Mechanical And Metal Trades Handbook 3rd Edition 2011 3b1w2q

EUROPA-TECHNICAL BOOK SERIES for the Metalworking Trades

Ulrich Fischer Roland Gomeringer

Max Heinzler Roland Kilgus

Friedrich Näher Stefan Oesterle

Heinz Paetzold Andreas Stephan

Mechanical and Metal Trades Handbook 3rd English edition

Europa-No.: 1910X

VERLAG EUROPA LEHRMITTEL · Nourney, Vollmer GmbH & Co. KG Düsselberger Straße 23 · 42781 Haan-Gruiten ·

Original title: Tabellenbuch Metall, 45th edition, 2011 Authors: Ulrich Fischer Roland Gomeringer Max Heinzler Roland Kilgus Friedrich Näher Stefan Oesterle Heinz Paetzold Andreas Stephan

Dipl.-Ing. (FH) Dipl.-Gwl. Dipl.-Ing. (FH) Dipl.-Gwl. Dipl.-Ing. (FH) Dipl.-Ing. Dipl.-Ing. (FH) Dipl.-Ing. (FH)

Reutlingen Meßstetten Wangen im Allgäu Neckartenzlingen Balingen Amtzell Mühlacker Marktoberdorf

Editor: Ulrich Fischer, Reutlingen Graphic design: Design office of Verlag Europa-Lehrmittel, Ostfildern, The publisher and its s 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 s 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, . 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, http://www.europa-lehrmittel.de Translation: Eva Schwarz, 76879 Ottersheim, ; www.technische-uebersetzungen-eva-schwarz.de Typesetting: Satz+Layout Werkstatt Kluth GmbH, D-50374 Erftstadt, ; www.slw-kluth.de Printed by: ­­M. P. Media-Print Informationstechnologie GmbH, D-33100 Paderborn,

3

Preface

1 Mathematics

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 educational 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

2 Physics 29 – 50

3 Technical Drawing

Notes for the The 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 several 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 edition In the present edition, we have updated the cited standards and restructured, updated, enhanced or added the following chapters in line with new developments in engineering: – Fundamentals of technical – PAL programming system mathematics for NC turning and NC – Strength of materials milling – Plastics – Steel types – Production management – Material testing – Forming – Machining processes – Welding – Injection molding (new) – GRAFCET Acknowledgement Special 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

9 – 28

Authors and publisher

51 – 110

4 Material Science 111 – 200

5 Machine Elements 201 – 268

M

P

TD

MS

ME

6 Production Engineering PE 269 – 366

7 Automation and Information Tech367 ­– 424 nology

A

8  International Material Comparison Chart, Standards 425– 434

S

4

Table of Contents 1  Mathematics (M) 1.1

9

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 . . Formulas, equations, graphs . . . . . . . . . . . . . Transformation of formulas . . . . . . . . . . . . . . Quantities and units . . . . . . . . . . . . . . . . . . . . Calculation with quantities . . . . . . . . . . . . . . Percentage and interest calculation . . . . . . 1.3 Angels and triangels Types of angels, sum of angels in a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem of intersecting lines, Pythagorean theorem . . . . . . . . . . . . . . . . . . . Functions of right triangles . . . . . . . . . . . . . . Functions of oblique triangles . . . . . . . . . . . .

 13  14  15  16  17  17

 18  18  19  19

1.4 Lengths Division of lengths . . . . . . . . . . . . . . . . . . . . . .  20 Spring wire lengths . . . . . . . . . . . . . . . . . . . . .  21 Rough lengths . . . . . . . . . . . . . . . . . . . . . . . . .  21 1.5 Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . Triangle, polygon, circle . . . . . . . . . . . . . . . . . Circular sector, circular segment . . . . . . . . . Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

2  Physics (P)

29

2.1 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 . . . . . . . . . . . Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic power transmission . . . . . . . . . . . Pressure intensification, flow velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . Changes of state in gases . . . . . . . . . . . . . . .

 22  23  24  24

 38  38  38  39  39

2.6 Strength of materials Load cases, types of loading . . . . . . . . . . . . . Material properties, stress limits . . . . . . . . . Strength calculation, safety factors . . . . . . Allowable stresses, modulus of elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensile and compressive stress, surface pressure . . . . . . . . . . . . . . . . . . . . . . . Shear, bending and torsional stress . . . . . . Moments of area, section moduli . . . . . . . .

 42  43  44

2.7 Thermodynamics Temperature, linear expansion, shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantity of heat . . . . . . . . . . . . . . . . . . . . . . . . Heat flux, heat of combustion . . . . . . . . . . . . Net calorific values . . . . . . . . . . . . . . . . . . . . .

