EUROPA-TECHNICAL BOOK SERIES for the Metalworking Trades
Ulrich Fischer
Max Heinzler
Friedrich Näher
Heinz Paetzold
Roland Gomeringer
Roland Kilgus
Stefan Oesterle
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, 45 th 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 Neckartenzli ngen 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 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 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 i n the classroom since they are unchanged, except for some corrections to typographical errors and slight changes in standards.
ISBN 13 978-3-8085-1914978-3-8085-1914-1 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 Translation: Ottersheim,, Germany; www.technische www.technische-uebersetzun -uebersetzungen-eva-schwarz gen-eva-schwarz.de .de Typesetting: Satz+Layout Werkstatt Kluth GmbH, D-50374 Erftstadt Erftstadt,, Germany; www.slw-kluth.de Printed by: M. P. Media-Pri Media-Print nt Informati Informationstechnologie onstechnologie GmbH, D-33100 Paderborn, Germany
3
Preface
1 Mathematics
M
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
9 – 28
2 Physics
•
P
•
29 – 50
• • • • •
3 Technical Drawing
• •
Notes for the user 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 3 rd 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
Authors and publisher
TD 51 – 110
4 Material Science
MS 111 – 200
5 Machine Elements
ME 201 – 268
6 Production Engineering PE 269 – 366
7 Automation and Information Technology 367 – 424
A
8 International Material Comparison Chart, Standards 425– 434
S
4
Table of Contents 1 Mathematics (M)
9
1.1
Units of measurement SI base quantities and base units. . . . . . . . . . 10 Derived quantities and their units. . . . . . . . . 11 Non-SI units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4
Lengths Division of lengths . . . . . . . . . . . . . . . . . . . . . . 20 Spring wire lengths . . . . . . . . . . . . . . . . . . . . . 21 Rough lengths . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2
Formulas Formula symbols, mathematical symbols . . Formulas, equations, graphs . . . . . . . . . . . . . Transformation of formulas . . . . . . . . . . . . . . Quantities and units. . . . . . . . . . . . . . . . . . . . . Calculation with quantities. . . . . . . . . . . . . . . Percentage and interest calculation . . . . . .
1.5
Areas Angular areas . . . . . . . . . . . . . . . . . . . . . . . . . . Triangle, polygon, circle . . . . . . . . . . . . . . . . . Circular sector, circular segment . . . . . . . . . Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
1.6
1.7
1.8
Volume and surface area Cube, cylinder, pyramid. . . . . . . . . . . . . . . . . . Truncated pyramid, cone, Truncated cone, sphere . . . . . . . . . . . . . . . . . . Volumes of composite solids . . . . . . . . . . . . . Mass General calculations. . . . . . . . . . . . . . . . . . . . Linear mass density . . . . . . . . . . . . . . . . . . . . Area mass density . . . . . . . . . . . . . . . . . . . . . .
2.2
2.3
2.4
2.5
2.6
Forces Representation, addition and resolution of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Types of forces . . . . . . . . . . . . . . . . . . . . . . . . . 33 Torque and levers, torque in gear drives. . . 34 34 35 35 36
Friction Friction force . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Coefficients of friction. . . . . . . . . . . . . . . . . . . 37 Coefficients of rolling friction. . . . . . . . . . . . . 37 Pressure in liquids and gases Pressure, definition and types . . . . . . . . . . . . Buoyancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydraulic power transmission. . . . . . . . . . . . Pressure intensification, flow velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . Changes of state in gases. . . . . . . . . . . . . . . .
26 27 27 27 27
29
Motion Uniform and accelerated motion . . . . . . . . . 30 Speeds on machines . . . . . . . . . . . . . . . . . . . . 31
Work, power, efficiency Mechanical work . . . . . . . . . . . . . . . . . . . . . . . Simple machines and energy. . . . . . . . . . . . . Potential and kinetic energy. . . . . . . . . . . . . . Power and efficiency. . . . . . . . . . . . . . . . . . . .
25
Centroids Centroids of lines . . . . . . . . . . . . . . . . . . . . . . . 28 Centroids of plane areas . . . . . . . . . . . . . . . . 28
2 Physics (P) 2.1
22 23 24 24
38 38 38 39 39
2.7
2.8
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 . . . . . . . . .
40 40 41 41 42 43 44
Thermodynamics Temperature, linear expansion, shrinkage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantity of heat. . . . . . . . . . . . . . . . . . . . . . . . . Heat flux, heat of combustion . . . . . . . . . . . . Net calorific values . . . . . . . . . . . . . . . . . . . . .
45 45 46 46
Electricity Quantities and units. . . . . . . . . . . . . . . . . . . . . Ohm’s law, electrical resistance . . . . . . . . . . Current density, resistor circuit . . . . . . . . . . . Types of current . . . . . . . . . . . . . . . . . . . . . . . . Electrical work and power . . . . . . . . . . . . . . .
