Mechanical and Metal Trades Handbook 3rd Edition 2011

March 25, 2018 | Author: Mathias | Category: Screw, Alloy, Hertz, Stress (Mechanics), Steel
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EUROPA-TECHNICAL BOOK SERIESfor 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 · Germany 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, 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, Germany http://www.europa-lehrmittel.de Translation: Eva Schwarz, 76879 Ottersheim, Germany; www.technische-uebersetzungen-eva-schwarz.de Typesetting: Satz+Layout Werkstatt Kluth GmbH, D-50374 Erftstadt, Germany; www.slw-kluth.de Printed by: ­­M. P. Media-Print Informationstechnologie GmbH, D-33100 Paderborn, Germany Units are not specified in the legends for the formulae if several units are possible. we have updated the cited standards and restructured. Changes in the 3rd edition In the present edition. 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. machine building. updated. Subject Index and Standards Index. including Tables of Contents. the calculation examples for each formula use those units normally applied in practice. especially in practical work or curricula and continuing education programs. types. dimensions and standard values for their subject areas. The Subject Index at the end of the book (pages 435 – 444) is extensive. 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 user The contents of this book include tables and formulae in eight chapters. The tables contain the most important guidelines. His assistance has been extremely valuable. tooling. However.3 Preface 1 Mathematics The Mechanical and Metal Trades Handbook is well-suited for shop reference. Standards 425– 434 S . 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. for his input into the English translation of this book. more familiar standards to new ones. Vocational Training ­Specialist – Tool and Die. designs. maintenance and as a general book of knowledge. 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. The Standards Index (pages 425 – 434) lists all the current standards and regulations cited in the book. Ontario. In many cases previous standards are also listed to ease the transition from older. It is also useful for educational purposes. . . . . Calculation with quantities . . .3 Work. . . . . . . . . . .  25  Truncated pyramid. . . . addition and resolution of forces . . . . . . . . .  32 Types of forces . . . . . . . . . .  47 Current density. . . . . cone. . . . . . . . . . . . . . . . . . . . . . . cylinder. . .  27 1. . . . . . . . . . . . . . . . . . . . .  30 Speeds on machines . . . .  36 2. . . . . . . . . . . . Quantity of heat . . .  28 2  Physics (P) 29 2. . . . . . . . . . . . .  27 1. . . . . . . . . . . . . . . . . . . . . . . . . 34 Simple machines and energy . . . . . . . . . . . . . . . . . . . . . . . . . .4 Table of Contents 1  Mathematics (M) 1. .8 Centroids Centroids of lines. . . . . . . . . . . . . . . .  22  23  24  24  38  38  38  39  39 2. . Net calorific values . . . . . . . . . . . . Moments of area. . . . . . . . . . . . . .3 Angels and triangels Types of angels. . . . . . . . . . . . . . . . . . . . . . .  28 Centroids of plane areas. . . . . . . . . circular segment . . . . . . definition and types . .  45  45  46  46  40  40  41  41 2. . . . . . . . . . .6 Volume and surface area Cube. . . Allowable stresses. . . . . 10 Derived quantities and their units . . . . . . . . Triangle. . . . . .  21 1. . . . . polygon. . . . . . .5 Pressure in liquids and gases Pressure. . . . . . . . . sum of angels in a triangle . . . . pyramid . . . . . . Theorem of intersecting lines. . . . . . . . . . . . . . . . . . Material properties. . . . . . . . . . . . modulus of elasticity . . . . . . . . . . . . . . . . . . . . . Transformation of formulas . . . . . .  37 Coefficients of rolling friction . .  49 Electrical work and power . . . types of loading .  27 Linear mass density . Shear. . . . .  12 1. . . . . . . . . . . . . . . . . . . . . . . Hydraulic power transmission . . . .  20 Spring wire lengths . . . . . . . . .7 Mass General calculations. . . . . . .  42  43  44 2. . . . . . . . . . . . . . . . .1 Motion Uniform and accelerated motion . .7 Thermodynamics Temperature. . . . . . . . . . resistor circuit . . . stress limits . . . 1. . efficiency Mechanical work . . shrinkage . sphere . . . . . electrical resistance . . . . . . . . .  31 2. . section moduli . . . . . . . . .5 Areas Angular areas . . . . . . . . .  37 Coefficients of friction . .  26 Volumes of composite solids . . . . . . . Strength calculation. . Ellipse . . .  33 Torque and levers. . . . . . . . . . . . . . . . . . . . Buoyancy . . . . . . . Quantities and units . . . linear expansion. . . . . mathematical symbols . . . . . . . . . . circle . . .8 Electricity Quantities and units . . . . . . . . . . . . . .  13  14  15  16  17  17  18  18  19  19 1. . . . . . . . . . . .4 Friction Friction force . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensile and compressive stress. . . . . . . . .  48 Types of current . . . . . . Functions of right triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure intensification. . . .6 Strength of materials Load cases. . . . . . . . . . . . . . . . . Heat flux. . . . . . . . . Truncated cone. . . . . . . . . . . . . . . . . Formulas. . . . . . . . . . . . . . . . . . . . . . Pythagorean theorem . . . . . . . . . . . . . . . . . . . . . . graphs . . . .  