.27)

cos λi = cosθ cos ϕ i sinψ i − cosψ i sin ϕ i

(2.28)

cos µ i = cos ϕ i sinθ

(2.29)

Using simple trigonometry, angle δ i may be related to the complementary angle of µ i .

cos δ i = 1 − (cos µ i ) 2

(2.30)

δ i = π / 2 − µi

(2.31)

Even though Eqs. (2.30) and (2.31) are equivalent, only Eq. (2.31) can indicate on
which side of the housing plane (positive or negative Z iH ) the trunnion is located (Figure
2.18 and Figure 2.19).

35

d @degD

20
15
10
5

1

2

3

4

5

6

y @radD

Figure 2.18. Angle δ, per Eq. (2.30)

d @degD

20
10

1

2

3

4

5

- 10
- 20
Figure 2.19. Angle δ, per Eq. (2.31)

36

6

y @radD

When the spherical roller is tilted only a fraction of the distance traveled along the
housing ball bore will cause rolling (Figure 2.20). More tilting means less rolling. In the
case of constant inclination, the arch length displaced at the outer diameter of the
spherical roller can be obtained from Eq. (2.32), which is valid only if there is contact, all
the time, between spherical roller and housing ball bore.

s i = Z i cos δ i

(2.32)

Figure 2.20. Partial rolling of ball due to tilting

37

However, in a tripode joint, angle δ i is not constant (Figure 2.19) during the whole

Z i displacement. Therefore, a differential approach has to be followed and Eq. (2.32)
can be rewritten as Eq. (2.33). If Eqs. (2.29) and (2.30) are substituted into Eq. (2.33)
then Eq. (2.34) is obtained.

dsi = dZi cosδi

(2.33)

ds i = dZ i 1 − (cos ϕ i cos θ ) 2

(2.34)

Variable Z i is a function of angular displacement ψ i and joint angle θ. Let’s
consider a simple case for the sake of clarity. Assume that the initial position of the
spider in Figure 2.20 is the position where the origin of the spider reference frame
coincides with the origin of the housing reference frame. This can only happen when
joint angle θ is zero. Then, without rotating the joint (ψ i = ϕ i = 0 ) change joint angle θ
from 0 to θ o (if ϕ i = 0 , then δ i = θ o ). When the spherical roller starts to displace, tilting
angle will be very small and its effect in ball rolling could be discarded but as joint angle
gets closer to θ o tilting will impact ball rolling. If the actual ball rolling wants to be known
Eq. (2.34) has to be integrated from zero joint angle to actual joint angle, for a given
rotational (angular) displacement ψ i of the spider. Differential dZ i can be obtained from
Eqs. (2.10) and (2.11).

dZ i =

∂Z i
∂Z i
∂Z
dθ +
dψ i + i dθ o
∂θ
∂ψ i
∂θ o

(2.35)

38

Where dψ i = 0 , since dZ i will be evaluated for a specific displacement ψ i . Thus, Eq.
(2.34) becomes


1 − (cos ϕ i cos θ )2 dθ
0
0

θ o _ max  ∂Z i
2

+
 ∂θ 1 − (cos ϕ i cosθ (ψ i ,θ o )) dθ o

0
 o


∫

si

ds i =

∫

θ (ψ i ,θ o _ max )  ∂Z i


 ∂θ

(2.36)

∫

and the partial differentials contained in Eq. (2.36) are,

∂Z i 1
= a(2 cosψ i + (1 − cosθ ) cos 3ψ i ) sec 2 θ
∂θ
2
1
+ b sinθ + a cos 3ψ i sinθ tanθ
2

(2.37)

∂Z i
= −b sinθ o
∂θ o

(2.38)

Eq. (2.36) can be evaluated numerically to calculate s i . Displacement s i follows
the same trend that Z i , as expected (Figure 2.13 and Figure 2.21). On the other hand,
the difference between Z i and s i is maximum at the points where angle δi is maximum,
and zero (Figure 2.22) where δ i is zero, which is the condition when the spherical roller
doesn’t displace along the ball bore.

39

s1 @mmD

5

1

2

3

4

5

6

y @radD

-5

Figure 2.21. Actual arch displaced at ball outer diameter

Z1-s1 @mmD
0.2
0.1
0.0
- 0.1
- 0.2

1

2

3

4

5

6

y @radD

Figure 2.22. Difference in arch displacement without and with ball tilting effect

Before proceeding to calculate angles φij , there is an additional component that
must be taken into account due to the change of orientation of the spider axis within a

40

revolution. Let’s consider another simple case. When angular displacement ψ i = π / 2
the spherical roller will not move along the ball bore. However, there will be an angle
between the plane that passes through φi 1 = 0 and a plane parallel to the housing plane
that passes through the point of contact between ball and ball bore (Figure 2.23).
Needle j = 1 is the needle that is located at φi 1 when joint angle θ = 0 (Figure 2.16).

Figure 2.23. Graphical representation of angle ε

The angle formerly mentioned will be named ε i and can be obtained from
calculating the angle between vectors YiS and YiH . Thus, the dot product of such
vectors is

cos ε i = cosθ sin ϕ i sinψ i + cosψ i cos ϕ i

(2.39)

41

Evaluating ε i numerically, through one revolution, can be observed that only
positive angles are obtained (Figure 2.24). Even though this is mathematically correct, it
represents a disadvantage. Position of needles on the trunnion depends not only on the
magnitude of angle ε i . Let’s consider needle j =1. Position φi 1 of this needle on the
trunnion will be a positive number when ψ i = π / 2 and negative number when

ψ i = 3π / 2 , even though magnitude of ε i is the same at both ψ i positions (Figure 2.23).

e @degD

20
15
10
5

1

2

3

4

5

6

y @radD

Figure 2.24. Angle ε, per expression (2.39)

To overcome such problem a SIGN function, which depends on ψ i , can be used.
Angle ε i is redefined as angle ε ' i , which is phase corrected (Figure 2.25).

ε ' i = sign(sinψ i ) ⋅ arccos(cosθ sin ϕ i sinψ i + cosψ i cos ϕ i )

42

(2.40)

e' @degD

20
10

1

2

3

4

5

6

y @radD

- 10
- 20
Figure 2.25. Angle ε', per expression (2.40)

Then, after considering the effects of ball tilting and the change in spider
orientation, the angular displacement of the contact point between spherical roller (ball)
and ball bore, with respect to the local trunnion reference frame, is defined as η i .

η i = ε 'i −

2
si
mo

(2.41)

The second term of Eq. (2.41) comes from dividing the arch length displaced, s i ,
calculated from Eq. (2.36), by the outer radius of the spherical roller. Eq. (2.41) can also
be evaluated numerically and plotted (Figure 2.26).

43

h1 @degD

30
20
10
0
- 10
- 20
- 30

1

2

3

4

5

6

y @radD

Figure 2.26. Angle η, per expression (2.41)

Relative angular motion between spherical rollers and trunnions will result in
needle angular displacements. It is convenient to establish a relationship between the
angular displacement of the ball and the angular displacement of the needles, with
respect to the local trunnion cylindrical coordinate system U i - Φ i -Vi (Figure 2.16).
Assuming contact and no sliding between trunnion, needle rollers and spherical roller
(ball), consider the case where a needle roller displaces from position (B-C) to position
(B’-C’) as a result of an angular displacement η i of the ball, from point (A) to point (A’).
Since no needle sliding around the trunnion was assumed, it can be said (based on
kinematic principles of relative motion) that the arch-length of segment (C-C’) equals
arch-length of segment (B’-D) (Figure 2.27).

44

Figure 2.27. Angular displacements and positions in trunnion-rollers set

Diametrical clearance in the trunnion-rollers set can be neglected (as stated in the
beginning of this section) for kinematic analysis purposes. Clearance is three to four
orders of magnitude smaller than the trunnion diameter. This makes the inner diameter
of the ball similar to the trunnion diameter plus two times the needle roller diameter.
Thus, the following identities can be generated when initial needle position is zero,

mi ≈ t + 2d

(2.42)

45

t
t

φij =  + d (η i − φij )
2
2


(2.43)

Isolating position φij from Eq. (2.43) and adding a term to take into account the initial
conditions of needles that are not in a zero position it is obtained,

t

 +d 
η i + ( j − 1) 2π
φij =  2
n
 t +d 





(2.44)

for j = 1, 2, 3, … , n

Evaluating Eq. (2.44), numerically, and plotting for the case i =1 and j = 1,

f11 @degD
20

10
0
- 10

1

2

3

4

5

6

y @radD

Figure 2.28. Angular position of needle j = 1, per expression (2.44)

46

It must be noted that when j = 1, position and displacement are equal in
magnitude. Therefore, φi 1 will be used to refer to needle displacement as well.

2.7 Critical joint angle
Critical joint angle is an important design parameter. In a tripode joint, it is defined
as the joint angle that causes a total needle displacement equal to the angular spacing
between needles. In other words, critical joint angle is when fatigue cycles duplicate due
to more than one needle passing over the same trunnion area in a joint revolution.
Given the complexity of the equations derived in previous sub-sections, it is difficult to
obtain a closed form solution for critical angle. Such problem can be tackled
numerically, using a numeric method that iterates joint angle until total needle
displacement equals needle spacing.

2π / n − φi 1(θ o ,ψ i )max − φi 1(θ o ,ψ i )min = 0

(2.45)

A computer program was developed to solve Eq. (2.45) for θ o . The calculated
critical angle of the joint used for the experimental portion of this work, whose main
characteristics were previously described, is θ cr = 5.965°

47

2.8 Critical joint angle, when ball tilting is neglected and ψ = ϕ
In most commercial applications, joints will spend most of its life running between
3° and 5°. In some applications, continuous running angles may be as high as 11° but
that is uncommon. Tripode joints are designed to have a critical angle in the range of 5°
to 9°, typically. It was discussed in previous sub-sections the joint angle variation due to
the small variations in input and output displacements. Such joint angle variation
reduces as the nominal joint angle decreases. Figure 2.29 compares the deviation from
nominal angle, in a revolution, at two different nominal angles. The solid line represents
a nominal angle of 23°, maximum deviation is 0.342° or 1.49%. The dashed line
represents a nominal angle of 6°, maximum deviation is 0.022° or 0.37%.

Dq @degD

0.30
0.25
0.20
0.15
0.10
0.05
1

2

3

4

5

6

y @radD

Figure 2.29. Deviation from nominal angle in a revolution at θ o = 23°, 6°

48

Figure 2.30 shows the maximum proportional deviation from nominal angle as a
function of nominal angle. It can be observed that the behavior is almost linear and that
even at angles as high as 15° the relative error is still less than 1%. Of course the
maximum deviation will be different for different joint dimensions but magnitude will be
of the same order. In this work a relative error of joint angle variation of less than 1% will
be considered acceptable when predicting critical angle. Therefore, it can be assumed
that ψ i = ϕ i , and consequently θ = θ o , for the purpose of calculating critical joint angle.

Dqêqo

0.015

0.010

0.005

5

10

15

20

25

qo @degD

Figure 2.30. Maximum proportional deviation from nominal angle vs. θ o

Given the assumption of equal input and output displacements, many of the
equations used to calculate needle roller displacement can be simplified. Let’s start by
defining φi 1 * as needle displacement and η i * as angular ball displacement, when

ψ i = ϕi .

49

t

 +d 
η i *
φi1* =  2
 t +d 





(2.46)

Similarly, ε ' i * and s i * are defined when ψ i = ϕ i ,

ηi * = ε 'i * −

2
si *
mo

(2.47)

It was mentioned in the previous section that the effect of ball tilting at low joint
angles could be neglected.

Figure 2.31 shows the difference between Z i and s i , in a revolution, for a joint
angle of 6°, while Figure 2.32 shows the relative error, with respect to s i , of assuming

s i = Z i . The absolute error in arch length displaced is in the order of microns, while the
relative error is less than 0.2%, including joint angle variation (ψ i ≠ ϕ i ). If joint angle
variation is ignored, the relative error becomes 0.37%, at 6° (Figure 2.33).

50

Z1-s1 @mmD
0.004
0.002
0.000
- 0.002

1

2

3

4

5

6

y @radD

Figure 2.31. Difference in arch displacement without and with ball tilting, at 6°

HZ1-s1Lês1

0.0015
0.0010
0.0005

1

2

3

4

5

6

y @radD

Figure 2.32. Relative error of ignoring ball tilting, at 6°

Figure 2.33 shows the maximum relative error caused by ignoring ball tilting and
ignoring joint angle fluctuation. It can be observed that the behavior is highly non-linear

51

at large angles. However, at low angles the error is in the order of 1% or lower. Such
error will be considered acceptable in this work and it will be assumed that Z i ≈ s i for
the purpose of calculating critical joint angle.

HZ1-s1Lês1

0.10
0.08
0.06
0.04
0.02
5

10

15

20

25

qo @degD

Figure 2.33. Relative error of arch-length displacement, ignoring tilting, vs. θ o

Once the two main assumptions for a simplified formulation of critical angle have
been established, Eqs. (2.10), (2.40), (2.46) and (2.47) can be reduced and
reformulated. Another secondary assumption, nevertheless important, is that initial
stroke c o = 0 . Therefore,

a
s i * ≈ Z i * =  (2 cosψ i + (1 − cos θ )cos 3ψ i ) tanθ o
2

(2.48)

ε ' i * = sign(sinψ i ) ⋅ arccos 1 − sin 2 θ o ⋅ sin 2 ψ i 



(2.49)





52

η i * = sign(sinψ i ) ⋅ arccos 1 − sin 2 θ o ⋅ sin 2 ψ i 




2
−
mo



a
 (2 cosψ i + (1 − cos θ o )cos(3ψ i )) tan θ o
2

t

 +d 
(sign(sinψ i ) ⋅ arccos 1 − sin 2 θ o ⋅ sin 2 ψ i 
φi1* =  2


 t +d 






a
−
(2 cosψ i + (1 − cos θ o )cos(3ψ i )) tanθ o )
mo

(2.50)

(2.51)

Eq. (2.45) represents the simplified formulation for needle roller displacement,
which is valid only at low joint angles, as previously described. Figure 2.34 shows a
comparison between φi 1 and φi 1 * at 23° joint angle. The solid line represents the
needle roller displacement including joint angle fluctuation and ball tilting effect. The
dashed line represents the simplified version.

Eq. (2.45) can be rewritten as Eq. (2.52) and solved numerically for θ o , using the
same joint parameters.

2π / n − φi 1 * (θ o ,ψ i )max − φi 1 * (θ o ,ψ i )min = 0

Then, the resulting critical angle is θ cr * = 5.981°

53

(2.52)

f11 @degD
20

10
0
- 10

1

2

3

4

5

6

y @radD

Figure 2.34. Needle roller displacement, complete vs. simplified solution

Relative error of θ cr * , with respect to θ cr , is calculated and is found to be equal
to 0.27%. It was mentioned before that this difference will change from joint to joint but
the order of magnitude will be the same. Therefore, Eqs. (2.51) and (2.52) will be
considered as fairly good approximations to calculate critical angle. To finalize this
section, it will be made reference to an expression, formerly used by a C/V joint
manufacturer [14], to calculate critical angle in tripode joints. Such expression was
developed considering only one position of ψ, which was ψ = 0.

 2π mo (t + d ) 

⋅

 n bcd ⋅ mi 

θ cr ' = arctan


(2.53)

54

Evaluating Eq. (2.53) with previously used joint parameters, it is found that the
critical angle predicted by this expression is θ cr ' = 7.83°. This represents a
considerable difference with respect to the exact and the approximate solutions derived
in this work. The source of this difference, as previously explained, is the fact that Eq.
(2.53) was derived without considering full joint rotation, which makes it incomplete.

55

3. QUASI-STATIC ANALYSIS OF A STROKING TRIPODE JOINT

3.1 Introduction
This section deals with the way internal forces are distributed in a tripode joint.
Forces can be dynamic or static and be influenced by the friction in the system. Many
authors [2-3, 11-15] have dealt with this problem from different perspectives. Small
variations between input and output angular displacements may cause non constant
velocity (CV) behavior in a tripode joint. Such variations were discussed in Section 2.
The reader may refer back to Eq. (2.1) and Figure 2.10. The first major assumption in
this section will be to neglect the small variations between input and output angular
displacements, as well as the joint angle fluctuation caused by them (Figures 2.6 and
2.7), and consider the tripode joint as a true constant velocity (CV) joint. Urbinati and
Pennestri [11] performed a complete dynamic analysis of the tripode joint. They
concluded that in steady state or constant input velocity the effect of joint angle
fluctuation (source of the deviation from constant velocity) on reaction forces inside the
joint can be neglected.

The second major assumption made in this section is relative to inertial forces. In
most commercial applications tripode joints operate at relatively low speeds. Mariot,
K’Nevez and Barbedette [12] verified in their work that for a constant input velocity,
dynamical effects are negligible versus static effects.

56

The third major assumption is relative to rolling and sliding friction between
spherical roller and needle rollers on the trunnion. Sliding between spherical roller and
needle rollers occurs when the spherical roller displaces along the trunnion. Serveto,
Mariot and Diaby [13] modeled analytically and numerically (ADAMS model) the
Generated Axial Force (GAF) in tripode joints. GAF is the main generator of shudder
and vibrations in the joint and is a function of internal forces. They concluded that
friction between spherical roller and needle rollers on the trunnion, when the joint is
lubricated, has little influence in GAF and its effect can be neglected. Lee [14] in his
doctoral dissertation measured the sliding friction force between spherical rollers and
needles and found the average to be very close to zero, within a joint revolution, under
quasi-static conditions.

Published studies on GAF [12-14] identify the friction between spherical roller
and ball track as the dominant element in the disturbances generated by the joint.
Friction forces at this contact are also larger than friction forces between spherical roller,
needle rollers and trunnion, by one order of magnitude, even in the absence of
lubricant.

3.2 Normal forces at spider trunnions
Input torque is applied to the tripode at the housing axis. This torque is
transmitted through three loads to the spider. Loads on the spider generate torque at
the spider axis, equal in magnitude to the input torque. If sliding and rolling friction

57

between spherical rollers and trunnions are ignored, it can be assumed that the forces
that generate torque in the spider axis are all located on the spider plane X ' - Y ' , pass
through the center of the spherical roller and are perpendicular to the trunnion axes
(Figure 3.1).

