ABSTRACT
A.1 Single Fourier Series The Fourier series are indispensable aids in the analytical treatment of many problems in the field of applied mechanics. The representation of periodic functions using trigonometric series is commonly called the Fourier series expansion. A function f(x) defined in the interval (–L,L) and determined outside of this interval by f(x + 2L) = f(x) is said to be periodic, of period 2L. Here L is a nonzero constant; for example, for sin x there are periodic 2 2 4π π π, , ,....− The Fourier expansion corresponding to f(x) is of the form [1]
f x a
a n x
L b
n x Ln nn
( ) cos sin= + +
=
π π (A.1)
in which a0, an, and bn are the Fourier coefficients. To determine the coefficients an, we multiply both sides of Equation A.1 by cos
( / ),m x Lπ integrate over the interval of length 2L, and make use of the orthogonality relations:
cos cos
cos sin
= ≠ =
L
m x L
n x L
dx m n L m n
m x L
π π
π
n x L
dx m n
m x L
n x L
dx m n
L mL L
π
π π
=
= ≠ −∫
for all ,
sin sin =
n .
