Bonfiglioli Riduttori S.p.A. Gruppo Bonfiglioli Bonfiglioli Riduttori S.p.A. (Eds.) Gear Motor Handbook With contributions by D. W. Dudley, J. Sprengers, D. Schroder, H. Yamashina With 325 Figures , Springer Bonfiglioli Riduttori S.p.A. (Eds.) Via Giovanni XXIII, 7/A Lippo di Calderara di Reno, Bologna / Italy Darle W. Dudley Prof Bon.-Eng. Jacques Sprengers 17777 Camino Murrillo, Rue E. Mahaim 98, San Diego, CA 92128, USA B-4100 Seraing, Belgium Prof Dierk Schroder Prof Bajime Yamashina Technische Universitat Munchen, Uji-Shi Lehrstuhl fUr Elektrische Antriebstechnik, Nanryo-Cho ArcisstraBe 21, 5 Chome 7 D-80333 Munchen, FRG Kyoto, Japan ISBN-13:978-3-642-79591-6 e-ISBN-13:978-3-642-79589-3 DOl: 10.1007/978-3-642-79589-3 This. work is subject to copyright. AIl rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only pennitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Springer-Verlag Berlin Heidelberg 1995 Softcover reprint of the hardcover 1st edition 1995 The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publishers cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Typesetting: camera-ready by author SPIN: 10495689 62/3020 -5 4 3 2 1 0 -Printed on acid-free paper Darle W. Dudley, Manager of Gear Technology Jacques Sprengers, President Isorrc 60; Development with Dudley Technical Group, Inc.; Chevalier de l'Ordre de Leopold, Commandeur Honorary Member of AGMA de l'Ordre de Leopold II, Insigne d'argent de Laureat du Travail Prof. Dr.-Ing. Dr.-Ing. h.c. Dierk SchrOder, Prof. Hajime Yamashina, Chair for Production Chair for Power Engineering and Electric Drives, Eng. at the Department of Precision Mechanics, Technical University of Munich Kyoto University, and Affiliate Professor at London Business School Preface In these years of constant growth and further development for our company, research and development has become more and more important, and has allowed us to be at the forefront in our business sector, where innovation is the obvious and decisive factor. It has therefore been consistent with our everyday business philosophy to involve ourselves deeply in writing and printing this handbook, which is designed to recognize the capacity and hard work of all employees working successfully in the Bonfiglioli Group. The book is intended to be a concrete contribution by Bonfiglioli Riduttori S.p.A. to the development and application of power transmissions. The book is addressed to all who have technical dealings with power transmissions, from university students to engineers active in the workplace. For this reason we have invited the cooperation of four prestigious professionals - Darle W. Dudley, Jacques Sprengers, Dierk SchrOder, and Hajime Yamashina - in the knowledge that only through the cooperation of the leading specialists in the field of power transmissions could we develop a truly useful and helpful handbook. It has been hard work, but we are sure the reader's appreciation will amply reward our efforts. Clementino Bonfiglioli President Table of Contents Part I Editor: Dade W. Dudley 1 Introduction................................................................................................ 3 1.1 History of the Art of Gear Making .... ...... ..... .... ... ........ ............... ........ ... 3 1.2 Development of Gear Technology... .... .... ............ ....................... ........ ... 7 1.3 Development of Gear Manufacturing Organizations ............................. 7 1.4 Developments at Bonfiglioli Riduttori................................................... 8 1.5 The content of This Technical Handbook......... ... ... ................... ..... ... .... 11 Part II Editor: Jacques Sprengers 1 The Dynamics of Solids.. ..... ... ........ .... ......... ......... .............. ................ ........ 21 1.1 Introduction ........................................................................................... 21 1.2 Gravity Centre of Solids ....................................................................... 21 1.3 Movements of Solids .......... ...... .................. ...... .............. .......... ............ 23 1.4 Rectilinear Translating Movement .................... .............. ........ .............. 23 1.5 Movement of Rotation........................................................................... 23 1.6 Mechanical Energy, Work and Power ................................................... 23 1.6.1 Introduction................................................................................... 23 1.6.2 Potential Energy............. ............ .................... ............................... 23 1.6.3 Kinetic Energy .............................................................................. 24 1.6.4 Work in the Uniform Rectilinear Movement ................................ 26 1.6.5 Work in the Uniform Movement of Rotation................................ 26 1.6.6 Power in the Uniform Rectilinear Movement ............................... 27 1.6.7 Power in the Movement of Rotation ............................................. 27 1.7 Inertia..................................................................................................... 28 1.7.1 Introduction................................................................................... 28 1.7.2 Inertia of Solids During Translation............................................. 28 1.7.3 Balance of Rotors. Imbalances...................................................... 30 1.7.4 Inertia During Accelerated Rotation. Moments of Inertia of Solids 31 X Table of Contents 1.8 Friction .................................................................................................. 34 1.8.1 Fundamental Equation .................................................................. 34 1.8.2 Angular Value of the Coefficient of Friction. ............................... 34 1.8.3 Wedge Effect ................................................................................ 35 1.8.4 Friction of a Roller on its Axis...................................................... 37 1.8.5 Friction of a Belt around its Pulley............................................... 38 1.8.6 Adhesion ....................................................................................... 40 1.8.7 Friction on disks............................................................................ 41 1.8.8 Values of some Coefficients of Friction........................................ 42 1.9 Rolling Resistance..... ..... ..... ... ...... ...... ..... ......... ..... .......... ...................... 43 1.10 Torque Necessary for a Roller of Translation...................................... 44 1.11 Cranking Torque.................................................................................. 44 1.12 Kinematic Chain. ..... .............. ...... ................. ........... ..... ..... ..... ..... ... ..... 45 1.12.1 Definition... .................... ..... ......... ........................ ..... ..... ..... ..... ... 45 1.12.2 Torques ....................................................................................... 45 1.12.3 Moments of Inertia...................................................................... 