Trans RINA, Vol 152, Part A2, Intl J Maritime Eng, Apr-Jun 2010
forces exactly from the Navier-Stokes equations, the components of the hydrodynamic forces and moments acting on the hull and on each appendage are usually treated as a function of the motion state variables, in terms of ‘hydrodynamic coefficients’ (HDCs) [2, 4, 5].
In general, the most important part of developing the
dynamic motion equations is in expressing the external force and moment vector {FE} properly. These forces /moments may be split into physically meaningful components corresponding
to hydrostatic terms), propulsion (gravity-
dependent) forces, fluid inertia (i.e. added mass, or acceleration-dependent terms), fluid (velocity-dependent
and
control surface forces, which are the most crucial for initiating any manoeuvre of the body.
The HDCs may be denoted using another form of subscripts, where subscripts denote the motion variable they are multiplied with. For example, for a small sway velocity v, the hydrodynamic force component arising from that motion is expressed as Yvv, where Yv is the HDC for the motion variable v in the equation of motion for the Y-force. All coefficients can be made non- dimensional using appropriate combinations of vessel length and initial axial velocity.
In order to generalise the various mathematical models available for expressing {FE} in terms of HDCs, we can consider {FE} as the product of various HDCs (as matrix [B] of size 6 x n) with respective motion variables (as matrix [C] of size n x 6), where n depends on the particular model and may typically vary from 6 to 15.
Thus, {FE} = diag ([B].[C])
(14)
i.e. the vector {FE} is formed by the diagonal terms of the 6 x 6 matrix [B].[C], which also includes terms corresponding to hydrostatic forces and control forces.
Various models developed for specific applications can be conveniently expressed in the above form. For example, only linear terms may be retained in some models, or cross-coupling between various motions may be neglected, or special terms may be added to account for cross-flow drag, or empirical corrective terms may be included.
Three mathematical models for external hydrodynamic force have been considered in this study, as described below.
3.2 (a) First Mathematical Model
For the first model [9, 13], we consider the following structure
hydrodynamic forces. of matrices representing the external
damping forces forces,
where, vw w v
KZ Y qr =−
=− r q MX Z uw u w
K NM K Y − Z =−
Nuv v u
Kvq r q wr = − r q = Y − X
= Y + Z (16)
The relations of (14) above hold good for any model, since the combination of some acceleration-dependent HDCs are merely being represented in terms of the state variables with which they are multiplied in the respective force / moment equation.
In the first model considered, we have ⎡ u
⎢
⎢ vv w w ⎢
2 2
⎢ wvv 2
⎢ rr pr ⎢ ⎢
2 2
r
() 0 0 ⎢() 0 0
⎣rs 0 ⎥δθ ⎦
⎢δθφcs s ⎢
vr δδ δr s
s
2 2
. 0
where cθ denotes cosθ, sθ denotes sinθ, and δr and δs denote rudder and stern plane deflection, respectively.
3.2 (b) Second Mathematical Model
The second model considered here is a partly linearised version of the first model, dropping squared velocity terms and retaining only those second order terms which are cross-products of velocity components. Thus, we have:
[B2] =
⎡ 00 0 ZY ⎤wv 0 0 ⎢XY Z Z
0
⎢ uv w q ⎢ZY Xwv u
⎢ ⎢ ⎢ ⎢ ⎢
K K KK K K KzW M MN Y
vw p ⎣NN Zq
uw w uv
v
00 G
vq
0 −− −Zq Mq
r r Yr
qr wr δδr Mq Nr
s
M s N r
δ δ
00 0 00 0 0
−− −
Zq
Yr Yr
Y r Z s
δ δ
0 −
−zGW⎥ ⎥
⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦ (18) A - 74 ©2010: The Royal Institution of Naval Architects δ
δr ⎥ ⎥
q r
ur ⎥ ⎥ ⎥
q
wr δs
vp uq
pq ⎥ ⎥
[C1] = ⎢ qpq ⎢
p vp vq
⎢ wq wp uq qr ⎢
w w w
ur w vw uw uv ⎤ ⎥
rp wp ⎥ ⎥
v v ⎥
v ⎥ ⎥
(17)
[B1] = uu
⎡XX X ⎢
ww ⎢ ⎣ p vw uv
⎢XY Yuv w ⎢
vv
⎢wv u ⎢KK Kvq uw w
ZZ Y X ww vv ⎢NN N Zq v
−− −
Yr Z Yr K Yr Mq
vv ww w qZ −−Yv Zq q
Z Yr Z
K s
δ δ
⎢M MM N Y Z r−− − q ⎢
Kqr wr δδ zGW r
Y r Z s K r q
Nr N r
δ δ
M M z W ⎥ ⎥
Xδδ δδ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
s s X r r
00 00 0
s
− G 0
(15)
⎥ ⎦
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