(Part A) Machinerys Handbook 31st Edition Pages 1-1484

Machinery's Handbook, 31st Edition

A REFERENCE BOOK for the M anufacturing and M echanical E ngineer , D esigner , D rafter , M etalworker , T oolmaker , M achinist , H obbyist , E ducator , and S tudent

Machinery’s Handbook 31 st Edition

B y E rik O berg , F ranklin D. J ones , H olbrook L. H orton , H enry H. R yffel , and C hristopher J. M c C auley

L aura B rengelman , E ditor

2020 I ndustrial P ress , I nc .

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Machinery's Handbook, 31st Edition

INDUSTRIAL PRESS, INC. 32 Haviland Street, Suite 3 South Norwalk, Connecticut 06854 U.S.A. Phone: 203-956-5593 Toll-Free: 888-528-7852 Fax: 203-354-9391 Email: info@industrialpress.com

Title: Machinery’s Handbook, 31st Edition Authors: E rik Oberg, Franklin D. Jones, Holbrook L. Horton, Henry H. Ryffel, and Christopher J. McCauley Library of Congress Control Number: 2019954863 COPYRIGHT 1914, 1924, 1928, 1930, 1931, 1934, 1936, 1937, 1939, 1940, 1941, 1942, 1943, 1944, 1945, 1946, 1948, 1950, 1951, 1952, 1953, 1954, 1955, 1956, 1957, 1959, 1962, 1964, 1966, 1968, 1971, 1974, 1975, 1977, 1979, 1984, 1988, 1992, 1996, 1997, 1998, 2000, 2004, 2008, 2012, 2016, 2020 © by Industrial Press, Inc. ISBN 978-0-8311-3731-1 (Toolbox, Thumb-Indexed, 4.6 × 7 in., 11.7 × 17.8 cm) ISBN 978-0-8311-3631-4 (Large Print, Thumb-Indexed, 7 × 10 in., 17.8 × 25.4 cm) ISBN 978-0-8311-3831-8 (Digital Edition) ISBN 978-0-8311-3931-5 (Digital Edition Upgrade) ISBN 978-0-8311-4131-8 (Toolbox and Digital Edition Combo, 4.6 × 7 in., 11.7 × 17.8 cm) ISBN 978-0-8311-4031-1 (Large Print and Digital Edition Combo, 7 × 10 in., 17.8 × 25.4 cm) No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval While every possible effort has been made to ensure the accuracy of all information presented herein, the publisher expresses no guarantee of the same, does not offer any warrant or guarantee that omissions or errors have not occurred, and may not be held liable for any damages resulting from use of this text. Readers accept full responsibility for their own safety and that of the equipment used in conjunction with this text. Printed and bound by Thomson Press. MACHINERY’S HANDBOOK 31ST EDITION First Printing system, without written permission from the publisher. Limits of Liability and Disclaimer of Warranty

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Machinery's Handbook, 31st Edition

MACHINERY’S HANDBOOK, 31ST EDITION, TEAM Editorial Advisory Board

Steve Heather, an acclaimed mechanical engineer, worked in the defense, aircraft, auto- mobile, and lighting industries for more than 30 years. More recently, he taught mechani- cal engineering and computer-aided design (CAD) at the college level to engineering and architectural students. He is an expert in AutoCAD ® , CNC programming, multi-stage press tool design, and precision machining, as well as author of AutoCAD ® 3D Modeling and Engineers Precision Data Pocket Reference and coauthor of Beginning AutoCAD ® Exercise Workbook, Advanced AutoCAD ® Exercise Workbook, and AutoCAD ® Pocket Reference, all published by Industrial Press. He is an invaluable engineering advisor and contributor to material throughout the Machinery’s Handbook , as well as a skilled technical illustrator, whose work enhances the current edition . David O. Kazmer is professor of plastics engineering at the University of Massachusetts Lowell, as well as associate research professor at the University of Massachusetts Amherst. He is the recipient of 19 recognition awards, including the Ishii-Toshiba Design for Manu- facturing Award, an inventor with over 20 patents, and author of more than 200 publications and vital text in the Machinery’s Handbook . His academic work is motivated by industry experiences, most recently as director of research and development at Dynisco HotRunners. His teaching and research encompass process development, product and machine design, and design methodologies, including polymer processing, design for manufacturing, opti- mization, simulation, process control, and technology strategy. His ongoing research has contributed to development of new manufacturing processes with improved real-time con- trol and robust design tools. Howard Kuhn most recently served as adjunct professor at the University of Pittsburgh, Swanson School of Engineering, where he taught courses in manufacturing, additive man- ufacturing, product realization, and engineering entrepreneurship and performed research on additive manufacturing. He currently is a technical advisor to America Makes (National Additive Manufacturing Innovation Institute), where he previously was acting deputy direc- tor. Specializing in advanced technology implementation, he has engaged in the design and application of multiple additive manufacturing technologies for major government clients and private industry. He has developed undergraduate and graduate courses in engineering design, failure analysis, deformation processing, and powder and mechanical metallurgy; conducts tailored training courses on additive manufacturing for government agencies, trade organizations, and companies; and contributed authoritative revisions to this edition. Jennifer Marrs has worked as a mechanical engineer for more than 20 years. An accredited Professional Engineer, she earned a master of science degree in mechanical engineering at Northeastern University, holds a patent, and is a registered US patent agent. Her areas of expertise include manufacturing engineering, design and analysis of machinery, tool and fixture design, machinery safety, fluid systems, industry standards, and intellectual property issues. She has worked with companies of all sizes and ran a successful consulting prac- tice for nearly a decade. She is the author of the Industrial Press text Machine Designers Reference , as well as key technical material in the Machinery’s Handbook . Contributors Viktor P. Astakhov earned his Ph.D. in mechanical engineering from Tula State Polytech- nic University, Tula–Moscow, USSR–Russia, in 1983. He was awarded a DSci designation (Dr. Habilitation, Docteur d’État) in 1991 and the title “State Professor of Ukraine” in 1991 for outstanding service rendered during his teaching career and the profound impact his work had on science and technology. An internationally recognized educator, researcher, and mechanical engineer, he has won a number of prestigious awards. In 2011, he was elected to the SME College of Fellows, and, in 2018, he became a member of the European Union x

