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 |  موضوع: كتاب Mechanics of Mechanisms and Machines السبت 22 أغسطس 2020, 12:31 am | |
| 
أخوانى فى الله
أحضرت لكم كتاب
Mechanics of Mechanisms and Machines
Ilie Talpasanu and Alexandru Talpasanu
و المحتوى كما يلي :
Contents
Preface xvii
Acknowledgments xix
Authors . xxi
1. Background 1
1.1 Links and Links Numbering .1
1.2 Joints and Joints Labeling 1
1.3 Graph Representation of a Mechanism .4
1.3.1 Graph .4
1.3.2 Labeling of Nodes and Edges 5
1.3.3 Digraph 6
1.3.4 Paths .6
1.3.5 Open Paths 7
1.3.6 Closed Paths (Cycles) .7
1.3.7 Tree and Spanning Tree 9
1.3.8 Matrix Description of a Digraph . 11
1.3.9 Incidence Nodes–Edges Matrix, G 11
1.3.10 Reduced Incidence Nodes–Edges Matrix, G 12
1.3.11 The Path Matrix, Z . 13
1.3.12 Spanning Tree Matrix, T . 15
1.3.13 Cycle Basis Incidence Matrix, C . 15
1.3.14 Cycle Matroid Fundamentals 17
1.4 Number of Independent Cycles in a Mechanism for Planar
and Spatial Mechanisms 18
1.5 Mobility of Planar Mechanisms 19
1.6 Mobility of Spatial Mechanisms . 19
References . 19
2. Kinematics of Open Cycle Mechanisms 21
2.1 Link and Joint Labeling, Frames, Home Position of
Mechanism, Mechanism’s Digraph, and Mechanism’s Mobility 21
2.1.1 Link and Joint Labeling 21
2.1.2 Home Position of Mechanism 23
2.1.3 Mechanism’s Digraph .23
2.1.4 Mechanism’s Mobility . 24
2.2 Direct and Inverse Analysis: Frame Orientation and Position
for the Spatial Open Cycle Mechanisms 24
2.3 Incidental and Transfer (IT) Notation 25
2.3.1 Notation for Joint Displacement Based on Incidental
Digraph’s Edge .25viii Contents
2.3.2 Notation for Frames Based on Digraph’s Nodes .25
2.4 Relative Frames Orientation and Relative Rotation Matrix 26
2.4.1 Relative Rotation Matrix about x
m-Axis: IT Notation .26
2.4.1.1 Relative Rotation Matrix about x
m .27
2.4.2 Relative Rotation Matrix about ym-Axis: IT Notation .27
2.4.2.1 Relative Rotation Matrix about ym-Axis 28
2.4.3 Relative Rotation Matrix about z
m-Axis: IT Notation .29
2.4.3.1 Relative Rotation Matrix about z
m-Axis 30
2.4.4 Properties of a Relative Rotation Matrix 31
2.5 Open Cycle Mechanisms: The Relative Rotation Matrices
along the Tree 32
2.5.1 Direct and Inverse Relative Rotation Matrix . 32
2.6 Additional Frames on the Same Link 33
2.7 The Absolute Rotation Matrix for Links’ Orientation .33
2.8 Direct Links’ Orientation Analysis for Open Cycle
Mechanisms .34
2.8.1 Example of Direct Links’ Orientation for a Spatial
Mechanism with 4 DOF 37
2.9 Inverse Links’ Orientation Analysis along a Closed Path
(Cycle): The Matroid Method .44
2.9.1 Independent Equations Generated from Entries in
Cycle Basis Matrix (Cycle Matroid) .45
2.9.2 The Task Orientation Matrix 46
2.9.3 Example of Inverse Orientation Analysis for an
Open Cycle Spatial Mechanism with 4 DOF .47
2.9.4 Example of Inverse Orientation Analysis for an
Open Cycle Spatial Mechanism with 5 DOF .48
2.9.5 Example of Inverse Orientation Analysis for an
Open Cycle Spatial Mechanism with 6 DOF .50
2.10 Direct Positional Analysis: Governing Equations for Open
Cycle Mechanisms 53
2.10.1 Transformation of Vector Components between
Frames 53
2.10.2 Linear Displacement at Prismatic, Cylindrical, and
Helical Joints .54
2.10.3 Linear Displacement at Revolute, Spherical, and
Meshing Joints 54
2.10.4 Constraint Equations for Angular and Linear
Displacements 55
2.10.5 Translation Vectors between Frame Origins and
Position Vectors for Frame Origins along the
Open Path 55
2.10.6 The End-Effector Position Vector .58
2.10.7 Equations for Direct Positional Analysis .58
2.10.8 Joint Position Matrix, r 59Contents ix
2.10.9 COM Position Vectors .59
2.10.10 Example of Direct Positional Analysis for a Spatial
Mechanism with 4 DOF 59
2.10.11 Simulations for a Spatial Mechanism with 4 DOF 64
2.10.11.1 Input SW Simulation 64
2.10.11.2 Output from SW Simulation 64
2.10.11.3 Output from Engineering Equation Solver
(EES) Calculation .66
2.10.12 Example of Direct Positional Analysis for a Spatial
Mechanism with 5 DOF 69
2.10.13 Example of Direct Positional Analysis for a Spatial
Mechanism with 6 DOF 70
2.11 Inverse Positional Analysis: Governing Equations for Open
Cycle Mechanisms: The Task Position Vector .71
2.11.1 Example of Inverse Positional Analysis for a Spatial
Mechanism with 4 DOF 72
2.11.2 Example of Inverse Positional Analysis for a Spatial
Mechanism with 5 DOF 73
2.12 The System of Combined Equations for Inverse Orientation
and Positional Analysis 73
2.13 The Matroid Method: Equations Based on Latin Matrix and
Cycle Matroid Entries .75
2.13.1 The Latin Matrix 75
2.13.2 Algorithm for Automatic Generation of Latin Matrix
Based on Digraph Matrices 77
2.13.3 Example of Matroid Method on Inverse Positional
Analysis for a Spatial Mechanism with 4 DOF .80
2.13.3.1 Equations for Inverse Positional Analysis 81
2.13.4 Example Solution for Inverse Orientation and
Positional Equations for a 4 DOF Mechanism .84
2.13.4.1 Using EES for Inverse Orientation and
Positional Analysis of the TRRT 4 DOF
Robotic Mechanism .84
2.13.4.2 EES for Direct Positional Analysis of the
TRRT 4 DOF Robotic Mechanism 91
2.13.5 Conclusions .93
2.14 The IT Method of Relative Homogeneous Matrices:
Combined Equations for Direct Orientation and Positional
Analysis 93
2.14.1 Absolute Homogeneous Matrix for Link
(Node Digraph), Rm 94
2.14.2 Relative IT Homogeneous Matrix for Joint
(Edge Digraph), ITYZ 94
2.14.3 Relations between Absolute and Relative