27 Area mass density. . . . . . . . . . . . Circular sector. . bending and torsional stress . . . . . . . . . flow velocity . . .  34 2. . Changes of state in gases . Percentage and interest calculation . . . . . . surface pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . . . . . . . . . . . safety factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  37 2. . . . . . . 50 . . . . . . . . . . . torque in gear drives . . . . . . . power.2 Forces Representation. . . . . . . .  47 Ohm’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . heat of combustion . . . . . . . . . . . . . . . . . . . . .2 Formulas Formula symbols. .  35 Potential and kinetic energy . .4 Lengths Division of lengths . . . . .  21 Rough lengths . . . . . . .  11 Non-SI units . . .1 9 Units of measurement SI base quantities and base units . . . . . . . . . . . . . . . .  35 Power and efficiency . . . . . . . . . . . . . . . . . . . . . equations. . . . . . . . . . . . . . . . . . . . . . . . . . . Functions of oblique triangles . . . . . .9 Surfaces Hardness specifications in drawings . . .3 Elements of drawing Fonts . . . . . . . . . knurls. . . . . . . . . Steels for bright steel products . . . . Retaining rings. . . .   58   59   60   62 3.1 Materials Material characteristics of solids . Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   78 3. . . .  178  181  182  186 . . . . . . . . . . . . . . . . . . . . . involute curves. . . . . . . .   91 3. . . . . . . .  113 Periodic table of the elements . . . . . .9 Heavy non-ferrous metals Overview and designation system . .   73 Tolerance specifications . . . General tolerances . . . free cutting steels . .  161 4. . . . . . . . . . . . . . . . . . . . . . . . .  138 Profiles . . . undercuts . . . . . . . . . . . Case hardened steels. . . . . . representation Gear types . . . scales . . . . . Thermoplastics. . . . . . . . . . elastomers . . . . . . .   92 Form deviations. . . . . . . . . . . . . . . . . . . . . . radii. . . . . . . . . . . . . . . . . . . . . . Hatching . . . Line types . . . . . . . 159 Malleable cast iron. . . . . . Tangents. . . .10 Other metallic materials . . . . . . . . . Sectional views . . . . Views . . . . . . . . . . . . . . . . . . . . parabolas . . . . . . . . . . . . . . . . . . . . . Unalloyed steels . . . . . . . . . . . . . . . . . . . .8 Light alloys Overview of aluminum alloys . . . . . . tubes .  118 4. . . Center holes. . . . . . . . . . . . . . . . . . . . . object edges . . . . . . . . . . . . .  174 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Classification . . . . . . . . . . . . .5 Table of Contents 3  Technical Drawing (TD)51 3. . . .  72 Elements of drawing . . . . . . . . . . . . . . . . . . .4 Steels. . . . . Thread runouts. . . . . . . . . . Tool steels. . . . . . . . . . . Preferred numbers.7 Foundry technology .   64   66   68   70 3. . . . . Inscribed circles. . . . .  163 Wrought aluminum alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   94 3. . . . . . . . . . . . . . . circular arcs . . . . . . . .   83   84   85   86 3. . . . . . . . .  153 Hardening of aluminium alloys . . . . .2 Basic geometric constructions Lines and angles . . . cast steel . . . . . finished products Sheet and strip metal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic hole system. . . . . . . . . . ellipses . . . . . . . . . .5 Dimensioning drawings Dimensioning lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 ISO tolerances and fits Fundamentals . . .  112 Material characteristics of liquids and gases . . .8 Welding and soldering Graphical symbols . . . .   52 Graph types . . . . . . . Drawing layout. . .  123  126  129  131  132  137 4. . . . . . . . . . . .   71 Dimensioning rules . . .  158 Cast iron . . . . . . . . . . . . . . .   53 3. . . . . . . . . . . .  151 4. . . . . . . . . . . . . . . .  117 Designation system for steels . . . . . . . . . . . . . .  114 4. . . . . . . . . . . . . . . . . . testing of plastics . . . . Cycloids. . . roughness . bills of materials . . . . . . . . . . . . . . . . .   88 Dimensioning examples . . . 171 4.  172 Copper and refined zinc alloys . . . . . . . . . . . . . . . .   98  102  106  106  107  108 4  Materials Science (MS)111 4. . . . . .  142 Linear mass density and area mass density . . . . . . . . . . . . . . . dimension values . . . .5 Heat treatment Iron-carbon phase diagram .  176 4. . . . . . . . screw joints . . . . . . .  156 4. . . . Nitriding steels.   93 Surface testing. . . . . . . . . Geometric tolerancing . . . . . . .6 Machine elements. . . Roller bearings . . . . . . . . quenched & tempered steels . . . . . . . . . . . . . . Plastics processing. . .   75 Types of dimensioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fit recommendations . . . . . . . . . . . . . . . . . . . . . . . . . basic shaft system . . . . .6 Cast iron materials Designation and material codes . .2 Steels.  152 Heat treatement of steels . steel types Overwiew of steel products . . . . . . . . . . .   79   80   81   82 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermoset plastics . . . . . . . . . . .7 Object elements Bosses. . . . . . . .4 Representation Projection methods . . . . . . . . . . . . . . .11 Plastics Overview and designation . . . . . . . . . . . . . . . . . . . . . . . . . . .  