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Machinery's Handbook, 31st Edition

MACHINERY’S HANDBOOK, 31ST EDITION, TEAM

Academy of Science. As a professor, he has supervised graduate students at Michigan State University. He currently serves as a research professor at St. Petersburg State Polytechnic University in Russia, as well as the tool research and application manager of the General Motors Business Unit of PSMI. Vukota Boljanovic received his B.S., M.S., and Ph.D. in mechanical engineering and has more than 45 years of experience in applied engineering in the aircraft and automotive indus- tries, including serving as vice president for research and development with a major aircraft company. He has taught aerospace engineering, among other subjects, and has performed extensive research in development and manufacturing engineering, including the impact of design modification on tools, dies, and processes selection; aircraft assembly; and inspec - tion. He is the author of numerous technical papers and books, including the Industrial Press titles Applied Mathematical and Physical Formulas , Metal Shaping Processes , Sheet Metal Forming Processes and Die Design , and Sheet Metal Stamping Die Designs, and has been widely recognized by both academia and industry for his contributions to manufacturing. Charles “Wes” Cross has been a weld engineer for more than 25 years. He holds a degree in weld engineering from Le Tourneau University in Longview, Texas, and has held many posi- tions in local American Welding Society (AWS) chapters. His industry experience encom- passes weld applications, quality control programs, shop supervision, and weld program implementation and maintenance. He is an expert in weld processes and adhering to ASME, AWS, and NAVSEA standards and codes, as well as an AWS Certified Welding Inspector and Certified Welding Educator. He recently established All Welding Services, a consulting company for weld shops and weld code programs, working with companies around the world. Brad Dulin is a senior metrologist and has worked in the research and development in aero- space, biomedical, and other technical fields for more than 30 years. He has collaborated with such technological giants as Hughes Helicopters and Space and Communications divisions, the Jet Propulsion Laboratory, and Raytheon, as well as technology companies worldwide. He has served as a metrology consultant with BAE for the Kuwait Air Force and, most recently, as operations manager for the Kuwait Green Energy Company. He currently lives in Kuwait and is collaborating on the development of advanced hydrogen/solar-based alternative energy solutions. Arief Era is a graduate of Columbia University, where he received his master’s degree in mechanical engineering. As a structural analysis engineer, he worked on various commer- cial aircrafts for the Boeing Company. At Consolidated Edison of New York in the Gas Engi- neering department, he has developed fittings, valves, and piping systems for gas delivery while performing root cause analysis for field failures. More recently, he has led the Maps and Records Team for the gas delivery infrastructure of Manhattan. His experience and extensive work on developing, refining, and working with industry standard and related tech - nical information has been crucial to the current edition of the Handbook. Charles Gillis has over 24 years of machine design experience. He received his bachelor of science degree from Worcester Polytechnic Institute and a master’s degree in mechanical engineering from Northeastern University. Currently serving as a mechanical design engi- neer for the Gillette Company, designing automated machinery for manufacturing blade and razor products, he has been training practicing engineers in geometric dimensioning and tolerancing, print reading, and related mechanical analysis, design, and documentation topics for the last decade. A licensed Professional Engineer in Massachusetts, he holds a Geometric Dimensioning and Tolerancing Professional Certificate– Senior Level (GDTP- S) from ASME. In addition to authoring the bestselling Hammer’s Blueprint Reading Basics, 4th Edition, he also contributed to the Industrial Press titles Machine Designers Reference and The CAM Design and Manufacturing Handbook . Edmund Isakov earned his Ph.D. in technical sciences at the Novocherkassk Polytechnic Institute (Soviet Union). He is known for his work in research, development and applications of cutting tools for milling, turning, and boring. During nearly two decades at Kennametal, he became a noted authority on the technical analysis of cutting tools and processes. He holds 7 U.S. patents and 10 U.S.S.R Inventor’s Certificates pertaining to carbide and diamond

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Machinery's Handbook, 31st Edition