47 47 48 49 50
5
Table of Contents
3 Technical Drawing (TD)
51
3.1
Graphs Cartesian coordinate system. . . . . . . . . . . . 52 Graph types . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2
3.3
3.4
3.5
3.6
Basic geometric constructions Lines and angles . . . . . . . . . . . . . . . . . . . . . . 54 Tangents, circular arcs . . . . . . . . . . . . . . . . 55 Inscribed circles, ellipses . . . . . . . . . . . . . . 56 Cycloids, involute curves, parabolas . . . . . 57
Machine elements, representation Gear types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . 80 Seals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Retaining rings, springs . . . . . . . . . . . . . . . . 82
3.7
Elements of drawing Fonts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Preferred numbers, radii, scales. . . . . . . . . 59 Drawing layout, bills of materials . . . . . . . . 60 Line types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Object elements Bosses, object edges . . . . . . . . . . . . . . . . . . 83 Thread runouts, thread undercuts . . . . . . . 84 Threads, screw joints . . . . . . . . . . . . . . . . . . 85 Center holes, knurls, undercuts . . . . . . . . . 86
3.8
Welding and soldering Graphical symbols . . . . . . . . . . . . . . . . . . . . . 88 Dimensioning examples . . . . . . . . . . . . . . . . 91
3.9
Surfaces Hardness specifications in drawings . . . . 92 Form deviations, roughness. . . . . . . . . . . . . 93 Surface testing, surface indications . . . . . 94
Representation Projection methods . . . . . . . . . . . . . . . . . . . . 64 Views . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 Sectional views . . . . . . . . . . . . . . . . . . . . . . . 68 Hatching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Dimensioning drawings Dimensioning lines, dimension values. . . . Dimensioning rules . . . . . . . . . . . . . . . . . . . . Elements of drawing . . . . . . . . . . . . . . . . . . . Tolerance specifications . . . . . . . . . . . . . . . Types of dimensioning . . . . . . . . . . . . . . . . . Simplified presentation in drawings. . . . . .
71 72 73 75 76 78
3.10 ISO tolerances and fits Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . Basic hole system, basic shaft system . . . General tolerances. . . . . . . . . . . . . . . . . . . . . Roller bearing fits . . . . . . . . . . . . . . . . . . . . . . Fit recommendations. . . . . . . . . . . . . . . . . . . Geometric tolerancing . . . . . . . . . . . . . . . . . .
4 Materials Science (MS) 4.1
111
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. . . . . . . . . .
4.4
98 102 106 106 107 108
123 126 129 131 132 137
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 . . . . . . . . . Classification . . . . . . . . . . . . . . . . . . . . . . . . . . Cast iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Malleable cast iron, cast steel . . . . . . . . . .
157 158 159 160
4.7
Foundry technology . . . . . . . . . . . . . . . . . . . . 1 6 1
4.8
Light alloys Overview of aluminum alloys. . . . . . . . . . . . Wrought aluminum alloys. . . . . . . . . . . . . . . Aluminum profiles . . . . . . . . . . . . . . . . . . . . . Magnesium and titanium alloys . . . . . . . . .
4.9
163 165 168 171
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) 5.1
5.2
5.3
5.4
5.5
Threads Types of threads, overview . . . . . . . . . . . . . Metric ISO threads . . . . . . . . . . . . . . . . . . . . Other types of threads. . . . . . . . . . . . . . . . . . Thread tolerances . . . . . . . . . . . . . . . . . . . . . Bolts and screws Overview, designations, property classes Hexagon head bolts & screws . . . . . . . . . . Other bolts & screws . . . . . . . . . . . . . . . . . . Screw joint calculations . . . . . . . . . . . . . . . . Locking fasteners, bolt and screw drive systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201 202 202 205 207 208 211 214 220
Pins and clevis pins Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Grooved pins, grooved drive studs, clevis pins . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
5.7
Shaft-hub connections Keys, gib-head keys. . . . . . . . . . . . . . . . . . . . Feather keys, splined shaft joints . . . . . . . . Metric tapers, morse tapers, steep tapers Tool holding fixtures . . . . . . . . . . . . . . . . . . .
238 239 240 241
Other machine elements Tension, compression and disc springs . Grub screws, thrust pads, knobs . . . . . . . . T-slots and accessories . . . . . . . . . . . . . . . . Quick-release drilling fixture . . . . . . . . . . . .
