Abstract
This paper presents an alternative formulation of the boundary value problem for linear seakeeping and added resistance analysis based on a body-fixed coordinate system. The formulation does not involve higher-order derivatives of the steady velocity potential on the right-hand side of the body-boundary condition, i.e., the so-called m _{ j }-terms in the traditional formulation when an inertial coordinate system is applied. Numerical studies are made for a modified Wigley I hull, a Series 60 ship with block coefficient 0.7, and the S175 container ship for moderate forward speeds where it is thought appropriate to use the double-body flow as basis flow. The presented results for the forced heave and pitch oscillations, motion responses, and added resistance in head-sea waves show good agreement with experiments and some other numerical studies. A Neumann–Kelvin formulation is shown to give less satisfactory results, in particular for coupled heave and pitch added mass and damping coefficients.
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Appendices
Appendix 1
The transformation matrix \( {\mathbf{R}}_{b \to i}^{{}} \) that transforms the representation of a vector in the body-fixed oxyz system to its representation in the inertial OXYZ system is defined as
Here \( s_{i} = \sin (\alpha_{i} ) \) and \( c_{i} = \cos (\alpha_{i} ) \) with i = 4, 5, 6, \( {\mathbf{R}}_{b \to i} \) can be approximated as
with
Appendix 2
The linear force \( \vec{F}^{(1)} \), linear moment \( \vec{M}^{(1)}, \) and the added resistance \( \vec{F}^{(0)}, \) defined with respect to the inertial coordinate system, are written, respectively, as
Here CW_{0} is the mean waterline. \( \vec{r} \) is the position vector of a point on the ship surface relative to the center with respect to which the moments are defined. \( \vec{n}^{(0)} = (n_{1} ,n_{2} ,n_{3} ) \) and \( \vec{n}^{(k)} = {\mathbf{R}}_{b \to i}^{(k)} \vec{n}^{(0)} \) (k = 1, 2) are the components of the normal vector on the body. \( n_{1}^{(1)} \) is the first component of \( \vec{n}^{(1)}. \) \( \vec{m}^{(0)} = \vec{r} \times \vec{n}^{(0)} \) and \( \vec{m}^{(1)} = {\mathbf{R}}_{b \to i}^{(1)} (\vec{r} \times \vec{n}^{(0)} ) \) are the generalized normal vectors. \( \phi_{I}^{(k)} \) (k = 1, 2) is the kth order velocity potential of the incident wave described in the inertial coordinate system. \( p^{(0)} \) and \( p^{(1)} \) are pressure components giving zeroth-order and first-order contributions to the forces and moments. They are defined, respectively, as
\( \vec{U}'^{(1)} \) has been defined in Eq. 10.
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Shao, YL., Faltinsen, O.M. Linear seakeeping and added resistance analysis by means of body-fixed coordinate system. J Mar Sci Technol 17, 493–510 (2012). https://doi.org/10.1007/s00773-012-0185-y
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Keywords
- Seakeeping
- Added resistance
- Body-fixed coordinate system
- m _{ j }-Terms
- Boundary element method