1 a Semi-Analytical Load Distribution Model of Spline

A SEMI-ANALYTICAL LOAD DISTRIBUTION MODEL OF SPLINE JOINTS

DISSERTATION

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

Jiazheng Hong, B.S.

Graduate Program in Mechanical Engineering

The Ohio State University

2015

Dissertation Committee:

Professor Ahmet Kahraman, Advisor

Professor Robert Siston

Professor Soheil Soghrati

Professor Sandeep Vijayakar

Jiazheng Hong

2015

ABSTRACT

While spline joints are commonly used in power transmission devices and drivetrains of

most automotive, aerospace and industrial systems, the level of design knowledge of them is far

lower than other components such as gears, shafts and bearings. This study proposes a family of

semi-analytical models to predict load distribution of clearance-fit (side-fit), major diameter-fit and minor diameter-fit spline joints with the intention of enhancing spline design practices.

These models include all major components of spline compliance stemming from deformations associated with bending, shear and base rotation of the teeth as well as contact and torsional deformations. For clearance-fit splines, only drive side tooth surfaces are allowed to contact while top and root lands of the external spline are also chosen as potential contact zones in case of major diameter-fit and minor diameter-fit splines, respectively. Any helix mismatch or interference conditions are also handled by allowing contacts on back side tooth surfaces as well.

All of these models are formulated for any general loading condition consisting of torsion, radial forces and tilting moments, such that loading conditions of gear-shaft splines can be modeled conveniently.

Since contacting spline tooth surfaces are conformal, the potential contact zone covers all of the tooth surfaces, whose direct load distribution solution might require significant computational time. A new multi-step discretization solution scheme is devised and implemented in the semi-analytical models to reduce the computational time significantly such that they can be used as convenient design tools. Meanwhile, accuracy of the predictions of the proposed models

is demonstrated through comparisons to those from a detailed deformable-body contact model.

As afforded by their computational efficiency, proposed models are used to perform extensive

parameter studies to quantify influences of loading conditions, misalignments, tooth

modifications and tooth indexing errors on spline load distribution. The results indicate that load

and contact stress distributions are impacted by these factors significantly. Furthermore major and

minor diameter fit spline joints show much better self-centering capability under nominal gear

loading conditions in comparison to that of side-fit spline joints. A comprehensive statistical

analysis methodology is also implemented to relate spline quality level defined by manufacturing

tolerance class to the resultant probability distributions of tooth-to-tooth load sharing and contact pressure.

At the end, as an application of the semi-analytical models, a general analytical stiffness formulation of splines joints is proposed. It defines a fully populated stiffness matrix of a spline joint including radial, tilting and torsional stiffness values as well as off-diagonal coupling terms.

A blockwise inversion method is proposed and implemented with this formulation to reduce computational time required. A detailed parametric study is performed to demonstrate the sensitivity of the spline stiffness matrix to loading conditions, torque level, tooth modifications, misalignments, and tooth indexing errors.

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DEDICATION

Dedicated to the memory of my beloved father

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to my advisor, Prof. Ahmet Kahraman, for his invaluable knowledge, intellectual guidance, patient support and endless encouragement throughout the course of my Ph.D. study at the Ohio State University. I am grateful to Prof.

Robert Siston, Prof. Soheil Soghrati and Prof. Sandeep Vijayakar in serving on my dissertation committee. A special thank you goes to Prof. Sandeep Vijayakar for generously providing a license of the Helical 3D software package and supporting me in the development of a finite element model of involute spline joints.

I would like to thank Dr. David Talbot for sharing his technical expertise, providing unending support and reviewing my dissertation. I also would like to thank Mr. Jonny Harianto for developing a user interface for the model proposed in this study and packaging them into a spline load distribution program SplineLDP. Additionally, I would like to thank Mr. Sam Shon,

Dr. Sheng Li, Dr. Hyun Sik Kwon and all my lab mates for their friendship and support throughout my work at Gear Lab.

Last, but never least, I wish to thank my mother, siblings and uncles for their unending love and encouragement throughout the years, without which I would not have made it this far.

VITA

Sep. 1989 ········································· Born in Ezhou, Hubei, China

Jun. 2011 ········································· B.S. in Mechanical Engineering, Huazhong University of Science and Technology Wuhan, Hubei, China

Sep. 2011- Present ······························ Graduate Research Associate Gear and Power Transmission Research Laboratory The Ohio State University Columbus, Ohio, USA

PUBLICATIONS

1. Hong, J., Talbot, D. and Kahraman, A., 2014, “Load Distribution Analysis of Clearance-Fit Spline Joints Using Finite Elements,” Mechanism and Machine Theory, 74, 42-57.

2. Hong, J., Talbot, D. and Kahraman, A., 2014, “A Semi-Analytical Load Distribution Model for Side-Fit Involute Splines,” Mechanism and Machine Theory, 76, 39-55.

3. Hong, J., Talbot, D. and Kahraman, A., 2015, “Effects of Tooth Indexing Errors on Load Distribution and Tooth Load Sharing of Splines under Combined Loading Conditions”, Journal of Mechanical Design, 137(3), 032601.

4. Hong, J., Talbot, D. and Kahraman, A., 2013, “Load Distribution Analysis of Clearance-Fit Spline Joints Using Finite Elements”, International Conference on Gears, Munich, Gearmany.

5. Hong, J., Talbot, D. and Kahraman, A., 2014, “Semi-Analytical Modelling of Load Distribution of Side-Fit Involute Splines”, International Gear Conference, Lyon, France.

FIELDS OF STUDY

Major Field: Mechanical Engineering

TABLE OF CONTENTS

Page ABSTRACT ·································································································· ii DEDICATION ······························································································ iv ACKNOWLEDGMENTS ·················································································· v VITA ········································································································· vi LIST OF TABLES ··························································································· x LIST OF FIGURES ························································································· xii NOMENCLATURE ······················································································· xxi

CHAPTERS:

1. Introduction ······························································································ 1

1.1 Background and Motivation ···································································· 1 1.2 Literature Review ················································································ 4 1.3 Scope and Objectives ············································································ 8 1.4 Dissertation Outline ············································································· 9 References for Chapter 1 ·············································································· 10

2. A Semi-Analytical Load Distribution Model of Clearance-Fit Spline Joints ··························································································· 16

2.1 Introduction ······················································································ 16 2.1.1 Objectives and Scope ································································· 18 2.2 Semi-Analytical Spline Contact Model ······················································ 22 2.2.1 Determination of the Potential Contact Zones ····································· 23

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2.2.2 Discretization of the Contact Zones ················································· 24 2.2.3 Compatibility Conditions ····························································· 24 2.2.4 Equilibrium Conditions ······························································· 29 2.2.5 Governing Equations ································································· 32 2.3 Definition of Spline Compliance Matrix ····················································· 33 2.3.1 Tooth Bending and Shear Deformations ··········································· 34 2.3.2 Base Rotation and Base Translation Deflections ·································· 40 2.3.3 Contact Deformations································································· 44 2.3.4 Torsional Deflections ································································· 47 2.4 Multi-Step Discretization Solution Method ·················································· 49 2.5 Semi-analytical versus FE-based Deformable-body Models ······························ 68 2.6 Parameter Studies ··············································································· 78 2.6.1 Influence of Loading Conditions ···················································· 78 2.6.2 Effect of Spline Misalignment ······················································· 88 2.6.3 Effect of Lead Crown Modifications················································ 88 2.7 Summary ························································································· 93 References for Chapter 2 ·············································································· 93

3. Effect of Tooth Indexing Errors on Spline Load Distribution and Tooth Load Sharing ···························································································· 99

3.1 Introduction ······················································································ 99 3.2 Application of Indexing Errors to the Load Distribution Model ························ 101 3.3 Numerical Results and Discussions ························································· 105 3.3.1 Combined Torsion and Radial Forces ············································ 105 3.3.2 Combined Torsional, Radial Loads and Tilting Moment ······················· 121 3.4 Summary ······················································································· 134 References for Chapter 3 ············································································ 134

4. A Load Distribution Model for Major or Minor Diameter-Fit Splines and Mismatched Splines ············································································· 137

4.1 Introduction ···················································································· 137

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4.2 Contact Formulation ·········································································· 138 4.2.1 Major Diameter-Fit Splines ························································ 138 4.2.2 Minor Diameter-Fit Splines ························································ 146 4.2.3 Mismatched Splines ································································· 149 4.3 Results and Discussions ······································································ 154 4.3.1 Major and Minor Diameter-Fit Splines ··········································· 154 4.3.2 Mismatched Spline ·································································· 167 4.4 Summary ······················································································· 171 References for Chapter 4 ············································································ 171

5. A Semi-Analytical Stiffness Formulation for Spline Joints ····································· 173

5.1 Introduction ···················································································· 173 5.2 Stiffness Formulation ········································································· 175 5.2.1 Analytical Method ·································································· 177 5.2.2 Numerical Method ·································································· 188 5.3 Parametric Studies ············································································ 195 5.3.1 Effects of Tooth Surface Modifications ·········································· 195 5.3.2 Effects of Misalignments ··························································· 197 5.3.3 Effects of Indexing Errors ·························································· 201 5.4 Summary ······················································································· 204 References for Chapter 5 ············································································ 204

6. Conclusion ···························································································· 206

6.1 Summary ······················································································· 206 6.2 Contributions ·················································································· 208 6.3 Conclusions ···················································································· 209 6.4 Recommendations for Future Work ························································ 210

Bibliography ······························································································· 212

LIST OF TABLES

Table Page

2.1 Parameters of an example spline design used in this study ····································· 52

2.2 CPU time required for analysis of the example spline of Table 2.1 at different contact grid densities using the direct discretization solution of the proposed semi-analytical model and alternate multi-step discretization solutions. Also provided in the last column is the CPU time for the Helical3D solution with QQ= 2 and PP= 2 ············································································· 64

3.1 Parameters of an example spline design used in the indexing error study ·················· 107

4.1 Parameters of an example major diameter-fit spline joint used in the study ················ 155

4.2 Parameters of an example minor diameter-fit spline joint used in the study ··············· 156

4.3 Parameters of an example side-fit spline joint used in the study ····························· 168

5.1 Comparison of CPU time for analytical stiffness calculation using direct inversion and blockwise inversion for a spline loaded in pure torsion ······················ 182

5.2 Computational time for analytical stiffness calculation using direct inversion for splines under helical gear loading ··························································· 183

5.3 Computational time for analytical stiffness calculation using direct inversion for misaligned splines ············································································· 185

5.4 Parameters of an example side-fit spline joint used in the study ····························· 187

5.5 Comparison of K of the example spline joint under pure torsional loading condition; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad) ················································································ 191

5.6 Comparison of K of the example spline joint under pure torsional loading condition; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad) ················································································ 193

5.7 Comparison of K of the example spline joint under torsional loading with misalignments; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad) ········································································· 194

LIST OF FIGURES

Figure Page

1.1 Schematic side view of an example (a) side-fit spline; (b) major diameter-fit spline and (c) minor diameter-fit spline ···························································· 2

2.1 (a) An example clearance-fit spline joint consisting of an external spline and internal spline, and (b) side view of a pair of engaging spline teeth and corresponding potential contact area ······························································ 20

2.2 A spline with different loading conditions; (a) pure torsion loading, (b) spur gear loading, and (c) helical gear loading ························································· 21

2.3 Discretization of the potential contact zone of an engaging spline tooth pair into a set of contact grid ············································································ 25

2.4 (a) The initial separation between an arbitrary pair of contact points along their normal vector in unloaded condition, (b) influence of elastic deformations induced by external force, and (c) final gap between this pair of contact points when considering rigid body approach along their normal vector ····························· 26

2.5 Schematic representation of reaction load components of an external spline undergoing an arbitrary combined loading condition ··········································· 31

2.6 (a) A schematic representation of tooth bending and shear deformation, and (b)

Yau’s tapered plate model [2.27] with a concentrated load Fx(,000 y ,) z ··················· 35

2.7 (a) A schematic representation of the tooth base rotation deflections, and (b) a schematic representation of the tooth base translation deflections····························· 41

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2.8 Schematics of (a) the moment distribution and (b) the shear force distribution

at tooth base induced by a concentrated load Fx(,000 y ,) z ···································· 43

2.9 (a) A schematic representation of contact deformations caused by a concentrated force F and (b) mapping of contact grids in the xyz coordinate system to the ξη coordinate system ······························································ 45

2.10 A schematic representation of torsional deflections of (a) an external spline, and (b) an internal spline ············································································ 48

2.11 Contact stress distributions of the example spline under pure torsion at

M z = 2260 Nm predicted by the semi-analytical model with different number of contact grid cells in axial (Q) and profile direction (P). ····································· 53

2.12 Mapping of the load distribution on a tooth to a contact stress distribution plot ············· 54

2.13 Contact force distribution on each tooth section along the spline face width of the example spline under pure torsion loading. (a) Influence of the number of contact grid cells P in the profile direction with Q = 8 and (b) influence of the number of contact grid cells Q in the axial direction with P = 5 ······························· 55

2.14 Contact stress distributions of the example spline predicted by the semi- analytical model under spur gear loading condition using different number of

contact grid cells in axial (Q) and profile direction (P) at M z = 2260 Nm ·················· 57

2.15 Influence of (a) number of contact grid cells P in profile direction with Q = 8 and (b) number of contact grid cells Q in axial direction with P = 5 on tooth force distributions of splines under spur gear loading ··········································· 58

2.16 Contact stress distributions of the example spline predicted by the semi- analytical model under helical gear loading condition using different number of

contact grid cells in axial (Q) and profile direction (P) at M z = 2260 Nm ··················· 59

2.17 Influence of (a) number of contact grid cells P in profile direction with Q = 8 and (b) number of contact grid cells Q in axial direction with P = 5 on tooth force distributions of splines under helical gear loading ········································ 60

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2.18 Comparison of direct and multi-step discretization solutions of the semi- analytical model for the example spline under (a) pure torsion, (b) spur gear,

and (c) helical gear loading. P0 =10, Q0 =12, P1 = 5, and Q1 = 4 ·························· 65

2.19 Sensitivity of the maximum contact stress to the discretization parameters P and Q for four variations of the example spline; (a) no profile modification, no face width offset and Q =12 ; (b) slight profile modification, 20 mm tooth face width offset and P =10 ; (c) slight profile modification, no face width offset and Q =12 ; (d) slight profile modification, no face width offset and P =10 .

M z = 2260 Nm ······················································································ 66

2.20 Spline finite element model; (a) spline interface, (b) internal spline, (c) external spline and shaft, (d) potential contact area and contact elements ······························ 73

2.21 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of pure torsion loading ················· 75

2.22 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of spur gear loading ···················· 76

2.23 Comparison of spline tooth forces predicted by the semi-analytical model and

deformable-body model for the case of spur gear loading at (a) M z =1130 Nm

and (b) M z = 2260 Nm ············································································· 77

2.24 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of helical gear loading ·················· 79

2.25 Comparison of spline tooth forces predicted by the semi-analytical model and

deformable-body model for the case of helical gear loading at (a) M z =1130

Nm and (b) M z = 2260 Nm········································································ 80

2.26 Load distribution of a spline under pure torsion loading at different torque levels ·································································································· 81

2.27 Load distribution of a spline under spur gear loading at different torque levels ············· 83

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2.28 Tooth loads of a spline under spur gear loading at different torque levels ··········· 84

2.29 Load distribution of a spline under helical gear loading at different torque levels, gear helix angle, β =15 ··································································· 86

2.30 Load distribution of a spline under helical gear loading with different helix

angles at M z = 2260 Nm ··········································································· 87

2.31 Load distribution of a misaligned spline having different misalignment angles,

θx , at M z = 2260 Nm ··············································································· 89

2.32 Load distribution of a misaligned spline having different lead crown  modification magnitudes, g, at M z = 2260 Nm and θx = 0.06 ······························· 90

2.33 Load distribution of a spline under helical gear loading having different lead  crown modification magnitudes, g, at M z = 2260 Nm, and β = 20 ························· 92

3.1 Schematic representation of tooth indexing errors of (a) external spline teeth and (b) internal spline teeth ······································································· 103

3.2 Schematic side view of the external spline under combined loading condition at four sample rotational positions of (a) 0°, (b) 90°, (c) 180° and (d) 270° ·················· 108

3.3 Load distributions of the example spline having no tooth error under combined torsional and radial loads at different rotational positions of (a) 0°, (b) 90°, (c)

180° and (d) 270°. M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN ····················· 109

3.4 Load sharing factors of (a) tooth #5, (b) tooth #10, (c) tooth #15 and (d) tooth #20 as a function of rotational position for the example spline having no tooth

error under combined torsional and radial loads. M z = 8000 Nm,

Fx = −105 kN and Fy = 38 kN ·································································· 110

3.5 (a) Indexing error sequence, and load distributions of the example spline having single tooth indexing error under combined torsional and radial loads at different rotational positions of (b) 0°, (c) 90°, (d) 180° and (e) 270°.

M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN ············································· 112

3.6 Load sharing factors of (a) tooth #10, (b) tooth #13, and (c) tooth #15 as a function of the rotational position for the example spline having single tooth * indexing error (λ13 = 20 μm) in comparison to the no tooth error case under

combined torsional and radial loads. M z = 8000 Nm , Fx = −105 kN and

Fy = 38 kN ························································································· 114

3.7 (a) Indexing error sequence, and load distributions of the example spline having random tooth indexing errors under combined torsional and radial loads at different rotational positions of (b) 0°, (c) 90°, (d) 180° and (e) 270°.

M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN ············································ 116

3.8 Load sharing factors of (a) tooth #6, (b) tooth #14, (c) tooth #17 and (d) tooth #18 as a function of rotational position for the example spline having a random tooth indexing error sequence in comparison to the no tooth error case under

combined torsional and radial loads. M z = 8000 Nm , Fx = −105 kN and

Fy = 38 kN ························································································· 118

3.9 Load sharing factors of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 at different rotational positions for 100 sets of random tooth indexing errors under combined torsional and radial loads.

M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN ············································ 120

3.10 Probability distribution of load sharing factor of the critical tooth of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 under

combined torsional and radial loads. M z = 8000 Nm , Fx = −105 kN and

Fy = 38 kN ························································································· 122

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3.11 Load distributions of the example spline having no tooth error under combined torsional, radial loads and tilting moment, at different rotational positions of (a)

0°, (b) 90°, (c) 180° and (d) 270°. M z = 8000 Nm , Fx = −105 kN ,

Fy = 39 kN and M x = −2050 Nm ······························································ 124

3.12 Load sharing factors of (a) tooth #5, (b) tooth #10, (c) tooth #15 and (d) tooth #20 as a function of rotational position for the example spline having no tooth error under combined torsional, radial loads and tilting moment.

M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ······················ 125

3.13 (a) Indexing error sequence, and load distributions of the example spline having single tooth indexing error under combined torsional, radial loads and tilting moment loading at different rotational positions of (b) 0°, (c) 90°, (d)

180° and (e) 270°. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and

M x = −2050 Nm ·················································································· 127

3.14 Load sharing factors of (a) tooth #10, (b) tooth #13, and (c) tooth #15 as a function of rotational position for the example spline having single tooth * indexing error (λ13 = 20 μm) in comparison to the no tooth error case under

combined torsional, radial loads and tilting moment. M z = 8000 Nm ,

Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ··········································· 128

3.15 (a) Indexing error sequence, and load distributions of the example spline having random tooth indexing error under combined torsional, radial loads and tilting moment, at different rotational positions of (b) 0°, (c) 90°, (d) 180° and

(e) 270°. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ·········· 129

3.16 Load sharing factors of (a) tooth #6, (b) tooth #14, (c) tooth #17 and (d) tooth #18 as a function of rotational position for the example spline having a random tooth indexing error sequence in comparison to the no tooth indexing error case

under combined torsional, radial loads and tilting moment. M z = 8000 Nm ,

Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ·········································· 131

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3.17 Load sharing factors of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6, at different rotational positions for 100 sets of random tooth indexing errors under combined torsional, radial loads and tilting moment.

M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ······················ 132

3.18 The probability distribution of load sharing factor of the critical tooth of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 under

combined torsional, radial loads and tilting moment. M z = 8000 Nm ,

Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm ··········································· 133

4.1 (a) Schematic side view of an example major diameter-fit spline and its potential contact zones, and (b) discretization of the potential contact zones ·············· 139

4.2 (a) Schematic side view of an example minor diameter-fit spline and its potential contact zones, and (b) discretization of the potential contact zones ·············· 147

4.3 (a) Schematic side view of a spline joint having circumferential interference and its potential contact zones, and (b) discretization of the potential contact zones at the tooth driving side and back side ··················································· 150

4.4 Load distributions of (a) major diameter-fit (b) minor diameter-fit and (c) side-

fit splines under combined torsional and radial loads. M z = 8000 Nm,

Fx = −105 kN and Fy = 38 kN ·································································· 157

4.5 Comparison of tooth-to-tooth load sharing of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined

torsional and radial loads. M z = 8000 Nm, Fx = −105 kN and Fy = 38 kN ·············· 159

4.6 Relative rigid body translation along the x axis, ux , between the external and internal member of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined torsional and radial loads at different torque levels ············································································· 160

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4.7 Load distributions of examples of (a) major diameter-fit (b) minor diameter-fit and (c) side-fit spline under combined torsional, radial loads and tilting

moment. M z = 8000 Nm, Fx = −105 kN, Fy = 39 kN and M x = −2050 Nm ··········· 162

4.8 Comparison of tooth-to-tooth load sharing of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined

torsional, radial loads and tilting moment. M z = 8000 Nm, Fx = −105 kN,

Fy = 39 kN and M x = −2050 Nm ······························································ 163

4.9 Relative rigid body rotation about x axis, θx , between the external and internal member of (a) major diameter-fit and side-fit splines, and (b) minor diameter- fit and side-fit splines under combined torsional, radial loads and tilting moment at different torque levels ································································ 165

4.10 Influence of radial clearance on the relative rigid body rotation about x axis,

θx , between the external and internal member of the example major diameter- fit spline under combined torsional, radial loads and tilting moment at different torque levels ························································································ 166

4.11 (a) Schematic representation of intentional mismatch, δim , of a spline tooth on the pitch circle; (b) load distribution of an intentionally mismatched spline

having various δim values at different torque levels ·········································· 169

4.12 Maximum contact stress versus mismatch magnitude, δim , at different torque levels ································································································ 170

5.1 Schematic representation of (a) reaction load components of a spline joint, and (b) relative rigid body displacements between the external spline and internal spline ································································································ 176

5.2 Flowchart of the procedure for stiffness calculation using the analytical stiffness formula ··················································································· 180

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5.3 Influence of face width contact grid density on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of a spline joint under pure torsion at different torque levels with P =10 cells along the profile direction ··································· 186

5.4 Influence of profile contact grid density on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of a spline joint under pure torsion at different torque levels with Q = 24 contact cells along the face width direction ····················· 189

5.5 Effects of profile crown modification on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of the example spline under pure torsion at different torque levels ············································································· 196

5.6 Effects of lead crown modification on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of the example spline under pure torsion at different torque levels ························································································ 198

5.7 Effects of misalignment about x axis, θx , on (a) torsional stiffness, (b) radial stiffness along the x axis, (c) radial stiffness along the y axis, (d) tilting stiffness about the x axis and (e) tilting stiffness about the y axis, for the example spline at different torque levels ························································ 199

5.8 Effects of (a) random tooth indexing error sequence on (b) torsional stiffness, (c) radial stiffness along the x axis, (d) radial stiffness along the y axis, (e) tilting stiffness about the x axis and (f) tilting stiffness about the y axis of the

example spline having 0.04° misalignment at M z = 4520 Nm at different rotational positions ················································································ 202

NOMENCLATURE

Symbol Definition

a Face width of spline tooth

Aij Taper plate model coefficients

AART, Coefficients for base rotation and translation calculation A First fundamental form of surface b Tooth height

Bij Taper plate model coefficients B Second fundamental form of surface

Cij Taper plate model coefficients

CCRT, Coefficients for base rotation and translation calculation C Compliance matrix of spline joint

d p Pitch circle diameter of gear

D0 Flexural rigidity of taper plate at the base

De Major diameter of external spline

Di Minor diameter of internal spline

Dre Minor diameter of external spline

Dri Major diameter of internal spline e Unit vector E Young’s modulus

Fa Reaction force along axial direction ˆ FFbb, Shear force per unit width and total shear force at tooth base

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FFij, Contact force at contact point i and j

F k Sum of contact force carried by spline tooth k

Fn Gear mesh force

FFrt, Reaction forces along radial and tangential directions, respectively

FFxy, Reaction forces along x and y axis, respectively F Vector of contact forces g Magnitude of lead crown modification

GG12, Shear modulus of external and internal splines, respectively G Geometric matrix of spline h0 Tooth root thickness ht Tooth tip thickness

JJ12, Torsional moment of inertia of external and internal spline k Index for spline tooth number kR Tooth base rotational stiffness ks Shear correction factor used in taper plate model kT Tooth base translational stiffness

KK(1), (2) Relative principle normal curvatures K Stiffness matrix of a spline joint M Reaction moment N Normal distribution ˆ MMbb, Moment per unit length and total moment at tooth base n Number of contact grid cells over the spline interface nn123, , n Number of contact grid cells over the spline interface in step 1, 2 and 3 n Unit normal vector P Number of contact grid cells along profile direction P Number of elements along profile direction in finite element model

P0 Number of refined contact grid cells along profile direction

P1 Number of coarse contact grid cells along profile direction

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pp12, Coefficients for base rotation and translation calculations P Vector of external load components on the external spline q Position vector of some points inside the body far beneath tooth surface Q Number of contact grid cells along face width direction

Q Number of elements along face width direction in finite element model

Q0 Number of refined contact grid cells along face width direction

Q1 Number of coarse contact grid cells along face width direction Q Submatrix of blockwise inversion r Position vector at the common contact point

rr12, Position vectors of contacting surface 1 and 2, respectively

Rb Base circle radius of spline

Rij Vector of admissible function corresponding to φi and ψ j

Rb Vector of all elements being base circle radius Rb s Curvilinear parameter

ss12, Curvilinear parameters of contacting surface 1 and 2, respectively S Stiffness matrix of taper plate model t Curvilinear parameter

tt12, Curvilinear parameters of contacting surface 1 and 2, respectively

tt(1), (2) Unit normal vectors in the principal directions

(1) (2) ttjj, Unit normal vectors in the principal directions of surface Σ=j (j 1, 2) T Total CPU time of the multi-step solution

T0 CPU time required using the direct solution with refined grid cells

TT123, , T CPU time required in step 1, step 2 and step 3 of the multi-step solution u Displacement field of taper plate along the x axis

u0 Mid-plane displacement field of taper plate along the x axis

un Inward normal displacement component

uuxy, Relative rigid body translation between spline members along the x and y axes, respectively

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U Strain energy U Vector of coefficients of admissible functions v Displacement field of taper plate along the y axis v0 Mid-plane displacement field of taper plate along the y axis V Potential energy of external force w Displacement field of taper plate along the z axis w0 Mid-plane displacement field of taper plate along the z axis W Submatrix of blockwise inversion Y Final gap between a pair of contact points Y Vector of final gaps z z coordinate of a contact point Z Number of spline teeth Z Vector of artificial variables α Taper angle of plate

αn Normal pressure angle of gear β Gear helix angle

γγγxy, yz, zx Shear strain components

δδ12, Elastic deformation of a contact point on external and internal spline

δim Intentional lead mismatch magnitude

δT Base translation

δr Radial clearance of major diameter-fit spline δ Elastic deformation vector

δδ12, Elastic deformation vectors of external and internal splines, respectively ε Initial separation between a pair of contact points

εεxx, yy Normal strain components ε Initial separation vector ζ Rigid approach component ζ Rigid approach vector η Coordinate of face width length

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η0 Half of the face width length of a contact grid cell

θR Base rotation

θθθxyz, , Relative rigid body rotation between spline members about the x , y and z axes, respectively  Tooth load sharing factor

κκ(1), (2) Principal normal curvatures

(1) (2) κκjj, Principal normal curvatures of surface Σ=j (j 1, 2)

12 λλkk, Indexing errors of external and internal spline teeth k, respectively * λk Combined indexing error of external and internal spline tooth k

λλ12, Vector of tooth indexing errors of external and internal splines

λ* Vector of combined tooth indexing error of external and internal spline Π Total potential energy σ Standard deviation

ΣΣ12, Contacting surfaces 1 and 2

ττ(1), (2) Unit vectors in the direction of the principal normal relative curvatures

(1) (2) ττjj, Unit vectors in the direction of the principal normal relative curvatures of

surface Σ=j (j 1, 2)

υ Poisson’s ratio ξ Coordinate of profile arc length

ξ0 Half of the profile arc length of a contact grid cell

φi Mode shape of a free-free beam

ϕϕ12, Angles between specific vectors defined in Ref. [2.19]

Φi The i-th element of vector Φ Φ Vector of rigid approaches

ψ j Mode shape of a clamped-free beam yyxy, Rotation of taper plate about the x and y axes, respectively Ω Eigenvector

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Subscripts

b Tooth back side bb Compliance of back side induced by contact force at back side bd Compliance of back side induced by contact force at driving side c Tooth contact deformation cc1, 2 Tooth contact deformation of external and internal spline c6, c7 Tolerance class 6 and 7 d Tooth driving side db Compliance of driving side induced by contact force at back side dd Compliance of driving side induced by contact force at driving side dr Compliance of driving side induced by contact force at root land dt Compliance of driving side induced by contact force at top land f Tooth base flexibility deformation ff1, 2 Tooth base flexibility deformation of external and internal splines, respectively i , j Indices for contact point number

k Index for tooth number l Local tooth deformation including tooth bending and shear, tooth base flexibility and contact deformation max Maximum value min Minimum value p Tooth bending and shear deformation pp1, 2 Tooth bending and shear of external and internal splines, respectively r Tooth root land rd Compliance of root land induced by contact force at driving side rr Compliance of root land induced by contact force at root land t Tooth top land td Compliance of top land induced by contact force at driving side tt Compliance of top land induced by contact force at top land T Torsional deformation

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TT1, 2 Torsional deformation of external and internal splines, respectively x x direction y y direction z z direction

Superscripts

( ˆ ) Contact point pairs that are in contact ( ) Mean value b Tooth back side c6, c7 Tolerance class 6 and 7 d Tooth driving side ie Indexing error k Index for tooth number t Tooth top land r Tooth root land sm Surface modification

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CHAPTER 1

INTRODUCTION

1.1 Background and Motivation

Involute splines are widely used in mechanical drive systems to transfer rotary motion and torsion from one rotating component to another, say from a shaft to a gear or vice versa. A spline joint consists of an external and internal spline of the same number of teeth that are tightly fitted together along the same rotational axis. The spline teeth have similar profile shapes to that of involute gear teeth, while they typically have higher strength than gear teeth due to their larger pressure angle and smaller tooth height-to-tooth thickness ratio. When a spline joint is loaded, multiple spline tooth pairs are engaged simultaneously to carry the load, leading to significantly higher load carrying capacity compared to other forms of connections such as conventional keyway joints. In addition, involute spline joints with certain lead crown modifications can also effectively accommodate some angular misalignment between two mating shafts.