242 245 247 248
5.8
221
Countersinks Countersinks for countersunk head screws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Counterbores for cap screws . . . . . . . . . . . 224 Nuts Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Designations, property classes. . . . . . . . . . Hexagon nuts . . . . . . . . . . . . . . . . . . . . . . . . . Other nuts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6
225 226 228 230
Washers Overview, flat washers . . . . . . . . . . . . . . . . . 232 Washers for HV bolts . . . . . . . . . . . . . . . . . . . 234
5.9
5.10 Bearings Plain bearings, overview . . . . . . . . . . . . . . . Plain bearing bushings . . . . . . . . . . . . . . . . . Antifriction bearings, overview . . . . . . . . . . Ball bearings . . . . . . . . . . . . . . . . . . . . . . . . . . Roller bearings . . . . . . . . . . . . . . . . . . . . . . . . Sealing elements . . . . . . . . . . . . . . . . . . . . . . Lubricating oils . . . . . . . . . . . . . . . . . . . . . . . . Lubricating greases. . . . . . . . . . . . . . . . . . . .
6 Production Engineering (PE) 6.1
Quality management Standards, terminology. . . . . . . . . . . . . . . . . Quality planning, quality testing . . . . . . . . . Statistical analysis . . . . . . . . . . . . . . . . . . . . . Statistical process control . . . . . . . . . . . . . . Quality management of processes. . . . . . .
Drive elements Belts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Transmission ratios . . . . . . . . . . . . . . . . . . . . 256 257 258 259 261 262 266 267 268
269 270 272 273 275 276
6.2
EC Machinery Directive EC Machinery Directive . . . . . . . . . . . . . . . . 278 CE marking . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
6.3
Production organization Overview; product breakdown structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 Work planning . . . . . . . . . . . . . . . . . . . . . . . . . 282 Cost accounting . . . . . . . . . . . . . . . . . . . . . . . 286
6.4
Maintenance, repair & overhaul (MRO) Maintenance, inspection, repair, improvement . . . . . . . . . . . . . . . . . . . . . . . . . . 289 MRO concepts . . . . . . . . . . . . . . . . . . . . . . . . 290 Documentation system . . . . . . . . . . . . . . . . . 292
6.5
Machining processes Overview; forces and power . . . . . . . . . . . . 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 Joining Welding processes . . . . . . . . . . . . . . . . . . . . Weld preparation . . . . . . . . . . . . . . . . . . . . . . Gas-shielded welding . . . . . . . . . . . . . . . . . . Arc welding . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam cutting . . . . . . . . . . . . . . . . . . . . . . . . . . Gas cylinders, identification. . . . . . . . . . . . . Brazing, bonded joints. . . . . . . . . . . . . . . . . .
6.11 Workplace and environmental protection Safety colors . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Warning signs, safety signs . . . . . . . . . . . . . 360 Sound and noise . . . . . . . . . . . . . . . . . . . . . . . 366
7 Automation and Information Technology (A) 7.1
7.2
7.3
7.4
7.5
Control engineering, basic terminology Basic terminology, code letters, symbols Analog controllers . . . . . . . . . . . . . . . . . . . . . Discontinuous and digital controllers . . . . Binary logic . . . . . . . . . . . . . . . . . . . . . . . . . . . Numbering systems. . . . . . . . . . . . . . . . . . . . Information processing. . . . . . . . . . . . . . . . .
368 370 371 372 373 374
Electrical circuits Circuit symbols . . . . . . . . . . . . . . . . . . . . . . . . Designations in circuit plans . . . . . . . . . . . . Circuit diagrams . . . . . . . . . . . . . . . . . . . . . . . Sensors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety precautions. . . . . . . . . . . . . . . . . . . . .
375 377 378 379 380
GRAFCET Important basic terms . . . . . . . . . . . . . . . . . . Steps, transitions . . . . . . . . . . . . . . . . . . . . . . Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Branchings. . . . . . . . . . . . . . . . . . . . . . . . . . . .
382 383 384 386
Programmable logical controllers PLC PLC programming languages, overview . . Ladder diagram (LD). . . . . . . . . . . . . . . . . . . . Instruction list (IL).. . . . . . . . . . . . . . . . . . . . . PLC programming languages, comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . Programming example . . . . . . . . . . . . . . . . .
Subject Index
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
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 . . . . . . . . .
388 389 390 391 392
International material comparison chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
367
7.6
8 Material Chart, Standards (S) 8.1
343 345 346 348 350 352 354
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 terms 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
A standard is the published work of standardization, e. g. the selection of particular fits in DIN 7157.
Part
DIN 30910-2
Standards can comprise several parts associated with each other. The part 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.
DIN VDE
VDI Guidelines
VDI
Verein Deutscher Ingenieure e.V., Düsseldorf (Association of German Engineers)
VDE printed publications
VDE
Verband Deutscher Elektrotechniker e.V., Frankfurt (Association for Electrical, Electronic & Information Technologies)
DGQ publications
DGQ
Deutsche Gesellschaft für Qualität e.V., Frankfurt (German Society for Quality)
Recommendations in the area of quality technology.
REFA sheets
REFA
Association for Work Design, Industrial Organization and Corporate Development REFA e.V., Darmstadt
Recommendations in the area of production and work planning.
