Civil Engineering Reference
In-Depth Information
its limitations. In order to simplify, we will use a 1-dof system for the presentation,
as extending the method to N-dof systems does not raise any specific problem.
Given a non-linear oscillator with a motion equation:
2
2
dx/dt
h x, dx/dt Ȗ t
[8.57]
J(t) being stationary, we are interested in the proven solution for [8.57], for example,
we wish to replace this oscillator with an
equivalent linear oscillator
of an equation:
2
2
dy/dt
2Ȧİdy/dt Ȧ y Ȗ t
[8.58]
eq
eq
2
eq
The y(t) displacement, the solution to [8.58], does not verify [8.57], but we want
to adjust Z
eq
and H
eq
so that it is as close as possible to its solution x(t). The
adjustment is done by minimizing, in the mean squares sense, the difference
between the non-linear term and the equivalent linear term both applied to the exact
solution of the problem (equation [8.57]). The Z
eq
and H
eq
thus obtained are such that
the variance of the response calculated with [8.58] corresponds quite well to the
solution value of [8.57].
Let us call *(t), X(t) and Y(t) the random processes associated with J(t), x(t) and
y(t).
Unfortunately, the statistic characteristics of the X(t) “non-linear response”
process are not known
a priori
. In practice, we will therefore try to minimize the e(t)
gap between the non-linear term and its linear equivalent applied to the Y(t)
“equivalent linear response” process. This approximation explains the low efficiency
of the method in some cases:
2
et
hY,dY/dt
2ȦİdY/dt+Ȧ Y
[8.59]
eq
eq
eq
Z
eq
and H
eq
are constants given by the equations:
ª
2
º
ª
2
º
2
w
Eet
/ Ȧ
w
w
Eet
/ İȦ 0
w
[8.60]
«
»
eq
«
»
eq
eq
¬
¼
¬
¼
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