160 4. . . . . . . . . .3 Steels. . . . . . . . . . . . . . . . . . . . . . . . . . . Roller bearing fits . . . . . . . . . . . . . . .   76 Simplified presentation in drawings . . . designation system Definition and classification of steel .   54   55   56   57 3. . . . . . . . . . . . . . . Threads. . . . . . . . . .1 Graphs Cartesian coordinate system . springs . . .  165 Aluminum profiles . thread undercuts . . . . . . . surface indications .  168 Magnesium and titanium alloys . . . . . .  116 Standardization of steel products . . . . . . . . . stainless steels . . . . . . quality testing . . . . . . . . . . . . . . . . . . .  256 5. . . . . . . . . . . Designations. . . . . . flat washers . . . . . . . . . . . . . . . . . . . .6 Table of Contents 4.12 Material testing Overview. . . . . . . . . . . . . . . . . .6 Pins and clevis pins Overview . . . . . . . . . . . . . . . Statistical process control . . . . . . . . . property classes  208 Hexagon head bolts & screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quality management of processes . . . . . . . . . . . . . . . . . . . . . . . . .  323 Process parameters in EDM erosion . . . . .  241 5. . . . . . . . . . . . Grinding. . . . . . . . . . . . . . . . . . . .  279  270  272  273  275  276 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . overview . . . . . .10 Bearings Plain bearings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sealing elements . . . . . . . . . . . . . . . . . inspection. . . . . . designations. . . . . . . . . . Cooling lubrication . . . . . . . . . . . 6. .6  293  298  301  305  308  311  315  317 Removal operations Electric discharge machining . .4 Maintenance. Turning . . . . . . . . . . . Lubricating greases . . . . . . . . tensile test . . . . . . . . . . . . . gib-head keys . . . . . . . . . . . . . . . . . .3 Countersinks Countersinks for countersunk head screws . . . . . . forces and power . . .  220 Locking fasteners. . . . . . Lubricating oils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  282 Cost accounting . . . . . . . . . . . . . .  192 Hardness test . . . . .  234 5. . . . . . . .  289 MRO concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  250 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  224 5. . . property classes . . . . . . . . . . . . . . . . . . . . . . . . . Indexable inserts . . . . . . . . . . . . . . . . . Quality planning. . repair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . repair & overhaul (MRO) Maintenance.  280 Work planning . . .  253 Transmission ratios . . . . . . . . . . . . . . . . .  247 Quick-release drilling fixture . . . . . . . .8 Other machine elements Tension. . . . . . . . . . 6. . . . . . . . . .  225  226  228  230 5. . . . . . . . . . . . . . . . . . . . . . .2 EC Machinery Directive EC Machinery Directive . . . .  278 CE marking . . . . . . . . . . . Plain bearing bushings . . . . . . . . . . . . . . . . . . . . . . . bolt and screw drive systems . . . . . . . . . . . . . . . .1 Threads Types of threads. . . . .  214 Screw joint calculations . . . . . . . . . . . . . . . . . . . . . . . . .  290 Documentation system . . . . . . . . . . . . . . . . . . thrust pads. . .  324 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . morse tapers. . . . . . honing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cuttin tool materials . . . Other types of threads .  189 Rotary bending test . . . . . . . . . . . Hexagon nuts .  197 5  Machine Elements (ME)201 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Milling . . . . . . . . . . . . . . . . . . . . . . .7 Shaft-hub connections Keys. . Other nuts . . .3 Production organization Overview. . . . . . . . . . .  232 Washers for HV bolts . . splined shaft joints . . . .9 Drive elements Belts . . .  202  202  205  207 5. . . . . . . . . . . . . . .  242 Grub screws.5 Washers Overview. . . . overview . . . . . . . . . improvement . . . . . . . . . . . . . . . . . . . .  248 5. . . .5 Machining processes Overview. Drilling . . . . . . . . . . . . . . . . . Antifriction bearings. .  239 Metric tapers. . . . . . . . . . terminology . . . . . . . . . . . compression and disc springs . . .  235 Grooved pins. . .  286 6. .1 Quality management Standards. . . . . . Thread tolerances . . .  193 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . Metric ISO threads . . . . . .  223 Counterbores for cap screws . . . . . . . . . grooved drive studs. Statistical analysis . . 211 Other bolts & screws . product breakdown structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 5. . . . .  237 5. . . . . . . . . . . . hazardous materials Corrosion . . . . Ball bearings . . . 245 T-slots and accessories . . . . . . .  238 Feather keys.  292 6. . . . . . . . . . . . . . . . . Roller bearings . . clevis pins . . . . . . . . . . . . . . . .13 Corrosion.  196 Hazardous materials . . overview . . . . . . . . . . .4 Nuts Overview . .  257  258  259  261  262  266  267  268 6  Production Engineering (PE)269 6. . . . . . . . . . . . . . knobs . . . . . . . . . . . . . . . steep tapers  240 Tool holding fixtures . . . . .2 Bolts and screws Overview. . . . . . . . . 