EXPERIMENTAL AND MATHEMATICAL INVESTIGATION INTO ASPECTS OF

SPATIAL INVOLUTE GEARING

Thesis submitted in fulfilment of the requirements for the degree of

MASTER OF ENGINEERING (HONOURS)

By

Michael Killeen BE (Mech)

School of Engineering and Industrial Design

University of Western Sydney

18 April, 2005

ii

STATEMENT OF ORGINALITY

Unless otherwise noted by way of references the material presented in this thesis in the areas

of inspection of gears and development of mathematical models is the work of the author. The

mathematics used to develop the gear bodies to be inspected and mathematical models within

this thesis, however, were based on the theory and explanatory geometry of Phillips gleaned

through many discussions and readings of his publications.

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ACKNOWLEDGEMENTS

Many people contributed to this work. Without these people’s contribution, I would not have

been able to pull it all together. My academic supervisors, Dr John Gal and Dr Jack Phillips,

played an important role in guiding me through this and assisting me in developing my

understanding of both this material and the fundamental kinematic theories that tie this

material together. I also thank John for his faith in my ability to keep myself motivated and be

self-directing. Although Jack and I worked on the same problems, it was often in a different

‘language’. He was, however, able to ask probing questions that often, if not solving the

problem, at least gave me an insight into a new approach. Had it not been for the many hours

of discussion with Jack, I would undoubtedly still be wandering up some blind alley.

I would also like to thank Dr Fred Sticher. Fred speaks my language and provided a lot of

guidance for me in the early stages of my work in this area. Fred has an uncanny knack of

reducing two or three pages of my laborious derivations into half a page. This usually

involves, what in hind-sight is, a blindingly obvious observation at about line two.

I also have many people to thank for their efforts in assisting me with the practical aspects of

this work. Mr Paul Toner & Mr Andrzej Hudyma of UWS assisted in the construction of the

equipment used in functional inspection of the gear bodies. Paul also did much of the

machining of the equipment required for this functional inspection. Mr Richard Turnell and

Mr Chris Chapman of UTS provided guidance on determining the suitability of data

acquisition and virtual instrumentation for functional inspection whilst Mr Ian Gibson of UTS

provided his time and guidance in setting up the analytical inspection of the gear body.

iv

This is now also the second time my family have had to endure me whilst I devote all of my

spare time to a thesis. Although they did not contribute directly to the development of the

thesis, it was their support that made this possible.

v

ABSTRACT

This thesis is a small part of a much larger work, the aim of which is to continue the transition

from gear theory to gear practice. The thesis deals with some aspects of the testing and

theoretical development of equiangular and plain polyangular gears respectively. Initial

prototypes of the equiangular spatial involute gearing, a small subset of a general spatial

involute gear set, developed in previous works are to be tested for both function and form.

The tests, based on the principles of the single flank gear tester, investigate constancy of

transmission ratio and use both electronic and mechanical means. The former of these

highlights the shortcomings of some aspects of the experimental set up. Algebraic expressions

are also developed for plain polyangular gearing, a more general form of spatial involute

gearing. These equations demonstrate the links to the underlying kinematic principles and are,

consequently, more robust. This is verified by their application to both the equiangular and

plain polyangular cases. The expressions were checked by comparing their results to

graphical and numerical models developed concurrently with the algebraic expressions. Initial

investigations are also undertaken into turning the mathematical theory into gear machining

theory.

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TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION ....................................................................................................................... 1

1.1 THESIS OUTLINE AND STRUCTURE......................................................................................................... 2

1.2 NOTES ON NOMENCLATURE .................................................................................................................. 5

CHAPTER 2 A SURVEY OF THE LITERATURE........................................................................................ 9

2.1 A REVIEW OF THE LITERATURE ON GEAR THEORY, DESIGN AND MANUFACTURE................................... 9

CHAPTER 3 THE MATHEMATICS OF EQUIANGULAR AND PLAIN POLYANGULAR

INVOLUTE GEARING............................................................................................................ 13

3.1 THE KINEMATIC FOUNDATIONS........................................................................................................... 17

3.2 THE ARCHITECTURE OF PLAIN POLYANGULAR GEARS ......................................................................... 24

3.3 THE RELATIONSHIP BETWEEN THE LINES OF ACTION AND THE GEAR AXES ....................................... 30

3.4 A SPECIAL CASE OF THE PLAIN POLYANGULAR – EQUIANGULAR GEARING.......................................... 33

3.5 THE RESULTING ENTITIES GENERATED FROM THE PLAIN POLYANGULAR ARCHITECTURE ................... 37

3.6 ALGEBRAIC PROOF OF THE RELATIONSHIPS BETWEEN THE SLIP TRACK AND THE BASE HELIX............. 41

3.7 THE MATHEMATICS OF SOME FUNDAMENTAL PLANES ........................................................................ 43

3.8 DISCUSSION ........................................................................................................................................ 51

CHAPTER 4 GEAR INSPECTION................................................................................................................ 53

4.1 FUNCTIONAL INSPECTION ................................................................................................................... 53

4.2 ANALYTICAL INSPECTION ................................................................................................................... 55

CHAPTER 5 FUNCTIONAL INSPECTION OF EQUIANGULAR SPATIAL INVOLUTE GEARS

USING ENCODERS ................................................................................................................. 59

5.1 BACKGROUND .................................................................................................................................... 59

5.2 AIM..................................................................................................................................................... 60

5.3 METHOD ............................................................................................................................................. 60

5.4 RESULTS ............................................................................................................................................. 65

5.5 DISCUSSION ........................................................................................................................................ 69

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CHAPTER 6 FUNCTIONAL INSPECTION OF EQUIANGULAR SPATIAL INVOLUTE GEARS

USING DIAL INDICATORS ................................................................................................... 77

6.1 AIM..................................................................................................................................................... 77

6.2 METHOD ............................................................................................................................................. 77

6.3 RESULTS ............................................................................................................................................. 80

6.4 DISCUSSION ........................................................................................................................................ 86

6.5 CONCLUSION ...................................................................................................................................... 95

CHAPTER 7 PROFILE MEASUREMENT OF EQUIANGULAR SPATIAL INVOLUTE GEARS...... 97

7.1 A REVIEW OF THE THEORETICAL TOOTH PROFILE................................................................................ 97

7.2 AIM................................................................................................................................................... 100

7.3 METHOD ........................................................................................................................................... 100

7.4 RESULTS ........................................................................................................................................... 103

7.5 DISCUSSION ...................................................................................................................................... 104

CHAPTER 8 CONCLUSION ........................................................................................................................ 123

8.1 A UNIVERSAL THEORY OF GEARING .................................................................................................. 123

8.2 A BRIDGE BETWEEN THE KINEMATIC THEORY AND GEAR MACHINING THEORY................................. 124

8.3 FUNCTIONAL TESTING....................................................................................................................... 125

8.4 ANALYTICAL INSPECTION ................................................................................................................. 126

REFERENCES 128

BIBLIOGRAPHY ............................................................................................................................................. 132

APPENDIX A GLOSSARY OF TERMS AND NOTATIONS ..................................................................... 136

KINEMATIC DEFINITIONS: TERMS RELATING TO THE RELATIVE POSITION OF AXES .......................................... 137

KINEMATIC DEFINITIONS: TERMS RELATING TO MATING GEARS...................................................................... 137

KINEMATIC DEFINITIONS: TERMS RELATING TO RELATIVE SPEEDS .................................................................. 137

KINEMATIC DEFINITIONS: TERMS RELATING TO PITCH AND REFERENCE SURFACES ......................................... 137

TOOTH CHARACTERISTICS: TERMS RELATING FLANKS AND PROFILES ............................................................. 138

TOOTH CHARACTERISTICS: TERMS RELATING TO PARTS OF THE FLANK........................................................... 138

GEOMETRICAL AND KINEMATICAL NOTIONS USED IN GEARS: TERMS RELATING TO GEOMETRICAL LINES ....... 139

viii

GEOMETRICAL AND KINEMATICAL NOTIONS USED IN GEARS: TERMS RELATING TO GEOMETRICAL SURFACES 140

GEOMETRICAL AND KINEMATICAL NOTIONS USED IN GEARS: OTHER TERMS................................................... 141

CYLINDRICAL GEARS AND GEAR PAIRS: TERMS RELATING TO CYLINDERS AND CIRCLES ................................. 142

CYLINDRICAL GEARS AND GEAR PAIRS: TERMS RELATING TO HELICES OF HELICAL GEARS ............................. 142

CYLINDRICAL GEARS AND GEAR PAIRS: TERMS RELATING TO TRANSVERSE DIMENSIONS................................ 143

CYLINDRICAL GEARS AND GEAR PAIRS: TERMS RELATING TO GENERATING TOOLS AND ASSOCIATED FEATURES

143

APPENDIX B A PROCESS MAP FOR THE DEVELOPMENT OF PLAIN POLYANGULAR

ARCHITECTURE AND SURFACE ENTITIES.................................................................. 144

APPENDIX C A NUMERICAL EXAMPLE ................................................................................................. 146

APPENDIX D DETAILED MATHEMATICAL DEVELOPMENTS......................................................... 148

DETAILED WORKINGS THAT PRODUCED V12 AND THE BDL............................................................................... 148

THE DETAILS OF THE CREATION OF THE LOA FROM THE BDL ........................................................................... 149

DETERMINING THE BASE RADIUS (RB) AND THE ANGLE THE LINES OF ACTION SUBTEND TO THE GEAR AXES ... 152

APPENDIX E SURFACE MEASUREMENT................................................................................................ 155

APPENDIX F THE DESIGN PROCESS AND TEST RIG FUNCTIONAL SPECIFICATION.............. 157

THE DESIGN PROCESS ...................................................................................................................................... 157

THE FUNCTIONAL SPECIFICATION FOR THE TEST RIG ....................................................................................... 160

THE DESIGN OF THE GEAR BODY TEST RIG ....................................................................................................... 161

APPENDIX G DRAWINGS OF THE GEAR BODY TEST RIG ................................................................ 163

APPENDIX H LABVIEW PANEL AND VIRTUAL INSTRUMENT FOR DATA COLLECTION

DURING FUNCTIONAL TESTING..................................................................................... 166

APPENDIX I USEFUL STATISTICAL ANALYSIS................................................................................... 169

SIMPLE LINEAR REGRESSION AND CORRELATION ............................................................................................ 169

HYPOTHESIS TESTING...................................................................................................................................... 171

APPENDIX J THE MATHEMATICS OF OBJECT TRANSFORMATION............................................ 174

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APPENDIX K A COMPARISON OF THE CYCLOIDAL AND INVOLUTE PROFILES...................... 178

THE CYCLOIDAL PROFILE ................................................................................................................................ 180

THE RADIUS OF THE GENERATING CIRCLE........................................................................................................ 181

THE INVOLUTE PROFILE .................................................................................................................................. 182

APPENDIX L PRINCIPLES FOR AND EXAMPLES OF THE CALCULATION OF AXIAL

DISTANCES ............................................................................................................................ 185

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LIST OF FIGURES

Figure 1: A comparison of Phillips’ and Killeen’s architecture............................................... 16

Figure 2: The equivalent RSSR mechanism to a pair of spatial involute gears; the RSSR ..... 18

Figure 3: An illustration of the first law of gearing ................................................................ 21

Figure 4: The parabolic hyperboloid defined by GA1 & GA2 and the Transversal viewed

along GA2 .............................................................................................................. 23

Figure 5: The architecture of spatial involute gearing showing the Lines of Action............... 30

Figure 6: The cosine of cosγb ................................................................................................... 32

Figure 7: The right triangle of cosγb......................................................................................... 35

Figure 8: An illustration of the slip track for the case where rb = 51.58 and γb = 1.909 rad.... 39

Figure 9 : Graph of the coordinates of the slip track................................................................ 39

Figure 10: The angle between the slip track and the base helix for the general plain

polyangular case ..................................................................................................... 42

Figure 11: Two points on a general involute helicoid viewed along the axis of the base

helix ........................................................................................................................ 43

Figure 12: The triad of planes defined by the involute helicoid .............................................. 44

Figure 13: The tangent plane of r at (u0, v0) represented by its normal, n. .............................. 45

Figure 14: Graph of the unit tangent vector components......................................................... 48

Figure 15: Graph of the unit normal vector of slip track 11 .................................................... 49

Figure 16: A graph of the binormal vector for slip track 11 ................................................... 50

Figure 17: A spatial involute gear body................................................................................... 56

Figure 18: The real tooth profile and the graph of the profile.................................................. 57

Figure 19: A schematic of a single flank gear tester ................................................................ 60

Figure 20: A picture of the electrical functional testing arrangement...................................... 61

Figure 21: The ideal response of the output signals of the encoders ....................................... 69

Figure 22: A graph of a sample of the response of Lines 0 and 1 from test 1 ......................... 70

Figure 23: A graph of the transmission ratio for test 1 ............................................................ 72

Figure 24: An illustration of the sampling rate as determined by the operating system.......... 74

Figure 25: Graph of transmission ratio for test 1 ..................................................................... 90

Figure 26: Graph of transmission ratio for test 2 ..................................................................... 90

Figure 27: Graph of transmission ratio for test 3 ..................................................................... 91

Figure 28: A general involute helicoid..................................................................................... 98

Figure 29: A graphical solution to the relationship between µ and t7 .................................... 100

xi

Figure 30: The arrangement of the gear body on the CMM .................................................. 101

Figure 31: A plan view of the gear body (looking down on the tooth surface) ..................... 102

Figure 32: A view of Gear Body 2 defined by Gear Axis 2 and the Centre Distance Line ... 108

Figure 33: A graph of the profile for selected values of z for Gear Body 1 .......................... 109

Figure 34: A graph of the profile for selected values of z for Gear Body 2 .......................... 110

Figure 35: Ball radius compensation – the planar case .......................................................... 114

Figure 36: Ball radius compensation - the spatial (3-D) case ................................................ 116

Figure 37: All solutions to the common normal to the sphere and involute helicoid given

by the x component .............................................................................................. 119

Figure 38: All solutions to the common normal to the sphere and involute helicoid given

by the y component .............................................................................................. 120

Figure 39: Graph of OPx, OPy, OPz and theta versus t7.......................................................... 121

Figure 40: A diagram of the helix angle (a) and lead angle (b) ............................................. 142

Figure 41: The components of v12 shown with v12 at the origin ............................................ 150

Figure 42: A schematic of laser-based surface roughness measurement ............................... 156

Figure 43: The six phases of design ....................................................................................... 157

Figure 44: A pair of gear bodies for the set u=0.6, Σ=50° and a=80mm. .............................. 162

Figure 45: The labview front panel used for functional inspection data collection............... 167

Figure 46: The Labview VI used to collect the data for the functional inspection ................ 168

Figure 47: The principle of conjugate action ......................................................................... 179

Figure 48: The involute created by a line rolling on a circle ................................................. 182

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LIST OF TABLES

Table 1: A list of the commonly used symbols .......................................................................... 6

Table 2: A comparison of Phillips' and Killeen's symbolism .................................................... 8

Table 3: Cross reference between type of gear set and location of Qx..................................... 24

Table 4: Comparison of the results of the complex and simple expressions for γb21 ............... 36

Table 5: Settings for the Labview PC+ DAC........................................................................... 62

Table 6: Connections between encoders and Labview PC+ card ............................................ 63

Table 7: Results of electronic functional inspection for ‘as designed’ configuration.............. 66

Table 8: Results of electronic functional inspection for modified geometry (a reduced by

1.5mm) ................................................................................................................... 67

Table 9: Results of electronic functional inspection for modified geometry (Σ reduced) ....... 68

Table 10: A selection of the results of test 1 represented as change of state data.................... 71

Table 11: A count of the rising and falling edges and the resulting transmission ratio for

the selected data range of test 1.............................................................................. 71

Table 12: Effective outside pulley diameters ........................................................................... 80

Table 13: Results for macro mechanical testing of transmission ratio. ................................... 80

Table 14: Results for detailed mechanical testing of transmission ratio.................................. 83

Table 15: Comparison of internal consistency of the macro mechanical inspection via

error analysis .......................................................................................................... 88

Table 16: Analysis of internal consistency of the macro mechanical inspection..................... 88

Table 17: Summary of macro mechanical inspection results................................................... 89

Table 18: Summary of the detailed mechanical inspection results .......................................... 91

Table 19: A comparison of the observed and critical f-statistic............................................... 94

Table 20: Data points created by the CMM measurements of Gear Body 1 ......................... 103

Table 21: Data points created by the CMM measurements of Gear Body 2 ......................... 104

Table 22: Raw data from the CMM for Gear body 1............................................................. 105

Table 23: Transformed data for Gear body 1......................................................................... 109

Table 24: Sample data points from Gear body 2 for analysis of data consistency ................ 112

Table 25: Model 1 solving the involute function for Point 1 ................................................. 112

Table 26: Model 2 solving the involute function for Point 2 ................................................. 112

Table 27: Model 3 solving the involute functions by forcing rb to be equal.......................... 113

Table 28: A numerical solution for the set of equations defining the common tangent plane

to the sphere and involute helicoid....................................................................... 120

xiii

Table 29: The impact of ball radius on the accuracy of CMM values .................................. 122

Table 30: Values for the numerical example ......................................................................... 146

Table 31: Contents and drawing cross-reference for the test rig design ................................ 163

1

CHAPTER 1 INTRODUCTION

If many of the gear publications are any indication, much of the focus of current commercial

work on gears is on analysing existing problems out of gears. Considerable advances have

been made in methods of gear manufacturing, gear inspection, gear vibration analysis, and

computerised gear design. There have also been significant developments in gear materials,

gear noise analysis, evaluation and gear lubrication methods. Commercial gear publications

and standards, for example, discuss means of modifying the profiles of gears to address

problems with over-constraint. These problems, however, are caused by a lack of attention to

the fundamental underlying theory itself and ‘solutions’ such as crowning and end-relief treat

the symptoms rather than the cause.

What is not explored in gear publications is a solution that addresses the root cause of the

problems with commercial gears; there is relatively little work done on synthesising a

completely new and exact fundamental underlying theory. The work that has been done,

rather than resulting in a grand unifying theory of gears, approaches the issue from a number

of different perspectives and it has been rare for the approaches to converge. Where the

approaches do converge, it appears to be by accident rather than intent. As a result, a grand

unifying theory of gears, a theory that takes the basic functions of a gear pair and, through the

application of kinematic fundamentals comes up with a gear wheel, is, as yet, undeveloped

although Phillips (2003, pp349-406) has formulated the basis for producing such a gear

wheel.

Developing a grand unifying theory is, however, in itself, the first step in its eventual use and

application. The theory needs to be applied to a test case, the outcomes of the test case need to

2

be compared to the outcomes the theory predicted and any causes of errors identified and

integrated into the theory. Mechanisms then need to be developed for implementing the

theory in a commercial environment. Although one approach to the final step is, as Gleason

did, to develop a machine for executing the theory, it is preferable, especially in an

environment of fiscal restraint, to utilise existing equipment and processes.

This thesis builds on earlier work by Phillips (1984, 1995 & 2003) and Killeen (1996) aimed

at achieving the ultimate goal of gear design, a complete unified theory of gearing that ties all

aspects of gear design together and yet satisfies all of the existing partial theories; a theory

that has clearly demonstrable foundations in three-body kinematics and screw theory; a theory

that provides a solid foundation on which to base gear design rather than the ‘modern

tendencies in the theory of gearing…directed at the skilful use of computers and display of

the results by computer graphics, to improve gear design and manufacture’ Litvin (1992,

p1.1); a theory that can be applied in practice using current gear manufacturing techniques.

1.1 Thesis outline and structure

Previous work by Killeen (1996) focussed on theoretical aspects of gear design and to a

limited extent gear body construction. These were largely synthesis activities based on

Phillips’ (2003, pp159-233) theories for equiangular involute gearing. No checking of the

claims of Phillips (2003, pp159-233) with regard to surfaces thus generated and constancy of

transmission ratio was undertaken in the previous work by the author. From the perspective of

Demming’s Plan-Do-Check-Act (PDCA) cycle (Samson 1995), work thus far has been on the

Plan-Do phases.

The objectives of this thesis are to close that loop i.e. to:

• continue the development of a ‘universal theory of gearing’

3

• start to bridge the gap between the theory and the practice of gear manufacture.

• compare the theoretical transmission ratio to the actual transmission ratio for a gear

body pair designed using example 1 in Phillips’ (2002, p200) theory and process

The thesis is in two parts. The first part is a synthesis of new mathematical and theoretical

material for the next stage in the development of a general involute theory. The second

involves the inspection of various aspects of the existing gear pair built by the author during

his undergraduate work (Killeen 1996). Although the gear bodies were constructed prior to

the development of the work performed in synthesising the mathematical and theoretical

material in the first part of this thesis, the first part provides the reader with an understanding

on how the gear bodies tested were developed.

Chapter 2 comprises a survey of the current literature in the two areas, gear inspection and

testing and gear theory and mathematics. A range of works are discussed from more

theoretical ones to very practical ones. The chapter focuses attempts to identify the

applicability of the text to the theory of gear design.

Chapter 3 introduces the topic of spatial involute gearing through mathematical synthesis of

expressions describing the surface entities that will ultimately define the gears. After closing

some of the outstanding issues from previously published work in the mathematics of

equiangular spatial involute gearing, it looks at some major simplifications to the previously

intractable equations describing the equiangular equivalent to the Revolute Spherical

Spherical Revolute (RSSR) mechanism, a mechanism instantaneously equivalent to the

spatial involute gear.

4

Chapter 3 then goes on to develop a mathematical model for generating and expressing

mathematically the architecture and surface geometry of the plain polyangular gear. Previous

work in Killeen (1996) focussed on a very specific subset of Phillips theory, the equiangular

gear. Continuing the move toward a ‘grand unifying theory’, Chapter 3 details both the

process used to develop, and, a more general set of equations applicable to both plain

polyangular and equiangular spatial involute gears. To demonstrate this principle and check

how robust the new equations are, the chapter also demonstrates the applicability of the more

general plain polyangular equations to the specific equiangular geometry.

With a mathematical foundation in place, Chapter 3 begins the transition from mathematical

theory to gear machining. In line with the attempt to bridge the gap between the theory and

the practical aspects of gear machining, the chapter transitions the mathematical analysis into

an investigation of the key planes and their characteristics. Specifically, mathematical

expressions for the surfaces are used to determine behaviour of the normal, osculating and

rectifying planes of the involute helicoid. Understanding the behaviour of these planes will be

central to developing practical machining techniques.

Chapter 4 is background on what is to follow. It provides an overview of the two different

types and general principles of gear body inspection. The chapter also explains factors that

influence the results of the various inspections.

Chapters 5 and 6 detail the aims, method, results and an analysis of the results of a functional

inspection of the equiangular involute gear bodies constructed in Killeen (1996), a gear set

equivalent to example 1 in Phillips (2003, p200). Chapter 5 provides details of an approach

utilising optical encoders providing inputs to a Data Acquisition Card (DAC) and Labview

5

Virtual Instrument (VI). It then goes on to investigate why the results varied from the

theoretical results. Chapter 6 details a simpler and more robust approach to the same problem,

an approach using pulleys and dial indicators. The experimental results are then analysed

using statistics and an estimate is made of the probability of experimental outcomes matching

theoretical predictions.

Building on the explanation contained in Chapter 4, Chapter 7 investigates the principles of

profile measurement as applied to equiangular involute gearing. Following a definition of the

theoretical surfaces, the chapter then explains the equipment and processes used to check

whether the actual/as-machined tooth profile matched the theoretical profile of the

aforementioned gear bodies. There is then a detailed discussion of the results, including

analysis of the causes of experimental error. A number of tools are used to perform the

analysis including detailed algebraic analysis of the surface and numerical solvers.

Chapter 9 summarises the initial objectives of this thesis and ties these objectives to specific

outcomes achieved in the preceding chapters. It also details the significant findings of the

thesis in the process of developing a grand unifying theory of gears. Suggestions for further

work in the field are also made.

1.2 Notes on nomenclature

The mathematical nature of the work contained in this thesis necessitates the early definition

of what will turn out to be the more commonly used symbols. These symbols, summarised in

Table 1, are based on AS2075 – 1991 where possible. The Australian Standard, however, was

written using the current paradigm on gear design and in some instances AS definitions are

regarded by the author as inadequate. In cases where the Phillips’ symbolism was not given

6

another meaning in AS2075-1991, the Phillips’ symbolism was adopted as the ‘standard’ in

this thesis.

Table 1: A list of the commonly used symbols

Symbol Term Definition

a Centre distance The distance between the axes of a gear pair is measured

along a line perpendicular to the [both] axes.

Σ Shaft angle The smallest angle through which one of the axes must be

rotated to bring the axes into coincidence (gear pair with

intersecting axes), or must be swivelled to bring the axes

parallel (gear pair with non-parallel non-intersecting axes),

so as to cause their directions of rotation to be opposite.

Suffix 1 Pinion That gear of the pair which has the smaller number of teeth

Suffix 2 Wheel That gear of the pair which has the larger number of teeth

z Nbr teeth The number of teeth on a gear

u Gear ratio The ratio of the number of teeth of the wheel to that of the

pinion

i Transmission

ratio

The ratio of the angular speed of the first driving gear of a

gear train to that of the last driven gear

β Helix angle The acute angle between the tangent to a helix and the

straight generator of the cylinder on which the helix lies.

γ Lead angle The acute angle between the tangent to a helix and a plane

perpendicular to the axis of the cylinder on which the helix

lies.

’

(apostrophe)

Working A qualification applicable to every term defined from the

pitch surface of a gear in a gear pair.

d Diameter A generic symbol representing any diameter

r Radius A generic symbol representing any radius

Subscript b Base Relating to the base circle

t Parametric

coordinate

In defining various lines using parametric equations it was

necessary to select a parameter. The parameter chosen was

‘t’ and each t, t1 to tn is a parameter specific to the line being

defined. This is not an Australian Standard definition.

Note: Appendix A contains a comprehensive list of nomenclature and definitions

7

System for abbreviations: Rather than carrying around large complicated expressions for

either points or vectors, it has been necessary in this thesis to reduce these expressions to

symbols. The following system is used to achieve this:

• Points are simplified by taking the symbol for the point, usually an upper case letter,

and making its component a subscript. For example, the point F is represented by its

three coordinate components, Fx, Fy, and Fz.

• Vectors are:

o Represented in bolded all lower case or with a ‘hat’. For example, the vector

of the transversal would be represented as transversal whilst the vector ω

would be represented as ω .

o Simplified by making the direction a subscript of the symbol for the vector.

Consider for example, the vector 12v where the x component would be

represented by v12x.

• Lines are represented in bold with the first letter capitalised. For example, the line

formed by placing the transversal at a given point, say J, is the Transversal.

• Entities are usually represented in lower case non-bolded text. The base helix and slip

track are examples.

A comparison of relevant symbolism ‘standards’: There are some significant differences

between this work and that of Phillips’ despite the fact the work here is based on Phillips’

theories. Not the least significant of these differences is symbolism. Table 2 provides an

overview of these differences. Phillips uses symbolism that he developed over a series of

publications such as Phillips (1984, 1990) Freedom in machinery Volumes 1 & 2 respectively.

This symbolism was developed to explain and convey the theory.

8

Table 2: A comparison of Phillips' and Killeen's symbolism

Killeen’s

symbol

Description Phillips’

symbol

a Centre distance: The distance between the axes of a gear pair

as measured along a line perpendicular to the [both] axes.

C

Σ Shaft angle: The smallest angle through which one of the

axes must be rotated to bring the axes into coincidence (gear

pair with intersecting axes), or must be swivelled to bring

the axes parallel (gear pair with non-parallel non-intersecting

axes), so as to cause their directions of rotation to be

opposite.

Σ

u Gear ratio: The ratio of the number of teeth of the wheel to

that of the pinion

k

d’ Pitch diameter: The diameter of the pitch circle 2ε

γb Base lead angle: The lead angle of the base helix of an

involute gear α

rb Base circle radius: For an involute cylindrical gear, the

radius of the circle from which the tooth flank involute is

derived

a

9

CHAPTER 2 A SURVEY OF THE LITERATURE

The following chapter details work of previous authors in the fields of gear theory, design and

manufacture. There is also a section on literature in the field of thinking and problem solving.

2.1 A review of the literature on gear theory, design and manufacture

Probably the most significant single work on conventional gear theory is Townsend et al

(1992). This work, an amalgamation of the works of 26 gear specialists, covers a variety of

topics, some of which will be investigated in the following sections along with details of other

resources in the area of gear theory.

There is a range of literature on planar involute theory with works by Litvin, Shigley (1986)

and Mabie and Reinholtz (1987) to name a few. Mabie and Reinholtz (1987) provides an

excellent introduction to planar involute theory. There is also a range of internet resources

including a page explaining how to create the mathematical expressions for rack movements

for a helical gear written in Maple V.

The Australian Standard 2075 – 1991 describes specific aspects of a number of key types of

gearing including cylindrical, bevel and hypoid as well as providing the reader with some

basic kinematic and geometric definitions pertaining to these types of gearing. These

definitions provide a starting point for the development of a universally applicable set of

definitions and nomenclature. As previously mentioned, however, the standard falls short of

providing definitions applicable to general spatial involute gearing.