Homogeneous Matrices 96x Contents
2.14.4 Direct Orientation and Positional Combined
Equations on Open Cycle Mechanisms 96
2.14.5 End-Effector Absolute Homogeneous Matrix .98
2.14.6 Example for Combined Equations on a 4 DOF
Open Cycle Mechanism: Method of Relative
Homogeneous Matrices 99
2.15 Inverse Orientation and Positional Combined Equations
along a Closed Path: The Homogeneous Matrix Method . 106
2.15.1 The Direct and Inverse Sign of Relative IT Matrices 107
2.15.2 The Inverse of Homogeneous Matrix . 107
2.15.3 The Task Absolute Homogeneous Matrix 108
2.15.4 Example for Orientation and Positional Analysis
of a 4 DOF Robotic Mechanism without Vision:
Introduction to Robot Programming 109
2.15.5 Orientation and Positional Analysis of a Robotic
Mechanism with Vision 124
2.15.6 Example for Orientation and Positional Analysis of a
4 DOF Robotic Mechanism with Vision .125
2.16 Direct Orientation and Positional Analysis for Planar Open
Cycle Mechanisms 128
2.16.1 Governing Equation for Links’ Orientation for
Planar Open Cycle Mechanisms . 128
2.16.2 Relations between Absolute and Relative Angular
Displacements 129
2.16.3 The Path Matrix and Its Transposed, ZT . 130
2.16.4 Example of Simulation for Planar Open Cycle
Manipulator with 3 DOF 131
2.16.4.1 Link and Joint Labeling . 131
2.16.4.2 Home Position of Mechanism 131
2.16.4.3 Mechanism’s Digraph for Open Cycle is a
Spanning Tree . 132
2.16.4.4 Notation for Frames Based on Digraph’s
Nodes . 133
2.16.4.5 Mechanism’s Mobility . 133
2.16.4.6 Constraint Equations for Angular and
Linear Displacements 134
2.16.4.7 Relation between Absolute and Relative
Angular Displacements . 135
2.16.4.8 The Relative and Absolute Rotation
Matrices .136
2.16.5 Direct Positional Analysis: Position Vectors of Frame
Origins and End-Effector 138
2.16.5.1 Position Vector Matrix . 139
2.16.5.2 End-Effector Position Vector . 139
2.16.6 Center of Mass Position Vectors . 140Contents xi
2.16.7 The Matroid Method: Equations for Inverse
Orientation and Positional Analysis for Planar Open
Cycle Mechanisms . 141
2.16.7.1 Equations for Inverse Orientation . 141
2.16.7.2 Equations for Inverse Positional Analysis 142
2.16.7.3 Solution of Nonlinear System of Equations . 144
2.16.8 Application for Inverse Analysis: The Required
Manipulator’s Joint Displacements to Place the EndEffector E in Three Task Orientation Positions . 144
2.16.8.1 EES for Inverse Orientation and Positional
Analysis of the TRT 3 DOF Manipulator 145
2.16.8.2 EES for Direct Positional Analysis of the
TRT 3 DOF Manipulator 148
2.16.9 The IT Method of Relative Homogeneous Matrices:
Absolute Homogeneous Matrix and Relative
Homogeneous Matrix for Planar Mechanisms . 152
2.16.9.1 Absolute Homogeneous Matrix for Planar
Mechanisms, Rm . 152
2.16.9.2 IT Relative Homogeneous Matrix for Planar
Mechanisms, ITYZ . 152
2.16.9.3 Planar Mechanisms: Relations between
Absolute and Relative Homogeneous
Matrices . 153
2.16.10 Example for Orientation and Positional Analysis of a
3 DOF Planar Manipulator . 153
2.16.10.1 The Inverse of Homogeneous Matrix 158
2.17 Velocity Analysis . 159
2.17.1 Direct Angular Velocity Analysis for Open Cycle
Spatial Mechanisms . 159
2.17.2 Automatic Generation of Mobile Links’ Angular
Velocities from the Path Matrix . 162
2.17.3 Example of Direct Angular Velocity Analysis for
Open Cycle TRRT Spatial Mechanism with 4 DOF 162
2.17.4 Example of Direct Angular Velocity Analysis for
Open Cycle TRRTR Spatial Mechanism with 5 DOF . 166
2.17.5 Example of Direct Angular Velocity Analysis for
Open Cycle TRRTRT Spatial Mechanism with 6 DOF . 169
2.17.6 The Matroid Method for Inverse Angular Velocity
Analysis on a Closed Path 172
2.17.7 Cycle Basis Matrix (Matroid) 173
2.17.8 The Relative Angular Velocity Matrix, ? j 173
2.17.9 Inverse Angular Velocity Analysis Equations for
TRRTR Spatial Mechanism 176
2.17.10 Inverse Angular Velocity Analysis Equations for
TRRTRT Spatial Mechanism 178xii Contents
2.17.11 Direct Linear Velocity Analysis for Open Cycle
Spatial Mechanisms 179
2.17.12 Automatic Generation of All Mobile Links’ Linear
Velocities from the Path Matrix .184
2.17.13 Example of Direct Linear Velocity Analysis for Open
Cycle TRRT Spatial Mechanism with 4 DOF 184
2.17.14 Inverse Linear Velocity Analysis of Open Cycle
Mechanisms with Equation Functions of Absolute
Angular Velocities .190
2.17.15 Example of Inverse Velocity Analysis for Open
Cycle TRRT Spatial Mechanism with Equation
Functions of Absolute Angular Velocities .194
2.17.16 The Inverse Linear Velocity Analysis: Governing
Equation Functions of Relative Angular Velocities 196
2.17.16.1 The Spanning Tree Matrix, T 198
2.17.16.2 The Analogy to Moment of a Force and
Couple from Statics .199
2.17.16.3 The Jacobean Matrix from the Combined
Equations for Inverse Angular and Inverse
Linear Velocities .200
2.17.16.4 Example of Inverse Velocity Analysis for
Open Cycle TRRT Spatial Mechanism
with Equation Functions of Relative
Angular Velocities 200
2.17.17 Combined Equations for Inverse Velocity Analysis
Based on Twists .203
2.17.17.1 Twists for Joints with Single and Multiple
DOF 203
2.17.17.2 Geometric Jacobean Based on Twists along
the Path in Tree .207
2.17.17.3 Example of Inverse Velocity Analysis for
Open Cycle TRRT Spatial Mechanism
with Equation Based on Twists 208
2.17.18 Example of Velocity Analysis for Planar Open Cycle
Mechanisms with Equation Functions of Absolute
Velocities .209
2.17.18.1 Inverse Velocity Analysis of the TRT 3
