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 Title:
 Error compensation in radial profile grinding
 Series Title:
 Error compensation in radial profile grinding
 Creator:
 Dalrymple, Timothy Mark
 Publisher:
 Timothy Mark Dalrymple
 Publication Date:
 1997
 Language:
 English
Subjects
 Subjects / Keywords:
 Axes of rotation ( jstor )
Cams ( jstor ) Circles ( jstor ) Control systems ( jstor ) Coordinate systems ( jstor ) Error rates ( jstor ) Geometric angles ( jstor ) Grinding ( jstor ) Grinding wheels ( jstor ) Human error ( jstor )
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 University of Florida
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 University of Florida
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 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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 028009992 ( alephbibnum )
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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 InProcess Inspection........................ 10
Error Compensation Based on PostProcess 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 PointofInterest ...............................................................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
31 Cam Profile Nomenclature...................................................................... 16
41 Machining and Inspection Axes of Rotation ............................................38
42 Nomenclature for Process Geometry ......................................................... 42
43 Timing Angle and Bias Results for Type I Nonconcentricity Error ..............60
44 Timing Bias Results for Type II Nonconcentricity Error .............................61
51 Grinding Model Nomenclature................................................................... 72
61 Modified Grinding Model Nomenclature for Superfinishing Model...............92
62 Process Repeatability..............................................................................96
71 Nomenclature for Asperity Detection Method............................................. 101
81 Digital Filtering Nomenclature............................................................... 115
91 Lift Control System Nomenclature... .....................................................139
92 Repeatability for Grinding, Superfinishing and Gauging.......................... 156
93 Controller Gains Determined by Simulation for Noisy Conditions.......... 157
101 Gains for Profile Grinding................................................................... 164
102 Gains for Parts Inspected after Superfinishing.......................................173
103 Gains for Profile Grinding: Flat Follower.............................................. 182
103 Gains for Profile Grinding: Flat Follower.............................................. 181
LIST OF FIGURES
Figure ............................................ ......... ..... ................ ..... .............
21 Plunge G rinding M odel ......................................................... .................. 9
31 Camshaft Coordinate System: Side View.......................... ............ 15
32 Shaft Coordinate System ..................... ................................................ 15
33 Lobe Coord. System ........................... ....... .............................................. 15
34 Camshaft Coordinate System : CW and CCW Convention..........................17
35 Lobe Coordinate System: Nose +1800 Convention ....................................17
36 Lobe Coordinate System: Nose Convention............................................. 18
37 Typical Manufacturers Cam Lift Data Specification..................................... 18
38 Translational Roller Follower Coordinate Transform................................... 21
39 Coordinate Transformation: Translational Flat to Roller .................................26
310 Coordinate Transformation: Translational Roller to Flat ............................28
41 Steady Rest and Three Grinding with Jaw Chuck: AJAOR..........................37
42 Inspection of Nonconcetric Nominally Round Part.......................................43
43 Introduction of Phase Shift ....................................................................... 46
44 Measurement of a rj Phase Error for an Eccentric............................. ..47
45 Typical Cam Lobe Lift and Geometric Velocity.............................................50
46 Frequency Content for Typical Exhaust Cam Profile........................................51
47 Cam Lobe with a Timing Error of 86 ........................................................ 51
48 Sim ulated Cam Lift Error ......... ...............................................................54
49 Existing Data Reduction Method for Profile Data ......................................55
410 Form of Timing and Nonconcentricity Errors...........................................56
411 Proposed Data Reduction Method for Profile Data ....................................58
412 Effect of Nonconcentricity Error in Grinding Wheel Motion on Follower
Motion ............... ........................................ ....... .........59
413 Effect of Journal Bearing Nonconcentricity Error .......................................60
414 Inspecting a Nonconcentric Surface with a Radiused Follower ................62
415 Flat Follower Approximation for Nonconcentricity Errors...........................68
51 Cylindrical Grinding Model Modified for Cam Profile Grinding ...................69
52 Cam Profile Grinding Model ............................................................... ..75
53 Typical Grinding Cycle: Rough, Finish and Sparkout.................................. 81
54 Grinding Model Results in Grinding Wheel Motion Coordinates................ 85
55 Grinding Model Results in Grinding Wheel Motion Coordinates ................ 86
56 Linearized Profile Grinding Process Model:k, = 0.001, K4.0.00008 .............87
57 Linearized Profile Grinding Process Model:k, = 0.002, K,.0.00016 ............88
61 Camshaft Superfinishing Operation.............................................................90
62 Camshaft Superfinishing Model............... ................................................ 91
63 Typical Lift Error for Profile Grinding: No Compensation .............................93
64 Typical Lift Error for Profile Grinding and Superfinishing : .................... 94
65 Mean Lift Error: Nonconcentricity Removed............................................... 94
66 Statistics of Lift Error: Nonconcentricity Removed ..................................... 95
71 Deleted Residual M odel ....................................................... ................ 100
72 Lift Inspection Data with Spurious Inspection Points................................ 105
73 Modified Studentized Deleted Residuals............................................. 106
81 Nominal Lift and Error Data for a Typical Injector Lobe ...........................110
82 Fourier Transform of Typical Exhaust Lobe Nominal Lift Data.................. 110
83 Effect of BandLimiting the Nominal Lift Data................................... 111
84 Modeled and Measured Lift Inspection Data................. ..... ...................111
85 Frequency Content of Modeled and Actual Lift Error ............................... 112
86 Frequency Response for a Zero Phase Linear Smoothing Filter ........... 114
87 Filtering Results for a Zero Phase Linear Smoothing Filter ..................... 114
88 Simulated Journal Bearing Inspection Data ............................................... 116
89 Simple Two Pad and WrapAround Three Pad Steady Rest.................... 120
810 Mapping of Bearing Roundness Error on Profile................................. 121
811 Effect of Lobing Error .................................................................. .....124
812 Nominal Lift and Lift Error: Coordinate System from Figure 38............. 126
813 Removal of Nonconcentricity and Lobing Effects.................................132
91 Cam Profile Grinding Control System............................ .................. 134
92 Cam Profile Grinding with Superfinishing Control System .......................134
93 Lift Control System Including Process Model............................................. 139
94 Simulated Control of Profile Grinding Error: k,=0.001.............................. 143
95 Simulated Control of Profile Grinding Error: k,=0.002..............................143
96 The Interacting Profile Grinding Process Model......................................... 147
97 Simulation of Noninteracting Controller: Symmetrical Lobe.................... 149
98 Noise in Noninteracting Compensation.................. ... ..................150
99 Grinding Process Simplified Control System..................................... 151
910 Rearranged Block Diagram for Disturbance Rejection............................. 153
911 Simulated Control of Lift at an Arbitrary Point Over N Parts................... 156
912 Grinding and Superfinishing Simplified Control System......................... 157
913 Base Circle Radius Controller ............................................................. 158
914 Simulated Control of Base Circle Size Error........................................... 159
915 Base Circle Nonconcentricity Controller............................................... 159
916 Simulated Control of Nonconcentricity Error ......................................... 160
917 Lobe Timing Angle Controller ....... .... ..................... ....................... 161
918 Sim ulated Tim ing Controller ........................................................ ......... 162
101 Uncompensated Error: Part 1 after Grinding: Lobes 16....................... 167
102 Compensated Error: Part 2 after Grinding: Lobes 16............................ 168
103 Compensated Error: Part 3 after Grinding: Lobes 16..............................169
104 Compensated Error: Part 4 after Grinding: Lobes 16............................ 170
105 Standard Deviation of Lift Error after Grinding....................................... 171
106 Base Circle Size and Timing after Grinding ............................................ 172
107 Uncompensated Error: Part 1 after Superfinishing: Lobes 16................. 175
108 Compensated Error: Part 2 after Superfinishing: Lobes 16................... 176
109 Compensated Error: Part 3 after Superfinishing: Lobes 16.................. 177
1010 Compensated Error: Part 4 after Superfinishing: Lobes 16................. 178
1011 Standard Deviation of Lift Error after Superfinishing..............................179
1012 Base Circle Size and Timing after Superfinishing................................ 180
1013 Uncompensated Lift Error: Part 1 after Grinding Lobes: 16 ............... 184
1014 Compensated Lift Error: Part 6 after Grinding: Lobes 16.................... 185
1015 Standard Deviation of Lift Error after Grinding ..................................... 186
1016 Total Lift Error after Grinding.................................................................. 187
1017 Lobe Timing After Grinding .................................................................. 188
A1 Deletion of the Point of Interest............................................................. 192
A2 Computation of the Variance for a Sliding Window.................................. 192
B1 Error due to Journal Bearing Nonconcentricity ........................................209
C1 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, nonconcentricity and timing errors. Experimental resultsobtained in an
industrial environmentshow 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 resultscorrected for bias using
methods developed herewith 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, peertopeer 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 highspeed 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 laserscanner 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 quasistatic 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 highspeed 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 phaselag 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 lowfriction 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 highproduction,
highprecision 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
industrywide 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 shownadapted here for external radial profile
grindingin Figure 21.
S'j Feed
SSlide
Machine Bed '
Figure 21 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 21
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. GalTzur, 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 InProcess 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 outofphase 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 inprocess 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 outofphase.
Therefore, the inprocess 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 PostProcess Inspection
Various researchers11'12'13 have investigated the general problem of error
compensation in discrete manufacturing processes using postprocess
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 bspline
tensorproduct 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 31 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 32, 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 31.
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 33,. 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 31 Camshaft Coordinate System: Side View
Typical Lobe
Typical Lobe
Timing Datum
c
Figure 32 Shaft Coordinate System Figure 33 Lobe Coord. System
Direction of
Rotation
Iiiii"I
cx C W E2
Table 31 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 kN
iUrection of
Rotation
C12
10.C
C, X,CE2
x
Direction of
Rotation
.; .. .
Cy1 C
CXC *~rE1
Figure 34 Camshaft Coordinate System : CW and CCW Convention
Figure 35 Lobe Coordinate System: Nose 1800 Convention
_ I
18
Irf I r
X 
rb rb
Direction \ Direction
of Rotation of Rotation Y
Figure 36 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 37 Typical Manufacturers Cam Lift Data Specification
Figure 34 shows the camshaft coordinate system for both clockwise
(CW) and a counterclockwise (CCW) directions of rotation. Figure 35 and
Figure 36 show the lobe coordinate systems used to program machine tools.
Lastly, Figure 37 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 35 and Figure 36 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 31. 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 38 the radial distance from the cam axis of rotation to the
14
follower center is
r = rb + + rf
Equation 31
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 38 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 32
yo = rj sin(Oj)
Equation 33
where xj and yc are the XY coordinates of the center of the follower. From
Figure 38, 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 34
The partial derivative of Equation 34 with respect to 0 is
aF(x, y,) dx dy c
2(x xj) de 2(yj Ycj de =
Equation 35
Equation 34 and Equation 35 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 36
dx., [dx 2 dy. 2 d
yj = y l +r +dj
Equation 37
The derivative of Equation 32 with respect to 0 is
dxc, drj
de r sin(Oe)+ cos(Oj)
Equation 38
The derivative of Equation 33 with respect to 0 is
dyc, dr
d = rj cos(Oj)+ Osin(9,)
Equation 39
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 310
The first difference equation for N evenly spaced points is
dr1 r1+ r,_1
dO 2
Equation 311
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 312
For N evenly spaced points, the second difference is
d2r rj+ 2 r + r_,
d02 2 2
IN
Equation 313
24
The polar coordinates of the point of contact is
rp oc= x,)2 +(y,)2
Equation 314
and the two argument inverse tangent function is used to guarantee the correct
quadrant forca as
ai = ATan2(y, xj)
Equation 315
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 316
y j = y + r'j sin(O,) yj)
Y j i f
Equation 317
The transform from Cartesian to polar coordinates is
r = )2+
Equation 318
Again, the two argument inverse tangent function is used to guarantee the
correct quadrant and 0 is
0; = ATan2(ycj,x,)
Equation 319
Flat Follower to Roller Follower
For completeness, the transform from flat follower to roller follower motion
coordinates, illustrated in Figure 39, is presented. For a flat follower, the XY
coordinates of the point of contact are
dr
xi = rj cos(O) sin(9O)
Equation 320
yj = r, sin( j)+ dr cos(9j)
Equation 321
The coordinates of the follower center are
x = x +rf cos(6j)
Equation 322
and
y = yj +rf sin(O,)
Equation 323
And, as before, Equation 318 and Equation 319 are used to transform the
results to from Cartesian to polar coordinates.
Figure 39 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 310
09 = ATan2(y,, yj, xr x)
Equation 324
and, as before
ai = ATan2(y,,x )
Equation 325
Where, as before, the coordinates of the cam surface are calculated from
Equation 36 and Equation 37. Again, from Figure 310 rj is
r,'= Vx?+y? cos(a 0;)
Equation 326
Finally, for the flat follower the lift is
s; = r rb
Equation 327
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 310 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 310
0; = ATan2(y1 y,, x, xi)
Equation 328
and, as before
aj = ATan2(yj, xj)
Equation 329
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 36 and Equation 37. Again, from Figure 310 rj is
r = x + y? cos(a, 0;)
Equation 330
Finally, for the flat follower the lift is
si = r rb
Equation 331
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 d2
Equation 332
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 j1
Equation 333
and from Equation 313, the second divided difference is
d2 rpo 1 'pocj+.1 rp. j r'opj_
dCa2 aj+1 a.jI 1 C j I
Equation 334
Substituting the difference expressions into Equation 332 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 aj1 L + a i j1 aj1 j ,j ji1
Equation 335
In the case where the data are known at evenly spaced points, Equation 335
becomes
3
2 V
p 2 + 2
poc +2 27 poCI 2 2
N N
Equation 336
Interpolation
When transforming motion coordinates from one follower to another, 0 is
not, in general, equal to Oj. The difference is illustrated in Figure 38, Figure 3
9, and Figure 310. 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 337
where
C, = r
Equation 338
c2,j = Lj
Equation 339
[e., ,e1] r Lj
C3J= AO c4,j A
Equation 340
Lj +L,1 2 [0,0,o;] r
C4,j = A0;
Equation 341
A9 =9, 9;
Equation 342
[ 0 1 6, '] r i l l 
Equation 343
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 93 + A'_2)
...... 0 A0,_,
AON_2
2(AL3
2(AO,_2 + A9,_,)
Equation 344
where
b, = 3(AO;[o;_,,;] r + AO;,[O;,O,] r')
Equation 345
"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
!N2 0 0 ...... N2 2(AON3 + AN2) O 3 bN2
.,N1 AN2 0 ...... 0 AO 2(At,_2 + AO,,) L,
Equation 346
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 ShermanMorrison 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 337 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 interlobe 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 41. 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 41.
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
41.
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 41.
Figure 41 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 41 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 34 and Figure 36. In Figure 34, 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 36, 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 41. 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 35 and Figure 36, 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 42. 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 42 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 42 Inspection of Nonconcetric Nominally Round Part
The Fourier series is28
Ni 21. N 'b sin27
ri =R+a, acos j n + bn sin j n
n=l N n=l N
Equation 41
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 42. For a flat follower, the DC term of the Fourier series is
the average part radius, R, and is defined as
1 N1
R= j orJ
RN
Equation 42
The cosine coefficients for n = 1 to N1 are
2 N 12 xn n
an N CN os( Nj
j=o
Equation 43
The sine coefficients for n = 1 to N1 are
bn = 2 r sin 2j n
b N j=J N
Equation 44
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 41, Equation 43 and Equation 44 become
r, =R+a1 cos j +b, sin(N jN
Equation 45
2 N1 2x7
a, N Zr, cos( N j
j=o
Equation 46
2 N1 f2.s
b1=Nrsn
Equation 47
where al and bi represent the Xp and Yp components of the nonconcentricity
respectively. These components are illustrated in Figure 42. For the phasing
of the nonconcentricity shown, the follower motion is the sum of a sine wave and
a constant as shown in Figure 43 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 43, 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 48
where 6rj = r" r,
Equation 49
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 44 for 65 equal to ten degrees. From Equation 48, 6r is
recognized to be of the form
dr
rj= dO
Equation 410
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
phaseshifted 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 411
Direction of Rotation
Shown at 0 = 0
S1
Figure 43 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 44 Measurement of a rj Phase Error for an Eccentric
Substituting Equation 411 into Equation 49 gives
6r = r. S.860 rj
+ N
Equation 412
The first order forward first difference equation29 for point 0 is
dr r S r
dO 60
Equation 413
Rearrange Equation 413, substitute it into Equation 412 and solve for 6e to get
6r1
60 dr}
Equation 414
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 415
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
N1 dr
A d (r r)
k=
N_dr]2
Equation 416
Lastly, substitute the expression for 8r (v)j given in Equation 415 for 6rj in
Equation 414 and solve for 68 to get
65 = k
Equation 417
Therefore, the timing error, 56, is
N1 dr
A d (r' r)
N [dr 2
Equation 418
A
where 60 is the least sum of the squares estimate of the parameter 65. It is
important to note Equation 418 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 415 with Equation 48 and
dr
recognize that, for an ideal round feature, k and d in Equation 415
dO
correspond to ai and cos N ij) respectively in Equation 48.
Obviously, if all that is desired is to determine the timing of round part
features, then nothing is gained from Equation 414: the timing error can be
more directly determined using the Fourier transform. However, Equation 414
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 45 are analyzed. Figure 46 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 45 for circular features are invalid for
noncircular cam profiles. However, since Equation 418 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 45 Typical Cam Lobe Lift and Geometric Velocity
Again, consider the cam profile and ideal inspection process shown in
Figure 47. The follower lift and geometric velocity for a roller follower, with 60
equal to zero, are shown in Figure 45. 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 45 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 46 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 47.
Timing Direction of Rotation
Reference Pin Shown at 0, = 00
Figure 47 Cam Lobe with a Timing Error of 68
1 11
For an ideal lobe with an error only in phase of 60, from Equation 412 the
measured lift error is
r, = r 360 rj
I N
Equation 419
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 45. Recall that for the eccentric, the form of r
is that of a sine wave and the error term associated with a phaseshift is a cosine
wave or simply the derivative or geometric velocity. Therefore, while Equation 4
5 is invalid for noncircular features, Equation 418 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 420
Finally, it should be noted that Equation 420 requires that r. j.e0 ri
J+ N
dr
approximate d well. If 60 is large, then this approximation is poor and
d9
Equation 420 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 420 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 phaseshifted
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 45. 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 48 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 48 Simulated Cam Lift Error
From Figure 48, 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 420 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 49.
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 49 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 49. For the simulated
inspection data shown in Figure 48 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 43 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 420 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 410. 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 420 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 420.
Figure 410 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 411 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 48, the
nonconcentricity error is removed and the timing error is calculated from
Equation 420 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 overconstrained during cam profile grinding due to
the combination of the centers and steady rests as shown Figure 41. In this
overconstrained 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 411 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 412. The distortion of the
nonconcentricity error in the follower motion coordinates is of the form of the
dr
timing error d From Equation 420 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
S0.002
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 412 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 413. 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 413 is 0.0389. The timing angle bias with the base circle
nonconcentricity is 0.00540. These results are summarized Table 44.
Effect of Journal Bearing Nonconcentricity Error
0 01
0.008
0006
0 004
E 0.002
E
2 0.002____
0.004  Nonconcentricity
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 413 Effect of Journal Bearing Nonconcentricity Error
In conclusion, from the results summarized Table 43 and Table 44 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 43 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 44 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 414 Inspecting a Nonconcentric Surface with a Radiused Follower
Figure 414 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 414 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 421
a, cos(, )+b sin(,) (rb + rf )cos( )r = 0
Equation 422
Equation 421 and Equation 422 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 423
Equation 423 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 424
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 425
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 426
Therefore, if the coefficients ai, bl, and rb in Equation 425 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 427
Typically, the sum of square of the error, e, is minimized to determine the
coefficients in Equation 425. Some authors31 propose a general least sum of
the squares solution to determine a,, bl, and rb using the NewtonRaphson
iterative method. However, this is not generally required for two reasons that will
be demonstrated. First, the unique form of Equation 425 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 45, which
expresses the displacement for a flat follower displacement, s (lift), to estimate
the follower motion for a radiused follower. Equation 45 is repeated in a slightly
more convenient as
si =a, cos(Oj)+b, sin(9j)
Equation 428
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 429
Equation 426 can be rewritten as
s(e) = f() + g(6)
Equation 430
where
f(9)= a cos(O)+ b sin(6)
Equation 431
and
g(e) = rb r, + (rb + r)2 (a2 + + ab sin(2e))
Equation 432
Substituting Equation 431 and Equation 432 into Equation 429 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 433
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 428. 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 434
With the bias know, the base circle can be calculated from the DC Fourier
coefficient, R, as
rb = R + bias
Equation 435
where R is defined in Equation 42. And finally
1 N1 N1
rb i r, +N(rb +rf (rb + r) (a2 +b + sin(20e)))
Sj=0 j=0
Equation 436
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 436 for a single parameter, rb.
Finally, the second pointthe roundness error introduced due to the flat
follower assumption is small for typical nonconcentricity errorsis considered.
Recall that Equation 428 and Equation 426 give the follower motion for flat and
a radiused follower respectively. If the relationship for the flat follower, Equation
428, is used to approximate the motion of the flat follower that occurs due to a
nonconcentric round partexactly represented by Equation 426then 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 432. 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 415. Clearly, the roundness error
introduced is insignificant: the bias for rb is approximately 106 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
2.0E06 0.015
1.8E06 .*  Estimation Error
1.6E06  Nonconcentricity 0.01
oE
S 1.4E06 0.005
1.2E06 a
SE 1.0E06 \ 0 E
E
S 8.0E07 a
V 6.0E07 / 00
WL 4.0E07 n.
; ". / 0.01 5
2.0E07
0.0E+00 0.015
0 30 60 90 120 150 180 210 240 270 300 330 360
Theta (degrees)
Figure 415 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 51.
Figure 51 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 51
32
The general grinding relationship is
V, + Vs + = v
Equation 52
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 51.
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 53
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 54
where the equivalent diameter, De, for external cylindrical grinding is
D, D,
D +D
SD + Ds
Equation 55
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 56
where kth is a constant related to the tendency of the grinding wheel and
workpiece material to rub instead of cut.
Table 51 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 51 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 57
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 52 with the subscripted quantities.
Direction
of Rotation
Direction
of Rotation
Figure 52 Cam Profile Grinding Model
With these modifications, Equation 52 through Equation 57 become
Vwjp+s+ rjp = Vp
Equation 58
wrpp (Fnj Fthj)
Vwjp 27 rr
Y"L 2xC1
Equation 59
kwp v1i
wrp = D0.14
ei
Equation 510
FthJ = khi Dej
Equation 511
Fp, cos(y,)
2 pj R,
De p, + Rs
Equation 512
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 513
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 pocj1
pi 2
2 pocjl rOocj_ 1 1oci rpOj rc o pocj1
poj ?i ajl a, Jpoci + o.aj_1 j jaa a,_
r +2 r
pocji + 2 j+1 'j1 poc ,j1 Xj+1 j ( I j1
Equation 514
Since the radius of curvature of the cam profile is infinite at points of inflection, it
is noteworthy that Equation 512 is well behaved at such points, and
consequently, the grinding model does not break down. This is verified by
taking the limit of Equation 512 as the radius of curvature of the surface, pi,
goes to infinity as
2 pi Rs
lim D 
Pj) Pi Rs
Equation 515
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 516
Simplifying Equation 516 gives
2 R,
lim Dj RS
jPj
Equation 517
or that
lim Dj = 2 R,
pj oo
Thus the limit is finite and the model is, therefore, well behaved.
Equation 513 must be modified to account for the fact that in cam profile
grinding the angular velocity of contact, ad, shown in Figure 51, 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 513 becomes
vw.