The work of Beam (1954) in explaining beveloid gearing attempts to tie all existing gear

theory together. It discusses how beveloid gearing can mesh with a range of other types of

10

gears and is insensitive to errors in manufacture. He also provides an excellent graphic

detailing the angles in beveloid gearing. The article, however approaches beveloid gearing

more from a gear theory and manufacturing perspective than a kinematic perspective. There is

no indication of the kinematic fundamentals upon which the gears are based.

Although there is a variety of gear literature on general gear topics and conventional gearing,

there is less on the mathematics of general spatial involute gearing. The vast majority of work

in this area has been undertaken by Phillips. His initial works ‘Freedom in machinery 1’

(1984) and ‘Freedom in machinery 2’ (1990) provide a comprehensive explanation of many

of the fundamental geometrical and kinematic theories that underpin the concepts presented in

this thesis. There is also a range of excellent graphics supporting the explanations.

The majority of the work Phillips completed in this field is contained in Phillips (2003).

Phillips has also published a number of papers. These include Phillips (1995), Phillips

(1999a) and Phillips (1999b). Phillips (1995) and (1999a) are an excellent introduction to the

links between three-body kinematic theory and gear design in the spatial arena. Phillips

(1999b) details the more practical aspects of gear design and development in an investigation

of the truncation of some theoretical gear tooth profiles developed using the theories detailed

in the previous two papers.

Other authors in the field investigate specific aspects of general spatial involute gearing

theory. Litvin in Townsend et al (1992) cites a number of his own publications. (They

constitute 14 of the 24 references he cited!). The theory presented, however, fails to draw the

whole discussion together. The discussion on pages 1.38-1.39, for example, highlights some

of the work occurring in the area of contact between surfaces and replicates the work of

11

Timoshenko and Goodier (1970), but fails to demonstrate how this would be applied to

gearing from a theoretical perspective.

There is a variety of literature available on topics such as analytical and functional gear

inspection. Townsend et al (1992) presents an overview of the whole topic of gear inspection

and focuses on specific types of gear inspection, providing schematics of inspection

arrangements. Bonfiglioli (1995) looks at some of the limitations of analytical inspection and

investigates advanced profile measurement techniques involving tracing the path of an

involute to a circle along the flank of a tooth. He then investigates ways to compare the as

measured profile to the theoretical profile.

Doebelin (1990) and Floyd (1992) provide an insight into electrical measurement systems

used for functional inspection and how these devices work. The discussion of instrumentation

theory and practice is comprehensive. Wells (1995) provides and overview of computer based

instrumentation and guidance on writing programmes for Labview-based ‘instruments’. This

book is written for the novice data acquisition and Labview user and, although it does not

replace actually doing it, Wells provides a good basic understanding of the principles of

virtual instrumentation. There are also a number of useful examples.

In the area of numerical analysis and statistics, Gerald and Wheatley (1989) cover a range of

topics and numerical analysis tools such as numerical differentiation whilst Walpole and

Myers (1993) provide an excellent understanding of the statistical theory required to analyse

and interpret analytical inspection data. There is also a plethora of material investigating

statistical tools on the internet. The Institute of Phonetic Sciences (IFA) (1999), for example,

provides a high level investigation of the suitability of particular tests to certain types of data.

12

Given the general mathematical nature of the work on developing a plain polyangular theory,

it is worth surveying the mathematical literature. Anton (1995) provides an explanation on a

range of topics in the areas of calculus, analytic geometry and vector-valued functions.

Importantly, he also provides insight into the application of the theory in the computer-

assisted environment. Edwards and Penney (1990) also provide coverage of a good range of

topics in pure mathematics suitable for application to gear theory.

In the area of gear manufacture, Townsend et al (1992) again provides a good overview of

gear manufacturing techniques. Woodbury (1976) contains an interesting investigation into

the history of gear manufacturing and the development of new techniques. It looks at some of

the thinking that preceded modern gear manufacturing and, in doing so, covers some basic

gear manufacturing theory.

13

CHAPTER 3 THE MATHEMATICS OF EQUIANGULAR AND PLAIN

POLYANGULAR INVOLUTE GEARING

Although there are dozens of types of gears and methods of manufacture, screw theory

provides the basis for their operation. Unfortunately, however, links to these fundamentals

are either invisible or have been foregone for ease of manufacture. The result is that there is

no link between the fundamental requirements of the gear set and the resulting gear

architecture and tooth form and modifications to tooth forms are required to address the

symptoms created by this lack of kinematic purity.

Phillips (2003) proposes a new approach to designing gears, an approach that has its basis in

screw theory and carries this theory into the design of the tooth form. This approach will

ultimately tie kinematicians, gear theoreticians and gear manufacturers together. The ‘string’

to achieve this is a comprehensive mathematical statement that ultimately takes, as its inputs,

the customer’s requirements and uses kinematics and gear theory to produce gear

architectures and surface geometries that manufacturers can turn into useable gears.

The objective of this work is to develop a mathematical expression that replicates the

geometry of Phillips (2003) for two special cases of the general spatial involute gear theory,

the equiangular and general plain polyangular case. This expression is to take as its inputs,

four fundamental values:

• the shaft angle (Σ),

• the transmission ratio (u),

• the centre distance (a) and

• the distance from the Centre Distance Line (CDL) to where meshing is to occur (rf).

14

It is to provide as its outputs key components of the architecture and surface geometry of the

equiangular and plain polyangular gear pair.

In keeping with the approach of previous work undertaken in this area, no complicated

mathematical theories, such as Plückers coordinates, are utilised to achieve these objectives.

Rather the level of mathematics used to develop the models is commonly seen in second and

third year engineering courses. The intention is to demonstrate that:

• despite the far reaching implications of Phillips’ theory and geometry and apparent

complexity of many ‘non-universal’ theories, it has solid foundations and

• these are easily developed from a mathematical perspective where the mathematics

required to achieve the outcomes shown below is within the grasp of gear

manufacturers.

A word of caution: The temptation for the reader is to compare the numerical results of the

various references of Phillips with the results in the following section. As many of the

examples to be used below are based on those of Phillips, one would expect the numbers to be

the same. It will become evident, however, that this is not the case due to the frame of

reference on which the works are based. In Phillips (2003) a geometrical frame of reference

was used where, after starting off with a specific set of axes to determine the answers to

fundamental kinematic questions, a different set of axes was adopted to develop the

architecture and surfaces of the gear bodies. The work below, being mathematically based,

was not afforded that liberty as it was necessary to keep the same axes throughput the

development for reasons of transparency. The set of axes used to develop answers to the

kinematic question were stuck to religiously for the development of the gear architecture and

surface geometry.

15

Figure 1 provides an overview of the architecture and shows the difference between the two

axis systems. The respective pitch circle radii of Phillips and Killeen are equivalent to each

other, however they are the mirror image of each other. The mirror plane is the plane defined

by the Transversal, the line created by joining the points defined by the intersection of the

plane y=r and the gear axes, and the y axis rotated 90 degrees about the y axis i.e. the mirror

plane contains the y axis and has the transversal as its normal.

The following mathematical development should therefore not be viewed as contradictory to

any of those appearing in Phillips (2003) but rather confirms, using algebraic means, the

universal application of the underlying geometry to all types of gears. Caution should,

however, be exercised when comparing numerical values.

16

Figure 1: A comparison of Phillips’ and Killeen’s architecture

Σ/2

Transversal

J

K

J

K

K

J

Σ/2

Transversal

17

Appendix B contains a process map on what is to follow. Each of the subheadings in the

following section involves three or four steps within the process map. A numerical

representation of the following algebraic work is shown in Appendix C.

3.1 The kinematic foundations

Phillips (2003, p78) visualises the kinematical arrangement of the tooth surfaces of a spatial

involute gear pair as two different involute helicoids1 mounted on their respective Gear Axes

touching at point in space. Figure 2 shows a general arrangement of two such involute

helicoids. The smaller involute helicoid (the pinion) acts on the larger involute helicoid (the

wheel) thereby driving the larger involute helicoid. In practice:

• A section of the involute helicoid would define the shape of the surface of the one side

of the tooth of a gear wheel

• the relative rotational speeds of the involute helicoids (ω1 and ω2) can be expressed as

function of the number of teeth on the gear wheels (z1 and z2) according to:

(1)

where:

ω1 =

ω2 =

z1 =

z2 =

the angular speed of the pinion

the angular speed of the wheel

the number of teeth on the pinion

the number of teeth on the wheel

1 The involute helicoid is defined in Appendix A.

uz

z==

2

1

1

2

ωω

18

Figure 2: The equivalent RSSR mechanism to a pair of spatial involute gears; the RSSR

At any instant the mating involute helicoids can be replaced by an equivalent Revolute,

Spherical, Spherical, Revolute or RSSR mechanism. The length of the RS linkages is

equivalent to the respective base circle radii (rb) of the helix used to generate the involute

helicoid. The transmission ratio, i, for the RSSR linkage and therefore for the spatial involute

gear pair can be shown to be:

(2)

as adapted from Phillips (1995) where:

rb1 and rb2 are the radii of the base circles

γ1 and γ2 are the angles the line-of-action subtends to the respective Gear Axes

(A modified version of Phillips 2002 figure 3.08)

22

11

sin

sin

γγ

b

b

r

ri =

19

Substituting (1) into (2) produces

(3)

This equation is dependent only on the base circle radii, rb, and the angles γ thereby

supporting Phillips (2003) claims that the transmission ratio is “independent of errors or

intentional variations made… in the angle between the input and output links [rb1] and [rb2].”

Similarly, equation (3) supports the hypothesis that the transmission ratio is also independent

of the errors in assembly such as variations in the centre distance, a.

Referring to Figure 1, a set of axes are defined such that the y-axis bisects the shaft angle Σ

and the z-axis is collinear with the CDL. Killeen (1996) showed that the equations of the gear

axes (GA), shown in Figure 1, are:

GA1 xGA1 = - t1.sinΣ/2

yGA1 = t1.cosΣ/2

zGA1 = a/2

GA2 xGA2 = t2.sinΣ/2

yGA2 = t2.cosΣ/2

zGA2 = -a/2

The key kinematic values, the relative angular velocity of the gears, ω12, the relative screwing

of the two gears, h12, the location of the screwing axis, z12, and the orientation of the screwing

axis, δ’, with respect to these axes, were formulated geometrically by and presented in

Phillips (1990) and the corresponding mathematical representations were developed in

Killeen (1996). The mathematical representations are restated here for convenience.

11

22

sin

sin

γγ

b

b

r

ru =

20

(4)

(5)

(6)

For the kinematic case, sense is important and u = -u. The ‘screwing axis’ is actually the

Pitch Line or the Instantaneous Screw Axis (ISA) as defined in Appendix A. The unit vector

of the Instantaneous Screw Axis (ISA) of the two gear bodies can be located from:

(7)

The first law of gearing re-stated: Phillips (1995) and Phillips (2003) states the first law of

gearing must apply for constancy of angular velocity ratios in gears. The first law of gearing

as stated in Phillips (2003, p42) is:

“…the contact normal must at all stages of the meshing be located in a such away that q tanφ

remains a constant, namely, p. The parameters q and φ are the shortest distance and the angle,

respectively, between the contact normal and the pitch line”. This is illustrated in Figure 3, a

copy of Phillips (2003, p42) and stated mathematically as:

(8)

1cos2

sin.212 +Σ−

Σ=

uu

uah

)1cos2(2

)1(2

2

12 +Σ−−−

=uu

uaz

+−

= Σ

)1(

)1(cotanarctan' 2

u

uδ

p = q tanφ

j'sini'cosˆ12 δδω +=

21

Figure 3: An illustration of the first law of gearing

Equation (8) can be re-stated using terms more in line with AS2075-1991 symbolism as

(9)

and could be derived from the fundamental equation of the linear velocity of any point in

body two with respect to body one,

(10)

where:

h12 = The pitch of the pitch line and is given by equation (4)

q = rISA-n = the minimum distance between the pitch line and the contact normal

i.e. the magnitude of the vector

φ = The angle between the pitch line and the contact normal

v12 = The vector representing the relative velocity of a point in body 2 with respect to

body 1

ωωωω12 = The unit rotational vector representing the relative rotation between the two

h12 = rISA-n tan φ

v12= h12ωωωω12 + ωωωω12 x r12

(Phillips 2003 p42)

22

bodies given by (6) and (7).

r12 = The vector representing the distance between the pitch line and the point under

consideration

Taking the fourth fundamental quantity, rf, a plane y=rf can be generated. This plane will

contain the point of contact between the two gear bodies, Qx, and can be visualised as a circle

of infinite radius centred on the y axis. A line, the Transversal, can then be generated by

joining the points where each of the gear axes intersect the plane y=rf. Killeen (1996) showed

these points, J and K, as shown in Figure 1, are:

J x = - rf .tanΣ/2

y = rf

z = a/2

K x = rf .tanΣ/2

y = rf

z = -a/2

Giving the Transversal the following parametric equation:

xTrans. = -2.rf.t4.tanΣ/2 + rf. tan Σ/2

y Trans.= rf

z Trans.= a.t4 - a/2

where:

t4 is a parameter describing the distance along the Transversal measured from J

The three lines, Gear Axis 1 (GA1), Gear Axis 2 (GA2) and the Transversal define a

surface in space called a parabolic hyperboloid and is shown in Figure 4.

23

Figure 4: The parabolic hyperboloid defined by GA1 & GA2 and the Transversal viewed along GA2

There are three cases for synthesis. These three cases, starting from the specific and

progressing to the general are shown in Table 3. One condition implicit in all of these

syntheses, however, is that the paths of the point-of-contact continue to intersect at Qx

(although, in general, this does not have to be the case).

Pitch circle 2

Pitch circle 1

Parabolic hyperboloid defined

by GA1, GA2 and the

transversal

Qx

Base circle11

Base circle21

24

Table 3: Cross reference between type of gear set and location of Qx

Type of gear set Location of Qx Reference within this

document

Plain

polyangular

Qx lying on the Transversal at a

position other than i times JK along

the Transversal measured from J. Qx

is on the surface of the parabolic

hyperboloid defined by GA1 & 2 and

the Transversal. A point at other than

i times JK along the Transversal

measured from J is F.

This scenario will be

investigated in Section 3.2.

Equiangular Qx is located i times JK along the

Transversal measured from J. ‘i’

times JK along the Transversal is E.

This scenario will be

investigated in Section 3.4.

General

polyangular

Qx is located at a general point on the

hyperbolic paraboloid but not on the

Transversal.

This scenario is not

investigated here.

3.2 The architecture of plain polyangular gears

Consider the more general case, the plain polyangular. Recall that, for the equiangular case,

the chosen point of contact Qx was at E and E lay on the Transversal at the points derived

above (Phillips 1995) i.e. ‘i’ times JK along JK. The polyangular case means the point Qx is

not at E as it was for the equiangular architecture but is elsewhere along the interval JK.

Qx could, theoretically, lie anywhere on the hyperbolic paraboloid defined by the Gear Axes

and the Transversal, shown in Figure 4, however ‘let us agree that the removal and

repositioning of Qx might first and most conveniently be made by moving it to and fro along

the said transversal JEYK’ (Phillips 2003 p238) i.e. the point of intersection of the paths of

the points of contact, Qx, is at a point on JK such that JQx: QxK u. This case will be the

subject of the following mathematical development.

Moving Qx along the Transversal towards J, decreases the size of the smaller wheel.

Alternately moving Qx towards K will increase the diameter of the smaller wheel until it

25

eventually is larger than the originally larger wheel. The movement along the Transversal

changes the ratio JQx: QxK, a ratio defined henceforth as εd’ where ε and d’ are defined by

AS2075 (1991) as ‘ratio’ and ‘pitch diameter’ respectively, although Phillips uses the symbol

‘j’. The location of Qx is therefore given when t4 = 1/(1+εd’) and the plain polyangular

equivalent of E, F, is given by:

(11)

The centres of the pitch circles for the plain polyangular case can then be stated as:

O’1

x = -rf.tanΣ/2(1+εd’.cosΣ)/( εd’+1)

y = rf.(1+ εd’.cosΣ)/( εd’+1)

z = a/2

O’2

x = rf.tanΣ/2( εd’+ cosΣ)/( εd’+1)

y = rf.( εd’+cosΣ)/( εd’+1)

z = -a/2

and the radii of the pitch circles, r’1 and r’2 can then be stated as:

22

22

'

2

22

22

'

'1

sin41

1'

sin41

'

arr

and

arr

f

d

f

d

d

+⋅+

=

+⋅+

=

Σ

Σ

ε

εε

The following points can be drawn from these equations:

• Setting εd’ equal to u produces the equiangular case.

• The sum of the pitch circle radii is 22

22sin4 ar f +⋅ Σ as it was for the equiangular

case.

F

)1(

)1(.

2

)1(

)1(tan.

'

'

'

'2

+−

−=

=+−

= Σ

d

d

d

d

az

ry

rx

εε

εε

26

By reverting to the first law of gearing and invoking (8) and (10) almost all of the values

required to calculate φ and v12 are known or can be determined easily. The only unknown is

r12. The distance between the Pitch Line and the point F is represented by r12, that is the

distance between the points where the Pitch Line passes through the plane

x cosδ + y sinδ = Fx cosδ + Fy sinδ

and F. Phillips (2003, p90) gives the point on the Pitch Line the symbol N and:

• in the absence of an equivalent Australian Standard symbol for this point

• Australian Standard 2075 not using ‘N’ for anything else

N will also be used here.

The point where the Pitch Line passes through this plane is Fx cosδ + Fy sinδ along the Pitch

Line and, from the parametric equation of the Pitch Line:

(12)

from which r12 can be determined.

Invoking (9) produces:

(13)

Similarly, all components of v12 are now known, that is,

v12 = h12 ωωωω12 + ωωωω12 x r12

becomes

(14)

r12 = < Fx (cos2δ−1)+ Fy cosδ.sinδ, Fy.(sin

2δ−1)+ Fx. cosδ.sinδ, z12-Fz>

12

12

ˆarctan

r

h=φ

δδδδδδ cos.sin.,cos)(sin.,sin)(cos.ˆ1212121212 yxzz FFFzhFzhv −−−−+=

27

from which v12 can be determined. The Polar Plane i.e. the plane which has as its normal

vector the relative velocity vector v12 and passes through the point Qx, is now also fully

specified. (The detailed calculations are shown in Appendix D.)

The best drive line, bdl, can now also be determined. The bdl is:

• The line along which the components of the relative velocity vectors in the two

involute helicoids are equal and in the same direction. (Phillips 2003 p88)

• Perpendicular to both v12 and the transversal. Calculating the cross product produces:

(15)

(The details of this calculation are in Appendix D.)

The two lines destined to become the paths of the points of contact, henceforth defined as the

lines of action are the bdl rotated + and -α (thereby creating two lines of action) within the

polar plane as described in Phillips (2003), ‘On either side of the bdl and within the polar

plane, set out the two paths of the points of contact Q’. Phillips (2003, p189) calls the angle

between the paths of contact and the bdl the angle of obliquity, however, the angle is the

pressure angle in planar terminology hence the symbol α.

Rotations about the normal to the polar plane (v12) are considerably more complicated for the

plain polyangular case than for the equiangular case as the polar plane, defined by v12, no

longer lies parallel to the plane defined by the y-axis. The rotation therefore comprises a

series of rotations about the z, x and y axis respectively where these rotations are defined by

the x, y and z components of v12.

)cos)(sin(tan.2

),cos.sin.(tan.2]cossin)[(),cos)(sin.(ˆ

12122

212121212

δδ

δδδδδδ

zf

yxfzz

Fzhr

FFrhzFaFzhaldb

−−

−−−−−−=

Σ

Σ

28

If:

• ρvz is the angle of inclination of the bdl to the yz-plane, where ρvz can be expressed as

a function of vi, vj and vk as shown in Figure 41 in Appendix D.

• ρvx is the angle of inclination of the bdl to the xz plane, where ρvx can be expressed as

a function of vi, vj and vk as shown in Figure 41.

α is the pressure angle

• the lines of action have the symbol loa, the components of the lines of action are

kji loaloaloa

…then the equation of the lines of action are:

29

kji loaloaloa = ×ldb ˆ

+−+−−++−+++−+−−+−++

vxvxvxvzvzvxvxvzvzvx

vzvxvzvxvxvxvzvzvxvxvzvz

vzvxvzvxvxvzvzvxvxvxvzvz

ραραρραρραρραρρρααρρραρρραρρααρρρρααρρραρρρραραρραρ

22

22222

22222

coscossin))1(cossincossinsin(cos))1(cossinsinsin(coscos

sinsin)1(cossin(coscos)cossin(coscoscossinsinsin)cos1(coscossin

)sinsin)1(cossin(sincos)cos1(cossincossinsin)sincos(cossincoscos

(16)

30

The lines of action are now fully defined as functions of the four fundamental design

parameters:

• the shaft angle (Σ),

• the transmission ratio (u),

• the centre distance (a) and

• the distance from the Centre Distance Line (CDL) to where meshing is to occur (rf).

3.3 The relationship between the Lines of Action and the Gear Axes

Figure 5 shows the relationship between the Best Drive Line and each Line of Action i.e. the

best drive line and each line of action passing through Qx.

Figure 5: The architecture of spatial involute gearing showing the Lines of Action

To determine the surface entities that will define the gear bodies the relationships between the

LOA and the Gear Axes must be known. These relationships are defined by the distance (rb)

between the Lines of Action and the Gear Axes and the angle (γb) the lines of action subtend

to the gear axes. For Gear Axis 1,

31

(17)

Similarly, for Gear Axis 2,

(18)

Rotating the radius, rb, around Gear Axis 1 and Gear Axis 2 generates two base circles for

each of the Lines of Action. The resulting base circles generated from Line of Action 2 were

also shown in Figure 5. The base circles generated from Line of Action 1 have been omitted

for clarity. The centre of the base circles lie t*1n and t

*2n along their respective Gear Axes

where:

t*1n represents the distance between the intersection of the CDL and Gear Axis 1 and the

centre of base circle 1n

t*2n represents the distance between the intersection of the CDL and Gear Axis 1 and the

centre of base circle 2n

The angle the Lines of Action subtend to Gear Axis 1 is given by:

(19)

(The detailed algebraic manipulations used to derive these equations are in Appendix D.) To

calculate the transmission ratio, equation (2) requires the sine of γb rather than the cosine of

γb. In the above equation, there is enough information to determine sinγb. As cosine is the

222

2

2222

'

',1

)cossin(

)cossin(sin

1

2

ΣΣ

ΣΣΣ

⋅+⋅+

⋅+⋅+⋅⋅

+=

ijk

ija

kf

d

d

nb

loaloaloa

loaloaloarr

εε

222

2

ˆ

2222

'

,2

)cossin(

)cossin(sin

1

2

ΣΣ

ΣΣΣ

⋅−⋅+

⋅−⋅−⋅⋅

+−

=ijk

ija

kf

d

nb

loaloaloa

loaloaloarr

ε

2

ˆ

2

ˆ

2

ˆ

22

,1

cossincos

kji

ji

nb

loaloaloa

loaloa

++

⋅+⋅−=

ΣΣ

γ

32

adjacent side over the hypotenuse of a right triangle, the equation can be verified graphically

as in Figure 6.

Figure 6: The cosine of cosγb

Using the theorem of Pythagoras, PR can be shown to be

2222 )sincos( kji loaloaloa +⋅+⋅ ΣΣ from which:

(20)

Similarly the angle each Line of Action subtends to Gear Axis 2 is given by:

(21)

and, again this can be rearranged to produce:

222

22

,2

cossincos

kji

ji

nb

loaloaloa

loaloa

++

⋅+⋅=

ΣΣ

γ

222

2222

,1

)sincos(sin

kji

kji

nb

loaloaloa

loaloaloa

++

+⋅+⋅=

ΣΣ

γ

γb1,n

222

kji loaloaloa ++

22 cossin ΣΣ ⋅+⋅− ji loaloa

P Q

R

33

(22)

Substituting and simplifying, (2) becomes

(23)

3.4 A special case of the plain polyangular – equiangular gearing

As stated in Table 3, for the equiangular case, Qx is on the Transversal and is given the

designation E. The location of E is determined by letting t4 = 1/(1+u). The coordinates of E

are:

E

)1(

)1(.

2

)1(

)1(tan. 2

+−

−=

=+−

= Σ

u

uaz

ry

u

urx

f

f

This result can be compared to the fundamental kinematic equations. Consider the equation of

the location of the pitch line, (5) and let Σ equal zero, which is equivalent to assuming a

planar set then (5) becomes:

)1(2

)1(

)1)(1(2

)1)(1(

12

12

−+−

=

−−−+−

=

u

uaz

uu

uuaz

This is almost the same as the above equation for zE. The difference, of course, lies in the fact

that the sense of u changed from the kinematic case to the architectural case in Killeen (1996).

In the kinematic case u = -u however in the architectural case u = u. Making this

substitution produces:

222

2222

,2

)sincos(sin

kji

kji

nb

loaloaloa

loaloaloa

++

+⋅−⋅=

ΣΣ

γ

)cossin(sin

)cossin(sin(

sin

sin

2222

2222

22

11

ΣΣΣ

ΣΣΣ

⋅−⋅+⋅⋅−

⋅+⋅+⋅⋅=

⋅⋅

ija

kf

ija

kf

bb

bb

loaloaloar

loaloaloar

r

r εγγ

34

)1(2

)1(12 +

−−=

u

uaz

This equation can, in turn, be back substituted into the z component of the Transversal to

determine t4 and t4 subsequently used to determine xTrans and yTrans. This process, of course,

produces the same result as above.

With the benefit of hindsight, a number of aspects of the characteristics of the Equiangular

gear set investigated in Killeen (1996) have become clearer. This is particularly evident in the

area of mathematical proofs. The Lines of Action which are destined to become the

generators of the surface of the hyperboloid, as opposed to the generators of the base involute

helicoid, are inclined at γb to the Gear Axes, as illustrated in Figure 5. Killeen (1996) then

goes on to state the value of the cosγb to be:

(24)

(25)

Herein lies the problem. The transmission ratio of the RSSR mechanism equivalent to the

gear bodies, given by (2), is expressed in terms of sinγ whilst the equations above are

expressed in terms of cosγ. This is not a problem for numerical analysis, however

completeness requires an explicit value of sinγ.

The simplest way to determine sinγ, is to construct a right triangle such that in accordance

with cos γ = adjacent/hypotenuse, the numerator in both equations (24) and (25) is the

adjacent and the denominator is the hypotenuse as shown below. Determining sinγ can then

be determined by calculating the opposite side and the ratio opposite/hypotenuse.

2222

22

,2

2222

22

,1

tan4

)tansin2cos(sincos

tan4

)tansin2cos(sincos

Σ

ΣΣ

Σ

ΣΣ

+

+=

+

+−=

ra

ra

ra

ra

nb

nb

ααγ

ααγ

35

Figure 7: The right triangle of cosγb

From the theorem of Pythagoras, PR2 = QR

2-PQ

2, the opposite side can be determined from:

)sinsin1(tan4tansincossin4)sincos1(

)tansin2cos(sintan4

222

222

222

2222

222

22

2222

ΣΣΣΣΣ

ΣΣΣ

−+−−=

+−+=

αααα

αα

rraa

raraPR

Consider, as Fred Sticher did, the following substitution,

)-osA cos ϕαγ (= c

where A and φ are arbitrary constants, then

)sinsincosA(cos cos ϕαϕαγ +=

Equating coefficients of cosα and sinα with those in equations (24) and (25)

Summing the squares and adding produces

Α=sinΣ/2

and dividing Asinϕ by Acosϕ produces

( )

( ) 222

2

22

222

2

22ra

tan

sintAsin

tan

sinAcos

Σ

ΣΣ

Σ

Σ

+=

+=

ra

ra

anϕ

ϕ

√(a2+4r

2tan

2Σ/2)

sinΣ/2(a.cosα+2r.sinα.tan

Σ/2)

γb

P Q

R

36

(26)

therefore

where

The hypotenuse of the aforementioned triangle is then equal to one and:

(27)

where again

Sticher’s substitution considerably simplifies the previous equations for γ and, as Table 4

shows, produces the same result. Consider the case of Gear Axis 2 and Line of Action 2 with

parameter values of Σ=50o, a=80mm, r=140mm, u=0.6 and α=-20

o,

Table 4: Comparison of the results of the complex and simple expressions for γb21

Equation (24) (27))

cosγb21 √(1− cos2γb21) (=sinγb21) sinγb21

0.08422928 0.9964464… 0.9964462…

The transmission/gear ratio resulting from the equivalent RSSR mechanism, from equation

(3), can therefore be stated in terms of the fundamental design quantities, r, a and Σ by

substituting into (2). It is:

)(cossin1sin 22

2 ϕαγ −−= Σ

a

r 2tan2tan

Σ=ϕ

)cos(sincos 2 ϕαγ −= Σ

a

r f 2tan2tan

Σ

=ϕ

a

r

f 2tan2tan =ϕ

37

(28)

where:

rb2n and rb1n are the lengths of the RS linkages 1 and 2 (from Figure 2) respectively

φ is given by(26)

Equation (27) provides an equivalent expression to equations (24) and (25) in terms of sinγ

rather than cosγ. An explicit expression is therefore available for the transmission ratio, i, by

substituting equation (27) into equation (2). This provides i as a function of fundamental

design parameters such as the perpendicular distance between the axes, the angle between the

axes i.e. it algebraically links the transmission ratio to the fundamental design parameters.

