Estimation of Rolling Motion of Ship in Random Beam Seas by Efficient Analytical and Numerical Approaches

Abstract

A steady-state roll motion of ships with nonlinear damping and restoring moments for all times is modeled by a second-order nonlinear differential equation. Analytical expressions for the roll angle, velocity, acceleration, and damping and restoring moments are derived using a modified approach of homotopy perturbation method (HPM). Also, the operational matrix of derivatives of ultraspherical wavelets is used to obtain a numerical solution of the governing equation. Illustrative examples are provided to examine the applicability and accuracy of the proposed methods when compared with a highly accurate numerical scheme.

Appendices

Appendix 1: Ultraspherical polynomials

The ultraspherical polynomials are special types of Jacobi polynomials that are associated with the real parameter $\left(\alpha >\frac{-1}{2}\right)$. They are orthogonal polynomials on the interval (− 1, 1), with respect to the weight function $w(x)={\left(1-{x}^2\right)}^{\alpha -\frac{1}{2}},$ and defined by

$$ {\int}_{-1}^1{\left(1-{x}^2\right)}^{\alpha -\frac{1}{2}}{C}_m^{\left(\alpha \right)}(x){C}_n^{\left(\alpha \right)}(x) dx=\left\{\begin{array}{cc}0,& m\ne n\\ {}{h}_n& m=n\end{array}\right. $$

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where

$$ {h}_n=\frac{\pi {2}^{1-2\alpha}\varGamma \left(n+2\alpha \right)}{n!\left(n+\alpha \right)\varGamma {\left(\alpha \right)}^2}, $$

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are the eigenfunctions of the following singular Sturm-Liouville equation

$$ \left(1-{x}^2\right){\ddot{\phi}}_m(x)-\left(2\alpha +1\right)x{\ddot{\phi}}_m(x)+m\left(m+2\alpha \right){\phi}_m(x)=0. $$

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The following integral formula and the theorem that follows are needed to establish the convergence of the expansion of the ultraspherical wavelets

$$ \int {C}_n^{\left(\alpha \right)}(x)w(x) dx=-\frac{2\alpha {\left(1-{x}^2\right)}^{\alpha +\frac{1}{2}}}{n\left(n+2\alpha \right)}{C}_{n-1}^{\left(\alpha +1\right)}(x),n\ge 1. $$

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Theorem 3 The following inequality holds for ultraspherical polynomials

$$ {\left(\sin \theta \right)}^{\alpha}\left|{C}_n^{\left(\alpha \right)}\left(\cos \theta \right)\right|<\frac{2^{1-\alpha}\varGamma \left(n+\frac{3\alpha }{2}\right)}{\varGamma \left(\alpha \right)\varGamma \left(n+1+\frac{\alpha }{2}\right)},0\le \theta \le \pi, 0<\alpha <1.\kern1em $$

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Appendix 2: Shifted ultraspherical polynomials

The shifted ultraspherical polynomials are defined on [0, 1] by

$$ {\overset{\sim }{C}}_n^{\left(\alpha \right)}(x)={C}_n^{\left(\alpha \right)}\left(2x-1\right). $$

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All properties of ultraspherical polynomials remain valid for the shifted polynomials.

The orthogonality relation for ${\overset{\sim }{C}}_n^{\left(\alpha \right)}(x)$ with respect to the weight function $\overset{\sim }{\omega }(x)={\left(x-{x}^2\right)}^{\alpha -\frac{1}{2}}$ is given by

$$ \int {\left(x-{x}^2\right)}^{\alpha -\frac{1}{2}}{\overset{\sim }{C}}_m^{\left(\alpha \right)}(x){\overset{\sim }{C}}_n^{\left(\alpha \right)}(x) dx=\left\{\begin{array}{cc}0,& m\ne n\\ {}\frac{\pi {2}^{1-4\alpha}\Gamma \left(n+2\alpha \right)}{n!\left(n+\alpha \right){\left(\Gamma \left(\alpha \right)\right)}^2},& m=n.\end{array}\right. $$

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For more properties of ultraspherical polynomials, see Rainville (1960).

Appendix 3: Basic idea of HPM

Consider the nonlinear differential equation

$$ A(u)-f(r)=0,r\in \varOmega, $$

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with the boundary condition

$$ B\left(u,\frac{du}{dr}\right)=0,r\in \Gamma, $$

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where A, B, f(r), and Γ are a general differential operator, a boundary operator, a known analytical function, and the boundary of the domain Ω, respectively. Expressing A(u) as the sum of linear (L) and nonlinear (N) parts, Eq. (74) becomes

$$ L(u)+N(u)-f(r)=0.\kern1.5em $$

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The homotopy technique begins by defining

v(r, p) : Ω × [0, 1] → R, such that

$$ H\left(v,p\right)=\left(1-p\right)\left[L(v)-L\left({u}_0\right)\right]+p\left[A(u)-f(r)\right]=0, $$

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where p ∈ [0, 1] is an embedding parameter and u₀ is an initial approximation of Eq. (74) that satisfies boundary conditions (Eq. 75). Evidently, Eq. (77) implies that

$$ H\left(v,0\right)=L(v)-L\left({u}_0\right)=0,\kern0.5em $$

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$$ H\left(v,1\right)=A(v)-f(r)=0. $$

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As p changes from 0 to 1, v(r, p) changes from u₀ to u_r, a process known as a homotopy. The solution of Eq. (77) may be expressed in terms of a power series in the form:

$$ v={v}_0+p{v}_1+{p}^2{v}_2+\cdots . $$

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An approximate solution to Eq. (77) is given by the following:

$$ u=\underset{p\to 1}{\lim }v={v}_0+{v}_1+{v}_2+\cdots . $$

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