 45  45  46  46

 40  40  41  41

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

5

Table of Contents

3  Technical Drawing (TD)51 3.1 Graphs Cartesian coordinate system . . . . . . . . . . .   52 Graph types . . . . . . . . . . . . . . . . . . . . . . . . . . .   53 3.2 Basic geometric constructions Lines and angles . . . . . . . . . . . . . . . . . . . . . . Tangents, circular arcs . . . . . . . . . . . . . . . . Inscribed circles, ellipses . . . . . . . . . . . . . . Cycloids, involute curves, parabolas . . . . .

  54   55   56   57

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

  58   59   60   62

3.4 Representation Projection methods . . . . . . . . . . . . . . . . . . . . Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sectional views . . . . . . . . . . . . . . . . . . . . . . . Hatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  64   66   68   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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Retaining rings, springs . . . . . . . . . . . . . . . .

  79   80   81   82

3.7

Object elements Bosses, object edges . . . . . . . . . . . . . . . . . . Thread runouts, thread undercuts . . . . . . Threads, screw ts . . . . . . . . . . . . . . . . . . Center holes, knurls, undercuts . . . . . . . . .

  83   84   85   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 . . . . . . . . . . . . . . . . . . . . . . . . . Basic hole system, basic shaft system . . . General tolerances . . . . . . . . . . . . . . . . . . . . Roller bearing fits . . . . . . . . . . . . . . . . . . . . . Fit recommendations . . . . . . . . . . . . . . . . . . Geometric tolerancing . . . . . . . . . . . . . . . . . .

  98  102  106  106  107  108

4  Materials Science (MS)111 4.1 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 . . . . . . . . . . . . Unalloyed steels . . . . . . . . . . . . . . . . . . . . . . Case hardened steels, quenched & tempered steels . . . . . . . . . . . . . . . . . . . . . . . Nitriding steels, free cutting steels . . . . . . Tool steels, stainless steels . . . . . . . . . . . . . Steels for bright steel products . . . . . . . . .

 123  126  129  131  132  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 . . . . . . . . . . . . . . Thermoset plastics . . . . . . . . . . . . . . . . . . . . Thermoplastics, elastomers . . . . . . . . . . . . Plastics processing, testing of plastics . .

 178  181  182  186

6

Table of Contents 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

5  Machine Elements (ME)201 5.1 Threads Types of threads, overview . . . . . . . . . . . . . Metric ISO threads . . . . . . . . . . . . . . . . . . . . Other types of threads . . . . . . . . . . . . . . . . . Thread tolerances . . . . . . . . . . . . . . . . . . . . .

 202  202  205  207

5.2 Bolts and screws Overview, designations, property classes  208 Hexagon head bolts & screws . . . . . . . . . . 211 Other bolts & screws . . . . . . . . . . . . . . . . . .  214 Screw t 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Designations, property classes . . . . . . . . . Hexagon nuts . . . . . . . . . . . . . . . . . . . . . . . . . Other nuts . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 225  226  228  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 ts . . . . . . . .  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 . . . . . . . . . . . . . . . Plain bearing bushings . . . . . . . . . . . . . . . . . Antifriction bearings, overview . . . . . . . . . Ball bearings . . . . . . . . . . . . . . . . . . . . . . . . . . Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . Sealing elements . . . . . . . . . . . . . . . . . . . . . . Lubricating oils . . . . . . . . . . . . . . . . . . . . . . . . Lubricating greases . . . . . . . . . . . . . . . . . . .

 257  258  259  261  262  266  267  268

6  Production Engineering (PE)269 6.1

Quality management Standards, terminology . . . . . . . . . . . . . . . . Quality planning, quality testing . . . . . . . . . Statistical analysis . . . . . . . . . . . . . . . . . . . . Statistical process control . . . . . . . . . . . . . . Quality management of processes . . . . . .

6.2

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

 270  272  273  275  276

6.3 Production organization Overview; product breakdown structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  280 Work planning . . . . . . . . . . . . . . . . . . . . . . . .  282 Cost ing . . . . . . . . . . . . . . . . . . . . . . .  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 . . . . . . . . . . . . Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indexable inserts . . . . . . . . . . . . . . . . . . . . . . Cooling lubrication . . . . . . . . . . . . . . . . . . . . Cuttin tool materials . . . . . . . . . . . . . . . . . . . Grinding, honing . . . . . . . . . . . . . . . . . . . . . . . 6.6

 293  298  301  305  308  311  315  317

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

7

Table of Contents 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 ing Welding processes . . . . . . . . . . . . . . . . . . . . Weld preparation . . . . . . . . . . . . . . . . . . . . . . Gas-shielded welding . . . . . . . . . . . . . . . . . . Arc welding . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam cutting . . . . . . . . . . . . . . . . . . . . . . . . . Gas cylinders, identification . . . . . . . . . . . . Brazing, bonded ts . . . . . . . . . . . . . . . . . 6.11

 343  345  346  348  350  352  354

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

7  Automation and Information Technology (A)367 7.1

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 . . . . . . . . . . . . . . . . . . . . . . . . Designations in circuit plans . . . . . . . . . . . . Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety precautions . . . . . . . . . . . . . . . . . . . .