ajp Aa,
Nwij AEp
Equation 518
Also, the general industrial convention is to specify Nj.p as
Nwjt = Nwp wj
Equation 519
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 59 and
Equation 519 into Equation 518 gives the depth of cut as
wrpp (Fnj, Fth.i)
ajpA A
Np wj AOj 2x rj
Equation 520
At this point, it is useful to develop an expression for the velocity of the
point of contact on the work surface. From Figure 52 the velocity is
Aa1
vN = Nwp 27 rpo
Equation 521
The comparison of the denominator of Equation 520 with Equation 521, the
substitution of Equation 510 for the work removal parameter, wrp, and
combining constants yields
k ( ( k )
jp o.84 00.14 FnjP kth
w JP e j
Equation 522
where, of course, Equation 522 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 51 and
Figure 52
act rj, = rjp +Arjp =u jp Us ,p
Equation 523
The normal grinding force per unit width at lobe rotation angle, O for pass, p, is
Fnjp = ke Ar,p
Equation 524
where the system deflection is
P P1
Arj = fp aj
p=1 p=1
Equation 525
The substitution of Equation 525 into Equation 524, gives the normal grinding
force as
P P1
Fnj, = ke fp aj
p=1 p=1
Equation 526
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 514,
Equation 522. Equation 524, and Equation 526, 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 53. 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 53 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 527
In the finish grinding phase, the steady state error is
j,p=finih, ss = WfT
Equation 528
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 520, it is clear that for zero depth of cut
F Fth =O
Equation 529
or
Fnj.p = F',
Equation 530
and the deflection at every point j does not change with the pass number. In
Equation 529 and Equation 530, 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 52. 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 531
Substituting Equation 510 into Equation 531 gives
kth DejCOS( j)
Aj,p=sparkout k
e
Equation 532
Combining the constants gives
Arj,p=sparkout =kc j cos(y )
Equation 533
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 533 gives
j2 p R,
Arj,p=sparkout =k l ,
Equation 534
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 535
From Equation 534 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 534 the residual sparkout profile grinding error can be
simulated with k selected to provide a best fit with experimental data. Figure 54
and Figure 55 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 nonconcentricity 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 54 and Figure 55, 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 54 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 55 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 535, 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 536
This equation may be rewritten in a form that is more convenient for later use as

Full Text 
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f 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
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Copyright 1997 by Timothy Mark Dalrymple
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r 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 t o Mike Taylor at Dana Corporation for his help in coordinating the experimental trials Also I would like t o acknowledge the continued support and enthus i asm of John Andrews at Andrews Products Lastly I would thank my advisor John Ziegert His confidence encouragement and patience have made the difference. 111
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1 ~ f I I l TABLE OF CONTENTS ACKNOWLEDGMENTS .. .... .... ................. ........ ........ . ... ... ................. ... . . ...... iii LIST OF TABLES . ............... .. ... .. .......... .. ............... ............................ ....... .. .... vii LIST OF FIGURES ..... ... ... . .. .... ....... ..... ...... . . .. . . ..... .. ........ .. .. .. ... .... .. .. ... .. . viii ABSTRACT ...... . ......... .. .. .. ........................ .............. .. ... .. .... ........... ..... .. .. .. xiii CHAPTERS 1 INTRODUCTION ....... ... ... ... ..... .. .................... .. .......... . . . ... ..... ... ......... ... 1 Scope of the Problem 1 Profile Gr i nding Technology .............. ... ......... Profile Inspection Technology Potential for Improvement . .. .... .. .. .... .. .. . ............ . ... . .. . . . . ...... . .. .. .. . ... 4 2 REVIEW OF THE LITERATURE ................... ... . ... ..... .. ....... .. ...... .................. 8 Error Avoidance .. . . . . ... . .... ... ......... ............. ............................. .. . . ..... .. .. .. 8 Error Compensation Based on InProcess Inspection .... .... ..... .... ................ 1 o Error Compensat i on Based on Pos t Process Inspection ........ .. ...... . .. .......... 12 3 CAMSHAFT GEOMETRY ...... . .............. ............ ..... .... .. .................... .. .. .. 13 Introduction . .. ... . .. . .. ........ . . .. ... ... .... . ......... ... .... . ... .. . .. ........ .. . .. ........ .. .. 13 Coupling of Timing and Profile Errors ....... . .... .. ... .. ... .. ...... . .... .. .. . .... ........ . 13 Follower Motion Coordinate Transformations ... ............ . ............. .... .. ... 19 Radius of Curvature of the Cam Profile ................... .. ....... .................... 29 I nterpolation ...... .. ....... .. ... .... .............................. .. ....... .. ... .. ...... ... ... .......... 31 4 CAMSHAFT GEOMETRY : IMPLICATIONS FOR THE GRINDING AND INSPECTION PROCESSES ... ... . ....... ... ..... .. . ... .. ........... ........ .. .. ... ........ 34 Introduction .. .. .. .. . .... ...... .... . ..... .... .. ............... .. .. ............ .. . .............. ....... 34 Camshaft Geometry and the Grinding P r ocess .... ... ... ... .............................. 35 The Camshaft Timing Datum .... ... ....... .... . ... ... .. .... .. .... .. .. .. .... ...... ..... ... 35 The Machining Axis Of Rotation ......... .. ..... . . .. .. .. . .. .. ... . ....... .... ... .. . ... 35 Camshaft Geometry and the Inspection Process ... .. ... ............... .. ...... ... .... 37 I V ~~ 
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Basic Elements of Cam Geometry . .... ...... ..... .. .. .. . .................. ... . .. ..... 39 Timing of Round Features ........ .... ... ................. .. .. .... ....... .. .... ............ .. 41 Measurement of Cam Lobe Timing .. ............. . .. ........ ...... ................. .. ... 49 Comparison of Methods ... .............................. ... .................. ............. .... . 53 The Effect of Radiused Followers on the Nonconcentricity Calculation ... .... .. .... ..... ... ........ .. ... .. .... ... .. .................. .. ....... .. . .... ... 61 5 MODELING OF THE PROFILE GRINDING PROCESS . .. ....... .. ..... ... .. ........ 69 Introduction ................. . ..... .. ... .. ... ........... ..... .... .... ..... .................... .. ... ...... 69 Modeling Approach and Requirements .......... ... .. .. ................................ ....... 69 Physical Model of the Cam Grinding Process: Depth of Cut.. .. .. ...... .. ....... 7 4 Residual Sparkout Errors ........................................... .. .................... .... .. 81 Linearization of the Process Model .... ....... ....... .............................. ....... 86 6 THE SUPERFINISHING PROCESS ..... ........... .... ... ... ...... ..... .. . .... .. ....... .. .. 89 Introduction ........................................... . ................ .. ..................... .... ..... ..... 89 Description of Process ............................... .................................................. 90 Experimental Results . .... .. ......... ... ..... ........ ....... . . . ................................ .... 92 Discussion of Experimental Results ............ ........... .... ................................. 95 t 7 DETECTION OF SPURIOUS ROUNDNESS AND LIFT DATA ..................... 97 Introduction ................................. ..... ........... .. ...... ...... .. . .... .. .......... ... ..... .... 97 Description of Method ............................................................... ........ .. . ...... 98 Linear Surface Model and Residuals ............... .. ............................... ...... 98 Deletion of the PointofInterest . ....... .. ....... .. ............ ...... .... .... ........ ...... 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 I r 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 V _J
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9 CONTROL SYSTEM ............. ..... .. .................. ... ...................... ............... 133 The Controlled Process .. .. .... ... ......... ........ .. .. ... .... ... ....... ... .... ....... .. ...... 133 Physical Implementation ........ ......... .. .. ... ... .. ........ .. ... .... .. ......... ........ .. .. 135 Control System Design ....... ...... .... .. .. ..... .... ... . . ...... .. .. ........ .. ........ .. .... ... . 136 Parameter Interaction and the Design of the Lift Control System : Grinder Only ................. .. ... ..... .. ....... ......... .. .......... .. .. .. .. .. ......... . 136 Simulation of the Simple Lift Controller for Grinding Under Noisy Conditions ..... .... ................... . ... ........ ......... . ... .... .......... .. ......... .. ... 150 Base Circle Radius Controller ...... .. ....... .. ... . ..... ....... ........ .... ..... ........ .... 158 Base Circle Nonconcentricity Controller .... ...... ..................................... 159 Lobe Timing Ang l e Controller .. ........... ..... ....................................... .. .... 160 Summary ......................................... ... . ....... ....... .... ... ............................ .... 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 Work ......... ......... .. ........................ .. .. .. .. ......... .. .. .. .... ......... . .. ... .. 190 APPENDICES A VARIANCE FOR A SLIDING WINDOW .............. .... .............. .. ......... .......... 192 1 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 l VI J
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. ..... " ~~ ~ LIST OF TABLES Table .... .. .... . .. ......... .... ........ .... ... .... ........................ .... ....... .... .... ..... .... ..... .. 31 Cam Profile Nomenclature ..... ...... ...................... ..................................... .. ... 16 41 Machining and Inspection Axes of Rotation .......... ....................................... 38 42 Nomenclature for Process Geometry ........................................................... 42 43 Timing Angle and Bias Results for Type I Nonconcentricity Error .. . .. .... . ... 60 44 Timing Bias Results for Type II Nonconcentricity Error .... .......................... 61 51 Grinding Model Nomenclature ......... ............................ ....... ....... .. ..... ... ........ 72 61 Modified Grinding Model Nomenclature for Superfinishing Model ............ ... 92 62 Process Repeatability ....................... ... ...... ......................... ............ .. ....... ... 96 7 1 Nomenclature for Asperity Detection Method .............. .. ... .... ...... ..... .... ..... 101 81 Digital Filtering Nomenclature .................................................................... 115 91 Lift Control System Nomenclature ................................. .. ........................... 139 92 Repeatability for Grinding, Superfinishing and Gauging ............................ 156 93 Controller Gains Determined by Simulation for Noisy Conditions .............. 157 101 Gains for Profile Grinding ...... .... ........... ... ...... ....... ................. .. ...... ....... ... 164 102 Gains for Parts Inspected after Superfinishing ......................................... 173 103 Gains for Profile Grinding: Flat Follower .... ........................ ...... ............... 182 103 Gains for Profile Grinding: Flat Follower .... ................................ ........ .... 181 vii
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LIST OF FIGURES Figure ... .. ...... ... .. ... .......... . . .... .............. ........................... ....... . ......... .. ... . ... 21 Plunge Grinding Model .. . .... ... .... .. ... .. . .. ...... .......... ..... .. .. .. .. ... .... . ................ 9 31 Camshaft Coordinate System: Side View .. . ......... ... .. ... ........... .. ... ... .... . ... 15 32 Shaft Coordinate System ................. .. .. .. ... .. . . ..... ......................... .. ..... .. .. 15 33 Lobe Coard. System ... . .............. .... ........... . .. .. ......... ................... ... .... . .. .. 15 34 Camshaft Coordinate System : CW and CCW Convention ....... .. .............. .. 17 35 Lobe Coordinate System: Nose Convention ......................... .. .. .. . .... 17 36 Lobe Coordinate System: Nose Convention ........ .. ...... .. .... ...... . .... . ......... 18 37 Typical Manufacturers Cam Lift Data Specification . ..... .... ......................... 18 38 Translational Roller Follower Coordinate Transform ................... . ........ ...... 21 39 Coordinate Transformation: Translational Flat to Roller .............. ...... ........ 26 310 Coordinate Transformation: Translational Roller to Flat ... .............. .. ... .. .. 28 41 Steady Rest and Three Grinding with Jaw Chuck: AJAOR ..... .. ................. 37 42 Inspection of Nonconcetric Nominally Round Part .. .. .. ..... ..... . ....... .. .. . .... . 43 43 Introduction of Phase Shift .................. .... ........ ......... .. .... .......................... 46 44 Measurement of a ri Phase Error for an Eccentric ..... .. ... ........ ........ ..... ... ... 47 45 Typical Cam Lobe Lift and Geometric Velocity ..................... ......... ....... ... .. 50 46 Frequency Content for Typical Exhaust Cam Profile .. ... ................ .. ............ 51 47 Cam Lobe with a Timing Error of 80 ................... ... .. ..... .. .... .. ..... .. ..... ...... ..... 51 viii
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1 48 Simulated Cam Lift Error .. .. . ..... . .... ............ . ... .. ...... ........ ...... ..... . ... ... .. .. 54 49 Existing Data Reduction Method for Profile Data .. ............... ...... ....... .... .. .... 55 41 O Form of Timing and Nonconcentricity Errors ...... .. .. ............ . ... .... .... ....... 56 411 Proposed Data Reduction Method for Profile Data .. ............ .................... 58 412 Effect of Nonconcentricity Error in Grinding Wheel Motion on Follower Motion . ........ ........................................... ......... .. .......... .. ... .................... 59 413 Effect of Journal Bearing Nonconcentricity Error .. ... ........... .. ...... ... .... ....... 60 414 Inspecting a Nonconcentric Surface with a Radiused Follower ............. .. ... 62 415 Flat Follower Approximation for Nonconcentricity Errors ...... ....... ... ....... .. 68 51 Cylindrical Grinding Model Modified for Cam Profile Grinding .... .... ...... .... 69 52 Cam Profile Grinding Model ......................................................................... 75 53 Typical Grinding Cycle: Rough, Finish and Sparkout.. ......... .. ...................... 81 54 Grinding Model Results in Grinding Wheel Motion Coordinates .................. 85 55 Grinding Model Results in Grinding Wheel Motion Coordinates .................. 86 56 Linearized Profile Grinding Process Model:kc = 0.001, KL .. 0.00008 .. ....... ... 87 57 Linearized Profile Grinding Process Model:kc = 0 002, .. 0.00016 ............. 88 61 Camshaft Superfinishing Operation ......... .......... ....... ....... . .. .. ... ....... ........ 90 62 Camshaft Superfinishing Model .................................................................... 91 63 Typical Lift Error for Profile Grinding: No Compensation ............................. 93 64 Typical Lift Error for Profile Grinding and Superfinishing t: ........................... 94 65 Mean Lift Error: Nonconcentricity Removed ................................................. 94 66 Statistics of Lift Error: Nonconcentricity Removed ........ .............................. 95 IX
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~~~~ 7 1 Deleted Residual Model ..... ...... ... ..... ....... ... .......... .. .. .. .. ..... .. . .. ........... 100 72 Lift Inspection Data with Spurious Inspection Points .. ........ .... .. . . .. . . ...... 105 73 Modified Studentized Deleted Residuals ........ ........ ... .. ... ... ... .... .... . .... .. .. 106 81 Nominal Lift and Error Data for a Typical Injector Lobe ............ .... .... .. ...... 110 82 Fourier Transform of Typical Exhaust Lobe Nominal Lift Data .. ............. ... 110 83 Effect of BandLimiting the Nominal Lift Data ............................................. 111 84 Modeled and Measured Lift Inspection Data ..... ... .. .... ............ .... ..... .. ....... 111 85 Frequency Content of Modeled and Actual Lift Error ................................. 112 86 Frequency Response for a Zero Phase Linear Smoothing F i lter ......... .. .... 114 87 Filtering Results for a Zero Phase Linear Smoothing Filter .......... ...... .... .. 114 88 Simulated Journal Bearing Inspection Data ... .. ...... .. .. . ......... .. ... ...... . ...... 116 89 Simple Two Pad and WrapAround Three Pad Steady Rest.. ..... ... ...... .. .... 120 810 Mapping of Bearing Roundness Error on Profile ........ ......... . ......... ......... 121 811 Effect of Lobing Error ........................ ... . ................................................. 124 812 Nominal Lift and Lift Error: Coordinate System from Figure 38 ............... 126 813 Removal of Nonconcentricity and Lobing Effects .. ....... ..... ... ... ............. 132 91 Cam Profile Grinding Control System .. ... ... .. .. .......... .... .... . ............. ......... ... 134 92 Cam Profile Grinding with Superfinishing Control System ......................... 134 93 Lift Control System Including Process Model ........ ....... ................... .. ....... 139 94 Simulated Control of Profile Grinding Error : kc=0.001 ................................ 1 43 95 Simulated Control of Profile Grinding Error: kc=0 002 .... ....... ..................... 143 96 The Interacting Profile Grinding Process Model. ........................................ 147 X
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~ 97 Simulation of Noninteracting Controller : Symmetrical Lobe ... . .. ... .. ..... ... 149 98 Noise in Noninteracting Compensation ..... . .. ... .. ...... .. . . .. ... .. ..... ... ... ... .. 150 99 Grinding Process Simplified Contro l System . .. ...... .... . . ....... .. ...... ... ... 151 910 Rearranged Block Diagram for Disturbance Rejection .. .. ..... . ... .. ... ...... .. 153 911 Simulated Control of Lift at an Arbitrary Point Over N Parts ...... .. .. .. ... .. .. 156 912 Grinding and Superfinishing Simpl i fied Control System . ..... .. ...... . .. . . . 157 913 Base Circle Radius Controller .. . .. . ... .. . .. .... ......................................... 158 914 Simulated Control of Base Circle Size Error .... .............. . .. . ........... .... . . 159 915 Base Circle Nonconcentricity Controller ..................... ....... .. .. ... .. . ... .. . 159 916 Simulated Control of Nonconcentricity Error .... ... ............ ... .... .. ... .. .... 160 917 Lobe Timing Angle Controller . .......... .................... ............... ....... ......... 161 918 Simulated Timing Controller ..... .. . .. .. ....... ........... . ..... . ..... ..... .... .... . 162 101 Uncompensated Error : Part 1 after Grinding: Lobes 16 ... .... .. ......... .. .. 167 102 Compensated Error: Part 2 after Gr i nding: Lobes 16 ... . .. ...... ... .... . ... .. .. 168 103 Compensated Error: Part 3 after Grinding: Lobes 16 . ... .. ....... .. . .. .. ..... 169 104 Compensated Error: Part 4 after Grinding: Lobes 16 ... . .... .. ... .. .. ... .. .. 170 105 Standard Deviation of Lift Error after Grinding .. . ... .... ....... ..... .. ... . ... . . 171 106 Base Circle Size and Timing after Grinding .. .. ... .. .. .. ... . .. . . ... .. ........... 172 107 Uncompensated Error: Part 1 after Superfinishing: Lobes 16 ... . ......... . 175 108 Compensated Error: Part 2 after Superfinishing: Lobes 16 ............. ....... 176 109 Compensated Error: Part 3 after Superfinishing: Lobes 16 ........ .......... . 177 101 O Compensated Error: Part 4 after Superfinishing: Lobes 16 ......... .. ....... 178 xi
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1011 Standard Deviation of Lift Error after Superfinishing .. .... ................ ... . 179 1012 Base Circle Size and Timing after Superfinishing . .. . .. ...... ... .. .... .... .. 180 1013 Uncompensated Lift Error: Part 1 after Grinding Lobes : 16 ... ............ . 184 1014 Compensated Lift Error: Part 6 after Grinding: Lobes 16 ........... ........ . 185 1015 Standard Deviation of Lift Error after Grinding ....... .. ... .. . .. . .. ...... .. ..... . 186 1016 Total Lift Error after Grinding .. .. ... . . ... .... . .. .. ..... .. .. .. ... .................... 187 1017 Lobe Timing After Grinding . .. .. ........ ..................................................... 188 A1 Deletion of the Point of Interest.. .. .... .... ..... ..... ..... .... .. ........... ................ 192 A2 Computation of the Variance for a Sliding Window . ... .............. .... ....... .. 192 B1 Error due to Journal Bearing Nonconcentricity ..... . . ..... ......... .. ............... 209 C1 Radial Distance to the Point of Contact .................. ........... .. .... .. .. .. ..... ... 211 XII _J
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfi ll ment of the Requirements for the Degree of Doctor of Philosophy ERROR COMPENSATION IN RADIAL PROFILE GRINDING By T i mothy Mark Dalrymple August 1997 Chairman : John C Z i egert Major Department: Mechanical Enginee r ing To take advantage of the existing computer numerical control technology in radial profile grinding and inspection a closed loop control scheme is proposed In th i s scheme the commanded part geometry is modified based on the errors observed in previous parts in an effort to minimize the lobe profile size nonconcentricity and timing errors Experimental resultsobtained in an industrial environmentshow 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 Th i s 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 i s developed and equations are presented to simulate the complete grinding cycle From this model the predicted error is compared with experimental resultscorrected for bias using methods developed herewith good agreement. xi i i
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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 w i th objectives other than min i mizing the profile error in mind For example i f 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 peertopeer network xiv
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I L 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 highspeed assembly equipment 1 Also cams are critical components of diverse machines such as blood separation units fuel injection pumps internal combustion engines power steering pumps and automated l aserscanner 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 1
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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 2 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 Technology 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 quasistatic positioning .I
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3 errors and the dynamic follow i ng errors result i n a profile error Additionally the static and dynamic synchronization errors between the work spindle rotat i onal axis and grinding whee l infeed axis result in a profile error Other and more i mportant 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 vary 2 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 significant l y In addition to causing a profile error the footprint speed variation may lead to burning in highspeed 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 "
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~ nonrepeatable errors that is examined in detai l i n this research A lso 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 4 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 i nstalled in engine manufacturing plants worldwide 3 Potential for Improvement Computerized cam gauges that produce digital inspection r esu l ts have been available for over two decades. However t he po t ential offered by these gauges for error compensation is poorly utilized. Given the flexibility of CNC profile grinders and if the repeatable profile t i ming, nonconcentricity and size errors can be identified then it should be possible to correct for errors in subsequent parts by man i pulating 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 s i gnificant marketing difficulty 
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l 5 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 t he 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 4 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
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... ~! 6 With the ti ming profile and nonconcentricity errors successfully decoupled the observed profile errors can be related to the commanded part profile A s is shown in Chapter 5 a s i mple model of the profile gr i nding process which successfully predicts the form of the profile error can be deve l oped With the form of the model known the coeff i cients of the model can readily be determined from experimental data. In Chapter 6 the repeatability o f the superfinishing process is examined Experimental results show that the profile error due to the superfinis h ing 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 i n the cam lobe pr i or to the superfinishing process This error is of the form such that after the part is superfinished the profile error i s minimized. The processes to be controlled exist in industrial environments. Consequently grinding swart 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 ope r ator can usually recognize results that are caused by contaminates and can pass or reinspect the part as appropriate. However for the case where the inspec t ion 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 J
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7 Since it is possible only to compensate for the repeatable portion of the measured error i t 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 t he problem it also allows the data to be filtered without introducing phaselag into the control problem This topic is presented in Chapter 8 Finally in Chapters 9 and 1 O the control system is described and experimental results are reported for a variety of industrial situations
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~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 lowfriction machine ways and axes. While these i mprovements have resulted in significant gains in machine accuracy, they have not done so without great expense. Today a typical h i ghproduction highprecision 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 industrywide 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 8
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9 geometry The literature review is restricted to methods proposed for correct i ng for these types of errors The basic model of the external cylindrical grinding process is described by Hahn 5 and Lindsay 6 and shownadapted here for external radial profi l e grindingin Figure 21 ~ Machine Bed Feed Slide Figure 21 Plunge Grinding Model Adapted from King, R. I., and Hahn R. s Handbook of Modern 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 = Ow O s e D +D w s Equation 21
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where D w i s the workpiece diameter and D s i s the diameter o f the grinding whee l. 1 0 Hahn 7 also specifically discusses the systematic process errors f or i nternal radial profile grinding. For radial profile grinding t he 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, t he 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. GalTzur, Shpitaini and Malkin 8 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 InProcess Inspection In their work at Cranfield Institute of Technology Cooke and Perkins 10 proposed an error compensation scheme for profile grinding based on in process workpiece inspection In this scheme a measurement probe is installed 180 outofphase with the grinding wheel. The authors proposed sampling the profile error during the grinding operation and using the 180 phase lag between the gr i nding wheel and inspection probe to compute a corrected position command for the servo controller on the next pass. While such inprocess inspection and control schemes appear attractive they have not
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I l ______ _____ ..,......been implemented in radial profile g r inders. Wh il e the reasons for the not i mplementing this system are not known there appear to be several technical difficulties associated w i th this approach. 11 One major difficulty is the integration of the inspection process into the grinding environment. The grinding environment is contaminated by swart from the grinding wheel. The environment is also subjected to relatively large temperature variations Machine vib r ations 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 t o the system that results from the grinding process mode l discussed earlier. Recall that the variation in the depth of cut which resu l ts 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. Consequen t ly, 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 i nterest and the defection due to the grinding force 180 outofphase. Therefore the inprocess 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 PostProcess Inspection Various researchers 11 1 2 13 have investigated the general problem of error compensation in discrete manufacturing processes using postprocess
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1 2 inspection results Yang and Menq 11 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 bspline tensorproduct 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 work 4 in this area
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CHAPTER 3 CAMSHAFT GEOMETRY Introduction 4 14 15 ,1 6 1 7 18 1 9 20 Various authors and companies use 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 i ndustrial convention the cam profile is not specified directly, but rather the motion of a cam follower is specified. In cam manufacturing and i nspection 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 part i cularly 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 rad i us is distorted when the profile error i s descr i bed 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 31 shows a typical camshaft coordinate system The coordinate system is attached to the camshaft with the origin fixed by the camshaft ax i s of rotation The c X axis which is the camshaft timing datum is fixed by the camshaft timing reference 13
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1 4 The c Z datum is chosen as the driven end of the camshaft. As shown in Figure 32 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 31. The lobe timing angle c cpi, the axial position of the lobes, czxi, and the axial position of the journal bearings c zxi, are specified in the camshaft coordinate system The cam profile i s not specified directly in the camshaft coordinate system Rather it is specified in terms of follower lift in t h e polar coordinate system that is attached to the cam lobe shown This coordinate system is shown in Figure 33 ,. Here, this system is referred to as the lobe coordinate system The lobe coordinate system is rotated with respect to the camshaft coordinate system by c xi ( or c xi 180) and translated along the c Z b C axis y zxi 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 180 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. 
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Camshaft Driven End Figure 31 Camshaft Coordinate System: Side View Direction of Timing Reference Timing Datum Profile Datum X 1 5 r Direction of Rotation Lobe Timing \ Datum 'I of Rotation ,, Figure 32 Shaft Coordinate System Figure 33 Lobe Coard. System ~
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~ ~1 6 Table 31 Cam Profile Nomenclature J Subscriptr on quantities tha t vary with the lobe rotation angle 8 N Number of discrete points used to specify follower lift (generally 360 ) r i Radius of follower pitch curve at point j r i Radius of follower pitch curve for alternate follower or cutting t ool at point j rb Base circle radius r t Roller follower radius r t Alternate follower or cutting t ool radius rpocj Radial distance from lobe axis to point of contact j S i Lift for follower with radius rf at point j S i Lift for alternate f ollower with radius rf at point j x i X coordinate of point of contact of follower and lobe at point j X Ci X coordinate of follower center at point j x :i X coordinate alternate follower center at point j X Lobe coordinate system x axis ex Camshaft coordinate system x axis Y i Y coordinate of point of contact of follower and lobe at point j Y Ci Y coordinate of follower center at point j y ; i Y coordinate alternate follower center at point j Y i Y coordinate of point of contact of follower and lobe at point j y Lobe coordinate system y axis cy Camshaft coordinate system y axis cz Camshaft coordinate system z axis C 2 m i Axial position of feature (bear i ng or lobe) xi: xis the feature type ; i is the feature index (X i Lobe contact angle for point j c
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17 ce1 Figure 34 Camshaft Coordinate System: CW and CCW Convention .. . { _,;t\ Direction Direction of Rotation of Rotation X X Figure 35 Lobe Coordinate System: Nose 180 Convention
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\ Direction of Rotation of Rotation Figure 36 Lobe Coordinate System: Nose Convention Closing side Opening side (Opening side) (Closing side) Direction of Rotation / ( Direction of Rotation) Lift Data Table: Exhaust . ...... rb = 13.2081 r 1 = 10.8585 ~ i .. (c:l ~g} .i ..... ~ i ... (r:ri.r:ri} .. : ..... s j (r.r.i rnL ....... .. ...... ........ : . . 9Pf3. rii lJ9 ..... ...... c:.19.i;;i_r:1g ...... i _(~lc,sing) ( opening) 0 8.0264 8 0264 1 8.0242 8.0242 2 ii : a1 ~io1 is .... a : 0064 8 : 6cie 4 '"'' 1 ggoa 1: gg os 1 sioa 1:9ias 7 9464 7.9175 ....... 7 8841 7 8463 ......... .. ............... .. .. 7 8040 Figure 37 Typical Manufacturers Cam Lift Data Specification ~1 8
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c1 9 Figure 34 shows the camshaft coordinate system for both clockwise ( CW) and a counterclockwise (CCW) directions of rotation F i gure 35 and Figure 36 show the lobe coordinate systems used to program machine tools Lastly Figure 37 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 35 and Figure 36 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 31 These transformations are used to calculate the commanded motion of the grinding wheel and milling cutters and also to compare inspection results performed with
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a different follower radius In the following development all parameters are specified in the lobe coordinate system 20 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 du r ing the grinding process w i ll 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 rr to Roller Follower with Radius r; From Figure 38 the radial distance from the cam axis of rotation to the 14 follower center is Equation 31 where rb is the base circle radius rr is the follower radius and s i is the follower lift at point j The curve described by the r adial distance ri, is known as the pitch curve _J
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~~follower translational axis* ., follower t r anslational ax i s '"\ .. r r Direction of Rotation Figure 38 Translational Roller Follower Coordinate Transform The radial distance r i, is resolved into XY components in the system attached to the cam lobe as x ci = r i cos(8 ) Equation 32 Equation 33 21 where xc i and y c i are the XY coordinates of the center of t h e follower From Figure 38 a right triangle is formed by the line segment from point x ci, y ci to the
PAGE 36
point of contact x i Yi and this segment s projections onto both the X and Y 1 6 axes The segment is of length r r and for this triangle Equation 34 The partial derivative of Equation 34 with respect to e is 8 F(x,y 0) =2( )dx ci 2( )dy ci = 0 ae x i X e i d0 y i y c i de Equation 35 Equation 34 and Equation 35 are solved simultaneously for the inner cam envelope to determine the XY coordinates o f the point of contact as Equation 36 Equation 37 The derivative of Equation 32 with respect to e is Equation 38 22
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23 The derivative of Equation 33 with respect to 8 is dye dr dS 1 = ri cos(8i) + d~ sin(8i) Equation 39 dr. For the discrete data, the derivative, d~ is approximated from difference equations. Often, ri is not known at evenly spaced points of 8 i and the divided difference formulation must be used. In other cases, ri 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 dr 1 r 1 + 1 r 1 1 d8::::: 8 8 j+1 j1 Equation 31 O The first difference equation for N evenly spaced points is Equation 311 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
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d 2 r 1 [ r 1 r r r 1 ] ) ) ) de 2 e 1 e 1 e 1 e e e 1 J J J J J J Equation 312 For N evenly spaced points, the second difference is Equation 313 24 The polar coordinates of the point of contact is Equation 314 24 and the two argument inverse tangent function is used to guarantee the correct quadrant for a i as Equation 315 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 rt. The XY coordinates of the center of the cutting tool or follower are 14 rt ( ) x. = x +r cos(8 )x. CJ J r J J J f Equation 316 ~~
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Equation 317 The transform from Cartesian to polar coordinates i s r i = ( x : i) 2 + ( Y: i) 2 Equation 318 Again the two argument i nverse tangent function is used to guarantee the correct quadrant and 8 ; i s Equation 319 Flat Follower to Roller Follower 25 For completeness t he transform from flat follower to roller follower motion coordinates illustrated in Figure 39 is presented. For a flat follower the XY coordinates of the point of contact are Equation 320 Equation 321
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The coordinates of the follower center are Equation 322 and Equation 323 And as before Equation 318 and Equation 319 are used to transform the results to from Cartesian to polar coord i nates. Roller Follower Translational Axis Flat Follower Translational Axis Direction of Rotation Figure 39 Coordinate Transformation: Translational Flat to Roller 26 ~
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Roller Follower to Flat Follower The transformation from a translating roller follower to a translating fl at follower is now developed. From Figure 310 Equation 324 and as before Equation 325 Where, as before the coordinates of the cam surface are calculated from Equation 36 and Equation 37 Again from Figure 3 10 r / is Equation 326 Finally, fo r the f l at follower the lift is Equation 327 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 e i ther direction Similar expressions for other types of 14 cams are given by Chen
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Roller Follower Translational Axis Flat Follower Translational Axis Direction of Rotation \ _.,,,,,,,. Figure 310 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 310 Equation 328 and, as before Equation 329 28 I J
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Where as before the coordinates of the cam surface are calculated from Equation 36 and Equation 37 Again f rom Figure 310 ri is Equation 330 Finally, for the flat follower the lift is Equation 331 29 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 8 i, the radius of curvature for the surface of the cam profile at the 16 point of contact j, is 3 [ 2 ( drpoc ) 2 ] 2 rpoc + d i a Equation 332 I i I I __ ___ J 
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30 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 312, the first divided difference is r r poc j + 1 poc 1 1 a, j + 1 a, j 1 Equation 333 and from Equation 313 the second divided difference is Equation 334 Substituting the difference expressions into Equation 332 gives Equation 335 In the case where the data are known at evenly spaced points Equation 335 becomes ~~ ~ ~
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[ ] 2 r r 2 2 poc H poc j 1 r poc + 2 rpoc J 1t J 2 N Equation 336 rpoc 2rpo c + rpoc J 1 J J 1 ( ~r Interpolation 31 When transforming motion coord i nates from one follower to another e ; i s not in general equal to ei. The difference is illustrated in Figure 38 F i gure 39 and Figure 310. Consequently the calculated values of ri do not occur at evenly spaced integer degree values of e; To use the va l ues of r i convenien tl y in programming most machine tools it is necessary to interpolate ri to integer degree spacing of e ;. The interpolation is generally performed using cubic 1 8 25 26 splines as follows The piecewise cubic interpolating polynomial is r.(e) = c, + C2 (88 ~)+ C3 (88~)2 +C4 (8 8~) 3 J J ,J J ,J J ,J J Equation 337 where Equation 338 C2 = L J J Equation 339
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~ 32 Equation 340 Equation 341 Equation 342 Equation 343 The vector L is the vector of free slopes and has N rows For cubic spline interpolation of periodic data 2( .'.\8~ 1 + il8~ ) 118~_ 0 0 L\8 ~ L o b o 110; 2( L\8~ + L\8; ) L\8~ 0 0 L b 0 '18; 2 (Ll0; + ~e; ) ~e; 0 = 0 0 L\0~ 2 2( .'.\8~ 3 + il8~ 2 ) L\8~ 3 2 ~ 2 L\8~2 0 0 L\ 8 ~ 1 2( .'.\8~ 2 + L\0~ ,) bN 1 Equation 344 where Equation 345 ~~
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33 Symbolically L i s solved for as L o 2( ~0~ 1 + 10~ ) '10~ 1 0 0 ..'l0~ 1 b o L ..'i8; 2( ~0~ + '10 ; ) ..'l0~ 0 0 b, 0 e 2 2 (~e; + ..'le; ) '10; 0 = L N2 0 0 ..'i0~ 2 2( ~0~ 3 + e ~ 2 ) ..'l0~ 3 ~ 2 ~ 1 ~e~ 2 0 0 !i0~ 1 2( ~0~_ 2 + ~ e~ 1 ) ~ 1 Equation 346 The matrix of 0 ; 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 ShermanMorr i son formula The ShermanMorrison 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 1 i, c 2 i c 3 i and c 4 i are calculated and substituted into Equation 337 to calculate the interpolated value of r ; (8) For common cam geometries cubic spline interpolation re t urns 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 ; (8) to integer degree spacing, then transforming the grinding wheel motion back to the design follower motions and then interpolating r i (8) to integer spacing
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i l __ 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 rotat i on 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 inspec t ion 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 i s the error predicted by the gr i nding model. Th is mode l is developed i n Chapter 5 34
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L 35 Camshaft Geometry and the Gr i nding 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 in t erlobe t iming, 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 essent i ally a clamping error and not related to the accuracy of the mach i ne 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 i n profile
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36 grinding since the radia l grinding force varies w i th the r adius of curvature of the cam profile This variation i n force leads to a variation i n the part deflection within a single revolution and consequently a profile error Therefore to minimize the deflection t he 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, i t 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 i s shown in Figure 41 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 addit i onal 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 i s 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 41. 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 41 ~
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37 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 41 Timing Reference Locating F i xture Three Jaw Chuck Grinding I Wheel Grinding Wheel Direct i on of Rotation ==} Part Direction of Rotat i on Figure 41 Steady Rest and Three Grinding with Jaw Chuck: AJAOR Camshaft Geometry and the Inspection Process To analyze geometr i cal 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. __ __J
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3 8 Table 41 Machining and Inspection Axes of Rotation Centers Axis of Rotation (CAOR) Machining Axis Of Rotation MAOR Continuous axis defined by fit of part and grinder centers Adjacent Segmented axis defined adjacent Journal journal bearing surfaces steady Axis of rests and chuck or second center. Rotation (AJAOR) End Journals Axis of Rotation (EJAOR) with three jaw chuck with second center Continuous axis defined by end journal bearing surfaces steady rest and chuck Inspection Axis of Rota t ion IAOR Continuous axis defined by the fit of part and gauge cente r s Segmented ax i s defined by the best fit of circle to adjacent journal bearings. Continuous axis defined by t he best fit of circle to end journal bearings The coupling of the l i ft and timing measurements is readily apparent from Figure 34 and Figure 36 In Figure 34, 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 36 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 t i ming and profile measurements. In this chap t er an alternative method for reducing the 7
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i nspection data into timing and li ft components i s presented Thi s proposed method i s shown to more effectively d ecoup l e l i ft and timing errors in the presence of nonconcentr i city errors 39 Prior to demonstrating this method the basic equation to calculate the lobe t i ming is developed through analogy with the standard method of determining the timing or p hase of a c i rcular feature: Fourier analysis 2 8 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 c i rcle 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 f rom the measured lift values so that the average l i ft 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 c i rcle center with a reference axis. The axes referred to in this section are summarized in Figure 41. 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). 
