Graphics Programs Reference
In-Depth Information
The differentialequationsofmotion of the mass-spring systemare
k
(
−
2
u
1
+
u
2
)
=
m u
1
k
(
u
1
−
2
u
2
+
u
3
)
=
3
m u
2
k
(
u
2
−
2
u
3
)
=
2
m u
3
where
u
i
(
t
) is the displacementofmass
i
fromits equilibrium position and
k
is
the spring stiffness.Substituting
u
i
(
t
)
=
y
i
sin
ω
t
, weobtain the matrix eigenvalue
problem
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
2
−
1
0
y
1
y
2
y
3
1 00
0 3 0
002
y
1
y
2
y
3
2
m
ω
−
12
−
1
k
−
0
12
Determine the circular frequencies
ω
and the corresponding relative amplitudes
y
i
of vibration.
14.
u
1
u
n
u
2
k
3
k
n
k
1
k
2
m
m
m
The figure shows
n
identical masses connectedbyspringsofdifferentstiffnesses.
The equationgoverning free vibration of the systemis
Au
=
ω
2
u
, where
ω
m
is the
circular frequency and
⎡
⎣
⎤
⎦
k
1
+
k
2
−
k
2
0
0
···
0
−
k
2
k
2
+
k
3
−
k
3
0
···
0
0
−
k
3
k
3
+
k
4
−
k
4
···
0
A
=
.
.
.
.
.
.
.
.
.
.
.
.
0
···
0
−
k
n
−
1
k
n
−
1
+
k
n
−
k
n
0
···
0
0
−
k
n
k
n
k
1
k
2
···
k
n
T
Giventhe spring stiffness array
k
=
,writeaprogramthatcomputes
the
N
lowest eigenvalues
λ
=
m
ω
2
and the corresponding eigenvectors.Runthe
programwith
N
=
4 and
400
400 200
T
k
=
400
400
0
.
2 400
kN/m
Note that the systemis weakly coupled,
k
4
being small.Do the resultsmake sense?
15.
L
x
n
1
2
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