Depending on the locations of contact between the external and internal spline teeth, involute splines are typically classified in three types of fits: (i) side-fit (or clearance-fit), (ii) major diameter-fit, and (iii) minor diameter-fit. Figure 1.1(a) shows schematically the side view of a side-fit spline, where contacts occur on the sides of the teeth only. In this fit, driving sides of

(a) Internal Spline

Major Diameter Minor Diameter Dri De D Di External Spline re

(b) Internal Spline

Minor Diameter Major Diameter Dre Di Dri De External Spline

Minor Diameter External Spline Major Diameter Dre Di Dri De

Fig. 1.1 Schematic side view of an example (a) side-fit spline; (b) major diameter-fit spline and (c) minor diameter-fit spline.

the teeth transmit the load as well as center the mating splines. Tooth backside contacts are also

possible to occur in some applications where circular clearance is very small or an intentional

interference at the tooth backside is introduced.

Figure 1.1(b) shows a major diameter-fit spline joint, where contacts between the mating components occur at both tooth sides and the major diameter along the top land of the external teeth. In this type of spline, the sides of the teeth transmit the torsional load, while contacts at the major diameters of both splines pilot them radially. Relative motions between the external and internal splines are constrained by the minimal radial clearance in between, leading to a good self-centering performance. This type of spline is commonly used in applications where the alignment of shaft axes is critical, for perhaps minimizing the run-out error of a gear supported by the splined shaft. Likewise, Fig. 1(c) shows a minor diameter-fit spline joint, which is similar to major diameter-fit except the additional contacts are along the minor diameter of each member.

All three types of splines are frequently used based on their self-centering and load carrying capacity as demanded by the application in hand. Common failure modes of splines are surface wear, fretting corrosion fatigue and tooth bending fatigue. These failures of a spline joint within its design life cannot be thoroughly understood unless distribution of load along and amongst its teeth is known. Without a model to conveniently predict spline load distribution, it is typically assumed that a portion of spline teeth carry the load that is evenly distributed along the spline tooth face width. While providing a convenient and rough design guideline, it fails to predict the actual spline load distribution, which is far from uniform. Complicated effects of combined loading conditions, tooth surface modifications, shaft misalignments, and tooth manufacturing errors are also not properly accounted for. In order to better address these spline durability issues,

3 a computationally efficient spline load distribution model that captures these effects accurately is of great interest for design of power transmission systems.

Aside from their durability performance, another important aspect of spline joints is the influence of their stiffness on the dynamic behavior of mechanical drive systems. Spline joints are typically assumed to be rigid in dynamic models of gearboxes, transmissions and drive trains.

However, rotor dynamic instability and nonsynchronous whirl associated with the stiffness of a spline joint occurs frequently. For the fidelity of dynamic analysis of mechanical drive systems, it is critical to have a convenient and accurate stiffness formulation of spline joints.

This dissertation focuses on the development of a family of accurate and computationally efficient spline load distribution models. The goal here is to develop semi-analytical models for prediction of load distribution for all three types of splines and verify them through comparisons to predictions of a finite elements based deformable-body model. In addition, a general stiffness formulation for spline joints will also be proposed based on the semi-analytical load distribution model.

1.2 Literature Review

Various experimental studies on spline durability performance have been reported in the literature. Ku and Valtierra [1.1] studied wear of splines experimentally, demonstrating that misalignments of splines significantly increased wear of a spline joint. Brown [1.2] reported accelerated wear of involute spline couplings in aircraft accessory drives primarily due to spline misalignment and undesirable lubrication conditions. Limmer et al [1.3] investigated the fretting fatigue performance of gas turbine spline couplings using a simplified fretting fatigue apparatus.

They found that increased contact pressure and shearing force resulted in decreased fatigue life of

spline couplings. Leen et al [1.4, 1.5] and Ratsimba et al [1.6] performed fatigue tests on an

aerospace engine spline coupling, and found that major cycle torque overload contributed to plain

fatigue failure in the root-fillet region of the external spline at the side where the torque is applied.

Wavish et al [1.7] devised an experimental methodology to mimic the fretting fatigue condition

of a spline coupling under combined torque, axial load and tilting moment. They found contact

pressure distribution and local sub-state fatigue stress were critical to fretting fatigue cracking of

splines. More recently, Cuffaro et al [1.8-1.12] measured the contact pressure distribution in a

spline coupling and investigated the fretting wear damage of misaligned spline couplings

experimentally by monitoring debris in the lubricant and inspecting the changes of tooth surface

topography. They demonstrated that misalignment of spline couplings increased fretting wear damage considerably. The abovementioned experimental studies on splines were instrumental in

defining and documenting the failure modes in spline joints. Yet their contributions to the

understanding of spline failure mechanisms were limited without knowing the load distribution

along the spline contact interfaces.

Review of literature reveals only a few analytical spline load distribution models, all of

which were limited to simple loading conditions. This is due to the complexity of the conformal

contact over a wide area along engaged spline tooth pairs and the fact that all spline tooth pairs

are potentially in contact simultaneously. Volfson [1.13] proposed a rough estimation of contact

force distribution of splines under pure torsion or pure bending (tilting moment) loading

conditions. Effects of torsional and tilting moment loading on contact force distribution were

well captured qualitatively by his model, while the accuracy of this model was rather limited

since an empirical load concentration factor was used. Tatur and Vygonnyi [1.14] developed an

analytical model to estimate torque distribution along the face width of spline teeth for a spline

joint carrying pure torsion. Barrot et al [1.15-1.17] formulated the spline tooth torsional stiffness

in the model of Ref. [1.14] by analyzing spline tooth deflections due to tooth bending, shear,

compression and base rotation, and calculated the load distribution along axial direction of splines.

They also proposed a two-dimensional model in attempt to predict load distribution along spline

tooth profile direction. Yet, their models were again limited to pure torsion loading. Chase et al

[1.18] devised a statistical methodology to predict tooth engagement sequences of splines having

certain manufacturing errors under pure torsion loading. Influences of manufacturing errors on

resultant tooth load sharing were investigated. Cura et al [1.19] proposed an analytical model

capable of predicting tooth load sharing of spline joints under pure torsion loading with some

parallel offset misalignments. Wink and Nakandakar [1.20] accounted for the influence of spur

gear loads, i.e. combined torsion and radial loads, on tooth load sharing of splines. Models

proposed in Refs. [1.18-1.20] demonstrated some level of improvements in capturing complicated effects of manufacturing error, parallel offset misalignment and combined loading conditions,

while they were all limited to calculating only the tooth-to-tooth load sharing while failing to

predict the actual load distributions and contact stresses on spline teeth. Effects of tooth surface modifications were also not considered in these models. In summary, the review of analytical models clearly indicates that available analytical spline load distribution models are rather limited in terms of their accuracy as well as capability.

Aside from these analytical models, computational models using finite element (FE) or boundary element (BE) methods were also commonly used in spline load distribution analysis.

Limmer et al [1.3], Hahn-Jetter and Wright [1.21] and Tjernberg [1.22, 1.23] used commercial FE packages to predict spline load distributions under pure torsion loading, while the last two accounted for effects of certain manufacturing errors as well. FE models for helical spline couplings were proposed by Leen et al [1.4, 1.5, 1.24] and Ding et al [1.25-1.27] for splines under combined torsional and axial loading. Adey at al [1.28] proposed a BE model for analysis of

spline joints. Their model was capable of handling combined torsional and bending loading in

the presence of certain manufacturing errors. Using Adey’s model, Medina and Olver [1.29, 1.30]

studied load distribution of misaligned splines as well as the impact of spline pitch errors and lead

crown modifications. Hong et al [1.31] adopted a FE based deformable-body model of Vijayakar

[1.32] to analyze gear-shaft spline joints. They investigated the influences of combined loading conditions, shaft misalignments, tooth modifications and tooth indexing errors. While these computational models are superior to analytical models in terms of their capabilities and accuracy, they require considerable computational time for each analysis at a single kinematic position. In the presence of asymmetric loading conditions and tooth manufacturing errors, load distribution analysis of a spline joint has to be repeated at multiple rotational positions over an entire revolution. Considering that countless tooth indexing error sequences can occur within the tolerance range of a spline designed to certain quality level, the same contact analyses over an entire revolution have to be repeated for many different tooth manufacturing error sequences in order to obtain statistically meaningful spline load distribution results. In such cases, the number of contact analyses required would be very large, which makes the use of these computational models rather impractical, if not impossible. Also note that none of these computational models were capable of calculating the load distributions of major diameter-fit or minor diameter-fit splines.

Despite the above analytical and computational studies on spline load distribution, there have been a very limited number of publications on the stiffness of spline joints, which was reported to substantially influence dynamic behavior of mechanical drive systems [1.33-1.36].

Ku et al [1.37] measured the tilting stiffness of a spline joint experimentally. Cura and Mura

[1.38] measured the torsional stiffness of a spline joint in both aligned and misaligned conditions to show that misalignment of spline joints would reduce their torsional stiffness. In addition to

these experimental studies, Marmol et al [1.33] and Barrot et al [1.15, 1.16] proposed analytical formulae to calculate torsional and tilting (rotational) stiffness values of spline joints. These studies assumed that load distribution was identical for all spline teeth, which limited their models to pure torsion only. Effects of tooth modifications, shaft misalignments, combined loading conditions and tooth manufacturing errors were also neglected in these theoretical studies.

1.3 Scope and Objectives

An accurate prediction of spline load distribution is of great importance to the powertrain industry for addressing spline durability issues. At the same time, a comprehensive knowledge of the stiffness of spline joints is critical to the fidelity of dynamic analyses of drive trains having splines. However, the spline load distribution models reviewed above were either over-simplified

(analytical models), or very demanding computationally (computational models) to qualify as accurate design tools. In addition, none of these models were capable of dealing with load distributions of major diameter-fit or minor diameter-fit splines. Aside from these load distribution studies, published models on stiffness of spline joints were all limited to the simplest pure torsion case, failing to account for effects of multi-directional loading conditions, tooth

modifications, misalignments and manufacturing errors. With these, the main objectives of this

dissertation are listed as follows:

• Develop a semi-analytical load distribution model for side-fit spline joints with

significantly reduced computational time such that the model qualifies as a design tool.

Conduct a comprehensive parameter study using the semi-analytical model and compare

its results to a FE based deformable body model for its verification.

• Modify the baseline semi-analytical spline load distribution model to investigate the

influence of tooth indexing errors on load distribution and tooth load sharing

characteristics of side-fit spline joints. Implement a statistical analysis methodology to

relate the spline quality level to the resultant probability distribution of tooth load

sharing and contact pressure distributions.

• Extend the side-fit spline joint model to include load distribution analysis of (i)

intentionally mismatched splines and (ii) major or minor diameter-fit splines. Conduct a

comprehensive investigation of load distribution and tooth load sharing characteristics as

well as self-centralization performance of these types of splines.

• Propose a general stiffness formulation to efficiently define the fully populated stiffness

matrix of a spline joint including radial, tilting and torsional stiffness values as well as

coupling between these motions represented by the off-diagonal terms of the stiffness

matrix. Perform a parametric study to investigate influences of tooth modifications,

misalignments and tooth indexing errors on the stiffness matrix.

The ultimate goal of this research is to arrive at a verified semi-analytical spline load distribution model, which is highly efficient and capable of analyzing all three types of splines, at the same time capturing effects of tooth modifications, loading conditions, misalignments and manufacturing errors and providing convenient stiffness predictions.

1.4 Dissertation Outline

In Chapter 2, a new semi-analytical load distribution model for side-fit spline joints will be proposed and verified via comparison to a FE-based deformable-body model. A simplex

9 algorithm based multi-step discretization solution scheme will be devised to reduce computational time significantly while not jeopardizing the accuracy of the predictions.

In Chapter 3, effects of various types of tooth indexing errors on spline load distribution and tooth load sharing will be investigated using the semi-analytical model proposed in Chapter 2.

Given the numerical efficiency provided by the semi-analytical model, a robustness study will be performed to relate spline quality level to the resultant statistical distribution of tooth load sharing and contact pressure distributions.

Chapter 4 will be devoted to modelling of (i) intentionally mismatched splines, (ii) major diameter-fit splines and (iii) minor diameter-fit splines. The semi-analytical model proposed in

Chapter 2 will be modified to allow such types of contact analyses. A parametric study on these types of splines will also be presented.

Chapter 5 will focus on the stiffness matrix of spline joints. A general stiffness formulation based on the semi-analytical load distribution model will be presented. An analytical method for stiffness calculation will be proposed and verified through comparison against a conventional finite difference based numerical method. Parametric studies will then be performed to investigate the effects of various factors such as torque level, tooth modifications, misalignments and indexing errors. Finally, the main conclusions and major contributions of this dissertation as well as future research recommendations will be presented in Chapter 6.

References for Chapter 1:

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Engine Spline Coupling,” Journal of Strain Analysis, 37(6), 565-582.

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[1.14] Tatur, G. K. and Vygonnyi, A. G., 1969, “Irregularity of Load Distribution Along a

Splined Coupling,” J. Russ Eng, XLIX: 23–7.

[1.15] Barrot, A., Paredes, M. and Sartor, M., 2006, “Determining Both Radial Pressure

Distribution and Torsional Stiffness of Involute Spline Couplings,” Proc. IMechE., Part

C: J. Mechanical Engineering Science, 220, 1727-1738.

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and Second Moment of Area Calculations for an Analytical Model of Spline Coupling

Behaviour,” Proc. IMechE., Part C: J. Mechanical Engineering Science, 222, 891-902.

[1.17] Barrot, A., Paredes, M. and Sartor, M., 2009, “Extended Equations of Laod Distribution

in The Axial Direction in a Spline Coupling,” Engineering Failure Analysis, 16, 200-211.

[1.18] Chase, K. W., Sorensen, C. D. and DeCaires, B. J. K., 2010, “Variation Analysis of

Tooth Engagement and Loads in Involute Splines,” IEEE Transactions on Automation

Science and Engineering, 7(4), 54-62.

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with Parallel Offset Misalignment,” Proc. IMechE., Part C: J. Mechanical Engineering

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[1.20] Wink, C. H. and Nakandakar, M., 2014, “Influence of Gear Loads on Spline Couplings,”

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[1.21] Kahn-Jetter, Z. and Wright, S., 2000, “Finite Element Analysis of an Involute Spline,”

Journal of Mechanical Design, 122, 239-244.

[1.22] Tjernberg, A., 2000, “Load Distribution in the Axial Direction in a Spline Coupling,”

Engineering Failure Analysis, 8, 557-570.

[1.23] Tjernberg, A., 2001, “Load Distribution and Pitch Errors in a Spline Coupling,”

Materials and Design, 22, 259-266.

[1.24] Leen, S. B., Richardson, I. J., McColl, I. R., Williams, E. J. and Hyde, T. R. , 2001,

“Macroscopic Fretting Variables in a Splined Coupling Under Combined Torque and

Axial Load,” Journal of Strain Analysis, 36(5), 481-497.

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Simulation of Fretting Wear-Fatigue Interaction In Spline Couplings,” Tribology-

Materials, Surface and Interfaces, 2(1), 10-24.

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[1.27] Ding, J., McColl, I. R. and Leen, S. B. , 2007, “The Application of Fretting Wear

Modeling to a Spline Coupling,” Wear, 262, 1205-1216.

[1.28] Adey, R. A., Baynham, J. and Taylor, J. W., 2000, “Development of Analysis Tools for

Spline Couplings,” Proc Instn Mech Engrs, Part G, Journal of Aerospace Engineering,

214, 347-357.

[1.29] Medina, S. and Olver, A. V., 2000, “Regimes of Contact in Spline Coupling,” Journal of

Tribology, 124, 351-357.

[1.30] Medina, S. and Olver, A. V., 2002, “An Analysis of Misaligned Spline Coupling,” Proc

IMechE., Part J: J Engineering Tribology, 216, 269-279.

[1.31] Hong, J., Talbot, D. and Kahraman, A., 2014, “Load Distribution Analysis of Clearance-

Fit Spline Joints Using Finite Elements,” Mechanism and Machine Theory, 74, 42-57.

[1.32] Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution for a

Three-Dimensional Contact Problem,” International Journal for Numerical Methods in

Engineering, 31, 525-545.

[1.33] Marmol, R. A., Smalley, A. J. and Tecza, J. A., 1980, “Spline Coupling Induced

Nonsynchronous Rotor Vibrations,” Journal of Mechanical Design, 102, 168-176.

[1.34] Park, S. K., 1991, “Determination of Loose Spline Coupling Coefficients of Rotor

Bearing Systems in Turbomachinery,” PhD Thesis, Texas A&M University, College

Station, TX.

[1.35] Al-Hussain, K. M., 2003, “Dynamic Stability of Two Rigid Rotors Connected by a

Flexible Coupling with Angular Misalignment,” Journal of Sound and Vibration, 266,

217-234.

[1.36] Sekhar, A. S. and Prabhu, B. S., 1995, “Effects of Coupling Misalignment on Vibrations

of Rotating Machinery,” Journal of Sound and Vibration, 185(4), 655-671.

[1.37] Ku, C. P. R, Walton, J. F. Jr. and Lund, J. W., 1994, “Dynamic Coefficients of Axial

Spline Couplings in High-Speed Rotating Machinery,” Journal of Vibration and

Acoustics, 116, 250-256.

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Stiffness in Spline Coupling Including Angular Misalignment,” Mechanical Systems and

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CHAPTER 2

A SEMI-ANALYTICAL LOAD DISTRIBUTION MODEL OF CLEARANCE-FIT

SPLINE JOINTS

2.1 Introduction

Splines are widely used in mechanical drive systems to transfer rotary motion and torque between a shaft and a gear or between two rotating components. Failures of splines joints due to fretting wear, fretting corrosion fatigue and tooth breakage occur frequently. In the absence of a model to conveniently predict load distribution of splines, addressing spline durability issues is often based on trial-and-error or component tests.

Review of literature reveals only a few analytical models on splines, all of which were limited to simple loading conditions due to the complexity of the contact in the spline interface.

Volfson [2.1] proposed a rough estimation of contact force distribution along the axial direction of splines under pure torsion or pure bending loading conditions. Tatur and Vygonnyi [2.2] developed an analytical model to estimate torque distribution along the face width direction of spline teeth for the case when the spline joint carries pure torsion. This simplified model required a user-defined spline torsional stiffness. Barrot et al [2.3-2.5] formulated the spline tooth

16 torsional stiffness in the model of Ref. [2.2] by analyzing spline tooth deflections due to bending, shear, compression and foundation rotation. They adopted the model of Tatur and Vygonnyi [2.2] and calculated the load distribution along axial direction of splines under purely torsional loading condition. These analytical models provided an estimate of load distribution along the face width direction of the spline under simple loading conditions, but they failed to predict the load distribution across the spline tooth profile direction. Furthermore, they fell short of handling load distribution of splines under combined loading conditions as is the case for gear-shaft spline joints. Other complicating effects such as spline tooth surface modifications and spline shaft misalignments were also not considered in these models.

Finite Element (FE) or Boundary Element (BE) based computational models were also used in spline analyses. FE models by Limmer et al [2.6], Kahn-Jetter and Wright [2.7] and

Tjernberg [2.8, 2.9] used commercial FE packages to predict spline load distributions under pure torsion loading, while the last two accounted for effects of certain manufacturing errors as well.

FE models for helical spline couplings were proposed by Leen et al [2.10-2.12] and Ding et al

[2.13-2.15] for splines under combined torsional and axial loading. Meanwhile, Adey et al [2.16] developed a BE model for spline analysis. This model had the capability to investigate combined torsional and bending loading in presence of certain manufacturing errors. Using Adey’s model,

Medina and Olver [2.17, 2.18] studied load distribution of misaligned splines, and the impact of spline pitch errors and lead crown modifications.

Conventional FE based models have various difficulties in modeling splines where contact areas are dependent on loading conditions, misalignments, spacing errors or tooth modifications.

There are commercial contact solvers designed primarily for gear contacts that can be adapted to splines. One such model will be introduced in this chapter for verification of the proposed semi-

analytical model. While these FE based gear contact models have the capability in handling load

distribution of splines under various loading conditions with different types of tooth

modifications, manufacturing errors and shaft misalignment, their use as a design tool is limited

due to their computational demand. In some situations, analysis of a spline joint requires multiple contact analyses at different rotational positions, spanning an entire revolution. In such cases, the

same contact analysis must be repeated at a large number of rotational positions, which makes the

use of the computational models, at least in certain conditions, rather impractical. Therefore, a

new accurate spline load distribution model that can reduce computational time significantly is

needed.

2.1.1 Objectives and Scope

This chapter aims at developing and formulating a framework for a semi-analytical spline

load distribution model with a high computational efficiency. Specific objectives of this chapter are as follows:

• Develop a new semi-analytical model formulation for predicting load distribution of

clearance-fit spline joints.

• Devise a solution scheme to minimize the computational time required by the semi-

analytical model without jeopardizing the accuracy of the predictions.

• Compare results of the semi-analytical model to those from a computational model to

assess their accuracy for verification purposes in the absence of validation data.

• Use the semi-analytical model to characterize the impact of loading types and torque as

well as misalignments on the resultant load distributions.

First, the semi-analytical spline load distribution model will be presented. Next, main

components of the spline compliance formulation will be described. A new multi-step

discretization solution scheme will be proposed. Finally, predictions of the model will be

compared to those from the contact model of Vijayakar [2.19] for the verification of the model.

At the end, parametric studies will be presented to show the impact of loading condition and

misalignments on the resultant load distributions.

The splines considered in this chapter will be clearance-fit (side-fit) type as shown in Fig.

2.1 with contact happening only on one side of the tooth flanks and no contact on the other tooth

flank, tooth tip and root regions. The spline types allowing such contacts will be studied by using

a modified version of this model in Chapter 4.

While the spline model proposed is generic to handle any clearance-fit spline under any

loading condition, three specific loading conditions will be focused on. In the first case, a

moment (torsion) M z is applied to the end of the shaft, as shown in Figure 2.2(a), while the cylindrical disk having the internal spline is constrained along its perimeter to represent purely torsional loading of the spline with no radial force and tilting moment. The second loading case,

shown in Fig. 2.2(b), represents a spline supporting a spur gear where the applied torque, M z , is

balanced by the mesh force Fnz= 2 Md (p cosα n ) ( αn and d p are the normal pressure angle

and pitch circle diameter of the spur gear, respectively), and Fn acts in the transverse plane of the gear along the line of action, in the process applying a torque M z to the spline interface. The

third loading condition considered in this study represents the case of a spline supporting a helical

(a)

External Spline

Internal Spline

(b)

An Internal Spline Tooth Potential Contact Zone

An External Spline Tooth

Fig. 2.1 (a) An example clearance-fit spline joint consisting of an external spline and internal spline, and (b) side view of a pair of engaging spline teeth and corresponding potential contact area.

y (a)

M z

x M z

y (b) y F (c) n F r Ft

z z Fa

x x M M z z

Fig. 2.2 A spline with different loading conditions; (a) pure torsion loading, (b) spur gear loading, and (c) helical gear loading.

21 gear as shown in Fig. 2.2(c). In this case, the gear mesh force Fn in the plane of action in the normal direction has a tangential component Ft= 2 Md zp , a radial component

FFrt= tana n cos β and an axial component FFat= tan β , where αn , β and d p are the normal pressure angle, helix angle and pitch circle diameter of the gear, respectively. This results in a torsion M z about the rotational axis (z) of the shaft, radial forces FFxt= , FFyr= − , and a tilting moment MMxz= tan β about the x axis. In this case, the moment M x must be applied in addition to Fx and Fy to the gear body while M z applies to the input end of the shaft. In order to encompass all loading cases, the model formulation will be generic to include a torque, two radial force components and two twisting moment components.

2.2 Semi-Analytical Spline Contact Model

The contact between spline joints under combined torsion, radial loads and tilting moments can be interpreted as a general elastic contact problem with spline bodies having multiple degrees of freedom. A mathematical programming method proposed by Conry and Seireg [2.20, 2.21] and Conry [2.22] has been shown to be effective and efficient in simulating elastic bodies in contact. However, their model was limited to normal surface loading conditions with each elastic body having only one degree of freedom. This section aims at developing a new spline load distribution model as a general elastic contact problem that can be solved mathematically.

Major assumptions of the proposed spline contact formulation are as follows:

. All deformations are elastic and the total elastic deformation is the sum of various

elastic deformation components.

. The elastic deformations are small compared to the size of splines such that surface

curvatures of splines over the contact zone can be assumed to remain unchanged.

. It is assumed that contact forces on one tooth will not cause any type of tooth

deformations on other spline teeth except torsional deflections.

. The spline bodies and supporting shaft are assumed to be solid or hollow cylinders

when computing their torsional deformations.

. Effect of any friction between engaged spline tooth pairs is assumed to be negligible.

With above assumptions, contact analysis of splines is carried out in several steps. The first step is to determine the potential contact zones between mating spline tooth pairs. Next, the potential contact zones are discretized into a set of contact grids, with each grid represented by a contact point. Then, compatibility conditions for all contact points in the contact zone are established. Finally, equilibrium conditions on the spline bodies are enforced. The compatibility and equilibrium conditions constitute the governing equations for spline joint contacts.

2.2.1 Determination of the Potential Contact Zones

The potential contact zones between elastic bodies in contact under no load can be estimated by inspecting the geometry of the contacting surfaces. In case of spline joints, as shown in Fig. 2.1(a), the external spline and internal spline have the same number of teeth, module, and pressure angle. In addition, the center distance between them is zero. Thus, the external and internal spline bodies have the same surface curvature along the surfaces of the engaged teeth. For this, potential contact zones over multiple tooth pairs are considered that cover the entire tooth mating surfaces, starting from the minor diameter of the internal spline and extending to the major diameter of the external spline as shown in Fig. 2.1(b).

2.2.2 Discretization of the Contact Zones

With the potential contact zones defined, the next step is to discretize them into a set of

contact grids along both face width and profile directions. In the profile direction, the contact

zone of a tooth pair is divided into P grids of equal arc length. In the face width direction, the

same potential contact zone is divided into Q grids of equal length. With this, the potential

contact zone between each engaged spline tooth pair is discretized into PQ× contact grids as

shown in Fig. 2.3. Denoting the number of teeth of the splines as Z, the total number of contact grids over the whole spline interface becomes n=×× Z PQ. Provided that contact grids representing the potential contact zone are refined (i.e. each grid cell is small compared to the size of the contact zone), contact pressure within each contact grid cell can be assumed to be constant and represented by a concentrated force acting at the center of the contact grid cell. Spline load distribution can be determined by solving for the contact forces at all of the contact points

(centers of grid cells).

2.2.3 Compatibility Conditions

In order to solve for contact forces at all contact points over the potential contact zone, equations related to the contact forces have to be set up. The first type of equations relating to contact forces can be derived from compatibility conditions for all contact points over the potential contact zones. Compatibility conditions can be established by examining the final gaps between all contact point pairs along their normal vectors after the spline is loaded.

An arbitrary contact point pair i in a contact zone is selected and the influence of elastic deformations and rigid body approaches on the behavior of the contact point pair along its normal vector is shown schematically in Fig. 2.4. In an unloaded condition, initial separation between

Face Width Direction An Internal Q Spline Tooth … … …

25 A Contact

Point … An External 2 Spline Tooth 1 Profile 1 2 3 … … P Direction

Fig. 2.3 Discretization of the potential contact zone of an engaging spline tooth pair into a set of contact grid.

(a) (b) (c)

Internal Spline

2 δi

12 Yiii=++−εδδ i ζ i ε 12 i εδδii++ i 26

1 δi

ζi External Spline P

Fig. 2.4 (a) The initial separation between an arbitrary pair of contact points along their normal vector in unloaded condition, (b) influence of elastic deformations induced by external force, and (c) final gap between this pair of contact points when considering rigid body approach along their normal vector.

the two points of an arbitrarily selected contact point pair along the normal direction is εi , which is a function of spline tooth modifications and manufacturing errors. When external load P is

1 2 applied, contact forces between engaged spline teeth induce elastic deformations δi and δi at the two points forming the contact pair along their normal direction. Meanwhile, rigid body approach ζi along the normal direction also occurs due to the external load. The final gap Yi at

this arbitrarily selected contact point pair i becomes

12 Yiii=++−εδδ i ζ i. (2.1)

Since only elastic deformations are allowed, the final gap must always be nonnegative. If the

final gap is positive, then the contact point pair is not in contact and the contact force is zero. If

the final gap is zero, then the contact point pair is in contact and the contact force is nonnegative.

For all contact point pairs, the compatibility conditions can be written as

Y =++−εδ12 δ ζ, (2.2a)

YFii>=0, for 0,  in∈ [1, ] . (2.2b) YFii= 0, for ≥ 0, where Y denotes the vector of final gaps between all contact point pairs, ε denotes the vector of

initial separations between all contact point pairs, δ1 and δ2 are the elastic deformation vectors of

the external and internal spline surfaces, respectively, and ζ represents the vector of rigid body approaches along the contact normal vectors.