These guidelines give an account of the current state of the art in specific subject areas and contain, for example, concrete procedural guidelines for the performing calculations or designing processes in mechanical or electrical engineering.
Table of Contents
9
1 Mathematics Unit Quantity Symbol Name Symbol
Œ
Lengths
meter
= p
·d ·h + 2 ·
p
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 . . . . . . . . . . . .
m
Surface area As
1.1
· d 2 4
Lateral surface area
AM = p · d · h
1.3 sine
=
opposite side hypotenuse
cosine
=
adjacent side hypotenuse
tangent
=
opposite side adjacent side
Angels and triangels Types of angels, sum of angels in a triangle . . . . . Theorem of intersecting lines . . . . . . . . . . . . . . . . . Functions of right triangles . . . . . . . . . . . . . . . . . . . Functions of oblique triangles. . . . . . . . . . . . . . . . .
13 14 15 16 17 17
18 18 19 19
22 23 24 24
1.6 Volume and surface area Cube, cylinder, pyramid . . . . . . . . . . . . . . . . . . . . . 25 Truncated pyramid, cone, truncated cone, sphere . 26 Volumes of composite solids . . . . . . . . . . . . . . . . . 27
' in
m
kg m
1.7 Mass General calculations . . . . . . . . . . . . . . . . . . . . . . . . 27 Linear mass density . . . . . . . . . . . . . . . . . . . . . . . . . 27 Area mass density . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1 m
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
d
y
1.8 Centroids Centroids of lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Centroids of plane areas . . . . . . . . . . . . . . . . . . . . . 28
S S2 S1 s
xs
S
y
x
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
Thermodynamic temperature
Amount of substance
Luminous intensity
Base units
meter
kilogram
second
ampere
kelvin
mole
candela
m
kg
s
A
K
mol
cd
Unit symbol 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 Name
Symbol
Relationship
Symbol
Length, Area, Volume, Angle Length
Œ
meter
m
1m 1 mm 1 km
Area
A, S
MS Volume
V
square meter
m2
are hectare
a ha
cubic meter
m3
liter
ME
Remarks Examples of application
Plane angle (angle)
Solid angle
a, b, g …
—, L
= 10 dm = 100 cm = 1000 mm = 1000 µm = 1000 m
1 inch = 25.4 mm In aviation and nautical applications the following applies: 1 international nautical mile = 1852 m
1 m2
= 10 000 cm2 Symbol S only for cross-sectional = 1 000 000 mm2 areas 1a = 100 m2 1 ha = 100 a = 10 000 m2 Are and hectare only for land 100 ha = 1 km2 1 m3
= 1000 dm3 = 1 000 000 cm3 — 1 = 1 L = 1 dm3 = 10 — d = 3 0.001 m 1 — m = 1 cm3
Mostly for fluids and gases
radian
rad
degrees
°
minutes seconds
* +
1 rad = 1 m/m = 57.2957…° 1 rad is the angle formed by the = 180°/ p intersection of a circle around the center of 1 m radius with an arc of 1 p rad = 60* 1° = m length. 180 In technical calculations instead of 1* = 1°/60 = 60+ a = 33° 17* 27.6+, better use is a = 1+ = 1 */60 = 1°/3600 33.291°.
≈
steradian
sr
1 sr
m
kilogram gram
kg g
1 kg 1g
megagram metric ton
Mg t
= 1 m2 /m2
PE
An object whose extension measures 1 rad in one direction and perpendicularly to this also 1 rad, covers a solid angle of 1 sr.
Mechanics Mass
A
1 metric t = 1000 kg = 1 Mg 0.2 g = 1 ct
Mass in the sense of a scale result or a weight is a quantity of the type of mass (unit kg). Mass for precious stones in carat (ct).
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
= 1000 g = 1000 mg
r
11
1.1 Units of measurement
Units of measurement Quantities and Units (continued) Quantity
Symbol
Unit Name
M Relationship
Symbol
Remarks Examples of application
Mechanics Moment of inertia, 2nd Moment of mass
J
kilogram x square meter
kg · m2 The following applies for a homogenous body: J = r · r 2 · V
Force
F
newton
N
kg · m J The force 1 N effects a change in 1N =1 =1 s2 m velocity of 1 m/s in 1 s in a 1 kg mass. 3 1 MN = 10 kN = 1 000 000 N
newton x meter
N·m
kg · m2 1N·m=1 s2
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
Weight Torque Bending mom. Torsional mom. Momentum
Pressure Mechanical stress
F G, G M M b T
s, t
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 Second moment of area
Energy, Work, Quantity of heat Power Heat flux
I
meter to the fourth power centimeter to the fourth power
E, W joule
m4
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.
MS 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.
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.
P
watt
W
t
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.
