Mechanical and Metal Trades Handbook

Bearbeitet vonUlrich Fischer, Roland Gomeringer, Max Heinzler, Roland Kilgus, Friedrich Näher, Stefan Oesterle, Heinz

Paetzold, Andreas Stephan

1. Auflage 2013. Taschenbuch. 444 S. PaperbackISBN 978 3 8085 1914 1

Format (B x L): 15,2 x 21,5 cmGewicht: 598 g

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EUROPA-TECHNICAL BOOK SERIESfor the Metalworking Trades

Ulrich Fischer Max Heinzler Friedrich Näher Heinz PaetzoldRoland Gomeringer Roland Kilgus Stefan Oesterle Andreas Stephan

Mechanical and Metal Trades Handbook3rd English edition

Europa-No.: 1910X

VERLAG EUROPA LEHRMITTEL · Nourney, Vollmer GmbH & Co. KG

Düsselberger Straße 23 · 42781 Haan-Gruiten · Germany

Original title:Tabellenbuch Metall, 45th edition, 2011

Authors:Ulrich Fischer Dipl.-Ing. (FH) ReutlingenRoland Gomeringer Dipl.-Gwl. MeßstettenMax Heinzler Dipl.-Ing. (FH) Wangen im AllgäuRoland Kilgus Dipl.-Gwl. NeckartenzlingenFriedrich Näher Dipl.-Ing. (FH) BalingenStefan Oesterle Dipl.-Ing. AmtzellHeinz Paetzold Dipl.-Ing. (FH) MühlackerAndreas Stephan Dipl.-Ing. (FH) Marktoberdorf

Editor:Ulrich Fischer, Reutlingen

Graphic design:Design office of Verlag Europa-Lehrmittel, Ostfildern, Germany

The publisher and its affiliates have taken care to collect the information given in this book to the best of their ability. However, no responsibility is accepted by the publisher or any of its affiliates regarding its content or any statement herein or omission there from which may result in any loss or damage to any party using the data shown above. Warranty claims against the authors or the publisher are excluded.

Most recent editions of standards and other regulations govern their use. They can be ordered from Beuth Verlag GmbH, Burggrafenstr. 6, 10787 Berlin, Germany.

The content of the chapter „Program structure of CNC machines according to PAL“ (page 412 to 424) complies with the publications of the PAL Prüfungs- und Lehrmittelentwicklungsstelle (Institute for the development of training and testing material) of the IHK Region Stuttgart (Chamber of Commerce and Industry of the Stuttgart region).

English edition: Mechanical and Metal Trades Handbook 3rd edition, 2012

6 5 4 3 2 1

All printings of this edition may be used concurrently in the classroom since they are unchanged, except for some corrections to typographical errors and slight changes in standards.

ISBN 13 978-3-8085-1914-1

Cover design includes a photograph from TESA/Brown & Sharpe, Renens, Switzerland

All rights reserved. This publication is protected under copyright law. Any use other than those permitted by law must be approved in writing by the publisher.

© 2012 by Verlag Europa-Lehrmittel, Nourney, Vollmer GmbH & Co. KG, 42781 Haan-Gruiten, Germanyhttp://www.europa-lehrmittel.de

Translation: Eva Schwarz, 76879 Ottersheim, Germany; www.technische-uebersetzungen-eva-schwarz.deTypesetting: Satz+Layout Werkstatt Kluth GmbH, D-50374 Erftstadt, Germany; www.slw-kluth.dePrinted by: M. P. Media-Print Informationstechnologie GmbH, D-33100 Paderborn, Germany

3

Preface

The Mechanical and Metal Trades Handbook is well-suited for shop reference, tooling, machine building, maintenance and as a general book of knowledge. It is also useful for ed-ucational purposes, especially in practical work or curricula and continuing education programs.

Target Groups• Industrial and trade mechanics• Tool & die makers• Machinists• Millwrights• Draftspersons• Technical Instructors• Apprentices in above trade areas• Practitioners in trades and industry• Mechanical Engineering students

Notes for the userThe contents of this book include tables and formulae in eight chapters, including Tables of Contents, Subject Index and Standards Index. The tables contain the most important guidelines, designs, types, dimensions and standard values for their subject areas. Units are not specified in the legends for the formulae if sev- eral units are possible. However, the calculation examples for each formula use those units normally applied in practice. The Table of Contents in the front of the book is expanded further at the beginning of each chapter in form of a partial Table of Contents.The Subject Index at the end of the book (pages 435 – 444) is extensive.The Standards Index (pages 425 – 434) lists all the current standards and regulations cited in the book. In many cases previous standards are also listed to ease the transition from older, more familiar standards to new ones.

Changes in the 3rd editionIn the present edition, we have updated the cited standards and restructured, updated, enhanced or added the follow-ing chapters in line with new developments in engineering:

AcknowledgementSpecial thanks to Alexander Huter, Vocational Training Specialist – Tool and Die, Ontario, for his input into the English translation of this book. His assistance has been extremely valuable.

November 2012 Authors and publisher

M1 Mathematics

9 – 28

P2 Physics

29 – 50

TD3 Technical

Drawing51 – 110

MS4 Material

Science111 – 200

ME5 Machine

Elements201 – 268

PE6 Production

Engineering269 – 366

A7 Automation and

Information Tech-nology 367 – 424

S8 International Material

Comparison Chart, Standards 425– 434

– Fundamentals of technical mathematics

– Strength of materials– Plastics– Production management– Forming– Welding

– PAL programming system for NC turning and NC milling

– Steel types– Material testing– Machining processes– Injection molding (new)– GRAFCET

4

Table of Contents

2 Physics (P) 292.1 Motion Uniform and accelerated motion . . . . . . . . . 30 Speeds on machines . . . . . . . . . . . . . . . . . . . . 31

2.2 Forces Representation, addition and resolution of

forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Types of forces . . . . . . . . . . . . . . . . . . . . . . . . . 33 Torque and levers, torque in gear drives . . . 34

2.3 Work, power, efficiency Mechanical work . . . . . . . . . . . . . . . . . . . . . . . 34 Simple machines and energy . . . . . . . . . . . . . 35 Potential and kinetic energy . . . . . . . . . . . . . . 35 Power and efficiency . . . . . . . . . . . . . . . . . . . . 36

2.4 Friction Friction force . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Coefficients of friction . . . . . . . . . . . . . . . . . . . 37 Coefficients of rolling friction . . . . . . . . . . . . . 37

2.5 Pressure in liquids and gases Pressure, definition and types . . . . . . . . . . . . 38 Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Hydraulic power transmission . . . . . . . . . . . . 38 Pressure intensification,

flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Changes of state in gases . . . . . . . . . . . . . . . . 39

2.6 Strength of materials Load cases, types of loading . . . . . . . . . . . . . 40 Material properties, stress limits . . . . . . . . . 40 Strength calculation, safety factors . . . . . . . 41 Allowable stresses, modulus of

elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Tensile and compressive stress,

surface pressure . . . . . . . . . . . . . . . . . . . . . . . 42 Shear, bending and torsional stress . . . . . . . 43 Moments of area, section moduli . . . . . . . . . 44

2.7 Thermodynamics Temperature, linear expansion,

shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Quantity of heat . . . . . . . . . . . . . . . . . . . . . . . . . 45 Heat flux, heat of combustion . . . . . . . . . . . . 46 Net calorific values . . . . . . . . . . . . . . . . . . . . . 46