Neglecting inertial effects the analysis can be simplified to a simple static
analysis. Thus, the three equations that define the sum of forces and moments in the
spider plane are:

p1 cosψ + p2 cos(ψ + 2π / 3 ) + p3 cos(ψ + 4π / 3 ) = 0

(3.1)

p1 sinψ + p2 sin(ψ + 2π / 3 ) + p3 sin(ψ + 4π / 3 ) = 0

(3.2)

p1r1 + p2 r2 + p3 r3 = tψ

(3.3)

Solving Eqs. (3.1), (3.2) and (3.3) simultaneously,

p1 = p2 = p3 = p =

tψ

(3.4)

r1 + r2 + r3

Let’s remember that ri can be obtained from Eqs. (2.2) and (2.14),

e=

a (1 − cos θ )
2 cos θ

(2.2)

ri =

a  (1 + cos θ ) 2(1 − cos θ )

+
cos 2ψ i  = a + e(1 + 2 cos 2ψ i )

2  cos θ
cos θ


(2.14)

58

for i = 1, 2, 3

Then, the sum of r1, r2 , and r3 can be expressed as,

r1 + r2 + r3 = 3(a + e ) =

3 
1 
a 1 +

2  cos θ 

(3.5)

Therefore,

p1 = p2 = p3 = p =

tψ
2
1 
3 
a 1 +

 cos θ 

(3.6)

3.3 Normal forces at ball bores
Forces q i are defined to be perpendicular to ball bore axes. Forces pi are
components of forces q i when projected on the spider plane. Wagner [2] derived
expressions to estimate forces q i as a function of torque through graphical methods, for
only two phase angles of the joint (ψ = 0 and ψ = π / 2). Orain [15] derived expressions
for a full revolution of the joint, which yield numerical values similar to those obtained
from Wagner’s expressions at ψ = 0, ψ = π / 2 and mirror positions. Lee [14] measured
values of q i and pi in a joint lubricated with grease under quasi-static conditions and

59

found them to be very close, where pi behaves like a constant within a revolution and

q i has little fluctuation, following a trend similar to that predicted by Orain [15].

Figure 3.1. Forces on spider plane

Experimental data, computer simulations (ADAMS) and analytical formulations
presented by Mariot, Serveto and Diaby [13] for GAF show that the influence of
rotational speed in GAF can be neglected, within the normal operation range in most
commercial applications. GAF is a function of friction and internal loads. The formerly
mentioned authors simplified the analysis at low joint angles by making q i ≈ pi to
develop GAF expressions. The equations used in this work to calculate GAF are

60

equivalent to expressions derived by Lee [14], Wagner [2] and Mariot, Serveto and
Diaby [13].

3

GAF =

∑w i

(3.7)

i =1

w i = µ b q i sin δ i sign(sin δ i sinψ i ) + µ r q i sign(sinψ i )

(3.8)

where angle δ i can be obtained from Eqs. (2.31) and (2.29), derived in the previous
section,

δ i = π / 2 − µi

(2.31)

cos µ i = cos ϕ i sinθ

(2.29)

After a break-in period, friction in the joint is assumed to be constant. Therefore,
peak GAF will be proportional to the normal load applied on the ball bore, where the
constants of proportionality are the friction coefficients.

This being said, equations for q i can be developed making sure quasi-static
equilibrium of forces is maintained.

(3.9)

(q1 + q2 + q3 )a = tψ = tϕ

61

The input torque and the distance between housing axis and ball bore axes are
constants. Therefore, the sum of normal forces acting on housing ball bores is an
invariant. In the position shown in Figure 3.2 the normal force on the ball bore has the
same direction of the normal force on the trunnion. In other words, q i = pi at ψ i = 0.

Figure 3.2. Rotating coordinate systems and joint orientation when ψ i = 0

When the joint has rotated 90° and 270°, the relationship between q i and pi can
be represented graphically as shown in Figure 3.3.

62

Figure 3.3. Graphical representation of angle ε and forces q i and pi

It will be noticed that pi has the same direction that YiS and q i has the same
direction that YiH . Therefore, the angle ε between pi and qi is equivalent to angle ε i ,
from Eq. (2.39), for i = 1.

cos ε i = cosθ sin ϕ i sinψ i + cosψ i cos ϕ i

(2.33)

Since pi is a projection of qi its relationship can be expressed as

q i = pi / cos ε i

(3.10)

63

3.4 Numerical examples and GAF experimental results
Following the process developed in the previous section, a computer program was
written to perform calculations. The code was debugged verifying the outputs with the
numerical results of examples published by Wagner [2] and Orain [15]. Graphics and
numerical values in this section were obtained using the characteristics the tripode joint
used to do the experimental portion of this work. The joint characteristics and torque
applied are:
Nominal BCD

=

40.8mm

Shaft length

=

300mm

Maximum functional angle

=

23° (design limit)

Trunnion diameter

=

20.97mm

Needle roller diameter

=

1.9975mm

Number of needles (n)

=

36

Spherical roller outer diameter =

34.942mm

Input torque

2118Nm (design limit)

=

Figure 3.4 compares the magnitude of force p1 assuming a constant joint angle
(dashed line) and including joint angle variation (solid line). It can be observed that the
frequency is the same as joint angle variation calculated in section two. The maximum
and minimum values in a revolution, at the design limits of the joint, are: p1max =
33,175.3N and p1min = 33,131.2N. The magnitude of the variation, with respect to the
maximum value of p1 is 0.13%. This small variation justifies the assumption of
neglecting joint angle fluctuation, as indicated at the beginning of the section.

64

p1
33170
33160
33150
33140

1

2

3

4

5

6

y @radD

Figure 3.4. Magnitude of p1 with (solid line) and without joint angle variation

Similarly, q1 is calculated and plotted in Figure 3.4 with and without joint angle
variation effects. The maximum and minimum values for each case are,

q1max (constant joint angle) = 36,040.4N

q1min (constant joint angle) = 33,175.3N
q1max (joint angle variation) = 36,062.9N

q1min (joint angle variation) = 33,175.3N

The relative error associated with ignoring joint angle variation to calculate the
maximum value of q1 is 0.06%, at extreme joint conditions. This verifies the assumption
of neglecting joint angle variation and considering the tripode joint as a CV joint through

65

the rest of this work. The reader will also notice that the amplitude of q1 is small with
respect to its medium value, even at maximum joint angle of 23° (Figure 3.5). It must be
considered that at lower joint angles the amplitude to mean value ratio will reduce.

amplitude
(36,062 − 33,175)N / 2
=
= 0.0417
mean _ value (36,062 + 33,175)N / 2

q1
36000
35500
35000
34500
34000
33500
1

2

3

4

5

6

y @radD

Figure 3.5. Magnitude of q1 with (solid line) and without joint angle variation

GAF was measured in joints that have the geometric characteristics of the joint
simulated in this section. The same greases that Lee [5] used in his experiments are
used in this work. Table 3.1 summarizes the main characteristics of the greases.

A total of sixteen tripode joints were tested. Only the average of peak GAF values
(third order components) are reported in this work and compared against the theoretical
calculations using Eqs. (3.7) and (3.8). The full set of measurements can be found in

66

referenced Nexteer Automotive Test Activities [17]. Such experiments were performed
in a machine specifically designed and created to measure GAF in CV joints. Details of
the machine are not disclosed in this work since they are proprietary to Nexteer
Automotive.

Figures 3.6 and 3.7 show the graphs corresponding to Eqs. (3.8) and (3.7). Load
used to generate such figures was 600Nm, at 7.5°, using grease A, with µ b = 0.04 and

µ r = 0.001. For grease B µ b = 0.08 and µ r = 0.001. The coefficients of friction used in
this work are slightly lower than coefficients of friction reported by Lee [14]. Hardware
used by Lee to do his experiments was made by manufacturing processes less
accurate than the processes used to make the hardware used for the experimental
portion of this work.

______________________________________________________________________
Properties

Grease A

Grease B

Type

PU (polyurea)

PU (polyurea)

Performance

Low Friction

High Durability

NLGI grade

2

2

Base oil

Mineral / synthetic

Mineral

Viscosity @ 100ºC η (mPa-s)

12

12

Representative additives

Organo-molybdenum

Solid EP*

Color

Green

Blue

Table 3.1. Properties of CVJ greases. (*) stands for “Extreme Pressure”

67

w1
60
40
20
1

2

3

4

5

6

y @radD

- 20
- 40
- 60
Figure 3.6. Tangential force w1, at 600Nm, 7.5°, grease A

Figures 3.8 and 3.9 compare theoretical calculations against experimental values,
for two levels of torque, two different greases and a set of joint angles that range from
2.5° to 17°.

GAF
100
50
0
- 50

1

2

3

4

5

6

y @radD

Figure 3.7. Generated axial force, at 600Nm, 7.5°, grease A

68

It can be observed that theoretical calculations do not match experimental data
perfectly but clearly follow the same trend. The purpose of this work is not to develop an
accurate GAF model, which could be achieved by defining coefficients of friction as
functions of torque and joint angle instead of constants. Such approach was already
used by Lee [5] and is outside of the scope of this work.

GAF, Tripode @ 300Nm, 200rpm
250

Peak Force [N]

200
Grease A (exp)
Grease B (exp)
Grease A (theo)
Grease B (theo)

150
100
50
0
0

2.5

5

7.5

10

12.5

15

17.5

Joint Angle [deg]
Figure 3.8. Generated axial force, at 300N, experimental vs. theoretical values

The purpose of comparing theoretical GAF calculations against real experimental
data is to justify the validity of the assumptions described at the beginning of this section

69

to calculate normal loads on trunnions. Such loads will be the basis to estimate load
distribution at the contacts between needle rollers and trunnion and determine the state
of stress, in the next sections.

GAF, Tripode @ 600Nm, 200rpm
450
400
Peak Force [N]

350
300

Grease A (exp)
Grease B (exp)
Grease A (theo)
Grease B (theo)

250
200
150
100
50
0
0

2.5

5

7.5

10

12.5

15

17.5

Joint Angle [deg]
Figure 3.9. Generated axial force, at 600N, experimental vs. theoretical values

70

4. LOAD DISTRIBUTION ON A TRUNNION

4.1 Introduction
This section deals with the way contact forces are distributed on trunnions of a
tripode joint. Three areas in the joint where mating surfaces are subjected to contact
forces are:
a) Contact between spherical roller and ball bore in the housing
b) Contact between needle rollers and spherical roller (ball)
c) Contact between needle rollers and trunnion

Typically, the contact region to present the first signs of wear is the contact area
between trunnion and needle rollers. Countless validation tests of different sizes of
joints throughout the past 30 years at Nexteer Automotive support the statement. This is
the reason to consider that the wear life of a tripode joint will be defined by the wear
performance of trunnion surface. It is possible for needles to wear first. However, this
condition occurs rarely in tripode joints.

Methods used to analyze roller bearings will be employed to study load
distribution in spider trunnions. The following assumptions are made:
a) Needle rollers and trunnions are made of steel, similar elastic properties and
homogeneous material.

71

b) Unless otherwise indicated, it is assumed in this section that plastic
deformation doesn’t occur at the contacts.
c) Unless otherwise indicated, residual stresses are assumed to be zero in this
section.

4.2 Normal load distribution on needles (axial direction)
The forces transmitted from housing to spider are shown to be normal to the
trunnion axis in Section 3. The spherical roller rotates and slides with respect to the
trunnion. The spider assembly can be treated as a set of three roller bearings, where
the trunnions function as inner races, the needles are equivalent to simple cylindrical
rollers and the spherical rollers or balls function as outer races, one per roller bearing.

In traditional roller bearings the rollers are fully supported in both inner and outer
races. This is not the case in a tripode joint, where needles are not fully supported on
the sphericall roller with axial relative motion between them. Three main load cases are
identified, as shown schematically in Figures 4.1, 4.2a and 4.2b. It must be noted that,
for the sake of clarity, the sketches represent only one needle and the concentrated
force pij is the equivalent concentrated force acting on that particular needle and not
the total force acting between spherical roller and ball bore.

72

Figure 4.1. Case 1: Spherical roller fully supported, no offset

Figure 4.1 depicts Case 1, which is the ideal case. The equator of the spherical
roller is aligned with the transversal plane of symmetry of the needles. Since there is no
offset between such planes, the load transmitted by the needles can be considered
uniformly distributed along the effective contact length. Figure 2.14 was presented to
show radial position of the spherical roller center from the spider center, measured
along the trunnion axis. Case 1 will occur at one radial position. Typically, this position
will be ri ≈ a. The reader is reminded that a = BCD / 2. Solving Eq. (2.17) for ri ≈ a, it is
found that Case 1 will occur four times per joint revolution.

73

Figure 4.2a. Case 2: Positive offset (Λ)

Figure 4.2b. Case 3: Negative offset (–Λ)
74

Figure 4.2a and Figure 4.2b depict Case 2 and Case 3, respectively, which occur
twice per revolution. Case 2 represents the condition where the ball spends more time
in a joint revolution. Offset Λi is a direct function of ball axial displacement. Therefore, it
can be expressed as,

Λi = Λo + fi = Λo + e(1+ 2 cos 2ψ i )

(4.1)

for i = 1, 2, 3

where Λo is the offset at zero degree joint angle. In other words, Λo is an initial
condition. The ends of the needle rollers have the clearance Θ o , with respect to roller
retainer and ring spacer, which allows some additional axial motion of the needles with
respect to the trunnion. This motion is very small if compared to the length of the needle
but not very small if compared to the ball displacement at low angles. It will be assumed
that the coefficient of friction between needle and ball is larger than the coefficient of
friction between needle and trunnion. Therefore, the needle will displace together with
the ball while there is clearance between the top end of the needle and the roller
retainer in one direction, and the low end of the needle and the ring spacer in the
opposite direction. Once the clearance becomes zero, the needle will stop displacing
and the ball will continue its motion. This is shown graphically in Figure 4.3.

75

Figure 4.3. Clearance between needle, roller-retainer and spacer ring

Such displacement can be considered as a lash that allows the needles to oscillate
to the same frequency of the ball axial displacement, twice per revolution. Thus, the
following piecewise function is defined to take into account needle axial displacement,

3e >

Θo
2

Θi = −

; 0 ≤ψ i <

Θo
2

Θo
2

;

2

; π ≤ψ i <

3π
2

;

fi − 3e +

Θ i = fi − 3e +

3e >

π

π
2

Θo
2

;

≤ψ i < π ;

fi − 3e +

3π
≤ ψ i < 2π
2

76

Θo

Θ
<− o
2
2

Θo

Θ
≥− o
2
2

Θi =

Θo
2

+ς ;

Θ i = fi + e −

fi + e −

Θo
2

+ς ;

fi + e −

Θo
2

Θo
2

>

Θo

≤

Θo

2

2

ς = Θ o − 4e ;

4e < Θ o < 6e

ς = 0;

Θ o < 4e ; Θ o > 6e

3e ≤

Θo
2

; 0 ≤ ψ i < 2π

Θ i = fi
for i = 1, 2, 3

(4.2)

Using the same joint characteristics employed in Section 2 and Section 3,
Expression (4.2) is plotted for two different joint angles, 6° and 23°, for a nominal value

Θ o = 0.452mm.

Q1 @mmD

0.15
0.10
0.05
0.00

1

2

3

4

5

6

y @radD

Figure 4.4a. Needle axial displacement at 6° joint angle

77

Figure 4.4a shows that the loaded needles follow the same axial motion of the ball.
Amplitude of the motion is less than the clearance Θ o . Therefore, it is expected that
the motion of the needles will not be limited by lack of the clearance. On the other hand,
Figure 4.4b shows the truncation of the amplitude of axial motion of needles beyond the
clearance Θ o . This clearly indicates that the needle displacement will be limited by the
contact between roller retainer and spacer ring during a revolution.

Q1 @mmD

0.2
0.1
0.0
- 0.1
- 0.2

1

2

3

4

5

6

y @radD

Figure 4.4b. Needle axial displacement at 23° joint angle

78

L1 @mmD

2.0
1.5
1.0
0.5
0.0
- 0.5

1

2

3

4

5

6

y @radD

Figure 4.5. Offset between ball equator and needles symmetry plane

Then, Eq. (4.1) can be re-written by adding a term for needle motion, resulting in
Eq. (4.3). It is plotted in Figure 4.5, for Λo = –0.206mm.

Λi = Λo + fi − Θ i = Λo + e(1 + 2 cos 2ψ i ) − Θ i

(4.3)

for i = 1, 2, 3

Once the offset between ball and needles has been defined, the analysis can
continue to estimate load distribution on needles. Average loads per unit of length in ball
and trunnion, are respectively,

pbmij =

pij

(4.4)

cb

79

ptmij =

pij

(4.5)

ct

Crowning of needle rollers used in tripode joints is very small and its effect will be
neglected in this section. Effective contact length between needle and trunnion c t is a
constant, while the effective contact length between needle and ball c b is a function of
offset Λi and is defined by Eq. (4.6) through Eq. (4.8). The relationship is plotted in
Figure 4.6,

c b = c bu + c bd

(4.6)

c bu = c buo

ct
− (c buo + Λi ) ≥ 0
2

c
c bu = t − Λi
2

ct
− (c buo + Λi ) < 0
2

c bd = c bdo

ct
− (c bdo − Λi ) ≥ 0
2

c
c bd = t + Λi
2

ct
− (c bdo − Λi ) < 0
2

80

(4.7)

(4.8)

cb @mmD
9.5
9.0
8.5
8.0
7.5
1

2

3

4

5

6

y @radD

Figure 4.6. Effective contact length between needle roller and ball, at 23°

Orain’s [15] assumption of linear load distribution will be considered in this section
and moments trying to rotate the ball in any direction perpendicular to trunnion axis will
be neglected. Therefore, the equivalent concentrated force applied on the ball will be
collinear with the equivalent concentrated reacting force on the trunnion. Under these
conditions, the following relationship can be established for the load distribution
between needle and ball,

 pbuij + pbdij 
 = pbmij c b = pij
cb 


2



(4.9)

Simplifying,

pbdij = 2 pbmij − pbuij

(4.10)

81

The following paragraphs will describe the load distribution at the contact between
needle roller and ball for the three cases previously identified.

