Theory of Mechanisms and Machines

Theory of Mechanisms and Machines

Theory of Mechanisms and Machines C.S. SHARMA Formerly Professor Department of Mechanical Engineering Jai Narain Vyas University Jodhpur KAMLESH PUROHIT Professor Department of Mechanical Engineering Jai Narain Vyas University Jodhpur New Delhi-110001 2011

THEORY OF MECHANISMS AND MACHINES C.S. Sharma and Kamlesh Purohit 2006 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher. ISBN-978-81-203-2901-0 The export rights of this book are vested solely with the publisher. Fourth Printing October, 2011 Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus, New Delhi-110001 and Printed by Rajkamal Electric Press, Plot No. 2, Phase IV, HSIDC, Kundli-131028, Sonepat, Haryana.

Chapter 15 Contents Preface... xiii Chapter 1 Introduction to Mechanisms and Machines 1 15 1.1 Introduction... 1 1.2 Kinematic Link... 2 1.3 Kinematic Pairs...2 1.3.1 Nature of Contact... 3 1.3.2 Nature of Relative Motion... 3 1.3.3 Nature of Mechanical Constraints... 4 1.4 Kinematic Chain... 5 1.5 Mechanism... 6 1.6 Four Bar Chain... 7 1.7 Inversions of Kinematic Chain... 9 1.7.1 Four Bar Chain... 9 1.7.2 Slider-crank Chain... 10 1.7.3 Double Slider-crank Chain... 13 Exercises... 15 Chapter 2 Kinematic Analysis of Mechanism 16 71 2.1 Introduction... 16 2.2 Velocity Analysis... 17 2.3 Relative Velocity Method... 17 2.3.1 Slider-crank Mechanism... 19 2.3.2 Four Bar Mechanism... 19 2.3.3 Crank and Slotted Lever Mechanism... 20 2.4 Instantaneous Centre Method... 32 2.4.1 Arnold Kennedy Theorem... 34 2.4.2 Locating Instantaneous Centres... 35 2.5 Acceleration Analysis... 40 2.5.1 Procedure to Draw Acceleration Polygon of a Mechanism... 42 2.6 Klein s Construction... 49 v

vi Contents 2.7 Coriolis Acceleration... 51 2.8 Analytical Method... 58 Exercises... 63 Multiple Choice Questions... 69 Chapter 3 Synthesis of Mechanisms 72 103 3.1 Introduction... 72 3.2 Number Synthesis... 72 3.2.1 Degree of Freedom of a Mechanism... 72 3.2.2 Grashof s Criterion of Movability... 77 3.2.3 Limit Positions, Dead Centres and Transmission Angle... 80 3.3 Dimensional Synthesis... 82 3.4 Synthesis by Relative Pole Method... 82 3.4.1 Synthesis of Four Bar Mechanism... 84 3.4.2 Synthesis of Slider-Crank Mechanism... 86 3.5 Synthesis of Mechanism by Inversion Method... 87 3.5.1 Synthesis of Four Bar Mechanism... 88 3.5.2 Synthesis of Slider-Crank Mechanism... 89 3.6 Slider-Crank Mechanism for Given Quick Return Ratio... 90 3.7 Synthesis of Mechanism by Analytical Method... 91 3.7.1 Freudenstein s Method of Synthesis... 92 3.7.2 Approximate Synthesis Using Least Square Technique... 93 3.7.3 Bloch s Method... 94 Exercises... 99 Multiple Choice Questions... 101 Chapter 4 Lower Pair Mechanisms 104 135 4.1 Introduction... 104 4.2 Straight Line Motion Mechanism... 104 4.2.1 Exact Straight Line Motion Mechanism... 104 4.2.2 Approximate Straight Line Motion Mechanism... 109 4.3 Pantograph... 112 4.4 Engine Indicator Mechanisms... 114 4.5 Automobile Steering Mechanism... 120 4.5.1 Davis Steering Mechanism... 122 4.5.2 Ackermann Steering Mechanism... 124 4.6 Hooke s Joint... 126 Exercises... 133 Chapter 5 Friction 136 183 5.1 Introduction... 136 5.2 Laws of Dry Friction... 137