MACHINERY’S HANDBOOK, 31ST EDITION, TEAM

tools. A senior member of SME, he has authored numerous articles and papers, as well as the Industrial Press titles Cutting Data for Turning of Steel , Engineering Formulas for Metal- cutting , and International System of Units (SI) . Melissa Klingenberg holds a master’s degree in manufacturing systems engineering and a Ph.D. in materials engineering. She has more than 25 years of experience in inorganic finishing operations, specializing in development, technology evaluation, and implemen - tation of innovative coatings and surface finishing processes to improve engineering prop - erties and address environmental issues. Known for research in wear-resistant coatings and replacement technologies for defense applications, she has co-authored many publi- cations. A research and development engineer at Pennsylvania State University’s Applied Research Laboratory, she also has served as AESF Foundation president and on the AESF Council, Emerging Technologies Committee, NASF Research Board, and Sur/Fin Tech- nical Committee, and as organizer and chair for the Surface Engineering for Defense and Aerospace Applications Conference. While a principal advising engineer at Concurrent Technologies Corporation, she received the 2015 National Association for Surface Finish- ing’s Scientific Achievement Award. Kathleen McKenzie is an educator, writer, editor, and copy editor specializing in science and mathematics. With a bachelor’s degree in chemistry and a master’s in mathematics, she has been a career mathematics educator— primarily at Binghamton University, SUNY, where she teaches undergraduate calculus. She has worked for major publishers and com- position houses nationwide, and has helped develop and add materially to a number of best selling Industrial Press texts. Among these, she contributed to Technical Shop Mathematics, and served as development editor on The Handbook of PVC Pipe Design and Construction. In the last and particularly the current edition of the Machinery’s Handbook, she helped make significant revisions in the mathematics and measuring units sections, as well as refining other key topics. Merwan Mehta is a professor in the College of Engineering and Technology at East Caro- lina University in Greenville, North Carolina. He has taught engineering economics at the undergraduate and graduate levels since 2004. Prior to joining academia, he spent more than 20 years in the manufacturing industry as a partner in business, vice president, project director, manager, industrial and manufacturing engineer, and machine tool design engi- neer. A Certified Manufacturing Engineer and Certified Six Sigma Black Belt, he conducts workshops internationally on various engineering and operational excellence topics and has served as an examiner for the Missouri Quality Award, based on the Baldrige Criteria. His Applied Engineering Economics Using Excel is a top Industrial Press text. David R. Quinonez has over 25 years of experience in welding, welding inspection, and nondestructive testing. His impressive experience in nondestructive testing began with nuclear submarines and aircraft carriers. Subsequent positions included nondestructive test- ing (NDT) and welding inspection on Rolls- Royce gas turbine engines, F- 22 stealth fighter airframes, missile defense, commercial/military rockets, pipeline, and structural steel for private sector and public works projects. He is a Certified Welding Inspector performing visual welding inspection, dimensional verification, and NDT. His Level II certifications include UT, MT, and PT; past certifications have included Level II RT, ET, and ASNT Level III MT and PT. He is the author of the highly instructive Industrial Press title 1,001 Questions & Answers for the CWI Exam: Welding Metallurgy and Visual Inspection Study Guide . Peter Smid is a professional consultant, educator and speaker, with many years of practical, hands-on experience. He consults to manufacturing industry and educational institutions on use of CAD/CAM software, CNC technology, part programming, advanced machin- ing, tooling, and setup. His comprehensive industrial background in CNC programming, machining, and company- oriented training has assisted hundreds of companies, and he has developed and delivered thousands of customized educational and industrial programs to instructors, students, manufacturing companies, private sector organizations, and others. He is author of numerous, definitive articles and publications on the subject of CNC and CAD/CAM, with six titles for Industrial Press, including the CNC Programming Handbook and the CNC section of the Handbook.

xii

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Machinery's Handbook, 31st Edition

PREFACE In continuous publication since 1914, Machinery’s Handbook has served as the principal reference work in metalworking, engineering, design, and manufacturing facilities, and in technical schools, colleges, and universities throughout the world for more than a century. Throughout this period, the Handbook editors have strived to create a comprehensive and practical resource, combining the most basic and crucial aspects of sophisticated manufac- turing practice. The Handbook is an invaluable tool, to be used in much the same way as other tools, to design, make, repair, and maintain products of highest quality, at the lowest cost, and in the shortest time possible. The essential basics, material of proven and everlasting worth, always must be included, if the Handbook is to continue to serve the needs of the manufacturing and mechanical engineering community. Yet it is difficult to select suitable material from the vast supply of data related to these traditional yet rapidly evolving fields. An ongoing challenge is to provide valuable information for design and production departments in manufacturing plants and workshops of all sizes, as well as for product and system designers, job shops, hobbyists, and instructors and students in general, trade, technical, and engineering schools. The editorial team relies on conversations and written communications with users of the Handbook and experts in techni- cal fields for guidance on topics to be introduced, revised, lengthened, shortened, or omitted. The original Handbook was designed to fit inside a standard toolbox. At the request of users, in 1997, the large print or “desktop” edition of the Handbook was introduced. The large print edition is identical to the traditional toolbox edition, only its size is increased by 140 percent, making it an easier-to-read reference. (Note that the type is standard reference size, not a larger font designed for visually impaired readers.) Other than size, there is no difference between the toolbox and large print editions. In 1998, Christopher McCauley developed and launched the first Machinery’s Hand- book CD- ROM, containing the complete content of the printed book, with added indexes and hundreds of pages of archival material restored from earlier editions. Continued as the Machinery’s Handbook 31st Digital Edition , this versatile format offers rapid searching and navigation aids in the form of clickable links and cross references that take you quickly to pages referenced. The growing family of Machinery’s Handbook products also includes the Guide, Pocket Companion , and attractive combination packages. Longtime users of the Handbook will note many changes in recent editions, but an endur- ing goal of the editors is to make this encyclopedic reference easier to use. The Handbook continues to incorporate time- saving thumb tabs, much requested by users. In addition to the front table of contents, sectional contents beginning each major section, introduced in the 25th edition, also have proven useful to readers. In the 31st edition, these sectional contents have been expanded to provide even more detailed navigation aids. Overall, this edition has been edited, updated, and reset. Incorporating thousands of indi vidual changes and more than 250 new and revised tables and figures, it has expanded by nearly 100 pages, to 2,992 pages. Among major revisions of existing content and new mate- rial are the following: First and foremost, hundreds of specific references and pieces of key information based on the most current ANSI, ASME, and ASTM standards have been updated throughout the Handbook. Of all the reasons to purchase the 31st edition , these timely updates make this a must- have resource. To examine other improvements starting at page 1, while the core concepts of MATH- EMATICS remain unchanged, this baseline material has benefited from reorganization, expansion, and elucidation by subject experts and educators to reflect current terminology and teaching. Among other formula additions in the Handbook, new calculations for Toler- ance Analysis and Assignment can be found on page 684. Also see MEASURING UNITS , on page 2827, for added information on International System of Units (SI), related, base, and derived units, names, and prefixes. Expert revisions have been made throughout MECHANICS AND STRENGTH OF MATE- RIALS, beginning on page 156, regarding forces, strength, testing, and related analysis. The subsection PROPERTIES OF BODIES is now more specifically defined as RIGID BODY PARAMETERS. vii

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Machinery's Handbook, 31st Edition

