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
