• TABLE OF CONTENTS
HIDE
 Title Page
 Copyright
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Review of the literature
 Camshaft geometry
 Camshaft geometry: implications...
 Modeling of the profile grinding...
 The superfinishing process
 Detection of spurious roundness...
 Digital filtering of inspection...
 Control System
 Experimental results
 Conclusions and recommendation...
 Appendix
 Reference
 Biographical sketch
 Copyright














Title: Error compensation in radial profile grinding
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Permanent Link: http://ufdc.ufl.edu/UF00089744/00001
 Material Information
Title: Error compensation in radial profile grinding
Series Title: Error compensation in radial profile grinding
Physical Description: Book
Language: English
Creator: Dalrymple, Timothy Mark
Publisher: Timothy Mark Dalrymple
Publication Date: 1997
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Bibliographic ID: UF00089744
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Table of Contents
    Title Page
        Page i
    Copyright
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
        Page ix
        Page x
        Page xi
        Page xii
    Abstract
        Page xiii
        Page xiv
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Review of the literature
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
    Camshaft geometry
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
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        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
    Camshaft geometry: implications for the grinding and inspection process
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
        Page 39
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        Page 66
        Page 67
        Page 68
    Modeling of the profile grinding process
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
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        Page 85
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        Page 87
        Page 88
    The superfinishing process
        Page 89
        Page 90
        Page 91
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
    Detection of spurious roundness and lift data
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
    Digital filtering of inspection data
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
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        Page 127
        Page 128
        Page 129
        Page 130
        Page 131
        Page 132
    Control System
        Page 133
        Page 134
        Page 135
        Page 136
        Page 137
        Page 138
        Page 139
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    Experimental results
        Page 163
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    Conclusions and recommendations
        Page 189
        Page 190
        Page 191
    Appendix
        Page 192
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        Page 222
        Page 223
    Reference
        Page 224
        Page 225
        Page 226
        Page 227
        Page 228
    Biographical sketch
        Page 229
        Page 230
        Page 231
    Copyright
        Copyright
Full Text








ERROR COMPENSATION IN RADIAL PROFILE GRINDING


By


TIMOTHY MARK DALRYMPLE














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1997






















Copyright 1997

by

Timothy Mark Dalrymple












ACKNOWLEDGMENTS
Like all undertakings, completion of this work would not have been
possible without the help of others. I am particularly grateful for the support of
Addison Cole and Chuck Dame at Adcole Corporation. I am also grateful to

Mike Taylor at Dana Corporation for his help in coordinating the experimental
trials. Also, I would like to acknowledge the continued support and enthusiasm
of John Andrews at Andrews Products.

Lastly, I would thank my advisor John Ziegert. His confidence,
encouragement and patience have made the difference.












TABLE OF CONTENTS

page
ACKNOWLEDGMENTS ...................................................................................... iii

L IS T O F T A B L E S .................................. ........... ........... .................................. vii

LIST O F FIG U R E S ..................................................................................... . viii

A B S T R A C T .......... . ..... . ................ .......... ....... ...... ........... ........ .....................xiii

CHAPTERS

1 IN T R O D U C T IO N ........................................................................................... 1

Scope of the Problem .................. ...........................................1
Profile Grinding Technology.................................................... ...................... 2
Profile Inspection Technology..................................... .. .................... 4
Potential for Improvement ............................................................ .................4

2 REVIEW O F THE LITERATURE.....................................................................8

Error A voidance............. ............................................................................. .... 8
Error Compensation Based on In-Process Inspection........................ 10
Error Compensation Based on Post-Process Inspection .............................. 12

3 CAMSHAFT GEOMETRY ........................................................................... 13

Introd uctio n ............................................................................................. . 13
Coupling of Timing and Profile Errors .................................... ........... .. 13
Follower Motion Coordinate Transformations.....................................19
Radius of Curvature of the Cam Profile .................................................29
Interpo latio n ......................................................................................... . 3 1

4 CAMSHAFT GEOMETRY: IMPLICATIONS FOR THE GRINDING AND
INSPECTION PROCESSES....................................................34

Introd uctio n ......................................................... .................................. . 34
Camshaft Geometry and the Grinding Process........................................35
The Cam shaft Tim ing Datum ..................................................................35
The M achining Axis Of Rotation ..............................................................35
Camshaft Geometry and the Inspection Process................................... 37








Basic Elements of Cam Geometry ........ ........................................39
Timing of Round Features ............................................ ..................41
Measurement of Cam Lobe Timing....................................................... 49
Com prison of M ethods .................................................. ..................... .53
The Effect of Radiused Followers on the Nonconcentricity
Calculation........ .. ...........................61

5 MODELING OF THE PROFILE GRINDING PROCESS................................69

Introduction n .................................... ............ ...................................... . 69
Modeling Approach and Requirements......................................................69
Physical Model of the Cam Grinding Process: Depth of Cut.........................74
Residual Sparkout Errors............. ................. ...................................... 81
Linearization of the Process Model.....................................................86

6 THE SUPERFINISHING PROCESS............................................................89

Introduction........................ ......... ...................................................... 89
Description of Process .......... ....................... .................................. .... 90
Experim ental Results.................................................. .............................. 92
Discussion of Experimental Results.................................................. 95

7 DETECTION OF SPURIOUS ROUNDNESS AND LIFT DATA................ 97

Introduction.......... ......... ......... .........................................................................97
Description of Method .......................................................................... 98
Linear Surface Model and Residuals.....................................................98
Deletion of the Point-of-Interest ...............................................................99
Deletion of a Window of Points......... .......................... .................... 100
Studentized Deleted Residuals.................................. ...................... 101
The Modified Studentized Deleted Residuals..................................... 102
Computational Efficiency .................................................................... 104
Preliminary Experimental Results........ ................ ..................... ...... 104

8 DIGITAL FILTERING OF INSPECTION DATA ......................................... 107

Measurement and Process Noise ........................... ............................. 107
Recursive Filters for Lift, Timing, Size, and Nonconcentricity..................... 107
Smoothing Inspection and Compensation Data ........................................ 108
Control Signal Noise due to Journal Bearing Roundness Error................. 116
Sources of Roundness Error in External Cylindrical Grinding ............. 117
Effect of Journal Bearing Roundness Error on Cam Profile................. 119
Design and Implementation of the Digital Lobing Filter...................... 119
Identification and Removal of Lobing Effect.......................................122
Effect of Lobing on Timing Error Noise........ ............. ..................... 132








9 CONTROL SYSTEM ......... ........... ...... ............................ 133

The C controlled Process........................................................................ 133
Physical Implementation ............ ................................................ 135
C control S system D esign ................... ........................ .............................. 136
Parameter Interaction and the Design of the Lift Control System:
G rinder O nly ............ ............. .... .......... ........... ... 136
Simulation of the Simple Lift Controller for Grinding Under Noisy
C onditions............................................... ........................ 150
Base Circle Radius Controller....................................... ................... 158
Base Circle Nonconcentricity Controller.......................................... 159
Lobe Timing Angle Controller ........................................ .................. 160
Sum m ary ............ ........ .. .. .. .. . ...... ........ ..... .... ................. 162

10 EXPERIMENTAL RESULTS......................... ............................ 163

Description of Experimental Setup ........................................ ................. 163
Control of the Radial Grinding Process Alone........................................... 163
Control of the Radial Grinding Process and the Superfinishing Process.... 173
Discussion of Results from First Round Testing ....................................... 181
Final Round Testing .......... .. ........... .......... ... ........... ............... 181
Discussion of Results........................ ....... .......... .. ................ 182

11 CONCLUSIONS AND RECOMMENDATIONS......................................... 189

Implementation of the Control System.............. .... .......................... 190
Future W ork ....................... ........................................................ 190

APPENDICES

A VARIANCE FOR A SLIDING WINDOW....... ....... ............................... 192

B NONCONCENTRICITY OF BASE CIRCLE RELATIVE TO BEARING
JOURNAL ......... ..... ............................... 209

C RADIAL DISTANCE TO A POINT OF CONTACT......................................211

D PROGRAM DATA: GRINDER AND GAUGE ........................................215

REFERENCES .................... ........................ ............................... 224

BIOGRAPHICAL SKETCH.................................. .. ... ..........................229












LIST OF TABLES

T a b le ............................................................................................................. a g e

3-1 Cam Profile Nomenclature...................................................................... 16

4-1 Machining and Inspection Axes of Rotation ............................................38

4-2 Nomenclature for Process Geometry ......................................................... 42

4-3 Timing Angle and Bias Results for Type I Nonconcentricity Error ..............60

4-4 Timing Bias Results for Type II Nonconcentricity Error .............................61

5-1 Grinding Model Nomenclature................................................................... 72

6-1 Modified Grinding Model Nomenclature for Superfinishing Model...............92

6-2 Process Repeatability..............................................................................96

7-1 Nomenclature for Asperity Detection Method............................................. 101

8-1 Digital Filtering Nomenclature............................................................... 115

9-1 Lift Control System Nomenclature... .....................................................139

9-2 Repeatability for Grinding, Superfinishing and Gauging.......................... 156

9-3 Controller Gains Determined by Simulation for Noisy Conditions.......... 157

10-1 Gains for Profile Grinding................................................................... 164

10-2 Gains for Parts Inspected after Superfinishing.......................................173

10-3 Gains for Profile Grinding: Flat Follower.............................................. 182

10-3 Gains for Profile Grinding: Flat Follower.............................................. 181












LIST OF FIGURES


Figure ............................................ ......... ..... ................ ..... .............

2-1 Plunge G rinding M odel ......................................................... .................. 9

3-1 Camshaft Coordinate System: Side View.......................... ............ 15

3-2 Shaft Coordinate System ..................... ................................................ 15

3-3 Lobe Coord. System ........................... ....... .............................................. 15

3-4 Camshaft Coordinate System : CW and CCW Convention..........................17

3-5 Lobe Coordinate System: Nose +1800 Convention ....................................17

3-6 Lobe Coordinate System: Nose Convention............................................. 18

3-7 Typical Manufacturers Cam Lift Data Specification..................................... 18

3-8 Translational Roller Follower Coordinate Transform................................... 21

3-9 Coordinate Transformation: Translational Flat to Roller .................................26

3-10 Coordinate Transformation: Translational Roller to Flat ............................28

4-1 Steady Rest and Three Grinding with Jaw Chuck: AJAOR..........................37

4-2 Inspection of Nonconcetric Nominally Round Part.......................................43

4-3 Introduction of Phase Shift ....................................................................... 46

4-4 Measurement of a rj Phase Error for an Eccentric............................. ..47

4-5 Typical Cam Lobe Lift and Geometric Velocity.............................................50

4-6 Frequency Content for Typical Exhaust Cam Profile........................................51

4-7 Cam Lobe with a Timing Error of 86 ........................................................ 51








4-8 Sim ulated Cam Lift Error ......... ...............................................................54

4-9 Existing Data Reduction Method for Profile Data ......................................55

4-10 Form of Timing and Nonconcentricity Errors...........................................56

4-11 Proposed Data Reduction Method for Profile Data ....................................58

4-12 Effect of Nonconcentricity Error in Grinding Wheel Motion on Follower

Motion ............... ........................................ ....... .........59

4-13 Effect of Journal Bearing Nonconcentricity Error .......................................60

4-14 Inspecting a Nonconcentric Surface with a Radiused Follower ................62

4-15 Flat Follower Approximation for Nonconcentricity Errors...........................68

5-1 Cylindrical Grinding Model Modified for Cam Profile Grinding ...................69

5-2 Cam Profile Grinding Model ............................................................... ..75

5-3 Typical Grinding Cycle: Rough, Finish and Sparkout.................................. 81

5-4 Grinding Model Results in Grinding Wheel Motion Coordinates................ 85

5-5 Grinding Model Results in Grinding Wheel Motion Coordinates ................ 86

5-6 Linearized Profile Grinding Process Model:k, = 0.001, K4.0.00008 .............87

5-7 Linearized Profile Grinding Process Model:k, = 0.002, K,.0.00016 ............88

6-1 Camshaft Superfinishing Operation.............................................................90

6-2 Camshaft Superfinishing Model............... ................................................ 91

6-3 Typical Lift Error for Profile Grinding: No Compensation .............................93

6-4 Typical Lift Error for Profile Grinding and Superfinishing : .................... 94

6-5 Mean Lift Error: Nonconcentricity Removed............................................... 94

6-6 Statistics of Lift Error: Nonconcentricity Removed ..................................... 95








7-1 Deleted Residual M odel ....................................................... ................ 100

7-2 Lift Inspection Data with Spurious Inspection Points................................ 105

7-3 Modified Studentized Deleted Residuals............................................. 106

8-1 Nominal Lift and Error Data for a Typical Injector Lobe ...........................110

8-2 Fourier Transform of Typical Exhaust Lobe Nominal Lift Data.................. 110

8-3 Effect of Band-Limiting the Nominal Lift Data................................... 111

8-4 Modeled and Measured Lift Inspection Data................. ..... ...................111

8-5 Frequency Content of Modeled and Actual Lift Error ............................... 112

8-6 Frequency Response for a Zero Phase Linear Smoothing Filter ........... 114

8-7 Filtering Results for a Zero Phase Linear Smoothing Filter ..................... 114

8-8 Simulated Journal Bearing Inspection Data ............................................... 116

