Page 193 - Practical Design Ships and Floating Structures

168

mooring system), SMS (spread mooring system) and DICAS systems (SMS with differentiated
compliance). These expressions are here applied to problems involved in the design of new ships, in
the conversion of a VLCC into a FPSO and in the omoading operation, with the objective of
furnishing procedures for the engineers to avoid the undesirable unstable dynamics of a floating
system.

2 EQUATIONS OF MOTION
To study the dynamics of a stationary floating body in the horizontal plane a system attached to the
ship and an earth-fixed system are used (see figure 1). The ship is exposed to a current with constant
intensity C and an angle of incidence a (a = 0" means current coming from astern). The ship is moored
to the sea floor by a schematic spread mooring system. Then, the external forces and moments acting
on the ship are the hydrodynamic action due to the current and the reaction due to the mooring system.
The hydrodynamic forces and moments are written as functions of the relative hull-current velocity
and acceleration according to the quadratic maneuvering model developed by Sphaier, Fernandes, and
Correa (1998, 1999, 2000a and 2000b) for stationary floating units. The mooring line forces are
expressed as a function of the distance between the fairleads and the anchor points, calculated from a
catenary's formulation and considering the drag forces on the mooring lines.

Figure 1 : Geometric definition of the mooring system

Setting together the external forces and moments and the inertial terms according to Newton's second
law, the equations of motion, expressed in the body-attached system, are given by:
x
+
+
+
+
1;
X,v.r
(m - X. = ~,,u m .u. r + ~ ~ , ~ I ~ ~ I ~~,~~~lv~lv X,r2 + T, (1)
..
(I - N.)r - N. v = N,v + N vlvl+ N,r + N rlrl+ N,r.v2 + N,r2v + N,u.v + N,u.r + T, (3)
VlYl 1/11
where the parameters rn and I are respectively the mass and the inertia of the ship. The components of
the velocities in the longitudinal axis, in the transversal axis and the angular velocity are respectively u,
v and r. The terms Xo, Yo and No are the hydrodynamics derivatives related to (). The dots over the
variables means time derivative and r = dyddt, and Vis the yaw angle. Tu, T, and T,, are the reaction
forces due to the mooring system and can be written as: