164 



THEORY OF SEAKEEPING 



in which {dN/d\p) acquires a hirge positive (i.e., desta- 

 bilizing) value because of the submergence of the bow and 

 emergence of the stern, while the ^'alue of the (dN/dS) is 

 reduced. 



The presence of the gyroscopic coupling terms on the 

 left-hand sides of equations (12), in addition to the less 

 evident couplings due to derivatives on the right-hand 

 sides of eciuations, spreads the effects of the rudder mo- 

 tions through all six equations. 



A limited problem of rudder-controlletl motion will be 

 discussed in the next section. 



2.33 Approximations based on a limited number of 

 degrees of freedom. In towing-tank tests, artificial 

 limitation of the degrees of freedom of model motion can 

 often be useful. This may permit a relatively simple 

 analysis of a few free modes and evaluation of certain 

 derivatives or wave-excited forces. However, the sub- 

 ject of this section will be a simplified analysis of the 

 motion oi free models in regular long-crested waves, based 

 on a more or less justifiable neglect of certain cross- 

 couplings or, in other words, an analysis based on the as- 

 sumption that the motion can be described by a few 

 selected degrees of freedom. 



(a) Two quasi-rational cases. First, two cases will be 

 cited which, although not mathematically rigorous, can 

 be accepted intuitively as mathematically and physically 

 compatible. In the first of these cases, it is assumed that 

 surging motion has no significant effect on other motion 

 modes. This is an extension of the practice of neglect- 

 ing the effect of surging on coupled heave-pitch motion. 

 The six equations (12) with 108 stability derivatives are 

 thereby reduced to five with 75 derivatives. An illus- 

 tration of practical application of the five-degrees-of-free- 

 dom analysis to a relatively simple airplane case will be 

 found in the paper by Westerwick (1957), in which fur- 

 ther references are also given. This paper is an ex- 

 cellent example of the application of the five ec(uations 

 of motions (with some derivatives omitted by judgment) 

 to the determination of the desirable automatic control 

 function to achieve a specific ol)jective. 



The second quasi-rational multi-mode case is obtained 

 by further assuming absence of rolling; i.e., <^ = <f> = 

 = 0.''' The system of ecjuations is thus reduced to four 

 with 48 derivatives on the right-hand sides. Gyroscopic 

 cross-coupling moments on the left-hand sides of equa- 

 tions (12) are also eliminated since p (i.e., <^) = 0. More- 

 over, a good deal of the difficulties in relationships be- 

 tween co-ordinate systems fixed in a ship and fixed in 

 water surface is eliminated, because the rolling angle 

 is the only angle which normally reaches large values. 

 The author feels that application of this simplified anal- 

 ysis to model experiments in oblique waves is one of the 

 most useful projects which can be undertaken with cur- 

 rently available theoretical and physical facilities. 



The case just outlined correspf)nds closely to physical 

 reality in the case of roll-stabilized ships. A'arious means 

 of roll-stabilization reduce the roll angles to such small 



'■* This case was suggested to the author by Prof. E. V. Lewis 

 of the Davidson Laboratory, Stevens Institute of Technology. 



values that the cross-coupling effects of rolling become 

 quantities of second order. Stabilizing water tanks evi- 

 dently have no effect on hydrod.ynamic derivati\'es. An- 

 tirolling fins, in a conventional arrangement near amid- 

 ships, would cause such a small lateral force and yawing 

 moment as to be negligible in comparison with other 

 forces and moments acting on a ship. 



After the solution for motions is completed with = 

 = (ji = 0, the \'alues of the \'ariables determined can 

 be inserted into the fourth equation of (12). This will 

 yield the amplitude and phase of the rolling moment 

 which must be developed by the roll-stabilization system; 

 i.e., will lead to a rational design of this system. 



(6) Arbitrary selection of motion modes. Often an 

 analysis limited to a few degrees of freedom will furnish 

 information on a particular aspect of the motion despite 

 the lack of physical reality in such a limitation. It is 

 necessary, however, to remember the nature of the 

 limitation, to be satisfied with an answer limited to cer- 

 tain conditions and not to expect a uni\'ersally valid solu- 

 tion. 



Rydill (1959) has undertaken such a limited approach. 

 His objective was to find the amplitude of yawing oscilla- 

 tion of a ship in long-crested waves with and without 

 rudder control, to evaluate the amount of rudder motion 

 and to appraise the effect of alternate rudder-control 

 functions with 8 = 5 (J^Tpdt, \p, xj/, \f). Two linearized 

 coupled equations in side sway and yaw, i.e., the second 

 and sixth equations of (12), were considered and the 

 possible effect of all others was neglected. Since the mo- 

 tions are limited in the linearized theory to small values 

 of the variables, the rudder forces and moments were 

 taken as proportional to the rudder angle d, and the (in 

 this case) second-order effect of the yawing velocity \p = 

 r on the rudder angle of attack was neglected. The 

 analysis was first made for long-crested regular waves and 

 was subsequently generalized to long-crested irregular 

 waves. This last step was based on the spectral analysis 

 methods which will be described in Section 3. 



The action of the rudder involves a chain of successive 

 events. First, there is a lag in a ship's response to wave 

 excitation which implies lag in the reading of the sensing 

 elements of a control mechanism. Next, there are vari- 

 ous lags of other responses to the signal before the de- 

 sired rudder motion is accomplished. Finally, there is a 

 lag in the ship's response to the rudder movement. 

 While these features can be incorporated in the dif- 

 ferential equations of motion, Rydill demonstrated how 

 they can be better analyzed b.y the servoniechanism 

 theory. 



Probably the most valuable result of t he paper is the 

 demonstration that ship responses to rudder movements 

 are large at low wave-encounter frequencies and become 

 small at high frecjuencies. The frequent and rapid rud- 

 der motions induced by conventional control systems in 

 head seas are, therefore, practically useless in reducing 

 ship-yawing motions. The motions with and without 

 rudder control are essentially the same in this case. The 

 rudder control becomes indispensable in quartering seas 