2.8 Electricity Quantities and units . . . . . . . . . . . . . . . . . . . . . 47 Ohm’s law, electrical resistance . . . . . . . . . . 47 Current density, resistor circuit . . . . . . . . . . . 48 Types of current . . . . . . . . . . . . . . . . . . . . . . . . 49 Electrical work and power . . . . . . . . . . . . . . . 50

1 Mathematics (M) 91.1 Units of measurement SI base quantities and base units . . . . . . . . . . 10 Derived quantities and their units . . . . . . . . . 11 Non-SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Formulas Formula symbols, mathematical symbols . . 13 Formulas, equations, graphs . . . . . . . . . . . . . 14 Transformation of formulas . . . . . . . . . . . . . . 15 Quantities and units . . . . . . . . . . . . . . . . . . . . . 16 Calculation with quantities . . . . . . . . . . . . . . . 17 Percentage and interest calculation . . . . . . 17

1.3 Angels and triangels Types of angels, sum of angels in

a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Theorem of intersecting lines,

Pythagorean theorem . . . . . . . . . . . . . . . . . . . 18 Functions of right triangles . . . . . . . . . . . . . . . 19 Functions of oblique triangles . . . . . . . . . . . . 19

1.4 Lengths Division of lengths . . . . . . . . . . . . . . . . . . . . . . 20 Spring wire lengths . . . . . . . . . . . . . . . . . . . . . 21 Rough lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Triangle, polygon, circle . . . . . . . . . . . . . . . . . 23 Circular sector, circular segment . . . . . . . . . 24 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6 Volume and surface area Cube, cylinder, pyramid . . . . . . . . . . . . . . . . . . 25 Truncated pyramid, cone,

Truncated cone, sphere . . . . . . . . . . . . . . . . . . 26 Volumes of composite solids . . . . . . . . . . . . . 271.7 Mass General calculations . . . . . . . . . . . . . . . . . . . . 27 Linear mass density . . . . . . . . . . . . . . . . . . . . 27 Area mass density . . . . . . . . . . . . . . . . . . . . . . 27

1.8 Centroids Centroids of lines . . . . . . . . . . . . . . . . . . . . . . . 28 Centroids of plane areas . . . . . . . . . . . . . . . . 28

5Table of Contents

4 Materials Science (MS) 1114.1 Materials Material characteristics of solids . . . . . . . . 112 Material characteristics of liquids

and gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Periodic table of the elements . . . . . . . . . . . 114

4.2 Steels, designation system Definition and classification of steel . . . . . 116 Standardization of steel products . . . . . . . . 117 Designation system for steels . . . . . . . . . . . 118

4.3 Steels, steel types Overwiew of steel products . . . . . . . . . . . . . 123 Unalloyed steels . . . . . . . . . . . . . . . . . . . . . . . 126 Case hardened steels, quenched &

tempered steels . . . . . . . . . . . . . . . . . . . . . . . 129 Nitriding steels, free cutting steels . . . . . . . 131 Tool steels, stainless steels . . . . . . . . . . . . . 132 Steels for bright steel products . . . . . . . . . . 137

4.4 Steels, finished products Sheet and strip metal, tubes . . . . . . . . . . . . . 138 Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Linear mass density and area mass

density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.5 Heat treatment Iron-carbon phase diagram . . . . . . . . . . . . . 152 Heat treatement of steels . . . . . . . . . . . . . . . 153 Hardening of aluminium alloys . . . . . . . . . . 156

4.6 Cast iron materials Designation and material codes . . . . . . . . . 157 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Cast iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Malleable cast iron, cast steel . . . . . . . . . . 160

4.7 Foundry technology . . . . . . . . . . . . . . . . . . . . 161

4.8 Light alloys Overview of aluminum alloys . . . . . . . . . . . . 163 Wrought aluminum alloys . . . . . . . . . . . . . . . 165 Aluminum profiles . . . . . . . . . . . . . . . . . . . . . 168 Magnesium and titanium alloys . . . . . . . . . 171

4.9 Heavy non-ferrous metals Overview and designation system . . . . . . . 172 Copper and refined zinc alloys . . . . . . . . . . 174

4.10 Other metallic materials . . . . . . . . . . . . . . . 176

4.11 Plastics Overview and designation . . . . . . . . . . . . . . 178 Thermoset plastics . . . . . . . . . . . . . . . . . . . . . 181 Thermoplastics, elastomers . . . . . . . . . . . . . 182 Plastics processing, testing of plastics . . . 186

3 Technical Drawing (TD) 513.1 Graphs Cartesian coordinate system . . . . . . . . . . . . 52 Graph types . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Basic geometric constructions Lines and angles . . . . . . . . . . . . . . . . . . . . . . 54 Tangents, circular arcs . . . . . . . . . . . . . . . . 55 Inscribed circles, ellipses . . . . . . . . . . . . . . 56 Cycloids, involute curves, parabolas . . . . . 57

3.3 Elements of drawing Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Preferred numbers, radii, scales . . . . . . . . . 59 Drawing layout, bills of materials . . . . . . . . 60 Line types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 Representation Projection methods . . . . . . . . . . . . . . . . . . . . 64 Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Sectional views . . . . . . . . . . . . . . . . . . . . . . . 68 Hatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.5 Dimensioning drawings Dimensioning lines, dimension values . . . . 71 Dimensioning rules . . . . . . . . . . . . . . . . . . . . 72 Elements of drawing . . . . . . . . . . . . . . . . . . . 73 Tolerance specifications . . . . . . . . . . . . . . . 75 Types of dimensioning . . . . . . . . . . . . . . . . . 76 Simplified presentation in drawings . . . . . . 78

3.6 Machine elements, representation Gear types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . 80 Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Retaining rings, springs . . . . . . . . . . . . . . . . 82

3.7 Object elements Bosses, object edges . . . . . . . . . . . . . . . . . . 83 Thread runouts, thread undercuts . . . . . . . 84 Threads, screw joints . . . . . . . . . . . . . . . . . . 85 Center holes, knurls, undercuts . . . . . . . . . 86

3.8 Welding and soldering Graphical symbols . . . . . . . . . . . . . . . . . . . . . 88 Dimensioning examples . . . . . . . . . . . . . . . . 91

3.9 Surfaces Hardness specifications in drawings . . . . 92 Form deviations, roughness . . . . . . . . . . . . . 93 Surface testing, surface indications . . . . . 94

3.10 ISO tolerances and fits Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . 98 Basic hole system, basic shaft system . . . 102 General tolerances . . . . . . . . . . . . . . . . . . . . . 106 Roller bearing fits . . . . . . . . . . . . . . . . . . . . . . 106 Fit recommendations . . . . . . . . . . . . . . . . . . . 107 Geometric tolerancing . . . . . . . . . . . . . . . . . . 108

6 Table of Contents

6 Production Engineering (PE) 2696.1 Quality management Standards, terminology . . . . . . . . . . . . . . . . . 270 Quality planning, quality testing . . . . . . . . . 272 Statistical analysis . . . . . . . . . . . . . . . . . . . . . 273 Statistical process control . . . . . . . . . . . . . . 275 Quality management of processes . . . . . . . 276

6.2 EC Machinery Directive EC Machinery Directive . . . . . . . . . . . . . . . . 278 CE marking . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

6.3 Production organization Overview; product breakdown

structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Work planning . . . . . . . . . . . . . . . . . . . . . . . . . 282 Cost accounting . . . . . . . . . . . . . . . . . . . . . . . 286