The initial separation vector ε is a function of spline tooth modifications, misalignments

and manufacturing errors. It can be determined if all design parameters are available. The elastic

deformations δ1 and δ2 relate to the contact forces through a compliance matrix as

δ12+= δ CF , (2.3)

where C is a nn× global compliance matrix of the spline joint, and F is a n ×1 vector of contact forces between all contact point pairs in the contact zone. The global compliance matrix C can be obtained by summing up all individual compliance matrices representing the deformations of each member and the contact. Definition of C will be covered in next section.

The n ×1 vector ζ contains rigid body approach components of all contact points along

their normal vectors. It is a function of rigid body approaches between spline bodies and surface

geometry of all contact points. At different locations, the rigid body approach components along

the surface normal vector of the contact points may vary. Friction between engaged spline teeth is assumed to be negligible and relative motion between the external and internal part of the spline joint in the axial direction has to be constrained. With this, there are five possible rigid body approaches of uuxyxy, , θθ, and θz between the external and internal member. For contact point pair i, the total rigid body approach along their unit normal vector ni can be obtained by summing up the contributions of each individual rigid approach component along the unit normal vector as follows:

zi=uu xxiyyien ⋅+ en ⋅− z ixyiθθ en ⋅+ z iyxibz en ⋅+ R θ (2.4)

where Rb is the base circle radius of the spline. The above equation can be repeated for all

contact point pairs to obtain

en⋅ en ⋅ −⋅z en zR en ⋅  z1 xy1 1 1 y 11 xb 1 ux      uy   z = en⋅ en ⋅ −⋅z en zR en ⋅ θ i xi yi iyi ixibx (2.5a)       θ y  z ⋅ ⋅−⋅ ⋅  n enxn en ynzzR nyn en nxn en bθz

ζ= GΦ (2.5b)

where ζi is the rigid approach component along the normal vector at contact point ii (∈ [1, n ]) ,

and zi is the z coordinate of the i-th contact point. Here ζ is the vector of the rigid approach

components of all contact points, G is a matrix that is related to the surface geometry, and Φ is

the vector of five rigid approach components.

Combining Eq. (2.3) and (2.5), the compatibility conditions of Eq. (2.2) are rewritten as

−CF + GΦYε +=, (2.6a)

YFii>=0, for 0,  in∈ [1, ] . (2.6b) YFii= 0, for ≥ 0,

2.2.4 Equilibrium Conditions

Using the compatibility conditions, n equations and n additional constraints are obtained as

shown above. It is noted here that there are n unknown contact forces Fii (∈ [1, n ]) , n unknown

T final gaps Yii (∈ [1, n ]) and five unknown rigid body approach terms Φ = uuxyxyz,,,,θθθ .

In order to solve for the contact forces, five additional equations are needed. A second set of

equations can be obtained by enforcing equilibrium conditions on the spline.

Since the system is constrained in the axial z direction, only the nominal force components in x and y directions and the nominal moment components about the x, y and z axes will be

considered. These arbitrary external load components are denoted as FFMxy,, x , M y and M z

as shown in Fig. 2.5. Equilibrium conditions require that contact forces on the spline teeth have

to balance these reaction load components from the supporting structures. The following five

equations can be derived from the equilibrium conditions described:

n n ∑ FFiine⋅= x x, ∑ FFiine⋅= y y, (2.7a,b) i=1 i=1

n n n ∑−zFiine i ⋅= y M x, ∑ zFiine i⋅= x M y, ∑ FRib= M z. (2.7c-e) i=1 i=1 i=1

Here, Fi is the contact force between the i-th contact point pair (in∈ [1, ]) , zi and Rb denote the

same parameters as in Eq. (2.5). The above equilibrium conditions can be rewritten in matrix

form as

enx⋅⋅⋅1  enxi en xn F Fx 1  en⋅⋅⋅ en en  F y1 yi yn  y −⋅zen −⋅ z en − zF en ⋅= M . (2.8a) 11y iy i ny n i x zzzen⋅⋅ en en ⋅ M y 11x ix i nx n  F RRbb R bn M z

M y z Fy

Fx x M x M z

Fig. 2.5 Schematic representation of reaction load components of an external spline undergoing an arbitrary combined loading condition.

Observing that the matrix on the left hand side is exactly the transpose of the geometry matrix G

in Eq. (2.5), the above equation of equilibrium conditions is rewritten as

GT F=P (2.8b)

where F is a vector of the contact forces Fii (∈ [1, n ]) , P is a vector containing the five

nominal load components experienced by the external spline.

2.2.5 Governing Equations

Combining the equilibrium equations and compatibility conditions, governing equations

and constraints for the spline contact problem can be written as

Z  I0GT 0YP  = , (2.9a) 0I− CGF ε Φ

YFii>=0, for 0,  in∈ [1, ] . (2.9b) YFii= 0, for ≥ 0,

where Z is a vector of artificial variables added to construct an identity matrix I required by the simplex-type algorithm. The vector P allows any combination of nominal loads, thus the above

equation has the capability to characterize load distribution of splines under various nominal

loading conditions, i.e. pure torsion, combined torsion and radial load condition representing a

spur gear loaded spline, and combined torsion, radial load and tilting moment representing a

helical gear loaded spline. In addition, the initial separation vector ε allows any type of tooth

surface deviations from a perfect involute profile, such as profile modification, lead modification,

and manufacturing errors if any. This provides the model with the capability to predict the effect

of tooth modifications and tooth profile errors on spline load distribution.

Aside from the above capabilities of the model, minor variations of the governing equations

(2.9) also make it possible to investigate the influence of any type of misalignment on spline load

T distribution. In Eq. (2.9), the components of the load vector P = FFMxyxyz M M

are required to be known as the input while the components of the rigid body approach vector

T Φ = uuxyxyzθθθ are unknowns. In the case of misaligned splines, rigid approach terms θx and θ y are known, while load components M x and M y are unknown. In this

situation, the misalignments can be considered to contribute to the initial separations and can be added to ε . The unknown load components resulting from these misalignments can be obtained when all contact forces are solved. Therefore, with minor variations to the governing equations, this model can also account for the effects of misalignments.

2.3 Definition of Spline Compliance Matrix

The compliance matrix C in Eq. (2.9) must be defined next. Four types of deformation of the spline teeth are considered: (i) tooth bending and shear deformations C p , (ii) tooth base

flexibility deformations C f , (iii) contact deformations Cc and (iv) torsional deflections CT .

With an individual compliance matrix representing each of these deformations, the total global

compliance matrix of a spline interface is given as

CC=p + C f ++ C cT C. (2.10)

These components of C will be formulated in the next four sections.

2.3.1 Tooth Bending and Shear Deformations

The first type of elastic deformations to be considered is tooth bending and shear deformations, which are illustrated schematically in Fig. 2.6(a). A spline tooth is essentially the same as a gear tooth. Therefore, currently available gear tooth models can be employed to

calculate tooth bending and shear deformations. Timoshenko’s formula [2.23] for the bending of

an infinite thin plate was widely used to approximate gear tooth bending. This model provided a

convenient estimate of tooth bending deformation, but its accuracy was limited due to the finite

width of a gear tooth. Wellauer and Seireg [2.24] proposed a semi-empirical formula for a finite

cantilever plate using the “moment image” method. This formula showed improvement for bending deflections at the tooth edges, but accuracy of the model was still limited due to the assumption of uniform thickness of gear tooth. Yakubek [2.25] developed a three-dimensional

(3D) Rayleigh-Ritz method based tapered plate model with varying tooth thickness. This tapered

plate model notably improved the accuracy in predicting gear tooth bending deformation.

However, tooth shear deformations were neglected, which can be considerable especially in the

case of small tooth height to tooth thickness ratio. Cornell [2.26] accounted for shear

deformations in his tapered gear tooth model, but this model was restricted to two dimensions.

Yau et al [2.27] modified Yakubek’s 3D tapered plate model to account for shear deformations by

including shear strain terms in the strain energy formula. This model was shown to be in good

agreement with finite elements predictions and experimental results [2.27, 2.28]. Here, the

procedure to formulate the spline tooth bending and shear compliance matrix will be based on the

tapered plate model of Yau et al [2.27], which will be described here briefly.

Figure 2.6(b) shows the tapered plate used in reference [2.27] to represent the gear tooth,

where h0, ht , ab, and α are the tooth root and tip thicknesses, tooth face width, tooth height

and taper angle, respectively. For a given concentrated transverse load F acting on the plate at

(a)

(b) a x

Fx(,000 y ,) z

휶 ℎ0 y ℎ푡 b z

Fig. 2.6 (a) A schematic representation of tooth bending and shear deformation, and (b)

Yau’s tapered plate model [2.27] with a concentrated load Fx(,000 y ,) z .

position ( xyz000,,), the displacement fields due to tooth bending and shear can be determined

using the Rayleigh-Ritz energy method. The principle of this method is to find an approximated displacement field that will minimize the total potential energy of the system. Total potential energy Π is the sum of the potential energy of the external force V and the strain energy U of the plate:

Π =VU + . (2.11)

Potential energy of the external force is negative of the work done by the force:

V= − Fw(, x000 y ,) z (2.12a)

where wx(,000 y ,) z denotes the displacement of point (,xyz000 ,) along the z axis. The strain energy of the system, assuming linear elastic deformation, can be written in terms of strain components as

E 2 21  222 U = ∫∫∫ εxx+2 υε xx ε yy + ε yy +−(1 υ ) γxy + γ xz + γ yz dv (2.12b) 2(1−υ2 ) { 2 }

where E and υ are the Young’s modulus and Poisson’s ratio, respectively. The strain

components relate to the displacement fields u, v and w as follows:

∂u ∂v ε = , ε = , (2.13a,b) xx ∂x yy ∂y

∂∂vu ∂∂wu γ = + , γ = + , (2.13c,d) xy ∂∂xy xz ∂∂xz

∂∂wv γ = + . (2.13e) yz ∂∂yz

With first order approximation, the displacement fields u, v and w can be represented in terms of

the corresponding middle-plane displacements and rotation components as [2.29]:

uxyz(, ,)= u0 () xy ,+ zy x (, xy ), (2.14a)

vxyz(, ,)= v0 (, xy ) + zy y (, xy ), (2.14b)

wxyz(, ,)= w0 (, xy ). (2.14c)

Here u0 , v0 and w0 denote the mid-plane displacements while ψ x and y y represent rotations of

normal to middle plane about the y and x axes, respectively. With the assumption that the

neutral plane remains un-stretched after deformation, u0 and v0 can be ignored. The remaining terms w0 (, xy ), y x (,xy ) and y y (,xy ) can be represented by a series of admissible functions

with unknown coefficients. In the model of Yau et al [2.27], characteristic functions of vibration

modes for uniform beams were chosen to represent the displacement and rotations:

w0 (, xy )= ∑∑ Aijφy i () x j () y  ij  y(,xy )= Bφy () x () y  x ∑∑ ij i j (2.15)  ij yy (,xy )= ∑∑ Cijφy i () x j () y   ij

where φi and ψ j are the mode shapes of a free-free and clamped-free beam, and Aij , Bij and

Cij are unknown coefficients to be solved. With the displacement fields represented by admissible functions with unknown coefficients, the potential energy of the external force and the

strain energy of the beam can be written as

VF= − ∑∑ Axijφy i()00 j () y (2.16) ij

3 2 −∂υφba∂y D0 (1 ) 2y tana j i U = 1−+∑∑Bijφy i ∑∑Cij j dxdy 4 ∫∫ h ∂∂yx 000 ij ij 22 ba 3    2y tana ∂φ ∂y j +−1 D 1 B i yφ +C  dxdy 2 0 ∫∫∑∑ij j ∑∑ ij i h0  ∂∂xy  00 ij ij  3 ba ∂φ  ∂y  2y tana  i  j  +−D0υ 1 ∑∑Bijyφ j ∑∑C ij i dxdy ∫∫ h  ∂∂xy  000  ij  ij  2 −υ ba∂φ 31Dk0 s ( ) 2y tana i +−1 ∑∑Bijφy i j +∑∑A ijy j dxdy 2 ∫∫ h ∂x h0 000 ij i j 2 −υ ba∂y 31Dk0 s ( ) 2y tana j +−1  ∑∑Cijφy i j + ∑∑Aijφ i dxdy 2 ∫∫ h ∂y h0 000 ij ij

(2.17a)

where ks is the shear correction factor, a and b are the face width and tooth height respectively, and D0 is the flexural rigidity of the plate at the supported edge defined as [2.27]

3 Eh0 D0 = . (2.17b) 12(1−υ2 )

In Eqs. (2.16, 2.17), V and U are both functions of unknown coefficients Aij , Bij and Cij such

that the total potential energy Π in Eq. (2.11) is also a function of the same coefficients.

Derivatives of the potential energy with respect to Aij , Bij and Cij yield a set of linear equations

11 mn SS11 11 U R  11 11 11 mn   SSU12R 12 12 12 = F , (2.18a)    11 mn UijR ij SSij ij   where

Aij φyij()xy00 ()   Uij= B ij , Rij = 0 . (2.18b,c)  0 Cij  

With the unknown coefficients obtained by solving Eq. (2.18a), displacement fields of the plate subject to the external force F at point (,xyz000 ,) can be determined by using Eqs. (2.14, 2.15).

Repeating this procedure for a unit contact force applied at every contact point, the compliance matrix for bending and shear deformation of a single tooth is obtained.

With the stated assumption that contact force on one tooth will not induce any tooth bending or shear on any other tooth, the overall tooth bending and shear compliance matrices of external and internal components of a spline joint with Z number of teeth are defined as

C011 0 C 0 0  pp12   22   0Cpp12 0 0C 0 CCpp12=   , =  (2.19a,b)  00       ZZ   0 0 0Cpp12 0 0 0C 

where the bending and shear compliance matrices of a single spline tooth k ( kZ∈[1, ] ) are

k k defined as Cp1 (for external) and Cp2 (for internal). With this, the total tooth bending and shear

compliance matrix of the spline joint is given as

Cpp =C12 +C p. (2.20)

2.3.2 Base Rotation and Base Translation Deflections

Additional elastic deformations occur due to the base flexibility of the spline tooth, allowing the tooth to rotate and/or translate about its base, as shown schematically in Fig. 2.7(a) and (b). These types of deflections can be sizable in comparison to tooth bending deformations, which are often small due to the small length-to-thickness ratio of spline tooth. Analytical solutions for base flexibility deflections of cantilever plates or beams in plane stress and plane strain conditions can be found in references [2.30-2.32]. However, due to the finite face width of a gear or spline tooth, accuracy of predictions using these models is marginal. Stegemiller and

Houser [2.33, 2.34] conducted an extensive parameter study on gear tooth base flexibility deflections using the FE method and proposed dimensionless formulae to approximate base flexibility deflections. As these approximations were shown to be superior to plane strain or plane stress solutions in terms of accuracy while also having high computational efficiency, they will be employed here to determine compliance matrix associated with spline tooth base flexibility.

Finite element parameter investigations of base flexibility of a two-dimensional (2D) gear tooth demonstrated that the base rotation θR and base translation δT were proportional to the

moment and the shear force at the base [2.33], respectively. The rotational stiffness kR and the

(a) F

(b)

Fig. 2.7 (a) A schematic representation of the tooth base rotation deflections, and (b) a schematic representation of the tooth base translation deflections.

translational stiffness kT were approximated based on FE results and the following equations for

2D tooth base flexibility deflections were proposed [2.33]:

MM FF θ =bb = C , δ =bb = (2.21a,b) RR2 TTC kR Eh0 kET

where CR and CT are constants obtained from FE results, E is Young’s modulus, h0 is the tooth

base thickness, and Mb and Fb are moment and shear force per unit width at the base,

respectively.

Successive FE investigations on 3D gear tooth base flexibility showed that formulas for 2D

gear tooth can be extended to 3D cases as follows [2.33, 2.34]:

Mz() Fz() θ ()zC= b , δ ()zC= b (2.22a,b) RR2 TT Eh0 E

where θR ()z and δT ()z are the base rotation and the base translation distributions of a 3D tooth along its face width, respectively, Mzb () and Fzb () are the moment and shear force

distributions at the tooth base along its face width caused by a concentrated load acting on the

tooth, as shown schematically in Fig. 2.8(a) and (b), respectively. Bell-shaped exponential curves were used as fits for the moment and shear force distribution along face width direction.

Dimensionless formulae for the base rotation and the base translation were proposed [2.33, 2.34], respectively, as

2 0.76C AMˆ 2( z− z ) θ ()zp= RR bexp 0 , (2.23a) R 3 1 ah aEh0 0

(a) Mzb ()

Fx(,000 y ,) z

y z

(b) Fzb ()

Fx(,000 y ,) z

y z

Fig. 2.8 Schematics of (a) the moment distribution and (b) the shear force distribution at tooth

base induced by a concentrated load Fx(,000 y ,) z .

2 0.76C AFˆ 2( z− z ) δ ()zp= TTbexp 0 . (2.23b) T 3 2 ah aEh0 0

Here CR , CT , AR , AT , p1, and p2 are coefficients determined by FE element results, a is the

ˆ ˆ face width of the gear, z0 is the z coordinate where the force is applied, and Mb and Fb are the

total moment and shear force at the tooth base induced by an arbitrarily concentrated force acting

on the tooth.

Using above dimensionless formulae, base rotation and translation deflection fields of a

tooth caused by a concentrated force at any given contact point on the tooth can be obtained.

Repeating this process with a unit contact force applied at all contact grid points yields the

k compliance matrices for the base flexibility of a single spline tooth k ( kZ∈[1, ] ) as C f 1 (for

k external) and Cf 2 (for internal). Using the same assembly procedure as used for the tooth bending and shear compliance matrix, the total base flexibility compliance matrix of the spline is obtained as

C011 0C 0 0 ff12   22  0Cff12 0 0C 0 C f =  + . (2.24)  00        ZZ  0 0 0Cff12 0 0 0C

2.3.3 Contact Deformations

Contact deformation of a gear tooth in contact, as shown schematically in Fig. 2.9(a), were

reported to be often comparable to tooth bending and base flexibility deformations [2.22, 2.26,

2.35]. The same can be expected for the contact deformation of spline teeth as well. Local

(a)

(b) Profile direction

Face width direction Fiii(,ξη )

Fjjj(,ξη ) η

Fig. 2.9 (a) A schematic representation of contact deformations caused by a concentrated force F and (b) mapping of contact grids in the xyz coordinate system to the ξη coordinate system.

contact deformation of gear teeth can be approximated by modified Hertzian contact formulae

[2.35]. However, such formulae are not applicable to spline teeth in contact because identical curvatures of engaging spline teeth at their interface violate Hertzian assumptions. For this reason, the Boussinesq solution [2.36] for contact deformation in an elastic half space is used here.

The contact grids in the xyz coordinate system are mapped to a ξη coordinate system, as shown

in Fig 2.9(b). According to the Boussinesq solution [2.36], the contact deformation along the normal direction at a point (,ξηii ) due to a unit normal load applied at another point (,ξηjj ) is

given as

1−υ2 δc = . (2.25a) 22 πE ( ξξij− )( +− ηη ij )

Denote the area of each contact grid in ξη coordinate system as 22ξη00× . The contact

deformation at a point (,ξηii ) due to a uniformly distributed unit load at a contact grid cell centered at point (,ξηjj ) can be obtained by integrating Eq. (2.25a) over the contact grid as

ξη 1100 −υ2 dc = ∫∫ ddξη . (2.25b) 4ξη00 πξ−ξξ+ 22 +− ηηη+ −−ξη00E [ij ()] [ i (j )]

Using this expression, contact deformation fields of a spline tooth induced by a contact force at

any given point can be obtained. Again, repeating this process with a unit contact force applied at

all contact grid points yields the compliance matrices for contact deformation of a single spline

k k tooth k ( kZ∈[1, ] ) as Cc1 (for external) and Cc2 (for internal). Using the same assembly

procedure as used for the tooth bending and shear compliance matrix, the total contact

deformation compliance matrix of the spline is obtained as

C011 0C 0 0 cc12   22  0Ccc12 0 0C 0 Cc =  + . (2.26)  00     ZZ  0 0 0Ccc12 0 0 0C

2.3.4 Torsional Deflections

One remaining main contributor to the compliance of a spline joint is the torsional deflection. For simplicity, the spline bodies are assumed to be solid or hollow cylinders when calculating the torsional deflection. As illustrated in the Fig. 2.10(a), the side where the torque is applied to the external spline is constrained. Torsional deflections at any point over the external spline due to a contact force F are given by [2.37]

 2 FRb z  δT10= for 0≤≤zz ≤ a ,  GJ11  (2.27) 2  FRb z0 δT10= for 0 ≤z ≤≤ za  GJ11

where Rb is base circle radius of the spline, a is face width of the spline, z is the coordinate of

selected point, z0 is the distance of the applied contact force to the origin of the z axis, G1 is the

shear modulus of external spline material and J1 is the torsional moment of inertia of the external

spline. Similarly, the torsional deflections at any point over the internal spline as shown in Fig.

2.10(b) can be written as

 2 FRb () a− z0 δT 20= for 0≤≤zz ≤ a ,  GJ22  (2.28) 2  FRb () a− z  δT 20= for 0 ≤z ≤≤ za  GJ22

F (a) z0

a M z z

(b) a

M z z0 F

Fig. 2.10 A schematic representation of torsional deflections of (a) an external spline, and (b)

an internal spline.

where G2 and J2 are the shear modulus and torsional moment of inertia of the internal spline, respectively.

Using the above equations, torsional deflection fields induced by a unit contact force can be determined. Repeating this procedure for a unit contact force at all contact points, the torsional compliance matrix of tooth i caused by contact force on tooth j is defined. Denoting the torsional

k1 k2 k1 k2 compliance matrices of tooth k1 caused by contact force on tooth k2 as CT1 and CT 2

(kk12,∈ [1, Z ]) for the external spline and internal spline tooth, respectively, the total torsional compliance matrix of external spline and internal spline can be assembled as

11CC 12 1 CZZ 11 C 12 C 1 C  TT11 T 1 T 2 T 2 T 2 21 22 2ZZ 21 22 2  CCTT11 C T 1 C T 2 C T 2 C T 2 CT =  + . (2.29)     Z12 Z ZZ Z1 Z 2 ZZ CCTT11 C T 1 C T 2 C T 2 C T 2

2.4 Multi-step Discretization Solution Method

The simplex type algorithm proposed in Ref. [2.38] is employed to solve Eq. (2.9).

Initially, Z and Y are in the basis in Eq. (2.9). In order to solve it, F and Φ will have to enter

the basis while removing Z and Y from the basis [2.38]. The global compliance matrix C must be updated each time an element Fii (∈ [1, n ]) of F and a rigid approach term Φ∈i (i [1, 5]) of Φ

enters the basis. As the dimension of C is usually very large for splines, the majority of CPU time is used for updating the compliance matrix. Entrance of all Fii (∈ [1, n ]) and Φ∈i (i [1, 5])

to the basis requires that C is updated (n + 5) times. As C is an nn× matrix, the number of

numerical operations required is roughly On()3 and the computational time is approximately proportional to On()3 .

An acceptable resolution to predict spline tooth load distributions requires a refined contact grid. One immediate approach is to solve the governing equations for such finely discretized

contact zones by employing the simplex algorithm mentioned above [2.38]. In this case of direct

discretization, computational time that is proportional to On()3 can be expected to be significant

especially for splines having a large number of teeth. For instance, with the potential contact zone discretized into 10 sections along profile and 12 sections along face width ( P =10, Q =12 ),

the CPU time required for analysis of a 60-tooth spline at a single kinematic position takes about

3 hours. In case of asymmetric loading or in the presence of tooth indexing errors, load

distribution analyses might be required to be performed at many time steps representing various

incremental rotational positions. As such, the potential for using this semi-analytical model as a design tool would be rather limited if the CPU time were to remain at these levels. A new solution scheme with significantly increased numerical efficiency is desirable. In order to achieve this goal, a multi-step discretization solution scheme is developed here.

Instead of solving for global spline load distribution directly in one single step, this new

multi-step discretization solution method obtains a refined global spline load distribution in

several consecutive steps. In the first step, a coarse contact grid is defined over potential contact

zones whose solution yields a coarse spline load distribution prediction. In the next step, the contact grid in the face width direction is refined and a set of governing equations for each individual tooth are set up (based on the load distributions obtained in the first step) and are solved to obtain a refined load distribution along the face width of each individual spline tooth.

In the third step, the contact grid along profile direction is refined and a set of governing

equations for each individual tooth are set up based on the load distribution obtained in the

second step. As in step two, a refined load distribution along profile direction of each individual

spline tooth can be obtained. Finally, all local spline tooth load distributions are assembled

together to form a refined overall spline load distribution. While this procedure is promising in

reducing CPU time, the key issue is its accuracy compared to the direct discretization scheme.

For this, the influence of contact grid density on the resultant load distributions must be

investigated first.

Consider an example clearance-fit spline joint that is designed according to ANSI B92.1-

1996 standard [2.39]. The parameters of this clearance-fit spline are listed in Table 2.1. A

parabolic profile modification (4 µm at bottom line of the potential contact zone, 0 µm at pitch

line and 3 µm at tooth tip) is applied to external spline teeth intending to eliminate edge contact along profile direction. Figure 2.11 shows load distributions of this example spline under pure torsion at M z = 2260 Nm using different contact grid sizes. Here load distribution along each tooth contact surface is mapped to a rectangular plane as depicted in Fig. 2.12 and the instantaneous contact distributions of all the spline teeth are displayed in Fig. 2.11 by lining up these individual load distributions in sequence. As expected, the contact grid density is observed

to influence contact stresses. However, the corresponding contact force distributions along the

spline face width are not necessarily influenced by the grid size. In order to demonstrate this, the

spline is divided into eight uniform sections along the face width direction, the total contact

forces acting on each section of the spline teeth are plotted in Fig. 2.13(a) for the example spline

under pure torsion at M z = 2260 Nm with the number of contact grid cells in axial direction fixed

as Q = 8 and the number of contact grid cells in profile direction varied as P = 5, 7 and 10.

Table 2.1 Parameters of an example spline design used in this study.

External spline Internal spline

Number of teeth 25

Spline Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.550 85.725

Minor diameter [mm] 73.025 76.200

Circular space width [mm] - 5.055

Circular tooth thickness [mm] 4.981 -

Inner rim diameter [mm] 45 -

Outer rim diameter [mm] - 150

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Q = 8 MPa P =10 118

100 Q = 8 P = 7 80

Q = 8 P = 5 60 53

40 Q =16 P = 5

Q = 24 P = 5 0

Fig. 2.11 Contact stress distributions of the example spline under pure torsion at M z = 2260 Nm predicted by the semi-analytical model with different number of contact grid cells in axial (Q) and profile direction (P).

M z

Profile direction

Color indicates Contact pressure

Face width direction

Fig. 2.12 Mapping of the load distribution on a tooth to a contact stress distribution plot.

1000 (a) Series1P = 5 Series2P = 7 Series3P = 10

Contact Force (N) 500

1000 (b) Series1Q = 8 Series2Q = 16 Series3Q = 24 Contact Force (N) 500

0 2 4 6 8 Tooth Sections along Face Width

Fig. 2.13 Contact force distribution on each tooth section along the spline face width of the example spline under pure torsion loading. (a) Influence of the number of contact grid cells P in the profile direction with Q = 8 and (b) influence of the number of contact grid cells Q in the axial direction with P = 5 .

Figure 2.13(b) shows the force distributions over each section for P = 5 with Q = 8 , 16 and 24.

It is observed that the density of the contact grid, either along profile or axial direction, has little influence on the overall contact force distribution on each tooth along the face width direction for splines under this pure torsion loading condition.

Influence of contact grid density on load distribution of splines under spur gear loading is shown in Fig. 2.14 for the same spline. As expected, contact grid density is shown to notably affect contact stress levels, while contact patterns and tooth load sharing characteristics among different spline teeth are similar in all cases. Figure 2.15(a) shows the tooth force distribution of the spur gear loaded spline for Q = 8 with P = 5, 7 and 10 while Fig. 2.15(b) shows the tooth

force distribution for P = 5 and with the number of contact grid cells in axial direction varied as

Q = 8 , 16 and 24. Again contact grid density has a rather negligible influence on the tooth force

distribution of spur gear loaded splines. The same conclusion was reached in the case of helical gear loading, shown in Fig. 2.16 and Fig. 2.17.

Assume that the number of contact grid cells required for an acceptable contact stress distribution resolution is n= ZP00 Q with T0 representing the CPU time required for solution

when using the direct discretization solution method. Then CPU time required when using a

multi-step discretization solution method can be estimated as follows.

Step 1: Consider a coarser contact grid ( n1= ZPQ 11 where QQ10< , and PP10< ) for the sole purpose of obtaining reasonably accurate spline tooth force distributions and rigid body

T approaches Φ = uuxyxyz,,,,θθθ . The computational time in this step can be estimated as

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Q = 8 MPa P =10 120

Q = 8 100 P = 7

80 Q = 8 57 P = 5 60

Q =16 40 P = 5

Q = 24 P = 5 0

Fig. 2.14 Contact stress distributions of the example spline predicted by the semi-analytical model under spur gear loading

condition using different number of contact grid cells in axial (Q) and profile direction (P) at M z = 2260 Nm.

6,000

(a) P = 5 P = 7 P = 10 4,000

F k (N)

2,000

0 6,000

(b) Q = 8 Q = 16 Q = 24 4,000

F k (N)

2,000

0 0 5 10 15 20 25 Tooth Number, k

Fig. 2.15 Influence of (a) number of contact grid cells P in profile direction with Q = 8 and (b)

number of contact grid cells Q in axial direction with P = 5 on tooth force distributions of splines under spur gear loading.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Q = 8 MPa P =10 160

140 Q = 8 P = 7 120

100 Q = 8 P = 5 80 59

60 Q =16 P = 5 40

20 Q = 24 P = 5 0

Fig. 2.16 Contact stress distributions of the example spline predicted by the semi-analytical model under helical gear loading

condition using different number of contact grid cells in axial (Q) and profile direction (P) at M z = 2260 Nm.