PREFACE The PLASTICS section, beginning on page 555, has received important updates, with new information on material characteristics, properties, and costs; calculations for elastic- ity, stress, strain, loads, and temperature effects; manufacturing and machining processes; design considerations; use of plastics for prototyping; and additive manufacturing (AM). Metal Additive Manufacturing , beginning on page 1555, has been redone, with illustra- tions and tables and updated text addressing the ongoing evolution of this dynamic field; key trends and recent developments in materials, processes and workflow, production consider - ations, and finishing; comparisons of AM costs, efficiencies, and capabilities with conven - tional processes for producing parts; and applicable ASTM and ISO standards. The related topic of POWDER METALLURGY , beginning on page 1522, also has received updates and added dimensional information. A new CORROSION section on page 548 discusses forms, causes, and methods of preven- tion; galvanic series, compatibility, and coupling; and common effects and mitigation meth- ods. Other surface-related information includes updates in FINISHING OPERATIONS , starting on page 1632, on electropolishing, passivation, and plating, and added Sheet Metal Mill Finishes on page 1448. Related sheet metal additions include Blanking Pressure on page 1413, Three- Roll Bending on page 1423, and Sheet Metal Gauge Sizes on page 1453. In MACHINING OPERATIONS , see page 1154 for new material on boring and indexable boring bars. The new NONTRADITIONAL MACHINING AND CUTTING section on page 1344 describes mechanical erosion and electro-thermal processes. And in CNC NUMER- ICAL CONTROL PROGRAMMING , the topic CAD/CAM , on page 1390, has been reintro- duced and revised. IRON AND STEEL CASTING has been renamed METAL CASTING, MOLDING, AND EXTRUSION , and this section, starting on page 1480, has been rewritten and expanded. Look for new information on working with iron, steel, and nonferrous casting metals; indus- try casting and molding processes; heating, pouring, flow, and fluidity; heat transfer, solidifi - cation, and cooling; and materials, applications, casting defects, design considerations, and computer modeling. The SOLDERING AND BRAZING and WELDING sections also have received important updates, including new and revised figures and tables, and an expanded section on Non destructive Testing, on page 1627. O-RINGS , on page 2666, has been rewritten and expanded, with 13 tables and details of standards, designations, selection, clearance gap, cramping, face seals, glands, grooves, installation, lubrication, squeeze, stretch, and equations. In UNIFIED SCREW THREADS, the latest Standard Series and Selected Combination tables, beginning on page 1951, incorporate numerous updates. TRANSMISSION CHAINS, on page 2616, has been revised, with meticulous updates to the Horsepower Ratings for Roller Chains tables, based on the latest standards. There are the countless other changes in tables, figures, calculations, and text resulting from the Handbook team’s extensive review of current industry standards. As in previous editions, we continue to include expanded, parallel US Customary and metric figures. Where possible, formulas are presented with equivalent metric expressions (some in bold). Addition of new and revised Handbook topics often requires removal of older topics to gain space. Materials removed from the print book generally appear in the Digital Edition , which contains added material not in the current print edition. Included in the Digital Edition are mathematical tables on topics such as logarithms and trigonometry; material on cement and concrete, adhesives and sealants, coloring and etching metals, forge shop equipment, silent chain, worm and other gears, keys and keyways; and many other extracts from past editions that may be of use and interest. Absent in the 31st print edition is a conversion table that previously appeared on page 3 on fractional and decimal inch to millimeter conversion; the same information can be found in tables in the MEASURING UNITS section, which starts on page 2831. Other material moved to the Digital Edition includes tabular data, such as constants involving uses of π, moments of inertia and section moduli for shafts, and an older screw thread table on European systems, as well as a short section on Bakelite, and Change Gears for Helical Mill- ing. Users requiring this information, or wishing to comment on these or other topics moved to the Digital Edition , are urged to contact the editors. viii

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Machinery's Handbook, 31st Edition

PREFACE The editors are greatly indebted to readers who call attention to possible errors and defects in the Handbook , offer suggestions for including new or revised material, or have technical questions about the content and applicability to manufacturing problems encountered in the shop. Such dialog helps identify topics that require clarification or expansion. Queries involving Handbook material usually entail an in-depth review of the topic in question, frequently resulting in improved or new material. We also welcome new contributors to each edition, joining the long line of erudite industry experts who have made the Handbook what it is today, and we invite topical experts to contact the editors. Our perpetual goal is to increase the usefulness of the Handbook as much as possible. We welcome your input and look forward to hearing from you. Acknowledgments and Special Thanks Machinery’s Handbook is indebted to the whole mechanical field for the data contained in this master reference work. On behalf of the Handbook editors past and present, we wish to express our appreciation to all who have assisted in furnishing data and contributed ideas, corrections, and other commentary on the Handbook. Most importantly, we thank the thousands of readers who have contacted us over the years with constructive criticism and suggestions regarding Handbook topics and presentation. Your comments on this edition, as well as past and future ones, are invaluable. Many of the American National Standards Institute (ANSI) standards that deal with mechanical engineering, extracts from which are included in the Handbook, are published by the American Society of Mechanical Engineers (ASME). The editors thank ASME for its exceptional collaboration in helping identify and bring essential data up to date, accord- ing to the latest, definitive, industry standards. Information concerning other standards and nomenclature also is included in the Handbook. Official standards and related publications are copyrighted by the issuing organizations; contact them directly for further information regarding current editions of standards and to purchase copies. On the following pages are brief biographies for the Machinery’s Handbook, 31st Edition team— an impressive roster of Editorial Advisory Board members and contributors. These esteemed colleagues have played a crucial role in guiding content decisions and advising on specific engineering questions and content challenges. Their lifelong educational and industry experience, impressive technical knowledge and expertise, and meticulous research have immeasurably enhanced the content of this edition. We also wish to thank those behind the scenes, our tremendous editorial and production team, without whom this edition would not be possible: the incomparable Jason Hughes, Abigail Parker, Billie Rothstein, and the rest of the remarkable team at Scribe; math maven Dan McKinney; eagle- eyed editorial team members Teresa Barensfeld, Cara Chamberlain, Gerald Murray, Julia Phelps, and Deborah Ring; and our masterful printing and binding partners at Thomson Press. In addition, longtime editor, mastermind, and retired leader of the Handbook team Christopher McCauley is recognized for his extensive additions to essential content over multiple editions. Accordingly, he has been added to the masthead of illustrious authors of the Handbook. The Handbook is indebted to many others in industry organizations and associations, educational institutions, commercial enterprises, and private practice. In this context, we cannot thank everyone who has written in, helped resolve a question, or has otherwise spent valuable time and effort assisting us improve past and present editions . Therefore, we have added a “Handbook Hall of Fame” to our website, where we can acknowledge and acclaim the most important participants in the community surrounding this legendary product. Again, we encourage you to send us your thoughts and feedback, and to share with us how the Machinery’s Handbook product family supports and enhances your involvement in this endlessly fascinating field. Laura Brengelman Editor ix

