文档介绍：Université de Versailles et Saint Quentin-en-Yvelines
LISV - Laboratoire d’Ingénierie des Systèmes de Versailles
Equipe MMS : Mécanique des Matériaux et des Structures
Bâtiment Descartes – RC27
45, Avenue des Etats-Unis
78035 Versailles – FRANCE
Master 2 SPI – DSME
Responsable: Paolo I (paolo.******@)
.: Mécanique pour l’Industrie du Pétrole
Cours: Offshore Engineering
Enseignant: P. i
. 2006-07
Examples of offshore structures calculation
1. Response of a single-degree of freedom structure to a white noise.
A white noise is pletely random signal; this means that the energy is equally distributed
in the frequency domain, . the spectrum is a constant, see the figures. We want to analyse
the response of a single-degree of freedom structure to a white noise.
The mean square of the response is
∞∞
σ 2(t) = S ( f )df = H (if ) 2 S ( f )df .
x ∫0 xx ∫0 FF
The transfer function of such a system is
1
H (iω) = ,
f 2
K − Mω f + i(Cω f )
and with the angular frequency f=ω/2π we get
1
H (if ) = .
K − 4π 2 f 2M + i2π f C
The spectrum of the force is of the type
SFF ( f ) = S,
so that
∞ S
σ 2(t) = df .
x ∫0 2
()K − 4π 2 f 2M + ()2π f C 2
To perform integration, it is worth to transform the above expression as
∞ S 1 K
2
σ x (t) = ∫dΩ
0 2 2 2 2 2π M
K ()1−Ω+ ()2ξΩ
with
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_______________________________________P. i – Examples of offshore structures calculation
2πf C
Ω= , ξ= .
K 2 MK
M
In the case, relevant for real structures, of small ξ, we get
2 S K S
σ x (t) = = .
8ξ K 2 M 4 C K
The standard deviation of the response is hence
1 S K 1 S
σ(t) = = .
x 2K 2ξ M 2 C K
The distribution of the response being Gaussian with zero mean, the probability that the
response be less than σx in absolute value is % (. for the % of the time the
absolute value of the response is less than σx), while it