DOF Manipulator 211
2.17.18.2 Singularities for Inverse Velocity Analysis 212
2.17.19 Example of Velocity Analyses for Planar Open Cycle
Mechanisms with Equation Functions of Twists 212
2.17.19.1 Capability of Motion for the TRT
Manipulator .213
2.17.19.2 EES for Direct Velocity Analysis of the
TRT 3 DOF Manipulator .214Contents xiii
2.18 Velocity Analysis of Planar Open Cycle Mechanisms with
All Revolute Joints . 218
Problems 218
References .256
3. Kinematics of Single and Multiple Closed Cycle Mechanisms . 257
3.1 Coordinate Systems for Planar Mechanism 257
3.2 Enumeration of Planar Mechanisms Based on the Number
of Cycles 258
3.2.1 Parallel Axes Gear Trains with Gear and Revolute
Joints 260
3.3 Position Analysis for Single-Cycle Planar Mechanisms
with Revolute Joints 262
3.3.1 The Incidence Nodes–Edges Matrix, G 264
3.3.2 The Cycle Basis Matroid Matrix, C 265
3.3.3 Joint Position Vectors Matrix, rj: .265
3.3.4 Digraph Joint Position Matrix, rc,j .266
3.3.5 Latin Matrix Method for Positional Analysis 267
3.3.6 Centers of Mass Position Vector Matrix, rGm 272
3.3.7 Center of Mass to Joint Position Matrix, LGj . 273
3.3.8 Absolute Links’ Orientation, ?n . 274
3.3.9 Relative Links’ Orientation, ?j 274
3.3.10 Relative Links’ Orientation from Digraph, ?c,j 276
3.3.11 Transmission Angle . 276
3.3.12 Input to Output Relation .277
3.3.13 Dead Centers 277
3.3.14 Coupler-Point Curves 278
3.3.15 Mechanism Branches 279
3.3.16 Grashof’s Criterion for the Four-Bar Mechanisms
and Mechanism Inversions 279
3.4 Single-Cycle Planar Mechanisms with Revolute and
Prismatic Joints’ Position Analysis .290
3.4.1 The Planar Crank Slider Mechanism 290
3.4.2 Example: The Planar RRTR Mechanism 304
3.5 Multiple-Cycle Planar Mechanisms with Revolute and
Prismatic Joints’ Position Analysis . 318
3.6 Planar Mechanisms with Cams 327
3.6.1 Background . 327
3.6.2 Input–Output Relation 330
3.6.3 Equations for Cam Contour .335
3.6.4 Cam with Constant Velocity Rise or Constant
Velocity Fall 338
3.6.5 Cam with Constant Acceleration Rise or Constant
Acceleration Fall .345
3.6.6 Cam with Harmonic Motion Rise or Fall .346xiv Contents
3.6.7 Cam with Cycloidal Motion Rise or Fall 347
3.7 Velocity Analysis of Single-Cycle Planar Mechanisms .347
3.7.1 Velocity Analysis for Single-Cycle Planar
Mechanisms with Revolute Joints: Example: The
Four Bar Mechanism . 352
3.7.2 Velocity Analysis for Single-Cycle Planar
Mechanisms with Revolute and Prismatic Joints:
Example: The Crank Slider Mechanism .359
3.8 Velocity Analysis for Multiple-Cycle Planar Mechanisms
with Revolute and Prismatic Joints 367
3.9 Gears . 375
3.9.1 Parallel Axes Epicyclic Gear Trains . 375
3.9.1.1 Mobility Formula for Gear Trains Based on
the Number of Links and Cycles .377
3.9.1.2 Equations Based on Absolute Angular
Velocities: The Matroidal Method for Gear
Trains 380
3.9.1.3 Gears’ Number of Teeth 384
3.9.2 Gear Trains with the Fixed Parallel Axes (GT) 384
3.9.2.1 Equations Based on Absolute Angular
Velocities: The Matroidal Method for GT . 387
3.9.2.2 GT Velocity Ratio 390
3.9.2.3 Gears’ Number of Teeth 390
3.9.3 Bevel Gear Trains . 391
3.9.3.1 Equations Based on Twists: The Matroidal
Method for BGT 391
3.9.3.2 Twist Velocity Matroidal Matrix 400
3.9.3.3 Absolute Angular Velocities of Gears,
Planets, and Carriers 405
3.9.3.4 Gears’ Number of Teeth 406
3.9.3.5 Automatic Generation of BGT Equations .408
Problems 409
References .441
4. Dynamic and Static Analysis of Mechanisms 443
4.1 Direct Angular Acceleration Analysis for Open Cycle
Mechanisms .443
4.1.1 The Joint Relative Angular Acceleration 443
4.1.2 The Joint Axial Angular Acceleration, ?Z .444
4.1.3 Constraint Equations for Axial Angular Acceleration .444
4.1.4 The Joint Complemental Angular Acceleration, ?Z com 445
4.1.5 Links’ Absolute Angular Acceleration Matrix, ?m 446
4.2 Governing Equation for Links’ Absolute Angular
Accelerations 446Contents xv
4.3 Matroid Method for Inverse Angular Acceleration Analysis
on Closed Cycle Mechanisms . 452
4.3.1 Cycle Basis Matrix Assigned to Relative Angular
Accelerations . 452
4.3.2 The Relative Angular Acceleration Matrix, ? j . 452
4.4 Governing Equation for Links’ Absolute Linear
Accelerations 454
4.4.1 Direct Linear Acceleration Analysis for Open Cycle
Spatial Mechanisms . 457
4.4.2 Example of Direct Linear Acceleration Analysis for
Open Cycle TRRT Spatial Mechanism with 4 DOF 458
4.4.3 The Matroid Method for Linear Acceleration
Analysis of Single- and Multiple-Cycle Planar
Mechanisms 464
4.4.4 Acceleration Analysis for Single-Cycle Planar
Mechanisms with Revolute Joints . 467
4.4.5 Acceleration Analysis for Multiple-Cycle Planar
Mechanisms 473
4.5 Governing Equations in Dynamics of Mechanisms .484
4.5.1 The Governing Force Equations for Open Cycle
Mechanisms 484
4.5.2 The Incidental and Transfer-IT Method on Dynamic
Forces and Differential Equations for Open Cycle
Mechanisms 486
4.5.3 The Incidence Nodes-Edges Matrix 487
4.5.4 The Reduced Incidence Nodes-Edges Matrix, G .487
4.5.5 The Path Matrix, Z .489
4.5.6 Relation for G and Z Matrices 489
4.5.7 Equations for Reaction Forces on Open Cycle
Mechanisms: The IT-Resistant Force .490
4.5.8 The Evaluation of IT Forces 491
4.6 The IT Method on Dynamic Moments and Differential
Equations for Open Cycle Mechanisms 492
4.6.1 Equations for Reaction Moment on Open Cycle
Mechanisms: The IT-Resistant Moment . 494
4.6.2 The Position Vector Skew-Symmetric Matrix,
?