 375  377  378  379  380

7.3 GRAFCET Important basic . . . . . . . . . . . . . . . . . Steps, transitions . . . . . . . . . . . . . . . . . . . . . . Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Branchings . . . . . . . . . . . . . . . . . . . . . . . . . . .

 382  383  384  386

7.4 Programmable logical controllers PLC PLC programming languages, overview . . Ladder diagram (LD) . . . . . . . . . . . . . . . . . . . Instruction list (IL) . . . . . . . . . . . . . . . . . . . . . PLC programming languages, comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . Programming example . . . . . . . . . . . . . . . . .

 388  389  390  391  392

8  Material Chart, Standards (S) 8.1 International material comparison chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Subject Index

7.5 Hydraulics, pneumatics Circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . Proportional valves . . . . . . . . . . . . . . . . . . . . Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . Pneumatic control . . . . . . . . . . . . . . . . . . . . . Electro-pneumatic control . . . . . . . . . . . . . . Electro-hydraulic control . . . . . . . . . . . . . . . Hydraulic fluids . . . . . . . . . . . . . . . . . . . . . . . Pneumatic cylinders . . . . . . . . . . . . . . . . . . . Hydraulic pumps . . . . . . . . . . . . . . . . . . . . . . Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 393  395  396  397  398  399  400  401  402  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 . . . . . . . . . . . . . . . . . . . . . . . Program structure . . . . . . . . . . . . . . . . . . . . . Tool offset and cutter compensation . . . . Program structure according to DIN . . . . . Program structure according to PAL . . . . . PAL functions for lathes . . . . . . . . . . . . . . . . PAL cycles for lathes . . . . . . . . . . . . . . . . . . PAL functions for milling machines . . . . . . PAL cycles for milling machines . . . . . . . . .

 407  408  409  410  412  413  414  417  418

425 – 434 8.2 Index of cited standards and other regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

435 – 444

8

Standards and other Regulations Standardization and standards Standardization is the systematic achievement of uniformity of material and non-material objects, such as components, calculation methods, process flows and services for the benefit of the general public. Standards term

Example

Explanation

Standard

DIN 7157

Part

DIN 30910-2

A standard is the published work of standardization, e. g. the selection of particular fits in DIN 7157. Standards can comprise several parts associated with each other. The part numbers 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 specifications. 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 commenting. 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.

Types of standards and regulations (selection) Type

Abbreviation

Explanation

Purpose and contents

International Standards (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.

European Standards (EN standards)

EN

European Committee for Standardization (Comité Européen de Normalisation), Brussels

German Standards (DIN standards)

DIN

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

DIN EN

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

Technical harmonization and the associated reduction of trade barriers for the advancement of the European market and the coal­escence of Europe. National standardization facilitates rationalization, quality assurance, environmental protection and common understanding in economics, technology, science, management and public relations.

DIN ISO

German standard for which an international standard has been adopted without change.

DIN EN ISO

European standard for which an international standard has been adopted unchanged and the German version has the status of a German standard. Printed publication of the VDE, which has the status of a German standard. Verein Deutscher Ingenieure e.V., Düsseldorf (Association of German Engineers)

DIN VDE VDI Guidelines

VDI

VDE printed publications

VDE

DGQ publications

DGQ

REFA sheets

REFA

Verband Deutscher Elektrotechniker e.V., Frankfurt (Association for Electrical, Electronic & Information Technologies) Deutsche Gesellschaft für Qualität e.V., Frankfurt (German Society for Quality) Association for Work Design, Industrial Organization and Corporate Development REFA e.V., Darmstadt

These guidelines give an of the current state of the art in specific subject areas and contain, for example, concrete procedural guidelines for the performing calculations or deg processes in mechanical or electrical engineering. Recommendations in the area of quality technology. Recommendations in the area of production and work planning.