430 435 – 444 . Designations in circuit plans . . . . . . . Program structure . . .  359 Warning signs. . . . . . . . . . . . . . comparison . . . . . . . . . . . . . . . . .11  343  345  346  348  350  352  354 Workplace and environmental protection Safety colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Table of Contents 6. . . . . . . . . . . . . . Proportional valves . . . . . . . . . . Steps. . . . . . . . .  406 7. . . . . .  366 7  Automation and Information Technology (A)367 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  334 6. . . . . . . . . . .8 Forming Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . transitions . . Instruction list (IL) . . . . . . . . . . . overview . . . . . Brazing. . . . . PAL functions for milling machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9 Injection molding Injection molding tools . . .  372 Numbering systems . . . . . Programming example .  404 Robot designs . . . . . . . . . . .  382  383  384  386 7. . . . . . . . . . . . Weld preparation . . safety signs . . . . Electro-pneumatic control . . .  373 Information processing . Hydraulic fluids . . . . . . . . . . . . . .  388  389  390  391  392 8  Material Chart. . . . .  370 Discontinuous and digital controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Subject Index 7. . Beam cutting . . . . . . . . . . . . Standards (S) 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bonded joints . . . . . . . . 6. . . . . . Pneumatic cylinders . . . .  374 7. . . . . . . . . . . . . . . .6 Handling and robot systems Coordinate systems and axes . Sensors . . . . . . . . . . . . . . . . . .  330 Deep drawing . . . Arc welding . . . . . .  360 Sound and noise . . . . . . . .1 Control engineering. batching . . . . identification . . . . . . . . .  375  377  378  379  380 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Electrical circuits Circuit symbols . . code letters. . . . . Gas cylinders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Programmable logical controllers PLC PLC programming languages. . . . . . pneumatics Circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 Separation by cutting Cutting force . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ladder diagram (LD) . . . . . job safety . . . . . . . . . . . . . . . . . . . .7 CNC technology Coordinate axes . . . . . . . . . . . . . . . . . . . . . . . . . Pneumatic control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PAL functions for lathes . . . . . . . . . . . . . . . . . PAL cycles for milling machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PLC programming languages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PAL cycles for lathes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . basic terminology Basic terminology. . . . .  393  395  396  397  398  399  400  401  402  403 7. . .5 Hydraulics. . . . . . . . . . . . .  341 6. . . . . . . . . . . . . . . . . . . .  407  408  409  410  412  413  414  417  418 425 – 434 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Branchings . . . . . . Actions . . . . . . . . . . . . . .3 GRAFCET Important basic terms . . .  371 Binary logic . .  326 6. . . . Circuit diagrams . . . . . . . . . .  325 Cutting tool . . . . . . . . . . . . . cooling. . . . . Program structure according to DIN . . . . . Hydraulic pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  405 Grippers. . . Program structure according to PAL . . . . . . . . . . . . . . . . . . . . Tool offset and cutter compensation . . .  338 Shrinkage. . . . . Gas-shielded welding . . . . . . . . . . . . . . . . . . . . . . . .10 Joining Welding processes . . . . . . . . . . . . . . . . Tubes . Safety precautions . . . . . . .2 Index of cited standards and other regulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . symbols  368 Analog controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electro-hydraulic control . .1 International material comparison chart . . calculation methods. whereas Part 3 and 4 deal with sintered materials for bearings and formed parts. this is the date at which time the standard becomes valid. technology. quality assurance. 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. contains application examples of load capacity calculations for shafts and axles described in DIN 743. Preliminary standard DIN V 66304 (1991-04) A preliminary standard contains the results of standardization. concrete procedural guidelines for the performing calculations or designing processes in mechanical or electrical engineering.V. Darmstadt These guidelines give an account of the current state of the art in specific subject areas and contain. Supplement DIN 743 Suppl. which has the status of a German standard. Recommendations in the area of quality technology. as well as cooperation in scientific. e. Düsseldorf (Association of German Engineers) DIN VDE VDI Guidelines VDI VDE printed publications VDE DGQ publications DGQ REFA sheets REFA Verband Deutscher Elektrotechniker e. Berlin (German Institute for Standardization) DIN EN European standard for which the German version has attained the status of a German standard. Industrial Organization and Corporate Development REFA e.V. DIN V 66304. for example. DIN 30910-2 describes sintered materials for filters for example. technical and economic areas. Recommendations in the area of production and work planning. Verein Deutscher Ingenieure e. Brussels German Standards (DIN standards) DIN Deutsches Institut für Normung e. . 