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Machinery's Handbook, 31st Edition

TABLE OF CONTENTS

PREFACE

vii ix

ACKNOWLEDGMENTS AND SPECIAL THANKS MACHINERY’S HANDBOOK TEAM

MATHEMATICS

• REAL NUMBERS AND THEIR OPERATIONS • ALGEBRA • GEOMETRY • TRIGONOMETRY: SOLUTION OF TRIANGLES • MATRICES • CALCULUS • STATISTICAL ANALYSIS OF MANUFACTURING DATA • ENGINEERING ECONOMICS MECHANICS AND STRENGTH OF MATERIALS • MECHANICS • VELOCITY, ACCELERATION, WORK, AND ENERGY • STRENGTH OF MATERIALS • RIGID BODY PARAMETERS • BEAMS • COLUMNS • PLATES, SHELLS, AND CYLINDERS • SHAFTS • SPRINGS • DISC SPRINGS PROPERTIES, TREATMENT, AND TESTING OF MATERIALS • THE ELEMENTS, HEAT, MASS, AND WEIGHT • PROPERTIES OF WOOD, CERAMICS, PLASTICS, METALS • STANDARD STEELS • TOOL STEELS • HARDENING, TEMPERING, AND ANNEALING • DRAFTING PRACTICES • ALLOWANCES AND TOLERANCES FOR FITS • MEASURING, INSTRUMENTS, AND INSPECTION METHODS • MICROMETER, VERNIER AND DIAL CALIPERS • SURFACE TEXTURE TOOLING AND TOOLMAKING • CUTTING TOOLS • CEMENTED CARBIDES • MILLING CUTTERS • REAMERS • TWIST DRILLS AND COUNTERBORES • TAPS • STANDARD TAPERS • ARBORS, CHUCKS, AND SPINDLES • BROACHES AND BROACHING • FILES AND BURS • KNURLS AND KNURLING • TOOL WEAR AND SHARPENING MACHINING OPERATIONS • NONFERROUS ALLOYS • CORROSION • PLASTICS DIMENSIONING, GAGING, AND MEASURING

154

365

618

829

1073

• CUTTING SPEEDS AND FEEDS • SPEEDS AND FEEDS TABLES • ESTIMATING SPEEDS AND MACHINING POWER • MICROMACHINING • MACHINING ECONOMETRICS • SCREW MACHINES, BAND SAWS, CUTTING FLUIDS • MACHINING NONFERROUS METALS AND NONMETALLIC MATERIALS • GRINDING FEEDS AND SPEEDS • GRINDING AND OTHER ABRASIVE PROCESSES • NONTRADITIONAL MACHINING AND CUTTING • CNC NUMERICAL CONTROL PROGRAMMING MANUFACTURING PROCESSES 1397 • SHEET METAL WORKING AND PRESSES • ELECTRICAL DISCHARGE MACHINING • METAL CASTING, MOLDING, AND EXTRUSION • POWDER METALLURGY • SOLDERING AND BRAZING • WELDING • FINISHING OPERATIONS

Each section has a detailed Table of Contents located on the page indicated.

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Machinery's Handbook, 31st Edition

TABLE OF CONTENTS

FASTENERS 1649 • TORQUE AND TENSION IN FASTENERS • INCH THREADED FASTENERS • METRIC THREADED FASTENERS • HELICAL COIL SCREW THREAD INSERTS • BRITISH FASTENERS • MACHINE SCREWS AND NUTS • CAP AND SET SCREWS • SELF-THREADING SCREWS • T-SLOTS, BOLTS, AND NUTS • RIVETS AND RIVETED JOINTS • PINS AND STUDS • RETAINING RINGS • WING NUTS, WING SCREWS, AND THUMB SCREWS • NAILS, SPIKES, AND WOOD SCREWS THREADS AND THREADING 1936 • SCREW THREAD SYSTEMS • UNIFIED SCREW THREADS • CALCULATING THREAD DIMENSIONS • METRIC SCREW THREADS • ACME SCREW THREADS • BUTTRESS THREADS • WHITWORTH THREADS • PIPE AND HOSE THREADS • OTHER THREADS • MEASURING SCREW THREADS • TAPPING AND THREAD CUTTING • THREAD ROLLING • THREAD GRINDING • THREAD MILLING GEARS, SPLINES, AND CAMS 2201 • GEARS AND GEARING • HYPOID AND BEVEL GEARING • WORM GEARING • HELICAL GEARING • OTHER GEAR TYPES • CHECKING GEAR SIZES • GEAR MATERIALS • SPLINES AND SERRATIONS • CAMS AND CAM DESIGN MACHINE ELEMENTS 2389

• PLAIN BEARINGS • BALL, ROLLER, AND NEEDLE BEARINGS • LUBRICATION • COUPLINGS, CLUTCHES, BRAKES • KEYS AND KEYSEATS • FLEXIBLE BELTS AND SHEAVES • TRANSMISSION CHAINS • BALL AND ACME LEADSCREWS • ELECTRIC MOTORS • ADHESIVES AND SEALANTS • O-RINGS • ROLLED STEEL, WIRE, SHEET METAL, WIRE ROPE • SHAFT ALIGNMENT • FLUID POWER MEASURING UNITS • SYMBOLS AND ABBREVIATIONS • MEASURING UNITS • US SYSTEM AND METRIC SYSTEM CONVERSIONS

2826

2876

Each section has a detailed Table of Contents located on the page indicated.