6,000 (a) P = 5 P = 7 P = 10 4,000

F k (N)

2,000

0 6,000 (b) Q = 8 Q = 16 Q = 24 4,000

F k (N)

2,000

0 0 5 10 15 20 25 Tooth Number, k

Fig. 2.17 Influence of (a) number of contact grid cells P in profile direction with Q = 8 and (b) number of contact grid cells Q in axial direction with P = 5 on tooth force distributions of splines under helical gear loading.

3 3 n1 PQ 11 TT10= =  T 0. (2.30) n PQ00

Step 2: Refine the contact grid in the axial direction to the desired level of Q0 while retaining the coarser grid in the profile direction P1 from Step 1, resulting in n2= Z PQ 10. In this step, instead of solving a global spline contact problem, a set of local contact problems for each individual spline tooth is set up and solved. As shown previously, contact grid density does not influence spline tooth force distributions, the sum of contact forces on all refined contact grid cells of each individual spline tooth can be assumed to be equal to the sum of forces on all coarse

contact grid cells along the same tooth. For involute splines, the torque over an individual tooth is proportional to the sum of forces on that tooth by a ratio of Rb , which is the base circle radius

of the spline tooth. Thus, the torque balance condition of each individual spline tooth remains

valid when a coarse contact grid is refined. However, there is no evidence that forces and

moments along the x and y axes will also remain unchanged for each individual tooth when the

contact grid is refined. Therefore, local contact equations for each individual spline tooth k

( kZ∈[1, ] ) can be set up with minor modification of Eq. (2.9) as follows

A  T k k 1 0Rb 0Y M z  = , (2.31a) k k k 0I− C Rb F ε   θz

kk YFjj>=0 for 0  j∈[1, PQ ] (2.31b) kk 10 YFjj= 0 for ≥ 0

k where Rb is a column vector with all elements being Rb , M z is the torque provided by contact

of individual spline tooth k, which can be calculated using coarse solution obtained in step 1, θz

is the relative rotation between the external spline and internal spline along shaft axis. Contact equations in this form have to be solved for all Z individual spline teeth, each of which has PQ10

contact grid cells. Thus, the CPU time in this step can be estimated to be

3 3 PQ 1 P TZ= 10 T= 1 T. (2.32a) 2 002  nPZ 0

Step 3: Refine contact grid in the profile direction next so that the number of contact grid cells becomes n3= n = ZP 00 Q . As in Step 2, a set of local contact equations are set up for each

individual spline tooth. Equations in the form of (2.31) are solved for all Z spline teeth

individually, each of which now has PQ00 contact grids. Thus, CPU time is estimated to be

3 PQ00 1 TZ3=  T 00= T. (2.32b) n Z 2

Contact forces obtained by solving these local contact equations for individual spline teeth are finally assembled together to form a global solution for spline load distribution. Total CPU time of this multi-step discretization solution is TTT=++123 T, that is found from Eq. (2.30) and

(2.32) as

33 PQ 11  P TT=++11  1 . (2.33) PQ22 P 0 00 ZZ  0

With Z = 25, P0 =10, P1 = 5, Q0 =16, and Q1 = 4 for instance, Eq. (2.33) yields TT0 = 0.004,

which represents a 250 times faster solution over the direct discretization scheme. Table 2.2

shows the actual CPU time using the proposed semi-analytical model with the direct and multi-

step discretization solution schemes (as well as requirements from a FE based deformable-body model that will be introduced later) to indicate that the multi-step discretization solution is about two orders of magnitude faster than the direct discretization scheme (requiring only 6 to 12 seconds), in the process qualifying the semi-analytical model as a design tool.

For the purpose of demonstrating that the multi-step discretization method does not adversely impact the accuracy of predicted contact stress distributions, Fig. 2.18 compares the direct and multi-step discretization solutions of the same example spline under pure torsion, spur gear loading and helical gear loading conditions. Here, no tangible difference is observed between the two sets of solutions to indicate that the accuracy of the predictions is not compromised by the multi-step discretization scheme. It is clear from Fig. 2.18 that this new multi-step discretization solution method is not only desirably fast but also as accurate as the direct discretization method.

As contact grid density influences the contact stress levels, especially for the cases when

severe edge loading is expected, a grid sensitivity study is performed here for proper selection of

the discretization parameters. First, the example spline joint that has edge contact conditions are

considered, including (i) edge contact along the profile direction, and (ii) edge contact along the

face width direction. Figure 2.19(a) shows the maximum contact stress of the example spline

having no profile modification under pure torsion. Here, grid density along the face width is kept

constant at Q =12 while parameter P defining the number of grid cells in the profile direction is

varied from 5 to 20. It can be seen that as the profile contact grid is refined, the maximum

Table 2.2 CPU time required for analysis of the example spline of Table 2.1 at different contact grid densities using the direct discretization solution of the proposed semi-analytical model and alternate multi-step discretization solutions. Also provided in the last column is the CPU time for the Helical3D solution with QQ= 2 and PP= 2 .

Spline Parameters CPU time [min]

Multi-step Z Q P Direct Discretization Helical3D Discretization

25 8 10 4 0.1 10

25 12 10 13 0.12 40

25 16 10 30 0.2 90

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 MPa direct 94 discretization 63 (a)

multi-step 31 discretization 0

MPa direct 108 discretization 72 (b)

multi-step 36 65

discretization 0

MPa 137 direct

discretization 91 (c) multi-step 46 discretization 0

Fig. 2.18 Comparison of direct and multi-step discretization solutions of the semi-analytical model for the example spline under (a)

pure torsion, (b) spur gear, and (c) helical gear loading. P0 =10, Q0 =12, P1 = 5, and Q1 = 4 .

150 (a) (b)

100

Maximum Contact Stress (MPa) 50

0 5 10 15 20 8 12 16 20 24 28 32 36 150 (c) (d)

100 Maximum Contact Stress (MPa)

0 5 10 15 20 8 12 16 20 24 28 32 36

P Q

Fig. 2.19 Sensitivity of the maximum contact stress to the discretization parameters P and Q for four variations of the example spline; (a) no profile modification, no face width offset and Q =12 ; (b) slight profile modification, 20 mm tooth face width offset and P =10 ; (c) slight profile modification, no face width offset and Q =12 ; (d) slight

profile modification, no face width offset and P =10 . M z = 2260 Nm

contact stress is increased from 64 to 121 MPa, demonstrating that the maximum contact stress

will increase as the contact grid size is reduced without settling to a converged value. This can be explained by the fact that edge contact condition along the profile direction occurs in the absence

of profile modification. The theoretical stress at the edge along profile direction could be

infinitely large to result in such numerical circumstances. Similarly, Fig. 2.19(b) shows the

maximum contact stress of the example spline having face width offset. Here, P =10 while Q is varied from 8 to 36. The same profile modification described earlier in sect. 2.4 is applied to external spline teeth to reduce the edge contact along the profile direction, while a 20 mm face width offset is applied to introduce edge contact along face width direction (the active face width in contact remains as 50.8 mm). Again, it is seen that the maximum contact stress keeps increasing as Q increases due to the edge contact along face width direction. Above two cases demonstrated that in the presence of edge contact, increasing the contact grid density does not guarantee an accurate prediction of the maximum contact stress.

The sensitivity of the maximum contact stress to the grid size demonstrated in Fig. 2.19(a,b)

is, however, not a concern since any spline design must avoid such edge loading conditions

through proper corrections to the contact surfaces in the form of profile and/or lead modifications.

As such, the practical question to answer is whether grid size influences contact stresses under no

edge loading conditions or not. For this, the example spline that has no edge contact will be

considered next. Here, the same profile modification used above is applied to external spline teeth to reduce edge contact along profile direction. Also, there is no face with offset between the external and internal spline causing edge loading along the sides. With Q =12 , Fig. 2.19 (c) shows the sensitivity of the maximum contact stress to grid size defined by P. It is evident that increasing P does not considerably influence the maximum contact stress. Similarly, Fig 2.19 (d)

shows influence of Q on the maximum contact stress where P =10 . Again, deviation of the

maximum contact stress for different Q is not significant. These two cases demonstrated that, in the absence of edge contact condition, a moderate contact grid density is sufficient to obtain accurate predictions. As such, P =10 and Q =12 will be used in subsequent case studies of this

chapter.

2.5 Semi-analytical versus FE-based Deformable-body Models

In this section, the predictions of the semi-analytical model with the multi-step discretization method will be compared to those from a finite-element based deformable-body model [2.19]. In the absence of experimental data, this comparison is intended to serve the purpose of verification of the semi-analytical model.

A commercial FE based contact mechanics model Helical3D (Advanced Numerical

Solutions, LLC) designed specifically for loaded contact analysis of helical gears is modified here to analyze spline joints. The core contact solver of this software (CALYX) is based on a formulation by Vijayakar [2.19], which combines the finite element method and surface integral method to represent contacting bodies, and calculates load distribution and rigid body

displacements by using the linear programming method.

The first phase of contact analysis is to determine the contact zone. CALYX estimates the

contact zone by using Hertz’s model after locating a set of “primary contact points” on the

contacting surfaces and determining relative principal curvatures and directions. The second

phase is to compute the compliance matrix and set up the contact equations to be solved by a

modified simplex method.

In order to locate the primary contact point, two contacting surfaces Σ1 and Σ2 are defined

in terms of their curvilinear parameters s and t as r1(,)st 11 and r2(,)st 22. The primary contact

points are determined and located when r1 and r2 become the closest to each other [2.19]. For

this, the surface r1(,)st 11 is discretized into a grid of points r1ij(,st 11 i j ). For each of these grid points, a primary contact point r2ij(,st 22 i j ) is determined such that rr1(,)st 11− 2 ( s 2 , t 2 ) is

minimum. This minimization is equivalent to solving the following system of nonlinear

equations [2.19]:

 ∂r (,st ) 222ij rr1ij−= 222(,st i j ) 0  ∂s2i  (2.34) ∂r (,st ) −=222ij rr1ij 222(,st i j ) 0  ∂t2 j

The Newton-Raphson method is used to solve this system of non-linear equations to obtain mating point r2ij on the second surface for each of the grid point r1ij . Then a refined grid is set up around the point r1ij such that the separation rr12ij− ij is the smallest. This process is repeated several times with progressively smaller grids to locate the principal contact point

[2.19].

The principal curvatures and principal directions of two surfaces at the common contact

point are determined in terms of the coefficients of the first and second fundamental form of the

surfaces. For a unit normal vector defined at the common contact point as

∂∂rr × n = ∂∂st, (2.35) ∂∂rr × ∂∂st

matrices

22 ∂∂rr ∂∂ rr ∂∂rr ⋅⋅ nn⋅⋅ ∂ 2 ∂∂st ∂∂ss ∂∂ st = s A = , B  (2.36a,b) ∂∂rr ∂∂ rr ∂∂22rr ⋅⋅ ⋅⋅ ∂∂ ∂∂ nn ts tt ∂∂ts ∂t2

contain the coefficients of the first and second fundamental form of the surface, respectively. The

corresponding eigen value problem

BΩ= κ AΩ (2.36c)

yields eigen values κ (1) and κ (2) that are the principal normal curvatures and corresponding

T eigenvectors Ω1 and Ω2 . With the eigenvectors normalized (i.e. Ωmm AΩ =1 m =1, 2 ), the two unit vectors in the principal directions corresponding to the principal curvatures are found as

[2.19]

 (1) T ∂∂r s t = Ω1   ∂∂r t  (2.37)  (2) T ∂∂r s t = Ω2   ∂∂r t

(1) (2) (1) (2) With the principal curvatures κ j and κ j and the principal directions t j and t j of

surface Σ j ( j =1, 2 ) defined at the common contact point, the principle normal relative

(1) (2) curvatures ( K (1) , K (2) ) and their directions ( τ , τ ) are given from

KK(1)+=+ (2) κκ(1) (2) ++ κκ (1) (2)  ( 11) ( 22)  (2.38a) KK(1)−=− (2) κκ(1) (2) cos 2 ϕκκ +−(1) (2) cos 2( ϕϕ −)  ( 11) 2( 22) 21

 (1) (1) (2) τt=11cosϕϕ22 + t sin  (2.38b) (2) (1) (2) τt=−+11sinϕϕ22 t cos

(1) (1) (1) (1) where the angle ϕ1 between t2 and t1 , and the angle ϕ2 between τ and t1 are defined as [2.19]

 tt(1)⋅ (2)  −1 21 ϕ1 = tan  tt(1)⋅ (1)  21  (2.38c) κκ(1)− (2) sin(2 ϕ )  − ( 22) 1 ϕ = 1 tan 1  2 2   κκ(1)− (2) +− κκ (1) (2) cos(2 ϕ )  ( 11) ( 22) 1

Once the primary contact points are located and the relative curvatures are obtained,

Hertzian theory is used to predict the size of the contact zone and consequently a grid of points in

the contact zone is laid out on both surfaces. Then a surface integral (si) method near the contact

zone and a finite element (fe) method away from the contact zone are combined to predict cross

compliance terms between the set of grid points.

The inward normal displacement component un(rr ij ;) of point r caused by a unit normal

force at surface grid point rij is given as [2.19]:

(si) (si) (fe) uun(;)rr ij=−+ n (;) rr ij u n (;) rq ij u n (;) rq ij . (2.39)

Here q is some location inside the body on a matching surface, sufficiently far beneath the tooth surface. The first two terms in this equation denote the relative deflection of r with respect to q, which is evaluated using the surface integral formulae. The third term denotes the displacement of q, which is computed using finite element method. The point q is chosen such that elastic half space assumption will be valid and the finite element prediction will not be significantly affected by local stresses on the surface. The surface integral and finite element solutions are combined along this matching surface interface, as described in detail in Ref. [2.19]. The combination of surface integral formulae and finite element method described above provides an accurate and numerically efficient way of obtaining the compliance matrix for the contacting bodies. Contact equations are set up using the compliance matrix and solved by a modified simplex algorithm of linear programming, which is described in detail in Ref. [2.38].

The Helical3D model of the example spline of Table 2.1 as shown in Fig. 2.20 uses a contact grid with P number of elements along the profile direction and Q number of elements in

the face width direction. Within each contact element, there are 2 contact grid cells in both the

face width direction and profile direction. Width of the contact cells is defined such that 2P

grids in the profile direction can capture all the contact on the tooth. With this, a spline joint with

Z teeth would have a total of ZPQ××22 grid cells defining the contacts along the drive flanks of the teeth. In the model of Fig. 2.20 (d), P = 5 and Q = 6 were deemed to provide a

reasonable load distribution resolution while keeping the computational time relatively short (a

single position load distribution analysis of the spline joint of Fig. 2.20 required 40 min. on a PC

with a 3.39 GHz processor).

(a) (b) y y

z z

x x

Fig. 2.20 Spline finite element model; (a) spline interface, (b) internal spline, (c) external spline and shaft, (d) potential contact area and contact elements.

Figure 2.21 shows the contact stress distributions of spline teeth under pure torsion loading

conditions at two different torque levels obtained by using the proposed semi-analytical model

with multi-step discretization and the deformable body model that employs the CALYX solver.

Here the discretization parameters for the semi-analytical model are P0 =10, P1 = 5, Q0 =12,

and Q1 = 4 . It can be observed that the semi-analytical model agrees well with Helical3D in terms of load distribution patterns and contact stress magnitudes. Both solutions show that all spline teeth have identical contact stress distribution that is biased towards the side where the torque is applied. In addition, maximum contact stresses in both cases at different torque levels

are also close to each other. For instance at M z = 2260 Nm, maximum contact stresses predicted

by Helical3D and the semi-analytical model are 91 and 94 MPa, respectively.

Next, predictions of spline load distribution under spur gear loading conditions are

compared in Fig. 2.22. Here, relevant spur gear tooth parameters are the normal pressure angle of

α αn = 20 and the pitch circle diameter of d p =150 mm. Spline tooth #1 is centered below (and closest to) the gear mesh. In Fig. 2.22, contact stress distributions predicted by the deformable- body and the semi-analytical model (using multi-step discretization with the same grid parameters as Fig. 2.21) are observed to agree well. Tooth-to-tooth variations of contact patterns are captured well by the semi-analytical model. Both models are shown to predict higher contact stresses on teeth #1-9 and teeth #20-25. Figure 2.23 compares the corresponding tooth force distributions obtained from the semi-analytical model and Helical3D. In line with the load sharing characteristics, both models predict that teeth #1-9 and teeth #20-25 share larger tooth loads, while teeth #10-19 share much less load. For instance, in Fig. 2.23(a) at M z =1130 Nm,

the nominal equal tooth load value is 1315 N while the maximum tooth load predicted by the semi-analytical model is 2692 N on tooth #1, indicating nearly 105% higher load than the

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa 94 M z =1130Nm

63 (a)

31 2260Nm

0 75

MPa 91 1130Nm

61 (b)

30 2260Nm

Fig. 2.21 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of pure torsion loading.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa 108 M z =1130Nm

72 (a)

36 2260Nm

0 76

MPa 117 1130Nm

78 (b)

39 2260Nm 0

Fig. 2.22 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of spur gear loading.

4,000 (a) Deformable-body model Semi-analytical model Nominal force

F k (N) 2,000

0 8,000 (b) Deformable-body model Semi-analytical model Nominal force

k F (N) 4,000

0 0 5 10 15 20 25 Tooth Number, k

Fig. 2.23 Comparison of spline tooth forces predicted by the semi-analytical model and

deformable-body model for the case of spur gear loading at (a) M z =1130 Nm and

(b) M z = 2260 Nm.

77 nominal tooth load due to the spur gear loading condition. At the same time, tooth #13 is predicted to carry only 278 N, representing a 79% reduction compared to the nominal tooth load.

Since the spur gear, and hence the spline joint, rotates, each tooth of this spline would experience a cyclic variation of its load between these two limiting values at the rotational frequency of the gear, in the process dictating the fatigue life of the spline joint.

One final comparison between the two models is presented in Fig. 2.24 for a helical gear loading condition with a gear helix angle β =15 and normal pressure angle and pitch circle diameter remaining the same as the spur gear above. As in previous loading cases, the essence of the contact stress distributions is captured by the proposed semi-analytical model. Differences between the contact stress patterns predicted by the two models are minute. The same is true for the tooth force distributions shown in Fig. 2.25. For design purposes, such secondary differences can be overlooked in favor of the computational efficiency provided by the semi-analytical model.

It is noted that each analysis of Fig. 2.24 took 40 minutes using Helical3D and only 7 seconds using the proposed semi-analytical model, as shown in Table 2.2.

2.6 Parameter Studies

2.6.1 Influence of Loading Conditions

In case of pure torsion loading of Fig. 2.2(a), one would expect identical load distributions on each spline tooth with load varying in the face width direction. Figure 2.26 shows the contact stress distribution on the spline teeth of the example system under pure torsion at torque levels of

M z = 565, 1130, 1695 and 2260 Nm. At M z = 565 Nm, maximum contact stress is predicted to be about 45 MPa that occurs at the edge on the input side where torque is applied to the shaft.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa 137 M z =1130Nm

91 (a)

46 2260Nm

0 79

MPa 150 1130Nm

(b) 100

50 2260Nm

Fig. 2.24 Comparison of contact stress distributions predicted by (a) the semi-analytical model and (b) deformable-body model for the case of helical gear loading.

4,000

(a) Deformable-body model Semi-analytical model Nominal force

F k (N) 2,000

0 8,000 (b) Deformable-body model Semi-analytical model Nominal force

k F (N) 4,000

0 0 5 10 15 20 25 Tooth Number, k

Fig. 2.25 Comparison of spline tooth forces predicted by the semi-analytical model and

deformable-body model for the case of helical gear loading at (a) M z =1130 Nm

and (b) M z = 2260 Nm.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa M z = 565Nm 94

1130Nm 63

81 47

1695Nm 31

2260Nm 0

Fig. 2.26 Load distribution of a spline under pure torsion loading at different torque levels.

Contact stresses reduce significantly with axial distance from this edge. Non-uniform load distributions become clearer with increased M z while the location of maximum stress remains at

the input-side edge. The maximum contact stresses are 66, 81 and 94 MPa at this locality for

M z =1130 , 1695 and 2260 Nm, respectively. Meanwhile, the maximum contact stresses along the mid-plane of the spline joint are only 22, 36, 45 and 52 MPa for M z = 565 , 1130, 1695 and

2260 Nm, respectively. It is noted that as the torque increases, the contact area gradually extends toward the edge of the spline teeth along the profile direction and the contact stress increases simultaneously, with the load distribution pattern remaining the same.

In a spur gear loading condition, splines are subject to combined torsion and radial loads.

This asymmetrical loading leads to different load distributions on different spline teeth. Figure

2.27 shows the resultant load distributions for the same torque values of M z = 565 , 1130, 1695 and 2260 Nm. Here, relevant spur gear tooth parameters are the normal pressure angle of

α αn = 20 , and the pitch circle diameter of d p =150 mm. Two main observations can be made

from this figure. First of all, load sharing along the face width direction is still biased towards the

side where torque applied. Secondly, load distributions are no longer identical for all spline teeth

since the loading is no longer axisymmetric. With tooth #1 centered below (and closest to) the

gear mesh, teeth #1-8 and #21-25 are shown to experience larger loads while teeth #9-20 bear less

load. For instance, the resultant maximum contact stresses at M z = 2260 Nm are 104, 86, 76 and

106 MPa for teeth #5, #10, #15 and #25, respectively, as a direct consequence of this unequal

loading. The resultant total spline tooth forces are shown in Fig. 2.28 for the same loading

condition and torque levels. With the nominal spline tooth force defined as F= Mzb() ZR

where Rb is the base radius of the spline joint and Z is the number of teeth, the ratio of actual

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa M z = 565Nm 108

1130Nm 72

54 83

1695Nm 36

2260Nm 0

Fig. 2.27 Load distribution of a spline under spur gear loading at different torque levels.

M z = 565 Nm 4 F k (kN) 2

0 6

M z =1130 Nm 4 F k (kN) 2

0 6

M z =1695 Nm 4 F k (kN) 2

0 6

M z = 2260 Nm 4 F k (kN) 2

0 0 5 10 15 20 25 Tooth Number, k

Fig. 2.28 Tooth loads of a spline under spur gear loading at different torque levels.

force carried by tooth kk (∈ [1, 25]) to nominal tooth force FFk are 1.61, 0.77, 0.21 and 1.98

for k = 5, 10, 15 and 25 , respectively, at torque level M z = 2260 Nm. This indicates that some teeth carry about twice their nominal (torsion only) share.

As stated earlier and shown graphically in Fig. 2.2(c), helical gear loading applies a

combination of torsion, M z , tangential force, Ft , radial force, Fr , and tilting moment, M x .

Figure 2.29 shows the resultant load distributions for the same torque values of M z = 565 , 1130,

 α 1695 and 2260 Nm with a gear helix angle β =15 , a normal pressure angle αn = 20 and a pitch circle diameter d p =150 mm . In comparison to Fig. 2.27 for spur gear loading, helical gear

loading maintains the same qualitative load sharing characteristics of spline teeth. It shows that

teeth #1-8 and #21-25 carry larger load, while teeth #9-20 loaded less. This can be explained by the fact that the spline interface transmits the same radial load in spur gear loading and helical gear loading conditions. However, a remarkable difference is observed that axial load distribution on teeth #19-25 and #1-2 is biased to the opposite side where the torque is applied.

This is due to the additional tilting moment M x transmitted over the spline interface, which

causes the load on some teeth to be biased to the other side as to balance it. Fig. 2.29 also

illustrates that an increase in torque would only extend the area of contact and increase contact

stress, not significantly affecting load distribution patterns.

Figure 2.30 shows contact stress distributions on the spline teeth at M z = 2260 Nm for

different gear helix angles of β =10 , 15 , 20aa and 25 . Normal pressure angle and pitch circle

diameter of these helical gears are the same as the spur gear above. It is observed that, as β

increases, the load on each tooth gets more concentrated on the side where the load is biased. For

instance, loads on teeth #3-10 are concentrated to the side where the torque is applied, while loads

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa M z = 565Nm 137

114

1130Nm 91

69 86

1695Nm 46

2260Nm 0

Fig. 2.29 Load distribution of a spline under helical gear loading at different torque levels, gear helix angle, β =15 .

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa β =10 379

316

 15 253

190 87

20 126

25 0

Fig. 2.30 Load distribution of a spline under helical gear loading with different helix angles at M z = 2260 Nm.

on teeth #20-25 are concentrated to the opposite side. The maximum contact stresses are 120,

137, 175 and 379 MPa for gear helix angle β =10 , 15 , 20aa and 25 , respectively. The corresponding tilting moments are M x = 398, 606, 823 and 1054 Nm, respectively. Load concentration increases significantly because of the larger resultant tilting moment on the spline when the gear helix angle increases.

2.6.2 Effect of Spline Misalignment

Misalignment of spline couplings has been recognized as harmful because it causes significant load concentration on spline teeth, and accelerates wear and fretting fatigue of splines.

Figure 2.31 shows the load distribution on the example spline with different misalignments of

  θx = 0 , 0.02 , 0.04 and 0.06 at a torque level of M z = 2260 Nm. It is observed here that the

load distribution is biased similar to that of the helical gear loading condition because the tilting

moment caused by misalignment has a dominant effect on the load distribution. It is also noted

that the increase of misalignment leads to a significantly larger load concentration at the end of

the spline in the axial direction, due to larger tilting moment resulting from increased

misalignment.

2.6.3 Effect of Lead Crown Modifications

Significant edge loading was observed in both misaligned splines and splines experiencing helical gear loading. A lead crown modification along the face width of the spline can be expected to relieve some this load concentration. Figure 2.32 shows load distribution of a spline

 having a misalignment of θx = 0.06 along with different lead crown modification magnitudes of

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

 MPa θx = 0 212

177

0.02 141

106 89

0.04 71

0.06 0

Fig. 2.31 Load distribution of a misaligned spline having different misalignment angles, θx , at M z = 2260 Nm.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa g = 0 μm 212

177

20 μm 141

106 90

40 μm 71

60 μm 0

Fig. 2.32 Load distribution of a misaligned spline having different lead crown modification magnitudes, g, at M z = 2260 Nm and  θx = 0.06 .

g = 0 , 20, 40 and 60 µm at M z = 2260 Nm under pure torsion loading. It is observed that the crown modification moves the load from the edge towards the center of the spline, in the process reducing maximum contact stresses significantly. The maximum contact stresses are 212, 139,

133, 137 MPa for lead crown magnitudes g = 0 , 20, 40, 60 µm, respectively, indicating that the optimum lead crown magnitude is 40 µm in this case. It is also noted that when the lead crown modification is increased to 60 µm, only a small portion of the spline tooth across the face width carries load. This indicates that a moderate lead crown modification can reduce load concentration effectively while maintaining the spline’s load carrying capacity, while excessive amounts of lead crown modifications would increase contact stress and reduce load bearing capacity of the spline.

Figure 2.33 shows load distribution on the splines with different lead crown modification

 magnitudes, g = 0 , 20, 40 and 60 µm for a helical gear loading with β = 20 at M z = 2260 Nm.

The maximum contact stresses are 175, 168, 179 and 203MPa for lead crown magnitudes g = 0,

20, 40 and 60 µm, respectively. Unlike that in the misaligned spline case, lead crown

modification under a helical gear loading condition neither reduces load concentration, nor moves

tooth load from the edge to the center. This occurs because the moment acting on the spline

remains constant for a given torque in helical gear loading despite the lead crown modification.

The biased load concentration exists no matter how much lead crown modification is adopted.

Meanwhile, in misaligned splines, the lead crown modification effectively reduces the tilting

moment acting on the spline thus reducing load concentration and improving load distribution of

the spline.

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

MPa g = 0 μm 203

169

20 μm 135

102 92

40 μm 68

34 60 μm 0

Fig. 2.33 Load distribution of a spline under helical gear loading having different lead crown modification magnitudes, g, at  M z = 2260 Nm, and β = 20 .

2.7 Summary

In this chapter, a semi-analytical model to predict load distributions of side-fit spline joints

has been proposed. The model includes all essential components of spline compliances

associated with the tooth bending and shear deformations, tooth base flexibility, contact deformations and torsional deformations. This model has the capability to predict load

distributions of splines undergoing different types of loading conditions. A new multi-step

discretization solution method has been developed and implemented in this semi-analytical model

to show that the computational time is reduced by nearly three orders of magnitude compared to a

corresponding deformable-body model. Numerical examples of spline load distribution in three different nominal loading conditions using the semi-analytical model are presented and compared to a commercially available FE-based contact model Helical3D to demonstrate the accuracy of the predictions of the semi-analytical model.

An extensive parameter study of variations of the baseline system was performed in order to quantify the impact of loading conditions, misalignment and spline tooth modifications on spline load distributions. Misalignment of the spline resulted in a similar load distribution pattern as the case of helical gear loading. Lead crown modifications were shown to reduce load concentrations of misaligned splines significantly, while they did not improve the load distribution of a spline undergoing helical gear loading.

References for Chapter 2:

[2.1] Volfson, B. P., 1982, “Stress Sources and Critical Stress Combinations for Splined

Shaft,” Journal of Mechanical Design, 104, 551-556.

[2.2] Tatur, G. K. and Vygonnyi, A. G., 1969, “Irregularity of Load Distribution Along a

Splined Coupling,” J. Russ Eng, XLIX: 23–7.

[2.3] Barrot, A., Paredes, M. and Sartor, M., 2006, “Determining Both Radial Pressure

Distribution and Torsional Stiffness of Involute Spline Couplings,” Proc. IMechE., Part

C: J. Mechanical Engineering Science, 220, 1727-1738.

[2.4] Barrot, A., Sartor, M. and Paredes, M., 2008, “Investigation of Torsional Teeth Stiffness

and Second Moment of Area Calculations for an Analytical Model of Spline Coupling

Behaviour,” Proc. IMechE., Part C: J. Mechanical Engineering Science, 222, 891-902.

