OrcaFlex Manual - Orcina

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Theory, 6D Buoy Theory
(due to added inertia) equal to -PW IaΩc where Ωc is the component in that direction of the cylinder angular
velocity vector (relative to the earth).
� If the angular acceleration of the local water isobar Ωf (relative to the earth) is non-zero then the cylinder
experiences a fluid acceleration moment equal to (If + PW Ia)Ωf. This can be written as IfΩf + PW IaΩf, where the
first term is a moment analagous to the Froude-Krylov force, and the second term is a moment analagous to the
added mass force.
Note: If the added inertia Ia specified is zero for a given direction (normal or axial), then the Froude-
Krylov moment term is omitted for that direction, so that no moment due to added inertia is
applied about that direction. This does not apply to the Froude-Krylov force terms – to suppress the
Froude-Krylov force you need to specify Cm = Ca (and set both to zero if you want no force at all due
to added mass.
For a spar buoy the cylinder axis direction is the buoy z-direction, so the components in the buoy axes directions of
the total force and moment due to added mass are therefore as follows.
where
Fx = CmnMfnAfx - CanMfnAcx
Fy = CmnMfnAfy - CanMfnAcy
Fz = CmaMfaAfz - CaaMfaAcz
Mx = (Ifn + PW Ian)Ωfx - PW IanΩcx or Mx = 0 if Ian = 0
My = (Ifn + PW Ian)Ωfy - PW IanΩcy or My = 0 if Ian = 0
Mz = (Ifa + PW Iaa)Ωfz - PW IaaΩcz or Mz = 0 if Iaa = 0
PW = proportion wet for this cylinder
Can, Caa = specified added mass coefficients specified for the normal and axial directions, respectively.
Cmn, Cma = specified inertia force coefficients specified for the normal and axial directions, respectively.
Mfn, Mfa, Ifn, Ifa = instantaneous values of reference fluid mass (Mf) and moments of inertia (If), for the normal
and axial directions respectively, as described below.
Acx, Acy, Acz = components in buoy axes directions, of the cylinder translational acceleration relative to earth.
Afx, Afy, Afz = components in buoy axes directions, of the fluid translational acceleration relative to earth.
Ωcx, Ωcy, Ωcz = components in buoy axes directions, of the cylinder angular acceleration relative to earth.
Ωfx, Ωfy, Ωfz = components in buoy axes directions, of the angular acceleration of the fluid local isobar relative
to earth.
For a Towed Fish the cylinder axis direction is instead the buoy x-direction. So for a Towed Fish the subscripts x and
z in the above equations are interchanged, so that the axial values of the coefficients and reference masses and
inertias are used in the equations for Fx and Mx, and the normal direction values are used in the equtions for Fz and
Mz.
Reference Fluid Mass and Inertia used for Added Mass
For motion normal to the cylinder axis the values used for the reference fluid mass Mf and inertia If are the mass
and moments of inertia of the fluid displaced by the submerged part of the whole of the cylinder cross-section. So if
the cylinder is hollow (i.e. inner diameter is non-zero) then for motion normal to the cylinder axis the reference
fluid mass and inertia used includes the fluid trapped inside the part of the cylinder that is below the surface.
For motion parallel to the cylinder axis, the reference fluid mass Mf and inertia If that are used depend on whether
the cylinder is hollow. If it is not hollow then they are the same as the values used for motion normal to the cylinder
axis, i.e. equal to the mass and moments of inertia of the fluid displaced by the submerged part of the cylinder. But if
the cylinder is hollow then the reference fluid mass Mf and inertia If used for motion parallel to the cylinder axis are
the mass and moments of inertia of the fluid displaced by the submerged part of just the cylinder annulus,
excluding the fluid trapped inside the part of the cylinder that is below the surface.
These values for the reference fluid mass and inertia are based on the assumption that for a hollow cylinder the
trapped fluid contents are free to translate and rotate axially relative to the cylinder, but not free to move normal to
the cylinder axis.