Influence of the GZ calculation method on parametric roll prediction

Abstract

Parametrically excited roll response in a container ship sailing in irregular head sea has been studied. Short-term predictions for roll have been made for a ship at a constant forward speed based on different hydrodynamic roll descriptions.

The first order reliability method (FORM) has been used to obtain the probability that the roll motion will exceed a given limiting angle. The results have been compared to the results by a Monte Carlo simulation. Using FORM the computational time is greatly reduced as compared to direct simulations, still retaining the probability of failure of the correct order of magnitude. Calculations have been made for different sea states and operational conditions leading to parametric roll resonance and for relevant maximum pre-defined roll angles. The outcome of this comparative study could be useful for the decisions in an early design phase or for application to on-board decision support systems, where a computationally efficient method is needed in order to have an operationally feasible time frame.

Introduction

The current IMO intact stability code does not account explicitly for phenomena related to the hull stability variation in waves. The need to improve the stability criteria related to novel ship designs with the latest achievements in the research into dynamic intact stability physical mechanisms has, however, been indicated. The IMO working group proposed a framework for the development of the new criteria on the meetings in 2007 (IMO, SLF 50/WP.2, 2007). A revision of the intact stability criteria has been discussed in e.g. the document submitted to SLF 51 (IMO, SLF/51/4/3, 2008). The new generation intact stability criteria should be expanded to include dynamic behavior of hulls in irregular sea, the updated work plan being agreed on during the SLF 52 session in 2010. The related phenomenon of major implications on ship intact stability is known as parametric roll, excited by the variation of the restoring term in the system (e.g. Shin et al., 2004, Umeda et al., 2004). Different numerical simulation methods for parametric roll prediction have been compared in a recent large benchmark study by Spanos and Papanikolaou (2009). Operational guidance to avoid large roll angles and the validity of the prediction tools have been discussed by e.g. Levadou and Gaillarde (2003).

In order to make a rational design, engineering judgment must comprise the information uncertainties (e.g. uncertainty and randomness in the load model). Thus the probabilistic reliability concept might be utilised as the method of analysis. The present hydrodynamic time domain roll models have been linked to a probabilistic tool and can easily be substituted by an alternative more or less complicated mathematical description. However, to evaluate the dynamic stability condition of an intact ship in the context of vulnerability criteria, the probability based integrated model should be relatively simple but physically well-founded (Bassler et al., 2009) since the probability of failure should be calculated for every sea state combination in the scatter diagram. Recently, the short-term criteria for operational guidance have been discussed by Shigunov et al. (2010), where the roll calculation has been based on the ROLLS model (Kröger, 1986).

Within the reliability theory, the probability of exceeding a threshold value of a system is searched for at an arbitrary time instant. Various structural reliability methods have been described in e.g. Ditlevsen and Madsen (1996). Probability estimation by use of Monte Carlo simulations gives almost an “exact” solution, provided a large number of simulations. Thus, if an elaborate non-linear structural model is applied, the large number of samples needed to satisfy a reasonably small coefficient of variation can make the application of Monte Carlo technique practically impossible. This is especially true of rare events (i.e. large roll limits).

In Jensen and Pedersen (2006) dealing with parametric rolling of ships in head sea the FORM approach was found to be about two orders of magnitude faster than the direct simulation for the realistic exceedance levels, i.e. for small failure probabilities. An outcome of the analysis is a critical, i.e. the most probable, wave episode that leads to the pre-defined response limit. FORM can be widely used for decision making since there is no direct requirement of linearity of an underlying system. However, the non-linear shape of the failure surface around the design point may influence the accuracy of the prediction, e.g. if an additional degree of freedom is considered in a hydrodynamic model, as discussed in Vidic-Perunovic and Jensen (2009a) where surge was added to roll.

In the present paper a container ship has been analysed. The aim is to investigate how accurate the restoring moment has to be calculated in order to get an acceptable accuracy in the probability of exceedance. Different hydrodynamic models for roll prediction in a long-crested irregular sea have been used, ranging from a very simple to a more detailed description of the GZ restoring arm. Results have been generated for two sea states relevant to parametric roll and in head sea condition, for a narrow range of ship velocities and preconditioned roll angles in the range of 0.3–0.5 rad. Predictions by use of FORM have been compared to results by the Monte Carlo method. For this purpose the general tool for probabilistic calculations PROBAN (DNV (Det Norske Veritas), 2003) has been used where both methods have been implemented (Tvedt, 2006). Results have been discussed with regard to practical implications.