6.4 Maintenance, repair & overhaul (MRO) Maintenance, inspection, repair,

improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 289 MRO concepts . . . . . . . . . . . . . . . . . . . . . . . . 290 Documentation system . . . . . . . . . . . . . . . . . 292

6.5 Machining processes Overview; forces and power . . . . . . . . . . . . 293 Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 Indexable inserts . . . . . . . . . . . . . . . . . . . . . . 308 Cooling lubrication . . . . . . . . . . . . . . . . . . . . . 311 Cuttin tool materials . . . . . . . . . . . . . . . . . . . . 315 Grinding, honing . . . . . . . . . . . . . . . . . . . . . . . 317

6.6 Removal operations Electric discharge machining . . . . . . . . . . . 323 Process parameters in EDM erosion . . . . . 324

5 Machine Elements (ME) 2015.1 Threads Types of threads, overview . . . . . . . . . . . . . 202 Metric ISO threads . . . . . . . . . . . . . . . . . . . . 202 Other types of threads . . . . . . . . . . . . . . . . . . 205 Thread tolerances . . . . . . . . . . . . . . . . . . . . . 207

5.2 Bolts and screws Overview, designations, property classes 208 Hexagon head bolts & screws . . . . . . . . . . 211 Other bolts & screws . . . . . . . . . . . . . . . . . . 214 Screw joint calculations . . . . . . . . . . . . . . . . 220 Locking fasteners, bolt and screw drive

systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

5.3 Countersinks Countersinks for countersunk head

screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Counterbores for cap screws . . . . . . . . . . . 224

5.4 Nuts Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Designations, property classes . . . . . . . . . . 226 Hexagon nuts . . . . . . . . . . . . . . . . . . . . . . . . . 228 Other nuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

5.5 Washers Overview, flat washers . . . . . . . . . . . . . . . . . 232 Washers for HV bolts . . . . . . . . . . . . . . . . . . . 234

5.6 Pins and clevis pins Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Grooved pins, grooved drive studs,

clevis pins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

5.7 Shaft-hub connections Keys, gib-head keys . . . . . . . . . . . . . . . . . . . . 238 Feather keys, splined shaft joints . . . . . . . . 239 Metric tapers, morse tapers, steep tapers 240 Tool holding fixtures . . . . . . . . . . . . . . . . . . . 241

5.8 Other machine elements Tension, compression and disc springs . 242 Grub screws, thrust pads, knobs . . . . . . . . 245 T-slots and accessories . . . . . . . . . . . . . . . . 247 Quick-release drilling fixture . . . . . . . . . . . . 248

5.9 Drive elements Belts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Transmission ratios . . . . . . . . . . . . . . . . . . . . 256

5.10 Bearings Plain bearings, overview . . . . . . . . . . . . . . . 257 Plain bearing bushings . . . . . . . . . . . . . . . . . 258 Antifriction bearings, overview . . . . . . . . . . 259 Ball bearings . . . . . . . . . . . . . . . . . . . . . . . . . . 261 Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . 262 Sealing elements . . . . . . . . . . . . . . . . . . . . . . 266 Lubricating oils . . . . . . . . . . . . . . . . . . . . . . . . 267 Lubricating greases . . . . . . . . . . . . . . . . . . . . 268

4.12 Material testing Overview, tensile test . . . . . . . . . . . . . . . . . . 189 Rotary bending test . . . . . . . . . . . . . . . . . . . . 192 Hardness test . . . . . . . . . . . . . . . . . . . . . . . . . 193

4.13 Corrosion, hazardous materials Corrosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Hazardous materials . . . . . . . . . . . . . . . . . . . 197

7Table of Contents

8.1 International material comparison chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

8.2 Index of cited standards and other regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

8 Material Chart, Standards (S) 425 – 434

Subject Index 435 – 444

7 Automation and Information Technology (A) 3677.1 Control engineering, basic terminology Basic terminology, code letters, symbols 368 Analog controllers . . . . . . . . . . . . . . . . . . . . . 370 Discontinuous and digital controllers . . . . 371 Binary logic . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Numbering systems . . . . . . . . . . . . . . . . . . . . 373 Information processing . . . . . . . . . . . . . . . . . 374

7.2 Electrical circuits Circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . 375 Designations in circuit plans . . . . . . . . . . . . 377 Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . 378 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Safety precautions . . . . . . . . . . . . . . . . . . . . . 380

7.3 GRAFCET Important basic terms . . . . . . . . . . . . . . . . . . 382 Steps, transitions . . . . . . . . . . . . . . . . . . . . . . 383 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 Branchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

7.4 Programmable logical controllers PLC PLC programming languages, overview . . 388 Ladder diagram (LD) . . . . . . . . . . . . . . . . . . . . 389 Instruction list (IL) . . . . . . . . . . . . . . . . . . . . . . 390 PLC programming languages,

comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Programming example . . . . . . . . . . . . . . . . . 392

7.5 Hydraulics, pneumatics Circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . 393 Proportional valves . . . . . . . . . . . . . . . . . . . . 395 Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . 396 Pneumatic control . . . . . . . . . . . . . . . . . . . . . 397 Electro-pneumatic control . . . . . . . . . . . . . . 398 Electro-hydraulic control . . . . . . . . . . . . . . . 399 Hydraulic fluids . . . . . . . . . . . . . . . . . . . . . . . . 400 Pneumatic cylinders . . . . . . . . . . . . . . . . . . . 401 Hydraulic pumps . . . . . . . . . . . . . . . . . . . . . . . 402 Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

7.6 Handling and robot systems Coordinate systems and axes . . . . . . . . . . . 404 Robot designs . . . . . . . . . . . . . . . . . . . . . . . . . 405 Grippers, job safety . . . . . . . . . . . . . . . . . . . . 406

7.7 CNC technology Coordinate axes . . . . . . . . . . . . . . . . . . . . . . . 407 Program structure . . . . . . . . . . . . . . . . . . . . . 408 Tool offset and cutter compensation . . . . . 409 Program structure according to DIN . . . . . 410 Program structure according to PAL . . . . . 412 PAL functions for lathes . . . . . . . . . . . . . . . . 413 PAL cycles for lathes . . . . . . . . . . . . . . . . . . . 414 PAL functions for milling machines . . . . . . 417 PAL cycles for milling machines . . . . . . . . . 418

6.7 Separation by cutting Cutting force . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Cutting tool . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

6.8 Forming Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 Deep drawing . . . . . . . . . . . . . . . . . . . . . . . . . 334

6.9 Injection molding Injection molding tools . . . . . . . . . . . . . . . . . 338 Shrinkage, cooling, batching . . . . . . . . . . . . 341

6.10 Joining Welding processes . . . . . . . . . . . . . . . . . . . . 343 Weld preparation . . . . . . . . . . . . . . . . . . . . . . 345 Gas-shielded welding . . . . . . . . . . . . . . . . . . 346 Arc welding . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 Beam cutting . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Gas cylinders, identification . . . . . . . . . . . . . 352 Brazing, bonded joints . . . . . . . . . . . . . . . . . . 354

6.11 Workplace and environmental protection Safety colors . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Warning signs, safety signs . . . . . . . . . . . . . 360 Sound and noise . . . . . . . . . . . . . . . . . . . . . . . 366

8

Standards and other RegulationsStandardization and standards terms

Standardization is the systematic achievement of uniformity of material and non-material objects, such as compo- nents, calculation methods, process flows and services for the benefit of the general public.

Types of standards and regulations (selection)

Standards term Example Explanation

Standard DIN 7157 A standard is the published work of standardization, e. g. the selection of particular fits in DIN 7157.