3.5 The resulting entities generated from the plain polyangular architecture

Returning to the plain polyangular case, the Gear Axes and the Lines of Action define three

geometric entities, the base helix, the base hyperboloid and the involute helicoid. The latter

two define a fourth geometric entity, a line entity representing the line of intersection of the

hyperboloid and helicoid, the slip track. The following section contains an analysis of the key

characteristics of these entities.

The base helix can be expressed as a function of the base circle radii (rb) and the angle of

inclination (γb) of the generator to the axis. Killeen (1996) showed the equations of the base

helix expressed as functions of rb and γb, are:

(29)

bbBH

bBH

bBH

trz

try

trx

γtan

sin

cos

7

7

7

===

)(cossin1

)(cossin1

22

2

1

22

2

2

φα

φα

−−

−−=

Σ

Σ

nb

nb

r

ru

38

Killeen (1996) also showed the two fundamental surfaces, the hyperboloid and the involute

helicoid, with their axes lying along the z axis and their throats in the x-y plane, defined as a

function of rb and γb:

(30)

and:

(31)

respectively, and that the equation of the line representing the intersection of these two

surfaces, the slip track, is:

(32)

where:

rb = the minimum distance between the Line of Action and the Gear Axis and is given by

(17) and (18)

γb= the angle the Line of Action subtends to the Gear Axis given by (19) and (21)

µ= a parameter describing the distance along the generator of the involute helicoid where

the point generated by the equation lies

t7= a parameter describing the rotation of the tangent to the helix or the involute roll

angle; a parameter describing the angle of rotation of the point on the slip track about

the z axis

By way of example, Figure 8 provides a three dimensional view of a general slip track (the

perpendiculars to the z axis shown in light blue provide a frame of reference) and Figure 9

illustrates the behaviour of the slip track by tracing the coordinates for various values of t7.

bb zryx γ2222 tan+=+

bbbIH

bbIH

bbIH

trz

ttry

ttrx

γµγγµγµ

sintan

coscossin

sincoscos

7

77

77

+=+=−=

b

b

ST

bb

b

bST

bb

b

bST

trz

ttr

try

ttr

trx

γ

γγ

γγ

2tan2

costan2tan2

sin

sintan2tan2

cos

7

7

7

7

7

7

7

=

+=

−=

39

Figure 8: An illustration of the slip track for the case where rb = 51.58 and γb = 1.909 rad

Figure 9 : Graph of the coordinates of the slip track

Graph of coordinates of a general slip track(r

b=51.58 γb

=1.909 rad)

-250

-200

-150

-100

-50

0

50

100

150

200

250

-3.0

-2.6

-2.2

-1.8

-1.4

-1.0

-0.6

-0.2

0.2

0.6

1.0

1.4

1.8

2.2

2.6

3.0

t7 (rads)

x y z Dist. f rom z axis

Slip track

Perpendicular

from the z

axis to the

slip track

40

It is extremely important to note that defining the surface and line entities in this way means

the frame of reference has changed. Previously the axes of the gear bodies were collinear with

the Gear Axes and the centre of the base circle lay on the respective Gear Axis thereby

defining the origin of the line/surface entity. Equations (29) to (32) describe entities with their

axes collinear with the z-axis, the plane defined by the base circle lies on the x-y axis and the

origin of the entities, defined by the centre of the base circle, lies on (0,0,0). This is clearly

illustrated by Figure 9 as z varies linearly with t7.

Given the location of F (Qx) with respect to the origin of the line/surface entities must stay the

same, the location of F with respect to the origin of the entities described by (29) to (32), is

given by one of two equations depending on which Gear Axis is under consideration.

For Gear Axis 1, the location of F is given by:

(33)

For Gear Axis 2, the location of F is given by:

(34)

where:

t*1/2n = the distance from the CDL to the centre of the throat circle of base hyperboloid

1/2n measured along GA1/2

−−

−

+−

−+− ΣΣ

ΣΣ

Σ

10

0010

0cos0sin

0sin0cos

1)1(

)1(.

2)1(

)1(tan.

1*

2

22

22

'

'

'

'2

na

d

d

f

d

d

f

t

arr

εε

εε

−−−

−

+−

−+− ΣΣ

ΣΣ

Σ

10

0010

0cos0sin

0sin0cos

1)1(

)1(.

2)1(

)1(tan.

2*

2

22

22

'

'

'

'2

na

d

d

f

d

d

f

t

arr

εε

εε

41

3.6 Algebraic proof of the relationships between the slip track and the base helix

Given equations (17) to (32), conclusive mathematical proofs can be determined for what

have hitherto been geometrically ‘obvious’. The following section sets out to investigate,

from a mathematical perspective, the relationships between key aspects of the relationships

between the base helix and the slip track. In doing so, it will validate Phillips’ (2002, pp72-

74) geometrically derived formulations these relationships.

The angle between the base helix and the slip track is:

STBH

STBHSTBH

trtr

trtr

)(')('

)(')('cos

•=−ς

where, given a general line in space defined by r(t), the tangent to the line at the point

determined by t, is r’(t) and similarly the second derivative is r”(t) i.e. if:

r(t) = x(t)i + y(t)j + z(t)k

then r(t) = x(t)i + y(t)j + z(t)k

and r(t) = x (t)i+y (t)j+z(t)k

Applying this to the base helix and slip track:

222222''''''

''''''cos

STSTSTBHBHBH

STSTSTBHBHBH

STBH

zyxzyx

zyxzyx

++++=−ς

where:

bbBH

bBH

bBH

rz

try

trx

γtan'

cos'

sin'

7

7

==

−=

and

(35)

b

b

ST

bb

b

bST

bb

b

bST

rz

tttr

try

tttr

trx

γ

γγ

γγ

2tan2

'

)sin(cos2tantan2

cos'

)cos(sin2tantan2

sin'

7777

7777

=

−+=

+−−=

42

Substituting and simplifying produces

(36)

Figure 10 is a graph of (36) for the aforementioned numerical example. It clearly shows a

localised maximum of π or 1800 at t7 = 0. The slip track and core helix are tangential to one

another at t7=0.

Figure 10: The angle between the slip track and the base helix for the general plain polyangular case

The key outcomes of (36) are:

• The graph intersects the y-axis at 3.14159… or π radians i.e. the slip track is parallel

to the base helix at t7=0. This conforms to Phillips (2003, p73) and is equivalent to the

conclusion that at Ω=0 [t7=0], the slip track and the core helix are tangential with one

another.

ANGLE BETWEEN BASE HELIX AND SLIP TRACK

0

0.5

1

1.5

2

2.5

3

3.5

-4 -3 -2 -1 0 1 2 3 4 5

ROTATION, (t, rads)

AN

GL

E,

(ra

d)

Gear set 11

Gear set 21

[ ])1tan(2tan)2tantan1()tan1(

2tantan1cos

22

7

24

122

12 ++++

+=−

bbbbb

bb

STBH

t γγγγγ

γγς

43

• The denominator approaches ∞ as t7 approaches ∞ i.e. the further the point under

consideration is from the throat circle, the closer the angle between the slip track and

the base helix is to π/2 rads.

3.7 The mathematics of some fundamental planes

Given the previously developed mathematical surfaces, now fully defined, it is possible to

explicitly determine the behaviour of various planes lying on these surfaces. Phillips (2003,

p75) defines a triad of planes defined by the involute helicoid called the curvature triad.

These planes, illustrated in Phillips’ figure 3.05 and 3.07, are the:

1. plane of adjacency, a tangent plane to the involute which contains one of the straight

lines that defines the surface of the involute

2. plane normal to the said ruling [of the involute helicoid]

3. plane containing the said ruling [of the involute helicoid] and the relevant ruling upon

the hyperboloid’

Figure 11 shows two points selected at random on a general involute helicoid.

Figure 11: Two points on a general involute helicoid viewed along the axis of the base helix

Base circle

Point 1

Point 2

Generator 1

Generator 2

Perpendicular 1

Perpendicular 2

44

The perpendicular is perpendicular to the generator at the selected point on the generator and

passes through the axis of the involute helicoid. The three aforementioned planes (1, 2 and 3)

can be constructed at the point of intersection of the generator of the involute helicoid and the

perpendicular. Figure 12 shows the planes (represented as triangles) and the normals to the

said planes.

Figure 12: The triad of planes defined by the involute helicoid

The plane of adjacency, plane 1, can be envisaged as the plane defining the face of a face-

milling cutter as it cuts the tooth surface. If the face of the face milling cutter was to have the

same orientation as the plane of the adjacency as it moved over a piece of metal, a gear tooth

would be formed.

The plane of adjacency is tangential to the surface of the involute helicoid so the orientation

of the face of the face milling cutter, as it moves over the surface, can be determined. If there

is a surface represented by the parametric equations r(u,v) and the partial derivatives

Perpendicular 1

Generator 1

Point 1

Normal to

plane 3

Normal to

plane 2 Normal to

plane 1

45

ur

∂∂ and

vr

∂∂ are both continuous over the region and non zero at any point (u0, v0), then

ur

∂∂ is the tangent to the u curve and

vr

∂∂ is the tangent to the v curve at the point (u0, v0),

0≠∂∂×∂

∂v

ru

r and

(37)

(Anton 1995 p795)

where

vz

vy

vx

uz

uy

ux

kji

v

r

u

r

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂=

∂∂

×∂∂

ˆˆˆ

and where the vector, n, is the normal defining the tangent plane at the point r(u0, v0). n is a

unit vector perpendicular to both u

r∂

∂ and v

r∂

∂ as shown in Figure 13.

Figure 13: The tangent plane of r at (u0, v0) represented by its normal, n.

Killeen (1996) demonstrated that the involute helicoid can be defined as:

r(µ, t7) = (rb .cost7 – µ .cosγb.sint7)i + (rb .sint7 + µ .cosγb.cost7)j + (rb .t7 . tanγb + µ .sinγb)k

(38)

v

r

u

r

v

r

u

r

n

∂∂

×∂∂

∂∂

×∂∂

=ˆ

(Anton 1995 p796)

46

The partial differentials are therefore:

(39)

(40)

Substituting these into (37) and evaluating produces:

kjttr

t

rbbbbb

ˆ)cos(ˆ)cossincos(i)sinsincos( 2

77

7

γµγγµγγµµ

−++−=∂∂

×∂∂

from which

b

r

t

r γµµ

cos7

=∂∂

×∂∂

and

(41)

The equation of the plane defined by the normal and a point in the plane, given by (41) is:

bbbbb

bbbbbbb

bbbbbb

trzytxt

trttrt

ttrtzytxt

γγγγγµγγγµγ

γµγγγγ

sincoscossinsinsin-

)sintan(cos)coscossin(cossin

)sincoscos(sinsin-coscossinsinsin-

777

7777

77777

−=−++−+

+−=−+

A point on the base helix lies in the above plane, therefore substituting the equation of the

base helix into the above equation and solving provides a cross-check.

bb

bbbbbbbbb

trRHS

trttrttrt

γγγγγγ

sin

sintancossincossincossinsin-LHS

7

777777

−=

−=−+=

bb

bb

bb

rt

z

ttrt

y

ttrt

x

γ

γµ

γµ

tan

sincoscos

coscossin

7

77

7

77

7

=∂∂

−=∂∂

−−=∂∂

b

b

b

z

ty

tx

γµ

γµ

γµ

sin

coscos

sincos

7

7

=∂∂

=∂∂

−=∂∂

kjtitn

kjt

it

n

bbb

b

b

b

bb

b

bb

ˆcosˆcossinˆsinsin-ˆ

ˆcos

cosˆcos

cossincos-ˆcos

sinsincos-ˆ

77

2

77

γγγ

γµγ

µγµγγµ

γµγγµ

−+=

−+=

47

∴ the equation of the plane is verified. Figure 13 shows that the vectors defining 2 and 3

above are, in fact, developed in the process of developing (41). Planes 2 and 3 are defined,

respectively, by calculating the cross product of (39) and (40) (restated here for convenience):

b

b

b

z

ty

tx

γµ

γµ

γµ

sin

coscos

sincos

7

7

=∂∂

=∂∂

−=∂∂

bb

bb

bb

rt

z

ttrt

y

ttrt

x

γ

γµ

γµ

tan

sincoscos

coscossin

7

77

7

77

7

=∂∂

−=∂∂

−−=∂∂

Phillips (2003, p75) then defines a fourth plane inclined at to the aforementioned triad. The

fourth plane is the rectifying plane of the Tangent vector-Normal vector-Binormal vector

(TNB) triad, defined by the normal vector to the slip track.

The normal vector and therefore the rectifying plane can be determined by determining the

tangent vector and the normal vector in turn. The unit tangent vector, T(t), to the slip track can

be determined from:

(42)

(Anton 1995)

where:

T(t) defines the normal plane to the line at the point under consideration

r(t) is given by (35)

)(' tr is the magnitude of r(t)

All of the components are theoretically calculable, however, determining )(' tr where r’(t) is

given by (35), will be a long and cumbersome process. Consequently the risk of making a

)('

)(')(

tr

trtT =

48

mistake will be high. In the first instance it is more practical to graph the numerical results.

Figure 14 shows the result of graphing the components of equation (42).

Figure 14: Graph of the unit tangent vector components

Figure 14 illustrates the characteristics of the tangent to the slip track. Phillips (2003, p351)

nominates the tangent as one of the lines defining the rack triad. The rack triad, he goes onto

explain, is ‘intimately related to the cutting mechanics on both sides of the phantom rack.’

The principal unit normal vector is determined from:

(43)

(Anton 1995)

where:

N(t) defines the rectifying plane

T(t) is the differential of equation (42)

)(' tT is the magnitude of T(t)

GRAPH OF UNIT TANGENT VECTOR VALUES FOR ST11

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4 5

Rotation (t7), RADS

x

y

z

)('

)(')(

tT

tTtN =

49

Calculating T’(t)

(44)

Again, it is theoretically possible to calculate )(' tT however it is more practical to simply

plot the numerical results. Figure 15 is a graph of equation (43).

Figure 15: Graph of the unit normal vector of slip track 11

The binormal vector determined from:

(45)

(Anton 1995)

B(t) defines the osculating plane

T(t) is the unit tangent vector given by (42)

N(t) is the normal vector given by (43)

GRAPH OF UNIT NORMAL VECTOR FOR ST11

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4 5

Rotation (t7), RADS

x

y

z

0"

2tantan)cossin2(2

sin"

2tantan)sincos2(2

cos"

7777

7777

=

+−−=

−−−=

ST

bb

b

bST

bb

b

bST

z

tttr

try

tttr

trx

γγ

γγ

)()()( tNtTtB ×=

50

Figure 16: A graph of the binormal vector for slip track 11

Therefore there are a total of six planes defined at any point on the slip track, three defined by

the involute helicoid passing through the slip track and three defined by the slip track itself.

Determining the corresponding equations of the planes at any given point on the involute

helicoid/slip track, however, is presently not possible. Equations (39) and (40) have µ as an

unknown.

Equating the ‘z’ components of the equation of the involute helicoid and the slip track,

equations (31) and (32), respectively and solving for µ produces:

(46)

The results of equation (46) can be back substituted into equations (39) and (40) to produce

the Phillips’ (2002, p75) curvature triad. For any given t7, the six planes are, therefore, fully

defined.

GRAPH OF BINORMAL VECTOR FOR ST11

-1.5

-1

-0.5

0

0.5

1

1.5

-4 -3 -2 -1 0 1 2 3 4 5

Rotation (t7), (RADS)

x

y

z

b

bbbrtγ

γγµ

sin

tan22tan

27

−=

51

3.8 Discussion

The key objectives of this chapter were to develop mathematical expressions for the plain

polyangular gear architecture and surface geometry of Phillips (2003, pp237-284) using basic

engineering mathematics. As equations (4) to (36) show, these objectives have been met.

Key results of the plain polyangular model included:

• the location of F is given by:

)1(

)1(.

2

)1(

)1(tan.

'

'

'

'2

+−

−=

=+−

= Σ

d

d

d

d

az

ry

rx

εε

εε

• the relative velocity vector is given by:

δδδδδδ cossin,cos)(sin,sin)(cosˆ1212121212 FFFF yxzzhzzhv −−−−+=

• the base radii of the gears are given by:

2ˆ2ˆ2

2

ˆ

ˆ2ˆ22ˆ2

'

'

,1

)cos(sin

)cos(sinsin

1

2

ijk

ija

k

d

d

nb

loaloaloa

loaloaloarr

ΣΣ

ΣΣΣ

++

++

+=

εε

and

2ˆ2ˆ2

2

ˆ

ˆ2ˆ22ˆ2

'

,2

)cos(sin

)cos(sinsin

1

2

ijk

ija

k

d

nb

loaloaloa

loaloaloarr

ΣΣ

ΣΣΣ

−+

−−

+−

=ε

where i

loa ˆ , j

loa ˆ and k

loa ˆ can be determined, using Equations (15) and (16), from the key

functional performance values of:

• centre distance

• shaft angle

• transmission ratio

• meshing radius.

The lead angles of the gears are given by:

52

2

ˆ

2

ˆ

2

ˆ

2

ˆ

22ˆ2ˆ

,1

)sincos(sin

kji

kji

nb

loaloaloa

loaloaloa

++

++=

ΣΣ

γ

and

2

ˆ

2

ˆ

2

ˆ

2

ˆ

22ˆ2ˆ

,2

)sincos(sin

kji

kji

nb

loaloaloa

loaloaloa

++

+−=

ΣΣ

γ

Equations (42) to (46) extended the applicability of the work by providing a preliminary

investigation into the orientations of key planes. Initial indications are that these planes will

be central the machining of plain polyangular involute gearing.

53

CHAPTER 4 GEAR INSPECTION

One of the objectives of this thesis is to ‘compare the theoretical transmission ratio to the

actual transmission ratio for a gear body pair designed using Phillips’ (2002) theory and

process’. To achieve this objective, a structured process needs to be followed. This chapter

provides an overview of the principles and methods of gear inspection. It is not intended to be

a detailed investigation of the respective types of gear inspection. Rather its aim is to provide

a background for the chapters to follow.

There are two broad categories of gear inspection:

1. Functional inspection and

2. Analytical inspection

The remainder of this chapter contains an overview of the principles of functional and

analytical inspection as they apply to gears generally.

4.1 Functional inspection

Functional inspection was the earliest form of testing (Townsend, Broglie, Smith 1992) and

involves the rolling of two gears, one of which may be a master, to determine whether the

gear is fulfilling its primary and secondary functions. Functional inspection in this thesis

focuses on the fulfilment of the primary function. The primary function of a gear is to

transmit motion from one axle to another axle at a predefined ratio i.e. to produce relative

rotational ‘speeds’ between the gears equal to the transmission ratio. Secondary functions

such as, to have variations in transmission ratio of less than 1% or to produce vibration levels

less than or equal to ‘x’Gs or composite variations of less than or equal to ‘y’ will be

investigated in less detail.

54

The basic idea behind functional inspection is, assuming the pinion is the driving gear in a

gear pair, that:

(47)

where:

i = the transmission ratio or the ratio of the angular speed of the first driving gear to that of

the last driven gear and can be calculated from ω2/ω1

u = the gear ratio or the ratio of the number of teeth on the wheel to that of the pinion and

can be calculated from z1/z2

which, when substituted, produces

(1)

where (stated again here for convenience):

ω1 =

ω2 =

z1 =

z2 =

the angular speed of the pinion

the angular speed of the wheel

the number of teeth on the pinion

the number of teeth on the wheel

By measuring this value using mechanical or electrical means, the theoretical gear ratio can

be compared with the actual transmission ratio graphically to determine the transmission

error. Houser (1992) points out that there are two forms of transmission error:

• Loaded Transmission Error (LTE) and Manufactured Transmission Error (MTE).

• LTE accounts for deflection of the teeth due to loading and is influenced by the

following factors:

o Mean tooth compliance, which is a constant component affected by profile

modifications

ui

1=

2

1

1

2

z

z=

ωω

55

o a combination of mesh stiffness variation and MTE producing a time varying

error.

• MTE is determined by testing an unloaded gear pair in a single flank gear tester. The

major influences on MTE are profile inaccuracies and gear tooth runout. Perfect

involute profiles produce little or very small but constant transmission error traces.

4.2 Analytical inspection

Analytical inspection involves the measurement of dimensional characteristics such as profile

geometry, tooth thickness, lead and spacing and it determines whether the product of the

manufacturing process conforms to the theoretical requirements. Analytical inspection:

• looks for the causes of functional failures or failure modes

• leads to modification of the manufacturing process to maintain tooth accuracy or

ensure the dimensions of the gear fall within a pre-defined error band.

The first three criteria for satisfactory operation of gears (Degarmo, Temple Black, Kohser

1990) relate to aspects of gearing determined by analytical inspection. These criteria are:

• The actual tooth profile must be the same as the theoretical tooth profile.

• The tooth spacing must be uniform and correct.

• The actual and theoretical pitch circles must be coincident and be concentric with the

axis of rotation of the gear

• The actual and theoretical tooth surfaces must be smooth and sufficiently hard to resist

wear and prevent noisy operation.

• Adequate shafts and bearings must be provided so that the desired centre distances are

retained under operational loads.

Analytical inspection would, for example, be applicable to the gear bodies shown in Figure

17. Various characteristics of the gear body could be tested including involute measurement

56

and lead measurement. Other characteristics, including tooth spacing, index measurement,

base tangent measurement and chordal tooth thickness require the use of a gear wheel.

Although analytical inspection would be of some value at this stage of the gear development,

further limitations are imposed on this form of inspection by errors generated in the

manufacturing process.

Improvements in analytical gear inspection have been facilitated by the development of

computer hardware and software specifically designed for the task. There is a wide variety of

gear inspection equipment and technical data available and measurements with accuracy of

less than 1µm are possible (Townsend, Broglie, Smith 1992). Similarly, developments in

hardware and software have made gear inspection equipment more flexible. Machines are

capable of measuring both different types of gears and different aspects of the same gear.

Figure 17: A spatial involute gear body

The profile generated by the manufacturing process, the actual profile, potentially differs from

the design or theoretical profile in a number of ways. The first of these possible differences is

the variation of the tooth’s normal profile. Under close inspection, using a stylus-based

roughness gauge for example, the profile may actually look like the one shown in the figure

57

below (Appendix E contains an explanation of surface roughness gauges). Note the variation

between the normal and theoretical profile overlay. This is further illustrated by graphing the

deviation measured against a length proportional to the position of the point of measurement

on the pressure line. This graph is also shown in Figure 18.

Figure 18: The real tooth profile and the graph of the profile

Figure 18 also clearly shows a series of lines on the flank. These lines are troughs and peaks

created by the manufacturing process. The error measured by the roughness gauge therefore

originates from two sources:

1. The manufacturing process itself, and

2. The measurement process. The ball on the gauge tends to jam in the troughs.

All of the above inspections provide a measure of tolerance. Tolerance is the magnitude of

“permissible variation of a dimension or control criterion for the specified value” (Smith

1992). The most significant factor affecting the selection of tolerances is that of function.

Tolerance selection must be performed in light of its effect on the transmission ratio and the

criticality of transmission ratio to the final application.

The selection of tolerances is also influenced by non-functional requirements such as cost of

manufacturing. Cost increases with increasing tightness on tolerances. To minimise the cost,

(Bonfiglioli 1995)

58

the tolerance should be set as low as is necessary to achieve the functional performance

criteria. On the other hand, an example of a non-functional manufacturing tolerance in Smith

(1992) is the tolerance placed on lateral runout. Although this is not critical to the gears’

performance per se, it may be necessary in order to obtain “functional tolerance on total

composite variation”. Some other non-functional requirements in the manufacture of gears

include tolerances applied to gear wheel mountings such as keyways, setscrews and boltholes.

59

CHAPTER 5 FUNCTIONAL INSPECTION OF EQUIANGULAR

SPATIAL INVOLUTE GEARS USING ENCODERS

This chapter applies the principles of functional inspection to the equiangular spatial involute

gear body pair produced by the author (Killeen 1996). Following a brief introduction to the

theoretical background of the gear body design and its implications on transmission ratio, this

chapter discusses the fundamental design principles upon which the gear body test rig design

was based and goes on to describe a functional test of the gear body pair using optical

encoders. The results of that test are then discussed in detail with particular attention paid to

the causes of errors encountered within the test.

5.1 Background

A device that provides a means of measuring the transmission ratio is the single flank gear

tester. The single flank gear tester, illustrated in Figure 19, uses optical encoders to measure

the angular velocity of the driven gear whilst the axis of the driving gear rotates at a constant

angular velocity. The principle is applicable to not only spur and helical gears but also spiral

bevel, worm and hypoid gears.

The remainder of this chapter investigates the claims of constancy of transmission ratio, i, for

the ‘as manufactured’ gear bodies and the impact of variations in a and Σ on the transmission

ratio using a single flank gear test rig designed to accommodate the spatial involute set. The

‘as manufactured’ gear bodies were previously constructed in Killeen (1996) with the basic

dimensions shown in Table 30 (in Appendix C) however principles identical to those in

Section 3.4 were applied.

60

Figure 19: A schematic of a single flank gear tester

5.2 Aim

To investigate the hypothesis of Phillips that spatial involute gearing produces constancy of

transmission ratio and that this constancy of transmission ratio is independent of small

variations in fundamental quantities used to design the gearing.

5.3 Method

To test the claims of constancy of transmission ratio, i, a test rig is required. The design of

this test rig is governed by the requirements of the rig. These requirements or the functional

specification and an explanation of the design methodology are in Appendix F. The former is

summarised below. The test rig shall:

• mount the gears in their meshing position and allow them to rotate about their

Instantaneous Screw Axes over the entire range of useable flank.

• facilitate the comparison of rotational displacements.

(Townsend, Broglie, Smith 1992 p23.12)

61

The aforementioned design methodology resulted in the mechanical hardware design

illustrated in Figure 20. Appendix G also contains material from the Hengstler encoders

catalogue. This material explains the principle of operation of the optical encoders.

The mechanical hardware, however, is only part of what is required for the acquisition and

processing of data. To determine the transmission ratio, the outputs from the optical encoders

must be collected and compared.

To achieve this the outputs from the encoders were acquired using a:

• Labview PC+ Data Acquisition Card (DAC)

• CB 50 I/O connector-block with 1.0m of cable (Part No-776164-02).

The outputs were then graphed and written to a spreadsheet using a programme written in

Labview Student Edition. The arrangement is shown in Figure 20.

Figure 20: A picture of the electrical functional testing arrangement

This arrangement was set up using the following method.

62

Testing of the optical encoders: A power supply was used to provide 5V to the input of the

optical encoders. The output of the encoders was then analysed using a Cathode Ray

Oscilloscope (CRO) to ensure all output channels of the encoders were producing 5V step-

change as the shaft rotated.

Installation and configuration of the DAC: The card was configured, using the dip

switches, as indicated in the table below and installed in an ISA slot of a 486 DX2/66

Personal Computer (PC) with 8MB of Random Access Memory running DOS 6.22 and

Windows 3.11.

Table 5: Settings for the Labview PC+ DAC

Labview PC+

board

Settings Hardware implementation

Base I/O address Hex 260 (factory) A9:1; A8:0; A7:0; A6: 1; A5;1

DMA channel Channel 3 (factory) W6 DACK 3

Interrupt level IRQ 5 (factory) W5 Row 5

Note: The above configuration is as per the factory settings listed in National Instruments

(1992)

The “NI-DAQ for PC compatibles for Windows” driver for the Labview PC+ card was then

installed in the PC. The driver changed both the Autoexec.bat and Windows System.ini file to

automatically load the driver for the card every time the computer booted and Windows

started.

After running the NIDAQCONF utility to confirm the Labview PC+ device number and that

the hardware settings matched the software settings, the computer was rebooted for all

settings to take effect. Individual data lines of the card were then tested to confirm the

operation of the hardware. The testing process comprised applying 5V from terminal 49 of the

63

Labview PC+ to various digital data lines for the port under test whilst viewing the status

indicator in the configuration programme.

The process identified a potential problem with the Labview PC+. Applying 5V to a digital

data line caused a ‘high’ signal on adjacent digital data lines. To ensure only those lines that

had signals applied to them would go high, all but the used digital lines were connected to

neutral.

Connection of the encoders to the Labview PC+ card: The outputs of the encoders were

connected to the terminals of the Labview PC+ in the following configuration.

Table 6: Connections between encoders and Labview PC+ card

Gear body Channel Device Port Digital

data line

Labview

PC+

terminal No.

2 (wheel) A 1 0 0 14

2 (wheel) R 1 0 1 15

1 (pinion) A 1 0 6 20

1 (pinion) R 1 0 7 21

Note: Channel N was not recorded as it is a marker pulse used for reference or positioning

purposes.

