Understanding and Prediction of Ship Motions 



APPENDIX B 



Systems Lacking Some Restoring Forces 



We want to treat here the case that, after a transient disturbance, some of 

 the a^(t) may not return to zero. As explained in the main text, we specifically 

 exclude the case that any a^(t) may continue to grow indefinitely, for then the 

 linear analysis must certainly break down and there is no meaning to applying 

 Fourier transforms to the equations of motion — even though they may be valid 

 in the initial stages of the motion. 



If a^(t) -> a^(oo) :|: 0, as t ^ CO, the Fourier transform of aj^ does not exist 

 in the classical sense. However, we can still treat it by using the concepts of 

 generalized function theory. (See Lighthill (1958).) Let 



^kCt) = [a^Ct) - aj^(oo) H(t)] + a^(oo) H(t) , 



where 



( for t < , 



H(t) = \ 



[ 1 for t > . 



The Fourier transform of the quantity in brackets exists in the usual sense and 

 is easily shown to be: 



3 {a^(t) - a^(oo) H(t)} = j^ [a^(oS) - a^(oo)] , 



where, for brevity, I have introduced the notation 

 We also note that 



'iu(O) 



GO 



= J i^(t) dt = a^(oo) 



In the language of generalized functions (see page 43 of Lighthill), 



Therefore, 



3{H(t)} = H(a)) = 1 S(co) + i- . 



_ 1 '^uC'^) 



Since we assume that (\(t) remains bounded, there is no difficulty about 

 taking transforms of the terms containing a^ or a.^ , but the convolution integrals 



75 