8-9 Simple Two Pad and Wrap-Around Three Pad Steady Rest.................... 120

8-10 Mapping of Bearing Roundness Error on Profile................................. 121

8-11 Effect of Lobing Error .................................................................. .....124

8-12 Nominal Lift and Lift Error: Coordinate System from Figure 3-8............. 126

8-13 Removal of Nonconcentricity and Lobing Effects.................................132

9-1 Cam Profile Grinding Control System............................ .................. 134

9-2 Cam Profile Grinding with Superfinishing Control System .......................134

9-3 Lift Control System Including Process Model............................................. 139

9-4 Simulated Control of Profile Grinding Error: k,=0.001.............................. 143

9-5 Simulated Control of Profile Grinding Error: k,=0.002..............................143

9-6 The Interacting Profile Grinding Process Model......................................... 147








9-7 Simulation of Noninteracting Controller: Symmetrical Lobe.................... 149

9-8 Noise in Noninteracting Compensation.................. ... ..................150

9-9 Grinding Process Simplified Control System..................................... 151

9-10 Rearranged Block Diagram for Disturbance Rejection............................. 153

9-11 Simulated Control of Lift at an Arbitrary Point Over N Parts................... 156

9-12 Grinding and Superfinishing Simplified Control System......................... 157

9-13 Base Circle Radius Controller ............................................................. 158

9-14 Simulated Control of Base Circle Size Error........................................... 159

9-15 Base Circle Nonconcentricity Controller............................................... 159

9-16 Simulated Control of Nonconcentricity Error ......................................... 160

9-17 Lobe Timing Angle Controller ....... .... ..................... ....................... 161

9-18 Sim ulated Tim ing Controller ........................................................ ......... 162

10-1 Uncompensated Error: Part 1 after Grinding: Lobes 1-6....................... 167

10-2 Compensated Error: Part 2 after Grinding: Lobes 1-6............................ 168

10-3 Compensated Error: Part 3 after Grinding: Lobes 1-6..............................169

10-4 Compensated Error: Part 4 after Grinding: Lobes 1-6............................ 170

10-5 Standard Deviation of Lift Error after Grinding....................................... 171

10-6 Base Circle Size and Timing after Grinding ............................................ 172

10-7 Uncompensated Error: Part 1 after Superfinishing: Lobes 1-6................. 175

10-8 Compensated Error: Part 2 after Superfinishing: Lobes 1-6................... 176

10-9 Compensated Error: Part 3 after Superfinishing: Lobes 1-6.................. 177

10-10 Compensated Error: Part 4 after Superfinishing: Lobes 1-6................. 178








10-11 Standard Deviation of Lift Error after Superfinishing..............................179

10-12 Base Circle Size and Timing after Superfinishing................................ 180

10-13 Uncompensated Lift Error: Part 1 after Grinding Lobes: 1-6 ............... 184

10-14 Compensated Lift Error: Part 6 after Grinding: Lobes 1-6.................... 185

10-15 Standard Deviation of Lift Error after Grinding ..................................... 186

10-16 Total Lift Error after Grinding.................................................................. 187

10-17 Lobe Timing After Grinding .................................................................. 188

A-1 Deletion of the Point of Interest............................................................. 192

A-2 Computation of the Variance for a Sliding Window.................................. 192

B-1 Error due to Journal Bearing Nonconcentricity ........................................209

C-1 Radial Distance to the Point of Contact...................................................211













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ERROR COMPENSATION IN RADIAL PROFILE GRINDING

By

Timothy Mark Dalrymple

August, 1997

Chairman: John C. Ziegert
Major Department: Mechanical Engineering

To take advantage of the existing computer numerical control technology
in radial profile grinding and inspection, a closed loop control scheme is

proposed. In this scheme, the commanded part geometry is modified based on

the errors observed in previous parts in an effort to minimize the lobe profile,
size, non-concentricity and timing errors. Experimental results--obtained in an
industrial environment--show that significant improvement in cam geometry can
be achieved using a properly designed control system and signal processing

algorithms.
In order to design a successful control system, the sources of the process
error are physically modeled. This analysis shows that the primary source of

profile error is the depth of cut variation that exists due to the curvature of the
cam profile. A physical model of this effect is developed and equations are
presented to simulate the complete grinding cycle. From this model, the
predicted error is compared with experimental results-corrected for bias using

methods developed here--with good agreement.

xiii









Also in this work, a camshaft superfinishing process in examined and

found to be sufficiently repeatable, in terms of the introduced profile errors, to
be included in the process control loop. Consequently, it is not necessary to

disrupt the normal process flow by inspecting parts between the grinding and
superfinishing operation.
The implemented control system is designed to be general in nature and
is not restricted to a particular camshaft geometry, a particular cam grinder or a

specific inspection gauge. The minimization of the profile error by error

compensation allows the potential for the grinding process to be optimized with

objectives other than minimizing the profile error in mind. For example, if a
sparkout cycle is included only to reduce profile errors, it can be eliminated and
thereby reduce the total grinding cycle time. Implementation of this system on

existing computer numerically controlled equipment is inexpensive. The

controller is implemented in software and the communication with the inspection
and production equipment is over a low cost, peer-to-peer network.












CHAPTER 1
INTRODUCTION


Scope of the Problem

Cams find application in a wide range of consumer and industrial
products. In machine tools, cams have long been used to produce precise high-

speed machine axis movements. Applications of cams in machine tool motion
control include both chip producing and dedicated high-speed assembly

equipment1. Also, cams are critical components of diverse machines such as
blood separation units, fuel injection pumps, internal combustion engines,

power steering pumps, and automated laser-scanner systems.

Recent advances in servo motors and computer numerical control (CNC)
technology have led to the replacement of cams in many industrial applications.

CNC based motion control increases flexibility and eliminates high cost, long
lead time, part specific cams and greatly reduces setup time. Thus, in most
applications, CNC motion control displaces cam based systems as
manufacturers update existing equipment. Still, cams will remain essential for
certain dedicated applications, such as in the internal combustion engines for
the foreseeable future.

Cams are used to control the valve timing in internal combustion engines.
In some diesel applications, cams are also used to control the fuel injection
timing. In both instances, the cam profile is crucial in controlling the combustion
process and the resulting level of exhaust emissions. In recent years, most
industrialized nations have tightened emission standards for internal combustion
engines. These new standards have simultaneously effected a change in cam









profile design philosophy and increased the required profile accuracy. The

change in design philosophy generally results in more radical cam profiles

(higher acceleration) which are inherently more difficult to manufacture.

In order to meet the accuracy and surface finish requirements of most
applications, the finished cam profile is produced by grinding. The grinding of

the cam profile is generally referred to as radial profile grinding or simply as cam

grindings. In applications where surface finish requirements are especially high,

the cam profile may be honed or superfinished after grinding. Superfinishing

operations generally improve the surface finish quality at the expense of the cam

profile accuracy.


Profile Grinding Technoloqy

The recent wide spread introduction of CNC technology into the radial
profile grinder has produced significant changes in the machine design. The

most significant changes, from the point of view of this research, are in the way

the nominal part geometry is specified and the method that the required relative
axial motions are generated. Prior to the introduction of CNC and high

bandwidth servo motors, the nominal part data were stored in the form of a

master cam. The cam stored the nominal part data and generated the relative

motion between the rotational and linear axes to produce the desired cam
profile. On CNC grinders the nominal part data are stored in digital memory and

are therefore readily modified. The discussions that follow and the

compensation strategy developed in this research are applicable only to CNC

radial profile grinders.

Numerous factors contribute to cam profile errors that result from the
grinding process. For servo controlled axes, both the quasi-static positioning









errors and the dynamic following errors result in a profile error. Additionally, the

static and dynamic synchronization errors between the work spindle rotational

axis and grinding wheel infeed axis result in a profile error. Other and more

important error sources arise due to the geometry of cam grinding. As the cam

lobe rotates during the grinding process, the radius of curvature of the cam

surface that contacts the grinding wheel varies. This effects a change in the

grinding threshold force and causes the depth of cut to vary2. Also the depth of

cut varies due to the change in the "footprint" speed. The footprint is the contact

zone between the grinding wheel and the workpiece. This variation in footprint

speed is due to the nature of the contact geometry for a cam profile and a

grinding wheel. This speed variation affects the depth of cut to produce a profile

error that is similar in form to the profile error caused by varying radius of

curvature. However, the magnitude of the two effects may differ significantly. In
addition to causing a profile error, the footprint speed variation may lead to

burning in high-speed zones is high. Therefore, many profile grinders vary the

workpiece speed of rotation within each revolution to minimize the footprint
speed variation.

The profile errors described above are systematic in the sense that they
repeat from part to part. Nonrepeatable errors are also present in the grinding
process. One source of nonrepeatable errors is the use of work supports,

known as steady rests. Though these supports take several forms, they

essentially support the camshaft by restraining the camshaft journal bearing

against deflections caused by the grinding force. In this setup if the journal
bearings are not perfectly round, then the axis of rotation of the camshaft shifts
due to the journal bearing roundness error. This shift in the axis of rotation

causes an error in the cam profile that is a function of both the form of the

roundness error and the geometry of the steady rest. This error is one of the









nonrepeatable errors that is examined in detail in this research. Also, profile,

size and timing errors which do not repeat from part to part are present in the

process due a variety of other sources. These sources include machine

vibrations, work spindle errors, and thermal deformations of the machine bed.


Profile Inspection Technology

To evaluate and control the profile grinding process, it is necessary to
reduce the inspected data into the parameters that are used to specify the

nominal part geometry in the profile grinder. The parameters are as follows: cam

profile, timing angle, base circle size, and base circle nonconcentricity.

Additionally, the inspection gauge must be sufficiently repeatable so that it does

not introduce an excessive amount of noise into the control system. Currently,

gauges of this standard are widely installed in engine manufacturing plants

worldwide3.


Potential for Improvement

Computerized cam gauges that produce digital inspection results have
been available for over two decades. However, the potential offered by these

gauges for error compensation is poorly utilized. Given the flexibility of CNC

profile grinders, and if the repeatable profile, timing, nonconcentricity and size

errors can be identified, then it should be possible to correct for errors in
subsequent parts by manipulating the commanded part geometry. At least one
reason this is not done is that builders of cam grinders are reluctant to promote

the idea that their machines do not stand alone, but rather require the feedback

of inspection results. Thus, despite the potential advantage of feedback, its

promotion by cam grinder builders presents a significant marketing difficulty.









However, as this research demonstrates, the compensation can effectively

reduce profile error. Currently, some grinder builders do supply manual
compensation utilities which the machine programmer can use to modify the

nominal part data by eye while examining a plot of the cam inspection data. This
technique is time consuming, error prone, and does not allow varying gains for

errors from different sources. Consequently, it is not very effective in reducing

profile error.

This research builds on previous research by the author to develop an
error compensation control system for cam profile grinding that works with

existing standard profile grinders and cam inspection gauges. In this work,

various error sources are identified in the grinding process and a control system
is designed to compensate for each of the repeatable errors. This system
simultaneously controls the geometric parameters of the cam profile grinding
process. These parameters are as follows: cam profile, cam profile base circle

size, base circle nonconcentricity to the workpiece axis, and cam profile timing.
Much of this research deals with issues that are essential to implementing the
control system in an industrial environment using existing inspection gauges and
grinding machines.

Timing and profile errors of a cam profile are coupled. Therefore, the
algorithm used to reduce the inspection data into profile and timing errors effects
magnitude of the timing error and the magnitude and shape of the reported
profile error. No standard algorithm exists and all of the methods currently in

use can lead to deceptive results under certain conditions. These results can
cause difficulty in identifying the source of the observed profile error. Therefore,
a new and more robust method of decoupling the profile inspection data into
profile and timing errors is proposed and tested in Chapter 4.









With the timing, profile and nonconcentricity errors successfully
decoupled, the observed profile errors can be related to the commanded part

profile. As is shown in Chapter 5, a simple model of the profile grinding process

which successfully predicts the form of the profile error can be developed. With

the form of the model known, the coefficients of the model can readily be
determined from experimental data.

In Chapter 6, the repeatability of the superfinishing process is examined.
Experimental results show that the profile error due to the superfinishing process

is repeatable. Therefore, the superfinishing process is included in the control

loop. In this arrangement, the control system modifies the commanded part

geometry to produce an intentional profile error in the cam lobe prior to the

superfinishing process. This error is of the form such that after the part is

superfinished, the profile error is minimized.

The processes to be controlled exist in industrial environments.
Consequently, grinding swarf or dirt on the part is common and often causes
erroneous measurement results. The effect of this type of contamination on the
measurement results is to add noise to the data. In general, dirt may cause a

point on the profile that lies out of the expected range of the profile error based

on the apparent randomness of the surface. This point, known as an outlier or
asperity, does not generally present significant difficulties when the inspection

process is used only as a quality monitoring process. A skilled gauge operator

can usually recognize results that are caused by contaminates and can pass or
reinspect the part as appropriate. However, for the case where the inspection
results are used to calculate a compensated commanded cam profile, care must

be taken to ensure that spurious inspection results are rejected from the control

signal. Therefore, a scheme to identify potential outliers in the profile inspection

data is developed. This topic is discussed in detail in Chapter 7.