Supplying power to the circuit: The outputs of a variable power supply were connected to

the 5V and neutral terminals of the encoders. Although it was possible to use the 5V supply

from the Labview PC+, a set-up that has a distinct advantage with regard to eliminating

floating neutrals, no means of switching is available meaning a separate switch is required.

Floating neutrals were eliminated by connecting the neutral of the Labview PC+ board to the

neutral of the power supply. The neutral of the power supply was then connected to the earth

terminal of the power supply thereby ensuring:

• The neutrals were tied together

64

• That neutral was, in fact, zero volts.

Note: The Labview PC+ 1 amp supply was protected by a 1-amp pico-fuse mounted on the

card. Accessing this fuse required removal of the PC’s power supply, cover and the Labview

PC+ card. To prevent the pico-fuse from blowing a fast blow fuse, with a rating of less than 1

amp, was placed in series with the switch when the Labview PC+ was used as the power

supply. Appendix G contains an electrical schematic of the above arrangement.

Installation of Labview and the construction of a Virtual Instrument (VI) for data

acquisition: Following the installation of Labview Student edition on the PC, the front panel

and VI shown in Appendix H were written to acquire the data. The front panel and VI

operated as shown in the following pseudo-code.

While ‘STOP’ is not pushed or timer has not timed out

Read while loop counter

Read clock counts

Read digital data line 0, port 0, device 1

Read digital data line 1, port 0, device 1

Read digital data line 6, port 0, device 1

Read digital data line 7, port 0, device 1

Combine data into single line array

Wait 55ms

Build a multi line array from the single line arrays

Write multi line array to spreadsheet file

With the following conditions

• No modification to the architecture of the gear bodies

65

• The centre distance, a, was reduced by nominally 1.5mm to 78.5mm, and

• 1.5mm nominal packers were placed under the pinion bearing blocks closer to the

CDL,

…data was acquired from the encoders for the following two cases:

• The wheel drove the pinion up

• The pinion drove the wheel down

5.4 Results

The tables on the following pages show the results captured by Labview for the first case of

each of the conditions above. To minimise the impact of inconsistencies in start up and

ending, results are shown for approximately 40 data points either side of the median data

point.

66

Table 7: Results of electronic functional inspection for ‘as designed’ configuration

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

1

Channel

6

Channel

7

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

1

Channel

6

Channel

7

181 9955 0 1 0 1 226 12430 1 0 0 1

182 10010 1 0 0 0 227 12485 0 1 0 0

183 10065 0 1 1 0 228 12540 1 1 1 1

184 10120 1 0 1 1 229 12595 1 1 0 0

185 10175 1 1 0 0 230 12650 1 0 0 1

186 10230 0 1 0 1 231 12705 0 1 0 0

187 10285 1 0 0 0 232 12760 1 0 1 0

188 10340 1 0 1 1 233 12815 0 1 1 1

189 10395 1 1 1 1 234 12870 0 0 0 1

190 10450 1 0 0 0 235 12925 1 0 1 1

191 10505 1 1 0 0 236 12980 0 1 0 1

192 10560 0 0 1 0 237 13035 1 0 0 0

193 10615 1 0 0 1 238 13090 0 1 0 0

194 10670 0 1 0 0 239 13145 1 0 0 0

195 10725 1 0 1 1 240 13200 1 0 0 0

196 10780 0 0 0 0 241 13255 0 0 1 1

197 10835 0 1 0 0 242 13310 1 0 0 0

198 10890 1 0 1 0 243 13365 0 1 1 1

199 10945 0 1 0 0 244 13420 0 1 0 0

200 11000 1 0 1 0 245 13475 1 1 1 1

201 11055 0 1 1 1 246 13530 1 1 0 0

202 11110 1 0 0 1 247 13585 0 0 0 0

203 11165 0 1 0 1 248 13640 1 0 0 1

204 11220 1 0 0 1 249 13695 0 0 1 1

205 11275 0 1 0 1 250 13750 1 1 1 0

206 11330 1 1 1 1 251 13805 0 0 1 1

207 11385 0 1 0 0 252 13860 1 1 0 1

208 11440 1 0 1 0 253 13915 0 0 0 1

209 11495 0 1 1 1 254 13970 0 1 0 1

210 11550 1 1 0 0 255 14025 0 1 1 1

211 11605 0 0 1 0 256 14080 1 1 0 0

212 11660 1 0 0 1 257 14135 0 1 1 1

213 11715 1 0 1 1 258 14190 1 0 0 1

214 11770 1 0 0 0 259 14245 0 1 0 0

215 11825 0 1 0 1 260 14300 1 1 1 1

216 11880 1 0 0 0 261 14355 1 0 0 1

217 11935 0 0 1 1 262 14410 0 1 0 0

218 11990 1 1 0 0 263 14465 1 0 0 1

219 12045 0 1 0 0 264 14520 1 1 0 0

220 12100 1 1 0 0 265 14575 1 0 1 1

221 12155 0 1 1 1 266 14630 0 1 1 1

222 12210 1 0 0 0 267 14685 1 0 0 0

223 12265 0 1 0 0 268 14740 0 1 0 0

224 12320 1 0 0 0 269 14795 0 0 1 1

225 12375 0 1 1 1 270 14850 1 0 0 0

67

Table 8: Results of electronic functional inspection for modified geometry (a reduced by 1.5mm)

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

1

Channel

6

Channel

7

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

1

Channel

6

Channel

7

161 8855 0 1 0 0 206 11330 0 1 0 0

162 8910 1 0 0 0 207 11385 1 0 1 0

163 8965 0 1 0 0 208 11440 0 1 1 1

164 9020 1 0 1 1 209 11495 1 1 0 0

165 9075 0 1 0 0 210 11550 1 1 0 0

166 9130 1 0 1 0 211 11605 1 0 0 0

167 9185 1 1 1 1 212 11660 1 0 1 1

168 9240 1 0 0 1 213 11715 1 1 1 1

169 9295 0 1 0 1 214 11770 0 0 0 0

170 9350 1 0 0 0 215 11825 0 1 0 0

171 9405 1 1 1 0 216 11880 1 0 0 0

172 9460 0 1 1 1 217 11935 1 0 1 0

173 9515 0 1 1 1 218 11990 1 0 1 1

174 9570 1 0 1 0 219 12045 1 0 0 0

175 9625 0 1 1 1 220 12100 0 0 1 1

176 9680 1 0 1 1 221 12155 1 1 0 1

177 9735 0 1 0 1 222 12210 0 0 0 0

178 9790 1 0 0 0 223 12265 0 1 0 0

179 9845 0 1 1 1 224 12320 0 1 1 1

180 9900 0 1 1 1 225 12375 0 0 1 1

181 9955 0 1 0 0 226 12430 0 1 0 0

182 10010 1 0 1 0 227 12485 1 1 0 0

183 10065 1 0 1 1 228 12540 0 0 1 0

184 10120 0 1 1 1 229 12595 1 0 0 1

185 10175 1 0 1 1 230 12650 0 0 0 0

186 10230 0 1 0 0 231 12705 0 0 0 0

187 10285 0 1 0 0 232 12760 1 1 1 0

188 10340 0 0 1 1 233 12815 1 0 0 0

189 10395 1 1 1 1 234 12870 1 0 1 1

190 10450 0 0 1 1 235 12925 1 1 0 1

191 10505 0 1 1 1 236 12980 1 0 0 0

192 10560 1 0 1 1 237 13035 1 0 1 0

193 10615 0 1 0 1 238 13090 1 0 1 1

194 10670 0 1 1 1 239 13145 0 1 0 1

195 10725 0 1 0 0 240 13200 1 0 0 1

196 10780 1 0 0 1 241 13255 0 1 0 1

197 10835 1 1 1 0 242 13310 1 1 1 1

198 10890 0 1 1 1 243 13365 0 0 0 1

199 10945 1 0 1 1 244 13420 0 0 0 0

200 11000 0 1 1 1 245 13475 0 1 0 0

201 11055 1 0 0 0 246 13530 0 1 1 0

202 11110 0 1 0 0 247 13585 1 0 1 1

203 11165 1 1 1 1 248 13640 0 1 0 1

204 11220 1 0 0 0 249 13695 0 1 0 0

205 11275 0 0 1 1 250 13750 0 1 1 1

68

Table 9: Results of electronic functional inspection for modified geometry (Σ reduced)

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

l

Channel

6

Channel

7

Data

pt

No.

Timer

value

(ms)

Channel

0

Channel

1

Channel

6

Channel

7

203 11165 0 0 0 0 248 13640 1 0 0 1

204 11220 0 0 0 1 249 13695 1 1 1 0

205 11275 0 0 0 1 250 13750 0 1 0 0

206 11330 1 0 1 1 251 13805 1 0 1 0

207 11385 1 0 1 0 252 13860 1 1 0 0

208 11440 1 0 1 0 253 13915 0 0 0 0

209 11495 1 1 0 0 254 13970 1 0 0 1

210 11550 0 1 1 1 255 14025 1 1 0 0

211 11605 1 0 0 0 256 14080 1 0 0 0

212 11660 0 1 1 0 257 14135 0 1 0 0

213 11715 0 0 1 1 258 14190 1 1 1 0

214 11770 1 1 0 1 259 14245 0 0 1 0

215 11825 1 0 0 1 260 14300 1 0 1 1

216 11880 1 1 0 0 261 14355 0 0 1 1

217 11935 1 0 0 0 262 14410 0 1 1 1

218 11990 0 1 1 0 263 14465 0 1 0 1

219 12045 1 0 1 1 264 14520 1 1 1 1

220 12100 0 1 1 0 265 14575 0 1 0 1

221 12155 1 0 1 1 266 14630 1 0 1 0

222 12210 0 1 0 0 267 14685 1 1 1 1

223 12265 1 1 1 1 268 14740 0 1 0 0

224 12320 0 1 0 0 269 14795 1 0 1 1

225 12375 1 0 1 1 270 14850 0 1 1 1

226 12430 0 1 0 1 271 14905 1 0 0 0

227 12485 0 0 0 0 272 14960 1 1 0 0

228 12540 1 1 1 1 273 15015 1 1 0 0

229 12595 0 1 0 0 274 15070 0 1 0 1

230 12650 0 1 1 1 275 15125 0 0 1 1

231 12705 0 1 0 1 276 15180 1 1 1 1

232 12760 0 1 1 1 277 15235 0 0 0 1

233 12815 0 0 1 0 278 15290 0 1 0 0

234 12870 1 0 0 1 279 15345 1 0 1 1

235 12925 0 1 0 1 280 15400 1 1 0 0

236 12980 0 1 1 1 281 15455 0 0 1 1

237 13035 0 0 0 1 282 15510 0 0 1 1

238 13090 1 1 0 0 283 15565 0 1 0 0

239 13145 0 0 1 1 284 15620 0 0 0 0

240 13200 1 1 1 0 285 15675 0 1 1 1

241 13255 0 1 0 0 286 15730 1 1 0 0

242 13310 1 1 0 0 287 15785 0 1 0 0

243 13365 0 0 0 0 288 15840 0 0 0 1

244 13420 0 1 1 1 289 15895 0 1 0 1

245 13475 0 1 0 1 290 15950 1 1 1 1

246 13530 1 0 0 0 291 16005 1 1 0 1

247 13585 0 0 0 0 292 16060 0 0 0 0

69

5.5 Discussion

Consider the typical ideal response, or output signals, of the encoders as shown in the

‘Quadrature: A & B’ diagram in Figure 21.

Figure 21: The ideal response of the output signals of the encoders

From Table 7 to Table 9, the channel pairs 0 and 1 and 6 and 7 are creating signals

representing the position of Gear Body 2 and 1 respectively. The signals created by Lines 0

and 1 are 90 degrees out of phase. Similarly the signals created by Lines 6 and 7 are 90

degrees out of phase. The ratio of the number of rising (and falling) edges created by Lines 0

and 1 to the number of rising (and falling) edges created by Lines 6 and 7 should be equal to

the transmission ratio. Figure 22shows a subset of the results of test 1 for Gear Body 2

(PCA p3)

70

GRAPH OF ENCODER RESPONSE

(TEST 1)

0

0.2

0.4

0.6

0.8

1

1.2

215 220 225 230 235

DATA POINT NUMBER

RE

SP

ON

SE

Line 0

Line 1

Figure 22: A graph of a sample of the response of Lines 0 and 1 from test 1

The data above represents the state of the signal rather than the number of rising or falling

edges. Subtracting the signal at data point n-1 from the signal at data point n and summing the

results automates the process of counting the edges. The data from Figure 22 is represented in

the table below as change of state data. A ‘1’ represents a rising edge, a ‘-1’ represents a

falling edge and ‘0’ represents no change at all between 2 consecutive data points.

71

Table 10: A selection of the results of test 1 represented as change of state data

Data pt No Channel 0 Channel 1 Channel 6 Channel 7

215 -1 1 0 1

216 1 -1 0 -1

217 -1 0 1 1

218 1 1 -1 -1

219 -1 0 0 0

220 1 0 0 0

221 -1 0 1 1

222 1 -1 -1 -1

223 -1 1 0 0

224 1 -1 0 0

225 -1 1 1 1

226 1 -1 -1 0

227 -1 1 0 -1

228 1 0 1 1

229 0 0 -1 -1

230 0 -1 0 1

231 -1 1 0 -1

232 1 -1 1 0

233 -1 1 0 1

234 0 -1 -1 0

235 1 0 1 0

As the data is in Excel, both rising and falling edges can be counted and the transmission ratio

can be calculated to be equal to:

(rising edges Channel 0 + rising edges Channel 1+falling edges Channel 0 + falling edges Channel

1)/ (rising edges Channel 6 + rising edges Channel 7+falling edges Channel 6 + falling edges

Channel 7)

Table 11: A count of the rising and falling edges and the resulting transmission ratio for the selected

data range of test 1

Channel 0 Channel 1 Channel 6 Channel 7

Rising edges 17 11 13 12

Falling edges 16 11 13 13

The above process and result can be extended over the full range of data. The results of doing

this are shown in Figure 23.

72

Figure 23: A graph of the transmission ratio for test 1

The results indicate that the transmission ratio ranges from less than 1 to over 1.5 and has an

average value of 1.2. There is an error of over 100% between this result and the theoretical

value for the case of test 1. Similarly the graphs below show a lack of precision and accuracy

for tests 5 and 9 respectively.

The apparent lack of constancy of transmission ratio is the result of one of two things:

The gear bodies are actually producing a transmission ratio as indicated in the graphs.

The instrumentation is not working properly and the gear bodies are producing a transmission

ratio different to that indicated in the graphs above.

The first of these hypotheses seems highly unlikely as the transmission ratio is experiencing

step changes of up to 0.4 (Refer to Test 1, data point 225). A physical phenomenon that could

cause the gear bodies to produce inconsistent results, such as machining errors, would

produce either a consistent error in transmission ratio or noise. The graphs above indicate

large step changes distributed randomly amongst inconsistent transmission ratio and noise.

GRAPH OF TRANSMISSION RATIO FOR TEST 1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 100 200 300 400 500

DATA POINT

TR

AN

SM

ISS

ION

RA

TIO

73

Consider the latter hypothesis and refer to Figure 21 and Figure 22. Figure 21 indicates

channels 0 and 1 should produce the same signals of equal numbers of high and low values

90o out of phase. Figure 22 indicates this did not occur. In this small sample of 20 data points

there are three instances of either Channels 0 and 1 not producing equal numbers of high and

low signals or dropping out of phase. The locations of these instances are data points 216, 219

and 233. There is up to 15% error in this small sample alone. Assuming this sample, chosen at

random is indicative of the testing population, instrumentation caused up to 15% error in the

experimental results.

Causes of instrumentation error: Consider the simple data acquisition system process flow.

The encoders (the transducer) create an electrical signal that the DAC samples i.e. the card

retrieves the signal created by the transducer at a frequency determined by the card’s clock

speed and configuration. The DAC could for example be multiplexing and the sampling

frequency of each channel is reduced for each additional channel added. The computer then

retrieves the signal from the DAC in accordance with the software instructions. As was the

case for the DAC, the computer retrieves the signal from the DAC at a frequency set by the

clock speed of the computer, operating system and software.

Typically the sampling frequencies of DACs are in the mega-Hertz range i.e. the DAC

samples a single digital data channel millions of times per second. Even if multiplexing is

occurring and the clock speed is 2MHz and there are, for example, four data channels, each

data channel is sampled 500 000 times per second. This hardly appears to be a problem

however there are serious limitations imposed by the operating system as the operating

system update frequencies are considerably lower. Section 5.3 Method contained a pseudo

code describing the operation of the Labview VI used in this experiment. Channel 9 contained

a statement, ‘Wait 55ms’. This is a limitation imposed by the operating system, in this case

74

Windows 3.11. The time between ticks of the clock on computers running Windows 3.X is

55ms (Wells 1995) and setting the ‘Wait’ to less than this caused the timer to behave

erratically.

Consider a specific example. The encoders used in the above experiment each had two data

channels producing 2500 pulses per revolution for a total of 5000 pulses per revolution per

encoder. To accurately measure the rotation of the encoder, the VI must sample the output at

the high and low point. The resulting sample, with channels joining the data points is shown

in Figure 24 with the sampling period imposed by the operating system.

Figure 24: An illustration of the sampling rate as determined by the operating system

As the encoder produced 5000 pulses per revolution, there will be 5000 transitions from 0 to 1

in one revolution. Between each 0 to 1 and 1 to 0 step, however, there must be 55ms or the

Labview VI will not ‘see’ the state. One revolution must therefore take 550 seconds or over

nine minutes.

It was therefore conceivable that there were instances when, the gear bodies moved too fast

for the acquisition cycle. This would result in a data point being missed and, for example, a

sample may appear as 01011 when it was 010101. This error appears to have occurred in

Table 10 at data points 218 to 221. Channel 0 continued to sample whilst the graph indicates

channel 1 stayed in the same state.

55ms 55ms

75

Alternately, manually moving the gear bodies, slow enough for acquisition creates a

significant risk, of inadvertently moving backwards (at 5000 pulses per revolution, a mere

0.72 degrees creates one pulse). The VI would interpret this as a change of state, however it

would be in the wrong sense.

These problems with instrumentation may be minimised or overcome by using an alternate

operating system such as Windows NT or Macintosh (17ms clock tick) or by using a DAC

with an in-built counter. Edge counts would then occur at the hardware level rather than at the

software level, as in the above experiment. A VI would be written to read the counter value at

the end of the experiment. Although the Labview PC+ DAC was fitted with three Oki 82C53

counter/timer chips (National Instruments 1994 p2-34):

• The above experiment required four counters, one for each input

• The Labview VI required to read a counter, the ‘Count events or time’ VI, “does not

work on boards with the 8253 chip, such as the Lab series.” (Wells 1995 p295)

Other causes of error: Ideally, as the introduction to this chapter indicated, the input gear is

driven at constant velocity by, say, a small DC motor. The single flank gear tester normally

applies to gear wheels and pinions comprising a full set of teeth that are able to operate ad

infinitum. This was not possible in the test above as there was only one gear body or one

tooth of the wheel and pinion. Consequently, it was impractical to set up a DC motor to

operate over such a small range and ‘constant velocity’ was provided by manual means.

This inevitably was less than ideal. Although every attempt was made to rotate the driven gear

body at constant velocity, the relatively low speed required meant, in some instances, there

were consecutive points whose value did not change. An example of this is Table 20, data

76

points 227 and 228. This shortcoming in the procedure was allowed for in the analysis as

changes in value, not absolute values, were used to calculate the transmission ratio.

An alternative: The discussion above highlights the significant problems associated with the

electrical functional inspection given limitations of the hardware and software employed. In

discussions with Dr John Gal and Dr Jack Phillips, an alternate mechanical functional

inspection apparatus was suggested. The following section explains this experiment.

77

CHAPTER 6 FUNCTIONAL INSPECTION OF EQUIANGULAR

SPATIAL INVOLUTE GEARS USING DIAL

INDICATORS

One of the initial objectives of this thesis was to verify the Phillips’ (2003) theory regarding

the constancy of transmission ratio between (equiangular) spatial involute gears when

designed according to the equations given in Phillips (2003). To this end, the previous chapter

detailed a functional inspection of the equiangular gear body pair using data acquisition. The

results of that inspection were inconclusive. A range of issues associated with the technology

employed indicated the actual transmission ratio was up to 100% in error with the theoretical

transmission ratio and there were random variations in transmission ratio of up to 25% in the

average.

This chapter now deals with a functional inspection of the gear body pair using a considerably

simpler and more robust experimental set up. It details the aims, method and results of a

functional inspection of the gear body pair using mechanical testing apparatus. It then goes on

to discuss the results of the inspection including a detailed error analysis.

6.1 Aim

To verify the results of the electronic measurement of transmission ratio using a method

independent of that employed in the experiment described in CHAPTER 5 i.e. a mechanical

test.

6.2 Method

As it is more convenient to measure linear rather than angular movement of bodies in space,

the mechanical hardware design was modified to facilitate the measurement of relative

angular displacement via linear movement of two masses attached to the gear bodies. Steel

pulleys having outside diameters (OD) of approximately 145mm were keyed to the extended

78

shafts, shown in the drawing in Appendix G. Grub screws fitted to the OD of the pulleys

provided a means of connecting the mass carriers to the pulleys with wire.

Each mass carrier carried 250g comprising 125g of tare mass and 125g of added mass. The

configuration of the mass carriers was such that the moment created by the mass on the carrier

attached to Gear Body 1 was in the opposite sense to the moment created by the mass on the

carrier attached to Gear Body 2 when viewed from the Centre Distance Line (CDL). This

configuration ensured the gear bodies were reacting against one another and minimised the

chances of gear bodies separating in the course of the experiment.

Dial indicators, with a range of 50mm linear movement and a resolution of 0.01mm, were

fixed to the frame of reference such that the centre of the range of motion of the gear bodies

corresponded to the centre of the range of motion of the dial indicators. The pointers of the

dial indicators were placed as close as possible to the centre of the mass carrier. This,

combined with the relatively large masses, minimised if not totally eliminated the influence of

the dial indicator on the movement of the mass.

A series of measurements were then taken comparing the relative displacements of the

masses, via the dial indicators, for both directions of travel over the full range of motion. Care

was taken to ensure the extremes of the range of motion were within the machined area of

gear bodies.

The experiment was conducted in two parts:

• A macro level inspection where the only the two extreme values were recorded.

• A detailed inspection where at least five values, spread roughly evenly between the

extremes, were recorded.

79

(A video is available which shows how the latter part was conducted. The video was to

facilitate increased measurement points via playback. This idea was abandoned, however,

when increasing the field of vision to allow a full view of the experiment, resulted in an

inadequate level of detail to accurately read the dial indicators.)

The relative angular displacements were then related to the linear movements. The equation

relating linear distance, l, the radius of the pulley, r and angular displacement, θ, is

from which it follows that

(48)

The following equations for Gear Bodies 1 and 2, respectively, follow

l1 = r1 θ1 and l2 = r2 θ2

and

∆l1 = r1 ∆θ1 and ∆l2 = r2 ∆θ2

Now transmission ratio, i, for a general gear pair equals the ratio of the angular speed of the

driving gear to the driven gear (AS2075 1991). This can be stated mathematically and

rearranged to produce a relationship between angular displacements as follows:

o

i

d

di

θθ

=

and thus

o

iiθθ

∆∆

=

Rearranging (48) to make ∆θ the subject for both input and output gear bodies and

substituting produces an equation for the transmission ratio based on the linear displacement

of the masses and radius of the pulleys as follows

(49)

l = r.θ

∆l = r.∆θ

io

oi

rl

rli

.=

80

Therefore, if the pinion or Gear Body 1 is the input, (49) becomes

12

21.

rl

rli =

6.3 Results

This section presents all measured values and resulting calculated values complete with error

analysis. Table 12 indicates the effective OD of the pulleys.

Table 12: Effective outside pulley diameters

Gear body 1 Gear body 2

Effective outside diameter 144.75mm 145.00mm

Error +/-0.025

+/- 0.017%

+/-0.025

+/- 0.017%

Note: The measurements were taken using Vernier callipers.

Macro level inspection: The following tables indicate dial indicator values, at the extremes

of travel, for Gear Bodies 1 and 2 for the respective directions of travel.

Table 13: Results for macro mechanical testing of transmission ratio.

Test 1: Gear body 1 driving gear body 2

Gear body 1 Gear body 2

Dial indicator value Error Dial indicator value Error

Start 26.778mm +/-0.005 36.314mm +/-0.005

Finish 48.439mm +/-0.005 23.024mm +/-0.005

∆l 21.66 +/-0.01

+/-0.05%

13.29 +/-0.01

+/-0.08%

Test 2: Gear body 2 driving gear body 1

Start 47.367mm +/-0.005 23.628mm +/-0.005

Finish 33.122mm +/-0.005 32.289mm +/-0.005

∆l 14.25 +/-0.01

+/-0.07%

8.66 +/-0.01

Substituting the results above, for the case of Gear body 1 driving Gear body 2, into (49),

whilst ignoring the division by 2 in both the numerator and denominator resulting from the

substitution of the diameter value, produces

81

003.0/632.1

%2.0/632.1

)%07.008.0017.005.0(/29.1375.144

00.14566.21

−+=−+=

+++−+=

i

i

x

xi

Similarly, for the second case of Gear body 2 driving Gear body 1,

001.0/607.0

001.0/6070.0

%2.0/6070.0

−+=−+=−+=

i

i

i

83

Detailed inspection: The following tables indicate the dial indicator calculated values for each of the three tests.

Table 14: Results for detailed mechanical testing of transmission ratio

Test 1: Gear body 1 driving gear body 2

Gear body 1 Gear body 2 Transmission ratio

Point Dial ind.

Value

(mm)

Error

(abs)

Error

(rel) ∆li Error

(abs)

Error

(rel)

Dial ind.

value

(mm)

Error

(abs)

Error

(rel) ∆li Error

(abs)

Error

(rel)

Numer-

ator

Denom-

inator

Dividend Error

(rel)

Error

(abs)

1 37.514 0.005 0.01% 37.83 0.005 0.01%

2 33.625 0.005 0.01% -3.89 0.01 0.3% 35.037 0.005 0.01% -2.79 0.01 0.4% -564 -404 1.39 2% 0.03

3 27.381 0.005 0.02% -6.24 0.01 0.2% 31.82 0.005 0.02% -3.22 0.01 0.3% -905 -466 1.94 2% 0.04

4 22.482 0.005 0.02% -4.90 0.01 0.2% 28.845 0.005 0.02% -2.98 0.01 0.3% -710 -431 1.65 2% 0.03

5 17.487 0.005 0.03% -5.00 0.01 0.2% 25.839 0.005 0.02% -3.01 0.01 0.3% -724 -435 1.66 2% 0.03

6 14.814 0.005 0.03% -2.67 0.01 0.4% 24.17 0.005 0.02% -1.67 0.01 0.6% -388 -242 1.60 2% 0.03

84

Test 2: Gear body 1 driving gear body 2 (Rpt)

Gear body 1 Gear body 2 Transmission ratio

Point Dial ind.

Value

(mm)

Error

(abs)

Error

(rel) ∆lI Error

(abs)

Error

(rel)

Dial ind.

value

(mm)

Error

(abs)

Error

(rel) ∆li Error

(abs)

Error

(rel)

Numer-

ator

Denom-

inator

Dividend Error

(rel)

Error

(abs)

1 34.815 0.005 0.01% 36.222 0.005 0.01%

2 25.412 0.005 0.02% -9.40 0.01 0.1% 30.646 0.005 0.02% -5.58 0.01 0.2% -1363 -807 1.69 2% 0.03

3 19.727 0.005 0.03% -5.69 0.01 0.2% 27.188 0.005 0.02% -3.46 0.01 0.3% -824 -501 1.65 2% 0.03

4 14.908 0.005 0.03% -4.82 0.01 0.2% 24.247 0.005 0.02% -2.94 0.01 0.3% -699 -426 1.64 2% 0.03

5 9.593 0.005 0.05% -5.32 0.01 0.2% 21.025 0.005 0.02% -3.22 0.01 0.3% -771 -466 1.65 2% 0.03

85

Test 3: Gear body 2 driving gear body 1

Gear body 1 Gear body 2 Transmission ratio

Point Dial ind.

