4. REGRESSION APPROACH
Prior to details of the regression analyses performed, some remarks are necessary. If the principal aim is to get regression formulas for the main damping coefficients which also include the influence of the aft body the question arises: On which part of the aft body the primary attention has to be focused? In the view of the author the parameter σa, Equation (1), is also only a global parameter, because details at the real aft body, like frame and bottom characteristics, are not described accordingly. In the field of wave-making resistance the aft shoulder, Fig. 2, plays a leading part. In compliance with this fact in the following the range between aft shoulder and stern is called aft body. When forming usual body parameters, like block coefficient, prismatic coefficient, etc., the real length between aft shoulder and stern is used. Differing from global parameters the aft body parameters show the additional index as. The parameter of Eq. (1) is accordingly modified, σas.
There are two reasons why the paper of Clarke and Horn (1997) was taken as archetype. First, their attempt was to devise empirical formulas, which are based on the theoretical forms of the generalised slender-body theory. Second, they endeavoured to present a solid modelling with only few characteristic parameters and without unreasonable parameter combinations.
The basic equations and regression formulas of Clarke and Horn (1997) are as follows:
Dominating in the approach of Clarke and Horn (1997) is Equation (6), in which the horizontal zero-frequency added mass coefficient at the stern of a vessel (CH)S is of particular interest. Clarke and Horn determined the added mass coefficient by simplifying the stern shape to a triangle, an ellipse or a rectangle attached to a thin skeg. The values of (CH)S were then obtained by analytic methods according to Clarke (1992).
In contrast to Clarke and Horn special attention, as indicated already, was focussed on the aft shoulder. Accordingly the corresponding added mass coefficient (CH)as was introduced and used. For the determination of two-dimensional lateral added mass coefficients different sources are known, e.g. Fig. 50 of Crane et al. (1989). In the present case however the calculations of Tamura (1961) for two-dimensional cylinders at zero-frequency, based on a theory presented by Grim (1956/57), were used, because Keil and Thiemann (1963) showed in an experimental investigation that the agreement between measured and calculated values within the range of finite frequencies was surprisingly good. Corresponding data for ω→0 depending on the frame section coefficient CMas and the ratio B/2T are given in Fig. 3.
Fig.3 Added Mass Coefficient(CH)as
For the identification of the linear damping coefficients either a multiple linear regression algorithm or, in case the parameters were already fixed, a usual least squares fitting was used. For the coefficient Y'vT the latter generated the following equation:
Compared to Eq. (6) it shows that the proportional coefficient as well as the two power coefficients totally differ from the approach of Clarke and Horn (1997). Main reason is the difference between (CH)S and (CH)as. Which agreement could be performed presents Fig.4. It shows the estimated data versus the corresponding data from HSVA's data base, usually called scatter plot. In case of total agreement all data lie on the dash-dotted half-court line. The scatter shown is acceptable. However, independent to the basic Clarke and Horn approach a different estimate was accomplished, which yielded the following result:
The corresponding scatter plot is given in Fig. 5. The differences to Fig. 4 are marginal. The proved standard deviations σ are nearly identical.