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The differentialequationsofmotion for 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.Determine the circular frequencies of vibration and the
corresponding modeshapes.
20.
L
L
L
L
i
2
i
3
i
4
i
1
i
2
i
1
i
3
i
4
C
C
/2
C
/3
C
/4
C
/5
Kirchoff'sequationsfor the circuit are
L
d
2
i
1
dt
2
1
C
i
1
+
2
C
(
i
1
−
+
i
2
)
=
0
L
d
2
i
2
dt
2
2
C
(
i
2
−
3
C
(
i
2
−
+
i
1
)
+
i
3
)
=
0
L
d
2
i
3
dt
2
3
C
(
i
3
−
4
C
(
i
3
−
+
i
2
)
+
i
4
)
=
0
L
d
2
i
4
dt
2
4
C
(
i
4
−
5
C
i
4
=
+
i
3
)
+
0
Find the circular frequencies of the currents.
21.
C
C
/2
C
/3
C
/4
i
1
i
4
i
2
i
3
i
2
i
1
i
3
i
4
L
L
L
L
L
Determine the circular frequencies of oscillation for the circuit shown, given the
Kirchoff equations
L
d
2
i
1
dt
2
dt
2
L
d
2
i
1
dt
2
d
2
i
2
1
C
i
1
=
+
−
+
0
L
d
2
i
2
dt
2
dt
2
L
d
2
i
2
dt
2
dt
2
d
2
i
1
d
2
i
3
2
C
i
2
−
+
−
+
=
0
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