9

Table of Contents

1 Mathematics Quantity Symbol Œ

Lengths

Unit Name Symbol meter

m

Surface area

p ·d2 As = p ·d · h + 2 · 4 Lateral surface area

AM = p · d · h

opposite side sine = hypotenuse adjacent side cosine = hypotenuse opposite side tangent = adjacent side

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 . . . . . . Formulas, equations, graphs . . . . . . . . . . . . . . . . . Transformation of formulas . . . . . . . . . . . . . . . . . . Quantities and units . . . . . . . . . . . . . . . . . . . . . . . . . Calculation with quantities . . . . . . . . . . . . . . . . . . . Percentage and interest calculation . . . . . . . . . . . .

 13  14  15  16  17  17

1.3

 18  18  19  19

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

1.6

kg m' in m

d y

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

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

S S2 ys

S1 xs

 22  23  24  24

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

1m

x

P

TD

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

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

M

MS

ME

PE

A

S

10

1.1 Units of measurement

Units of measurement M

P

SI1)

Base quantities and base units

cf. DIN 1301-1 (2010-10), -2 (1978-02), -3 (1979-10)

Base quantity

Length

Mass

Time

Electric current

Base units

meter

kilogram

second

ampere

kelvin

mole

candela

m

kg

s

A

K

mol

cd

Unit symbol

Thermodynamic temperature

Amount of substance

Luminous intensity

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.

Base quantities, derived quantities and their units Quantity

TD

Unit Symbol Name Symbol

Remarks Examples of application

Length, Area, Volume, Angle Length

Area

Œ

A, S

MS Volume

ME

Relationship

Plane angle (angle)

Solid angle

V

meter

m

1 m 1 mm 1 km

= 10 dm = 100 cm = 1000 mm = 1000 µm = 1000 m

square meter

m2

are hectare

a ha

1 m2 1 a 1 ha 100 ha

= 10 000 cm2 Symbol S only for cross-sectional = 1 000 000 mm2 areas = 100 m2 = 100 a = 10 000 m2 Are and hectare only for land = 1 km2

cubic meter

m3

liter

—, L

a, b, g  … radian

rad

1 inch = 25.4 mm In aviation and nautical applications the following applies: 1 international nautical mile = 1852 m

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

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°.

degrees

°

minutes seconds

* +

1 rad = 1 m/m = 57.2957…° = 180°/p 1° = p rad = 60* 180 1* = 1°/60 = 60+ = 1*/60 = 1°/3600 1+

≈

steradian

sr

1 sr = 1 m2/m2

An object whose extension measures 1 rad in one direction and perpendicularly to this also 1 rad, covers a solid angle of 1 sr.

m

kilogram gram

kg g

1 kg = 1000 g 1 g = 1000 mg

megagram metric ton

Mg t

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

PE Mechanics Mass

A

Mass for precious stones in carat (ct).

Linear mass density

m*

kilogram per meter

kg/m

1 kg/m = 1 g/mm

For calculating the mass of  bars, profiles, pipes.

Area mass density

m+

kilogram per square meter

kg/m2

1 kg/m2 = 0.1 g/cm2

To calculate the mass of  sheet metal.

kilogram per 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

S

1 metric t = 1000 kg = 1 Mg 0.2 g = 1 ct

r

11

1.1 Units of measurement

Units of measurement Quantities and Units (continued) Quantity

M

Unit Symbol Name Symbol

Remarks Examples of application

Relationship

Mechanics Moment of inertia, 2nd Moment of mass

J

kilogram x square meter

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

Force

F

newton

N

The force 1 N effects a change in m = 1 J 1 N = 1 kg ·2 velocity of 1 m/s in 1 s in a 1 kg mass. m s 1 MN = 103 kN = 1 000 000 N

newton x meter

N·m

1 N · m = 1 kg ·2m s

2

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

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 direction of the velocity.

p

pascal

Pa

newton per square millimeter

N/mm2

meter to the fourth power centimeter to the fourth power

m4

E, W

joule

P G

t

Weight Torque Bending mom. Torsional mom. Momentum

Pressure

FG, G M Mb T

Mechanical stress

s, t

Second moment of area

I

Energy, Work, Quantity of heat Power Heat flux

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.

1 Pa = 1 N/m2 = 0.01 mbar 1 bar = 100 000 N/m2 = 10 N/cm2 = 105 Pa 1 mbar = 1 hPa 1 N/mm2 = 10 bar = 1 MN/m2 = 1 MPa 1 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 )

1 m4 = 100 000 000 cm4

Previously: Geometrical moment of inertia

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.

