Understanding and Prediction of Ship Motions 



response being the sum of the responses to the various frequencies. Alternately, 

 one may use the f.r. function amplitudes obtained from one test, together with 

 the wave height spectrum in a second test, to predict the motion spectra in the 

 second test. Comparison of these predictions with measured spectra then pro- 

 vides an indication of the degree of validity of superposing responses. 



Let us be more specific. Suppose that the energy spectrum of the wave 

 height at the center of gravity of the ship* is given by \J^) and that the f.r. 

 function in heave is given by T^(co). Then the energy spectrum of the heave 

 motion will be: 



If both energy spectra are known, this formula yields the magnitude of Tj^(w).t 

 In a single test, the quantity \t^(co) | can always be found from this equation; 

 defining such a ratio of two energy spectra implies nothing about the physical 

 processes involved. However, in a second test with a different 'i'^^(co), the same 

 |Tj^(w)i will be obtained only if the ship responds "separately" and linearly to 

 each frequency component. Thus a simple direct means is provided for check- 

 ing the principle of superposition and thus for checking the linearity of the whole 

 process. 



Only the amplitude of T^(c.o , that is \t^(m) \ , is found by the above proce- 

 dure. Such a result is to be expected of course, for the calculation of an energy 

 spectrum from a given test record washes out all phase information. But for 

 some purposes it is necessary to know the actual complex value of T^^(co). For 

 example, if we want to predict bow emergence, the occurrence of slamming, 

 deck wetness, etc., we must be able to relate the instantaneous ship position and 

 attitude to the simultaneous free surface shape. 



The complete evaluation of Tj^(&.o can be made from random seas tests, 

 through measurement of cross- spectra. For example, if \^(co) is the cross- 

 power spectrum of heave and wave height, then 



Since ^^^('^) is a real quantity, this equation states that the cross- spectrum has 

 the same argument in the complex plane as the f.r. function. 



A remarkable series of such experiments has been performed at the David- 

 son Laboratory in recent years, in which the limits of validity of the linearity 

 hypothesis have been extended more and more. (See Dalzell (1962a,b).) 

 Long-crested random seas were created for a great range of degrees of sever- 

 ity. Figure 1, taken from Dalzell (1962b), shows the wave height power spectra 



*The spectrum must be properly adjusted so that "frequency" is really "frequency 

 of encounter." See St. Denis and Pierson (1953) for the frequency mapping. 



tin general, the f.r. function will depend on the angle of incidence of the waves, 

 as well as on co. I am now assuming long-crested waves, moving parallel to the 

 ship center plane. 



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