Contents vii 5.3 Angle of Friction... 137 5.4 Inclined Plane Friction... 138 5.4.1 Motion Up the Plane... 139 5.4.2 Motion Down the Plane... 141 5.5 Inclined Plane with Guide Friction... 145 5.6 Wedge Friction... 147 5.7 Screw Friction... 153 5.8 Pivot and Collar Friction... 158 5.8.1 Collar Bearings... 160 5.8.2 Pivot Bearing... 163 5.9 Friction Clutches... 165 5.9.1 Plate Clutches... 166 5.9.2 Cone Clutches... 168 5.10 Friction in Turning Pairs... 175 5.11 Friction Axis... 176 Exercises... 178 Multiple Choice Questions... 182 Chapter 6 Belts, Ropes and Chains 184 234 6.1 Introduction... 184 6.2 Belt Drive... 184 6.3 Mechanics of Belt Drive... 188 6.4 Velocity Ratio... 190 6.5 Length of Belt... 192 6.5.1 Open-belt Drives... 192 6.5.2 Cross-belt Drives... 194 6.6 Ratio of Belt Tensions... 195 6.6.1 Ratio of Tensions in V-Belt... 197 6.7 Effect of Centrifugal Force... 199 6.8 Power Transmission Capacity... 200 6.9 Stresses in Belt... 201 6.10 Elastic Creep... 203 6.11 Belt Materials... 204 6.12 Stepped Pulley Drive... 214 6.13 Rope Drive... 218 6.13.1 Wire Rope Construction... 218 6.13.2 Forces on a Wire Rope... 219 6.13.3 Breaking Strength... 221 6.14 Chain Drive... 224 6.15 Types of Chain... 225 6.16 Chain Length... 228 6.17 Chordal Action... 229 Exercises... 230 Multiple Choice Questions... 233

viii Contents Chapter 7 Brakes and Dynamometers 235 291 7.1 Introduction... 235 7.2 Block Brakes... 235 7.2.1 Block Brakes with Small Angle of Contact... 236 7.2.2 Block Brake with Large Angle of Contact... 239 7.3 Band Brakes... 250 7.3.1 Simple Band Brake... 251 7.3.2 Differential Band Brake... 252 7.4 Band and Block Brake... 262 7.5 Internal Expanding Shoe Brake... 265 7.6 Effect of Braking on Vehicle... 270 7.7 Dynamometers... 274 7.7.1 Absorption Dynamometers... 274 7.7.2 Transmission Dynamometers... 278 Exercises... 285 Multiple Choice Questions... 291 Chapter 8 Governors 292 330 8.1 Introduction... 292 8.2 Watt Governor... 292 8.3 Porter Governor... 295 8.4 Proell Governor... 300 8.5 Hartnell Governor... 305 8.6 Wilson-Hartnell Governor... 312 8.7 Spring-Controlled Gravity Governor... 316 8.8 Pickering Governor... 317 8.9 Characteristics of a Governor... 318 8.10 Effort of a Governor... 320 8.11 Power of a Governor... 321 8.12 Controlling Force... 322 Exercises... 326 Multiple Choice Questions... 329 Chapter 9 Cams 331 387 9.1 Introduction... 331 9.2 Classification of Cams and Followers... 331 9.3 Cam Terminology... 334 9.4 Analysis of Follower Motions... 335 9.4.1 Constant Velocity Motion... 336 9.4.2 Simple Harmonic Motion... 338 9.4.3 Constant Acceleration and Deceleration... 339 9.4.4 Cycloidal Motion... 343 9.4.5 Polynomial Motion... 344

Contents ix 9.5 Graphical Synthesis of Cam Profile... 346 9.6 Analytical Synthesis of Cam Profile... 355 9.7 Pressure Angle... 359 9.8 Cams with Specified Profile... 362 9.8.1 Tangent Cam with Roller Follower... 362 9.8.2 Circular Cam with Roller Follower... 372 9.8.3 Circular Cam with Flat Faced Follower... 374 Exercises... 384 Multiple Choice Questions... 386 Chapter 10 Gears 388 440 10.1 Introduction... 388 10.2 Gear Terminology... 389 10.3 Law of Gearing... 391 10.4 Velocity of Sliding... 393 10.5 Forms of Gear Teeth... 394 10.5.1 Involute Profile... 394 10.5.2 Cycloidal Profile... 396 10.6 Arc of Contact... 397 10.7 Interference... 405 10.8 Minimum Number of Teeth to Avoid Interference... 406 10.9 Minimum Number of Teeth on Pinion with Rack... 408 10.10 Force Analysis of a Spur Gear... 415 10.11 Helical Gear... 417 10.11.1 Formative Number of Teeth... 419 10.11.2 Forces on Helical Gear... 419 10.12 Spiral Gear... 420 10.12.1 Efficiency of Spiral Gear... 422 10.13 Worm Gear Set... 427 10.13.1 Efficiency of Worm Gear Set... 430 10.14 Bevel Gears... 432 Exercises... 436 Multiple Choice Questions... 439 Chapter 11 Gear Train 441 470 11.1 Introduction... 441 11.2 Simple Gear Train... 441 11.3 Compound Gear Train... 443 11.4 Epicyclic Gear Train... 445 11.4.1 Tabulation Method... 445 11.4.2 Relative Velocity Method... 446 11.5 Torque Transmitted by Epicyclic Gear Train... 447 Exercises... 464 Multiple Choice Questions... 469

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