Table Of ContentCam Design and Manufacturing, 2nd Edition
Cam Design
and
Manufacturing
Handbook
Second Edition
Robert L. Norton P. E.
Worcester Polytechnic Institute
http://www.me.wpi.edu/Norton/
and
Norton Associates Engineering
http://www.designofmachinery.com
Industrial Press
New York
Copyright 2009, Industrial Press Inc., New York, NY - http://industrialpress.com
Cam Design and Manufacturing, 2nd Edition
Library of Congress Cataloging-in-Publication Data
Norton, Robert L.
Cam Design and Manufacturing Handbook / Robert L. Norton—2nd edition; p. cm
Includes index and contributions by Ronald G. Mosier Ph.D.
ISBN: 978-0-8311-3367-2 (hardcover)
1. Design and construction. 2. Machinery, Kinematics of. 3. Machinery, Dynamics of. I. Title.
TJ206 .N67 2009
621.8'38—dc22 2009004571
All text, drawings, and equations in this book were prepared and typeset electronically, by the author, on a Macintosh®
computer using Freehand®, MathType®, and Pagemaker® desktop publishing software. The body text was set in Times
Roman, and headings set in Avant Garde. Printing press plates were made directly from the author's files.
Cover design by Janet Romano. Cover and title page photos courtesy of The Ferguson Company, St. Louis, MO.
Industrial Press, Inc.
989 Avenue of the Americas
New York, NY 10018
Copyright © 2009 by Industrial Press Inc., New York. Printed by Thomson Press.
All rights reserved. This book, or any parts thereof, may not be reproduced, stored in a retrieval
system, or transmitted in any form without the permission of the publisher.
10 9 8 7 6 5 4 3 2 1
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Cam Design and Manufacturing, 2nd Edition
This book is dedicated to the memory of:
Dr. Ronald G. Mosier
1938-2008
A gentleman, scholar, superb mathematician,
and a good friend. He will be greatly missed.
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Copyright 2009, Industrial Press Inc., New York, NY - http://industrialpress.com
Cam Design and Manufacturing, 2nd Edition
Copyright 2009, Industrial Press Inc., New York, NY - http://industrialpress.com
Cam Design and Manufacturing, 2nd Edition
FOREWORD
For many years, Professor Robert L. Norton has been known not only for his teaching
skills, but also for his ability to tie his academic work very closely to practical applica-
tions in industry. He has always made time to work in the field, building and maintain-
ing strong relationships with practicing engineers throughout industry. His teachings and
practice range from fundamentals to state-of-the-art developments.
Professor Norton’s newest book, Cam Design and Manufacturing Handbook, is the
most comprehensive work we know of on a subject that is so important to machine de-
sign. From cover to cover, the book provides valuable information for both students and
practicing engineers. It is a complete work, ranging from the basics of cam design and
manufacture to advanced topics such as spline functions and vibration analysis.
At Gillette and throughout industry, engineers are constantly challenged to create
mechanisms that perform faster and more efficiently than ever before. Sources of acous-
tic noise must be eliminated or minimized. Machines must run without failure and they
must run for long periods of time between routine maintenance. Designing and building
equipment that satisfy all of these criteria is essential to achieving high levels of produc-
tivity, product quality, and competitiveness. All of these challenges can be addressed
through proper attention to cam design.
By working closely with Professor Norton to learn and apply the principles covered
in this book, we have been able to manufacture new products with unprecedented levels
of productivity, quality, and reliability. We have also been able to improve the perfor-
mance of existing equipment.
I am confident that students and practicing engineers alike will find this book to be
an essential text and reference.
Thomas J. Lyden
Group Director, Manufacturing, Blades & Razors
The Gillette Company
Boston, Massachusetts
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Cam Design and Manufacturing, 2nd Edition
PREFACE
Cam-follower systems are an extremely important and ubiquitous component in all kinds
of machinery. It is difficult to find examples of machinery that do not use one or more
cams in their design. Cams are the first choice of many designers for motion control
where high precision, repeatability, and long life are required.* All automotive engines
depend on cams for their proper valve function. Most automated production machinery
uses cams extensively.
The design and manufacture of cams has changed dramatically in recent years. The
development and proliferation of computers in engineering design and of numerical con-
trol in manufacturing have completely changed the process of cam design and manufac-
turing, and very much for the better. Until about the late 1960’s cams were designed only
by manual graphical layout techniques, manufactured in low quantities by manually con-
trolled machining methods and in high quantities by analog duplication of a hand-dressed
master cam. The subtleties of the effects of higher derivatives of the cam’s chosen math-
ematical function were often ignored, due either to ignorance of their importance, or the
inability to accurately determine their effects given the lack of computational facilities
available at the time.
