starboard at once, instead of sideslipping to port as they do with an aft rudder. This makes it easier to avoid an obstacle appearing directly ahead. The disadvantage on the other hand is that a forward rudder is in a rather vulnerable position.
4. A MATHEMATICAL MODEL
Consider a vessel as in Figure 5, drawn with an arbitrarily shaped tall superstructure to suggest a high mass centre. The axes x, y, z are fixed in the vessel, x forward, y to starboard and z down. The origin is amidships on the waterline, and the mass centre is at x = xm, y = 0 and z = zm (<0). The vessel is assumed to have a fixed forward velocity u0.
Time-varying quantities of
interest (the “state variables”) are the lateral velocity v (defined at the origin), angular velocities p of roll and r of yaw, and the roll angle (assumed to be small). The mass of the vessel is m, and its metacentre is at xm and z = zmet (>zm to give the assumed instability in roll). The mass of the vessel is supposed to be contained in the x-z plane, so the relevant moments and products of inertia are Ixx, Izz and Ixz = Izx (defined with respect to the origin rather than the mass centre).
The hydrodynamic influences on its motion are a force Y along the y-axis and moments L and N about the x- and z- axes. The hydrostatic buoyancy is gm acting vertically upwards through the metacentre.
Axes x, y and z move with the vessel, so the equations of motion are (for the small-perturbation case):
m(v + u0r – zmp + xmr) = Y .
.
-mzm(v + u0r) + Ixxp – Ixzr = L + gm(zmet – zm) (2) mxm(v + u0r) – Izxp + Izzr = N
. . and = p . . . .
(3) (4)
For a linearised analysis, Y, L and N may be expressed in the usual way in terms of acceleration, velocity and rudder derivatives [13]:
Y = Yvdv + Ypdp + Yrdr + Yvv + Ypp + Yrr + Yδδ (5) L = Lvdv + Lpdp + Lrdr + Lvv + Lpp + Lrr + Lδδ (6) N = Nvdv + Npdp + Nrdr + Nvv + Npp + Nrr + Nδδ (7)
. . .
. . . . . .
Yv, but the form adopted here will, it is hoped, reduce the possibility of confusion with Yv etc.)
where δ is the rudder angle. .
The values of the derivatives are due partly to the hull and partly to the appendages. The hull components have been estimated using a method described in [14].
If the
hull cross-sections are assumed to be semi-elliptical, they depend solely on the longitudinal variation of the keel draught zk(x), as below, in “unnormalised” form:
B-4 (Yvd etc are usually written . . (1)
Yvd = /2 zk
2dx Yrd = Nvd = /2 zk Nrd = /2 zk 2x2dx
Yv = u0/2. zk (stern) Yr = u0/2. (zk
2 2x)(stern) Nv = u0/2{zk 2dx + (zk Nr = u0/2 {zk 2x)(stern)} 2xdx +(zk 2x2)(stern)}
The integrals are carried out in the +x direction, from stern to bow.
The roll-related derivatives have been
assumed to be negligible for the narrow hulls in question, and have been taken as zero for the hull, though the appendages contribute to them significantly.
Three of the appendages (rudder, fin and propeller-shaft support) may be regarded as hydrofoils and their contributions calculated accordingly (see Appendix 3). Allowance has been made for the fact that lateral motion at any point along the hull will induce a flow beneath it in the opposite direction, predictable from potential flow theory. This varies with depth, but a value averaged over the depth of each appendage has been used.
Contributions were also estimated for the propeller and its shaft.
Table 1 shows the estimates of the derivatives for one of the vessels mentioned above, having waterline length 4m and maximum beam of 0.5m, travelling at 2.2m/s with a displacement of 112kg. The contributions from hull and appendages are listed separately. It may be seen that for the velocity derivatives, with the exception of Nv, the appendages completely dominate the hull. The derivatives were used in their un-normalised form, as quoted, but normalised values are given too for the totals.
Other data relevant to the mathematical model are:
m = 112kg, xm = 0.263m, zm = 0.771m, Ixx = 107kgm2, Izz = 54.6kgm2, Ixz = 22.7kgm2, zmet – zm = 0.626m.
Eqns 1-7 can be manipulated (see Appendix 2) into a vector-matrix differential equation: x = Ax + bδ
. . . (9)
where xT is the 4-element vector [v p r ], A is a 4x4 matrix and b a 4-element vector. The bottom line of the equation is simply x4 = x2, i.e. = p, so a42 = 1 and a41 = a43 = a44 =b4 = 0. The other elements of A and b are rather more complicated functions of the data.
©2007: Royal Institution of Naval Architects (8) 2xdx
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