[2.5] Barrot, A., Paredes, M. and Sartor, M., 2009, “Extended Equations of Laod Distribution

in the Axial Direction in a Spline Coupling,” Engineering Failure Analysis, 16, 200-211.

[2.6] Limmer, L., Nowell, D. and Hills, D. A., 2001, “A Combined Testing and Modeling

Approach to the Prediction of the Fretting Fatigue Performance of Splined Shafts,” Proc

IMechE., Part G: Journal of Aerospace Engineering, 215, 105-112.

[2.7] Kahn-Jetter, Z. and Wright, S., 2000, “Finite Element Analysis of an Involute Spline,”

Journal of Mechanical Design, 122, 239-244.

[2.8] Tjernberg, A., 2000, “Load Distribution in The Axial Direction in a Spline Coupling,”

Engineering Failure Analysis, 8, 557-570.

[2.9] Tjernberg, A., 2001, “Load Distribution and Pitch Errors in a Spline Coupling,”

Materials and Design, 22, 259-266.

[2.10] Leen, S. B., Richardson, I. J., McColl, I. R., Williams, E. J. and Hyde, T. R., 2001,

“Macroscopic Fretting Variables in a Splined Coupling Under Combined Torque and

Axial Load,” Journal of Strain Analysis, 36(5), 481-497.

[2.11] Leen, S. B., Hyde, T. H., Ratsimba, C. H. H., Williams, E. J. and McColl, I. R., 2002,

“An Investigation of the Fatigue and Fretting Performance of a Representative Aero-

Engine Spline Coupling,” Journal of Strain Analysis, 37(6), 565-582.

[2.12] Leen, S. B., McColl, I. R., Ratsimba, C .H. H. and Williams, E. J., 2003, “Fatigue Life

Prediction For a Barreled Spline Coupling Under Torque Overload,” Proc IMechE., Part

G: Journal of Aerospace Engineering, 217, 123-142.

[2.13] Ding, J., Leen, S.B., Williams, E. J. and Shipway, P. H., 2008, “Finite Element

Simulation of Fretting Wear-Fatigue Interaction In Spline Couplings,” Tribology-

Materials, Surface and Interfaces, 2(1), 10-24.

[2.14] Ding, J., Sum, W. S., Sabesan, R., Leen, S. B., McColl, I. R. and Williams, E. J., 2007,

“Fretting Fatigue Predictions in a Complex Coupling,” International Journal of Fatigue

29, 1229-1244.

[2.15] Ding, J., McColl, I. R. and Leen, S. B., 2007, “The Application of Fretting Wear

Modeling to a Spline Coupling,” Wear, 262, 1205-1216.

[2.16] Adey, R. A., Baynham, J. and Taylor, J. W., 2000, “Development of Analysis Tools for

Spline Couplings,” Proc IMechE., Part G: Journal of Aerospace Engineering, 214, 347-

357.

[2.17] Medina, S. and Olver, A. V., 2000, “Regimes of Contact in Spline Coupling,” Journal of

Tribology, 124, 351-357.

[2.18] Medina, S. and Olver, A. V., 2002, “An Analysis of Misaligned Spline Coupling,” Proc

IMechE., Part J: J. Engineering Tribology, 216, 269-279.

[2.19] Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution for a

Three-Dimensional Contact Problem,” International Journal for Numerical Methods in

Engineering, 31, 525-545.

[2.20] Conry, T. F. and Seireg, A., 1971, “A Mathematical Programming Method for Design of

Elastic Bodies in Contact,” Journal of Applied Mechanics, 38, 387-392.

[2.21] Conry, T. F. and Seireg, A., 1973, “A Mathematical Programming Technique for the

Evaluation of Load Distribution and Optimal Modifications for Gear Systems,” Journal

of Engineering for Industry, 95 (4), 1115-1122.

[2.22] Conry, T. F., 1970, “The Use of Mathematical Programming in Design for Uniform Load

Distribution in Nonlinear Elastic Systems,” PhD thesis, The University of Wisconsin,

Madison, WI.

[2.23] Timoshenko, S. and Woinowsky-Krieger, S., 1959, “Theory of Plates and Shells,” 2nd

Edition, McGraw-Hill Book Company, Inc., New York.

[2.24] Wellauer, E. J. and Seireg, A., 1960, “Bending Strength of Gear Teeth by Cantilever-

Plate Theory,” Journal of Engineering for Industry, 82 (3), 213-222.

[2.25] Yakubek, D., 1984, “Plate Bending and Finite Element Analysis of Spur and Helical

Gear Tooth Deflections,” M.S. Thesis, The Ohio State University, Columbus, OH.

[2.26] Cornell, R. W., 1981, “Compliance and Stress Sensitivity of Spur Gear Teeth,” Journal

of Mechanical Design, 103 (2), 447-459.

[2.27] Yau, E., Busby, H. R. and Houser, D. R., 1994, “A Rayleigh-Ritz Approach to Modeling

Bending and Shear Deflections of Gear Teeth,” Computers & Structures, 50(5), 705-713.

[2.28] Yau, H., 1987, “Analysis of Shear Effect on Gear Tooth Deflections Using the Rayleigh-

Ritz Energy Method,” Master’s thesis, The Ohio State University, Columbus, OH.

[2.29] Reddy, J.N., 1984, “A Simple Higher-Order Theory for Laminated Composite Plates,”

Journal of Applied Mechanics, 51, 745-752.

[2.30] Muskhelishvili, N. I., 1963, Some Basic Problems of the Mathematical Theory of

Elasticity, 4th Edition, Groningen, P. Noorhoff.

[2.31] O’Donnell, W. J., 1963, “Stresses and Deflections in Built-In Beams,” Journal of

Engineering for Industry, 85, 265-272.

[2.32] O’Donnell, W. J., 1960, “The Additional Deflection of a Cantilever Due to the Elasticity

of the Support,” Journal of Applied Mechanics, 27, 461-464.

[2.33] Stegemiller, M. E., 1986, “The Effects of Base Flexibility on Thick Beams and Plates

Used in Gear Tooth Deflection Model,” M.S. Thesis, The Ohio State University,

Columbus, OH.

[2.34] Stegemiller, M. E. and Houser, D. R., 1993, “A Three-Dimensional Analysis of the Base

Flexibility of Gear Teeth,” Journal of Mechanical Design, 115, 186-192.

[2.35] Weber, C., 1949, “The Deformation of Loaded Gears and the Effect of Their Load

Carrying Capacity,” Report 3, British Dept. of Scientific and Industrial Research,

Sponsored Research (Germany).

[2.36] Johnson, K. L., 2004, Contact Mechanics, Cambridge University Press.

[2.37] Timoshenko, S. P., 1955, “Strength of Materials, Elementary Theories and Problems,” 3rd

Edition, D.Van Nostrand Company, Inc., New York.

[2.38] Vijayakar, S., Busby, H. and Houser, D., 1988, “Linearization of Multibody Frictional

Contact Problems,” Computers & Structures., 29, 569-576.

[2.39] STANDARD ANSI B92.1-1996, Involute Splines and Inspection, SAE, Warrendale, PA.

CHAPTER 3

EFFECT OF TOOTH INDEXING ERRORS ON SPLINE LOAD DISTRIBUTION AND

TOOTH LOAD SHARING

3.1 Introduction

Within manufacturing tolerances defined, teeth of manufactured splines are typically

subject to certain magnitudes of tooth indexing errors. Allowable indexing error magnitudes are

dependent on the quality class of the spline that dictates the allowable tolerance ranges. Within

these ranges, tooth indexing errors follow certain statistical distributions, which should dictate

similar statistical variations of the load distribution and tooth forces. In the absence of a model to

conveniently and efficiently predict spline load distribution, designers often assume only a

portion of spline teeth carry load that is evenly distributed along the active face width. While this might provide convenient design guidelines, it fails to reveal the actual spline load distribution.

For a high-fidelity design of splines, one must know the relationship between quality level chosen for a particular spline and excess loads and stresses that must be accounted for due to indexing errors allowed by that quality level. A higher accuracy spline requiring higher manufacturing costs would result in more modest increases in loads while a lower quality class spline with lower production cost might experience loads and stresses that are far beyond the nominal “no error”

conditions. This chapter aims at providing means to capture the impact of tooth indexing errors

to provide the spline designer with the knowledge to make an informed decision in regards to proper quality level for a spline.

A literature survey reveals that both analytical and computational models have been used to address spline load distribution problems. Analytical models [3.1-3.4] were rather limited in terms of their capabilities due to the complex conformal contact occurring over a wide area simultaneously at multiple spline tooth pairs. For instance, the models described in Ref. [3.1-3.3] provided an estimation of spline load distribution for a simple torsional loading condition, while neglecting effects of tooth manufacturing errors. The model proposed in Ref. [3.4] accounted for influences of tooth manufacturing errors, but was not able to handle load distribution of splines under more realistic combined loading conditions or misalignment.

Aside from these analytical models, computational models using finite element (FE) or boundary element (BE) methods [3.5-3.11] were more commonly used in analysis of spline load distribution while accounting for effects of tooth manufacturing errors. For instance, Medina and

Olver [3.9, 3.10] developed a BE based spline model and studied load distribution of misaligned splines, along with effects of spline pitch errors and lead crown modifications. Hong et al [3.11] developed a FE based deformable-body model of a gear-shaft spline and investigated influences of combined loading conditions, tooth surface modifications and effects of tooth indexing errors under limited loading cases. While these computational models are superior to analytical models in terms of their capabilities and accuracy, they require considerable computational time for each analysis. In the presence of axis-asymmetric loading conditions and tooth manufacturing errors, load distribution analysis of a spline joint has to be repeated at multiple rotational positions over an entire revolution of the spline joint. Also, in order to obtain the probability distribution of

100

contact stresses and tooth forces of a spline designed to a specific tolerance class, the same

contact analyses must be performed over a complete rotational cycle for many different tooth

manufacturing error combinations such that statistically meaningful distributions can be obtained.

In such cases, the number of total contact analyses required is very large, making use of these computational models rather impractical, if not impossible.

The semi-analytical spline load distribution model proposed in Chapter 2 allows analysis of splines under arbitrary combined loading conditions. In addition, the model captures effects of design variations and misalignments, at the same time, requiring a computational effort that is several orders of magnitude less than that required by computational models since a simplex algorithm based multi-step discretization solution scheme is used. In this chapter, first, certain revisions to the semi-analytical model of Chapter 2 will be proposed to include tooth indexing errors. This revised model will then be used to investigate effects of various forms of tooth indexing errors on load distribution and tooth load sharing of splines under different combined loading conditions. Given the numerical efficiency of the semi-analytical load distribution model, a comprehensive statistical analysis methodology will also be implemented to relate the spline quality level to the resultant probability distribution of tooth load sharing and contact pressure distributions. At the end, recommendations on how to account for indexing errors in design of splines will be provided.

3.2 Application of Indexing Errors to the Load Distribution Model

The governing equations and relevant constraints for load distribution prediction of all n potential contact point pairs of side-fit involute splines under arbitrary combined loading conditions were proposed in Chapter 2 in matrix form as

101

Z  I0GT 0YP  = , (3.1a) 0I− CGF ε Φ

YFii>=0, for 0,  in∈ [1, ] . (3.1b) YFii= 0, for ≥ 0,

The initial separation vector, ε , in Chapter 2 included only geometric deviations of tooth surfaces that are identical for each tooth of the spline joint, with the intention of accounting for nominal tooth surface (profile and lead) modifications denoted below by superscript sm. For any two

()sm () sm ()sm arbitrary teeth k1 and k2 , εεεkk12= = , ∀∈kk12, [1, Z ] . With this, the initial separation

vector ε was defined in Chapter 2 as

ε()sm    ε = ε()sm . (3.2)   ()sm ε

Here each vector ε()sm contains PQ× entries defining the combination of nominal tooth surface

modifications for both the internal and external spline. With identical entries for Z teeth, ε is of length n=×× Z PQ.

In this chapter, initial separations induced by tooth indexing errors will be included in the

semi-analytical model. Figure 3.1 shows schematically indexing errors of internal and external spline teeth, which are deviations of actual tooth position from the ideal tooth position. An

indexing error is defined as positive if it reduces the clearance between the two contacting sides

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(a) 1 1 λk λk−1 1 λk+1

Tooth k + 1 Tooth k Tooth k − 1

(b)

Tooth k Tooth k + 1 Tooth k − 1

2 λ2 λk+1 k 2 λk−1

Ideal tooth position Actual tooth position

Fig. 3.1 Schematic representation of tooth indexing errors of (a) external spline teeth and (b) internal spline teeth.

103

of a spline tooth pair, otherwise it is defined as negative. For a spline having Z number of teeth,

1 the indexing errors of tooth z of the external and internal spline members are denoted as λk and

2 λk ( kZ∈[1, ] ), respectively, and the combined indexing error of tooth pair z is denoted as

*12 λλλkkk= + . Then, the initial separations induced by tooth indexing errors (ie) are defined as

ε()ie 1   T ε()ie = ()ie , ε()ie = λλλ*** (3.3a,b) εk k kkk   ()ie εZ

()ie * where each sub-vector entry εk contains PQ× identical elements, λk . With this error

introduced, the initial separation vector ε including both tooth modification and indexing errors becomes

εε()sm + ()ie 1   ε = εε()sm + ()ie . (3.4) k  ()sm ()ie εε+ Z

It is also worthwhile to note here that other types of deviations such as tooth profile errors

specific to each tooth of the spline can also be accounted for in the model as additional terms in

Eq. (3.4).

104

3.3 Numerical Results and Discussions

Effects of tooth indexing errors on load distribution of splines under pure torsion have been

investigated in a number of studies [3.4, 3.7, 3.10, 3.11]. They showed that under pure torsion loading, spline teeth come to contact in a sequential manner based on their indexing error magnitudes. Spline teeth with larger indexing error have smaller clearance magnitudes, causing them to engage first. Spline teeth with smaller indexing error, meanwhile, have larger clearances such that they come into contact only when the torque is increased to a certain level to deflect the

teeth already in contact sufficiently. These studies were instrumental for pointing to the critical

importance of tooth indexing errors, but they were limited to purely torsional loading.

Accordingly, the loading conditions experienced by gear splines will be emphasized here, and

effects of tooth indexing errors on the load distribution and tooth load sharing of splines under

different combined loading conditions will be investigated.

3.3.1 Combined Torsion and Radial Forces

As stated in Chapter 2, a spline joint connecting a spur gear to its shaft is subject to

combined torsion and radial loads and this asymmetric loading condition results in considerable tooth-to-tooth variations of load distribution and load sharing. As the spline rotates to different kinematic positions, load distribution and load sharing of any individual spline tooth changes with rotation angle. Aside from effects of this asymmetric loading condition, spline tooth indexing errors also influence the resultant load distribution and tooth to tooth load sharing. In order to

distinguish the effects of loading condition from that of tooth indexing errors, analyses will be

performed here for three different configurations, i.e. no tooth indexing error, single tooth

indexing error and a random sequence of tooth indexing errors. In addition to these studies,

105

investigations of the robustness to random tooth indexing errors will also be performed aiming to

obtain the probability distribution of tooth load sharing of splines designed to a specific tolerance

class.

Case of No Tooth Indexing Error: Load distribution analysis of an example spline having

no tooth indexing error is repeated at 121 rotational positions evenly distributed from 0° to 360°

spanning a full revolution at an increment of 3°. The parameters of the example spline are shown

in Table 3.1. As it is not practical to show load distributions of each tooth of the spline at all 121

rotational position increments, a few sample rotational positions, namely 0°, 90°, 180° and 270° shown schematically in Fig. 3.2, are selected for demonstration purposes. Figure 3.3 shows load distributions of the example spline having no tooth error under combined torsional and radial loads of M z = 8000 Nm, Fx = −105 kN (negative sign indicates the force direction is along

negative x axis) and Fy = 38 kN. It can be observed that load distribution changes significantly

as the spline rotates to different positions. At each rotational position, load distribution also

varies from tooth to tooth due to the asymmetric loading condition. Spline teeth whose contact

force is nearly opposite the radial load direction carry more load, while those whose contact force

is nearly in the same direction as the radial load carry less. For instance, at rotational position 0°,

teeth #20-25 and teeth #1-5 carry more load while the remaining teeth carry less. As the spline

rotates to 180°, teeth #20-25 and teeth #1-5 carry lower loads while the remaining teeth carry

larger loads. This clearly demonstrates the rotational position dependent (time-varying) nature of

load distribution of a spline due to this loading condition.

As a consequence of the time-varying load distribution, the tooth-to-tooth load sharing of

the spline also varies as the spline rotates. Figure 3.4 shows load sharing factors  of a few

representative spline teeth, namely teeth #5, #10, #15 and #20, as a function of rotational position

106

Table 3.1 Parameters of an example spline design used in the indexing error study.

External spline Internal spline Number of teeth 25

Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.550 85.725

Minor diameter [mm] 73.025 76.200

Circular space width [mm] - 5.032

Circular tooth thickness [mm] 4.943 -

Inner rim diameter [mm] 0 -

Outer rim diameter [mm] - 127

Profile crown [µm] 8 0.0

107

y y (a) (b)

2 1 25 21 20 19 3 24 22 18 4 M 23 23 17 y 16 5 22 24 F M Fy M 15 6 y x 21 25 x 14 7 M z 20 1 M z F 19 x Fx 13 x 8 x 2 z z 12 9 18 3 10 17 4 11 11 16 10 5 9 12 13 14 15 6 7 8

y y (c) (d)

15 14 13 12 8 7 6 16 11 9 5 10 17 10 4 11 3 18 F 9 y M x 12 2 19 8 M z 13 M z 1 20 7 Fx x 14 x 21 6 25 z 15 z 22 5 24 16 23 4 23 17 24 3 21 22 25 1 2 18 19 20

Fig. 3.2 Schematic side view of the external spline under combined loading condition at four sample rotational positions of (a) 0°, (b) 90°, (c) 180° and (d) 270°.

108

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 MPa 400 (a) 350

300 (b) 250

200 109 (c) 150

100

(d) 50

Fig. 3.3 Load distributions of the example spline having no tooth error under combined torsional and radial loads at different

rotational positions of (a) 0°, (b) 90°, (c) 180° and (d) 270°. M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

109

3 (a)

2 5 1

0 3 (b) 2

10 1

0 3 (c)

15 1

0 3 (d) 2

20 1

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.4 Load sharing factors of (a) tooth #5, (b) tooth #10, (c) tooth #15 and (d) tooth #20 as a function of rotational position for the example spline having no tooth error under

combined torsional and radial loads. M z = 8000 Nm, Fx = −105 kN and

Fy = 38 kN .

110

angle. Here, tooth load sharing factor of a tooth k ( kZ∈[1, ] ) is defined as the actual force F k

carried by the tooth divided by the average (nominal) tooth force F , expressed mathematically as

F k  = , (3.5a) k F

Z ∑ k = Z . (3.5b) k=1

It is clear that for each of the teeth, load sharing factor changes as the spline rotates to different positions. For instance, tooth #5 shares about 1.6 times the average load at rotational position 0°, which drops gradually down to its minimum ( 5,min = 0.19 ) at 129° and then increases to its peak ( 5,max = 2.04 ) at 309°. It is also found that teeth #10, #15 and #20 have the same time-

varying cyclic load sharing characteristics as tooth #5, with each of them having a different phase

shift. This can be expected since there is no tooth indexing error.

Case of a Single Tooth Indexing Error: Similar load distribution analysis under the same

loading condition as in previous case is repeated here for a single tooth indexing error. In this

1 case, a 20 μm indexing error is applied to tooth #13 of the external spline ( λ13 = 20 μm ) while

2 1 all the other spline teeth have no indexing error ( λk = 0 for ∀∈k [1, 25] , λk = 0 for

k ∈∪[1, 12] [14, 25] ). Figure 3.5 shows this tooth indexing error sequence and corresponding

load distributions of the spline at four sample rotational positions of 0°, 90°, 180° and 270°.

Compared to the no error case, it is found that this indexing error significantly increases the

contact stress and contact area on tooth #13 at the same sample rotational positions. For instance,

the maximum contact stress of tooth #13 in case of no tooth error was 208, 249, 303 and 270 MPa

at rotational positions of 0°, 90°, 180° and 270°, respectively. In the presence of this tooth

111

20 (a) λ*(µm) 10

0 MPa 400 (b) 350

300

200 112

(d) 150

100

(e) 50

0 1 5 10 15 20 25 Tooth Number

Fig. 3.5 (a) Indexing error sequence, and load distributions of the example spline having single tooth indexing error under combined torsional and radial loads at different rotational positions of (b) 0°, (c) 90°, (d) 180° and (e) 270°.

M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

112

indexing error, the maximum contact stresses of tooth #13 increase to 309, 345, 389 and 355 MPa

at these rotational positions, respectively, representing 28-48% increase in maximum contact

stress. This is because the indexing error results in smaller clearance between spline tooth pair

#13, thus at any given position, causing tooth #13 to carry more load than that in the no error case.

Regardless of this, the remaining spline teeth maintain similar time-varying load distribution

patterns as that in the no tooth error case shown in Fig. 3.3. In other words, the only significant

adverse effect here is on tooth #13 which has the positive indexing error.

Figure 3.6 shows the corresponding time-varying load sharing factors of three

representative spline teeth, #10, #13 and #15, over a complete revolution, together with those for

the no error case. It is evident that, in the presence of the indexing error, load sharing factor of

tooth #13 becomes significantly larger than that in the no tooth error case. For example, at

rotational positions 60°, 120°, 180° and 240°, load sharing factors of tooth #13 were 13 = 0.31,

1.21, 2.00 and 1.69, respectively, for the case of no indexing error. As a direct result of the

indexing error on tooth #13, these values are seen to increase to 13 =1.67, 2.91, 3.75 and 3.43, respectively. In spite of this, load sharing of teeth #10 and #15 reduce only slightly in comparison to the no-tooth-error case. These results illustrate that the overall load distribution pattern and load sharing characteristic are determined by loading condition, while indexing error of a single tooth induces considerable variations to load distribution and load sharing of that tooth.

Case of Random Tooth Indexing Errors: The previous case of single tooth indexing error

(error only on one tooth while others remain in their perfect positions) is not realistic. It is more representative of real-life to have random indexing errors on both external and internal spline teeth. According to STANDARD ANSI B92.1 [3.12], the tooth index variation of the example

113

4 (a) No Error 3 Single Tooth Error

10 2

0 4 (b)

13 2

0 4 (c)

15 2

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.6 Load sharing factors of (a) tooth #10, (b) tooth #13, and (c) tooth #15 as a function of the rotational position for the example spline having single tooth indexing error * (λ13 = 20 μm) in comparison to the no tooth error case under combined torsional and

radial loads. M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

114

spline of Table 3.1 for a tolerance class of 7 must be within the band: λ ∈−[ 45,45] μm . This

2 index variation can be approximated by the normal distribution, λ~N (, λσλ ) , where the mean index error λ = 0 µm and the standard deviation σλ =15 µm. With this, two sets of random tooth

indexing error sequence λ1 (for external spline) and λ2 (for internal spline) can be generated.

The total indexing error sequence of all tooth pairs can be obtained as λλλ*12= + .

Figure 3.7(a) shows a representative random sequence λ* of total tooth indexing errors for the example spline for tolerance class 7. Corresponding load distributions of the spline at rotational positions of 0°, 90°, 180° and 270° are also shown in Fig 3.7(b-e). Here, loading conditions are the same as in Fig. 3.3. With random tooth indexing errors present, the load distribution patterns are seen to become more complicated. In this case, both loading condition and random tooth indexing errors have considerable effect on the load distribution. For the given combined torsional and radial loading condition, load distribution of the spline varies from tooth to tooth when there is no indexing error. In the presence of tooth indexing errors, clearance between each pair of spline teeth differs from other pairs, thus leading to additional sizable variations in load distribution. A larger total indexing error of a pair of spline teeth effectively

reduces the clearance in between, causing them to carry more load than their nominal share. For instance, at rotational position of 0°, teeth #17 and #18 experience lower contact stress in small areas when there is no tooth error as shown in Fig. 3.3. However, at the same rotational position, they exhibit much larger contact area and higher contact stresses as shown in Fig. 3.7 as a consequence of having large indexing errors. On the other hand, if the total indexing error of a pair of spline teeth is smaller, then the clearance between them becomes larger and they are likely to carry less or no load. For example, teeth #2 and #24 experience high contact stresses over a

115

45 (a)

* λ (µm) 0

-45 MPa 400 (b)

350

300 (c) 250 116 200

(d) 150

100

(e) 50

0 1 5 10 15 20 25 Tooth Number

Fig. 3.7 (a) Indexing error sequence, and load distributions of the example spline having random tooth indexing errors under combined torsional and radial loads at different rotational positions of (b) 0°, (c) 90°, (d) 180° and (e) 270°.

M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

116

wide area at rotational position 0° in the case of no tooth error as shown in Fig. 3.3, while they

are unengaged (totally unloaded) in Fig. 3.7 due to negative indexing errors. Similar explanations

can be extended to load distributions of other spline teeth at various rotational positions.

The above load distribution results show that spline teeth with larger positive tooth indexing errors are prone to having larger contact areas and higher contact stresses, making them the primary concern in terms of spline durability performance. In this case, spline teeth #6, #14,

#17 and #18 have large positive indexing errors and will be of concern. Figure 3.8 shows the corresponding time-varying load sharing factors of these teeth with a direct comparison to the no tooth error case under the same loading condition. It is found that load sharing factors of these spline teeth are increased significantly by indexing errors. For instance, load sharing factors of tooth #18 are 18 =1.44, 2.04, 1.50 and 0.43, respectively at rotational position 60°, 120°, 180°

and 240° in case of no tooth error. In the presence of the random indexing errors of Fig. 3.7, load

sharing factors of tooth #18 at these positions are increased significantly to 18 = 3.33, 4.27, 3.38

and 1.54, respectively. Similar observations can also be made for teeth #6, #14 and #17 as well.

Statistical Treatment of Random Tooth Indexing Errors: The results of the previous

section illustrate effects of random tooth indexing errors on load distribution and tooth load

sharing of splines. However, exact tooth indexing error sequences are usually unknown, or not of

particular interest, when designing splines. For splines designed to a certain manufacturing tolerance class, any random indexing error sequence that falls within the corresponding tolerance range is deemed acceptable. Yet, load distribution and tooth load sharing can vary considerably from one indexing error sequence to another. As such, a statistical robustness analysis of random

tooth indexing errors is of main interest from the design point of view. Here, statistical studies on

117

6 (a) No Error 4 Random Error

6 2

0 6 (b)

14 2

0 6 (c)

17 2

0 6 (d)

18 2

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.8 Load sharing factors of (a) tooth #6, (b) tooth #14, (c) tooth #17 and (d) tooth #18 as a function of rotational position for the example spline having a random tooth indexing error sequence in comparison to the no tooth error case under combined

torsional and radial loads. M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

118

the same example spline designed to tolerance classes of 7 and 6 according to STANDARD

ANSI B92.1 [3.12] will be presented.

From Ref. [3.12], for the example spline the tooth index variation range for tolerance class

7 is λc7 ∈−[ 45, 45]µm and for class 6 is λc6 ∈−[ 31.5, 31.5]µm. Unless actual distributions of

these tooth index errors from the manufacturing capability are known, they can be approximated

2 2 using Gaussian or normal distributions, i.e. λc7~(N λσ c7 , c7 )for class 7 and λc6~(N λσ c6 , c6 )

for class 6, where the mean tooth indexing errors are λλcc76= = 0 μm , and standard deviations are σc7 =15 μm and σc6 =10.5 μm . In this study, 100 sets of random tooth indexing error

sequences are generated for both the external and internal spline. The total tooth indexing error

sequences are then obtained by combining the two randomly generated sequences. Load distribution analyses at 121 rotational increments spanning a complete revolution as in previous cases are performed here for each of these 100 sets of random indexing error sequences.

Due to the sheer amount of results in this case, instead of presenting individual load distribution plots as in Fig. 3.7, load sharing of the critical spline tooth pair that has the largest total tooth indexing error, namely the smallest clearance in between, will be focused on. The time-varying load sharing factors of the critical spline tooth for the 100 random error sequences of tolerance class 7 and class 6 are predicted as shown in Fig. 3.9. In this figure, there are 100

critical  values at each rotational positon increment. For both tolerance class 7 and 6, the overall qualitative trend of load sharing factor is well maintained regardless of tooth errors. The load sharing factor of the critical tooth gradually drops as the spline rotates from 0° to about 180°, then it slowly rises as the spline rotates from 180° to 360°. It is also observed that, at every rotational position, load sharing factor of the critical tooth varies over a wide range for both

119

8 (a) 7

(c 7) 4

0 8

7 (b)

(c 6) 4 3

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.9 Load sharing factors of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 at different rotational positions for 100 sets of random tooth

indexing errors under combined torsional and radial loads. M z = 8000 Nm ,

Fx = −105 kN and Fy = 38 kN .

120

tolerance classes. Note that at any given rotational position, the mean load sharing factor of

tolerance class 6 is smaller than that of tolerance class 7.

The above results explicitly demonstrate that the load sharing factor of the critical spline

tooth not only changes with rotational position, but also varies significantly for different random

indexing error sequences. Assuming that 100 sets of random indexing errors are adequate as a

representative sample set, the probability distribution of load sharing factor of the critical spline

tooth over a full rotational cycle for tolerance class 7 and class 6 are obtained and shown in Fig.

3.10. The load sharing factor of the critical spline tooth for tolerance class 7 varies within

(c 7) (c 7) ∈[0, 8] with its mean value at  = 3.59 . About 95% of the load sharing factor values are within [1, 7], and over 81% are within [1.5, 5.5]. Meanwhile, for tolerance class 6, the load

(c6) sharing factor is within the range (c 6) ∈[0, 7] , and the mean value is  = 2.94 . More than

95% of the load sharing factor values fall into [0.5, 5.5], and about 81% are within [1, 4.5].