7


Since it is possible only to compensate for the repeatable portion of the
measured error, it is important to reduce noise in the measured data. As we

shall see, our control problem is multidimensional and we therefore can filter the

data in two directions. While this adds a certain amount of complexity to the

problem, it also allows the data to be filtered without introducing phase-lag into

the control problem. This topic is presented in Chapter 8.

Finally in Chapters 9 and 10 the control system is described and

experimental results are reported for a variety of industrial situations.












CHAPTER 2
REVIEW OF THE LITERATURE
In the area of cam profile grinding, much research and industrial effort
has focused on error avoidance. This research has produced improvements in

thermal stability of the machines, stiffer and more accurate work and grinding

spindles, and precise low-friction machine ways and axes. While these
improvements have resulted in significant gains in machine accuracy, they have

not done so without great expense. Today, a typical high-production,

high-precision CNC cam profile grinder sells for approximately one million
dollars.

In previous work,4 the positional errors in the current generation of radial
profile grinders are shown to be small compared with the errors inherent in the

radial profile grinding process. This is true since improvements in positioning

accuracy, both static and dynamic, of cam grinders has paralleled the

industry-wide improvement in machine ways and motion control technology.
Conversely, process errors which depend on factors such as machine and part

stiffness, the grinding wheel cutting stiffness, and the geometry of the grinding

process remain largely unaddressed and can exceed 40 meters.


Error Avoidance

As discussed, for radial profile grinding on high precision CNC cam
grinder, process errors are the dominant source of profile errors in the ground

parts. Process errors are those that exist even with perfect positioning of all
machine elements. These errors arise due to deflections in the machine

elements that result from grinding forces which change based on the part









geometry. The literature review is restricted to methods proposed for correcting

for these types of errors.

The basic model of the external cylindrical grinding process is described
by Hahn5 and Lindsay6 and shown-adapted here for external radial profile

grinding-in Figure 2-1.











S'j Feed
SSlide

Machine Bed '


Figure 2-1 Plunge Grinding Model Adapted from King, R. I., and Hahn, R. S.,
Handbook of Modem Grinding Technology, New York: Chapman and Hall, 1986.

As shown, the model includes the workpiece stiffness, the grinding
wheelhead structural stiffness, and also the wheel stiffness. The stiffness of the

system is important since the grinding depth of cut is a function of the difference

between normal grinding force and the threshold grinding force. The value of

the threshold force varies as a function of the equivalent diameter, De, which

depends on the diameter of the workpiece as


D, DS
De
e DW +D
Equation 2-1









where D, is the workpiece diameter, and Ds is the diameter of the grinding
wheel.

Hahn7 also specifically discusses the systematic process errors for

internal radial profile grinding. For radial profile grinding, the workpiece

"diameter" is replaced by twice the local radius of curvature of the cam surface.

The variation the radius of curvature produces a corresponding change in the

threshold force, and therefore, the depth of cut. Also, according to Hahn, the

depth of cut depends on the velocity of the point of contact between the cam

profile and the grinding wheel. For cam profiles, the velocity of the point of

contact varies significantly due to the cam geometry. Gal-Tzur, Shpitaini and

Malkin"'9 also discuss these error sources.

These relationships provide the basis for the development of a model of

cam grinding profile errors. The actual coefficients and thus the magnitude of

the most significant error must be determined from experimental results.

However, even while these process errors are the dominate error source, they

remain uncorrected.


Error Compensation Based on In-Process Inspection

In their work at Cranfield Institute of Technology, Cooke and Perkins'1
proposed an error compensation scheme for profile grinding based on in-

process workpiece inspection. In this scheme, a measurement probe is

installed 1800 out-of-phase with the grinding wheel. The authors proposed

sampling the profile error during the grinding operation and using the 1800
phase lag between the grinding wheel and inspection probe to compute a

corrected position command for the servo controller on the next pass. While

such in-process inspection and control schemes appear attractive, they have not









been implemented in radial profile grinders. While the reasons for the not

implementing this system are not known, there appear to be several technical

difficulties associated with this approach.

One major difficulty is the integration of the inspection process into the
grinding environment. The grinding environment is contaminated by swarf from

the grinding wheel. The environment is also subjected to relatively large

temperature variations. Machine vibrations are present during the grinding

operation and, furthermore, the relatively high rotation speed of the workpiece
during grinding would require high mechanical bandwidth for the measurement

probe.

Also, it is interesting to consider an additional complication to the system
that results from the grinding process model discussed earlier. Recall that the

variation in the depth of cut, which results in the process profile errors, is

caused by the deflection of the grinding system where the workpiece is generally

the most flexible component in the system. Consequently, the measurement of

the cam surface during grinding results not in a reading of the error at one point
on the surface, but rather in the superposition of the ground error at the point of
interest and the defection due to the grinding force 1800 out-of-phase.

Therefore, the in-process measurement accuracy of the cam profile is influenced

by the workpiece stiffness and the profile error cannot be measured directly. In

fact, as discussed in Chapter 5, the dominant profile error source for high

accuracy CNC cam grinders is the grinding system flexibility.


Error Compensation Based on Post-Process Inspection

Various researchers11'12'13 have investigated the general problem of error
compensation in discrete manufacturing processes using post-process






12


inspection results. Yang and Menq" describe a scheme for improving the form
of a sculpted milled surface. In this scheme, a part is milled and inspected. The

compensation is then calculated from interpolating the best fit cubic b-spline
tensor-product surface model. The modeled error is then used to compensate
the part program data. The authors reported an 73% improvement in form error
by their method. The error compensation approach in this work resembles the

approach take by Yang and Menq and is an extension of the author's previous
work4 in this area.












CHAPTER 3
CAMSHAFT GEOMETRY


Introduction

4. 14, 15,16 17, 18, 19, 20
Various authors 14, 1and companies 18 19 20use different cylindrical
coordinate systems to describe cam geometry. Implementation of a feedback

control system of the profile grinding process requires the transformation of cam

geometry data between the inspection, the grinder, and the design coordinate

systems. By industrial convention, the cam profile is not specified directly, but

rather, the motion of a cam follower is specified. In cam manufacturing and

inspection, it is often necessary to transform follower motion specifications for

one follower size or geometry to the equivalent specification for a different

follower size (or grinding wheel size) or geometry. These follower motion

transformations are particularly important for understanding the source of profile

errors. This is true since the form of the profile error ground into the cam with a
given grinding wheel radius is distorted when the profile error is described in

terms of a follower size that differs from the grinding wheel radius size.

Therefore, in this chapter, existing specification conventions are identified and

the follower motion coordinate transformation equations are developed.


Coupling of Timing and Profile Errors

Figure 3-1 shows a typical camshaft coordinate system. The coordinate
system is attached to the camshaft with the origin fixed by the camshaft axis of

rotation. ThecX axis, which is the camshaft timing datum, is fixed by the

camshaft timing reference.









The C z datum is chosen as the driven end of the camshaft. As shown in
Figure 3-2, the camshaft timing datum is located by a timing reference feature

such as a dowel pin, keyway, or eccentric. All nomenclature used in this section
is defined in Table 3-1.
The lobe timing angle, c , the axial position of the lobes, Cz,,, and the
axial position of the journal bearings, c zx, are specified in the camshaft

coordinate system. The cam profile is not specified directly in the camshaft

coordinate system. Rather, it is specified in terms of follower lift in the polar

coordinate system that is attached to the cam lobe shown. This coordinate

system is shown in Figure 3-3,. Here, this system is referred to as the lobe

coordinate system. The lobe coordinate system is rotated with respect to the
camshaft coordinate system byc (or c 4 1800) and translated along the cZ

axis by Czi.

The follower lift is then specified as a function of the angle 0 measured

from the X axis. The orientation of the X axis, referred to here as the lobe
profile datum is defined as the lobe timing datum, or the lobe timing datum
1800. The lobe timing datum, in turn, depends on the profile shape: it is

defined as the angle of maximum follower lift. While this specification presents

no complications for nominal profile data, the situation is quite different for

measured profile data. This complication occurs because the lobe timing datum
and the profile angle datum are functions of the form of the profile and must be

determined from the inspection data. Consequently, the profile and timing error

are coupled and the method used to determine the lobe datum affects both the
form of the reported profile error and the reported lobe timing error. This

coupling has many implications for this research and is discussed in more detail

in Chapter 4.














Camshaft
Timing
Reference


Driven
End
c X Direction
of Rotation
cF 3 m Cori Cte Z1 Ccs t c Z12 c ZE2i CS V
Figure 3-1 Camshaft Coordinate System: Side View


Typical Lobe


Typical Lobe
Timing Datum



c


Figure 3-2 Shaft Coordinate System Figure 3-3 Lobe Coord. System


Direction of
Rotation
Iiiii"I


cx -C W E2









Table 3-1 Cam Profile Nomenclature


J ISubscript on quantities that vary with the lobe rotation angle 0
N Number of discrete points used to specify follower lift (generally 360)
rj Radius of follower pitch curve at point j
'r Radius of follower pitch curve for alternate follower, or cutting tool at
point j
rb Base circle radius
r, Roller follower radius
rf" Alternate follower or cutting tool radius
rpoc Radial distance from lobe axis to point of contact j
si Lift for follower with radius rf at point j
s* Lift for alternate follower with radius rf" at point j
X, X coordinate of point of contact of follower and lobe at point j
Xj X coordinate of follower center at point j
Xj X coordinate alternate follower center at point j
X Lobe coordinate system x axis
Cx Camshaft coordinate system x axis
Yj Y coordinate of point of contact of follower and lobe at point j
YCj Y coordinate of follower center at point j
Yj Y coordinate alternate follower center at point j
YJ Y coordinate of point of contact of follower and lobe at point j
Y Lobe coordinate system y axis
C Y Camshaft coordinate system y axis
C Z Camshaft coordinate system z axis
CZmi Axial position of feature (bearing or lobe) xi: x is the feature type; i is
the feature index
a, Lobe contact angle for point j
c mi Lobe timing angle for lobe m,i where m is the feature type; I = intake, E
= exhaust, J = injector, O = other and i is the feature index (intake lobe
0 Lobe rotation angle in lobe coordinate system
6, Lobe rotation angle at point j in lobe coordinate system
eo Lobe rotation angle for alternate follower at point j
p* Radius of curvature of cam surface


t
All subscripts related to the angle of rotation, 0, are periodic and must be normalized as follows:
ifjk<0jk= j k +n;ifj+ kN, jk= j k-N










iUrection of
Rotation


C12
10.C


C, X,CE2
x


Direction of
Rotation
.; .. .


Cy1 C


CXC *~rE1


Figure 3-4 Camshaft Coordinate System : CW and CCW Convention


Figure 3-5 Lobe Coordinate System: Nose 1800 Convention


_ I





18






Irf I r

- -X - -





rb rb
Direction \ Direction
of Rotation of Rotation Y



Figure 3-6 Lobe Coordinate System: Nose Convention


Closing side Opening side Lift Data Table: Exhaust
(Opening side) (Closing side) / rb = 13.2081 rf =10.8585

S .( de.) .. (mm) i (mm)
.. \ \.... ..................... ( ...... ....... .... .....
o n.......................e...o.on .ng loi
Y O (closing) (opening)
0 8.0264 8.0264
1 8.0242 8.0242
2 8.0175 8.0175
ta end r 3 8.0064 8.0064
s- n ( start) 4 7.9908 7.9908
5 7.9708 7.9708
6 7.9464 7.9464
.. ........ ... ... .......... .. ......, .. .. .. ... ...
rb 7 7.9175 7.9175
k 8 7.8841 7.8841
Direction (Direction of 9 7.8463 7.8463
of Rotation Rotation ) 10 7.8040 7.8040


Figure 3-7 Typical Manufacturers Cam Lift Data Specification








Figure 3-4 shows the camshaft coordinate system for both clockwise

(CW) and a counterclockwise (CCW) directions of rotation. Figure 3-5 and
Figure 3-6 show the lobe coordinate systems used to program machine tools.

Lastly, Figure 3-7 shows a typical cam profile specification used in

manufacturers' drawings. Unless otherwise stated, results in this research are

presented using this convention. In this figure, "start" and "end" refer to the

beginning and the end of the base circle. The base circle is the region with zero
nominal lift.

The specification given is not unique in the sense that a follower of any

size radius could be used to specify the lobe profile. Follower sizes represent
different follower motions and if the cam profile is known for a given follower

size, then the follower motion can be calculated for any other follower size or

follower geometry. In the next section, the follower motion coordinate

transformations necessary to transform between various followers are
presented.


Follower Motion Coordinate Transformations

The conventions shown in Figure 3-5 and Figure 3-6 are used to develop
the follower motion coordinate transformations. In all the transforms, the
asterisk (*) represents parameters of the new follower, the grinding wheel, or the
milling cutter motion. Quantities without an asterisk represent both original

motion parameters and quantities that are invariant with respect to the follower.

The nomenclature used in this section is defined in Table 3-1. These
transformations are used to calculate the commanded motion of the grinding
wheel and milling cutters and also to compare inspection results performed with








a different follower radius. In the following development, all parameters are
specified in the lobe coordinate system.

Lift errors which are not a function of the commanded lift appear
differently depending on the follower size/geometry, for example, grinding a

camshaft where the camshaft runout during the grinding process will produce a
first harmonic error in the ground profile. However, this error is exactly the form

of the fundamental frequency of rotation in the grinding wheel motion

coordinates. The form of the fundamental frequency, as well as other harmonics

related to the runout of the camshaft axis of rotation, is distorted in follower

motion coordinates. The magnitude of distortion depends on the shape of the

cam profile and the relative size of the cam follower and the grinding wheel.