Case 1 occurs when pbuij = pbdij . If the ball is symmetric, c b = c buo + c bdo .
Case 2 occurs when offset Λi is positive. Then, summing the moments with respect to
point BD,

c
c 
pij c bdo − pbdij c b  b  − pbuij − pbdij b
2
 2 

(

)

2 
 cb  = 0
3 

(4.11)

Simplifying,

pij c bdo −

2
pbuij c b

3

−

2
pbdij c b

6

=0

(4.12)

Case 3 occurs when offset Λi is negative. Then, doing sum of moments with respect to
point BU,

c 2 
c 
− pij c buo + pbuij c b  b  + pbdij − pbuij b  c b  = 0
2 3 
 2 

(

)

Simplifying,

82

(4.13)

− pij c buo +

2
pbuij c b

6

+

2
pbdij c b

3

=0

(4.14)

Equations (4.10) and (4.12) can be solved simultaneously to obtain pbuij and

pbdij for Case 2,

pbuij =

6 pij c bdo
2
cb

pbdij = 4 pbmij −

− 2 pbmij

(4.15)

6 pij c bdo

(4.16)

2
cb

When pbdij ≤ 0 , the above solution is no longer valid since pbdij can not be
negative. Additionally, such condition is not desired during the operation of the joint
since the load would be concentrated on a very small area. Thus, making pbdij = 0 in
Eq. (4.16) and solving for c b it can be found the minimum contact length required for
ball support when Case 2 is present.

c b min =

3 pij c bdo
2 pbmij

=

3
c bdo
2

(4.17)

Using Eqs. (4.6), (4.7) and (4.8), the effective contact length between needle and
ball for Case 2 can be expressed as,

83

c
c b = t − Λi + c bdo
2

(4.18)

From Eqs. (4.1), (2.2) and (2.18), it is known that Λi is a function of joint angle and
reaches its maximum value when phase angle ψ i equals zero or π. The formerly
mentioned equations are combined with Eqs. (4.17) and (4.18), where c b = c b min and

Θ i ≈ 0 . Such conditions are equivalent to assume the needle does not displace axially
(which is the worst possible condition). Then, it can be solved for the maximum joint
angle where the full contact between needle and spherical roller exists, when offset Λi
is positive,

cos θ

=
bcr Λ +

3a
3a − 2Λo − c bdo + c t

(4.19)

The dimensions of the joint used for the experimental portion of this work are used
to evaluate Eq. (4.19) and θ

bcr Λ +

= 23.56°.

Similarly, Eqs. (4.10) and (4.14) can be solved simultaneously to obtain pbuij and

pbdij for Case 3,

pbuij =

6 
2 2

 − pij c buo + c b pbmij 
3

c2 

(4.20)

b

84

pbdij = 2 pbmij −

6 
2 2

 − pij c buo + c b pbmij 
2
3

c

(4.21)

b

When pbuij ≤ 0 , the above solution is no longer valid since pbuij cannot be
negative.

Additionally, similar to Case 2, such condition is not desired during the

operation of the joint since load would be concentrated on a very small area. Thus,
making pbuij = 0 in Eq. (4.20) and solving for c b it can be found the minimum contact
length required for ball support when Case 3 is present.

c b min =

3 pij c buo
2 pbmij

=

3
c buo
2

(4.22)

If the ball is symmetrical, c buo = c bdo , and Eq. (4.17) equals Eq. (4.22). Using
Eqs. (4.6), (4.7) and (4.8), the effective contact length between needle and ball for Case
3, can be expressed as,

c
c b = t + Λi + c buo
2

(4.23)

Similar to Case 2, Eqs. (4.22), (4.23), (4.1), (2.2) and (2.18) are combined for ψ i =

π / 2, c b = c b min and Θ i ≈ 0 . Then, it can be solved for the maximum joint angle, where
full contact between needle and spherical roller exists, when offset Λi is negative,

85

cos θ

=
bcr Λ −

a
a + 2Λo − c buo + c t

(4.24)

Evaluating Eq. (4.24) numerically, θ

bcr Λ −

= 35.77°.

Load distribution at the contact between needle and ball has been obtained, as
well as the joint angles where the assumption of linear load distribution stops being
valid. Now the analysis will be done for the load distribution between needle and
trunnion, following the same assumptions.

 ptuij + ptdij 
 = ptmij c t = pij
ct 


2



(4.25)

Simplifying,

ptdij = 2 ptmij − ptuij

(4.26)

Case 1 occurs when ptuij = ptdij and Λi = 0
Case 2 occurs when offset Λi is positive. Then, doing sum of moments with respect to
point TD,

c 2 
c

c 
pij  t + Λi  − ptdij c t  t  − ptuij − ptdij t  c t  = 0
2 3 
2

 2

(

)

86

(4.27)

Simplifying,

2
ptdij c t2
 ct
 ptuij c t
−
=0
pij  + Λi  −
3
6
2


(4.28)

Case 3 occurs when offset Λi is negative. Then, doing sum of moments with respect to
point TU,

c 2 
c

c 
− pij  t − Λi  + ptuij c t  t  + ptdij − ptuij t  c t  = 0
2 3 
2

2

(

)

(4.29)

Simplifying,

2
ptdij c t2
 ct
 ptuij c t
− pij  − Λi  +
+
=0
6
3
2


(4.30)

Equations (4.26) and (4.28) can be solved simultaneously to obtain ptuij and ptdij
for Case 2,

ptuij =

6 pij  c t

 + Λi  − 2 ptmij
2  2

ct

(4.31)

87

ptdij = 4 ptmij −

6 pij  c t

 + Λi 

c t2  2

(4.32)

When ptdij ≤ 0 , the above solution is no longer valid since ptdij cannot be
negative. Additionally, such condition is not desired during the operation of the joint
since load would be concentrated on a very small area. Thus, making ptdij = 0 in Eq.
(4.32) and solving for Λi it can be found the maximum offset allowed to have full needle
support when Case 2 is present.

Λ Λ+ =
cr

2 ptmij c t2
3 pij

c
c
− t = t
2
6

(4.33)

Combining Eqs. (4.1), (2.2) and (2.18) with Eq. (4.33), when ψ i = 0, Λi = Λ Λ +
cr
and Θ i ≈ 0 and solving for joint angle,

cos θ

=
tcr Λ +

3a

(4.34)

c
3a − 2Λo + t
3

The dimensions of the joint used for the experimental portion of this work are used
to evaluate Eq. (4.34) and θ

tcr Λ +

= 19.47°.

88

Similarly, Eqs. (4.26) and (4.30) can be solved simultaneously to obtain ptuij and

ptdij for Case 3,

ptuij =

6 
c
 2
 − pij  t − Λi  + c t2 ptmij
2
2
 3
ct 

ptdij = 2 ptmij −






(4.35)

6 
c
 2
 − pij  t − Λi  + c t2 ptmij
2
 2
 3
ct 






(4.36)

When ptuij ≤ 0 the above solution is no longer valid since ptuij cannot be
negative.

Additionally, similar to Case 2, such condition is not desired during the

operation of the joint since the load would be concentrated on a very small area. Thus,
making ptuij = 0 in Eq. (4.35) and solving for Λi , it can be found that the maximum
offset is allowed to have full needle support when Case 3 is present.

Λ Λ− = −
cr

2 ptmij c t2
3 pij

c
c
+ t =− t
2
6

(4.37)

Similar to Case 2, Eqs. (4.37), (4.1), (2.2) and (2.18), are combined for ψ i = π / 2,

Λi = Λ Λ + and Θ i ≈ 0 . Then, it can be solved for the maximum joint angle with the full
cr
contact between needle and trunnion when the offset Λi is negative,

89

cos θ

=
tcr Λ −

a

(4.38)

c
a + 2Λo + t
3

and

(

θ Λ = min θ Λ + ,θ Λ − ,θ Λ + ,θ Λ −
cr
bcr
bcr
tcr
bcr

Evaluating Eq. (4.38) numerically, θ
the joint used in this work is θ

tcr Λ +

)
tcr Λ −

(4.39)

= 28.84°. The lowest critical angle for

= 19.47° = θ Λ . To illustrate this, loads per unit
cr

length ptuij and ptuij are plotted for pij , which is assumed to be 1000N, Θ o = 0 and

θ o = 19.5°, as shown in Figure 4.7 and Figure 4.8.

ptu @N êmmD
200

150

100

1

2

3

4

5

6

y @radD

Figure 4.7. Load per unit length at upper end of needle, at 19.5°

90

ptd @N êmmD
150

100

50

1

2

3

4

5

6

y @radD

Figure 4.8. Load per unit length at lower end of needle, at 19.5°

Figure 4.8 shows that the load at the lower end of the needle gets slightly below
zero, verifying the formulation of the critical angles. Initial joint offset Λo may be
obtained from the geometry of the joint,



Λo = a −  espider + c spacer +


b

+ needle 
2
2 

Θo

(4.40)

91

Figure 4.9. Spider Geometry

4.3 Normal load distribution on needles (radial direction)
In most roller bearings, the outer race is fully supported along its perimeter. In a
tripode joint the outer perimeter of the spherical rollers is supported at only one contact
point in its matching ball bore. This difference in boundary conditions changes the load
distribution among needles due to the flexibility of the spherical roller (outer race). Harris
[18] studied load distributions in bearings loaded at two points, used in planetary gears,
Filleti and Rumbarger [19] also studied the case of a bearing supported at two points,
collecting experimental data to verify their theoretical results, Papadopoulos [20] studied
the load distribution in a roller bearing by experimental methods and Zhao [21] by finite
element methods.

92

This work will apply the method developed by Harris [18] to estimate the radial
load distribution on a trunnion. The main advantage of this method is its compactness
and the ability to be easily incorporated into the program written to calculate kinematic
and quasi-static variables in previous sections. The main disadvantage of the method is
the assumption of considering outer races (balls in our case) as “thin rings” to calculate
deflections. This assumption makes the method inaccurate when dealing with balls that
are “thick rings”. Ugural and Fenster [22] consider that for curved beams in which the
depth of the member is small relative to the radius of curvature, strain energy due to the
bending can be approximated by that of a straight beam. Then, the deflections in a “thin
ring” can be calculated under this assumption. The ratio used by Ugural and Fenster is
given by,

Rb
>4
hb

(4.41)

and the formulation in this section will apply only to joints with the balls that meet this
condition. Before proceeding to describe the method used to calculate the load
distribution on the trunnion, the properties of the area defined by the radial cross section
of the ball will be determined (Figure 4.10). The small chamfers cut in the ball to avoid
sharp corners will be neglected for the purposes of calculating areas and second
moments of area. The specifications of the geometry in Figure 4.10 are the following,

93

1
2
2
mo − c bw
2

y bA =

 c bw
 mo

α b = arcsin


2

(4.42)






(4.43)

2m  3
2
4
ab1 =  o  α b 1 − 0.2α b + 0.019α b
3 2 

(

)

(4.44)

m 

ab 2 = c bw  y bA − i 
2 


(4.45)

ab = ab1 + ab 2

(4.46)

y b1 =

(

(

mo
2
2
4
1 − 0.3α b 1 − 0.0976α b + 0.0028α b
2

))

(4.47)

y
m
y b 2 = bA + i
2
4

(4.48)

y a + y b 2 ab 2
R b = b1 b1
ab

(4.49)

4

m  7
2
4
I b1x = 0.01143 o  α b 1 − 0.3491α b + 0.0450α b
 2 

I b2 x =

(

m 
1

c bw  y bA − i 
12
2 


)

(4.50)

3

(4.51)

I bxx = I b1x + I b2 x + (R b − y b1 )2 ab1 + (R b − y b 2 )2 ab2

94

(4.52)

Figure 4.10. Cross section of spherical roller

Figure 4.11 shows schematically the loading on the spherical roller. Following the
Harris’ approach, only one half of the ball is analyzed due to the symmetry condition.
Superposition is used in conjunction with the method of influence coefficients to
calculate the radial deflections of the ball. The influence coefficient method implies,

pv

∆u = p Cuv pv

(4.53)

where, pv ∆u is the deflection at angle (position) φu due to the load pv and pCuv is the
influence coefficient. Similarly, the deflections q ∆u experienced by the ring at angle φu
due to the external load q i applied to the ball can be obtained from Eq. (4.54).

95

q ∆u = q Cu q i

(4.54)

Figure 4.11. Schematic view of loads

The influence coefficients published by Harris [18] that can be applied to the
tripode problem are given by Eqs. (4.55) and (4.56).



 cos(mφv )

cos(mφu )
p Cuv = −

2
πEY I bxx m = 2  2

 m −1


(4.55)



1


cos(mφu )
q Cu =

2
πEY I bxx m = 2  2

 m −1


(4.56)

3
2R b

3
2R b

∞

∑

(

)

∞

∑

(

)

Using the principle of superposition, the radial deflection of the ring at angle φu
can be obtained from Eq. (4.57),

96

∆u = q ∆u + ∑ pv ∆u = q Cu q i + ∑ p Cuv pv
v

(4.57)

v

It is assumed that the center of the trunnion will remain in its original position after
applying the load to the ball. Therefore, due to the contact deformations and clearance
in the trunnion-needle-ball assembly, the center of the ball will displace a distance δ b
as shown in Figure 4.11. Thus, the relative radial approach δ v between ball and
trunnion at angle φv can be obtained from Eq. (4.58),

δ u = δ b cos(φu )+ q ∆u + ∑ pv ∆u = δ b + q Cu q i + ∑ p Cuv pv
v

(4.58)

v

For the roller compressed between raceways Palmgren [18] gives the following
equation,

pv = k pυv n r

(4.59)

where k p is a material constant defined for this work, based on Palgrems’s
approach, using effective contact lengths at trunnion and ball, as,


1
kp = 

 1/ k pt 1 / n r + 1/ k pb 1 / n r


(

)

(

)

n
 r





(4.60)

97

k pt = 7.86 × 10 4 c t8 / 9

(4.61)

8
k pb = 7.86 × 10 4 c b / 9

(4.62)

Deformation υv is the difference between the relative radial approach δ u and
actual clearance between the trunnion and ball at angle φu . Therefore, the load at the
needle roller with index v can be expressed as,

n
1

 r
pv = k p  δ u − (mi − 2d − t ) − Ω v  ;
2



1


 δ u − (mi − 2d − t ) − Ω v  > 0
2



pv = 0 ;

1


 δ u − (mi − 2d − t ) − Ω v  ≤ 0
2



(4.63)

The assumption is that the first roller position of the n needle rollers is right at the
plane of symmetry of the trunnion. This needle roller is assigned the index v = 2. Thus,
angle φv is defined as,

φv =

2π
(v − 2 + Γ o )
n

(4.64)

where Γ o = 0 if the first roller is at the plane of symmetry. Otherwise, Γ o = 0.5.
Similarly,

98

φu =

2π
(v − 2 + Γ o )
n

(4.65)

The upper limit for both u and v is,

n

+ 2 − Γo 
2


Γ = Floor 

(4.66)

So far only the displacements and deformations have been defined. Equilibrium of
forces must be satisfied in the problem. Therefore, sum of forces on the trunnion must
equal zero.

Γ

qi −

∑ (Γv pv cos φv ) = 0

(4.67)

v =2

where Γv = 0.5 for φv = 0. Otherwise, Γv = 1. Then, we have a set of Γ equations
that is obtained combining and simplifying Eqs. (4.53) through (4.67). This set of
equations can be solved simultaneously for δ t and δ 2 through δ Γ using a numerical
method. Once the relative radial approaches have been determined the load at each
trunnion can be obtained from Eq. (4.63).

99

4.4 Numerical examples of load distribution on trunnions
The equations to calculate load distribution on the trunnion were programmed and
incorporated to the computer program developed and presented in Sections 2 and 3.
The dimensions of the tripode used to generate numerical values in the previous
sections (2 and 3) are used in this section as well. The computer program developed
allows the calculation of load distribution at any joint angle. To show the effect of
applied torque on load distribution and number of loaded needles, the tripode joint is
analyzed at a joint angle of 2± and at a rotational position of 0±. A perfect round trunnion
is assumed and the load distribution is calculated for different levels of applied torque.
Table 4.1 summarizes the loads, reported in Newtons, on each needle roller versus
percentage of yield torque (first row).

φv

100%

75%

50%

25%

10%

0°

7211

5302

3591

1810

763

10°

6233

4673

3124

1586

673

20°

4222

3189

2153

1112

471

30°

2479

1890

1287

665

251

40°

1298

990

658

303

35

50°

571

408

215

8

0

60°

168

51

0

0

0

70°

0

0

0

0

0

Table 4.1. Load distribution (N) at different proportions of yield torque

100

Figure 4.12 shows a graphical representation of the load distribution at 50% (solid
line) and 25% (dashed line) of the nominal yield torque in the joint. The reader will
notice that when a higher torque is applied more needles will be loaded. Ideally, a round
trunnion is “round” but in reality most trunnions have some type of lobing or waviness
occurred during the manufacturing processes. In center-less grinding processes the
trunnions tend to be tri-lobal or penta-lobal. The small deviations from the basic circle
that generate the lobes in the trunnion are in the order or microns and can only be
checked with specialized equipment. In later sections actual measurements of parts will
be shown. The lobing in the trunnion will cause needles that roll at the peaks of the
undulations to be more heavily loaded than those that roll at the valleys.

Figure 4.12. Load distribution at 25% and 50% of yield torque

For a single harmonic causing lobing in the trunnion the function that defines the
deviation from the basic circle can be written as,

101

Ω v = Ω o cos(hw φv )

(4.68)

With more than one frequency, amplitude and phase that need to be considered to
estimate the undulations in the trunnion, Eq. (4.68) can be expanded to Eq. (4.69) and
be substituted in Eq. (4.63).

∞

Ωv =

∞

∑ (Ωw cos(hw φv )) + ∑ (Ωw sin(hw φv ))
w =2

(4.69)

w =2

Figure 4.13 shows a comparison of load distribution at 50% of yield torque
between a perfect circular trunnion (solid line) and a penta-lobed trunnion (dashed line),
roundness error is in the order of 0.010mm. The shape of the penta-lobed trunnion
(phantom line) has been exaggerated for clarity purposes.

