Page 194 - Practical Design Ships and Floating Structures

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where n is the number of mooring lines, xu and yu are the longitudinal and the transversal points of the
fairlead, T, is the mooring line tension (Ti = f(L3). These expressions, developed for a single ship
system, can be used for a multiple-ship system by considering the coupling of the ships due to the
connection hawser in the terms Tu and T,.

The velocities in both coordinate systems are related by the following expressions:
c+ c = u. cos(ly) - v .sin( y/)
(7)
q = u. sin(w) + v . cos(^)
(8)
where 6 and 17 are the absolute displacement of the ship. The acceleration relation is obtained by the
time derivative of eqns. 7 and 8. Introducing eqns. 7 and 8 in eqns. 1,2 and 3, the equations of motion
can be expressed as function of the absolute accelerations, velocities and displacements of the ship as:

The final expression are extensive and are not included in the text, but can be obtained following the
steps presented above (see Matter, Sales and Sphaier, 2001)

3 THE STABILITY ANALYSIS
The stability analysis consists on the observation of the ship movements around the many possible
equilibria, the equilibrium positions of the system, for different incidence angles of the current. The
equations of motions (Eqns. 1, 2 and 3) are expanded in a Taylor's series around the equilibria and
linearized. The final expressions are similar to:

where the tb's are the resultant terms of the linearization. These terms can be obtained analytically as
shown in Matter et a1 (2001a). The linearized expressions (Eqns. 10, 11 and 12), can be written as:
x=A,x

where x = (q,z, w, 6 7, WIT and q, z and w are respectively the time derivatives of 6, 1;1 and This is the
classical eigenvalue problem with x = @!e"'. The eigenvalues d are obtained from the solution of a
polynomial like:
tan .R =o (14)
k=O
The index n is equal to the number of ships times twice the number of degrees of freedom for a single
ship. Sometimes, this polynomial can be solved analytically and then, it is possible to obtain the
eigenvalue derivatives in relation to each parameter of the system and perform a more detailed study of
the stability, including a bifurcation study. Extending the study to the offloading operation, the