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Application of linear filter and moment equation for parametric rolling in irregular longitudinal waves

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Abstract

Parametric rolling is one of the dangerous dynamic phenomena. To discuss the safety of a vessel when a dangerous phenomenon occurs, it is important to estimate the probability of certain dynamical behavior of the ship with respect to a certain threshold level. In this paper, the moment values are obtained by solving the moment equations. Since the stochastic differential equation (SDE) is needed to obtain the moment equations, the autoregressive moving average (ARMA) filter is used. The effective wave is modeled using the 6th-order ARMA filter. In addition, the parametric excitation process is modeled using a non-memory transformation obtained from the relationship between GM and wave elevation. The resulting system of equations is represented by the 8th-order Itô stochastic differential equation, which consists of a second-order SDE for the ship motion and a 6th-order SDE for the effective wave. This system has nonlinear components. Therefore, the cumulant neglect closure method is used as higher-order moments need to be truncated. Furthermore, the probability density function of roll angle is determined using moment values obtained from the SDE and the moment equation. Here, two types of the probability density function are suggested and have a good agreement.

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Acknowledgements

This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant Number 19H02360) and by Support for Pioneering Research Initiated by the Next Generation from Japan Science and Technology Agency (JST SPRING, Grant Number JPMJSP2138), as well as the collaborative research program / financial support from the Japan Society of Naval Architects and Ocean Engineers. This study was supported by the Fundamental Research Developing Association for Shipbuilding and Offshore (REDAS), managed by the Shipbuilders’ Association of Japan from April 2020 to March 2023.

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Authors and Affiliations

  1. Department of Naval Architecture and Ocean Engineering, Graduate School of Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka, Japan

    Yuuki Maruyama, Atsuo Maki & Naoya Umeda

  2. Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, 21043, Hamburg, Germany

    Leo Dostal

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  1. Yuuki Maruyama
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  2. Atsuo Maki
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  3. Leo Dostal
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  4. Naoya Umeda
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Correspondence to Yuuki Maruyama.

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Appendices

Appendix 1

In this study, we use the ARMA filter. Here, the 2nd, 4th, and 6th-order ARMA filter are compared. First, the ordinary differential equations of the 2nd and 4th-order ARMA filter are expressed as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \dfrac{\mathrm {d} x_{1}^{}}{\mathrm {d} t} = x_{2}^{} - \alpha _{1}^{} x_{1}^{} + k \sqrt{\pi } \dfrac{\mathrm {d} W(t)}{\mathrm {d} t} \\ \\ \displaystyle \dfrac{\mathrm {d} x_{2}^{}}{\mathrm {d} t} = - \alpha _{2}^{} x_{1}^{} \\ \end{array}\right. \end{aligned}$$
(29)
$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle \frac{\mathrm {d} x_{1}}{\mathrm {d} t}=x_{2}-\alpha _{1} x_{1} \\ \\ \displaystyle \frac{\mathrm {d} x_{2}}{\mathrm {d} t}=x_{3}-\alpha _{2} x_{1} + k\sqrt{\pi } \dfrac{\mathrm {d} W(t)}{\mathrm {d} t} \\ \\ \displaystyle \frac{\mathrm {d} x_{3}}{\mathrm {d} t}=x_{4}-\alpha _{3} x_{1} \\ \\ \displaystyle \frac{\mathrm {d} x_{4}}{\mathrm {d} t}=-\alpha _{4} x_{1} \end{array}\right. \,\,. \end{aligned}$$
(30)

Here, W denotes the standard Wiener process, and the notation for differentiation is represented by Lagrange’s notation. As discussed later, the coefficients \(\alpha _{i}^{} (i = 1, \cdots , 4 )\) and k are determined such that they agree with the effective wave spectrum. From Eqs. 29 and 30, the spectra of the 2nd and 4th-order ARMA process can be obtained as:

$$\begin{aligned} \displaystyle S_{2}(\omega ) = \dfrac{k^{2}\omega ^{2}}{ (\omega ^{2} - \alpha _{2}^{})^{2} + \alpha _{1}^{2}\omega ^{2} } \end{aligned}$$
(31)
$$\begin{aligned} \displaystyle S_{4}(\omega ) = \dfrac{k^{2}\omega ^{4}}{\left( \omega ^{4} - \alpha _{2}^{}\omega ^{2} + \alpha _{4}^{} \right) ^{2} + \left( \alpha _{1}^{}\omega ^{3} - \alpha _{3}^{}\omega ^{} \right) ^{2}} \,\,. \end{aligned}$$
(32)

Next, the coefficients \(\alpha _{i}^{}\) and k of Eqs. 10, 31, and 32 are determined by fitting the effective wave spectrum. Here, the stability criterion of their corresponding system explained in section2 is used. In Fig. 17, the effective wave spectrum, 2nd, 4th, and 6th-order ARMA spectra are plotted. From this figure, we can see that the 6th-order ARMA filter is better than the 2nd and 4th-order ARMA filters. Therefore, the 6th-order ARMA filter is used in this study.