1 per second
1/s
1/s
1 per minute
1/min
= 60/min = 60 min–1 1 1/min = 1 min–1 = 60 s
The number of revolutions per unit of time gives the revolution frequency, also called rpm.
meters per second meters per minute kilometers per hour 1 per second radians per second
m/s
1 m/s
Nautical velocity in knots (kn): 1 kn = 1.852 km/h
meters per second squared
m/s2
G
TD
Pressure refers to the force per unit area. For gage pressure the symbol p g is used (DIN 1314). 1 bar = 14.5 psi (pounds per square inch )
1 m4 = 100 000 000 cm4
cm4
P
ME
Time Time, Time span, Duration
Frequency
f, v
Rotational speed, Rotational frequency
n
Velocity
v
Angularvelocity Acceleration
w
a, g
= 60 m/min = 3.6 km/h m/min 1 m/min = 1 m 60 s 1m km/h 1 km/h = 3.6 s 1/s rad/s
w = 2 p · n
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. Symbol g only for acceleration due to gravity. g = 9.81 m/s2 fi 10 m/s 2
S
12
1.1 Units of measurement
Units of measurement M
Quantities and units (continued) Unit Name
Symbol
Quantity
Symbol
Remarks Examples of application
Relationship
Electricity and Magnetism
P
E
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
Electric current
I
Electromotive force Electrical resistance Electrical conductance Specific
resistance Conductivity
TD
MS
1
in
Ω · mm2
m
k k=
1 r
in
m Ω · mm2
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
1J =1W·s=1N·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: P cos j = U · I
The angle between current and voltage in inductive or capacitive load.
Elect. field strength
E Q C L
volts per meter V/m coulomb C F farad H henry
P
watt
Elect. charge inductance Power Effective power
W
Unit Name
Symbol
Quantity
F
Q
, C , Q 1 C = 1 A · 1 s; 1 A · h = 3.6 kC E Q U 1 F = 1 C/V 1 H = 1 V · s/A =
=
I
·
t
In electrical power engineering: Apparent power S in V · A
1 W = 1 J/s = 1 N · m/s =1V·A
Symbol
=
Remarks Examples of application
Relationship
Electricity and Magnetism Thermodynamic 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
Kelvin (K) and degrees Celsius (°C) are used for temperatures and temperature differences. t = T – T 0; T 0 = 273.15 K degrees Fahrenheit (°F): 1.8 °F = 1 °C
joule
J
1J = 1W·s=1N·m
1 kcal ‡ 4.1868 kJ
Celsius temperature
PE
S/m
r=
Frequency
Elect. capacitance
ME
g, k
The motion of an electrical charge is called current. The electromotive force is equal to the potential difference between two points in an electric field. The reciprocal of the electrical resistance is called the electrical conductivity.
Quantity of heat
Q
Net calorific value
H u
1 kW · h = 3 600 000 J = 3,6 MJ
joule per kilogram Joule per cubic meter
J/kg
1 MJ/kg = 1 000 000 J/kg
J/m3 1 MJ/m3 = 1 000 000 J/m3
Thermal energy released per kg fuel minus the heat of vaporization of the water vapor contained in the exhaust gases.
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
1 sq.yd
= 0.8361 m2
1 cu.yd
= 764.6 dm3
1t
= 1000 kg
1 kcal
= 4186.8 Ws
1 acre
= 4046.856 m2
1 gallon — 1 (US) = 3.785
1 short ton
1 kcal
= 1.166 Wh
= 907.2 kg
1 gallon — 1 (UK) = 4.546
1 Karat
= 0.2 g
1 yard (yd)
= 0.9144 m
1 nautical mile = 1.852 km 1 mile
S
Volume
Pressure
= 1.6093 km 1 bar
= 14.5 pound/in2
1 N/mm2 = 145.038 pound/in2
Mass
— 1 barrel = 158.8
Energy, Power
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
cf. DIN 1304-1 (1994-03) Formula symbol
Meaning
Meaning
Formula symbol
M
Meaning
Length, Area, Volume, Angle
Πw h s
r, R d, D A, S V
Length Width Height Linear distance
a, b, g
Radius Diameter Area, Cross-sectional area Volume
W
l
Planar angle Solid angle Wave length
P
Mechanics m m* m+
r
J p p abs p amb p g
Mass Linear mass density Area mass density Density Moment of inertia Pressure Absolute pressure Ambient pressure Gage pressure
F F W, W M T M b
s t e
E
Force Gravitational force, Weight Torque Torsional moment Bending moment Normal stress Shear stress Normal strain Modulus of elasticity
G µ, f W I
W, E W p, E p W k, E k P
n
Shear modulus Coefficient of friction Section modulus Second moment of an area Work, Energy Potential energy Kinetic energy Power Efficiency
TD
Time t T n
f, v v, u
Time, Duration Cycle duration Revolution frequency, Speed
w
Frequency Velocity Angular velocity
a g
a ·
Q, V, q v