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.

seconds minutes hours day year

s min h d a

1 min = 60 s 1 h = 60 min = 3600 s 1 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.

hertz

Hz

1 Hz = 1/s

1 Hz ‡ 1 cycle in 1 second.

cm4

P

TD

MS

ME

Time Time, Time span, Duration

Frequency

f, v

Rotational speed, Rotational frequency

n

Velocity

v

Angular­ velocity Acceleration

w

a, g

min–1

1/s = 60/min = 60 1/min = 1 min–1 = 1 60 s

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

1 m/s

= 60 m/min = 3.6 km/h m/min 1 m/min = 1 m 60 s km/h 1 km/h = 1 m 3.6 s 1/s w=2p·n rad/s

Nautical velocity in knots (kn): 1 kn = 1.852 km/h

m/s2

Symbol g only for acceleration due to gravity. g = 9.81 m/s2 fi 10 m/s2

1 per second

1/s

1 per minute

1/min

meters per second meters per minute kilometers per hour 1 per second radians per second meters per second squared

m/s

1 m/s2 = 1 m/s 1s

PE

A

miles per hour = 1 mile/h = 1 mph 1 mph = 1.60934 km/h For a rpm of n = 2/s the angular ve­locity w = 4 p/s.

S

12

1.1 Units of measurement

Units of measurement M

Quantities and units (continued) Unit Symbol Name

Quantity

Symbol

Remarks Examples of application

Relationship

Electricity and Magnetism

P

Electric current Electromotive force Electrical resistance Electrical conductance Specific resistance Conductivity

TD

MS

PE

ampere volt

A V

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

R

ohm

O

1 O = 1 V/1 A

G

siemens

S

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

r

ohm x meter siemens per meter

O · m 10–6 O · m = 1 O · mm2/m

g, k

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.

S/m

2 1 Ω mm2 1 Ω ·· mm r r= = in in k m k m m 1 m k= = 1 in in k 2 r Ω ·· mm r Ω mm2

Frequency

f

hertz

Hz

1 Hz = 1/s 1000 Hz = 1 kHz

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

Electrical energy

W

joule

J

1 J = 1 W · s = 1 N · m 1 kW · h = 3.6 MJ 1 W · h = 3.6 kJ

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

Phase difference

j

–

–

for alternating current: cos j = P U·I

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

Elect. field strength Elect. charge Elect. capacitance inductance

E Q C L

volts per meter V/m C coulomb farad F henry H

F Q 1 C = 1 A · 1 s; 1 A · h = 3.6 kC E = , C = , Q = I · t Q U 1 F = 1 C/V 1 H = 1 V · s/A

Power Effective power

P

watt

1 W = 1 J/s = 1 N · m/s =1V·A

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

Relationship

Remarks Examples of application

W

Unit Symbol Name

Quantity

ME

I E

Symbol

Electricity and Magnetism Thermodynamic temperature Celsius temperature

T, Q

kelvin

K

0 K = – 273.15 °C

t, h

degrees Celsius

°C

0 °C = 273.15 K 0 °C = 32 °F 0 °F = – 17.77 °C

Quantity of heat

Q

joule

J

1 J = 1 W · s = 1 N · m 1 kW · h = 3 600 000 J = 3,6 MJ

1 kcal ‡ 4.1868 kJ

Net calorific value

Hu

joule per kilogram Joule per cubic meter

J/kg

1 MJ/kg = 1 000 000 J/kg

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

J/m3

1

MJ/m3

= 1 000 000

Kelvin (K) and degrees Celsius (°C) are used for temperatures and temperature differences. t = T – T0; T0 = 273.15 K degrees Fahrenheit (°F): 1.8  °F = 1 °C

J/m3

Non-SI units

A

Length

Area

1 inch (in) = 25.4 mm

1 sq.in

= 6.452 cm2

1 cu.in

= 16.39 cm3

1 oz

= 28.35 g

1 PSh

= 0.735 kWh

1 foot (ft) = 0.3048 m

1 sq.ft

= 9.29 dm2

1 cu.ft

= 28.32 dm3

1 lb

= 453.6 g

1 PS

= 735 W

= 1000 kg

1 kcal

= 4186.8 Ws

1 kcal

= 1.166 Wh

1 yard (yd)

= 0.9144 m

1 nautical mile = 1.852 km 1 mile

S

Volume

Mass

Energy, Power

1 sq.yd = 0.8361 m2

1 cu.yd = 764.6 dm3

1 t

1 acre

1 gallon 1 (US) = 3.785 —

1 short ton

= 907.2 kg

1 Karat

= 0.2 g

= 4046.856 m2

Pressure

= 1.6093 km 1 bar 1

= 14.5 pound/in2

N/mm2 =

45.038 1 pound/in2

1 gallon 1 (UK) = 4.546 — 1 barrel = 158.8 —

1 pound/in3 = 27.68 g/cm3

1 kpm/s = 9.807 W 1 Btu

= 1055 Ws

1 hp

= 745.7 W

13

1.2 Formulas

Formula symbols, Mathematical symbols Formula symbols Formula symbol

Meaning

cf. DIN 1304-1 (1994-03) Formula symbol

Meaning

Formula symbol

M

Meaning

Length, Area, Volume, Angle

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 W l



F Force FW, W Gravitational force, Weight M Torque T Torsional moment M b 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, E k Kinetic energy P Power n Efficiency