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40 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 i n Figure 35 and Figure 36 the cam l i ft 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 nomina l 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 d i rection. This is true since the geometric veloc i ty 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 L _________
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41 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 concentr i c 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 42. 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 . 
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42 Table 42 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 ei, 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 8 k Phase Shift (88) ,... LSS estimate of k 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 Qp Least sum of the squares center of a round part rb Base circle radius r, Follower radius ri Radius of follower pitch curve at point j Measured radius of follower pitch curve at point j ri R Nominal radius for round part inspected with flat follower. Biased estimate of radius for nonconcentric round part inspected w i th radiused follower Si Follower lift MeasS i Measured follower li ft upr Undulations per revolution V Geometric velocity X X axis of lobe coordinate system X axis of qauqe coordinate system Xp X axis of part coordinate system y Y axis of lobe coordinate system YG Y axis of qauqe coordinate system Yp Y axis of part coordinate system 8ri, Roundness error for circular part ( same as e) 88 Phase shift in 8i 80 LSS estimate of phase shift in 8i Lobe timing angle (also used for follower pressure angle in Figure 414 and associated equations and discussions) 0i Lobe angle of rotation
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Timing Reference Pin Axis of Rotation Figure 42 Inspection of Nonconcetric Nominally Round Part The Fourier series is 28 N 1 (2 N 1 (2 ri = R+ ~an cos : j n) + ~bn sin : j n) Equation 41 43 Direction of Rotation Shown at 8 1 = o 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 42. For a flat follower, the DC term of the Fourier series is the average part radius, R, and is defined as
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44 Equation 42 The cosine coefficients for n = 1 to N1 are Equation 43 The sine coefficients for n = 1 to N1 are Equation 44 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 41 Equation 43 and Equation 44 become Equation 45 Equation 46 Equation 47
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~~ ~ .. 45 where a1 and b 1 represent the Xp and Y P components of the nonconcentricity respectively These components are illustrated in Figure 42 For t he phasing of the nonconcentricity shown the fo ll ower motion is the sum of a sine wave and a constant as shown in Figure 43 as the design curve. For clockwise rotation, the angle 8 is measured counterclockwise in the part coordinate system For the configuration shown R is the part radius a1 is zero and b1 is equal to the magnitude of the nonconcentricity If a phase shift of 88 degrees is introduced as shown in Figure 43, then both a1 and b1 are nonzero. If this phase shift is considered to be an error then the error in the follower motion or is ( 27t 1 or i = a, cos NJ ) Equation 48 Equation 49 and r i is defined as the measured value of r at angle 8 while r i is the nominal value at angle 8 i (i.e the value of r with no phase error) The effect of this e r ror is illustrated in Figure 44 for 88 equal to ten degrees From Equation 48 or is recognized to be of the form Equation 410
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46 d r This is true since the cosine is the derivative of the sine The term de 1s generally called the geometric velocity ; Also if the inspection data for the phaseshifted part r* is compared with the nominal data then for a pure phase error of 88 expressed in degrees r* =r oe 360 J J +N Equation 411 i i i i I I Direction of Rotation Shown at e i = o ~1 l, I Figure 43 Introduction of Phase Shift :t: dr dr drd e The term geometric is used to distinguish this term from time velocity elf which is elf =
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. Follower Motion due to an Eccentric Q) 0.8 s:: 0 6 0 0.4 0 u. C 0 2 Design 11/easured Error 'J IJ ,'' I, .. ,. ... 0 .Q O ...... ... .. ,,._ . Q) N 0 =::'.? 0.2 t'a .. ~ E 0.4 .... 0 0 6 z 0.8 1 180 150 120 90 60 30 0 30 60 90 120 1 50 180 Angle (degrees) Figure 44 Measurement of a ri Phase Error for an Eccentric Substituting Equation 411 into Equation 49 gives ori = r 68 360 ri J+N Equation 412 The first order forward first difference equation 29 for point 8 is r 68360 ri J + N88 Equation 413 47 Rearrange Equation 413, substitute it into Equation 412 and solve for 88 to get or 60 = (:iJ Equation 414 ~,
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~ 48 Where for the idealized case this expression is invariant over the range of 0 except in the region were :; is zero and thus 0 8 is undefined. It is important to remember that the form of the error term o r i s restricted to that of a one undulation per revolution sine wave which is the geometric velocity If the f orm 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 cSr(v)i = dS k Equation 415 where or(v) is the component of the measured error that has the form of the A 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 as 30 Equation 416 Lastly substitute the expression for or (v)i given in Equation 415 for o r i in Equation 414 and solve for 08 to get 50 = k Equation 417 I I 
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Therefore the timing error 8 8 is N 1 dr . L d8 ( r i r i ) 80 = J = _O ~[~ ] 2 j= O d0 Equation 418 where 80 is the least sum of the squares estimate of the parameter 80. I t is important to note Equation 418 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 415 with Equation 48 and recognize that for an ideal round feature k and ~; in Equation 4 15 correspond to a1 and cos ( 2 : j) respectively in Equation 48 49 Obviously, if all that is desired is to determine the timing of round part features, then nothing is gained from Equation 414: the timing error can be more directly determined using the Fourie r transform. However Equation 414 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 45 are analyzed. Figure 46 shows the Fourier transform of the followe r lift data. From this figure it is clear that a broad frequency spectrum approximately 30 harmonics or undulations per revolution (UPR) in this case i s
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I 50 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 no t possible to directly infer the t i ming of the cam lobe from the phase of the Fourier transform. Therefore, the assumptions that lead to the development of Equation 45 for c i rcu l ar features are invalid for noncircular cam profiles However since Equation 418 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 7 /.., '\ 0 15 I \ 1' I follower radius= 19.00 mm 6 0 1 "" I C) I Q) 5 0 05 32 E I E E E 4 ___ ... 0 ,= 0 05 '5 :::J 3 I Lift 0 2 I 0 1 Q) Velocity I > I 0 15 0 +..;"',+++___;;'+=+++'0 2 180 1 50 120 90 60 30 0 30 60 90 1 20 150 Angle (degrees ) Figure 45 Typical Cam Lobe Lift and Geometric Velocity Again consider the cam profile and ideal inspection process shown in Figure 47 The follower lift and geometric velocity for a roller follower with 88 equal to zero are shown in Figure 45. For consistency with the previous discussion, a flat follower is considered here. This presents no difficulty s i nce, as discussed in the Chapter 3, the radial follower cam motion specification given in Figure 45 may be readily transformed to a flat follower motion specification
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1 8 ,.. 1 6 E E 1.4 .._,, 1 2 C: Q) C: 1 0 () >0 8 u C: 0 6 Q) ::J g 0.4 'LL 0 2 0 0 11 11 I I I I I I I I I I I \ I 30 Harmonic Content for Exhaust Lift Data 60 Frequency Content Scale on left y axis Frequency Content Scale on right y axis 90 Frequency (UPR) 120 150 Figure 46 Frequency Content for Typical Exhaust Cam Profile 51 0 01 0 008 E E .._,, C: Q) 0 006 c: 0 () 0 004 >, u C: Q) ::J 00 002 Q) 'LL 0 180 To develop the timing measurement algorithm, again consider the effect of introducing an arbitrary timing error, 80, into inspection data for a cam lobe is considered. This situation is illustrated in Figure 47 Timing Reference Pin Figure 47 Cam Lobe with a Timing Error of 60 Direction of Rotation Shown at ei = 0
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L~ 52 For an ideal lobe with an error only in phase o f 80, from Equation 412 the measured lift error is 8 r i = r oe. 360 r i J N Equation 419 Now however, the form of 8r is not a single sine wave but rather i ts form is a function of the nominal data : specifically the error has the form of the geometric velocity which is shown in Figure 45 Recall that for the eccentric the form of r is that of a sine wave and the error term associated with a phaseshift is a cosine wave or simply the derivative or geome t ric velocity. Therefore while Equation 45 is invalid for noncircular features Equation 418 is valid independent of the form and the timing error for the lobe is 80 = j~1 [~]2 i = O d0 Equation 420 Finally, it should be noted that Equation 420 requires that r 60 360 ri J +N approximate :; well. If 80 is large then this approximation is poor and Equation 420 must be solved iteratively Still even for larger angle errors the solution converges rapidly. For typical angle errors r oe. 360 r i approximates J +N dr de well and no iterations are required. J
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L 5 3 Comparison of Methods T he actual value of timing angle calculated from Equation 420 depends on the method used to apply the equat i on to the i nspection data. Since this research is primarily concerned with understanding and controlling the manufacturing process t he 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 phaseshifted 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 t he proposed method no increased risk of accepting bad parts exists since the reported lift error will always be larger t han or equal to the error reported by the exis t ing method Timing errors may be larger or smaller depending on the relative phase and magnitude of the various lift error components. The s i mulated lift error is for the sl i ght l y asymmetrical exhaust lobe shown in Figure 45 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 48 and represents the superposition of the following components : a 20 m residua l sparkout e r ror, a 10 m nonconcentricity error and a 0.4 phase shift or timing error. The residual sparkout error grinding model used here to simulate the lift error is presented in Chapter 5 ~ ~ ~ 
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5 4 Effect o f T i m i ng Sparkout and Nonconcentr i city Errors 0 08 0 06 0 04 ,..._ E 0 02 E 0 ... 0 ::: 0 02 w 0 04 In cludes liming Erro r Effect 0.06 Exc l udes liming Error Effect 0 08 180 1 50 12 0 90 60 30 0 Angle (degrees ) Figure 48 Simulated Cam Lift Error From Figure 48 it is clear that when a significant timing error exis t s 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 420 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 A l gorithm 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 i s not removed prior to calculating the timing angle Nonconcentricity errors common l y account for 50% of the total lift error on high precision CNC cam profile gr i nde r s and the timing bias due to nonconcentricity can significantly distort the form of the reported lift error as shown in Figure 49 ~
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Profile Error Reduction : Existing Method 0. 02 ~,,,, 0 015 0 01 E E 0 005 ._,. .... e oF+_c_~=~H1~ uj 0 005 0 01 Existing Wethod Error ', ' : 0 015 '' .' _______ __. 180 1 50 1 20 90 60 30 0 30 60 90 120 150 1 80 Angle (degrees) Figure 49 Existing Data Reduction Method for Profile Data 5 5 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 nomina l data If the timing angle calculation i s 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 49. For the simulated inspection data shown in Figure 48 the timing angle error is calculated as 0.3423 Recall that the simulated angle error is 0.4000 and thus the timing bias is 0 0577 This bias is due to several independent factors summarized in Table 43 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
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56 nonconcentricity error The effect is greatest for the phasing shown in Figure 410. Here the nonconcentricity error and the timing error are both odd functions and have similar forms Equation 420 can be used to calculate the component of the nonconcentricity error that is of the form of timing error :; This component is the timing bias of the nonconcentricity error for the existing method and is 0.0540 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 410 Due to this asymmetry the sparkout error has a dr component of the form de For the simulated data the t i ming bias due to the sparkout lift error is calculated from Equation 420 as 0 0037 Thus, the tota l 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 420 Profile Error due to Timing and Nonconcentricity Error 0 02 ,, 0 015 0 01 E S 0.005 .... 0 0 .... .... w 0 005 0 01 Timing : j Nonconcentricity 0. 015 .L..::::::=======__J 1 80 1 50 120 90 6 0 3 0 0 Angle (degrees) Figure 410 Form of Timing and Nonconcentricity Errors
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57 Proposed Data Reduction Algorithm Figure 411 shows the reduced profile error for t he 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 80 and added back to the profile error. So, for the data shown in Figure 48, t he nonconcentricity error is removed and the t i ming error is calculated from Equation 420 as 0 3899 While the bias is significantly reduced it is still nearly three times larger than the expected sparkout bias of 0 0037 which remains unchanged. The remaining 0.0064 bias occurs since the nonconcentricity error is not actually a s i ngle sine wave with a frequency of one undulation per revolution but rather it is a s i ne 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 i naccurate 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 overconstra i ned during cam profile grinding due to the combination of the centers and steady rests as shown Figure 41 In this overconstrained 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
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types of nonconcentricity error produces a different distort i on and again t hese sources are considered separately below Profile Error Reduction 0.02 0 015 0.01 E E 0 005 .._, .... 0 0 .... .... w 0 005 Reduced Data 0 01 Error 0. 015 '' 180 15 0 1 20 90 60 30 0 3 0 60 90 1 20 150 1 80 Angle (d egrees) Figure 411 Proposed Data Reduction Method for Profile Data 5 8 The first nonconcentricity error described above the runout of the part during cam profile grinding, produces a nonconcentricity error that i s a sing l e sine wave superimposed on the commanded nominal mot i on of the grinding wheel. However when th i s nonconcentricity error is expressed in terms of the follower motion i t is no longer exactly a sine wave but rather it is distorted in the region of the flanks as shown in Figure 412 The distortion of the nonconcentricity error in the follower motion coordinates i s of the form of the timing error~; From Equation 420 the angle bias due to this error is calculated as 0 0064
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Grinding VVheel Axis of Rotation and Nonconcentricity Errors 0 01 0 008 0 006 0 004 E 0 002 E 0 .... 0 0 002 .... .... ;"', ._ ..... .,UJ 0 004 Grinding wheel motion coordinates 0.006 Follow er wheel motion coordinates 0 008 Difference 0 01 .L==============::...~.....l 1 80 1 50 120 90 60 30 0 30 60 90 120 150 1 80 Angle (degrees) Figure 412 Effect of Nonconcentricity Error in Grinding Wheel Motion on Follower Motion 59 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 413 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 413 is 0 0389. The timing angle bias with the base circle nonconcentricity is 0 0054 These results are summarized Table 44
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0. 0 1 0. 00 8 0 006 ,.. 0 0 04 E 0 00 2 g 0 ._ 0 ._ 0 002 ._ w 0 004 0 006 0 008 0 01 180 Effect of Journal Bearing Nonconcentricity Error Nonconcentr i city ... . Sine w ave Res i dual 1 50 1 20 90 60 30 0 Angle ( degrees) Figure 413 Effect of Journal Bearing Nonconcentricity Error 60 In conclusion from the results summarized Table 43 and Table 44 the nonconcentricity error timing angle bias i s reduced sign i ficantly 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 43 Timing Angle and Bias Results for Type I Nonconcentricity Error Summary of Results for Simulated Timing Error Existing Proposed Method Method Simulated Timing Error 0.4000 0.4000 Calculated TiminQ AnQle Error 0.3423 0.3990 Sparkout Error Timing Angle Bias 0 0037 0.0037 Nonconcentricity Error Timing Angle Bias : Source I 0.0540 0 0064 Total T i ming Angle Bias 0.0577 0 0101 Sum of Timing Angle Error and Bias 4 0000 4 0000
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61 Table 44 Timing Bias Results for Type II Nonconcentricity Error Summary of Bias due to Source II Nonconcentricity Error Existing Proposed Method Method Nonconcentricity Error Timing Angle Bias : Source II 0 0389 0 0054 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 nonconcentric i ty 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 ~
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J Y, Direction of Rotation 62 Figure 414 Inspecting a Nonconcentric Surface with a Radiused Follower Figure 414 shows a nominally round part that is rotated about an inspection axis, 0 0 which is not coincident with the actual part center, Op. From Figure 414 the vector loop equation for the outlined triangle is written in Xo and Ya components as Equation 421
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. , Equation 422 Equation 421 and Equation 422 can be comb i ned to get r / 2 ( a 1 cos(8 i) + b 1 sin( 8 i ) ) r i + ( a 1 cos( 8 i) + b 1 s i n(8 i ) )2 + ... ( a 1 sin(8 i ) + b 1 cos(8 i ) }2( r b + r ti ) = O Equation 423 6 3 Equation 423 is quadratic in r and selecting the appropriates sign the so l ution i s Equation 424 This can be written in terms of the relative follower displacement s as Equation 425 Expanding the second term under the radical and simplifying gives Equation 426 Therefore if the coeff i cients a 1 b 1 and rb in Equation 425 are know then the contribution of the nonconcentricity error to the i ndicated relative follower displacement s can be calculated and then subtracted from the indicated values of the follower displacement s to determ i ne the actual roundness error _.!
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Also since the part is not perfectly round a measured d i splacement M eass 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 Meass as 2 8 Equation 427 64 Typically the sum of square of the error e is minimized to determ i ne the coefficients in Equation 425. Some authors 31 propose a general least sum of the squares solution to determine a1, b1 and rb using the NewtonRaphson iterative method However this is not generally required for two reasons that will be demonstrated First the unique form of Equation 425 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 45 which expresses the displacement for a flat follower displacement s (l ift), to estimate the follower motion for a radiused follower Equation 45 is repeated in a slightly more convenient as Equation 428
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6 5 Two fu nctions f(8) and g( 8) are said to be orthogonal over the range from a to b i f 1 0 fat(8)g(8)d8 = 0 Equation 429 Equation 426 can be rewritten as s(e) = f(8) + g(8) Equation 430 where f(8) = a 1 cos(8) + b 1 sin(8) Equation 431 and Equation 432 Substituting Equation 431 and Equation 432 into Equation 429 integrating over one rotation of the part gives s: 71 ( rb rf + rrf ( a ~ + b~ + a,b sin(2e) ) (a 1 cos(8) + b, sin(e))de = 0 Equation 433 for all values of a,, b1 rr and rb Thus the functions f(8) and g(8) 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. 1 Consequently the coefficients a1 and b1 that
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, . .. ____ _____ L give the least sum of the squares best f it of the i nspection data to the follower displacement s can be determined direct l y from t he least sum of t he squares best fit of the i nspection data to Equation 428 Which is of course just the Fourier coefficients of the fundamen t al f requency 6 6 With a1 and b1 known the problem is reduced to determining the bes t fit value for the base circle radius rb First i t i s noted that g(0) is composed on a constant and a sinusoid. Consequently i t 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 i s just the sum of g(0) over a single rotation or Equation 434 With the bias know the base circle can be calculated from the DC Fourier coefficient R as rb = R + bias Equation 435 where R is defined in Equation 42. And fi na ll y Equation 436
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The problem of reducing inspection data for a nonconcentric nom i nally round part is now reduced to formulating the l east sum of the squares so l ut i on to Equation 436 for a single parameter rb 67 Finally, the second pointthe roundness error i ntroduced due to the flat follower assumption is small for typical nonconcentricity errorsis considered. Recall that Equation 428 and Equation 426 give the follower mot i on for flat and a radiused follower respectively. If the relationship for the flat follower, Equation 428, is used to approximate the motion of the flat follower that occurs due to a nonconcentric round partexactly represented by Equation 426then an approximation error in the lift s will result. By comparing these two equations the approximation is recognized to be negative g(8). 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 432 A value of base circle nonconcentricity of 0.014 mm with an arbitrary phase angle of 45 is selected This gives a 1 and b 1 both equal to 0 010 mm Additionally, a follower radius r t 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 415 Clearly, the roundness error introduced is insignificant: the bias for rb i s approximately 1 o s 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.
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~ .... e .... w C: o E :.:: E ro....,, E w Error Using a Flat Follower Approximation to Identify Nonconcentricityfor Parts Measured with a Radiused Follower 2 0E06 1. BE06 1 6E06 1.4 E06 1 2E06 1 0E06 8.0E07 6.0E07 4.0E07 2.0E07 0 OE+OO ~_ _ _ _ _ _ _ _ ~ 0 015 Estirration Error ,,. , Nonconcentricity ,/ ,,:' :' I \ .: 0 01 E E 0 005 ._ :;::' Q) C: 3: Q) o .2 E 0 0 005 u.. a. (/l 0.01 0 L ______ ~::;:::1C_..______ _: :.:.., ......:.: .,":;_' _j_ 0 015 0 30 60 90 120 150 180 21 0 240 270 300 330 3 60 Theta (degrees) Figure 415 Flat Follower Approximation for Nonconcentricity Errors 68
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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 contro l 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 7 The general cylindrical plunge grinding model is described by Hahn and 32 Lindsay and is represented in Figure 51 Feed Slide ~ s Vs Machine Bed Figure 51 Cylindrical Grinding Model Modified for Cam Profile Grinding Adapted from Principles of Grinding Lindsay R P ., Handbook of Modern Grinding Technology. Editors : King R. I. Hahn R S New York : Chapman and Hall 1986 69
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... 7 0 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, i s defined as Equation 51 32 The general grinding relationship is Equation 52 where v w is the rate of decrease of the workpiece radius, vs 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 51 The basic grinding model assumes that the workpiece and the grinding wheel mutually machine each other However for the cubic boron n i tride (CBN) superabrasive grinding wheel used in this research the rate of decrease of the grinding wheel radius vs is negligible. The rate of decrease of the workpiece 32 radius which is analogous to the material removal rate is give by Lindsay as Equation 53 _ _J
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L 71 where wrp is the work removal parameter F ~ is the normal grinding force per unit width and F ; h 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 O w is the diameter of the workpiece The work 32 removal parameter wrp is given by Lindsay as wrp= [ v ] 3 ~ 9 [ 2C ] 1 11 k _y_ 1+ L 1 19 V wrp v 3L s s 43 / 5 ,: 27 / o 1 304 volo 41 d 1 3s R 19 e C Equation 54 where the equivalent diameter, 0 e, for external cylindrical grinding is 0 = _o_w_o_s_ e Ow Os Equation 55 where the plus is for convex surfaces and the minus is for concave surfaces 3 2 Also according to Lindsay the threshold force is approximately related to the workpiece equivalent diameter as Equation 56 where kth is a constant related to the tendency of the grinding wheel and workpiece material to rub instead of cut. ~r~
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7 2 Table 51 Grinding Model Nomenclature AOR Axis of rotation a a i, a i. P Depth of cut per revolution C Diametral depth of dress d Grinding wheel abrasive size De De i Equivalent diameter Os Grinding wheel diameter ow Workpiece diameter fp Wheelhead infeed F~, F~iJl Normal grinding force per unit width F;h F;h F;h J J,P Threshold grinding force per unit width F~n, F ~ n Difference between normal and threshold grinding forces J P j Subscript indicating discrete points of contact at angle 8 i ka Stiffness of grinding wheel and workpiece at contact zone kc Combined sparkout constant ke Effective cutting stiffness of the grinding system kl Sparkout constant for linearized grinding model ks Stiffness of the wheel head k,h Threshold grinding force constant kw Stiffness of the workpiece supports kwpr, k~ Work removal parameter constant lumped constant L Grinding wheel dress lead Nw,Nwp NW jp Nominal angular velocity of workpiece for pass p NWjp Instantaneous angular velocity at angle e i on pass p p Subscript indicating grinding pass (rotation) number rgw Radius of grinding wheel ri Nominal radial distance from follower center to AOR at point j riJl Commanded radial distance from follower center to AOR at point i for pass p ri,p Commanded radial distance from follower center to AOR at point j for pass p act r Actual radial distance from follower center to AOR at point j J, P for pass p r r j, rj.p Rate of deflection of the grinding system
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7 3 Table 51 Continued rpoc j Distance from camshaft axis to point of contact at angle 8 i Re Rockwell hardness of workpiece Rs Grinding wheel radius Us' Us X position of grinding wheel AOR in machine coordinate J, P frame uw,u w. J, P X position of cam lobe AOR in machine coordinate frame Vr Feedrate of the cross slide vff vff p Finish gr i nd feedrate Rough grind feedrate vrr vr r ,p vsor Sparkout feedrate ( equal to zero by definition) v s Rate of decrease of the grinding wheel radius vs Surface velocity of the grinding wheel Rate of decrease of workpiece radius Vw Vw Vw J J P vw' VW j Velocity of the point of contact on the workpiece vol Volume factor for grinding wheel w i Normalized angular velocity of workpiece at angle 0 i wrp wrpi Work removal parameter WrP in wrpo X X axis: lobe coordinate system y Y axis: lobe coordinate system XM X axis: Machine coordinate system YM X axis : Machine coordinate system ~ai Change in lobe contact angle a, for point j ~e j Change in lobe angle 0 for point j ai Lobe contact angle for point j ai Time rate of change of a i t. r l P Error in commanded grind i ng wheel position Yi Grinding wheel contact angle for point j Pi Radius of curvature of workpiece at angle a i 't 't j Time constant of grinding process at angle a i ej Lobe rotation angle for point j ej Time rate of change of 8 i
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7 4 32 Finally from Lindsay t he depth of cut per revolution is called a and is defined as Equation 57 where N w 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 cylindr i cal 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 e i The second modification is to introduce the subscript p to for parameters that change depending on t he pass or rotation number. The third change is to replace diametral values with the i r equivalent radial representations Th i s is done since the diameter' of a cam profile is not particularly meaningful. However, the radius is directly analogous to the radius of curvature P i or the radia l distance to the point of contac t, rpoc i" Fourth multiple constants in expressions are lumped to simply t he rela ti onships The grinding model is shown in Figure 52 with the subscripted quantities
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, rgw 1/ D i rect i on of Rotat i on Figure 52 Cam Profile Grinding Model y Direction of Rotation W i th these modificat i ons Equation 52 through Equation 57 become Equation 58 Equation 59 k V0 1 6 wrp w j p 0 0 14 e l Equation 510 Equation 511 75 I I __J
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. . . 7 6 Equation 512 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 5 pi
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77 Also recall from Chapter 3 that the r adius of curvature of the cam surface at point of contact j, is ( ] 2 [ ( ]] r r r r r r 2 poc j + 1 p oc j 1 1 poc i '1 poc j poc j poc j 1 rpoc + 2 rpoc J a. 1a. J a. 1a. a. 1a. a, a, 1 J + J J JJ + J J J Equation 514 Since the radius of curvature of the cam profile is infinite at points of inflection it is noteworthy that Equation 512 is well behaved at such points and consequently, the grinding model does not break down This is verified by taking the limit of Equation 512 as the radius of curvature of the surface pi, goes to infinity as Equation 515 The limit is determined by dividing all terms in the numerator and denominator by pi to get Equation 516
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Simplifying Equat i on 51 6 gives Equation 517 or that l im D 8 i= 2 R s pj 00 Thus the limit is finite and the model is therefore well behaved 7 8 Equation 513 must be modified to account for the fact that in cam profile grinding the angular velocity of contact cii, shown in F i gure 5 1 is not equal to the angular veloc i ty of rotation ei. To i n clude this effect the ratio of the change /xJ., in the contact angle to the change in the rotation angle NJ '. can be introduced J In this case Equation 5 1 3 becomes Equation 518 Also the general industrial convention is to specify N w as J P Equation 519
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where w i i s the normalized angular velocity at l obe rotation angle ei, and N w,p is the nominal angular velocity for pass p Substituting Equation 59 and Equation 519 into Equation 518 gives the depth of cut as Equation 520 At this point it is useful to develop an expression for the velocity of the point of contact on the work surface From Figure 52 the velocity is Equation 521 The comparison of the denominator of Equation 520 with Equation 521 the substitution of Equation 510 for the work removal parameter wrp and combining constants yields aJ ,P == O 84 kwp O 14 (Fn. j",p kth i[)e ) V D '1 ,_,e ; w j p e j Equation 522 where of course Equation 522 is valid only when F~ k 1 h i s J, P '\J ,._, e ; nonnegative The normal grinding force F ~ i s induced by the wheel head J,P 7 9 infeed and the effective grinding system stiffness ke. The commanded grinding wheel position r i, P, is in error due to the system deflection caused by the l ~ ~_l
