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Acta Mechanica
Acta Mechanica 186, 221227 (2006)
DOI 10.1007/s00707-006-0312-0
S. K. Das and S. N. Das, Pune, India
Received October 24, 2005Published online: May 5, 2006 Springer-Verlag 2006
Summary. The paper deals with the prediction of coupled sway-roll-yaw motions of a ship with constant forward speed, subject to regular wave force. The governing equations derived in time domain after balancing the hydrodynamic and exciting forces are solved analytically while considering the motion variables, wave frequency and external force are complex quantities. Numerical experiments were carried out for a Panamax container ship under the action of a sinusoidal wave of periodicity 11.2 sec with the variations of speed and wave height. The eect of various parameters on motion responses and system stability is investigated.
1 Governing equations and method of solution
Usually in ship motion studies, frequency response analysis corresponding to a Fourier approach can be conveniently applied [1]. Owing to complex interactions between the hull and ship generated waves, the governing equations are represented in the form of integrodierential equations posing enormous diculty in solving [2]. Such diculty can be conveniently avoided if one considers the ship motion in regular waves. This reduces the integro-dierential equation to an ordinary dierential equation with frequency dependent coecients. In this paper, we consider a Cartesian co-ordinate system (x, y, z) xed with respect to the mean position of the ship with the z-axis acting in the vertical upward direction. The details of motion denition are described in the work of Das and Das [3]. In Fig. 1, g1, g2, g3,
g4, g5, and g6 indicate surge, sway, heave, roll, pitch and yaw, respectively.
The governing equations of motion for sway, roll and yaw can be described in compact form as [4]
di2g2t di4g4t di6g6t Fit; i 2; 4; 6; 1
where the operator dij is given by
dij Mij
d2 dt2 Bij
ddt Cij; i; j 2; 4; 6: 2
Fit, i 2, 4, 6 are the exciting wave forces or moments, Mij Mij Aij is the virtual mass or mass moment of inertia; Mij is the mass or mass moment of inertia; Aij, Bij and Cij are the hydrodynamic coecients like added mass, damping and restoring in the direction i due to any
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