MAE 326 Spring 2008 HW 11, due Wed., 16 April
Problems 14 are for review purposes and are optional extra credit, but recommended if
you had trouble with problems 1 or 3 in Prelim 2.
1.
A
=
ˆ
6
2
0

2
!
(1)
Find the eigenvalues and eigenvectors of
A
. Do the calculation by hand.
2.
B
=
1
2
3
0
2

1
0
0
1
(2)
One of the eigenvalues of this matrix is
λ
= 2. The associated eigenvector is
v
=
2
1
0
Find all solutions,
x
to the equation
B
x
= 2
x
.
3. For the matrix
A
from problem 1, if ˙
x
=
A
x
, sketch the trajectories of solutions in the
neighborhood of the origin.
4. Consider two masses,
m
1
= 1
kg
and
m
2
= 0
.
2
kg
connected by a spring with stiffness
k
= 1
N/m
as shown in figure 1.
The masses are immersed in a viscous fluid that
provides damping,
b
1
=
b
2
= 0
.
05
N/
(
m/s
). The masses and springs are constrained to
move in one dimension.
Input to this system is a force
f
(
t
) applied to mass 1 in the direction that would cause
the spring to initially go into compression. Outputs of this model are the position and
velocity of mass 1.
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 Spring '08
 PSIAKI
 1Kg, 0.2kg, 9.81m, 0.1m/s

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