Kinematics and Dynamics
of Mechanical Systems
Implementation in MATLAB®
and Simmechanics®
Kinematics and Dynamics
of Mechanical Systems
Implementation in MATLAB®
and Simmechanics®
Second Edition
Kevin Russell, Qiong Shen, Raj S. Sodhi
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Library of Congress Cataloging-in-Publication Data
Names: Russell, Kevin (Mechanical engineer), author. | Shen, Qiong
(Mechanical engineer), author. | Sodhi, R. S. (Raj S.), author.
Title: Kinematics and dynamics of mechanical systems : implementation in
MATLAB and SimMechanics / Kevin Russell, Qiong Shen, Raj S. Sodhi.
Description: Boca Raton : CRC Press, Taylor & Francis, 2018. | Includes index.
Identifiers: LCCN 2018022923 | ISBN 9781138584044 (hardback)
Subjects: LCSH: Machinery, Kinematics of--Computer simulation. | Machinery,
Dynamics of—Computer simulation. | MATLAB. | SimMechanics.
Classification: LCC TJ175 .R87 2018 | DDC 621.01/5311—dc23
LC record available at https://lccn.loc.gov/2018022923
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface............................................................................................................................................ xiii
Authors......................................................................................................................................... xvii
1. Introduction to Kinematics...................................................................................................1
1.1 Kinematics.......................................................................................................................1
1.2 Kinematic Chains and Mechanisms...........................................................................2
1.3 Mobility, Planar, and Spatial Mechanisms.................................................................3
1.4 Types of Mechanism Motion........................................................................................5
1.5 Kinematic Synthesis......................................................................................................7
1.6 Units and Conversions..................................................................................................8
1.7 Software Resources........................................................................................................9
1.8 Summary.........................................................................................................................9
References................................................................................................................................ 10
Additional Reading................................................................................................................ 11
2. Mathematical Concepts in Kinematics............................................................................. 13
2.1 Introduction.................................................................................................................. 13
2.2 Complex Numbers and Operations.......................................................................... 13
2.2.1 Complex Number Forms............................................................................... 13
2.2.2 Complex Number Addition........................................................................... 15
2.2.3 Complex Number Multiplication and Differentiation.............................. 17
2.3 Vector and Point Representation............................................................................... 20
2.4 Linear Simultaneous Equations, Matrices, and Matrix Operations.....................22
2.4.1 Linear Simultaneous Equation Systems and Matrices..............................22
2.4.2 Matrix Transpose, Addition, Subtraction, and Multiplication................. 23
2.4.3 The Identity Matrix and Matrix Inversion.................................................. 26
2.5 Intermediate and Total Spatial Motion..................................................................... 29
2.6 General Transformation Matrix................................................................................. 33
2.7 Summary....................................................................................................................... 36
References................................................................................................................................ 37
Additional Reading................................................................................................................ 37
Problems................................................................................................................................... 37
3. Fundamental Concepts in Kinematics.............................................................................. 41
3.1 Types of Planar and Spatial Mechanisms................................................................ 41
3.1.1 Planar Four-Bar Mechanism......................................................................... 41
3.1.2 Slider-Crank Mechanism............................................................................... 41
3.1.3 Geared Five-Bar Mechanism.........................................................................42
3.1.4 Planar Multiloop Six-Bar Mechanisms........................................................44
3.1.5 Spatial Four-Bar Mechanisms....................................................................... 45
3.2 Links, Joints, and Mechanism Mobility.................................................................... 46
3.3 Number Synthesis........................................................................................................ 49
3.4 Grashof’s Criteria and Transmission Angle............................................................. 50
3.5 Circuit Defect................................................................................................................ 53
v
vi
Contents
3.6 Mechanism Inversion..................................................................................................54
3.7 Passive Degree of Freedom and Paradoxes.............................................................. 55
3.8 Summary....................................................................................................................... 56
References................................................................................................................................ 57
Problems................................................................................................................................... 58
4. Kinematic Analysis of Planar Mechanisms.....................................................................63