45 1.12.4 Elastic Constants of Shafts.. ..... ......... ... ................ ..... .......... ........ 46 1.12.5 Application of a Lifting Winch on Bridge Crane ........................ 47 2 Strength of Materials.... ..... .............. ...... ..... ......... ........................ ... .......... 48 2.1 Single Traction or Compression.... ..... ......... ... ................ ..... ..... ..... ........ 48 2.2 Shearing ................................................................................................. 49 2.3 Simple Bending ..... .............. ......... ..... ...... ..... .... ..... ..... ..... .................. .... 50 2.3.1 Plane Bending ............................................................................... 50 2.3.2 Left Bending ................................................................................. 52 2.4 Torsion .................................................................................................. 53 2.5 Buckling ................................................................................................ 54 2.6 Composition of Simple Stresses... ...... ..... ..... .... ..... ..... ..... ...................... 56 Part III Editor: Jacques Sprengers 1 Stresses and Materials .............................................................................. 61 1.1 Introduction .. ... ..... ... ... ..... ...... ................. ...... ... ..... ...... ........ ....... ......... ... 61 1.2 Stresses .................................................................................................. 61 1.3 Materials: Properties and Features......................................................... 62 1.3.1 Permissible Static Stress ............................................................... 62 1.3.2 Fatigue. Endurance Limit.............................................................. 63 1.4 Materials................................................................................................ 69 Table of Contents XI 1.4.1 General Features ........................................................................... 69 1.4.2 Ferrous Materials .......................................................................... 69 1.4.3 Bronzes ......................................................................................... 73 2 Geometry and Gears .................................................................................. 74 2.1 General Features.................................................................................... 74 2.2 Involute to a Circle ............................................................................... . 75 2.3 Elementary Geometric Wheel. Meshing Theory ................................... . 77 2.4 Standard Basic Rack Tooth Profile ....................................................... . 80 2.5 Geometrical Wheel -Basic Rack Profile .............................................. . 80 2.6 Tooth Thickness ................................................................................... . 82 2.7 Influence of the Rack Shift Factor. Undercut and Meshing Interference 84 2.8 Reference Centre Distance. Modified Centre Distance ......................... 86 2.9 Contact Ratio ......................................................................................... 88 2.10 Calculation of an Angle Starting from its Involute ............................. . 91 3 Mechanical Gears...................................................................................... 92 3.1 General Spur Features ........................................................................... 92 3.2 Cylindrical Wheels ................................................................................ 92 3.2.1 Definition ...................................................................................... 92 3.2.2 Geometry....................................................................................... 92 3.2.3 Standardized Basic Profile and Modules ...................................... 93 3.2.4 Facewidth...................................................................................... 94 3.2.5 Undercut........................................................................................ 94 3.2.6 Tooth Thickness............................................................................ 94 3.2.7 Internal Toothing .......................................................................... 95 3.3 Parallel Spur Gears ................................................................................ 96 3.4 Helical Gears. Involute Helicoid ........................................................... 98 3.4.1 Helical Wheel............................................................................... 100 3.4.2 Helical Parallel Gear Pairs ............................................................ 105 3.4.3 Sliding ........................................................................................... 110 3.4.4 Forces............................................................................................ 115 3.4.5 Gear Ratio -Speed Ratio .............................................................. 117 3.4.6 Normal Pressure............................................................................ 118 3.4.7 Rack Shift Factor .......................................................................... 119 3.4.8 Gears that Reduce Speed/Gears that Increase Speed .................... 120 3.4.9 Tooth Modifications...................................................................... 121 3.4.10 Mechanical Losses in Gears........................................................ 123 3.5 Bevel Gear Pair ..................................................................................... 124 3.5.1 Definition ...................................................................................... 124 3.5.2 Equivalent Gears. Trestgold's Hypothesis .................................... 127 3.5.3 Constant Tooth Depth Bevel Gear ................................................ 129 xn Table of Contents 3.5.4 Constant Tooth Depth Gear .......................................................... 130 3.5.5 Forces on Bevel Gear Pairs........................................................... 131 3.5.6 Dimensions of the Blanks Forming the Gears ............................... 132 3.6 Worms and Worm Wheels..................................................................... 135 3.6.1 Definition ...................................................................................... 135 3.6.2 Main Dimension............................................................................ 135 3.6.3 Efficiency of Worms and Worm Wheels. Reversing .................... 138 3.6.4 Tooth Profile................................................................................. 139 3.6.5 Forces............................................................................................ 140 3.7 Single Planetary Gear Train................................................................... 142 3.7.1 Description.................................................................................... 