Higher-Order Spectral Estimation of Drift Force QFRF's for Random Waves
Authors: Kim S.B. , Powers E.J. , Miksad R.W. , Kim C.H. , Boo S.Y. 
Published: October 2008

ABSTRACT
A preliminary study of the statistical estimation of the quadratic frequency response function (QFRF) using higher-order spectral analysis is carried out and compared with an analytical QFRF. The QFRF is estimated via digital higher-order spectral analysis from simulation data of sea wave elevation and slow sway drift wave force on a ship in a unidirectional random sea. The initial result shows that the two QFRF's are in good agreement in the difference-interaction region where the slow drift force dominates. The estimated QFRF is able to "predict" the simulated drift force within 1 % mean square error.

1. Introduction
The concept of QFRF (quadratic frequency response function) has been shown to be very useful in that it provides a convenient and systematic way to model the nonlinear relationship between the drift force exerted on large floating bodies[l, 2] (such as ships, TLP's, etc.) by various sea states. Most previous experimental determinations of the QFRF have been carried out by determining the drift force corresponding to the excitation of the structure by two monochromatic waves[3]. The frequency of the two waves is varied to map out the QFR, F over the relevant portion of the two-dimensional bifrequency plane. This is a fairly time consuming task to carry out, especially experimentally, because of the repetitive experiments associated with the large number of frequency pairs that must be considered. 1
Recently, spectral techniques for experimental determination of the QFRF under random sea conditions was developed in order to analyze the time series data for random wave excitation and drift force[4, 5]. The approach rests upon a frequency domain Volterra model of a quadratically nonlinear system[6, 7]. Specifically, digital higher-order spectral analysis is used to estimate the appropriate higher-order auto- and cross-spectral moments in order to calculate the quadratic frequency response function from the input/output time series records. This is a very powerful technique since it is a black-box approach in that one need not have all the information about the physical system. The linear and nonlinear features are quantified in terms of linear and nonlinear frequency response functions, the output response is accurately predictable, and the approach is applicable to any experimental configuration for which time series of the sea wave excitation and its response can be measured.
Estimation of the QFRF using higher-order spectral analysis techniques has been shown to be quite promising in modeling quadratic drift phenomena. However, questions concerning the uniqueness of the QFRF and its ability to predict the drift forces for various sea states have been raised[8]. As a first step in investigating these issues, we report on an investigation to compare the statistically estimated QFRF using simulation data to an analytically determined QFRF by Kim and Boo[9]. The time series data were obtained from a time domain simulation of sway drift force of a 500 ft ship in 60 degree quartering random seas of significant wave height 18 ft.