Value

(mm)

Error

(abs)

Error

(rel) ∆lI Error

(abs)

Error

(rel)

Dial ind.

value

(mm)

Error

(abs)

Error

(rel) ∆li Error

(abs)

Error

(rel)

Numer-

ator

Denom-

inator

Dividend Error

(rel)

Error

(abs)

1 14.818 0.005 0.03% 24.16 0.005 0.02%

2 19.492 0.005 0.03% 4.67 0.01 0.2% 27.071 0.005 0.02% 2.91 0.01 0.3% 421 678 0.622 2% 0.01

3 24.8 0.005 0.02% 5.31 0.01 0.2% 30.145 0.005 0.02% 3.07 0.01 0.3% 445 770 0.578 2% 0.01

4 29.705 0.005 0.02% 4.91 0.01 0.2% 33.088 0.005 0.02% 2.94 0.01 0.3% 426 711 0.599 2% 0.01

5 34.611 0.005 0.01% 4.91 0.01 0.2% 36.202 0.005 0.01% 3.11 0.01 0.3% 451 711 0.634 2% 0.01

6 37.54 0.005 0.01% 2.93 0.01 0.3% 37.995 0.005 0.01% 1.79 0.01 0.6% 260 425 0.611 2% 0.01

Note: The values in the columns titled ‘Dial ind. value’ and ‘Error (abs)’ are raw data values whilst all other values are calculated values. The

calculations used to produce the values in the Transmission section are identical to those detailed in the macro mechanical test above. Test 3 is a

repeat of Test 1 and was carried out as a result of the large standard deviation in the results in Test 1.

86

6.4 Discussion

Comments on the method: This section discusses the sources of error in the experimental

methodology and their potential impact.

The first potential source of error was elongation of the wire used to connect the mass carrier

and the pulley. Consider the equation for Young’s modulus

eA

PLE =

where:

E = the modulus of elasticity or Young’s modulus (Pa)

P = the force applied (N)

E = the elongation of the specimen (m)

A = the x-sectional area of the specimen (m)

L = the length of the wire (m)

Assuming elastic behaviour and given the Modulus of Elasticity of 70GPa (aluminium), the

length of wire (200mm) and the wires x-sectional area (0.75mm2), the elongation of the wire

was approximately 9.3x10-6

m or 9.3µ. This value however is constant over the duration of the

test therefore e is constant over the duration of the experiment i.e. all elasticity was removed

before the first dial indicator value was recorded.

An increasing ‘unsupported’ length of wire as it winds off the pulley may have some effect.

The extent of the effect can be determined by considering the increased length. Consider the

data from Table 14. The effect of increasing ‘unsupported’ length (assuming aforementioned

mechanical characteristics) is approx 1µ. The effect of elasticity can therefore be ignored.

87

The second potential source of error was the tipping of the mass carrier caused by the force

exerted by the dial indicator pointer. The effects of this source of error were extremely small

for the following reasons:

• The dial indicator applied a relatively small force to the mass carrier that was

relatively small compared to the weight of the carrier.

• This acted at a distance of less than 25mm from the centre of mass of the carrier.

• The force exerted by the dial indicator was constant over the range of movement. This

means any tipping of the mass carrier was constant over the duration of the

experiment.

• As readings were taken with a motionless system, all readings were equally affected

and the effects caused by tipping of the mass can be ignored.

Comments on the internal consistency of the macro inspection: Each of the above

experiments consisted of two stages of testing, a ‘driving’ test and a ‘driven’ test with respect

to the pinion, as illustrated. This provides a means of determining the internal consistency of

the test results. For the results to be internally consistent:

• the result of the driving test should equal the inverse of the results of the driven test.

As this theory applies to both the pinion and the wheel, it is more applicable to

consider an actual value and a calculated value where the calculated value is the

inverse of the actual value for the alternate test result i.e. iactual should equal1/icalculated.

• the error bands for iactual and 1/icalculated should overlap.

Table 15 indicates the result of applying these operations to the test results.

88

Table 15: Comparison of internal consistency of the macro mechanical inspection via error analysis

Test Transmission

ratio (iactual)

Error iactual Max

Min/Max

1/icalculated Error 1/icalculated

Max/Min

1 1.632 +/-0.003

+/-0.2%

1.635

1.629

1.65

(1/0.607)

+/-0.2%

+/-0.01

1.66

1.64

2 0.607 +/-0.001

+/-0.2%

0.608

0.606

0.613

(1/1.632)

+/-0.2%

+/-0.001

0.612

0.614

Clearly, for the most part, the error bands do not overlap. Consider the following table

comparing the driving to the inverse of the driven case and vice versa. The calculated values

in the table provide some indication of the internal consistency of the testing. The column

labelled 1/icalculated is the inverse of the transmission ratio of the alternate test result. The

column labelled ‘Difference’ indicates the absolute value of the difference between the value

iactual and 1/icalculated whilst the column labelled ‘% error’ is the value ‘Difference/iactual’.

Table 16: Analysis of internal consistency of the macro mechanical inspection

Test Transmission

ratio (iactual)

Error 1/icalculated Difference %

error

1 1.632 +/-0.003

+/-0.2%

1.65 (1/0.607) 0.018 1.1%

2 0.607 +/-0.001

+/-0.2%

0.612 (1/1.632) 0.006 0.98%

In summary, in:

• Test 1, the difference between iactual and 1/icalculated is approximately 1% and rounding up

values for iactual to give them the same number of significant figures as those in icalculated,

the Maximum of iactual equals the minimum of icalculated.

• Test 2, the difference between iactual and 1/icalculated and the error bands is less than 1% of

the maximum iactual value.

The macro mechanical inspection results are therefore internally consistent within a 1% error

band.

89

Comments on external consistency of the macro mechanical inspection: For simplicity

and completeness the above results are summarised below and compared to the theoretical

value rounded to the same number of significant figures as the ‘Actual’ value. Absolute and

relative errors, calculated as a percentage of theoretical are also shown, the theoretical value

having been determined from (48)

Table 17: Summary of macro mechanical inspection results

Test Transmission

ratio (actual)

Error Theoretical Difference % error

1) Gear body 1

driving Gear body

2

1.632 +/-0.003

+/-0.2%

1.667 0.035 2.1%

2) Gear body 2

driving Gear body

1

0.607 +/-0.001

+/-0.2%

0.600 0.007 1%

In summary, in:

• Test 1, the difference between iactual and itheoretical is approximately 2.1%.

• Test 2, the difference between iactual and itheoretical is just greater than 1% of the itheoretical

value.

The macro mechanical inspection results are therefore externally consistent within a

maximum error band of 2.1%.

Analysis of the internal and external consistency of the detailed mechanical inspection:

With five to six data points available for analysis, there are more analytical options available.

The analysis can therefore be more rigorous. The graphs below contain plots of the data

points, adjusted for the difference in pulley diameters, whilst the table summarises the results.

The equations of the lines of best fit and correlations shown on the graphs are generated from

a least squares linear model as discussed in Appendix I.

90

Figure 25: Graph of transmission ratio for test 1

Figure 26: Graph of transmission ratio for test 2

GRAPH OF TRANSMISSION RATIO

(GEAR BODY 1 DRIVING GEAR BODY 2)

y = 1.689x - 26.115

R2 = 0.9988

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

Gear body 1 (mm)

Gear

bo

dy 2

(m

m)

GRAPH OF TRANSMISSION RATIO

(GEAR BODY 1 DRIVING GEAR BODY 2)

y = 1.6474x - 24.971

R2 = 0.9998

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

Gear body 1 (mm)

Gear

bo

dy 2

(m

m)

91

Figure 27: Graph of transmission ratio for test 3

Table 18: Summary of the detailed mechanical inspection results

Test Transmission ratio

(actual)

(theory)

Difference

Avg/ls

% error

Avg Std dev Lst sqrs R2

1 1.65 0.2 1.69 0.999 1.67 0.02/-0.02 1%/1%

2 1.66 0.02 1.66 1 1.67 0.01/0.01 1%/1%

3 0.609 0.021 0.606 0.999 0.600 0.009/0.006 2%/1%

Note:

• The value in the ‘Transmission ratio - Avg’ column of the above table is the average

of the ‘Dividend’ values in the Transmission section of the detailed mechanical

inspection table. The value in the ‘Std dev’ column is the standard deviation of the

data used to calculate the aforementioned average.

• The results of test 1 clearly indicate that conducting this second test was justified

since:

• The standard deviation of test 1 is ten times that of the result of test 2.

• The difference between the transmission ratio measured by the average method and

the least squares method is 2.4% for test 1 compared to zero and 0.5% of tests 2 and 3,

respectively.

GRAPH OF TRANSMISSION RATIO

(GEAR BODY 1 DRIVING GEAR BODY 2)

y = 0.6031x + 15.266

R2 = 1

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25 30 35 40

Gear body 1 (mm)

Gear

bo

dy 2

(m

m)

92

• Figure 27 shows points five and six on opposite sides of the line of best fit and the

difference between the line of best fit value and the actual value is up to ten times that

for tests 2 and 3.

As R2>0.99 or 99%, over 99% of the variation in the values of the driven gear body are

accounted for by a linear relationship with the driving gear body. For the case of test 1, the

theoretical equation of the line, µY|x = α + βx, is estimated to be Y = A + Bx, where A is –

0.361 and B is 1.692. It is, therefore, appropriate to propose the hypothesis (Appendix I):

1. H0: β=1.666…

2. H1: β≠1.666...

To test this hypothesis, assume the difference between the point given by the theoretical

regression line and the experimental value, the error, is Ei i.e. YI = α +βxi + Ei and:

• Ei has the same variance σ2 for all i

• E1, E2, …, En are independent from run to run in the experiment

• Ei is normally distributed

Thus Yi is also normally distributed with a probability distribution n(yi, α + βxi, σ), A and B

are also normally distributed with probability distributions n(a, α, σΑ) and n(b, βxi, σΒ).

Under the normality assumption, the statistic

(50)

has a t distribution with n-2 degrees of freedom. The number of sample points, n, can be used

to construct a (1-α)100% confidence interval for β where:

xxSS

Bt

β−=

93

s is an unbiased estimate of the variance σ and is calculated in accordance with the

equation of Appendix I.

Sxx. is the sum of the differences between the actual value of x, xi, and the mean value of x

squared. This is shown in more detail in Appendix I.

Applying this to the proposition of testing the hypothesis that β=1.66, against the alternative

that β≠1.66 for test 1:

1. H0: β=1.666…

2. H1: β≠1.666...

gives:

Sxx s tα/2α/2α/2α/2 DoF Tails P

0.026738 0.004768 0.86485 4 2 0.4359

where:

Sxx and s are calculated, from the above data, in accordance with Appendix I.

tα/2 is given by (50) assuming a 0.05 level of significance

P is calculated using the ‘TDIST’ function in MS Excel

Decision: There is strong evidence (P≈0.436 and P>>α) that β=1.666…

Similarly for tests 2 and 3 there is strong evidence (P≈0.426 and P≈0.218) that β equals

1.666… and 0.6 respectively.

94

The results of analysis using the f-distribution confirm the results obtained above using a t-

distribution. Table 19 indicates the observed f-statistic and compares it to the critical f-

statistic.

Table 19: A comparison of the observed and critical f-statistic

Test Observed f-statistic Critical f-statistic

1 3366.597 7.71

2 68892.1 9.28

3 22695.9 7.71

Note:

• All tests assume a 95% confidence interval.; tests 1 and 3 assume 4 degrees of freedom

whilst test 2 assumes 3 degrees of freedom.

• Critical f statistic is from Walpole, Myers (1993)

In all cases the observed value is a lot greater than the critical value confirming the results of

the t statistic.

A comment on the validity of the t and f statistic: As stated above, the analysis relies on the

assumption that the errors, Ei, are normally distributed. As there are insufficient data points to

test this assumption (30 is considered the minimum for a normal distribution), it could be

suggested a non parametric statistic may be more appropriate. IFA Services (2000) state that

when comparing the t-test to its non-parametric equivalent, the Wilcoxon matched pairs

signed ranks test, ‘for small numbers with unknown distributions this test is even more

sensitive than the Student t-test’. Consider for example the non-parametric equivalent to the

correlation coefficient (r), the Spearman rank correlation coefficient

)1(

6

12

1

2

−−=

=

nn

d

r

n

i

i

s

95

where:

di = is the difference between the ranks assigned to xi and yi

n = the number of pairs of data

The rank correlation coefficient produces a number between –1 and +1 and is interpreted in

the same manner as the correlation coefficient. A value of +1 or –1 indicates perfect

correlation between X and Y. (Walpole, Myers 1993)

When determining di however, the analyst ranks the X and Y values, giving the smallest X

value the rank 1 and the largest the rank ni. The same is done for the Y values. The ranks are

then subtracted creating the values di. Applying this process to the data in the above

experiment produces di=0 and rs = 1 because the values of X and Y both increase. The

correlation coefficient was therefore considered to be a worst case scenario. Similarly, it was

decided to use the t and f statistic to test the hypotheses.

6.5 Conclusion

Example 1 in Phillips (2003) presented the raw data for a gear set and estimated the

transmission ratio of the set, constructed in accordance with the methodology presented there,

to be 0.6 or 1.666… depending on which is the driver and driven gear.

Regression analysis for all experiments, showed that:

• at least 99.88% of the variation in the values of the rotation of the driven gear body

can be explained by the variation in the values of the driving gear body.

• the transmission ratio, for a pair constructed and located in accordance with this

methodology, to be within 2% of the theoretical or estimated values.

• Statistical analysis of the results, using t-distribution and f-distribution, indicated NOT

to reject the hypothesis that the transmission ratio was either 1.666 or 0.6 despite the

96

fact that the aforementioned results produced an error of 2%. There is a probability of

least 22% and as much as 43% that the transmission ratios were 0.6 and 1.666…

respectively.

The testing of gear bodies, developed using the underlying theories and equiangular synthesis

developed in Phillips (2003), illustrate that the outcomes predicted by Phillips were achieved

in the experiment. Moreover, as this was an arrangement chosen at random and constructed

using the underlying theory of Phillips (2003), from an engineering and scientific point of

view, the experimental evidence indicates that, in all probability, the results would be the

same for any pair of equiangular involute gears constructed in accordance with the theory of

Phillips (2003)

97

CHAPTER 7 PROFILE MEASUREMENT OF EQUIANGULAR

SPATIAL INVOLUTE GEARS

The previous chapters detailed a functional inspection of a pair of manufactured gear bodies.

It was determined by mechanical inspection that transmission errors of approximately 1%

existed and that there was a 40% chance that, despite this error, the actual transmission ratio

equalled the theoretical transmission ratio.

This chapter investigates potential causes of that error or profile related failure modes of the

gears. Following a review of the theoretical profiles, the chapter will detail the approach to

and results of an analytical inspection of the spatial involute gear bodies used in the previous

functional inspections.

7.1 A review of the theoretical tooth profile

During the development of the theory of spatial involute gearing Phillips (1999a) and Killeen

(1996) showed in geometric and mathematical discussions, respectively, that although the

reference surface is a cylinder for all involute gears, the tooth surface is a general involute

helicoid. The parametric equations for the planar involute are a special case of the spatial

involute, the involute helicoid, that is, a “section by a plane perpendicular to the axis of the

cylinder is an involute to a circle.” (AS2075 1991)

Figure 28 shows four views of a general involute helicoid with the following properties:

rb

Pitch

γb

= 50mm

= 200 mm/turn from which

= 32.4816… degrees

98

The mauve lines represent the architecture of the helix and the blue lines represent the

generators of the involute helicoid.

Figure 28: A general involute helicoid

The parametric equations of the involute helicoid with its axis aligned along the z-axis from

Killeen (1996), are:

(51)

where:

rb = the radius of the base circle

t7 = a parameter describing the rotation of the tangent to the helix or the involute roll angle

γb = the angle the tangent to the helix subtends to the transverse plane

µ = a parameter describing the position of the point along the tangent to the helix.

bbbIH

bbIH

bbIH

trz

ttry

ttrx

γµγγµγµ

sin.tan

cos.cos.sin

sin.cos.cos

7

77

77

−=−=+=

base circle

(z) axis

helix

involute

helicoid

99

From (51), µ.cosγb is the projected length of the helicoid generator onto the plane defined by

the throat circle. This can be stated mathematically by letting zIH equal zero and isolating rbt7

producing,

rbt7=µ.cosγb

The parametric equation, therefore, for the section at z = 0 is

(x = rbcost7+ rbt7.sint7, y = rbsint7- rbt7.cost7)

or

(x = rb(cost7+ t7.sint7), y = rb(sint7- t7.cost7))

Transposing these equations by rotating them through 90o and mirroring them across the yz-

plane, as discussed in Appendix J, produces the parametric equations

(52)

These equations are identical to those of Litvin (1994) with the exception of the substitution

of φ for t7. Appendix K contains further investigation into the equations of the planar involute.

Taking a section at a general z, say z equals z*, through the above involute helicoid produces

the line in the plane z = z*, having the following parametric equations

Thus, the parameters µ and t7 of any plane, defined by z = z*, in the involute helicoid are

related according to

(53)

This can be expressed graphically as is shown in Figure 29.

bbb

bbIH

bbIH

trz

ttry

ttrx

γµγ

γµγµ

sin.tan

cos.cos.sin

sin.cos.cos

7

*

77

77

−=

−=+=

x = rb(sint7-t7.cost7) and y = rb(cost7+t7.sint7)

µ = rb.t7/cosγb- z*/sinγb

100

Figure 29: A graphical solution to the relationship between µ and t7

7.2 Aim

The aim of this chapter is to investigate the actual profile of the gear bodies and compare

them to the theoretical profile paying attention to the relationship between respective sections

of the involute helicoid and the planar involute.

7.3 Method

Following calibration of the ruby ball on the Coordinate Measuring Machine (CMM) using

the steel ball to the left of the figure below, each gear body was set up on the table of the

Ferranti Mercury CMM as shown.

t7 µ

arctan(rb/cosγb) -z*/sinγb

101

Figure 30: The arrangement of the gear body on the CMM

The face representing z=z*min of the involute helicoid, for each gear body, was aligned to

within 0.005mm of the y-axis of the CMM. The point of intersection of the planes

representing the z=z*min, the tooth root and the upper most face of the gear body was set as

the point (0,0,0) as illustrated below.

102

Figure 31: A plan view of the gear body (looking down on the tooth surface)

Ball radius compensation was turned off and for nine values of z* a series of values

representing the coordinates of the point of contact between the CMM and the gear body,

were recorded across the 50mm facewidth. Care was taken to ensure each of the points was on

the machined surface representing the involute helicoid. To allow generation of a reference

surface three non-collinear points on the top land were also recorded.

(0,0,0)

x

z y

103

7.4 Results

The diameter of the ruby was found to be 3.993mm (compared to 4.000mm theoretical).

Table 20 and Table 21 show the results obtained for each gear body.

Table 20: Data points created by the CMM measurements of Gear Body 1

X Y Z X Y Z

POINT 20.457 -2.002 -6.952 POINT 19.742 -27 -9.062

POINT 22.57 -2.002 -7.317 POINT 22.637 -27 -9.58

POINT 25.087 -2.002 -8.15 POINT 25.1 -27 -10.47

POINT 27.372 -2.002 -9.18 POINT 26.855 -27 -11.312

POINT 28.96 -2.002 -10.037 POINT 29 -27 -12.557

POINT 30.837 -2.002 -11.237 POINT 30.557 -27 -13.63

POINT 32.795 -2.002 -12.672 POINT 32.695 -27 -15.325

POINT 34.1 -2.002 -13.752 POINT 20.027 -32 -9.52

POINT 20.2 -6.997 -7.375 POINT 22.262 -32 -9.912

POINT 22.922 -7 -7.872 POINT 24.422 -32 -10.655

POINT 25.977 -7 -8.995 POINT 26.88 -32 -11.807

POINT 28.557 -6.997 -10.297 POINT 28.885 -32 -12.992

POINT 30.687 -6.997 -11.647 POINT 30.35 -32 -14

POINT 32.717 -6.997 -13.157 POINT 32.52 -32 -15.725

POINT 34.142 -6.997 -14.335 POINT 19.805 -37.002 -9.915

POINT 20.505 -12 -7.822 POINT 22.257 -37.002 -10.352

POINT 22.62 -11.997 -8.235 POINT 24.797 -37.002 -11.27

POINT 25.64 -11.997 -9.307 POINT 27.107 -37.002 -12.412

POINT 28.425 -11.997 -10.717 POINT 28.557 -37.002 -13.29

POINT 30.302 -11.997 -11.907 POINT 29.865 -37 -14.172

POINT 32.23 -11.997 -13.31 POINT 31.402 -37.002 -15.345

POINT 33.67 -11.995 -14.495 POINT 32.572 -37.002 -16.327

POINT 19.707 -17 -8.18 POINT 20.082 -42 -10.372

POINT 22.28 -17 -8.587 POINT 22.075 -42 -10.745

POINT 24.607 -17 -9.35 POINT 24.202 -42 -11.475

POINT 26.99 -17 -10.42 POINT 26.065 -42 -12.33

POINT 29.067 -17 -11.6 POINT 28.482 -42 -13.735

POINT 30.397 -17 -12.477 POINT 30.192 -42 -14.92

POINT 32.025 -16.997 -13.685 POINT 31.795 -42 -16.225

POINT 33.28 -16.997 -14.712 POINT 19.852 -47 -10.77

POINT 19.812 -22.002 -8.637 POINT 22.432 -47 -11.292

POINT 22.29 -22.002 -9.032 POINT 24.865 -47 -12.227

POINT 24.942 -22.002 -9.942 POINT 27.455 -46.997 -13.597

POINT 27.447 -22 -11.152 POINT 29.612 -46.997 -15.027

POINT 29.062 -22 -12.1 POINT 31.662 -47 -16.667

POINT 30.802 -22 -13.29

POINT 32.265 -22 -14.412

POINT 33.617 -22 -15.572 Three non collinear points

POINT 35.63 -5.035 -18.37

POINT 34.72 -21.635 -26.295

POINT 33.43 -44.915 -20.495

104

Table 21: Data points created by the CMM measurements of Gear Body 2

7.5 Discussion

Comments on the orientation of the axes: The raw data indicated measurements for various

values of y, that is, the results from the CMM would produce an involute helicoid aligned to

the y-axis. The equations of the involute helicoid, (51), however apply to an involute helicoid

with its axis aligned to the z-axis.

X Y Z X Y Z

POINT 19.137 -2 -20.017 POINT 19.195 -27 -17.882

POINT 22.115 -2 -20.885 POINT 23.68 -27 -19.202

POINT 24.622 -2 -21.892 POINT 26.982 -27 -20.642

POINT 27.255 -2 -23.175 POINT 29.282 -27 -21.865

POINT 29.08 -2 -24.197 POINT 32.935 -27 -24.16

POINT 31.287 -2 -25.6 POINT 36.707 -27 -26.972

POINT 33.237 -2 -26.985 POINT 18.93 -32 -17.39

POINT 35.622 -2 -28.842 POINT 21.912 -32 -18.15

POINT 37.617 -2 -30.562 POINT 24.122 -32 -18.927

POINT 17.982 -7.002 -19.352 POINT 26.907 -32 -20.13

POINT 20.747 -7.002 -20.002 POINT 29.087 -32 -21.265

POINT 23.745 -7.002 -21.055 POINT 31.212 -32 -22.522

POINT 26.3 -7.002 -22.205 POINT 33.577 -32 -24.08

POINT 28.637 -7.002 -23.452 POINT 36.175 -32 -25.997

POINT 31.207 -7.002 -25.045 POINT 38.687 -32 -28.097

POINT 33.04 -7.002 -26.307 POINT 18.957 -37 -16.965

POINT 35.405 -7.002 -28.102 POINT 22.58 -37 -17.915

POINT 37.09 -7.002 -29.537 POINT 26.54 -37 -19.492

POINT 17.867 -12.002 -18.882 POINT 29.947 -37 -21.26

POINT 21.467 -12.002 -19.785 POINT 32.905 -37 -23.102

POINT 24.195 -12.002 -20.78 POINT 35.802 -37 -25.177

POINT 27.087 -12.002 -22.117 POINT 38.382 -37 -27.265

POINT 29.485 -12.002 -23.457 POINT 40.315 -37 -28.982

POINT 31.725 -12.002 -24.867 POINT 18.927 -42 -16.527

POINT 33.35 -12.002 -26.005 POINT 21.907 -42 -17.26

POINT 35.265 -12.002 -27.452 POINT 25.147 -42 -18.42

POINT 37.082 -12.002 -28.947 POINT 28.17 -42 -19.817

POINT 38.322 -12.002 -30.06 POINT 30.767 -42 -21.257

POINT 18.912 -17 -18.677 POINT 33.207 -42 -22.812

POINT 21.987 -17 -19.5 POINT 35.782 -42 -24.622

POINT 24.612 -17 -20.487 POINT 37.407 -42 -25.897

POINT 27.097 -17 -21.657 POINT 39.672 -42 -27.837

POINT 29.755 -17 -23.112 POINT 18.94 -47 -16.095

POINT 32.882 -17 -25.14 POINT 22.697 -47 -17.062

POINT 36.017 -17 -27.5 POINT 26.962 -47 -18.757

POINT 38.202 -17 -29.382 POINT 30.682 -47 -20.702

POINT 18.737 -22 -18.207 POINT 34.345 -47 -23.055

POINT 20.807 -22 -18.697 POINT 36.63 -47 -24.722

POINT 23.547 -22 -19.607 POINT 39.425 -47 -27.055

POINT 26.892 -22 -21.067

POINT 29.297 -22 -22.362 Three non collinear points

POINT 32.482 -22 -24.352

POINT 35.097 -22 -26.252 POINT 39.43 -1.55 -36.477

POINT 36.937 -22 -27.707 POINT 40.535 -20.055 -33.092

POINT 38.555 -22 -29.102 POINT 41.935 -42.882 -38.787

105

“We are dealing here with a single flank on one side only of a real tooth on a real wheel. It is

important to understand that in gears the axis of the said involute helicoid is under all

circumstances collinear with the axis of the wheel.” (Phillips 2003 §3.24) The measured

surface of the gear body is actually a piece of the involute helicoid, similar to the involute

helicoid shown in Figure 28. In the case of the gear bodies, however, the axis of the involute

helicoid, shown in Figure 28, is aligned to the gear axis, not the z-axis. There is an involute

helicoid mounted on both GA1 and GA2 of Figure 1. It is, therefore, a fundamental property

of the measured surface that the axis of rotation of the gear bodies is aligned with the axis of

the involute helicoid.

For the results to correlate to the equation of the involute helicoid the data must be

transformed (Appendix J details the principles of object transformation.) such that:

• z axis of the CMM is collinear with the theoretical axis of rotation and

• the origin of the CMM is coincident to the centre of the base circle of the gear body.

Consider the arrangement indicating the orientation of the axes of the CMM relative to the

Gear body and, in Table 22, the resulting data for Gear body 1 for y=-2.

Table 22: Raw data from the CMM for Gear body 1

Data X Y Z

POINT 20.457 -2.002 -6.952

POINT 22.57 -2.002 -7.317

POINT 25.087 -2.002 -8.15

POINT 27.372 -2.002 -9.18

POINT 28.96 -2.002 -10.037

POINT 30.837 -2.002 -11.237

POINT 32.795 -2.002 -12.672

POINT 34.1 -2.002 -13.752

106

To align the y-axis of the involute helicoid on the CMM with the z-axis of the involute

helicoid of (51), requires two steps:

1. A 90o rotation of the gear bodies about the x-axis thereby making the z-axis of the

CMM parallel to the z-axis of the theoretical involute helicoid. This can be expressed

mathematically as the transformation matrix

2. Translation of the centre of the base circle to the origin of the CMM making the z-axis

of the CMM collinear to the z axis of the theoretical involute helicoid and the centre of

the base circle and CMM origin coincident. There are three components to this

translation. The centre of the gear body boss must be moved:

a. in the positive x-direction, a distance equal to the distance between the gear

axis and the origin of the CMM.

b. in the negative y-direction, half the thickness of the gear body (measured to the

top land).

3. Toward the base circle a distance gx,mn,E, given by the distance between the centre of

the base circle, e-circle and gear body. This distance can be seen in Figure 32, a view

defined by the Gear Axis and Centre Distance Line. It can be shown (Appendix L) that

this distance is, for Gear Axis 1:

gx,1n,z* = t*mn + xesinΣ/2 − yecosΣ/2 + (gear body width)/2

and for Gear Axis 2:

(54)

where:

−

1000

0100

0090cos90sin

0090sin90cos

gx,2n,z* = t*mn - xe.sinΣ/2 − ye.cosΣ/2 + (gear body width)/2

107

t*mn = the distance from the CDL to the centre of the base circle for gear body

n measured along the Gear Axis m.

xe = the x coordinate of the point E

ye = the y coordinate of the point E

Gear body width = the width of the gear body from drawing A30006 Revision A or A30005

Revision A in Appendix G.

108

Figure 32: A view of Gear Body 2 defined by Gear Axis 2 and the Centre Distance Line

For the Gear Body under consideration these values are 26.66, -20 and g*mn, respectively.

This produces the transformation matrix

that when applied to the data in Table 22 produces the following.

− 1*2066.26

0100

0010

0001

mng

gear body 2

axis

centre distance line

gear body 1

axis

e-circle base circle

gx,mn,E

gear body 2

109

Table 23: Transformed data for Gear body 1

Data X Y Z

POINT 47.117 -13.048 21.1639

POINT 49.23 -12.683 21.1639

POINT 51.747 -11.85 21.1639

POINT 54.032 -10.82 21.1639

POINT 55.62 -9.963 21.1639

POINT 57.497 -8.763 21.1639

POINT 59.455 -7.328 21.1639

POINT 60.76 -6.248 21.1639

The figures below illustrate the result of applying the process to Gear bodies 1 and 2

respectively for selected values of z.