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... . . ~8 0 gr i nding f orce T he actua l v a l ue o f r iP i s designated ac\ P. F rom F i gure 5 1 and Figure 52 act A rJ.P = r J P + Ll r J. P = U w. U s. , J, P J, P Equation 523 The normal grinding force per unit width at lobe rotation angle e i, for pass p is F ~ = k e ~ r 1 P J, P Equation 524 where the system deflect i on is P P 1 ~ r i P = LfP La iP p = 1 p = 1 Equation 525 The substitution of Equation 525 into Equation 524 gives the normal grinding force as Equation 526 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 : t heir estimation is outside the scope of this work ~ 
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81 The f ocus here is to deve l op t he depth of cut model. U sing Equation 514 Equation 522 Equation 524 and Equation 526 i t is possible to si mu late the complete cam profile grinding operation if the values o f 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 mode l, t he 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 r elationship between the commanded infeed and the actual plunge grinding cycle for gr i nding is shown in Figure 53 In most cases the grinding cycle proceeds from a relatively high feedrate u 1 during rough grind to a slower feedrate of u 2 during finish grind and finally to a feedrate of zero during the sparkout phase E Typical Commanded and Actual Grinding lnfeed VSOf = 0 ke f th I 140r;::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..:::..::: =..__, ..... + 1 20 1 00 c onmanded r adial in f eed I effective radial infeed I .. 80 60 40 sparkout 2 0 147""' :....._ ___ roughing ___ _._,.. ___ finishing 0 .i,.q.......,..::. + ...+.......... f,'........... .,........ ,J 0 5 10 1 5 20 2 5 30 35 40 4 5 time (seconds) Figure 53 Typical Grinding Cycle: Rough, Finish and Sparkout Adapted f r om: Malkin S . G rind i ng T echnology : T heory and Applicat i on of Machining with Abrasives E llis Horwood Chichester UK 1989 50
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8 2 In the rough f i nish and sparkou t phases a steady s t ate error between t h e commanded radial distance and the effective rad i al d i stance ex i sts due t o th e deflections in the system I n the rough grind i ng p h ase t he steady s t ate error f or 3 3 a specific point j on the cam surface is Equation 527 In the finish grinding phase the steady state error is .i:irj p = finish s s = vf f "C j Equation 528 where 1 i is the time constant for the process at the po i nt of contact j Sparkout i s the dwell cycle that occurs at the end of the grind i ng operation At sparkout the feedrate is v f p is zero for all po i nt and the subscript j is dropped During the sparkout cycle the residual system deflect i on varies due to the variation in threshold force which var i es w i th the radius of curvature of the cam surface The residual error is the error that r emains when t he process reaches equilibrium and the depth of cu t a i.i is zero for a ll j i f p is suffic i ent ly large. Therefore f rom Equation 520 it i s c l ear that for zero depth of cu t F~ F ; h = 0 J,J))
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steady state conditions for t he f i rst order process From Figure 5 2 the deflection of the grinding system i n t he sens i t i ve direct i on at t h e completion of t he sparkout cycle i s F n J, P + "' ~ r j, p = s parkou t = k e F ;h j cos( y i ) = k cos( y i ) e Equation 531 Substituting Equation 51 0 into Equa t ion 531 gives ~ r j, p = sparkout = k 1 h c os( yi) Equation 532 Combining the constants gives Equation 533 83 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 i, is less than 3 5 degrees. Therefore it is appropriate to approximate cos(y i ) 1 Using this approximation and substituting Equation 515 into Equation 533 gives ~r j p = sparkout = k C Equation 534 2 l pj l R s l pj l R s 7
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And finally the effective radial distance to the follower center at the end of the sparkout cycle is Equation 535 84 From Equation 534 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 534 the residual sparkout profile grinding error can be simulated with k selected to provide a best fit with experimental data. Figure 54 and Figure 55 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 nonconcentricity and lobing effects, which distort the process error due to the data reduction algorithm are removed from the measured data. I I ~~_J
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85 From the results presented in F i gure 54 and F i gure 55 it is clear that the residual sparkout model successful l y 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 rocess and thereb the simulation of the control s stem. E E ,_ 0 ,_ ,_ w Comparison of Modeled and Experimental Profile Lift Errors O.D1 0.008 0 006 0 004 0 002 0 0 002 0 004 0 006 180 I IVodel I M:!as ured I ' ... ,. ,. ~'~ ,', .. : .IO ~ 1 50 1 20 90 60 30 0 ' ' ' ' ' ,_ ... ,, &.. ... . 30 60 90 1 20 1 50 1 80 Angle (degrees) Figure 54 Grinding Model Results in Grinding Wheel Motion Coordinates
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E S .... 0 .... .... w Comparison of Modeled and Experimental Profile Lift Errors 0 01 0 008 0.006 0 004 0 002 0 0 002 0 004 0.006 180 rvbdel . .. IVleasured 1 ., .. J ' . , .,. 1 50 1 20 90 60 30 0 ,. . r . . ~ 30 60 90 12 0 150 Angle (d egrees) 86 Figure 55 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 535 can be approximated as linear function by substituting the geometric acceleration of the commanded grinding wheel dr. 2 radial distance 1 2 for the radical term With this approximation Equation 5d0 35 becomes Equation 536 This equation may be rewritten in a form that is more convenient for later use as
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87 Equation 537 The agreement between Equation 535 and Equation 536 for grinding wheel motion coordinates with kc equal to 0 001 and 0.002 is shown in Figure 56 and Figure 57 respectively, In general this approximation improves as the radius of the grinding wheel approaches infinity and deteriorates for smaller grinding wheel radii and for more radical cam profiles From Figure 56 and Figure 57 it is apparent that the linearized model is a reasonable approximation to the actual sparkout model for these specific cam profiles. In Chapter 9 the linearized recess model is used to anal ze the effect of the model on the control desi n Comparison of Nonlinear and Linear Process Models 0 009 0 008 Modeled Error 0 007 E 0 006 Linearized Modeled Error g 0 005 I0 I0 004 Iw 0 003 .::: 0 002 :::i 0 001 0 0 001 1 80 150 12 0 90 60 30 0 Angle (d egrees) Figure 56 Linearized Profile Grinding Process Model:kc = 0.001, KL=0.00008
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88 Comparison o f Non l inear and Linear Process Models 0 0 1 8 0 016 l'vbdeled Error 0 014 Li near iz ed l'vbde le d Error 0 0 1 2 E E 0 01 .... 0 008 0 .... .... w 0 006 = 0.004 :.:J 0 002 0 0 002 180 1 50 1 20 90 60 30 0 3 0 6 0 90 1 20 1 50 1 80 Angle ( degrees ) Figure 57 Linearized Profile Grinding Process Model:kc = 0.002, KL=0.00016
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CHAPTER 6 THE SUPERFINISHING PROCESS Introduction Researchers have shown that the surface finish of a cam lobe affects the 36 37 28 38 39 wear rate and engine noise Specifically with improved surface finish cam life is extended engine noise is reduced and a requisite breakin period is 40 eliminated As intensive international and domestic competition in the automotive and diesel engine business has forced engine manufacturers to improve engine reliability extend engine life, and reduce engine noise the use of superfinishing has become standard in camshaft production Superfinishing improves the surface finish of a workpiece while removing a relatively small amount of material. The term superfinishing, and alternate l y the term microfinishing describe several different processes which improve both the surface finish and roundness of cylindrical surfaces: for the case of cam profiles the surface finish is improved at the expense of the geometry In this chapter the repeatability of the superfinishing process errors are examined to determine the feasibility of including the superfinishing process in the control loop. Clearly this is advantageous since if the superfinishing process is in the loop it is possible to correct for the profile error that results from the superfinishing process. Therefore both the repeatable and nonrepeatable geometric profile errors that result from the superfinishing process are considered here 89
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Description of Process The superfinishing process used in this research is described by 4 1 Puthanangady and Malk i n for cylindrical workpieces. In this process shown in Figure 61 for cam lobes the part is rotated at a constant speed while a stone containing abrasive grains is held against the workpiece by a pneumatic cylinder with a constant air pressure As the workpiece rotates the stone oscillates parallel to the workpiece axis where the number of oscillations per rotation is not an integer Typically the superfinishing process has a rough and a finish cycle. 4 1 Puthanangady and Malkin report that the mechanisms of the superfinishing process is similar to those found for grinding and bore honing. Figure 61 Camshaft Superfinishing Operation For superfinishing of cam lobes the stroke length of the pneumatic cylinder is large as compared with cylindr i cal superfinish i ng. The normal superfinishing force F~ varies due to the inertial forces the sign of the velocity J (direction of friction force) and the superfinishing tool angle of contact y i. Also the tool angle of contact y i, may be large compared to the angle in grinding
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91 since the radius of the superfinishing stone Rs, is smal l compared to the radius of the grinding wheel Rs, Therefore, from Figure 62, the normal superfinishing force at lobe angle of rotation 8 i, lobe angle of contact a i, and point of contact j is Equation 61 where Fi is the inertial force of the cylinder per unit width and F; is the frictional J J 41 force of the system per unit width. Again from Puthanangady and Malkin, t he stock removal is a function of the normal force and it is clear from this one effect alone that the amount of material removed varies around the cam surface Consequently an additional lift error results from the superfinishing process. Additionally, the nominal rotational velocity, the axial oscillation frequency and 41 the local surface finish affect the amount of material removed. Figure 62 Camshaft Superfinishing Model
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~~9 2 Table 61 Modified Grinding Model Nomenclature for Superfinishing Model F ~ c Axial force exerted by pneumatic cylinder per unit width F ~ri The normal superfinishing fo r ces in excess of the threshold force k ~c The stiffness of the pneumatic cylinder R sr Radius of the superfinishing stone us J, P X position of the superfinishing stone X s F x coordinate of the superfinishing machine system YsF y coordinate of the superfinishing machine system Y i Superfinishing stone angle of contact at angle of rotation 8 i Experimental Results In order to determine the repeatable and nonrepeatable errors associated with the superfinishing process used in this research eleven sixcylinder heavy diesel camshafts with integral diesel injection lobes were ground inspected superfinished, and then reinspected. The workpiece material was SAE 1513 fine grain steel carburized to an effective depth of 1 5 mm with a minimum hardness of the 58 Rockwell C The parts were ground on Landis 3L Series cam profile grinders equipped with steady rest type work supports The grinding wheel was a Beck 812649R0200110 81 (CBN) with a 225 mm radius. The wheels were dressed with a rotary diamond dressing attachment. The parts were inspected on an Adcole Model 911 cam inspection gauge The manufacturing and inspection program sequenc e s are included as Appendix D The superfinishing was performed on a custom built superfinishing machine using Darmann GC60045V2S green silicon carbide superfinishing stones Figure 63 shows the lift error of exhaust lobe one for eleven different parts that are inspected directly after profile grinding. Though it presents no complication for the current discussion it should be noted that in addition t o the expected grinding process error that occurs due to the res i dual sparkout ~
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1 93 deflection a large nonconcentricity error i s also present in this l obe Additionally though the topic is not directly discussed in the current research this graph also shows the repeatability of the cam profile grinding process which is essential to the successful implementation of any control strategy based on post process inspection with feedback. Th i s subject is discussed in detail in 4 previous work. Finally Table 62 shows the repeatab i lity errors for both, t he grinding process alone and the combined grinding and superfinishing processed. The repeatability error is given here as six standard deviations (cr) of the parameter measured for the eleven workpieces These values are used later in the design and simulation of the control system. Lift Error after Grinding Multiple Runs 0 015 0 01 0 005 E E ._, 0 '0 ''w 0.005 0 01 0 015 180 150 12 0 90 60 30 0 Angle (degrees) Figure 63 Typical Lift Error for Profile Grinding: No Compensation t Figure 64 shows the inspection results of the same lobe for the same camshafts after superfinishing. As with the grinding process the superfinishing is highly repeatable t Results here are expressed in terms of a different follower size than for all other results is this research This has no impact on the current discussion ~~~~
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9 4 Lift Error after Superfinishing : Multiple Runs 0 015 0 01 E 0 005 E 0 0 UJ 0 005 0 01 0 015 180 1 50 120 90 60 30 0 30 60 90 120 150 180 Angle (degrees) Figure 64 Typical Lift Error for Profile Grinding and Superfinishing t: Figure 65 shows the mean grinding and superfinishing lift error where nonconcentricity and the timing bias are removed. Figure 66 shows the means and standard deviations of the same data Mean Lift Errors: NonConcentricity Bias Removed 0.015 ~, 0 01 E 0 005 E 0 UJ 0.005 Ground 0 01 1 Superfinished 0 015 ~~ 180 1 50 120 90 60 30 0 30 60 90 12 0 1 50 1 80 Angle (degrees) Figure 65 Mean Lift Error: Nonconcentricity Removed i I _ j
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E E .... 0 .... .... w 0 01 0.008 0 006 0 004 0 002 0 0 002 0.004 0 006 0 008 Mean and Standard Deviation of Lift Error 11./ean Ground 11./ean Superfinished Std Deviation Ground I Std Deviation Superfinished I '' , 0.01 '' 180 150 1 20 90 60 30 0 30 60 90 120 1 50 180 Angle (degrees) 95 Figure 66 Statistics of Lift Error: Nonconcentricity Removed Discussion of Experimental Results As stated initially, the purpose of this chapter is to consider the feasibility of including the superfinishing operation in the control loop. Its inclusion is entirely dependent on the signaltonoise ratio of the lift error introduced by the superfinishing process This may be quantified somewhat by calculating the 3cr average process lift error repeatability for the lift error after superfinishing and grounding. For the eleven runs, the average process variability for the ground parts is 3 6 m. For the same parts after superfinishing, the variability is 4.4 m. While this is a significant increase, approximately 20% the variability is still low enough that the signaltonoise ratio is sufficiently high so that the effects of the mean superfinishing error can be compensated for with an appropriate controller and filter gain settings The repeatability of the grinding and superfinishing process is summarized in Table 62. _J
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~96 Table 62 Process Repeatability RepeatBase + Non Lift Timing Timing Timing ability circle concentric i ty to lobe to dowel to (combined) size one keyway ( cr) mm mm mm degrees degrees degrees Grind and 0.016 0 003 0 0018 0 12 0.025 0.7 Gauge Superfinish Grind and 0.013 0.003 0.0022 0.13 0.025 n a Gauge Finally, while the development of a model for the cam profile superfinishing process is outside the scope of this work it is nevertheless of interest to discuss factors which should be considered in developing such a model. First the instantaneous velocity between the stone and workpiece varies in relative terms much more that it does in grinding. This is true since the surface velocity of the grinding wheel dominated the relative velocity for grinding, while in superfinishing, the stone contributes only a small axial component to the relative velocity of the workpiece and the stone Second the variation in the angley i for superfinishing is large compared to grinding and correspondingly the change in the normal superfinishing force F~ due to this J factor is large Variations in F ~i also result from the inertial and friction forces Third the change in the part radius of curvature or the equivalent diameter is expected to be a significant factor as it is for grinding. t For worst case : lobe closest to workhead. I I ~~J
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CHAPTER 7 DETECTION OF SPURIOUS ROUNDNESS AND LIFT DATA Introduction Inspection of cylindrical surfaces and cam lobes is often performed under conditions only moderately cleaner than the production environment. Consequently, measurement results are adversely affected by the contaminants present i n the environment. Generally detection of spurious results due to contaminants requires the experience of a gauge operator to examine the graphical presentation of the inspection results This method is generally adequate where the i nspection process is used solely as a monitoring tool. However when the inspection results are used to modify the commanded part geometry in the manufacturing process an automatic method to detect these results is desired The approach presented here fits linear models to short sections of the inspection data. The differences between the modeled surface and the actual measurement data are then analyzed for potential as outlying data points (asperities) The scheme requires the selection of three parameters and testing for a fourth These parameters are as follows : the number of points included in the linear model of a smooth surface (P) the number of points deleted from the surface model (2m+1 ) the number of points deleted from t he variance calculation (2q+1) and finally the number of standard dev i ations used as the detection limit. These parameters must be established by investigating the nature of the outliers and the desired sensi t ivity of the detection scheme 97
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9 8 Description of Method Linear Surface Model and Residuals For measurement data from a smooth surface such as those produced by 2 3 grinding, consider a linear smoothing formula of the form 1 Ysmoothed (j) = 5 (Y j2 + Y j 1 + Y i+ Y j + 1 + Y j + 2} Equation 71 where y smoothed (j) is the smoothed inspect i on data y i is the Inspection data j is the index for pointofinterest which ranges from O to N 1 where N is the number of data points It is convenient to rewrite this expression in a more compact form as P 1 j+1 2 Y smoothed (j) = p L Y i . P 1 l=J2 Equation 7 2 where i is a dummy index and P is the number of points in the linear surface model. And of course all indices are normalized between O and N 1 here and throughout this discussion Now the residual e is defined as Equation 7 3 ~
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99 Deletion of the PointofInterest The smooth ng formula is analogous to performing a least sum of the squares best fit of a linear model to the data and calculating the center point using this model. In the proposed method the model is applied to a section of the part surface and the probability that a given point is an outlier is evaluated. Since point j is the point of interest it is intuitively appealing to remove or delete point j from the data used to determine its smoothed value. In this way, the value of the smoothed data point is not influenced by the potential outlying point. Therefore Equation 7 1 is modified accordingly and the term y smoothed is replaced with the more appropriate term y mooet to get 1 Y model(j) = 4 (Yi 2 + Y j1 + Yi 1 + Yi + 2) Equation 7 4 where y modetrn is the modeled value for point j The deleted residual di, is defined as di = Yi Y model(j) Equation 7 5 This model is shown in Figure 71 The idea of removing the pointofinterest from the data prior to calculating the modeled value is referred to as the "method of deleted 42 residuals." This method generally improves the ability to identify outliers through inspection of the residuals. This effect is illustrated in Figure 71
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100 The Effect of Deleting the Point of Interest from the Resiudal Computation point of __ ., ___ ____ __ _ ,. __ i nterest ys (smoothed) ym (model) f I j2 j1 j+1 j+2 Figure 71 Deleted Residual Model Deletion of a Window of Points Legend I Raw data mil Smoothed data X Modeled data e Residual d Deleted residual If this idea is extended to detect adjacent outlying points then the deletion of multiple points centered at point j must be allowed This is necessary since the outliers may not be electrical or other noise in the system but rather physical objects such as dirt. Consequently, the outliers have "length" and thus may span a number of data points Thus Equation 74 is generalized to get J + [ P 1 l 1 j1m 2 Y model(j) = p (2m + 1) L Yi + L Yi .. P 1 l = J +1+ m l = /2 Equation 76 where the nomenclature is defined in Table 71. The number of points included in the model is P (2m + 1).
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.. 1 01 Table 71 Nomenclature for Asperity Detection Method d i Deleted residuals d '. Studentized deleted residuals l d : Modified Studentized deleted residuals l d i Average deleted residual over the range q + j + 1 i Nq + j 1 i Dummy index for summation of points i Index for pointofinterest MSE Mean square error m 2 m + 1 is the number of points deleted N The number of data points p The number of points in the linear surface model prior to deletion q 2 q + 1 is the number of points deleted from variance calculation V Variance vd Deleted variance Y i Inspection data y model(j) Modeled values for smooth surface Studentized Deleted Residuals In order to apply a statistical test to the residuals it is convenient to normalize the residuals by dividing by the square root of the mean squared error (MSE). The MSE for our linear model formulation would strictly speaking, be defined as P 1 j 1 m d.2 i + 2 d.2 MSE(j) = L 1 + L _,_ .. P1P2 i = i + 1 + mP 2 l = J 2 Equation 77 However this is not very robust as it computes the MSE based on only a few points (P2m+1 ) Rather the computation of the MSE is based on all N residuals with an N1 weighting. With these changes the MSE becomes I ~J
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~~ ,_. 1 ~N 1 2 MSE = N 1 .L. i = O d i Equation 78 42 Recall that the formula for the variance of d is Equation 79 1 02 Comparing Equation 7 8 and Equation 7 9 it is recognized that this formulation of the MSE is simply the variance of d since for all the models considered the average of the deleted residuals d is zero Therefore, the variance v is substituted for the MSE. The Studentized deleted residuals are then defined as Equation 710 The Studentized deleted residuals have a t" distribution The Modified Studentized Deleted Residuals Since an outlying point as shown in Appendix A can significantly inflate the value of the variance the effectiveness of the scheme to discriminate between outliers and surface/measurement noise can be improved by a final modification to the variance computation. From Equation 710, it is recognized that decreasing the value of the variance in the region of a outlying point will increase the value of d .. Similarly increasing the variance in all other regions increases the discrimination of the scheme. To effect this increase in the
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103 discrimination the variance v i s replaced with the deleted variance vd. The deleted variance is defined as the variance of the residuals where the data points centered near the pointofinterest are deleted. Therefore to delete 2q+1 points the limits on the summation are changed and the subscript j is i ntroduced. With these modifications the deleted variance vdi is 1 Nq + j 1 2 2 vdi = N( 2 1 )_ 1 Lidi (N(2q+1))di q + l = q + j+1 Equation 711 where q 2q + 1 is the number of points deleted di is the average of the deleted residuals over the range q+ j + 1 $ i $ Nq+ j1 and is not necessarily zero. It is defined as 1 N q + j 1 dj=Ldi N(2q + 1) i = q + i + 1 Equation 712 Finally, the modified Studentized deleted residuals d .. i s defined as Equation 713 Since a large number of points (N(2q+1 )) are used to calculate the variance d .. has an approximately normal distribution That is d .. i indicates the number of standard deviations that point j lies from its expected or modeled value __ ___J
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104 Therefore only the number of standard deviations to use as an outlier lim i t for the test needs to be selected Computational Efficiency This formulation requires N computations of the variance for N(2q+1) points However it possible to calculate the variance only once and then use a recursive formulation for calculating the variance at the remaining N1 points. The required equations are developed in Appendix A Preliminary Experimental Resu l ts To demonstrate the effect of the algorithm on cam lift inspection data the lift error shown in Figure 72 is considered From this plot, it is clear that an asperitylikely due to contamination on the part surfaceexists at approximately 90 degrees. It is also noteworthy that on this cam lobe, which was superfinished prior to inspection a large negative spike in the lift data occurs at zero degrees This negative spike is typical of injector cam lobes that are superfinished after gr i nding This negative spike resu l ts from the superfinishing process sinceas discussed in Chapter 6the superfinishing operation removes a greater amount of material where the radius of curvature is smaller The modified Studentized deleted residuals of these inspection data are calculated using Equation 75 and Equation 7 6 and Equation 7 11 through Equation 7 13. The results are plotted in Figure 7 3 for P m and q equal to seven one and three respectively These results show that while the algorithm successfully amplifies the actual asperity that occurs at 90 degrees i t also amplifies the negative spike that occurs at zero degrees Therefore care must
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. , ~ ~~ 1 05 be taken i n the asperity identification algorithm to identify only tho se points where the asperity is due to a positive metal condition Still the algorithm shows that it is effective in identifying spurious data Clearly the number of standard deviations that a points lies away from zero must be chosen as a test to determine if a point is an asperity While this choice is somewhat subjective it should be made such that the number of false positives is minimized while the likelihood that a spurious data point will be detected is maximized Further testing o f the algorithmnot shown here indicates that a value of approximately five standard deviations provides good results. Still asperities by there very nature occur i nfrequently and consequently, only a small number of samples were available for testing. Additional testing is required to determine the effectiveness of the algorithm under a range of conditions Also testing of this method on round surfaces is given in Appendix A 0 015 0 01 E 0 005 E .._,, 0 .._ 0 .._ .._ w 0. 005 0. 01 0.015 1 80 150 Asperities in Cam Lift Inspection Data 1 20 90 60 Asperity (c ontamination) Rubbed off nose ( actual shape) 30 Angle (degrees) Figure 72 Lift Inspection Data with Spurious Inspection Points I
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Modified Deleted Residuals Test for Cam Lobe Inspection Data 15 ,,=============,;, '8 1 0 c:: Q 5 iii 5 I njector Lobe w I Asperrties Inject or Lob e w I no Asperrt i es I ', : ', 0 ~'~"r1 ~ 1,,...., .c4,
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CHAPTER 8 DIGITAL FILTERING OF INSPECTION DATA Measurement and Process No i se Noise from the measurement and grinding process contribute to fluctuations in the output of the controlled system. Systems which attempt to correct for nonrepeatable processes or measurements amplify the i nherent process variability Recognition of this fact has lead to the implementation of Statistical Process Control (SPC) i n many facilities which manufacture discrete parts. SPC represents a form of deadband controller in which no control action is taken unless the measured error exceeds a specified threshold. While this system is useful in limiting the amplification of the inherent process noise i t also fails to correct for developing trends until the threshold is exceeded. In this research three different types of noise are of considered. The first is the random variability in a parameter over a series of parts The second is the spatial noise or highfrequency component of the profile inspection data for a single part T he third is noise that is introduced into the profile grinding due to the roundness error of the bearing journals Recursive F i lters for Lift, T i ming, Size, and Nonconcentricity The difference equation for the discrete firstorder recursive filter is Fittc = a Actc + ( 1 a) F i ttc n n n 1 Equation 81 107
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1 08 where a is the filter gain Act e n i s the actual measured value for part n and R 1 e n is the filtered value for part n The Z transform for the filter is Fitt c(Z) a G1(Z) = Act c(Z) = 1+( a 1) Z 1 Equation 82 This filter is used to reduce the noise in lift, timing, nonconcentricity and size 4 errors. In previous work it is shown that the use of this type of filter successfully reduces the process noise while introducing a phase lag into the inspection data. The filter coefficients are chosen together with the control l er gains based on the observed signal to noise ratios These choices are discussed in the Chapter 9. Smoothing Inspection and Compensation Data As noted while the filtering of lift data over a series of parts reduces the process noise it also introduces an undesirable phase lag. Fortunately, in the case of the lift data it is possible to take advantage of the smooth nature of the part surface to spatially filter the data This spatial filtering reduces no i se by averaging the inspection data at adjacent points while not introducing any lag to the controller In this section a linear zerophase smoothing filter is designed to limit the noise in the lift inspection data. While noise exists in the both the measurement data and the grinding process, the grinding process noise dominates. For successful feedback control the noise in the feedback signal needs to be limited so that the noise or high frequency content in the inspection signal is not amplified. As in most filter design, the distinction between signal and noise requires engineering judgment.