1 A supplement contains information for a standard..V.. Draft E DIN 743 (2008-10) Draft standards are made available to the public for examination and commenting. DIN ISO German standard for which an international standard has been adopted without change. science. Frankfurt (German Society for Quality) Association for Work Design. Types of standards and regulations (selection) Type Abbreviation Explanation Purpose and contents International Standards (ISO standards) ISO International Organization for Standardization. which sets undercuts for metric ISO threads has been valid since June 2004 for example. The part numbers are appended to the main standard number with hyphens. National standardization facilitates rationalization. Technical harmonization and the associated reduction of trade barriers for the advancement of the European market and the coal­escence of Europe.. European Standards (EN standards) EN European Committee for Standardization (Comité Européen de Normalisation). DIN 76-1.. however no additional specifications. has been published since October 2008 as Draft E DIN 743. Electronic & Information Technologies) Deutsche Gesellschaft für Qualität e. Standards can comprise several parts associated with each other. The planned new version of DIN 743 on load-bearing calculations of shafts and axes. 1. for example. the selection of particular fits in DIN 7157. The supplement DIN 743 Suppl. g. for example. Frankfurt (Association for Electrical. discusses a format for exchange of standard part data for computer-aided design. Geneva (O and S are reversed in the abbreviation) Simplifies the international exchange of goods and services.8 Standards and other Regulations Standardization and standards terms Standardization is the systematic achievement of uniformity of material and non-material objects.. environmental protection and common understanding in economics. process flows and services for the benefit of the general public.V.V. management and public relations. Issue date DIN 76-1 (2004-06) Date of publication which is made public in the DIN publication guide. which have not been released as a standard because of certain provisos. for example. Standards term Example Explanation Standard DIN 7157 Part DIN 30910-2 A standard is the published work of standardization. such as components. Printed publication of the VDE. . . . . . . . . . Circular sector. . .4 Lengths Division of lengths . .3  18  18  19  19 Angels and triangels Types of angels. . . . . . . . . . . . . Theorem of intersecting lines . .  27 1m x P TD 1. . . . . . . . . . . . . . . . . . . . . . equations. . . . . . . . . . . . . . Transformation of formulas . . . . . . . Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . .1 Units of measurement SI base quantities and base units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. . . . . Functions of oblique triangles . . .6 kg m' in m d y Volume and surface area Cube. . . . . .  21 Rough lengths . . . .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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Triangle. . . . .  27 Area mass density. . . . . . . . . . . . . . . . . . .  27 Linear mass density. . . . . . . M MS ME PE A S . . . . . . . .2 Formulas Formula symbols. . . cylinder. . . . . . . . . . . . .5 Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . circular segment . . . .  25 Truncated pyramid. . . . . . . . . . . . . . . . . . . . polygon. . . . . . . . . . . . . . . . . . . . . . . graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  21 1. sphere. . . . . . . . . . . . . . circle . . . . . . . . . . . . . .  20 Spring wire lengths . . . . . . . . . mathematical symbols . . . . . . Percentage and interest calculation . . . . . . .8 Centroids Centroids of lines. . . . . . . . . . . . . . . .  13  14  15  16  17  17 1. . . . . . . cone. . . . . . . . . Functions of right triangles . . . . . . 10 Derived quantities and their units .  12 1. . . . . . . . . . . . . . . . . . . . Quantities and units . . . . . . .  11 Non-SI units . . . . . Formulas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26 Volumes of composite solids. sum of angels in a triangle . . . .  27 1. . . . Calculation with quantities . . . . . pyramid . .  28 S S2 ys S1 xs  22  23  24  24 1. . . . . . . .7 Mass General calculations .  28 Centroids of plane areas. . truncated cone. . . . . . . . . . . degrees ° minutes seconds * + 1 rad = 1 m/m = 57. derived quantities and their units Quantity TD Unit Symbol Name Symbol Remarks Examples of application Length.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— = 0.1 Units of measurement Units of measurement M P SI1) Base quantities and base units cf. 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).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.10 1.2 g = 1 ct r . PE Mechanics Mass A Mass for precious stones in carat (ct). L a. Base quantities. DIN 1301-1 (2010-10). covers a solid angle of 1 sr. 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. In technical calculations instead of a = 33° 17* 27. Linear mass density m* kilogram per meter kg/m 1 kg/m = 1 g/mm For calculating the mass of  bars.6+. 