Fig. 17
figure 17

Comparison among effective wave spectrum \(S_{\mathrm {eff}}^{}\), 2nd-order ARMA spectrum \(S_{2}\), 4th-order ARMA spectrum \(S_{4}\), and 6th-order ARMA spectrum \(S_{6}\). Here, the sea condition is \(T_{01}=9.99[s]\) and \(H_{1/3}=5.0[m]\)

Full size image

Appendix 2

From Eq. 24, the following eight first-order moment equations are obtained:

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_1^{}\right] = \mathbb {E}\left[ X_2^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_2^{}\right] = - \mathbb {E}\left[ G(X_1^{},X_2^{})\right] - \mathbb {E}\left[ F(X_3^{})X_1^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_i^{}\right] = \mathbb {E}\left[ X_{i+1}^{}\right] - \alpha _{i-2}^{} \mathbb {E}\left[ X_3^{}\right] \,\,\,(i=3\cdots 7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_8^{}\right] = - \alpha _6^{} \mathbb {E}\left[ X_3^{}\right] \end{array} \end{aligned}$$
(33)

Moreover, thirty-six second-order moment equations are obtained as:

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_1^2\right] = 2 \mathbb {E}\left[ X_1^{}X_2^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_1^{}X_2^{}\right] = \mathbb {E}\left[ X_2^2\right] - \mathbb {E}\left[ G(X_1^{},X_2^{})X_1^{}\right] - \mathbb {E}\left[ F(X_3^{})X_1^2\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_1^{}X_i^{}\right] = \mathbb {E}\left[ X_2^{}X_i^{}\right] + \mathbb {E}\left[ X_1^{}X_{i+1}^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad - \alpha _{i-2}^{} \mathbb {E}\left[ X_1^{}X_3^{}\right] \,\,\,(i=3\cdots 7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_1^{}X_8^{}\right] = \mathbb {E}\left[ X_2^{}X_8^{}\right] - \alpha _6^{} \mathbb {E}\left[ X_1^{}X_3^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_2^2\right] = - 2 \mathbb {E}\left[ G(X_1^{},X_2^{})X_2^{}\right] - 2 \mathbb {E}\left[ F(X_3^{})X_1^{}X_2^{}\right] \end{array} \end{aligned}$$
$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_2^{}X_i^{}\right] = - \mathbb {E}\left[ G(X_1^{},X_2^{})X_i^{}\right] - \mathbb {E}\left[ F(X_3^{})X_1^{}X_i^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad + \mathbb {E}\left[ X_2^{}X_{i+1}^{}\right] - \alpha _{i-2}^{} \mathbb {E}\left[ X_2^{}X_3^{}\right] \,\,\,(i=3\cdots 7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_2^{}X_8^{}\right] = - \mathbb {E}\left[ G(X_1^{},X_2^{})X_8^{}\right] - \mathbb {E}\left[ F(X_3^{})X_1^{}X_8^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad - \alpha _6^{} \mathbb {E}\left[ X_2^{}X_3^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_3^2\right] = 2\mathbb {E}\left[ X_3^{}X_4^{}\right] - 2\alpha _1^{}\mathbb {E}\left[ X_3^2\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_3^{}X_{i}^{}\right] = \mathbb {E}\left[ X_4^{}X_{i}^{}\right] - \alpha _1^{}\mathbb {E}\left[ X_3^{}X_{i}^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad + \mathbb {E}\left[ X_3^{}X_{i+1}^{}\right] - \alpha _{i-2}^{}\mathbb {E}\left[ X_3^2\right] \,\,\,(i=4\cdots 7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_3^{}X_8^{}\right] = \mathbb {E}\left[ X_4^{}X_8^{}\right] - \alpha _1^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] - \alpha _6^{}\mathbb {E}\left[ X_3^2\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_4^2\right] = 2\mathbb {E}\left[ X_4^{}X_5^{}\right] - 2\alpha _2^{}\mathbb {E}\left[ X_3^{}X_4^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_4^{}X_i^{}\right] = \mathbb {E}\left[ X_5^{}X_i^{}\right] - \alpha _2^{}\mathbb {E}\left[ X_3^{}X_i^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad + \mathbb {E}\left[ X_4^{}X_{i+1}^{}\right] - \alpha _{i-2}^{}\mathbb {E}\left[ X_3^{}X_4^{}\right] \,\,\,(i=5\cdots 7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_4^{}X_8^{}\right] = \mathbb {E}\left[ X_5^{}X_8^{}\right] - \alpha _2^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] - \alpha _6^{}\mathbb {E}\left[ X_3^{}X_4^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_5^2\right] = 2\mathbb {E}\left[ X_5^{}X_6^{}\right] - 2\alpha _3^{}\mathbb {E}\left[ X_3^{}X_5^{}\right] + \pi k_{}^{2} \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_5^{}X_i^{}\right] = \mathbb {E}\left[ X_6^{}X_i^{}\right] - \alpha _3^{}\mathbb {E}\left[ X_3^{}X_i^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad + \mathbb {E}\left[ X_5^{}X_{i+1}^{}\right] - \alpha _{i-2}^{}\mathbb {E}\left[ X_3^{}X_5^{}\right] \,\,\,(i=6,7) \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_5^{}X_8^{}\right] = \mathbb {E}\left[ X_6^{}X_8^{}\right] - \alpha _3^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] - \alpha _6^{}\mathbb {E}\left[ X_3^{}X_5^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_6^2\right] = 2\mathbb {E}\left[ X_6^{}X_7^{}\right] - 2\alpha _4^{}\mathbb {E}\left[ X_3^{}X_6^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_6^{}X_7^{}\right] = \mathbb {E}\left[ X_7^2\right] - \alpha _4^{}\mathbb {E}\left[ X_3^{}X_7^{}\right] \\ \displaystyle \quad \quad \quad \quad \quad \quad + \mathbb {E}\left[ X_6^{}X_8^{}\right] - \alpha _5^{}\mathbb {E}\left[ X_3^{}X_6^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_6^{}X_8^{}\right] = \mathbb {E}\left[ X_7^{}X_8^{}\right] - \alpha _4^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] - \alpha _6^{}\mathbb {E}\left[ X_3^{}X_6^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_7^2\right] = 2\mathbb {E}\left[ X_7^{}X_8^{}\right] - 2\alpha _5^{}\mathbb {E}\left[ X_3^{}X_7^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_7^{}X_8^{}\right] = \mathbb {E}\left[ X_8^2\right] - \alpha _5^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] - \alpha _6^{}\mathbb {E}\left[ X_3^{}X_7^{}\right] \\ \\ \displaystyle \frac{\mathrm {d}}{\mathrm {d}t}\mathbb {E}\left[ X_8^2\right] = - 2\alpha _6^{}\mathbb {E}\left[ X_3^{}X_8^{}\right] \end{array} \end{aligned}$$
(34)

