SHIP MOTIONS 



177 



Resonance UJ -4.21 



Resonance uj = 3,95 



2 4 6 

 a) CSec"')- 



Fig. 21 Energy spectra of a ship model's rolling in natural ir- 

 regular waves. Results of direct analysis of rolling records are 

 shown by dotted lines. Results of wave-energy spectrum multi- 

 plied by model's frequency-response functions are shown by 

 solid lines 



quency-response function) were measured in a towing 

 tank. Fig. 21 shows the comparison of the spectra ob- 

 tained by analysis of the model's rolling with the spectra 

 computed on the l>asis of waxe spectra and frecjuency- 

 response functions. The over-all agreement is very 

 good, although there are certain differences in the forms 

 of the spectra. In particular, the lower two diagrams in- 

 dicate that the spectrum of actual motion is more nar- 

 rowly concentrated aroiuid the natural frequency than 

 has been indicated by the linear superposition theory. 

 As far as the mean amplitude of the ^/s highest rolls is 

 concerned, the agreement was excellent as shown by 

 citation of the five ca.ses in Table 2. 



Table 2 Rolling Amplitude Obtained 



Attention should be called to the fact that the ex- 

 cellent agreement shown in Table 2 was for the predic- 

 tion of average ship rolling which is based on the spectral 

 area (zero moment). Were the inverse problem at- 

 tempted, that of evaluating ship-response fimctions from 

 the wave and rolling records, the results would have been 

 quite unsatisfactory. This is evident from a comparison 

 of the ordinates of the solid and dotted lines at various 

 specific frequencies. Heavy damping in heaving and 

 pitching motions made the inverse problem practical 

 as far as those modes were concerned. It appears to the 

 author that the inadequacies of the superposition theory 



make solution of this inverse prolilem impossil:)le in 

 rolling. 



It should be emphasized, furthermore, that the fore- 

 going experiments were limited to beam seas, zer(j for- 

 ward speed, antl artificially restrained model. The 

 method of calculations used, which was based on St. 

 Denis and Pierson (1953), cannot be expected to apply to 

 a ship at sea with forwartl speed and in oljliciue wave di- 

 rections. Under these conditions the rolling of a ship 

 will be excited by: 



1 Direct wave action. 



2 Ship yawing caused by waves and wind. 



3 Rudder action. 



4 Rolling moment caused by wind. 



Although rudder movements are made in response to 

 wave and wind yawing distiu'bances, they are not 

 uniquely connected with these. A certain indetermined 

 response function is involved here. Furthermore, the 

 rudder causes not only a yawing moment but also a 

 lateral force. Analyses of ships' turning indicate the im- 

 portance of considering this lateral force in the Ijalance 

 of forces and moments. 



Kato, Motora, and Ishikawa j^rovided some data on the 

 rolling due to wind. They indicated that if wind and 

 waves were not correlated, the rolling excited by each 

 could be computed and the results of both added to- 

 gether. Under storm conditions, however, one should 

 expect a certain correlation between waves and wind."^ 



If an attempt were made to derive frequency-response 

 functions from wave and ship-rolling records alone, the 

 motion resulting from several separate causes would be 

 artificially attributed to one. It appears to be clear that 

 no consistent results could be expected under these condi- 

 tions. In this connection, attention should be called to 

 the fact that Press and Tukey (1956, equations 45 and 46) 

 indicated briefly a method of cross-spectral analysis for 

 multiple disturbances. Within the author's knowledge, 

 this method has not yet been applied to the analysis of a 

 ship's rolling. 



A certain confirmation of the difficulties with rolling 

 (which may have been cau.sed by transient and multiple 

 responses) can be found in the paper by Cartwright and 

 Rydill (1956). These authors applied cross-spectral 

 analysis to the ship's pitching and rolling records in con- 

 junction with wa\-e records and compared the resultant 

 responses with theoretically computed ones. In this 

 comparison, the derived pitching ampfitude responses 

 either agreed with or showed a systematic deviation from 

 theoretical cur\-es with a reasonable scatter of data. 

 The derived rolling amplitude responses, on the other 

 hand, scattered widely and irregularly about theoreti- 

 cally computed curves and "agreement" could be estab- 



'^ The reader will find the wind-tunnel data on side force and 

 heeling moments acting on several ships in steady side wind in the 

 paper by Kinoshita and Okada (1957). Kato, Motora, and 

 Ishikawa (1957) established the near-Gaussian distribution of 

 wind pressure. Firsoff and Fedyaevsky (1957) developed theo- 

 retical formidas antl furnished necessary empirical coefficients for 

 calculating the impulsive lateral force and heeling moment caused 

 by a sudden change of wind velocity and direction. The data on 

 gust distribution in storms are, however, still lacking. 