Part DIN 30910-2 Standards can comprise several parts associated with each other. The part num-bers are appended to the main standard number with hyphens. DIN 30910-2 describes sintered materials for filters for example, whereas Part 3 and 4 deal with sintered materials for bearings and formed parts.

Supplement DIN 743 Suppl. 1

A supplement contains information for a standard, however no additional specifi- cations. The supplement DIN 743 Suppl. 1, for example, contains application examples of load capacity calculations for shafts and axles described in DIN 743.

Draft E DIN 743 (2008-10)

Draft standards are made available to the public for examination and comment-ing. The planned new version of DIN 743 on load-bearing calculations of shafts and axes, for example, has been published since October 2008 as Draft E DIN 743.

Preliminary standard

DIN V 66304 (1991-04)

A preliminary standard contains the results of standardization, which have not been released as a standard because of certain provisos. DIN V 66304, for example, discusses a format for exchange of standard part data for computer-aided design.

Issue date DIN 76-1 (2004-06)

Date of publication which is made public in the DIN publication guide; this is the date at which time the standard becomes valid. DIN 76-1, which sets undercuts for metric ISO threads has been valid since June 2004 for example.

Type Abbreviation Explanation Purpose and contents

InternationalStandards(ISO standards)

ISO International Organization for Standardization, Geneva (O and S are reversed in the abbreviation)

Simplifies the international exchange of goods and services, as well as cooperation in scientific, technical and economic areas.

EuropeanStandards(EN standards)

EN European Committee for Standardi- zation (Comité Européen de Normalisation), Brussels

Technical harmonization and the associated reduction of trade barriers for the advance-ment of the European market and the coa- l escence of Europe.

GermanStandards(DIN standards)

DIN Deutsches Institut für Normung e.V., Berlin (German Institute for Standardization)

National standardization facilitates rational-ization, quality assurance, environmental protection and common understanding in economics, technology, science, manage-ment and public relations.

DIN EN European standard for which the German version has attained the status of a German standard.

DIN ISO German standard for which an inter- national standard has been adopted without change.

DIN EN ISO European standard for which an international standard has been adopted unchanged and the Ger-man version has the status of a Ger-man standard.

DIN VDE Printed publication of the VDE, which has the status of a German standard.

VDI Guidelines VDI Verein Deutscher Ingenieure e.V., Düsseldorf (Association of German Engineers)

These guidelines give an account of the cur-rent state of the art in specific subject areas and contain, for example, concrete procedu-ral guidelines for the performing calculations or designing processes in mechanical or electrical engineering.

VDE printed publications

VDE Verband Deutscher Elektrotechniker e.V., Frankfurt (Association for Electrical, Electronic & Information Technologies)

DGQ publications

DGQ Deutsche Gesellschaft für Qualität e.V., Frankfurt (German Society for Quality)

Recommendations in the area of quality technology.

REFA sheets REFA Association for Work Design, Indus-trial Organization and Corporate Development REFA e.V., Darmstadt

Recommendations in the area of produc-tion and work planning.

9Table of Contents

1 Mathematics

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Quantity SymbolUnit

Name Symbol

Lengths Πmeter m

1.1 Units of measurement SI base quantities and base units . . . . . . . . . . . . . . .10 Derived quantities and their units . . . . . . . . . . . . . . 11 Non-SI units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Formulas Formula symbols, mathematical symbols . . . . . . . 13 Formulas, equations, graphs . . . . . . . . . . . . . . . . . . 14 Transformation of formulas . . . . . . . . . . . . . . . . . . . 15 Quantities and units . . . . . . . . . . . . . . . . . . . . . . . . . 16 Calculation with quantities . . . . . . . . . . . . . . . . . . . 17 Percentage and interest calculation . . . . . . . . . . . . 17

1.3 Angels and triangels Types of angels, sum of angels in a triangle . . . . . 18 Theorem of intersecting lines . . . . . . . . . . . . . . . . . 18 Functions of right triangles . . . . . . . . . . . . . . . . . . . 19 Functions of oblique triangles . . . . . . . . . . . . . . . . . 19

1.4 Lengths Division of lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Spring wire lengths . . . . . . . . . . . . . . . . . . . . . . . . . 21 Rough lengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Triangle, polygon, circle . . . . . . . . . . . . . . . . . . . . . . 23 Circular sector, circular segment . . . . . . . . . . . . . . . 24 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6 Volume and surface area Cube, cylinder, pyramid . . . . . . . . . . . . . . . . . . . . . 25 Truncated pyramid, cone, truncated cone, sphere . 26 Volumes of composite solids . . . . . . . . . . . . . . . . . 27

1.7 Mass General calculations . . . . . . . . . . . . . . . . . . . . . . . . 27 Linear mass density . . . . . . . . . . . . . . . . . . . . . . . . . 27 Area mass density . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.8 Centroids Centroids of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Centroids of plane areas . . . . . . . . . . . . . . . . . . . . . 28

sine = opposite side hypotenuse

cosine = adjacent side hypotenuse

tangent = opposite side adjacent side

xs x

ys

y

S1

S2S

m' inkg m

1m

d

A d hd

s = +pp

· · ··

24

2

Surface area

AM = p · d · h

Lateral surface area

10 1 .1 Units of measurement

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Units of measurementSI1) Base quantities and base units cf . DIN 1301-1 (2010-10), -2 (1978-02), -3 (1979-10)

Base quantities, derived quantities and their units

Length, Area, Volume, Angle

Mechanics

Base quantity Length Mass Time

Electric current

Thermo- dynamic

temperature

Amount of substance

Luminous intensity

Base units meter kilo-

gram second ampere kelvin mole candela

Unit symbol m

1) The units for measurement are defined in the International System of Units SI (Système International d’Unités) . It is based on the seven basic units (SI units), from which other units are derived .

kg s A K mol cd

Quantity Symbol

Œ meter m 1 m = 10 dm = 100 cm = 1000 mm1 mm = 1000 µm1 km = 1000 m

1 inch = 25 .4 mmIn aviation and nautical applications the following applies:1 international nautical mile = 1852 m

Unit Name Symbol

Relationship Remarks Examples of application

Length

A, S square meter

arehectare

m2

aha

1 m2 = 10 000 cm2

= 1 000 000 mm2

1 a = 100 m2

1 ha = 100 a = 10 000 m2

100 ha = 1 km2

Symbol S only for cross-sectional areas

Are and hectare only for land

Area

V cubic meter

liter

m3

—, L

1 m3 = 1000 dm3

= 1 000 000 cm3

1 — = 1 L = 1 dm3 = 10 d— = 0 .001 m3

1 m— = 1 cm3

Mostly for fluids and gases

Volume

a, b, g … radian

degrees

minutesseconds

rad

°

*+

1 rad = 1 m/m = 57 .2957…° = 180°/p

1° = p rad = 60* 1801* = 1°/60 = 60+1+ = 1*/60 = 1°/3600

1 rad is the angle formed by the intersection of a circle around the center of 1 m radius with an arc of 1 m length .In technical calculations instead of a = 33° 17* 27 .6+, better use is a = 33 .291° .

Plane angle(angle)

≈ steradian sr 1 sr = 1 m2/m2 An object whose extension measures 1 rad in one direction and perpendicu-larly to this also 1 rad, covers a solid angle of 1 sr .