This effect will be considered in Chapter 4. With this background stated, the

follower motion coordinate transformations are now developed.

Roller Follower with Radius r, to Roller Follower with Radius rf

From Figure 3-8 the radial distance from the cam axis of rotation to the
14
follower center is


r = rb + + rf

Equation 3-1

where rb is the base circle radius, r, is the follower radius and Sj is the follower

lift at point j. The curve described by the radial distance, rj, is known as the


pitch curve.











follower follower
translational translational
axis* A axis





^~~---4^^ /
rf

x .y


xo\, Yc .(*-nJ .:



Ssrb
start


.-- ... ,--- end
Direction of
Rotation
X

Figure 3-8 Translational Roller Follower Coordinate Transform

The radial distance, rj, is resolved into XY components in the system

attached to the cam lobe as
xcj = rj cos(Oj)

Equation 3-2

yo = rj sin(Oj)

Equation 3-3

where xj and yc are the XY coordinates of the center of the follower. From

Figure 3-8, a right triangle is formed by the line segment from point xcj, Yc to the








point of contact, xj, yj and this segment's projections onto both the X and Y
axes. The segment is of length r, and for this triangle


F(xy,) = (xi xj)2 +(yj yc)2 -r2 = 0

Equation 3-4

The partial derivative of Equation 3-4 with respect to 0 is


aF(x, y,) dx- dy c
-2(x xj) de -2(yj Ycj de =

Equation 3-5

Equation 3-4 and Equation 3-5 are solved simultaneously for the inner cam
envelope to determine the XY coordinates of the point of contact as

1
dy= dxCj,2 dyc2 2
ix = x rfYe +I
X1 r dO d) dOG

Equation 3-6


dx., [dx 2 dy. 2- d
yj = y l +r +dj


Equation 3-7

The derivative of Equation 3-2 with respect to 0 is


dxc, drj
de -r sin(Oe)+ -cos(Oj)


Equation 3-8









The derivative of Equation 3-3 with respect to 0 is


dyc, dr
d = rj cos(Oj)+ Osin(9,)

Equation 3-9

dr.
For the discrete data, the derivative, is approximated from difference
de
equations. Often, r, is not known at evenly spaced points of 0j, and the divided

difference formulation must be used. In other cases, rj is given at evenly spaced

points and the difference equations can be simplified. Since both of these cases

occur frequently, both formulations are presented here. The first divided
21,22,23
difference is defined as


drj rj, rj
dO ()j+1 --0j_1

Equation 3-10

The first difference equation for N evenly spaced points is


dr1 r1+ -r,_1
dO 2


Equation 3-11

It is convenient at this point to define the difference equations for the second

derivative as they are needed in later developments. The second divided

difference is








d2ri 1 rJ- ri ri rj1
dOe2 0 -0j 0j-_jl

Equation 3-12

For N evenly spaced points, the second difference is


d2r rj+ -2 r + r_,
d02 2 2
IN

Equation 3-13

24
The polar coordinates of the point of contact is


rp oc= x,)2 +(y,)2

Equation 3-14

and the two argument inverse tangent function is used to guarantee the correct
quadrant forca as


ai = ATan2(y, xj)

Equation 3-15

With the XY coordinates of the cam lobe surface known, the cam profile can be
described in any follower, grinding wheel or milling cutter coordinate system with
radius r'. The XY coordinates of the center of the cutting tool or follower arel4


x, = x + (rj cos(.)- x)
Cj


Equation 3-16








y j = y + r'-j sin(O,)- yj)
Y j i f

Equation 3-17

The transform from Cartesian to polar coordinates is


r = )2+

Equation 3-18

Again, the two argument inverse tangent function is used to guarantee the
correct quadrant and 0 is


0; = ATan2(ycj,x,)

Equation 3-19

Flat Follower to Roller Follower
For completeness, the transform from flat follower to roller follower motion
coordinates, illustrated in Figure 3-9, is presented. For a flat follower, the XY
coordinates of the point of contact are

dr
xi = rj cos(O)- sin(9O)

Equation 3-20



yj = r, sin( j)+ dr cos(9j)


Equation 3-21









The coordinates of the follower center are


x = x +rf cos(6j)

Equation 3-22


and
y = yj +rf sin(O,)


Equation 3-23


And, as before, Equation 3-18 and Equation 3-19 are used to transform the

results to from Cartesian to polar coordinates.


Figure 3-9 Coordinate Transformation: Translational Flat to Roller


Flat Follower
Translational Axis


Roller Follower
Translational Axis

xl





,\


Direction of
Rotation








Roller Follower to Flat Follower

The transformation from a translating roller follower to a translating flat
follower is now developed. From Figure 3-10


09 = ATan2(y,, yj, xr x)

Equation 3-24

and, as before

ai = ATan2(y,,x )

Equation 3-25

Where, as before, the coordinates of the cam surface are calculated from
Equation 3-6 and Equation 3-7. Again, from Figure 3-10 rj is


r,'= Vx?+y? cos(a -0;)

Equation 3-26

Finally, for the flat follower the lift is


s; = r -rb

Equation 3-27

With the equations presented here, nominal and measured lift data may
readily be transformed between roller followers with different radii and between
flat and roller followers in either direction. Similar expressions for other types of
14
cams are given by Chen



































Figure 3-10 Coordinate Transformation: Translational Roller to Flat


Roller Follower to Flat Follower

The transformation from a translating roller follower to a translating flat

follower is now developed. From Figure 3-10


0; = ATan2(y1 y,, x, -xi)

Equation 3-28

and, as before

aj = ATan2(yj, xj)


Equation 3-29


Flat Follower
Translational Axis


Roller Follower
Translational Axis


rf





Sj


Direction of
Rotation









Where, as before, the coordinates of the cam surface are calculated from
Equation 3-6 and Equation 3-7. Again, from Figure 3-10 rj is


r = x + y? cos(a, -0;)

Equation 3-30

Finally, for the flat follower the lift is


si = r -rb

Equation 3-31

With the equations presented here, nominal and measured lift data may
readily be transformed between roller followers with different radii and between
flat and roller followers in either direction. Similar expressions for other types of
14
cams are given by Chen.


Radius of Curvature of the Cam Profile

In the development of the cam grinding model that is presented in
Chapter 5, the radius of curvature of the cam profile is required. For the lobe
rotation angle 0j, the radius of curvature for the surface of the cam profile at the
16
point of contact, j, is

3
r+dr 2]2
oci da __
S +2 drpo 2 d2r
prcj da ), l d-2


Equation 3-32









where the angle of contact, a, is not evenly spaced and therefore, the first and

second divided differences are used for the derivative terms. From Equation 3-

12, the first divided difference is


drpoc rpocj+l rpocj_l
da a -a j-1

Equation 3-33

and from Equation 3-13, the second divided difference is


d2 rpo 1 'pocj+.1 rp. j r'opj_
dCa2 aj+1 -a.jI 1 -C j I

Equation 3-34

Substituting the difference expressions into Equation 3-32 gives


3
po -poc 1
Sr j, 1 r- r r -
P0.r +2 rl --pc- 2 I

2 pcj1 rpoc \ 'poc pocj rpocj pocj_
p i K aj+1 -aj-1 L + a i j-1 aj-1 j ,j j-i1


Equation 3-35

In the case where the data are known at evenly spaced points, Equation 3-35

becomes








3


2 V


p 2 -+ 2
poc +2 27 poCI 2 2
N N

Equation 3-36


Interpolation

When transforming motion coordinates from one follower to another, 0 is
not, in general, equal to Oj. The difference is illustrated in Figure 3-8, Figure 3-

9, and Figure 3-10. Consequently, the calculated values of ri' do not occur at
evenly spaced integer degree values of 90. To use the values of rj' conveniently
in programming most machine tools, it is necessary to interpolate rj'to integer
degree spacing of 0*. The interpolation is generally performed using cubic
1825 26
splines as follows. The piecewise cubic interpolating polynomial is


rj'() = c, j + c2,j(e- ;) +C3,j(e- ;)2 + C4,j (- ;)3

Equation 3-37

where
C, = r

Equation 3-38

c2,j = Lj

Equation 3-39









[e., ,e1] r Lj
C3J= AO c4,j A

Equation 3-40

Lj +L,-1 -2 [0,0,o;] r
C4,j = A0;


Equation 3-41

A9 =9, 9;

Equation 3-42


[ 0 1 6, '] r i l l -


Equation 3-43


The vector L is the vector of free slopes and has N rows. For cubic spline

interpolation of periodic data


2(A9,_ + AG;)
AO;
0


2(AO + AO;)
AO;


0
AO;
2(A9; + AO;)


AG,


A0,_, 2(A 9-3 + A'_2)
...... 0 A0,_,


AON-_2


2(AL-3
2(AO,_2 + A9,_,)


Equation 3-44


where
b, = 3(AO;[o;_,,;] r + AO-;,[O;,O,] r')


Equation 3-45


"L

L,









Symbolically, L is solved for as


Lo 2(AO _, + AO;) AO9 0 0 ...... A bo
L, Ae 2(AO + AO;) AO, 0 ...... 0 b,
:0 Ae 2(AO + AO;) Ae; ...... 0

!N-2 0 0 ...... N-2 2(AON3 + AN-2) O 3 bN-2
.,N1 AN-2 0 ...... 0 AO 2(At,_2 + AO,,) L,



Equation 3-46

The matrix of Q0 values is of the form known as cyclic tridiagonal. This

matrix form occurs frequently with periodic data and difference equations. Cyclic

tridiagonal equations are solved efficiently by LU decomposition with forward
27
and back substitution and the Sherman-Morrison formula The Sherman-

Morrison formula adds a correction to the LU decomposition solution to account

for the nonzero upper right hand and lower left hand elements. With the value of
L known, the coefficients c,,, c2,, c3,and c4, are calculated and substituted into

Equation 3-37 to calculate the interpolated value of r,(0). For common cam

geometries, cubic spline interpolation returns the original data to an accuracy of

better than 6 parts in a million. This is shown by transforming design follower
motion to a typical grinding wheel radius, interpolating r;'() to integer degree

spacing, then transforming the grinding wheel motion back to the design follower
motions and then interpolating rj(6) to integer spacing.












CHAPTER 4
CAMSHAFT GEOMETRY: IMPLICATIONS FOR THE GRINDING AND
INSPECTION PROCESSES


Introduction

For successful modeling and feedback control of the profile grinding
process, it is important to establish consistency between the grinding process

and the inspection method. In this chapter, the machining axis of rotation
(MAOR) and the inspection axis of rotation (IAOR) are defined and identified as

two parameters that must be in agreement for successful process control.

Additionally, the different work holding methods are identified and the effects on

the machining axis of rotation is considered.

Also in this chapter, algorithms for reducing inspection lift data are
considered. While the specification of nominal camshaft geometry is

straightforward, many of the parameters used are coupled and must therefore,

be decoupled in the inspection process. Currently, different gauge builders
reduce the inspection data differently, and consequently, report different lift

errors which often do not faithfully reflect the process errors. The differences are

most pronounced when a nonconcentricity error of the cam base circle to the

camshaft axis of rotation exists. This distortion is a problem since it causes the

reported lift error to differ from the expected errort. Therefore, existing data

reduction methods are examined and a new method is proposed. The proposed
method is tested and shown to more faithfully decouple cam profile, timing and

nonconcentricity errors.


t The expected error is the error predicted by the grinding model. This model is developed in Chapter 5

34








Camshaft Geometry and the Grinding Process

The fixturing used in the cam profile grinder establishes the machining
axis of rotation and locates the camshaft timing datum. The type of fixturing
affects both the stability of the position of the axis of rotation and the

repeatability of the timing datum. These two factors account for most of the

nonrepeatable error in the cam grinding operation and are therefore considered
in detail in this section.


The Camshaft Timing Datum

On cam profile grinders, the camshaft timing datum is mechanically
located with respect to the grinding machine's timing datum. The repeatability

with which the timing datum is located depends on the design of the camshaft

timing datum and the design of the grinder fixture. The camshaft timing datum is
most commonly a keyway, a dowel pin, or an eccentric. The camshaft timing
datum is located relative to the machine's timing datum by a fixture on the
grinding machine chuck or by a drive dog. While the timing of the lobes relative

to the timing datum is unrepeatable, the more important inter-lobe timing,
defined here as the timing of all lobes relative to lobe one, is unaffected by the
timing datum and are generally an order of magnitude more repeatable than the
timing relative to a keyway.4 Therefore, the error is essentially a clamping error

and not related to the accuracy of the machine axes.


The Machining Axis Of Rotation

Typical camshafts for gasoline and diesel engines are relatively flexible in
the radial direction. In fact, the part can be several orders of magnitude more
flexible than the rest of the grinding system. System stiffness is critical in profile









grinding since the radial grinding force varies with the radius of curvature of the

cam profile. This variation in force leads to a variation in the part deflection

within a single revolution, and consequently, a profile error. Therefore, to

minimize the deflection, the camshaft is often supported along its length during

the grinding operation.