Currently, it is virtually universal and also very economical to use computer-aided
engineering and design techniques to create cam geometry, including proper consider-
ation of the effects of higher derivatives, and also to make the cam with high precision
using continuous numerically controlled milling, grinding, or electrical discharge ma-
chining (EDM) equipment. A significant number of fundamental research results on the
subject of cam design and manufacture have been published in recent years. This book
is intended to provide a definitive reference for the design and manufacturing of cam-
follower systems by bringing up-to-date cam design technology and cam research to-
gether between a single set of covers for the benefit of the design and manufacturing
engineering community.
The book takes the subject from an introductory level through advanced topics need-
ed to properly design, model, analyze, specify, and manufacture cam-follower systems.
Beginning with a description of "how not to design a cam" in order to point out pitfalls
that may not be obvious to the beginner, the proper way to design a cam for multiple and
single-dwell situations is developed in detail. All the acceptable (and some unaccept-
able) classical cam functions are described and their mathematics defined for the com-
mon double-dwell application. Polynomial functions are introduced and used for both
double- and single-dwell examples. Problems with polynomial cams are defined in de-
* Some machines use tail and ways to design around these problems are discussed. Spline functions are intro-
pneumatic devices for duced as a class of cam motion functions that can solve the most difficult cam design
motion control rather than problems. Many examples are developed to show how splines, especially B-splines, can
cams. The former have
solve otherwise intractable cam design problems.
poor dynamics and
accuracy compared to cam- The issues of cam pressure angle and radius of curvature are fully addressed for
driven machines. various types of cams and followers: radial, barrel, globoidal, translating, and oscillat-
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Cam Design and Manufacturing, 2nd Edition
ing, roller and flat-faced. The dynamics of the cam-follower system are introduced along
with techniques for modeling the follower system as lumped parameters. Both the in-
verse dynamic (kinetostatic) and forward dynamic solutions are developed for a multi-
plicity of models of various degrees of freedom. The extensive literature on these topics
is referenced in the bibliography. Residual vibrations in the follower train are addressed
along with a number of cam functions that can reduce the level of vibration. Polydyne
and splinedyne cams are defined and methods for their calculation described.
Calculations for the cam contour of radial, barrel, and globoidal cams with oscillat-
ing or translating roller or flat followers are defined. Cutter compensation algorithms
and cam surface generation are defined for all common cam-follower configurations.
Conjugate cam calculation is defined as well. Cam materials and manufacturing tech-
niques are described and recommendations made.
Stress analysis of the cam-follower joint is presented in detail along with methods
to determine the failure modes of typical cam/follower materials in surface contact un-
der time-varying loads. Lubrication and wear of the cam and follower is also addressed.
Methods for the experimental measurement of acceleration, velocity and displace-
ment of cam-follower systems are described, and examples of such measurements taken
on operating machinery are shown. Case studies from automotive and automated man-
ufacturing machinery are presented.
The second edition adds treatment of globoidal cams, updates the topics of torque-com-
pensation cams, and adds development of multi-degree-of-freedom models of cam-follower
systems including follower impact, all based on work done by the author's graduate students
and colleagues in industry since the first edition appeared.
The author would like to express his sincere appreciation to Dr. Ronald G. Mosier who
wrote Chapter 5 on spline functions and checked many of the equations in other chapters.
Also, Dennis Klipp of Klipp Engineering, Waterville, ME, Paul Hollis of Tyco Electronics
Corporation, Harrisburg, PA, R. Alan Jordan of Delta Engineering, Muncie IN, and Dr. Tho-
mas A. Dresner, Mountain View, CA provided welcome and helpful comments on the book
during its development. Many other people reviewed sections of the book or supplied data,
illustrations, and information used in the book. I would like to especially thank Gregory
Aviza, Arthur Borgeson, Al Duchemin, Charles Gillis, Robert Gordon, Joel Karsberg, Donald
Loughlin, Thomas Lyden, Corey Maynard, Edwin Ryan, Edward Swanson, and John Wash-
ington, all now or formerly with P&G Gillette, Boston, MA. Finally, the author thanks his
editors at Industrial Press, John Carleo and Janet Romano, for making this the most pleasant
and productive book development process yet experienced.
Every effort has been made to ensure that the material in this book is technically
correct. All known errors in the first edition have been corrected in this edition. If any er-
rors remain, the author takes full responsibility, and will greatly appreciate their being
pointed out to him. Please send an email to norton@ designofmachinery.com or
[email protected] or if you discover any errors in the text or in the programs.