Figure 4.13. Load distribution at 50% of yield torque, perfect vs. lobed trunnion

102

It can be observed that even though the load in some of the needles is lower due
to the pentagon shape the central needle will be more heavily loaded than those in the
circular case. This difference in the load magnitude will increase the contact stresses
and reduce the wear life of the part. The contact stresses and estimated fatigue life will
be studied in subsequent sections of this work.

The shape of the trunnion can also be made different from a perfect circle to have
a better load distribution and/or to reduce the highest load in the distribution. “Ellipticallike” shapes have been used by Harris [18] in roller bearings and Orain in tripode joints
[15]. Such “elliptical” shapes may not match the mathematical definition of an ellipse but
“elliptical” is a generic name that has been used in industry.

Figure 4.14 shows a comparison at 50% of yield torque between a perfect circular
trunnion (solid line) and a perfect elliptical trunnion (dashed line), where the difference
between the semi-major axis and the semi-minor axis is 0.031mm. The minor axis is
parallel to the spider axis. The radial deviations of the ellipse with respect to its
minimum circumscribed circle are given by Eq. (4.70). It can be combined with Eq.
(4.69) to form Eq. (4.71) and take into account the shape of the ellipse and its waviness
due to manufacturing. Then, Eq. (4.71) can be substituted in Eq. (4.63) to calculate the
load distribution on the trunnion.

103



2

2

π
1
π 
t


Ω v = − 1 − cos φv +  +
sin φv +  
2
2
2
2 


et






2

2



π
π 
t
1


Ω v = − 1 − cos φv +  +
sin φv +  
2
2
2 


et2




∞

(4.71)

∞

w =2

+

(4.70)

w =2

∑ (Ωw cos(hw φv )) + ∑ (Ωw sin(hw φv ))

Variable et is the elliptical ratio of the trunnion, defined as the ratio of the major to
the minor axis. More terms could be added to Eq. (4.71) and virtually any possible
trunnion shape could be analyzed.

Figure 4.14. Load distribution at 50% of yield torque, circular vs. elliptical trunnion

104

4.5 Making trunnion load distribution a function of joint angle
After finding out how loads are distributed in a tripode joint it is necessary to relate
them to the displacements of the internal components, caused by the rotation of the
joint at angle, in order to estimate the fatigue cycles. It is assumed that the load profile
on a trunnion, such as that shown in Figure 4.12, will change as a function of the normal
load between ball and ball bore. The instantaneous position of the rollers may also
affect the load distribution when they are not symmetrically positioned. However, the
variation in load distribution profile due to non-symmetrical conditions is assumed to be
small and will be neglected. The loads when the rollers are located at non-symmetrical
positions will be obtained interpolating within the load distribution profile obtained from
the symmetrical case. Observation of tripode joints tested in the Nexteer Automotive
validation labs suggests wear tends to start at the intersection of the trunnion surface
and spider plane, which is the region where a roller will see the highest possible loads
and more fatigue cycles. The reader may refer to Figure 4.15 which shows an example
of initial wear in a spider trunnion [23].

Figure 4.15. Example of initial wear in a trunnion

105

It is necessary to define the position of a roller with respect to the plane of
symmetry of the load distribution. This position is defined by angle χ ij , which can be
obtained from Eq. 4.72 and is shown in Figure 4.16.

χ ij = φij − ε i

(4.72)

where i is the trunnion index and j is the needle index

Figure 4.16. Position of roller within load profile

106

Once χ ij is known, the load acting on a roller at any position within the load profile
may be calculated from Eqs. (4.73) and (4.74). Eq. (4.73) is an interpolation function
that uses positions φv and loads pv , calculated from Eqs. (4.63) and (4.64), as data
points.

Ψ = Interpolation[{{φv , pv },..., {φΓ , pΓ } ]

(4.73)

where v = 2, 3, …, Γ

[ ]

pij = Ψ χ ij

(4.74)

c @degD

15
10
5
0
-5
- 10
- 15

1

2

3

4

5

6

y @radD

Figure 4.17. Roller position with respect to symmetry plane of loading profile

107

The angle χ11 , obtained using Eq. (4.72), is plotted in Figure 4.17 at 23±
(continuous line) and 6± (dashed line). The reader is asked to remember that the
calculated critical angle in Section 2 was very close to 6±. That is why the peak-valley
amplitude of χ11 at 6± is very similar to the needle spacing 360± / n, where n = 36.

Figure 4.18 shows the load profile at a joint angle of 6± and zero angular
displacement ψ, when an input torque of 1059Nm is applied. Eqs. (4.73) and (4.74)
were used to obtain the load profile. The type of interpolation chosen is a third order
type.

Pij@N D
3500æ
3000

æ

2500
2000
1500

æ

æ

1000
æ

500

æ
æ

0.5

1.0

æ æ

æ

æ æ

1.5

æ

2.0

æ æ

2.5

æ æ

æ

3.0

Figure 4.18. Load profile at joint angle of 6±

108

c @radD

5. STATE OF STRESS AT THE ROLLER-TRUNNION CONTACT

5.1 Introduction
This section deals with the state of stress at the roller-trunnion contact. The first
introductory part of this section explains the state of stress using a Hertzian formulation
for line contacts. In the second part, the contact stress problem is solved using a
numerical technique based on the Boussinesq solution for point loads. The following
assumptions are made:
a) The yield limit of the material is not exceeded. In other words, all deformation
occurs in the elastic range.
b) Loading is perpendicular to the surface.

The contact area dimensions are small compared to the radii of curvature of the
roller and trunnion. In the third part of this section it is studied the effects of needle roller
crowning and the stress distribution when the applied load is non-symmetrical. In the
fourth part sub-surface stress calculations are described.

In Section 4 it was stated that the contact region to show the first signs of wear is
the contact between trunnion and needle rollers. Contact stresses between balls and
needle rollers and pressure distribution between balls and housing ball bores are not
studied in this work.

109

5.2 Hertizan formulation for line (cylindrical) contacts
The Hertz solution to a general contact problem is not discussed here. The
reader is referred to contact stress and elasticity books such as those written by Ugural
[22] and Johnson [24].

A line contact exists, as its name clearly states, when the contact between two
bodies is along a line and not at a single point. Line contact is an ideal condition where
two bodies of cylindrical shape are in contact and have the same length. A schematic
representation of a line contact is shown in Figure 5.1. The area of contact is a narrow
rectangle of width 2bh and length l h . Pressure distribution is elliptical. The half width

bh is given by Eq. (5.1), the elliptical pressure distribution σ h is given by Eq. (5.2) and
the maximum pressure σ h max is given by Eq. (5.3) [18].

bh =

(

)

(

)

2
2
4 pij 1 − ν s EY + 1 − ν s EY
l hπ
1/ d + 1/ t

 x
1+ 
σh =
b
bh l hπ
 h
2 pij

σ h max =






(5.1)

2

(5.2)

2 pij

(5.3)

bh l hπ

In this work it is assumed that the Poisson’s ratio of steel is 0.3. Both needle roller and
spider are made of steel. Therefore, Eq. (5.1) can be simplified to,

110

bh = 1.522267

pij

d ⋅t
l h EY d + t

(5.4)

Figure 5.1. Ideal cylindrical contact

Experimental evidence in roller bearings [18] suggests that surface fatigue is
originated at points below the stressed surface. Experimental evidence shows that CV
joints operating under normal conditions will fail due to sub-surface fatigue (also named
“spalling”) [25]. Therefore, it is of interest in this work to know the state of stress in the
contact region below the surface. Eqs. (5.5) through (5.11) define the sub-surface stress
components [24], along the Z-axis, shown in Figure 5.1.

111



2


 z 
 1 + 2

b 

 h  −2 z 

σ hx = −σ h max

2
bh 
 z 




 1+  b 

 h



(5.5)



2
 z 

z 
σ hy = −2ν s σ h max  1 +   −

b 
bh 

 h



(5.6)

σ h max

σ hz = −

 z
1+ 
b
 h






(5.7)

2

σ h1 = σ hy

0 ≤ z ≤ 0.436bh

(5.8)

σ h1 = σ hx

z > 0.436bh

(5.9)

σ h3 = σ hz
τh =

(5.10)

σ h1 − σ h3

(5.11)

2

The stress components and the maximum principal shear are plotted and shown in
Figure 5.2, where the continuous line represents principal shear, the small-dashed line
represents σ hy , the medium-dashed line represents σ hx and the large-dashed line
represents σ hz . It is noted that the principal stress σ h1 = σ hy changes to σ h1 = σ hx at

z = 0.436bh . The peak value of the maximum shear occurs at z = 0.786bh , with a value
of 0.3σ h max .

112

s êsmax
1.0

0.8
0.6
0.4
0.2

0.0

0.5

1.0

1.5

2.0

2.5

3.0

z êb

Figure 5.2. Subsurface stresses along the plane of symmetry of the contact

Hertz provided an analytical solution for the load-deflection relationship of a point
contact but not for a line contact. Others have established load-deflection relationships
based on some experiments. Teutsch and Sauer [26] compiled and compared most of
the published load-deflection relationships for rollers in contact. The contact roller to
trunnion in this work is equivalent to the contact between a roller and an inner race.
Teutsch found that Palmgren’s relationship for a roller to inner race contact has
deviated somewhat with respect to more modern and sophisticated relationships. The
main advantage of Palmgren’s load-deflection relationship is its closed form and
simplicity. It is described as Eq. (5.12).

α ij = 3.84 × 10

−5

0
pij .9

(5.12)

c t0.8

113

5.3 Non-Hertizan formulation for line (cylindrical) contacts
Several authors, Conry and Seireg [27], Singh and Paul [28], Oh and Trachman
[29] and Hartnett [30], have tried to solve Non-Hertzian contact problems using
numerical techniques based on the Boussinesq solution for a concentrated load, applied
normally to a homogeneous elastic space. Other authors, Harris [18] and Teutsch and
Sauer [26], have tried to solve the pressure and force distribution in line contacts using
slicing techniques.

The numerical solution published by Hartnett [30] has the advantage of being
three dimensional. This solution divides the contact regions in rectangles and
superposition can be used to calculate sub-surface stresses. Hartnett’s method was
programmed and its convergence was verified in this work with a relatively simple
problem: Contact of a sphere on a plane. The validity of Hartnett’s numerical solution to
line contact problems has been verified experimentally by Hartnett and Kannel [31].

The contact of two elastic bodies is shown in Figure 5.3. The origin of the
coordinate system is located at the first point of contact. If two arbitrary points are
brought into contact by the displacement α of body 1 to body 2, the displacements v1
and v 2 , and the initial separations z1 and z2 of the points inside the contact region,
can be described by Eq. (5.13), and by Eq. (5.14) outside of the contact region.

v 1 + v 2 + z1 + z2 = α

(5.13)

114

v 1 + v 2 + z1 + z2 ≥ α

(5.14)

Figure 5.3. Two bodies in contact

If the area of contact is divided in r rectangles and it is assumed that each
rectangle has a uniform pressure s n , then, the sum of pressures times the area of the
rectangular element equals the applied load pij . This is expressed as Eq. (5.15), where
the product 4ab is the area of a single rectangular element.

r

4ab

∑ sn = pij

(5.15)

n =1

115

It is assumed that negative pressures may not occur on the contact surface.
Therefore,

sn ≥ 0

(5.16)

The pressure s n on an element n will cause a deflection at the element n and a
deflection at an element m in the contact region and vice versa. Then, using
superposition, the total displacement at an element m is given by Eq. (5.17).

r

∑ sn fmn = α − z1m − z2m

(5.17)

n =1

and the influence coefficients fmn are given by Eq. (5.18), which comes from integrating
the Boussinesq solution for a normal point load.

116



(x + b )ln (y + a ) +
fmn = κ


 (y − a ) +



 (x + b ) + (y + a )2
(y + a )ln

 (x − b ) + (y + a )2


(y + a )2 + (x + b )2 

+
(y − a )2 + (x + b )2 


+ (x + b )2 

+
+ (x − b )2 

 (y − a ) + (y − a )2 + (x − b )2 

(x − b )ln
+

 (y + a ) + (y + a )2 + (x − b )2 



 (x − b ) +
(y − a )ln

 (x + b ) +


(5.18)

(y − a )2 + (x − b )2 


(y − a )2 + (x + b )2 



where κ is a material constant, given by Eq. (5.19), when the contacting bodies are
made of the same material.

κ =2

(1−ν s2 )

(5.19)

EY π

Thus, the problem consists of solving the set of linear equations, given by Eq.
(5.15) and Eq. (5.17), constrained by Eq. (5.16), for α and s n . A combination of the
Newton-Raphson and the Gauss-Seidel methods is used in this work to solve the set of
equations.

A sphere-plane contact was evaluated and compared to the exact solution to
debug the code. The radius of the sphere is 10mm and the applied load is 100N. It was

117

assumed both contacting bodies are made of steel. The exact solutions for contact
patch radius, maximum stress and displacement are [32]:

1

 2r w  3
aball = 0.881 ball 
 E



Y

σ ball max

(5.20)

1
2 3
E w
= 0.616 Y 

(5.21)

 4r

 ball 
1

 w2
α ball = 1.55
 2E 2 r
 Y ball

3




(5.22)

where, w is the applied load at the contact. Using Eq. (5.20) it is found that the radius of
the contact patch is 0.1873mm. Therefore, a rectangular mesh of 0.4mm x 0.4mm was
used. When it is not possible to obtain an initial approximation of the size of the contact
patch, a coarse mesh can be used to get a rough estimate of the size of the contact
area. Then, the mesh can be refined. Table 5.1 shows the convergence of the
approximate solution as the number of elements is increased and compares to the
exact solution.

118

Mesh

σ ball max [MPa] α ball [mm]

3x3

1443.7

0.003431

5x5

1367.4

0.003463

7x7

1366.3

0.003486

9x9

1366.5

0.003499

11 x 11

1364.7

0.003500

15 x 15

1364.4

0.003505

41 x 41

1362.4

0.003505

Exact

1362.3

0.003504

Table 5.1. Convergence of maximum pressure and deflection

The reader may observe that with a relatively small number of elements a good
approximation may be obtained.

Figure 5.4. Pressure distribution in a sphere-plane contact

119

Figure 5.4 shows the pressure distribution in the contact, calculated using a 41 x
41 mesh. Once the program has been verified it can be proceeded to analyze the roller
to trunnion contact.

In Section 4 it was found that the load on a needle roller varies within one
revolution of the tripode joint. It was also found that the applied load is not always
symmetrical. This section deals with the contact problem only. Therefore, an arbitrary
load and an arbitrary position of the load will be used to exemplify the pressure
distribution in the contact. In a later section of this work, the stress distributions will be
calculated for different positions of the tripode joint to estimate fatigue life.

The peak load on the trunnion at 25% of the yield torque of the joint, from Table
4.1 is selected arbitrarily. It is assumed an initial effective contact length c t of 9.9mm,
based on the geometry of the needle. The semi-width of the contact area, maximum
contact pressure and deflection, per Hertz solution, may be obtained from Eq. (5.4), Eq.
(5.3) and Eq. (5.12), respectively. Such values are,

bh = 0.0432mm

σ h max = 2694.1MPa

α j = 0.00524mm

120

For the first numerical solution it was assumed no crowning on the needle and a
sharp transition at the end of the contact length. The discontinuity caused by the sharp
edge generates a stress concentration, which is captured by the numeric model. The
reader will observe in Table 5.2 that the more refined the mesh the higher the stress at
the edge of the contact. This makes sense since the stress at a discontinuity, assuming
the material doesn’t yield, tends to infinite. The size of the grid was 0.1mm x 9.9mm.

Mesh

σ h max [MPa] σ hcenter [MPa] α [mm]

9 x 11

2910.3

2623.1

0.006477

13 x 13

2942.0

2605.9

0.006464

21 x 21

3058.2

2604.5

0.006448

41 x 41

3296.9

2597.7

0.006430

Table 5.2. Convergence of maximum pressure and deflection

The difference in the stress value with respect to the Hertzian formulation can be
explained due to the stress concentrations at the edge. The difference in deflection can
be explained due to the lack of crowning since Palgrem’s equation was experimentally
developed for rollers with a specific type of crowning on a flat surface. Figure 5.5 shows
the pressure distribution in the line contact for the 41 x 41 mesh.

121

Figure 5.5. Pressure distribution in a cylinder contact

5.4 Non-Hertizan formulation for contact of rollers with crowning
Needle rollers are typically made by grinding its diameter to a given size.
Subsequently, a tumble polishing process is used to remove sharp edges. The polishing
process causes crowning at the end of the needle rollers. Crowning is defined in this
work as a convex deviation from a perfect cylindrical form. Figure 5.6 shows the
dimensions of the needle roller considered and used in this work. Eq. (5.23) and Eq
(5.24) are the mathematical representation of the crowning in such roller.

Crowning starts at a distance of 2mm from the end of the needle. This distance will
vary depending on the length and diameter of the needle roller.

122

Figure 5.6. Needle roller dimensions

zcw = 0.002(exp(y ) − 1)

for

y ≤ 1.47237

(5.23)

zcw = 0.4067033 − 0.4 2 − (y − 1.46881)2

for

y > 1.47237

(5.24)

where the origin of the reference frame is at the gage height of 2mm, as shown in
Figure 5.6. It must be noted that crowning is symmetrical. The portion of the roller
between the 2mm gage heights is assumed to be a perfect cylinder. Thus, if Eq. (5.23)
and Eq. (5.24) are rewritten with respect to roller’s transversal plane of symmetry,
crowning can be expressed as,

zcw = 0

for

y ≤ 3.405

(5.25)

zcw = 0.002(exp( y − 3.405 ) − 1)

for

3.405 < y ≤ 4.87737

(5.26)

for

y > 4.87737

(5.27)

zcw = 0.4067033 − 0.4 2 − ( y − 4.87381)

2

123

Figure 5.7 shows a graphical representation of crowning, based on Eqs. (5.25)
through (5.27), for the needle roller used in this work.

Zcw
0.030
0.025
0.020
0.015
0.010
0.005
-6

-4

-2

0

2

4

6

y

Figure 5.7. Crowning of 10.81mm needle roller

The formerly mentioned equations are incorporated into the contact model and the
pressure distribution is recalculated. Such distribution is shown in Figure 5.8a and
Figure 5.8b. The reader will notice that there is still a slight “edge contact” effect. Such
effect could be reduced or eliminated by introducing more crowning. Table 5.3 shows
the convergence of the solution as the mesh was refined.

