Mechanical Engineer's Handbook Academic Press Series in Engineering Series Editor J. David Irwin Auburn University This a series that will include handbooks, textbooks, and professional reference books on cutting-edge areas of engineering. Also included in this series will be single- authored professional books on state-of-the-art techniques and methods in engineer- ing. Its objective is to meet the needs of academic, industrial, and governmental engineers, as well as provide instructional material for teaching at both the under- graduate and graduate level. The series editor, J. David Irwin, is one of the best-known engineering educators in the world. Irwin has been chairman of the electrical engineering department at Auburn University for 27 years. Published books in this series: Control of Induction Motors 2001, A. M. Trzynadlowski Embedded Microcontroller Interfacing for McoR Systems 2000, G. J. Lipovski Soft Computing & Intelligent Systems 2000, N. K. Sinha, M. M. Gupta Introduction to Microcontrollers 1999, G. J. Lipovski Industrial Controls and Manufacturing 1999, E. Kamen DSP Integrated Circuits 1999, L. Wanhammar Time Domain Electromagnetics 1999, S. M. Rao Single- and Multi-Chip Microcontroller Interfacing 1999, G. J. Lipovski Control in Robotics and Automation 1999, B. K. Ghosh, N. Xi, and T. J. Tarn Mechanical Engineer's Handbook Edited by Dan B. Marghitu Department of Mechanical Engineering, Auburn University, Auburn, Alabama San Diego San Francisco New York Boston London Sydney Tokyo This book is printed on acid-free paper. Copyright # 2001 by ACADEMIC PRESS All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Requests for permission to make copies of any part of the work should be mailed to: Permissions Department, Harcourt, Inc., 6277 Sea Harbor Drive, Orlando, Florida 32887-6777. Explicit permission from Academic Press is not required to reproduce a maximum of two ®gures or tables from an Academic Press chapter in another scienti®c or research publication provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press A Harcourt Science and Technology Company 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com Academic Press Harcourt Place, 32 Jamestown Road, London NW1 7BY, UK http://www.academicpress.com Library of Congress Catalog Card Number: 2001088196 International Standard Book Number: 0-12-471370-X PRINTED IN THE UNITED STATES OF AMERICA 010203040506COB987654321 Table of Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Contributors xv CHAPTER 1 Statics Dan B. Marghitu, Cristian I. Diaconescu, and Bogdan O. Ciocirlan 1. Vector Algebra 2 1.1 Terminology and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Product of a Vector and a Scalar . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Zero Vectors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Resolution of Vectors and Components . . . . . . . . . . . . . . . . . . 6 1.8 Angle between Two Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.9 Scalar (Dot) Product of Vectors . . . . . . . . . . . . . . . . . . . . . . . 9 1.10 Vector (Cross) Product of Vectors . . . . . . . . . . . . . . . . . . . . . . 9 1.11 Scalar Triple Product of Three Vectors . . . . . . . . . . . . . . . . . . 11 1.12 Vector Triple Product of Three Vectors . . . . . . . . . . . . . . . . . . 11 1.13 Derivative of a Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2. Centroids and Surface Properties 12 2.1 Position Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Centroid of a Set of Points . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Centroid of a Curve, Surface, or Solid . . . . . . . . . . . . . . . . . . . 15 2.5 Mass Center of a Set of Particles . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Mass Center of a Curve, Surface, or Solid . . . . . . . . . . . . . . . . 16 2.7 First Moment of an Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.8 Theorems of Guldinus±Pappus . . . . . . . . . . . . . . . . . . . . . . . 21 2.9 Second Moments and the Product of Area . . . . . . . . . . . . . . . . 24 2.10 Transfer Theorem or Parallel-Axis Theorems . . . . . . . . . . . . . . 25 2.11 Polar Moment of Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.12 Principal Axes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3. Moments and Couples 30 3.1 Moment of a Bound Vector about a Point . . . . . . . . . . . . . . . . 30 3.2 Moment of a Bound Vector about a Line . . . . . . . . . . . . . . . . . 31 3.3 Moments of a System of Bound Vectors . . . . . . . . . . . . . . . . . 32 3.4 Couples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 v 3.5 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.6 Representing Systems by Equivalent Systems . . . . . . . . . . . . . . 