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Kinematics of MachineryThrough HyperWorks

J.S. Rao

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Dedicated to the memory of my parents

Jammi Chikka RaoJammi Ramanamma

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Beginnings of the Theory of Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Beginning of the Wheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Archimedes (287–212 BC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Water Wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Wind Mills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Renaissance Engineers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Industrial Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 The Nature of This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Planar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Basic Kinematic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Elementary Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Grübler’s Criterion for Planar Mechanisms . . . . . . . . . . . . . . . . . . . . . 182.4 Four-Link Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Kinematic Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.6 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Kinematic Analysis of Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.1 Velocities by the Centro Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Relative Velocity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.2.1 Rotation of a Rigid Link about a Fixed Axis . . . . . . . . . . . . . . 473.2.2 Relative Velocity Equation of Two Points on a Rigid

Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2.3 Relative Velocity Equation of Two Coincident Points

Belonging to Two Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . 543.3 Relative Acceleration Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3.1 Rotation of a Rigid Link about a Fixed Axis . . . . . . . . . . . . . . 593.3.2 Relative Acceleration of Two Points on a Rigid Body . . . . . . 60

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3.3.3 Relative Acceleration Equation of Two Coincident PointsBelonging to Two Rigid Bodies . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Acceleration Analysis of Reciprocating Engine Mechanism . . . . . . . 743.4.1 Klein’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.2 Ritterhaus Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.4.3 Bennet’s Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.5 Analytical Determination of Velocity and Acceleration of thePiston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.5.1 Harmonic Analysis for Velocity and Acceleration of the

Piston . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.6 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4 Straight Line Motion and Universal Coupling . . . . . . . . . . . . . . . . . . . . . 854.1 Condition for Exact Straight Line Motion . . . . . . . . . . . . . . . . . . . . . . 864.2 Exact Straight Line Motion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . 87

4.2.1 Paucellier Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.2 Hart Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2.3 Scott–Russel Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Approximate Straight Line Motion Mechanisms . . . . . . . . . . . . . . . . . 894.3.1 Modified Scott–Russel (Grasshopper) Mechanism . . . . . . . . . 894.3.2 Watt Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.3.3 Tchebicheff Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.3.4 Robert Straight Line Mechanism . . . . . . . . . . . . . . . . . . . . . . . 934.3.5 Pantograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.6 Beam Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.7 Richards Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3.8 Crosby Indicator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.3.9 Dobbie–McInnes Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.4 Steering Gear Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.4.1 Davis Steering Gear Mechanism. . . . . . . . . . . . . . . . . . . . . . . . 974.4.2 Ackermann Steering Gear Mechanism . . . . . . . . . . . . . . . . . . . 101

4.5 Hooke’s (Cardan, Universal) Joint or [Universal Coupling] . . . . . . . . 1014.5.1 Double Hooke’s Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.7 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5 Cams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1 Types of Cams and Followers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.2 Displacement Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.3 Disk Cam with Knife-Edge Follower . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.4 Translating Roller Follower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.5 Translating Flat Follower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1515.6 Oscillating Flat Follower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.7 Cams of Specified Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575.8 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Contents ix

5.9 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

6 Spur Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.1 Classification of Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1876.2 Types of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1926.3 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1946.4 Law of Gear Tooth Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1986.5 Involute as a Gear Tooth Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1996.6 Layout of an Involute Gear Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2016.7 Producing Gear Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.8 Meshing Gears and Line of Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.9 Interference of Involute Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2076.10 Minimum Number of Teeth to Avoid Interference . . . . . . . . . . . . . . . . 2106.11 Contact Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2126.12 Cycloidal Tooth Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.13 Cycloidal and Involute Tooth Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.14 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.15 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

7 Helical, Spiral, Worm and Bevel Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.1 Involute Helicoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.2 Helical Gear Tooth Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2297.3 Contact of Helical Gear Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.4 Helical Gear Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.5 Spiral [Crossed Helical] Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2357.6 Worm Gearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2367.7 Bevel Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2397.8 Formation of Bevel Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2407.9 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2427.10 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

8 Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.1 Classification of Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2498.2 Simple Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2508.3 Compound Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2518.4 Synthesis of Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528.5 Gear Train Applications to Machine Tools . . . . . . . . . . . . . . . . . . . . . . 2538.6 Epicyclic Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2578.7 Inversions of Epicyclic Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2588.8 Differential Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2618.9 Torque Distribution in Epicyclic Trains . . . . . . . . . . . . . . . . . . . . . . . . 2628.10 Example of an Epicyclic Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2638.11 Coupled Epicyclic Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2648.12 Wilson Four-Speed Automobile Gear Box . . . . . . . . . . . . . . . . . . . . . . 267

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8.13 Solved Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2688.14 Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

Preface

The Theory of Machines was borne as a subject with the Industrial Revolution andthe birth of Reciprocating Steam Engine nearly 230 years ago. The reciprocatingsteam engine was the main work horse for just over hundred years during the 19thcentury and gradually lost its place in history by the turn of 20th century. It is thenthe turn of Internal Combustion Engines and Rotating Machinery.