124

Figure 5.8a. Pressure distribution of needle roller with crowning

Figure 5.8b. Lateral view of pressure distribution of needle roller with crowning

125

Mesh

σ h max [MPa] σ hcenter [MPa] α [mm]

9x9

2980.5

2869.9

0.007492

13 x 13

3035.2

2837.2

0.007346

21 x 21

2993.7

2841.7

0.007384

41 x 41

3083.9

2840.3

0.007379

41 x 81

3066.1

2840.2

0.007379

Table 5.3. Convergence of maximum pressure and deflection

So far the assumption has been that the needle roller axis is parallel to the
trunnion axis. This occurs only when there is no clearance in the trunnion-roller-ball
assembly. Manufacturing processes have variation and interference is not desired in the
design. Therefore, the trunnion-roller-ball assembly is designed with clearance. Such
clearance allows “skewing” of the needle roller. When the roller skews the “edge
contact” effect is eliminated since the ends of the needle are not in direct contact with
the trunnion. Figure 5.9 shows a needle skewed with respect to the trunnion.

Figure 5.9. View of needle skewing

126

The clearance at the ends of the needle due to the skewing can be modeled as
additional crowning. Such equivalent crowning is given by Eq. (5.28) and Eq. (5.29).

2

t d
t d
rsk =  +  − (y tan(ζ ))2 −  + 
2 2
2 2

(5.28)





 y tan(ζ )  
 r
zsk = cos arctan
t d   sk


+



 2 2 


(5.29)

where y is the distance from the initial contact point. It may be noticed that the sign of y
does not influence the value of the equivalent crowning. Skew angles of needle rollers
in tripode joints are typically around 4º. Eq. (5.28) and Eq. (5.29) were added to the
contact model and the pressure distribution was recalculated considering a skew angle
of 4º. A mesh of 41 x 41 elements was used. Figure 5.10a and Figure 5.10b show the
pressure distribution including skewing effect.

Figure 5.10a. Pressure distribution of needle roller with crowning and skewing

127

Figure 5.10b. Lateral view of pressure distribution of needle roller with crowning and
skewing

It is clear that the “edge contact” effect is not present and pressure has a “barrel”
type pressure distribution. On the other hand, given the symmetry conditions, the
pressure at the center of the contact equals the maximum pressure. In practice the
needle roller will see perfect alignment with the trunnion only twice per joint revolution
(the reader may refer to Section 2 for details). Therefore, analysis of pressure
distribution in subsequent portions of this work will assume a skew angle of 4º, unless
otherwise indicated.

Sometimes “undercuts” on the cylindrical surface of the trunnion are required due
to the manufacturing constraints. An example of a undercut is shown in Figure 5.11.
When this type of discontinuity is present in the cylindrical surface of the trunnion an

128

edge contact effect will take place. The edge contact may be minimized by having extra
crowning in the needle. However, the additional crowning will concentrate the load more
at the center, thus, causing a higher maximum stress. To run experiments, parts without
undercut will be used to avoid adding an extra variable.

Mesh

σ h max [MPa] σ hcenter [MPa] α [mm]

13 x 13

3233.8

3233.8

0.008857

21 x 21

3233.3

3233.3

0.008863

41 x 41

3233.3

3233.3

0.008868

Table 5.4. Convergence of maximum pressure and deflection

Figure 5.11. Undercut on trunnion surface

Figure 5.12a and Figure 5.12b show the pressure distribution when a 2mm
undercut is present and the needle has been skewed 4º. The maximum stress in the
distribution is 3324MPa. The edge effect will be more pronounced in the absence of
needle skewing.

129

Figure 5.12a. Pressure distribution of skewed needle roller with undercut on trunnion

Figure 5.12b. Lateral view of pressure distribution of skewed needle roller with undercut
on trunnion

So far perfect symmetry has been assumed at the contact. When a equivalent
concentrated load on the needle roller is not centered, the pressure distribution will

130

change. An example of an asymmetrical load distribution is shown in Figure 5.13. Other
cases can be reviewed in Section 4.2. Rotation of the contact elements is not part of the
model. Thus, the slope of the load distribution must be transformed into an equivalent
slope of misalignment to model the asymmetry.

The variables ptuij and ptdij represent the load per unit of length at the ends of
the load distribution. If a unit length is assumed and the formerly mentioned variables
substitute pij in Eq. (5.12) then Eq. (5.30) and Eq. (5.31) are obtained.

Figure 5.13. Case of asymmetrical load, positive offset

131

0.9
α uij = 3.84 × 10 −5 ptuij

(5.30)

0.9
α dij = 3.84 × 10 −5 pduij

(5.31)

Now, the slope of the equivalent misalignment caused by the asymmetrical load
distribution can be calculated along the effective contact length c t , yielding Eq. (5.32).

ξ ij =

α uij − α dij
ct

= 3.84 × 10

 p 0.9 − p 0.9
duij
− 5  tuij




ct







(5.32)

The slope is defined as positive when the equivalent concentrated load is above
the plane of symmetry of the needle or there is a positive offset Λi , as shown in Figure
5.13. The additional equivalent crowning due the asymmetrical load can be estimated
from Eq. (5.33)

zξ = yξ ij

(5.33)

The previous expression, combined with the expressions for crowning and
equivalent crowning due to skewing can be combined and used to accurately describe
the pressure distribution at the contact between needle and trunnion, at any position of
the tripode joint. Thus,

zt = zcw + zsk + zξ

(5.34)

132

Some numeric examples, to illustrate the effect of an asymmetrical load
distribution, are carried out for an arbitrary offset Λi of –1mm. The same concentrated
load magnitude used for previous numeric examples is applied and the calculated
pressure distribution assuming no skewing is shown if Figure 5.14a and Figure 5.14b.
The maximum pressure in the distribution is 3420MPa.

Figure 5.14a. Pressure distribution of needle roller with crowning and a load offset Λi of
–1mm, no skewing

133

Figure 5.14b. Lateral view of pressure distribution of needle roller with crowning and a
load offset Λi of –1mm, no skewing

Figure 5.15a. Pressure distribution of needle roller with crowning and a load offset Λi of
–1mm, skewing of 4º

134

Figure 5.15b. Lateral view of pressure distribution of needle roller with crowning and a
load offset Λi of –1mm, skewing of 4º

If a skewing of 4º is added, while keeping the same offset, a maximum pressure of
3286MPa is calculated. The corresponding pressure distribution is shown in Figure
5.15a and 5.15b.

5.5 Sub-surface stress distribution
Once the surface pressure in each one of the rectangular elements that compose
the contact area is known, it can be proceeded to calculate the sub-surface stress
distribution. The approach to calculate the stress distribution under the surface is
superposition. The solution for stress components in a semi-infinite solid due to the
uniform pressure over a rectangle is published as the appendix in Trachman’s [29]

135

work.

They were originally published by Love [33] and are shown in this work for

reference purposes. Consider the potential functions V and Χ for each element n. Then,

V =

s n dξdη
r

∫A ∫

Χ =∫

A

(5.35)

∫ s n log(z + r )dξdη

(5.36)

where,
r = ( x − ξ )2 + (y − η )2 + z 2

(5.37)

Then, the stress components are given by,

σx =

1
2π

 ∂V
∂2Χ
∂ 2V
 2ν
− (1 − 2ν )
−z

∂z
∂x 2
∂x 2







(5.38)

σy =

1
2π

 ∂V
∂2Χ
∂ 2V
 2ν
− (1 − 2ν )
−z

∂z
∂y 2
∂y 2







(5.39)

σz =

1
2π

 ∂V
∂ 2V

−z
 ∂z
∂z 2







(5.40)

1 ∂ 2V
z
τ yz = −
2π ∂y∂z

(5.41)

τ xz = −

1 ∂ 2V
z
2π ∂x∂z

(5.42)

τ xy = −

1
2π

2
2 

 (1 − 2ν ) ∂ Χ + z ∂ V 

∂x∂y
∂x∂y 



(5.43)

136

where,



∂V
2π − arccos
= −s n


∂z






(b − x )(a − y )


(b − x )2 + z 2 (a − y )2 + z 2 




− arccos




(b − x )(a + y )


(b − x )2 + z 2 (a + y )2 + z 2 




− arccos




(b + x )(a − y )


(b + x )2 + z 2 (a − y )2 + z 2 




− arccos




(b + x )(a + y )


(b + x )2 + z 2 (a + y )2 + z 2 



(5.44)


 z (a − y ) 
a + y 
a − y 
= s n arctan
 + arctan
 − arctan
 α (b − x ) 

b−x
b−x
∂x 2
 1



∂2Χ

 z (a + y ) 
a − y 
a + y 
 + arctan
− arctan
 + arctan

 α (b − x ) 
b + x 
b+ x
 4


(5.45)

 z (a + y ) 
 z (a − y ) 
 − arctan
− arctan
 α (b + x ) 
 α (b + x ) 

 2

 3


∂2Χ


 z (b − x ) 
b−x
b−x
= s n arctan
 a − y  + arctan a + y  − arctan α (a − y ) 









∂y 2
 1



 z(b − x ) 
b+ x
b+ x
− arctan




 α (a + y )  + arctan a − y  + arctan a + y 





 4


(5.46)

 z (b + x ) 
 z (b + x ) 
 − arctan
− arctan
 α (a − y ) 
 α (a + y ) 

 2

 3


 (z + α 1 )(z + α 3 ) 
∂2Χ
= s n log

∂x∂y
 (z + α 2 )(z + α 4 )

(5.47)

137


b−x
= −s n 
2
 (b − x )2 + z 2
∂x


∂ 2V

a − y a + y 

 α + α 

4 
 1

(5.48)

 a − y a + y 
b+x


+
+
α 3 
(b + x )2 + z 2  α 2



a−y
= −s n 
 (a − y )2 + z 2
∂y 2


∂ 2V

b − x b + x

 α + α 

2 
 1

(5.49)

 b − x b + x 


+
+
α 3 
(a + y )2 + z 2  α 4


a+y


b−x
= sn 
2
 (b − x )2 + z 2
∂z

a − y
b+x

+
+
(b + x )2 + z 2  α 2

∂ 2V

+

a − y a + y 

 α + α 

4 
 1
a+y
a−y
+

α 3  (a − y )2 + z 2

b− x b+ x

 α + α 

 1
2 

(5.50)

 b − x b + x 


+
2
2  α
α 3 
(a + y ) + z  4

a+y


∂ 2V
z
= sn 
∂x∂z
 (b − x )2 + z 2


a − y a + y 

 α + α 

4 
 1

(5.51)

 a − y a + y 


−
+
α 3 
(b + x )2 + z 2  α 2


z

1
∂ 2V
1
1
1 
= sn  −
+
−

∂x∂y
α 1 α 2 α 3 α 4 

∂ 2V
z
= sn 
∂y∂z
 (a − y )2 + z 2


(5.52)

b− x b+ x

 α + α 

2 
 1

(5.53)

 b − x b + x 
z


−
+
α 3 
(a + y )2 + z 2  α 4



and

138

α1 =

(x − b )2 + (y − a )2 + z 2

(5.54)

α2 =

(x + b )2 + (y − a )2 + z 2

(5.55)

α3 =

(x + b )2 + (y + a )2 + z 2

(5.56)

α4 =

(x − b )2 + (y + a )2 + z 2

(5.57)

−

π
2

≤ arctan ξ ≤

π
2

and 0 ≤ arccos ξ ≤ π

(5.58)

Once the stress components are known, it is possible to calculate the principal
stresses, which are the roots σ p of the cubic equation,

3
2
σ p + σ Ι σ p + σ ΙΙ σ p + σ ΙΙΙ = 0

(5.59)

where, the stress invariants are,

σΙ = σx +σy +σz

(5.60)

2
2
2
σ ΙΙ = σ x σ y + σ y σ z + σ x σ z − τ xy − τ yz − τ xz

(5.61)

2
2
2
σ ΙΙΙ = −σ x σ y σ z − 2τ xy τ yzτ xz + σ zτ xy + σ xτ yz + σ y τ xz

(5.62)

and the maximum principal shear stress is,

139

τ max =

σ p1 − σ p3

(5.63)

2

Eqs. (5.38) through (5.63) were programmed in addition to the computer program
that calculates pressure distribution in a contact. Similar to the pressure distribution
calculation, the code was debugged comparing the numeric solution to the exact
solution of a sphere-plane contact. The radius of the sphere is 10mm and the applied
load is 100N. It was assumed both contacting bodies are made of steel.

From Eq. (5.20) is known that the radius aball of the contact patch is 0.1873mm
and from Eq. (5.21) is known that the maximum pressure in the contact is 1362.3MPa.
According to the expressions published by Johnson [24], the maximum principal shear
stress has a value of approximately 0.31σ ball max at a depth of 0.48aball . Thus,
maximum shear in the sphere-plane contact problem is 422.31MPa at a depth of
0.89904mm.

Figures 5.16a through 5.16d show the stress components and maximum shear as
a function of the depth. Such figures were generated from the numeric solution of the
sphere-plane contact problem using a 41x41 mesh. The maximum principal shear,
calculated at a depth of 0.9mm using the numeric procedure, is 422.27MPa.

140

sx
0
- 200
- 400
- 600
- 800
- 1000
- 1200

0.00

0.05

0.10

0.15

0.20

0.25

0.30

z

Figure 5.16a. Stress distribution along the depth of the contact, σ x component

sy
0
- 200
- 400
- 600
- 800
- 1000
- 1200

0.00

0.05

0.10

0.15

0.20

0.25

0.30

z

Figure 5.16b. Stress distribution along the depth of the contact, σ y component

141

sz

0.05

0.10

0.15

0.20

0.25

0.30

z

- 200
- 400
- 600
- 800
- 1000
- 1200
- 1400
Figure 5.16c. Stress distribution along the depth of the contact, σ z component

tmax
500

400
300
200
100

0.00

0.05

0.10

0.15

0.20

0.25

0.30

z

Figure 5.16d. Stress distribution along the depth of the contact, τ max

It must be noted that the previously described state of stress is at the element that
matches the position of the applied concentrated load. Shear stress components are

142

zero at the plane of symmetry. The numeric procedure allows calculation of the state of
stress at any element within the mesh. This will prove extremely useful when dealing
with needle rollers asymmetrically loaded.

As a second verification of the program, it is calculated the sub-surface stress
distribution for the contact problem of a crowned needle (no skewing), described in
Table 5.3 and shown in Figure 5.8a and Figure 5.8b. This problem was chosen since it
is a very good numerical approximation to a line contact without having the numeric
instability caused by an edge contact. It is known that the pressure at the center of the
needle roller is 2840MPa and the semi-width of the contact patch, at the plane of
symmetry, is approximately 0.048mm. According to the Hertzian subsurface stress
distribution shown in Figure 5.2, it is expected that at maximum shear stress of

τ max = 0.3smax = 852MPa

will

occur

at

a

depth

of

z = 0.786bh = 0.786(0.048 ) = 0.038mm . It is also expected that the stress components

σ x and σ y will be equal at approximately z = 0.436bh = 0.436(0.048 ) = 0.021mm .

Figures 5.17a through 5.17c show the stress components and maximum shear as
a function of the depth. Such figures were generated from the numeric solution in Table
5.3 for a 41x41 mesh.

143

sx and sy
0
- 500
- 1000
- 1500
- 2000
- 2500

0.00

0.02

0.04

0.06

0.08

0.10

z

Figure 5.17a. Stress distribution along the depth of the contact, σ x component
represented with a solid line, σ y component represented by a dashed line

sz

0.02

0.04

0.06

0.08

0.10

z

- 500
- 1000
- 1500
- 2000
- 2500
- 3000
Figure 5.17b. Stress distribution along the depth of the contact, σ z component

144

tmax
1000

800
600
400
200

0.00

0.02

0.04

0.06

0.08

0.10

z

Figure 5.17c. Stress distribution along the depth of the contact, τ max

From Figure 5.17a it can be verified, graphically, that the point where the stress
components σ x and σ y are equal happens around 0.02mm, which is very close to the
expected value of 0.021mm. From Figure 5.17c it can be obtained that the maximum
shear is approximately 850MPa at depth of 0.035mm, not very different from the
Hertzian calculation of 852MPa at 0.038mm.

At this point it is possible to calculate the state of stress at any location of the
needle-trunnion contact at any position of the joint, under any load conditions. The next
section will deal with the different yield and fatigue theories that will be used to predict
the wear performance of the tripode joint.

145

6. FATIGUE-LIFE THEORIES

6.1 Introduction
Studies on bearing life prediction started in the early twentieth century. The
dominant model since then has been the Lundberg-Palmgren equation, which is the
basis of the current ANSI/ABMA standards [34, 36]. Such standards can be easily
applied for traditional bearings under simple load conditions but not to Tripode joints.
Other authors such as Zaretsky [34, 35] and Ioannides and Harris [36, 37] have
proposed alternative or expanded versions of the Lundberg-Palgrem equation. Such
models are discussed in this section and a new model, specific to Tripode joints, will be
proposed.

6.2 Weibull Equation
Weibull proposed a statistical approach to determine the strength of solids and
observed that the dispersion in material strength for a homogeneous group of test
specimens would follow the following relation,

σ
  1 
ln ln   = ε ln w
 S 
 σ wβ
  







(6.1)

146

where, S is the survival (i.e. completed load cycles) to the fracture strength σ w . Eq
  1 
(6.1) can be rearranged to have the form of a straight line equation, and if ln ln   is
 S 
  
used as the ordinate and ln(σ w ) is used as the abscissa the slope of the line ε can be
defined as the Weibull slope, which is a measure of the dispersion of the data in a
statistical distribution. The scatter in the data is inversely proportional to the Weibull
slope, the lower the value of ε the larger the scatter.

  1 
ln ln   = ε ln(σ w ) − ε ln σ wβ
 S 
  

(

)

(6.2)

Weibull also proposed the idea of relating the probability of survival of a
component to the stressed volume. The larger the stressed volume, the lower the
strength or life of the component. This is mathematically expressed in Eq (6.3).