36 4. Equilibrium 40 4.1 Equilibrium Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.2 Supports. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.3 Free-Body Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5. Dry Friction 46 5.1 Static Coef®cient of Friction . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.2 Kinetic Coef®cient of Friction . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.3 Angles of Friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 References 49 CHAPTER 2 Dynamics Dan B. Marghitu, Bogdan O. Ciocirlan, and Cristian I. Diaconescu 1. Fundamentals 52 1.1 Space and Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.2 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.3 Angular Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2. Kinematics of a Point 54 2.1 Position, Velocity, and Acceleration of a Point. . . . . . . . . . . . . . 54 2.2 Angular Motion of a Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.3 Rotating Unit Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4 Straight Line Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.5 Curvilinear Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.6 Normal and Tangential Components . . . . . . . . . . . . . . . . . . . . 59 2.7 Relative Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3. Dynamics of a Particle 74 3.1 Newton's Second Law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2 Newtonian Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3 Inertial Reference Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.4 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5 Normal and Tangential Components . . . . . . . . . . . . . . . . . . . . 77 3.6 Polar and Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . 78 3.7 Principle of Work and Energy . . . . . . . . . . . . . . . . . . . . . . . . 80 3.8 Work and Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.9 Conservation of Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.10 Conservative Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.11 Principle of Impulse and Momentum. . . . . . . . . . . . . . . . . . . . 87 3.12 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . . . . . 89 3.13 Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.14 Principle of Angular Impulse and Momentum . . . . . . . . . . . . . . 94 4. Planar Kinematics of a Rigid Body 95 4.1 Types of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 Rotation about a Fixed Axis . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.3 Relative Velocity of Two Points of the Rigid Body . . . . . . . . . . . 97 4.4 Angular Velocity Vector of a Rigid Body. . . . . . . . . . . . . . . . . . 98 4.5 Instantaneous Center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.6 Relative Acceleration of Two Points of the Rigid Body . . . . . . . 102 vi Table of Contents 4.7 Motion of a Point That Moves Relative to a Rigid Body . . . . . . 103 5. Dynamics of a Rigid Body 111 5.1 Equation of Motion for the Center of Mass. . . . . . . . . . . . . . . 111 5.2 Angular Momentum Principle for a System of Particles. . . . . . . 113 5.3 Equation of Motion for General Planar Motion . . . . . . . . . . . . 115 5.4 D'Alembert's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 References 117 CHAPTER 3 Mechanics of Materials Dan B. Marghitu, Cristian I. Diaconescu, and Bogdan O. Ciocirlan 1. Stress 120 1.1 Uniformly Distributed Stresses . . . . . . . . . . . . . . . . . . . . . . . 120 1.2 Stress Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 1.3 Mohr's Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 1.4 Triaxial Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 1.5 Elastic Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 1.6 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 1.7 Shear and Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 1.8 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 1.9 Normal Stress in Flexure. . . . . . . . . . . . . . . . . . . . . . . . . . . 135 1.10 Beams with Asymmetrical Sections. . . . . . . . . . . . . . . . . . . . 139 1.11 Shear Stresses in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 1.12 Shear Stresses in Rectangular Section Beams . . . . . . . . . . . . . 142 1.13 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 1.14 Contact Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 2. De¯ection and Stiffness 149 2.1 Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 2.2 Spring Rates for Tension, Compression, and Torsion . . . . . . . . 