Frequency-Domain Analysis of Offshore Platform in Non-Gaussian Seas

A Fast Time–domain Integration Method for Computing Non-Stationary Response Histories of Linear Oscillators With Discrete-Time Random Forcing

“…ratio using the ensemble of error sample functions and integrals shown in Figures 5(a) and 5(b), compared to the theoretical relative integration error obtained by using equation (38). This comparison clearly shows that equation (38) gives a very good estimate of the O( f ) error and therefore from the asymptotic series properties for equation (23), and also gives a practical upper bound on the total relative truncation error. Both theory and simulation also show two important things: "rst, that the largest relative error arising in the use of equation (23) occurs as t!t P0, and second, that the relative error is a minimum for any integer multiple of the damped natural period, i.e., at t!t "k / B for k"1, 2, 2 .…”

“…These questions can be answered de"nitively "rst by using equations (28}30) for the derivatives needed in the leading error terms in the EMSF equation (23). Now at transitions equal to an integer multiple of the damped period, h(0)"h(k / B )"0 and hQ (0)"hQ (k / B ) +1, the corresponding derivative terms will be dominated by the magnitude of d L\ Q/dtL\.…”

“…Now for "nite duration transitions, the per-step errors can be broken down by using equation (25) in "rst order, second order terms, etc., and studied by using equation (80). Here, both "rst and second order error propagation are considered, but to generate the error terms in equation (23), it is necessary to obtain "rst, second, and third derivatives of the discrete excitation process, namely Q Q , Q G , and Q 2 , as required from equations (28)}(30) at respective transition times "t and t for the two speci"c cases (1) t!t "k / B and (2) t!t " . These di!erentials can be obtained in the frequency domain by using FFT.…”

“…Since the transition t!t will be many multiples of (possibly up to 1000 times), correction involves an insigni"cant amount of computation compared with the "rst summation. The asymptotic expansion properties of equation (23), enable the O( f ) error to be obtained by using the numerical integration formula (67) as…”

“…The FPK equation [21] also o!ers a route to transition probabilities for low dimension systems with normal or functionally related forcing, but the form of numerical solution usually required is often very demanding. Non-normal excitation can cause signi"cant prediction errors if normality is otherwise assumed [22,23], and several methods have been developed to handle systems driven by certain types of non-normal excitation. Two methods are of signi"cance, namely use of the space moments method [24] for systems driven by a "ltered train of pulses randomly distributed in time, and the maximum entropy method [25] for polynomial functions of normal forcing.…”