15.8 Random Response of SDF Systems
545
X (ω) = H (ω) F(ω)
(15.44)
where X (ω) and F(ω) are Fourier transforms of x(t) and F(t) and H (ω) is the
complex frequency response function. For a single degree freedom system having
damping and stiffness, we know from Eq. (3.111) that
H (ω) =
1
−mω 2 + icω + k
=
1/k
1 −
ω
p
2
+ i2ζ
ω
p
Taking complex conjugate of all the quantities in Eq. (15.44), we get
X ∗ (ω) = H ∗ (ω) F ∗ (ω)
(15.45)
Multiplying Eq. (15.44) by Eq. (15.45), we obtain
X (ω)X ∗ (ω) = H (ω)H ∗ (ω)F(ω)F ∗ (ω)
or
X (ω)X ∗ (ω) = |H (ω)|
2 F(ω)F ∗ (ω)
(15.46)
Dividing both sides of Eq. (15.46) by T, and taking limits as T → ∞, we get
lim
T →∞
1
T
|X (ω)|
2
= |H (ω)|
2 lim
T →∞
1
T
|F(ω)|
2
(15.47)
From the definition of power spectral density as given by Eq. (15.31), we can
write
S xx (ω) = lim
T →∞
1
T
|X (ω)|
2
(15.48)
and
S F F (ω) = lim
T → ∞
1
T
| F(ω)|
2
(15.49)
S xx (ω) and S F F (ω) are the spectral densities of the response and the excitation.
Combining Eqs. (15.47)–(15.49), we get
S xx (ω) = |H (ω)|
2 S F F (ω)
(15.50)
Equation (15.50) gives the relationship between the spectral densities of excitation
and response. In the remaining part of the chapter, we shall analyze the different
systems by frequency domain approach.