110
3 Forced Vibration of Single Degree of Freedom System
This response function is known as unit impulse response function. This response
function describes the response of the system in time domain, whereas the complex
frequency response function describes its response in the frequency domain.
By the definition of unit impulse, the Fourier transform of the unit impulse is
given by
F(ω) =
∞
− ∞
F(t) e
− iωt dt = 1
Therefore,
x (t) = h (t) =
1
2π
∞
− ∞
H (ω) F(ω) e
iωt dω
(3.132)
or
h (t) =
1
2π
∞
− ∞
H (ω) e
iω t dω
h(t) is thus the inverse Fourier transform of H (ω ) and therefore
H (ω ) =
∞
− ∞
h (t) e
− iω t dt
(3.133)
Thus, h(t) and H (ω ) are Fourier transform pairs.
3.17 Support Motion
Determination of the response of the system due to movement of the support rather
than the application of external force constitutes an important class of problems. Due
to earthquake, the supports of the structure move. The analysis due to these effects
can be performed by the displacement approach or the acceleration approach. Let us
consider both the approaches one after another.