Solid angle

m kilogramgram

megagrammetric ton

kgg

Mgt

1 kg = 1000 g1 g = 1000 mg

1 metric t = 1000 kg = 1 Mg0 .2 g = 1 ct

Mass in the sense of a scale result or a weight is a quantity of the type of mass (unit kg) .

Mass for precious stones in carat (ct) .

Mass

m* kilogramper meter

kg/m 1 kg/m = 1 g/mm For calculating the mass of bars, pro-files, pipes .

Linear mass density

m+ kilogramper square meter

kg/m2 1 kg/m2 = 0 .1 g/cm2 To calculate the mass of sheet metal .Area mass density

r kilogramper cubic meter

kg/m3 1000 kg/m3 = 1 metric t/m3 = 1 kg/dm3

= 1 g/cm3

= 1 g/ml

= 1 mg/mm3

The density is a quantity independent of location .

Density

111 .1 Units of measurement

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J kilogram xsquare meter

kg · m2 The following applies for a homogenous body:J = r · r2 · V

The moment of inertia (2nd moment of mass) is dependent upon the total mass of the body as well as its form and the position of the axis of rotation .

Moment of inertia, 2nd Moment of mass

Quantities and Units (continued)

Mechanics

F

FG, G

newton N 1 N = 1 kg · m = 1 J s2 m1 MN = 103 kN = 1 000 000 N

The force 1 N effects a change in velocity of 1 m/s in 1 s in a 1 kg mass .

Force

Weight

MMbT

newton x meter

N · m 1 N · m = 1 kg · m2

s21 N · m is the moment that a force of 1 N effects with a lever arm of 1 m .

TorqueBending mom .Torsional mom .

p kilogram x meter per second

kg · m/s 1 kg · m/s = 1 N · s The momentum is the product of the mass times velocity . It has the direc-tion of the velocity .

Momentum

p

s, t

pascal

newtonper square millimeter

Pa

N/mm2

1 Pa = 1 N/m2 = 0 .01 mbar1 bar = 100 000 N/m2

= 10 N/cm2 = 105 Pa1 mbar = 1 hPa1 N/mm2 = 10 bar = 1 MN/m2

= 1 MPa1 daN/cm2 = 0 .1 N/mm2

Pressure refers to the force per unit area . For gage pressure the symbol pg is used (DIN 1314) .1 bar = 14 .5 psi (pounds per square inch )

Pressure

Mechanicalstress

I meter to the fourth power centimeter to the fourth power

m4

cm4

1 m4 = 100 000 000 cm4 Previously: Geometrical moment of inertia

Second moment of area

E, W joule J 1 J = 1 N · m = 1 W· s = 1 kg · m2/s2

Joule for all forms of energy, kW · h preferred for electrical energy .

Energy, Work, Quantity of heat

P G

watt W 1 W = 1 J/s = 1 N · m/s = 1 V · A = 1 m2 · kg/s3

Power describes the work which is achieved within a specific time .

Power Heat flux

Time

t seconds minutes hours day year

sminhda

1 min = 60 s1 h = 60 min = 3600 s1 d = 24 h = 86 400 s

3 h means a time span (3 hrs .),3h means a point in time (3 o’clock) .If points in time are written in mixed form, e .g . 3h24m10s, the symbol min can be shortened to m .

Time, Time span, Duration

f, v hertz Hz 1 Hz = 1/s 1 Hz ‡ 1 cycle in 1 second .Frequency

n 1 per second

1 per minute

1/s

1/min

1/s = 60/min = 60 min–1

1/min = 1 min–1 = 1 60 s

The number of revolutions per unit of time gives the revolution frequency, also called rpm .

Rotational speed, Rotational frequency

v meters per secondmeters per minutekilometers per hour

m/s

m/min

km/h

1 m/s = 60 m/min = 3 .6 km/h

1 m/min = 1 m 60 s

1 km/h = 1 m 3 .6 s

Nautical velocity in knots (kn):1 kn = 1 .852 km/h

miles per hour = 1 mile/h = 1 mph1 mph = 1 .60934 km/h

Velocity

w 1 per secondradians per second

1/srad/s

w = 2 p · n For a rpm of n = 2/s the angular ve locity w = 4 p/s .

Angular-velocity

a, g meters per second squared

m/s21 m/s2 = 1 m/s 1 s

Symbol g only for acceleration due to gravity .g = 9 .81 m/s2 fi 10 m/s2

Acceleration

Quantity Sym-bol

Unit Name Symbol

Relationship Remarks Examples of application

12 1 .1 Units of measurement

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Electricity and Magnetism

Quantity Sym-bol

Unit Name

Relationship Remarks Examples of application

Units of measurement

IE

R

G

amperevolt

ohm

siemens

AV

O

S

1 V = 1 W/1 A = 1 J/C

1 O = 1 V/1 A

1 S = 1 A/1 V = 1/O

The motion of an electrical charge is called current . The electromotive force is equal to the potential difference be -tween two points in an electric field . The reciprocal of the electrical resistance is called the electrical conductivity .

Frequency of public electric utility: EU 50 Hz, USA/Canada 60 Hz

Electric currentElectromotive force Electrical resistanceElectrical conductance

r

g, k

ohm xmetersiemens per meter

O ·m

S/m

10–6 O · m = 1 O · mm2/m r

k

kr

=

=

1

1

2

2

inmmm

inmmm

Ω

Ω

·

·

rk

kr

=

=

1

1

2

2

inmmm

inmmm

Ω

Ω

·

·

SpecificresistanceConductivity

f hertz Hz 1 Hz = 1/s1000 Hz = 1 kHz

Frequency

In atomic and nuclear physics the unit eV (electron volt) is used .

W joule J 1 J = 1 W · s = 1 N · m1 kW · h = 3 .6 MJ1 W · h = 3 .6 kJ

Electrical energy

The angle between current and voltage in inductive or capacitive load .

j – – for alternating current:

cos j = P U · I

Phase difference

In electrical power engineering: Apparent power S in V · A

P watt W 1 W = 1 J/s = 1 N · m/s = 1 V · A

Power Effective power

EFQ

CQU

Q t= =, , = ·IEQCL

volts per metercoulombfaradhenry

V/mCFH

1 C = 1 A · 1 s; 1 A · h = 3 .6 kC1 F = 1 C/V1 H = 1 V · s/A

Elect . field strengthElect . chargeElect . capacitance inductance

Quantities and units (continued)

Electricity and Magnetism

Quantity Sym-bol

Unit Name

Relationship Remarks Examples of application

Sym-bol

Non-SI units

T, Q

t, h

kelvin

degrees Celsius

K

°C

0 K = – 273 .15 °C

0 °C = 273 .15 K0 °C = 32 °F0 °F = – 17 .77 °C

Kelvin (K) and degrees Celsius (°C) are used for temperatures and tempera-ture differences .t = T – T0; T0 = 273 .15 K degrees Fahrenheit (°F): 1 .8 °F = 1 °C

Thermo- dynamic temperatureCelsius temperature

Q joule J 1 J = 1 W · s = 1 N · m1 kW · h = 3 600 000 J = 3,6 MJ

1 kcal ‡ 4 .1868 kJQuantity of heat

Hu joule per kilogramJoule per cubic meter

J/kg

J/m3

1 MJ/kg = 1 000 000 J/kg 1 MJ/m3 = 1 000 000 J/m3

Thermal energy released per kg fuel minus the heat of vaporization of the water vapor contained in the exhaust gases .

