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CHAPTER 2
Problem 2.1 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.1. Consider that vector j V always lies along the real axis.
Figure P.2.1 Vector loop (3 vectors where j V 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 ( ) ( )
1 1 2 2
12
ii j V e V e
+ +
− + = V
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( )
( ) ( )
1 1 1 2 2 2
1 1 1 2 2 2
cos cos 0 sin sin 0 j VV VV
+ − + + =
+ − + =
V .
Problem 2.2 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.2. Consider that vector j V always lies along the real axis and vector 3 V is always perpendicular to the real axis.Solutions Manual for Kinematics and Dynamics of Mechanical Systems Implementation in MATLAB® and Simscape Multibody™, 1e by Kevin Russell, John Shen, Raj Sodhi (All Chapters, Ch 1 Missing) 1 / 4
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Figure P.2.2 Vector loop (4 vectors where j V 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 ( ) ( )
1 1 2 2
- 2 3
ii j V e V e
+ +
− + − − = VV
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( )
( ) ( )
1 1 1 2 2 2
1 1 1 2 2 2 3
cos cos 0 sin sin 0 j VV VV
− + + + − =
− + + + − =
V V .
Problem 2.3 Statement:
Calculate the first derivative of the vector loop equation solution from Problem 2.2. Consider angles 1
, 2
and vector j V from Problem 2 to be time-dependent.
Problem 2.3 Solution:
Differentiating the vector loop equation solution from Problem 2.2 produces the equation ( ) ( )
1 1 2 2
1 1 2 2
ii j i V e i V e
+ +
− + − = V
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( )
( ) ( )
1 1 1 1 2 2 2 2
1 1 1 1 2 2 2 2
sin sin 0 cos cos 0 j VV VV
+ − + − =
− + + + =
V
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Problem 2.4 Statement:
Calculate the second derivative of the vector loop equation solution from problem 2.2. Consider only angles 1
, 2
and vector j V from Problem 2 to be time-dependent.
Problem 2.4 Solution:
Differentiating the vector loop equation solution from Problem 2.3 produces the equation ( ) ( ) ( ) ( )
1 1 1 1 2 2 2 222
1 1 1 1 2 2 2 2
i i i i j V e i V e V e i V e
+ + + +
− − + − =V
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
22
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
22
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2
cos sin cos sin 0 sin cos sin cos 0 j
V V V V
V V V V
+ + + − + − + − =
+ − + − + + + =
V .
Problem 2.5 Statement:
Formulate an equation for the vector loop illustrated in Figure P.2.3.
Figure P.2.3 Vector loop (4 vectors) in 2-D complex space
Problem 2.5 Solution:
Taking the clockwise sum of the vector loop in Figure P.2.3 produces the equation ( ) ( ) ( )
331 1 2 20
1 2 3 0
iiii V e V e V e V e
+ + +
− + − =
.When expanded and separated into real and imaginary terms, the vector loop equation becomes 3 / 4
5
( ) ( ) ( )
( ) ( ) ( )
1 1 1 2 2 2 3 3 3 0 0
1 1 1 2 2 2 3 3 3 0 0
cos cos cos cos 0 sin sin sin sin 0
V V V V
V V V V
+ − + + + − =
+ − + + + − =
.
Problem 2.6 Statement:
Calculate the first derivative of the vector loop equation solution from Problem 2.5. Consider only angles 1
, 2
and 3 from Problem 5 to be time-dependent.
Problem 2.6 Solution:
Differentiating the vector loop equation solution from Problem 2.5 produces the equation ( ) ( ) ( )
331 1 2 2
1 1 2 2 3 3
iii i V e i V e i V e
+ + +
− + =
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( ) ( )
( ) ( ) ( )
1 1 1 1 2 2 2 2 3 3 3 3
1 1 1 1 2 2 2 2 3 3 3 3
sin sin sin 0 cos cos cos 0 V V V V V V
− + + + − + =
+ − + + + =
.
Problem 2.7 Statement:
Calculate the second derivative of the vector loop equation solution from Problem 2.5. Consider only angles 1
, 2
and 3 from Problem 5 to be time-dependent.
Problem 2.7 Solution:
Differentiating the vector loop equation solution from Problem 2.5 produces the equation ( ) ( ) ( ) ( ) ( ) ( )
3 3 3 31 1 1 1 2 2 2 22 2 2
1 1 1 1 2 2 2 2 3 3 3 3
iii i i i V e i V e V e i V e V e i V e
+ + + + + +
− + + − − + =
.When expanded and separated into real and imaginary terms, the vector loop equation becomes ( ) ( ) ( ) ( ) ( )
- )
( ) ( ) ( ) ( ) ( )
- )
- 2 2
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3
3 3 3 3
- 2 2
1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3
3 3 3 3
cos sin cos sin cos sin 0 sin cos sin cos sin cos 0
V V V V V
V
V V V V V
V
− + − + + + + + − +
− + =
− + + + + + − + − +
+ + =
.
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