Figure 5: Rear view as rolled, and elevation, illustrating mathematical model


Table 1: Estimates of hydrodynamic derivatives Hull


Acceleration derivatives (independent of u0) Yvd Ypd Yrd Lvd Lpd Lrd


55.3 0


+8.08 0 0 0


Nvd Npd Nrd


Yp Yr Lv Lp Lr


Nv Np Nr


+8.08 0


Velocity derivatives at 2.2m/s (proportional to u0) Yv


37.8 0.70 +1.39 0


0 0 0


120 0


+15.0


906 +204 +142 +237 63.3 17.8


Rudder derivatives at 2.2m/s (proportional to u0 Yδ Lδ Nδ


Equation 9 can be used 2)


+661 182 +843


to develop a computer


simulation. It can also be used to find transfer functions from rudder angle to the elements of x, since in s- transform terms:


x(s) = (sI – A) -1 b δ(s) (10)


For instance, the vessel to which the Table 1 data apply is calculated to have the following rudder-to-roll-angle transfer function at 2.2m/s:


(s) =


kgms-2 kgm2s-2 kgm2s-2


+17.1E-3 1.18E-3 +5.44E-3


–5.14 (s + 1.96)(s + 11.25) δ(s)


(s – 2.23)(s + 1.77)(s +10.7–j0.25)(s + 10.7+j0.25) (11)


while that from rudder angle to yaw-rate is: r(s) =


7.47(s–2.28)(s+1.78)(s +10.34) δ(s)


(s – 2.23)(s + 1.77)(s +10.7–j0.25)(s + 10.7+j0.25) (12)


The poles of these transfer functions (equivalent to the eigenvalues of the A-matrix) are:


©2007: Royal Institution of Naval Architects B-5


+5.72 4.45 +5.72 1.58 +1.43 -4.45 +1.43 34.3


Appendages 24.7


Total 80.0


+5.72


Units kg


+3.63 kgm +5.72 1.58 +1.43 +3.63 +1.43 72.1


907 +204 +144 +237 63.3 17.8


+142 +22.0 11.4 1312


11.4 1297


kgm kgm


kgm2 kgm2 kgm kgm2 kgm2


kgs-1


kgms-1 kgms-1 kgms-1 kgm2s-1 kgm2s-1 kgms-1 kgm2s-1 kgm2s-1


Normalised 2.50E-3


+44.7E-6 +28.4E-6 +44.7E-6 3.09E-6 +2.79E-6 +28.4E-6 +2.79E-6 141E-6


51.5E-3 +2.89E-3 +2.04E-3 +3.36E-3 225E-6 63.4E-6 +312E-6


40.4E-6 4.61E-3


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