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Furthermore, this book contains the latest, most important techniques and test data available The philosophy of the book is to present diversi ed applications for cams to aid the reader s understanding and design selection This volume also contains many new examples, a listing of computer software, and an extensive list of references It should be noted that cams, gearing, rolling-element bearings, and traction (friction) drives are of the same basic mechanical family, the study of which is generically referred to as contact mechanics These devices have in common rolling and/or sliding action coupled with reasonably high concentrated contact (Hertz) stresses In the eld of contact mechanics the failure mechanisms among these machine elements are similar and are related to material metallurgy and heat treatment, lubricant rheology and chemistry, surface topography and geometry, and applied contact load (stress) These machine components also have commonly related manufacturing requirements Hence, they share a similar manufacturing technology and engineering analysis The signi cance of cam pro le accuracy is emphasized A surface may appear smooth to the eye and yet have poor dynamic properties Depending on the application and loads the cam pro le accuracy and its surface nish may affect the life, vibration, and noise of a cam-follower system Poor manufacturing techniques can impede the functional ability of a mechanism or machine Also, the choice of a lubricant and the lubrication system is pertinent on all designs We express our respect for the inventors, designers, engineers, scientists, toolmakers, and machinists who produced cam systems in the past Our knowledge and experience are a result of the amazing contributions and successful machines these experts handed down over the years that they toiled This evidence of their work is our fortunate heritage We are surrounded with evidence of the high-quality performance of these machines, which was accomplished with less mathematical guidance, poor manufacturing facilities, and crude measurement tools All the while, nature revealed her secrets, very slowly, bit by bit It is to be noted that the Chinese exercise for health and combat, T ai Chi Ch uan, has a ritual that spiritually parallels the foregoing thought The training performance classes conclude with the loud clapping of hands, which is speculated to be in deference to all those who went before Last, I wish to express my appreciation to Heather Tinsley, Washington, DC, who edited and assisted in the production of this handbook Also, grateful acknowledgment is due the elite chapter contributors whose dedication to the handbook is most noteworthy I wish also to thank the many in academia and industry who were helpful in providing valuable guidance and information for the book, such as Samuel L Doughty, Consulting Engineer, Mount Ephraim, New Jersey; Arthur F Erdman, Professor of Mechanical Engineering, University of Minnesota, Minneapolis; Ralph Fusco, Consulting Engineer, Commack, New York; Grigor Kerdanyan, Consulting Engineer, Van Nuys, California; James A Kurth, Senior Analytical Engineer, Commercial Cam Company, Inc, Wheeling, Illinois; Bernard Roth, Professor of Mechanical Engineering, Stanford University, Palo Alto, California; Michael Savage, Professor Emeritus of Mechanical Engineering, University of Akron, Akron, Ohio; and James L Wiederrich, Manager, Modeling and Simulation, United Defense LP, Santa Clara, California Special recognition goes to Erwin V Zaretsky, Branch Chief of Mechanical Components Branch, NASA Lewis Research Center, Cleveland, Ohio, for the many inspirational hours we spent discussing the subject of cams Harold A Rothbart.

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a = acceleration in/rad2 A = constant acceleration in/sec2 ae = semi-major axis of ellipse, in A = acceleration in/sec2 be = semi-minor axis of ellipse, in C,C1,C2 = constants h = total rise, in h1 = displacement at transition point, in n = any number a ne = e ratio be V0 = initial velocity, in/sec y = follower displacement, in dy y = = follower velocity, dimensionless dq d2y y = = follower acceleration, dimensionless dq 2 3 d y y = 3 = follower jerk, dimensionless dq dy y= = w y = follower velocity dt 2 d y y = 2 = w 2 y = follower acceleration dt d3y = y = w 3 y = follower jerk dt 3 y1,y2 = displacement during acceleration periods 1 and 2, respectively, in b = cam angle for rise h, rad b1 = angle of rotation for rise h1, rad f = angle of rotation for cycloidal curve, rad

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40 Introduction 41 Antiderivatives and Indefinite Integrals 411 The Concept of Antiderivative 412 The Indefinite Integral 42 Area 43 Signed Area 44 The Area Between Two Curves 45 Rules of Integration 451 Linear Properties 452 Additivity Quiz

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In 1, we saw that it is possible to construct a cam by blending crude increments and observing its appearance This method is not acceptable in the design of today s cam machinery Now it is necessary to provide accurate mathematical information for the cam characteristics of displacement velocity, acceleration, and sometimes the jerk In so doing one can interpret and control the ultimate design performance Also, the higher the cam speed, the more critical is the investigation This is especially true of the acceleration data which is the important determining factor of the dynamic loads on cam-follower systems This chapter presents mathematically established basic curves which are the rst selection to establish the follower action They are easy to analyze and manipulate Cams can be designed for any acceptable curve or shape Appendix A lists various kinds of geometric curves that have been used in the past They include the ellipse, parabola, hyperbola, logarithmic spiral, and involute of a circle Alternative combinations of these curves combined with the circle and straight line have also been constructed (see 15)

120 120 120 121 124 134 140 144 144 144 145

As previously indicated, a cam can be considered similar to a wedge having a cyclical rise and fall which establishes the motion of the follower In all cams, the displacement of the follower is given by the mathematical relationship y = f (q ) in (21)

xL+x-2 (x + 2)(x = lm i x+l x-1 x+l x-1

where q = cam angle rotation in radians However, since the cam rotates at a constant angular velocity, the displacement can also be written as y = g(t ) in and q =wt (22) (23)

51 l'H pital's Rule 511 Introduction 512 l'H pital's Rule 52 Other Indeterminate Forms 521 Introduction 522 Writing a Product as a Quotient 523 The Use of the Logarithm 524 Putting Terms over a Common Denominator 525 Other Algebraic Manipulations 53 Improper Integrals: A First Look 531 Introduction

where t = time for cam to rotate through angle q, sec w = cam angular velocity, rad/sec By the use of Eq (21) the follower characteristics can be normalised (dimensionless) as follows: The cam pro le is usually given as a function of the angle q Thus y = follower displacement The instantaneous angular rate of change of displacement y = dy = follower velocity dq (25) (24)

148 148 148 154 154 154 155 156 158 160 160

The instantaneous angular rate of change of velocity y = d2y = follower acceleration dq 2 (26)

The instantaneous cam angle rate of change of acceleration y = d3y = follower jerk dq 3 (27)

532 Integrals with Infinite Integrands 533 An Application to Area 54 More on Improper Integrals 541 Introduction 542 The Integral on an Infinite Interval 543 Some Applications Quiz

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