AE 352: Aerospace Dynamics II, Fall 2008
Homework 3
Due Friday, September 19
Problem 1.
Find the free and forced responses of the following system:
d
2
y
dt
2
+
4
dy
dt
+
4
y
=
3
du
dt
+
2
u
where
u
(
t
)
=
0 for
t
<
0 and
u
(
t
)
=
e

3
t
for
t
≥
0.
Solution.
There are multiple ways to solve this problem.
We could use the
method of undetermined coe
ffi
cients, or the convolution integral (impulse re
sponse) method. To use the impulse response method, we must first find the free
response, that is, solve:
d
2
y
dt
2
+
4
dy
dt
+
4
y
=
0
To solve, assume a solution
y
(
t
)
=
Ce

at
. Substitute this solution (and its deriva
tives) back into the homogeneous ODE:
C
(
a
2

4
a
+
4)
e

at
=
0
Solving for
a
we get
a
=
2
,
2. When there is a repeated root, the solution becomes:
y
(
t
)
=
C
1
e

2
t
+
C
2
te

2
t
In terms of the initial conditions:
C
1
=
y
0
, and
C
2
=
v
0
+
2
y
0
. Thus the free response
of the system is:
y
h
(
t
)
=
y
0
e

2
t
+
(
v
0
+
2
y
0
)
te

2
t
Next, we must find the particular solution (forced response) of the system. We
will calculate the forced response using the convolution integral. First, simplify
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 Digital Signal Processing, Velocity, Impulse response

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