4.1 Introduction..................................................................................................................63
4.2 Numerical Solution Method for Two Simultaneous Equations............................64
4.3 Link Velocity and Acceleration Components in Planar Space..............................64
4.4 Four-Bar Mechanism Analysis.................................................................................. 66
4.4.1 Displacement Equations................................................................................ 66
4.4.2 Velocity Equations.......................................................................................... 67
4.4.3 Acceleration Equations................................................................................... 68
4.4.4 Kinematics of Coupler Locations of Interest............................................... 70
4.4.5 Instant Center, Centrodes, and Centrode Generation............................... 76
4.5 Slider-Crank Mechanism Analysis...........................................................................80
4.5.1 Displacement Equations................................................................................80
4.5.2 Velocity Equations.......................................................................................... 81
4.5.3 Acceleration Equations...................................................................................83
4.5.4 Centrode Generation...................................................................................... 89
4.6 Geared Five-Bar Mechanism Analysis..................................................................... 91
4.6.1 Displacement Equations................................................................................ 91
4.6.2 Velocity Equations.......................................................................................... 92
4.6.3 Acceleration Equations................................................................................... 93
4.6.4 Kinematics of Intermediate Link Locations of Interest............................. 95
4.7 Watt II Mechanism Analysis...................................................................................... 97
4.8 Stephenson III Mechanism Analysis....................................................................... 100
4.8.1 Displacement Equations.............................................................................. 100
4.8.2 Velocity Equations........................................................................................ 103
4.8.3 Acceleration Equations................................................................................. 104
4.8.4 Kinematics of Intermediate Link Locations of Interest........................... 105
4.9 Time and Driver Angular Velocity.......................................................................... 108
4.10 Mechanism Configurations...................................................................................... 108
4.11 Constructing Cognates............................................................................................. 109
4.12 Planar Mechanism Kinematic Analysis and Modeling in Simmechanics®....... 112
4.13 Summary..................................................................................................................... 117
References.............................................................................................................................. 118
Additional Reading.............................................................................................................. 118
Problems................................................................................................................................. 118
5. Dimensional Synthesis...................................................................................................... 131
5.1 Introduction................................................................................................................ 131
5.2 Branch and Order Defects........................................................................................ 133
5.3 Planar Four-Bar Motion Generation: Three Precision Positions......................... 135
5.4 Order- and Branch-Defect Elimination.................................................................. 140
5.5 Path Generation versus Motion Generation........................................................... 144
5.6 Stephenson III Motion Generation: Three Precision Positions............................ 145
5.7 Planar Four-Bar Function Generation: Three Precision Points........................... 149
Contents
vii
5.8
5.9
lanar Four-Bar Function Generation: FSPs and MSPs........................................ 153
P
Mechanism Dimensions: From Dimensional Synthesis to
Kinematic Analysis��������������������������������������������������������������������������������������������������� 158
5.10 Summary..................................................................................................................... 163
References.............................................................................................................................. 164
Additional Reading.............................................................................................................. 165
Problems................................................................................................................................. 165
6. Static Force Analysis of Planar Mechanisms................................................................. 175
6.1 Introduction................................................................................................................ 175
6.2 Static Loading in Planar Space................................................................................. 176
6.3 Four-Bar Mechanism Analysis................................................................................ 177
6.4 Slider-Crank Mechanism Analysis......................................................................... 182
6.5 Geared Five-Bar Mechanism Analysis................................................................... 185
6.6 Watt II Mechanism Analysis.................................................................................... 190
6.7 Stephenson III Mechanism Analysis....................................................................... 196
6.8 Planar Mechanism Static Force Analysis and Modeling in SimMechanics®..... 201
6.9 Summary..................................................................................................................... 204
References.............................................................................................................................. 205
Additional Reading.............................................................................................................. 206