Kinematics and Dynamics of Reciprocating and Rotating machinery is the fun-damental study to mechanical engineers before proceeding to stress design. Theseanalyses were done mainly by graphical methods for planar mechanisms as theygave good insight into the mechanism and repetitive history of velocities, accel-erations, static and dynamic forces, etc.

All machines are periodic in operation depending on thermodynamic cycles forexample one revolution in a two-stroke engine or two revolutions in a four-strokeengine. Besides thermodynamics as a fundamental mechanical engineering subject,Theory of Machines is the first subject a fresh entrant to mechanical engineeringstudies faces. The concepts of moving machine members over a period of thermo-dynamic cycle and the variation of displacements, velocities and accelerations formthe subject of Kinematics. Here we do not question what makes these machine mem-bers move but merely find the kinematics when there is a motion. When we ask thequestion of forces that make the motion, we are dealing kinetics; together we haveDynamics of Machinery. When we include the machinery aspects such as links,kinematic chains, mechanisms, etc., to form a given machine we have the subject ofTheory of Machines.

Usually this subject is introduced as a two-semester course, kinematics and ki-netics simultaneously with Thermodynamics or Heat Engines before the design ofmachine members begins. This book forms the material for first semester of Theoryof Machines.

As this subject is over 200 years old, there are several textbooks already available.The new books that appear from time to time take into account new techniquesavailable; the subject matter itself has not changed particularly for the entry level.What is the difference here then?

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xii Preface

This book attempts to bring in the machine live on to the screen and explain thetheory of machines concepts through animations and introduce how the problemsare solved in industry to get complete history in the shortest possible time ratherthan using graphical (or analytical) methods that are in vogue even today. Thus thestudent is introduced to the concepts through visual means and brings him closeto industrial applications by the end of the two semester program taking him wellequipped for design courses.

International Federation for the Promotion of Mechanism and Machine Science(IFToMM) has developed a standard nomenclature and notation on Mechanism andMachine Science and this book adopts these standards so that any communicationbetween scientists and teachers in classrooms across the world can be with the sameterminology causing no confusion.

This book adopts HyperWorks MotionSolve to perform the analysis and visual-izations, though the book is independent of the requirement of any software. Havingthis software helps in further studies and analysis. The avis in this book can be ac-cessed from extras.springer.com by using the ISBN; they are: Figs. 2.1, 2.3, 2.4, 2.5,2.6, 2.7, 2.8, 2.9, 2.10, 2.11b, 2.12a, 2.12b, 2.13, 2.20, 2.21, 2.22, 2.23, 2.24, 2.25,2.26, 2.27, 2.28, 2.29, 2.30, 2.31, 2.32, 2.33, 3.1, 3.2a, 3.2b, 3.2c, 3.5a, 3.13f, 3.20,3.28, 3.30, 3.38, 4.1, 4.2, 4.3, 4.4, 4.5,4.6, 4.8, 4.9, 4.10, 4.11, 4.12a, 4.12b, 4.12c,4.13, 4.14, 4.15,4.16, 4.18, 4.19a, 4.19b, 4.20, 4.22a, 4.22b, 4.22c, 5.1, 5.2, 5.3, 5.4,5.5a, 5.5b, 5.6a, 5.6b, 5.6c, 5.7, 5.17, 5.18a, 5.18b, 5.26, 5.29, 5.30, 5.31, 5.32-33,5.34-35, 6.1, 6.2a, 6.5a, 6.7b, 6.10, 6.18a, 6.18b, 6.23a, 6.23b, 6.23c, 6.24a, 6.24b,7.4c, 7.4d and 8.1.

The author acknowledges help given by various students and colleagues overfour decades. Of particular mention, I thank Professor J. Srinivas, Anil Sakhamuri,Uday M. Udapi and Sundar Nadimpalli. I am also thankful to Mr. Pavankumar andMr. Nelson Dias of Altair Engineering India.

Finally, I am ever so thankful to my beloved wife Indira for her understanding inmy work and cooperation.