These results indicate that load sharing of the critical spline tooth would fall into a certain

probability distribution due to manufacturing tolerances. Also, splines designed to a tighter

manufacturing tolerance class, i.e. tolerance class 6 in this example, would exhibit more favorable

load sharing characteristics, which usually represent better durability performance.

3.3.2 Combined Torsional, Radial Loads and Tilting Moment

Another common loading condition for spline joints is combined torsional, radial loads and

tilting moment, which is usually seen in applications of splines supporting helical and cross-axis

gears. In addition to effects of radial loads, additional tilting moment was also shown to

significantly influence load distribution and tooth load sharing characteristics in Chapter 2.

121

0.15 (a)

0.1

0.05

0 Probability 0.15 (b)

0.1

0.05

0 0 2 4 6 8 Load Sharing Factor,  Fig. 3.10 Probability distribution of load sharing factor of the critical tooth of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 under combined

torsional and radial loads. M z = 8000 Nm , Fx = −105 kN and Fy = 38 kN .

122

Combined effects of tooth indexing errors under this loading condition will be investigated here using the same procedure as the previous section.

Case of No Tooth Indexing Error: Figure 3.11 shows load distributions of the example spline with no tooth indexing error under combined torsional, radial loads and tilting moment at rotational positions 0°, 90°, 180° and 270°. Here a torque of M z = 8000 Nm , radial load components of Fx = −105 kN and Fy = 39 kN , and a tilting moment of M x = −2050 Nm are applied simultaneously to the spline joint. As in spur gear loading, it can be seen that load distribution changes with some phase offset as the spline rotates to different positions. At each position, the load distribution varies significantly from tooth to tooth due to this particular asymmetric loading condition. Compared to spur gear loading, the additional tilting moment M x in this case results in higher peak contact stresses, 303 MPa in Fig. 3.3 versus 395 MPa in Fig.

3.11. Aside from this, a significant difference is observed that the load distribution along the face width is biased toward one side for some teeth while toward the other side for the remaining teeth.

For instance, at rotational position 0°, axial load distributions on teeth #3-15 are biased to the side where torque is applied, while those of the remaining teeth are biased to the opposite side. This occurs due to the additional tilting moment transmitted through the spline interface, which requires a biased load distribution to balance it.

As a consequence of the time-varying cyclic load distribution, the tooth load sharing factor is also expected to exhibit a cyclic pattern as the spline rotates. Figure 3.12 shows the corresponding time-varying load sharing factors of a few representative teeth, namely #5, #10,

#15 and #20, over a full rotational cycle. These teeth have identical time-varying load sharing patterns with a certain phase shift with one another. The peak load sharing factor is increased

123

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 MPa 540 (a) 480

420

(b) 360

300

240 124 (c) 180

120

(d) 60

Fig. 3.11 Load distributions of the example spline having no tooth error under combined torsional, radial loads and tilting moment,

at different rotational positions of (a) 0°, (b) 90°, (c) 180° and (d) 270°. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and

M x = −2050 Nm .

124

3 (a)

2 5 1

0 3 (b) 2

10 1

0 3 (c)

2 15 1

0 3 (d) 2

20 1

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.12 Load sharing factors of (a) tooth #5, (b) tooth #10, (c) tooth #15 and (d) tooth #20 as a function of rotational position for the example spline having no tooth error under

combined torsional, radial loads and tilting moment. M z = 8000 Nm ,

Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm .

125

from 2.04 in Fig. 3.4 for the spur gear loading case to 2.20 in this case, due to the additional

tilting moment.

* Case of a Single Tooth Indexing Error: Next, consider a spline having λ13 = 20 μm on

* tooth #13 with all other teeth in their perfect “no error” position (all other λk = 0 ) under the same helical gear loading condition. The tooth indexing error sequence and the corresponding

load distributions at rotational positions of 0°, 90°, 180° and 270° are shown in Fig. 3.13 in the

same format as Fig. 3.5. The overall load distribution pattern is similar to that of Fig. 3.11 while

the introduction of indexing error causes larger contact area and higher contact stress on tooth

#13. As a result of this, 13 values are increased in Fig. 3.14. These results demonstrate again

that the overall load distribution and load sharing characteristic is determined by the loading

condition, and indexing error of a specific tooth would significantly influence its load distribution

and load sharing while having minor effects on other spline teeth.

Case of Random Tooth Indexing Errors: Using the same random sequence of indexing

errors considered in Fig. 3.7, Fig. 3.15 displays predicted load distributions of the example spline

under the same combined torsional, radial loads and tilting moment at different rotational

positions. Compared to the no tooth error case in Fig. 3.11, it can be seen that the overall biased

load distribution pattern is somewhat maintained regardless of the random indexing errors. For instance, at rotational position 0° axial load distribution of teeth #3-15 is biased to the side where torque is applied, while for the remaining teeth it is biased toward the opposite side. In spite of this, spline teeth that have large positive indexing errors tend to have larger areas of contact and higher contact stresses, while those having negative indexing errors exhibit smaller (or no) contact areas and lower (or zero) contact stresses. For instance, teeth #17 and #18 have large

126

20 (a) * λ (µm) 10

0 MPa 540 (b) 480

420

300 127

240 (d) 180

120

(e) 60

0 1 5 10 15 20 25 Tooth Number

Fig. 3.13 (a) Indexing error sequence, and load distributions of the example spline having single tooth indexing error under combined torsional, radial loads and tilting moment loading at different rotational positions of (b) 0°, (c) 90°, (d) 180° and

(e) 270°. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm .

127

4 (a) No Error 3 Single Tooth Error

10 2

0 4 (b)

13 2

0 4 (c)

15 2

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.14 Load sharing factors of (a) tooth #10, (b) tooth #13, and (c) tooth #15 as a function of rotational position for the example spline having single tooth indexing error * (λ13 = 20 μm) in comparison to the no tooth error case under combined torsional,

radial loads and tilting moment. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and

M x = −2050 Nm . 128

45 (a)

* λ (μm) 0

-45

MPa 540 (b) 480

420

300 129

240 (d) 180

120

(e) 60

0 1 5 10 15 20 25 Tooth Number

Fig. 3.15 (a) Indexing error sequence, and load distributions of the example spline having random tooth indexing error under combined torsional, radial loads and tilting moment, at different rotational positions of (b) 0°, (c) 90°, (d) 180° and (e)

270°. M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm .

129

positive indexing errors and they exhibit larger contact areas and higher contact stresses at all

sample rotational positions in comparison to the no error case in Fig. 3.11. On the other hand,

teeth #2 and #19 have negative indexing errors and they are not in contact at all for these four

rotational positions shown. As such, spline teeth that have large positive indexing errors are again the concern here. Figure 3.16 shows load sharing factors of some spline teeth (i.e. teeth #6,

#14, #17 and #18), that have large indexing errors and the comparison to the no tooth error case.

It is observed that indexing errors of these teeth significantly increase their load sharing factors at all rotational positions. For example, in the case of no tooth indexing error, load sharing factors of tooth #18 at rotational position 60°, 120°, 180° and 240° are 18 =1.66, 2.15, 1.14 and 0.70,

respectively, while in the presence of random indexing error, these values are increased to

18 = 3.32, 4.12, 2.60 and 1.56, respectively. Similar observations can also be made for teeth

#6, #14 and #17 as well.

Statistical Treatment of Random Tooth Indexing Errors: The same type of statistical

analysis as that for the spur gear loading condition is repeated here, with an additional tilting

moment. Figure 3.17 shows the corresponding load sharing factors of the critical tooth of the

example spline designed to tolerance classes of 7 and 6, for the same sets of random tooth

indexing errors as in Fig. 3.9. Similar observations can be made here as in Fig. 3.9 in terms of

rotation dependency of the load sharing factor of the critical tooth. At each individual rotational

position, load sharing factors are spread over a range. Figure 3.18 shows the corresponding

probability distribution of load sharing factor of the critical spline tooth for tolerance classes 7

and 6. Load sharing factor values of the critical spline tooth for tolerance class 7 are within

(c 7) (c 7) ∈[0, 8] , with the mean value  = 3.05 . Meanwhile, for tolerance class 6,

130

6 (a) No Error 4 Random Error

6 2

0 6 (b)

14 2

0 6 (c)

4 17 2

0 6 (d)

18 2

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.16 Load sharing factors of (a) tooth #6, (b) tooth #14, (c) tooth #17 and (d) tooth #18 as a function of rotational position for the example spline having a random tooth indexing error sequence in comparison to the no tooth indexing error case under

combined torsional, radial loads and tilting moment. M z = 8000 Nm ,

Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm .

131

8 (a) 7

(c 7) 4

0 8 (b) 7

(c 6) 4 3

0 0 60 120 180 240 300 360 Angular Rotational Position [°]

Fig. 3.17 Load sharing factors of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6, at different rotational positions for 100 sets of random tooth indexing errors under combined torsional, radial loads and tilting moment.

M z = 8000 Nm , Fx = −105 kN , Fy = 39 kN and M x = −2050 Nm .

132

0.15 (a)

0.1

0.05

0 Probability 0.15 (b)

0.1

0.05

0 0 2 4 6 8 Load Sharing Factor, 

Fig. 3.18 The probability distribution of load sharing factor of the critical tooth of the example spline designed to (a) tolerance class 7 and (b) tolerance class 6 under combined

torsional, radial loads and tilting moment. M z = 8000 Nm , Fx = −105 kN ,

Fy = 39 kN and M x = −2050 Nm .

133

(c 6) (c6) ∈[0, 6.5] and  = 2.52 . These results demonstrate again that splines having tighter manufacturing tolerances exhibit better load sharing characteristics.

3.4 Summary

In this chapter, the semi-analytical model proposed in Chapter 2 was modified to investigate the influence of tooth indexing errors on load distribution and tooth load sharing characteristics of spline joints under different combined loading conditions. The load distribution pattern and tooth load sharing characteristic are found to be influenced by both the loading conditions and tooth indexing errors. Loading conditions determine the overall load distribution pattern and load sharing characteristics, while tooth indexing errors induce additional variations to the overall load distribution and tooth load sharing. Compared to the no tooth error cases, spline tooth pairs that have larger indexing errors tend to experience higher contact stress over a wider contact area, thus sharing more load. Aside from these, a more practical robustness study reveals that load sharing factor of the critical spline tooth falls into a domain with a certain probability distribution for a given manufacturing tolerance. Splines with tighter manufacturing tolerance are found to exhibit better load sharing characteristics.

References for Chapter 3:

[3.1] Volfson, B. P., 1982, “Stress Sources and Critical Stress Combinations for Splined Shaft,”

Journal of Mechanical Design, 104, 551-556.

134

[3.2] Barrot, A., Paredes, M. and Sartor, M., 2006, “Determining Both Radial Pressure

Distribution and Torsional Stiffness of Involute Spline Couplings,” Proc. IMechE., Part

C: J. Mechanical Engineering Science, 220, 1727-1738.

[3.3] Barrot, A., Paredes, M. and Sartor, M., 2009, “Extended Equations of Laod Distribution

in the Axial Direction in a Spline Coupling,” Engineering Failure Analysis, 16, 200-211.

[3.4] Chase, K. W., Sorensen, C. D. and DeCaires, B. J. K., 2010, “Variation Analysis of

Tooth Engagement and Loads in Involute Splines,” IEEE Transactions on Automation

Science and Engineering, 7(4) 54-62.

[3.5] Kahn-Jetter, Z. and Wright, S., 2000, “Finite Element Analysis of an Involute Spline,”

Journal of Mechanical Design, 122, 239-244.

[3.6] Tjernberg, A., 2000, “Load Distribution in the Axial Direction in a Spline Coupling,”

Engineering Failure Analysis, 8, 557-570.

[3.7] Tjernberg, A., 2001, “Load Distribution and Pitch Errors in a Spline Coupling,”

Materials and Design, 22, 259-266.

[3.8] Ding, J., Leen, S.B., Williams, E. J. and Shipway, P. H., 2008, “Finite Element

Simulation of Fretting Wear-Fatigue Interaction in Spline Couplings,” Tribology-

Materials, Surface and Interfaces, 2(1), 10-24.

[3.9] Medina, S. and Olver, A. V., 2000, “Regimes of Contact in Spline Coupling,” Journal of

Tribology, 124, 351-357.

135

[3.10] Medina, S. and Olver, A.V., 2002, “An Analysis of Misaligned Spline Coupling,” Proc

IMechE., Part J: J. Engineering Tribology, 216, 269-279.

[3.11] Hong, J., Talbot, D. and Kahraman, A., 2014, “Load Distribution Analysis of Clearance-

Fit Spline Joints Using Finite Elements,” Mechanism and Machine Theory, 74, 42-57.

[3.12] STANDARD ANSI B92.1-1996, Involute Splines and Inspection, SAE, Warrendale, PA.

136

CHAPTER 4

A LOAD DISTRIBUTION MODEL FOR MAJOR OR MINOR DIAMETER-FIT

SPLINES AND MISMATCHED SPLINES

4.1 Introduction

The semi-analytical model proposed in Chapter 2 was shown to exhibit modeling flexibility combined with exceptional computational efficiency for load distribution analysis of side-fit spline joints. One major assumption of the model proposed in Chapter 2 was that spline contacts can occur only at driving sides of tooth surfaces. In other words, it did not allow contacts along other surfaces such as the tooth top land, tooth root land or tooth back side. In this chapter, spline joints allowing such types of contacts will be analyzed.

As stated in Chapter 1, major or minor diameter-fit splines are commonly used in applications where radial alignment of the shaft axes is critical, for perhaps minimizing run-out error of a gear supported by a splined shaft. In such splines, aside from contact at the tooth driving side, there is also potential contact occurring at the tooth top land (for a major diameter-fit spline) or tooth root land (for a minor diameter-fit spline). Slight radial clearance between contact surfaces at the tooth top or root land tightly constrains relative rigid body motions between the external and internal member of a spline joint, providing good centralization capability.

137

Under high-torque or reverse-torque conditions, spline joints are commonly designed to

have a certain amount of intentional mismatch by introducing a slight helix angle to the external

spline while the internal spline teeth are straight. This mismatch is typically larger than what can

be accommodated by the circumferential clearance, leading to a torsional wind-up as a preload.

In this type of spline, contact potentially occurs at both the tooth driving sides and back sides.

Most published spline load distribution models are limited to side fit splines only,

neglecting contacts at other tooth surfaces. Hong et al [4.1] investigated load distribution of

intentional mismatched splines under pure torsion using the FE based deformable-body model of

Vijayakar [4.2, 4.3], which accounts for back side contact. Other than this, all spline load

distribution models reviewed in previous chapters ignored back side contact. Moreover, literature

lacks any model on load distribution of major or minor diameter-fit splines.

This chapter aims at extending the semi-analytical model of Chapter 2 to include load distribution analysis of major and minor diameter-fit splines, as well as intentionally mismatched splines. First, modifications to the baseline model of Chapter 2 will be presented to account for contact at tooth surfaces other than the tooth driving side. With the revised model, parametric studies will be performed to investigate load distribution characteristics of major and minor

diameter-fit splines, as well as intentionally mismatched splines.

4.2 Contact Formulation

4.2.1 Major Diameter-Fit Splines

Figure 4.1(a) schematically shows the side view of an example major diameter-fit spline

joint. In this type of fit, the major diameter of internal spline is marginally larger than that of the

138

(a) Internal Spline Contact zone at tooth driving side

Major Diameter 35 D D ri e Contact zone at Minor Diameter tooth top land Dre Di External Spline 30

(b) Face Width Face Width 25 Direction Direction -15 -10 -5 0 5 10 15

Qd Qt

… … …

d t Fid, i [1, n ] Fjt, j [1, n ]

nd Z  P d  Q d nt Z  P t  Q t

… … …

2 Profile 2 Circumferential 1 Direction 1 Direction … … … … 1 2 3 Pd 1 2 3 Pt Contact zone at Contact zone at tooth driving side tooth top land

Fig. 4.1 (a) Schematic side view of an example major diameter-fit spline and its potential contact zones, and (b) discretization of the potential contact zones.

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external spline, leading to a very small radial clearance between the external and internal member

at the major diameter circle. As such, contact can potentially occur at the tooth driving side and tooth top land. Discretization of these potential contact zones is shown in Fig. 4.1(b). The contact zone at the tooth driving side is discretized into Pd cells with equal arc length along the

profile direction and Qd cells with equal length along the face width direction. Similarly, the tooth top land is divided into Pt cells with equal arc length along the circumferential direction

and Qt cells with equal length along the face width direction. Denoting the number of spline teeth as Z, the total number of contact cells at tooth driving sides and tooth top land over the spline interface are nd=×× ZP dd Q and nt=×× ZPQ tt, respectively. Providing that each grid cell is small compared to the size of the contact zone, contact pressure over each grid cell is

d represented by a concentrated contact force ( Fiid, ∈[1, n ] for tooth driving side, and

t Fjjt, ∈[1, n ] for tooth top land) acting at the center of each grid cell. As for side fit splines of

Chapter 2, equations related to these contact forces over a major diameter-fit spline joint can be

established by enforcing compatibility and equilibrium conditions.

For a contact point pair ii (∈ [1, nd ]) within the contact zone on the tooth driving side and a contact point pair jj (∈ [1, nt ]) within the contact zone on the tooth top land, the initial

d t separations of these two contact point pairs in an unloaded condition are denoted as εi and ε j ,

respectively. When external load is applied, elastic deformations occur due to the contact forces.

The sum of elastic deformations of both the external and internal members at these two point

d t pairs along their normal directions are denoted as di (driving side) and δ j (top land),

d t respectively. Meanwhile, rigid body approach ζi (driving side) and ζ j (top land) along their

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d t normal directions also occur. The final gaps, Yi (driving side) and Yj (top land), at these selected contact point pairs in a loaded condition can be written as

 dddd Yi=+−εdζ iii, in ∈ [1, d ],   (4.1)  ttt t Yjjjj=+−εdζ, jn ∈ [1, t ].

Since only elastic deformations are allowed, the final gaps must always be nonnegative. If the final gap is positive, then the potential contact pair is not in contact and the contact force is zero.

If the final gap is zero, then the potential contact point pair is in contact and the contact force is positive. For all contact point pairs, the compatibility conditions can be written in vector form as

Ydd εδζ  dd    =+−    , (4.2a) tt tt Y εδζ   

 dd YFii>=0, for 0,  in∈ [1, ] , (4.2b) dd d YFii= 0, for ≥ 0,

 tt YFjj>=0, for 0,  jn∈ [1, ] . (4.2c) ttt YFjj= 0, for ≥ 0,

d t In above equation, the vectors δ of dimension nd and δ of dimension nt denote elastic deformations of all contact points at the tooth driving side and tooth top land, respectively. They relate to the contact forces via a coupled compliance matrix as

dd  δFCCdd dt  = . (4.3) tt δFCCtd tt 

141

d t Here, vectors F of dimension nd and F of dimension nt are vectors of contact forces at the

tooth driving side and tooth top land, respectively. Sub-matrix Cdd represents compliance of contact points at the tooth driving side induced by contact forces at the tooth driving side. This sub-matrix can be obtained by including tooth bending and shear, tooth base flexibility, tooth contact and torsional compliance as described in Chapter 2. Sub-matrix Ctt represents compliance of contact points at the tooth top land induced by contact forces at the tooth top land.

In this case, contact forces at the tooth top land are perpendicular to the tooth tip so that tooth

bending and shear, tooth base flexibility and torsional compliance become negligible. Thus, the

sub-matrix Ctt will only include contact compliance, which can be obtained in a similar procedure using Eqs. (2.25, 2.26) as described in Chapter 2. Sub-matrix Cdt represents compliance of contact points at the tooth driving side induced by contact force at the tooth top

land, and Ctd represents compliance of contact points at the tooth top land induced by contact

force at the tooth driving side. Influence of these coupled compliance matrices are considered to

T be secondary in this study such that CC0dt= td = .

In Eq. (4.2), vectors ζd and ζt contain rigid body approach components along normal

directions of all contact points at the tooth driving side and tooth top land, respectively. Similarly

as in Chapter 2, these vectors relate to tooth surface geometry and relative rigid body motions

between the external and internal splines as follows

d ζ Gd = Φ . (4.4) t  ζ Gt

142

Here, the vector Φ contains five relative rigid body motions of the external spline with respect to

T the internal spline, i.e. Φ = uuxyxyzθθθ . The sub-matrices Gd and Gt relating to the tooth surface geometry are defined as

en⋅d en ⋅ d −⋅z dddd en zR en ⋅ xy1 1 1111 y xb      d d dddd Gd = enxi⋅ en yi ⋅ −⋅z iyi en zR ixi en ⋅ b, (4.5a)      d d d dd d enxy⋅ en ⋅−⋅zzR en y en xb ⋅ nd n d n d nn dd n d

en⋅t en ⋅ t −⋅zz tt en tt en ⋅ 0 xy1 1 1111 y x      t t tttt Gt = enxj⋅ en yj ⋅ −⋅zz jyj en jxj en ⋅ 0 (4.5b)      t t t tt t enxy⋅ en ⋅−⋅zz en y en x ⋅0 nt n t n t nn tt n t

d d where ni and zi ( in∈[1,d ] ) denote the unit normal vector and z coordinate of the i-th contact

t t point in the contact zone on the tooth driving side. Likewise, n j and z j ( jn∈[1,t ] ) denote the unit normal vector and z coordinate of the j-th contact point in the contact zone on the tooth top land. ex and ey are the unit vectors along the x and y axes, respectively, and Rb is the base

circle radius of the spline.

With Eq. (4.3) and (4.4) in hand, the compatibility conditions of Eq. (4.2) are rewritten as

d  dd   CCdd dt FY   Gd  ε  −  +{Φ} +=   , (4.6a)  t  tt CCtd tt FY   Gt  ε 

143

 dd YFii>=0, for 0,  in∈ [1, ], (4.6b) dd d YFii= 0, for ≥ 0,

 tt YFjj>=0, for 0,  jn∈ [1, ] . (4.6c) ttt YFjj= 0, for ≥ 0,

These compatibility conditions yield ()nndt+ equations and ()nndt+ constraints. While

d t there are ()nndt+ unknown contact forces, Fiid(∈ [1, n ]) and Fijd(∈ [1, n ]) , ()nndt+

d t unknown final gaps, Yiid(∈ [1, n ]) and Yjjt(∈ [1, n ]) , as well as five unknown rigid body

T approach terms Φ = uuxyxyzθθθ . Five additional equations needed to solve this problem are obtained from the static equilibrium conditions. As in Chapter 2, arbitrary external load components applied to the external spline are denoted as Fx , Fy , M x , M y and M z .

Contact forces of the external spline teeth must balance the external load components, yielding

dd tt ∑∑Fine i⋅+ x FF jj ne ⋅= x x, (4.7a)

dd tt ∑∑Fine i⋅+ y FF jj ne ⋅= y y, (4.7b)

d dd t tt ∑∑−zi F ine i ⋅+− yzF jjj ne ⋅= y M x, (4.7c)

d dd t tt ∑∑zi F ine i⋅+ x zF jjj ne ⋅= x M y, (4.7d)

d ∑ FRib= M z. (4.7e)

144

These equations can be written in matrix form as

d TTF (GG) ( ) = P (4.8) dtt F

T T where (Gd ) and (Gt ) are the transpose of matrices Gd and Gt defined in Eq. (4.5a) and

(4.5b), respectively, and P is a vector containing the five nominal load components experienced

T by the external spline, i.e. P = []FFMxyxyz M M . Note that contact forces at the tooth top land help counterbalance radial loads and tilting moments transmitted over the spline joint.

Combining the equilibrium and compatibility conditions, governing equations and constraints for the major diameter-fit spline contact problem can be written as

Z  Yd TT I00 G G 0P ( dt) ( ) t Y d  0I0−− Cdd C dt G d   =ε  , (4.9a) d 00I−− C C G F t  td tt t ε Ft  Φ

 dd YFii>=0, for 0,  in∈ [1, ] , (4.9b) ddd YFii= 0, for ≥ 0,

 tt YFjj>=0, for 0,  jn∈ [1, ] . (4.9c) ttt YFjj= 0, for ≥ 0,

145

where Z is a vector of artificial variables added to construct an identity matrix I required by the

simplex-type algorithm. The above equations have the same structure as that of the governing

equations for side fit splines described in Chapter 2, such that the same solution algorithm described in Chapter 2 can be applied here to obtain load distribution of major diameter-fit splines.

4.2.2 Minor Diameter-Fit Splines

Figure 4.2(a) shows a schematic side view of a minor diameter-fit spline. The radial clearance between the external and internal member of this type of spline at the minor diameter circle is very small such that there will be potential contact at the tooth root land of the external spline teeth in addition to potential contact at tooth driving sides. Figure 4.2(b) shows discretization of these potential contact zones. The contact zone at the tooth driving side has the same discretization as that of major diameter-fit splines shown in Fig. 4.1(b). The contact zone at the tooth root land is divided into Pr cells with equal arc length along the circumferential direction and Qr cells with equal length along the face width direction. For a spline having Z

teeth, the total number of contact cells at tooth driving sides and tooth top land over the spline

interface are nd=×× ZP dd Q and nr=×× ZP rr Q, respectively. Equations related to contact

d r forces ( Fiid, ∈[1, n ] for tooth driving side, and Fjjr, ∈[1, n] for tooth root land) at all contact grid cells over a minor diameter-fit spline joint can be obtained by enforcing compatibility and equilibrium conditions in a similar procedure as described in the previous section. Governing equations and respective constraints for the minor diameter-fit spline contact problem are then obtained as follows

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45 Contact zone at (a) Internal Spline tooth driving side

Major Diameter D 35 ri De Minor Diameter Contact zone at tooth root land External Spline Dre Di

(b) -15 Face Width-10 -5 0 Face5 Width 10 15 Direction Direction

Qd Qr

… … …

d r Fid, i [1, n ] Fjr, j [1, n ]

nd Z  P d  Q d nr Z  P r  Q r

… … …

2 Profile 2 Circumferential 1 Direction 1 Direction … … … … 1 2 3 Pd 1 2 3 Pr Contact zone at Contact zone at tooth driving side tooth root land

Fig. 4.2 (a) Schematic side view of an example minor diameter-fit spline and its potential contact zones, and (b) discretization of the potential contact zones.

147

Z  Yd T T  I00 G G 0P ( dr) ( ) r Y d  0I0−− Cdd C dr G d   =ε  , (4.10a) d 00I−− C C G F r  rd rr r ε Fr  Φ

 dd YFii>=0 for 0,  in∈ [1, ] , (4.10b) dd d YFii= 0 for ≥ 0,

 rr YFjj>=0 for 0,  jn∈ [1, ] . (4.10c) rr r YFjj= 0 for ≥ 0,

Above set of equations is analogous to Eq. (4.9), with the same symbols representing the same parameters. For instance, P is the vector of nominal load components experienced by the

d d external spline and F is the vector of contact forces ( Fiid, ∈[1, n ] ) at the tooth driving side.

Also note that in Eq. (4.10) the superscript or subscript r denotes respective parameters corresponding to contact points in the contact zone at the tooth root land. For instance, Fr is the

r r vector of contact forces ( Fjjr, ∈[1, n ] ) at the tooth root land, ε is the vector containing initial

r separations ( ε jr, jn∈[1, ] ) of contact point pairs at the tooth root land, and Gr is a matrix relating to tooth surface geometry at the tooth root land, which can be defined similarly as Gt of

Eq. (4.5b).

In regards to the sub-matrices of compliance in Eq. (4.10), similar definitions and assumptions as in the previous section for major diameter fit splines also apply here. Sub-matrix

Cdd includes tooth bending and shear, tooth base flexibility, tooth contact and torsional

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compliance, meanwhile sub-matrix Crr contains only contact compliance at the tooth root land.

The coupled compliance sub-matrices are again considered to have minor effects and are assumed

T to be trivial, i.e. CC0dt= td = . With these, load distribution of minor diameter-fit splines can be solved similarly as for major diameter-fit splines.

4.2.3 Mismatched Splines

Intentional mismatch of splines, introducing a slight helix angle to the external spline while none to the internal spline, is commonly used in high torque or reverse torque applications. This mismatch can induce a certain amount of circumferential interference at the tooth back side under unloaded conditions as shown schematically in Fig. 4.3(a). Due to circumferential interference, there are potential contact zones at the tooth driving side as well as tooth back side. Figure 4.3(b) shows discretization of these potential contact zones. Discretization of the contact zone at the

tooth driving side remains the same as that of Fig. 4.1(b). The contact zone at the tooth back side

is similarly discretized into Pb cells of equal arc length along the profile direction and Qb cells

of equal length along the face width direction. For a spline having Z teeth, the total number of

contact points at the tooth driving side and tooth back side are nd=×× ZP dd Q and

nb=×× ZP bb Q, respectively. Given that the area of a grid cell is small enough compared to the

contact zone, contact pressure within each grid cell can be assumed to be uniform and can be

d represented by a concentrated contact force, Fiid( ∈[1, n ]) at the tooth driving side, or

b Fjjb( ∈[1, n ]) at the tooth back side. By enforcing compatibility and equilibrium conditions,

governing equations and relevant constraints for splines potentially having back side contact can

be obtained as follows

149

Contact zone at (a) Internal Spline tooth driving side

Major Diameter

Dri De 35 Contact zone at Minor Diameter tooth back side Dre Di External Spline

(b)

-15 -10 -5 0 5 10 15 Face Width Face Width Direction Direction

Qd Qb

… … …

d b Fid, i [1, n ] Fjb, j [1, n ]

nd Z  P d  Q d nb Z  P b  Q b

… … …

2 Profile 2 Profile 1 Direction 1 Direction … … … … 1 2 3 Pd 1 2 3 Pb Contact zone at Contact zone at tooth driving side tooth back side

Fig. 4.3 (a) Schematic side view of a spline joint having circumferential interference and its potential contact zones, and (b) discretization of the potential contact zones at the tooth driving side and back side.