G ( m a × L) and Za m × (L?) . 494
4.6.3 The Evaluation of IT Moments 497
4.6.4 Dynamic Force and Moment Reactions from Joint
Constraints 498
4.6.5 Review on Computation of IT Equations for
Dynamics of Open Cycle Mechanisms 505
4.7 Closed Cycle Mechanisms: The IT Method for Dynamic
Forces and Differential Equations .506xvi Contents
4.7.1 The Incidence Nodes-Edges Matrix: The Reduced
and Row Reduced Matrix .507
4.7.2 The Weighting Matrix, W . 511
4.7.3 The Cut-Set Matroid . 511
4.7.4 Governing Dynamic Force Equations for Closed
Cycle Mechanism . 512
4.7.5 The Cut-Set Reaction Forces in Joints . 514
4.7.6 The IT-Resistant Force . 515
4.7.7 Reaction Forces in Arcs Tree Expressed from
Reaction Forces in Cut-Edges . 515
4.7.8 Force Equations for Multiple-Cycle Mechanisms . 516
4.8 Closed Cycle Mechanisms: The IT Equations for Dynamic
Moment Reactions and Differential Equations 517
4.8.1 The Governing Equations to Evaluate the Reactions
in Cut-Joints 520
4.8.2 The Cut-Set Reaction Moment in Tree’s Joints: The
Cut-Set Matroid Method . 521
4.8.3 Reaction Moment Equations for Closed Cycle
Mechanisms: The Resistant Moment 521
4.8.4 Review on Computation of IT Equations for
Dynamics of Closed Cycle Mechanisms 521
4.8.5 Examples of Mechanisms with Single and Multiple
Cycles: Singularity Coefficient . 526
4.8.6 Example: Dynamic Reactions and Differential
Equation for a Mechanism with Gears .550
4.9 Statics of Mechanisms and Machines 559
4.9.1 Background . 559
4.9.2 The Governing Equations for Closed Cycle
Mechanisms 559
4.9.3 Example: Static Reactions and Break Torque
Calculation for a Mechanism with Gears 560
4.10 Conclusions 565
Problems 566
References . 573
Index .
575
Index
Page numbers followed by f indicate figures.
A
absolute acceleration, 443
absolute and relative angular
displacements, 129–130, 135
absolute angular acceleration
governing equation for links,
446–452
matrix, 446
absolute angular velocities
bevel gear trains (BGT), 395
equations based on, 380–384
of gears, planets, and carriers,
405–406
gear trains with fixed parallel
axes, 388
parallel axes epicyclic gear trains, 382
absolute homogeneous matrix
for link, 94
for planar mechanisms, 152, 153
relative and, 96
absolute linear accelerations
direct linear acceleration analysis
example, 458–464
for open cycle spatial
mechanisms, 457–464
governing equation for links,
454–484
single- and multiple-cycle planar
mechanisms
acceleration analysis, 467–484
matroid method, 464–466
absolute links’ orientation
matrix, 274
planar crank slider mechanism, 298
versus time, 287, 288f, 299–300f,
312–313f
absolute rotation matrix
for links’ orientation, 33–34
for mobile links, 397
absolute velocity, 159
acceleration analysis
for multiple-cycle planar
mechanisms, 473–484
for single-cycle planar mechanisms
with revolute joints, 457–473
acceleration matroidal matrix, 464–466
acceleration matroidal vector, 454–457
angular displacements, 55
absolute and relative, 129–130, 135
constraint equations for, 55, 134–135
arcs, 45
axial angular acceleration, 443
constraint equations for, 444–445
B
bevel gear trains (BGTs), 377
absolute angular velocities of gears,
planets, and carriers, 405–406
absolute angular velocity matrix, 395
automatic generation of equations,
408–409
equations based on twists, 391–400
gears’ number of teeth, 406–407
twist velocity matroidal matrix,
400–405
bottom dead center (BDC) position,
302, 304
C
cam; see also planar mechanism, with
cams
for constant velocity follower, 441
joint, 3, 3f
cam contour, equations for, 335–338
center of mass acceleration, 458
center of mass position vectors, 140–141
matrix, 272–273
center of mass to joint position
matrix, 273576 Index
chord, 9, 44
closed cycle mechanisms
gears
bevel gear trains (BGT), 391–409
gear trains with fixed parallel
axes (GT), 384–391
parallel axes epicyclic gear trains
(EGT), 375–384
inverse angular acceleration analysis
on, 452–454
IT equations for dynamic moment
reactions and differential
equations, 517–558
cut-set matroid method, 521
gears, examples, 550–558
governing equations to evaluating
reactions in cut-joints, 520–521
IT equations computation, 521–526
resistant moment, 521
singularity coefficient, examples,
526–550
IT method for dynamic forces and
differential equations, 506–517
cut-set matroid, 511–512
cut-set reaction forces in
joints, 514
force equations for multiple-cycle
mechanisms, 516–517
governing dynamic force
equations, 512–514
incidence nodes-edges matrix,
507–510
IT-resistant force, 515
reaction forces in arcs tree,
515–516
reduced and row reduced matrix,
507–510
weighting matrix, 511
kinematics of single and multiple,
257–441
multiple-cycle planar mechanisms
with revolute and prismatic joints
position analysis, 318–327
with revolute and prismatic joints
velocity analysis, 367–375
planar mechanism
coordinate systems for, 257–258
enumeration, 258–262
parallel axes gear trains, 260–262
planar mechanism, with cams
background, 327–330
constant acceleration motion
during follower’s rise/fall,
345–346
constant velocity motion during
follower’s rise/fall, 338–345
cycloidal motion during follower’s
rise/fall, 347
equations for cam contour,
335–338
harmonic motion during
follower’s rise/fall, 346
input–output relation, 330–334
problems, 409–423
single-cycle planar mechanisms,
see single-cycle planar
mechanisms
statics of mechanisms, governing
equations for, 559–560
closed paths (cycles), 7–8, 45
complemental angular acceleration, 443,
445–446
complemental linear acceleration,
455–456
complex link, 1, 2f
COM position vectors, 59
computer-aided design (CAD)
drawings, 46
constant acceleration motion
during follower’s rise/fall, 345–346
constant velocity motion
during follower’s rise/fall, 338–345
Coriolis acceleration, 456, 465
coupler-point curves, 278–279, 290f
crank-rocker mechanism
coordinates for joints for, 431, 434
versus Latin matrix, 430
mobile links’ orientation in time, 432
crank slider mechanism, 359–367, 361f
