Trans RINA, Vol 152, Part A2, Intl J Maritime Eng, Apr-Jun 2010
reference, and by its position and orientation in an inertial frame of reference (Figure 2).
Let F denote the external force on the body, and
origin o
components are denoted as, F ={X,Y,Z}
of
and M = {K,M,N}
M denote the moment of the external force about the
the body-fixed system oxyz. Their .
The mass of the vehicle is m and the moments of inertia about the body-fixed system oxyz are denoted as Iij where the indices i,j correspond to x,y,z coordinates. The coordinates of the body’s center of gravity (CG) with respect to the body system oxyz are {xG, yG, zG}.
O Y Z Pitch Yaw (ψ) g z
Figure 2 Definition diagram of symbols for orientation; linear and angular velocities
Heave (w, r)
The governing 6-DOF equations of motion can be written as follows for the body-fixed frame of reference [5, 6, 13]:
Surge:
m[u - vr +wq- x (q +r )+ y (pq- r)+z (pr +q)] = X Sway:
22 GG G
m[v- wp+ur - y (r + p )+z (qr - p)+x (qp+r)] = Y Heave:
22 GG G
m[w- uq+vp- z (p +q )+x (rp- q)+ y (rq+ p)] = Z Roll:
22 GG G
I p+(I - I )qr - (r + pq)I +(r - q )I +(pr - q)I +m[y (w - uq+vp) - z (v - wp+ur)] = K
xxzz
Pitch: yy
Yaw: zz
22 yy GG
I q+(I - I )rp - (p+qr)I +(p - r )I +(qp - r)I +m[z (u - vr + wq) - x (w- uq+vp)] = M
22 xx zz GG
I r +(I - I )pq - (q+rp)I +(q - p )I +(rq - p)I +m[x (v - wp+ur) - y (u - vr + wq)] = N
22 yy xx GG
The above six dynamic motion equations can be written in matrix form [13] as:
[]{ } { A IEVF F−= } { }
where the external generalised forces are denoted by {} {
FE XY Z K M N}T = , , , , , (7) zy xy zx (6) yx zx yz (5) xz yz xy (4) (1) (2) (3) X Roll (ϕ) (θ)
Surge x
(u, p)
Horizontal reference
y Sway (v, q)
and the terms on the left hand side of equation (7) are matrices of size 6x6, defined as follows.
The extended mass matrix (including added mass terms) is given by:
[] MMA = [] [ '] + where mass matrix is given by:
⎡mmz 000 GG ⎢ mmz00 −
⎢ [M] =
⎢ 00 mmy mx ⎢ ⎢
− 0
⎢ mz ⎢
⎢−− −I zy ⎣ my mx
⎡−X ⎢
u
GG GG
mz my 0
I yx I zx
and added mass matrix is taken as: 00 0
[M ]′ = ⎢
⎢ ⎢
u
⎢ vp ⎢
0
00 0 00 0
⎢ vp ⎢
00 0 00 0
vp
GG Ixx mx
I yy
−− − −− − 0
0
−− − −−
−− − −−
YY Y ZZ
wq 0 ⎥ ⎥
r ⎥
⎤ ⎥
KK K MM
⎣00 0−− −NN N ⎦r
wq 0 ⎥ ⎥
r ⎥ ⎥
The subscript notation represents partial differentiation, so
XX u∂ , etc. = ∂ /
Acceleration vector is given by: {} {
V u v w pqr = }T (12)
The inertial force terms, independent of modelling of external hydrodynamic forces, are given by
{FI}= ⎧
⎪ ⎪ ⎪ ⎪
⎪ ⎨
⎪ xz
I −+ + yz ⎪ (I −+ + zx
yx xx yyzy xy
⎪() ( ⎪
yy zzI qr I pq I q r I pr myG (vp uq mzG (ur wp) ⎪ ⎪
mvr wq x q r y pq z pr mwp ur y r p z qr x qp muq vp z p q x rp y rq −
[( ) [( ) [( ) −
−+ + − − −+ + − − −+ + − −
⎪(II ++ −
⎩⎭ (13)
zz xx ) () () ( − )(22) − zx
I rp I qr I r p I qp mzG wq vr mxG vp uq) ⎪ ⎪
22 22
− − yz pq I rp I p q I rq mxG (ur wp my wq vr)⎪
− −
Thus far, there are mathematical model, but
no − ) + approximations G ( − in the the external forces and
moments are yet to be modelled. 3.2 VARIANTS OF MATHEMATICAL MODEL
The external forces and moments due to hydrodynamic (and hydrostatic) loads
(8) are complicated functions of
many factors, including water density, viscosity, surface tension, pressure, vapour pressure, and motions of the body. Rather
than attempting to obtain hydrodynamic ©2010: The Royal Institution of Naval Architects A - 73 −
) + +
−
GG G GG G GG G ) − xy
22 22 22
− −
⎪ ⎬
⎫ ⎪ ⎪ ⎪ ⎪
(11)
GG GG I xy
0
mx 0
I zz
−my ⎤ ⎥ ⎥ ⎥ ⎥
I yz
I xz ⎥ ⎥ ⎥ ⎥
⎦ (9)
(10)
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64