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MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
Department
of
Mechanical
Engineering
2.004
Dynamics
and
Control
II
Fall
2007
Problem
Set
#2
Solution
Posted:
Friday,
Sept.
21,
’07
1.
In
class,
we
showed
in
two
diﬀerent
ways
that
the
torque
constant
of
a
DC
motor
equals
the
back–EMF
constant,
K
m
=
K
v
.
Verify
from
the
deﬁnitions
of
these
constants,
K
m
i
=
T
and
K
v
ω
=
v
e
,
respectively,
that
the
units
associated
with
these
constants
are
consistent
as
well.
Answer:
From
the
deﬁnition
of
K
m
and
K
v
,
[
N
·
m
]
[
J
]
[
K
m
]=
=
,
[
A
]
[
A
]
and
[
V
]
[
W/A
]
[
W
·
s
]
[
J
]
[
K
v
=
=
=
.
[1
/s
]
[1
/s
]
[
A
]
[
A
]
Therefore
they
have
the
same
units.
2.
Rework
Problem
2
of
Problem
Set
#1
(the
motor–shaft
system
of
Lecture
2
with
non–zero
initial
condition)
by
using
the
Laplace
transform.
You
should
arrive
at
the
same
step
response
as
you
did
in
Problem
Set
#1.
Answer:
The
equation
of
motion
is
dω
b
T
0
u
(
t
)
+
ω
=
.
dt
J
J
Using
the
Laplace
transform,
we
can
write
as
b
T
0
s
Ω(
s
)
−
ω
0
+
Ω(
s
)=
.
J
Js
and
it
can
be
solve
as
T
0
1
Ω(
s
+
ω
0
sJ
s
+
b/J
ω
0
T
0
=
+
s
+
b/J
sJ
(
s
+
b/J
)
ω
0
T
0
1
1
=
+
−
.
s
+
b/J
b
s
s
+
b/J
1
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³
²
³
´
By
taking
the
inverse
Laplace
transform,
we
end
up
with
ω
(
t
)=
ω
0
e
−
t/τ
+
T
0
1
−
e
−
t/τ
,
b
where
τ
=
J/b
.
The
result
is
identical
to
the
solution
we
derived
in
the
lecture.
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 Fall '08
 DerekRowell
 Mechanical Engineering, Equations

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