ABSTRACT
The governing equations of motion for such a marine craft (assumed to be rigid body) are derived from the Newton's laws of motion and can be conveniently written as follows:
Sway m [6 + ru — pw + zg(qr — 15) + x g(qp + 0] = E Y (1.a) Roll Izp-mzg(v + ru — pw) — mx g zg(pq + i.) = E K (1.b) Yaw Izi• + (I y — I z )pq + Trix g(V + ru — pw) + mz g x g(rq — 13) = E N (1.c)
Surge m [ii + qw — ry — x g(q2 + r2) + zg(pr + 0] = Ex (1.d) Heave m kb + pv — qu — zg(p2 + q2) + x g(r p — q)] = E Z (1.e) Pitch Igq + (Ix — I z )r p + m zg (V + qw — rv) — mx g (tb + pv — qu)+
mxg zg(p2 — r) = E M (1.f) where the notation used is the one adopted by the SNAME. The terms on the left hand side of the equations (1) relate to the kinematics of the motion and the terms on the left hand side relate to the external forces and moments acting on the craft which actually cause the motion. The set of equations 1 can be further simplified by assuming that the origin is at the centre of gravity, hence:
Sway m(h + ru — pw)= E Y (2.a) Roll izp= EK (2.b) Yaw /zi• + (ig — I r )pq = E N (2.c)
Surge m(ii + qw — rv) = E X (2.d) Heave m(th + pv — qu) = E Z (2.e) Pitch /gq + (ix — I z)rp = E m (2.f)
3. HEAVE AND PITCH MOTION
The motion in surge, sway, yaw and roll have been subject of numerous investigations (e.g. Mikelis[8], Pourzanjani[12] and Hirano et al[3]) and hence will not be discussed in this paper. It is also a valid to assume that for small motion the equations in the horizontal plane (xy plane, i.e sway, roll and yaw) can be decoupled from the motion in the vertical plane (xz plane, i.e surge, heave and pitch). This would result in the following equations of motion in the vertical `xz' plane:
mu = E x (3.a) meth — qu) -= E Z (3.b) I yq -=- E m (3.c)
= — k
The hydrostatic restoring moment can be written as
M2 = kh39 (11)
and similarly the damping moment as
M3 = kd26
(12)
and the moment of the forcing functions as
M4 = B sin(wet) (13)
The complete equations for pitching motion can be written as
o(iy ki) = ka2(ui — u20) + kh30 kd20 B sin(wet) (14)
The coefficients involved in both the force and moment equations can be calculated using strip theory for a given ship and sea conditions. Any change from these conditions will require a new set of coefficients, which will have to be calculated. To consider changes of the sea state and operating conditions of the craft during a simulation run, a look up table of the coefficients can be produced and interpolation carried out for any changes from the initial settings.
