ESE 35101, Fall 2017
Homework Set #9, due Nov. 7
General note on sketches: It is OK to use a computer to do your “sketches”, but by
hand is also OK if it catches the key behaviors.
1
. Consider the following sets of input and transfer functions.
a
( )
r t
( )
=
δ
t
( )
H s
( )
=
1
s
+
1
b
( )
r t
( )
=
1
t
( )
H s
( )
=
1
s
+
1
c
( )
r t
( )
=
t
⋅
1
t
( )
H s
( )
=
1
s
+
1
d
( )
r t
( )
=
δ
t
( )
H s
( )
=
1
s
2
+
2
s
+
5
e
( )
r t
( )
=
1
t
( )
H s
( )
=
1
s
2
+
2
s
+
5
f
(
)
r t
( )
=
sin2
t
(
)
1
t
( )
H s
( )
=
1
s
+
2
For each case, use the Final Value Theorem, if applicable, to find the steadystate value
of
y
(
t
). If the FVT is not applicable, state why.
2.
Mukai 10.7.6. Skip part (h).
3.
Mukai 10.7.7. Skip part (h).
4.
Mukai 10.7.8. Skip part (h). (Note: this system is not BIBO stable, but apply frequency
analysis as usual anyway. Notice anything funny with the gain plot?)
5.
The picture below depicts the situation of an unbalanced rotating mass.
y
K
B
m
0
Total mass =
M
y
r
e
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ESE 35101, Fall 2017
Homework Set #9, due Nov. 7
The rotating mass,
m
0
, is at a radius of
e
from the center of rotation. The vertical distance
from that center is
y
r
, which can be described as
e
cos
ω
r
t
. where an angle of 0 means
straight up.
Looking at only vertical motion, we can derive an equation of motion.
M
−
m
0
(
)
!!
y t
( )
=
−
m
0
!!
y t
( )
+
!!
y
r
t
( )
⎡
⎣
⎤
⎦
−
Ky t
( )
−
B
!
y t
( )
where
!!
y
r
=
−
e
ω
r
2
cos
ω
r
t
M
!!
y t
( )
+
B
!
y t
( )
+
Ky t
( )
=
m
0
e
ω
r
2
cos
ω
r
t
≡
u t
( )
(a) Find the transfer function,
H
(
s
) =
Y
(
s
)/
U
(
s
).
(b) Find the complex frequency response,
H
(
i
ω
) and its magnitude, 
H
(
i
ω
).
(c) Given
u t
( )
=
m
0
e
ω
r
2
cos
ω
r
t
, find the magnitude of the steadystate response, 
y
ss
(
t
).
(d) We define “natural frequency”,
ω
n
, damping ratio,
ζ
, and the frequency ratio,
r
, as
follows.
ω
n
=
K
M
ζ
=
1
2
B
MK
,
or
B
M
=
2
ζω
n
r
=
ω
r
ω
n
Write the normalized magnitude,
M

y
ss
(
t
)/(
m
0
e
), as a function only of
r
and
ζ
.
(e) Sketch the normalized magnitude from part (d) for these values of
ζ
: 0.25, 0.5 and
0.75, and for 0 <
r
< 5. (You may put them all on one sketch.)
6.
Mukai 10.7.14 modified. Parts (a)(f) only. Part (d): sketch only; no
comments/answers.
7.
Mukai 10.7.17 modified. Parts (a)(f) only. Part (d): sketch only; no
comments/answers.
8
. Consider the
weighted
2point moving average below.
y k
( )
=
2
3
u k
( )
+
1
3
u k
−
1
(
)
Repeat Mukai 10.7.12, parts (a)(d) only, without comments, for this system. (Note: there
is not some easy trick to factor out a real term, like in the previous two problems. You
will need Euler’s formula to do part (c). I suggest a computer for part (d); I used Excel.)
ESE 35101, Fall 2017
Homework Set #9, due Nov. 7
Solutions
1
. Consider the following sets of input and transfer functions.
a
( )
r t
( )
=
δ
t
( )
H s
( )
=
1
s
+
1
b
( )
r t
( )
=
1
t
( )
H s
( )
=
1
s
+
1
c
( )
r t
( )
=
t
⋅
1
t
( )
H s
( )
=
1
s
+
1
d
( )
r t
( )
=
δ
t
( )
H s
( )
=
1
s
2
+
2
s
+
5
e
( )
r t
( )
=
1
t
( )
H s
( )
=
1
s
2
+
2
s
+
5
f
(
)
r t
( )
=
sin2
t
(
)
1
t
( )
H s
( )
=
1
s
+
2
For each case, use the Final Value Theorem, if applicable, to find the steadystate value
of
y
(
t
). If the FVT is not applicable, state why.
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 Spring '14
 Fuhrmann