106
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.32 Plot of (π k |x n |/2 A) versus n
3.15 Response Due to Non-periodic Excitation
In many physical problems such as earthquake and wave loading, the period of
forcing function cannot be observed. As such they are non-periodic. For this class of
problems, Fourier transform method is used. Further, for certain category of problems, it is more convenient to perform frequency domain analysis than time domain
analysis.
The Fourier series concept can be extended to incorporate the non-periodic
loading. An arbitrary non-periodic loading is shown in Fig. 3.33. The coefficient
C n of this loading is given by Eq. (3.106) which can be obtained in the interval 0
< t < T, if it is made periodic as shown by the dotted line. However, this repetitive
loading has been assumed to be imaginary and can be eliminated by extending the
loading period to infinity. So, the Fourier series expression is reformulated to take
into account this infinite time range.
Use the notation defined below:
1
T
=
ω 0
2π
(3.118)
As T approaches infinity, ω 0 =
2π
T
tends to zero. Hence, each component of the
series is nearly equal to the adjacent one. In other words, the smallness of the quantity
Fig. 3.33 Non-periodic loading
3 Forced Vibration of Single Degree of Freedom System
Fig. 3.32 Plot of (π k |x n |/2 A) versus n
3.15 Response Due to Non-periodic Excitation
In many physical problems such as earthquake and wave loading, the period of
forcing function cannot be observed. As such they are non-periodic. For this class of
problems, Fourier transform method is used. Further, for certain category of problems, it is more convenient to perform frequency domain analysis than time domain
analysis.
The Fourier series concept can be extended to incorporate the non-periodic
loading. An arbitrary non-periodic loading is shown in Fig. 3.33. The coefficient
C n of this loading is given by Eq. (3.106) which can be obtained in the interval 0
< t < T, if it is made periodic as shown by the dotted line. However, this repetitive
loading has been assumed to be imaginary and can be eliminated by extending the
loading period to infinity. So, the Fourier series expression is reformulated to take
into account this infinite time range.
Use the notation defined below:
1
T
=
ω 0
2π
(3.118)
As T approaches infinity, ω 0 =
2π
T
tends to zero. Hence, each component of the
series is nearly equal to the adjacent one. In other words, the smallness of the quantity
Fig. 3.33 Non-periodic loading