150

Z  Yd TT I00GG0P ( db) ( ) b Y d  0I0−− Cdd C db G d   =ε  , (4.11a) d 00I−− C C G F b  bd bb b ε Fb  Φ

 dd YFii>=0 for 0,  in∈ [1, ] , (4.11b) dd d YFii= 0 for ≥ 0,

 bb YFjj>=0 for 0,  jn∈ [1, ] . (4.11c) bb b YFjj= 0 for ≥ 0,

It is clear that the above equations have the same structure and format as that of Eq. (4.9).

The superscript or subscript b denotes respective parameters corresponding to contact points in

the contact zones at the tooth back side. For instance, Fb is the vector of contact forces

b b b ( Fjjb, ∈[1, n ] ) at the tooth back side, ε is a vector containing initial separations (ε jb, jn∈[1, ] )

of contact point pairs at the tooth back side.

While definitions of most sub-matrices and sub-vectors of the above equations are the same

as that of Eq. (4.9), a few sub-matrices associated with surface geometry and compliance have

slightly different definitions. One such sub-matrix is Gb , which is related to the surface

geometry at tooth back side contact. Since the relative rotation about the z axis between the external and internal spline will increase the final gaps of a contact point pair at the tooth back side by a ratio of the base circle radius of a spline, this sub-matrix is defined as

151

en⋅b en ⋅ b −⋅zz bb en bb en ⋅ − R xy1 1 1111 y x b     b b bb bb Gb = enxj⋅ en yj ⋅ −⋅zz jyj en jxj en ⋅ − R b. (4.12)     b b b bb b enxy⋅ en ⋅−⋅zz en y en x ⋅− R b nb n b n b nn bb n b

b b where n jb (jn∈ [1, ]) and zjjb (∈ [1, n ]) denote unit normal vector and z coordinate of the j-th contact point of the contact zone at the tooth back side, respectively.

Regarding the compliance, only sub-matrix Cdd has the same definition as that of Eq. (4.9).

Sub-matrix Cbb represents the compliance of contact points at the tooth back side induced by contact forces at the tooth back side. Similarly to Cdd , this sub-matrix also includes tooth bending and shear, tooth base flexibility, tooth contact and torsional compliances. Providing that tooth back side has the same discretization as that of the tooth driving side ( PPdb= and

QQdb= ), this sub-matrix will be identical to that of the driving side, i.e. CCbb= dd .

Sub-matrix Cdb denotes the compliance of contact points at the tooth driving side induced by contact forces at the tooth back side. Compliance associated with tooth bending and shear

   ( C p ), tooth base flexibility ( C f ), and torsional deformations ( CT ) will be considered, while

 contact deformations ( Cc ) will be neglected since the driving side is far from the back side.

With this, the sub-matrix Cdb is defined as:

 Cdb=++ CCC p f T (4.13)

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Note that the direction of elastic deformation caused by contact forces at back side contact

is opposite to that caused by contact forces at the driving side, also contact forces at back side of one internal spline tooth are identical to that at back side of an adjacent internal spline tooth.

Sight modification to the methods described in Chapter 2 is required to define these compliance matrices. Considering identical discretization of tooth driving side and tooth back side, the

  compliance matrices C p and Cb are defined as:

C12 0 0 0 0C 0 0 pp12  0C2 0 0     p1    Z −1 C = −−    00 Cp2 0, (4.14a) p    ZZ−1 0 0 Cpp12 0 0 0 0C     Z 1  0 0 0Cpp12 C 0 0 0

C12 0 0 0 0C 0 0 ff12  0C2 0 0     f 1    Z −1 C = −−    00 Cf 2 0. (4.14b) f    ZZ−1 0 0 Cff12 0 0 0 0C     Z 1  0 0 0Cff12 C 0 0 0

where all sub-matrices in Eqs. (4.14a, b) have the same definition as in Eqs. (2.19, 2.24). The

 matrix CT is negative to the torsional compliance (CT ) defined in Eq. (2.29) of Chapter 2:

 CCTT= − (4.14c)

The last sub-matrix Cbd represents the compliance of contact point pairs at the tooth back side induced by contact forces at the tooth driving side. Due to symmetry of the compliance

T matrix, this sub-matrix is the transpose of the previous one, i.e. CCbd= db .

153

With abovementioned sub-matrices properly defined, load distribution of mismatched splines, or splines having any circumferential interference can be analyzed by solving Eq. (4.11) using the same algorithm described in Chapter 2.

4.3 Results and Discussions

In this section, the model presented in Sect. 4.2 will be used to study load distribution of

major and minor diameter-fit spline joints under combined loading conditions. These results will be compared to those of a side-fit spline joint. Load distribution of an example spline designed with different intentional mismatch magnitudes at various torque levels will also be studied.

4.3.1 Major and Minor Diameter-Fit Splines

Tables 4.1 and 4.2 list the parameters of example major and minor diameter-fit spline joints, respectively. Despite their differences of major and minor diameters, these two example spline joints are intentionally designed with the same number of teeth, module, pressure angle, tooth thickness and radial clearance. The profile lengths of the potential contact zones at the tooth driving side of these two spline joints are also identical. With these, performance of the example major and minor diameter-fit spline under two common loading conditions, namely (i) combined torsional and radial loads representative of spur gear loading and (ii) combined torsional, radial loads and tilting moment representative of helical gear loading, will be investigated and compared to that of the example side-fit spline joint of Table 3.1. At the end, influence of radial clearance on the performance of the major diameter-fit spline will also be studied.

Combined Torsional and Radial Loads: Figure 4.4 shows load distributions of the

example major and minor diameter-fit as well as side-fit splines under combined torsional and

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Table 4.1 Parameters of an example major diameter-fit spline joint used in the study.

External spline Internal spline Number of teeth 25

Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.547 82.550 Minor diameter [mm] 73.025 76.200

Circular space width [mm] - 5.032

Circular tooth thickness [mm] 4.943 -

Inner rim diameter [mm] 0 -

Outer rim diameter [mm] - 127

Profile crown [µm] 8 0.0

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Table 4.2 Parameters of an example minor diameter-fit spline joint used in the study.

External spline Internal spline

Number of teeth 25

Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.550 85.725

Minor diameter [mm] 76.197 76.200

Circular space width [mm] - 5.032

Circular tooth thickness [mm] 4.943 -

Inner rim diameter [mm] 0 -

Outer rim diameter [mm] - 127

Profile crown [µm] 8 0.0

156

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Driving Side MPa 306 (a)

Top 255 Land

204

Driving Side 153 157

(b) 102 Root Land 51

0 Driving (c) Side

Fig. 4.4 Load distributions of (a) major diameter-fit (b) minor diameter-fit and (c) side-fit splines under combined torsional and radial

loads. M z = 8000 Nm, Fx = −105 kN and Fy = 38 kN.

157

radial loads of M z = 8000 Nm, Fx = −105 kN and Fy = 38 kN . In addition to contact at the

tooth driving side, there is also additional contact at the tooth top land of major diameter-fit spline,

and at the tooth root land of minor diameter-fit spline. In both cases, additional contact is

observed to occur at the top or root land of spline teeth #1-9, whose contact forces help

counterbalance the radial loads. As the contact at the tooth top or root land relieves the

asymmetric loading induced by radial loads, unequal tooth to tooth load distribution at tooth

driving sides is notably reduced compared to that of the side-fit spline. For instance, the driving

sides of teeth #8-20 of the major and minor diameter-fit splines experience slightly higher contact stresses over a wider area than that of the side-fit spline. Meanwhile, the remaining teeth experience reduced contact stress over a smaller area compared to that of the side-fit spline. This leads to a more uniform tooth to tooth load distribution for the major and minor diameter-fit splines under the combined loading condition. As a direct result of a more uniform load distribution, major and minor diameter-fit splines also exhibit more favorable tooth-to-tooth load sharing characteristics as shown in Fig. 4.5. It is observed that both major and minor diameter-fit splines exhibit smaller deviations of tooth to tooth load sharing factor, k (k ∈ [1, 25]) , with a considerably reduced peak value. Also note that there is a marginal difference between load distribution and tooth load sharing of major and minor diameter-fit splines, because their basic design parameters are identical.

In spite of their better load distribution and tooth load sharing characteristics, major and

minor diameter-fit splines are often favored primarily for their centralization capability to

minimize shaft axes alignment errors. In the above case of combined torsional and radial loads,

radial load along the x axis, Fx , is significant and is expected to induce sizeable relative rigid

body translation along the x axis, ux , between spline members. Figure 4.6 shows the predicted

158

2.5 (a) Major Diameter Fit 2 Side Fit Nominal

1.5

k 1

0.5

0 2.5 (b) Minor Diameter Fit 2 Side Fit Nominal 1.5

k 1

0.5

0 0 5 10 15 20 25 Tooth Number, k

Fig. 4.5 Comparison of tooth-to-tooth load sharing of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined torsional and

radial loads. M z = 8000 Nm, Fx = −105 kN and Fy = 38 kN.

159

0 (a) Major Diameter Fit -2 Side Fit

-4

ux (μm) -6

-8

-10

-12 0 (b) Minor Diameter Fit -2 Side Fit

-4

ux (μm) -6

-8

-10

-12 2000 3000 4000 5000 6000 7000 8000 M (Nm) z

Fig. 4.6 Relative rigid body translation along the x axis, ux , between the external and internal member of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined torsional and radial loads at different torque levels.

160

ux of the major and minor diameter-fit splines as well as the side-fit spline. Here, torque M z

varies from 2000 to 8000 Nm. At each torque level, the radial loads FFxy and also change

proportionally with torque such that it represents a spur gear loading condition at various torque levels. For instance, at the torque of M z = 8000 Nm the radial loads are Fx = −105 kN and

Fy = 38 kN , as the torque reduces to M z = 4000 Nm the radial loads become Fx = −52.5 kN

and Fy =19 kN . It is observed that, as the torque level increases, magnitudes of ux for all three

types of splines grow larger. For any given torque, both major and minor diameter-fit splines are

shown to significantly reduce the magnitude of ux compared to that of side fit splines, evidently

dictating their superior centralization capability.

Combined Torsional, Radial Loads and Tilting Moment: Figure 4.7 shows load

distributions of the major and minor diameter-fit splines together with the side-fit spline under

combined torsional, radial loads and tilting moment of M z = 8000 Nm, Fx = −105 kN,

Fy = 39 kN and M x = −2050 Nm . Again it is observed that the tooth top land of the major diameter-fit spline and the tooth root land of the minor diameter-fit spline are in contact. Contact forces along the tooth top land or root land counterbalance a certain amount of radial loads and tilting moment, such that the tooth driving side will carry less and will have a more uniform tooth to tooth load distribution than that of side-fit splines. For instance, teeth #10-18 of the side-fit

spline barely carry any load in this loading condition. However, in case of the major or minor

diameter-fit spline, teeth #10-18 have a larger area of contact and carry more load such that load concentration of the remaining teeth is considerably reduced. The resultant tooth-to-tooth load sharing of these example splines are compared in Fig. 4.8. Major and minor diameter-fit splines are observed to undergo smaller variations of tooth-to-tooth load sharing factor, k (k ∈ [1, 25]) ,

161

Tooth Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Driving Side MPa 395 (a)

Top Land 329

263

Driving Side 198 162

(b)

132 Root Land 67

0 (c) Driving Side

Fig. 4.7 Load distributions of examples of (a) major diameter-fit (b) minor diameter-fit and (c) side-fit spline under combined

torsional, radial loads and tilting moment. M z = 8000 Nm, Fx = −105 kN, Fy = 39 kN and M x = −2050 Nm.

162

2.5 (a) Major Diameter Fit 2 Side Fit Nominal

1.5

k 1

0.5

0 2.5 (b) Minor Diameter Fit 2 Side Fit Nominal 1.5

k 1

0.5

0 0 5 10 15 20 25 Tooth Number, k

Fig. 4.8 Comparison of tooth-to-tooth load sharing of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined torsional,

radial loads and tilting moment. M z = 8000 Nm, Fx = −105 kN, Fy = 39 kN and

M x = −2050 Nm.

163

with significantly lower peak values, again demonstrating their favorable load sharing characteristics compared to that of side-fit splines. Relative rigid body rotations about the x axis,

θx , of the example major and minor diameter-fit splines, as well as the side-fit splines under this loading condition are compared in Fig. 4.9. Similarly as in Fig. 4.6, the radial loads and tilting moment change proportionally with the torque to represent a nominal helical gear loading condition. It is observed that as the torque increases the magnitude of θx becomes larger for all

three types of splines. For a given torque, major and minor diameter-fit splines are shown to

experience considerably reduced magnitudes of θx , again demonstrating their stronger centralization capability.

Effects of Radial Clearance: Referring to Fig. 4.1(a), radial clearance of a major diameter

fit spline is defined as δ =1 DD − , where D and D are the major diameters of internal r2 ( ri e ) ri e

and external spline, respectively. By intentionally changing the major diameter of the external

spline, a set of different radial clearance values are generated for the example major diameter-fit

spline. Repeating the same analysis as Fig. 4.9(a) for each radial clearance value, relative rigid

body rotations about the x axis (θx ) for different δr values at various torque levels are predicted,

as shown in Fig. 4.10. Here, δr ∈[0,40] µm with δr →∞ representing the example side-fit

spline since there will be no contact at the tooth top land. Here the magnitude of θx increases

with torque for any radial clearance value. At a given torque level, the magnitude of θx changes

as δr varies. Within certain clearance ranges, smaller δr values result in smaller magnitudes of

θx . Beyond these ranges, the magnitude of θx is not impacted by δr . For instance, at a torque

  level of M z = 3000 Nm , the magnitude of θx increases from 0.007 to 0.024 as the radial

clearance increases from 0 to 15 µm. However, the magnitude of θx at this torque level remains

164

0 (a)

-0.02

-0.04

 θx () -0.06

Major Diameter Fit -0.08 Side Fit

-0.1 0 (b)

-0.02

-0.04  θx () -0.06

Minor Diameter Fit -0.08 Side Fit

-0.1 2000 4000 6000 8000 M (Nm) z

Fig. 4.9 Relative rigid body rotation about x axis, θx , between the external and internal member of (a) major diameter-fit and side-fit splines, and (b) minor diameter-fit and side-fit splines under combined torsional, radial loads and tilting moment at different torque levels.

165

δr (μm) 0 -0.02

 10 θx ()

-0.04 15

30 35 40 -0.06 ∞

2000 4000 6000 8000 M (Nm) z

Fig. 4.10 Influence of radial clearance on the relative rigid body rotation about x axis, θx , between the external and internal member of the example major diameter-fit spline under combined torsional, radial loads and tilting moment at different torque levels.

166

constant within δr ∈[15,40] µm since there is no contact at the tooth top land due to large radial

clearance (i.e. the spline joint acts as a side-fit joint). Depending on the maximum allowed

magnitude of θx of the application, a proper δr value can be selected for a design torque.

4.3.2 Mismatched Splines

In this section, spline joints potentially having contact at tooth back side, specifically,

intentionally mismatched splines, are analyzed. Table 4.3 lists the parameters of an example

side-fit spline intended for a lead mismatch, where the circular gap width of the internal spline is

equal to circular tooth thickness of external spline to introduce zero backlash. By slightly

changing the helix angle of the external spline (or by applying a lead slope modification), a

certain amount of intentional mismatch, δim , can be introduced as shown schematically in Fig.

4.11(a). Due to this mismatch, a spline joint is preloaded with torsional wind-up when the

external part is pressed into the internal part.

In Fig. 4.11(b), load distributions of the example spline with different mismatch values of

δim = 0, 5, 10, 15, 20, 25 and 30 µm under pure torsion of M z = 0, 2000, 4000, 6000 and 8000

Nm are shown. Since load distribution of each spline tooth is identical in pure torsion, load

distribution on the driving side (A) and the back side (B) of a single spline tooth is selected in this

figure to represent the entire spline. It is observed that contact occurs on both the driving and

back sides for lower M z values, and contact stress increases as δim increases. For instance, the maximum contact stresses under zero torque are 87, 123 and 157 MPa for δim =10, 20 and 30µm, respectively. It is also noted that, back side contact vanishes at higher M z values and load

concentration on the driving side moves from one side to the other as δim increases. Figure 4.12

167

Table 4.3 Parameters of an example side-fit spline joint used in the study.

External spline Internal spline

Number of teeth 25

Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.550 85.725

Minor diameter [mm] 73.025 76.200

Circular space width [mm] - 4.943

Circular tooth thickness [mm] 4.943 -

Inner rim diameter [mm] 0 -

Outer rim diameter [mm] - 127

Profile crown [µm] 5 0.0

168

(a) (b) δim = 0 μm 5 µm 10 µm 15 µm 20 µm 25 µm 30 µm Side A δim MPa M z = 0 Nm 210

175 2000 Nm

140

4000 Nm 105 169

70 6000 Nm

8000 Nm 0 δ Side B im A B A B A B A B A B A B A B

Fig. 4.11 (a) Schematic representation of intentional mismatch, δim , of a spline tooth on the pitch circle; (b) load distribution of an

intentionally mismatched spline having various δim values at different torque levels.

169

220

200

180

Maximum Contact 160 Stress (MPa)

140

M z (Nm) 4000 120 6000 8000

100 0 5 10 15 20 25 30 δ (μm) im

Fig. 4.12 Maximum contact stress versus mismatch magnitude, δim , at different torque levels.

170

is a plot of corresponding maximum contact stress versus the magnitude of intentional lead

mismatch at different torque levels. The results illustrate that there is an optimum mismatch

magnitude for a given design torque that reduces the maximum contact stress by providing an

evenly distributed load. For instance, at M z = 6000 Nm the maximum contact stresses are 179,

170, 156, 148, 161, 173 and 193 MPa for δim = 0, 5, 10, 15, 20, 25 and 30 µm, respectively,

indicating that the optimum mismatch value is about 15 µm.

4.4 Summary

In this chapter, the semi-analytical model proposed in Chapter 2 was modified to allow

load distribution analysis of major and minor diameter-fit and lead mismatched splines. Using

the revised model, performances of example major and minor diameter-fit splines were studied.

In nominal gear loading conditions, major and minor diameter-fit splines demonstrated more

favorable load distribution and load sharing characteristics as well as superior centralization

performance compared to that of side-fit splines. In addition, radial clearance was shown to

significantly influence centralization capability of a major diameter-fit spline. Within certain

ranges of design torque, a smaller radial clearance typically resulted in better centralization performance. Regarding mismatched splines, application of an optimum intentional mismatch was shown to result in an even load distribution along the full length of spline teeth by cancelling out the torsional wind up of the spline shaft.

References for Chapter 4:

[4.1] Hong, J., Talbot, D. and Kahraman, A., 2014, “Load Distribution Analysis of Clearance-

Fit Spline Joints Using Finite Elements,” Mechanism and Machine Theory, 74, 42-57.

171

[4.2] Vijayakar, S., Busby, H. and Houser, D., 1988, “Linearization of Multibody Frictional

Contact Problems,” Computers & Structures., 29, 569-576.

[4.3] Vijayakar, S., 1991, “A Combined Surface Integral and Finite Element Solution for a

Three-Dimensional Contact Problem,” International Journal for Numerical Methods in

Engineering, 31, 525-545.

172

CHAPTER 5

A SEMI-ANALYTICAL STIFFNESS FORMULATION FOR SPLINE JOINTS

5.1 Introduction

Due to lack of knowledge in terms of their flexibility and deformation, spline joints are

often assumed to be rigid in dynamic models of gearboxes, transmissions and drivetrains. It was reported by several studies that compliance of a spline joint can substantially influence dynamic behavior, leading to rotor dynamic instability and nonsynchronous whirl [5.1-5.4]. Properly capturing the compliance of spline joints might be critical to the fidelity of dynamic analyses of drive trains.

Only a few experimental studies have been reported in the literature to determine stiffness of spline joints. Among them, Ku et al [5.5] determined the rotational stiffness (termed as tilting stiffness in this chapter) of a spline joint by measuring the bending moments and angular deflections transmitted across an axial spline with a nonrotating shaft excited by an external shaker. They showed that the tilting stiffness decreases as the force and frequency of the excitation increases. Cura and Mura [5.6] measured torsional stiffness of a spline joint in both

173

aligned and misaligned conditions to show that misalignment of a spline coupling would reduce

torsional stiffness. These experimental studies provided experimental data of spline stiffness for the simplest torsional loading cases.

On the theoretical side, Marmol et al [5.1] proposed an analytical model to calculate the

torsional, radial and tilting (rotational) stiffness values of spline joints based on the assumption

that all spline teeth carry the same amount of load and the load is evenly distributed along the

pitch line across face width. This assumption is often invalid since actual load distributions of

splines usually vary from tooth to tooth and load on a specific tooth is usually unevenly

distributed over a wide area as demonstrated clearly in previous chapters. Recently, Barrot et al

[5.7-5.8] proposed an analytical formula to calculate torsional stiffness of a spline joint. The

model they employed for this purpose was limited to pure torsional loading since they also

assumed that load distributions of all spline teeth are identical. Effects of tooth modifications,

misalignments, combined loading conditions and tooth indexing errors were also not accounted

for in these theoretical studies.

This chapter aims at proposing a general stiffness formulation for spline joints based on the

semi-analytical spline load distribution model developed in Chapter 2. The goal here is to define

a fully populated stiffness matrix of a spline joint including radial, tilting and torsional stiffness

values as well as coupling between these motions defined by the off-diagonal terms of the

stiffness matrix. With this stiffness formulation in place, a parametric study will be performed to

investigate the influences of tooth surface modifications, misalignments and tooth indexing errors

on the spline stiffness matrix.

174

5.2 Stiffness Formulation

The stiffness of a spline joint relates reaction loads (forces and moments) acting on it to

relative rigid body displacements between the external and internal components. In this study,

relative displacement along the axial z direction ( uz ) between the external and internal spline is constrained and the axial load component Fz is excluded. As such, there are five reaction load components (two radial forces and three moments) depicted by the reaction load vector

T P = FFMxyxyz M M and five relative displacement components (two radial

displacements and three rotational displacements) depicted by the relative rigid body

T displacement vector Φ = uuxyxyzθθθ as shown in Fig. 5.1. Then the stiffness

∂P matrix of a spline joint is defined as K = which is written explicitly as ∂Φ

∂∂∂∂∂FFFFFxxxxx ∂∂∂∂∂θθθ uuxyxyz ∂∂∂∂∂FFFFF yyyyy ∂∂∂∂∂uuxyxyzθθθ  ∂∂∂∂∂MMMMMxxxxx K = . (5.1) ∂∂∂∂∂uuθθθ xyxyz ∂∂∂∂∂MMMMMyyyyy  ∂∂∂∂∂uuxyxyzθθθ  ∂∂∂∂∂MMMMMzzzzz ∂∂∂∂∂uuxyxyzθθθ

Load and displacement vectors P and Φ in Eq. (5.1) are related implicitly through the

semi-analytical load distribution model proposed in Chapter 2. The governing equations of this

semi-analytical model were derived as

175

y (a) M y z Fy

x M M x z

y (b) θ y z uy

x θ θz x

Fig 5.1 Schematic representation of (a) reaction load components of a spline joint, and (b) relative rigid body displacements between the external spline and internal spline.

176

Z  I0GT 0YP  =  , (5.2a) 0I− CGF ε Φ

YFii>=0 for 0  in∈ [1, ] . (5.2b) YFii= 0 for ≥ 0

With this semi-analytical model, two methods, namely an analytical method and a numerical

method, can be devised to compute the stiffness matrix K of a spline joint.

5.2.1 Analytical Method

Load distribution of a spline joint varies as a function operating load conditions, resulting

from changes of P and Φ . As such, K is dependent on operating load conditions requiring it be evaluated at each operating condition individually. For a given operating condition, spline load distribution is solved first using Eq. (5.2) that yields the contact force vector F. A subset of F

ˆˆ ˆ ˆT ˆ containing the positive contact force elements is defined as F = FFF1 inˆ , Fi > 0 ,

(i∈≤ [1, nnnˆˆ ], ) , which represents nˆ contact point pairs in contact. With the contact point pairs

determined, the compatibility constraints of Eq. (5.2b) can be eliminated, in the process reducing the governing equations to the following:

GFˆ T ˆ = P, (5.3a)

−CFˆˆˆ += GΦεˆ , (5.3b)

177

where Cˆ and Gˆ are the corresponding subsets of the compliance (C), and geometric (G)

matrices, and εˆ is the corresponding subset of the initial separation vector ε . From the compatibility condition of Eq. (5.3b), the contact force vector can be written as

ˆ =ˆˆ−1  − FC GΦεˆ . (5.4)

Back substituting the above equation into the equilibrium condition of Eq. (5.3a) yields

= ˆˆT −1  ˆ − P GC GΦεˆ . (5.5)

Here εˆ is a constant vector. Furthermore, Cˆ and Gˆ are both independent of Φ . Consequently, the stiffness matrix of the spline coupling can be obtained by taking partial derivative of P with respect to Φ as

∂P K= = GCˆˆT −1 G ˆ. (5.6) ∂Φ

In this equation, the matrices Cˆ and Gˆ depend on contact point pairs that are actually in contact,

indicating that the stiffness of a spline joint may vary as spline load distribution changes. Factors

that change the spline load distribution, such as torque level, tooth modifications and indexing

errors, will possibly influence the stiffness. In addition, contact grid density also affects these

matrices and consequently can influence the stiffness calculation. Investigations on effects of

these factors will be performed using a numerical procedure described subsequently.

In the above analytical stiffness formula, computing the inverse of the compliance matrix

subset Cˆ might require considerable computational time depending of the size of the compliance

matrix, especially if the number of spline teeth is large and all or most spline teeth carry load.

178

The flowchart of computation shown in Fig. 5.2 provides two avenues: one to handle splines

having any arbitrary load distributions and another to handle a special case where all spline teeth

have identical contact conditions, i.e. all spline teeth have the same respective contact point pairs

in contact as it is the case for pure torsional loading with no indexing errors. In this special case,

the compliance sub-matrix Cˆ has a unique structure

CCˆˆ+ C ˆ C ˆ Tl T T ˆ ˆˆ ˆ CT CC Tl+  C =  (5.7)   ˆ ˆ ˆˆ CT C T CC Tl+

ˆ ˆ where CT is the corresponding subset of the torsional compliance and Cl is corresponding subset of the local tooth compliance including tooth bending and shear, tooth base flexibility and contact compliance as described in Chapter 2. For a spline with Z number of teeth, the matrix Cˆ

will include ZZ× sub-matrices. As these sub-matrices have identical size and all the off-

diagonal sub-matrices are identical, the inverse of this matrix can be proven to be

QW+ Q Q  Q QW+  Cˆ −1 =  (5.8a)    Q Q QW+

where

−1 =−+ˆ ˆ ˆˆ−1 = ˆ −1 QZ CT C l CC Tl, WCl . (5.8b,c)

179

Load Distribution Analysis

Identical Contact No Conditions on All Teeth?

Yes

Blockwise Inversion of Matrix Cˆ

Calculation of Stiffness Calculation Direct Inversion Matrix Gˆ K= GCGˆˆˆT -1 of Matrix Cˆ

Fig. 5.2 Flowchart of the procedure for stiffness calculation using the analytical stiffness formula.

180

Accordingly, Cˆ −1 can be constructed from Q and W, requiring evaluation sub-matrices

−1 ˆˆ+ ˆ −1 ZCCTl and Cl . This can be expected to reduce computational time drastically

compared to direct inversion of Cˆ . Table 5.1 shows the CPU times taken for stiffness calculation

using this blockwise inversion method the direct inversion method for a few example cases. In

this table, Z is the number of spline teeth, and P and Q denote the number of contact cells along the profile and face width directions, respectively. It is observed that as Z increases, the size of

Cˆ becomes larger and stiffness computation using the direct inversion approach takes much longer. However, if the blockwise inversion is used, the CPU time required becomes negligibly small. For instance, as the number of teeth increases from 25 to 60, CPU time using direct inversion increases by more than 10 times from 3.3 seconds to 42.5 seconds, while CPU time using blockwise inversion only slightly increases from 0.03 seconds to 0.04 seconds.

The above blockwise inversion method applies only to cases where all spline teeth have

identical contact conditions, such as pure torsion loading (with or without any nominal tooth

modifications). In cases where misalignments or tooth indexing errors are present as well as cases when the spline joint is under a gear loading condition, load distributions vary from tooth to tooth such that blockwise inversion of Cˆ in the form of Eq. (5.8) is no longer valid and the entire

Cˆ matrix must be inverted directly. Fortunately, in these cases, many point pairs are not in

contact so that the size of Cˆ is reduced significantly, making computational demand for this still acceptable. Table 5.2 shows CPU time required to compute the K matrix of a helical gear loaded spline joint having Z = 60 teeth. As helix angle increases, more contact point pairs are separated

and the size of Cˆ becomes much smaller. Consequently, the CPU time is reduced for larger gear helix angles. For instance, the CPU time required drops from 4.12 seconds to 0.81 seconds as the

181

Table 5.1 Comparison of CPU time for analytical stiffness calculation using direct inversion and blockwise inversion for a spline loaded in pure torsion.

Spline Discretization Parameters CPU Time (second) Size of Cˆ Z P Q Direct Inversion Blockwise Inversion

25 10 12 3000×3000 3.34 0.03

45 10 12 5400×5400 18.39 0.03

60 10 12 7200×7200 42.51 0.04

182

Table 5.2 Computational time for analytical stiffness calculation using direct inversion for splines under helical gear loading.

Spline Discretization Parameters Helix Angle CPU Time Size of Cˆ (degrees) (second) Z P Q

60 10 12 15 3092×3092 4.12

60 10 12 20 2262×2262 1.93

60 10 12 25 1193×1193 0.81

183

gear helix angle is increased from 15° to 25°. Sensitivity of computational time to misalignments

of the spline is similar as shown in Table 5.3. As misalignment increases from 0.01° to 0.03°,

required CPU time is reduced from 1.55 seconds to 0.73 seconds, solely because the size of Cˆ is

decreased.