Acceleration Gravitational acceleration Angular acceleration Volumetric flow rate
Electricity Q E C I
Electric charge, Quantity of electricity Electromotive force Capacitance Electric current
L R
r g, k
X Z
Inductance Resistance Specific resistance Electrical conductivity
j
N
Reactance Impedance Phase difference Number of turns
MS
Heat T, Q
DT, Dt, Dh t, h a—, a
Thermodynamic temperature Temperature difference Celsius temperature Coefficient of linear expansion
Q
l a
k
· , Q G
Heat, Quantity of heat Thermal conductivity Heat transition coefficient Heat transmission coefficient
a c H net
Heat flow Thermal diffusivity Specific heat Net calorific value
ME
Light, Electromagnetic radiation E
Illuminance
f n
Focal length Refractive index
LP
Acoustic pressure level Sound intensity
I
Q , W
Luminous intensity Radiant energy
Acoustics p c
Acoustic pressure Acoustic velocity
I
N LN
Mathematical symbols
Math. symbol
fi ‡
… 6 =
Ï
def
== <
‰
>
›
+
– · –, /, :, ÷ S
Spoken approx. equals, around, about equivalent to and so on, etc. infinity equal to not equal to is equal to by definition less than less than or equal to greater than greater than or equal to plus minus times, multiplied by over, divided by, per, to sigma (summation)
Loudness Loudness level
PE
cf. DIN 1302 (1999-12)
Math. symbol
,
an
0 3 0 3 æxæ o ø n
º º
º @ ™ 9 Dx º
% ‰
Math. symbol
Spoken
proportional a to the n-th power, the n-th power of a square root of n-th root of
log lg ln e
logarithm (general) common logarithm natural logarithm Euler number (e = 2.718281…)
absolute value of x perpendicular to is parallel to parallel in the same direction
sin cos tan cot
sine cosine tangent cotangent
parallel in the opposite direction angle triangle congruent to
(), [], { }
Spoken
delta x (difference between two values) percent, of a hundred per mil, of a thousand
p AB
£ AB
a*, a+ a1, a2
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. v c 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
The formula symbols (page 13) are wildcards for quantities. When solving mathematical problems, the known quantities with their units are filled in the formulas. Before or during the calculation process, the units are converted in a way that
Formula for cutting velocity
v c = p · d · n
• 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 v c in m/min for d = 200 mm and n = 630/min? v c = 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.
What is the torque M of an electrical motor with a driving power of P = 15 kW and a speed of n = 750/min? M =
M =
9550 · P n
Designated unit
Example:
ME
Numerical value equation for torque
9550 · P 9550 · 15 = 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 =0.5 x +1
x y
2 y
b
–2
S
e u n e v e r r o s t s o c
–1 –1
800000 $ 600000
s s l o
0
Linear function
0 1
2 2
3 2.5
2
2nd example: Cost function and revenue function C t = 60 $/piece · Q + 200 000 $ R = 110 $/piece · Q
=1
3
x
revenue
Q C t R
f i t p r o
total costs
200000 0
1
break-even point (BEP)
400000
y = f(x)
y = m · x + b
m = 0.5
1
A
–2 0
Assigned function
variable costs fixed costs
2000 4000 pieces 6000 quantity
C t Q C f C v
R
0 200 000 0
4 000 440 000 440 000
6 000 560 000 660 000
total costs ∫ dependent variable quantity ∫ independent variable fixed costs ∫ y coordinate section variable costs ∫ gradient of the function revenue ∫ dependent variable
Examples: Cost function
C t = C V · Q + C f 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 Example: formula L = Œ1 + Œ2, transformation to find
1 L = Œ1 + Œ2
æ– Œ1
2 L – Œ1 = Œ1+ Œ2 – Œ1
Œ2
TD
subtract Œ1
3 L – Œ1 = Œ2
perform subtraction
4
invert both sides
Œ2 = L – Œ1
transformed formula
Transformations of products
MS
Example: formula A = Œ · b , transformation to find
1 A = Œ · b
2
æ: b
A Œ · b = b b
Œ
divide by b
3
A =Πb
invert both sides
cancel b
4
Œ=A
transformed formula
b
ME Transformations of fractions Example: formula n =
1 n =
2
Œ
Œ1 + s
Œ Œ1 + s , transformation to find s æ· (Œ1 + s )
Œ · (Œ1 + s ) n · (Œ1 + s ) = (Œ1 + s )
3 n · Œ1 + n · s = Œ
æ– n · Œ1
subtract divide by n
multiply by (Œ1 + s )
4 n · Œ1 – n · Œ1 + n · s = Œ – n · Œ1
cancel (Œ1 + s ) on the right side solve the term in brackets
5 s · n = Œ – n · Œ1 n n
cancel n
subtract – n · Œ1
6 s = Œ – n · Œ1 n
