SERVOMOTOR SIZING AND APPLICATION by Gary Kirckof, P.E.
Basic Contents About the Author xvii Introduction xix 1 Kinematics 1 Introduction 1 Rectilinear Motion 2 Position and Distance 2 Velocity and Speed 3 Acceleration and Deceleration 4 Jerk 4 Constant of Integration Four Basic Motion Cases 6 Velocity Profile 12 Common Motion Profiles 14 Rotational Motion 49 Angular Position 50 Angular Velocity 51 Angular Acceleration 52 Angular Jerk 52 Vectors 53 Rotational Motion Summary 53 and Initial Values 4 2 Advanced Kinematics 55 Introduction 55 Harmonic Profile 56 v
General Modified Sine Profile 83. Servomotor Sizing and Application Scaling Normalized Profiles 57 Cycloid Profile 58 Constructing Profiles 59 Time Scaling 60 Distance Scaling 62 Offsets 63 Profile Construction Rules 63 Profile Construction Process 67 Modified Trapezoid Profile 68 General Modified Trapezoid 76 Modified Sine Profile 80 Poly 5 87 Poly 5 Example 89 Poly 5 as a Segment 91 Electronic Camming 92 Pick and Place Example 92 Changing Profiles 101 Tracking Example 103 Recovery 109 User Profiles 110 Linear Interpolation 111 Cubic Spline Interpolation 113 Summary 114 Duty Cycle 115 Introduction 115 Peak Velocity 115 Peak Torque 116 Continuous Torque 116 Duty Cycle 117 Duty Cycle Example 118 Adding or Subtracting Friction 121 Effective Torque 121 Effective Torque Example 123 Combining Effective Torques 124 Torque versus Velocity Curves 126 Motor Selection 128 Torque Margin 129
Contents NonDutyCycle Conditions 131 Dynamic Braking 133 Rectifier and DC Bus 133 Inverter 134 Dynamic Braking Circuit 134 Sizing the Dynamic Braking Resistor 135 Minimum Stopping Time 139 Sharing the DC Bus 140 Motor Sizing Steps 140 4 Profile Comparison 143 Introduction 143 Effective Acceleration 143 General Trapezoid 144 Scaling Example 146 S Curve Segment 147 General Bell 148 Harmonic 148 Cycloid 149 Modified Trapezoid 149 General Modified Trapezoid 149 Modified Sine 149 General Modified Sine 149 Poly 5 as a Profile 150 Profile Comparison 151 Recommendations 154 Constant Speed 155 Coordinated Motion 156 Conclusion 156 5Inertia 157 Introduction 157 What is Inertia? 159 System of Particles 160 Inertia of a Hoop 162 Inertia of a Cylinder 163 Inertia of a Hollow Cylinder 165 Inertia of a Solid Cone 168 Inertia ofa Solid Sphere 169 vii
Friction Servomotor Sizing and Application Parallel Axis Theorem 171 Example 173 Units of Inertia 176 Perpendicular Axis Theorem 177 Inertia of a Cylinder about a Perpendicular Axis 178 Inertia through a Gear Ratio 181 Inertia of a Belt and a Load on a Belt 183 Total Reflected Inertia 184 Inertia of a Lead Screw 185 Servomotor Sizing Procedure 187 Conclusion 187 Object Inertia Summary 189 6 Gear Ratios 199 Introduction 199 Gear Ratio Physics 199 Position, Speed and Acceleration Changes 200 Torque Changes 202 Inertia Changes 204 Selecting a Gear Ratio 205 Speed Matching 206 Optimal Gear Ratio 208 Stability and Inertia Ratio 212 Selection Process 216 Backlash 224 Common Mechanism Types 226 Inline Planetary Gearboxes 226 Right Angle Gearboxes 227 Worm Gears 229 Timing Belts and Pulleys 230 Lead Screws 236 Rack and Pinions 238 Roller Chain 240 j 7 247 Introduction 247 Coulomb Friction 248 Vibration 251 Normal Force 252 viii
Contents Rolling Friction 254 Viscous Friction 256 Lead Screws 258 Ball Screws 263 Disk Brakes 264 Stopping Time 266 Thrust Bearings 268 Journal Bearings 271 Bushings 274 Seals 276 Loading 278 8 Sizing Steps 281 Sizing Procedure 281 9 Horizontal Pusher Example 291 Introduction 291 Application Data 293 Product Kinematics 295 Load Conditions 296 Load Kinematics 297 Inertia 297 Frictional Torque and Torque Due to Additional Loading 298 Preliminary Torque 299 Prepare Gearbox Data 300 Prepare Servomotor Data 301 Select a Servomotor 302 Finalize Calculations 304 Special Conditions 305 Dynamic Braking 306 10 Vertical Elevator Example Introduction 311 Gravitational Force 312 Power flow 313 Application 316 Application Data 316 Product Kinematics 319 Load Conditions 320 311 ix