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1 09 However t he model of the prof i le grinding process developed earlier can be used to estimate the expected profile error form and determine the harmonic content that is necessary to synthesize the expected error shape for a given cam profile From the required harmonic content a zerophase lowpass smoothing filter is readily designed The purpose of the filter is to pass the frequency content necessary t o accurately represent the nominal shape while eliminating the frequencies associated with the process and inspection noise. However examination of the spectrum of the nominal and inspection data shows they have nearly the same frequency content. Therefore it is easier to filter the reduced error data and the corresponding compensation that is applied to the nominal lift data. To understand the significance of this, examine the nominal lift data in the frequency domain First consider the most extreme case ( i e. the lobe shape with highest harmonic content) for an internal combustion engine : the diesel injector lobe The lift curve for a typical injector lobe is shown in Figure 81. Figure 82 shows that lift has s i gnificant frequency content up to 100 th undulations per revolution (upr) Where by significant it is meant the frequency content that is required to represent the lift data to a precision of less than 1 The frequency content required to represent the lift to the desired precision can be estimated by inverse Fourier transforming the bandl i mited lift data (all harmonics (upr) above a given harmonic set equal to zero) and comparing the result with the original data. This comparison is shown i n Figure 83 for a cutoff frequency of 100 upr. For this cam profile the 100 upr cutoff frequency gives the desired precision. Thus since the nominal lift data has higher harmonic content it is impossible to design a filter which direct l y removes noise from the signal without significantly altering the signa l as well. Therefore, the approach is modified and the error as opposed to the actual value is filtered
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Nominal Lift and Inspection Data for Injector Lobe 16 .......1 6 14 12 10 E a .s 6 5 4 2 2 follower radius = 19.00 mm Lift Error I I I I I base circle radius = 59 875 mm 14 12 2 4 ...__ ________________________ ___._ 4 1 80 1 50 12 0 90 60 30 0 30 60 90 1 20 1 50 1 80 Angle (degrees) Figure 81 Nominal Lift and Error Data for a Typical Injector Lobe Fourier Transform of Lift Data 1ttt. E o 9 e o.8 ~ 0 7 ';:' 0.6 CJ) 0 5 0.... 0.4 2 g 0 3 0.2 co I 0.1 I f I I \ I I I I I \ 'V '1' \ Lift Error i ,_"' ,.,,. Q +=""..=,.:::_w__:\.iL:::_::::..::!11.!::.,:...,,._,,....:=====::::::::::__J 0 15 30 45 60 75 90 105 120 135 150 165 180 Undulations per Revolution ( UPR ) Figure 82 Fourier Transform of Typical Exhaust Lobe Nominal Lift Data 11 0
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111 Effect of Band Limiting Lift Data 0 6 0.4 E 0.2 0 '<..l I 0 '0 0 2 ''w 0 4 0.6 180 150 120 90 60 30 0 30 60 90 120 150 180 Angle (degrees) Figure 83 Effect of BandLimiting the Nominal Lift Data 0 008 0 006 0 004 E E 0 002 '0 0 ''w 0 002 0 004 0 006 180 Modeled and Measured Lift Error Modeled Error Measured Error 150 12 0 90 60 30 ,~ "" ... _._,, .. : I\ I 0 30 60 Angle (degrees) Figure 84 Modeled and Measured Lift Inspection Data 90 120 1 50 180 To see the difference this approach makes consider the nominal error predicted by the residual sparkout model and the measured error shown in Figure 84 From the frequency content of these signals shown in Figure 85 it is apparent that frequency content above about the 25 th harmonic is not s i gnificant to represent the modeled data while the measured data shows frequency content up to the 90 th harmonic. Therefore using the model as a
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1 12 guideline for the actual s i gnal content of the repeatable measured data that is to be compensated the filter can be designed Frequency Content of Modeled and Actual Lift Error 0 8 r=======i 0 7 Modeled E o.6 Actual ..:;!; 0 5 (I) 0.4 5J 0.3 ro 0 2 0 1 0 5 1 0 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 Frequency (upr) Figure 85 Frequency Content of Modeled and Actual Lift Error Thus a lowpass zerophase filter with a cutoff frequency of approximately 25 upr is chosen The difference equation for this type of 23 smoothing filter is Equation 83 where Fi 11 L\ri is the filtered lift at point j, Actl\r i is the actual measured lift and the ak .' s are the filter coefficients This expression can be generalized as ~ ~
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113 K "Act~ra Jk k F i ll Ar __ k = _K ___ Ll j K Z:ak k =K Equation 84 where 2 K+1 represents the number of terms in the filte r. Taking the Z transform of Equation 84 gives K "z k ak Fitt~r(z) H() _ k = K __ z = Act~r(z) = K Z:ak k = K Equation 85 The frequency response of the spatial filter in terms of the number of undulations per revolution is K j( upr ) k ( j(upr) ) L e N ak H e N= _k = _K _ K __ Z:ak k = K Equation 86 where for this expression only j is ~. For all ak s equal to one k equal to two and passing the data through the filter twice gives a filter which is computationally efficient relatively flat in nopass band and a cutoff frequency of 23 upr. The frequency response fo r the filter is shown in Figure 86 The result of this filter on typical lift inspection data is shown in Figure 87 This is the filter implemented in the control system and further demonstrated in the chapter on experimental results. The two passes of the filter are implemented so
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that the first pass is on the inspection data and the second pass is on the commanded compensation. Frequency Response for Smoothing Filter 1 0 9 0 8 Q,) 0 7 "O 0 6 C: 0 5 Cl ctl 0 4 0 3 0 2 0 1 0 114 0 15 30 45 60 75 90 1 05 120 135 150 165 1 80 Frequency (UPR) Figure 86 Frequency Response for a Zero Phase Linear Smoothing Filter Measured and Filtered Lift Error Data 0 008 ~~ 0 006 0 004 E E 0 002 .... e o1:,r,~_,,~_.~~ ' '\t~#'14,~~~~~ .... w 0 002 0.004 l'v'easured Error Filtered Error 0 006 ....._ ________________________ __, 180 150 120 90 60 30 0 30 60 00 120 150 180 Angle (degrees) Figure 87 Filtering Results for a Zero Phase Linear Smoothing Filter
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1 15 Table 81 Digital Filtering Nomenclature a First order recursive filter qain ak Smoothinq filter coefficient an Fourier coefficients of cosine terms al Fourier coefficients of cosine term for lobino frequency bn Fourier coefficients of sine terms bl Fourier coefficients of sine term for lobino freauency b Vector of Fourier coefficients for nonconcentricity and lobing e rrors b Least sum of the squares estimate of the vector b Bcos Bias due to cosine Bsin Bias due to sine ActCr, Actual measured value of variable c for part n rlnCn Filtered value of variable c for part n end End of base circle (i value) k Number of terms in the smoothing filter n Part number nl Lobing frequency in uor N Number of lift inspection points r i Radius of follower oitch curve rb Base circle radius MeaSrb Measured base circle radius r....., Grindino wheel radius R Fourier Coefficient for DC (constant) term start Start of base circle (j value) s Follower lift s Vector of N measured follower l i ft values MeSSS lj Measured lift at point i upr Undulations per revolution X Matrix of measured lift values on base circle XM X axis of machine coordinate svstem XCn i Cosine element n, i of x matr i x XSn i Sine element n j of x matrix YM Y axis of machine coordinate system z Z Transform operator Act .1.ri Actual measured value of pitch curve radius Filt,1.ri Filtered value of pitch curve radius a i Lobe angle of contact Y, Yi Grinding wheel angle of contact ei Lobe angle of rotation
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1 16 Control Signal Noise due to Journal Bearing Roundness Error Typical journal bearings produced by external cylindrical grinding exhibit a roundness error of the form shown in Figure 88 While errors of this type are often removed by superfinishing, the superfinishing operation is generally not 44 performed until after the cam profiles are ground This means that significant roundness error may be present during the cam profile grinding operation For camshafts ground using steady rests this roundness error contributes significantly to the nonrepeatable cam profile error This nonrepeatable error is a source of noise in the process control signal which is not removed by the lowpass smoothing filter designed earlier. Therefore, a method to selectively remove this low frequency noise component from the inspection data is developed in this section. The development is preceded by a discussion of the source and randomness of the roundness error. 0 01 E o 005 E .._,, lg W 0 005 0.01 Simulated Journal Bearing Inspection Data 0 015 ' 180 150 120 90 60 30 0 30 60 90 1 20 1 50 180 Angle (degrees) Figure 88 Simulated Journal Bearing Inspection Data
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1 17 Sources of Roundness Error i n External Cylindrical Grinding Typically camshaft journal bearings are ground sequentially on external cylindrical grinders Due to the camshaft s flexibility the center journal bearing is generally ground first and i ts ground surface supported by a steady rest prior to grinding the remaining bearings. The use of the steady rests increase the stiffness of the grinding system and reduces the amplitude of the vibrations 45 3 3 present in the system which produce roundness errors Srinivasan Malkin 46 and Snoeys and Brown i dentify the types of vibrations present in cylindrical grinding as forced and self excited. Self excited vibrations fall into two distinct categories: those due to wheel regenerative chatter and those due to workpiece regenerative chatter. Forced vibrations exist i n cylindrical grinding due to persistent disturbing forces that generally originate within the grinding system Typical sources of disturbing forces are unbalanced grinding wheel and inaccurate grinding spindles or spindle drives The extent to which these forces lead to chatter depends on the magnitude and frequency of the force and the frequency response of the grinding system. Chatter due to forced vibrations may be fairly repeatable in terms of the magnitude and lobing frequency if the exciting force does not change. However the phase of lobing will vary from part to part since the rotation of the grinding wheel spindle the most like l y source of the f orcing function and the workhead are not synchronous Furthermore the amplitude of vibrations due to unbalanced wheels changes as the wheel wears So while the forced vibration is clearly not a random process nevertheless it produces an apparently random effec t on the profile grinding operation Regenerative chatter exists in cylindrical grinding due the regrinding of the same surface in successive rotat i onal passes Undulations are created on ~
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1 18 either the surface of the grinding wheel or workpiece These undulations which begin due to transient or shortlived disturbances. are then regenerated on t he subsequent passes over the same surface I f the chatter loop described by 46 Snoeys and Brown i s unstable then the undulations grow i n successive passes and eventually create distinct chatter on the workpiece surface. The effect is strongest in plunge grinding where the overlap in the surface on successive passes is complete. In both wheel and work regenera t ive chatter the finite contact length formed by the compliance of the grinding wheel and the workpiece acts as a spatial low pass filter This filter attenuates the effects of the high frequency undulations present on the workpiece or the grinding wheel surfaces The relative magnitudes of the angular velocity of the workpiece and the angular velocity of the grinding wheel differ substantially Therefore the undulation wave lengths on the workpiece are short and do not regenerate due to the attenuation of the spatial filter. However due to the high speed of the grinding wheel the undulation wave lengths on the grinding wheel surface are long and the spatial filter has little effect. When wheel regenerat i ve chatter is present the undulations on the grinding wheel surface grow slowly and can be removed by dressing the wheel. The roundness error produced due to regenerative chatter is random in the sense that the bearing journal error varies from journal to journal and part to part depending on the magnitude frequency and phasing of the vibrations which produce the error For the process studied the lobing frequency is invariant for all journal bearings on the camshaft even though the natural frequency of the grinding system changes once the center bearing is ground and a steady rest is moved into place This consistency of the lobing frequency and therefore the frequency of the source is due to a forced vibration. Also the lobing frequency is relatively
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3 3 lowthree undulations per revolutionwhich according to Malkin i s more typical of forced vibrations 1 19 Additionally lobing errors of such l ow frequenc i es occur i n cylindrical grinding due to geometrical effects o f the part and mach i ne centers. Kato and 47 Nakano show that different lobing frequencies can be attribu t ed to the error i n the centers and other factors Effect of Journal Bearing Roundness Error on Cam Profile During the cam profile grinding process the journal bearings run on the steady rest pads as shown in Figure 89 The intended function of the steady rests is to increase the stiffness of the grinding system The increase in stiffness is desirable as it greatly reduces the grinding cycle time and the residual sparkout grinding error Unfortunately the steady rests also map the bearing roundness error into the ground cam profile since the machining axis of rotation now shifts due to the roundness error This error is analogous to work spindle error and it produces the same sort of error in the ground cam profile From the perspective of the profile grinding operation this profile error due to the bearing roundness is a nonrepeatable error and therefore it is noise that needs to be filtered from the control signal. Design and Implementation of the Digital Lobing Filter In order to remove the component of the cam profile error that is due to the bearing roundness error it is necessary to consider how the bearing error is mapped into the cam profile by the s t eady rests Figure 89 illustrates two different steady rest designs. The two pad steady rest is mechanically adjusted to align the camshaft axis of rotation and does not automatically compensate for
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7 journal bearing size variation The threepad wraparound is frequently designed to compensate for journal bearing size error Clearly, these two designs cause different shifts in the machining axis of rotation due to the roundness error In this research only the first case is considered Direction o Rotation Direction of Figure 89 Simple Two Pad and WrapAround Three Pad Steady Rests 1 20 In the first design the steady rest used two pads located 90 to each other. For typical CBN cam profile grinders the grinding wheel radius r gw, is an order of magnitude larger that the cam base circle radius rb. Consequently the contact angle y, shown in Figure 810 is generally less than 2 degrees and the profile grinding operation is insensit i ve to shifts of the machining axis of rotation in the Y M direction Therefore only shifts in the XM direction are important and these map onetoone into grinding wheel motion coordinate errors by shifting the machining axis of rotation in the XM direction Identification and Removal of Lobing Magnitude and Phase If the lobing error is a regular shape and its magnitude and phase can be estimated then the lobing error can be directly removed from the inspection ~ ~ 
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121 data However the identification of the lobing error is complicated by the shape of the cam profile. If the cam were nominally round then the lobing errors could be calculated directly as the Fourier coefficients. This approach does not work with the cam since the nominal cam shape has a wide band of harmonic content. Therefore it is necessary to modify this approach using the specifics of the problem to identify the lobing. The most straightforward method is to calculate the Fourier coefficients using only the base circle data. To use this method it is essential that the lobing frequency be known in advance and that the observation window be chosen so that the lobing is exactly periodic within this window By selecting the window in this way and correcting for the bias that exists in the inspection data due to the zeroing of the average base circle lift the lobing phase and magnitude can be accurately calculated. The restriction that the lobing frequency be known presents no complication as it is easily identified by Fourier transforming the journal bearing inspection data. y X Direction of Rotation Figure 81 O Mapping of Bearing Roundness Error on Profile
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1 22 Grinding Wheel Motion Coordinates As discussed previously t he f orm of error for the follower motion coordinates is different than for the grinding wheel motion coordinates due to the difference in the point of contact for different size followers. Consequently the lobing error can be modeled as a pure sine wave for grinding wheel motion coordinates. In follower motion coordinates the form of the lobing error is significantly distorted as shown in F i gure 811 If the lobing error is removed directly from the follower motion coordinates then a residual distortion remains as shown in Figure 811 In Chapter 4 the effect of removing the nonconcentricity error from follower motion coordinates was discussed and a distortion effect similar to that shown in Figure 811 was demonstrated. The removal of the nonconcentricity error in the grinding wheel mot i on coordinates was not considered since this would make the part inspect i on dependent on process parameters. Here the situation is quite different as the lobing effect is not to be filtered from the reported error data used to evaluate the process but it is only filtered from the feedback signal. Therefore i n cases where the lobing error is significant i t is necessary to transform the inspection data to grinding wheel motion coordinates identify and remove the lobing effect and then transform the data back to design follower motion coordinates The transformations are performed as discussed in Chapter 3 The lobing magnitude and phasing are identified and removed as described in the following section. Identification and Removal of Lobing Effect For nominally round objects the removal of lobing effects of any frequency is a straightforward proposition. For round objects the lobing frequency, amplitude and phase are readily determined by examination of t he frequency content of the inspection data. The frequency content for N I J
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I L i nspection points of the measured part r adius r i s given as Equat i on 87 Equation 88 and Equa t ion 89 Equation 87 1 23 In Equation 87 R is the DC or constant term of the Fourier transform. I f the bias effect due to a radiused follower discussed in Chapter 4 is neglected then R is the average part radius The cosine coefficients for n = 1 to N1 are 2 j = N 1 ( 27t ) a n = L Measr j cos j n N j = O N Equation 88 The sine coefficients for n = 1 to N1 are Meas 7t 2 j = N 1 ( 2 ) b n = N r j Sin N J n Equation 89 These expressions may i n general be directly applied to roundness data without any of the complications genera l ly associated with Fourier transform methods such as leakage and aliasing. Leakage is not generally a problem since roundness inspection data for nominally round objects are by definition exactly periodic within the observation window Aliasing is also relatively easy to avoid since the inspection probe can be designed to provide the spatial filtering necessary to ensure that the mechan i ca l frequency response of the gauge is less than half the sampling frequency and thereby satisfies the Nyquist criterion
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1 24 Effect of Journal Bearing Lobing Error 0 01 ,,......,,....., 0.008 0 006 ,..... 0 004 0 002 0r\.. l.'4.,'l'" c,,,..+~ g 0 002 W 0.004 0 006 0.008 0 01 _,___ _____ ........,. _______ .........,....._ _______ .......,._......, 180 1 50 120 90 60 30 0 30 60 90 12 0 1 50 1 80 Angle (degrees) I 3 U PR i n Grinding Wheel Motion Coordinates I 3 UPR Removed Directly from Follower Motion 3 UPR in Follower Motion Coordinates Figure 811 Effect of Lobing Error The identification of lobing effects from inspection data of objects that are not nominally round, such a cam profiles is more difficult. The complication arises as it does in the case of the smoothing the actual cam inspection data from the fact that a high harmonic content is required to represent the nominal cam shape Naturally one of these harmonics is equivalent to the l obing frequency Therefore it i s necessary to use only the inspection data from the base circle to identify the lobing magnitude and phase While this approach eliminates the harmonic content of the nominal cam shape it reduces the data available for analysis to less than a complete period since the data repeat with a perfect gauge, every 360 or N data points If the inspection data for the base circle are simply Fourier transformed to determine the lobing, the inspection data are not periodic in the observation window and the signal "leaks" and the spectrum is smeared in frequency : i e. the power due to a single lobing frequency will leak into adjacent frequencies in the form of side lobes. A general discussions of Fourier transform and methods to reduce or eliminate leakage is
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1 25 4 8 49 presented by Kay and Marple Renders et al. present an unbiased f ast Fourier transform method for spectral analysis that applies when less than one period of data is availab l e In many of the methods discussed by Kay and Marple the basic idea used to reduce leakage is to exploit existing knowledge of the form of the signal to develop a more specific signal model. Therefore in the following section the form of the reported base circle roundness or lift error are discussed in detail. The Base Circle Lift ( Roundness) Data Model To develop a model of the reported base circle measurement data it is useful to consider how the measured base circle radius is computed. The measured base circle radius Measrb is e nd N1 L Meas r j + L Meas r j Meas r = i = O j= slart b N + end + 1 start Equation 810 where N is the total number of inspection points over the entire cam surface The indexes for the starting and end i ng points of the cam base circleregion of zero nominal liftare designated by start and end" respectively The starting point refers to the transition from the closing side of the ramp to the base circle and the end point refers to the transition point from the base circle to the opening side ramp. These transition points are illustrated in Figure 37 and Figure 38. The nominal lift and simulated i nspection data are shown in Figure 812 ~ ~ . ~" ~
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126 Identification of Base Circle NonConcentriciy and Lobing Errors 8 ,~ 0 015 7 Es E ....,, (fl 5 :.J 4 3 2 z ,. .. . ~ .. Lift Error ', ~ . ' .... Base Circle 0 01 E 0 005 E ,, ....__,, ... 0 O ... ... w 0 005 :E _J 0 01 0 L..loo<..~+............ """'+++'0 015 0 30 60 90 120 150 180 210 240 270 300 330 360 Angle ( degrees) Figure 812 Nominal Lift and Lift Error: Coordinate System from Figure 38 With the base circle size defined the reported lift at point j, Meas s i, is defined as Meas S Meas r Meas f jj b Equation 811 Solving Equation 811 for Meas ri and then substituting this value into Equation 810 gives ~d ~1 L( Meas sj +Measrb) + L ( Meas sj+Measrb) Meas f = j = O j= start b N + end + 1 start Equation 812 rewriting gives end ~1 N 1 N 1 L Meassj + L Meass j L Measrb + L Measrb Meas f __ j = O j = slart j= start j= slart _.:;.__ ___ ....:..._ ___ +b N + end + 1 start N + end + 1 start Equation 813
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The l ast two term are just Mea s rb and there f ore en d N 1 L M eas S j + L Me a s S i = Q j = O j = sta rt Equation 814 Equation 814 shows that the sum of the measured lift over the base circle is by definition zero However if the l i ft data are considered to be t he sum of a series of harmonically related sine waves as Equation 815 1 27 where the coefficients a 0 and b 0 are determined by performing a least sum of the squares best fit of the inspection data to the model. A least sum of the squares of the error best fit to the form of the function in Equation 815 is exactly 22 equivalent to the Fourier transform The component of the l i ft s at a given frequency n i s Equation 816 Summing Equation 816 over N data points gives Equation 817 
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Equation 817 i s equal to zero for all n since there are exactly n periods of the lift s(n) in N data points and the sum of a sine wave over an integer number of periods i s always equal to zero. In general this condition must be satisfied to use the Fourier transform on a signal without experiencing leakage. However in the case of cam lobe inspection only the base circle data are available to estimate an and bn and with the summation executed only on the base circle data under these conditions Equation 817 becomes Equation 818 Clearly, Equation 818 is not in genera l equal to zero since the summation of the sine waves starts and ends at the start and the "end" of the base circ l e Therefore the sine functions are not generally summed over an integer number of periods and in this cases purely periodic effects such as lobing and nonconcentricity error cause a size bias to exist in the calculation of the base circle. Therefore in order to correctly calculate the lobing and nonconcen t ricity of the base circle it is necessary to modify the signal model to account for the size bias effect. For the case of cam base circ l e inspection data with lobing frequencies that are integer multiples of the part rotation frequency, the signal model for the base circle is the measured lift on the cam s base circle. Using the notation developed in Chapter 3 and the coordinate system shown in Figure 38 the measured lift at point j, Meas s j I is ~~ 1 J
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Equation 819 where e nd ( 27t N1 ( 27t l Lcos Nj n ) + L cos Nj n ) B ( ) J = O J = start cos n = N + end + 1start Equation 820 and en d ( 27t l N 1 ( 27t ) L sin N j n ) L sin N j n B ( ) J= O J=s tart sin n = N + end + 1 start Equation 821 1 29 where the Bcos(n) and Bsin(n) terms represent the signa l bias due to the cosine and sine components of the lobing and nonconcentricity error respectively For lobing errors n is the frequency of the lobing error expressed in terms of undulations per revolution The case with n equal to one is not lobing but rather the special case of the nonconcentricity error. For cam base circle with lobing, the inspection s i gnal consists of a single known lobing frequency, nL the fundamental frequency due to the nonconcentricity error and high frequency terms. In th i s case, Equation 819 becomes ,..,. si = a, { cos ( 2 : j ) B= (1)} + b, { sin ( 2: j ) B,m (1) } + a, { cos ( 2: j n, ) B(n, )} + b, { sin(2: j n, ) B,.(n, ) } Equation 822
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1 30 It is significant to note that these four sinusoidal functions are not generally summed over complete periods and therefore are not orthogonal. Recall from the discussion in Chapter 4 that functions are orthogonal over the range from 81 to 82 if f 2 f(8)g(8)d8 = 0 e Equation 823 Or in the discrete case orthogonality i s established if N 1 If(0j)g(0 j ) = o j = O Equation 824 Applying Equation 824 to the lobing and nonconcentricity frequencies over the range of the base circle inspection data gives Equation 825 The implication of the lack of orthogonality is that the coefficients a1, b1, al and bl in Equation 822 are not independent. This is contrasted with the Fourier transform where the coefficients are independent and the Fourier coefficients may be calculated separately for each frequency. This is true for the Fourier transform since the signal record length must be an integer multiple of periods and all harmonics of a sinusoid are orthogonal over the an integer multiple of 2 2 periods of the fundamental frequency
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1 31 With the lack of orthogonality in mind a formulation for the least sum of the squares solution can be formulated For convenience Equation 822 can be rewritten in matrix form as s=Xb Equation 826 where xc1 start XS 1 start xcn start L' xsn start L' XC1 start T1 XS 1, start 1 xc n start + 1 L' xsn start + 1 L' 81 xc1 N 1 xs1 N 1 XCn L, N 1 XSnL N 1 b= b1 X = an xc1 o xs, O xcn 0 xsn 0 L L' L' bn L Send XC, end XS, end XCn end L' where ( 21t 1 xcn j = cos NJn ) Equation 827 and Equation 828 The solution of Equation 826 for b the least sum of the squares estimates of 42 the vector b, is well known and given as Equation 829
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~1 32 Thus. using Equation 829 it is possible to determine the least sum of the squares coefficients for the nonconcentricity and lobing signals. The effectiveness of the algorithm in removing a three undulation per revolution lobing error is shown in Figure 813. The error is removed in the grinding wheel motion coordinates and reported in the follower motion coordinates. 0 012 0 01 0.008 E E 0 006 ,_ 0 0 004 ,_ ,_ w 0 002 = :.:J 0 0 002 0 004 180 Removal of Lobing and Nonconcentricity Error nspection D:ita l\bnconcentricity and Lobing Effects Rerroved 1 50 1 20 90 60 30 0 ;, 30 60 Angle (degrees) 90 1 20 150 180 Figure 813 Removal of Nonconcentricity and Lobing Effects Effect of Lobing on Timing Error Noise Finally it is interesting to consider the random timing error that occurs due to the lobing effect. The timing angle bias due to the profile error shown in Figure 811 is calculated from Equation 420 as 0.0281 Since the bias can occur in either direction depending on the relative phasing of the lobing error and the cam lobe the total timing noise is twice this value or 0.0562
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CHAPTER 9 CONTRO L SYSTEM The Controlled Process In camshaft production the grinding of the journal bearing, the grinding of the cam profile and the superfinishing of both the profiles and bearings all affect the finished part geometry All three of these processes contain repeatable and nonrepeatable errors T he design objective of the control system is to correct for the repeatable errors while not excessively amplifying the nonrepeatable ones. The controlled process for cam manufacturing is shown schemat i cally in Figure 91 and Figure 92 for the two different cases considered in this work In Figure 91 the control loop in closed around the cam profile grinding operation. In this case, it is necessary to inspect the ground parts prior to optional superfinishing or polishing operations that follow the grinding process. This approach isolates the lift errors due to the cam grinding process from those due to the superfinishing process While this approach eliminates t he superfinishing noise from the feedback signal it also eliminates the repeatable portion of the lift error due to superfinishing. Therefore even if the parts are ground with zero lift error the finished parts will have an error due to the geometrical errors due t o the superfinishing process. 1 33 ~... 1
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CNC Cam Profile Grinder 1 Computer : Int e l x86 OS : Windows 3 11 or Motorola Chip Set OS : Proprietary Compensated Data via Disk or Network I System Controller Computer : Intel x86 OS : Windows 95/NT Ground Workpieces Inspection Oat ~ via Network Figure 91 Cam Profile Grinding Control System CNC Cam Profile Grinder Computer : Intel x86 OS : Windows 3 11 or Motorola Chip Se t OS : Proprietary i Compensated Data via Disk or Netwo rk System Controller Computer : Intel x86 OS: Windows 95/NT Ground .i Workpieces Inspection Data via Network l CNC Cam Gauge Computer : Intel x86 OS : MSDOS Cam Superfinisher Simple Sequence Controller I Superfinished Workpieces CNC Cam Gauge Computer : Intel x86 OS : MSDOS Figure 92 Cam Profile Grinding with Superfinishing Control System 134 Alternatively, as shown in Figure 92, the superfinishing operation can be included in the control loop and the system tuned to accommodate the noise due to the superfinishing process. In fact, even though the superfinishing process does contribute a source of noise to the feedback signal, it also eliminates some highfrequency noise in the profile inspection signal. This reduction is directly due to the improvement of surface finish, and reduction in chatter that 
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. ,, ___ .. ... .. .,_, ,, _ __ _____ ______ __ _ 135 superfinishing produces. The primary benefit of i ncluding the superf i nish i ng operation inside the process loop is so that the repeatable portion of the lift error due to the superfinishing can be compensated. In this approach a deliberate lift error is ground into the finished lobe shape such that after superfinishing the lift error is eliminated Controllers for both arrangements are developed in the following sections Physical Implementation The control system is designed to be compatible with existing CNC cam profile grinders and camshaft inspection gauges The i mplemented system automatically modifies the part program files that are used by the cam profile grinder to compute the internal commands for the individual servo axes. As described in Chapter 3 the cam lobe profile is specified for a given lobe type as the lift, si, of a cam follower of radius rr relative to the base circle radius, rb, for N discrete points such that O j N1. It is the industrial convention to specify this information in the lobe data file. Separately the lobe type and the axial positions and the angular orientation of the lobes are specified relative to axial and timing reference features in the part data file. Several other files are a l so necessary to completely specify the grinding process These files include the lobe workspeed file and the grinding parameter file. The workspeed file is the file used to specify the instantaneous angular velocity, 8 i The grinding parameter file specifies all relevant grinding settings such as the feedrate dress frequency, wheelhead speed etc. In typical industrial environments multiple cam grinders operate simultaneously while the parts produced on any cam grinder may be inspected on any one of a number of cam inspection gauges. Consequently, the designed
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r_ .. ~control system must allow the processes on multiple machine to be controlled simultaneously The number of machines controlled is limited only by t he inspection cycle time Control System Design 1 36 The control system is designed to treat the profile lift the lobe nonconcentricity the base circle size and the lobe timing as independent parameters As discussed previously i t is convenient to decouple these parameters since the decoupled parameters are used to descr i be the cam profile in both the CNC grinder and also the CNC cam inspection gauge This method also allows for the separation of physical error sources as discussed in Chapter 4 It is particularly useful to separate noisy error sources from r epeatable sources since, with the signals separated the control gains can be selected to best control the error due to each source. This situation arises when the la r ge and often nonrepeatable timing error is superimposed on a much smaller and highly repeatable profile error In this case the decoupling of these parameters allows for the design of both a stable control system for the noisy tim i ng measurement and tight control on the profile lift. Parameter Interaction and the Design of the Lift Control System : Grinder Only From the model of the cam profile grinding process as developed in Chapter 5 it is possible to simulate the effectiveness of a simple feedback control system where the lift of the follower or grinding wheel is controlled separately at N discrete points. Here N is the number of points used by the machine builder to specify the cam profile and is generally 360
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1 37 To simulate the response of the system the relevant equations used to describe the sparkout error are collected here. Recall from Equation 535 that the modeled radial distance at the end of the sparkout cycle is Equation 91 2 l pj l R s \ Pj l Rs where Moc\.n is the modeled radial distance for part n and Comr j, n is the commanded radial distance for part n. Also recall from Equation 331 that the radius of curvature of the cam profile surface at the point of contact j is Equation 92 From Appendix C Equation C14 the radial distance to the point of contact can be written as Equation 93 From these equations it is possible to simulate the interaction effects of the grinding process The control system and the process model are shown i n
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138 Figure 93. Here the direct and cross transfer functions are shown as G 0 G 0 1 and G 1 0 From the description of the grinding process given i n the above nonlinear equations it is apparent the functions G 0 G 0 1 and G 1. o cannot be represented directly Nevertheless the block diagram gives a hint as i s shown later to the relationship between the commanded lift at adjacent discrete points of contact. The form of this relationship is directly due to the difference equations that occur in the calculation of the location of the point of contact and the radius of curvature, r pocHowever for now this is simply a convenient way to represent the process and run the simulation. Simulation of a Simple Lift Control System For the simulation a proportional plus integral (Pl) controller with a one 5035 part delay is chosen The Z domain transfer function is G (z) Com~r(Z) K z 1 Ki z 1 C = Meas~r(Z) = p + 1z 1 Equation 94 where Z is the complex Z transform operator and Kp and Ki, are the proportional and integral gains respectively The one unit delay term Z1 arises since the inspection results for the current part are used to modify the commands for the next part. Taking the inverse Z transform of Equation 94 and solving for the discrete control action com ~r i, n gives Com ~r = Com ~r + K (Meas ~r Meas ~r ) + K. Meas i1r J n J,n 1 p J,n 1 J,n 2 1 j,n 1 Equation 95 where the subscript n refers to the dh part.