transformed formula
æ: n
PE
Transformations of roots
A
1222222
Example: formula c = a 2 + b 2, transformation to find a a 2 + b 2 1 c = 122222
æ( )2
square equation
4 a 2 = c 2 – b 2
2 c 2 = a 2 + b 2
æ– b 2
subtract b 2
5
3 c 2 – b 2 = a 2 + b 2 – b 2
subtract, invert both sides
6
2 = c 2 – b 2 12a 2 1222222
1222222
a = c 2 – b 2
æ12 extract the root simplify the expression transformed formula
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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
2 = 5 — — = 5 · 10 — 500 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
— = 5 · 10–1 — = 5 — 0.5 d (deciliters)
c
centi
10–2
hundredth
0.25 m = 25 · 10–2 m = 25 cm (centimeters)
m
milli
10–3
thousandth
0.375 A = 375 · 10–3 A = 375 mA (milliamperes)
µ
micro
10–6
millionth
0.000 052 m = 52 · 10–6 m = 52 µm (micrometers)
n
nano
10–9
billionth
0.000 000 075 m = 75 · 10–9 m = 75 nm (nanometers)
p
pico
10–12
trillionth
0.000 000 000 006 F = 6 · 10–12 F = 6 pF (picofarads)
Conversion of units Calculations with physical units are only possible if these units refer to the same base in this calculation. When solving mathematical problems, units often must be converted to basic units, e. g. mm to m, s to h, mm 2 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 = 10 mm = 1000 mm = = 1 km 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 1 = 100 mm = 100 cm = 2 2 1 cm 1 dm
Angle
1 = 60’ = 60’’ = 3600’’ = 1° 1° 1’ 1° 60 s
Volume
1 = 1000 mm = 1000 cm = 3 1 cm 1 dm3
Inch
1 inch = 25.4 mm; 1 mm =
PE
3
3
1 inches 25.4
1st example:
Convert volume V = 3416 mm3 to cm3. Volume V is multiplied by a conversion factor. Its numerator has the unit cm3 and its denominator the unit mm3.
A V =
3416 mm3 =
1 cm3 · 3416 mm3 3416 cm3 = = 3.416 cm 3 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
= 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, L =
Œ2 = 18 mm, Œ3 = 44 mm; L = ?
Multiplying powers
124 mm + 18 mm – 44 mm = (124 + 18 – 44) mm = 98 mm
P
a 2 · a 3 = a 2+3 • Multiplying and dividing
The numerical values and the units correspond to the factors of products. Dividing powers
Example: F 1 · Œ1 = F 2 · Œ2 mit F 1 = 180 N, Œ1 = 75 mm, Œ2 = 105 mm; F 2 = ? F 2 =
F 1 · Œ1
Œ2
a 2 = a 2–3 3 a
180 N · 75 mm N · mm = = 128.57 = 128.57 N 105 mm mm
• Multiplying and dividing powers
Special cases
Powers that have the same base are multiplied or divided by adding or subtracting their exponents.
a –2 =
1 a 2
Example:
W =
MS
A · a 2 with A = 15 cm2, a = 7.5 cm, e = 2.4 cm; W = ? e
W =
TD
a 1 = a
a 0 = 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. P r percentage rate, in percent
P v percent value
Percent value P v
B v base value
=
ME
B v · P r
100%
Example:
Weight of raw part: 250 kg (base value); material loss of 2 % (percentage rate); material loss in kg = ? (percent value) P v
=
· P r 100 %
Bv
=
250 kg · 2 % 100 %
=
PE
5 kg
Interest calculation P A
principle amount accumulated
I r
t interest interest rate per year
=
$ 2800.00; r $ 2800.00 · 6
I
=
=
6
% a
100%
% a
; t
· 0.5 a
=
1/
2
a; I
=
=
$ 4800.00; r
$ 4800.00 · 5.1
=
I
=
5. 1 ; t a
% a
$ 84.00
100 % · 360
=
· 50 d d a
100% · 360
A
1 interest year (1 a) = 360 days (360 d)
%
=
P · r · t =
?
360 d = 12 months 1 interest month = 30 days
2nd Example: P
Interest
I
1st Example: P
period in days, interest period
=
50 d; I ?
$ 34.00
=
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18
1.3 Angels and Triangels
Types of angles, Theorem of intersecting lines, Angles in a triangle, Pythagorean theorem M
Types of angles straight line g 1, g 2 parallel straight lines g
¿ g2
¶ å
P
a, b b, d a, d a, g
Corresponding angles
a = b
corresponding angles
Opposite angles
opposite angles
b = d
alternate angles adjacent angles
Alternate angles
©
g1 g
If two parallels are intersected by a straight line, there are geometrical interrelationships between the resulting angles.
a = d Adjacent angles
a + g = 180°
Theorem of intersecting lines
TD
tto tti
If two intersecting lines are intercepted by a pair of parallels, the resulting segments form equal ratios.
outer torsional stress inner torsional stress
Theorem of intersecting lines d
a2
a 1
a1
a 2
Example:
=
b 1
=
b 2
2
D 2
†to
D = 40 mm, d = 30 mm, tta = 135 N/mm2; tti = ?