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TABLE OF CONTENTS MATHEMATICS Machinery's Handbook, 31st Edition

REAL NUMBERS AND THEIR OPERATIONS 3 Real Numbers 3 Properties of Real Numbers 3 Integers (Signed Numbers) 4 Order of Operations 5 Fractions and Mixed Numbers 6 Adding and Subtracting 7 Multiplying 8 Dividing 8 Decimal Numbers 9 Ratio and Proportion 10 Percentage 11 Powers and Roots 11 Properties of Exponents 12 Scientific Notation 13 Factorial Notation 13 Permutation 13 Combination 13 24 Definitions 24 EvaluatingAlgebraic Expressions 24 Combining Like Terms 25 Solving an Equation for an Unknown 26 Rearrangement and Transposition of Terms in Formulas 27 Algebraic Operations 27 Properties of Monomials and Exponents 27 Properties of Radicals 28 Polynomials 28 Operations on Polynomials 29 Factoring Polynomials 31 Equation Solving 31 System of Linear Equations 32 Second-Degree (Quadratic) Equation 34 Completing the Square 34 Using the Quadratic Formula 34 Cubic Equation 35 Functions 35 Graphs of Functions 36 Logarithms 36 Meaning 36 Properties 37 Common Prime Factorization of Numbers ALGEBRA

ALGEBRA

(Continued) 37

Natural

38 38

Using Calculators to Solve Logarithms Solving an Equation Using Logarithms

39 Arithmetic Sequence 39 Geometric Sequence GEOMETRY 42 Analytic Geometry 42

Rectangular Coordinate System Slope of a Line Lines and Line Segments Equation Forms of a Line

42 43 44 49 51 54 55 57

Circle Ellipse

Four-Arc Oval Approximating an Ellipse

Sphere Parabola

58 Hyperbola 59 Complex Numbers 59 61 Pure Geometry 61 71 Area and Volume 71 59 66

Imaginary Number Forms of a Complex Number Propositions of Geometry Geometric Constructions Prismoidal Formula Pappus-Guldinus Rules Finding Area of a Surface of Revolution Area of Irregular Plane Figure Areas Enclosed by Cycloidal Curves Contents of Cylindrical Tanks at Different Levels

71 72 72 73 73

75 Dimensions of Plane Figures 81 Polygons 83 Segments of a Circle 84 Segments of a Circle for Radius = 1 86 Diameters of Circles and Sides of Squares of Equal Area 87 Diagonals of Squares and Hexagons 88 Volumes of Solids

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Machinery's Handbook, 31st Edition

TABLE OF CONTENTS MATHEMATICS

TRIGONOMETRY: SOLUTION OF TRIANGLES 94 Terminology 94 Degree and Radian Angle Measure 94 Trigonometric Ratios of Essential Angles 95 Functions of Angles 95 Right Triangle Ratios 96 Law of Sines 96 Law of Cosines 96 Trigonometric Identities 98 Solution of Right Triangles 99 Solution and Examples of Right Triangles 100 Solution and Examples of Oblique Triangles 102 Rapid Solution of Triangles 103 Conversion Tables of Angular Measure 105 Trigonometric Functions 106 Trigonometry Tables 111 Using a Calculator to Find

CALCULUS

130 Derivatives 130 Formulas 131 Rules 131 Integrals (Antiderivatives) 132 Integral Rules 132 Newton’s Method for Solving Equations Formulas for Differential and Integral Calculus 135 Series Representation of a Function STATISTICAL ANALYSIS OF MANUFACTURING DATA 136 Statistics Theory in Brief 136 Probability 137 Normal Distribution Analysis 139 Applying Statistics 139 Minimum Number of Test or Data Points 139 Comparing Products with Respect to Average Performance ENGINEERING ECONOMICS 143 Interest 143 Variables 144 Simple Interest 144 Compound Interest 144 Determining Principal, Rate, or Time 145 Nominal versus Effective Interest Rates 146 Cash Flow and Equivalence 146 Present Value and Discount 146 Annuities 147 Sinking Funds 147 Cash Flow Diagrams 149 Depreciation 149 Straight Line 149 Sum of the Years Digits 149 Double Declining Balance 149 Statutory Depreciation 150 Evaluating Alternative Investments 150 Net Present Value 151 Capitalized Cost 152 Equivalent Uniform Annual Cost 153 Rate of Return 153 Benefit-Cost Ratio 153 Payback Period 133

Trigonometric Function Values Versed Sine and Cosine Sevolute Functions

111 111 111

Involute Functions 116 Spherical Trigonometry 116 Right-Angle Spherical Trigonometry 118 Oblique Spherical Trigonometry 120 CompoundAngles 122 Interpolation MATRICES 124 Matrix Operations 124 Addition and Subtraction 124 Multiplication 125 Transpose 125 Determinant of a Square Matrix 125 Minors and Cofactors 126 Adjoint of a Matrix 126 Singularity and Rank 126 Inverse 127

Solving a System of Equations

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Machinery's Handbook, 31st Edition

