STRUCTURAL RESPONSE STATISTICS: PART I
177
With this last resuit, équation (7.48) becomes
Sv(w) = ^lff(w)|25Pi(w)
(7.50)
This last remarkably simple and useful resuit relates the power spectral
density of pj (t) to the power spectral density of v(t) through the complex frequency response function. As shown in Example Problem 7.2, Spl(u>) is given
by équation (7.26) in terms of S^Çco) and G(u>). With this and the modulus of
the harmonie response function given by équation (5.63), the response spectral
density of équation (7.50) for the linear structural model of équation (7.27) is
deduced as
Sv(üj) =
|G(^)I2W
(Aj - mw2)2 + cfro2
(7-51)
The variance of the response is then
|CMI2S,M
(ki — ma>2)2 + Cju;2
(7.52)
which follows from its définition given by équations (7.22) and (7.51).
These results are summarized. To calculate av, the model parameters ki, ci,
and m are identified. Then G(w) is calculated by the methods discussed in
Chapter 4 where linear small-amplitude wave theory is assumed. After a design
wave height spectrum is chosen, such as the Pierson-Moskowitz form of équation
(6.21) or the JONSWAP form of équation (6.24), then crv is calculated from
équation (7.52) in which the limits of intégration (0, oo) can be replaced for
practical purposes by (0.16, 1.60) rad/sec.
It is important to note that for a linear structure, if r/(t) and pi(t) are Gaussian, then the response v(t) is also Gaussian (Newland, 1975). Thus, with the
assumption that the excitation is Gaussian, then the probability that v(t) will
exceed the calculated ± 3
this type of calculation does not exclude the possibility of local material failure
by fatigue.
The following comprehensive example brings together many of the basic ideas
elaborated on in this and preceding chapters. In working through such problems,
the reader is reminded of the many and sometimes subtle assumptions involved
in this analysis and is cautioned to temper the interprétation of numerical results
accordingly.
Example Problem 7.3. The free, undamped latéral motion of the threelegged jackup rig shown in Figure 2.17 has already been investigated in Example
Problem 5.4- Now include light damping, and subject this structure to steady,
unidirectional linear waves with a significant wave weight H, = 15 m and with a
distribution given by the Pierson-Moskowitz spectrum, équation (6.21). Based
177
With this last resuit, équation (7.48) becomes
Sv(w) = ^lff(w)|25Pi(w)
(7.50)
This last remarkably simple and useful resuit relates the power spectral
density of pj (t) to the power spectral density of v(t) through the complex frequency response function. As shown in Example Problem 7.2, Spl(u>) is given
by équation (7.26) in terms of S^Çco) and G(u>). With this and the modulus of
the harmonie response function given by équation (5.63), the response spectral
density of équation (7.50) for the linear structural model of équation (7.27) is
deduced as
Sv(üj) =
|G(^)I2W
(Aj - mw2)2 + cfro2
(7-51)
The variance of the response is then
|CMI2S,M
(ki — ma>2)2 + Cju;2
(7.52)
which follows from its définition given by équations (7.22) and (7.51).
These results are summarized. To calculate av, the model parameters ki, ci,
and m are identified. Then G(w) is calculated by the methods discussed in
Chapter 4 where linear small-amplitude wave theory is assumed. After a design
wave height spectrum is chosen, such as the Pierson-Moskowitz form of équation
(6.21) or the JONSWAP form of équation (6.24), then crv is calculated from
équation (7.52) in which the limits of intégration (0, oo) can be replaced for
practical purposes by (0.16, 1.60) rad/sec.
It is important to note that for a linear structure, if r/(t) and pi(t) are Gaussian, then the response v(t) is also Gaussian (Newland, 1975). Thus, with the
assumption that the excitation is Gaussian, then the probability that v(t) will
exceed the calculated ± 3
this type of calculation does not exclude the possibility of local material failure
by fatigue.
The following comprehensive example brings together many of the basic ideas
elaborated on in this and preceding chapters. In working through such problems,
the reader is reminded of the many and sometimes subtle assumptions involved
in this analysis and is cautioned to temper the interprétation of numerical results
accordingly.
Example Problem 7.3. The free, undamped latéral motion of the threelegged jackup rig shown in Figure 2.17 has already been investigated in Example
Problem 5.4- Now include light damping, and subject this structure to steady,
unidirectional linear waves with a significant wave weight H, = 15 m and with a
distribution given by the Pierson-Moskowitz spectrum, équation (6.21). Based