Figure 33: A graph of the profile for selected values of z for Gear Body 1

RESULTS OF GEAR BODY 1 PROFILE

MEASUREMENT

-14

-12

-10

-8

-6

-4

-2

0

40 45 50 55 60 65

X VALUE (mm)

Y V

AL

UE

(m

m)

Z=21.164

Z=1.164

Z=-18.83

110

Figure 34: A graph of the profile for selected values of z for Gear Body 2

An assessment of and comments on the internal consistency of the data: The relative

smoothness of the curves in the above figures indicates that the data is internally consistent.

The accuracy, however, of this judgement is severely limited by the fact that the curves are at

least 0.5mm thick and the marks representing the data points are two to three millimetres

across. In comparison to the 0.001mm level of accuracy of the CMM, these values are,

figuratively speaking, potentially miles apart.

An accurate method of comparing the consistency of the data points is to utilise the fact that

each of the above lines represents a section of the planar involute. It was shown above and in

Appendix K, that the planar involute has the parametric equation

(55)

The values of x and y can be taken directly from the data above leaving two unknowns for

each point, rb and t7. As each of the curves in the above graphs represents a planar section of

the involute, all points in a given curve, for example z=-34.87, should have a common rb

RESULTS OF GEAR BODY 2 PROFILE

MEASUREMENT

-6-4-202468

1012

80 85 90 95 100 105

X VALUE (mm)

Y V

AL

UE

(m

m)

Z=-34.87

Z=-14.87

Z=--5.132

x = rb(cost7+ t7.sint7), y = rb(sint7- t7.cost7)

111

whilst the value of t7, representing the rotation of the involute generator, will vary from point

to point.

The mechanics of this process however, are considerably more complicated than is indicated

here. As can be seen, equation (55) is highly non-linear. Eliminating rb, for example, by

calculating x/y leaves:

(56)

, an equation that is impossible to solve by other than numerical means.

Sticher was able to demonstrate this expression could be stated as the well-known involute

function in λ. By inverting and dividing the top and bottom by cost7, the above expression

becomes:

77

77

tan1

tan

tt

tt

x

y

+−

=

which is now in the form:

BA

BABA

tantan1

tantan)tan(

+−

=−

allowing the substitution

)tan( 7 λ−= tx

y

where

7

1tan t−=λ

therefore

)(tan

tantan 7

1

7

1

λλλ inv

ttx

y

=−=

−= −−

777

777

cossin

sincos

ttt

ttt

y

x

−+

=

112

This expression still needs to be solved numerically however it is a neater and more widely

recognised form of (56). Utilising Microsoft Excel Solver, a number of models were set up

with the objective of determining rb and t7 necessary to solve the above equations for various

values of x and y. The Left Hand Side (LHS) of the model is the value x or y from the CMM

whilst the Right Hand Side (RHS) is the expression rb(cost7…) or rb(sint7…) respectively. A

sample model is shown in the tables below.

Table 24: Sample data points from Gear body 2 for analysis of data consistency

X Y Z

93.127 -1.257 -5.132

95.567 -2.812 -5.132

Table 25: Model 1 solving the involute function for Point 1

X Y rb t7

LHS 93.127 1.257 87.86369 0.351567

RHS 93.127 1.257

LHS-RHS 8.63E-12 1.41E-12

Table 26: Model 2 solving the involute function for Point 2

X Y rb t7

LHS 95.567 2.812 86.75498 0.463158

RHS 95.567 2.812

LHS-RHS -1.9E-07 -3.8E-08

113

The results in the above tables indicate that rb for two points on the same planar involute is

not the same. An alternate approach is to use the Solver to force rb from Model 1 to equal

Model 2.

Table 27: Model 3 solving the involute functions by forcing rb to be equal

Involute

point 1

X Y rb t7

LHS 93.127 1.257 86.75498 0.353071

RHS 91.99502 1.257

LHS-RHS 1.131983 -1.1E-09

Involute

point 2

X Y rb t7

LHS 95.567 2.812 86.75498 0.463158

RHS 95.567 2.812

LHS-RHS -1.2E-08 -2.5E-09

Herein lies the problem. From the above analysis it would appear the data is not internally

consistent. If the models allow different base radii, the base radii of the same planar involute

are different by approximately 1.2 or 1.4%. Alternately if the base radii are forced to be equal

not all conditions of the model can be satisfied. The difference between the data point value

of y and the calculated value of y in Point 1 is approximately 1.3 or 1.4%. Although, in

absolute terms, these errors may appear to be acceptable, the error in the set up was just

0.005mm and the accuracy of the CMM was a mere 0.007mm with a repeatability of

114

0.005mm. (Ferranti Mercury) The total error is of the magnitude of 0.017mm. There is

therefore another source of error.

Another source of error: The investigation above indicates there is a source of error with an

order of magnitude of 1 to 1.5mm. The cause of this error may well lie in the fact that the

CMM’s ball radius compensation capabilities were turned off. So what is the order of

magnitude of the error resulting from ball-radius compensation? The following section

establishes an estimate of the magnitude of the error caused by ball-radius compensation

being turned off.

Consider the planar case of a circle, with a radius rball, approaching a line that is inclined to

the horizontal at an angle of theta (θ). At the point of contact, the said line forms a tangent to

the circle and the line joining the centre of the circle (xm, ym) to the point of contact (xa, ya)

therefore subtends 90o to the said line. This is illustrated below.

Figure 35: Ball radius compensation – the planar case

115

Assuming a standard right handed Cartesian coordinate system,

xa = xm + rball sinθ and ya = ym + rball cosθ

Now

θ = 90 – t7

which when substituted produces

(57)

The accuracy of (57) can be verified mentally by imagining the errors resulting from

measuring vertical and horizontal surfaces respectively. Bringing the CMM probe into contact

with a vertical surface (t7=0) would result in an error of rball in the x direction and zero in the y

direction and vice versa for a horizontal surface.

Alternately, relating θ to the gradient of the involute requires the product rule to be applied in

two stages. Firstly, to determine the gradient from its constituent equations,

And secondly, applying the product rule to the two components of (55), in turn produces

dy/dt7=rbt7cost7

and

dt7/dx=1/(rbt7sint7)

Back substitution then produces the following result

dy/dx=1/(tant7)

a result that is obvious when one looks at the form of the equation of the involute as a

function of the base radius (Appendix K).

The error or the difference between xa and xm, for example, is rball cost7. The error therefore

varies non-linearly depending on the location of the CMM probe on the involute surface and

dx

dt

dt

dy

dx

dy 7

7

=

xa = xm + rball cost7 and ya = ym + rball sint7

116

is different for the x and y values. Substituting the values from the models in the above tables,

for example, produces an error in Model 1 of 1.885 in the x direction and 0.874 in the y

direction. Using the planar estimate, it is possible from the comparison of the relative sizes of

the errors as measured and those likely to be caused by the absence of ball radius

compensation.

The above analysis is limited to the planar case. The spatial case places further restrictions on

the accuracy of the profile measurement using the CMM.

Now consider the spatial case. The diagram below represents the ball of the CMM.

Figure 36: Ball radius compensation - the spatial (3-D) case

P

x

y

z

φ

θ

O

117

Again the point O represents the centre of the ball corresponding to the measured values (xm,

ym, zm) and the point P, located on the surface of the ball, which represents the actual values

(xa, ya, za). The distance OP is equal to the radius of the ball, rball. The values φ and θ represent

the angles the line OP subtends to the z and x axes, respectively.

The relationship between this point, P, and the point given by the CMM, O, is

xa = xm + OPx ya = ym + OPy za = zm + OPz

The distances OPx, y, z can be determined from the given geometry.

OPx = rball sinφ cos θ OPy = rball sinφ sin θ OPz = rball cos φ

(The reader would notice that the components of OP are, in fact, the equations of the spherical

coordinate system with rball substituted for ρ.)

Substituting the values for OP produces,

(58)

From the above equations, the following generally applies:

• The closer the surface is to vertical, the greater the difference between xa and xm and

ya and ym. This error varies as a function of the sine of the angle.

• The closer the surface is to vertical, the smaller the difference between za and zm. This

error varies as a function of the cosine of the angle.

• The closer the surface is to being parallel to the yz plane, the greater the difference

between xm and xa and less the difference between ya and ym and vice versa.

An additional unknown has, however been introduced. To get an estimate of the error created,

in the spatial case, by the absence of radius compensation, it is now also necessary to know

(or have an estimate of) φ.

xa = xm + rball sinφ cos θ

ya = ym + rball sinφ sin θ

118

It is possible to obtain an estimate of φ by considering the relationship between the ball of the

CMM and the surface it is measuring. At the point of contact with the surface of the involute

helicoid, P, the surface of the sphere and the surface of the involute helicoid share a common

tangent plane i.e. their tangent planes are coplanar. The normal to this tangent plane passes

through O and P i.e. the direction of the radius represents the ‘radius vector’ (Anton 1995), r

= xi + yj + zk where, from Figure 36:

x = sinφ cos θ

y = sinφ sin θ

z = cos φ

Restating r,

(59)

At some point on the involute helicoid, there is a tangent plane with a unit normal equal to the

above equation. All that remains is to find the location of this unit normal. Recall that:

(41)

defined the tangent plane to the involute helicoid. The tangent plane of the sphere and

involute helicoid at the point of contact are co-planar therefore the normals are collinear.

Equating x, y, & z components of (59) and ∂(51)/∂µ produces:

x component -sinγb . sint7 = sin φ . cos θ

y component sinγb . cost7 = sin φ . sin θ

z component -cosγb .= cos φ

Some general comments on the above result:

r(φ, θ) = sinφ cos θ i + sinφ sin θ j + cos φ k

kjttn bbbˆcosˆcossinisinsin-ˆ

77 γγγ −+=

119

• The solution is represented by the intersection of the two surfaces given by the x and y

components and the line given by the z part. The surfaces, assuming constant γb, are

shown in Figure 37 and Figure 38.

Figure 37: All solutions to the common normal to the sphere and involute helicoid given by the x

component

φ is calculable from the z component and, if γb is acute, 90 < φ < 270. From the spherical

coordinate system, the point of contact is therefore below the equator of the ball, a logical

location.

γb is known from the initial geometry leaving three equations and three unknowns. This

system of three (highly) non-linear equations can be solved numerically using Microsoft

Excel Solver, for example. One possible solution is shown in Table 28.

0

0.6

28

1.2

57

1.8

85

2.5

13

3.1

42

0

1.6760

0.5

1

1.5

2

2.5

3

3.5

θθθθ

t7

φφφφ

GRAPH OF θθθθ v. t7 AND φφφφ(X Component)

3-3.5

2.5-3

2-2.5

1.5-2

1-1.5

0.5-1

0-0.5

120

Figure 38: All solutions to the common normal to the sphere and involute helicoid given by the y

component

Table 28: A numerical solution for the set of equations defining the common tangent plane to the

sphere and involute helicoid

Angle Value (rads)

γb 1.232891

φ 1.908702

θ 1.570797

t7 6.3E-07

Thus from (58) and assuming rball = 2mm

OPx = -1.22679E-06 OPy= 1.886902 OPz = -0.66302

xa = xm + -1.22679E-06 ya = ym + 1.886902 za = zm - 0.66302

Note: The values OPx, y & z provide a check on the accuracy of the equations. The square root

of the sum of the squares of OPx, y &z should (and does) equal 2 (to within 0.001%).

Of course, when γb is known, the z component can be used to determine φ i.e.

φ = cos-1

(-cosγb)

from which θ can be determined by:

0

0.6

28

1.2

57

1.8

85

2.5

13

3.1

42

0

1.152

2.304

-1.5

-1

-0.5

0

0.5

1

1.5

θθθθ

t7

φφφφ

GRAPH OF θθθθ v. t7 AND φφφφ(Y Component)

1-1.5

0.5-1

0-0.5

-0.5-0

-1--0.5

-1.5--1

121

• rearranging the X and Y components to make a substitution for sinφ

• substituting φ into the X and Y components respectively.

The equations resulting from these actions are:

)sin

cos(tan

7

71

t

t−= −θ

or

)2

(-or 2

)cos(sinor )(cossin

77

7

1

7

1

tt

tt

−−=

−= −−

ππθ

θ

Thus, θ is a function of t7. As t7 represents the rotation around the helix, the impact of moving

around the helix on the values OPx, y & z can be shown graphically, as it is in Figure 39.

Figure 39: Graph of OPx, OPy, OPz and theta versus t7

The two equations produce the same results. Theta (sin) peaks at π whilst Theta (tan) equals

zero at the same value of t7.

GRAPH OF OPx, OPy, OPz & THETA v. t7

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 1 2 3 4

t7 (rads)

DIS

T (

mm

)

-2

-1

0

1

2

3

4

TH

ET

A (

rad

s) OPx

OPy

OPz

sin(theta)

tan(theta)

122

Figure 39 shows that the impact on xa, ya and za is significant compared to the combined

errors of the CMM of 0.017mm. Taking a best case of the maximum xm, ym and zm from the

CMM and comparing them to the errors shown above produces the following:

Table 29: The impact of ball radius on the accuracy of CMM values

x y z

Max measured value 60.802 -13.048 -22.002

A sample error 1.22679E-06 1.886902 -0.66302

% of ordinate value 2.018x10-6

% 14.46% 3.0134%

These results highlight the shortcomings of this process. Table 29 shows the error is

potentially a lot greater than the required error of 1.4% determined in Table 27. The

magnitude of the error in the above example is approximately 15%. The above process also

assumed γb was accurately replicated on the gear bodies however this is one of the values to

be verified. Given the objective of profile measurement is to determine how accurately the

machined surface matches the theoretical surface, it is clearly desirable to use measured

values only.

From the above data and analysis it is apparent that due to the complexity of the surface, and

limitations of the measurement equipment and process, profile measurement in this case has

been of limited value in assessing the accuracy of the machining process and resulting

surface.

123

CHAPTER 8 CONCLUSION

This work started out with three aims. They were:

• in the broadest sense, to close the Plan-Do-Check-Act cycle on the equiangular gear

body development. Specifically, the aim of the work was to test the validity of

Phillips’ claims with regard to constancy of transmission ratio for the as manufactured

gear tooth pair.

• to move the development of a ‘universal theory of gearing’ to the second stage.

• to start to bridge the gap between the theory and the practice of gear manufacture.

8.1 A universal theory of gearing

After some refinement of the equiangular models afforded by an insight that Dr F.C.O.Sticher

provided, a completely new set of mathematical models of gearing were developed. These

models drew on the fundamental principles from three-body kinematics leading into the

development of gear architecture and surface geometry. Furthermore they were applicable to a

wider range of spatial involute gears, the plain polyangular set, of which the equiangular case

with the corresponding equations previously developed, are a subset. The design process for

the plain polyangular set also required an additional user selectable value, d, that defined the

location of the point of contact, Qx.

As would be expected, substitution of equiangular data into the polyangular model produced

the same results as the equiangular model detailed in Killeen (1996), thereby proving the

validity of the model. To reinforce this point, a numerical model was developed concurrently

in MS Excel and its results compared to the results of the algebraic model. In all cases the

results correlate.

124

The work also highlighted a significant difference between the architecture standards adopted

here and those of Phillips and the robustness of the theories. Phillips used a geometrical frame

of reference with a specific set of axes to determine the answers to fundamental kinematic

questions and then switched to a different set of axes to develop the architecture and surfaces

of the gear bodies. This work employed the same set of axes in both the development of

answers to the kinematic question as well as for the formulation of the gear architecture and

surface geometry. The results were shown to be the same.

To highlight the simplicity and logic behind the theory and algebraic development, the work

also presented a 15-step process map that defined the steps undertaken in creating the

algebraic expressions. Following these steps leads to the development of the aforementioned

numerical model applicable to plain polyangular spatial involute gears.

The algebraic and numerical models also conclusively proved the validity of some aspects

that have hitherto, been shown only geometrically. Specifically, for example the key

outcomes were that ‘…at Ω=0 [t7=0], the slip track and the core helix are tangential with one

another…’ This confirms the work of Phillips (2003).

8.2 A bridge between the kinematic theory and gear machining theory

The algebraic models developed also provided a solid foundation for the transition from

kinematic analysis to machining special involute gears in practice. Following the creation of a

solid foundation in the understanding of the principles of gear machining, this work showed

how the equations previously developed could be applied to gear machining. Six of the planes

Phillips discusses were also fully mathematically defined. The behaviour of three of these

planes was presented graphically. Understanding the behaviour of these planes and the

expressions will be central to the development of machining techniques for a complete plain

polyangular gear set.

125

8.3 Functional testing

Mechanical functional inspection showed, despite small variations in:

• the surface roughness and texture of the gears caused by machining

• the location of the gears

that the error, measured over three tests, was 1% for two of those tests and less than 2% for

the third. The lines of best fit had a correlation of greater than 0.999. Furthermore, the results

of the testing also indicated that there was in excess of 40% chance the transmission ratio as

measured was exactly equal to as designed.

This directly confirms Phillips’ geometry that the transmission ratio of the gears designed

using a general spatial involute theory of gearing is given by:

ur

r

bb

bb =11

22

sin

sin

γγ

for the case of equiangular spatial involute gears i.e. that the transmission ratio is independent

of small variations in the location of the gears.

Initial attempts to obtain ‘instantaneous’ transmission ratios were unsuccessful. The

arrangement based on a Labview virtual instrument and the following hardware:

• optical encoders mounted on the gear body axes that fed a binary signal to

• a DAC mounted in a PC using a Labview virtual instrument

to interrogate the card was unsuccessful due to the limitations imposed by the operating

system of the Personal Computer. Windows 3.1 can only acquire data every 55ms. As the

encoders used produced 5000 pulses per revolution, the minimum time to complete one

revolution was approximately nine minutes. This effect was compounded by the fact that

there were two encoders attached to the same DAC. Although data acquisition was

undertaken considerably faster than this, re-running the tests at the required speed was not

126

practicable due to the risk of moving in the wrong direction. Despite this, the use of optical

encoders and PC based instrumentation proved itself in principle as a more flexible and robust

method of measurement. Further, the creation of electronic data increases flexibility in terms

of analysis.

8.4 Analytical inspection

Efforts aimed at developing a more accurate picture of both the as-machined surface and

‘instantaneous transmission ratio’ were largely unsuccessful due to limitations with the

measuring equipment. The results of surface measurement using a Coordinate Measuring

Machine (CMM) could not be correlated with the theoretical tooth profile. The magnitude of

internal error using the CMM was approximately 1.5% whilst the actual ‘as-measured’ error

was approximately 15%. The primary cause of this was the impact of the gradient(s) of the

surface on the calculation of the actual position of the point versus the measured position of

the same point. This was substantiated by analysis of the results that showed the larger the

gradient(s) the higher the error.

This work has achieved the original objectives in that it proved the validity of Phillips theories

by demonstrating, through experimentation, the high correlation between theoretical and

actual transmission ratios. The development of mathematical expressions and a numerical

model for the plain polyangular gear bodies, a model which is more robust than the previous

equiangular model of Killeen (1996), provides a basis for a designing such gears in practice.

Picking up from the leads alluded to in the previous two paragraphs, significant opportunities

exist for converting the geometric theories of Phillips (2003) and the mathematical

expressions of this thesis into working gears. Although it is possible to machine these gear

bodies using a five-axis machine, efforts should be directed toward the use of high volume

127

manufacturing processes such as hobbing would allow the spatial involute gear to be mass

produced.

With a complete gear set, work can then begin on aspects of gear design including wear,

lubrication and the deformation of the tooth surface under load. Investigators in this field will

be assisted considerably by the fact that Phillips (2003) established a tooth surface that can be

defined algebraically thereby making the work of Timoshenko and Goodier (1970, p414)

applicable and minimising the need for exclusively developing solutions through the ‘skilful

use of computers’.

128

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136

APPENDIX A GLOSSARY OF TERMS AND NOTATIONS

This glossary contains all terms and symbols used in this thesis. The terms and symbols where

possible are identical to those in AS2075-1991. AS2075-1991, in turn, uses ‘ISO 701

International gear notation’, ‘ISO 1122 Glossary of gear terms’, ‘ISO 1122.1 Part 1:

Geometric definitions’ and ‘ANSI/AGMA Gear nomenclature-definitions of terms with

symbols’ as its references. (AS2075 1991)

This thesis and the work of Phillips is a foundation for years of gear development work and,

as they were written from a more mathematical/geometric point of view respectively, it has

become necessary to enhance or add to the standard symbolism. A reader of this thesis will

therefore find that the symbolism used is starting to deviate from Australian Standard

symbolism.

137

Kinematic definitions: terms relating to the relative position of axes

Term Symbol Definition

Toothed gear - Any toothed member designed to transmit motion to another

one or receive motion from it by means of successively

engaging teeth

Gear pair - An elementary mechanism consisting of two meshing gears

Centre distance a The distance between the axes of a gear pair is measured

along a line perpendicular to the axes.

Shaft angle Σ The smallest angle through which one of the axes must be

rotated to bring the axes into coincidence (gear pair with

intersecting axes), or must be swivelled to bring the axes

parallel (gear pair with non-parallel non-intersecting axes),

so as to cause their directions of rotation to be opposite.

Kinematic definitions: terms relating to mating gears

Term Symbol Definition

Mating gear - Either gear of a pair, considered in relation to the other one

Pinion Suffix 1 That gear of the pair which has the smaller number of teeth

Wheel Suffix 2 That gear of the pair which has the larger number of teeth

Driving gear That gear of a pair which turns the other gear

Driven gear That gear of a pair which is turned by the other gear

Kinematic definitions: terms relating to relative speeds

Term Symbol Definition

Gear ratio u The ratio of the number of teeth of the wheel to that of the

pinion

Transmission

ratio

i The ratio of the angular speed of the first driving gear of a

gear train to that of the last driven gear

Kinematic definitions: terms relating to pitch and reference surfaces

Term Symbol Definition

Pitch surface - The geometrical surface described by the instantaneous axis

of movement of the mating gear in relation to the gear under

consideration in a given gear pair

138

Term Symbol Definition

Pitch line The instantaneous axis of movement of the mating gear in

relation to the gear under consideration (inferred from the

above AS2075 1991 definition). Phillips (1990) defined the

pitch line as the line of the Instantaneous Screw Axis (ISA)

of body 3 with respect to body 2 (ISA23). The ISA is the axis

about which the twisting of the body at the instant is

occurring (Phillips 1984).

Pitch point The point of intersection between the ISA and the Centre

Distance Line (CDL). In the planar case, the pitch point

corresponds to the point of intersection between the two

pitch circles.

Reference

surface

- A conventional surface with reference to which the tooth

dimensions of a gear, considered alone, are defined

Reference… - A qualification applicable to every term defined from the

reference surface of a gear

Working… Suffix w or

sign’

(apostrophe)

A qualification applicable to every term defined from the

pitch surface of a gear in a gear pair. The apostrophe may be

replaced by the suffix w and vice versa

Pitch plane of

a gear pair

- The plane tangent to the pitch surfaces of a gear pair.

Tooth characteristics: terms relating flanks and profiles

Term Definition

Tooth profile The line of intersection of the tooth flank with any defined surface

Transverse profile The line of intersection of a tooth flank by a plane surface perpendicular

to the straight generator of the reference surface. For a cylindrical gear,

this plane surface is perpendicular to the axis of the gear.

Normal profile The line of intersection of a tooth flank by a plane surface perpendicular

to the tooth trace.

Axial profile The line of intersection of a tooth flank by a plane containing the axis of

the gear

Design profile The theoretical profile, inclusive of any tip or root relief, required of a

tooth flank.

Tooth characteristics: terms relating to parts of the flank

Term Definition

Active flank That portion of a tooth flank of a gear which contacts the tooth flank of

a mating gear

139

Geometrical and kinematical notions used in gears: terms relating to geometrical lines

Term Definition

Helix On a cylinder of revolution, a curve with tangents that is inclined at a

constant angle to the generators of a cylinder.

Helix angle, β

suffix

The acute angle between the tangent to a helix and the straight generator

of the cylinder on which the helix lies.

Lead angle, γ

suffix

The acute angle between the tangent to a helix and a plane

perpendicular to the axis of the cylinder on which the helix lies.

Lead, pz The distance between two consecutive intersections of a helix by a

straight generator of the cylinder on which it lies

Cycloid A plane curve described by a point on a circle (the generating circle)

which rolls without slip on a fixed straight line (the base line)

Epicycloid A plane curve described by a point on a circle (the generating circle)

which rolls without slip on the outside of a fixed circle (the base circle)

Hypocycloid A plane curve described by a point on a circle (the generating circle)

which rolls without slip on the inside of a fixed circle (the base circle)

Involute to a circle A plane curve described by a point on a straight line (the generating

line) which rolls without slip on a fixed circle (the base circle)

Spherical involute On a surface of a sphere, the curve described by a point on a great circle

(the generating circle) which moves over the sphere by rolling without

slip on a fixed small circle of the sphere (the base circle)

Involute roll angle For a given point on an involute curve the angle subtended, at the centre

of the base circle, by the intersection of the involute with the base circle

and the point where the tangent of the base circle from a given point

touches the base circle.

Polar angle For a given point on an involute curve the angle subtended at the centre

of the base circle, by the point and the intersection of the involute with

the base circle.

Parametric

coordinate, tn

In defining various lines using parametric equations it was necessary to

select a parameter. The parameter chosen was ‘t’ and each t, t1 to tn is a

parameter specific to the line being defined. This is not an Australian

Standard definition.

140

Geometrical and kinematical notions used in gears: terms relating to geometrical

surfaces

Term Definition

Normal plane A plane generally perpendicular to the traces of a tooth.

In a helical rack it is perpendicular to all tooth traces of all the teeth it

intersects.

In a helical gear pair, however, a plane can be perpendicular at the point

of intersection, to only one tooth trace of one tooth. Such a plane, or a

plane perpendicular at the point of intersection to any other line lying in

the reference or pitch surface in a direction corresponding with that of a

tooth trace, is a normal plane.

Important lines, other than a tooth trace, to which a normal plane may

be perpendicular, include the line lying in the reference or pitch surface

and denoting the centre of a tooth or of a tooth trace.

Axial plane A plane containing the axis of a gear.

Axial plane of a

pair2

The plane containing the gear axes.

Plane of rotation A plane perpendicular to the axis of the gear.

Transverse plane A plane perpendicular to a generator of the reference surface of a gear.

Involute helicoid The surface generated by a straight line inclined at a constant angle to

the axis of a cylinder of revolution (the base cylinder) and rolling

without slip on the surface of that cylinder (i.e. constantly tangential to a

helix of the cylinder)

2 This definition does not apply to spatial involute gearing because the entity that contains the axes ie axes of the

gear pair, is (generally) not a plane but a special type of ruled surface called a parabolic hyperboloid with axes

defined by the gear axes and the centre distance line. Phillips (2002) defines four parabolic hyperboloids of

which ‘the axial parabolic hyperboloid of a pair’ is a rectilinear parabolic hyperboloid (RPH) 4 shown below.

Gear Axis 3

Gear Axis 2

Parabolic hyperboloid defined

by GA2, GA3 and CDL

Pitch circle 3

Pitch circle 2

Centre Distance Line

x

y

141

Term Definition

Spherical involute

helicoid

The surface generated by a straight line inclined at a constant angle to

the axis of a cone of revolution (a base cone) and rolling without slip on

the surface of that cone.

A section by a sphere having its centre at the apex of the cone is a

spherical involute.

Geometrical and kinematical notions used in gears: other terms

Term Definition

Instantaneous axis3 In a gear pair with parallel or intersecting axes, the imaginary line

around which occurs the relative instantaneous rotation of a gear in

relation to its mating gear.

In a gear pair with non-parallel non-intersecting axes, the imaginary line

around which occurs the relative instantaneous helical movement of a

gear in relation to its mating gear.

Slide/roll ratio The ratio of the instantaneous sliding and rolling velocities at the

contacting surfaces of meshing gears and any point during the mesh. It

is usually considered only at the extremes of the path of contact.

