Theoretical estimation of roll acceleration in beam seas using PDF line integral method
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ORIGINAL ARTICLE
Theoretical estimation of roll acceleration in beam seas using PDF line integral method Atsuo Maki1 · Leo Dostal2 · Yuuki Maruyama1 · Masahiro Sakai1 · Kei Sugimoto3 · Yusuke Fukumoto3 · Naoya Umeda1 Received: 20 May 2020 / Accepted: 14 September 2020 © The Japan Society of Naval Architects and Ocean Engineers (JASNAOE) 2020
Abstract The prevention of excessive roll acceleration is one of the fundamental requirements of an oceangoing vessel at sea. In this paper, aiming at this requirement, we propose a new theoretical method for calculating roll angular acceleration, called “PDF line integral method.”. This paper presents the derivation of this method and a numerical comparison with Monte Carlo simulation (MCS) results; also, its validity is shown. Although the utilized GZ curve (restoring curve) exhibits strong asymmetricity, the proposed method can provide for such a condition. With only the information of roll and roll rate joint probability density function (PDF), this method can approximately calculate the roll angular acceleration, which is a highorder differential property. In addition, a relatively good agreement is achieved between the theory and MCS results. Keywords Irregular beam Seas · Roll angular acceleration · Excessive acceleration · PDF line integral method
1 Introduction The most fundamental requirement for an oceangoing vessel at sea is to prevent capsizing. Considerable number of researches for estimating ship roll in irregular beam seas has been conducted from numerical aspect [1] as well as theoretical aspect [2, 3]. In the past decade, Dostal et al. have developed methods for estimating ship roll in beam and quartering seas [4, 5]. Their approaches were mainly based on an energy-based stochastic averaging technique. In their methods, the diffusion and drift included in the Fokker–Plank–Kolmogorov equation for the energy level were calculated using the stochastic averaging theory, and then, Ito’s equation for the energy level was rigorously solved. On the other hand, Maki [6, 7] developed a method for estimating the ship roll performance in beam seas, based on Kimura et al.’s method [8]. In this method, the non-Gaussian joint * Atsuo Maki [email protected]‑u.ac.jp 1
Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering, Osaka University, 2‑1 Yamadaoka, Suita, Osaka 565‑0971, Japan
2
Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, 21043 Hamburg, Germany
3
Hull Rules Development Department, Class NK, 3‑3, Kioi‑cho, Chiyoda‑ku, Tokyo 102‑0094, Japan
probability density function (PDF) form for roll and roll rate was approximated through the nonlinearization technique proposed by To [9]. These analytical or semi-analytical methodologies seem to sufficiently agree with the Monte Carlo simulation (MCS) results, but further advanced theories and their validation are still necessary for developing more practically reliable estimation methods. In addition, excessive roll acceleration is one of the fi
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