Adjacent Journal Axis of Rotation. In addition to supporting the camshaft,
the work holding method also determines the machining axis of rotation. For

long flexible camshafts, it is generally necessary to support the camshaft at

multiple points along its length. Typically, work supports, know as steady rests,

are located on the bearing journals and the part is clamped with a three jaw
chuck. This arrangement is shown in Figure 4-1. For camshafts ground using

this arrangement, the accuracy of the ground profile is a limited by the

roundness of the journal bearings as they serve as additional work axis
bearings. Consequently, the out of roundness of these bearings causes a

corresponding profile error to be ground into the cam lobe. For this method, the

machining axis of rotation is defined as the adjacent journal axis of rotation

(AJAOR) and is physically the mating of the machine chuck and journal
bearings, the steady rests and journal bearings, and the male and female

centers. This and other axes of rotation are shown schematically in Table 4-1.

Centers Axis of Rotation. For short, stiff camshafts, grinding between
centers often provides a sufficient stiff system. In this case, the machining axis

of rotation is defined by the work spindle, the male machine centers and the

female part centers. This method eliminates the influence of the bearing error on
the axis of rotation and it is the most accurate method for sufficiently stiff parts

with accurate part centers. Again, the method is shown schematically in Figure

4-1.









End Journal Axis of Rotation. This method is same as the adjacent
journal method only here not all the journals are supported. This method is also

illustrated in Figure 4-1.


Figure 4-1 Steady Rest and Three Grinding with Jaw Chuck: AJAOR


Camshaft Geometry and the Inspection Process


To analyze geometrical cam errors, gauge builders typically separate or
reduce inspection data into several basic geometrical elements and compare
these elements with the nominal part geometry. While, these basic geometric
elements are fairly well standardized, the data reduction techniques used to
calculate the individual elements vary between different gauge builders, cam

grinder builders and cam manufacturers. Of particular interest to this work is the
method used to decouple the lobe timing from the lift data. The method is
important since it determines the form of the reported lift error, and
consequently, the model of the grinding process.








Table 4-1 Machining and Inspection Axes of Rotation


Machining Axis Of Rotation Inspection Axis of Rotation
(MAOR) (IAOR)
Centers Continuous axis defined by fit of Continuous axis defined by the
Axis of part and grinder centers, fit of part and gauge centers.
Rotation
(CAOR)


Adjacent Segmented axis defined adjacent Segmented axis defined by the
Journal journal bearing surfaces, steady best fit of circle to adjacent
Axis of rests and chuck or second center. journal bearings.
Rotation
(AJAOR) .- ...


with three jaw chuck



--with second center-----
with second center


End
Journals
Axis of
Rotation
(EJAOR)


Continuous axis defined by end
journal bearing surfaces, steady
rest, and chuck.


Continuous axis defined by the
best fit of circle to end journal
bearings.


The coupling of the lift and timing measurements is readily apparent from
Figure 3-4 and Figure 3-6. In Figure 3-4, the lobe timing angle, (, is measured

between the camshaft timing datum and the X axis of the lobe coordinate
system. Since, as shown in Figure 3-6, the X axis of the lobe coordinate system
is also the datum for the angle of rotation, 0, the method used to determine the 0

datum affects both the reported timing angle and the reported cam profile.
Currently several method are used to decouple the timing and profile

measurements. In this chapter, an alternative method for reducing the








inspection data into timing and lift components is presented. This proposed
method is shown to more effectively decouple lift and timing errors in the

presence of nonconcentricity errors.

Prior to demonstrating this method, the basic equation to calculate the
lobe timing is developed through analogy with the standard method of
determining the timing or phase of a circular feature: Fourier analysis.28 The

presented method provides an intuitive method for decomposing the inspection
data.


Basic Elements of Cam Geometry

The standard elements of cam geometry that are of interest in this work
are as follows: base circle radius size, base circle nonconcentricity, lobe timing
and lift error.

Base Circle Radius. The base circle size is calculated as the average
value of the measured lift values for the section of the lobe that defines the base
circle (i.e. the region of zero nominal lift). The algorithm then subtracts the
average base circle radius from the measured lift values so that the average lift
of the base circle is zero.

Base Circle Nonconcentricity Gauges calculate the base circle
nonconcentricity by comparing the least sum of the squares ( LSS ) base circle
center with a reference axis. The axes referred to in this section are

summarized in Figure 4-1. While this error has two components: magnitude and
phase, generally only the magnitude is reported. The reference axis may be
defined either as the functional part axis of rotation or the gauge axis of rotation.
The functional part axis can be defined two ways. First, the reference axis may
be defined by the male gauge centers and the female part centers (CAOR).









Second, the functional part axis may be defined as a segmented part axis which
connect the least sum of the squares centers of the adjacent bearing journals
located on either side of the cam lobe. This case is referred to as data reduction

relative to adjacent journals. Third, the functional axis can be defined by

connecting the least sum of the squares centers of the two end bearing journals.


Lobe Timing Datum. As show in Figure 3-5 and Figure 3-6, the cam lift is
specified as a function of the angle 0. Nominally, the angle 0 is measured from
the lobe timing datum which is defined by the point of maximum lift. For

inspection, this means that the datum of the angle 0 is a function of the form of
the profile and must be determined from the inspection data. The nominal
definition, while straightforward, is not useful for reducing the inspection data.

Its implementation would require the location of the timing datum to be measured
in an insensitive gauging direction. This is true since the geometric velocity has
a point of inflection in the region of the datum and therefore the gauge follower
displacement reading changes little for a relatively large angle of rotation. Also,

implementation of the definition is further complicated by measurement noise
and surface finish of the cam surface. Currently, various methods are used to
determine the phase or timing of cam lobes. The use of different methods
makes comparison of results from different gauges difficult. One popular

method performs a least sum of the squares best fit of the inspection data to
nominal lift values in regions where the geometric velocity exceeds some
minimum threshold. This method is adversely effected by base circle
nonconcentricity and does not work for some highly asymmetrical cams where
the geometric velocity is always below the minimum threshold for one flank of

the cam lobe. Alternatively, the timing may be calculated by performing a least

sum of the squares best fit of the inspection data to the nominal data by









comparing the values for only a few specified points. This method is particularly
susceptible to noise or asperities in the inspection data. Lastly, some gauge
builders measure lobe timing by identifying the maximum lift point on the cam.

This method suffers from the inherent insensitivity of the gauging process at this

point. That is, cams have a point of zero geometric velocity (slope) at the
maximum lift point, therefore the lift values may only change slightly for relatively
large angles of rotation.

Lift Error. With the lobe timing established, the inspected profile data are
shifted in phase by the observed timing error, interpolated to integer degree
spacing and then compared with the nominal data. The difference between the
actual and the nominal is reported as the lift or profile error.

Other features such as taper, velocity error and cylindricity are also
reported. However, these are not of direct relevance in this work and do not
affect the calculation of the two dimension parameters of interest.


Timing of Round Features

To illustrate the proposed timing method, consider the case of
determining the phase of an eccentric (a round bearing which is not concentric
to the axis of rotation) which is oriented with respect to some observable part
feature such as a timing pin. This arrangement is illustrated in Figure 4-2. If the

part is rotated through 360 degrees, then the output of the ideal linear
measurement axis for an ideal round part will be a pure sine wave when plotted
against the angle of rotation. The DC term of the Fourier series of the inspection
data is the size while the term representing the fundamental frequency is the
nonconcentricity.









Table 4-2 Nomenclature for Process Geometry


an Fourier cosine coefficient for frequency n
AJAOR Adjacent journals axis of rotation
bn Fourier sine coefficient for frequency n
CAOR Centers axis of rotation
DC Constant term in Fourier transform
el, e Lift or roundness error
EJAOR End journals axis of rotation
IAOR Inspection axis of rotation
j index on parameters that vary with the lobe angle of rotation 6
k Phase Shift (56)
kI LSS estimate of k
LSS Least sum of the squares
MAOR Machining axis of rotation
n Frequency in undulations per revolution (upr)
N Number of data points
OG Gauge axis of rotation
Op Least sum of the squares center of a round part
rb Base circle radius
r_ Follower radius
rJ Radius of follower pitch curve at point j
ri Measured radius of follower pitch curve at point j
R Nominal radius for round part inspected with flat follower.
Biased estimate of radius for nonconcentric round part inspected with
radiused follower.
i Follower lift
"sj Measured follower lift
upr Undulations per revolution
v Geometric velocity
X X axis of lobe coordinate system
XG X axis of gauge coordinate system
Xp X axis of part coordinate system
Y Y axis of lobe coordinate system
YG Y axis of gauge coordinate system
Yp Y axis of part coordinate system
8ri, Roundness error for circular part (same as e)
60 Phase shift in 6,
86 LSS estimate of phase shift in 9j
Lobe timing angle (also used for follower pressure angle in Figure 4-
14 and associated equations and discussions)
8j Lobe angle of rotation








Timing
Reference Pin


Direction of Rotation
,Shown at H = 0


Figure 4-2 Inspection of Nonconcetric Nominally Round Part

The Fourier series is28

N-i 21. N 'b sin27
ri =R+a, acos j n + bn sin j n
n=l N n=l N

Equation 4-1

where N is the number of evenly spaced data points per revolution and j is the
index on the inspection point number. The nomenclature for this chapter is
summarized in Table 4-2. For a flat follower, the DC term of the Fourier series is
the average part radius, R, and is defined as








1 N-1
R= -j orJ
R-N

Equation 4-2

The cosine coefficients for n = 1 to N-1 are

2 N 12 xn n
an N CN os( Nj
j=o

Equation 4-3

The sine coefficients for n = 1 to N-1 are


bn = 2 r sin 2j n
b N j=J N

Equation 4-4

However, for a perfectly round feature which is not concentric to an ideal axis of

rotation, all terms with n greater than one are equal to zero. In this case,

Equation 4-1, Equation 4-3 and Equation 4-4 become


r, =R+a1 cos j +b, sin(N jN

Equation 4-5


2 N-1 2x7
a, N Zr, cos( N- j
j=o

Equation 4-6


2 N-1 f2.s
b1=Nrsn


Equation 4-7









where al and bi represent the Xp and Yp components of the nonconcentricity
respectively. These components are illustrated in Figure 4-2. For the phasing
of the nonconcentricity shown, the follower motion is the sum of a sine wave and
a constant as shown in Figure 4-3 as the "design" curve. For clockwise rotation,
the angle 0 is measured counterclockwise in the part coordinate system. For the
configuration shown, R is the part radius, a, is zero and bi is equal to the
magnitude of the nonconcentricity. If a phase shift of 60 degrees is introduced
as shown in Figure 4-3, then both a, and bi are nonzero. If this phase shift is
considered to be an error, then the error in the follower motion, 5r, is


ar- = a, cos( njj

Equation 4-8

where 6rj = r" -r,

Equation 4-9

and rj is defined as the measured value of r at angle 0 while rj is the nominal
value at angle 0, (i.e. the value of r with no phase error). The effect of this error

is illustrated in Figure 4-4 for 65 equal to ten degrees. From Equation 4-8, 6r is
recognized to be of the form


dr
rj-= dO


Equation 4-10









dr
This is true since the cosine is the derivative of the sine. The term is
dO

generally called the geometric velocity.* Also, if the inspection data for the

phase-shifted part, r*, is compared with the nominal data, then for a pure phase

error of 60 expressed in degrees,


r *j =r -360
J N


Equation 4-11


Direction of Rotation
Shown at 0 = 0


S1


Figure 4-3 Introduction of Phase Shift


d r d r dr de
The term geometric is used to distinguish this term from time velocity, which is = -. Of
dr dr de
course, W loses proportionality to -K when varies within a single revolution. This is generally true
de dr
in cam profile grinding and is minimum in regions where -- is maximum. Thus, limits are imposed
dr
on due to dynamic considerations of the work axes and grinding process parameters.











Follower Motion due to an Eccentric


0.8 Design
0.6 -_ -- -Measured
0.4-- ---------.,Error^..
0.4 2
....-.. Error --
0.2 .. //


-04 -X /
-0.6
-0.8 0.6

-180 -150 -120 -90 -60 -30 0 30
Angle (degrees)


60 90 120 150 180


Figure 4-4 Measurement of a rj Phase Error for an Eccentric


Substituting Equation 4-11 into Equation 4-9 gives


6r = r. S.860 rj
+ N

Equation 4-12


The first order forward first difference equation29 for point 0 is



dr r S r
dO 60

Equation 4-13


Rearrange Equation 4-13, substitute it into Equation 4-12 and solve for 6e to get


6r1
60 dr}



Equation 4-14


-2
0
LL
u


z









Where for the idealized case, this expression is invariant over the range of 0
dr
except in the region were -is zero and thus, 68 is undefined. It is important to
dO
remember that the form of the error term, 5r is restricted to that of a one

undulation per revolution sine wave which is the geometric velocity. If the form

of the actual measured error is different, then this expression is invalid.

