3.15 Response Due to Non-periodic Excitation
109
Fig. 3.35 Variation of F(ω) with ω
or
x (t) =
1
2π
∞
− ∞
X (ω) e
iωt dω
(3.128)
where
X (ω) = H (ω) F(ω)
(3.129)
3.16 Relationship Between Complex Frequency Response
Function and Unit Impulse Response Function
For a unit impulse applied at t =0, the response from Eq. (3.89) is given by
x (t) =
sin pt
mp
(3.130)
In this case
∞
− ∞
F (t) dt = 1
For a damped system, the response due to unit impulse is given by
h(t) =
1
m
p 2 − n 2
exp (− nt) sin
p 2 − n 2 t
(3.131)
109
Fig. 3.35 Variation of F(ω) with ω
or
x (t) =
1
2π
∞
− ∞
X (ω) e
iωt dω
(3.128)
where
X (ω) = H (ω) F(ω)
(3.129)
3.16 Relationship Between Complex Frequency Response
Function and Unit Impulse Response Function
For a unit impulse applied at t =0, the response from Eq. (3.89) is given by
x (t) =
sin pt
mp
(3.130)
In this case
∞
− ∞
F (t) dt = 1
For a damped system, the response due to unit impulse is given by
h(t) =
1
m
p 2 − n 2
exp (− nt) sin
p 2 − n 2 t
(3.131)