Problems................................................................................................................................. 206
7. Dynamic Force Analysis of Planar Mechanisms..........................................................223
7.1 Introduction................................................................................................................223
7.2 Dynamic Loading in Planar Space.......................................................................... 224
7.3 Four-Bar Mechanism Analysis................................................................................ 224
7.4 Slider-Crank Mechanism Analysis......................................................................... 230
7.5 Geared Five-Bar Mechanism Analysis................................................................... 233
7.6 Watt II Mechanism Analysis.................................................................................... 238
7.7 Stephenson III Mechanism Analysis....................................................................... 243
7.8 Mass Moment of Inertia and Computer-Aided Design Software....................... 248
7.9 Planar Mechanism Dynamic Force Analysis and Modeling in
Simmechanics®���������������������������������������������������������������������������������������������������������� 250
7.10 Summary..................................................................................................................... 256
References.............................................................................................................................. 256
Additional Reading.............................................................................................................. 256
Problems................................................................................................................................. 256
8. Design and Kinematic Analysis of Gears...................................................................... 271
8.1 Introduction................................................................................................................ 271
8.2 Gear Types.................................................................................................................. 272
8.3 SPUR-Gear Nomenclature and Relationships of Mating Gears......................... 274
8.3.1 Spur-Gear Nomenclature............................................................................ 274
8.3.2 Pressure Angle and Involute Tooth Profile............................................... 277
8.3.3 Gear Center Distance and Contact Ratio................................................... 280
8.3.4 Gear-Tooth Interference and Undercutting.............................................. 282
8.3.5 Backlash.......................................................................................................... 283
8.4 Helical-Gear Nomenclature.....................................................................................284
8.5 Gear Kinematics......................................................................................................... 287
viii
Contents
8.5.1 Spur Gears and Gear Trains........................................................................ 287
8.5.2 Planetary Gear Trains.................................................................................. 291
8.5.3 Rack and Pinion Gears................................................................................. 295
8.5.4 Helical Gears................................................................................................. 296
8.5.5 Bevel Gears.................................................................................................... 298
8.5.6 Worm Gears................................................................................................... 301
8.6 Summary.....................................................................................................................305
References..............................................................................................................................305
Additional Reading..............................................................................................................306
Problems.................................................................................................................................306
9. Design and Kinematic Analysis of Disk Cams............................................................. 311
9.1 Introduction................................................................................................................ 311
9.2 Follower Types............................................................................................................ 312
9.3 Follower Motion......................................................................................................... 313
9.3.1 Rise, Fall, and Dwell..................................................................................... 313
9.3.2 Displacement, Velocity, Acceleration, and Jerk......................................... 315
9.3.3 Constant Velocity Motion............................................................................ 315
9.3.4 Constant Acceleration Motion.................................................................... 317
9.3.5 Simple Harmonic Motion............................................................................ 320
9.3.6 Cycloidal Motion........................................................................................... 322
9.3.7 Polynomial Motion....................................................................................... 325
9.4 Disk Cam Design and Pressure Angle................................................................... 331
9.5 Summary..................................................................................................................... 336
References.............................................................................................................................. 337
Additional Reading.............................................................................................................. 337
Problems................................................................................................................................. 338
10. Kinematic Analysis of Spatial Mechanisms..................................................................343
10.1 Introduction................................................................................................................343
10.2 RRSS Mechanism Analysis......................................................................................344
