Numerical and experimental motion simulations of nonsymmetric ship collisions

Abstract

A calculation model to simulate nonsymmetric ship collisions, implying an arbitrary impact location and collision angle, is described in the paper. The model that is introduced is based on the time integration of twelve equations of motion, six for each ship. The motions of the ships are linked together by a mutual contact force. The contact force is evaluated as an integral over the surface tractions at the contact interface. The calculation model provides full time histories of the ship motions and the acting forces. Physical understanding of the underlying phenomena was obtained by a series of model-scale experiments in which a striking ship collided with an initially motionless struck ship. In this paper, numerical simulations of four nonsymmetric collisions are presented and the calculations are validated with the results of the experiments.

Appendices

Appendix 1: Scalar form of equations of motion

For numerical integration, the equation of motion

$$ \left[ {\begin{array}{*{20}c} {\left[ {\mathbf{M}} \right]} & 0 \\ 0 & {\left[ {\mathbf{I}} \right]} \\ \end{array} } \right]\,\left\{ {\begin{array}{*{20}c} {{\dot{\mathbf{u}}}} \\ {{\dot{\mathbf{\Upomega }}}} \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {\left[ {\mathbf{M}} \right]{\varvec{\Upomega}} \times {\mathbf{u}}} \\ {{\varvec{\Upomega}} \times \left[ {\mathbf{I}} \right]{\varvec{\Upomega}}} \\ \end{array} } \right\} - {\mathbf{F}}_{\text{R}} - {\mathbf{F}}_{\mu } = \left\{ {\begin{array}{*{20}c} {\mathbf{F}} \\ {\mathbf{G}} \\ \end{array} } \right\} - {\mathbf{F}}_{\text{R}} - {\mathbf{F}}_{\mu } $$

(38)

is rearranged to obtain a more convenient form:

$$ \left[ {{\mathbf{M}}_{\mu } } \right]\left\{ {\begin{array}{*{20}c} {{\dot{\mathbf{u}}}} \\ {{\dot{\mathbf{\Upomega }}}} \\ \end{array} } \right\} + \left[ {{\mathbf{M}}_{\mu }^{\Upomega } } \right]\left\{ {\begin{array}{*{20}c} {\mathbf{u}} \\ {\varvec{\Upomega}} \\ \end{array} } \right\} - \left( {\left[ {\mathbf{T}} \right]^{\text{T}} \left[ {\mathbf{K}} \right]\begin{array}{*{20}c} {\left\{ {\mathbf{R}} \right\}} \\ {\left[ {\varvec\varphi} \right]} \\ \end{array} + \left[ {\mathbf{K}} \right]\begin{array}{*{20}c} {\left\{ {{\text{d}}{\mathbf{x}}} \right\}} \\ {\left[ {{\text{d}}{\varvec\varphi} } \right]} \\ \end{array} } \right) = \left\{ {\begin{array}{*{20}c} {\mathbf{F}} \\ {\mathbf{G}} \\ \end{array} } \right\} - {\mathbf{F}}_{\text{R}} - {\mathbf{F}}_{\mu } $$

(39)

where the matrices have the following component form:

$$ [{\mathbf{M}}_{\mu } ] = \left[ {\begin{array}{*{20}c} {m + a_{x} } & 0 & 0 & 0 & 0 & 0 \\ 0 & {m + a_{y} } & 0 & {a_{yx} } & 0 & {a_{yz} } \\ 0 & 0 & {m + a_{z} } & 0 & {a_{zy} } & 0 \\ 0 & {a_{xy} } & 0 & {I_{x} + a_{xx} } & 0 & { - I_{xz} } \\ 0 & 0 & {a_{yz} } & 0 & {I_{y} + a_{yy} } & 0 \\ 0 & {a_{zy} } & 0 & { - I_{xz} } & 0 & {I_{x} + a_{zz} } \\ \end{array} } \right], $$

(40)

$$ \left[ {{\mathbf{M}}_{\mu }^{\Upomega } } \right] = \left[ {\begin{array}{*{20}c} 0 & { - (m + a_{x} )r} & {(m + a_{x} )q} & 0 & 0 & 0 \\ {(m + a_{y} )r} & 0 & { - (m + a_{y} )p} & 0 & 0 & 0 \\ { - (m + a_{z} )q} & {(m + a_{z} )p} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & {I_{45} } & { - I_{46} } \\ 0 & 0 & 0 & { - I_{45} } & 0 & {I_{56} } \\ 0 & 0 & 0 & {I_{46} } & { - I_{56} } & 0 \\ \end{array} } \right] $$