150 2.3 De¯ection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.4 De¯ections Analysis Using Singularity Functions . . . . . . . . . . . 153 2.5 Impact Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 2.6 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 2.7 Castigliano's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 2.8 Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 2.9 Long Columns with Central Loading . . . . . . . . . . . . . . . . . . . 165 2.10 Intermediate-Length Columns with Central Loading . . . . . . . . . 169 2.11 Columns with Eccentric Loading . . . . . . . . . . . . . . . . . . . . . 170 2.12 Short Compression Members . . . . . . . . . . . . . . . . . . . . . . . . 171 3. Fatigue 173 3.1 Endurance Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.2 Fluctuating Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.3 Constant Life Fatigue Diagram . . . . . . . . . . . . . . . . . . . . . . . 178 3.4 Fatigue Life for Randomly Varying Loads. . . . . . . . . . . . . . . . 181 3.5 Criteria of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 References 187 Table of Contents vii CHAPTER 4 Theory of Mechanisms Dan B. Marghitu 1. Fundamentals 190 1.1 Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 1.2 Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 1.3 Kinematic Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 1.4 Number of Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . 199 1.5 Planar Mechanisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 2. Position Analysis 202 2.1 Cartesian Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 2.2 Vector Loop Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 3. Velocity and Acceleration Analysis 211 3.1 Driver Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3.2 RRR Dyad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 3.3 RRT Dyad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.4 RTR Dyad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 3.5 TRT Dyad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 4. Kinetostatics 223 4.1 Moment of a Force about a Point . . . . . . . . . . . . . . . . . . . . . 223 4.2 Inertia Force and Inertia Moment . . . . . . . . . . . . . . . . . . . . . 224 4.3 Free-Body Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.4 Reaction Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 4.5 Contour Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 References 241 CHAPTER 5 Machine Components Dan B. Marghitu, Cristian I. Diaconescu, and Nicolae Craciunoiu 1. Screws 244 1.1 Screw Thread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 1.2 Power Screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 2. Gears 253 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 2.2 Geometry and Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . 253 2.3 Interference and Contact Ratio . . . . . . . . . . . . . . . . . . . . . . . 258 2.4 Ordinary Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 2.5 Epicyclic Gear Trains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 2.6 Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 2.7 Gear Force Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 2.8 Strength of Gear Teeth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 3. Springs 283 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.2 Material for Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 3.3 Helical Extension Springs . . . . . . . . . . . . . . . . . . . . . . . . . . 284 3.4 Helical Compression Springs . . . . . . . . . . . . . . . . . . . . . . . . 284 3.5 Torsion Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 3.6 Torsion Bar Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 3.7 Multileaf Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 3.8 Belleville Springs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 viii Table of Contents 4. Rolling Bearings 297 4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 4.2 Classi®cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4.3 Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 4.4 Static Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 4.5 Standard Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 4.6 Bearing Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 5. Lubrication and Sliding Bearings 318 5.1 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 5.2 Petroff's Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 5.3 Hydrodynamic Lubrication Theory . . . . . . . . . . . . . . . . . . . . 