10.2.1 Displacement Equations..............................................................................344
10.2.2 Velocity Equations........................................................................................346
10.2.3 Acceleration Equations................................................................................. 347
10.3 RSSR Mechanism Analysis....................................................................................... 350
10.3.1 Displacement Equations.............................................................................. 350
10.3.2 Velocity Equations........................................................................................ 351
10.3.3 Acceleration Equations................................................................................. 352
10.4 Four-Revolute Spherical Mechanism Analysis...................................................... 355
10.5 Planar Four-Bar Kinematic Analysis Using RRSS and RSSR Kinematic
Equations�������������������������������������������������������������������������������������������������������������������� 359
10.6 Spatial Mechanism Kinematic Analysis and Modeling in Simmechanics®....... 362
10.7 Summary.....................................................................................................................364
References.............................................................................................................................. 365
Problems................................................................................................................................. 365
11. Introduction to Robotic Manipulators............................................................................ 373
11.1 Introduction................................................................................................................ 373
11.2 Terminology and Nomenclature............................................................................. 374
Contents
ix
11.3 Robotic Manipulator Mobility and Types.............................................................. 375
11.4 The General Transformation Matrix....................................................................... 377
11.5 Forward Kinematics.................................................................................................. 381
11.5.1 Definition and Application.......................................................................... 381
11.5.2 P-P-P................................................................................................................ 381
11.5.3 R-P-P................................................................................................................ 383
11.5.4 R-R-P............................................................................................................... 385
11.5.5 R-R-R............................................................................................................... 387
11.5.6 R-R-C............................................................................................................... 389
11.6 Inverse Kinematics..................................................................................................... 392
11.6.1 Definition and Application.......................................................................... 392
11.6.2 P-P-P................................................................................................................ 392
11.6.3 R-P-P................................................................................................................ 394
11.6.4 R-R-P............................................................................................................... 396
11.6.5 R-R-R............................................................................................................... 397
11.6.6 R-R-C............................................................................................................... 399
11.7 Robotic Manipulator Kinematic Analysis and Modeling in
Simmechanics®��������������������������������������������������������������������������������������������������� 401
11.8 Summary..................................................................................................................... 402
References.............................................................................................................................. 402
Additional Reading.............................................................................................................. 403
Problems................................................................................................................................. 403
Appendix A: User Information and Instructions for MATLAB®..................................... 409
A.1 Required MATLAB Toolkits..................................................................................... 409
A.2 Description of MATLAB Operators and Functions.............................................. 409
A.3 Preparing and Running Files in MATLAB and Operations in
SimMechanics������������������������������������������������������������������������������������������������������� 409
A.4 Rerunning MATLAB and SimMechanics Files with Existing *.csv Files......... 413
A.5 Minimum Precision Requirement for Appendix File User Input....................... 413
Appendix B: User Instructions for Chapter 4 MATLAB® Files......................................... 415
B.1 Planar Four-Bar Mechanism.................................................................................... 415
B.2 Planar Four-Bar Fixed and Moving Centrode Generation.................................. 415
B.3 Slider-Crank Mechanism.......................................................................................... 417
B.4 Geared Five-Bar Mechanism (Two Gears).............................................................. 419
B.5 Geared Five-Bar Mechanism (Three Gears)........................................................... 419
B.6 Watt II Mechanism..................................................................................................... 420
B.7 Stephenson III Mechanism.......................................................................................422
Appendix C: User Instructions for Chapter 6 MATLAB® Files........................................425
C.1 Planar Four-Bar Mechanism....................................................................................425
C.2 Slider-Crank Mechanism..........................................................................................425
C.3 Geared Five-Bar Mechanism (Two Gears).............................................................. 428
C.4 Geared Five-Bar Mechanism (Three Gears)........................................................... 429