frames for, 257, 258f
versus Latin matrix, 433
links’ orientation angles for, 436
mobile links’ orientation in time, 435
cut-joints
governing equations to evaluating
reactions in, 520–521
cut-set matroid method, 511–512, 521
cut-set reaction forces, in joints, 514Index 577
cycle base matroid, 350
for bevel gear trains (BGT), 391
cycle basis matrix
assigned to relative angular
accelerations, 452
incidence matrix (C), 15–17
independent equations generated
from entries in, 45–46
cycle basis matroid matrix, 78, 173, 265,
332
bevel gear trains (BGT), 394–395
planar crank slider mechanism,
292–293
planar RRTR mechanism, 307
two-cycle planar mechanism, 477
cycle matroid, 45–46
fundamentals, 17–18
cycles, 44, 106
closed paths, 7–8, 45
cycloidal motion, during follower’s rise/
fall, 347
cylinder–plane joint, 3f, 4
D
dead center position, 277, 278f
planar crank slider mechanism, 302
planar RRTR mechanism, 314
differential equation of motion
from actuating forces, 504
from actuating torques, 505
digraph, 6, 6f
bevel gear trains (BGT), 391
cut-sets, 512
epicyclic gear trains (EGT), 375
gear trains with fixed parallel axes
(GT), 386
joint position matrix, 266–267
matrix description of, 11
open cycle mechanism’s, 23–24
planar crank slider mechanism, 292
planar RRTR mechanism, 306
relative links’ orientation matrix
from, 276
direct analysis
orientation and positional analysis of
mechanisms, 24
direct and inverse relative rotation
matrix, 32
direct and inverse sign of relative IT
matrices, 107
direct angular acceleration analysis
joint axial angular acceleration, 444
constraint equations for, 444–445
joint complemental angular
acceleration, 445–446
joint relative angular acceleration,
443
links’ absolute angular acceleration
matrix, 446
for open cycle mechanisms, 443–446
direct angular velocity analysis
for open cycle spatial mechanisms,
159–161
example with DOFs, 162–172
directional angles, 27
directional cosines, 27
direct linear acceleration analysis
for open cycle spatial mechanisms,
457–464
example, 458–464
direct linear velocity analysis
for open cycle spatial mechanisms,
179–184
example with DOF, 184–190
direct orientation analysis
example for spatial mechanism with
4 DOF, 37–44
governing equations for, 36, 42–43
for open cycle mechanisms, 34–44
direct positional analysis
COM position vectors, 59
end-effector position vector, 58
equations for, 58
examples for spatial mechanism
with 4 DOF, 59–69
with 5 DOF, 69–70
with 6 DOF, 70–71
governing equations for open cycle
mechanisms, 53–71
joint position matrix, r, 59
linear displacements
constraint equations, 55
at joints, 54
translation vectors between frame
origins, 55–58
vector components transformation
between frames, 53–54578 Index
disjoint internal regions, 7, 7f
dynamics of mechanisms
evaluation of IT forces, 491–492
governing equations in, 484–492
incidence nodes-edges matrix, 487
IT-resistant force, 490–491
open cycle mechanisms
governing force equations for,
484–485
incidental and transfer-IT method
for, 486–487
path matrix, 489–490
reduced incidence nodes-edges
matrix, 487–488, 489–490
E
edge digraph (ITYZ)
relative IT homogeneous matrix for
joint, 94–96
edge/node labeling, 5
end-effector absolute homogeneous
matrix, 98–99
end-effector coordinates
from EES and SW, 248, 255
for TRRT spatial mechanism, 245
for TRT planar manipulator, 247
end-effector frame’s orientation, 35, 36,
38–39f
end-effector linear acceleration, 463, 464f
end-effector position vector, 58, 62,
139–140
for home position, 63
end-effector velocity
components for TRRT mechanism,
191f
Engineering Equation Solver (EES)
calculation
for direct positional analysis, 91–93,
148–152
for direct velocity analysis, 253
for inverse orientation and positional
analysis, 84–91, 145–148
parametric table, 246–247,
249–252, 254
for planar mechanisms, 284
simulations, 66–69
epicyclic gear trains (EGT), see parallel
axes epicyclic gear trains (EGT)
Euler angles, 37
Euler formula, 8, 19
external region, 8, 8f
F
fictitious equilibrium, 497
fictitious joint, 44
force equations, 560
four-bar mechanism, 352–359, 353f
frame for, 257, 258f
four-cycle IT dynamic equations,
525–526
frame orientation and position, for
spatial open cycle mechanisms,
24–25
incidence nodes–edges matrix, 11–12
reduced incidence nodes–edges
matrix, 12–13
G
gear joint, 3, 3f
gear ratios
bevel gear trains, 402
gear trains with fixed parallel axes,
389
matroidal method for gear trains,
382–383
gears
bevel gear trains (BGTs)
absolute angular velocities of
gears, planets, and carriers,
405–406
automatic generation of equations,
408–409
equations based on twists,
391–400
gears’ number of teeth, 406–407
twist velocity matroidal matrix,
400–405
dynamic reactions and differential
equation, example, 550–558
gear trains with fixed parallel axes
(GT), 384–391
equations based on absolute
angular velocities, 387–389
gears’ number of teeth, 390–391
velocity ratio, 390Index 579
parallel axes epicyclic gear trains,
375–384
equations based on absolute
angular velocities, 380–384
gears’ number of teeth, 384
mobility formula, 377–380
static reactions and break torque
calculation, example, 560–565
gear trains with fixed parallel axes (GT),
384–391
equations based on absolute angular
velocities, 387–389
gears’ number of teeth, 390–391
GT velocity ratio, 390
geometric Jacobian for angular
velocities, 174
graph theory, 4–5
Grashof’s criterion, 279–290
Gruebler formula, 19
GT velocity ratio, 390
H
harmonic motion, during follower’s
rise/fall, 346
Hartemberg and Denavit (HD) notation,
95
home position
with cam, 329
direction angles at, 43–44
joint position matrix for, 63
of mechanism, 23, 23f, 131–132
planar crank slider mechanism, 294
state, homogeneous matrix method,
110–113, 111f
homogeneous matrix method
direct and inverse sign of relative IT
matrices, 107
home-position state, 110–113, 111f
inverse of, 107–108
inverse orientation and positional
combined equations
along closed path, 106–128
