|
|
In a Nut Shell: A Fourier Series is an infinite series of
sine and cosine terms used
to represent any
periodic function. Suppose f(t)
is a periodic function. Then
the
Fourier Series expansion
of f(t) is
|
∞
f(t)
= ao
/2 + ∑ an cos
nt
+ bn
sin nt
n = 1
|
|
|
The objective is to determine the Fourier
coefficients, ao, an and
bn .
The Fourier Series
representations of the function, f(t),
may contain only
cosine terms, only sine
terms, or both cosine terms and sine terms depending on
whether the function,
f(t), is an even function, an odd function, or neither even
nor odd. More information on details on these
terms follows.
|
|
Why discuss Fourier Series?
One reason is that it
provides a way to represent more complicated (more realistic)
forcing functions as,
for example, with applications to vibration problems with a
forcing function,
f(t). Let x
= x(t) where x(t) is the displacement of the
mass, m,
with time t, c is the damping constant, and k is
the spring rate. The differential
equation of motion for
forced vibrations of mass, m, is:
m d2x/dt2 +
c dx/dt +
k x = f(t)
|
|
What is a periodic function? If P is the period of the function, f(t),
then the value
of f(t)
repeats for every period. In
other words, f(t) =
f (t+ P). The period of
a function, f(t), can
take on any value.
Case
1: f(t) has a period, P, of 2π. The Fourier series expansion of f(t) is then
|
∞
f(t)
= ao
/2 + ∑ an cos
nt
+ bn
sin nt
n = 1
where ao, an, and
bn are the Fourier coefficients.
|
π π
π
ao
= (1/π) ∫ f(t) dt , an =
(1/π) ∫ f(t) cos nt dt , bn
= (1/π) ∫ f(t) sin
nt dt
˗π
˗π
˗π
|
Click here to continue
with discussion of Fourier Series.
|