CONTENTS
Preface …………………………………………...……………………………………….. 1
Chapter 2 Mathematical Concepts in Kinematics ……………………………………….. 2
Chapter 3 Fundamental Concepts in Kinematics ……………………………………….. 8
Chapter 4 Kinematic Analysis of Planar Mechanisms …………………………………… 25
Chapter 5 Dimensional Synthesis ………………………………………………………... 86
Chapter 6 Static Force Analysis of Planar Mechanisms …………………………………. 195
Chapter 7 Dynamic Force Analysis of Planar Mechanisms ……………………………… 252
Chapter 8 Design & Kinematic Analysis of Gears ……………………………………….. 330
Chapter 9 Design & Kinematic Analysis of Disk Cams …………………………………. 370
Chapter 10 Kinematic Analysis of Spatial Mechanisms ………………………………….. 409
Chapter 11 Introduction to Robotic Manipulators ………………………………………… 455
, PREFACE
Because all the computed solutions in this manual were produced using the 2021 version of
MATLAB™ (specifically version R2021b, win64). any version of MATLAB™ post 2020 is also
suitable to run the MATLAB™ and Simscape Multibody™ files associated with this textbook
(provided all the required toolkits listed in Appendix A.1 are installed).
As presented in Chapter 5 (in the textbook), there is an infinite number of solutions for a
given dimensional synthesis problem. Because the solution values for a dimensional synthesis
problem depend on the dyad angles specified, the solutions given here for Chapter 5 (which were
calculated using arbitrary dyad displacement angles) are intended to serve as a guide to the
solution calculation process and not as a benchmark for evaluating student solutions.
We encourage and look forward to any feedback you may have. For e-mail correspondence,
we can be reached at . Thank you.
K. Russell
Q. Shen
R.S. Sodhi
1
, CHAPTER 2
Problem 2.1 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.1. Consider that vector V j
always lies along the real axis.
Figure P.2.1 Vector loop (3 vectors where V j changes length) in 2-D complex space
Problem 2.1 Solution:
Taking the clockwise sum of the vector loop in Figure P.2.1 produces the equation
V1ei( 1 +1 ) − V2 ei( 2 +2 ) + V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
V1 cos ( 1 + 1 ) − V2 cos ( 2 + 2 ) + V j = 0
.
V1 sin ( 1 + 1 ) − V2 sin ( 2 + 2 ) = 0
Problem 2.2 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.2. Consider that vector V j
always lies along the real axis and vector V3 is always perpendicular to the real axis.
2
, Figure P.2.2 Vector loop (4 vectors where V j changes length) in 2-D complex space
Problem 2.2 Solution:
Taking the clockwise sum of the vector loop in Figure P.2.2 produces the equation
−V1ei( 1 +1 ) + V2 ei( 2 +2 ) − V3 − V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
−V1 cos ( 1 + 1 ) + V2 cos ( 2 + 2 ) − V j = 0
.
−V1 sin ( 1 + 1 ) + V2 sin ( 2 + 2 ) − V3 = 0
Problem 2.3 Statement:
Calculate the first derivative of the vector loop equation solution from Problem 2.2. Consider
angles 1 , 2 and vector V j from Problem 2 to be time-dependent.
Problem 2.3 Solution:
Differentiating the vector loop equation solution from Problem 2.2 produces the equation
−i1V1ei( 1 +1 ) + i 2V2ei( 2 +2 ) − V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
1V1 sin ( 1 + 1 ) − 2V2 sin ( 2 + 2 ) − V j = 0
.
−1V1 cos ( 1 + 1 ) + 2V2 cos ( 2 + 2 ) = 0
3
Preface …………………………………………...……………………………………….. 1
Chapter 2 Mathematical Concepts in Kinematics ……………………………………….. 2
Chapter 3 Fundamental Concepts in Kinematics ……………………………………….. 8
Chapter 4 Kinematic Analysis of Planar Mechanisms …………………………………… 25
Chapter 5 Dimensional Synthesis ………………………………………………………... 86
Chapter 6 Static Force Analysis of Planar Mechanisms …………………………………. 195
Chapter 7 Dynamic Force Analysis of Planar Mechanisms ……………………………… 252
Chapter 8 Design & Kinematic Analysis of Gears ……………………………………….. 330
Chapter 9 Design & Kinematic Analysis of Disk Cams …………………………………. 370
Chapter 10 Kinematic Analysis of Spatial Mechanisms ………………………………….. 409
Chapter 11 Introduction to Robotic Manipulators ………………………………………… 455
, PREFACE
Because all the computed solutions in this manual were produced using the 2021 version of
MATLAB™ (specifically version R2021b, win64). any version of MATLAB™ post 2020 is also
suitable to run the MATLAB™ and Simscape Multibody™ files associated with this textbook
(provided all the required toolkits listed in Appendix A.1 are installed).
As presented in Chapter 5 (in the textbook), there is an infinite number of solutions for a
given dimensional synthesis problem. Because the solution values for a dimensional synthesis
problem depend on the dyad angles specified, the solutions given here for Chapter 5 (which were
calculated using arbitrary dyad displacement angles) are intended to serve as a guide to the
solution calculation process and not as a benchmark for evaluating student solutions.
We encourage and look forward to any feedback you may have. For e-mail correspondence,
we can be reached at . Thank you.
K. Russell
Q. Shen
R.S. Sodhi
1
, CHAPTER 2
Problem 2.1 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.1. Consider that vector V j
always lies along the real axis.
Figure P.2.1 Vector loop (3 vectors where V j changes length) in 2-D complex space
Problem 2.1 Solution:
Taking the clockwise sum of the vector loop in Figure P.2.1 produces the equation
V1ei( 1 +1 ) − V2 ei( 2 +2 ) + V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
V1 cos ( 1 + 1 ) − V2 cos ( 2 + 2 ) + V j = 0
.
V1 sin ( 1 + 1 ) − V2 sin ( 2 + 2 ) = 0
Problem 2.2 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.2. Consider that vector V j
always lies along the real axis and vector V3 is always perpendicular to the real axis.
2
, Figure P.2.2 Vector loop (4 vectors where V j changes length) in 2-D complex space
Problem 2.2 Solution:
Taking the clockwise sum of the vector loop in Figure P.2.2 produces the equation
−V1ei( 1 +1 ) + V2 ei( 2 +2 ) − V3 − V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
−V1 cos ( 1 + 1 ) + V2 cos ( 2 + 2 ) − V j = 0
.
−V1 sin ( 1 + 1 ) + V2 sin ( 2 + 2 ) − V3 = 0
Problem 2.3 Statement:
Calculate the first derivative of the vector loop equation solution from Problem 2.2. Consider
angles 1 , 2 and vector V j from Problem 2 to be time-dependent.
Problem 2.3 Solution:
Differentiating the vector loop equation solution from Problem 2.2 produces the equation
−i1V1ei( 1 +1 ) + i 2V2ei( 2 +2 ) − V j = 0 .
When expanded and separated into real and imaginary terms, the vector loop equation becomes
1V1 sin ( 1 + 1 ) − 2V2 sin ( 2 + 2 ) − V j = 0
.
−1V1 cos ( 1 + 1 ) + 2V2 cos ( 2 + 2 ) = 0
3
