Ogilvie 



to this function and has also made clear the way to get the impulse function from 

 a free damping test. 



Mr. Smith and Dr. Cummins insisted in their paper that the step compulsory 

 input is not applicable to get the impulse response function, because the step 

 function includes components of very wide range of frequency, theoretically 

 from zero to infinitive, and this incurs the noise which comes from the reso- 

 nance of model itself, guide, restraining frame or others to high frequency 

 component of input. However, if we are careful on a few points, this author be- 

 lieves, we can obtain the impulse response even from the inclining test, espe- 

 cially if the response has very low natural frequency as that of rolling. This 

 author obtained successfully tiie very complicated frequency response function 

 of a ship with Flume type anti-rolling tank as the Fourier inversion of the im- 

 pulse response obtained by free damping test. The results show that this method 

 is a useful way to guess the frequency response just by a simple free oscillation 

 test. The imp;ilse response function is also useful in this example, and here 

 some topics related to the statistical estimation of the impulse response will be 

 described. 



I 



When the system is linear and time invariant, 



^OD -CO 



y(t) = h(T) x(t-T) dr = h( t - r) x(t) dr = h(t)*x(t) 



J - oa »" - 00 



where * represents the convolution operation. The impulse response function 

 h(r) and the frequency response function H(w) are related to each other by 

 Fourier transformation as 



H(w) = r h(T) e'^'^Mr, h(T) = ^1 Uico) e^^'^dc^. 



= ( h(T) e-^'^Mr, h(r) = ^f 



J - m "^ - 



The Fourier transforms of output and input are connected by the frequency re- 

 sponse H(w) as 



Y(cS) = H(aj) -XCco) . 



Through manipulation, we will get 



n 



Kvv<^^) = Hf^) h(v) R^^(T-fj.+ v)di^dix 



00 "^ - CO 



= h(r)*h(T) *R^^(T) .; 



This corresponds to the relation in frequency domain 



Syy(a)) = H(a;) H(a)) S^^(c^) = |H(aj)| S^^(^) 



In the same way 



112 