Planar angle Solid angle Wave length

P

Mechanics

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

TD

Time t T n

Time, Duration Cycle duration 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



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



X Reactance Z Impedance j Phase difference N Number of turns



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

MS

Heat T, Q Thermodynamic temperature DT, Dt, Dh Temperature difference t, h Celsius temperature a —, a Coefficient of l­ inear 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



f n

Focal length Refractive index



I Q, W



LP I

Acoustic pressure level Sound intensity



ME

Light, Electromagnetic radiation

E Illuminance

Luminous intensity Radiant energy

Acoustics

p c

Acoustic pressure Acoustic velocity

N Loudness LN Loudness level

Mathematical symbols Math. symbol

Spoken

cf. DIN 1302 (1999-12) Math. symbol

Spoken

Math. symbol

, a n 03 n 03

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



æxæ o ø

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

sin cos tan cot

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



– · –, /, :, ÷ S

Dx % ‰

º ºº

minus times, multiplied by over, divided by, per, to sigma (summation)

PE



log lg ln e

º parallel in the opposite direction (), [], { } @ angle ™ triangle p 9 congruent to delta x (difference between two values) percent, of a hundred per mil, of a thousand

AB £ AB a*, a+ a1, a2

Spoken logarithm (general) common logarithm natural logarithm Euler number (e = 2.718281…) sine cosine tangent cotangent

A

parentheses, brackets open and closed pi (circle constant = 3.14159 …) line segment AB arc AB a prime, a double prime a sub 1, a sub 2

S

14

1.2 Formulas

Formulas, Equations, Graphs M

Formulas 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 …

P

Formula for cutting velocity

vc = p · d · n

The formula symbols (page 13) are wildcards for quantities. When solving m ­ athematical 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

TD

Example: 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

1 1m 1 m = p · 200 mm · · 630 = 395.84 min 1000 mm min min

Numerical value equations

MS

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.

Numerical value equation for torque

M = 9550 · P n Designated unit

Example:

ME

What is the torque M of an electrical motor with a driving power of P = 15 kW and a speed of n = 750/min? 9550 · P 9550 · 15 M= = N · m = 191 N · m n 750

Designation

Unit

M Torque

N·m

P

Power

kW

n

Speed

1/min

Equations and graphs

PE

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. 3

1st example: y = 0.5 x + 1

example: y = 0.5x+1

x y

2 y

m = 0.5

1 –2

S

costs or revenue

A

–1 –1

800 000 $ 600 000

break-even point (BEP)

3

revenue

200 000

s

los

0

it

f pro

total costs

400 000

0

2 x

variable costs fixed costs

2000 4000 pieces 6000 quantity

Q Ct R

0 200 000 0

y = f(x) Linear function

0 1

2 2

3 2.5

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

b =1 1

– 2 0

Assigned function

4 000 440 000 440 000

6 000 560 000 660 000

Ct total costs ∫ dependent variable Q quantity ∫ independent variable Cf fixed costs ∫ y coordinate section Cv variable costs ∫ gradient of the function R revenue ∫ dependent variable

y=m·x+b

Examples: Cost function

Ct = CV · Q + Cf Revenue function

R = R/piece · Q

15

1.2 Formulas

Transformation of formulas Transformation of formulas

M

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

Formula

P  = F · s t left side right side of the = of the formula formula

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

P

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

Transformations of sums

TD

Example: formula L = Œ1 + Œ2, transformation to find Œ2 1 L = Œ1 + Œ2

æ– Œ1

2 L – Œ1 = Œ1+ Œ2 – Œ1

subtract Œ1

3 L – Œ1 = Œ2

invert both sides

perform subtraction

4 Œ2 = L – Œ1

transformed formula

Transformations of products

MS

Example: formula A = Œ · b, transformation to find Œ 1 A = Œ · b

æ: b

2 A= Œ·b b b

divide by b

3 A=Πb

invert both sides

cancel b

4 Œ=A b

transformed formula

Transformations of fractions Example: formula n = Œ Œ1 + s

Œ , transformation to find s Œ1 + s multiply by (Œ1 + s)

