3. PROCEDURE FOLLOWED
As a first step, common for both the techniques used, a sensitivity analysis was performed (paragraph 3.1), in order to find the coefficients which affect more the manoeuvres. As already mentioned, the manoeuvres chosen are the usual ZigZag and Turning Circle because of the high number of data available. In particular, it has been decided to consider 10°/10° ZigZag and 35° Turning Circle because low rudder angle ZigZag is a manoeuvre which should be influenced mostly by linear coefficients, while in the Turning Circle manoeuvre effect of non-linear coefficients should be higher.
After the set of most influent coefficients has been identified, the two procedures have been applied separately on the selected manoeuvres, keeping fixed and equal to the initial value the remaining coefficients. It is believed, infact, that their Influence should not affect significantly the results.
The two procedures have been initially applied on "simulated" manoeuvres in order to verify the validity of the method; in the case of the filtering technique a white noise has been added to the simulated manoeuvres to reproduce the measurement noise, while for the optimization technique it was not necessary since just the main parameters of the manoeuvre are used.
After this first set of calculations which allowed to adjust the procedures, the two identification techniques have been applied directly to the experimental data available from sea trials, in order to evaluate in both cases the effect of disturbances such as uncertainties, experimental errors, weather effects (wind and current) and necessary difference between the model utilised and reality.
3.1 Sensitivity Analysis
In order to evaluate the influence of the different hydrodynamic coefficients on the manoeuvring behaviour of the ship, the following procedure has been used:
- Hydrodynamic coefficients of equations (2) were estimated using SIMSUP regression formulae;
- Using these coefficients a first simulation of the studied manoeuvres was performed, collecting the main macroscopic parameters (see the objective function used, which are reported in the following);
- A series of simulations were performed using the optimization program assuming each time a variation of 20% of one hydrodynamic coefficient, maintaining the original value for the remaining, in order to identify its influence on the objective functions.
The objective functions utilized for the sensitivity analysis represent the difference between the main parameters of the first simulated manoeuvre and the correspondent resulting from the variations of value of each coefficient, as follows:
For Turning Circle manoeuvre:
A = Advance
T = Transfer
DT = Tactical Diameter
DF = Steady Turning Diameter
VR = Final Rotative Velocity
T90 = Time corresponding to 90° heading variation
T180 = Time corresponding to 180° heading variation
For ZigZag manoeuvre:
Te = Time of rudder execution (1st, 2nd, 3rd)
TO = Time required for yaw checking (1st, 2nd, 3rd)
A = Overshoot yaw angle (1st, 2nd, 3rd)
In both the expressions the subscript "S" identifies the values of the parameters as obtained from the first calculation made by SIMSUP program with the initial set of hydrodynamic coefficients.
The results obtained are illustrated in figures 1 and 2, where the deviations "normalized" are represented:
where n is the total number of the coefficients analysed (equal to 28).
The deviations have been normalized in order to obtain a total value of 100% therefore the values are not directly connected to a percentage of error in the manoeuvre parameters and are used just to evaluate the relative importance of each parameter.
From the diagram it can be seen that the most influent coefficients for Turning Circle manoeuvre are Yd, Nr, Nv, Yr, Yv and Yddd, while for ZigZag manoeuvre they are Nr, Yd, Nv, Yv, Nrdot, Yr. As expected, therefore, the linear coefficients of hull and rudder result the most important parameters for both manoeuvres, with the addition of a non-linear effect on the rudder force in the 35° Turning Circle (consistently with its non-linear nature) and with the addition of the inertial parameter for the ZigZag manoeuvre (consistently with its transient nature).
It was also possible, therefore, to identify a set of coefficients significant for both the manoeuvres considered, which were object of the procedure of identification:
Nr, Yd, Nv, Yr, Yv, Nrdot, Yddd
35°Turning Circle - Sensitivity to 20% variation of each hydrodynamic coefficient
10°/10° ZigZag-Sensitivity to 20% variation of each hydrodynamic coefficient