Generator A point, line or figure which, by its motion, traces out a line, surface or

other parameter

3 AS2075 1991 duplicates the definition of ‘Instantaneous axis’. ”The most general form of instantaneous

relative movement between two bodies is not rotational but a screwing.“ (Hunt 1978 p58) and Phillips (1984

p49) states, “An instantaneous motion …any relative motion between a pair of rigid bodies-can be seen as a

screwing, or a twisting motion, about an axis with a pitch.” The former of the two definitions of instantaneous

axis in AS2075 (1991) is therefore simply a special case of the latter. Hunt (1978) defines the former as an

Instantaneous Rotational Axis (IRA) and the latter is an Instantaneous Screw Axis (ISA) respectively

142

Cylindrical gears and gear pairs: terms relating to cylinders and circles

Term Definition

Reference cylinder The reference surface of a cylindrical gear. It is the pitch surface of

engagement with the basic rack

Pitch cylinder The pitch surface of a cylindrical gear in a gear pair with parallel axes

Reference circle The line of intersection of the reference cylinder by a plane

perpendicular to the axis of the gear

Pitch circle The line of intersection of the pitch cylinder by a plane perpendicular to

the axis of the gear

Reference

diameter, d

The diameter of the reference circle

Pitch diameter, d’ The diameter of the pitch circle

Base circle For an involute cylindrical gear, the circle from which the tooth flank

involute is derived

Base cylinder The cylinder, coaxial with the gear, having the base circle as the

transverse section

Base diameter, db The diameter of the base circle and the base cylinder

Facewidth, b The width over the toothed part of the gear, measured along a straight

line generator of the reference circle. (Consider the facewidth of a rack

as being that of a gear of infinitely large diameter)

Cylindrical gears and gear pairs: terms relating to helices of helical gears

Term Definition

Reference helix The helix which contains a tooth trace of a helical gear

Pitch helix The helix which contains the intersection of a toothed flank with the

pitch cylinder of a helical gear

Base helix In an involute helical gear, the intersection of the involute helicoid of a

flank with the base cylinder

Helix angle, β The helix angle of the reference helix of a helical gear. (Consider the

helix angle of a rack as being that of a gear of infinite diameter)

Base helix angle,

βb

The helix angle of the base helix of an involute helical gear

Lead angle, γ The lead angle of the reference helix of a helical gear. (Consider the

lead angle of a rack as being that of a gear of infinite diameter)

Base lead angle, γb The lead angle of the base helix of an involute helical gear

Figure 40: A diagram of the helix angle (a) and lead angle (b)

143

Cylindrical gears and gear pairs: terms relating to transverse dimensions

Term Definition

Transverse

pressure angle, αt

The transverse pressure angle at the point where the profile cuts the

reference surface.

Transverse pitch or

circular pitch, pt

The length of the arc of the reference circle lying between two

consecutive corresponding profiles (for a rack the linear pitch is the

plane of translation of the rack).

Transverse

module, mt

The quotient obtained by dividing the transverse pitch by π (or the

quotient obtained by dividing the reference diameter by the number of

teeth).

Cylindrical gears and gear pairs: terms relating to generating tools and associated

features

Term Definition

Rack type cutter A generating cutting tool in the form of a rack with suitable flank relief

and cutting profile

Pinion type cutter A generating cutter tool, in the form of a cylindrical gear, with suitable

flank relief and cutting profile

Hob A generating cutter tool, in the form of a cylindrical worm, suitably

gashed and relieved to provide a series of cutting profiles

Nominal pressure

angle, αno

The normal pressure angle of the basic rack of the gear cut by the tool

Nominal pitch of

the cutter, pno

The normal pitch of the basic rack of the gear cut by the tool

Cutter module, mno The quotient obtained by dividing the nominal pitch of the cutter by π

144

APPENDIX B A PROCESS MAP FOR THE DEVELOPMENT OF PLAIN

POLYANGULAR ARCHITECTURE AND SURFACE

ENTITIES

The process maps are presented in the Integrated Definition for Function Modelling format

(IDEF0). In this format:

• inputs are shown entering from the left

• outputs are shown leaving from the right

• constraints are shown entering from the top

• mechanisms are shown entering from the bottom

145

XX REMOVE THIS PAGE AND REPLACE WITH A3 OF PROCESS MAP

146

APPENDIX C A NUMERICAL EXAMPLE

With the exception of some specific numerical examples, the reader has had little exposure to

the mechanics of performing the calculations outlined in CHAPTER 3. This appendix

provides a numerically based case study that:

• provides a complete numerically based case study

• allows the reader to check their calculations should they be inclined to apply the

values to the formulas of the aforementioned chapter

• would form the basis of any future programming

The numerical example resides on an MS Excel spreadsheet and is based on plain polyangular

architecture for the values shown in Table 30.

Table 30: Values for the numerical example

Dimension Symbol Value

Centre distance a 80mm

Radial distance r 140mm

Shaft angle Σ 50o

Gear ratio u 0.6

Pressure angle 1 α1 20o

Pressure angle 2 α2 -20o

Location of meshing point (given as

a ratio of the distance between the

points of intersection of the

Transversal and Gear Axes)

ε 0.7

147

XX REMOVE THIS PAGE AND REPLACE WITH THE SPREADSHEET

148

APPENDIX D DETAILED MATHEMATICAL DEVELOPMENTS

This appendix contains detailed mathematical developments that underpin many of the results

presented in the chapter, ‘The mathematics of plain polyangular involute gearing’. The intent

of this appendix is to provide sufficient detail to meet the requirements of scientific rigour and

satisfy the curiosity of the mathematically inclined reader without diverting the non-

mathematically inclined reader from the key message.

The appendix contains a number of subsections, each containing the development of the key

finding of the relevant section in the aforementioned chapter.

Detailed workings that produced v12 and the bdl

Restating equation (12)

r12 = < Fx (cos2δ−1)+Fy.cosδ.sinδ, Fy.(sin

2δ−1)+ Fx.cosδ.sinδ, z12-Fz>

and equation (7)

ωωωω12=cosδi+sinδj

All components of v12 are now known, that is,

v12 = h12 ωωωω12 + ωωωω12 x r12

after substituting equation (7), becomes

FFFF

ijkk

kji

kji

yxzzhzzhv

rrrrhh

rrr

kji

hh

rrrxhv

δδδδδδ

δδδδδδ

δδδδ

δδδδ

cossin),(cossin),(sincosˆ

sincos,cos.,sin0,sin,cos

0sincos

ˆˆˆ

0,sin,cos

,,0,sin,cos0,sin,cos.ˆ

1212121212

121212121212

121212

1212

1212121212

−−−−+=

−−+=

+=

+=

149

The best drive line can now also be determined. The best drive line is perpendicular to both

the relative velocity and the transversal (Phillips) and can therefore be determined by

calculating the cross product. Calculating the cross product produces:

)(cossin(tan2

),cos(sintan2]cossin)[(')),(cossin('

'0tan2

cossin)(cossin)(sincos

ˆˆˆ

ˆ

12122

212121212

2

12121212

F

FFFF

FFFF

zzhr

yxrhzzazzha

ar

yxzzhzzh

kji

ldb

−−−−−−−−

=

−−−−−+=

Σ

Σ

Σ

δδδδδδδδ

δδδδδδ

The details of the creation of the loa from the bdl

Rotating the bdl +/-α to produce the loas for the plain polyangular case is not simply a matter

of rotating the bdl about the y axis as it was for the equiangular case. In the equiangular case

v12 was parallel to the y axis so these were equivalent actions. Furthermore given the

limitations of standard matrix manipulations, which require rotation about one of the x, y or z

axes, rotating an object about a line that is not parallel to any one of the axes requires the:

1. line to be transformed so it does lie parallel to one of the axes

2. desired rotation to be carried out

3. original orientation to be restored by reversing the first step.

Giving the x, y and z components of v12, the symbols vi, vj and vk respectively, and the

orientation of v12 is as shown in Figure 41, The process to rotate the bdl about v12, is:

1. Rotate the bdl around the z axis )tan(ˆ

ˆ

j

i

vzv

va=ρ to place it in the yz plane

2. Rotate the bdl about the x axis )tan(2

ˆ

2

ˆ

ˆ

ji

k

vx

vv

va

+=ρ to place it on the y axis

3. Rotate the bdl about the y axis +α and -α to determine the lines of action.

4. Give the bdl and the two new lines, the lines of action, their original orientation to the

xy plane by doing the negative of step 2.

150

5. Give the bdl and the two new lines, the lines of action, their original orientation to the

yz plane by doing the negative of step 1.

(Dealing with vectors, rather than lines, simplifies the operation too, as there is no need to

move the line back to (0,0,0).)

Figure 41: The components of v12 shown with v12 at the origin

The transformations described above can be expressed respectively as 4x4 matrices.

v12

X axis

Z axis

Y axis

151

−

−

−

−

−

1000

0100

00cossin

00sincos

1000

0cossin0

0sincos0

0001

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

1000

0100

00cossin

00sincos

1ˆˆˆvzvz

vzvz

vxvx

vxvx

vxvx

vxvxvzvz

vzvz

kjibdlbdlbdl

ρρρρ

ρρρρ

αα

αα

ρρρρρρ

ρρ

which after expanding and simplifying produces: kji loaloaloa = Xbdlbdlbdlkji ˆˆˆ

+−+−−++−+++−+−−+−++

vxvxvxvzvzvxvxvzvzvx

vzvxvzvxvxvxvzvzvxvxvzvz

vzvxvzvxvxvzvzvxvxvxvzvz

ραραρραρραρραρρρααρρραρρραρρααρρρρααρρραρρρραραρραρ

22

22222

22222

coscossin))1(cossincossinsin(cos))1(cossinsinsin(coscos

sinsin)1(cossin(coscos)cossin(coscoscossinsinsin)cos1(coscossin

)sinsin)1(cossin(sincos)cos1(cossincossinsin)sincos(cossincoscos

152

Determining the base radius (rb) and the angle the lines of action subtend to the gear

axes

The distance between two skew lines can be determined using the expression for the distance

between a point P0(x0, y0, z0)and a plane ax+by+cz+d=0 given by:

(60)

(Anton 1995)

Whilst the angle the lines of action subtend to the gear axes can be determined using the dot

product:

(61)

(Anton 1995)

Taking each in turn, the vector value of the distance between the lines

kloaloajloailoa

loaloaloa

kji

r

ijkk

kji

nb

ˆ)cossin(ˆsinˆcos

0cossin

ˆˆˆ

ˆ

2222

22,1

ΣΣΣΣ

ΣΣ

⋅+⋅−⋅+⋅=

−=

The plane defined by this vector and the point (0,0,a/2) is

)cossin()cossin(sincos0 2ˆ2ˆ22ˆ2ˆ2ˆ2ˆΣΣΣΣΣΣ ⋅+⋅+⋅+⋅−⋅+⋅=

ija

ijkkloaloaloaloazyloaxloa

222

000

cba

dczbyaxD

++

+++=

vu

vu •=θcos

153

Substituting the constituent elements into (60), the distance between this plane and the point Qx (F) is:

222

2

2222211

22211

2

,1

)cossin(

)cossin()cossin)((sincos)(tan

ΣΣ

ΣΣΣΣ+

−ΣΣ+

−Σ

⋅+⋅+

⋅+⋅+⋅+⋅−⋅⋅+⋅⋅⋅=

ijk

ija

ija

kfkf

nb

loaloaloa

loaloaloaloaloarloarr

εε

εε

After some simplification, this becomes

(62)

2

ˆ

2

ˆ

2

ˆ

2ˆ2ˆ22ˆ

,1

)cossin(sin

1

2

kji

ija

kf

nb

loaloaloa

loaloaloarr

++

⋅+⋅+⋅⋅

+=

ΣΣΣ

εε

154

To determine the angle the loa subtends to ga 1, invoking (61) produces:

222

22

22

22222

22

,1

cossin

cossin

0cossincos

kji

ji

kji

kji

nb

loaloaloa

loaloa

loaloaloa

loaloaloa

++

⋅+⋅−=

+++

−=

ΣΣ

ΣΣ

ΣΣ

γ

155

APPENDIX E SURFACE MEASUREMENT

One of the keys to increased reliability, particularly in cars, is the improvement in surface

finish leading to reduced friction. Plateau surfaces, surfaces with the peaks knocked off, have

valleys that can hold lubricant thereby producing a low friction surface.

Surface roughness has come down from approximately 700µm to approximately 330µm in

the last decade (Aronson 1997). This is due not only to machine tools that provide a better

surface finish but also to the instruments that check that surface finish. Although the majority

of these instruments are stylus based, there is also another type of gauge, the light beam

gauge.

Stylus based gauges are contact instruments where a diamond stylus moves at a constant rate

across the surface perpendicular to the lay pattern. The rise and fall of the stylus is measured

by an LVDT and is either recorded on strip chart or used as the basis for electronic processing

to produce, via software, Ra or AA values directly. Indeed it is the software that has

undergone the development in this area.

One of the problems with stylus based gauges is they are 2-D instruments requiring a number

of parallel passes to create a topographic map. This is time consuming and may take 15-20

minutes, a long time if the device is used for Quality Assurance on a production line.

Furthermore, they are limited by the diameter of the radius of the tip of the stylus. Extreme

caution is required when the magnitude of the tip of the stylus approaches the magnitude of

the features to be measured.

156

Beam or laser-based gauges use light to measure surface roughness. Their principle is

illustrated below. Laser based systems are faster, providing “a 512 by 512 data point array

that shows the topography of a 1mm2 area in less than 10 sec[onds]” (Aronson 1997), have

similar accuracy to stylus gauges and do not need to contact the surface to measure roughness.

Some however are influenced by the materials reflectivity and the values produced by laser-

based systems, SN, have an entirely different meaning to the values produced by stylus gauges

such as Ra, Rpm (average peak height) and Rt (surface scratching).

Figure 42: A schematic of laser-based surface roughness measurement

(Degarmo et al 1990)

157

APPENDIX F THE DESIGN PROCESS AND TEST RIG FUNCTIONAL

SPECIFICATION

The design process

Design is the formulation of a plan for the satisfaction of a need; it is the steps between the

recognition of the need and the presentation of the physical reality. Specifically mechanical

design involves the design of systems with a mechanical nature and often utilises the natural

sciences such as mathematics, physics or materials science. It is important to note the design

process, illustrated below, does not produce the correct answer as there is no correct answer,

rather it should produce a good solution from one of the many.

Figure 43: The six phases of design

The phases in the design process are:

Definition of problem

Recognition of need

Synthesis

Analysis and

Presentation

Evaluation

(Shigley 1986)

158

Recognition of need: The first step is the recognition and definition of a need that, in this

case, is well defined. The statement of the problem is “We require a rig to mount the gear

bodies in and test their relative motions”.

Definition of problem: The second step, the definition of the problem, includes all of the

specifications for the thing to be designed. It states the outputs of the thing and the inputs

required to turn the thing into a reality. The definition of the problem places restrictions on

things as diverse as the method of manufacture, the accuracy to which the thing is

manufactured as well as the thing’s capabilities. The result of the definition of the problem is

a list of specifications or a functional specification for the thing.

Synthesis and analysis: These two steps must occur concurrently as each synthesis must be

analysed to determine if the performance conforms to the specifications. If the design fails the

analysis, the synthesis must begin again.

Evaluation and presentation: Evaluation proves the successful design by testing of the

prototype and demonstrates the design really satisfies the need as well as investigating its

reliability or ease of manufacture.

The final phase is the presentation phase. This is the marketing of the thing as a new idea to

solve the problem. It is essentially an exercise in communicating verbally, graphically and in

writing the benefits of the thing to gain user acceptance.

Pahl and Beitz (1984) and Ohsuga (1989) both propose alternate models in their publications.

The Pahl and Beitz model comprises four stages:

159

1. Clarification of the task. This stage requires the designer to collect information on the

design requirements and describe them in a specification.

2. Conceptual design. This stage involves the identification and development of suitable

solutions.

3. Embodiment design. This stage involves refining the conceptual solution to resolve

problems and eliminate the weaknesses.

4. Detail design. In this stage the dimensions, tolerances, material and form of the

individual components are specified for manufacture.

Although Ohsuga (1989) also describes the process in stages, he generalises various design

processes into a common form in which models are developed, evaluated and subsequently

refined before proceeding to the detailed design stage.

These models present design as a linear process; each stage is more or less complete before

proceeding to the following stage. The pressure to reduce product design and development

lead times, however, results in design, development, analysis and preparation of

manufacturing information being done in parallel. This process is termed simultaneous

engineering. (McMahon, Brown 1993)

One of the tools that assisted simultaneous engineering is Computer Aided Design (CAD).

CAD integrates many of the steps. Initial three dimensional (3D) models can be developed

using the geometric requirements of the specification and subsequently analysed using tools

such as Finite Element Analysis (FEA) and Computational Fluid Dynamics (CFD). The

details can then be presented for manufacture directly from the preliminary design.

160

These features are used to develop many of the products in commercial use today including

consumer products such as vacuum cleaners and automobiles. Specifically the design of the

gear body test rig, using the geometric requirements of the functional specification, via CAD

is an example of this.

The functional specification for the test rig

The test rig shall:

• Mount the gear bodies in a position as intended in their design within 0.01mm linearly

and 0.1o angularly.

• Allow the gear bodies to move unhindered through the full range of movement

available to the useable flanks.

• Allow some flexibility in the location of the gears. This is required to assess Phillips

(2003) hypothesis that the transmission ratio is independent of gear location for small

errors in assembly.

• Be made from commercially available materials.

• Be constructed with an accuracy equivalent to that in a commercial gearbox. This

means the mounting surfaces will have to be machined and consequently the structure

will have to be strong enough to support that surface during the machining process.

• Minimise the effects of the out of balance created by the large masses located away

from the ISA.

• Include some form of encoding device for determining constancy of transmission

ratio. This shall be an electrical output that can be collected and manipulated by

computer hardware and software.

161

The design of the gear body test rig

The designer was presented with three things, the aforementioned functional specification and

two gear bodies located in space as per Killeen (1996). Figure 44 shows the gear bodies

mounted on their axes.

Preliminary designs were developed in a 3D model in Cadkey 98 version 1.04. The process

was as follows:

1. Locate suitable bearing mounting surfaces whilst simultaneously identifying and

locating suitable bearings

2. Construct a frame around the bearing mounting surfaces such that the frame does not

inhibit the movement of the gear bodies through their range of motion. The material

was selected from Blackwoods (1999) to ensure it was commercially available

3. Identify and locate suitable encoders using the ‘RS’ catalogue

4. Design encoder mounting blocks

5. Design the gear body shafts to facilitate the connection of the shafts to the input of the

encoder

(Note: the above steps were presented as bullet points, rather than numbers, as there was some

iteration in the process as shown in Figure 43.)

The preliminary model was then converted to a detailed model using Cadkey’s layout mode.

Various views of the preliminary model were imported onto a drawing sheet and subsequently

detailed with dimensions, material specifications and machining instructions. Similarly, views

of other components such as the encoder mounting blocks were imported and detailed.

162

Figure 44: A pair of gear bodies for the set u=0.6, Σ=50° and a=80mm.

163

APPENDIX G DRAWINGS OF THE GEAR BODY TEST RIG

The following pages contain the final design of the gear body test rig.

Table 31: Contents and drawing cross-reference for the test rig design

Contents of drawing Drawing number

General assembly of the gear body test rig A30004

Detailed drawing of gear body 1e A30006

Detailed drawing of gear body 2e A30005

Detailed drawing of the gear body mounting frame A30007

Electrical schematic of the gear body test rig A30009

Detailed drawing of the gear boy axle A40001

Detailed drawing of the gear body encoder mounting bracket A40002

Specification sheet for the encoders NA

164

REPLACE THIS SHEET WITH THE DRAWINGS AND SPEC SHEET

165

166

APPENDIX H LABVIEW PANEL AND VIRTUAL INSTRUMENT FOR DATA

COLLECTION DURING FUNCTIONAL TESTING

Computer usage in industry and academia has skyrocketed. Computers are more flexible than

standard instruments allowing instrumentation to be reconfigured at the push of a button

rather than at the twist of a screwdriver saving both time and money.

Programming languages have also advanced significantly. The previous DOS based

programming languages for devices such as Programmable Logic Controllers (PLC) required

almost as much effort to navigate and fault find as would wiring the instrument it was

designed to replace. Now graphical programming permeates both industry and academia

providing a rich programming environment and user friendly features.

Laboratory Virtual Instrument Engineering Workbench (LabVIEW®) is one of the new breed

of computer based instruments based on graphical programming. The virtual instruments

(VIs) are designed to present the important values to the user in a customisable front panel or

Graphical User Interface (GUI) whilst keeping the ‘wiring’ hidden. The three parts of a VI are

covered in more detail below:

The front panel is the user interface and represents what the user would see if the instrument

was laid out on the panel in front of her/him. The instrument can contain inputs (such as

pushbuttons) and outputs (such as gauges and lights). The front panel for the acquisition of

data from the encoders is shown in Figure 45.

The graphical code or block diagram comprising the VI’s source code. Icons represent sub-

VIs and are joined by wires that indicate flow. The graphical code for the acquisition of data

from the encoders is shown in Figure 46.

167

The icon and connector allow I/O from one VI to pass to another VI. The connector defines

the inputs and outputs of the VI. (Wells 1995)

Figure 45: The labview front panel used for functional inspection data collection

168

Figure 46: The Labview VI used to collect the data for the functional inspection

169

APPENDIX I USEFUL STATISTICAL ANALYSIS

Simple linear regression and correlation

Often there is a need to determine whether there is any inherent relationship between a

number of variables. Statistically the objective is to determine a best estimate of the

relationship between the variables. The variables are generally Y, the single dependent

variable, a value that is uncontrolled in the experiment. Y’s response depends on one or more

independent regressor variables, x1, x2,…, xk.

For a single Y and x it is simply a matter of a regression of Y against x. The term linear

regression implies µY|x is linearly related to x. The two are related by the simple linear

regression equation

µY|x=α+βx

Y|x is the random variable Y corresponding to a fixed value x and its mean and variance are

µY|x and σ2Y|x. α and β are to be estimated from the experimental data. The estimates of α and

β are a and b respectively whilst y represents the estimate of µY|x. the fitted regression line is

therefore:

y = a + b.x

where a and b represent the y intercept and the slope respectively.

The method of least squares is used to determine a and b. The sum of least squares method

aims to minimise the error about the regression line and is given the symbol SSE i.e.

=

=

= ==

−−−=∂

∂

−−−=∂

∂

−−=−==

n

i

iii

n

i

ii

n

i

n

i

iiii

n

i

i

xbxay

bxay

bxayyyeSSE

1

1

1 1

22

1

2

)(2b

(SSE)

)(2a

(SSE)

b and a respect to with SSE atingDifferenti

)()ˆ(

170

Setting the partial derivatives to zero and rearranging produces the normal equations

i

n

i

i

n

i

i

n

i

i

n

i

i

n

i

i

yxxbxa

ybxbna

===

==

=+

=+

11

2

1

11

Which can be solved simultaneously to produce

n

xby

a

and

xx

yyxx

b

n

i

i

n

i

i

n

i

i

n

i

ii

==

=

=

−=

−

−−=

11

1

2

1

)(

))((

(Walpole, Myers 1993)

The second aspect of the statistical analysis is the correlation analysis. Correlation analysis

attempts to determine the strength of the relationship between the two variables and represent

it as a single number called the correlation coefficient. The measure of linear association

between two variables is estimated by the sample correlation coefficient, r, where:

yyxx

xy

yy

xx

SS

S

S

Sbr ==

where:

=

−=n

i

ixx xxS1

2)(

=

−=n

i

iyy yyS1

2)(

=

−−=n

i

iixy yyxxS1

))((

and x and y are the mean or average of the x and y values and is given by xi/n.

(Walpole, Myers 1993)

171

(Also, for completeness, the sample standard deviation, S, is the positive square root of the

sample variance. The sample variance is given by

1

)(1

2

2

−

−=

=

n

XX

S

n

i

ii

(Walpole, Myers 1993)

The sample correlation coefficient is of limited value however. Two values of r, 0.3 and 0.6,

mean there are two positive correlations, one stronger than the other, not a linear relationship

that is twice as good.

It is usually preferable to use the value sample coefficient of determination, r2 where:

yyxx

xy

SS

Sr

2

2 =

The value r2

• “represents the proportion of the variation of Syy explained by the regression of Y on x”

(Walpole, Myers 1993 p405)

• “expresses the proportion of the total variation in the value of the variable Y that can be

accounted for or explained by a linear relationship with the values of the random variable

x. Thus a correlation of 0.6 means… 36% of the total variation of the values in Y in our

sample is accounted for by a linear relationship with the values x” (Walpole, Myers 1993

p406)

Hypothesis testing

Often after reviewing the data resulting from an experiment, an engineer postulates or

conjectures something about the data. The conjecture is put in the form of a hypothesis. The

engineer then sets about proving or disproving the hypothesis based on the results of the

172

experiment. The following section summarises the principles of testing a statistical

hypothesis. It is not intended to be a complete reference to the concept, but rather an overview

to allow the reader get a general idea. Readers wanting a full explanation are referred to

Walpole, Myers (1993 Chapter 10).

There are two types of hypothesis the null hypothesis (H0) and the alternative hypothesis (H1).

The null hypothesis is stated to specify an exact value of the parameter such as ‘H0: p=0.5’.

The alternative hypothesis may be stated as either a single tailed in equality such as ‘H1:

p>0.5’ or ‘H1: p<0.5’ or a double tailed inequality such as ‘H1: p≠0.5’ or ‘H1: p<0.5 or p>0.5’

Over time it has become customary to accept or reject a hypothesis based on the level of

significance, α, of 0.05 or 0.01. The acceptance or rejection of H0 is then based on the critical

region where the border/s between the acceptance region and the critical region is/are given

by the value of the parameter to be tested based on the respective distribution. As an example,

selecting a two-tailed test with a 0.05 level of significance and the normal distribution, the

critical region is z>1.96 and z<-1.96. 1.96 comes from either a table of distributions such as

those in Walpole, Myers (1993 Appendix A) or a distribution algorithm such as those found

in MS Excel or HP48 series calculators.

As an alternative to the accept/reject format applied statistics, the P-value approach tells the

analyst if the actual value falls well into the critical region. If, for example, z is 2.73 and the

test is two tailed, the probability is 2x0.0032 (0.0064) where again 0.0032 comes from the

table of normal distributions. The z value is considerable less than the level of significance

0.05 and that a value of z=2.73 is extremely rare statistically occurring 64 times in 10000

experiments. Of course this process can be applied to other distributions such as the ‘t’, ‘f’

and ‘Chi2’ as appropriate.

173

So, in summary, the procedure for hypothesis testing is:

1. State the null hypothesis H0 that θ=θ0.

2. Choose an appropriate alternative hypothesis H1 from one of the alternatives θ<θ0,

θ>θ0 or θ≠θ0.

3. Choose a significance level of α.

4. Select the appropriate test statistic and establish the critical region. Alternately use the

P value.

5. Compute the value of the test statistic from the sample data.

6. Decision: Reject H0 if the test statistic has a value in the critical region or if the

computed P-value is less than or equal to the desired significance level α. (Walpole,

Myers 1993)

174

APPENDIX J THE MATHEMATICS OF OBJECT TRANSFORMATION

The process of object transformation involves moving an object with respect to a stationary

coordinate system. There are three main operations;

translation or linear movement. The mathematical process is a vector translation; p’ = p - t

rotation about the origin. The mathematical process is a matrix multiplication; p’ = p B and

scaling with respect to the origin. Scaling is also a matrix multiplication.

It is more convenient however to express all operations in terms of multiplication. This is

achieved using homogeneous coordinates. Homogeneous coordinates change the 1 x 3 row

matrix x, y, z into a 1 x 4 row matrix x, y, z, 1 allowing the object transformation to be

expressed in matrix form. The transformation matrix can then be any combination of

translation, rotation and scaling that, when multiplied, produces one matrix equivalent to the

combination.

Object translation: As an example consider the point p x, y, z, 1 translated to p’ by

dx in the x direction

dy in the y direction

dz in the z direction

This becomes in matrix terms using homogenous coordinates

[ ]

p x y z p A x y z

dx dy dz

x dx y dy z dz

' ' , ' , ' , , , ,

, , ,

= = =

= + + +

1 1

1 0 0 0

0 1 0 0

0 0 1 0

1

1

175

Object rotation: An object rotates in a positive sense about an axis if, when viewed looking

toward rather than along the vector of the axis, the rotation is Counter Clock Wise (CCW).

Generally, the Right Hand Rule determines the positive sense.

The coordinates of an object rotated about the z axis φ° rotates a further +θ° become

x’=rcos(φ + θ)

y’=rsin(φ + θ)

z’=z

but x = rcos φ and y = rsin φ therefore expanding the above two equations produces

x’=xcos θ-ysin θ

y’=xsin θ+ycos θ

z’=z

an expression that can be stated in terms of matrix transformation as

[ ] [ ]p x y z xyz' ' ' '

cos sin

sin cos= =

−

1 1

0 0

0 0

0 0 1 0

0 0 0 1

θ θθ θ

Similarly the matrices for rotation about the x axis and y axis respectively are

1 0 0 0

0 0

0 0

0 0 0 1

0 0

0 1 0 0

0 0

0 0 0 1

cos sin

sin cos

cos sin

sin cos

θ θθ θ

θ θ

θ θ

−

−

and

(The above was re-presented from Killeen (1996))

176

Applying this to the first of the data points (from Table 20) for gear body 1, (20.457, -2.002, -

6.952), produces in parametric form, the point

Which when multiplied by the transformation matrix for 90 degree rotation about the x axis,

Produces

Now from measurement off drawing A3006A the movements required to place the centre of

the base circle on the origin of the CMM are 26.8173 in the x direction, 20 in the negative y

direction and gx,11,Qx or 23.1659 in the z direction. This produces the transformation matrix

And, in turn, multiplying the point above by the transformation matrix produces

Now equation (31) is based on the involute starting at (0,rb) whilst the above line matrix

places the start of the involute on (rb,0). Therefore rotating the above through 90o using the

matrix

Produces

1952.6002.2457.20 −−

−1000

0010

0100

0001

− 11659.23208173.26

0100

0010

0001

11639.21048.13274.47 −

1002.2952.6457.20 −

−

1000

0100

0001

0010

11639.21274.47048.13

177

The final transformation, a mirror about the yz plane aligns the involute above with the

involute in equation (31) making equation (31) applicable to the data points. This

transformation

Produces, finally,

−

1000

0100

0010

0001

11639.21274.47048.13−

178

APPENDIX K A COMPARISON OF THE CYCLOIDAL AND INVOLUTE

PROFILES

The underlying principle behind all gearing is to transmit constant rotary motion or provide

conjugate action (Shigley 1986) between shafts at a specified distance from and angle to each

other. By limiting the investigation to shafts that are parallel to each other, this could be

achieved using friction wheels. The relative diameters of these friction wheels would

represent the transmission ratio between the axles.

