tion of time, it is necessary to measure the smaller 

 magnitudes in order to have a sufficiently repre- 

 sentative sample for the statistical analysis. Too 

 large a degree of truncation of the data would 

 make it difficult to establish the distribution pat- 

 terns with acceptable accuracy. 



In engineering application very large values 

 (extremes) of stress and motion, although they 

 occur very seldom, may be of considerable im- 

 portance. The reliability of predicting the mag- 

 nitude of these extreme, but rare, values is of ne- 

 cessity very poor; nevertheless it would be de- 

 sirable to have experimental data in this range if 

 such predictions are to be made. In any case, 

 if statistical predictions are to be made, it is neces- 

 sary that measurements of the lower magnitudes, 

 which occur relatively more often than the larger 

 values be available. However, if rare extremes 

 are not of particular interest, the reliability of 

 theoretical predictions based on experimental 

 data which do not include these rare extremes will 

 not be affected appreciably by lack of observations 

 of these extreme values. 



Summary 



The analysis of the voluminous data accumu- 

 lated during the past several years has indicated 

 that the pitch, roll, and heave motions of ships 

 as well as the hull-girder stresses, follow the same 

 general distribution pattern as do the heights of 

 ocean waves. 



It has been shown that the frequency distribu- 

 tions of pitch, roll, and heave motion of ships, as 

 well as the over-all ship-girder stresses induced by 

 the sea, can be represented by the one-parameter 

 Rayleigh distribution when the environmental 

 conditions of the sea, ship speed, and course are 

 constant. Furthermore, the ship responses can 

 be represented by the two-parameter logarithmi- 

 cally normal distribution when the environmental 

 conditions are allowed to vary over the range that 

 ships encounter over an extended period of 

 service. 



These logarithmically normal distributions thus 

 are the result of the physical summation of a large 

 number of Rayleigh distributions, each of which is 

 weighted in accordance with the relative proba- 

 bility with which the environmental conditions 

 pertaining to it are encountered in service. 



It also has been indicated that the long-term 

 distribution of wave heights and wave lengths 

 may be adequately represented by the log-normal 

 distribution. A study of wave observations taken 

 regularly over a period of 6 years at 14 stations 

 in the North Atlantic shows that distribution 

 based on the observations taken over 1 year differs 

 little from that obtained over 6 years. This fact 



suggests that inasmuch as ship motions and 

 stresses are induced by the waves, 1 year is a 

 sufficiently long test period for establishing a 

 reasonably good statistical distribution pattern of 

 ship motions and stresses. 



The patterns which have been found to fit the 

 experimental data can be specified in terms of one 

 or two numbers which provides a considerable 

 simphfication of the problem when it is required to 

 state what a ship is expected to do. The presen- 

 tation of the Rayleigh distribution has been simpli- 

 fied by utilizing a change of variables, such that 

 all Rayleigh distributions may be represented by 

 means of a single straight line, as in Fig. 3. In 

 addition to the basic distribution pattern, the 

 form of the distribution of the extreme values of 

 the ship response to the sea also has been specified. 



It is desirable, from the standpoint of savings in 

 time, and cost, that the actual distribution ap- 

 plicable to particular ships be determined in the 

 future by means of either theory or model tests. 

 There are far too many unknowns to make the 

 purely theoretical approach practical. Model 

 tests would seem to offer the most direct method 

 that could be developed into a useful tool within 

 the immediate future. It will first be necessary 

 to compare the distribution obtained from model 

 tests with those obtained from full-scale sea trials 

 under comparable conditions. The required full- 

 scale data are at hand and it should not be too 

 long before preliminary comparisons can be made. 

 The method of St. Denis and Pierson for com- 

 puting the power spectrum of the ship response, 

 given the power spectrum of the sea as well as the 

 speed and course of the ship, is directly applicable 

 for the calculation of the frequency distribution 

 of the ship motions or bending moments, pro- 

 vided certain simplifying assumptions made in the 

 derivation of their method are acceptable. Their 

 procedure should be of great help in making it 

 possible to use simplified model test results in regu- 

 lar waves for checking the validity of the model 

 test procedure for predicting fuU scale distribu- 

 tions. 



There are many practical applications of the fre- 

 quency-distribution patterns. In the determina- 

 tion of the capacity of ship and shipboard stabili- 

 zation equipment, it is necessary to have a reliable 

 estimate of the probabiUty of exceeding given 

 angles of roll or pitch. The design of aircraft 

 landing gear, rocket launchers, and fire-control 

 apparatus requires a knowledge of the ship mo- 

 tions expected in service. The ability to land 

 planes on a carrier in a given sea can be predicted 

 on the basis of frequency distributions such as 

 given herein. 



In the design of the ship structure, endurance 



39 