Appendix 3

The detailed formula about the relations between moments and cumulants can be shown in the electronic supplementary material (ESM).

Appendix 4

In the case of using the second-order cumulant neglect closure, the stationary solutions of roll angle \(X_{1}^{}\) and roll velocity \(X_{2}^{}\) are obtained. Here, these solutions are oscillating, as shown in Fig. 11. We consider the question whether the second-order cumulant method can sufficiently reflect the nonlinearity. Therefore, we examine the effect of considering higher-order cumulants. Thus, the third-order cumulant neglect closure is used as well. Thereby, one hundred twenty third-order moment equations, derived from Eq. 24, are additionally needed. In other words, one hundred sixty-four moment equations are used in the corresponding numerical calculation. Furthermore, it is necessary that 4th and higher-order moments are represented by first, second, and third-order moments. These relations can be derived as shown in Appendix 3.

Figure 18 shows the results when solving these moment equations numerically. Compared with the result of the second-order cumulant neglect closure in Figs. 10 and 11, it is clear that the steady state solutions are not oscillating. Furthermore, the moment values in the third-order cumulant neglect closure are closer to the moment values obtained from the numerical result of the SDE than the second-order cumulant neglect closure results, as shown in Table 7. We can therefore conclude that the result can reflect nonlinearity and be close to the actual value using higher-order cumulants and moments.

Table 7 Moment values obtained by solving the ordinary differential equations of moment equation with third-order cumulant neglect closure method
Full size table
Fig. 18
figure 18

Second order moment of \(X_{1}^{} \, \text {and} \, X_{2}^{}\) obtained by solving the ordinary differential equations of moment equation with third-order cumulant neglect closure method

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Maruyama, Y., Maki, A., Dostal, L. et al. Application of linear filter and moment equation for parametric rolling in irregular longitudinal waves. J Mar Sci Technol 27, 1252–1267 (2022). https://doi.org/10.1007/s00773-022-00903-8

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