Unfortunately, this is severely limited as a relatively small amount of torque would cause the

wheels to slip. It is therefore necessary to pick a profile that can provide conjugate action.

Conjugate action requires that the line of action intersects the same point, P say, on the line

joining the centres of the circles representing the pitch circles of the as yet undefined surfaces.

This is illustrated in Figure 47.

Alternately this can be stated, in the general spatial case, that “for constant angular velocity

ratio in gears, the contact normal must at all stages of the meshing be located in such a way

that q.tanφ remains a constant namely p” (Phillips 1998). The parameters q, φ and p have the

following meanings, q is the shortest distance between the contact normal and the pitch line, φ

is the angle between the contact normal and the pitch line and p is the relative screwing

between the pinion and wheel.

179

Figure 47: The principle of conjugate action

Of course for the planar case, the equations

and

)1cos2(2

)1('2

2

12 +Σ−−−

=uu

uaz

(Killeen 1996)

where:

h12 = the pitch of the pitch line or Phillips’ p

z12 = the location of ISA12 relative to the midpoint of the centre distance line or the

distance from the midpoint of the centre distance line to the point of intersection of the

pitch line and centre distance line

a = the distance between the centre(lines) of the pinion and wheel

u = the gear ratio

Σ = the shaft angle

reduce to

1cos2

sin'212 +Σ−

Σ=

uu

uah

012 =h

Pitch circle 1

Pitch circle 2

Base circle 2

Base circle 1

180

and

)1(2

)1('12 −

+−=

u

uaz

i.e. the value of p is zero. The value of φ on the other hand is π/2 as the line of action is in the

plane of the page and the pitch line is perpendicular to the plane of the page. The value of q

must therefore be zero. This results in the planar law of gearing, “the contact normal must at

all stages of meshing pass through the pitch point” (Phillips 1998).

This can be achieved with almost any profile by cutting one gear’s teeth and then effectively

cutting the mating gear with the first gear. Generally however there are only two profiles

which have been standardised, the cycloidal profile and the involute profile.

The cycloidal profile

The cycloid is a transcendental curve (Hunt 1978) “described by a point on a circle (the

generating circle) which rolls without slip on a fixed straight line (the base line)” (AS2075

1991 p18). There are a number of special cases of the cycloid including:

the epicycloid. In this case the generating circle rolls on the outside of a fixed circle (the base

circle).

the hypocycloid. In this case the generating circle rolls on the inside of the base circle.

the extended (curtate) and ordinary (prolate) cycloid where the location of the point traced is

outside of, or within the radius of the pinion respectively.

All of the aforementioned special cases can be represented mathematically by the following

formulas. These formulas are parametric equations of the trajectory of the point.

181

(Rewritten from Hunt 1978)

where:

r1 = the radius of the generating circle

r2 = the radius of the base circle

r = the radius of the point being traced and

θ = the parameter representing angle of rotation of the generating circle

The cycloid was the first profile used for gears however its applications were limited to

watches and clocks. Despite the fact that involutes have largely replaced the cycloid for

gearing today, cycloids have some distinct advantages over involutes which means they are

still used in watches and clocks and have important applications in industry including Roots

blowers and pumps.

These advantages include, generally stronger teeth because of the spreading flanks of the

cycloidal as opposed to the radial flanks of the involute teeth and less sliding therefore less

wear than involute teeth. Unfortunately there are significant disadvantages of cycloidal

gearing. One of these is the fact they have only one theoretically correct location at which

they will transmit the constant angular velocity. The manufacturing process is also more

expensive than for involute as although hobbing is possible, the rack does not have straight-

sided teeth as does an involute rack (Mabie Reinholtz1987).

+−

+=

−

+−

+=

θθ

θθ

)1(sin)(sin)(

)1(cos)(cos)(

2

1

2

112

2

1

2

112

r

rr

r

rrry

rr

rr

r

rrrx f

182

The involute profile

The involute is the locus of a point on a straight line that is rolling without sliding on a circle

This definition, of course, makes the involute a special form of epicycloid in which the circle

being rolled has an infinite radius. Litvin (1994) represents this pictorially as shown.

Developing the equation of the involute profile from this geometry produces:

(63)

where, from the aforementioned figure:

α = ∠MOP measured in radians

invα = polar angle, ∠MoOM measured in radians

Figure 48: The involute created by a line rolling on a circle

From the above it is obvious that

OM.cosα=OP

And due to rolling without sliding

PM = rbφ

(63) can be manipulated to produce the equation of the involute as a function of the base

b

b

b

b

r

rr

r

rrinv

22

1

22

tan−

−−

= −α

invα = tanα -α

183

circle radius, rb, and variable r in the following equation from Killeen (1996)

Or in parametric form from Litvin (1994) as

(64)

where:

x = the x coordinate of M

y = the y coordinate of M

φ = the involute roll angle, ∠MoOP. φ is equivalent to t7 in the equations of Killeen (1996)

Alternately, taking a geometric perspective, consider the two circles of Figure 47. Let these

circles represent the base circles and the lines tangential to both circles represent a taut piece

of string cutting the centre distance line at P. Tracing a point on the string on a piece of

cardboard say, attached to the wheel, creates an involute to the wheel. The involute profile

traced on the cardboard is equivalent to an involute tooth profile.

From this it can also be seen that the tangent or string represents the direction of the resultant

force between the gears. The angle this tangent subtends to the horizontal line passing through

the point P is the pressure angle, α. This is not to be confused with Litvin’s involute α.

Although Euler (1707-83) was the first to propose that the involute profile (and epicycloid for

that matter) satisfied the principle of common tangents and had the proper design necessary to

avoid friction, it was not until the 1850s that the transition was made from the mathematics to

machining.

Woodbury (1972) credits this transition with three men, G. Grant, O. Beale and in particular

E. Sang. Sang’s publication in 1852 of New general theory of the teeth of wheels lead the way

x = rb(sinφ-φ.cosφ) and y = rb(cosφ+φ.sinφ)

184

to successful generation of teeth. Under the heading of ‘entomy of wheels’ he proposed four

methods of generating teeth, only two of which became important, the formed cutter and rack

cutter. Sang, however thought the practical problems of the rack cutter outweighed its

usefulness.

The involute had a number of advantages of the cycloidal profile not the least of which was,

as was mentioned previously, the teeth on the basic rack had straight sides whereas the teeth

on the basic rack for the cycloidal profile had rounded sides. As may also be deduced from

the string analogy the transmission ratio is not affected by small variations in centre distance

as, although the diameters have effectively changed, this merely brings different parts of the

involute into contact.

Despite this there are some disadvantages of the profile. As was demonstrated there is

theoretical point contact between the profiles for the planar case. Unfortunately extruding the

planar case to create a solid spur gear produces line contact or over-constraint between the

profiles. This makes it necessary to modify the profile by either crowning or end relief to

localise the bearing contact.

The involute profile is also prone to interference. Interference occurs when contact between

the teeth occurs below the base circle. It was shown that the involute form starts at the base

circle i.e. it is not possible for involute profile to exist below the base circle. If contact

between teeth occurs below the base circle, the involute profile is meshing with a non-

involute profile and the non-involute profile will be undercut. This could significantly reduce

the contact ratio.

185

APPENDIX L PRINCIPLES FOR AND EXAMPLES OF THE CALCULATION

OF AXIAL DISTANCES

The sections of the Gear Body, as measured by the CMM, are defined with respect to the edge

of the Gear Body. To compare these sections with theoretical sections, the location of the

respective sections of the gear body must be determined with respect to the striction (base)

circle of the involute helicoid rather than the edge of the gear body. If a section is z* from the

base circle, z* can be substituted into the z component of equation (31) to produce a

theoretical section. The following procedure details the method used to determine z*.

Consider two planes one defined by the relevant E circle and one by the relevant Base Circle.

The plane defined by the E circle cuts the Gear Body exactly in half and has within it the

point E. (The point E is contained in the planes defined by both E-circle 1 and 2.) Similarly

the plane defined by the Base Circle contains the centre of the Base Circle by definition.

Anton (1995) gives, as the expression for the distance between a plane and a point where:

a, b, c & d are the coefficients of the x, y and z components respectively of the plane

under consideration and d is the constant

xo, yo and zo are the x, y and z coordinates of the point

As both planes have a common normal, the centre of the Base Circle could be considered as

‘the point’ whilst E and the gear axis define ‘the plane’.

The distance between the E circle, the plane containing E, and the base circle defined by Gear

Axis 1 and Line of Action 2 is found by substituting the relevant values of a, b and c along

with the coordinates of E into the above equation. The plane defined by gear axis 1 and E(xE,

yE, zE), therefore, is:

222 cba

dczbyaxD

ooo

++

+++=

186

sinΣ/2.x + cosΣ/2.y - sinΣ/2.xE - cosΣ/2.yE = 0

The distance from the Base Circle to the Centre Distance Line (t*mn), measured along the

respective Gear Axis, is given by equations 41 and 42 of Killeen (1996). The centre of the

Base Circle12, for example, has the coordinates

(t*12 sinΣ/2, t*12 cosΣ/2, a/2)

so the expression for the distance between the centre of the E circle and the centre of the Base

Circle12, gx,12,E is

(65)

Similarly it can be shown for Gear Axis 2

(66)

From the above results and the xy plane view of the architecture, a negative gx,mn,E means the

move from the E-circle to the Base circle is toward the CDL. Therefore, locating the

measurement plane on the base circle requires the gear body to move:

Along the Gear Axis away from the CDL a distance of half the gear body width. The centre

of the end of the boss and centre of the E circle are now the same point.

Toward the base circle a distance gx,mn,E. The centre of the end of the boss and the centre of

the base circle are coincident i.e. they are now the same point.

This process can be stated mathematically as:

For Gear Axis 1:

gx,1n,z* = t*mn + xesinΣ/2 − yecosΣ/2 + (gear body width)/2

and for Gear Axis 2:

gx,2n,z* = t*mn - xesinΣ/2 − yecosΣ/2 + (gear body width)/2

2212*

,12,

22

22

222212*

2212*

,12,

cossin

cossin

cossincoscossinsin

ΣΣ

ΣΣ

ΣΣΣΣΣΣ

⋅−⋅+=

+

⋅−⋅+⋅⋅+⋅⋅=

EEEx

EE

Ex

yxtg

yxttg

2222*

,22, cossin ΣΣ ⋅−⋅−= EEEx yxtg

187

Alternately moving toward the CDL a distance z* where z

* represents the axial distance from

the face of the gear body boss to the measurement plane of the CMM.

This process can be stated mathematically as:

For Gear Axis 1:

gx,1n,z* = t*mn + xesinΣ/2 − yecosΣ/2 + (gear body width)/2-z*

and for Gear Axis 2:

gx,2n,z* = t*mn - xesinΣ/2 − yecosΣ/2 + (gear body width)/2-z*

Fundamental data

a' 80 mmr 140 rf 139.058275 mmΣ 50 degs or 0.87266463 radsu 0.6 :1 0.6αf 20 degs or 0.34906585 rads

αr -20 degs or -0.3490659 rads

OD 50 mmδ' -1.454743 rads or -83.35065 degs -58.350648370

E( -16.21098 139.05828 10 )

h12 -17.25208

z12 12.011194Synthesis of equiangular gear architecture

gear axis 1 -0.422618 0.9063078 0 gear axis 2 0.42262 0.906308 0Gear Axis 1 intercepts the z axis at Gear Axis 2 intercepts the z axis at

( 0 0 40 ) ( 0 0 -40∴Gear Axis x = -0.4226183 t1 + 0 Gear Axis 2 x = 0.422618 t2 +

y = 0.9063078 t1 + 0 y = 0.906308 t2 +

z = 0 t1 + 40 z = 0 t2 +

i j kfaxl 0 1 0

Polar Plane0 x +

1 y +

0 z =

139.058

The points J and K are the intersections of the the Polar Plane and the Gear Axes.

Solving for the parameter, t2 and t3, produces J and K respectively

t2 = t1 = 153.43383

J ( 64.843939 139.05828 -40 ) The interval JK is 152.3776 mm longK ( -64.84394 139.05828 40 )

The Transversal therefore has the parametric coordinates

x = -129.68788 t3 + 64.84

y = 0 t3 + 139.1

z = 80 t3 + -40

which is the point E when t4 = 1/(1+u 0.625E ( -16.21098 139.05828 10 )

therefore the plane containing e circle 1 is

-0.422618 x + 0.9063078 y + 0 z = 132.881

Substituting the equations x, y and z for Gear Axis 2 and solving for t2 to find the centre of e circle 2

t1' = 132.88066 The radius of e1 circle

xe1-centre = -56.15779 re1 = 53.31726962 mm

ye1-centre = 120.43077

ze1-centre = 40

and the plane containing e circle 2 is

0.422618 x + 0.9063078 y + 0 z = 119.179

Substituting the equations x, y and z for Gear Axis 1 and solving for t1 to find the centre of e circle 1

t2' = 119.17854 The radius of e2 circle

xe2-centre = 50.367027 re2 = 88.86211604 mm

ye2-centre = 108.01244

ze2-centre = -40

The best drive line is perpendicular to both the faxl and the transversalThe best drive line= i j k

0 1 0-129.68788 0 80

The best drive line is 80 0 129.7 0

The Best Drive Linex = 80 t4 + -16.21

y = 0 t4 + 139.1

z = 129.68788 t4 + 10

The Line of Action is the Best Drive Line rotated about the Faxl or the y axis.

There are two Lines of Action:

Line of Action 1 is the Best Drive Line rotated 20 ° in the positive direction.Line of Action 2 is the Best Drive Line rotated -20 ° in the negative direction.

The transformation matrices for these operations are:

for Line of Action 1 for Line of Action 20.939693 0 -0.3420201 0 0.939692621 0 0.34202 0

0 1 0 0 0 0 0 00.34202 0 0.9396926 0 -0.342020143 0 0.939693 0

0 0 0 1 0 0 0 1

producing line of action 1 producing line of action 2119.5313 0 94.50513 0 30.81954319 0 149.2284 0

Line of Action 1 therefore Line of Action 2 therefore

x = 119.53128 t5 + -16.210985 x = 30.8195 t6 + -16.211

y = 0 t5 + 139.058275 y = 0 t6 + 139.058

z = 94.50513 t5 + 10 z = 149.228 t6 + 10

The important components of the base hyperboloids are the striction or base circle radii (rbi) and the

angle the generator subtends to the axes (γbi). The angle subtended by:

Gear Axis 1 & Line of Action 1 Gear Axis 2 & Line of Action 1γb11 = 1.908702 rad γb21 = 1.232890605 rad

or 109.36057 degrees or 70.63942826 degrees19.360571743

Gear Axis 1 & Line of Action 1 Gear Axis 2 & Line of Action 2γb12 = 1.6563767 rad γb22 = 1.485215965 rad

or 94.903394 degrees or 85.09660648 degrees-4.903393516

The line representing the shortest distance between the Line of Action and the indicatedGear Axis has the vector: for Gear Axis 1 & Line of Action 1 for Gear Axis 2 & Line of Action 185.65074 39.939594 -108.33213 85.65 -39.9395938 -108.33

The radius is therefore The radius is thereforerb11 = 51.581723 rb21 = 85.96953785

for Gear Axis 1 & Line of Action 2 for Gear Axis 2 & Line of Action 2135.2468 63.066627 -27.931992 135.2 -63.06662715 -27.932

The radius is therefore The radius is thereforerb12 = 48.843462 rb22 = 81.40577067

The equations of the base hyperboloids, base helices and involute helicoids thus created and locatedat (0, 0, 0) and aligned with the z axis are:

for Gear Axis 1 & Line of Action 1 for Gear Axis 2 & Line of Action 1

Base hyperboloid 11 Base hyperboloid 21x2 + y2 = 2660.6741 + z2 135.871266 x2 + y2 = 7390.76 + z2 8.09917

Involute helicoid 11 Involute helicoid 21x = 51.581723 cost7−µ -0.331512 sint7 x = 85.9695 cost7−µ 0.33151

y = 51.581723 sint7−µ -0.331512 cost7 y = 85.9695 sint7−µ 0.33151z = -146.7966 t7 + µ 0.94345101 z = 244.661 t7 + µ 0.94345

Slip track 11 Slip track 21x = 51.581723 cost7 − t7 -58.847608 sint7 x = 85.9695 cost7 − t7 -98.079

y = 51.581723 sint7 − t7 -58.847608 cost7 y = 85.9695 sint7 − t7 -98.079

z = 20.678007 t7 z = -34.463 t7

for Gear Axis 1 & Line of Action 2 for Gear Axis 2 & Line of Action 2Base hyperboloid 12 Base hyperboloid 22

x2 + y2 = 2385.6838 + z28.09916555 x2 + y2 = 6626.9 + z2

135.871Involute helicoid 12 Involute helicoid 22

x = 48.843462 cost7−µ -0.0854759 sint7 x = 81.4058 cost7−µ 0.08548y = 48.843462 sint7−µ -0.0854759 cost7 y = 81.4058 sint7−µ 0.08548z = -569.3381 t7 + µ 0.99634024 z = 948.897 t7 + µ 0.99634

Slip track 12 Slip track 22

x = 48.843462 cost7 − t7 -49.205611 sint7 x = 81.4058 cost7 − t7 -82.009

y = 48.843462 sint7 − t7 -49.205611 cost7 y = 81.4058 sint7 − t7 -82.009

z = 4.2213447 t7 z = -7.0356 t7

The entities created above are located at the intersection of the Gear Axis and the base circle radii.The origin for these entities is tn* along the Gear Axis n.Restating the Gear Axes and Lines of Action in unit vector form

Gear Axis 1 Gear Axis 2x = -0.422618 t1 + 0 x = 0.42262 t2 + 0

y = 0.9063078 t1 + 0 y = 0.90631 t2 + 0

z = 0 t1 + 40 z = 0 t2 + -40

Line of Action 1 Line of Action 2x = 0.784441 t6 + -16.210985 x = 0.20226 t7 + -16.211

y = 0 t6 + 139.058275 y = 0 t7 + 139.058

z = 0.6202034 t6 + 10 z = 0.97933 t7 + 10

The distance along Gear Axis 1 the base circle The distance along Gear Axis 2 the base circleradius created by Gear Axis 1 and Line of radius created by Gear Axis 2 and Line of Action 1 radius intersects Gear Axis 1 is: Action 1 intersects Gear Axis 2 is:

t11* = 137.62192 t21* = 127.081∆ = -0.331519 ∆ = 0.33152

The distance along Gear Axis 1 the base circle The distance along Gear Axis 2 the base circleradius created by Gear Axis 1 and Line of radius created by Gear Axis 2 and Line ofAction 2 radius intersects Gear Axis 1 is: Action 2 radius intersects Gear Axis 2 is:

t12* = 131.04654 t22* = 116.122∆ = -0.085478 ∆ = 0.08548

The following matrices transfer the geometric entities to the origin defined by a distance tn* along Gear Axis n.

For Gear Axis 1 and Line of Action 1 For Gear Axis 2 and Line of Action 1

0.906308 0.4226183 0 0 0.906307787 -0.4226 0 00 0 1 0 0 0 1 0

0.422618 -0.906308 0 0 -0.422618262 -0.9063 0 0-58.16154 124.72782 40 1 53.7066024 115.174 -40 1

For Gear Axis 1 and Line of Action 2 For Gear Axis 2 and Line of Action 2

0.906308 0.4226183 0 0 0.906307787 -0.4226 0 0

0 0 1 0 0 0 1 00.422618 -0.906308 0 0 -0.422618262 -0.9063 0 0-55.38266 118.7685 40 1 49.0751441 105.242 -40 1

The distanc between the centre of Base Circle 11 The distanc between the centre of Base Circle and E circle 1, measured along Gear Axis 1 is: and E circle 2, measured along Gear Axis 2 is

gx,11,E = 4.7412647 gx,21,E = 7.90211

The distanc between the centre of Base Circle 12 The distanc between the centre of Base Circle and E circle 1, measured along Gear Axis 1 is: and E circle 2, measured along Gear Axis 2 is

gx,12,E = -1.834114 gx,22,E = -3.0569

Synthesis of surface geometry

These matrices transform the entities into new positions defined by the Lines of Action and theGear Axes. The entites lying along Gear Axis 1 are:For Line of Action 1 -

The base hyperboloid is:(x' 0.90631 +y' 0.422618 )2 + (z' - 40 )2 =

2660.67 +(-x' 0.422618 +y' 0.90631 - 137.6219 )28.09917

The involute helicoid is:x'IH11 = 0.90631 (cost7(1,1) 51.58172 -µ1,1 0.94345 sint7(1,1)) −

0.422618 ( -146.8 t7(1,1) + µ1,1 0.943451 + -137.62

y'IH11 = 0.42262 (cost7(1,1) 51.58172 -µ1,1 -0.3315 sint7(1,1)) + -0.90631 ( -146.8 t7(1,1)+µ1,1 0.943451 + -137.62

z'IH11 = 51.5817 sint7(1,1) + -0.33151 µ1,1 cost7(1,1)+ 40

The slip track is:x'ST11 = 46.7489 (cost8(1,1) - t8(1,1) sint8(1,1) -1.1409 ) + 0.422618 (t8(1,1) 20.678

+ -137.62) y'ST11 = 21.7994 (cost8(1,1) - t8(1,1) sint8(1,1) -1.1409 ) + -0.906308 (t8(1,1) 20.678

+ -137.620 z'ST11 = 51.5817 sint8(1,1) + -58.8476 t8(1,1).cost8(1,1) + 40

0

-40For Line of Action 2 -The base hyperboloid is:

(x' 0.90631 + y' 0.422618 )2 + (z' 40 )2 =2385.68 + ( - x' 0.422618 + y' 0.90631 - 131.0465 )2 0.59152

The involute helicoid is:x'IH12 = 0.90631 (cost7(1,2) 48.84346 -µ1,2 0.79267 sint7(1,2)) −

0.422618 ( -569.34 t7(1,2) + µ1,2 0.99634 + -131.05

y'IH12 = 0.42262 (cost7(1,2) 48.84346 -µ1,2 -0.0855 sint7(1,2)) + -0.90631 ( -569.34 t7(1,2)+µ1,2 0.99634 + -131.05

z'IH12 = 48.8435 sint7(1,2) + -0.08548 µ1,2 cost7(1,2)+ 40

The slip track is:

x'ST12 = 44.2672 (cost8(1,2) - t8(1,2) sint8(1,2) -1.0074 ) + 0.422618 (t8(1,2) 4.22134+ -131.05

y'ST12 = 20.6421 (cost8(1,2) - t8(1,2) sint8(1,2) -1.0074 ) + -0.906308 (t8(1,2) 4.22134+ -131.05

z'ST12 = 48.8435 sint8(1,2) + -49.2056 t8(1,2).cost8(1,2) + 40

These matrices transform the entities into new positions defined by the Lines of Action and the G

Axes. The entites lying along Gear Axis 2 are:

For Line of Action 1 -

The base hyperboloid is:

(x" 0.90631 - y" 0.422618 )2 + (z"+ 40 )2 =7390.76 + ( - x" 0.422618 - y' 0.90631 + 127.0806 )2 8.09917

The involute helicoid is:x"IH21 = 0.90631 (cost7(2,1) 85.96954 - µ2,1 0.33151 sint7(2,1)) +

0.422618 ( -244.66 t7(2,1) - µ2,1 0.943451 + 127.081

y"IH21 = -0.4226 (cost7(2,1) 85.96954 - µ2,1 0.33151 sint7(2,1)) + 0.906308 ( -244.66 t7(2,1) - µ2,1 0.943451 + 127.081

z"IH21 = 85.9695 sint8(2,1) + 0.331512 µ1,2 cost7(1,2)- 40The slip track is:

x"ST21 = 77.9149 (cost8(2,1) - t8(2,1) sint8(2,1) -1.1409 ) - 0.422618 (t8(2,1) -34.463- 127.081

y"ST21 = 36.3323 (-cost8(2,1) - t8(2,1) sint8(2,1) -1.1409 ) - 0.906308 (t8(2,1) -34.463- 127.081

y"ST21 = 85.9695 sint8(2,1) + -98.0793 t8(2,1).cost8(2,1) - 40

For Line of Action 2 -The base hyperboloid is:

(x" 0.90631 - y" 0.422618 )2 + (z"+ 40 )2 =6626.9 + ( - x" 0.422618 - y' 0.90631 + 116.1217 )2

135.871

The involute helicoid is:

x'IH22 = 0.90631 (cost7(2,2) 81.40577 - µ2,2 0.08548 sint8(2,2)) − 0.422618 ( 948.897 t8(2,2) + µ2,2 0.99634 - 116.122

y"IH22 = -0.4226 (cost7(2,2) 81.40577 - µ2,2 0.08548 sint7(2,2)) + 0.906308 ( -948.9 t7(2,2) - µ2,2 0.99634 + 116.122

z"IH22 = 81.4058 sint8(2,2) + 0.085476 µ2,2 cost7(2,2)- 40The slip track is:x"ST22 = 73.7787 (cost8(2,2) - t8(2,2) sint8(2,2) -1.0074 ) + -0.422618 (t8(2,2) -7.0356

- 116.122y"ST22 = 34.4036 (-cost8(2,2) - t8(2,2) sint8(2,2) -1.0074 ) - 0.906308 (t8(2,2) -7.0356

- 116.122z"ST22 = 81.4058 sint8(2,2) + -82.0094 t8(2,2).cost8(2,2) - 40

The base hyperboloids

Substituting the coordinates of E into the equations of the base hyperboloids as a check producesGear Axis 1 and Line of Action 1 Gear Axis 2 and Line of Action 1

LHS RHS LHS-RHS LHS RHS LHS-RHS2842.7 2842.74 -0.008809 7896.48 7896.5 -0.02447

Gear Axis 1 and Line of Action 2 Gear Axis 2 and Line of Action 2

LHS RHS LHS-RHS LHS RHS LHS-RHS2842.7 2842.75 -0.019829 7896.48 7896.531 -0.055081

The involute helicoids

Solving the equations of the involute helicoid numerically by combining x and y and using the newequation with z to produce t7* and µ*. For the entites lying along Gear Axis 1 are:for Line of Action 1 -

0 = 514.37829 Solver model 1,1 FALSE0 = -10.19789 2

t7,11* µ11* FALSE4.60925 325.5085 100

for Line of Action 2 -0 = -1769.683 Solver model 1,2 FALSE0 = 10.9377 2

t7,12* µ12* FALSE-2.7211 99.404675 100

Solving the equations of the involute helicoid numerically by combining x and y and using the newequation with z to produce t7* and µ*. For the entites lying along Gear Axis 2 are:for Line of Action 1 -

0 = 27.293562 Solver model 2,1 FALSE0 = -15.09035 2

sint7 t7,21* µ21* FALSE

cost7 -4.5905 1250.9822 100

sint7 for Line of Action 2 -

cost7 0 = -10905.85 Solver model 2,2 FALSE

0 = 50.084078 2

t7,22* µ22* FALSE7.79154 3531.8436 100

The slip tracks

Solving the equations of the slip track numerically by changing t8 and rotating the hyperboloids locin the standard position produces, for the entites defined by Gear Axis 1 and Line of Action 1 -

sint7cost7 Equation 57 produces 0 = 0 Equation 56z produces 0 = 0

when θz1,1= -0.57105 rad

sint7 t8(1,1)*= 0.22929 rad Solver model 1,1 TRUE

cost7 0

FALSE100

in the standard position produces, for the entites defined by Gear Axis 1 and Line of Action 2 -

Equation 57 produces 0 = 0 Equation 56z produces 0 = -1E-08

when θz1,2= -0.57571 rad

t8(1,2)*= -0.43449 rad Solver model 1,2 FALSE

2TRUE

100

Solving the equations of the slip track numerically by changing t8 and rotating the hyperboloids locin the standard position produces, for the entites defined by Gear Axis 2 and Line of Action 1 -

eEquation 59 produces 0 = -1E-14 Equation 58z produces 0 = -2E-07

when θz2,1= 0.571045 rad

t8(2,1)*= -0.22929 rad Solver model 2,1 FALSE2

e FALSE100

Equation 59 produces 0 = 0 Equation 58z produces 0 = 1.5E-08

when θz2,2= 0.575708 rad

t8(2,2)*= 0.434486 rad Solver model 2,2 FALSE2

TRUE100