(41)

with

$$ \begin{gathered} I_{45} = (I_{zz} + a_{zz} )r - (I_{zx} + a_{zx} )p - I_{zy} q, \hfill \\ I_{46} = (I_{yy} + a_{yy} )q - I_{yz} r - I_{yx} p, \hfill \\ I_{56} = I_{xx} + a_{xx} p - I_{xy} q - I_{xz} + a_{xz} r \hfill \\ \end{gathered} $$

and

$$ [{\mathbf{K}}] = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & { - \rho gA_{W} } & 0 & {\rho gA_{W} x_{F} } & 0 \\ 0 & 0 & 0 & { - GM_{\text{T}} gm} & 0 & 0 \\ 0 & 0 & {\rho gA_{W} x_{F} } & 0 & { - GM_{\text{L}} gm} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] $$

where m is the structural mass, a _i and a _ii are the translational and rotational added masses, A _W is the waterplane area, x _F is the longitudinal centre of flotation, ρ is the water density, g is the gravitational acceleration, GM _T is the transverse metacentric height, and GM _L is the longitudinal metacentric height. The subscript characters in the mass and inertia terms follow the common notation; a single character refers to a value involved with translational motions, and two characters refer to rotational motion or to a coupling between two motion components.

The restoring force is divided into a constant part $\left. {{\mathbf{F}}_{\text{R}} } \right|_{{t_{j} }}$ evaluated at the beginning of the time increment at t = t _j , and into the change $\left. {{\text{d}}{\mathbf{F}}_{\text{R}} } \right|_{{t_{j + 1} }}$ in the force during the increment:

$$ \left. {{\mathbf{F}}_{\text{R}} } \right|_{{t_{j} }} + \left. {{\text{d}}{\mathbf{F}}_{\text{R}} } \right|_{{t_{j + 1} }} = [{\mathbf{T}}]^{T} [{\mathbf{K}}]\begin{array}{*{20}c} {\{ {\mathbf{R}}\} } \\ {[{\varvec\varphi} ]} \\ \end{array} + [{\mathbf{K}}]\begin{array}{*{20}c} {\{ {\text{d}}{\mathbf{x}}\} } \\ {[{\text{d}}{\varvec\varphi} ]} \\ \end{array} $$

(42)

This split is necessary, as the restoring force depends on the ship’s position with respect to the inertial coordinate system given by the position vector R and Euler’s angles. The increase in the force during the time increment is still evaluated via the displacements in the local coordinate system, but because of small angular displacements the error will be negligible.

During the time increment, the matrices and vectors on the left-hand side of Eq. 39 are updated several times within the increment, while the right-hand side is kept constant.

Appendix 2: Test matrix for second and third sets

 

Test	β (deg)	Bulb	L C (m)	mA (kg)	mB (kg)	u0 (m/s)	F AC,x (N)	F AC,y (N)	ED,P (J)
201	90	1	0.82	28.5	30.5	0.87	226	25	3.91
202	90	1	0.83	28.5	30.5	0.71	179	17	2.36
203	90	1	0.83	28.5	30.5	0.38	91	13	0.75
204	90	1	0.45	28.5	30.5	0.91	300	36	6.30
205	90	1	0.48	28.5	30.5	0.38	115	6	0.95
206	90	1	0.38	28.5	30.5	0.71	221	13	3.43
207	90	1	0.80	28.5	20.5	0.90	200	27	4.2
208	90	1	0.41	28.5	20.5	0.89	229	33	4.92
301	120	1	0.37	28.5	20.5	0.87	172	47	4.20
302	120	1	0.32	28.5	20.5	0.30	59	14	0.51
303	120	1	0.3	28.5	44.5	0.84	204	52	6.50
304	120	1	0.38	28.5	44.5	0.37	89	21	1.01
305 (sliding)	145	1	0.32	28.5	20.5	0.34	44	29	0.54
306 (sliding)	145	1	0.44	28.5	20.5	0.87	115	65	3.91
307 (sliding)	145	1	0.38	28.5	44.5	0.84	118	67	5.47
308	145	1	0.34	28.5	44.5	0.28	45	27	0.52
309 (sliding)	145	3	0.46	28.5	20.5	0.87	120	86	3.19
310 (sliding)	145	2	0.44	28.5	20.5	0.88	142	94	3.19
311	120	3	0.42	28.5	20.5	0.88	217	69	4.25
312	120	2	0.41	28.5	20.5	0.86	313	104	4.64
313	60	1	0.29	28.5	20.5	0.76	177	41	3.14
314	60	1	0.32	28.5	20.5	0.36	80	16	0.81
315	60	1	0.38	28.5	44.5	0.75	202	52	4.35
316	60	1	0.4	28.5	44.5	0.43	104	24	1.17

1. Absolute maximum values are presented for F ^A_C,x and F ^A_C,y
2. m, ship mass; u⁰, contact velocity; E_D,P, plastic deformation energy