Civil Engineering Reference
In-Depth Information
f
³
2
Variance of X (t) = ıȡ0
S (f) df = m
0 order spectral moment
0
0
Physically, for the signal, the PSD represents the energy distribution according to
the frequency. The notion of dX/dt(t) derivative process will also be used; it is
defined as a quadratic mean by:
^
`
2
ª
º
Expected value: E
X t+İ Xt /İ dX/dt (t)
o
0
¬
¼
İ
o
0
For a stationary process, the derivative process exists if d
2
U/dW
2
exists for W = 0.
The derivative process is “orthogonal” to the initial process for:
>
@
2
E X (t) dX/dt (t)
dȡ/dIJ (0) = 0, and its PSD is equal to (2ʌf) S (f).
Thus, the variance of the derivative process for a zero- average signal is:
2
2
³
2
2
ı
2ʌ fSf df =2ʌ m
[8.12]
der
2
0
where m
2
= second order spectral moment.
8.4.3.
Response of a linear system to random stress
Generally we know how to express the second order characteristics of the
response of a linear system characterized by its transfer functions to stress in the
shape of a space-time random field that is also second order characterized.
In order not to make this chapter too heavy, we will develop the formulation for
the case described previously of a structure loaded with an imposed acceleration of
its anchoring.
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