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 —. better use is a = 33. b.1 g/cm2 To calculate the mass of  sheet metal. -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). pipes. profiles. -2 (1978-02).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. It is based on the seven basic units (SI units). g  … radian rad 1 inch = 25. Area mass density m+ kilogram per square meter kg/m2 1 kg/m2 = 0. from which other units are derived.291°. Angle Length Area Œ A. Area. Volume. 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. kW · h preferred for electrical energy. 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. If points in time are written in mixed form.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. e. Torsional mom. 1 Pa = 1 N/m2 = 0. hertz Hz 1 Hz = 1/s 1 Hz ‡ 1 cycle in 1 second.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. Duration Frequency f. the symbol min can be shortened to m. also called rpm. v Rotational speed.1 N/mm2 Pressure refers to the force per unit area. t Second moment of area I Energy. 3h24m10s. Time span.60934 km/h For a rpm of n = 2/s the angular ve­locity w = 4 p/s.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. 1 m/s = 60 m/min = 3. G M Mb T Mechanical stress s. g.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. 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. cm4 P TD MS ME Time Time.11 1.6 km/h m/min 1 m/min = 1 m 60 s km/h 1 km/h = 1 m 3. 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. Momentum Pressure FG. 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. Work. 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. 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. For gage pressure the symbol pg is used (DIN 1314). g = 9. S .852 km/h m/s2 Symbol g only for acceleration due to gravity. 3h means a point in time (3 o’clock).6 s 1/s w=2p·n rad/s Nautical velocity in knots (kn): 1 kn = 1. Rotational frequency n Velocity v Angular­ velocity Acceleration w a. W joule P G t Weight Torque Bending mom. 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. 1 bar = 14. 6 MJ 1 W · h = 3. t = T – T0.7 W . T0 = 273. The reciprocal of the electrical resistance is called the electrical conductivity. Elect.35 g 1 PSh = 0. h degrees Celsius °C 0 °C = 273.3048 m 1 sq.852 km 1 mile S Volume Mass Energy.5 pound/in2 N/mm2 =  45.68 g/cm3 1 kpm/s = 9.6 kC E = .2 g = 4046.856 m2 Pressure = 1.6 dm3 1 t 1 acre 1 gallon 1 (US) = 3.1 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.15 °C t. 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.8 Ws 1 kcal = 1.in = 6. field strength Elect.735 kWh 1 foot (ft) = 0.8  °F = 1 °C J/m3 Non-SI units A Length Area 1 inch (in) = 25. k The motion of an electrical charge is called current.546 — 1 barrel = 158.ft = 28. Power 1 sq.785 — 1 short ton = 907.6 MJ 1 kcal ‡ 4.in = 16. Q kelvin K 0 K = – 273. 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. USA/Canada 60 Hz Electrical energy W joule J 1 J = 1 W · s = 1 N · m 1 kW · h = 3.807 W 1 Btu = 1055 Ws 1 hp = 745. J/m3 1 MJ/m3 = 1 000 000 Kelvin (K) and degrees Celsius (°C) are used for temperatures and temperature differences.15 K 0 °C = 32 °F 0 °F = – 17. The electromotive force is equal to the potential difference be­­ tween two points in an electric field.32 dm3 1 lb = 453.ft = 9.yd = 764.6093 km 1 bar 1 = 14.452 cm2 1 cu. charge Elect.6 kJ In atomic and nuclear physics the unit eV (electron volt) is used.8361 m2 1 cu. Phase difference j – – for alternating current: cos j = P U·I The angle between current and voltage in inductive or capacitive ­load.15 K degrees Fahrenheit (°F): 1.8 — 1 pound/in3 = 27.6 g 1 PS = 735 W = 1000 kg 1 kcal = 4186.2 kg 1 Karat = 0.39 cm3 1 oz = 28.9144 m 1 nautical mile = 1. 1 A · h = 3.77 °C Quantity of heat Q joule J 1 J = 1 W · s = 1 N · m 1 kW · h = 3 600 000 J = 3.1868 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. 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.yd = 0.29 dm2 1 cu. C = .166 Wh 1 yard (yd) = 0.038 1 pound/in2 1 gallon 1 (UK) = 4.4 mm 1 sq.12 1. Cross-sectional area V Volume a. symbol Spoken cf. Q Heat flow a Thermal diffusivity c Specific heat Hnet Net calorific value f n Focal length Refractive index I Q.13 1. DIN 1302 (1999-12) Math. brackets open and closed pi (circle constant = 3. R Radius d. 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. V. h Celsius temperature a —. g W l F Force FW. equals. of a hundred per mil. Speed f. Energy Wp. 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. a n 03 n 03 proportional a to the n-th power. S Area. a2 Spoken logarithm (general) common logarithm natural logarithm Euler number (e = 2. Area. multiplied by over. to sigma (summation) PE log lg ln e º parallel in the opposite direction ().718281…) sine cosine tangent cotangent A parentheses. W Gravitational force. Duration Cycle duration Revolution frequency.14159 …) line segment AB arc AB a prime. Volume. b. of a thousand AB £ AB a*. Quantity of electricity E Electromotive force C Capacitance I Electric current MS Heat T. u Velocity w Angular velocity a Acceleration g Gravitational acceleration a Angular acceleration · Q. D Diameter A. Angle Œ Length w Width h Height s Linear distance r. qv Volumetric flow rate L Inductance R Resistance r Specific resistance g. a double prime a sub 1. Mathematical symbols Formula symbols Formula symbol Meaning cf. Ep Potential energy Wk. E Work. W LP I Acoustic pressure level Sound intensity ME Light. a+ a1. Q Thermodynamic temperature DT. Dh Temperature difference t. 