IT relative homogeneous matrices,
93–106
for planar mechanisms, 152–153
orientation and positional analysis
of robotic mechanism without
vision, 109–123
of robotic mechanism with vision,
124–128
part-approach state, 113–115, 113f
part-grasped state, 115–117, 115f
robot programming, 109–123
target-approach state, 117–120, 118f
target-reached state, 120–123, 121f
task absolute homogeneous matrix,
108–109
I
incidence links–joints matrix, 78, 332
planar crank slider mechanism, 292
planar RRTR mechanism, 306
incidence matrix, partitions of, 513
incidence nodes–edges matrix, 11–12,
264, 391–394, 487, 507–510
incidental and transfer (IT) notation
on dynamic moments and
differential equations
for closed cycle mechanisms,
506–517
for open cycle mechanisms,
492–506
forces, evaluation of, 491–492,
497–498
for frames based on digraph’s
nodes, 25
for joint displacement based on
incidental digraph’s edge, 25
moments, evaluation of, 497–498
for open cycle mechanisms, 486–487
relative homogeneous matrices,
93–106
for planar mechanisms, 152–153
relative rotation matrix
about x
m-axis, 26–27, 241
about ym-axis, 27–29, 242
about z
m-axis, 29–31, 242
-resistant force, 490–491, 494, 515
incidental matrix, 95
independent cycles, 9–10, 10f
for planar and spatial mechanisms,
18–19
inertial frame, 257
input joint rates and output end-effector
linear and angular velocities
for TRT manipulator, 254580 Index
input–output relation
planar crank slider mechanism, 301
planar mechanism with cams,
330–334
planar RRTR mechanism, 314
single-cycle planar mechanisms, 277,
289f
input parameters, 36
inverse analysis
application for, 144–152
orientation and positional analysis of
mechanisms, 24
inverse angular velocity analysis
equations
for TRRTR spatial mechanism,
176–177
for TRRTRT spatial mechanism,
178–179
twists
example, 208–209
geometric Jacobean, 207–208
for joints with single and multiple
DOF, 203–207
inverse linear velocity analysis, of open
cycle mechanisms
with equation functions of absolute
angular velocities, 190–194
example, 194–196
with equation functions of relative
angular velocities, 196–201
analogy to moment of force and
static couples, 199–200
example, 200–203
Jacobean matrix from combined
equations, 200
redundant and nonredundant
system, 203
spanning tree matrix, 198–199
twists
example, 208–209
geometric Jacobean, 207–208
for joints with single and multiple
DOF, 203–207
inverse links’ orientation analysis
along closed path (cycle), 44–53
example for spatial mechanism
with 4 DOF, 47–48
with 5 DOF, 48–50
with 6 DOF, 50–53
independent equations, 45–46
task orientation matrix, 46–47
inverse of homogeneous matrix, 107–108
inverse of relative matrix, 32
inverse positional analysis
equations for, 81–84
examples for spatial mechanism
with 4 DOF, 72, 73–74, 80–84
with 5 DOF, 73, 74–75
governing equations for open cycle
mechanisms, 71–73
J
Jacobean matrix, for inverse angular
and inverse linear velocities,
200
joint axial angular acceleration, 444
constraint equations for, 444–445
joint complemental angular
acceleration, 445–446
joint coordinates, 64
joint linear accelerations, 457–458, 466
joint position matrix, 59, 60–62
for home position, 63
planar crank slider mechanism, 297
planar RRTR mechanism, 311
vectors matrix, 265–266
joint relative angular acceleration, 443
joints
and labeling, 1–4
relative constraints matrix, 76
twists for, 203–227
K
kinematics
of closed cycle mechanisms, see
closed cycle mechanisms
of open cycle mechanisms, see open
cycle mechanisms
Kirchhoff’s Matrix Tree Theorem, 12
Kutzbach formula, 19, 507
L
Latin matrix, 75–77
algorithm for automatic generation
of, 77–79Index 581
cam–follower mechanism, 333
entries for crank-rocker
mechanism, 430
entries for TRT manipulator, 251
entries versus time for crank slider
mechanism, 433
entries versus time for RRTR planar
mechanism, 437
equations for columns, 143–144
equations for rows, 143
gear trains with fixed parallel
axes, 387
method for positional analysis,
267–272
parallel axes epicyclic gear
trains, 380
planar RRTR mechanism, 307
two-cycle planar mechanism, 477
vector components for, 293
and velocity matroidal matrix,
348–350
Latin velocity vector, 180
between frame origins along closed
path, 192f
linear displacements
constraint equations for, 55, 134–135
at joints, 54
link(s)
constraints matrix, 76
and joint labeling, 21–22
and numbering, 1
link and joint numbering
bevel gear trains (BGT), 391
epicyclic gear trains (EGT), 375
gear trains with fixed parallel axes
(GT), 386
M
matroid method
for bevel gear trains (BGT), 391–400
cut-set reaction moment in tree’s
joints, 521
cycle basis matrix, 173
example on inverse positional
analysis
equations, 81–84
for spatial mechanism with 4
DOF, 80–84
example solution for inverse
orientation and positional
equations
for 4 DOF mechanism, 84–92
for gear trains, 380–384
with fixed parallel axes, 387–389
for inverse angular acceleration
analysis
on closed cycle mechanisms,
452–454
cycle basis matrix, 452
relative angular acceleration
matrix, 453–454
for inverse angular velocity analysis
on closed path, 172
inverse links’ orientation analysis
along closed path, 44–53
Latin matrix, 75–77
algorithm, 77–79
for linear acceleration analysis
of single- and multiple-cycle
planar mechanisms, 464–466
for planar open cycle mechanisms
equations for inverse orientation,
141–142
equations for inverse positional
analysis, 142–144
equations for Latin matrix,
143–144
solution of nonlinear system of
equations, 144
for velocity equations of mechanisms
with cycles, 350–351
matroids, 17–18
mechanism absolute rotation matrix, 35,
36, 41–42
mechanism branches, 279
mechanism, graph representation of;
see also specific entries
closed paths (cycles), 7–8
cycle basis incidence matrix, C, 15–17
cycle matroid fundamentals, 17–18
digraph, 6
matrix description of, 11
graph, 4–5
incidence nodes–edges matrix, G,
11–12
labeling of nodes and edges, 5
open paths, 7582 Index
mechanism, graph representation of
(cont.)