Obviously, the case of a pure velocity or timing error is theoretical and never

occurs in practice. However, this presents no difficulty, as it is always possible

to determine the component of the error data which is of the form


dr
6r(v) = k
S- dO

Equation 4-15

where 6r(v) is the component of the measured error that has the form of the

geometric velocity. The estimate of parameter k is designated as k and is given

by a least sum of the squares fit of the inspection data as30

N-1 dr
A d (r -r)
k=
N-_dr]2


Equation 4-16

Lastly, substitute the expression for 8r (v)j given in Equation 4-15 for 6rj in
Equation 4-14 and solve for 68 to get


65 = k


Equation 4-17








Therefore, the timing error, 56, is

N-1 dr
A -d (r' r)

N [dr 2


Equation 4-18
A
where 60 is the least sum of the squares estimate of the parameter 65. It is

important to note Equation 4-18 is valid independent of the form of the

inspection data and the form of the nominal shape. To reinforce the analogy

with the Fourier coefficients, compare Equation 4-15 with Equation 4-8 and
dr
recognize that, for an ideal round feature, k and d- in Equation 4-15
dO

correspond to ai and cos N ij) respectively in Equation 4-8.

Obviously, if all that is desired is to determine the timing of round part

features, then nothing is gained from Equation 4-14: the timing error can be

more directly determined using the Fourier transform. However, Equation 4-14

can be used to determine the timing of cam lobes since it is valid independent of

the nominal shape.


Measurement of Cam Lobe Timing

In this section, the more interesting problem of reducing the inspection

data of a cam lobe is considered. To understand the complications that arise in

using the Fourier series approach on a cam lobe, the cam lift data shown in

Figure 4-5 are analyzed. Figure 4-6 shows the Fourier transform of the follower

lift data. From this figure, it is clear that a broad frequency spectrum,

approximately 30 harmonics or undulations per revolution (UPR) in this case, is









required to represent the data to the number of significant digits needed for

production and inspection programming. Also, cam lobes are often

asymmetrical and it is not possible to directly infer the timing of the cam lobe

from the phase of the Fourier transform. Therefore, the assumptions that lead

to the development of Equation 4-5 for circular features are invalid for

noncircular cam profiles. However, since Equation 4-18 developed in the

previous section is valid for all shapes, it can be directly applied.



Follower Lift and Geometric Velocity for a Typical Exhaust Lobe
8 0.2
/- \ base circle radius = 59.875 mm 0.15
7 0.15
S\ follower radius = 19.00 mm
6 / 0.1 0
E5 /
E 4 ----------- - - ----- --- 0 E
3-- -0.05
Lift \
2 V -0.1
- - Velocity -01
1 \ \ -0.15
0 -0.2
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150
Angle (degrees)

Figure 4-5 Typical Cam Lobe Lift and Geometric Velocity


Again, consider the cam profile and ideal inspection process shown in

Figure 4-7. The follower lift and geometric velocity for a roller follower, with 60

equal to zero, are shown in Figure 4-5. For consistency with the previous

discussion, a flat follower is considered here. This presents no difficulty since,

as discussed in the Chapter 3, the radial follower cam motion specification

given in Figure 4-5 may be readily transformed to a flat follower motion

specification.










Harmonic Content for Exhaust Lift Data

1.8 I 0.01

E E
E 1.4 Frequency Content 0.008
1.2 Scale on left y axis
1 - Frequency Content 0.006
0 0
0.8 Scale on right y axis O
o 0.004 o
S0.6 4.
0.4 7
0.002
S0.2 _
0 0
0 30 60 90 120 150 180
Frequency (UPR)



Figure 4-6 Frequency Content for Typical Exhaust Cam Profile


To develop the timing measurement algorithm, again consider the effect

of introducing an arbitrary timing error, 68, into inspection data for a cam lobe is

considered. This situation is illustrated in Figure 4-7.


Timing Direction of Rotation
Reference Pin Shown at 0, = 00


Figure 4-7 Cam Lobe with a Timing Error of 68


1 11









For an ideal lobe with an error only in phase of 60, from Equation 4-12 the

measured lift error is


r, = r 360 rj
I N

Equation 4-19


Now, however, the form of 6r is not a single sine wave, but rather its form is a

function of the nominal data: specifically, the error has the form of the geometric

velocity which is shown in Figure 4-5. Recall that for the eccentric, the form of r

is that of a sine wave and the error term associated with a phase-shift is a cosine

wave or simply the derivative or geometric velocity. Therefore, while Equation 4-

5 is invalid for noncircular features, Equation 4-18 is valid independent of the

form and the timing error for the lobe is



A (rE r,)
^ I= -
j=?1 dr 2
C de -


j=0o dOl

Equation 4-20

Finally, it should be noted that Equation 4-20 requires that r. j.e0 ri
J+ N
dr
approximate d well. If 60 is large, then this approximation is poor and
d9

Equation 4-20 must be solved iteratively. Still, even for larger angle errors, the
solution converges rapidly. For typical angle errors r. ..36o rj approximates
J+ N
dr
Swell and no iterations are required.
dO








Comparison of Methods

The actual value of timing angle calculated from Equation 4-20 depends

on the method used to apply the equation to the inspection data. Since this

research is primarily concerned with understanding and controlling the
manufacturing process, the method which most successfully decouples lift and

timing errors, and thereby reveals the physical cause of the lift error, is desired.

In this section, two different algorithms, the existing one used on most cam

inspection gauges and a proposed modified version, are tested on phase-shifted

simulated lift error data. The proposed method is shown to be superior in

decoupling the lift and timing errors. The decoupling of errors makes it easier

for grinding machine operators and engineers to identify error sources and to

model the grinding process. Also, in the proposed method, no increased risk of

accepting bad parts exists since the reported lift error will always be larger than

or equal to the error reported by the existing method. Timing errors may be

larger or smaller depending on the relative phase and magnitude of the various

lift error components.

The simulated lift error is for the slightly asymmetrical exhaust lobe shown
in Figure 4-5. In this simulation, major repeatable profile lift grinding errors are
included in order to evaluate the effects each has on the data reduction method.

The simulated lift error is shown in Figure 4-8 and represents the superposition

of the following components: a 20 pim residual sparkout error, a 10 Pm

nonconcentricity error, and a 0.40 phase shift or timing error. The residual
sparkout error grinding model, used here to simulate the lift error, is presented in
Chapter 5.










Effect of Timing, Sparkout, and Nonconcentricity Errors
0.08
0.06
0.04
E 0.02 .,-.,
E

-0.02
-0.04 Includes Tiuing Error Effect
-0.06 - - Excludes Timing Error Effect
-0.08
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Angle (degrees)

Figure 4-8 Simulated Cam Lift Error


From Figure 4-8, it is clear that when a significant timing error exists the

reported lift is dominated by the timing error effect. Therefore, it follows that the

angle error must be determined and the lift data phase shifted by the correct

amount before it can be compared to the nominal lift data in a meaningful way.

The existing method of calculating the timing angle with Equation 4-20 is

strongly biased in the presence of nonconcentricity errors. This bias distorts the

reported lift error and obscures the physical source of the error. The proposed

modification to the method significantly reduces the bias.

Existing Data Reduction Algorithm

In the existing method, reducing the inspection data with respect to the

machining axis of rotation, leads to a large angle bias in the presence of

nonconcentricity errors. This is true since the nonconcentricity error is not

removed prior to calculating the timing angle. Nonconcentricity errors commonly

account for 50% of the total lift error on high precision CNC cam profile grinders

and the timing bias due to nonconcentricity can significantly distort the form of

the reported lift error as shown in Figure 4-9.









Profile Error Reduction: Existing Method

0.02
0.015
0.01
E 0.005
g 0
-0.005 _., :
--- Existing Method
-0.01 Error
------- Error
-0.015
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Angle (degrees)


Figure 4-9 Existing Data Reduction Method for Profile Data


In this figure, the reduced lift error is different than expected. This

difference results from a bias in the timing angle calculation. Before discussing

the source of this bias, it is useful to consider the how the lift and timing errors

are calculated.

In both the existing and proposed methods, the measured lift data are

phase shifted by the calculated timing angle error before the measured lift is

compared with the nominal data. If the timing angle calculation is biased, then

the data are not shifted by the proper angle and a lift error will be reported due

to the bias alone. It is the superposition of the lift error due to the bias and the

simulated lift error that gives the result shown in Figure 4-9. For the simulated

inspection data shown in Figure 4-8 the timing angle error is calculated as

0.34230. Recall that the simulated angle error is 0.40000 and thus the timing

bias is 0.05770. This bias is due to several independent factors, summarized in

Table 4-3 and considered below singly.

The first source of timing bias is the form of the profile error due the

nonconcentricity error. This bias arises for certain phasing of the









nonconcentricity error. The effect is greatest for the phasing shown in Figure 4-

10. Here, the nonconcentricity error and the timing error are both odd functions

and have similar forms. Equation 4-20 can be used to calculate the component
dr
of the nonconcentricity error that is of the form of timing error, -. This
dO

component is the timing bias of the nonconcentricity error for the existing method

and is 0.05400.

The second source of a timing bias is the sparkout lift error. For the

simulated data, the asymmetrical lobe produces an asymmetrical sparkout lift

error as shown in Figure 4-10. Due to this asymmetry, the sparkout error has a
dr
component of the form -. For the simulated data the timing bias due to the
dO

sparkout lift error is calculated from Equation 4-20 as 0.00370. Thus, the total

bias is the sum of the sparkout lift error bias and the nonconcentricity bias. This

bias is equal to the difference between the simulated timing error and the timing

error originally calculated using Equation 4-20.


Figure 4-10 Form of Timing and Nonconcentricity Errors


Profile Error due to Timing and Nonconcentricity Error
0.02
0.015
0.01
S0.005 '' ,
,0 0
-0.005 .,,,-"'1

- - Nonconcentricity i
-0.015
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Angle (degrees)








Proposed Data Reduction Algorithm

Figure 4-11 shows the reduced profile error for the proposed method. In
this method the nonconcentricity error is removed and the timing is calculated.

After the timing is calculated, the nonconcentricity error is shifted in phase by 60

and added back to the profile error. So, for the data shown in Figure 4-8, the

nonconcentricity error is removed and the timing error is calculated from

Equation 4-20 as 0.38990. While, the bias is significantly reduced, it is still

nearly three times larger than the expected sparkout bias of 0.00370 which

remains unchanged. The remaining 0.00640 bias occurs since the

nonconcentricity error is not actually a single sine wave with a frequency of one

undulation per revolution, but rather it is a sine wave that is distorted in the

regions of the cam flanks. To understand the nature of this distortion, it is

necessary to briefly consider the details of this error source.

For process control, cam lift data are reduced relative to the
manufacturing axis of rotation and only nonconcentricity errors to this axis are

considered. Nonconcentricity errors relative to this axis are caused in two ways.

First, for parts ground between centers, a nonconcentricity error can result from
radial error motion of the profile grinder work spindle or inaccurate part centers

or a badly designed drive fixture. Second, for parts ground using steady rests

and centers, a nonconcentricity error can occur due to the radial error motion of

the work spindle on the cylindrical grinder used to grind the journal bearing. This

occurs since the camshaft is over-constrained during cam profile grinding due to
the combination of the centers and steady rests as shown Figure 4-1. In this
over-constrained condition, it is possible that the journal bearings and steady

rests do not remain in contact during part rotation. Consequently, the cam lobe

is not ground concentrically to the journal bearing which defines the machining

axis of rotation and a nonconcentricity error results. Each of these two different









types of nonconcentricity error produces a different distortion and again these

sources are considered separately below.



Profile Error Reduction
0.02

0.015

0.01
E
E 0.005


-0.005
Reduced Data "'v
-0.01 \ ^
-0.01 - Error i
-0.015 1
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Angle (degrees)

Figure 4-11 Proposed Data Reduction Method for Profile Data


The first nonconcentricity error described above, the runout of the part

during cam profile grinding, produces a nonconcentricity error that is a single

sine wave superimposed on the commanded nominal motion of the grinding

wheel. However, when this nonconcentricity error is expressed in terms of the

follower motion, it is no longer exactly a sine wave, but rather it is distorted in the

region of the flanks as shown in Figure 4-12. The distortion of the

nonconcentricity error in the follower motion coordinates is of the form of the
dr
timing error d From Equation 4-20 the angle bias due to this error is
dO


calculated as 0.0064.









Grinding Wheel Axis of Rotation and Nonconcentricity Errors
0.01
0.008
0.006
0.004
E 0.002
S- -
0
S-0.002
LUI -. ---- Grinding w heel motion coordinates
-0.006 ------- Follow er w heel motion coordinates
-0.008 - - Difference
-0.01
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Angle (degrees)

Figure 4-12 Effect of Nonconcentricity Error in Grinding Wheel Motion on
Follower Motion


The second error described above, the runout of the journal bearing

relative to the machining axis of rotation, produces the nonconcentricity error

shown in Figure 4-13. Note that the form of this error is not that of a single sine

wave. This distortion results from the nonlinear change in the lobe angle, 0, at

which the follower contacts a given point on the cam surface and the fact that

the axis of rotation may not lie along the follower axis of rotation. The geometry

for this case is discussed in Appendix C.

Therefore, just as for the first type of nonconcentricity error, the entire

timing angle bias is greatly reduced, but not eliminated, by simply removing the

base circle nonconcentricity. The timing angle bias due to the nonconcentricity

error show in Figure 4-13 is 0.0389. The timing angle bias with the base circle

nonconcentricity is 0.00540. These results are summarized Table 4-4.









Effect of Journal Bearing Nonconcentricity Error
0 01
0.008
0006
0 004
E 0.002
E

2 -0.002____
-0.004 -- Non-concentricity
-0.006 ------ Sine wave
-0.008 -- Residual
-0.01
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Angle (degrees)


Figure 4-13 Effect of Journal Bearing Nonconcentricity Error


In conclusion, from the results summarized Table 4-3 and Table 4-4 the

nonconcentricity error timing angle bias is reduced significantly by the proposed

method. This method produces inspections results which more faithfully

represent the process errors. This method will be used to compare experimental

results with the process model discussed in Chapter 5.