†ti
MS
b
D d
1
d
t
ti
2
b
t
=
D
to
t
⇒ tti =
to
a 1
· d
D
b 1
135 N/mm2 · 30 mm =
=
40 mm
=
a 2
b 1
b 2
d
=
b 2 D
101.25 N/mm2
Sum of angles in a triangle a, b, c a, b, g
ME ©
b
a
a = 21°, b = 95°, g = ? � = 180° – a – b = 180° – 21° – 95° = 64°
c
PE
a + b + g = 180°
Example:
¿
å
Sum of angles in a triangle
sides of the triangle angles in the triangle
In every triangle, the sum of the interior angles equals 180°.
Pythagorean theorem
2
a
2
b
b
a
c
2
c
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
c 2 = a 2 + b 2
c
=
a2
2 + b
2nd example:
CNC programm with R = 50 mm and I = 25 mm. K = ?
P2
c2
G 0 3
S
R
P1 K
I
=
a2
R 2 = I 2
X Z
K
=
2 + b + K
2
R 2 – I 2
=
K = 43.3 mm
Length of the sides meeting the right angle a
=
b
=
c 2 – b 2
502 mm2 – 25 2 mm2
c 2 – a 2
19
1.3 Angels and Triangels
Functions of triangles Functions of right triangles (trigonometric functions) hypotenuse (longest side) sides, – b is the adjacent side of a – a is the opposite side of a angles in the triangle, g = 90° notation of sine notation of cosine notation of tangent sine of angle a
c
c hypotenuse
a adjacent side of å
å
a, b, g
b adjacent side of å c hypotenuse
¿
a, b
a adjacent side of ¿
sin cos tan sin a
M Trigonometric functions sine
=
opposite side hypotenuse
cosine
=
adjacent side hypotenuse
tangent
=
opposite side adjacent side
P
Relations applying to angle a:
b opposite side of ¿ 1st example
sin a =
L1 = 150 mm, L2 = 30 mm, L3 = 140 mm; angle a = ?
L3 =140mm
tan a = m m 0 5 1 =
L
L1 + L2 L3
180 mm =
140 mm
= 1.286
Angle � = 52°
1
L
å
ß
m m 0 3 =
F
2
2nd example L1 = 150 mm, L2 = 30 mm, a = 52°; Length of the shock absorber L = ? L =
L1 + L2
L
sin a
180 mm =
sin 52°
=
a b a cos a = tan a = c c b
Relations applying to angle b: sin b =
b a b cos b = tan b = c c a
TD
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.
MS
228.42 mm
Functions of oblique triangles (law of sines, law of cosines) ©
b
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. ∫ opposite angle sin a Side a ∫ opposite angle sin b Side b There are many transformation Hypothenuse c ∫ opposite angle sin g options:
a ¿
å
c
Example:
Fd
Fz
F = 800 N, a = 40°, b = 38°; F z = ?, F d = ?
4 0 }
The forces are calculated with the help of the forces diagram.
1 2 }
F
=
sin a
F
F z =
Diagram of forces
ß = 3
⇒ F z =
F · sin b
sin a
=
F d
sin g
a = b · sin a = c · sin a sin b sin g a c b = · sin b = · sin b sin a sin g
PE
a b c = · sin g = · sin g sin a sin b
= 766.24 N
sin40°
sin a F d =
sin b
800 N · sin38°
F
F
F z
ME
⇒ F d =
800 N · sin102°
Law of cosines F · sin g
sin a
a 2 = b 2 + c 2 – 2 · b · c · cos a b 2 = a 2 + c 2 – 2 · a · c · cos b c 2 = a 2 + b 2 – 2 · a · b · cos g
A
= 1217.38 N
sin40°
8}
å
Fd
} 2 0 1 =
= 4 © 0 }
Fz
1 2 }
The calculation of an angle in degrees (°) Transformation, e. g. or as a circular measure (rad) is done with b 2 c 2 a 2 cos a = + – the help of inverse trigonometric functions, 2 · b · c e. g. arcsine.
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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
s saw cutting width bar length 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
Example: Torsion spring
a
angle at center d diameter
Arc length
Example:
PE
Composite length D outside diameter d m mean diameter Œ1, Œ2 section lengths a angle at center
A
d inside diameter t thickness L composite length
Example (composite length, picture left):
D = 360 mm; t = 5 mm; a = 270°; Œ2 = 70 mm; d m = ?; L = ?
S
Composite length
L = Œ1 + Œ2 + …
