Newman 



with undetermined coefficients of damping, added mass, etc. Alternatively we 

 may formulate beforehand a physically realistic but mathematically tractable 

 idealized model for the ship and fluid, and proceed in a systematic manner 

 through the analysis thereof. In general the first approach is expedient but dan- 

 gerous, while the second is elegant but less productive. 



In the first approach, following Abkowitz (1964), one assumes that the hy- 

 drodynamic force and moment at any instant of time are analytic functions of 

 the linear and angular acceleration, velocity and displacement of the hull at that 

 same instant and that they are independent of any other details of the motion ex- 

 cept for the geometrical properties of the ship and the physical properties of 

 the water. It follows that the six components of the force and moment can be 

 expanded as Taylor series in powers of the above variables, and this leads di- 

 rectly to a set of linear and nonlinear terms in the equations of motion. Of 

 course this does not furnish immediately the desired solution, since the coeffi- 

 cients of these terms remain to be determined either through analytical or ex- 

 perimental techniques; usually the latter are employed at this point, so that the 

 approach serves only as a method of curve-fitting. However it does offer the 

 very practical advantage of a basis for generalizing captive model experimental 

 results to free maneuvering problems. 



In principle there are fundamental objections to the assumption that the 

 forces and moments are analytical functions of the above mentioned variables. 

 In recent years Cummins (1962) and Brard (1964) have called attention to the 

 "memory" associated with the effects of the free surface and of vorticity, re- 

 spectively, and the resulting necessity to represent the hydrodynamic forces 

 arising from a transient ship motion in terms of a convolution integral over the 

 entire time history of the motion. This situation has been recognized for many 

 years in the field of unsteady aerodynamics and in fact it was also pointed out 

 by earlier workers in ship hydrodynamics, e.g., Haskind (1946). Thus we should 

 properly consider the six components of the hydrodynamic force and moment to 

 be of the form 



F.(t) = K.[Uj(T), t-T]dT (i =: 1,2, . . . , 6) , 



J-co 



where Kj is a kernel function which depends in general on all six velocity com- 

 ponents Uj , and on the retarded time t - r. In the linearized case we have the 

 more familiar and simple form 



6 ^t 



j = 1 J_(D 



Fi(t) = 2^ U.(T) Ki.(t-T)dT, 



j = 1 J_ 00 



where the kernel Kj depends only on the retarded time and on the geometrical 

 properties of the ship and the physical properties of the fluid. As an example of 

 the necessity for this representation, we note that in the case of a captive model 

 which is given a short "pulse" disturbance and then returned to its original 

 steady restrained condition, an unsteady fluid motion (visible especially in the 

 free surface disturbance) and associated force and moment will persist there- 

 after, in principle ad infinitum. 



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