 1
ln  = f (σ w )dV
S  V

∫

(6.3)

According to Zaretsky [34], Weibull proposed the following probability of survival
equation to predict bearing life

c
f (σ w ) = τ cr N ε

(6.4)

which if substituted in Eq (6.3) yields,

147

 1
N ∝
τ
 cr






c /ε

1/ ε

 1
 
V 

(6.5)

From Hertz theory V and τ cr can be expressed as a function of the maximum
pressure in the contact σ h max , and if a proportionality constant is used Eq (6.5) takes
the form

 1
L = C1
τ
 cr






c /ε

1/ ε

 1
 
V 

∝

1

(6.6)

n
σ hh
max

where nh is reported to be 10.2 for a line contact, for ε = 1.11 and c / ε = 9.3 [38].

6.3 Lundberg-Palmgren Equation
Lundberg and Palmgren applied Weibull analysis to the prediction of rollingelement bearing life, including the depth to the maximum critical shear stress in a
Hertzian contact. Then Eq (6.4), the probability of survival function, takes the form
c

τ N
f (σ w ) = cr
zϑ
cr

ε

(6.7)

which, if substituted in Eq (6.3), and assuming a Hertzian contact, can be expressed as,

148

 1
L = C1
τ
 cr






c /ε

1/ ε

 1
 
V 

ϑ
zcr/ ε ∝

1

(6.8)

n
σ hh
max

where nh is reported to be 8.1 for a line contact, for ε = 1.11, c / ε = 9.3 and ϑ = 2.33
[38].

6.4 Ioannides-Harris Equation
Ioannides and Harris introduced a fatigue-limiting stress to the Lundberg-Palmgren
equation, based on reported endurance data of some bearing applications that
demonstrated virtually infinite fatigue life. It is reported that such bearings were
accurately manufactured from “clean” steel [37]. Then Eq (6.7), takes the form

f (σ w

(τ
) = cr

− τ u )c N ε

(6.9)

ϑ
zcr

Similarly, if replaced in Eq (6.3),



1
L = C1
τ −τ 

u
 cr

c /ε

1/ ε

 1
 
V 

ϑ
zcr / ε ∝

1

(6.10)

n (τ u )
σ h max

149

Ioannides and Harris use the same values of Lundberg-Palmgren for ε , c, and ϑ .
However, the exponent n is a function of (τ cr − τ u ) . If τ u equals 0, the Ioannides-Harris
equation is equivalent to the Lundberg-Palmgren equation.

6.5 Zaretsky Equation
In the Weibull and Lundberg-Palmgren equations the critical shear stress exponent c
and the Weibull slope ε are related, making c / ε the effective critical shear stress-life
exponent, implying that the life of a bearing also depends on the scatter of the life data
used to calculate the exponents. Zaretsky states that most stress-life exponents vary
from 6 to 12 and appear to be independent of the scatter of the data [34]. Following this
observation he modified Weibull’s probability of survival function making the exponent c
independent of the Weibull slope ε . Such equation has the form,

cε
f (σ w ) = τ cr N ε

(6.11)

Similarly, if replaced in Eq (6.3),

 1
L = C1
τ
 cr

c

  1 1/ ε
1
  
∝
 V
n
σhh
  
max

(6.12)

Zaretsky assumes a value for the stress-exponent c = 9 and

150

nh = c +

1

(6.13)

ε

The proportionality constant C1 in any of the previous fatigue-life equations
accounts for several adjustment factors, which may include: reliability factors, material
factors, geometry factors, lubrication factors and temperature factors. These adjustment
factors vary depending on the bearing manufacturer and have been refined over time.
There have also been efforts to standardize them and both Zaretsky and Harris
recognize its importance given all the possible permutations in bearing applications [36,
38].

6.6 Tripode Equation
Lundberg and Palmgren assumed that once initiated, the time a crack takes to
propagate to the surface and form a fatigue “spalled” area, similar to the “spalled” area
shown in Figure 4.15, is a function of the depth to the critical shear stress [38]. This
would imply that fatigue life in a bearing surface is crack propagation time dependent.
However, Zaretsky categorizes the life of rolling elements as a “high-cycle fatigue
problem” [34]. Zaretsky also supports the idea of crack propagation being an extremely
small time fraction of the total running time of the bearing [38].

In the author’s experience, Tripode joints can operate for hundreds, sometimes
thousands of hours, depending on the load conditions, without showing any signs of pits
or spalling. Once the first pit shows on the surface it takes a few hours to get a large

151

spalled area, supporting the idea of crack propagation being a very small fraction of the
total life of a Tripode joint.

Following Zaretsky’s approach for bearings, it is proposed to decouple the life of a
Tripode from crack propagation and the relation, Eq (6.14), to the depth of the maximum
critical shear stress, used in Lundberg-Palmgren model and Ioannides-Harris models, is
dispensed.

ϑ
L ∝ zcr / ε

(6.14)

Based on the author’s observation of the life behavior of Tripode joints, the
following empirical model is proposed:



 1
L = C1C 2 
τ
 eff






1/ ε



1


t 

 l eff φeff 
2


c

(6.15)

Where, τ eff is the effective shear stress, which depends on the static failure
theory used. l eff is the effective contact length, which can be obtained from the nonHertzian contact analysis discussed in Section 5 and is dependent on the crowning of
the needle roller and applied load. φ eff is the effective trunnion arch traveled by a
needle roller during a joint revolution, which times the trunnion radius t/2 yields the
effective arch-length traveled by the roller. Thus, rather than dealing with a stressed

152

volume the Tripode model deals with a projected area subject to contact stress. The
constant C1 is an empirical life adjustment factor for different greases and surface
textures generated by the manufacturing processes used to make the joint and the
constant C2 accounts for the duplication of load cycles when the critical joint angle,
discussed in Section 2, is exceeded.


 φ

C2 =  Ceiling  eff  


 2π / n  


−1

(6.16)

Where the function Ceiling[x] is defined as the smallest integer greater than or equal to
x.

6.7 Testing and prototypes to validate the Tripode model
Once a fatigue-life model has been proposed for the Tripode joint it is necessary to
perform actual experiments to verify the validity of the model. The experiments were
conducted in an industrial lab at Nexteer Automotive Headquarters. The test benches,
an example shown in Figure 6.1, can simulate different speeds, torques and angles that
a Constant Velocity Joint can experience in a vehicle.

153

Figure 6.1. CVJ wear test bench

For the purpose of validating the model, considering and subject to time and
budget constraints, three tests were performed at a fixed joint angle and constant
power. In Section 2 it was found that the critical angle in the Tripode joint used for the
experimental portion of this work is 6°. Testing at an angle higher than 6° allows to
duplicate load cycles. On the other hand, 7° is an angle commonly used in industry to
validate CVJ designs. Thus, a running angle of 7° was chosen. Table 6.1 summarizes
the conditions of the three tests.

154

Test/Schedule 1 (used in parts 1 to 4)
Step
Torque
Speed
Time
01
272 Nm 1200 rpm 20 minutes
02
-80 Nm 300 rpm
1 minutes
Power: 34.18 kW

Angle
7deg
0-8deg @ 0.5Hz

Test/Schedule 2 (used in parts 5 to 8)
Step
Torque
Speed
Time
01
466 Nm 700 rpm 20 minutes
02
-80 Nm 300 rpm
1 minutes
Power: 34.16 kW

Angle
7deg
0-8deg @ 0.5Hz

Test/Schedule 3 (used in parts 9 to 12)
Step
Torque
Speed
Time
01
816 Nm 400 rpm 20 minutes
02
-80 Nm 300 rpm
1 minutes
Power: 34.18 kW

Angle
7deg
0-8deg @ 0.5Hz

Table 6.1. Summary of wear tests conditions

The steps with negative torque were introduced to avoid “grease migration” in the
joint and keep functional lubrication conditions during the test. The tests schedules were
repeated several times until spalling occured in at least one trunnion of one spider.
Periodic inspections were performed to try to find the spalling in its initial stage. Before
discussing the results of the tests it is noted that controlled prototype parts were made
to perform the three tests, using production manufacturing processes. The purpose of
making controlled prototypes was to try to eliminate possible sources of noise in the
test, such as geometric variation from component to component. Some of the key
characteristics of the prototype parts are listed:
1. All parts were made from the same batch of material
2. All parts were made using the same machine in a single lot

155

3. At least four parts of the lot were carefully inspected using a Coordinate
Measurement Machine (CMM)
4. At least four parts were carefully inspected in the trunnion area for roundness.
Surface finish was also inspected.
5. All parts were visually checked for any possible defects (surface dents) or
corrosion that could generate premature spalling.

Figure 6.2. Example of roundness check

Roundness in the trunnions, which is the most critical characteristic for fatigue-life
in a trunnion, given the changes it produces in load distributions, was found to be in the
order of two microns in the inspected parts. Figure 6.2 shows an example of a

156

roundness check. The reader is asked to refer to the appendices to see surface texture
inspection and the effect of filtering in roundness inspection reports.

6.8 Load and stress conditions per test schedule
After knowing the geometrical characteristics of the prototype parts it is possible to
calculate the displacements, loads and the state of stress in the contacts between the
trunnion and the needle rollers. The following average dimensions were obtained from
the prototype inspection.
Housing, BCD

=

40.906mm

Spider shaft length, b

=

300mm

Ball outer diameter, mo

=

34.992mm

Ball inner diameter, mi

=

24.994mm

Ball to needle contact length, cb =

10.39mm

Number of needles, n

=

36

Needle roller diameter, d

=

1.9975mm

Needle roller length, bneedle

=

10.81mm

Trunnion diameter, t

=

20.975mm

Axial clearance of needle, Θ o

=

0.452mm

Height of shoulder, espider

=

13.921mm

Spacer thickness, cspacer

=

1.05mm

Applied torques, tϕ

=

272Nm, 466Nm, 816Nm

157

It is reported that there was part to part variation, as expected with any
manufacturing process, but the variation was so small, with respect to typical tolerances
for Tripode joints, that it can be neglected for the purpose of calculating loads and
stresses. Using the dimensions of the components and the computer programs
developed in previous sections, the kinematic and quasi-static behavior of the joint at
different input torque levels can be known.

From the kinematic analysis of Section 2 it is possible to know the maximum roller
displacement on the trunnion within a revolution. Figure 6.3 shows such displacement at
a joint angle of 7°. The peak to valley height for the exact solution of roller displacement
(considering joint angle fluctuation due to orbiting motion) and the linearized solution
are: φ eff = 10.76° and φ eff = 10.75°, respectively.

f11 @degD
6

4
2
0
-2
-4

1

2

3

4

5

Figure 6.3. Needle displacement

158

6

y @radD

Figure 6.4 shows the offset between the symmetry plane of the ball and the
symmetry plane of the needle within a revolution. It can be observed that in this
particular set of prototypes the amplitude of the offset variation is about 2% of its mean
value.

L1 @mmD

0.0835
0.0830
0.0825
0.0820

1

2

3

4

5

6

y @radD

Figure 6.4. Load offset

Figure 6.5a shows the load variation for one of the torque levels corresponding to
the test schedules. It can be appreciated that the amplitude of the force variation is

159

extremely small with respect to the mean force value. Something similar happens with
the other two torque levels, which are plotted together in Figure 6.5b, where the solid
line corresponds to an input torque of 272Nm, the mid-dashed line to an input torque of
466Nm and the small-dashed line to a torque of 816Nm.

The zero position for rotation is defined at the plane formed by the housing axis
and the spider axle axis, as shown in Figure 6.6. The virtual load variation between a
needle roller and the trunnion at the spider plane can be modeled within a joint
revolution. The load variation is refered as virtual because an actual load at a specific
point on the trunnion only exists when the needle roller is located on that specific point.
Once the needle roller displaces, because of the joint rotation, the load at that specific
point on the trunnion becomes zero until the needle passes through that point again.

q1
4450
4445
4440
4435
4430
4425
4420
1

2

3

4

5

6

y @radD

Figure 6.5a. Variation of load on ball bore, housing plane at 272Nm

160

q1
14000
12000
10000
8000
6000
4000
1

2

3

4

5

6

y @radD

Figure 6.5b. Loads on ball bore, housing plane

Figure 6.6. Plane formed by housing axis and spider axis, zero plane for rotation

Hence, the plots of Figures 6.7a and 6.7b show load variation on the trunnion at
the spider (symmetry) plane and were generated assuming a constant contact between
the needle and the trunnion at such plane. The purpose of performing this virtual load
evaluation was to identify the possible loads on the area of the trunnion that is most
susceptible to fatigue. In Figure 6.7b the solid line corresponds to an input torque of

161

272Nm, the mid-dashed line to an input torque of 466Nm and the small-dashed line to a
torque of 816Nm.

p1 @N D
960
950
940
930
920
910
50

100

150

200

250

300

350

y @degD

Figure 6.7a. Virtual load of needle to trunnion at spider plane, at 272Nm

p1 @N D
3000
2500
2000
1500
1000
0

50

100

150

200

250

300

350

y @degD

Figure 6.7b. Virtual load of needle to trunnion at spider plane

162

The virtual needle to trunnion load was also calculated at 4° and 10° joint angle at
the loads of 272Nm. The results are summarized in Table 6.2 and the maximum and
minimum loads for different torque inputs are summarized in Table 6.3.

θo

φ eff

P1max [N]

P1min [N]

P1mean [N]

P1amp [N]

P1amp/P1mean

4°

6.70°

963.56

943.45

953.51

20.11

0.0211

7°

10.75°

961.31

904.28

932.80

57.03

0.0611

10°

16.83°

956.85

849.27

903.06

107.58

0.1191

Table 6.2. Needle-trunnion load as a function of joint angle at 272Nm

tϕ [Nm]

φ eff

P1max [N]

P1min [N]

P1mean [N]

P1amp [N]

P1amp/P1mean

272

10.75°

961.31

904.28

932.80

57.03

0.0611

466

10.75°

1601.03

1504.38

1552.70

96.65

0.0622

816

10.75°

2771.80

2598.85

2685.33

173.80

0.0647

Table 6.3. Needle-trunnion load as a function of applied torque

The actual load between a needle roller and the trunnion at the spider plane, within
a joint revolution, could be any value between P1max and P1min. Data in Table 6.2
suggests that the probability of having actual lower loads at the spider plane is higher as
the joint angle increases. Thus, just looking at this data, a higher joint angle would
suggest a longer wear life, which is not what happens in reality. In the author’s

163

experience, the higher the joint angle the lower the life of a Tripode Joint. The higher the
joint angle, the higher the area over which the needle roller is moving, thus increasing
1/ ε





1

the probability of a subsurface crack forming. The term 
t 

 l eff φeff 
2


of the proposed

model, Eq (6.15), accounts for this.

On the other hand, significant changes in life occur anytime the critical angle is
exceeded because the load cycles are duplicated. If the joint running angle is high
enough to exceed two times the needle spacing, the load cycles are triplicated. In the
prototype joints used for this work the cycle triplication occurs when φ eff = 20° at θ o =
11.86°. The proposed fatigue model characterizes this cycle multiplication in C2, Eq
(6.16).

It has been explained how the Tripode model captures the effects of joint angle.
Now it needs to be described how the model will capture the effects of the applied
torque and geometry, such as needle crowning and trunnion roundness, in the term

 1

τ
 eff

c


 .



The reader will observe, from Table 6.2, that the maximum possible load at the
spider plane is virtually insensitive to joint angle. P1max occurs at 0° and 180° of rotation.
Referring to Figure 6.4 it can be observed that the maximum load offset occurs at 180°.

164

Thus, this rotational position is what will be used to calculate the state of stress at the
contact. Using the equations of Section 4 and Section 5 it is possible to define the state
of stress in the contact. Table 6.4 summarizes the inputs to calculate the non-Hertzian
pressure distribution.

tϕ [Nm]

Λ × 10 −2

P1max [N]

Ptm [N/mm]

Ptu [N/mm]

Ptd [N/mm]

ξ × 10 −5

272

8.3556

961.31

97.10

102.02

92.18

2.173

466

8.3556

1601.03

161.72

169.91

153.53

3.439

816

8.3556

2771.80

279.98

294.16

265.80

5.635

Table 6.4. Inputs to calculate non-Hertzian contacts

The same approach of Section 5 was used to calculate the pressure distribution at
the contact region on the trunnion. It was checked for convergence starting with a
coarse mesh and making it finer until the maximum pressure and the deformation in the
contact converged to five digits. The pressure distributions for the three different torque
levels are shown in Figure 6.8, Figure 6.9 and Figure 6.10.

165

Figure 6.8. Pressure distribution, input torque of 272Nm

Figure 6.9. Pressure distribution, input torque of 466Nm

Figure 6.10. Pressure distribution, input torque of 816Nm

166

Table 6.5 summarizes some characteristics of the non-Hertzian pressure
distribution, where s0 is the pressure at the origin of the local coordinate reference
frame, smax is the maximum pressure found in the distribution, α is the relative radial
displacement of the roller to the trunnion, n×m is the mesh and l eff is the length of the
area of contact. It is noticed that the point where the maximum pressure occurs is at
upper portion of the trunnion, for this specific set of prototypes. Thus, it is expected that
spalling will be most likely to start at the upper portion of the contact.

tϕ [Nm]

s0 [MPa]

smax [MPa]

α [mm]

n ×m

leff [mm]

272

2135.5

2315.9

0.004311

21×41

8.87

466

2688.0

2910.7

0.006652

27×41

9.39

816

3435.0

3729.7

0.010544

27×41

9.82

Table 6.5. Key characteristics of non-Hertzian contacts

For reference purposes, the Hertzian solution is also tabulated, in Table 6.6, where

σ h max is the maximum pressure in the contact, bh is the semi-width of the contact
area, zcrh is the depth at which τ h max is located, which is the maximum principal shear
stress. From Section 5 it is known that the maximum principal shear occurs at 0.786 bh ,
with a value of 0.3 σ h max . The reader may be able to compare the differences in
pressure, from both solutions. Once the pressure distribution is known, the effective
sub-surface shear stress can be calculated.

167

tϕ [Nm]

σ h max [MPa]

bh [mm]

zcr [mm]

272

1963.4

0.0315

0.0248

589.0

466

2533.9

0.0406

0.0319

760.2

816

3334.0

0.0535

0.0421

1000.2

τ h max

[MPa]

Table 6.6. Key characteristics of Hertzian contacts

Two static failure theories for ductile materials will be considered: The Maximum
Shear Stress Theory (Tresca), Eq (5.63), and the Maximum Distortion-Energy Theory
(Von Mises-Hencky / Octahedral), Eq (6.17).

