1、SERVOMOTOR SIZING AND APPLICATIONSERVOMOTOR SIZING AND APPLICATIONby Gary Kirckof, P.E.Servomotor Sizing and ApplicationCopyright 2012 International Society of Automation67 Alexander DriveP.O. Box 12277Research Triangle Park, NC 27709All rights reserved.Printed in the United States of America. 10987
2、65432ISBN 978-1-936007-84-4No part of this work may be reproduced, stored in a retrieval system, or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording or otherwise, withoutthe prior written permission of the publisher.NoticeThe information presented in this publi
3、cation is for the general education of the reader.Because neither the author(s) nor the publisher has any control over the use of the informa-tion by the reader, both the author(s) and the publisher disclaim any and all liability of anykind arising out of such use. The reader is expected to exercise
4、 sound professional judgmentin using any of the information presented in a particular application. Additionally, neither theauthor(s) nor the publisher has investigated or considered the effect of any patents on theability of the reader to use any of the information in a particular application. The
5、reader isresponsible for reviewing any possible patents that may affect any particular use of the infor-mation presented.Any references to commercial products in the work are cited as examples only. Nei-ther the author(s) nor the publisher endorses any referenced commercial product. Any trade-marks
6、or tradenames referenced belong to the respective owner of the mark or name. Neitherthe author(s) nor the publisher makes any representation regarding the availability of any ref-erenced commercial product at any time. The manufacturers instructions on use of any com-mercial product must be followed
7、 at all times, even if in conflict with the information in thispublication.Library of Congress Cataloging-in-Publication Data in process v CONTENTSAbout the Author xviiIntroduction xix1 Basic Kinematics 1Introduction 1Rectilinear Motion 2Position and Distance 2Velocity and Speed 3Acceleration and De
8、celeration 4Jerk 4Constant of Integration and Initial Values 4Four Basic Motion Cases 6Velocity Profile 12Common Motion Profiles 14Rotational Motion 49Angular Position 50Angular Velocity 51Angular Acceleration 52Angular Jerk 52Vectors 53Rotational Motion Summary 532 Advanced Kinematics 55Introductio
9、n 55Harmonic Profile 56SERVOMOTOR SIZING AND APPLICATION vi Scaling Normalized Profiles 57Cycloid Profile 58Constructing Profiles 59Time Scaling 60Distance Scaling 62Offsets 63Profile Construction Rules 63Profile Construction Process 67Modified Trapezoid Profile 68General Modified Trapezoid 76Modifi
10、ed Sine Profile 80General Modified Sine Profile 83Poly 5 87Poly 5 Example 89Poly 5 as a Segment 91Electronic Camming 92Pick and Place Example 92Changing Profiles 101Tracking Example 103Recovery 109User Profiles 110Linear Interpolation 111Cubic Spline Interpolation 113Summary 1143 Duty Cycle 115Intro
11、duction 115Peak Velocity 115Peak Torque 116Continuous Torque 116Duty Cycle 117Duty Cycle Example 118Adding or Subtracting Friction 121Effective Torque 121Effective Torque Example 123Combining Effective Torques 124Torque versus Velocity Curves 126Motor Selection 128Torque Margin 129CONTENTS vii Non-D
12、uty-Cycle Conditions 131Dynamic Braking 133Rectifier and DC Bus 133Inverter 134Dynamic Braking Circuit 134Sizing the Dynamic Braking Resistor 135Minimum Stopping Time 139Sharing the DC Bus 140Motor Sizing Steps 1404 Profile Comparison 143Introduction 143Effective Acceleration 143General Trapezoid 14
13、4Scaling Example 146S Curve Segment 147General Bell 148Harmonic 148Cycloid 149Modified Trapezoid 149General Modified Trapezoid 149Modified Sine 149General Modified Sine 149Poly 5 as a Profile 150Profile Comparison 151Recommendations 154Constant Speed 155Coordinated Motion 156Conclusion 1565 Inertia
14、157Introduction 157What is Inertia? 159System of Particles 160Inertia of a Hoop 162Inertia of a Cylinder 163Inertia of a Hollow Cylinder 165Inertia of a Solid Cone 168Inertia of a Solid Sphere 169SERVOMOTOR SIZING AND APPLICATION viii Parallel Axis Theorem 171Example 173Units of Inertia 176Perpendic
15、ular Axis Theorem 177Inertia of a Cylinder about a Perpendicular Axis 178Inertia through a Gear Ratio 181Inertia of a Belt and a Load on a Belt 183Total Reflected Inertia 184Inertia of a Lead Screw 185Servomotor Sizing Procedure 187Conclusion 187Object Inertia Summary 1896 Gear Ratios 199Introductio
16、n 199Gear Ratio Physics 199Position, Speed and Acceleration Changes 200Torque Changes 202Inertia Changes 204Selecting a Gear Ratio 205Speed Matching 206Optimal Gear Ratio 208Stability and Inertia Ratio 212Selection Process 216Backlash 224Common Mechanism Types 226Inline Planetary Gearboxes 226Right
17、Angle Gearboxes 227Worm Gears 229Timing Belts and Pulleys 230Lead Screws 236Rack and Pinions 238Roller Chain 2407 Friction 247Introduction 247Coulomb Friction 248Vibration 251Normal Force 252CONTENTS ix Rolling Friction 254Viscous Friction 256Lead Screws 258Ball Screws 263Disk Brakes 264Stopping Tim