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1 39 '<. ..i G o 1 7 r I M e asM I Com6 r I + Com Act J 1 J ln J 1 n y r J 1 n G o + y + r J 1 n ... ~ r .; G e I ., t_ 1 +t G I l G 1 o i / G o1 r i Meas 6 f Co m 6 f + Com r + I + Act f J n J n J n :f J n G e G o + //. G 1 o / G o 1 I r j ... 1 M e as 6 r eom 6 Com A ct J + 1 n r j+1, n + r J~1, n + + r j~1, n G e G o + G 1 o / Figure 93 Lift Control System Including Process Model Table 91 Lift Control System Nomenclature a Recursive filter ain Control transfer function Gp 0 (Z) Control transfer function to eliminate effects of Gp 0 1 (Z) Control transfer function to eliminate effects of Gp 1 0 (Z) Magnitude of gr i nding step disturbance at point j for part n
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140 Table 91 Continued Dsf i n Magnitude of superfinishing step disturbance at point j for part n G Matrix of interacting transfer functions Gc(Z) t Controller Z domain transfer function Gf(Z) I Recursive filter Z domain transfer function Gcf(Z) Cascaded recursive filter and controller Gi(Z) I Gauge transfer function Gm(Z) i Combined machine tool and machine controller transfer function Gmt(Z) I Machine tool process transfer function Gmtc(Z), Machine tool internal controller Gsf(Z) I Superfinishing process transfer function G o(Z)' Direct process transfer function Go ,1( Z) i Cross process transfer function for interaction of camr i1.n and Ac\ 1 .n G1o(Z) Cross process transfer function for interaction of cam ri + 1 .n and Act ri.n j Subscript on quantities that vary with the lobe rotation angle, 0 i i k Dummy variable: offset from subscript i kc Combined sparkout constant kl Sparkout constant for linearized grinding model Ki Integral controller gain KP Proportional controller gain n Part number N Number of data points nc Nominal base circle nonconcentricity error (zero) Actnc n Actual base circle nonconcentricity error Comnc n Commanded base circle nonconcentricity FiftnC n Filtered base circle nonconcentricity Measnc n Measured base circle nonconcentricity rb Nominal base circle radius size rpoc i Radial distance to point of contact Rs Radius of grinding wheel rf Follower radius Act r bn Actual base circle radius size t Z is omitted in all diagrams for simplicity _J
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. ~~ ~141 Table 91 Continued Comr bi n Commanded base circle radius size Measr bn Measured base circle radius size ri Nominal radial distance Act r Actual radial distance J, n Actr Vector of actual radial distance Comr Commanded radial distance J,n Comr Vector of commanded radial distance Measr. Measured radial distance J,n Mod r i .n Modeled radial distance Simr Simulated radial distance J, n Si Follower lift u(nk) Unit step function delayed by k parts Wii n Disturbance signal due to inspection process Wpi,n Disturbance signal due to grinding process Wsfi,n Disturbance signal due to superfinishing process z Complex operator of Z transform Ui Lobe angle for point of contact Comi:\r J n Commanded radial compensation Fill,M Filtered radial error J n Meas~r Measured radial error J n Mod i:\r Modeled radial error J,n Sim~r Simulated radial error J n Com L\r Vector of commanded radial compensation eom~r bn Commanded base circle s i ze compensation Fill~r bn Filtered base circle size error Meas~r bn Measured base circle size error Meas~n Measured lobe timing angle error Comi:\q>n Commanded lobe timing angle difference Nominal lobe timing angle Act q>n Actual lobe timing angle Comq>n Commanded lobe timing angle ___ J
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142 Table 91 Continued F i lt
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. .1 43 The results for k c equal to 0 00 1 and k c equal to 0 002 are shown in Figure 9 4 and Figure 95 respect i vely 0 009 0 008 0 007 0 006 E 0 005 E 0 004 .... 0 .... 0 003 .... w 0 002 0.001 0 0 001 180 Simulation of Contro l System Under Ideal Conditions Si mulated Error 1 st Part Si mulated Er r o r 2 nd Part 1 50 1 20 90 60 3 0 0 3 0 6 0 Angle (degrees) 90 1 20 1 50 180 Figure 94 Simulated Control of Profile Grinding Error: kc=0.001 0 016 0 014 0 012 0 01 E g 0 008 .... 0 .... 0.006 .... w 0 004 0 002 0 0 002 180 Simulation of Control Sys t em Under Ideal Conditions Si mulated E r ror 1 st Part Simulated E rr or 2 nd Pa rt S i mu l ated E rr o r 3rd Part 1 50 12 0 90 60 30 0 30 Angle ( degrees ) 6 0 90 1 20 1 50 18 0 Figure 95 Simulated Control of Profile Grinding Error: kc=0.002
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The simulation shows that for the given cam geometry with kc equa l to 0.001 the error in the radial distance Act r i. n, i s less than the experimental l y 1 44 observed grinding process noise on the second part. For the case with kc equal t o 0.002 the error in the radial distance r i s reduced t o approximately the actual process noise on the second part. On the third part the error is less than the expected process noise The reason that the error is not completely eliminated on the first control loop closure is due to the interaction of the controlled parameters. By interaction it is meant that it is not possible to manipulate the parameters the radial distance Act ri n, at adjacent points independently. This is due to the relationship between the model radial distance Mod r i.n and the commanded radial distance From these equations it i s clear this relationship is complex and highly nonlinear Therefore it is useful to consider the linearized process model to understand this effect and develop a control strategy. Recall the linearized process model from Chapter 5 and change the notation to indicate the simulated lift si mr i, n, to get Equation 96 Comr + J 1,n Co mr j 1 n From Equation 96 it is apparent that s i mr i n depends on the commanded values f th d I d t Com Co m d Com h I Sim o e ra Ia Is ances ri n, ri 1 .n, an ri 1 n w I e errors In r i, n are compensated for by manipulating only the commanded value of co m r i, n. Thus the actual values of the controlled parameters ( simulated in this case) simri n, Sim r i1 .n and sim r i 1 .n are said to interact. Therefore it is not possible to l
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"~~~ ~1 45 manipulate the parameters independently using the simple control system of Figure 93. The design of a control system that decouples the commanded radial distancesa noninteracting controlleris the subject of the next sect i on However as the results of the simulations in this section show the complication of such a noninteracting system is only justified if the interaction effects are l arger than the process noise. Nevertheless it is of interest to investigate the i mplementation of such a system The Grinding Process Model and a Noninteracting Controller Design From the linearized grinding process model of residual sparkout error i t is possible to design a control system that eliminates the i nteraction among the interacting variables. With the interaction transfer functions known it is possible to determine the control equation for the noninteracting control elements that allow independent control of the radial distance r i at point j From Figure 96 and Equation 96 the interaction of the controlled parametersthe lift at adjacent pointscan be describe in terms of the cross transfer functions G 0 G 01 and G1. 0 The function can be written by inspection as kl G G =1 0 0 1 ( 2: Y Equation 97 and G 0 = 1Equation 98 I ~ ~~ ~ ~j
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1 46 The control prob l em can be reformulated in matrix form From Figure 96 the ideal plant is assumed and the process can be described as Equation 99 where Act r o G o G o 0 0 G o C om r o A c'r, G o, G o G o 0 0 C omr A ctr = G= 0 G o 1 G o G o 0 Comr = A ctr N2 0 0 G o G o G o Co m r N1 Act G to 0 0 G o Go Co mr r N 1j N2 Equation 910 
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L ~ 147 i D o , i G o , I [ Com t'J. I r J1 r / 1 n r 1 1 Meas t'J. Com t'J. + Com + i+ Act r J 1 n r J 1 n J I + + I r J 1 n G o r J 1 n ... G e __; e D 0 + l r I + i '+ I , G I =.'.J D , o G , o I i' / )x' D o., G o , r comt'J.r r i Meas t'J.r Co m t'J.r + J J,n Comr. + + Act r J,n G e in Do J,n G o J n + / D , o G o / D o G o C omt'J. r j+1 Comt'J. + r j+,r j+ , n Com Ad r i ~ 1 n + + r j + 1 n + + r j+1 n G e Do + G o + D o G , o + Figure 96 The Interacting Profile Grinding Process Model To solve for the vector eomr realize that the model only corrects for the errors that are due to the sparkout model. Since the model does not perfectly predict the ground profile errors, the remaining lift errors must be compensated for by the feedback of reom~r. Therefore the output, Ac1r, is set equal to rcom~r Also the output of Equation 99 i s set equal to the desired output. Solving for eomr gives ~~~ ~ I
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1 48 Equation 911 where C om f o n ~ f N 2 n f o n C om_1 f N 1n ' Equation 912 The matrix G is N by N and cyclic tridiagonal. Therefore Equation 911 may be solved efficiently by LU decomposition with forward and back substitution and the ShermanMorrison formula 27 Simulation of the Noninteracting Control System The performance of the noninteracting control system i s tested by simulating the grinding process using the same method as before but now Equation 911 introduced to correct for the interaction effects. Clearly, if the linearized model perfectly approximated the actual process model then all profile error should be eliminated from the simulation results. Howeve r, the linearized model only approximates the actual process model which of course only approximates the actual process Consequently some errors are not corrected by the noninteracting element of the control system and these errors must be compensated for by the feedback elemen t s Figure 97 shows the results of process simulation for the noninteracting control system that is shown in Figure 96. The simulation is run using the same grinding process parameters and the same cam profile as for the simulation of the simple controller shown in Figure 95. These results f or the noninteracting ~~ ~~
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149 controller Figure 97 can be directly compared with the results for the simple controller Figure 95. From this comparison it is evident that the performance of the noninteracting controller is inferior to that of the simple controller. E E ,._ 0 ,._ ,._ w 0.02 0 015 0 01 0 005 0 0 005 Simulation of Noninteracting Controller I 1st part i 2nd Part II II ,, I \ I I I I I : \ ,' \ I I / I "L,.., . \. ,' 1 ,.J ~ > '7" ~ : ., .,, ., .. .. v ... ... .... +,. ,. ,, .. ._ , ..,. ... . , \ I I , 11 ' o,: ... 11 II I I : .. . .. ~ II : : '. \ . 0 01 ~' 180 150 120 90 60 30 0 30 60 90 120 150 180 Angle (degrees) Figure 97 Simulation of Noninteracting Controller: Symmetrical Lobe The degradation in performance that occurs with the noninteracting controller under simulation is due to two factors The first factor is the use of the linearized process model to replace the actual sparkout out model. The linearized model is used to make the solution to the control equation a tractable problem. The second factor is the numerical noise that arises in the solution of the control equations. Figure 98 shows the raw and filtered commanded compensation for the noninteracting controller. This figure shows that commanded compensation is corrupted by noise. This noise is occurs in the solution of Equation 911 and must be filtered before it can be used to update the commanded profile. While, the noise is a narrowband component and can be filtered from the compensation signal., it nevertheless corrupts the commanded compensation to some degree. ~ ~ 
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~1 50 ~1 Noise and Filtering for Noninteracting Compensation 0 015 ,, 0 02 Filtered 0 025 '' 180 15 0 12 0 90 6 0 30 0 3 0 60 90 120 1 50 1 80 Angle (d egrees) Figure 98 Noise in Noninteracting Compensation In conclusion it is apparent that the noninteracting controller as tes t ed does not perform better than the simple controller. Recall that the simple controller effectively reduces the residual sparkout error to a level below the process noise on the next part. Therefore the noninteracting controller i s no t pursued further Simulation of the Simple Lift Controller for Grinding Under Noisy Conditions In practice all processes and sensors have a nonrepeatable error component. In the first simulation runs no noise was assumed to be present in the process or in the measurement of the controlled variables In this case an integral control element with the gain K, equal to one effectively eliminated the effect of a step disturbance on the next part However under noisy conditions an integral controller amplifies the noise in the output compared to the open loop 52 process noise The amplification of the process noise can be reduced through the introduction of a discrete first order recursive filter However along with the
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l reduction in noise amplification a delay is introduced into the system From 4 3 Chapter 8 the Z transform of the filter i s FittM(Z) a Gf(Z) = MeasM(Z) = 1+( a1)z 1 Equation 913 Inverse Z transforming Equation 913 and solving for the filter difference equation gives Filt,1r. = a Meas,1r + (1a) Filt,1r, J, n J,n J,n1 Equation 914 1 51 The system diagram shown in Figure 93 can now be redrawn as Figure 99, where the filter Gt, is added in series with the process controller, Ge, and the interaction effects are replaced by a disturbance Wpi.n The inspection disturbance, Wi i,n, is added for completeness. The disturbance signal model of the inspection process, Wi i, n is a random component alone The process transfer function is separated into the transfer function for the machine tool controller, Gmtc, and the transfer function for the machine tool Gmt I I Wp j n Meas Ar. Fil l Ar C om Ar Com r r ; f , n ; t, n + + r~ &1 ~ Measr. + _, ,, n Wi i n Figure 99 Grinding Process Simplified Control System i
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, The disturbances Wp i ll and Wi i n are Equation 915 Wi ill = Yi i ll Equation 916 where Dp i is the magnitude of a step input into the grinding process. The process and inspection processes have random components Y p i, n and Yi ill respectively The random signals Ypi n and Yiill are independent white noise signals which are bandlimited by the Nyquist frequency of 0.5 cycles/part 1 52 For the control system shown in Figure 910 the filtered lift error FinMi,n, replaces the measured lift error Meas ~ri,n as the input to the controller Ge, and the controller difference equation given earlier as Equation 95 becomes Com A Com A K ( Filt A Fi lt A ) k Filt A u r i. n = u r j, n 1 + p urj n 1 uri n 2 + i u r i,n 1 Equation 917 Alternatively, the transfer function for the recursive filter Equation 913 and the transfer function for the proportional plus integral controller Equation 94 can be cascaded to get the combined controller transfer function G cr( Z) Again the filtered error variable, Fifl~ri,n replaces the measured error variable Meas~r ill in Equation 94 With this substitution the combined controller transfer function is Filt~r(Z) Co m ~ r(Z) Com~r(Z) Gcf (Z) = Gr (Z)Gc (Z) = Meas ~ r(Z) Filt ~r(Z) = Meas ~r(Z) Equation 918 
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Substitution of the individual transfer functions into Equation 918 gives com~r(Z) ( K )( a ) G c f(Z)= Meas~r(Z) = Kpz 1 + 1~ 1 z 1 1 + (a1)z 1 Equation 919 The block diagram in Figure 99 can be rearranged to give the relationship between the process disturbance Wpi n, and the output Act ri.n. I f 153 Gmtc, Gm1, and Gi are assumed equal to one then Figure 99 can be rearranged as shown in Figure 910. 1 1 + G 1 (Z)Gc (Z) Figure 910 Rearranged Block Diagram for Disturbance Rejection From Figure 910 the transfer function between the disturbance and the output is Ac\(Z) 1 =Wpi(Z) 1 + Gt(z)G 0 (Z) Equation 920 Substituting Equation 919 into Equation 920 and simplifying gives Ac\(Z) 1+(a2)z 1 +(1a)z 2 Wpi(Z) = a+{a(kp + k i +1)2}z 1 + {1a(1+kp)}z 2 Equation 921 51 For an ideal system with no noise present Bollinger and Duffie show that the choice of control parameters of a =1 ki =1 and kp =0 is optimal for
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1 54 rejection of a step disturbance T his is easy to verify by substituting these values into Equation 921 to get Act r (Z) J 1 z 1 Wp i (Z) = Equation 922 Taking the inverse Z transform and solving for the system response gives Equation 923 If the process disturbance Wp i,,, is equal to a step input with magnitude Dpi,,, then from Equation 923 it is apparent the effect of the disturbance Wpi,,, is completely eliminated on the next part The system response is that of a deadbeat controller However if the process disturbance contains a noise component i n addition to the step the integral controller given by Equation 922 amplifies the process noise. In the previous simulations it was convenient to ignore the effects of noise in the system in order to study the interaction effects independently However it is clear from comparing the weakness of the interaction effects demonstrated in Figure 94 and Figure 95 with the process repeatability errors given in Table 92 that the error due to the noise is greater than the error due to neglecting the interaction effects Therefore, the control system is designed to eliminate the repeatable error the step disturbance while not excessively amplifying the nonrepeatable error. As noted above the standard deviations for the lift error, as well as all other controlled errors are given in Table 92. This table gives the process I _ __ J
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___._ __ _ 1 55 repeatability errors in terms of the measured standard deviation cr. The values in this table were determined from a series of experimental trials The tables give the repeatability of the following process or combinations of processes: the inspection process the combined grinding and inspection processes and finally the combined grinding, inspection and superfinishing processes. With the process model as cast in Figure 99 the function of the controller is to reject process disturbances. Specifically, the controller should reject the repeatable portion of the disturbancethe stepwhile not excessively amplifying the nonrepeatable portion of the disturbance signal. With the introduction of the filter delay into the system it is necessary to select a nonzero proportional control term kp, to ensure that the system does not oscillate about the control point. Thus, three system parameters need to be chosen : the recursive filter gain a; the proportional controller gain kp, and the integral controller gain, k, While it is possible to use different methods to select the best values for these control system parameters they can be readily selected by trial and error through simulation of the control system using Equation 919. This is the approach taken in this research To select the controller and filter gains the ability of the controller to reject disturbance of the form given by Equation 915 is simulated. The control of the lift at a specific point on the cam profile is simulated using the observed process nonrepeatability error Table 92 gives the magnitude of the nonrepeatable errors for the processes considered here. The magnitude of the step function used for the simulation trials is based on observed lift errors This simulation assumes no interaction between controlled parameters i.e. lift at adjacent discrete points. The simulation results shown in Figure 911 indicate that the controller, given in Equation 919 successfully eliminates the initial offset on the first part. Furthermore the simulation shows that the process noise I I ~ ~ _J
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156 is moderately amplified compared with the open loop system. The selected controller gains are given in Table 93. Table 92 Repeatability for Grinding, Superfinishing and Gauging RepeatBase + Non Lift Timing Timing Timing ability circle concentri city to lobe to dowel to (combined) size one keyway (cr} mm mm mm degrees degrees degrees Grind and 0 016 0.003 0018 0 12 0.025 0.7 Gauge Superfinish Grind and 0.013 0 003 0 0022 0.13 0.025 n.a. Gauge Gauge .002 < 0.0005 0.0005 < 0 01 < 0.01 0.25 Simulated Control of Lift 25 04 r, I 25 03 25 02 o Q. 25 01 C1l 25 :.0 24 99 C1l = 24 98 :::i 24 97 +,.....__ __.._ ___________________ __, 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1 8 19 20 Part Number Figure 911 Simulated Control of Lift at an Arbitrary Point Over N Parts :j: For worst case: lobe closest to workhead _J
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1 57 Table 93 Controller Gains Determined by Simulation for Noisy Conditions Gain Base circle NonLift Timing to T i ming to Timing to size concentri city lobe one dowel keyway kp 0.45 0.45 0.45 0.7 0 35 0.35 k1 1 0 1 0 1 0 0.7 0.2 0.2 a 0 7 0.7 0 7 0.6 0 2 0 2 Including Superfinishing in the Control Loop It is also of interest to consider the inclusion of superfinishing process in the control loop. In this case the parts are inspected only after the superfinishing operation. The block diagram for this setup is shown in Figure 912. Meas r. + W I l i, n Figure 912 Grinding and Superfinishing Simplified Control System Again the error due to the superfinishing process i s modeled as a disturbance. As with the grinding process the superfinishing process disturbance Wsfi 11 can be described as a step input plus a random component as Equation 924 where Dsfi is the magnitude of the lift error due to superfinishing at point j. The random signal Ysf i, n is white noise that is bandlimited by the Nyquist frequency of 0.5 cycles/part The random component of the superfinishing process disturbance Ysfi n is spec i fied in terms of 3cr in Table 92. Again the gains for
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, 158 this system can be chosen through trial and error from simulation From Table 92 it is clear that the process repeatability for the combined superfinishing and grinding operation is the same as the process repeatability for the grinding process alone and therefore the gains selected in the previous simulation apply. Base Circle Radius Controller The block diagram for the control of the base circle radius is shown in Figure 913. Again, the problem is that of rejecting a disturbance that consists of a step plus a random component. The terminology follows that of the lift controller closely and is defined explicitly Table 91. Mea~r bn + Meas!'. + bn Comr + bn Figure 913 Base Circle Radius Controller Wp" Act!'. bn Simulation of the Base Circle Radius Controller Under Noisy Conditions For process and sensor repeatability errors given in Table 92 the control system parameters are selected from simulation results The simulation results are shown in Figure 914 for the controller settings given in Table 93 and an initial disturbance of 0.1 O mm. The results of the simulation, show that the controller successfully eliminates the initial offset on the first part and that, again the process noise is moderately amplified compared with the open loop system.
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1 59 Base Circle Nonconcentricity Controller The block diagram for the control of nonconcentricity error is shown in Figure 915 . Again the problem is that of rejecting a disturbance that consists of a step plus a random component. Note the command feedforward loop is eliminated since the commanded value of the nonconcentricity is always zero. The terminology follows that of the lift controller closely and is defined explicitly in Table 91 Simulated Control Base Circle Size Error 25 1 ,....._ 25 08 E 25 06 E Q) 25 04 N i:i5 25.02 Q) 2 5 u ... c3 24 98 Q) 24 96 (/) co CD 24 94 24.92 2 3 4 5 6 7 8 9 10 11 12 13 1 4 15 16 17 18 19 2 0 Part Number Figure 914 Simulated Control of Base Circle Size Error WP n Com nC C ~ I n r+ 1 f1 G m1c 1 G ~ r~< ., ~Meas nc + 'r ~' n I I Wi n Figure 915 Base Circle Nonconcentricity Controller
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1 60 Simulation of the Nonconcentr i city Controller Under Noisy Conditions For these conditions the control system parameters are selec te d based on process simulation Th e simulation results are shown in Figure 916 for the controller settings given in Table 93 where the initial disturbance is 0.0 1 mm The results of the simu l ation indicate that the controller successfully elim i nates the initial offset and that the process noise is not amplified. Simulated Control of Nonconcentricity Error 0 01 0 008 ,,...... 0.006 E E 0 004 ._, ,.__ 0 0 002 ,.__ ,.__ w 0 0 002 0.004 1 2 3 4 5 6 7 8 9 1 0 11 12 13 1 4 15 16 17 18 19 20 Part Number Figure 916 Simulated Control of Nonconcentricity Error Lobe Timing Angle Controller The block diagram for the control of the lobe timing angle , is shown in Figure 917. In this chapter the lobe number subscript on that was introduced in Chapter 3 is dropped for the sake of notational simplicity The control problem here is one of rejecting disturbances for the timing angle for each lobe over a number of parts. The notation used closely follows that of the lift controller and is defined explicitly in Table 91 ___ J
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r161 W Pn ,.. 1 Me as A,l,. Filt t,,.. Com t,~ Com,.. l 'I' I '>'f'n ci 'l'n n + 'l'n + + .,. .i G 1 1 1 G e~ G mtc+ G mt + L'+ ., ~~~ ., G ~ + i 'Meas ~n Figure 917 Lobe Timing Angle Controller S i mulation of the Lobe Timing Angle Controller Under Noisy Conditions The timing angle controller is designed to accommodate both the repeatability of the lobe timing angles relative to lobe one and the lack of repeatability of the timing angles relative to the camshaft timing datum Generally the repeatability of the t i ming angle to the camshaft timing referencesuch as a keyway or dowe l pinis poor compared with the repeatabil i ty of the lobes relative to one another. Therefore t iming of lobe one is controlled relative to the camshaft timing reference while the timing of all other lobes is controlled relative to lobe one In this section the timing controller is simulated f or three different situations : timing relative to a keyway timing relative to a dowel pin and timing relative to lobe one The simulation results are shown in Figure 918 for the controller settings given i n Table 93 and the process noise values given in Table 92 The initial errors for the timing relative one relative to the dowel pin and relative to the keyway are 0 01 0 25 and 1 2 respectively ~ ~
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1 62 Simulated Control of Lobe Timing Angle 1 .2 _._ liming Angle Relative to An U) 0 8 &liming Relative t o Lobe 1 Q.l 0 6 Q.l .... C) 0.4 Q.l "C 0 2 .... .... 0 .... 0 .... LU 0 2 0.4 0.6 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Part Number Figure 918 Simulated Timing Controller Summary In this chapter the various controllers are investigated and the control gains are selected. In Chapter 10, the results of industrial trials, based on the control system designs discussed here are presented. ~
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CHAPTER 10 EXPERIMENTAL RESULTS Description of Experimental Setup The experimental results presen t ed in this chapter were obtained a t Dana Corporation Perfect Circle Division s camshaft manufacturing plant in Russellville Arkansas. The parts were ground on various Landis 3L T M Series cam profile grinders equ i pped with steady rest type work supports The gr i nding wheel was a Beck B12649RO200110 81 (CBN) with a 225 mm radius The wheels were dressed with a rotary diamond dressing attachment. The parts were inspected on an Adcole Model 911 cam inspection gauge A sixcylinder heavyduty diesel camshaft with integral diesel injection lobe was the test workpiece for the first two series of tests The workpiece material is SAE 1513 fine grain steel carburized to an effective depth of 1 5 mm with a minimum hardness of 58 Rockwell C. The manufacturing and inspection program sequences are included as Appendix D. The initial trials are of the simple control system tuned as determined by the simulation trials presented in the Chapter 9. Initially the profile grinding process alone is controlled Then in a separate trial, the profile grinding process and the superfinishing process are both included inside the control loop Control of the Radial Grinding Process Alone In this trial, the camshafts are inspected immediately after profile grinding The control system settings are as shown Table 101 1 63 ~ ~
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1 64 Table 101 Gains for Profile Grinding Gain Base circle NonLift Timing to Timing to lobe size concentricity dowel one k p 0.45 0.45 0.45 0 35 0 7 k, 1.0 1 0 1. 0 0.2 0.7 a 0 7 0 7 0 7 0 2 0 6 Figure 101 shows the measured lift error for the part ground with uncompensated or nominal part program data. From these inspection plots four distinct errors sources are immediately visible. The first error is the residua l sparkout grinding process error which is of the form shown in Chapter 5 The second error is the nonconcentricity error. This error is largest for t he lobes nearest the workhead sp i ndle and is recognized as a periodic error at the fundamental frequency of one upr The third error is also periodic but with a frequency of three upr and is most easily observed in the base circle region. As discussed previously this error results from the axis shift that occurs due to lobed journal bearings running in the steady rest work supports Finally the forth error is the high frequency error component. Also, while not actually an error the reported process error is dis t orted due to a bias in the reported timing angle for lobes where significant nonconcentricity errors are present. Th i s effect is visible for the first exhaust injector and intake lobes Figure 102 shows the results for part two : the first part ground with compensated data After the first loop closure the lift error is reduced to nearly the level of the process noise The residual process noise that remains i s in the region where the process error was originally the greatest. Figure 103 shows the results for part three : the part ground based on t he inspection results from the first and second part The results for the exhaust and injector lobes are better wh i le the results for the intake lobes are slightly worse l ___________________
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165 Figure 104 shows that the results for part four are generally worse than for part three. Furthermore by comparing the results for successive parts it is recognized that the system is oscillatory for lobes with significant error. This oscillation is most obvious for intake lobe number five on the third and fourth parts. Also, as expected for the implemented control system design, the magnitude of the lobing error is amplified. Figure 105 summarizes the results shown in Figure 101 through Figure 104. These plots show the standard deviation crr of the measured lift error, Meas~. The standard deviation of the lift is defined here as 30 (5 = r N 1 2 2:(Meas~rj ~r) j = O N1 Equation 101 where N 1 ~r = L Meas ~rj j = O Equation 102 Total Lift Error = Maximum(~r) Minimum(~r) Equation 103 Figure 105 clearly shows the general trend of the control response From the plot, it is apparent that most of the initial error is eliminated on the first part. It is also apparent that the standard deviation increases after part number three. This excessive amplification of the process noise is due to the fact that the controller gains are too aggressive for the existing lobing error. The gains used in this trial were selected based on repeatability tests for parts that did not exhibit the large nonrepeatable lobing error of the journal bearings seen here. 
