Evaluation of Motions of Marine Craft in Irregular Seas 



several degrees of freedom) what might be termed a "lumped" transfer function 

 (or response to waves of discrete frequencies) and from this to calculate for- 

 mally a correspondingly lumped impulsive response function which, when con- 

 voluted with the given wave record, yields the instantaneous motion. Although 

 this procedure has been shown to work exceedingly well (see section on applica- 

 tions) questions arise as to the character of these impulsive response functions 

 primarily because the analysis has not been sufficiently lucid. At the risk of 

 appearing pedantic, the elementary theory is reexamined in the following pages 

 in the hope of providing a firmer foundation for the more-or-less mechanical 

 procedures used in arriving at the instantaneous response of a hull within, or 

 upon, the surface of a long-crested sea which is arbitrarily specified.* 



Heaving and Pitching In or Under Regular Waves 



Korvin-Kroukovsky [9] (1955) was quick to realize that the combined heav- 

 ing and pitching of responses of a ship are the solutions to a coupled pair of or- 

 dinary second-order, linear differential equations with coefficients which vary 

 with imposed frequency. Following his notation, the differential equations of 

 motion are, for the case of simple harmonic forcing functions: 



az + bz + cz + d -e + e^ + g^ = F^ e^'^* (1) 



Ad + B0 + Cd + B -'i + Ei + Gz = M^ e^"* (2) 



z is the heave displacement from equilibrium, 



9 is the pitch displacement from equilibrium, 



a,b,c are the virtual mass, damping and spring coefficients for pure 

 heaving, 



d, e, f are corresponding cross coupling coefficients due to pitch, 



A,B,C are the virtual mass moment of inertia, damping and spring 

 coefficients for pure pitching, 



D,E,G are corresponding cross coupling coefficients due to heave, 



F^ and M^ are the complex force and moment excitations for a regular 

 wave of amplitude 1 77^ | with the understanding that only the 

 real parts of the right-hand sides of (1) and (2) are to be ulti- 

 mately retained. 



(The dot notation is used to represent a total derivative with respect to time.) 



where 



'''The severity of the given wave trace (as a function of time) must be such as to 

 permit application of linear system analysis. 



463 