142 3.7.2 Kinematic Relation ....................................................................... 143 3.7.3 Forces............................................................................................ 144 3.7.4 Advantages Derived from the Use of Planetary Train Gears ........ 145 4 Gear Measuring.......................................................................................... 146 4.1 Measurement of the Tooth Thickness .................................................... 146 4.1.1 Base Tangent Measurement .......................................................... 146 4.1.2 Normal Chordal Tooth Thickness Measurement .......................... 148 4.1.3 Measuring by Balls and Pins......................................................... 150 4.2 Pitch Measurement ................................................................................ 151 4.3 Profile Measurements ............................................................................ 154 4.4 Measurement of the Helix Deviation ..................................................... 157 4.5 Composite Tangential Deviation ........................................................... 158 4.6 Composite Radial Deviation (Measurement on the two Flanks)............ 159 4.7 Eccentricity ............................................................................................ 161 4.8 Precision in Gear Blank......................................................................... 162 4.9 Centre Distance Deviation and Axis Parallelism ................................... 163 4.10 Information on Standards .................................................................... 163 4.11 Backlash .............................................................................................. 164 4.12 Contact Pattern Measurement .............................................................. 167 5 Gear Calculation ........................................................................................ 168 5.1 Aspects of Teeth After Operation .......................................................... 168 5.2 Parallel Gear and Bevel Gear Pair Load Capacity................................. 169 5.2.1 Prevention of Pitting (Contact Stress) ........................................... 169 5.2.2 Stress Applied to the Tooth Root .................................................. 170 5.3 Standard ISO 6336 (or DIN 3990) ........................................................ 172 5.3.1 Introduction ................................................................................... 172 5.3.2 Influence Factors........................................................................... 173 5.3.3 Contact Stress Calculation ............................................................ 180 5.3.4 Calculation of the Tooth Bending Strength ................................... 185 5.3.5 Allowable Stress Number.............................................................. 191 Table of Contents XIII 5.4 Standard ISO 10300 (Bevel Gear Pairs)................................................ 191 5.4.1 Introduction................................................................................... 191 5.4.2 Influence Factors........................................................................... 192 5.4.3 Contact Area ................................................................................. 193 5.4.4 Contact Stress Calculation ............................................................ 195 5.4.5 Calculation of Tooth Bending Strength......................................... 198 5.4.6 Materials ....................................................................................... 199 5.5 Calculation with a Load Spectrum ......................................................... 200 5.5.1 Complex Calculation ..................................................................... 200 5.5.2 Approximate Calculation .............................................................. 200 5.6 Worm and Worm Wheel Calculation .................................................... 201 5.6.1 General Features ........................................................................... 201 5.6.2 Load Capacity for Contact Stress .................................................. 201 5.6.3 Wear Prevention ........................................................................... 203 5.6.4 Tooth Strength .............................................................................. 204 5.6.5 Standard Information .................................................................... 205 5.6.6 Reversing and Efficiency .............................................................. 205 6 Shafts ........................................ ................................................................... 207 6.1 Definition and Function ............... .......................................................... 207 6.2 Shafts and Forces in Gears .................................................................... 207 6.3 Study of the Theoretical Stresses............................ ..... .......................... 207 6.3.1 Bending ......................................................................................... 207 6.3.2 Torsion .......................................................................................... 212 6.3.3 Normal Stress................................................................................ 213 6.3.4 External Compression ................................................................... 213 6.4 Real Stresses (Stress Concentration Factor -Form Factor) ................... 213 6.5 Static Allowable Stress ................................................ ....................... ... 214 6.5.1 Static Breaking Stress due to Traction .......................................... 214 6.5.2 Elastic Limit ................................................................................. 214 6.5.3 Allowable Static Stress ................................................................. 215 6.6 Allowable Stress for an Unlimited Duration .......................................... 215 6.6.1 Endurance (Endurance Factor yd ................................................ 215 6.6.2 Factor Modifying Endurance ........................................................ 216 6.6.3 Allowable Stress for Unlimited Life Duration .............................. 217 6.7 Allowable Stress .............. ........................ ......... ........... .......................... 217 6.8 Stress Concentration Factor ................................................................... 218 6.8.1 Definition ...................................................................................... 218 6.8.2 Notch Sensibility Factor ................................................................ 219 6.8.3 Notch Gradient.............................................................................. 219 6.8.4 Notch Factor ................................................................................. 219 6.8.5 Values of the Allowable Stress Concentration Coefficient and Notch Gradient ....................................................................... 220