Table 4-3 Timing Angle and Bias Results for Type I Nonconcentricity Error

Existing Proposed
Summary of Results for Simulated Timing Error Method Method

Simulated Timing Error 0.40000 0.4000

Calculated Timing Angle Error 0.34230 0.39900

Sparkout Error Timing Angle Bias 0.00370 0.00370

Nonconcentricity Error Timing Angle Bias: Source I 0.05400 0.00640

Total Timing Angle Bias 0.05770 0.01010

Sum of Timing Angle Error and Bias 4.0000 4.00000









Table 4-4 Timing Bias Results for Type II Nonconcentricity Error

Existing Proposed
Summary of Bias due to Source II Nonconcentricity Error Method Method

Nonconcentricity Error Timing Angle Bias: Source II 0.03890 0.00540


The Effect of Radiused Followers on the Nonconcentricity Calculation

Inspection of nominally round parts such bearings or parts with sections
that are nominally round, such as cam lobe base circles, is considered in this

section. Here, the scope of the discussion is limited to the two aspects of the

inspection process. First, the effect of using a radiused follower versus a flat

follower to identify nonconcentricity errors of the surface relative to an inspection

axis is considered. Second, the closely related issue of using the flat follower

approximation to remove the nonconcentricity effect from the inspection data is

discussed. These two aspects of are relevant to this work since the removal of

base circle nonconcentricity and lobing effects from cam lobe base circle

inspection data are important issues in process control. The method of

removing these effects from inspection data is discussed in detail in Chapter 8.

However, the discussion presented in this section gives the justifications for
certain assumptions that are implicit in the discussions in Chapter 8.








XGT









\ Ijrf

Direction of
Rotation

r = si +rb+rf "



YG -
brb




Figure 4-14 Inspecting a Nonconcentric Surface with a Radiused Follower

Figure 4-14 shows a nominally round part that is rotated about an
inspection axis, O0, which is not coincident with the actual part center, Op.

From Figure 4-14 the vector loop equation for the outlined triangle is written in
XG and YG components as


a, sin(j)+ b, cos(,j)-(rb +rf)sin(),) = 0


Equation 4-21








a, cos(, )+b sin(,) (rb + rf )cos( )-r = 0

Equation 4-22

Equation 4-21 and Equation 4-22 can be combined to get


rI2- 2(a, cos(O,)+b, sin(j))rj +(a, cos(ej)+b, sin(,j)) +...
(a, sin(Oj) + b cos(Oj)) -(rb +r ) = 0

Equation 4-23

Equation 4-23 is quadratic in r and, selecting the appropriates sign, the solution
is
r = a, cos( )+b, sin(OJ)+ (rb +rf)2 -(a1 in(O)+b, cos(0j))2

Equation 4-24

This can be written in terms of the relative follower displacement, s, as


s =acos(Oj)+blsin(j)-rb rf + (rb +r,)2 -(a in(O)+b cos(6))2

Equation 4-25

Expanding the second term under the radical and simplifying gives


si = a cos(6) + bsin( )- rbf + f(r.r +r a b -sin(29j)

Equation 4-26

Therefore, if the coefficients ai, bl, and rb in Equation 4-25 are know, then the
contribution of the nonconcentricity error to the indicated relative follower
displacement, s, can be calculated and then subtracted from the indicated
values of the follower displacement, s, to determine the actual roundness error.









Also, since the part is not perfectly round, a measured displacement, Me"s, will

differ from the displacement, s, for an ideal part. The roundness error, e, is

generally defined as the difference between follower motion, s, due to the

nonconcentricity and the actual indicated follower motions, M"s as2"


SMeas
e= s sj

Equation 4-27

Typically, the sum of square of the error, e, is minimized to determine the

coefficients in Equation 4-25. Some authors31 propose a general least sum of

the squares solution to determine a,, bl, and rb using the Newton-Raphson

iterative method. However, this is not generally required for two reasons that will

be demonstrated. First, the unique form of Equation 4-25 can be exploited to

simplify the solution. Second, the difference between the motion of a radiused

follower and a flat follower for a nominally round part may be neglected when the

nonconcentricity is small.

To consider the first point, consider the use of Equation 4-5, which

expresses the displacement for a flat follower displacement, s (lift), to estimate

the follower motion for a radiused follower. Equation 4-5 is repeated in a slightly

more convenient as


si =a, cos(Oj)+b, sin(9j)


Equation 4-28








Two functions, f(O) and g(O) are said to be orthogonal over the range from a to b
if 10
j (e)g(e)d =0

Equation 4-29

Equation 4-26 can be rewritten as


s(e) = f() + g(6)

Equation 4-30

where
f(9)= a cos(O)+ b sin(6)

Equation 4-31

and
g(e) = -rb r, + (rb + r)2 (a2 + + ab sin(2e))

Equation 4-32

Substituting Equation 4-31 and Equation 4-32 into Equation 4-29 integrating
over one rotation of the part gives


Jo2- rb- rf + (rb + r,)2 (a, +b + a,b, sin(20))(a cos(O) + b sin())dO = 0

Equation 4-33

for all values of ai, bi, rf and rb. Thus, the functions f(O) and g(O) are orthogonal.
For orthogonal functions, the coefficients of the best least sum of the squares fit
for a linear combination of orthogonal functions can be determined
independently for each function.10 Consequently, the coefficients al and bi that









give the least sum of the squares best fit of the inspection data to the follower

displacement, s, can be determined directly from the least sum of the squares

best fit of the inspection data to Equation 4-28. Which is, of course, just the

Fourier coefficients of the fundamental frequency.

With a, and bl known, the problem is reduced to determining the best fit

value for the base circle radius, rb. First, it is noted that g(O) is composed on a

constant and a sinusoid. Consequently. it is not orthogonal to any constant

signal. Therefore, the DC component of the Fourier transform gives a biased

estimate of the base circle radius. The size bias is just the sum of g(O) over a

single rotation or



JN-
bias = r, + r, +r, 2 (a2 + b2 + sin(290))
j=0

Equation 4-34

With the bias know, the base circle can be calculated from the DC Fourier

coefficient, R, as


rb = R + bias

Equation 4-35

where R is defined in Equation 4-2. And finally

1 N-1 N-1
rb i r, +N(rb +rf -(rb + r) -(a2 +b + sin(20e)))
Sj=0 j=0


Equation 4-36








The problem of reducing inspection data for a nonconcentric nominally round

part is now reduced to formulating the least sum of the squares solution to

Equation 4-36 for a single parameter, rb.

Finally, the second point-the roundness error introduced due to the flat

follower assumption is small for typical nonconcentricity errors-is considered.

Recall that Equation 4-28 and Equation 4-26 give the follower motion for flat and

a radiused follower respectively. If the relationship for the flat follower, Equation

4-28, is used to approximate the motion of the flat follower that occurs due to a

nonconcentric round part--exactly represented by Equation 4-26--then an

approximation error in the lift, s, will result. By comparing these two equations,

the approximation, is recognized to be negative g(O). Therefore, the error that

results from the use of the flat follower formulation to separate the

nonconcentricity effects from the roundness effects may be calculated from

Equation 4-32. A value of base circle nonconcentricity of 0.014 mm with an

arbitrary phase angle of 450 is selected. This gives a, and bl both equal to

0.010 mm. Additionally, a follower radius, rf of 25 mm and a base circle radius
of rb of 30 mm are selected. The follower motion, s, due to the nonconcentricity

and the estimation error is shown in Figure 4-15. Clearly, the roundness error

introduced is insignificant: the bias for rb is approximately 10-6 mm. This will

generally be the case for all centered, nominally round parts. For calculations

performed in the motions of the grinding wheel coordinates, the errors will be
smaller still since the radius of the grinding wheel approaches the infinite radius

of curvature of a flat follower. Appendix B shows a related analysis of

nonconcentricity errors.







68



Error Using a Flat Follower Approximation to Identify
Nonconcentricity for Parts Measured with a Radiused Follower
2.0E-06 0.015
1.8E-06 .*- ------- Estimation Error
1.6E-06 -- Nonconcentricity 0.01
oE
S 1.4E-06 0.005
1.2E-06 a
SE 1.0E-06 \ 0 E
E
S 8.0E-07 a
V 6.0E-07 / -00
WL 4.0E-07 n.
; ". -/ -0.01 5
2.0E-07
0.0E+00 -0.015
0 30 60 90 120 150 180 210 240 270 300 330 360

Theta (degrees)


Figure 4-15 Flat Follower Approximation for Nonconcentricity Errors












CHAPTER 5
MODELING OF THE PROFILE GRINDING PROCESS


Introduction

In this chapter, the profile grinding model is developed in an effort to

understand the physical causes of the typical cam profile error. While modeling

the cam profile grinding process is not the main thrust of this work, the process

model is useful in the development and justification of a control strategy. With

the process model known, the interaction of the cam profile at adjacent points

can be determined and an appropriate control strategy developed.


Modeling Approach and Requirements

The general cylindrical plunge grinding model is described by Hahn7 and
32
Lindsay and is represented in Figure 5-1.


Figure 5-1 Cylindrical Grinding Model Modified for Cam Profile Grinding
Adapted from "Principles of Grinding," Lindsay, R. P., Handbook of Modem Grinding Technology, Editors: King, R. I., Hahn,
R. S., New York: Chapman and Hall, 1986.









The lumped parameter model combines the workpiece flexibility and the

flexibility of the work holding method into a single parameter kw. The grinding

wheel head stiffness and machine stiffness are represented as ks. The contact

stiffness between the workpiece and the grinding wheel is designated ka. The

effective grinding system stiffness, ke, is defined as


1 1 1 1
ke ka kw k

Equation 5-1

32
The general grinding relationship is


V, + Vs + = v

Equation 5-2

where vW is the rate of decrease of the workpiece radius, V. is the rate of

decrease of the grinding wheel radius, r is the time rate of change of the system
deflection and vf is the feedrate. The nomenclature used in this chapter is

summarized in Table 5-1.

The basic grinding model assumes that the workpiece and the grinding

wheel mutually machine each other. However, for the cubic boron nitride (CBN)

superabrasive grinding wheel used in this research, the rate of decrease of the
grinding wheel radius, v,, is negligible. The rate of decrease of the workpiece
32
radius, which is analogous to the material removal rate, is give by Lindsay as


wrp (F, F )
r D,


Equation 5-3









where wrp is the work removal parameter, Fn is the normal grinding force per

unit width, and Fth is the threshold grinding force per unit width. The threshold

grinding force is the minimum force necessary for grinding to occur: below this
force only rubbing occurs. Dw is the diameter of the workpiece. The work
32
removal parameter, wrp, is given by Lindsay as


[v 19 r 2C 19 v
kL v L1+L- L 19vS
wrp = 343 - 7
De304 VOl0.47 d38 R7 19

Equation 5-4

where the equivalent diameter, De, for external cylindrical grinding is


D, D,
D +D
SD + Ds

Equation 5-5

where the plus is for convex surfaces and the minus is for concave surfaces.
32
Also, according to Lindsay the threshold force is approximately related to the

workpiece equivalent diameter as


F, = kt,,h D

Equation 5-6

where kth is a constant related to the tendency of the grinding wheel and

workpiece material to rub instead of cut.








Table 5-1 Grinding Model Nomenclature


AOR Axis of rotation
a, aj, ap Depth of cut per revolution
C Diametral depth of dress
d Grinding wheel abrasive size
De, Dei Equivalent diameter
Ds Grinding wheel diameter
Dw Workpiece diameter
fp Wheelhead infeed
Fn, Fn' Normal grinding force per unit width
Fe Ft Fhj Threshold grinding force per unit width
F', Fj' Difference between normal and threshold grinding forces
gn gnj,p
j Subscript indicating discrete points of contact at angle 01
ka Stiffness of grinding wheel and workpiece at contact zone
ko Combined sparkout constant
ke Effective cutting stiffness of the grinding system
kL Sparkout constant for linearized grinding model
k, Stiffness of the wheel head
kt Threshold grinding force constant
k, Stiffness of the workpiece supports
kwpr,, k Work removal parameter constant, lumped constant
L Grinding wheel dress lead
NNw, Nw Nw Nominal angular velocity of workpiece, for pass p
Nwjp Instantaneous angular velocity at angle Oj on pass p
p1 Subscript indicating grinding pass (rotation) number
r,, Radius of grinding wheel
rj Nominal radial distance from follower center to AOR at point j
rjp Commanded radial distance from follower center to AOR at
point j for pass p
rjp Commanded radial distance from follower center to AOR at
point j for pass p
t rjp Actual radial distance from follower center to AOR at point j
_for pass p
S, i,, rp Rate of deflection of the grinding system








Table 5-1 Continued


rpoc Distance from camshaft axis to point of contact at angle 0j
Ro Rockwell hardness of workpiece
Rs Grinding wheel radius
us, uSj X position of grinding wheel AOR in machine coordinate
frame
u, uW, X position of cam lobe AOR in machine coordinate frame
v, Feedrate of the cross slide
ff, Vffp Finish grind feedrate
vrf, vrf Rough grind feedrate
vsof Sparkout feedrate ( equal to zero by definition)
vs Rate of decrease of the grinding wheel radius
vs Surface velocity of the grinding wheel
vw Vwj vwjp Rate of decrease of workpiece radius
vW, vw Velocity of the point of contact on the workpiece
vol Volume factor for grinding wheel
wi Normalized angular velocity of workpiece at angle 0j
wrp wrpj Work removal parameter
wrp,pX wrpp X axis:
X X axis: lobe coordinate system
Y Y axis: lobe coordinate system
XM X axis: Machine coordinate system
YM X axis: Machine coordinate system
Aaj Change in lobe contact angle, a, for point j
A6j Change in lobe angle, 0, for point j
(a Lobe contact angle for point j
6i Time rate of change of a,
Ar1,p Error in commanded grinding wheel position
Yj Grinding wheel contact angle for point j
pj Radius of curvature of workpiece at angle a,
T, t Time constant of grinding process at angle a,
0i Lobe rotation angle for point j
0e Time rate of change of ,j








32
Finally, from Lindsay the depth of cut per revolution is called "a" and is defined

as

a=
NW

Equation 5-7

where N, is the angular velocity of the workpiece. From these basic

relationships, the depth of cut model and the residual sparkout error model are

developed in the following section.


