Civil Engineering Reference
In-Depth Information
When we introduce the following notations:
2
2
2
2
mȦ K , mȦ k , mhȦ k
[4.21]
s
h
h
ș
ș
and then eliminate u
0
and T between the three previous equations, we obtain:
ª
2
2
2
º
2
ZZ
[
Z[ Z
[
12i
12i
[4.22]
12i
u
u
«
»
ZZ[Z[ Z
g
2
2
12i
2
12i
2
¬
¼
s
h
h
T
T
s
Taking into account the fact that [, [
h
, [
T
1, the previous equation becomes:
ª
2
2
2
º
2
ZZ
Z
Z
[4.23]
12i
[
12i 2i)
[[
12i 2i)u
[[
u
«
»
2
2
h
2
T
2
g
ZZ
Z
Z
¬
¼
s
h
T
s
Let us now consider a simple oscillator with 1 degree of freedom and the same
mass m, with its characteristic pulsation
~
and damping
~
submitted to the
harmonic displacement
~
with a pulsation Z at its base (case of a structure
embedded at its base). The harmonic response of the oscillator is given by:
g
2
2
§
·
Z
Z
12
i
[
u
u
[4.24]
¨
¸
¨
¸
g
2
2
Z
Z
©
¹
The equivalent oscillator will have the same response as the structure in Figure
4.11 if the following equations are verified:
1 111
ZZZZ
[4.25]
2
2
2
2
s
h
T
2
2
2
ZZ Z
[
[ [ [
ZZ Z
[4.26]
h
-
2
2
2
s
h
T
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