Parametric rolling prediction in irregular seas using combination of deterministic ship dynamics and probabilistic wave theory

Abstract

In recent years there have been reports of serious accidents of parametric rolling for modern container ships and car carriers. For avoiding such accidents, a prediction method of parametric rolling in irregular seas is required. Since parametric rolling is practically non-ergodic, repetitions of numerical simulations or experiments could be not feasible to ascertain the behaviour. Therefore, in this paper, a method combining a stochastic approach with a deterministic approach in order to estimate the probabilistic index without such simple repetitions is developed. The ship's response in regular seas is estimated by solving an averaged system of the original 1-DoF roll model, and random waves necessary for occurrence of parametric rolling is achieved by using Longuet-Higgins’s or Kimura’s wave group theory. As a result, a fast and robust computation method of the probabilistic index is established. Finally, it is concluded that the proposed method is considered to be one of the useful tools for discussing the new IMO Intact Stability Code.

Appendix: Prediction of several parameters concerning with the wave group

In this section, we briefly describe the calculation method of using several parameters governing the wave group.

1.1 Appendix 1. Parameters according to encounter wave spectrum

Throughout this paper the lower and upper cut-off frequency at ω = 0.5ω _p and ω = 1.5ω _p , (truncated spectrum), is utilized in calculating some of the parameters, i.e. the moment of the wave spectrum. Longuet-Higgins examined several truncations, and the truncation spectrum within the closed set [0.5ω _p , 1.5ω _p ] was well validated. Since Takaishi et al. [19] followed Longuet-Higgins’s approach, this truncation is likewise employed in this study. In the case of an encounter wave, and the spectrum is represented as follows:

$$ S\left( {\omega_{\rm e} ,\,\chi } \right) = S\left( \omega \right)/\left| {1 - \frac{2\omega V\cos \chi }{g}} \right| $$

(28)

Here, an encounter frequency is of course given by:

$$ \omega_{\rm e} = \left| {\omega - \omega^{2} V\cos \chi /g} \right| $$

(29)

1.2 Appendix 2. The joint density in which the correlation is taken into account

The joint probability density function in which the correlation is taken into account, regarding adjacent wave-amplitudes H ₁ = 2ρ ₁ and H ₂ = 2ρ ₂, are represented as follows:

$$ p\left( {\rho_{1} ,\,\rho_{2} } \right) = \frac{{\rho_{1} \rho_{2} }}{{\mu_{0}^{2} \left( {1 - \kappa^{2} } \right)}}\exp \left[ { - \frac{{\rho_{1}^{2} + \rho_{2}^{2} }}{{2\mu_{0} \left( {1 - \kappa^{2} } \right)}}} \right]I_{0} \left[ {\frac{{\kappa {\kern 1pt} \rho_{1} \rho_{2} }}{{\mu_{0} \left( {1 - \kappa^{2} } \right)}}} \right] $$

(30)

where H ₁ or H ₂ denotes a certain wave height, whereas ρ ₁ or ρ ₂ denotes its amplitude. When the value κ of zero then p(ρ ₁, ρ ₂) reduces to the product of two Rayleigh distributions: p(ρ ₁) × p(ρ ₂), where:

$$ p\left( \rho \right) = \frac{\rho }{{m_{0} }}\exp \left( { - \frac{{\rho^{2} }}{{2m_{0} }}} \right) $$

(31)

where m _n denotes the n-th moment of the wave spectrum. In Eq. 30, I ₀(z) denotes the modified Bessel function of order zero:

$$ I_{0} \left( z \right) = \frac{1}{2\pi }\int\limits_{0}^{2\pi } {{\rm e}^{z\cos \theta } d\theta } . $$

(32)

κ in Eq. 30 can be calculated as:

$$ \kappa^{2} = \left| {\frac{1}{{m_{0} }}\int\limits_{0}^{\infty } {S\left( \omega \right)\cos \omega \tau {\rm d}\omega } } \right|^{2} + \left| {\frac{1}{{m_{0} }}\int\limits_{0}^{\infty } {S\left( \omega \right)\sin \omega \tau {\rm d}\omega } } \right|^{2} , $$

(33)

Longuet-Higgins sets a constant time interval τ as:

$$ \tau \equiv T_{01e}\; \left( { = 2\pi m_{e0} /m_{e1} } \right) $$

(34)

In this study, however, τ is set equal to the mean zero-up-crossing period:

$$ \tau \equiv T_{02e} \;\left( { = 2\pi \sqrt {m_{e0} /m_{e2} } } \right) $$

(35)

where m _en : n-th moment of the encounter wave spectrum, T _01e: mean wave period for encounter wave and T _02e: mean zero-up-cross period for encounter wave.