Physical Model of the Cam Grinding Process: Depth of Cut


The expressions given above are formulated for cylindrical grinding. In

cam profile grinding it is helpful to restate these relationships in a modified

discrete form. The first modification is to introduce the subscript j to parameters
that change with the lobe rotation angle, ij. The second modification is to

introduce the subscript p to for parameters that change depending on the pass

or rotation number. The third change is to replace diametral values with their

equivalent radial representations. This is done since the "diameter" of a cam

profile is not particularly meaningful. However, the radius is directly analogous
to the radius of curvature, pj, or the radial distance to the point of contact, rcj.

Fourth, multiple constants in expressions are lumped to simply the relationships.

The grinding model is shown in Figure 5-2 with the subscripted quantities.






















Direction
of Rotation


Direction
of Rotation

Figure 5-2 Cam Profile Grinding Model


With these modifications, Equation 5-2 through Equation 5-7 become


Vwjp+s+ rjp = Vp

Equation 5-8

wrpp (Fnj Fthj)
Vwjp 27 rr
Y"L 2xC1

Equation 5-9

kwp v1i
wrp = D0.14
ei

Equation 5-10

FthJ = khi Dej

Equation 5-11


Fp, cos(y,)








2 pj R,
De p, + Rs

Equation 5-12

where the absolute value is introduced to handle the sign change from positive
to negative on the radius of curvature that occurs at points of inflection between
concave and convex cam profile regions. The negative sign is selected for
oo pj < -Rs, or concave cam profile sections. The positive sign is chosen for

0 < pj +oo or convex cam profile sections. And for -Rs < pj < -0, an undercut

condition exists and the equivalent diameter is undefined. The depth of cut for
pass p is


V~jp
wjp

Equation 5-13

In the application of the subscripts to the preceding equations, both the
change in the grinding wheel radius, R, and the change in the radius of
curvature, p,, as a function of the number of grinding passes, p, are assumed to

have negligible effects on the value of the equivalent diameter. This assumption
for the grinding wheel radius is justifiable since the percent change is small
during the grinding of a single cam lobe. Also, the change in the radius of
curvature of the workpiece is small since the total depth of cut is small compared
with the value of radius of curvature.








Also, recall from Chapter 3 that the radius of curvature of the cam surface
at point of contact, j, is
3
r -r 2
2 r po cj-.1 pocj-1

pi 2
2 pocjl rOocj_ 1 1oci- rpOj rc- o pocj1
poj ?i ajl- a, Jpoci + o.-aj_1 j ja-a -a,_
r +2 -r-
pocji + 2 j+1 'j-1 poc ,j-1- Xj+1 j ( I j-1

Equation 5-14

Since the radius of curvature of the cam profile is infinite at points of inflection, it
is noteworthy that Equation 5-12 is well behaved at such points, and
consequently, the grinding model does not break down. This is verified by
taking the limit of Equation 5-12 as the radius of curvature of the surface, pi,

goes to infinity as


2 pi Rs
lim D -
Pj--) Pi Rs

Equation 5-15

The limit is determined by dividing all terms in the numerator and denominator
by pj to get

2 IRS
Pi
lim D -
p, pji Rs
Pj PJ


Equation 5-16









Simplifying Equation 5-16 gives


2 R,
lim Dj RS
jPj


Equation 5-17

or that
lim Dj = 2 R,
pj oo

Thus the limit is finite and the model is, therefore, well behaved.

Equation 5-13 must be modified to account for the fact that in cam profile
grinding the angular velocity of contact, ad, shown in Figure 5-1, is not equal to

the angular velocity of rotation, Oj. To include this effect, the ratio of the change
j
in the contact angle to the change in the rotation angle, can be introduced.

In this case, Equation 5-13 becomes


vw.
ajp Aa,
Nwij AEp

Equation 5-18

Also, the general industrial convention is to specify Nj.p as



Nwjt = Nwp wj

Equation 5-19








where wi is the normalized angular velocity at lobe rotation angle, Oj, and
N,p is the nominal angular velocity for pass, p. Substituting Equation 5-9 and

Equation 5-19 into Equation 5-18 gives the depth of cut as


wrpp (Fnj,- Fth.i)
ajpA A
Np wj AOj 2x rj

Equation 5-20

At this point, it is useful to develop an expression for the velocity of the
point of contact on the work surface. From Figure 5-2 the velocity is


Aa1
vN = Nwp 27 rpo


Equation 5-21

The comparison of the denominator of Equation 5-20 with Equation 5-21, the
substitution of Equation 5-10 for the work removal parameter, wrp, and

combining constants yields


k ( ( k )
j-p o.84 00.14 Fnj-P kth
w JP e j

Equation 5-22

where, of course, Equation 5-22 is valid only when F jp kth SDe is

nonnegative. The normal grinding force, Fnj,p is induced by the wheel head

infeed and the effective grinding system stiffness, ke. The commanded grinding
wheel position, r,p, is in error due to the system deflection caused by the






80


grinding force. The actual value of rj is designated c'rp. From Figure 5-1 and

Figure 5-2


act rj, = rjp +Arjp =u jp -Us ,p

Equation 5-23

The normal grinding force per unit width at lobe rotation angle, O for pass, p, is


Fnjp = ke Ar,p

Equation 5-24

where the system deflection is

P P-1
Arj = fp aj
p=1 p=1

Equation 5-25

The substitution of Equation 5-25 into Equation 5-24, gives the normal grinding

force as



P P-1
Fnj, = ke fp aj
p=1 p=1

Equation 5-26

In the forgoing development, different parameters which are constant
during the profile grinding process constants are lumped together as the

constant kwrp. This is done to simplify the presentation of the grinding model and

also because the actual values of these constants for the grinding wheel and the

machine tool are unknown: their estimation is outside the scope of this work.









The focus here is to develop the depth of cut model. Using Equation 5-14,

Equation 5-22. Equation 5-24, and Equation 5-26, it is possible to simulate the

complete cam profile grinding operation if the values of the constants are known.


Residual Sparkout Errors


While it is possible to run the complete simulation of the grinding process based

on the depth of cut model, the error in the finished ground cam profile may be

estimated more simply. For a properly designed grinding cycle, the profile errors

will be only those due to the residual deflections of the grinding system that exist

at the end of the sparkout cycle. The typical first order relationship between the

commanded infeed, and the actual plunge grinding cycle for grinding is shown

in Figure 5-3. In most cases, the grinding cycle proceeds from a relatively high

feedrate, ul, during rough grind to a slower feedrate of u2 during finish grind, and

finally, to a feedrate of zero during the sparkout phase.



Typical Commanded and Actual Grinding Infeed -
vsof =0 ke Fth
140 | .
--- commanded radial infeed = "
120 ....... effective radial infeed -.- -"
100
a so 'f, slope
vrf T
S60 A
40
sparkout
20 .a-- roughing --- finishing

0 5 10 15 20 25 30 35 40 45 50
time (seconds)


Figure 5-3 Typical Grinding Cycle: Rough, Finish and Sparkout
Adapted from: Malkin, S., Grinding Technology: Theory and Application of Machining with Abrasives, Ellis Horwood,
Chichester, UK, 1989.








In the rough, finish and sparkout phases, a steady state error between the

commanded radial distance and the effective radial distance exists due to the

deflections in the system. In the rough grinding phase, the steady state error for
33
a specific point j on the cam surface is3
Arj,p=rough, ss = rfTj

Equation 5-27

In the finish grinding phase, the steady state error is
j,p=finih, ss = WfT

Equation 5-28

where rj is the time constant for the process at the point of contact, j.

Sparkout is the dwell cycle that occurs at the end of the grinding
operation. At sparkout, the feedrate is v,p is zero for all point and the subscript j

is dropped. During the sparkout cycle, the residual system deflection varies due

to the variation in threshold force which varies with the radius of curvature of the

cam surface. The residual error is the error that remains when the process
reaches equilibrium and the depth of cut, a,,, is zero for all j if p is sufficiently

large. Therefore, from Equation 5-20, it is clear that for zero depth of cut
F Fth =O

Equation 5-29

or
Fnj.p = F',

Equation 5-30

and the deflection at every point j does not change with the pass number. In

Equation 5-29 and Equation 5-30, the subscript p is shown approaching infinity.

In practice, only a time equal to a few time constants are necessary to achieve









steady state conditions for the first order process. From Figure 5-2. the

deflection of the grinding system in the sensitive direction at the completion of

the sparkout cycle is


F, j,p_ Fth
rj,p=sparkout k cos(yj) cos(y )
ke ke

Equation 5-31

Substituting Equation 5-10 into Equation 5-31 gives


kth DejCOS( j)
Aj,p=sparkout k
e

Equation 5-32

Combining the constants gives


Arj,p=sparkout =kc j cos(y )

Equation 5-33

For the usual situation in cam profile grinding, the radius of the grinding wheel is

more than an order of magnitude larger that the profile lift and the grinding wheel
contact angle, y is less than 3.5 degrees. Therefore, it is appropriate to

approximate cos(y,) 1. Using this approximation and substituting Equation 5-

15 into Equation 5-33 gives



j2 p R,
Arj,p=sparkout =k l ,


Equation 5-34









And finally, the effective radial distance to the follower center at the end of the
sparkout cycle is


rjp=sparkout =rj +k
ipj+ Rs

Equation 5-35

From Equation 5-34 it is possible to calculate the form of the residual
sparkout error that is ground into the cam surface. The transformation relations

developed in Chapter 3 can then be used to calculate the follower lift error due

to this effect. In practice, the observed error will differ from the sparkout error

due to a number of factors. First, the cutting sharpness of the CBN grinding
34
wheel used in these trials varies with use and dress frequency Consequently,

steady state conditions may not, in every case, be reached during sparkout.
Second, the data reduction algorithms used on cam gauges affect the form of

the reported profile error as discussed in Chapter 4. Third, errors from previous

processes, such as journal bearing grinding where vibration in the grinding

system can cause the journal bearing to become lobed. This effect is discussed
in Chapter 8.

From Equation 5-34 the residual sparkout profile grinding error can be
simulated with k selected to provide a best fit with experimental data. Figure 5-4

and Figure 5-5 show the results of the best fit of the model to two different runs

performed with identical grinder setups. The details of the grinding conditions
are discussed in Chapter 10. In both these figures, the non-concentricity and
lobing effects, which distort the process error due to the data reduction

algorithm, are removed from the measured data.









From the results presented in Figure 5-4 and Figure 5-5, it is clear that

the residual sparkout model successfully predicts the form of the cam grinding

process error. Unfortunately, the constants in the model are not generally

known and thus the incorporation of the model directly into the cam grinder's

controller is precluded unless the workpiece, the grinding wheel and the grinding

machine are accurately characterized in terms of the system constants.

Nevertheless, as is discussed in Chapter 9, this model is quite useful in

designing the control system since it allows the simulation of the grinding

process and thereby the simulation of the control system.

Comparison of Modeled and Experimental Profile Lift Errors

0.01
0.008 i Model
0.006 ........ Measured!
E 0.004
S0.002
0 0-

-0.002
-0.004
-0.006
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180
Angle (degrees)


Figure 5-4 Grinding Model Results in Grinding Wheel Motion Coordinates





86



Comparison of Modeled and Experimental Profile Lift Errors
0.01
0.008 Model
--....... Measured
0.006
E 0.004
E
0.002
0

-0.002
-0.004
-0.006
-180 -150 -120 -90 -60 -30 0 30 60 90 120 150
Angle (degrees)


Figure 5-5 Grinding Model Results in Grinding Wheel Motion Coordinates



Linearization of the Process Model


Since, ultimately, interest in the development of the process model is

directly related to the usefulness of the model in designing a control system,
35
linearization of the model should be considered. Experience shows that the

process model, given as Equation 5-35, can be approximated as linear function

by substituting the geometric acceleration of the commanded grinding wheel
dr2
radial distance, for the radical term. With this approximation, Equation 5-
d0

35 becomes


rj+1 2 r + r1
rp=sparkout = rj + kL -2r

3N

Equation 5-36


This equation may be rewritten in a form that is more convenient for later use as




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