3. HYDRODYNAMIC DERIVATIVES
Hydrodynamic derivatives on Esso Osaka are reported by many research institutes in the world. But there is a scattering in derivatives of the same ship Esso Osaka. Many possible differences among these research institutes, such as on experimental apparatus, on measurement systems, on length of models, on captive conditions, on forcing method and motion range of captive model test, on analysis method and etc., cause this scattering of their measured derivatives. In this section, influencing factors, which affect on accuracy of identification for hydrodynamic derivatives, and their characteristics are discussed.
In this study, following three factors, which should be considered to identify hydrodynamic derivatives, are investigated, as they may possibly be main factors.
(1) The influence of the experimental range
(2) The influence of the analyzing procedure
(3) The influence of propeller loading
3.1 The experimental range on estimation of maneuverability
The motion range of captive model tests is defined by the maximum rate of turn and the maximum drift angle in predicting motion. This motion range should be based on the maximum quantitative state of motion of an estimating object. But, it is considered that the identification of derivatives by means of captive model tests with limited motion ranges, which is caused by constraint of experimental apparatus, is carried out.
In this section, the influence of the experimental range of captive model tests on hydrodynamic derivatives is investigated by reanalyzing the results of captive tests on 6.0m model. Hydrodynamic forces and moments acting on the hull, such as lateral forces and yaw moments are obtained by subtracting rudder normal forces and hullrudder interaction hydrodynamic forces induced by steering from the measured hydrodynamic forces and moments of captive tests on 6.0m models with rudders. Figure 1 shows the results of captive tests on 6.0m model.
In order to investigate the influence of the experimental range on hydrodynamic derivatives, following two kinds of reanalysis on the results of captive model tests were carried out. One is the analysis on the limited motion range data and the other is on the whole motion range data. Hydrodynamic derivatives were identified respectively. Simulations based on the identified hydrodynamic derivatives were also carried out respectively.
Figure1 Hydrodynamic forces and moment (6.0m model)
Figure 11 lateral force
Figure 12 yaw moment
(a) Derivatives based on the limited motion range data (hereinafter referred to as limited range data)
The limited motion range is 0.4≦r'≦0.4 and 10≦β≦10°.
(b) Derivatives based on the whole motion range data (hereinafter referred to as whole range data)
The whole motion range is 0.8≦r'≦0.8 and 30≦β≦30°.
Identified derivatives on each motion range data are shown in table 1.
On identified linear derivatives, both variation of Nv and Nr are about 10%, and Y+ and Yv are about 5% or less. On identified nonlinear derivatives, a variation of each derivative is different. Especially, the variation of Nvvv is large and its value varied from minus to plus. Variations of coupling derivative on turning and drifting are also large. On the basis of the above, it is clear that the motion range of captive model tests affects remarkably on accuracy of identification for nonlinear derivatives and coupling derivative of hydrodynamic forces and moments.
Table1 Hydrodynamic derivatives (The influence of the experimental range)

Whole Range 
Limited Range 
Limited/Whole 
Yv 
0.02225 
0.02448 
1.088 
Yr 
0.00649 
0.00658 
1.014 
Yvvv 
0.08293 
0.00073 
0.009 
Yrrr 
0.00192 
0.00585 
3.046 
Yvrr 
0.02529 
0.02689 
1.063 
Yvvr 
0.01119 
0.00002 
0.002 
Nv 
0.00924 
0.00885 
0.957 
Nr 
0.00320 
0.00345 
1.078 
Nvvv 
0.00177 
0.00027 
0.152 
Nrrr 
0.00112 
0.00012 
0.107 
Nvrr 
0.00271 
0.00358 
1.321 
Nvvr 
0.01970 
0.01181 
0.599 

A simulation of the turning test with rudder angle 35 degree was carried out using each identified derivatives. Simulated trajectories are shown in figure 2. The dotted line shows estimated results based on whole range data, and the chain line shows those on limited range data. The solid line in the figure shows a free running test result of the 2.5m model.
Figure 2 
The influence of experimental range of motion on turning motion 
By comparing the trajectory of turning motion and the time history of limited range data with them of whole range data, the influence of the experiment range on hydrodynamic derivatives are discussed.
On trajectories of turning motion, the difference of both simulations in the beginning of the turning is small. But the difference of both simulations become significant as the turning proceeds after the heading is more than 60 degree.
On U/UO, drift angle and r' of time histories, both simulation results in the beginning of the turning agree closely. But the difference of r' on both simulations becomes significant as the turning proceeds.
In a steady state of the turning, both simulated drift angles of limited range data and whole range data are 18 degree. The value of simulated r' by whole range data is about 0.28, and the value by limited range data is about 0.32. The difference of them is about 15%. Simulated r' by limited range data is similar to measured r' in free running test results comparing with that by whole range data. But simulated turning trajectory by whole range data is similar to the trajectory of free running test results comparing with that by limited range data. It seems that both estimated speed reduction ratios of whole range data and limited range data are larger than free running test results. In accordance with the above, it is clear that an accuracy of identification for longitudinal hydrodynamic forces on the motion with a large speed reduction become important.
