6.003
Homework
6
Due
at
the
beginning
of
recitation
on
Wednesday,
March
17,
2010
.
Problems
1.
Secondorder
systems
The
impulse
response
of
a
secondorder
CT
system
has
the
form
h
(
t
) =
e
−
σt
cos(
ω
d
t
+
φ
)
u
(
t
)
where
the
parameters
σ
,
ω
d
,
and
φ
are
related
to
the
parameters
of
the
characteristic
polynomial
for
the
system:
s
2
+
Bs
+
C
.
a.
Determine
expressions
for
σ
and
ω
d
(not
φ
)
in
terms
of
B
and
C
.
b.
Determine
–
the
time
required
for
the
envelope
e
−
σt
of
h
(
t
)
to
diminish
by
a
factor
of
e
,
–
the
period
of
the
oscillations
in
h
(
t
)
,
and
–
the
number
of
periods
of
oscillation
before
h
(
t
)
diminishes
by
a
factor
of
e
.
Express
your
results
as
functions
of
B
and
C
only.
c.
Estimate
the
parameters
in
part
b
for
a
CT
system
with
the
following
poles:
−
10
100
−
100
s
plane
The
unitsample
response
of
a
secondorder
DT
system
has
the
form
n
h
[
n
] =
r
0
cos(Ω
0
n
+ Φ)
u
[
n
]
where
the
parameters
r
0
,
Ω
0
,
and
Φ
are
related
to
the
parameters
of
the
characteristic
polynomial
for
the
system:
z
2
+
Dz
+
E
.
d.
Determine
expressions
for
r
0
and
Ω
0
(not
Φ
)
in
terms
of
D
and
E
.
e.
Determine
n
–
the
length
of
time
required
for
the
envelope
r
0
of
h
[
n
]
to
diminish
by
a
factor
of
e
.
–
the
period
of
the
oscillations
in
h
[
n
]
,
and
–
the
number
of
periods
of
oscillation
in
h
[
n
]
before
it
diminishes
by
a
factor
of
e
.
Express
your
results
as
functions
of
D
and
E
only.
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2
6.003
Homework
6
/
Spring
2010
0
.
938
0
.
149
z
plane
f.
Estimate
the
parameters
in
part
e
for
a
DT
system
with
the
following
poles:
2.
Maximum
gain
For
each
of
the
following
systems,
find
the
frequency
ω
m
for
which
the
magnitude
of
the
gain
is
greatest.
1
a.
2
1 +
s
+
s
b.
1 +
s
s
+
s
2
2
s
c.
2
1 +
s
+
s
Compare
the
ω
m
for
these
systems
and
explain
qualitatively
any
similarities
or
differences.
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 Spring '11
 DennisM.Freeman
 #, Shock absorber, Suspension, 2 Pole

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