path matrix, Z, 13–14
paths, 6–7
reduced incidence nodes–edges
matrix, G, 12–13
spanning tree matrix, T, 15
tree and spanning tree, 9–10
meshing joint, 54
mobility
formula for epicyclic gear trains
(EGT), 377–380
gear trains with fixed parallel axes
(GT), 386
planar crank slider mechanism, 292
of planar mechanism, 19, 133–134
planar RRTR mechanism, 306
of spatial mechanisms, 19, 24
moment equations, 560
motion
differential equations of, 535–550
joint and link constraints for, 210
multiple-cycle mechanisms
force equations for, 516–517
multiple-cycle planar mechanisms
acceleration analysis for, 473–484
with revolute and prismatic joints
position analysis, 318–327
with revolute and prismatic joints
velocity analysis, 367–375
multiple joined links, 4, 4f
N
node digraph (Rm)
absolute homogeneous matrix for
link, 94
node/edge labeling, 5
nonlinear system of equations for
positional analysis, 269–272
planar crank slider mechanism, 294
planar RRTR mechanism, 308
null rotation, 32
O
open cycle mechanisms
absolute rotation matrix
for links’ orientation, 33–34
additional frames on same link, 33
direct and inverse analysis, 24–25
direct angular acceleration analysis
joint axial angular acceleration,
444–445
joint complemental angular
acceleration, 445–446
joint relative angular acceleration,
443
links’ absolute angular
acceleration matrix, 446
direct links’ orientation analysis for,
34–44
direct positional analysis
COM position vectors, 59
end-effector position vector, 58
equations for, 58
examples for spatial mechanism,
59–71
joint position matrix, r, 59
linear displacements, 54–55
translation vectors between frame
origins, 55–58
vector components transformation
between frames, 53–54
equations based on Latin matrix and
cycle matroid entries, 75–93
equations for inverse orientation and
positional analysis, 73–75
governing force equations for,
484–485
home position of mechanism, 23, 23f
homogeneous matrix method
direct and inverse sign of relative
IT matrices, 107
home-position state, 110–113, 111f
inverse of, 107–108
inverse orientation and positional
combined equations along
closed path, 106–128
orientation and positional
analysis, examples, 109–128
part-approach state, 113–115, 113f
part-grasped state, 115–117, 115f
robot programming, 109–123
target-approach state, 117–120, 118f
target-reached state, 120–123, 121f
task absolute homogeneous
matrix, 108–109Index 583
inverse links’ orientation analysis
along closed path (cycle), 44–53
example for spatial mechanism,
47–53
independent equations, 45–46
task orientation matrix, 46–47
inverse positional analysis
equations for, 81–84
examples for spatial mechanism,
72–75, 80–84
governing equations for, 71–73
IT method for, 25, 492–506
dynamic force and moment
reactions from joint
constraints, 498–505
on dynamic forces and
differential equations for,
486–487
equations for reaction moment,
494
IT equations computation,
505–506
IT moments evaluation, 497–498
position vector skew-symmetric
matrix, 494–497
relative homogeneous matrices,
93–106
kinematics of, 21–255
Latin matrix, 75–79
link and joint labeling, 21–22
Matroid method, 44–53
mechanism’s digraph, 23–24
mobility, 24
planar open cycle mechanisms
absolute and relative angular
displacements, 129–130
absolute homogeneous matrix,
152, 153
application for inverse analysis,
144–152
center of mass position vectors,
140–141
direct orientation and positional
analysis for, 128–158
direct positional analysis, 138–140
end-effector position vector,
139–140
example for orientation and
positional analysis, 153–158
example of simulation with 3
DOF, 131–138
governing equation for links’
orientation for, 128–129
IT relative homogeneous matrix,
152–153
matroid method, 141–144
path matrix and its transposed
matrix, 130
position vector matrix, 139
relative homogeneous matrices,
152–153
problems, 218–241
reaction forces equations on,
490–491
relative homogeneous matrices, IT
method of, 93–106
relative rotation matrix
relative frames orientation, 26–32
along tree, 32–33
spatial TRRT mechanism, 21, 22f
velocity analysis, see velocity analysis
open paths, 7
orthogonality
between cycle base matroid and cutset matroid, 512
orthonormal matrix (D), 31
P
parallel axes epicyclic gear trains (EGT),
375–384
absolute angular velocity matrix, 378
classification, 427–429
gears and carrier absolute rotation
matrices, 379
gears’ number of teeth, 384
matroidal method for gear trains,
380–384
mobility formula for, 377–380
relative angular velocity matrix, 378
relative rotation matrices, 379
unit vectors for axes of rotation, 380
part-approach position, 64, 88, 89f
part-grasped position, 64, 88–89, 90f
path matrix (Z), 13–14, 509
automatic generation of mobile
links’ angular velocities from,
162, 184584 Index
path matrix (Z) (cont.)
bevel gear trains (BGT), 394
dynamics of mechanisms, 489
and its transposed matrix, 130
and reduced incidence nodes-edges
matrix, 489–490
paths, 6–7, 24, 24f
peripheral cycle, 8, 8f
planar mechanisms
with cams, 327–347
background, 327–330
constant acceleration motion
during follower’s rise/fall,
345–346
constant velocity motion during
follower’s rise/fall, 338–345
cycloidal motion during follower’s
rise/fall, 347
equations for cam contour,
335–338
harmonic motion during
follower’s rise/fall, 346
input–output relation, 330–334
classification, 424–426
coordinate systems for, 257–258
crank slider mechanism, 290–304
enumeration, based on number of
cycles, 258–262, 259f
parallel axes gear trains, 260–262
independent cycles in mechanism
for, 18–19
mobility of, 19
planar RRTR mechanism
coordinates for joints for, 438
example, 304–318
versus Latin matrix, 437
links’ orientation angles for, 439
planar open cycle mechanisms
absolute and relative angular
displacements, 129–130
application for inverse analysis,
144–152
EES for direct positional analysis,
148–152, 247–252
EES for inverse orientation and
positional analysis, 145–148
center of mass position vectors,
140–141
direct positional analysis, 138–140
end-effector position vector,
139–140
position vector matrix, 139
example for orientation and
positional analysis
of 3 DOF planar manipulator,
153–158
inverse of homogeneous
matrix, 158
example of simulation with 3 DOF
absolute and relative angular
displacements, 135
constraint equations for angular
and linear displacements,
134–135
home position of mechanism,
131–132
link and joint labeling, 131
mechanism’s digraph, 132–133
mechanism’s mobility, 133–134
notation for frames based on
digraph’s nodes, 133
relative and absolute rotation
matrices, 136–138
governing equation for links’
orientation for, 128–129
matroid method
equations for inverse orientation,
141–142
equations for inverse positional
analysis, 142–144
solution of nonlinear system of
equations, 144
path matrix and its transposed
matrix, 130