Net calorific value

Length Area Volume Mass Energy, Power

1 inch (in) = 25 .4 mm

1 foot (ft) = 0 .3048 m

1 yard (yd) = 0 .9144 m

1 nautical mile = 1 .852 km

1 mile = 1 .6093 km

1 sq .in = 6 .452 cm2

1 sq .ft = 9 .29 dm2

1 sq .yd = 0 .8361 m2

1 acre = 4046 .856 m2

Pressure

1 bar = 14 .5 pound/in2

1 N/mm2 = 145 .038 pound/in2

1 cu .in = 16 .39 cm3

1 cu .ft = 28 .32 dm3

1 cu .yd = 764 .6 dm3

1 gallon 1 (US) = 3 .785 —

1 gallon 1 (UK) = 4 .546 —

1 barrel = 158 .8 —

1 oz = 28 .35 g

1 lb = 453 .6 g

1 t = 1000 kg

1 short ton = 907 .2 kg

1 Karat = 0 .2 g

1 pound/in3 = 27 .68 g/cm3

1 PSh = 0 .735 kWh

1 PS = 735 W

1 kcal = 4186 .8 Ws

1 kcal = 1 .166 Wh

1 kpm/s = 9 .807 W

1 Btu = 1055 Ws

1 hp = 745 .7 W

Sym-bol

131 .2 Formulas

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Formula Meaning symbol

ΠLength w Width h Height s Linear distance

r, R Radius d, D Diameter A, S Area, Cross-sectional area V Volume

a, b, g Planar angle W Solid angle l Wave length

Formula Meaning symbol

Formula Meaning symbol

Formula symbols, Mathematical symbolsFormula symbols cf . DIN 1304-1 (1994-03)

Length, Area, Volume, Angle

t Time, Duration T Cycle duration n Revolution frequency, Speed

f, v Frequency v, u Velocity w Angular velocity

a Acceleration g Gravitational acceleration a Angular acceleration Q,

·V, qv Volumetric flow rate

Time

Q Electric charge, Quantity of electricity E Electromotive force C Capacitance I Electric current

L Inductance R Resistance r Specific resistance g, k Electrical conductivity

X Reactance Z Impedance j Phase difference N Number of turns

Electricity

T, Q Thermodynamic temperature

DT, Dt, Dh Temperature difference t, h Celsius temperature a—, a Coefficient of linear

expansion

Q Heat, Quantity of heat l Thermal conductivity a Heat transition coefficient k Heat transmission

coefficient

G, ·

Q Heat flow a Thermal diffusivity c Specific heat Hnet Net calorific value

Heat

E Illuminance f Focal length n Refractive index

I Luminous intensity Q, W Radiant energy

Light, Electromagnetic radiation

p Acoustic pressure c Acoustic velocity

fi approx . equals, around, about ‡ equivalent to … and so on, etc . 6 infinity = equal to Ï not equal to ==def is equal to by definition < less than

‰ less than or equal to > greater than › greater than or equal to + plus

– minus · times, multiplied by –, /, :, ÷ over, divided by, per, to S sigma (summation)

, proportional an a to the n-th power, the n-th power of a 03 square root of

n 03 n-th root of

æxæ absolute value of x o perpendicular to ø is parallel to parallel in the same direction

º parallel in the opposite direction @ angle ™ triangle 9 congruent to

Dx delta x (difference between two values) % percent, of a hundred ‰ per mil, of a thousand

log logarithm (general) lg common logarithm ln natural logarithm e Euler number (e = 2 .718281…)

sin sine cos cosine tan tangent cot cotangent

(), [], parentheses, brackets open and closed p pi (circle constant = 3 .14159 …)

AB line segment AB AB£ arc AB a*, a+ a prime, a double prime a1, a2 a sub 1, a sub 2

LP Acoustic pressure level I Sound intensity

N Loudness LN Loudness level

Acoustics

m Mass m* Linear mass density m+ Area mass density r Density J Moment of inertia p Pressure pabs Absolute pressure pamb Ambient pressure pg Gage pressure

F Force FW, W Gravitational force, Weight M Torque T Torsional moment Mb Bending moment s Normal stress t Shear stress e Normal strain E Modulus of elasticity

G Shear modulus µ, f Coefficient of friction W Section modulus I Second moment of an area W, E Work, Energy Wp, Ep Potential energy Wk, Ek Kinetic energy P Power n Efficiency

Mechanics

Mathematical symbols cf . DIN 1302 (1999-12)

Math. Spoken symbol

Math. Spoken symbol

Math. Spoken symbol

ººº

14 1 .2 Formulas

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Formulas, Equations, GraphsFormulas

In most cases, the calculation of physical quantities is done with the help of formulas . They consist of:

•  Formula symbols, e. g. vc for cutting velocity, d for diameter, n for speed•   Operators (calculation rules), e. g. · for multiplication, + for addition, – for subtraction 

and –– (fraction line) for division•  Constants, e. g. p (pi) = 3 .14159 …•  Numbers, e. g. 10, 15 …

The formula symbols (page 13) are wildcards for quantities . When solving mathematical problems, the known quantities with their units are filled in the formulas . Before or during the calculation process, the units are converted in a way that

•  the calculation becomes feasible or•  the result comprises the required unit.

Most quantities and units are standardized (page 10) .

The result is always a numerical value accompanied by a unit, e . g . 4 .5 m, 15 s

What is the cutting velocity vc in m/min for d = 200 mm and n = 630/min?

vc = p · d · n = p · 200 mm · 630

1min

= p · 200 mm ·

1 m1000 mm · 630

1min

= 395.84

mmin

Example:

What is the torque M of an electrical motor with a driving power of P = 15 kW and a speed of n = 750/min?

M = 9550 · Pn

= 9550 · 15750

N · m = 191 N · m

Example:

vc = p · d · n

Formula for cutting velocity

M = 9550 · P

n

Numerical value equation for torque

y = f(x)

Assigned function

y = m · x + b

Linear function

Ct = CV · Q + Cf

Examples:Cost function

R = R/piece · Q

Revenue function

Numerical value equations

Equations and graphs

Numerical value equations or numerical equations are formulas in which the typical conversions of units have already been integrated . The following should be noted when using equations:The numerical values of the individual quantities may only be used in combination with the designated unit .•  The units are not carried along in the calculation .•  The unit of the quantity to be obtained is predetermined .

In functional equations, y is the function of x, with x as an independent and y as a dependent variable . The number pairs (x, y ) of a value table form a graph in the x-y system of coordinates .

Designated unit

Designation Unit

M Torque N · m

P Power kW

n Speed 1/min

800 000

600 000

400 000

200 000

02000 4000 6000pieces0

cost

s or

rev

enue

m = 0.5

b = 1

example:y=0.5x+1

y

–2 –1 1 2

3

–1

1

2

x3

quantity

break-even point (BEP)

loss

profit

total costs

fixed costs

variable costs

revenue$

1st example:y = 0 .5 x + 1

x – 2 0 2 3y 0 1 2 2 .5

2nd example:Cost function and revenue functionCt = 60 $/piece · Q + 200 000 $R = 110 $/piece · Q

Q 0 4 000 6 000Ct 200 000 440 000 560 000R 0 440 000 660 000

Ct total costs ∫ dependent variableQ quantity ∫ independent variableCf fixed costs ∫ y coordinate sectionCv variable costs ∫ gradient of the

functionR revenue ∫ dependent variable

151 .2 Formulas

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Example: formula L = Œ1 + Œ2, transformation to find Œ2

Example: formula A = Œ · b, transformation to find Œ

Example: formula n = Œ

Œ1 + s, transformation to find s

Example: formula c = a 2 + b 2, transformation to find a

Transformation of formulasTransformation of formulas

Transformations of sums

Transformations of products

Transformations of fractions

Transformations of roots

Formulas and numerical equations are transformed so that the quantity to be obtained stands alone on the left side of the equation . The value of the left side and right side of the formula must not change during the transformation . The following rule applies to all steps of the formula transformation .

