544
15 Random Vibrations
15.8.1 Time Domain Analysis
Indicating x(t) as the response of the structure, Duhamel’s integral given by Eq. (3.91)
can be written in terms of unit impulse response function of Eq. (3.132), as
x(t) =
∞
−∞
h(τ )E[F(t − τ )]dτ
(15.40)
where h(τ ) is the unit impulse response function. If x(t) and F(t) are stationary
random processes, then the expectation of both sides of the integral is
E[x(t)] =
∞
−∞
h(τ )E[F(t − τ )]dτ
(15.41)
If m x and m F denote the mean values of x(t) and F(t), then Eq. (15.41) can be
written as
m x = m F
∞
−∞
h(τ )dτ
(15.42)
Equation (15.42) gives the relationship between the mean values of the input and
output of the process.
If we consider the autocorrelation R F F (τ ) for the excitation and R xx (τ ) for the
response, then
R xx (τ ) = E[x(t)x(t + τ )] =
∞
−∞
∞
−∞
h(θ 1 )h(θ 2 )E[F(t − θ 1 )
F(t + τ − θ 2 )]dθ 1 dθ 2
=
∞
−∞
∞
−∞
h(θ 1 )h(θ 2 )R F F (τ + θ 1 − θ 2 )dθ 1 dθ 2
(15.43)
15.8.2 Frequency Domain Analysis
The frequency domain analysis is much more popular than time domain analysis. If
x(t) is the response of the system due to an excitation F(t), then from Eq. (3.129),
we know