SHIP MOTIONS 



173 



Scps Filter 



0.245 



0614 

 17T/Te [Sec"'] 



858 



Fig. 1 5 Analog-filter computation of roll spectrum of a Liberty 

 ship in a state of 5 sea (head seas, 4.5 knots) 



5cps Filter 



0,3G8 0491 0.614 0.73G 



O0e=2'7^/Te [Sec"'] 



0.853 



0.961 



Fig. 1 6 Analog-filter computation of pitch spectrum of a liberty 



ship in a state 5 sea (head seas, 4.5 knots) (from Marks and 



Strausser, 1959) 



tablished that ship motions, Hke waves, can be repre- 

 sented by a stationary random process so that ship-mo- 

 tion spectra may be interpreted in the same way as are 

 wave spectra. 



The obvious starting place for evahiating ship motions 

 is the energy spectrum $(aj,). If the linear superposi- 

 tion hypothesis as applied by St. Denis-Pierson is used, 

 ^(w,) results directly from computation (model tests may 

 be used to obtain hydrodynamic coefficients). Towing- 

 tank tests of the type shown in Fig. 10 also result in 

 $(co,) directly. If the deterministic approach of Fuchs 

 and MacCamy is used, a motion record E{t) results which 

 may be converted to $(a),) by the numerical or analog 

 methods described in Section 1-8. The same applies to 

 motion measurements at sea and to motion measurements 

 in irregular waves in model tanks. In any case, $(aj,) is 

 obtainable and the statistics of a single ship-motion pa- 

 rameter derive from $(co,) in accordance with the pro- 

 cedure outlined in Section 1-8.6. 



Although not directly pertinent, it would be scarcely 

 fitting to end a section dealing with ship-motions spectra 

 without showing some typical examples. This obliga- 

 tion is fulfilled in Figs. 15 and 16. It should be remem- 

 bered that it is incumbent upon the investigator to show 

 that such spectra result from stationary, random, Gaus- 

 sian processes, before the statistics derived from them 

 can be interpreted properly. 



3.6 Cross-Spectrum Analysis. Section 3.5 dealt with 

 the analysis of single variables and C9n,sequently there was 

 no question of phase relationships. However, true insight 

 into ship-motion behavior can only be derived through a 

 more intimate understanding of how the input spectrum 

 (waves) and transfer function combine to produce a .ship- 

 response spectrum. In this case, one can study as before, 



the effect of the seaway upon the ship in eliciting a motion 

 response. In addition, the phase of the response, relative 

 to each freciuency component of the wave spectrum, can 

 be studied. This will depend on the point of wave meas- 

 urement so that distances between observation points 

 become important. One may al.so investigate the rela- 

 tionship say between hea\-e and pitch, without regard to 

 waves. The mechanism for providing information on 

 phase as well as amplitude is the cross-spectrum. 



Consider any two ship-motion parameters y{t) and c(t) 

 recorded simultaneously. If, for example /y(0 represents 

 the wave input and z{t} represents the ship response to 

 ^(0, then equation (23) suggests the necessary response 

 amplitude operator (here the terminology of St. Denis 

 and Pierson is more appropriate than the label ''transfer 

 function"). If, however, phase responses are desired, it 

 is necessary to consider the response amplitude operator 

 and the motion spectrum as complex C|uantities. 



*i,z(".) = c„.(we) + iq.Ao^,) 



(30) 



The real part of equation (30) is called the co-spectrum 

 and indicates the energy associated with the in-phase 

 components of the response, while the imaginary part is 

 called the ciuadrature spectrum and indicates the 90-deg 

 out-of-phase components of the response. The magni- 

 tude of the cross spectrum is the amplitude spectrum 



*(coj = {K(co,)]^+ hM)V]"' 



and the phase lag is 



tan" 



-g„-(^,) ' 



(31) 



(32) 



The ratio of the cross spectrum of any two seakeeping 

 variables to the scalar wave spectrum together with equa- 