Changes applied to the left formula side

= Changes applied to the right formula side

To be able to trace each step of the transformation, it is useful to mark it to the right next to the formula:

æ· t ∫ both sides of the formula are multiplied by t .

æ: F ∫ both sides of the formula are divided by F .

Formula

P = F · s

tleft side right sideof the = of the formula formula

1222222

122222

1

1

1

1

3

3

4

4

2

2

2

2

3

3

4

4

5

5

6

6

L = Œ1 + Œ2 æ– Œ1

A = Œ · b æ: b

n = Œ

Œ1 + s æ· (Œ1 + s)

c = a2 + b2 æ( )2

L – Œ1 = Œ2

Ab

= Œ

n · Œ1 – n · Œ1 + n · s = Œ – n · Œ1 æ: n

a2 = c2 – b2 æ12

L – Œ1 = Œ1+ Œ2 – Œ1

Ab

= Œ · bb

n · (Œ1 + s) = Œ · (Œ1 + s)(Œ1 + s)

c2 = a2 + b2 æ– b2

n · Œ1 + n · s = Œ æ– n · Œ1

c2 – b2 = a2 + b2 – b2

Œ2 = L – Œ1

Π= A

b

s · nn

= Œ – n · Œ1n

a2 = c2 – b2

s = Œ – n · Œ1

n

a = c2 – b2

subtract Œ1

divide by b

multiply by (Œ1 + s)

square equation

invert both sides

invert both sides

subtract divide by n

extract the root

perform subtraction

cancel b

cancel (Œ1 + s ) on the right sidesolve the term in brackets

subtract b2

subtract – n · Œ1

subtract, invert both sides

transformed formula

transformed formula

cancel n

simplify the expression

transformed formula

transformed formula

122 1222222

1222222

16 1 .2 Formulas

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Conversion of units

Quantities and units

Decimal multiples or factors of units cf . DIN 1301-1 (2004-10)

Conversion factors for units (excerpt)

Numerical values and units

Prefix Power of ten

Mathematical designation Examples

Symbol Name

T tera 1012 trillion 12 000 000 000 000 N = 12 · 1012 N = 12 TN (teranewtons)

G giga 109 billion 45 000 000 000 W = 45 · 109 W = 45 GW (gigawatts)

M mega 106 million 8 500 000 V = 8 .5 · 106 V = 8 .5 MV (megavolts)

k kilo 103 thousand 12 600 W = 12 .6 · 103 W = 12 .6 kW (kilowatts)

h hecto 102 hundred 500 — = 5 · 102 — = 5 h— (hectoliters)

da deca 101 ten 32 m = 3 .2 · 101 m = 3 .2 dam (decameters)

– – 100 one 1 .5 m = 1 .5 · 100 m

d deci 10–1 tenth 0 .5 — = 5 · 10–1 — = 5 d— (deciliters)

c centi 10–2 hundredth 0 .25 m = 25 · 10–2 m = 25 cm (centimeters)

m milli 10–3 thousandth 0 .375 A = 375 · 10–3 A = 375 mA (milliamperes)

µ micro 10–6 millionth 0 .000 052 m = 52 · 10–6 m = 52 µm (micrometers)

n nano 10–9 billionth 0 .000 000 075 m = 75 · 10–9 m = 75 nm (nanometers)

p pico 10–12 trillionth 0 .000 000 000 006 F = 6 · 10–12 F = 6 pF (picofarads)

Quantity Conversion factors, e . g . Quantity Conversion factors, e . g .

Length 1 =

10 mm1 cm

=

1000 mm1 m

=

1 m1000 mm

=

1 km1000 m Time 1 =

60 min1 h

=

3600 s1 h

=

60 s1 min

=

1 min60 s

Area 1 =

100 mm2

1 cm2 =

100 cm2

1 dm2 =

Angle 1 =

60’1°

=

60’’1’

=

3600’’1°

=

1°60 s

Volume 1 =

1000 mm3

1 cm3 =

1000 cm3

1 dm3 =

Inch 1 inch = 25 .4 mm; 1 mm =

125 .4

inches

Calculations with physical units are only possible if these units refer to the same base in this calculation . When solving mathematical problems, units often must be converted to basic units, e . g . mm to m, s to h, mm2 to m2 . This is done with the help of conversion factors that represent the value 1 (coherent units) .

Physical quantities, e . g . 125 mm, consist of a

•  numerical value, which is determined by measurement or calculation, and a •  unit, e . g . m, kg

Units are standardized in accordance with DIN 1301-1 (page 10) .

Very large or very small numerical values may be represented in a simplified way as decimal multiples or factors with the help of prefixes, e . g . 0 .004 mm = 4 µm .

Physical quantity

10 mm

Numerical Unit value

Convert volume V = 3416 mm3 to cm3 .

Volume V is multiplied by a conversion factor . Its numerator has the unit cm3 and its denominator the unit mm3 .

V = 3416 mm3 = 1 cm3 · 3416 mm3

1000 mm3 = 3416 cm3

1000 = 3.416 cm3

The angle size specification a = 42° 16’ is to be expressed in degrees (°) .

The partial angle 16’ must be converted to degrees (°) . The value is multiplied by a conversion factor, the numerator of which has the unit degree (°) and the denominator the unit minute (’) .

a = 42° + 16’ · 1°60’

= 42° + 16 · 1°60

= 42° + 0 .267° = 42.267°

1st example:

2nd example:

171 .2 Formulas

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Calculation with quantities, Percentage and interest calculation

Interest calculation

IP · r · t

=100% · 360

InterestP principle I interest t period in days,A amount accumulated r interest rate per year interest period

1 interest year (1 a) = 360 days (360 d) 360 d = 12 months 1 interest month = 30 days

P r t I

I

= = = =

=

$ ; / ; ?

$

%2800.00;

2800.00

6 12a

a

· 6 · 0.5a

100%84.00

4800.00;

%

$

$ .

a =

= =P r 5 1%%

%

; ; ?

$

a

a

d50

4800.00 · 5.1 · 50 d

t I

I

= =

=1100% · 360

34.00da

= $

1st Example:

P r t I

I

= = = =

=

$ ; / ; ?

$

%2800.00;

2800.00

6 12a

a

· 6 · 0.5a

100%84.00

4800.00;

%

$

$ .

a =

= =P r 5 1%%

%

; ; ?

$

a

a

d50

4800.00 · 5.1 · 50 d

t I

I

= =

=1100% · 360

34.00da

= $

2nd Example:

Percentage calculation

PB P

vv r=

·%100

Percent valueThe percentage rate indicates the part of the base value in hundredths .The base value is the value from which the percentage is to be calculated .The percent value is the amount representing the percentage of the base value .

Pr percentage rate, in percent Pv percent value Bv base value

Weight of raw part: 250 kg (base value); material loss of 2 % (percentage rate); material loss in kg = ? (percent value)

Example:

Pv 5 kg= = =B Pv ·100

2100%

250 kg · %%

r

Calculation with quantities

Physical quantities are mathematically treated as products .

•  Adding and subtracting

Numerical values that have the same unit are added or subtracted and the unit is carried over to the result .