C.5 Watt II Mechanism.....................................................................................................430
C.6 Stephenson III Mechanism....................................................................................... 431
Appendix D: User Instructions for Chapter 7 MATLAB® Files........................................ 435
D.1 Planar Four-Bar Mechanism.................................................................................... 435
D.2 Slider-Crank Mechanism.......................................................................................... 435
x
Contents
D.3
D.4
D.5
D.6
G
eared Five-Bar Mechanism (Two Gears).............................................................. 437
G
eared Five-Bar Mechanism (Three Gears)........................................................... 438
W
att II Mechanism..................................................................................................... 439
S
tephenson III Mechanism....................................................................................... 441
Appendix E: User Instructions for Chapter 9 MATLAB® Files.........................................447
E.1 S, V Profile Generation and Cam Design: Constant Velocity Motion................ 447
E.2 S, V, A Profile Generation and Cam Design: Constant Acceleration
Motion����������������������������������������������������������������������������������������������������������������� 447
E.3 S, V, A, J Profile Generation and Cam Design: Simple Harmonic Motion......... 449
E.4 S, V, A, J Profile Generation and Cam Design: Cycloidal Motion....................... 451
E.5 S, V, A, J Profile Generation and Cam Design: 3-4-5 Polynomial Motion.......... 451
E.6 S, V, A, J Profile Generation and Cam Design: 4-5-6-7 Polynomial Motion...... 451
Appendix F: User Instructions for Chapter 10 MATLAB® Files....................................... 453
F.1 RRSS Mechanism....................................................................................................... 453
F.2 RSSR Mechanism....................................................................................................... 453
Appendix G: User Instructions for Chapter 11 MATLAB® Files...................................... 457
G.1 R-P-P Robotic Manipulator Forward Kinematics.................................................. 457
G.2 R-R-P Robotic Manipulator Forward Kinematics................................................. 457
G.3 R-R-R Robotic Manipulator Forward Kinematics................................................. 457
G.4 R-R-C Robotic Manipulator Forward Kinematics................................................. 459
G.5 R-P-P Robotic Manipulator Inverse Kinematics.................................................... 460
G.6 R-R-P Robotic Manipulator Inverse Kinematics.................................................... 460
G.7 R-R-R Robotic Manipulator Inverse Kinematics.................................................... 461
G.8 R-R-C Robotic Manipulator Inverse Kinematics................................................... 462
Appendix H: User Instructions for Chapter 4 MATLAB® and
SimMechanics® Files.......................................................................................................... 465
H.1 Planar Four-Bar Mechanism....................................................................................465
H.2 Slider-Crank Mechanism..........................................................................................465
H.3 G
eared Five-Bar Mechanism (Two Gears).............................................................. 467
H.4 G
eared Five-Bar Mechanism (Three Gears)........................................................... 469
H.5 W
att II Mechanism..................................................................................................... 470
H.6 S
tephenson III Mechanism....................................................................................... 472
Appendix I: User Instructions for Chapter 6 MATLAB® and SimMechanics® Files.........475
I.1
Planar Four-Bar Mechanism.................................................................................... 475
I.2
Slider-Crank Mechanism.......................................................................................... 475
I.3
Geared Five-Bar Mechanism (Two Gears).............................................................. 479
I.4
Geared Five-Bar Mechanism (Three Gears)........................................................... 479
I.5
Watt II Mechanism.....................................................................................................480
I.6
Stephenson III Mechanism....................................................................................... 482
Appendix J: User Instructions for Chapter 7 MATLAB® and SimMechanics® Files.........487
J.1
Planar Four-Bar Mechanism.................................................................................... 487
J.2
Slider-Crank Mechanism.......................................................................................... 488
J.3
Geared Five-Bar Mechanism (Two Gears).............................................................. 490
J.4
Geared Five-Bar Mechanism (Three Gears)........................................................... 494
J.5
Watt II Mechanism..................................................................................................... 494
J.6
Stephenson III Mechanism....................................................................................... 497
Contents
xi
Appendix K: User Instructions for Chapter 10 MATLAB® and
SimMechanics® Files.......................................................................................................... 501
K.1 RRSS Mechanism....................................................................................................... 501
K.2 RSSR Mechanism....................................................................................................... 501
Appendix L: User Instructions for Chapter 11 MATLAB® and
SimMechanics® Files............................................................................................................ 505
L.1 R-P-P Robotic Manipulator Forward Kinematics.................................................. 505
L.2 R-R-P Robotic Manipulator Forward Kinematics.................................................505
L.3 R-R-R Robotic Manipulator Forward Kinematics................................................. 506
L.4 R-R-C Robotic Manipulator Forward Kinematics................................................. 507
Index.............................................................................................................................................. 509