Servomotor Sizing and Application Load Kinematics 321 Load Inertia 321 Friction and Additional Torque 322 Preliminary Torque 322 Prepare Gearbox Data 324 Prepare Servomotor Data 324 Select a Servomotor 325 Finalize Calculations 327 Special Conditions 329 Emergency Stopping 329 Dynamic Braking 331 Motor Brake 333 Efficiency Correction 335 Motor Brake Revisited 335 Motoring versus Regenerating 336 Efficiency Variables 336 Group 1 337 Group 2 339 Group 3 340 Comparison 341 Nip Roller Example 343 Introduction 343 Application Data 344 Product Kinematics 348 Constant Velocity Case 348 Normal Acceleration Case 349 Emergency Stopping Case 350 Jogging Case 350 Load Conditions 351 Load Kinematics 352 Load Inertia 352 Load Friction and Additional Torque 353 Preliminary Torque 353 Prepare Gearbox Data 354 Prepare Motor Data 354 Select a Servomotor 355 Finalized Calculations 356
Contents Dynamic Braking 356 Registration Control 357 12 Vectors 363 Introduction 363 Position Vectors 363 Vector Addition 365 Vector Subtraction 367 Polar Representation 367 Arc Tangent 368 Velocity Vector 370 Acceleration Vector 373 Loop Closure Method 375 Cross Product 377 Torque Example 380 Solving for Forces and Torques 382 Link Forces and Torques 382 Joint Forces 383 Solving for Forces and Torques for a System of Linkages Summary 384 13 Crank Arm 387 Introduction 387 Position Vectors 387 Quadratic Angle Equation 391 Crank Angle 394 Arm Angle 395 Velocity Vectors 395 Relative Velocity 398 Acceleration Vectors 400 Relative Acceleration 404 Forces and Torques 405 Crank FreeBody Diagram 406 Arm FreeBody Diagram 408 Slider FreeBody Diagram 410 Joint Forces 411 Motor Torque 412 Common Configurations 413 Forward Kinematics 413 Test 417 xi
Two Servomotor Sizing and Application Angles 418 Angular Velocities 418 Accelerations 420 Peak Torque 421 RMS Torque 422 General Design Considerations 423 Linkage Length 423 Friction 425 14 Crank Arm Example 427 Introduction 427 Application Data 427 Load Kinematics 431 Reverse Kinematics and Torque 436 Gear Box Selection 437 Peak and RMS Torque 439 Motor Conditions 440 15 Axis Parallel Robot 443 Introduction 443 Mechanism 444 Position Vectors 445 Position Limits 450 Preventing Envelope Violations 453 Velocity Vectors 457 Velocities 459 Acceleration Vectors 461 Accelerations 463 Supporting Links 465 Forces and Torques 467 Link 2 FreeBody Diagram 468 Link 3 FreeBody Diagram 470 Link 5 FreeBody Diagram 472 Link 4 FreeBody Diagram 473 Link 10 FreeBody Diagram 475 Link 8 FreeBody Diagram 477 Link 7 FreeBody Diagram 478 Link 6 FreeBody Diagram 480 OffCenter Loads 482 xii
Two Gather Determine Calculate Calculate Calculate Contents Joint Forces 483 System of Equations 484 Torque Solution for Motor A 486 Torque Solution for Motor E 488 Motor Sizing Process 490 Step 1 Step 2 Step 3 Step 4 Step 5 Summary 492 Application Data 490 Load Kinematics 491 Reverse Kinematics 491 Accelerations 491 Motor Torques 491 16 Axis parallel Robot Example 493 Introduction 493 Application Data 494 Robot Mechanics 495 Product 496 Kinematics 496 Tool Points 496 Motion Profiles 499 Reverse Kinematics and Torque 507 Gearbox and Motor Selection 510 Appendix 1 Units and Symbols 513 Unit Abbreviations 516 SI Unit Prefixes 517 Greek Alphabet 518 Unit Conversion 519 Angular Position 519 Energy 520 Force 520 Inertia 520 Length 520 Mass and Weight 521 Moment (Torque] 521 Power 521 Temperature 522 Volume 523 Weight and Force 523 xiii
Appendix 2 Shapes Triangle 525 Circle 526 Trapezoid 527 Ellipse 528 Parabola 529 Cylinder 530 Cone 531 Sphere 532 Toroid 532 Paraboloid 533 525 Appendix 3 Common Integrals and Differentials 535 Terminology 535 Integrals of X 535 Derivatives of X 537 Trigonometric Integrals 539 Trigonometric Derivatives 540 Exponentials 541 Chain Rule 541 Appendix 4 Trigonometric Identities 543 Basic Relationships 543 Pythagorean Identities 544 Sum of Angles 544 Sum to Product 545 Product to Sum 545 Power Reduction 545 Half and Double Angle 546 Phase Shift 546 Negation 547 Law of Sines 547 Law of Cosines 548 Law of Tangents 548 Inverse Functions 548 Power Series 548 Complex Plane 549 Quadratic (1) 549 Quadratic (2) 550
Cubic Contents Appendix 5 Timing Diagram Example 551 Appendix 6 Spline 553 Introduction 553 Creating a Cubic Spline 555 Reduce Quantity of Unknowns 556 Coefficient Summary 558 Governing Equation 558 Boundary Conditions 559 Solving System of Equations 566 Natural Cubic Spline Example 572 Glossary 579 Index 589 xv