•  Multiplying and dividing

The numerical values and the units correspond to the factors of products .

•  Multiplying and dividing powers

Powers that have the same base are multiplied or divided by adding or subtracting their exponents .

L = Œ1 + Œ2 – Œ3 mit Œ1 = 124 mm, Œ2 = 18 mm, Œ3 = 44 mm; L = ?

L = 124 mm + 18 mm – 44 mm = (124 + 18 – 44) mm = 98 mm

F1 · Œ1 = F2 · Œ2 mit F1 = 180 N, Œ1 = 75 mm, Œ2 = 105 mm; F2 = ?

F2 = F1 · Œ1Œ2

= 180 N · 75 mm

105 mm = 128 .57

N · mmmm

= 128.57 N

W =

A · a2

e with A = 15 cm2, a = 7 .5 cm, e = 2 .4 cm; W = ?

W =

15 cm2 · (7 .5 cm)2

2 .4 cm =

15 · 56 .25 cm2+2

2 .4 cm1 = 351 .56 cm4–1 = 351.56 cm3

Example:

Example:

Example:

Rules for raising to powers

a basem, n … exponents

a2 · a3 = a2+3

Multiplying powers

a2

a3 = a2–3

Dividing powers

a1 = a a0 = 1

a –2 = 1a2

Special cases

å ¿

b a

c

©

Sum of angles in a triangle

¿

¶

å

©

g

g2

g1

18 1 .3 Angels and Triangels

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b2 a2

c2

ab c

G03P1 X

Z

P2

R K I

Types of angles, Theorem of intersecting lines, Angles in a triangle, Pythagorean theoremTypes of angles

Theorem of intersecting lines

Pythagorean theorem

D d

a2

†to

†ti

a1

b 1

b 2

g straight lineg1, g2 parallel straight lines

a, b corresponding angles

b, d opposite angles

a, d alternate angles

a, g adjacent angles

If two parallels are intersected by a straight line, there are geometrical interrelationships between the resulting angles .

If two intersecting lines are intercepted by a pair of parallels, the resulting segments form equal ratios .

a = bCorresponding angles

b = dOpposite angles

a = dAlternate angles

a + g = 180°

Adjacent angles

ab

ab

1

1

2

2= b

dbD

1 2=

aa

bb

d

D1

2

1

2

2

2

= =

Theorem of intersecting lines

a + b + g = 180°

Sum of angles in a triangle

In every triangle, the sum of the interior angles equals 180° .

D = 40 mm, d = 30 mm, tta = 135 N/mm2; tti = ?

Example:

a = 21°, b = 95°, g = ?

g = 180° – a – b = 180° – 21° – 95° = 64°

Example:

tt

ttti

toti

to= ⇒ =

=

dD

dD·

135 N/mm · 30 mm4

2

00 mm= 101. N/mm225

tto outer torsional stress

tti inner torsional stress

a, b, c sides of the trianglea, b, g angles in the triangle

c a b

c a

==

35 mm; = 21 mm; = ?

= mm)2 2b – ( –2 35 ((21 mm)2 = 28 mm

1st example:

c2 = a2 + b2

Length of the hypotenuse

In a right triangle the square on the hypotenuse is equal to the sum of the squares on the sides meeting the right angle .a sideb sidec hypotenuse

CNC programm with R = 50 mm and I = 25 mm . K = ?

2nd example:

c a bR K

K R

2 2 2

2 2 2

2 2 2 250 25

= += += ==

I

I– –mm mm2 2

K 43.3 mm

c a b= +2 2

Square on the hypotenuse

a c b= 2 2–

b c a= 2 2–

Length of the sides meeting the right angle

191 .3 Angels and Triangels

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Functions of triangles

c hypotenuse

b adjacent side of å

a adjacent side of åå

b opposite side of ¿

aadjacent side of ¿

¿c hypotenuse

F

L3= 140mm

L

L1=

150mm

L2=

30mm

å

ß

©

å ¿

b a

c

FdFz

F

4012

ß=38

å=40 12

©=102

Fd

Fz

F

Diagram of forces

Functions of right triangles (trigonometric functions)

Functions of oblique triangles (law of sines, law of cosines)

sine = opposite side hypotenuse

cosine = adjacent side hypotenuse

tangent = opposite side adjacent side

Trigonometric functionsc hypotenuse (longest side)a, b sides, – b is the adjacent side of a – a is the opposite side of aa, b, g angles in the triangle, g = 90°sin notation of sinecos notation of cosinetan notation of tangentsin a sine of angle a

According to the law of sines, the ratios of the sides correspond to the sine of their opposite angles in the triangle . If one side and two angles are known, the other values can be calculated with the help of this function .Side a ∫ opposite angle sin aSide b ∫ opposite angle sin bHypothenuse c ∫ opposite angle sin g

The calculation of an angle in degrees (°) or as a circular measure (rad) is done with the help of inverse trigonometric functions, e . g . arcsine .

L1 = 150 mm, L2 = 30 mm, L3 = 140 mm; angle a = ?

Angle a = 52°

1st example

tan .a =+

= =L L

L1 2

31 286

180 mm140 mm

L1 = 150 mm, L2 = 30 mm, a = 52°; Length of the shock absorber L = ?

2nd example

L =+

= =L L1 2

sina180 mmsin 52°

228.42 mm

F = 800 N, a = 40°, b = 38°; Fz = ?, Fd = ?

The forces are calculated with the help of the forces diagram .

Example:

F FF

Fsin sin

· sinsina b

ba

= ⇒ =

=

zz

800 N · sin38°Fz ssin40°

= 766.24 N

F FF

Fsin sin

· sinsina g

ga

= ⇒ =

=

dd

800 N · sin102Fd

°°sin40°

= 1217.38 N

Relations applying to angle a:

sin a =

ac

cos a =

bc

tan a =

ab

Relations applying to angle b:

sin b =

bc

cos b =

ac

tan b =

ba

There are many transformation options:

a = b · sin a

sin b =

c · sin a

sin g

b = a · sin b

sin a =

c · sin b

sin g

c = a · sin g

sin a =

b · sin g

sin b

Law of sines

a : b : c = sin a : sin b : sin g

asin a

=

bsin b

= csin g

Law of cosines

a2 = b2 + c2 – 2 · b · c · cos ab2 = a2 + c2 – 2 · a · c · cos bc2 = a2 + b2 – 2 · a · b · cos g

The calculation of an angle in degrees (°) or as a circular measure (rad) is done with the help of inverse trigonometric functions, e . g . arcsine .

Transformation, e . g .

cos a =

b2 + c2 – a2

2 · b · c

20 1 .4 Lengths

Division of lengths, Arc length, Composite length

Πtotal length n number of holesp spacing

Spacing

Spacing

Sub-dividing lengths

Arc length

L = Œ1 + Œ2 + …

Composite length

D outside diameter d inside diameterdm mean diameter t thicknessŒ1, Œ2 section lengths L composite lengtha angle at center

Composite length

D = 360 mm; t = 5 mm; a = 270°; Œ2 = 70 mm; dm = ?; L = ?

Example (composite length, picture left):

Edge distance = spacing

Edge distance ) spacing

Example:

Πtotal length n number of holesp spacing a, b edge distances

Example:

Number of pieces

Subdividing into pieces Œ bar length s saw cutting widthz number of pieces Œr remaining lengthŒs piece length

Example:

Œr = Œ – z · (Œs + s)

Remaining length

Example: Torsion spring Œa arc length a angle at centerr radius d diameter

Example:

Arc length

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