REAL NUMBERS

REAL NUMBERS AND THEIR OPERATIONS Real Numbers

Most mathematical computation is performed in the real number system . The universal set of the “reals” includes the subsets: naturals, whole numbers, integers, rationals, and irrationals . The naturals (also called counting numbers ): {1, 2, 3, . . .} are included in the whole numbers: {0, 1, 2, 3, . . .}, which are included in the integers (or signed whole num- bers): {. . . ,– 2, – 1, 0, 1, 2, . . .}. And all of these subsets are included in the rationals. Rational numbers, including integers, can be written in fraction form. Since all frac- tions can be divided numerator by denominator, their decimal form either terminates or repeats. Examples of rational numbers: – 4/1, 3/5 = 0.6, 1/3 = 0.333. . . . The only set in the real numbers larger than the naturals that does not contain any of the other sets is the irrationals. These are not expressible as ratios. An irrational number’s decimal representation does not terminate and it has no pattern of repetition. Examples of irrational numbers are roots that cannot be simplified, such as 6 70 3 and , as well as quantities like π and the natural log base e . The entire real number set is the union of the rationals and the irrationals. Properties of Real Numbers.— Though often obvious and followed almost automati- cally, the properties of real numbers are critical to mathematical reasoning. These prop- erties justify various steps in solving algebraic problems, such as those in this Handbook. Equivalence properties (symmetry, reflexivity, transitivity) and operational properties of numbers are summarized here. Equivalence Properties: The properties of equivalence relations are the basis of equa- tion solving. Reflexive: a = a . Symmetric: If a = b , then b = a . Transitive: If a = b and b = c , then a = c . Substitution: If a = b , then a may be replaced by b in any equation or expression. Operational Properties: These concern addition, subtraction, multiplication, and divi- sion, as summarized in the table below. Property Addition Multiplication Commutative: a + b = b + a a × b = b × a Associative: ( a + b ) + c = a + ( b + c ) ( a × b ) × c = a × ( b × c ) Identity: a + 0 = 0 + a = a 1 × a = a × 1 = a Inverse: a + (–a ) = 0 a × 1/ a = 1 Other Properties: Distributive of multiplication over addition: a × ( b + c ) = ( a × b ) + ( a × c ) ( a + b ) × c = ( a × c ) + ( b × c ) Zero property of multiplication: If a × b = 0, then either a = 0 or b = 0 Zero property of division: If a / b = 0, then a = 0 ( b ≠ 0) Integers (Signed Numbers).— Positive whole numbers extend to the right of zero on the number line. Negative whole numbers extend to the left of zero. Together with zero, these make up the integers (sometimes called signed numbers ): {. . . ,– 2, – 1, 0, 1, 2, . . .}. The sciences (as well as economics and other fields) deal with negative as well as non- negative quantities. Temperature is an obvious example; so is land altitude, which can be

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Machinery's Handbook, 31st Edition

4 INTEGERS above, at, or below sea level. Angles can be negative, too, as explained in TRIGONOME- TRY . Calculators facilitate computation that involves integers (signed numbers). Knowing the rules of integer operations prevents errors that might occur when a calculator is used. Absolute Value: A number’s absolute value, sometimes called its magnitude , is the num- ber’s distance from zero on the number line. Whether a number is positive or negative, its absolute value is positive. For example, the absolute value of both 5 and – 5 is 5. The absolute value of n is notated | n |; thus, |5| = 5 and |– 5| = 5. Absolute value helps explain the rules of signed number addition and subtraction. Real Number Line: The real number line is generally shown with only the integers marked off (though all numbers are included). A number line is useful for conveying how signed numbers are added or subtracted. Operations on Signed Numbers: The following rules of operations apply to rational and irrational numbers as well. For simplicity, only integers are given as examples. Addition and Subtraction: Adding a negative number is equivalent to subtracting its absolute value. When a larger number is subtracted from a smaller number, the result is negative. The rules for adding and subtracting integers are illustrated with an example using four values: 7, 11, –7, and –11. The following examples illustrate the rules: Examples, Addition Examples, Subtraction 7 + 11 = 18 7 – 11 = –4 7 + (–11) = 7 – 11 = –4 7 – (–11) = 7 + 11 = 18 (–7) + 11 = 11 + (–7) = 11 – 7 = 4 (–7) – (–11) = (–7) + 11 = 11 + (–7) = 11 – 7 = 4 (–7) + (–11) = –18 –7 – 11 = –18 Multiplication and Division: Multiplication or division of numbers with the same sign results in a positive answer. Opposite signed numbers result in negative answers when multiplied or divided. The following examples illustrate the rules: Examples, Multiplication Examples, Division 5 × 2 = 10 12 ÷ 3 = 4 5 × (–2) = –10 (–12) ÷ 3 = –4 (–5) × 2 = –10 (12) ÷ (–3) = –4 (–5) × (–2) = 10 (–12) ÷ (–3) = 4 Order of Operations.—Mathematical operations are performed on numbers in a par- ticular order, commonly referred to as PEMDAS, which stands for “ P arentheses, E xpo- nents, M ultiplication, D ivision, A ddition, S ubtraction.” First, when there are no parenthe- ses or other grouping symbols, multiplication and division are done before addition and subtraction. Then, proceeding from left to right, the addition and subtraction are done in the order they appear. For example: 100 – 26 + 7 × 2 – 100 ÷ 4 = 100 – 26 + 14 – 25 = 74 + 14 – 25 = 88 – 25 = 63 Parentheses ( ) and brackets [ ]—called grouping symbols —indicate if addition and subtraction are to occur before multiplication and division. The operations are performed from the innermost to the outermost grouping symbols. For example: [6 × (15 – 7)] ÷ 2 = [6 × 8] ÷ 2 = 48 ÷ 2 = 24 Exponents are a multiplication operation, but unless parentheses or brackets are present, exponents are applied before multiplication. For example: 4 × 9 2 = 4 × 81 = 324 Also, when parentheses are present next to a multiplication, the × can be omitted: 5(8 – 3) = 5(5) = 25

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Machinery's Handbook, 31st Edition

