SHIP MOTIONS 



165 



at low frequencies of wave encounter. Various charac- 

 teristics of the rudder-control function 5 = 5 (J'4'dt, 4', 

 xj/) were brought out, verifying and extending the previous 

 wurlx of Schiff and Gimprich (1949). 



While giving a large amount of useful information, the 

 results demonstrated at the same time the a|3parcnt in- 

 adequacy of the linear theory and of neglecting coupling 

 with the pitching motion. The amplitude of ship yaw- 

 ing in wa\'es found by the above analysis appears to be too 

 small judging by observations at sea. The tendency of a 

 ship to yaw in quartering and following seas appears too 

 weak in the theoretical results and does not explain the 

 steering difficulties encountered on ships. In particular 

 theoretically found yawing does not lead to broaching. 

 The conclusions therefore appear doubtful at oje — *- 0. 



As stated earlier, the author believes that practical 

 steering conditions at sea depend primarily on the 

 cyclical variation of the derivatives dN/d4' with bow and 

 stern immersion and emersion. From Rydiil's findings, 

 the effect of this variation becomes more important with 

 the decrease of frequency in following seas. These ef- 

 fects can be brought out by a nvmierical step-by-step in- 

 tegration of four coupled eriuations of motions; i.e., neg- 

 lecting surging and rolling. The work of Davidson 

 (1948) on broaching can be cited as an. example of the 

 limiting case oje = 0, and the use of step-by-step integra- 

 tion. The equation for surging was not formally con- 

 sidered, but the action of a wave in increasing the speed 

 of a ship poi.sed on its advancing flank was taken into ac- 

 count. 



The aim of Rydiil's project was determination of the 

 most advantageous rudder-control method. Rydill called 

 attention to the fact that De Santis and Russo (1936) 

 and Allan (li)15) cited sea observations indicating the in- 

 fluence of rolling on the directional control of ships. 

 Nevertheless this effect was considered less important 

 than other yawing moments, and rolling was neglected. 

 Other limited groupings of degrees of freedom may be 

 chosen in order to bring out the rolling characteristics 

 of ships. Rolling is recognized as one of the most dis- 

 turbing motions of ships at sea and its more pronounced 

 manifestations were forcefully described by Captain 

 Patterson (1955). While rolling can seldom be relieved 

 by reduction of speed, it frequently delays ships by 

 necessitating course changes. Even if it were assumed 

 that most of the modern passenger liners will be equipped 

 with antiroUing devices, most of the cargo ships prob- 

 ably will not and improvement of rolling characteristics 

 must remain an important item of research. 



In Section 2.2 analysis was made of rolling in side 

 waves at zero speed and in these conditions many coupling 

 effects are inactive. They all come into play with in- 

 crease of a ship's forward .speed and with obliiiueness of a 

 ship's heading to wave crests. The couplings may act 

 as added excitation forces and moments or as restraining 

 ones. These latter may cause more rapid attenuation of 

 rolling amplitudes than can be expected from the damp- 

 ing in roll alone. From observations at sea and during 

 tests in oblique waves in a towing tank, the author has 



gained the strong impression that rolling motion was 

 controlled (rather than excited) by waves and differed in 

 nature from the free rolling in smooth water. An in- 

 vestigation of coupling effects in a ship's rolling motif)n 

 appears, therefore, necessary for a realistic presentation 

 of rolling at sea. The coupling effects result not only 

 from the hydrodynamic derivatives on the right-hand 

 sides of equations (12), but also from the gyroscopic 

 couplings indicated on the left-hand sides of these equa- 

 tions. These latter are suspected to be important in \'iew 

 of the generally small damping in pure rolling and the 

 small moment of inertia /^. 



It is known from empirical experience in connection 

 with ship behavior at sea that cross-coupling of rolling 

 and yawing motions is important. The significance of 

 strong yaw-heel (and therefore yaw-roll) coupling has 

 been forcefully brought out by the experience of gyro- 

 stabilizing the SS Conte-di-Savoia (De Santis and Russo, 

 1936). The stabilization system was not adef|uate to 

 control rolling of the ship as designed, but apparently 

 became much more effective when a skeg was added, in- 

 creasing the directional stability of the ship. 



The investigation of rolling of ships in waves can be ap- 

 proached by two methods. In the first method, it may 

 be assumed that rolling is strongly influenced by other 

 motions, but that it has little eft'ect on these other mo- 

 tions. Analysis of a four-mode motion (five degrees of 

 freedom including the rudder motion) can first be made 

 as outlined in Section 2.33 (a); i.e., omitting surging and 

 rolling. The resulting values of all time variables can 

 subsequently be inserted into the fourth of equations 

 (12), expressing the equilibrium of moments about the 

 :r-axis. The solution of the differential equation in will 

 describe the rolling motion. The author believes this ap- 

 proach to be a close approximation to physical reality. 



A second and reasonably realistic method of roll deter- 

 mination may consist of omitting the equations in surg- 

 ing and heaving and considering an assumed four-mode 

 motion in side sway, rolling, pitching and yawing, only. 

 The author considers this approach less realistic and at 

 the same time much more difficult than the first because 

 of the complications connected with various co-ordinate 

 systems at large angles of rolling. 



The author has already em]:)hasizcd the important ef- 

 fect of the ship liow submersion on yawing motion; 

 strong roll excitation by the yawing can also be expected 

 in this case. The usefulness of the small-displacement 

 solution of the differential e(]uations of motions, i.e., with 

 constant coefficients, is therefore questioned. The 

 analyses are expected to yield realistic results if step-by- 

 step integration is used with variable coefficients appro- 

 priate to the state of motion at each particular instant. 



Among other aspects of realistic motion in waves, the 

 described analyses will yield information for estimating 

 the probability of a ship's capsizing in following or quar- 

 tering seas. Deep bow submergence and stern emer- 

 gence create strong yawing instability simultaneously 

 with decreased rudder effectiveness. Molent yawing mo- 

 tion can therefore develop leading to a large yaw-induced 