τ oct =

2
2
2
(σ x − σ y )2 + (σ y − σ z )2 + (σ x − σ z )2 + 6(τ xy + τ yz + τ xz )

Figure 6.11. Max Shear and Octahedral shear at center of contact

168

(6.17)

Figure 6.12. Max Shear and Octahedral shear at point of maximum pressure

Figure 6.11 compares the maximum principal shear versus the octahedral shear
stress at the center of the contact, while Figure 6.12 does the same thing at the point of
maximum pressure, both for an input torque of 272Nm. The analysis is repeated for
466Nm and 816Nm.

tϕ [Nm]

zmax_shear [mm]

272

0.027

642.4

0.024

561.3

466

0.034

809.1

0.030

706.6

816

0.043

1035.1

0.039

903.6

τ max

[MPa]

zoct [mm]

τ oct

[MPa]

Table 6.7. Sub-surface Critical shear stresses at the center of the contact

169

The results for the maximum critical shear stresses τ max and τ o , and the depth at
which they occur are summarized in Table 6.7 and Table 6.8.

tϕ [Nm]

zmax_shear [mm]

272

0.029

707.5

0.026

613.3

466

0.036

888.7

0.033

770.9

816

0.047

1138.3

0.042

988.0

τ max

[MPa]

zoct [mm]

τ oct

[MPa]

Table 6.8. Sub-surface Critical shear stresses at the point of maximum pressure

Up to this point, the assumption of working in the elastic range of the materials
used to make the prototypes has been made. Before continuing the discussion about
the proposed tripode fatigue-life model, it needs to be reviewed if plasticity should be
considered in this work.

The characteristics of the materials used to make the tripode joints for the
experimental portion of this work are listed below.

Spider
Material: SCM-420H (similar to SAE 4118)
Heat treatment: Carburized, quenched and temper
Surface Hardness: 58HRc MIN

170

Needle Roller
Material: AISI 52100
Surface Hardness: 64HRc MIN

The minimum surface hardness of 58HRc on the trunnions of a tripode spider
corresponds to a Vickers hardness of 653Hv, per ASTM E-140, for non-austenitic
steels. The depth of the point where the maximum shear stresses occur is very close to
the surface and for now it will be assumed that the hardness at the region where the
maximum shear stress occur is 653Hv.

Pavlina and Van Tyne [39] recently published linear functions that relate tensile
strength and yield strength to hardness, in non-austenitic steels. Such functions are
expressed as Eq (6.18) and Eq (6.19), respectively.

TS = −99.8 + 3.734Hv

(6.18)

YS = −90.7 + 2.876Hv

(6.19)

The corresponding strength values to a hardness of 653Hv are TS = 2338.5MPa
and YS = 1787.3MPa. Applying the Maximum Shear Stress Theory, Eq (6.20), and the
Maximum Distortion-Energy Theory, Eq (6.21), the sub-surface critical stresses at the
applied torques can be compared to the yield strength of the material.

171

YS
SFmax_ shear = 2

(6.20)

τ max

SFoct =

2YS
3

(6.21)

τ oct

Thus, evaluating Eq (6.20) and Eq (6.21) it can be know if plasticity will occur in
the contacts due to the applied loads. Results are summarized in Table 6.9.

tϕ [Nm]

SFmax_ shear

SFoct

272

1.26

1.37

466

1.01

1.09

816

0.79

0.85

Table 6.9. Ratio of yield shear strength to critical shear stress

Pavlina and Van Tyne [39] reported that the standard error for Eq (6.18) and Eq
(6.19) were 112MPa and 102MPa, respectively. Then, it is clear that plasticity will occur
under a load of 816Nm and possibly at a load of 466Nm when the spider is made to its
minimum hardness of 58HRc.

The plastic behavior in contacts has been studied by different authors using Finite
Element Analysis (FEA) [40] and attempts to solve the problem analytically have also
been made [41]. Even though good correlation between FEA and analytical methods

172

has been reported there is not enough published information on the material constants
used in the corresponding plasticity models for a wide range of materials, specifically,
carburized steel SCM-420H or SAE 4118.

The problem of elastic-plastic behavior in line contacts has also been studied
using simpler approaches, such as that of Johnson, which is described and adopted in
this work [24]. The study of elastic-plastic line contacts is based on the idea that in
applications where rolling contact occurs and the elastic limit of the material is exceeded
the first pass of a roller on a point will induce some plastic deformation and the
corresponding residual stresses, when the second pass occurs the state of stress will
be defined by the combined action of contact stresses and residual stresses. If the
elastic limit is still exceeded more plastic deformation will take place and so on until the
residual stresses have build up to the point where the applied load only causes elastic
deformation. This process of cyclic deformation until the steady state is purely elastic is
known as “shakedown”. Johnson’s approach is based on Melan’s theorem for
shakedown (also known as static lower-bound theorem) which states:

Shakedown occurs whenever a system of residual stress, that satisfies equilibrium
requirements, acts in conjunction with applied loads in such a manner that the yield
criterion is not violated.

If the shakedown limit of a material is exceeded plastic flow will continue until
failure occurs. Johnson’s method assumes an elastic cylinder rolling freely on an

173

elastic-perfectly-plastic half-space [24]. If the elastic limit is not exceeded the contact
area and the contact pressure can be obtained by the methods described in Section 5,
which assume an elastic cylinder rolling freely on an elastic half-space. The assumption
of plane strain (zero strain in the axial direction) is necessary to simplify the problem.
Figure 6.13 shows the coordinate reference frame used. The sub-index r is used to
denote residual stress components.

Then, the assumption of plane strain eliminates the residual shear stresses τ xyr
and τ yzr and makes the remaining components independent of y.

Figure 6.13. Max Shear and Octahedral shear at point of maximum pressure

174

If it is assumed that the plastic deformation is steady and continuous then the
surface of the half-space will remain flat and the residual stresses can be assumed to
be independent of x. The surface is traction free, given the assumption of free rolling.
Therefore, the residual stresses σ zr and τ zxr are also eliminated and the system of
residual stresses reduces to Eq (6.22),

σ xr = f ( z )
σ yr = g ( z )

(6.22)

σ zr = τ xyr = τ yzr = τ zxr = 0

Thus, the principal stresses, due to the combination of contact and residual
stresses are given by Eq (6.23), Eq (6.24) and Eq (6.25).

σ p1 =

1
(σ x + σ xr + σ z ) + 1
2
2

(σ x + σ xr

2
− σ z )2 + 4τ zx

(6.23)

σ p2 =

1
(σ x + σ xr + σ z ) − 1
2
2

(σ x + σ xr

2
− σ z )2 + 4τ zx

(6.24)

σ p3 = ν s (σ x + σ z ) + σ yr

(6.25)

It is also known from Hooke’s law and from the plane strain assumption that

εy =

1
σ y − ν s (σ x + σ z ) = 0
EY

(

)

(6.26)

175

When σ y = σ yr the steady state residual stress in the axial direction becomes,

σ yr = ν s (σ x + σ z )

(6.27)

If the Maximum Shear (Tresca) criterion is used and if σ yr is such that σ p3 is the
intermediate principal stress, the following expression must be satisfied

(σ p1 − σ p2 ) ≤ YS

(6.28)

If Eq (6.28) is expressed in terms of the stress components it takes the form

2
1
2
(σ x + σ xr − σ z )2 + τ zx ≤ YS
4
4

(6.29)

Expression (6.29) can not be satisfied if τ zxr >

governs the shakedown limit [24]. When τ zxr =

YS
. According to Johnson, τ zxr
2

YS
the residual steady state stress in
2

the tangential direction becomes,

σ xr = σ z − σ x

When τ zxr <

(6.30)

YS
the residual steady state stress is defined by
2

176

2
σ xr = σ z − σ x + YS 2 − 4τ zx

(6.31)

Where the stress components in the tangential (x) and radial (z) directions correspond
to the state of stress at the depth of interest and the tangential location where σ xr is
maximized, which ensures shakedown. Table 6.10 shows the state of stress at the
depth where the maximum principal elastic shear stress occurs, z = 0.047mm. The
critical tangential locations are the plane of symmetry of the contact and at the point
where τ zxr is maximum. The residual stresses, are calculated using Eq (6.27) and Eq
(6.31)

Component

x = 0.000mm

x = 0.056mm

σx

–650.0MPa

–821.9MPa

σz

–2926.5MPa

–1434.7MPa

τ xz

0MPa

877.8MPa

σ xr

–489.2MPa

–277.7MPa

σ yr

–1073.0MPa

–677.0MPa

Table 6.10. Elastic stress and residual stress components at z = 0.047mm

Figure 6.14 shows the variation of shear stress τ zxr at the depth where the
maximum elastic shear stress occurs. Then, superposing the residual stresses to the
stress

field

the

principal

stresses

are

177

computed

to

σ p1 = −1139.2 MPa,

σ p 2 = −2926.5 MPa and σ p3 = −2146 MPa. Hence, the assumption of σ p3 being the
intermediate principal stress is verified and the maximum principal shear stress at a
depth of 0.047mm is τ max = 893.7MPa.

Figure 6.14. Shear stress τ zxr at a depth of 0.047mm

If the Maximum Distortion-Energy (Von Mises-Hencky / Octahedral) criterion is
used, the following expression must be satisfied

(σ p1 − σ p2 )2 + (σ p2 − σ p3 )2 + (σ p3 − σ p1 )2 ≤ 2YS 2

(6.32)

Substituting Eq (6.23) through Eq (6.25) and Eq (6.27) into Eq (6.32), simplifying
and solving for σ xr it is obtained,

178

σ xr =

1
(σ z − 2σ x (1 − ν ) + 2νσ z )
2

(

(

1
2
2
2
+
4YS 2 − 3 σ z (1 − 2ν )2 − 4ν (1 − 2ν )σ zσ x + 4 τ zx + ν 2σ x
2

))

(6.33)

Similarly, the state of stress at the plane of symmetry and at the point where τ zxr
is maximum is evaluated and summarized in Table 6.11. The depth where the
octahedral shear is maximum, z = 0.042mm, is considered. The residual stresses are
calculated using Eq (6.27) and Eq (6.33).

Component

x = 0.000mm

x = 0.050mm

σx

–780.8MPa

–870.6MPa

σz

–3046.9MPa

–1684.4MPa

τ xz

0MPa

899.4MPa

σ xr

–226.0MPa

+128.0MPa

σ yr

–1148.3MPa

–1277.5MPa

Table 6.11. Elastic stress and residual stress components at z = 0.042mm

Finally, superposing the residual stresses to the stress field the principal stresses
are computed to σ p1 = −1006.8 MPa, σ p 2 = −3046.9 MPa and σ p3 = −2296.6 MPa.
Hence, the octahedral shear stress at depth of 0.042mm is τ oct = 842.5MPa.

179

It was indicated at the beginning of this Section that the fatigue life problem in
Tripode joints is considered as a “high-cycle fatigue problem”. Therefore, the proposed
model applies only when the joint is operating in the elastic region of the material or
when shakedown occurs in the studied contacts, which is possible only if the square
roots of Eq (6.31) and Eq (6.33) are real, under the assumption of an elastic-perfectly
plastic material model.

6.9 Test results
Bearings are characterized by the L10 life, which is the life or number of cycles
that can be completed by 90% of a population of bearings. The same L10 definition is
adopted for tripode joints. Given there are twelve trunnions per test it was expected that
one or two trunnions would show pits or spalling before the rest. Therefore, the life of a
tripode on the conducted tests is defined as the number of cycles that ten out of twelve
or eleven out of twelve trunnions can complete without presenting spalling.

The tests were run per the schedules shown in Table 6.1. A conservative a priori
life estimation was done and joints were inspected on regular intervals until pits or
spalling were found in at least one trunnion. In two of the tests the spalled parts were
replaced by new ones and the evaluation continued to verify the expected dispersion in
life data. The test results are summarized in Table 6.12. The naming convention for the
tested spiders during the rest of this work will consist of using a letter to indicate the test
and the number the part had in the test (i.e. Part B-2).

180

Test / Torque

Part 1

Part 2

Completed

Part 3

Part 4

Completed

Completed

605hrs: 2mm of

605hrs: 1 pit on

808hrs:

Clean

808hrs:

Clean

spalling

one trunnion

surface

(test

surface

(test

2

A – 272Nm

pits

and

on

2

Completed

suspended)

one

suspended)

trunnion
Completed

Completed
2

Completed

246hrs: 5mm of

B – 466Nm

246hrs: 1 pit on

120hrs:

spalling

one trunnion

of

pits

and

on

5

Completed
2

246hrs:

Clean

on

surface

(test

56mm

spalling

one trunnion

suspended)

Completed

one

Completed

trunnion
Completed

C – 816Nm

96hrs:

Completed
2

Clean

96hrs: 26mm of

96hrs:

(test

spalling on one

surface

trunnion

suspended)

surface
suspended)

Clean
(test

96hrs:

Clean

surface

(test

suspended)

Table 6.12. Test results summary

Pictures of the spalled and/or pitted surfaces are shown in Figures 6.15 through
6.17.

Figure 6.15. Parts A-1 and A-2

181

Figure 6.16. Parts B-3 and B-1

Figure 6.17. Part C-2

It can be observed in the previous pictures that the location of the pits or spalling is
in the upper portion of the trunnion. The exception would be Part B-3 where the exact

182

location of the point of spalling initiation can not be determined given the large area.
Analysis using traditional Hertzian line contacts, uniform pressure along the length of
contact, would have not been able to predict the region of the trunnion where the
spalling was most likely to initiate. The non-Herztian contact analysis used in this work
allows so.

Part B-2 was checked under the confocal microscope. It was removed from test
when only a small pit was present, indicating the spalling had just initiated. Figure 6.18a
and Figure 6.18b show the scanned surface, which given the curvature of the trunnion
was obtained scanning a clean portion of the surface and then scanning and subtracting
the collected data from the region with the pit.

Figure 6.18a. Part B-2 (units are microns)

183

Figure 6.18b. Part B-2 (units are microns)

The depth of the pit has approximately the same value of the calculated depth
where the critical shear stress occurs. This fact supports the assumption of the spalling
being generated by a crack that starts under the surface at the region where the critical
shear stress occurs.

It is assumed in this work that the Weibull slope representing the scatter of the life
data for tripode joints is the same Weibull slope present in bearings, which is ε = 1.11
[38]. Table 6.13 summarizes the actual life in cycles and the geometric variables that

184

are input into Eq (6.15) to do curve fitting, using least-squares, for the exponent c and
for the constant C1 .



 1
L = C1C2 
τ
 eff






1/ ε



1


t 

 l eff φeff 
2


c

(6.15)

The trunnion diameter t and the constant C 2 used in Eq (6.15) are 20.975mm and
0.5, respectively. The time under load in Table 6.13 was obtained by multiplying the
time on test by a factor 20/21, which is based on the schedules from Table 6.1. A time
of twenty minutes of load and one minute of cooling and grease distribution per every
twenty one minutes of testing.

Torque

Speed

Time on

Time under

Life ×106

[Nm]

[rpm]

test [Hr]

Load [Hr]

[cycles]

272

1200

605

576

41.49

10.75°

8.87

466

700

120

114

4.80

10.75°

9.39

816

400

96

91

2.19

10.75°

9,82

φ eff

l eff

[mm]

Table 6.13. Inputs to best-fit life equation

Table 6.14 summarizes the effective shear stresses used in Eq (6.15) to do curvefitting, the fitted values of c and C1 and the predicted life after the curve fitting.

185

c = 9.462

Torque
[Nm]

c = 9.146

Life ×106

Life ×106

Life ×106

C1 =1E30

C1 = 3.42E28

[cycles]

[cycles]

[cycles]

Actual

τ max

τ oct

τ max

[MPa]

τ oct

[MPa]

272

707.5

613.3

41.49

41.32

41.49

466

888.7

770.9

4.80

4.54

4.87

816

893.7

842.5

2.19

4.13

2.07

Table 6.14. Life prediction summary

It can be observed from Table 6.14 that the values of the best-fit constant c are
within the range of stress-life exponents reported for bearings, which is 8.4 to 12 [38].
The octahedral shear criterion seems to be more appropriate to predict the life of
Tripode joints. Figure 6.19 compares the life predictions versus the actual life in a loglog scale, where the circles represent the actual life, the squares represent the
maximum principal shear criterion and the triangles represent the octahedral criterion.

Figure 6.19. Actual versus theoretical life

186

Residual stresses were checked experimentally by X-Ray diffraction in one of the
trunnions of Part C-4. An unloaded area was inspected at various depths to quantify the
residual stresses produced by the grinding process used to manufacture the prototypes.
Figure 6.20 shows the residual stress gradient, where the circles represent the stresses
in the circumferential direction and the triangles represent the stresses in the axial
direction. Residuals stresses were also checked at the surface and at a depth of
0.042mm in the area where the maximum contact pressure was calculated to occur,
after test. Squares in Figure 6.20 represent the steady state circumferential residual
stress and diamonds represent the steady state axial residual stress.

Figure 6.20. Residual stresses in Spider C-4

The reader may refer to the Appendices to see more data relative to the residual
stress analysis by X-Ray diffraction. The residual stresses experimentally determined
were superposed to the calculated elastic stresses. New critical shear stresses were

187

calculated and the corresponding curve-fitting was repeated. Results are summarized in
Table 6.15

C = 9.345

Torque
[Nm]

c = 9.700

Life ×106

Life ×106

Life ×106

C1 =1.02E29

C1 = 2.89E29

[cycles]

[cycles]

[cycles]

Actual

τ max

τ oct

τ max

[MPa]

τ oct

[MPa]

272

601.1

529.6

41.49

41.47

41.50

466

747.7

664.6

4.80

5.13

4.36

816

850.8

693.3

2.19

1.47

2.78

Table 6.15. Life prediction summary, residual stresses included

Figure 6.21. Actual versus theoretical life after superposing residual stresses

Figure 6.21 compares the life predictions after including the experimentally
determined residual stresses. The circles represent the actual life, the squares

188

represent the maximum principal shear criterion and the triangles represent the
octahedral criterion.

