3.15 Response Due to Non-periodic Excitation
107
(n + 1) ω 0 − nω 0 =
2π
T
(3.119)
is limited by the size of T.
The difficulties of this procedure could be removed by introducing minor
adjustments, that is
ω = ω 0 =
2π
T
and nω 0 = nω = ω n
(3.120)
Further, introducing the notation
F (nω 0 ) = T C n
(3.121)
or
ω · F (nω 0 ) = 2π C n
Equations (3.106) and (3.105) can be written as
F (ω n ) = T C n =
T /2
− T /2
F (t) exp (− iω n t) dt
(3.122)
and
F (t) =
ω
2π
∞
n =−∞
F (ω n ) exp (iω n t)
(3.123)
It may be noted that limits of the integral are arbitrary, so long its one complete
period is considered.
Now, we consider the loading period to be tending to infinity, which suggests
that the frequency increment ω tends to dω. The discrete frequencies ω n become a
continuous function of ω.
Thus, in the limit, the Fourier series expression of Eq. (3.123) becomes the
following Fourier integral
F (t) =
1
2π
∞
− ∞
F (ω) e
iω t dω
(3.124)
Equation (3.122) can be written as
107
(n + 1) ω 0 − nω 0 =
2π
T
(3.119)
is limited by the size of T.
The difficulties of this procedure could be removed by introducing minor
adjustments, that is
ω = ω 0 =
2π
T
and nω 0 = nω = ω n
(3.120)
Further, introducing the notation
F (nω 0 ) = T C n
(3.121)
or
ω · F (nω 0 ) = 2π C n
Equations (3.106) and (3.105) can be written as
F (ω n ) = T C n =
T /2
− T /2
F (t) exp (− iω n t) dt
(3.122)
and
F (t) =
ω
2π
∞
n =−∞
F (ω n ) exp (iω n t)
(3.123)
It may be noted that limits of the integral are arbitrary, so long its one complete
period is considered.
Now, we consider the loading period to be tending to infinity, which suggests
that the frequency increment ω tends to dω. The discrete frequencies ω n become a
continuous function of ω.
Thus, in the limit, the Fourier series expression of Eq. (3.123) becomes the
following Fourier integral
F (t) =
1
2π
∞
− ∞
F (ω) e
iω t dω
(3.124)
Equation (3.122) can be written as