Gerritsma and Beukelman 



Similar expressions are valid for the pitching motion. The determination 

 of the damping coefficients b and B and the damping cross-coupling coefficients 

 e and E is straightforward: for a given frequency these coefficients are propor- 

 tional to the quadrature components of the forces or moments for unit amplitude 

 of motion. For the determination of the added mass, the added mass moment of 

 inertia, a and A, and the added mass cross- coupling coefficients d and D it is 

 necessary to know the restoring force and moment coefficients c and C, and the 

 statical cross- coupling coefficients g and G. 



The statical coefficients can be determined by experiments as a function of 

 speed at zero frequency. For heave the experimental values show very little 

 variation with speed; they were used in the analysis of the test results. 



In the case of pitching there is a considerable speed effect on the restoring 

 moment coefficient C. C decreases approximately 12% when the speed increases 

 from Fn = 0.15 to 0.30. This reduction is due to a hydrodynamic lift on the hull 

 when the shipmodel is towed with a constant pitch angle. Obviously this lift ef- 

 fect also depends on the frequency of the motion. Consequently, the coefficient 

 of the restoring moment, as determined by an experiment at zero frequency, 

 may differ from the value at a given frequency. 



As it is not possible to measure the restoring moment and the statical 

 cross- coupling as a function of frequency, it was decided to use the calculated 

 values at zero speed. This is an arbitrary choice, which affects the coefficients 

 of the acceleration terms: for harmonic motions a decrease of c by AC results 

 in an increase of A by Ac/w^ when C is used in the calculation. 



The results for the whole model are given in the Figs. 5 and 6. The results 

 for the heaving motion were already published in [13]; they are presented here 

 for completeness. 



Results for the Sections 



The components of the forces on each of the seven sections were determined 

 in the same way as for the whole model. As only the forces and no moments on 

 the sections were measured two equations remain for each section: 



Heave 



(a* + PS/*) z^ + b*z^ + c*z^ = F* sin (wt + a*) , 



Pitch (4) 



(d* + PV*x.) 9 + ee + gd = -F* sin («t + S*) , 



where /ov*x. is the mass-moment of the section i with respect to the pitching 

 axis. The star (*) indicates the coefficients of the sections. The section co- 

 efficients divided by the length of the sections give the mean cross-section co- 

 efficients, thus: 



226 