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 – · –. v Frequency v. k Electrical conductivity X Reactance Z Impedance j Phase difference N Number of turns Electricity Q Electric charge.2 Formulas Formula symbols. around. a sub 2 S . divided by. DIN 1304-1 (1994-03) Formula symbol Meaning Formula symbol M Meaning Length. []. Electromagnetic radiation E Illuminance Luminous intensity Radiant energy Acoustics p c Acoustic pressure Acoustic velocity N Loudness LN Loudness level Mathematical symbols  Math. 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. about ‡ equivalent to … and so on. Quantity of heat l Thermal conductivity a Heat transition coefficient k Heat transmission coefficient · G. symbol . /. f Coefficient of friction W Section modulus I Second moment of an area W. :. per. a Coefficient of l­ inear expansion Q Heat. Dt. { } @ angle ™ triangle p 9 congruent to delta x (difference between two values) percent. etc. ÷ S Dx % ‰ º ºº minus times. symbol Spoken Math. Most quantities and units are standardized (page 10).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.5 x + 1 example: y = 0. with x as an independent and y as a ­dependent variable. 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 = 395.5 m. p (pi) = 3. e. 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. e. 10. g.5x+1 x y 2 y m = 0. 15 … P Formula for cutting velocity vc = p · d · n The formula symbols (page 13) are wildcards for quantities. + for addition. the known quantities with their units are filled in the formulas. The result is always a numerical value accompanied by a unit. Before or ­during the calculation process. g. g. – for subtraction and –– (fraction line) for division •  Constants. •  The unit of the quantity to be obtained is predetermined. 4.5 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 .14 1. g. When solving m ­ athematical problems. the calculation of physical quantities is done with the help of formulas.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. n for speed •  Operators (calculation rules). g. d  for diameter. e.14159 … •  Numbers. The number pairs (x. •  The units are not carried along in the calculation.2 Formulas Formulas. e. y ) of a value table form a graph in the x-y system of coordinates. 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. They consist of: •  Formula symbols. Graphs M Formulas In most cases. e. the units are converted in a way that •  the calculation becomes feasible or •  the result comprises the required unit. 3 1st example: y = 0. vc for cutting velocity. Equations. y is the function of x. · for multiplication. 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. 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. transformation to find Œ 1 A = Œ · b æ: b 2 A= Œ·b b b divide by b 3 A=Œ b invert both sides cancel b 4 Œ=A b transformed formula Transformations of fractions Example: formula n = Œ Œ1 + s Œ . Transformations of sums TD Example: formula L = Œ1 + Œ2. The value of the left side and right side of the formula must not change during the transformation. 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. 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. The following rule applies to all steps of the formula transformation. invert both sides 6 2 – b2 a = 12 c22 222 transformed formula 3 c2 – b2 = a2 + b2 – b2 æ12 extract the root S . 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. 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.15 1. æ: F ∫ both sides of the formula are divided by F. consist of a Physical quantity •  numerical value. which is determined by measurement or calculation.5 MV (megavolts) k kilo 103 thousand 12 600 W = 12. g.5 m = 1. Decimal multiples or factors of units Prefix TD Units are standardized in accordance with DIN 1301-1 (page 10). 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 = 8.16 1.5 · 106 V = 8. 0.25 m = 25 · 10–2 m = 25 cm (centimeters) m milli 10–3 thousandth 0. g. e. Unit Power of ten Name cf. the ­numerator of which has the unit degree (°) and the denominator the unit minute (’). A Volume V  is multiplied by a conversion factor. g.5 — = 5 · 10–1 — = 5 d— (deciliters) c centi 10–2 hundredth 0. 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. s to h. m. Quantity Conversion factors. e.6 kW (kilowatts) h hecto 102 hundred 500 — = 5 · 102 — = 5 h— (hectoliters) da deca 101 ten 32 m = 3. e. and a •  unit. 125 mm.004 mm = 4 µm.2 Formulas Quantities and units M Numerical values and units Physical quantities. V = 3416 mm3 = 1 cm3 · 3416 mm3 3416 cm3 = = 3. mm to m. The value is multiplied by a conversion factor.000 052 m = 52 · 10–6 m = 52 µm (micrometers) n nano 10–9 billionth 0.267° = 42.267° 60’ 60 . g. The partial angle 16’ must be converted to degrees (°). 1 mm = PE 1 inches 25.5 · 100 m d deci 10–1 tenth 0.4 1st example: Convert volume V = 3416 mm3 to cm3.375 A = 375 · 10–3 A = 375 mA (milliamperes) µ micro 10–6 millionth 0. This is done with the help of conversion factors that represent the value 1 (coherent units). When s­ olving mathematical problems. 