Order of Operations 5 As explained in Fractions , the horizontal line in a fraction implies division. The top number (called the numerator ) is divided by the bottom number (called the denomina- tor ). For example, In formulas, the multiplication sign ( × ) may be omitted (when letters—called variables—are multiplied) or replaced by parentheses, which serve the same purpose. A × B = AB , 6 × 4 = (6)(4), 8 × a = 8 a A multiplication dot ( ⋅ ) is also sometimes used. Fractions Rational numbers can be written as common fractions or as decimal fractions . Com- mon fractions are written as a b or a / b , where a (the numerator) and b (the denominator) 50 10 --- 50 10 ÷ 5 = = are integers (but b cannot be 0, since division by zero is not defined). The denominator represents the number of equal parts that a whole quantity is broken into. The numerator is the number of these parts under consideration. For example, 2 5 indicates the whole of something is broken into 5 equal parts, and 2 of these parts are being considered. Any integer is a fraction with a denominator of 1. For example, 6 1 6 = . The implied operation in a fraction is division. Thus, a b a b means ÷. Multiple: A multiple of a number n is the result of multiplying n by positive integer 1, 2, 3, . . . Thus, the multiples of 3 are 3, 6, 9, 12, . . . The least common multiple (LCM) of two or more num bers is the smallest multiple the numbers have in common. In the exam- ple below, the first few multiples of 6 and 20 are shown, with the LCM indicated in bold: 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 , 66, . . . 20: 20, 40, 60 , 80, . . . Thus, 60 is the LCM of 6 and 20. Factor: An integer a is a factor of n if there is no remainder when n is divided by a . That is, if the result of n ÷ a is an integer. For example, 3 is a factor of 12 because 12 ⁄ 3 = 4. The greatest common factor (GCF) of two or more numbers is the largest of their common factors. Thus, the common factors of 12 and 18 are 2, 3, and 6; 6 is the GCF. Unit Fraction: A fraction having the same numerator and denominator is the unit frac- tion, 1 (or “one whole”). For example, 2 ⁄ 2 , 4 ⁄ 4 , 8 ⁄ 8 , 16 ⁄ 16 , 32 ⁄ 32 , and 64 ⁄ 64 all equal 1. Proper Fraction: A fraction whose numerator is less than its denominator. 1 ⁄ 4 , 1 ⁄ 2 , and 47 ⁄ 64 are examples of proper fractions. The value of any proper fraction is less than 1. Improper Fraction: A fraction whose numerator is greater than its denominator. 3 ⁄ 2 , 5 ⁄ 4 , and –17 ⁄ 8 are examples of improper fractions. The absolute value of any improper fraction is greater than 1. Reducible Fraction: A reducible fraction is a common fraction in which numera- tor and denominator have a common factor and so can be reduced to lowest terms by dividing both numerator and denominator by this common factor. For example, in the fraction 12 ⁄ 18 , the numerator and denominator have a GCF of 6. Thus, 12 ⁄ 18 reduces to 2 ⁄ 3 by dividing each part of the fraction by 6. A fraction such as 16 ⁄ 21 cannot be reduced, since 16 and 21 do not have a common factor. Mixed Number: A mixed number is a combination of a whole number and a proper fraction. The implied operation between them is addition. For example, 4 2 9 2 9 4 means . + A mixed number is converted to an improper fraction by multiplying the whole number part with the denominator and adding the numerator to obtain the numerator of the final fraction; the denominator remains the same.

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Machinery's Handbook, 31st Edition

Fractions

Examples:

2 3

15 3

2 3

17 3

1 2

18 2

1 2

19 2

2 3

1 2

5 +

= + =

9 +

= + =

To convert mixed numbers to improper fractions, multiply the whole number by the de- nominator and add the numerator to obtain the new numerator. The de nominator remains the same. For example, 2 = = = An improper fraction is converted to its mixed number form by dividing the numerator by denominator and placing the remainder over the denominator. Sometimes the fraction part can be reduced, as the second example shows. 17 8 = 17 ÷ 8 = 2 1 2 -- 2 2+1 × 2 ----------- 7 16 --- 3 16+7 × 16 ------------ = 5 2 -- 3 55 16 --- 1 8

26 16

= 26 ÷ 16 = 1 = 1 10 16 5 8

Equivalent Fractions: A fraction raised to its equivalent form (“higher terms”) by mul- tiplying numerator and denominator by the same number (that is, by multiplying by a form of 1). For example, 1 ⁄ 4 × 4 ⁄ 4 = 4 ⁄ 16 and 3 ⁄ 8 × 4 ⁄ 4 = 12 ⁄ 32 . Any integer n can be expressed as a fraction with a chosen denominator value of m by simply writing n as n /1 and multiplying by m / m . Example: To express 4 as an equivalent fraction with a denominator of 16, write 4 ⁄ 1 × 16 ⁄ 16 = 64 ⁄ 16 Reciprocal: The reciprocal of any number a other than 0 is 1/ a . (0 has no reciprocal, since 1/0 is undefined.) The reciprocal also is called the multiplicative inverse , since a × 1/ a = 1. For example, the reciprocal of 8 is 1 ⁄ 8 ; the reciprocal of 4 ⁄ 7 is 7 ⁄ 4 . Least Common Denominator: Fractions cannot be added or subtracted without a com- mon denominator. For example, 2 5 inator in the answer is the same denominator seen in the fractions. In general, a c b c a b c + = + 1 5 3 5 + = = + 2 1 5 , a simple computation, since the denom- . But fractions with different denominators cannot be added or subtracted until they are converted to equivalent forms that have common denominators. This is done by rais- ing the fractions to higher terms (as explained previously). While any common multiple serves as a common denominator, it is preferable to use the least common multiple (LCM) of the denominator, referred to as the least common denominator (LCD). For example, 36 is the LCD of 2 9 5 6 and , since the LCM of 9 and 6 is 36. Raising each fraction to its equiva- lent form having a denominator of 36 is shown:

2 9

4 4

8 36

5 6

6 6

30 36

× =

and

Example: In the case of 9 11

7 10 and the LCD is the product of the denominators, 11 ×

10 = 110. Raising each fraction to its equivalent form is shown: 9 11 10 10 90 110 7 10 11 11 77 110 × × = = and

Adding and Subtracting Fractions and Mixed Numbers To Add or Subtract Common Fractions: 1) Convert each fraction to terms of the least common denominator; 2) add or subtract numerators; 3) if answer is an improper fraction, change it to a mixed number; and 4) reduce fraction part if necessary.

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