Math 334 Lecture #19
§
3.8: Periodically Forced Vibrations
Undamped Periodically Forced Motion.
This occurs when
γ
= 0:
mu
+
ku
=
F
0
cos
ωt,
where
F
0
is the forcing amplitude and
ω
is the forcing frequency.
[Recall that the natural frequency of undamped free motion is
ω
0
=
k/m
.]
Double Harmonic Motion
. This occurs when
ω
0
=
ω
, in which case a particular solution
has the form,
U
p
=
A
cos
ωt.
Substitution of
U
p
into the ODE gives

mω
2
A
cos
ωt
+
kA
cos
ωt
=
F
0
cos
ωt
⇒
A
=
F
0

mω
2
+
k
=
F
0

mω
2
+
mω
2
0
=
F
0
m
(
ω
2
0

ω
2
)
[
ω
0
=
k/m
⇒
k
=
mω
2
0
]
.
The general solution is
u
=
c
1
cos
ω
0
t
+
c
2
sin
ω
0
t
+
F
0
m
(
ω
2
0

ω
2
)
cos
ωt
which is the superposition of two simple harmonic motions of different frequencies and
amplitudes, what may be called
double harmonic motion
.
If
u
(0) = 0 and
u
(0) = 0, then the solution of the IVP is
u
=

F
0
m
(
ω
2
0

ω
2
)
cos
ω
0
t
+
F
0
m
(
ω
2
0

ω
2
)
cos
ωt
=
F
0
m
(
ω
2
0

ω
2
)
cos
ωt

cos
ω
0
t
=
2
F
0
m
(
ω
2
0

ω
2
)
sin
(
ω
0

ω
)
t
2
sin
(
ω
0
+
ω
)
t
2
[by a trig identity]
.
The term in the square brackets is a periodically varying amplitude, and the last sine
function is a sinusoidal vibration.
When the forcing frequency is close to the natural frequency, the sum
ω
0
+
ω
is much
larger than the difference
ω
0

ω
, so the resulting motion is a rapid sinusoidal vibration
with a slowly varying periodic amplitude, what is called a
beat
or amplitude modulation
(AM).
Example.
Solve the IVP
9
.
082
u
+ 890
.
0
u
= 0
.
5 cos 8
.
799
t,
u
(0) = 0
,
u
(0) = 0
.
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Since the natural frequency of the free motion is
ω
0
=
k/m
= 9
.
899, the forcing
frequency is
ω
= 8
.
799, and forcing amplitude is
F
0
= 0
.
5, the solution of the IVP is
(approximately)
u
=

0
.
02677 cos(9
.
899
t
) + 0
.
02677 cos(8
.
799
t
)
= 0
.
0590 sin(0
.
550
t
) sin(9
.
349
t
)
.
Here is the graph of this solution.
0.02
0.01
0.0
!
0.01
!
0.02
!
0.03
!
0.04
t
!
0.05
0.03
!
0.06
15.0
12.5
0.05
10.0
7.5
5.0
2.5
0.0
0.06
0.04
Resonant Motion
. This occurs when
ω
=
ω
0
.
Because
F
0
cos
ωt
is a solution of the associated homogeneous ODE, the form of a par
ticular solution is
U
p
=
At
cos
ω
0
t
+
Bt
sin
ω
0
t.
The general solution is
u
=
c
1
cos
ω
0
t
+
c
2
sin
ω
0
t
+
F
0
2
mω
0
t
sin
ω
0
t.
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 Fall '08
 DALLON
 Math, Differential Equations, Equations, Simple Harmonic Motion, Cos

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