4 n · Œ1 – n · Œ1 + n · s = Œ – n · Œ1 æ: n subtract divide by n

2 n · (Œ1 + s) = Œ · (Œ1 + s) (Œ1 + s)

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

5 s · n = Œ – n · Œ1 n n

cancel n

3 n · Œ1 + n · s = Œ

subtract – n · Œ1

6 s = Œ – n · Œ1 n

transformed formula

1 n=

ME

æ· (Œ1 + s)

æ– n · Œ1

PE

Transformations of roots

A

Example: formula c = 122222 a 2 + b 22, transformation to find a a2 + b2 1 c = 122222

æ( )2

square equation

4 a2 = c2 – b2

2 c2 = a2 + b2

æ– b2

subtract b2

5

2 – b2 a22 = 12 c22 12 222

simplify the expression

subtract, invert both sides

6

2 – b2 a = 12 c22 222

transformed formula

3 c2 – b2 = a2 + b2 – b2

æ12 extract the root

S

16

1.2 Formulas

Quantities and units M

Numerical values and units Physical quantities, e. g. 125 mm, consist of a

Physical quantity

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

10 mm Numerical value

P

Symbol

MS

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.

Decimal multiples or factors of units Prefix

TD

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

Unit

Power of ten

Name

cf. DIN 1301-1 (2004-10)

Mathematical designation

Examples

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)

pico

10–12

trillionth

0.000 000 000 006 F = 6 · 10–12 F = 6 pF (picofarads)

p

Conversion of units Calculations with physical units are only possible if these units refer to the same base in this calculation. When s­ olving 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).

ME

Conversion factors for units (excerpt) Quantity

Conversion factors, e. g.

Quantity

Conversion factors, e. g.

Length

1m = 1 km 1 = 10 mm = 1000 mm = 1 cm 1m 1000 mm 1000 m

Time

1 = 60 min = 3600 s = 60 s = 1 min 1h 1h 1 min 60 s

Area

2 2 = 100 cm = 1 = 100 mm 1 cm2 1 dm2

Angle

1 = 60’ = 60’’ = 3600’’ = 1° 1° 1’ 1° 60 s

Volume

3 3 = 1000 cm = 1 = 1000 mm 1 cm3 1 dm3

Inch

1 inch = 25.4 mm; 1 mm =

PE

1 inches 25.4

1st example: Convert volume V = 3416 mm3 to cm3.

A

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 3416 cm3 = = 3.416 cm3 1000 mm3 1000

2nd example: 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 (’).

S

a = 42° + 16’ ·

1° 16 · 1° = 42° + = 42° + 0.267° = 42.267° 60’ 60

17

1.2 Formulas

Calculation with quantities, Percentage and interest calculation Calculation with quantities

M

Physical quantities are mathematically treated as products.

Rules for raising to powers

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

a base m, n … exponents

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

P

Multiplying powers

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

a2 · a3 = a2+3

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

Dividing powers

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

a2 = a2–3 a3

N · mm F1 · Œ1 180 N · 75 mm = = 128.57 = 128.57 N 105 mm mm Œ2

• Multiplying and dividing powers

Special cases

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

a –2 =

Example: W= W=

TD

A · a2 with A = 15 cm2, a = 7.5 cm, e = 2.4 cm; W = ? e

a1 = a

1 a2

MS

a0 = 1

15 cm2 · (7.5 cm)2 = 15 · 56.25 cm2+2 = 351.56 cm4–1 = 351.56 cm3 2.4 cm 2.4 cm1

Percentage calculation The 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

Percent value

ME

B ·P Pv = v r 100 %

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

Pv =

PE

B v · Pr 250 kg · 2 % = = 5 kg 100 % 100 %

Interest calculation P principle A amount accumulated

I interest t r interest rate per year

1st Example: % ; t = 1/ 2 a; I = ? a % % P = $$ 2800.00; ; t = 1/ 2 a; I = ? 2800.00 · r6 =a 6· 0.5 a a I = = $ 84.00 % 100% $ 2800.00 · 6 a · 0.5 a I = % = $ 84.00 nd 2 P Example: 100% = $ 4800.00; r = 5.1 ; t = 50 d; I = ? a %.1 % ; t = 50 d; I = ? P = $$ 4800.00; =5 4800.00 · r5.1 · 50 d a a I = = $ 34.00 %d $ 4800.00 · 50 d 100%· ·5.1 360 a a I = = $ 34.00 d 100% · 360 a

period in days, interest period

Interest

I =

P·r·t 100% · 360

P = $ 2800.00; r = 6

A

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

S

18

1.3 Angels and Triangels

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

Types of angles g straight line g1, g2 parallel straight lines

¿

g2 ¶

corresponding angles

b, d

opposite angles

a, d

alternate angles

a, g

adjacent angles

©

P

å

a, b

g1 g

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

Corresponding angles

a=b Opposite angles

b=d Alternate angles

a=d Adjacent angles

a + g = 180°

Theorem of intersecting lines

TD

tto outer torsional stress

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

tti inner torsional stress

a2 a1

Example:

