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CHAPTER 12
FORCED OSCILLATIONS
12.1
More on Differential Equations
In Section 11.4 we argued that the most general solution of the differential equation
ay
by
cy
"'
+
+
=
0
11.4.1
is of the form
yA
f
x
Bg x
=
+
()
.
11.4.2
In this chapter we shall be looking at equations of the form
ay
by
cy
h x
(
)
.
+
+
=
12.1.1
If you look back at the arguments that led to the conclusion that equation 11.4.2 is the most general
solution of equation 11.4.1, you will be able to conclude that 11.4.2 is still a solution of equation
11.4.1, but it is not the only solution.
There is another function that is a solution, so that the most
general solution to equation 12.1.1 is of the form
f
x
Bg x
H x
=
+
+
12.1.2
The solution
H
(
x
) is called the
particular integral
, while the part
Af
(
x
)
+ Bg
(
x
)
is the
complementary function.
I shall be dealing in this chapter mainly with the particular integral,
though we shall not entirely forget the complementary function.
This is a book on classical mechanics rather than on differential equations, so I am not going into
how to obtain the particular integral
H
(
x
) for a given
h
(
x
).
There are several ways of doing it; for
those who know what they are and are in practice with them, Laplace transforms are among the
more attractive methods.
Some readers will already know how to do it.
They will doubtless want
to go back to equation 11.6.3 in the previous chapter and try their hand at finding the particular
integral for that.
Those who do not may be happy and content to take my word for the particular
integral in the sections that follow, or perhaps at least to differentiate it to verify that it is indeed a
solution.
12.2
Forced Oscillatory Motion.
We are thinking of a mass
m
attached to a spring of force constant
k
and subject to a damping
force
x
b
&
, but also subject to a periodic sinusoidal force
.
cos
ˆ
t
F
ω
The equation of motion is
,
cos
ˆ
t
F
kx
x
b
x
m
ω
=
+
+
&
&
&
12.2.1
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or, if we divide by
m
:
.
cos
ˆ
2
0
t
f
x
x
ω
=
ω
+
γ
+
&
&
&
12.2.2
Here
.
/
ˆ
ˆ
,
/
,
/
2
0
m
F
f
m
k
m
b
=
=
ω
=
γ
ω
is the forcing angular frequency and
ω
0
is the natural
frequency of mass and spring in the absence of damping.
One part of the general solution of
equation 12.2.2 is the complementary function, which we have dealt with at length in Chapter 11.
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