relative homogeneous matrices
absolute homogeneous matrix,
152, 153
IT relative homogeneous matrix,
152–153
plane–plane joint, 3, 3f
Plucker coordinates, 398
positional analysis; see also single-cycle
planar mechanisms
Latin matrix method for, 267–272
nonlinear system of equations for,
269–272
positional constraints, 210–211
position vectors, 55–58, 57fIndex 585
matrix, 139
skew-symmetric matrix, 494–497
prismatic joint, 2, 3f
R
reaction forces, in arcs tree, 515–516
reaction moment
equations on closed cycle
mechanisms, 521
equations on open cycle
mechanisms, 494
from joint constraints, 498–505
reduced and row reduced matrix,
507–510
reduced incidence nodes-edges matrix,
487–488
reduced matrix, partitions of, 509
redundant and nonredundant system,
201
reference axis, 26, 28
relative acceleration, 443
relative angular acceleration, 443,
453–454
cycle basis matrix assigned to, 452
relative angular velocity, 159
matrix, 173–176, 395–396
relative homogeneous matrices
direct orientation and positional
combined equations
example on 4 DOF, 99–106
on open cycle mechanisms, 96–98
edge digraph, 94–96
end-effector absolute homogeneous
matrix, 98–99
IT method of, 93–106
node digraph, 94, 96
for planar mechanisms, 152–153
relative linear acceleration, 456
relative links’ orientation matrix,
274–275
from digraph, 276
planar crank slider mechanism, 299
relative matrix, inverse of, 32
relative rotation matrix
bevel gear trains (BGT), 396–397
direct and inverse, 32
properties of, 31–32, 243–244
along tree, 32–33
about x
m-axis, 26–27, 241
about ym-axis, 27–29, 242
about z
m-axis, 29–31, 242
relative velocity, 159
resistant moment, 521
revolute joint, 2, 3f, 54
rotation; see also relative rotation matrix
about x
m-axis, 241
about ym-axis, 242
about z
m-axis, 242
row-reduced incidence matrix, 510
S
scaling factor, 94
screw joint, 2, 3f
screws at joints, 397–398
screw’s lead, 2, 3f
simple link, 1, 2f
single-cycle IT dynamic moment
equations, 523, 527–532
solution for, 533
single-cycle planar mechanisms
acceleration analysis for, 467–473
with revolute and prismatic joints
example, planar RRTR
mechanism, 304–318
planar crank slider mechanism,
290–304
with revolute joints
absolute links’ orientation, 274
center of mass to joint position
matrix, 273
centers of mass position vector
matrix, 272–273
coupler-point curves, 278–279
cycle basis matroid matrix, 265
dead centers, 277–278
digraph joint position matrix,
266–267
Grashof’s criterion, 279–290
incidence nodes–edges
matrix, 264
input to output relation, 277
joint position vectors matrix,
265–266
Latin matrix method, 267–272
mechanism branches, 279
position analysis for, 262–290586 Index
single-cycle IT dynamic moment
equations (cont.)
relative links’ orientation, 274–276
transmission angle, 276–277
velocity analysis for, 347–367
with revolute and prismatic joints,
359–367
with revolute joints, 352–359
singularities for inverse velocity
analysis, 212
singularity coefficient
examples of mechanisms with single
and multiple cycles, 526–550
skew-symmetric matrices, 494, 496, 518
sliding pair, 2, 3f
SolidWorks (SW) motion simulation,
64–69
end-effector’s path from, 151f
spanning tree, 9–10, 24f
bevel gear trains (BGT), 391, 394
epicyclic gear trains (EGT), 375
gear trains with fixed parallel axes
(GT), 386
matrix (T), 15, 198–199
planar crank slider mechanism, 292
planar RRTR mechanism, 306
spatial mechanisms
independent cycles for, 18–19
mobility of, 19
TRRT, 21–22
sphere–plane joint, 3f, 4
spherical joint, 3, 3f
springs, 1
states of the joint coordinates, 109
statics of mechanisms and machines
background, 559
example, 560–565
governing equations for closed cycle
mechanisms, 559–560
T
target-approach position, 64, 89, 90f
target-reached position, 64, 91, 91f
task absolute homogeneous matrix,
108–109
task orientation matrix, 46–47
task position vector, 71–73
task velocity vector, 193
teaching-by-doing approach, 109
three-cycle mechanisms
force equations for, 516–517
IT dynamic equations, 524–525
top dead center (TDC) position, 302,
303–304
translation matrix, 95
translation vectors, 55–58
transmission angle
for four-bar mechanism, 276–277,
289f
planar crank slider mechanism, 301
planar RRTR mechanism, 313
transpose matrix, 14
tree, 9–10
turning pair, 2, 3f
twists
combined equations for inverse
velocity analysis on
for cylindrical joint, 207
for prismatic joint, 206
for revolute joint, 203–205
for screw joint, 207
for spherical joint, 206–207
example of velocity analyses
capability of motion for TRT
manipulator, 213–214
EES for direct velocity analysis
of TRT 3 DOF manipulator,
214–218
for planar open cycle
mechanisms, 212–218
matroidal method for BGT, 391–400
velocity matroidal matrix, 400–405
two-cycle mechanisms
force equations for, 516–517
IT dynamic equations, 523–524
planar mechanism
acceleration analysis for, 473–484
links’ orientation and slider
displacement for, 440
U
unit screw
lower vector w.r.t. meshing joint,
399–400
upper vector w.r.t. fixed frame, 398
unit vectors for axes of rotationIndex 587
bevel gear trains (BGT), 397
epicyclic gear trains (EGT), 380
gear trains with fixed parallel axes
(GT), 380, 386
V
vector components, transformation
between frames, 53–54
velocity analysis
automatic generation of mobile links’
angular velocities, 162, 184
direct angular velocity analysis
example with DOFs, 162–172
for open cycle spatial
mechanisms, 159–161
direct linear velocity analysis
example with DOF, 184–190
for open cycle spatial
mechanisms, 179–184
example for planar open cycle
mechanisms
with equation functions of
absolute velocities, 209–212
with equation functions of twists,
212–218
inverse angular velocity analysis
equations
for TRRTR spatial mechanism,
176–177
for TRRTRT spatial mechanism,
178–179
inverse linear velocity analysis, of
open cycle mechanisms
with equation functions of
absolute angular velocities,
190–196
with equation functions of
relative angular velocities,
196–203
matroid method
cycle basis matrix, 173
for inverse angular velocity
analysis, 172
for multiple-cycle planar
mechanisms
with revolute and prismatic joints,
367–375
of planar open cycle mechanisms
with revolute joints, 218
relative angular velocity matrix,
173–176
of single-cycle planar mechanisms,
347–367
with revolute and prismatic joints,
359–367
with revolute joints, 352–359
twists, combined equations for
inverse velocity analysis on
example for open cycle TRRT
spatial mechanism, 208–209
geometric Jacobean, 207–208
for joints with single and multiple
DOF, 203–207
velocity matroidal matrix, 400–401
gear trains with fixed parallel axes,
387–388
Latin matrix and, 348–350
parallel axes epicyclic gear trains, 381
W
weighting matrix, 511
world frame, 257
Y
yaw–pitch–roll angles, 37
Z
zero-pitch screw, 205
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