The life prediction using the Octahedral Shear criterion fits better actual life values
than the prediction using the Maximum Shear criterion, where a better fit means a lower
value of the sum of squared errors. The better fit using the Octahedral Shear criterion
holds true when the residual stresses, caused by the manufacturing process used to
make the trunnions, are neglected or included.

A stress-life exponent of c = 9.7 was reported by Baughman [42] for rolling contact
fatigue of bearings, which is similar to best-fit c value determined in this work, using the
Octahedral shear criterion and including residual stresses. Then, Eq (6.15) can take the
form of Eq (6.34), which is the expression that will be used in the remaining portion of
this work.



 1
L = 2.89 × 10 29 C2 
τ
 eff








1


t 

 l eff φeff 
2


0.9

9.7 

(6.34)

Material analysis was performed in Part A-2, the actual chemical composition of
the batch of material used to make the prototype spiders can be reviewed in the
Appendices of this work. The average surface hardness before testing was measured to
be 61.5HRc. After testing, Part A-2 was sectioned close to the generated pit and a
hardness traverse check surface to core was performed, shown in Figure 6.22.

189

Hardness in the region of maximum shear was found to be 62HRc (743Hv), which
supports the idea of an elastic contact for the tests at 272Nm and 466Nm.

800
700

Hardness [Hv]

600
500
400
300
200
100
0
0

0.5

1

1.5

2

2.5

3

Depth [mm]

Figure 6.22. Hardness gradient in trunnion

190

3.5

4

7. TRUNNION FORM OPTIMIZATION

7.1 Introduction
It has been studied In the previous sections the way loads are transmitted inside
of the Tripode Joint (Section 3) after an input torque is applied, how those loads are
distributed among needle rollers (Section 4), the state of stress in the contacts because
of the loads (Section 5) and the relationship between the state of stress and the wear
fatigue-life of the joint (Section 6). One of the most effective ways of improving the life of
a Tripode Joint, while keeping the same materials, the same packaging and similar
manufacturing processes is to improve the load distribution among needles to reduce
the peak load on the most heavily loaded contact. The consequence of reducing the
peak load will be a reduced stress. Therefore, an increment in the number of cycles a
joint can survive before spalling. This Section deals with the optimization of the trunnion
form to minimize the peak load at the contacts.

7.2 Elliptical vs. Circular Trunnion
It was mentioned in Section 4 that a trunnion can be made with an elliptical
shape to have a better load distribution among the needle rollers. The elliptical form was
defined based on radial deviations from the minimum circumscribed circle. Eq (4.70)
describes such deviations, where et is the elliptical ratio, defined as the ratio of the
major to the minor axis of the ellipse.

191



2

2



π
1
π 
t


Ω v = − 1 − cos φv +  +
sin φv +  
2
2
2
2 


et


(4.70)



The author has tested joints made with elliptical trunnions and equivalent joints
made with circular trunnions. In the author’s experience parts made with elliptical
trunnions perform better than parts made with circular trunnions when tested on the
same durability schedule [43], which supports the idea of lower stresses due to a better
distribution of contact loads. For illustration purposes, Schedule 3 of Table 6.1 will be
used to show the estimated life improvement when an elliptical trunnion is used, with
respect to a perfect circular trunnion.

Table 7.1 shows a comparison of the load distributions on the trunnion at an
input torque of 816Nm and at a joint angle of 7°, while Table 7.2 shows a comparison of
the input variables for Eq (6.34), which is used to calculate the joint life at different
stress levels.



 1
L = 2.89 × 10 29 C 2 
τ
 eff








1


t 

 l eff φeff 
2


0.9

9.7 

(6.34)

192

Circular

Elliptical

[N]

[N]

0°

2772

2632

10°

2386

2257

20°

1661

1589

30°

991

1018

40°

494

610

50°

146

342

60°

0

87

70°

0

0

80°

0

0

φv

Table 7.1. Load distribution, circular vs. elliptical trunnion (et = 1.003)

The reader will notice in Table 7.1 that an extra needle carries load in the elliptical
trunnion and the peak load goes down by 5%. Form errors and waviness will be ignored
for the sake of illustrating the nominal differences between different trunnion
geometries. Also, for the purpose of illustrating differences between trunnion profiles, it
will be assumed that the trunnion is hard enough to neglect plastic deformation and only
elastic contacts will be considered.

193

It can be observed in Table 7.2 that octahedral shear stress in the elliptical
trunnion is lower than in the circular trunnion by 2.3% while there is an increment in
predicted life of 24.4%, a significant improvement.

Variable

Circular

Elliptical

P1max [N]

2772

2632

s0 [MPa]

3435.0

3355.9

smax [MPa]

3729.7

3646.9

α [mm]

0.010544

0.010095

φ eff

10.75°

10.75°

l eff [mm]

9.82

9.82

zoct [mm]

0.042

0.041

988.0

965.6

0.090

0.112

τ oct

[MPa]

L10 ×106
[cycles]

Table 7.2. Life comparison, circular vs. elliptical trunnion at 816Nm and 7°

7.3 Optimized vs. Elliptical Trunnion
The advantages of an elliptical trunnion over a circular trunnion are clear. Then, it
is necessary to find out if there is another trunnion form that will distribute contact loads
better than a perfect ellipse, an optimum shape. The computer program for the load
model described in Section 4 was modified to make it compatible with commercially

194

available optimization tools. Two tools were selected, the SOLVER function included in
a commercial spreadsheet software (Tool A) and a specialized optimization software
(Tool B) [44]. The optimization problem was set up the following way.

Define a load on each needle roller as pv , where v = 1, 2, 3, … , n. The number of
needle rollers is indicated by n. The angular location of the needle roller φv is
established with respect to the plane of symmetry of the load distribution, where the
maximum load pv max occurs when φv = 0° and v = 1. The form of the trunnion is
defined by the radial deviations Ω v from the trunnion base circle.

Figure 7.1. Load distribution on a trunnion

195

The SOLVER function of Tool A can handle optimization of only one variable.
Therefore, the objective function was to minimize the difference of loads with respect to
the maximum load.

 int  n +1

 4 


 

(pv max − pv )2 
min
 v =2






∑

(7.1)

For n = 36. Subject to the trunnion having a symmetrical shape and to the constraints

pv max ≥ pv ≥ pv +1

(7.3)

Ω1 = 0.031
Ω10 = 0

(7.4)

Changing the values of Ω v for v = 2, …, 9. Constraint Ω1 = 0.031 mm defines the
“elliptical gap”, which is the same gap of the ellipse used to calculate Table 7.1. The
elliptical gap is defined as the radial deviation from the base circle of the trunnion at the
spider plane.

Tool B has the capability of doing multi-objective optimization. Therefore, two
variables were targeted to be minimized: The difference of loads with respect to the
maximum load and the maximum load.

196


 int  n +1
 4 




2
(pv max − pv )  ;
min
 v =2






∑

min(pv max )

(7.5)

Subject to the trunnion having a symmetrical shape and to the constraints

pv − pv +1 ≥ 0

(7.6)

Ω1 = 0.031
Ω10 = 0

(7.4)

Figure 7.2. Pareto Front

197

A Pareto Front was generated for Eq (7.5), shown in Figure 7.2. The results of the
two optimizations are shown and compared in Table 7.3.

Elliptical

Elliptical

Tool A

Tool A

Tool B

Tool B

[N]

φv

Ωv

[N]

Ωv

[N]

Ωv

[mm]

[mm]

[mm]

0°

2632

0.0310

1714

0.0310

1702

0.0310

10°

2257

0.0301

1714

0.0270

1698

0.0270

20°

1589

0.0274

1714

0.0181

1689

0.0179

30°

1018

0.0232

1714

0.0076

1650

0.0074

40°

610

0.0182

1315

0

1448

-0.0015

50°

342

0.0128

0

0.0082

0

0.0027

60°

87

0.0077

0

0.0047

0

0.0077

70°

0

0.0036

0

0.0044

0

0.0002

80°

0

0.0009

0

0.0014

0

0.0000

Table 7.3. Load distribution, circular vs. elliptical vs. optimized trunnion

Even though less needle rollers are loaded in the optimized profile the load is
more uniformly distributed and there is a significant reduction in the peak load with
respect to the elliptical trunnion, 35.3% when the best optimized solution (Tool B) is
considered. If the two optimum solutions are compared it is clear that the solution

198

obtained using Tool B features a lower peak load and a reduced difference between
adjacent loads and the peak load, which were the optimization targets. Table 7.4 shows
a comparison of the pressure, critical stress and predicted life between the circular, the
elliptical and the best optimized trunnion profile.

Variable

Circular

Elliptical

Optimized

P1max [N]

2772

2632

1702

s0 [MPa]

3435.0

3355.9

2763.4

smax [MPa]

3729.7

3646.9

3031.1

α [mm]

0.010544

0.010095

0.007001

φ eff

10.75°

10.75°

10.75°

l eff [mm]

9.82

9.82

9.54

zoct [mm]

0.042

0.041

0.034

988.0

965.6

802.8

0.090

0.112

0.690

τ oct

[MPa]

L10 ×106
[cycles]

Table 7.4. Life comparison, different trunnion forms at 816Nm and 7°

The advantages of an optimized trunnion over an elliptical trunnion are clear.
Reduction in maximum pressure of 16.8% and an increment in predicted life of more
than six times, at 816Nm. It must be noted that given the non-linearity of contact
stresses, with respect to torque, the optimum trunnion profile would be different if a
different input torque were used.

199

8. CONCLUSIONS

The kinematics of a Tripode Joint has been studied in this work. A Tripode Joint is
not a perfect constant velocity joint and the effective angular displacement of ball rollers
changes as a function of joint angle. A higher joint angle means a higher deviation from
the perfect constant-velocity behavior and an integral approach has to be followed to
calculate the angular displacement of rollers.

At joint angles where the Tripode Joint is tested and validated (i.e. 7°) the nonconstant velocity behavior causes very small changes in the loads that are transmitted
between balls and ball bores, less than 0.1%. Such changes can be neglected for the
purpose of calculating the loads that are transmitted to the needle rollers.

The effective arch φ eff traveled by a needle roller on a trunnion can be calculated
neglecting joint angle variation at joint angles where the Tripode Joint typically operates
and kinematic equations, such as those used to calculate roller displacement, can be
linearized.

At typical functional angles, 7° and lower, the Tripode Joint can be considered a
Constant Velocity Joint even though it exhibits small deviations from constant-velocity
behavior.

200

The main contributor to Generated Axial Force (GAF) in a Tripode joint is the
sliding friction between balls and ball bores. The rolling friction between trunnions,
needle rollers and balls is very small and allows modeling of the contacts between the
trunnion and the needle rollers as frictionless contacts.

The state of stress in the contact between a needle roller and a trunnion can be
approximated considering Hertzian line contacts. However, non-Hertzian contacts are a
more complete solution. Non-Hertzian contact analysis, which can be performed
meshing the contact area in small rectangles, assigning a load to each rectangle and
applying superposition, allows calculation and understanding of the pressure distribution
in the contact. Factors that may influence the pressure distribution in the contact
between a trunnion and a needle roller, such as needle crowning, needle skewing, edge
loading and asymmetrical loading, were considered in this work and its effects were
individually studied.

The area in the contact where maximum pressure was calculated to occur
matched the area where trunnion pitting in tested parts initiated, verifying the
advantages of non-Hertzian contact analysis over traditional line-Hertzian contacts.

The pressure distribution on a contact causes sub-surface stresses. Two static
failure theories for ductile materials were considered in this work to calculate the critical
shear stress under the surface: The Maximum Shear Stress Theory and the Maximum
Distortion-Energy Theory. The Maximum Shear Stress Theory predicts the critical shear

201

stress to be at a deeper point than the location predicted by the Maximum DistortionEnergy Theory.

Wear fatigue-life in a Tripode Joint is defined when spalling or pitting occurs on the
surface of a trunnion due to repeated rolling contact. A pit always precedes a large
spalled area. A pit forms due to a crack under the surface at the depth where the critical
shear occurs. In this work it was found that the depth of a newly generated pit was in
the same region where the critical shear stress was predicted to be, which validates the
assumption of a pit initiating under the surface.

Residual stresses were included as part of the contact stress analysis for life
prediction. Theoretical residual stresses due to shakedown of the material were
calculated considering an Elastic-Perfectly Plastic model and were found to be
compressive. Residual stresses in the trunnion were measured using X-Ray diffraction
techniques and found to be compressive after testing in the contact area, which
confirms the existence of shakedown in the material.

Theoretical residual stresses were directionally correct when compared to
experimentally determined residual stresses and in a similar order of magnitude. The
use of more sophisticated plasticity models would allow a better prediction of residual
stresses.

202

Measured residual stresses were superposed to the calculated elastic state of
stress. These stresses, in combination with the corresponding actual life of Tripode
Joints obtained from testing, were used as data points to best-fit a stress-life exponent
and a proportionality constant for the Tripode Equation proposed in this work.

The critical stress values determined using the Maximum Distortion-Energy Theory
provided a better fit than the values obtained using the Maximum Shear Stress Theory.
Therefore, the Maximum Distortion-Energy Theory is preferred to predict wear fatiguelife in Tripode Joints.

The final form of the Tripode Equation proposed in this work is

1/ ε



 1
L = C1C2 
τ
 eff








1


t 

 l eff φeff 
2


c


 φ

C2 =  Ceiling  eff  


 2π / n  


−1

Where c = 9.7, ε = 1.11 and C1 = 2.89 × 10 29 . The model can be used to predict wear
life-fatigue in Tripode Joints.

The value of the constant C1 is specific to the type of grease and manufacturing
processes used to make the Tripode Joints used in this work, which cover a wide range

203

of automotive applications. A different grease or different manufacturing process would
require the experimental determination of a new constant C1.

The best way to improve wear life-fatigue in a Tripode joint is to minimize the peak
load at the contacts between the trunnion and the needle rollers, and at the same time
minimize the load differences between adjacent needles. Optimizing for these two
objective functions at the same time generated a Pareto front that converged to one
point, which uncovered that the objective functions are not in contraposition.

Elliptical trunnions distribute load among needle rollers better than circular
trunnions, which has the advantage of better durability.

An optimized trunnion shape can dramatically increase the wear life-fatigue in a
Tripode Joint, when compared to perfectly round and perfectly elliptical trunnions. The
improvement can be done without changing materials, processes and the size of the
joint.

The load analysis used in this work allows the incorporation of form errors and
trunnion waviness. This allows the possibility of optimizing the trunnion shape including
expected manufacturing variation.

During regular operation in a vehicle, a Tripode joint will see different loads and a
load history that will depend on the application (i.e. load history in city driving is different

204

from load history in freeway driving). The model developed in this work would allow to
extend the optimization problem previously discussed to include load history, where the
optimization of the trunnion is not done to minimize the contact loads for a specific
torque but to minimize the size of the trunnion or to maximize the life of the joint under a
series of different loads and the time the joint spends at each load. In other words,
optimize the size and/or the life of a Tripode Joint for a specific load history.

The results and models developed in this work can have an immediate and direct
application in the auto-industry.

205

APPENDICES

206

APPENDIX A
SURFACE FINISH INSPECTION BEFORE AND AFTER TEST
Ra = 0.18um

Figure A.1. Surface texture before testing

Ra = 0.09um

Figure A.2. Surface texture after testing

207

APPENDIX B
RESIDUAL STRESS ANALYSIS

Residual Stress, [MPa]
Depth,
[mm]

As
Measured

Corrected
for Stress
Gradient

Corrected
for Layer
Removal

Error,
±[MPa]

Peak
Width, [º]

1. Marked Location on Trunnion, Axial
0.000
0.025
0.032
0.042
0.055

− 375.5
−120.2
−164.4
−196.9
−204.1

− 417.6
−124.5
−140.0
−188.3
−201.7

− 417.6
−123.2
−138.6
−186.5
−199.4

21.2
16.9
23.3
32.0
21.7

5.95
6.16
6.40
6.41
6.47

2. Marked Location on Trunnion, Circumferential
0.000
0.025
0.032
0.042
0.055

−128.1
−236.1
−295.4
−279.7
−319.2

−112.5
−213.2
−282.6
−275.4
−305.9

−112.5
−212.8
−282.0
−274.5
−304.7

21.3
23.1
23.7
23.7
24.0

5.85
6.27
6.23
6.22
6.56

3. 90º from Original Location on Trunnion, Axial
0.000
0.042

−1206.6
−873.3

−1232.8
−900.4

−1232.8
−891.8

27.8
27.2

4.73
4.00

4. 90º from Original Location on Trunnion, Circumferential
0.000
0.042

−385.9
−595.2

−367.2
−576.7

−367.2
−574.8

Table B.1. Residual stresses

208

15.5
12.2

4.74
3.97

APPENDIX C
MATERIAL ANALYSIS OF PROTOTYPE MATERIAL

TEST RESULTS:
CHEMISTRY
Specification: JIS-G4052 SCM-420H
Actual: JIS-G4052 SCM-420H
Sample
Spider

C
0.20

Mn
0.80

P
0.027

S
0.009

Si
0.223

Ni
0.11

Cr
1.03

Sample
Spider

Al
0.035

Cu
0.142

Pb
0.0000

B
0.0001

Nb
0.043

Zr
0.001

Ti
0.004

Sample
Spider

Mo
0.16

V
Ca
0.010 0.00000
Table C.1. Chemical composition

HARDNESS
Trunnion B was sectioned transversely adjacent to the pitting in order to determine
the hardness profile. A microhardness traverse was performed from the surface to
the core with the readings taken in HV500g and converted to HRC.
Specification:
Finished trunnion surface hardness to be 58 HRC minimum
Finished trunnion case depth to 50 HRC to be 0.8mm minimum
As a reference: finished trunnion total case depth to be 1.0mm
minimum
Table C.2. Hardness

*Readings from taken microhardness traverse

209

*Actual:
62 HRC
1.1mm
2.0mm

APPENDIX D
TRUNNION ROUNDNESS INSPECTION REPORTS AND EFFECTS OF FILTERING

Figure D.1. Roundness with filter 0-15UPR

210

Figure D.2. Roundness with filter 0-50UPR

211

Figure D.3. Roundness with filter 0-150UPR

212

Figure D.4. Roundness with filter 0-500UPR

213

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214

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