18、e 266Thrust Bearings 268Journal Bearings 271Bushings 274Seals 276Loading 2788 Sizing Steps 281Sizing Procedure 2819 Horizontal Pusher Example 291Introduction 291Application Data 293Product Kinematics 295Load Conditions 296Load Kinematics 297Inertia 297Frictional Torque and Torque Due to Additional L
19、oading 298Preliminary Torque 299Prepare Gearbox Data 300Prepare Servomotor Data 301Select a Servomotor 302Finalize Calculations 304Special Conditions 305Dynamic Braking 30610 Vertical Elevator Example 311Introduction 311Gravitational Force 312Power flow 313Application 316Application Data 316Product
20、Kinematics 319Load Conditions 320SERVOMOTOR SIZING AND APPLICATION x Load Kinematics 321Load Inertia 321Friction and Additional Torque 322Preliminary Torque 322Prepare Gearbox Data 324Prepare Servomotor Data 324Select a Servomotor 325Finalize Calculations 327Special Conditions 329Emergency Stopping
21、329Dynamic Braking 331Motor Brake 333Efficiency Correction 335Motor Brake Revisited 335Motoring versus Regenerating 336Efficiency Variables 336Group 1 337Group 2 339Group 3 340Comparison 34111 Nip Roller Example 343Introduction 343Application Data 344Product Kinematics 348Constant Velocity Case 348N
22、ormal Acceleration Case 349Emergency Stopping Case 350Jogging Case 350Load Conditions 351Load Kinematics 352Load Inertia 352Load Friction and Additional Torque 353Preliminary Torque 353Prepare Gearbox Data 354Prepare Motor Data 354Select a Servomotor 355Finalized Calculations 356CONTENTS xi Dynamic
23、Braking 356Registration Control 35712 Vectors 363Introduction 363Position Vectors 363Vector Addition 365Vector Subtraction 367Polar Representation 367Arc Tangent 368Velocity Vector 370Acceleration Vector 373Loop Closure Method 375Cross Product 377Torque Example 380Solving for Forces and Torques 382L
24、ink Forces and Torques 382Joint Forces 383Solving for Forces and Torques for a System of Linkages Summary 38413 Crank Arm 387Introduction 387Position Vectors 387Quadratic Angle Equation 391Crank Angle 394Arm Angle 395Velocity Vectors 395Relative Velocity 398Acceleration Vectors 400Relative Accelerat
25、ion 404Forces and Torques 405Crank Free-Body Diagram 406Arm Free-Body Diagram 408Slider Free-Body Diagram 410Joint Forces 411Motor Torque 412Common Configurations 413Forward Kinematics 413Test 417SERVOMOTOR SIZING AND APPLICATION xii Angles 418Angular Velocities 418Accelerations 420Peak Torque 421RM
26、S Torque 422General Design Considerations 423Linkage Length 423Friction 42514 Crank Arm Example 427Introduction 427Application Data 427Load Kinematics 431Reverse Kinematics and Torque 436Gear Box Selection 437Peak and RMS Torque 439Motor Conditions 44015 Two Axis Parallel Robot 443Introduction 443Me
27、chanism 444Position Vectors 445Position Limits 450Preventing Envelope Violations 453Velocity Vectors 457Velocities 459Acceleration Vectors 461Accelerations 463Supporting Links 465Forces and Torques 467Link 2 Free-Body Diagram 468Link 3 Free-Body Diagram 470Link 5 Free-Body Diagram 472Link 4 Free-Bod
28、y Diagram 473Link 10 Free-Body Diagram 475Link 8 Free-Body Diagram 477Link 7 Free-Body Diagram 478Link 6 Free-Body Diagram 480Off-Center Loads 482CONTENTS xiii Joint Forces 483System of Equations 484Torque Solution for Motor A 486Torque Solution for Motor E 488Motor Sizing Process 490Step 1 Gather A
29、pplication Data 490Step 2 Determine Load Kinematics 491Step 3 Calculate Reverse Kinematics 491Step 4 Calculate Accelerations 491Step 5 Calculate Motor Torques 491Summary 49216 Two Axis parallel Robot Example 493Introduction 493Application Data 494Robot Mechanics 495Product 496Kinematics 496Tool Poin
30、ts 496Motion Profiles 499Reverse Kinematics and Torque 507Gearbox and Motor Selection 510Appendix 1 Units and Symbols 513Unit Abbreviations 516SI Unit Prefixes 517Greek Alphabet 518Unit Conversion 519Angular Position 519Energy 520Force 520Inertia 520Length 520Mass and Weight 521Moment (Torque) 521Po
31、wer 521Temperature 522Volume 523Weight and Force 523Appendix 2 Shapes 525Triangle 525Circle 526Trapezoid 527Ellipse 528Parabola 529Cylinder 530Cone 531Sphere 532Toroid 532Paraboloid 533Appendix 3 Common Integrals and Differentials 535Terminology 535Integrals of X 535Derivatives of X 537Trigonometric
32、 Integrals 539Trigonometric Derivatives 540Exponentials 541Chain Rule 541Appendix 4 Trigonometric Identities 543Basic Relationships 543Pythagorean Identities 544Sum of Angles 544Sum to Product 545Product to Sum 545Power Reduction 545Half and Double Angle 546Phase Shift 546Negation 547Law of Sines 54
33、7Law of Cosines 548Law of Tangents 548Inverse Functions 548Power Series 548Complex Plane 549Quadratic (1) 549Quadratic (2) 550CONTENTS xv Appendix 5 Timing Diagram Example 551Appendix 6 Cubic Spline 553Introduction 553Creating a Cubic Spline 555Reduce Quantity of Unknowns 556Coefficient Summary 558G
