Part I: theory
(Closed Book)
Question 1
(weight:3)
Given:
Consider a linear, time invariant, strictly stable, system
U
(
s
)
Y
(
s
)
G
(
s
)
?
with transfer function
G
(
s
)
. Suppose to supply as an input a sinusoidal input equal to:
u
(
t
) =
M
sin
ωt
of magnitude
M >
0
and frequency
ω
rad/sec.
Asked:
Prove that, waiting long enough, the output of the system for the given sinusoidal input
will be
y
(
t
)
≃
M

G
(
jω
)

sin(
ωt
+
∠
G
(
jω
)
)
I recall that:
L{
sin(
ωt
)
}
=
ω
s
2
+
ω
2
,
L
b
e
αt
B
=
1
s
−
α
,
G
(
s
∗
) =
G
(
s
)
∗
and the Euler formula
sin(
α
) =
e
jα
−
e
−
jα
2
j
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentPart II: exercises
(Open Book)
Question 2
(weight:3)
Given:
The input of the system illustrated below is the force
F
applied to the mass
m
1
.
F
k
x
1
x
2
m
1
m
2
b
1
b
2
The system is composed of two masses
m
1
and
m
2
, damper
b
1
connects mass
m
1
to the wall,
while damper
b
2
and spring
k
connect the two masses. The energy stored in the spring
k
is equal
to
E
(Δ
x
) =
1
2
·
k
1
·
Δ
x
2
+
1
4
·
k
2
·
Δ
x
4
where
Δ
x
=
x
1
−
x
2
. The damping effects can be modeled by the linear relation
F
b
=
b
·
v
where for each damper the the accompanying damping coef£cient
b
and velocity
v
needs to be
substituted.
Asked:
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 Taufik
 GC

Click to edit the document details