As the size of loaded areas on spline teeth impact K, the size of grid elements used to

discretize potential contact areas might have a critical influence on the solution. The analytical

stiffness calculation method of Fig. 5.2 will be used here to perform a study on the sensitivity of

K to grid density. Dynamic models are often concerned with diagonal terms of K, thus only diagonal stiffness terms will be presented here to demonstrate the influence of contact grid density. Figure 5.3 shows predicted torsional stiffness ( ∂∂M zzθ ), radial stiffnesses

( ∂∂Fii u, i = xy, ) and tilting stiffnesses ( ∂∂Miiθ , i = xy, ) of an example spline joint having

profile and lead modifications under pure torsion loading at different torque levels.

In Chapter 2, each active tooth surface was divided into P rows of grid cells in the profile direction and Q rows of grid cells in the lead direction as shown in Fig. 2.3. Contact grid density defined by the number of contact elements Q along the face width direction, is varied here first while keeping the number of contact elements along the profile direction as P =10 . The

parameters of the example spline joint are listed in Table 5.4. The modifications include a 3 µm

profile crown modification and a 10 µm lead crown modification, both applied to the external spline teeth. It is observed from Fig. 5.3 that, for a given torque, these diagonal stiffness terms exhibit slight deviation as the number of contact cells along the face width direction changes. For example, the torsional stiffness ( ∂∂M zzθ ) at torque M z = 2260 Nm are 9.18, 9.31, 9.22, 9.30 and 9.22 Nm/m rad for Q =12, 18, 24, 30 and 36, respectively. The same slight influence is seen for the other diagonal stiffness terms as well. These differences suggest that contact grid density

184

Table 5.3 Computational time for analytical stiffness calculation using direct inversion for misaligned splines.

Spline Discretization Parameters Misalignment CPU Time Size of Cˆ (degrees) (second) Z P Q

60 10 12 0.01 2045×2045 1.55

60 10 12 0.02 1244×1244 0.83

60 10 12 0.03 952×952 0.73

185

11 (a)

∂∂M zzθ 9 (106 Nm/rad) M z (Nm) 2260 8 3390 4520

7 12 (b) 11

∂∂Fuii 9 i= xy, 8 (109 N/m) 7

5 3 (c) 2.5

2 ∂∂Miiθ i= xy , 1.5 (106 Nm/rad) 1

0.5

0 12 18 24 30 36 Q

Fig. 5.3 Influence of face width contact grid density on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of a spline joint under pure torsion at different torque levels with P =10 cells along the profile direction.

186

Table 5.4 Parameters of an example side-fit spline joint used in the study.

External spline Internal spline

Number of teeth 25

Module [mm] 3.175

Pressure angle [°] 30

Base diameter [mm] 68.732

Face Width [mm] 50.8

Major diameter [mm] 82.550 85.725

Minor diameter [mm] 73.025 76.200

Circular space width [mm] - 5.080

Circular tooth thickness [mm] 4.897 -

Inner rim diameter [mm] 38 -

Outer rim diameter [mm] - 127

187

in the face width direction has a minor effect on the resultant diagonal terms of K. Similar

observations can be extended to the influence of grid density in the profile direction as shown in

Fig. 5.4 as varying P with Q = 24 results in minor variations of the same stiffness components.

As such, a moderate contact grid density represented by Q = 24 and P =10 will be used in subsequent stiffness calculations of this chapter.

5.2.2 Numerical Method

Aside from the analytical method described above, the stiffness matrix of a spline joint can also be calculated numerically using a finite difference approximation. Coefficients in each column of the stiffness matrix can be obtained by perturbation of a specific translational or rotational displacement. For instance, given a perturbation of displacement along the x axis, i.e.

T δδΦ = [ ux 0000] , the change of the reaction load vector due to this specific perturbation can be obtained by solving Eq. (5.2) as

T P(Φ00+−δ Φ ) PΦ () = δδδFFMxyxyz δ M δ M. (5.9)

Then the coefficients in the first column of the stiffness matrix K can be approximated as

∂FFδ ∂FFyyδ xx≈ , ≈ , (5.10a,b) ∂uuxxδ ∂uuxxδ

∂MMδ ∂MMyyδ ∂MMδ xx≈ , ≈ , zz≈ . (5.10c-e) ∂uuxxδ ∂uuxxδ ∂uuxxδ

Elements of the K matrix in other columns are obtained using this procedure with perturbations

applied to the other displacement terms. Using this first-order approximation, the load distribution

188

11 (a)

∂∂M zzθ 9 (106 Nm/rad)

M z (Nm) 2260 8 3390 4520

7 12 (b) 11

∂∂Fuii 9 i= xy, 8 9 (10 N/m) M z (Nm) 7 2260 3390 6 4520

5 3 (c) 2.5

2 ∂∂Miiθ i= xy , 1.5 (106 Nm/rad) 1 M z (Nm) 2260 0.5 3390 4520

0 5 10 15 20 P

Fig. 5.4 Influence of profile contact grid density on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of a spline joint under pure torsion at different torque levels with Q = 24 contact cells along the face width direction.

189

problem must be solved six times to define K entirely. Higher order finite difference

approximation formulas can also be used similarly, but they would require solving for the spline

load distribution many more times for different perturbations. In this study, only the first-order

approximation will be used since this numerical method is intended only for verification of the

fidelity of the analytical solutions of the previous section.

The contact grid density is selected to be the same as that used in analytical method for a

−10 direction comparison. A perturbation step of 10 m is selected for δux and δuy , and

−10 10 rad for δθxy, δθ and δθz . Splines under three different operating conditions of (i) pure torsional loading, (ii) combined loading and (iii) torsional loading with misalignments are considered for this comparison. The example spline is the same as used in the previous section.

Table 5.5 compares the K matrices of the example spline under pure torsion, obtained using the analytical and numerical methods. Here, M z = 2260 Nm , and all coefficients of the stiffness

matrix are given in SI units, e.g. units for ∂∂Fuxx, ∂∂Fxxθ and ∂∂M xxθ are N/m, N/rad, and Nm/rad, respectively. It is seen that K defined by the analytical method is symmetric, as

expected, since the spline joint is conservative, while the stiffness matrix predicted by the numerical method is not. In addition, the off-diagonal stiffness terms predicted by these two methods do not match well. These deviations can be attributed to perturbation around zero

relative rigid body displacement terms, i.e. uuxy= = 0 and θθxy= = 0 , in this pure torsion

loading condition. Regardless of this, diagonal stiffness terms of the matrix, which are often of

primary concern, predicted by these two methods agree well. The relative difference of the five

190

Table 5.5 Comparison of K of the example spline joint under pure torsional loading condition; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad).

(a) Analytical Stiffness Matrix Output (CPU time: 0.16 seconds)

∂ux ∂uy ∂θx ∂θ y ∂θz

∂Fx 9166181299 -233 -37192 -28242848 -6

∂Fy -233 9166181095 28242847 -37191 4

∂M x -37192 28242847 1514131 0.04 0

∂M y -28242848 -37191 0 1514130 0

∂M z -6 4 0 0 9230444

(b) Numerical Stiffness Matrix Output (CPU time: 6.77 seconds)

∂ux ∂uy ∂θx ∂θ y ∂θz

∂Fx 9005864321 -5024248 -1170 -26923781 -6

∂Fy 5023789 9005864120 26923781 -1169 4

∂M x 1740 26815815 1411583 -146 0.02

∂M y -26815815 1741 146 1411582 0.04

∂M z -6 4 0 0 8919947

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diagonal stiffness terms ranges between 1.7% and 6.6%. Note that the CPU time is 0.16 seconds

for the analytical method, while it takes 6.77 seconds when using the numerical method.

Table 5.6 provides a similar comparison for the combined loading defined by

M z = 2260 Nm , M x = 606 Nm , Fx = −29.6 kN and Fy =11.2 kN . Again, K defined by the

analytical method is symmetric while the numerical stiffness matrix is not. Compared to previous

pure torsion case, the off-diagonal stiffness terms predicted by these two methods have much

better correlation, since the relative rigid body displacement terms are no longer zero in this

combined loading condition. The diagonal stiffness terms predicted by these two methods also

appear very close to each other. The relative errors of the five diagonal stiffness terms are all

within 4.8%.

Similar observations are extended to the case of torsional loaded splines with

misalignments as shown in Table 5.7, where the torque M z = 2260 Nm and the misalignment

 terms include uuxy= =10 μm and θθxy= = 0.01 . Also note that in both cases, the CPU time required by analytical method is much smaller than that required by the numerical method.

The above comparisons clearly demonstrate the reliability and computational efficiency of the analytical method for stiffness calculation. Considering also the analytical method is

independent of parameters such as order of finite difference approximation and perturbation steps,

this analytical method will be employed in the following parametric studies for a thorough

investigation on stiffness of spline joints.

192

Table 5.6 Comparison of K of the example spline joint under combined loading; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad).

(a) Analytical Stiffness Matrix Output (CPU time: 0.73 seconds)

∂ux ∂uy ∂θx ∂θ y ∂θz

∂Fx 4374117585 207299978 -5014556 -11376255 -81703043

∂Fy 207300189 5683412487 15664329 -2016126 -10455031

∂M x -5014563 15664319 1244721 42332 -1834667

∂M y -11376257 -2016126 42332 1033768 251149

∂M z -81703041 -10455063 -1834667 251149 7843504

(b) Numerical Stiffness Matrix Output (CPU time: 4.91 seconds)

∂ux ∂uy ∂θx ∂θ y ∂θz

∂Fx 4246044271 219754976 -3831170 -10854281 -80219534

∂Fy 263664041 5535492743 14880519 -3011363 -8782681

∂M x -4159974 14450929 1229304 32986 -1807260

∂M y -10979061 -3206574 32653 982871 267624

∂M z -81434017 -4886119 -1855424 259934 7571259

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Table 5.7 Comparison of K of the example spline joint under torsional loading with misalignments; (a) analytical method and (b) numerical method. (SI units: N/m, N/rad, and Nm/rad).

(a) Analytical Stiffness Matrix Output (CPU time: 0.66 seconds)

ux uy x  y z

Fx 3533784254 -102489772 415665 -3584982 91255440

Fy -102489627 3682034179 16850621 -3408156 101983562

M x 415667 16850625 922090 -210510 792532

M y -3584992 -3408161 -210510 886298 17907

M z 91255454 101983566 792532 17907 6442642

(b) Numerical Stiffness Matrix Output (CPU time: 2.93 seconds)

Fx 3495407456 -78754136 952320 -3537528 89913719

Fy -116110698 3700812226 16919677 -3740819 100611601

M x 631399 16725236 915845 -203260 802457

M y -3518746 -3684865 -204954 865446 20779

M z 89783352 103064780 821951 22080 6357779

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5.3 Parametric Studies

5.3.1 Effects of Tooth Surface Modifications

As they influence load distribution of splines, tooth surface modifications should be

expected to impact the corresponding stiffness matrix as well. Using the same example spline

design of Table 5.4, influences of the tooth surface modifications (profile and lead modifications)

on K are investigated here. Figure 5.5 shows the variation of the diagonal torsional ( ∂∂M zzθ ),

radial ( ∂∂Fuii, i= xy, ) and tilting ( ∂∂Miiθ , i= xy, ) stiffness elements of K with torque for

four levels of external tooth profile crown values between 0 (no modification) and 9 µm. Here

the spline is under pure torsion and there is no lead modification. The torque range considered is

M z = 565 Nm to 5650 Nm. In this pure torsion loading case, the radial stiffness values along

the x and y axes are equal as shown in Fig. 5.5(b). The same is true in Fig. 5.5(c) for the tilting stiffness values about the x and y axes. It is seen that in the case of zero profile crown modification, all these diagonal stiffness terms remain constant regardless of the value of M z

since the contact conditions remain the same at all torque levels under this pure torsion loading

condition. However, in presence of positive profile crown modification, these stiffness terms are

seen to increase gradually as M z increases. For instance, for a 9 µm profile crown modification,

torsional stiffness values of the spline joint are 10.0, 10.2, 10.3 and 10.4 Nm/μrad , respectively, for M z =1130 , 2260, 3390 and 4520 Nm. The main reason for this is that more contact point

pairs come into contact as the torque increases. Aside from this, it is also observed that at a given

M z level, these diagonal stiffness terms decrease with an increase in profile crown modification.

For example, at M z = 2260 Nm, ∂M zz ∂=θ 11.1 , 10.5, 10.3 and 10.2 Nm/m rad for profile crowns equal to 0, 3, 6 and 9 µm, respectively. This can be explained by the fact that a larger

profile modification causes more contact point pairs to separate.

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11.5 (a)

10.5 ∂∂M zzθ 6 (10 Nm/rad) 10 Profile Crown (µm) 0 3 9.5 6 9

9 13 (b)

∂∂Fuii11 i= xy, Profile Crown (µm) (109 N/m) 10 0 3 9 6 9

8 4 (c)

3.5

∂∂Miiθ i= xy , 6 3 (10 Nm/rad) Profile Crown (µm) 0 2.5 3 6 9

2 0 1130 2260 3390 4520 5650 M z (Nm)

Fig. 5.5 Effects of profile crown modification on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of the example spline under pure torsion at different torque levels.

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Figure 5.6 shows variation of ∂∂M zzθ , ∂∂Fuii and ∂∂Miiθ ( i= xy, ) with torque

M z for four levels of external tooth lead crown values between 0 (no modification) and 15 µm

(no profile modifications). It is observed that the influence of lead crown modification is quite similar to that of profile crown modification: (i) stiffness terms are load-independent for the no

lead crown case; (ii) stiffness terms of modified splines become larger as torque increases; (iii) at

a given torque level, stiffness terms decrease with larger lead crown modifications.

5.3.2 Effects of Misalignments

In Chapter 2, misalignments of spline joints were shown to alter load distribution

significantly. Influence of misalignments on stiffness of spline joints is investigated here using

the same example spline. A nominal 10 µm lead crown modification is applied to the external

spline teeth in this case since it is common in application to reduce load concentration of

misaligned splines using lead crown modifications. Figure 5.7 shows variation of the diagonal

terms of K ( ∂∂M zzθ , ∂∂Fuxx, ∂∂Fuyy, ∂∂M xxθ and ∂∂M yyθ ) as a function of M z

applied to the spline. Here, spline misalignment is varied from 0° to 0.03° with an increment of

0.01°. Similar to previous cases, for a given misalignment value, these diagonal stiffness values increase as M z is increased for the same reason that the number of contact point pairs is increased. Meanwhile, at a given M z value, the change of the diagonal stiffness terms with misalignment becomes more complicated. At higher torque levels, say M z > 2260 Nm in Fig.

5.7, these stiffness values reduce as the misalignment magnitude is increased. For instance, at

M z = 3390 Nm, ∂M xx ∂=θ 2.23, 1.97, 1.62 and 1.44 Nm/m rad for misalignments of 0°, 0.01°,

0.02° and 0.03°, respectively. However, at low torque levels, say M z ≤ 2260 Nm, some of the

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10 (a)

9 ∂∂M zzθ 6 8 (10 Nm/rad) Lead Crown (µm) 0 7 5 10 6 15

5 13

12 (b) 11

∂∂Fuii10 i= xy, 9 9 (10 N/m) Lead Crown (µm) 8 0 7 5 10 6 15

5 4

0 0 1130 2260 3390 4520 5650 M z (Nm)

Fig. 5.6 Effects of lead crown modification on (a) torsional stiffness, (b) radial stiffness and (c) tilting stiffness of the example spline under pure torsion at different torque levels.

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11 (a)

9 ∂∂M zzθ 6  (10 Nm/rad) 8 θx () 0 7 0.01 0.02 0.03 6

12 (b) 10

8 ∂∂Fuxx (109 N/m) 6 0  0.01 θx () 4 0.02 0.03

2 12

11 (c)

∂∂Fuyy 9 (109 N/m) 8 0 7  0.01 θx () 0.02 6 0.03

5 0 1130 2260 3390 4520 5650 M z (Nm)

Continued

Fig. 5.7 Effects of misalignment about x axis, θx , on (a) torsional stiffness, (b) radial stiffness along the x axis, (c) radial stiffness along the y axis, (d) tilting stiffness about the x axis and (e) tilting stiffness about the y axis, for the example spline at different torque levels.

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Fig. 5.7 Continued

3 (d) 2.5

∂∂M xxθ 1.5 6 (10 Nm/rad)  θx () 1 0 0.01 0.5 0.02 0.03

0 3 (e) 2.5

∂∂M θ yy1.5 (106 Nm/rad) 1 0  0.01 θx () 0.5 0.02 0.03

0 0 1130 2260 3390 4520 5650 M z (Nm)

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stiffness terms become larger as misalignment increases. For example, at M z = 565 Nm,

∂M xx ∂=θ 0.30 , 0.45, 0.81, and 1.06 Nm/m rad for misalignments of 0°, 0.01°, 0.02° and 0.03°, respectively. These can be explained by the effects of the lead crown modification. At low torque lead crown modification results in a load distribution concentrated around the middle of the face width of the spline teeth. This load distribution pattern is poor at supporting tilting moment since the point pairs in contact are so close to the middle of the face width. As misalignment increases slightly, load distribution slightly moves away from the middle of the face width so that tilting stiffness also increases slightly in these low torque cases. Conversely, at high torque, the influence of lead crown modification is minor since point pairs in contact spread over almost the whole face width. In this case, as misalignment increases, more contact point pairs separate in the meantime lowering tilting stiffness as well as other diagonal stiffness terms.

5.3.3 Effects of Indexing Errors

In the presence of indexing errors, a spline joint was shown in Chapter 3 to exhibit time- varying (rotation dependent) load distribution characteristics. This should reflect in its stiffness matrix as well. The effects of tooth indexing errors will be demonstrated here using the same example spline joint with the external spline teeth having random indexing errors. In this case, the spline joint has no tooth modifications (purely involute), and an intentional misalignment of

0.04° about the x axis is applied to generate an asymmetric operating condition. Stiffness computation for this spline joint at a torque of M z = 4520 Nm is repeated at 60 different

rotational positions spanning a complete rotation at an increment of 6º. Figure 5.8(a) shows the indexing error sequence considered. The corresponding diagonal stiffness terms at different rotational positions are shown in Fig. 5.8(b-f). It is evident that as the spline rotates, all these

201

40 (a)

30 Indexing Error (µm) 20

0 0 5 10 15 20 25 Tooth Number

8.2 (b)

7.8 ∂∂M zzθ (106 Nm/rad) 7.6

7.4

7.2 6.5 (c)

∂∂Fuxx 5.5 (109 N/m)

4.5 0 60 120 180 240 300 360 Rotational Position (°)

Continued

Fig 5.8 Effects of (a) random tooth indexing error sequence on (b) torsional stiffness, (c) radial stiffness along the x axis, (d) radial stiffness along the y axis, (e) tilting stiffness about the x axis and (f) tilting stiffness about the y axis of the example spline

having 0.04° misalignment at M z = 4520 Nm at different rotational positions.

202

Fig 5.8 Continued

6 (d) 5.8

5.6

∂∂Fuyy 5.4 (109 N/m)

5.2

4.8 1.8 (e)

1.6

∂∂M xxθ (106 Nm/rad) 1.4

1.2 2 (f)

1.8

∂∂M yyθ 1.6 (106 Nm/rad)

1.4

1.2 0 60 120 180 240 300 360 Rotational Position (°)

203

stiffness terms change, resulting in a parametrically varying K at the rotational period of spline.

These variations are not insignificant. For instance, in Fig. 5.8(b), ∂∂M zzθ term varies

between 7.36 and 8.16 Nm/m rad , representing about a 10% peak-to-peak fluctuation over a full

rotational cycle. Other diagonal terms ∂∂Fuxx, ∂∂Fuyy, ∂∂M xxθ and ∂∂M yyθ in Fig.

5.8(c-f) exhibit even larger peak-to-peak fluctuations of 28%, 15%, 26% and 36%, respectively.

This indicates that the indexing errors of spline joints should be accounted for in high-fidelity drivetrain dynamic models.

5.4 Summary

A general analytical stiffness formulation of spline joints was proposed in this chapter based on the semi-analytical spline load distribution model of Chapter 2. This analytical stiffness formulation was verified through comparisons to a numerical perturbation method. The

analytical method was then used to perform a detailed parametric study on the impacts of tooth

modifications, torque levels, misalignments and tooth indexing errors. Profile and lead modifications were both found to reduce the torsional, radial and tilting stiffness of spline joints under pure torsion loading. Misalignments of spline joints were also observed to significantly influence the stiffness values. In the presence of random tooth indexing errors, stiffness terms of spline joints under asymmetric loading were shown to be time-varying with considerable peak to

peak variations as the spline rotates.

References for Chapter 5:

[5.1] Marmol, R. A., Smalley, A. J. and Tecza, J. A., 1980, “Spline Coupling Induced

Nonsynchronous Rotor Vibrations,” Journal of Mechanical Design, 102, 168-176.

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[5.2] Park, S. K., 1991, “Determination of Loose Spline Coupling Coefficients of Rotor

Bearing Systems in Turbomachinery”, PhD Thesis, Texas A&M University, College

Station, TX.

[5.3] Al-Hussain, K. M., 2003, “Dynamic Stability of Two Rigid Rotors Connected by a

Flexible Coupling with Angular Misalignment,” Journal of Sound and Vibration, 266,

217-234.

[5.4] Sekhar, A. S. and Prabhu, B. S., 1995, “Effects of Coupling Misalignment on Vibrations

of Rotating Machinery,” Journal of Sound and Vibration, 185(4), 655-671.

[5.5] Ku, C. P. R., Walton, J. F. Jr. and Lund, J. W., 1994, “Dynamic Coefficients of Axial

Spline Couplings in High-Speed Rotating Machinery,” Journal of Vibration and

Acoustics, 116, 250-256.

[5.6] Cura, F. and Mura, A., 2013, “Experimental Procedure for the Evaluation of Tooth

Stiffness in Spline Coupling Including Angular Misalignment,” Mechanical Systems and

Signal Processing, 40, 545-555.

[5.7] Barrot, A., Paredes, M. and Sartor, M., 2006, “Determining Both Radial Pressure

Distribution and Torsional Stiffness of Involute Spline Couplings,” Proc. IMechE., Part

C: J. Mechanical Engineering Science, 220, 1727-1738.

[5.8] Barrot, A., Sartor, M. and Paredes, M., 2008, “Investigation of Torsional Teeth Stiffness

and Second Moment of Area Calculations for an Analytical Model of Spline Coupling

Behaviour,” Proc. IMechE., Part C: J. Mechanical Engineering Science, 222, 891-902.

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CHAPTER 6

CONCLUSION

6.1 Summary

This research focused on development of a numerically efficient load distribution model of

various types of involute splines to be used as a potential design and analysis tool. In the first

phase, a semi-analytical load distribution model of clearance-fit (side-fit) involute splines was

formulated and verified via comparison to a finite element based deformable-body model. Using

the verified model, an extensive parameter study was performed, followed by a robustness study

of random tooth indexing errors. Then, the semi-analytical model was modified to facilitate load

distribution analysis of major and minor diameter-fit splines as well as splines having intentional

lead mismatch. At the end, a general stiffness formulation of involute spline joints based on the

semi-analytical load distribution model was proposed.

The semi-analytical model to predict load distribution of side-fit spline joints under arbitrarily combined loading included all essential components of spline compliance associated with tooth bending and shear deformations, tooth base flexibility, contact deformations and

206 torsional deformations. A multi-step discretization solution scheme was devised and implemented in this model to minimize the computational time required. Load distributions of an example spline under various loading conditions predicted by the semi-analytical model were compared to those predicted by a finite element type deformable-body model for verification of the semi- analytical model. Using the verified model, an extensive parameter study was performed to investigate effects of loading conditions, misalignments, tooth modifications and tooth indexing errors on tooth-to-tooth load sharing as well as contact stress distributions. Exploiting the computational efficiency of the model, a statistical analysis methodology was also devised to relate the spline quality level to the resultant probability distribution of tooth load sharing and contact pressure distributions.

In order to analyze other common types of splines, the baseline model of side-fit splines was expanded to account for additional contacts potentially occurring at the tooth top land, tooth root land or tooth back side. Using the revised model, load distribution, tooth-to-tooth load sharing behavior and self-centering performance of example major and minor diameter-fit spline joints were evaluated in comparison to those of side-fit spline joints. Moreover, load distribution of an example spline joint having various intentional mismatch magnitudes at different torque levels was also studied.

As an application of the semi-analytical load distribution model, a general analytical stiffness formulation for spline joints was proposed at the end. This analytical stiffness formulation resulted in a fully populated stiffness matrix of a spline joint including radial, tilting and torsional stiffness values as well as off-diagonal coupling terms. It was verified through comparison to a finite difference approximation method. Using the analytical stiffness formulation, a detailed parametric study of the stiffness of an example spline joint was performed

207

to investigate the effects of torque levels, tooth modifications, misalignments and tooth indexing

errors.

6.2 Contributions

As stated as part of the motivation for this study, load distribution analysis of spline joints

had long been neglected. While the tools for design and analysis of other power transmission

components such as gears and rolling element bearings have benefited significantly from new

tools for the last several decades, design and analysis of splines were still based on crude

assumptions and simple formulations. As stated in Chapter 1, deformable-body models also fell

short of elevating the state-of-the-art in spline design and analysis due to their computational

handicap. As such, the most significant contribution of this study can be identified as the semi-

analytical model itself. This model is novel as there has been no other spline analysis tool of this

kind. More importantly, it is flexible in handling all possible general loading conditions as well

as operational effects such as misalignments. Unified nature of the formulation behind the model

allows all four basic types of splines to be analyzed such that the designer can compare the

performance of different spline types to choose the most appropriate one for the application in

hand. Also critical is the fact that the proposed model requires only a few seconds in comparison

to several hours required by deformable-body models. This key feature of computational efficiency allows the proposed model to be used as the predictor for statistical treatment of splines having manufacturing errors. Finally, the accompanying stiffness formulation is also a

new contribution that has the potential to improve dynamic simulations of drivetrains having

spline joints significantly.

208

6.3 Conclusions

Based on parametric studies performed by using the three variations of the spline load distribution model, following specific observations and conclusions can be made:

• Load distribution and tooth-to-tooth load sharing of spline joints are influenced

significantly by loading conditions, misalignments and spline tooth modifications

(intentional tooth surface deviations from pure involute form). Combined loading

conditions representative of a spline supporting a gear and misalignments both result in

significant load concentrations across each tooth and unequal tooth-to-tooth load sharing.

A certain amount of lead crown modification relieves load concentration of misaligned

splines, while it does not help in combined loading conditions.

• Tooth indexing errors within manufacturing tolerances dictated by quality level of the

spline joint also affect load distribution and tooth-to-tooth load sharing characteristics of

spline joints in a substantial manner. Tooth indexing errors induce sizable variations to the

baseline load distribution and tooth load sharing characteristics determined by design

specifications and operating conditions. Spline tooth pairs having larger tooth indexing

errors, namely smaller clearances in between, usually experience increased contact stress

over a larger contact area, thus sharing more load.

• Load distribution and tooth load sharing of a spline designed to a specific manufacturing

tolerance class will typically fall into a certain probability distribution. Splines designed to

a tighter manufacturing tolerance (i.e. higher quality level) exhibit better load distribution

and tooth load sharing characteristics.

209

• In gear loading conditions, major and minor diameter-fit splines exhibit better load

distribution and tooth load sharing characteristics as well as superior centralization

capability, compared to that of side-fit splines.

• The stiffness matrix of a spline joint varies with any design or operation factor that changes

spline load distribution. Torque levels, tooth modifications, misalignments as well as tooth

indexing errors all influence the stiffness matrix of a spline joint. In the presence of tooth

indexing errors, the stiffness matrix of a spline joint under an asymmetric loading condition

typically exhibits a time-varying characteristic as a direct result of time-varying load

distributions of the spline joint.

6.4 Recommendations for Future Work

A number of recommendations for future research work to enhance the work presented in this study are listed as follows:

• Using the proposed model, investigate effects of other types of manufacturing errors, such

as profile or lead errors specific to individual spline teeth and out-of-roundness, on the

resultant load distribution of splines.

• Include effects of friction in the semi-analytical model, especially for major and minor

diameter-fit splines where a certain amount of friction forces at the tooth top land or tooth

root land are expected to influence load distributions to a certain extent.

• In the case of splines on hollow shafts, a tooth-to-tooth compliance can be included in the

semi-analytical model to account for effects of thin rim deflection, providing more accurate

predictions of spline load distribution in such cases.

210

• Develop a methodology to relate predicted load distribution and relative rigid body motions

to fretting wear and fretting fatigue of splines under various operating conditions.

• Based on load distributions predicted, develop a separate finite element model to efficiently

compute resultant root stress distributions of splines to facilitate designers in addressing

spline durability issues associated with tooth bending fatigue. This will also be essential

for experimental validation of the model.

• Validate the model presented in this study through comparisons to tightly-controlled

experiments. For this, an experimental set-up must be developed to measure root stress

distributions of splines under various loading conditions.

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BIBLIOGRAPHY

Adey, R. A., Baynham, J. and Taylor, J. W., 2000, “Development of Analysis Tools for Spline Couplings,” Proc IMechE., Part G: Journal of Aerospace Engineering, 214, 347-357.

Al-Hussain, K. M., 2003, “Dynamic Stability of Two Rigid Rotors Connected by a Flexible Coupling with Angular Misalignment,” Journal of Sound and Vibration, 266, 217-234.

Barrot, A., Paredes, M. and Sartor, M., 2006, “Determining Both Radial Pressure Distribution and Torsional Stiffness of Involute Spline Couplings,” Proc. IMechE., Part C: J. Mechanical Engineering Science, 220, 1727-1738.

Barrot, A., Paredes, M. and Sartor, M., 2009, “Extended Equations of Laod Distribution in the Axial Direction in a Spline Coupling,” Engineering Failure Analysis, 16, 200-211.

Barrot, A., Sartor, M. and Paredes, M., 2008, “Investigation of Torsional Teeth Stiffness and Second Moment of Area Calculations for an Analytical Model of Spline Coupling Behaviour,” Proc. IMechE., Part C: J. Mechanical Engineering Science, 222, 891-902.

Brown, H.W., 1979, “A Reliable Spline Coupling,” Journal of Engineering for Industry, 101, 421-426.

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