October 4, 2011
121
12. Forced Oscillations
We consider a massspring system in which there is an
external oscillating force applied.
One model for this is that the support of the top of
the spring is oscillating with a certain frequency.
The equation of motion becomes
m
¨
u
+
γ
˙
u
+
ku
=
F
0
cos
(
ωt
)
.
(1)
Let us ﬁnd the general solution using the complex func
tion method.
First assume that
γ
= 0; i.e., there is no friction.
The given equation is the real part of the complex
equation
m
¨
u
+
ku
=
F
0
e
i
ω
t
.
Let
ω
0
=
r
k
m
be the natural frequency of the unforced
equation. If
ω
6
=
ω
0
, then the general complex solution
is
u
(
t
) =
c
1
cos(
ω
0
t
) +
c
2
sin(
ωt
) +
Re
(
F
0
m
(
iω
)
2
+
k
e
i
ω
t
)
=
R
cos(
ω
0
t

δ
) +
Re
(
F
0
m
(
iω
)
2
+
k
e
i
ω
t
)
=
R
cos(
ω
0
t

δ
) +
F
0
m
(
k
m

ω
2
)
cos(
ωt
)
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122
=
R
cos(
ω
0
t

δ
) +
F
0
m
(
ω
2
0

ω
2
)
cos(
ωt
)
.
Resonance:
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