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7 1 66 With the existing controller design th e only way to reduce the noise amplification is to increase the filtering and adjust the integral and proportional gains accordingly This solution is undesirable as it makes the controller more sluggish The better solution as discussed in Chapter 8 is to filter lobing effects from the lift inspection data. Figure 106 shows the response of the control system over six parts for the base circle size, the timing to lobe one and the timing to the camshaft reference which is a dowel pin for this workpiece From the top plot in the figure it is seen that the base circle controller eliminates the large initial size erro r on the first pass. However, the spread in the results for the remaining parts exceed the expected process repeatability error of 0.006 mm The middle plot of Figure 106 shows the timing of all lobes relative to lobe one In the tested workpiece lobe one is first exhaust lobe. The initial error is small compared to the expected process repeatability error of 0.025 Again the control system seems to amplify the expected process error somewhat. Finally the bottom plot of Figure 106 shows the ti ming of all lobes relative to the camshaft timing reference. For this parameter all parts lie within the expected repeatability error of 0. 1 25 I I I I ____ __J
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. ~1 67 Uncompensated Exhaust Lobe Follower Lift Error : Part 1 0 02 0 015 0. 01 E 0 005 E g 0 0 005 LU . ;: .:. .. ~ __ .. \ \ ,, 0 01 0 015 1 I 3 4 5 6 \ '_, 0 02 180 1 50 1 20 90 60 30 0 3 0 60 90 1 20 1 50 1 80 Angle ( degrees ) Uncompensated Injector Lobe Follower Lift Error : Part 1 0 02 0 015 0 01 E 0.005 E ... 0 0 ... ... 0 005 LU .0 01 0 015 \ / ~, \ / 1 42 1 I 3 5 6 0 02 1 80 1 50 1 20 90 60 30 0 3 0 60 9 0 1 20 1 50 1 80 Angle ( degrees ) Uncompensated Intake Lobe Follower Lift Error: Part 1 0 02 0 015 0 01 E 0 005 E ... 0 g 0 005 LU 0 01 0.015 0 02 1 I 3 4 5 6 180 1 50 120 90 60 30 0 30 60 90 1 20 150 1 80 Angle ( degrees ) Figure 101 Uncompensated Error: Part 1 after Grinding: Lobes 16 ______.._ ~
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1 68 Compensated Exhaust Lobe Follower Lift Error : Part 2 0 02 0. 015 0.01 E 0 005 E .... 0 e w 0.0 05 0 01 1 .... ... 2 3 4 0. 015 .. ..... 5 6 0 02 ~' 180 15 0 12 0 90 60 30 0 3 0 6 0 90 120 15 0 1 80 Angle ( degrees ) Compensated Injector Lobe Follower Lift Error : Part 2 0.02 0.D15 0 01 E 0 005 E .... 0 0 .... .... 0 005 w 0 01 1 3 4 0 015 ....... 5 6 1 0 02 1 80 1 50 12 0 90 60 30 0 3 0 6 0 9 0 12 0 15 0 18 0 Angl e (d egrees ) Compensated Intake Lobe Follower Lift Error : Part 2 0 02 0 015 0 01 E 0 005 .s 0 e uj 005 .0 01 1 ... .. ... 3 4 0. 015 5 6 .0. 02 1 80 50 120 .9 0 60 30 0 30 60 90 12 0 15 0 1 80 Angle ( degrees ) Figure 102 Compensated Error: Part 2 after Grinding: Lobes 16 ~~ ~~~
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~ .__ ____ E E ._ e ._ w Compensated Exhaust Lobe Follower Lift Error: Part 3 0 02 0 015 0 01 0 005 0 \ .. ;.0 005 0 01 1 1 0 015 . .. .. 3 4 1 .... .. 5 6 1 0. 02 1 80 15 0 120 9 0 60 3 0 0 30 60 90 12 0 150 180 Angle (d egrees ) Compensated Injector Lobe Follower Lift Error : Part 3 0.02 ., 0 015 0 01 E 0 005 .s e .... .. w 0 005 0 01 0 015 1 _____ 1 1 1 3 ...... 5 6 1 0 02 ~' 180 15 0 12 0 9 0 6 0 3 0 0 3 0 60 90 12 0 15 0 1 80 Angle ( degrees) Compensated Intake Lobe Follower Lift Error : Part 3 0.02 0.D15 0 01 E 0 005 E 0 0 t:: 0 005 w 0 01 I 1 ..... 2 I 0 015 I 3 I ' 5 4 1 6 0 02 180 1 50 12 0 9 0 60 3 0 0 30 60 90 120 150 18 0 Angle (d egrees ) Figure 103 Compensated Error: Part 3 after Grinding: Lobes 16 1 69
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. .. 0.02 0.015 0 01 E 0 005 E ... 0 e w 0 005 0 01 0 015 0 02 180 0.02 0 015 0 01 E 0 005 E l5 0 ... w 0 005 0 01 0 015 0 02 180 0 02 0 015 0.01 E 0 005 .s 0 ... e w 0.005 0 01 0 015 0 02 180 Compensated Exhaust Lobe Follower Lift Error : Part 4 1 2 1 3 4 1 . .. . 5 6 1 150 120 90 60 30 0 30 Angle ( degrees) 60 90 120 150 Compensated Injector Lobe Follower Lift Error: Part 4 : , I 3 4 . . .... 5 6 150 120 90 60 .30 0 30 60 90 120 150 Angle (degrees ) Compensated Intake Lobe Follower Lift Error: Part 4 1 2 1 I ~.~ ~ .~ ~~ ~~: I 150 120 90 60 30 0 30 60 90 120 150 Angle (degrees) Figure 104 Compensated Error: Part 4 after Grinding: Lobes 16 170 180 180 180 ~~, ~ ~
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0 005 0 0045 E 0 004 __ 0. 0035 C: 0 0 003 .;::, (1J ;; 0 0025 Q) 0 0 002 (1J 0 0015 0 C: .l!! 0. 001 (/) 0. 0005 0 0 006 E 0 005 _. C: 0.004 0 ii; ;; 0 003 Q) 0 1:! 0 002 (1J 0 C: (1J in 0 001 0 0 006 E 0 005 .s C: 0 004 Q iii s: 0 003 Q) 0 E 0 002 (1J 0 C: (1J in 0 001 0 \. \. \. ... "Exhaust Lobe Standard Deviation of Lift Values 1 ... ..... 3 4 .. ... 5 6 1 . ~~ ~~~ ~~ :: ~ : ~~...: : ~: ~:: : = ;_:;_:_....;.?. ~:?:. : ;.=.: .: .,.,,' 2 3 4 5 6 Part Number Injector Lobe Standard Dev i ation of Lift Values ' '~ ' ' ' ... ... '~ 1 , I 3 4 ....... 5 6 .' . ....~ .. "'::::::. 2 3 4 5 6 Part Number Intake Lobe Standard Deviation of Lift Values ' .. 1 1 1 3 4 5 6 2 3 4 5 6 Part Number 17 1 Figure 105 Standard Deviation of Lift Error after Grinding 
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1 72 Base Circle Radius Size Error 0 06 0.05 0 04 E 0 03 ..s 0.02 .... 0 01 e .... l :::~ lJ.J 0 I "" Ii 0 01 ti I 0 02 .. 0 03 2 3 4 5 6 Part Number Timing Error of All Lobes Relative to Lobe 1 0 2 0.15 en
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~, Control of the Radial Grinding Process and the Superfinishing Process In this trial the camshafts are inspected after profile grinding and superfinishing. The control system setting are as shown in Table 102. Table 102 Gains for Parts Inspected after Superfinishing 173 Gain Base circle NonLift Timing to Timing to lobe size concentri city dowel one ko 0.45 0.45 0.45 0.35 0.7 ki 1 0 1.0 1 0 0.2 0 7 a 0.7 0.7 0.7 0.2 0.6 Figure 107 shows the measured lift error for the part ground with uncompensated or nominal part program data. From these inspection plots the four error sources cited earlier are still recognized, however an additional error due to the nonuniform material removal of the superfinishing process is now present. This error is most apparent at the lobe nose ( zero degrees}, where a steep downward spike is introduced. As discussed previously, the nose of the cam tends to be rubbed off by the superfinishing process Figure 108 shows the results for part two: the first part ground with compensated data After the first loop closure the lift error is reduced by approximately 60% yet the sparkout residual error is still visible above the process noise As before the residual process noise that remains, is in the region where the process error was originally greatest. Figure 109 shows the results for part three: the part ground based on inspection results from the first and second part. These results show further reduction in the lift error. Figure 1010 shows that the results for part four are generally better than for part three. ~,_____ ________..... 1
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1 74 Figure 1011 summarizes the results show in Figure 107 th rough Figure 1010 in terms of the standard deviation of all N discrete data points These results more c l early show the general trend of the con t rol response. From Figure 1011 it is easy t o recognize the continuous reduction in the lift error in all lobes except the first intake lobe On th i s lobe the standard deviation is increased due to an increase in the nonconcentricity error Figure 1012 shows the response of the control system over six parts for the base circle size the timing to lobe one and the timing to the camshaft reference which is a dowel pin for this part. From the top plot in the figure it is seen that the base circle controller overshoots on the first pass and then gradually converges on the control point. The spread in the results for the remaining parts is approximately equal to the expected process repeatability error of 0 013 mm The middle plot of Figure 1012 shows the timing of all lobes relative to lobe one. In the tested workpiece lobe one is the first exhaust lobe Again the initial error is small compared to the expected process repeatability error of this parameter which is 0.025 and as for the ground only parts the control system seems to amplify the expected process error. Finally the bottom p l ot of Figure 1012, shows the ti ming of all lobes relative to the camshaft timing reference. For this parameter the initial error is corrected on the first pass and all parts lie within the expected repeatability error of 0.125
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E E .... 0 .... .... w E E .... 0 .... w E E .... g w 0 02 0 015 0 01 0 005 0 0 005 0 01 0.015 Uncompensated Exhaust Lobe Follower Lift Error : Part 1 \. 1 1 3 4 1 1 6 1 :( : . : ~_,, ~::::;.! ,.. __ u \ ,,./ 0 02 ~'' 1 75 180 50 1 20 90 6 0 30 0 30 60 90 12 0 150 180 0 02 0 015 0.01 0 005 0 0 005 0 01 0 015 0 02 180 0 02 0 015 0 01 0 005 0 0 005 0 01 0 015 Angle ( degrees) Uncompensated Injector Lobe Follower Lift Error: Part 1 1 , I .. ...... 3 4 ... .. 5 .. .... . 1 50 0 0 0 I I . ' 0 0 30 60 90 12 0 150 Angle ( degrees ) Uncompensated Intake Lobe Follower Lift Error : Part 1 L 1 I 1 2 1 4 1 1 .. 5 6 1 ._ ._ \. __ .,_ 1 80 0 02 ~' 180 15 0 12 0 90 60 30 0 30 6 0 90 12 0 150 1 80 Angle (degrees) Figure 107 Uncompensated Error: Part 1 after Superfinishing: Lobes 16
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i 1 76 Compensated Exhaust Lobe Follower Lift Error : Part 2 0.02 0 015 0 01 E 0 005 __ 0 .... g 005 .. w ,. .._ i, 01 015 l =1 ....... 2 3 4 1 5 6 1 0 02 120 .9 0 30 0 30 60 90 1 20 1 50 180 Angle ( degrees) Compensated Injector Lobe Follower Lift Error : Part 2 0 02 0 015 0 01 E 0 005 /......_ ,' I J h ,,, \ E .... 0 g 0 005 w w,1 .. ~, l ... .. . ,~ ... '"' ':~. :_. 1r '" ,,t,,1"'0 01 1 1 2 1 .. . . .. 3 4 6 015 02 .,__ ______________________________ ___. 180 150 12 0 90 0 30 60 9 0 12 0 150 180 Angle (degrees) Compensated Intake Lobe Follower Lift Error : Part 2 0 02 0 015 0 01 E 0.005 __ 0 .... e w 0 005 .Q.01 0 015 '1 1 2 4 .. ... .. 3 \ 5 6 1 .0 02 180 .120 .90 .so 30 0 30 60 90 1 20 150 180 Ang l e (degrees) Figure 108 Compensated Error: Part 2 after Superfinishing: Lobes 16
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1 77 Compensated Exhaust Lobe Follower Lift Error : Part 3 0 02 0 015 0 01 E 0 005 E g 0 w 0 005 ,,,,,._,,,.1...: ,. ' .. 0 01 1 .. ...... 3 0.015 . ... 5 0.02 180 1 50 1 20 90 60 30 0 3 0 60 90 1 20 150 1 80 Angle ( degrees ) Compensated Injector Lobe Follower Lift E r ror: Part 3 0 02 0 015 0 01 E 0 005 E .... 0 0 l" r _..., : "' '.. .,. _ ,,...lt,..I .... .... 0 005 w 0.01 0 015  ' I 3 4 .. .. 5 6 0 02 180 1 50 1 20 90 60 3 0 0 30 60 90 1 20 150 180 Angle ( degrees ) Compensated Intake Lobe Follower Lift Erro r: Part 3 0 02 0 015 0 01 E 0 005 S 0 ... g 0 005 w !O J ,::.. .N..c... 0 01 1 2 1 .. .. ... 3 4 0 015 __ _ .. 5 ____ 6 0 02 L = := :=:= := := := := :..._ ___________________ __J 180 1 50 120 90 60 3 0 0 30 60 9 0 12 0 1 50 1 80 Angle ( degrees ) Figure 109 Compensated Error: Part 3 after Superfinishing: Lobes 16 L __ ____ ~~~ ~'
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0 02 0 015 0 01 E 0 005 E g 0 0 005 w 0 01 0. 015 0 02 18 0 0 02 0.015 0 01 E 0 005 _.. 0 .... e w 0 005 0 01 0 015 02 180 0 02 0 015 0 01 E 0 005 E .... 0 e .... 005 w 0.01 015 02 1 80 Compensated Exhaust Lobe Follower Lift Error : Part 4 1 .... .. 2 . . .. 3 4 .. . ... 5 6 150 12 0 9 0 0 3 0 0 3 0 6 0 90 120 1 50 180 An gle (d egrees ) Compensated Injector Lobe Follower Lift Error : Part 4 1 1 3 4 1 1 5 6 1 150 .12 0 .g o 6 0 30 0 3 0 60 9 0 12 0 15 0 18 0 Angl e ( degrees ) Compensated Intake Lobe Follower Lift Error: Part 4 ....... 2 1 3 4 1 5 6 1 .1 50 .12 0 .g o 3 0 0 30 Angle ( degrees ) : : 60 90 12 0 15 0 18 0 17 8 Figure 1010 Compensated Error: Part 4 after Superfinishing: Lobes 16 i ___ __J
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E E C 0 5 (1) 0 0 ... (IJ 0 C (IJ in E E C: 0 :.:, ('CJ 5 Q) D 0 .... ('CJ 0 C: ('CJ (/) E S C 0 .::, ca 5 Cl) 0 0 ... ca 0 C: en Exhaust Lobe Standard Deviation of Lift Values 0 006 0 005 0.004 0 003 0 002 0 001 0 0.006 0 005 0.004 0.003 0 002 0 001 0 0 006 0 005 0 004 0 003 0 002 0 001 0 ... l 1 3 ....... 5 .... 2 4 6 ... _;;~_ ... .. ~. _._:;:_ ~= ~:=~ :~ ~ ~ ... ... :; ... : ,;; # ~ ........ :. 2 3 4 5 Part Number Injector Lobe Standard Deviation of Lift Values .. .... ._ .. .. ........... , 1 1 3 . . 5 ! I 6 1 .. ..... ........... ,.. 411#"' .... ....... ...... ~ ,,, : ,.._:. ~;'? :..=:_. :. "'7 == ::::.~.... L~. . .... : : :_:.; ~. .. . . ..,,. . ..... .. .. ;:: :.:;: ' ... :. : 2 3 4 5 Part Number Intake Lobe Standard Deviation of Lift Values 2 3 Part Number 1 1 2 I 3 4 1 1 ..... 5 6 4 5 Figure 1011 Standard Deviation of Lift Error after Superfinishing 1 79
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1 80 Base C i rcle Rad i us Size E r ror 0.1 2 t 0 1 0 08 ~, E 0. 06 E g 0 04 0 02 w 0 0.02 0 04 !I I I i 2 3 4 5 Part Number Timing Error of All Lobes Relative to Lobe 1 0 2 (/) Q) 0 15 Ol 0 1 Q) ... 0 05 0 ... ... w Ol 0 C E 2 i= 0.05 t I 1111 0 1 1 2 3 4 5 Part Nu mber Timing Error of All Lobes Relative to Dowel Pin 0 1 (/) 0 05 i Q) Q) I ... Ol 0 Q) jf "O 0 05 ... e 0 1 I ... w Ol 0. 15 t C E i= 0 2 0.25 111 2 3 4 5 Part Number Figure 1012 Base Circle Size and Timing after Superfinishing
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1 81 Discussion of Results from First Round Testing The journal bearing of the camshafts in these trial were lobed as a result of vibrations present in the journal bearing grinding process From the perspective of process control based on postprocess inspection results lobing is a source of noise in the process control signal. The lobing error observed in this trial was not present in the original trials that were conducted to establish the process nonrepeatability levels. Consequently, the nonrepeatability values used to tune the control system understated the actual noise levels. Since lobing errors were not present during initial repeatability trials no provisions were made to separate this source of noise from the repeatable components of the inspection signal. Consequently the control system tuned for lower noise levels tends to overcompensate somewhat. However as previously discussed in Chapter 8 it is possible to filter out the effects of the journal bearing lobing on the profile error. This filtering technique was implemented for the final round of testing Final Round Testing Final testing was performed during the week of 19 May 1997 at Dana Corporation in Russellville, Arkansas The test part was a heavy duty twelvelobe camshaft used in a six cylinder truck engine The testing conditions differed from those of the previous tests, in that no significant camshaft bearing lobing was present. Therefore it was not possible to demonstrate the effectiveness of the lobing filtering technique developed in Chapter 8 to reduce the closed loop noise ~ ] 
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1 82 Also in previous trials the controller was evaluated for effectiveness in rejecting an error present in the first part In this round of testing the controller was again tested for rejection of a disturbance present on the first part however an additional test was added. After the fourth part was ground the steady rests on the grinding machine were repositioned. This repositioning changed the effective stiffness of the system and thereby superimposed a second step disturbance the initial disturbance. Consequently, the effectiveness of the system in rejecting a disturbance that occurs on a part other than the first is also demonstrated. The controller gains used in this trial are given in Table 103. The gains were selected based on simulation testing using the observed repeatability errors Note that the base circle size gains are indicated to be zero since in this trial the base circle size controller was disabled due to practical considerations. Table 103 Gains for Profile Grinding: Flat Follower Gain Base circle NonLift Timing to Timing to lobe size concentricity dowel one ko 0.0 0.45 0.45 0.85 0.85 ki 0.0 1.0 1 0 0.1 1 0 a 1.0 0.7 0.7 0.6 0.6 Discussion of Results Figure 1013 shows the initial uncompensated lift error for the intake and exhaust lobes of the camshaft The trials were conducted over six parts and the compensated error is shown for part 6 in Figure 1014. Figure 1015 and Figure 1016 clearly show the general trend of the control response. From the plot, it is apparent that most of the initial error is eliminated on the first part. As noted earlier a nonrepeatable lobing error was L __ __ 
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L 1 83 n ot present in these trial and the gains were selected based on the observed nonrepeatability levels. As expected the standard deviation in creased for part number five due to the adjustment of the steady rests. This increase is eliminated in the next part. Figure 1017 shows the response of the control system over six parts for t he timing to lobe one and the timing to the camshaft reference which is a dowel pin for this workpiece. The top plot of Figure 1017 shows the timing of all lobes relative to lobe one. The initial error is small compared to the expected process repeatability error of 0 025 However the control system does not amplify the expected process noise as in the previous trials. The bottom plot of Figure 1017 shows the timing of all lobes relative to the camshaft timing reference. Here, the initial error is eliminated on the second part and the process noise is not amplified. Results are not reported for the base circle size since, as noted previously, the base c i rcle size controller was disabled for these trials.
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184 Uncompensated Intake Lobe Follower Lift Error : Part 1 0. 02 0 015 I i. 0,01 E 0. 005 E 0 .... ... .~':' ~ : \, pa ,J,,. 0 .... .... 0 005 w 0. 01 1 0.015 0.02 3 4 .. .. .. 5 6 ( ..... , ,. 180 150 120 90 60 30 0 30 60 90 120 150 180 Angle (degrees) Uncompensated Exhaust Lobe Follower Lift Error. Part 1 0.02 0 015 0 01 E 0 005 E 0 .... 0 .... .... 0.005 w 0 01 1 1 ~ 1 0 015 1 3 I . .... 5 6 0. 02 180 1 50 120 90 60 30 0 3 0 60 90 120 150 180 Angle (degrees) Figure 1013 Uncompensated Lift Error: Part 1 after Grinding Lobes: 16
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1 85 Compensated Intake L obe Follower Lift Error : Part 6 0.02 0 015 0 01 E 0 005 s 0 ,._ ,. __ I' ~~ ,., + 0 ,._ ,._ 0. 005 w 0 01 1 0 015 3 4 1 .. ..... 5 6 1 0.02 1 80 150 1 20 90 60 3 0 0 30 60 90 1 20 1 50 180 Angle (degrees) Compensated Exhaust Lobe Follower Lift Error : Part 6 0 02 0 015 0 01 E 0 005 E 0 ,._ ... .... .. : 0 '_ ... ,._ 0 005 ,._ w 0 01 1 2 1 3 4 1 0 015 I .. .... 5 6 1 0 02 1=======__j 1 80 150 12 0 90 60 30 0 30 60 9 0 1 20 150 180 Angle (degrees) Figure 1014 Compensated Lift Error: Part 6 after Grinding: Lobes 16
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186 Exhaust Lobe : Standard Deviation of Lift Error Values r 0 008 E 1 ... ... 2 g 0 007 3 4 i C 0 006 2 ro 0 005 . .. .. 5 6 i s: 0.004 (I) 0 0 0 003 .... ..... . .. ell 0 002 0 C 0.001 ell (/) 0 ... ~: ~.: .. ::: ~ ............ ~.: ~~ ..:.~:__::...":. ~ .. . .. ..:.:..::..:~ 2 3 4 5 6 Part Number Intake Lobe : Standard Deviation of Lift Error Values r0008 E E 0 007 1 3 4 1 C 0.006 0 .. 0.005 co , ....... 5 6 j s: 0 004 (I) 0 0 0 003 .... co 0 002 0 C 0 001 ell ..:~. ~ '' .. r ,~ : _, .. ; ........ '~ .' .. . ~.. ; ..... .. .. __ __ .... ~,_ .... ~.:. .. .:::.""::: ..., ...... :.. "' : :... ... :: .:.,: :_. .. ..._ ..: .: :.'~ ::;~ .. ::. :.:.~.. (/) 0 2 3 4 5 6 Part Number Figure 1015 Standard Deviation of Lift Error after Grinding
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Exhaust L obe : T otal L i ft E r ror 0 035 ,========, 1 E o o3 3 4 .S 0. 025 5 6 1 ... 0 t: 0 02 LU :: 0 015 :.:::i ro 0 01 0 005 0+' 1 87 2 3 4 5 6 Part Number Intake Lobe : Total Lift Error 0 035 ,,========, E 0 03 _ 0 025 5 6 g 0 02 LU :: 0 015 :.:::i ca 0 01 0 005 ' '\ .. . . .... ., ... ~: '!, ") _~_ _ : : .'__ . _. _ ; .. ...... ... .. __ :.: : ;,_ ~ '..: .. ...;:_ ,.._ .... ...;. ~ ... ..... . .. _. ... _ .. .. :. . .. ' 0+2 3 4 5 6 P art Number Figure 1016 Total Lift Error after Grinding
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1 88 Timing Error of All Lobes Relative to Lobe 1 0.2 (ll 0 15 (I) (I) ... 0) 0.1 (I) ... 0 05 0 t:: LU 0) 0 c:: E 0 05 i= 0 1 1 2 3 4 5 6 Part Number liming Error of All Lobes Relative to Reference Pin 0 2 (ll 0 15 (I) (I) ... 0) 0 1 (I) ... 0 0 05 ... ... LU 0) 0 c:: E 0.05 i= 0 1 1 2 3 4 5 6 Part Number Figure 1017 Lobe Timing After Grinding 
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CHAPTER 11 CONCLUSIONS AND RECOMMENDATIONS In this work it was shown tha t the profile timing nonconcentricity and size errors that result from the cam grinding and superfinishing processes considered here have a l arge repeatable component that can be eliminated using error compensation. Furthermore it was shown that the method used to reduce the cam inspection data significantly affects the form of the reported l ift error. Analysis of data reduction algorithms show that in order to successfully decouple lift error base circle nonconcentricity size lobing effects and the lobe timing angle data reduction must be performed in the motion coordinates of the grinding wheel. With the data reduction performed in terms of the grinding wheel mot i on it was shown that the repeatable cam grinding lift error can be modeled in te r ms of residual sparkout error The residual sparkout error was shown to be a function of the grinding threshold force and the grinding system stiffness. The grinding threshold force in turn was shown to vary as a function of the radius of curvature of the cam profile Also, it was shown that the residual sparkout model can be used to develop a model for the interaction of the lift at adjacent angles. This model was then used to demonstrate that the interaction effects are weak and that i t is reasonable to neglect then in the controller design Additionally effective strategies to eliminate noise from a variety of sources were developed It was shown that spatial high frequency noise in cam inspection and compensation data is attenuated by a low pass filter while the removal of low frequency lobing 1 89
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effects was shown to require fitting the base circle inspection data to an appropriate signal model. 1 90 The control of the profile grinding was extended to show that it is possible to include the superfinishing operation inside the control loop. The inclusion of the superfinishing process inside the control loop is advantageous since this control strategy eliminated the repeatable portions of the superfinishing process errors that would otherwise remain in the finished part. Implementation of the Control System As a practical mater any new system introduced into an industrial environment should require relatively modest new knowledge to be developed by the end uses Unfortunately, the development of a system interface is inevitablya compromise between generality and simplicity The design philosophy taken here was to eliminate the need for any programming on the part of the end user by providing adequate default settings for all control parameters Furthermore the communication between the control system the inspection gauge and the cam grinder was automated to the point that the end user s role was limited to selecting the data files used to describe the nominal and actual part geometry. As a practical matter these files were selected using dialog boxes over the network that was used to provide the communication channel. The calculation of the actual compensation and the update of the commanded files used by the grinder was effected with a single command. Future Work As a result of this work the error compensation system was developed and successfully tested under industrial conditions. However as with all ~
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~ 1 91 research projects this work poses more questions than it answers One area that presents a range of interesting research is the general modeling of the profile grinding process. This problem is strongly coupled with the algorithms used to reduce the inspection data since the algorithm used to process the inspection data significantly affects the form of the reported profile. As the many trials performed in this research project have clearly shown the profile error is largely systematic and related to the variation of the radius of curvature of the cam surface Consequently it should be possible to integrate a model for the systematic error directly into the grinding machine controller and thereby eliminate the modeled error. The coefficients of the modelrepresenting the effective system stiffness and the grinding wheel threshold forcecould be determined by fitting the inspection data from a series of sample parts to the process model. Additionally, the area of modeling holds the potential of cycle time reductions since lengthy sparkout grinding cycles are generally included to reduce the profile error to the minimum. Thus, using the depth of cut model that underlies the residual sparkout model it may be possible to model the complete process so that the sparkout process is eliminated entirely This is a much more difficult task than simply modeling the residual sparkout error since this problem is transient and depends on the initial condition of the workpiece i I I I I I I ~~I
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APPENDIX A VARIANCE FOR A SLIDING WINDOW The Effect of Deleting the Point of Interest from the Resiudal Computation Legend rn~~~~~: .,. ; Raw data ys (smoothed) etYrYSi I i di=yrymi II Smoothed data ym (model) ( I .11 ' I I y j2 j1 j+1 j+2 Appendix Figure A1 Deletion of the Point of Interest Computation of the Variance for a Sliding Window I j (..+jl ) (1 +I) (l jq j1 j+1 j+q N1 : Window of 2q + 1 deleted points X Modeled data e Residua l d Deleted residual Appendix Figure A2 Computation of the Variance for a Sliding Window 192
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~3 0 The variance is defined as 1 [i =N 1 i v = N 1 d f N d 2 Appendix Equation A1 For the window centered at j with 2q+1 points deleted we define the Vd;, the deleted variance at point j, as Appendix Equation A2 where 1 i =Nq+j 1 d;=Ldi N (2q + 1) i= q+j + 1 Appendix Equation A3 1 93 It should be noted that the index ranges between i = q+j+1 and N1 +j1 where i is normalized to between O and N1. That is : if(O i N1) 1 = 1 if(i < 0) i = i + N if (i > N 1) i = i N Calculation of all N vd; requires approximately N 2 multiplies and 4N 2 additions if Appendix Equation A2 and Appendix Equation A3 are directly applied.
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However the number of calculations can be reduced to 5N additions and 4N multiplies by forming a recursive expression for the variance Substituting j+1 for j in Appendix Equation A2 gives Appendix Equation A4 Next Appendix Equation A2 can be rewritten as 1 [ i =N q + j 1 ] 2 2 2 Vdj = N(2 1)1 dq + j + 1 + L d i (N(2q + 1))dj q + l = q+J+2 Appendix Equation A5 Also Appendix Equation A4 can be rewritten as 1 [ i = N q + j 1 ] 2 2 2 Vd j+1 = N(2 1)1 Ld i + dNq + j (N(2q+1))d j+ 1 q + l = q + J + 2 Appendix Equation A6 Appendix Equation A5 can be solved for the summation as i= Nq j 1 "'"' 2 2 2 L.. di = (N (2q + 1)1)Vdi dq + i + 1 + (N(2q + 1))di i = q + j2 Appendix Equation A7 Appendix Equation A7can be substituted into Appendix Equation A6 to get 194
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1 95 1 [ 2 2 [ 2 2 ]] Vd i 1 = N ( 2 q + 1 ) 1 d Nq i d q i 1 + ( N ( 2q + 1) ) d i d H + Vd i Appendix Equation A8 where Appendix Equation A9 Now once the deleted variance is calculated for point zero using Appendix Equation A2 and Appendix Equation A3 then it may be read i ly calculated for the remaining N1 points using the recursive expressions given as Appendix Equation A8 and Appendix Equation A9 The following MathCad document illustrates the application of this technique for several different setting of the parameters m P and q The parameter settings should be based on the "length" of the typical asperity( i.e the number of data points that make up the asperity) Once identified a reasonable starting setting would be : 2m+1 = asperity length" (number of points) P = 2 x asperity "length" q = 2m + 1 +2 A recommended starting point for the detection limit is 4cr.
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I L ______ 1 96 Program Function : T est Asperity Detection Method with various Parameter Settings Date Created: 26 January 1995 Revision Date : 1 7 May 1 995 Correction Date : 9 June 1995 Replaced N1 with N in norm function Revision Date : 8 November 1995 documentation format only Program Name : D : \MyDocuments\adcole\asperity\del_res med Data Type: Reduced data for rod bearing Data Source : CEP #2 : startup mode Data Units : Appear to be micro inch Notation : Adapted from Applied Linear Statistical Models Neter Wasserman Kutner Programmer : Tim Dalrymple 3937 NW 23rd Drive Gainesville FL 326051600 9043351 51 5 9043351 513 FAX timd@grove ufl edu Terminology and Notation used in this document. Symbol Description Units/dimensions N Number of data points in data set y Inspection data (from rout. 13 in this case) ymodel Filtered inspection data array p Number of points in linear model m 2m+1 points are deleted in linear models d Deleted residual between ymodel and y MSE Mean Squared Error of d dstar Studentized deleted residual q 2q+1 points are deleted from variance dbar deleted average vd deleted variance dstarstar Modified studentized deleted residual J Subscript for data arrays (angle) i Subscript for sums : dummy vdq? Save deleted variance for plots dP?m? Save residuals for plots dstarp?m? Save Studentized deleted residuals dstarstarP?m?Save modified studentized deleted residuals Set parameters and import inspection data N = 360 Size (1) (Nx1) (Nx1) (1) ( 1) (Nx1) (1) (Nx1) ( 1) (Nx1) (Nx1) (Nx1) (1) ( 1) (Nx1) (Nx1) (Nx1) (Nx1} none " none none " none ""2 none none ""2 JJ
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__ ______ __.__. .. ..'~read in data for j = 0 to N1 j = angle in degrees for N=360 j =o .. N 1 yj = READ( cep 2r) Import data from disk To simplify the notation in the following sections define a function "n orm" to normalize the array subscript to the range O <= i ndex <= N1 norrr( index) = ifl: ( index .2 0) ( index ~ N l) in de x., ifl: index< o, index N index N)) The Reduced Inspection Data for a Rod Bearing Journal 197 2 00 ,r,r,,r,rr,.,, 150 JOO 5 0 Y o J 2000 30 6 0 90 120 1 50 180 2 10 240 2 70 3 00 3 30 360 J Note : Follower rod appears worn as indicated by two asperities located 180 degrees apart Detail of the Reduced inspection Data for a Rod Bearing Journal 2 00 150 JOO 50 v 0 J tso 100 150 20~ 60 265 2 70 275 280 2 85 290 ~
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1 98 We first consider a modified three point linear formulation for the model. In this formulation the center point is deleted thus the three point formulation is reduced to two points m = 2 p = 9 ymodel = J P ( 2 m + 1) d = y ymodel N MSE = var(d) N I dP3m0 = d d. dstar. = 1 1 ) MSE dstarP3m0 = dstar j I m v . p I t = J 2 Y norm ( i ) p I J 2 v ___J Y norm ( i ) i=j 1 m The Studentized Deleted Residuals 7 ,,.r,,...,......,~r, 6 5 4 3 d s tar 2 J 3 4~~~~~~~~~~~~~~ 0 30 6 0 90 12 0 15 0 180 2 10 2 40 2 70 30 0 33 0 3 6 0
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199 Comparison of the Deleted Residuals (d) the Data (y) and the Smooth Surface Model (ymodel) 200 .,~~.~~ d J +v . J 4<150 so 0 so vmodeJ. ~J 100 265 2 70 2 75 280 285 J Next consider a modified five point linear model: again the center point is deleted. m = o p = 5 j I m ymodel J I i \' = P (2m + 1) 1 L..J Ynorm(i) + . p I d = y ymodel N MSE = var(d) N I d dstar = 1 MSE 1= J 2 p I J +2 '\' Ynorm(i ) i=j + I + m 290
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~dstar. J 2 00 T he Studentized Deleted Residuals 9 ..............~......,, 8 7 6 5 l .i '.....__ _,_ __._ __ .....__.....__ _,_ __._ __ .....__......._ _._ __._ ___, 0 30 6 0 90 120 150 1 80 210 2 l0 270 300 330 360 J Comparison of the Deleted Residuals (d), the Data (y) and the Smooth Surface d. J +v . J ~V1Tiodel J .....J200 150 100 50 0 50 100 150 200 260 Model (ymodel) 265 270 2 75 280 285 Consider a seven point linearmodel with the three central points deleted. m = I p = 7 290
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I l y mode~ = P ( 2 m I) d = y ymodel N MSE = v ar(d) N I dP7ml = d d dstar = ,,J MSE dstarP7ml = dstar j I m \' . p I 1=J 2 Yno rrn (i ) P 1 J 1 i=j 1 m Yno rrn ( i ) The Studentized Deleted Residuals 201 9 ,......,,,,~...,~ 8 7 6 5 4 dstar 3 J 2 4......__.....___......,_ ___,__ ___. __ ....___.....___ ___._ ___._ ___.~_.,___.....___ __, 0 3 0 60 90 120 150 180 2 10 2 40 270 30 0 33 0 36 0 _________ J
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2 02 C omparison of the Deleted Residuals (d) t he Data (y) and the Smooth Surface Model (y model ) 20 0 1 50 100 d J 50 tv. 0 J ~1 mode l s o J 'I 100 150 2 0~60 26 5 270 275 2 8 0 2 85 J Consider an eleven point linearmodel with the five central points deleted. m = 2 P = 11 ymode~ s: 1 P (2 m 1 d = y y model N M SE v ar ( d ) N I dPl lm2 = d d dstar = MSE dstarP 11 m2 = dstar p 1 m 2 \ Y norrr. ( i T \ I ___,J p I ) 2 ~~~~ Y norm ( i m 2 90
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2 03 The Student iz ed Deleted Residuals 9 8 7 6 5 l dstar 3 J 2 0 i 3 4 0 30 6 0 90 120 150 180 2 10 24 0 270 3 00 33 0 36 0 Comparison of the Deleted Residuals (d), the Data (y) and the Smooth Surface Model (ymodel) 200 150 d. 10 0 J 50 tY J 0 4< yrnodel 50 J IJ100 150 io~6 o 26 5 270 2 75 280 2 85 Consider a five point linearmodel with the three central points deleted m = I p = 5 ymode~ = P (2m l) d = y y model j I m '\' __; Y norm ( i ) . p I 1= J 2 p I J + 2 \' __; Y norm ( i) i=j 1 m 290
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. . ........ 204 MSE = v ar( d ) N I dPllml = d d dstar = MSE The Studentized Deleted Residuals 9 ,..,.,,,,,.,, dstar J 8 7 6 5 4 4~~~~~~~~~~~ 0 3 0 6 0 90 1 2 0 150 180 2 10 2 40 2 70 3 00 3 30 3 60 Comparison of the Deleted Residuals (d) the Data (y) and the Smooth Surface Model (ymodel) 2 00 150 100 d. J 50 +Y J 0 4
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205 We now take this idea a step further and delete the data point(s)of i nterest from the computation of the variance The computation is implemented according to the scheme developed in the text. We first demonstrate this idea with the five points centered around the pointofinterest deleted. Note: it is not necessary or desirable to use the same number of points for calculating the deleted residuals or the deleted variance. q = 2 d = y ymodel Calculate deleted average dbar for the deletion window centered at zero N q I dbar = 0 N (2q + I) vd l......J 1 i = q + I J = 0 .. N I I dbar (N (2q + 1)) dbar d c ) d (N .) J norm J + q + 1 norm q + J = J + I (N (2q + !)) r N q 1 l vd = 1 l V d )2 (N (2q + ( dbar ; 2 1 o (N (2 q ,1) I) ___J i o i = q ,I I c 2 2 vdj + 1 = vdj N (2q + 1) 1 l\ dnorm(N q + j) / dno~ (q + j 1 ) ) ~ +(N (2 q '1)) dbar ( dbar J / J + I I vdq2 = vd We can also test the impact on the setting of q on the variance q = 3 dbar = 0 N (2 q + 1) N q I v L..J di i = q + I (N (2q + 1))dbar d d J norm(J 'q + 1) no rm ( N q + J ) dbar J I (N (2 q 1))
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v d 0 = (N (2 q l) l) >J q 1 '\' i= q l d I I 2 2 (N q l) dbar () I 2 v d = v d .d d 2 J .,__ 1 J N ( 2 q 1 ) 1 1 no rm ( N q J ) no~ ( q j 1 ) / 2 +(N (2q !)) dba~i / dba~ i + 1 1 J v dq3 = v d Further test the impact on the setting of q on the variance q = 5 N q I dbar = '\' d o N (2 q + l) ___J 1 i = q I ( N q 'l bar .d d J no rm (J .c. q 1 ) no rm ( N q ..... J) dbar = '~''_,c_ J + l (N q + 1)) vd =0 (N ( 2 q + I) I) q 1 \ d J2 (N(2 q 'l)) ( dbar 0 J2 i= q T I vd = vd .... 1 \ 1 d N .) ) 2 ( d ( ) ) 2 J + 1 J N ( 2 q 1 ) 1 norm ( q .... J norm q + J T 1 j = O .. N 1 vdq5 = vd +(N (2 q + [ ( dbar J 2 \ dbar j + 1 )2 ] 2 06 ~. . 