Information on book errata as discovered and demonstration versions of the comput-
Robert L. Norton
er programs mentioned throughout the text, along with files to run the examples, are down- Mattapoisett, Mass.
loadable from the author's website at http://www.designofmachinery.com. August, 2008
VII
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Cam Design and Manufacturing, 2nd Edition
Contents
Chapter 1 Introduction...........................................................................................1
1.0 Cam-Follower Systems......................................................................................................1
1.1 Fundamentals.....................................................................................................................1
1.2 Terminology.........................................................................................................................4
Type of Follower Motion...............................................................................................4
Type of Joint Closure....................................................................................................4
Type of Follower............................................................................................................5
Type of Cam..................................................................................................................6
Type of Motion Constraints..........................................................................................8
Type of Motion Program...............................................................................................8
1.3 Applications........................................................................................................................9
1.4 Timing Diagrams...............................................................................................................10
1.5 Cam Design Software.....................................................................................................10
1.6 Units....................................................................................................................................10
1.7 References........................................................................................................................16
Chapter 2 Unacceptable Cam Curves .............................................................17
2.0 Introduction......................................................................................................................17
2.1 S V A J Diagrams.............................................................................................................17
2.2 Double-Dwell Cam Design—Choosing S V A J Functions........................................19
2.3 The Fundamental Law of Cam Design........................................................................22
2.4 Simple Harmonic Motion (SHM).....................................................................................22
2.5 Constant Acceleration (Parabolic Displacement)....................................................25
2.6 Cubic Displacement.......................................................................................................25
2.7 Summary............................................................................................................................26
2.8 References........................................................................................................................26
Chapter 3 Double-Dwell Cam Curves...............................................................27
3.0 Introduction......................................................................................................................27
3.1 Cycloidal Displacement for Double dwells.................................................................27
3.2 Combined Functions for Double dwells.......................................................................30
Fall Functions...............................................................................................................41
3.3 The SCCA Family of Double-Dwell Functions..............................................................41
3.4 Polynomial Functions.......................................................................................................45
The 3-4-5 Polynomial...................................................................................................46
The 4-5-6-7 Polynomial ............................................................................................49
3.5 Fourier Series Functions...................................................................................................52
3.6 Summary ........................................................................................................................56
3.7 References........................................................................................................................56
Chapter 4 Single-Dwell Cam Curves.................................................................57
4.0 Introduction......................................................................................................................57
4.1 Single-Dwell Cam Design—Choosing S V A J Functions...........................................57
4.2 Single-Dwell Applications of Polynomials....................................................................61
Effect of Asymmetry on the Rise-Fall Polynomial Solution.......................................63
4.3 Summary ........................................................................................................................68
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Cam Design and Manufacturing, 2nd Edition
Chapter 5 Spline Functions..................................................................................69
5.0 Introduction......................................................................................................................69
5.1 Classical Splines...............................................................................................................70
Knots.............................................................................................................................70
Interpolation Equations..............................................................................................71
Smoothness Equations................................................................................................71
Boundary Conditions..................................................................................................72
Computation...............................................................................................................74
5.2 General Polynomial Splines............................................................................................82
5.3 B-splines.............................................................................................................................86
5.4 Bezier Curves....................................................................................................................97
5.5 Shape Preserving Splines................................................................................................98
Hermite Splines............................................................................................................99
Shumaker Quadratic Splines....................................................................................101
5.6 Knot Placement.............................................................................................................106
5.7 Periodic Splines..............................................................................................................115
5.8 Splines Other Than Polynomial Splines.......................................................................119
Trigonometric Splines................................................................................................119
Rational Splines ......................................................................................................120
5.9 Summary..........................................................................................................................121
5.10 Bibliography....................................................................................................................122
Chapter 6 Critical Path Motion Cam Curves..................................................125
6.0 Introduction....................................................................................................................125
6.1 Constant Velocity Motion............................................................................................125
Polynomials Used for Critical Path Motion..............................................................126
Half-Period Harmonic Family Functions..................................................................133
6.2 Combined Displacement and Velocity Constraints...............................................137
6.3 Summary..........................................................................................................................148
Chapter 7 Cam Size Determination .................................................................149
7.0 Introduction....................................................................................................................149
7.1 Pressure Angle—Radial cam with Translating Roller Follower................................150
Eccentricity................................................................................................................151
Choosing a Prime Circle Radius..............................................................................154
7.2 Pressure Angle—Barrel Cam With Translating Roller Follower................................155
7.3 Pressure Angle—Barrel Cam With Oscillating Roller Follower................................157
7.4 Overturning Moment—Radial Cam With Translating Flat-Faced Follower..........159
7.5 Pressure Angle—Radial Cam With Oscillating Roller Follower...............................160
7.6 Pressure Angle—Globoidal Cam With Oscillating Roller Follower........................163
7.7 Radius of Curvature—Radial Cam With Translating Roller Follower.....................164
7.8 Radius of Curvature—Radial Cam With Translating Flat-Faced Follower...........168
7.9 Radius of Curvature—Barrel Cam With Translating Roller Follower......................172
7.10 Radius of Curvature—Barrel Cam With Oscillating Roller Follower.......................172
7.11 Radius of Curvature—Radial Cam With Oscillating Roller Follower.....................173
7.12 Radius of Curvature—Radial Cam With Oscillating Flat-Faced Follower............173
Undercutting of Radial Cams with Oscillating Flat-Faced Followers...................174
7.13 Radius of Curvature—Globoidal Cam With Oscillating Roller Follower...............175
7.14 References......................................................................................................................176
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