326 5.4 Design Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 References 336 CHAPTER 6 Theory of Vibration Dan B. Marghitu, P. K. Raju, and Dumitru Mazilu 1. Introduction 340 2. Linear Systems with One Degree of Freedom 341 2.1 Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 2.2 Free Undamped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 343 2.3 Free Damped Vibrations. . . . . . . . . . . . . . . . . . . . . . . . . . . 345 2.4 Forced Undamped Vibrations . . . . . . . . . . . . . . . . . . . . . . . 352 2.5 Forced Damped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . 359 2.6 Mechanical Impedance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 2.7 Vibration Isolation: Transmissibility. . . . . . . . . . . . . . . . . . . . 370 2.8 Energetic Aspect of Vibration with One DOF . . . . . . . . . . . . . 374 2.9 Critical Speed of Rotating Shafts. . . . . . . . . . . . . . . . . . . . . . 380 3. Linear Systems with Finite Numbers of Degrees of Freedom . . . . . . . 385 3.1 Mechanical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 3.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 3.3 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 3.4 Analysis of System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 405 3.5 Approximative Methods for Natural Frequencies. . . . . . . . . . . 407 4. Machine-Tool Vibrations 416 4.1 The Machine Tool as a System . . . . . . . . . . . . . . . . . . . . . . 416 4.2 Actuator Subsystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 4.3 The Elastic Subsystem of a Machine Tool . . . . . . . . . . . . . . . 419 4.4 Elastic System of Machine-Tool Structure . . . . . . . . . . . . . . . . 435 4.5 Subsystem of the Friction Process. . . . . . . . . . . . . . . . . . . . . 437 4.6 Subsystem of Cutting Process . . . . . . . . . . . . . . . . . . . . . . . 440 References 444 CHAPTER 7 Principles of Heat Transfer Alexandru Morega 1. Heat Transfer Thermodynamics 446 1.1 Physical Mechanisms of Heat Transfer: Conduction, Convection, and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 Table of Contents ix 1.2 Technical Problems of Heat Transfer . . . . . . . . . . . . . . . . . . . 455 2. Conduction Heat Transfer 456 2.1 The Heat Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . 457 2.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 2.3 Initial, Boundary, and Interface Conditions . . . . . . . . . . . . . . . 461 2.4 Thermal Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 2.5 Steady Conduction Heat Transfer . . . . . . . . . . . . . . . . . . . . . 464 2.6 Heat Transfer from Extended Surfaces (Fins) . . . . . . . . . . . . . 468 2.7 Unsteady Conduction Heat Transfer . . . . . . . . . . . . . . . . . . . 472 3. Convection Heat Transfer 488 3.1 External Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . 488 3.2 Internal Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . 520 3.3 External Natural Convection. . . . . . . . . . . . . . . . . . . . . . . . . 535 3.4 Internal Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . 549 References 555 CHAPTER 8 Fluid Dynamics Nicolae Craciunoiu and Bogdan O. Ciocirlan 1. Fluids Fundamentals 560 1.1 De®nitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 1.2 Systems of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 1.3 Speci®c Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 1.4 Viscosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561 1.5 Vapor Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 1.6 Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 1.7 Capillarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562 1.8 Bulk Modulus of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . 562 1.9 Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 1.10 Hydrostatic Forces on Surfaces. . . . . . . . . . . . . . . . . . . . . . . 564 1.11 Buoyancy and Flotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 565 1.12 Dimensional Analysis and Hydraulic Similitude . . . . . . . . . . . . 565 1.13 Fundamentals of Fluid Flow. . . . . . . . . . . . . . . . . . . . . . . . . 568 2. Hydraulics 572 2.1 Absolute and Gage Pressure . . . . . . . . . . . . . . . . . . . . . . . . 572 2.2 Bernoulli's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573 2.3 Hydraulic Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 2.4 Pressure Intensi®ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 2.5 Pressure Gages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 2.6 Pressure Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580 2.7 Flow-Limiting Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592 2.8 Hydraulic Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 2.9 Hydraulic Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 2.10 Accumulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601 2.11 Accumulator Sizing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 2.12 Fluid Power Transmitted . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 2.13 Piston Acceleration and Deceleration. . . . . . . . . . . . . . . . . . . 604 2.14 Standard Hydraulic Symbols . . . . . . . . . . . . . . . . . . . . . . . . 605 2.15 Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606 x Table of Contents [...]... Bucharest 6-77206, Romania P K Raju, (339) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 xv 1 Statics DAN B MARGHITU, CRISTIAN I DIACONESCU, AND BOGDAN O CIOCIRLAN Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Inside 1 Vector Algebra 1. 1 1. 2 1. 3 1. 4 1. 5 1. 6 1. 7 1. 8 1. 9 1. 10 1. 11 1 .12 1. 13 2 Terminology and Notation 2 Equality 4 Product of... Vectors 11 Vector Triple Product of Three Vectors 11 Derivative of a Vector 12 2 Centroids and Surface Properties 2 .1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2 .10 2 .11 2 .12 3 Moments and Couples 3 .1 3.2 3.3 3.4 3.5 3.6 12 Position Vector 12 First Moment 13 Centroid of a Set of Points 13 Centroid of a Curve, Surface, or Solid 15 Mass Center of a Set of Particles 16 Mass Center of a Curve, Surface, or Solid 16 First... A .1 Differential Equations of Mechanical Systems 612 613 614 616 616 617 618 620 623 623 624 628 6 31 632 633 637 639 640 6 41 648 649 650 6 51 656 660 6 61 664 669 672 678 678 6 81 685 688 689 6 91 695 695 700 703 703 xii Table of... Horatiu Barbulescu, ( 715 ) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Bogdan O Ciocirlan, (1, 51, 11 9, 559) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Nicolae Craciunoiu, (243, 559) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Cristian I Diaconescu, (1, 51, 11 9, 243) Department of Mechanical Engineering,... head of the last vector (Fig 1. 4b) The sum v 1 Àv 2 is called the difference of v 1 and v 2 and is denoted by v 1 À v 2 (Figs 1. 4c and 1. 4d) Figure 1. 4 The sum of n vectors v i , i 1Y F F F Y n, n i 1 v i or v 1 v 2 Á Á Á v n Y is called the resultant of the vectors v i , i 1Y F F F Y n Statics 5 1 Vector Algebra 6 Statics Statics The vector addition is: 1 Commutative, that is, the... curves For each simple curve the centroid is known Figure 2 .11 represents a curve Figure 2.9 21 Statics 2 Centroids and Surface Properties Figure 2 .10 made up of straight lines The line segment L1 has the centroid C1 with coordinates xC 1 , yC 1 , as shown in the diagram For the entire curve 4 xC i 1 xCi Li L 4 Y yC i 1 yCi Li L X m Figure 2 .11 2.8 Theorems of Guldinus±Pappus The theorems of Guldinus±Pappus... If v 1 v1x i v1y j v1z k and v 2 v2x i v2y j v2z k, then the sum of the vectors is v 1 v 2 v1x v2x i v1y v2y j v1z v2z v1z kX 1. 8 Angle between Two Vectors Let us consider any two vectors a and b One can move either vector parallel to itself (leaving its sense unaltered) until their initial points (tails) coincide The angle between a and b is the angle y in Figs 1. 7a... Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Mircea Ivanescu, ( 611 ) Department of Electrical Engineering, University of Craiova, Craiova 11 00, Romania Dan B Marghitu, (1, 51, 11 9, 18 9, 243, 339) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Dumitru Mazilu, (339) Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Alexandru Morega,... 10 .5 Effects of Supplementary Poles and Zeros 10 .6 Design Example: Closed-Loop Control of a Robotic Arm 11 Design of Closed-Loop Control Systems by Frequential Methods 12 State Variable Models 13 Nonlinear Systems 13 .1 Nonlinear Models: Examples 13 .2 Phase Plane Analysis 13 .3 Stability... F F F of which all but one can be selected arbitrarily 1. 7 Resolution of Vectors and Components Let i1 , i2 , i3 be any three unit vectors not parallel to the same plane ji1 j ji2 j ji3 j 1X For a given vector v (Fig 1. 5), there exists three unique scalars v1 , v1 , v3 , such that v can be expressed as v v 1 i1 v2 i2 v3 i3 X Figure 1. 5 The opposite action of addition of vectors is the resolution . and Components 6 1. 8 Angle between Two Vectors 7 1. 9 Scalar (Dot) Product of Vectors 9 1. 10 Vector (Cross) Product of Vectors 9 1. 11 Scalar Triple Product of Three Vectors 11 1. 12 Vector Triple. 36849 Mircea Ivanescu, ( 611 ) Department of Electrical Engineering, University of Craiova, Craiova 11 00, Romania Dan B. Marghitu, (1, 51, 11 9, 18 9, 243, 339) Department of Mechanical Engineering,. . . . . 13 5 1. 10 Beams with Asymmetrical Sections. . . . . . . . . . . . . . . . . . . . 13 9 1. 11 Shear Stresses in Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 0 1. 12 Shear