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~~ 207 Comparison of the De l eted Variance for Different Values of m l00 v dq2 90 J v dq 3 J 8 0 v dq5 J 70 460 30 6 0 90 120 150 18 0 2 10 2 lO 2 7 0 3 0 0 33 0 J From this plot it is apparent that the deleted variance is substantially lower in the region of suspected asperities while essentially unchanged for areas with normal surface variation Consequently, dividing by the deleted variance will further increase the discrimination of this scheme We can substitute the deleted variance for the MSE in our previous models For example we will do this for the seven point linear model with the three central points deleted since it provides the best discrimination for the current data set. Note : m is different for j = o . N 1 dstar = dstarP7ml Recall the results from above for the studentized residuals Calculate the modified studentized deleted residuals dP7ml j dstarsta) = 3 6 0
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14 I~ 10 8 6 d s tar J 4 td s tar s tar 2 J 0 so 208 The Modified S t udentized Deleted Residuals 3 0 60 90 120 1 50 1 80 2 1 0 240 270 300 3 3 0 3 60 As implemented the deletion of points from the variance calculation boosts the ability to distinguish outliers by about seven standard deviations for the given data. However it is a simple matter to further boost the ab i lity to distinguish by modifying the exponent in the denominator or the variance. This strategy i s shown in the following graph Unfortunately the results no longer have units of standard deviations of a normal distribution but rather they represent a scaled standard deviation. Additionally, this strategy may fail for certain instances where, a region of the part surface has a poor surface finish, but in reality, no asperities or outliers are present. This feature may prove useful for detection of a region of chatter on a part surface. These strategies need to be tested to establish the appropriate parameter settings form and P. Insight into the selection of these values can be gained by first establishing the length (in the number of points) of the asperities which we are trying to detect.
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L __ APPENDIX B NONCONCENTRICITY OF BASE CIRCLE RELATIVE TO BEARING JOURNAL Appendix Figure 81 Error due to Journal Bearing Nonconcentricity From Appendix Figure 81 e2 = ex 2 + ev 2 Appendix Equation 81 = ATan2(ey e x) Appendix Equation B2 209
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L r. 2 + e 2 =Ne r 2 + 2 e r. Cos(8 n.) J J J J I' Appendix Equation 83 Ner i = .J r i 2 ..J... e 2 2 e r i Cos(8 i 13) Appendix Equation 84 Nc9 i =Atan2(cYi e y, c xie x ) Appendix Equation 85 210 Next the projection of Nc r i onto the follower line of action is needed. From the law of sines sin(e i 13) sin() = e Rearranging gives e sin(ei 13) sin() = Nc ri Appendix Equation 86 Solving for gives [ e sin(ei 13)] = arcs1n Ncr i Appendix Equation 87 Finally the projection of Ncri onto the follower s line of action is NCrr =N Cr j Cos() Appendix Equation 88 _J
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L APPENDIX C RADIAL DISTANCE T O POINT OF CONTACT Direction of Rotation follower translational axis Appendix Figure C1 Radial Distance to the Point of Contact Recall from Chapter 3 that xci = ri cos(8 i ) Appendix Equation C1 y ci = ri sin(8 i) Appendix Equation C2 Appendix Equation C3 211
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\ _ 212 Appendix Equation C4 dxc dr dei =risin(0i)+ d~cos(8 i) Appendix Equation C5 dy c dr de i = r i cos(0 i)+ d~ sin(8i) Appendix Equation C6 The term under the radical in Appendix Equation C3 and Append i x Equation C4 can be written as Appendix Equation C7 Expanding Appendix Equation C7 gives Appendix Equation C8 Simplifying Appendix Equation C7 gives 2 ( dri ) 2 ri + d0 Appendix Equation C9
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.. ~ Substituting Appendix Equation C1 Appendix Equation C2 and Appendix Equation C9 into Append i x Equation C 3 and Appendix Equation C4 gives Appendix Equation C10 Appendix Equation C11 21 3 From Appendix Figure C1 the radial distance to the point of contact rpoc i' can be written as Appendix Equation C12 Substituting Appendix Equation C10 Appendix Equation C 11 into Appendix Equation C12 gives
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+ r2 f + r2 f Appendix Equation C13 2 ( dri J 2 ri + d8 Simplifying Appendix Equation C13 gives Appendix Equation C14 2 14 I _J
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APPENDIX D PART PROGRAM DATA: GRINDER AND GAUGE P 278CBN 0 00 1 0 PRINT "SHAFT NAME 278CBN 20 PROFILE = 18N 30 ABSZ = 22 6709 40 ANGLE = 328.200 50 WFPSET = 4 60 WORKSPEED1 = 18N1" 68 WORKSPEED3 = WORK3" 7 0 PRINT "GRINDING NUMBER 6 INTAKE ; PROFILE 80 CPT 90 PROFILE = "32JR" 100 ABSZ = 24.1000 1 10 ANGLE=150.6167 120 WFPSET = 2 130 WORKSPEED1 = 32JR1" 138 WORKSPEED3 = "WORK3" 140 PRINT "GRINDING NUMBER 6 INJECTOR ; PROFILE 150 CPT 160 PROFILE= "16ER" 170 ABSZ = 25 7 457 180 ANGLE = 50 633 1 90 WFPSET = 5 200 WORKSPEED1 = "16ER1" 208 WORKSPEED3 = WORK3" 210 PRINT "GRINDING NUMBER 6 EXHAUST ; PROFILE .. Repeats for six cylinders END ADCOLE Inspection Rou t ine DDC278 OUT 1 1 .... One Cut inspection bottom to top. 1 1 .... Set Up path for error data 1 173 1 000000 1 000000 1 000000 1 000000 1 170 1.000000 1 000000 0.000000 1 83 2 84 7 000000 215 L __ 
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3 85 1 8.000000 3 1 ..._.... Following routines smooth lift and journal data 4 57 1 000000 4 000000 5 58 1 000000 4 000000 6 11 200 000000 7 34 0 000000 8 1 106 1 000000 99. 000000 1 000000 8 100 1 000000 0. 000000 0. 000000 1 000000 0 000000 9 111 99.000000 64.987000 013000 .013000 015000 1 0 112 99. 000000 006000 050000 7 000000 1 000000 11 113 000000 58 500000 500000 .050000 050000 12 113 1.000000 59 875000 .500000 .050000 050000 13 113 2.000000 59.250000 .500000 050000 050000 14 114 .000000 132.000000 228.000000 050000 .100000 15 114 1 000000 124. 000000 234 000000 050000 100000 1 6 114 2.000000 121 000000 270.000000 .1 00000 .100000 17 117 99. 000000 008000 008000 1 000000 008000 18 115 128.867000 129.867000 750000 1.000000 1.000000 22 14 0.000000 23 27 51 000000 000000 000000 24 1 ... 103 Clear plotting screen 24 103 25 1 ... Start journa l 7 ... 25 84 1 000000 26 81 7.000000 1 000000 1.000000 0 000000 27 48 50.000000 28 21 29 12 2.000000 0 000000 30 14 1.000000 31 13 1.000000 1.000000 32 40 1 000000 1 000000 21 000000 10. 000000 1000. 000000 41 48 50.000000 42 1 ... Start cam lobe 18 45 85 1 000000 46 82 18.000000 1.000000 1 000000 1.000000 47 21 48 12 2 000000 0.000000 49 14 1.000000 55 188 1 000000 1.000000 3.000000 2.000000 1 000000 60 48 50.000000 61 1 ... Start cam lobe 17 64 85 1 000000 65 82 1 000000 0. 000000 1 000000 1 000000 66 21 67 12 2.000000 0.000000 216 ~j
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" ...." 68 1 4 1 000000 7 4 188 1 000000 1 000000 3 000000 2 000000 1 000000 79 48 50.000000 80 1 _.. Start cam lobe 1 6 80 85 1 000000 81 82 1 000000 0. 000000 1 000000 1 000000 82 21 83 12 2. 000000 0 000000 84 14 1 000000 90 188 1. 000000 1 000000 3 000000 2 000000 1 000000 95 48 50 000000 96 1 _.. Start journal 6 98 84 1 000000 99 81 1 000000 0. 000000 1 000000 1 000000 100 21 101 12 2.000000 0.000000 102 14 1 000000 103 13 1 000000 1 000000 104 40 1 000000 1 000000 21 000000 10 000000 1000 000000 113 48 50 000000 Repeats for six cylinders 1000 50 ADCOLE Part Data File 278R19.DAT 7 18 24 350000 191 950000 355 950000 519 950000 683 950000 84 7. 950000 1008 350000 18Nr19.bin 16Er19.BIN 32Jr19 bin 71.200000 129 367 1 00 113.000000 29 384 2.00 145.500000 211.800 0.00 236 200000 9.367 1 00 278 000000 269 384 2 00 309 500000 91.800 0 00 400.200000 249.367 1 00 441 000000 149 384 2 00 473 500000 331 800 0 00 563 200000 69.367 1 00 217
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605.000000 329 384 2 00 638.500000 151.800 0 00 728.200000 189.367 1 00 769.000000 89 384 2.00 802.500000 271 800 0.00 892.200000 309.367 1 00 934 000000 209 384 2.00 965.500000 31 800 0.00 0.000000 0 000000 0.000000 0.000000 0.000000 0 000000 0.000000 0.000000 0 000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0 000000 1077 000000 000000 000000 19.000000 000000 0 000000 0 000000 0.000000 0.000000 0.000000 0.000000 0 0 0 0 0 0.0000 0 0000 0.0000 0 0000 0 0.0000 0.0000 0 0000 0 0000 4 150 0000 170.0000 240 0000 270.0000 0 0 0 0 0 0 0 0 0 0 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 00000 218
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219 00000 0 0000 c : \grinder\ journlxx.per profi lxx. per ddc27800 ppe I I I p P 001124163 60 p SERIES INJ 16ER 32JRI 18N 0.748031 0 748031 0 748031 1.178642 1 165453 1.151575 0 83 1 0 54 1 0 69 1 0 0 0 0 0 0 0 298676 0.480315 0.494185 0 298517 0.479835 0.493911 0 298184 0.478398 0.49309 0.297676 0.476 0.491721 0.296994 0.472642 0.489807 0 296137 0.468327 0.487347 0 295105 0.463051 0.484346 0.293898 0.456819 0.480805 0.292516 0.449626 0.476726 0.29096 0.441476 0.472113 0.289229 0.432366 0.466969 0 287325 0.422295 0.461299 0.285246 0.411268 0.455105 0.282992 0 39928 0.448393 0.280563 0.386421 0.441168 0.277958 0.373106 0.433436 0 275179 0 35972 0.425202 0 272228 0.346335 0.416473 0.269107 0.332949 0.407255 0.265817 0 319563 0.397557 0.26236 0 306177 0.387385 0.258736 0.292791 0.376749 0.254947 0.279406 0.365655 0.250992 0 26602 0.354114 0.246872 0.252634 0 342133 0 242591 0.239248 0.329724 0.238149 0.225862 0 316897 0 233547 0 212476 0.303659 0.228787 0.199091 0.290017 0 22387 0.185705 0 275976 0 218797 0.172319 0.261564 __ J
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2 20 0 213569 0 158933 0.246834 0 208188 0 145547 0.231877 0 202659 0.132169 0.216803 0.196986 0 119071 0 20 1 733 0 191173 0 106622 0 186793 0.185226 0.094921 0 172109 0.179149 0.083961 0 157812 0.172949 0 073732 0.144036 0 166632 0 06422 0.130895 0 160208 0.055417 0.118465 0.153686 0.047303 0.10678 0.147078 0 039866 0 095844 0 140397 0 033102 0.08564 0.133655 0.026996 0.07615 0 126869 0.021543 0.067352 0 120057 0.016728 0 059228 0 113236 0.012547 0.051764 0.106429 0 008996 0.044947 0.099659 0 006063 0.038765 0.092949 0.003752 0.033208 0 086325 0 002043 0 028268 0.079814 0.000913 0 023942 0 073443 0 000287 0.020232 0.067237 0 000039 0.017133 0 061221 55 235 1 0.014616 0 055421 0 0 0.012617 0 049859 0 0R 0.011034 0.044556 236 359 1 0.009746 0 039533 0 0 0 008629 0.034809 0.000079 0 007593 0.030399 0 000315 0.006581 0.026315 0.000709 0 005565 0.022565 0.001256 0 00453 0 019152 0.001965 0 003479 0 016076 0.002827 0.002456 0 013334 0 003846 0.001538 0.010918 0 005028 0.000801 0 008817 0.006362 0.000282 0.007015 0 007854 0.000004 0.00549 0.0095 70 291 1 0 004221 0.011307 0 0 0 003181 0.013272 0.000000R 0.002345 0 01539 292 359 1 0 001685 0.017669 0 0 0 001177 0 020102 0.000243 f l
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0 000795 0.000516 0 00032 0.000187 0.000102 0.00005 0 00002 0 000005 84 266 1 0 0 0.000000R 267 359 1 0 0 0.000019 0.000069 0.000153 0.000273 0.000428 0.000617 0.000836 0.001078 0.00134 0.001617 0.001908 0.00221 0.002524 0.002849 0.003186 0.003534 0.003893 0.004263 0.004647 0.005049 0.005479 0.005956 0.006503 0.007152 0.007943 0.008921 0 010138 0.011647 0.013503 0.015756 0.018449 0.021616 0 022693 0 025441 0 028346 0 031409 0 034626 0.038004 0.041539 0.045228 0.049075 0.053079 0.05724 0 061559 0 066035 0.070669 0 075457 0.080406 0 085508 0 090768 0.096185 0.10176 0.107492 0.113382 0.119429 0 12563 0 131988 0.138508 0.145181 0.152012 0.159 0.166146 0.173445 0 180866 0.188354 0 195862 0.203335 0.210728 0.218024 0.225217 0 232307 0 239299 0 246193 0.252988 0 259681 0.266272 0.272768 0 000788 0 001557 0 002481 0 003487 0 004525 0 005565 0.006591 0 0076 0.008623 0 009725 0.011008 0.012597 0.01461 0.017141 0.020246 0.023953 0.028272 0.033207 0.038763 0 044946 0 051765 0.05923 0.067354 0.076151 0.085637 0 095833 0.106764 0 118454 0 130899 0.144052 0 157829 0.172117 0.186795 0.201735 0 21681 0.23189 0.246845 0.261564 0.275966 0.290003 0.30365 0.316894 0 329725 0.342134 0 354114 2 21
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. '"' 2 22 0 02528 0.279157 0 365655 0 029453 0.285453 0.376749 0.034134 0.291646 0 387385 0.039308 0 297736 0.397556 0 044947 0.303732 0.407255 0.051013 0.309626 0.416472 0 057457 0.315417 0.425201 0 064226 0.32111 0.433435 0.071263 0.326705 0.441168 0.078512 0 332197 0 448393 0.085916 0 337591 0.455105 0.093423 0 342882 0.461298 0 100985 0.348075 0.46697 0 108558 0 353169 0.472114 0 116101 0.358161 0.476727 0.123584 0.363055 0.480806 0 130977 0.367846 0.484347 0.138258 0.372539 0.487348 0.145408 0.377134 0.489807 0.152414 0.381626 0.491721 0 159264 0 38602 0.493089 0.165951 0.390311 0.493911 0 172471 0.394504 0.178822 0.398598 E 0.185007 0.402591 0 191026 0.406484 0 196881 0.410276 0 202573 0.413969 0 208103 0.417563 0 213472 0.421055 0 21868 0.424449 0.22373 0.42774 0.228621 0.430933 0 233355 0.434028 0.237932 0.43702 0.242351 0.439913 0.246608 0.442705 0.250703 0.445398 0.254634 0.447992 0.2584 0.450484 0 262001 0.452878 0.265438 0.455173 0 26871 0.457366 0.271816 0.459457 0.274756 0.461449 L .. ____
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2 23 0. 277529 0 .46334 3 0 280134 0 .465 13 8 0 282571 0.466831 0 284839 0.46842 1 0.286938 0.469913 0 288867 0.47 1 307 0 290627 0.472602 0.292215 0.473795 0 293632 0.474886 0.294875 0.475882 0 295944 0.476772 0 296838 0.477567 0.297557 0.47826 0 2981 0.47885 0 298468 0.479346 0 29866 0.479736 0.480031 E 0.480224 E
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. REFERENCES [1] Jensen P W. Cam Design and Manufacture New York : Dekker 1987 [ 2] King, R. I. and Hahn R. S. Handbook of Modern Grinding Technology, New York : Chapman and Hall 1986. [3] Dalrymple T. M ., ADCOLE 910 Machine Acceptance Report ," Bosch Corporation Internal Report Charleston SC May 1985. [4] Dalrymple T. M. Improved Process Control in Camshaft Grinding Through Utilization of Post Process Inspection with Feedback ," Master s Thesis University of Florida 1993 [5) Hahn R. S. Influence of Threshold Forces on Size Roundness and Contour Errors in Precision Grinding, Annals of the CIRP Vol. 30 No. 1, 1981 p251254 [6] Lindsay R. P ., "On the Metal Removal and Wheel Removal Parameters Surface Finish Geometry and Thermal Damage in Precision Grinding, Ph D. Dissertation Worcester Polytechnic Institute 1971 [7] King, R. I. and Hahn, R. S. Part Processing by Grinding, Handbook of Modern Grinding Technology New York: Chapman and Hall 1986. [8) GalTzur Shpitaini M ., and Malkin S. Design and Manufacturing Analyses for Integrated CAD/CAM of Cams ," Journal of Engineering for Industry, Vol. 111, November 1989 p 307. [9] GalTzur Shpitaini M ., and Malkin S., "I ntegrated CAD/CAM System for Cams ," Annals of the CIRP Vol. 35 No 1 1986 p 99 [10) Cooke P and Perkins, D.R., A Computer Controlled Cam Grinding Machine ," Cams and Cam Mechanisms Ed J R. Jones London: Mechanical Engineering Publications 1978. [11) Yang, B D and Menq C H ., Compensation for Form Error of EndMilled Sculptured Surfaces Using Discrete Measurement Data ," Proceedings of the Japan/USA Symposium of Flexible and Automation ASME Vol. 1 San Francisco 1992, pp. 385392 224 
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2 25 ( 12] Roblee H W ., Chen Y L. Becker K. B and Fiedler K. H. Refinements in Postprocess Gauging with Feedback in the Production of Diamond Turned Optics ," Proceedings of Laser Interferometry and Computer Aided Interferometry IV SPIE Vol. 1553, The Hague Netherlands 1991 pp. 1 87191 (13] Eversheim V. W. K o nig, W W Weck M. and Pfeifer T. Productiontechnik auf dem Weg zu lntegrierten Systemen ," UDIZ No 6 June 1987 pp. 6165 (14] Chen F Y. Mechanics and Design of Cam Mechanisms New York: Pergamon 1982. [15] Tesar D ., and Matthew G. K. The Dynamic Synthesis, Analysis, and Design of Modeled Cam Systems, Lexington MA: Lexington Books 1976 (16] Wilson C E. and Sadler J P Kinematics and Dynamics of Machinery Second Edition New York: Harper Collins 1993. [17] "ADCOLE Model 911Computer Aided Camshaft and Piston Inspection Gage ," Adcole Corporation, Pub No 911 2M, Marlborough, MA, Feb. 1990 (18] "StateoftheArt 3L Series CNC Cam Grinding Systems, Litton i ndustrial Automation Systems, Pub. No 3L88 FR 3M, Waynesboro, PA, 1988. (19] "CF 41 CBN Camlobe Grinding Machine, Schaudt Machinenbau GmbH Publication Number 0496 en ADWERB/ES Stuttgart Germany : 1996 (20] "Andrews Cam Design with Graphics ," Andrews Products Inc. Rosemont IL undated. [21] James M L. Smith G. M ., and Wolford J. C. Applied Numerical Methods for Digital Computation with FORTRAN and CSMP, New York : Harper and Row, 1977. [22] Hamming, R. W., Numerical Methods for Scientists and Engineers, Second Edition New York: Dover Publications 1986. [23] Hildebrand, F. B ., Introduction to Numerical Analysis Second Edition, New York : Dover Publications 1987. [24] Craig, John, J., Introduction to Robotics Mechanics and Control Second Edition Reading, MA: AddisonWesley 1989.
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2 26 [25) Leventon W. Cam Production Increased by 16 Ax i s Control l er ," Design News Feb. 1 1993, pp. 75 76 [ 26) De Boor Carl A Practical Guide to Splines New York SpringerVerlag, 1978 [27) Press William H ., Teukolsky Saul A. Vetterling, William T. Flannery and Brian P ., Numerical Recipes in C : The Art of Scientific Computing, Second Edition Cambridge University Press Cambridge UK. [28) Whitehouse D. J. Handbook of Surface Metrology, Bristol UK: Institute of Physics Publishing, 1994 [29) James M L., Smith G M. and Wolford J. C Applied Numerical Methods for Digital Computation with FORTRAN and CSMP New York : Harper and Row 1977. [30) Mendenhall W ., Wackerly, D. D. and Scheaffer R. L. Mathematical Statistics with Applications, Belmont CA: Duxbury Press, 1990. [31) Doughty, S., Least Squares Applied to Circular Part Inspection ," SAE Technical Paper Series International OffHighway and Powerplant Exposition Milwaukee Wisconsin September 1995. [32] Lindsay, R. P. Principles of Grinding, Handbook of Modern Grinding Technology, Editors: King, R. I., Hahn, R. S. New York: Chapman and Hall 1986 [33] Malkin, S ., Grinding Technology : Theory and Application of Machining with Abrasives, Chichester UK: Ellis Horwood 1 989 [34] Lindsay R. P. Testing of CBN Wheels on 52100 ( RC 60) Steel ," The Center for Grinding Research and Development University of Connecticut Storrs Connecticut 1995. [35] Kuo 8. C. Automatic Control Engineering. Englewood Cliffs, New Jersey : Prentice Hall 1996, page 15. [36] Whitehouse D J ., Surface Topography and Quali t y and its Relevance on Wear ," Massachusetts Institute of Technology Conference on Wear 1978. [37] Tsukizoe T. The Effects of Surface Topography on Wear ," Massachusetts Institute of Technology Conference on Wear 1978. _J
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L [ 38] Ba y er R. G ., and Sirico J. L. The Infl uence of Surface Roughness on Wear ," Wear V ol. 35 19 75 pp 251260 2 27 [ 39] Microfinish i ng Silences Camshaft Chatter ," Tooling and Production June 1 991 p 75 ( 40] Excell M ., Smooth Operators ," Metalworking Production Vol. 139 Issue 9 September 1995. [ 41] Puthanangady T. K. and Malkin S. Experimental Investigation of the Superfinishing Process ," Wear Vol. 185 1995 pp. 173182. (42) Neter, J. Wasserman W. and Ku t ner M. H. Applied Linear Statistical Models 1 Third Edition Burr Ridge Illinois : I rwin 1990 [43] Strum R. D. and Kirk D. E ., The First Principles of Discrete Systems and Digital Signal Processing. New York : Addison Wesley 1988 [44] On the Surface ," Vol. 1 No 7 Lansing, Michigan: Industrial Metal Products Corporation undated. [45] Srinivasan K. Application of the Regeneration Spectrum Method to Wheel Regenerative Chatter in Grinding ," Journal of Engineering for Industry, Transactions of the ASME Vol. 104, No. 1 1982, pp. 4654. [46] Snoeys, R. and Brown D. Dominating parameters in Grinding Wheel and Workpiece Regenerative Chatter ," Proceedings of the 10 th International Machine Tool Design and Research Conference pp. 32548 New York: Pergamon 1969. [47] Kato H. and Nakano Y ., Transfer of Roundness Error from Center and Center Holes to Workpiece in Cylindrical Grinding and its Control ," CIRP Annals Vol. 34 No 1 1985. [48] Kay S M and Marple, S. L., Spectrum AnalysisA Modern Perspective ," Proceedings of the IEEE Vol. 69 No 11 November 1981 [49] Renders H ., Schoukens J. and Vilain, G. HighAccuracy Spectrum Analysis of Sampled Discrete Frequency Signals by Analytical Leakage Compensation, IEEE Transaction of Instrumentation and Measurement Vol. IM33 No. 4. December 1984 50] Franklin G. F Powell J. D ., and Workman M. L. Digital Control of Dynamic Systems 1 Second Edition New York : Addison Wesley, 1990.
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I l [51] Bollinger J G. and Duffie N A Computer Control of Machines and Processes New York : Addison Wesley 1988 228 [52] Koenig D M ., Control and Analysis of Noisy Processes Englewood Cl i ffs NJ: Prentice Hall 1991 _J
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BIOGRAPHICAL SKETCH The author was born on 24 February 1959 in Langdale Alabama. He grew up in the eastern United States where he attended public schools The author received his Bachelor of Mechanical Engineering from Auburn University in 1981. After graduation, he worked for Bosch Corporation in the U. S. and in Germany Prior to his return to graduate school at the University of Florida in 1991, he served as a volunteer in U. S. Peace Corps stationed in The Republic of Botswana. In Botswana he worked as a Lecturer of Mechanical Engineering at the national Polytechnic He received his Master of Science degree from the University of Florida in 1993. After graduation, he plans to pursue a career in manufacturing and metrology. 229
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l __ I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy Associate Professo =hanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy ~~24(\0, All A Seireg =' Ebaugh Professor of Mechanica l Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the deg ree of Doctor of Philosophy Sencer Yeralan Associate Professor of Industrial and Systems Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Associate Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy d~h;;:~eThomas E. Bullock Professor of Electrical and Computer Engineering I _J
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This dissertation was submitted t o the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy August 1997 Ll Winfred M Phillips Dean College of Engi Karen A Holbrook Dean Graduate School