34、overning Equation 558Boundary Conditions 559Solving System of Equations 566Natural Cubic Spline Example 572Glossary 579Index 589About the AuthorGary Kirckof, P.E. has spent the bulk of his thirty-year career working with auto-mated equipment and assembly lines. Servomotor Sizing and Application is a
35、direct result of that experience. He is both a registered mechanical and electricalengineer and works for Beckhoff Automation as an application engineer.His career started by writing and debugging the programs that ran the equipment.He would talk to the mechanical engineers who designed the equipmen
36、t to findout how it was supposed to work, the sensors they had specified, and so forth. Hewould then write the code for the control system and then bring the machines tolife. It became quite clear to him that after power was first applied to themachines, every mistake, omission, and problem that occ
37、urred during the con-cept, design, and assembly process became his problem. There were many prob-lems because the equipment was large custom machines and assembly lines. Hehad no desire to deal with these problems and had every desire to make his lifeeasier, so to eliminate as many potential problem
38、s as he could, he became involvedwith every step of the process. One of the problems he often encountered early in his career was undersized ser-vomotors. Servomotors were relatively new to industrial machinery and not manypeople were familiar enough with them to properly select them. He was discour
39、-aged at the lack of available information so he started to compile every piece ofinformation he could find. This book is the result of his research efforts and hisexperience working with servomotors, and he wishes for it to fill the still-existinginformation void and to be a useful resource. xix IN
40、TRODUCTIONServomotor sizing is the process of selecting a servomotor for a particular applica-tion. The size of a servomotor or how big a motor is can be judged in terms of howmuch torque it can provide because the torque output of a servomotor is generallyproportional to its physical size. A servom
41、otor is rated by how fast it can spin aswell as how much torque it can provide. The process of selecting a motor is todetermine how much speed and torque an attached mechanism needs, then topick a motor capable of providing both. This book brings together all the materialneeded to analyze an attache
42、d mechanism with the goal of selected a suitablymotor. Typical mechanisms are also discussed along the way to show how servo-motors are applied. Chapters are devoted to individual topics and several com-plete examples are provided to bring it all together.Torque is proportional to acceleration, so t
43、he required acceleration of the motormust be determined. This is found by performing kinematic calculations. Kinemat-ics is the study of motion and it provides the relationships between time, position,velocity and acceleration. You start by choosing how the load moves, then relate itto how the motor
44、 moves. The load is generally the part of the mechanism you areinterested in, such as the product to be transferred or the motion of a tool. Judgment on your part is needed because there are many ways to get from point Ato point B. How the load moves from one point to another is called its motion pr
45、o-file and there are many motion profiles to choose from. A profile can be chosen tomake the movement of the load as smooth as possible or to minimize the totalamount of energy required, for instance. The choice of motion profile impactsmotor size. It thus becomes a matter of balancing how you wish
46、to move the loadwith how big a motor you will have to use. To this end the most popular motionprofiles are discussed and compared to help you decide how to move your load.SERVOMOTOR SIZING AND APPLICATION xx The torque output of a servomotor varies over time as it moves its attached mech-anism and h
47、ow the torque varies with time, referred to as a torque profile, can bepredicted. Motor selection depends on knowing the peak and effective torquerequired to move the mechanism and both values can be found from the torqueprofile. The peak torque is the greatest value of torque and a motor must have
48、apeak torque rating greater than this value or the motor will simply not be able tomove the mechanisms as we desire it to. The temperature of a motor is propor-tional to torque output and its important that a motor never exceeds its tempera-ture rating. Short circuits in a motors copper windings are
49、 a common failuremode when the temperature rating is exceeded. Motor temperature increaseswhen the motor has a high torque output and the temperature decreases at loweroutput levels. The effective torque determined from the torque profile helps to pre-dict the temperature a motor will reach even when torque varies with time. Themaximum motor temperature is expressed as a continuous torque rating and amotor must have a continuous torque rating greater than the required effectivetorque. The chapter on duty cycles