†to †ti

tti d t ·d = ⇒ tti = to D tto D

b2

b1

D d

MS

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

=

Theorem of intersecting lines

d a1 b1 2 = = a2 b2 D 2 a1 a2 = b1 b2

b1 b2 = d D

135 N/mm2 · 30 mm = 101. 25 N/mm2 40 mm

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

ME ©

a

å

a = 21°, b = 95°, g = ?

c

PE

a + b + g = 180°

Example:

¿

b

Sum of angles in a triangle

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

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

Pythagorean theorem

2

a

b2 bc a

c2

A

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 side b side c hypotenuse

Length of the hypotenuse

1st example:

Square on the hypotenuse

c = 35 mm; a = 21 mm; b = ? b = c 2 – a 2 = (35 mm)2 – (21 mm)2 = 28 mm

c2 = a2 + b2

c = a2 + b2

2nd example: CNC programm with R = 50 mm and I = 25 mm. K=?

P2

c2

G0

S

3 R K

I

P1 X

Z

= a2

+ b2

Length of the sides meeting the right angle

R2 = I 2 + K 2

a = c 2 – b2

K = R 2 – I 2 = 502 mm2 – 252 mm2 K = 43.3 mm

b = c 2 – a2

19

1.3 Angels and Triangels

Functions of triangles Functions of right triangles (trigonometric functions) c hypotenuse

a adjacent side of å

å b adjacent side of å ¿

c hypotenuse

a adjacent side of ¿

c hypotenuse (longest side) a, b sides, – b is the adjacent side of a – a is the opposite side of a a, b, g angles in the triangle, g = 90° sin notation of sine cos notation of cosine tan notation of tangent sin a sine of angle a

M Trigonometric functions opposite side sine = hypotenuse adjacent side cosine = hypotenuse opposite side tangent = adjacent side

P

Relations applying to angle a:

b opposite side of ¿

1st example L1 = 150 mm, L2 = 30 mm, L3 = 140 mm; angle a = ?

L3 = 140mm

L

L1 = 150 mm

tan a =

å L2= 30 mm

ß F

L1 + L2 180 mm = = 1. 286 L3 140 mm

Angle a = 52° 2nd example L1 = 150 mm, L2 = 30 mm, a = 52°; Length of the shock absorber L = ? L=

L1 + L2 180 mm = = 228.42 mm sin a sin 52°

sin a = a cos a = b tan a = a c c b Relations applying to angle b: sin b = b cos b = a tan b = b c c a 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.

TD

MS

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

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

a

å

¿

b

c

a = b · sin a = c · sin a sin b sin g

Example:

Fz

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

40}

Fd

12}

Fz =

F

Fd =

ß=38

12}

} 40

å=

©=

10 2}

}

Fz

800 N · sin38° = 766.24 N sin 40°

F F F · sin g = d ⇒ Fd = sin a sin g sin a 800 N · sin102° = 1217.38 N sin 40°

PE

c = a · sin g = b · sin g sin a sin b

F F F · sin b = z ⇒ Fz = sin a sin b sin a

Diagram of forces

Fd

b = a · sin b = c · sin b sin a sin g

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

F

ME

Law of cosines a2

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

The calculation of an angle in degrees (°) Transformation, e. g. 2 2 2 or as a circular measure (rad) is done with cos a = b + c – a the help of inverse trigonometric functions, 2·b·c e. g. arcsine.

A

S

20

1.4 Lengths

Division of lengths, Arc length, Composite length M

Sub-dividing lengths Πtotal length p spacing

Edge distance = spacing









n number of holes

Spacing

n number of holes a, b edge distances

Spacing

Example:



P

Πtotal length p spacing

Edge distance ) spacing









Example:

TD







Œ bar length s saw cutting width z number of pieces Œr remaining length Œs piece length

Subdividing into pieces



MS



Number of pieces

Example: Remaining length



ME



Œr = Œ – z · (Œs + s)



Arc length Œa arc length r radius

a angle at center d diameter

Arc length



Example: Torsion spring



Example:



PE Composite length D outside diameter dm mean diameter Œ1, Œ2 section lengths a angle at center



A

Example (composite length, picture left):





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



S

d inside diameter t thickness L composite length

Composite length

L = Œ1 + Œ2 + …