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.6 · 103 W = 12. e.4 mm. mm2 to m2. S a = 42° + 16’ · 1° 16 · 1° = 42° + = 42° + 0. e.2 dam (decameters) – – 100 one 1. e.000 000 075 m = 75 · 10–9 m = 75 nm (nanometers) pico 10–12 trillionth 0. Its numerator has the unit cm3 and its denominator the unit mm3. g.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. units often must be converted to basic units.416 cm3 1000 mm3 1000 2nd example: The angle size specification a = 42° 16’ is to be expressed in degrees (°). ME Conversion factors for units (excerpt) Quantity Conversion factors. g.2 · 101 m = 3. I = ? a % % P = $$ 2800. F2 = ? F2 = a2 = a2–3 a3 N · mm F1 · Œ1 180 N · 75 mm = = 128. material loss of 2 % (percentage rate).4 cm.4 cm1 Percentage calculation The percentage rate indicates the part of the base value in hundredths. 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. e = 2. The percent value is the amount representing the percentage of the base value.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.00 d 100% · 360 a period in days. Œ3 = 44 mm.5 a a I = = $ 84.00 · r5.2 Formulas Calculation with quantities. Œ2 = 18 mm. Pr percentage rate.00. t = 50 d. I = ? 2800.00. Dividing powers Example: F1 · Œ1 = F2 · Œ2 mit F1 = 180 N.00 % 100% $ 2800. I = ? a %.00 %d $ 4800.00.56 cm3 2. 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).00 · r6 =a 6· 0. Œ1 = 75 mm.17 1.1 360 a a I = = $ 34. Percentage and interest calculation Calculation with quantities M Physical quantities are mathematically treated as products. a = 7. t = 1/ 2 a.5 cm)2 = 15 · 56. W = ? e a1 = a 1 a2 MS a0 = 1 15 cm2 · (7.00 · 50 d 100%· ·5. Œ2 = 105 mm.00 · 6 a · 0. a base m. 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.1 . r = 6 A interest year (1 a) = 360 days (360 d) 1 360 d = 12 months 1 interest month = 30 days S . interest period Interest I = P·r·t 100% · 360 P = $ 2800. 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: % .57 = 128.5 a I = % = $ 84.56 cm4–1 = 351. t = 50 d.5 cm.25 cm2+2 = 351. The base value is the value from which the percentage is to be calculated.1 · 50 d a a I = = $ 34. n … exponents Example: L = Œ1 + Œ2 – Œ3 mit Œ1 = 124 mm.1 % . a –2 = Example: W= W= TD A · a2 with A = 15 cm2. r = 5.00. I = ? P = $$ 4800. .4 cm 2.00 nd 2 P Example: 100% = $ 4800. =5 4800. t = 1/ 2 a. d alternate angles a. the sum of the interior angles equals 180°. b. d = 30 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. Angles in a triangle. 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.3 Angels and Triangels Types of angles. c sides of the triangle a. b g1 g If two parallels are intersected by a straight line. g adjacent angles © P å a. a side b side c hypotenuse Length of the hypotenuse 1st example: Square on the hypotenuse c = 35 mm. there are geometrical interrelationships between the resulting angles. 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. 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. g angles in the triangle ME © a å a = 21°. Pythagorean theorem M Types of angles g straight line g1. tta = 135 N/mm2. b = 95°. g2 parallel straight lines ¿ g2 ¶ corresponding angles b.18 1. b. 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 = 101.3 mm b = c 2 – a2 . Theorem of intersecting lines. the resulting segments form equal ratios. 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. d opposite angles a. a = 21 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. 25 N/mm2 40 mm Sum of angles in a triangle a. angle a = ? L3 = 140mm L L1 = 150 mm tan a = å L2= 30 mm ß F L1 + L2 180 mm = = 1.24 N sin 40° F F F · sin g = d ⇒ Fd = sin a sin g sin a 800 N · sin102° = 1217. 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. g. arcsine. e. b sides. TD MS Functions of oblique triangles (law of sines.19 1.38 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. a = 40°. 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. Length of the shock absorber L = ? L= L1 + L2 180 mm = = 228. – b is the adjacent side of a – a is the opposite side of a a. A S . 2·b·c e. 2 2 2 or as a circular measure (rad) is done with cos a = b + c – a the help of inverse trigonometric functions. If one side and two angles are known. 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.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. L2 = 30 mm. 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. g. Fz = ?.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. g. g angles in the triangle. 286 L3 140 mm Angle a = 52° 2nd example L1 = 150 mm. L3 = 140 mm. arcsine. e. law of cosines) © According to the law of sines. b. L2 = 30 mm. a = 52°. b = 38°. the other values a = b = c can be calculated with the help of this sin a sin b sin g ­function. Fd = ? 40} Fd 12} Fz = F Fd = ß=38 12} } 40 å= ©= 10 2} } Fz 800 N · sin38° = 766. Œ2 section lengths a angle at center  A Example (composite length. dm = ?. Œ2 = 70 mm. a = 270°.20 1. Arc length.4 Lengths Division of lengths. L = ?   S d inside diameter t thickness L composite length Composite length L = Œ1 + Œ2 + … . Composite length M Sub-dividing lengths Œ total length p spacing Edge distance = spacing     n number of holes Spacing n number of holes a. t = 5 mm. b edge distances Spacing Example:  P Œ total 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. picture left):     D = 360 mm.
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