SHIP MOTIONS 



163 



such a solution in the past.'- With the present avail- 

 ability of high speed eomputing machines the solution 

 probably can be accomplished. 



In application to a surface ship, it is necessary to 

 evaluate the complete trajectory of a ship moving in 

 waves, since the submersion of the bow oi of the gunwale, 

 for instance, must be found, as well as accelerations in 

 various directions. In the ca.se of airplanes the problem 

 was further simplified by applying the e(|uati()ns of mo- 

 tion to the study of stability rather than trajectory. 

 Furthermore, the most important practical airplane prob- 

 lem is stability in straight flight. For infinitesimal devia- 

 tions from straight flight, equations (12) separate into 

 two sets of three coupled e(iuations. The first set rep- 

 re.sents the motion in the plane of synmietry and in- 

 volves the first, third, and fifth equations. The second 

 set represents asymmetric motions governed by the 

 second, fourth, and sixth etiuations. Alternately, the 

 properties of steady three-dimensional motion, such as 

 spinning of airplanes, were investigated considering the 

 balance of forces, but not the stability. This elimi- 

 nated the dotted terms on the left-hand sides of e(|uations 

 (12). 



Despite the complications involved in application of 

 equations (12) to surface-ship motion in waves, the 

 author believes that an effort in this direction should be 

 made. The primary objective is to establish the relative 

 signiflcance of various derivatives and of the degree of 

 coupling between equatiojis. Large labor .saving in 

 future work will result from the elimination of any equa- 

 tion v.-hich will be demonstrated to have but a weak ef- 

 fect on other eciuations. Since oidy orders of magnitude 

 are required in this connection, the work can be at- 

 tempted despite the fact that many derivatives can be 

 only crudely estimated at present. A complete step-by- 

 step integration once accomplished for a typical shiji will 

 permit the de\'ising of simplified calculations in the 

 future by justifying neglect of certain cro.s.s-couplings and 

 of many derivatives. 



In using step-liy-step integration, equations (12) are 

 not limited to small disturbances, only each increment of 

 time dt must be small. Large total motion can be rep- 

 resented by assigning suitable values of coefficients for 

 each successive step. One of the most important ef- 

 fects in a six-component motion is the drastic cyclic \'ari- 

 ations of the coefficient dN/dtp (i.e., the yawing moment 

 due to the angle of yaw) caused by the bow submergence 

 and emergence. It becomes particularly important in a 

 following sea and at co -* it may pcjssibly leatl to broach- 

 ing. 



'^ Except in a few limited cases (Jones, 1933, p. 125). Krilciff 

 (1898) presented an example of a complete .solution for the mutiuns 

 of a cruiser at three forward speeds. This was liased on the 

 assimiption of the fore-and-aft mass (hstriliution synmietry and 

 on the "Froude-KrilofT hypothesis." Under tliis liypothesis all 

 external forces acting on a ship were assumed to l)e caused by 

 hydrostatic water pressures corrected for the pressure gradient in 

 waves (for the "Smith effect"). The distortion of the water How 

 by the presence of a ship was neglected. Kriloff's work is par- 

 ticularly important in that the trajectory of the motion was com- 

 puted and transient responses were included. 



2.32 A degree of freedom added by the use of a 

 rudder. For iH\'cstigations of the directional .stability 

 of .ships, sinusoidal motions of the rudder with a prede- 

 termined amplitude were sometimes used, producing a 

 sinus(jidal ship path and a harmonic rolling. The effect 

 of the rudder so u.sed can be represented in the equations 

 of motifin.s as an addetl exciting side-swaying force and a 

 yawing moment. Such a predetermined motion of the 

 rudder would no more assure the maintenance of the 

 specified mean heading in waves than would a stationary 

 rudder. 



To maintain a specified mean heading, the rudder must 

 be moved in respon,se to ship motions induced by \va\'es 

 and the nature of the rudder motions is not known in ad- 

 vance. (Jrtlinarily the rudder is uKn'ed only in response 

 to a yawing disturbance, but its motion generates a side- 

 swaying force as well as the desired yawing moment.'^ 

 Furthermore, the rudder neither is capable ot imnredi- 

 ately checking the yawing motion, nor as a rule, produces 

 the corrective yawing moment at the .same time the ex- 

 citing moment due to waves occurs. The time relation- 

 ship between the wave-excited yawing moment and the 

 rudder-produced corrective moment is a complicated 

 function involving the phase lag in the ship response to 

 waves as well as the properties of the rudder-controlling 

 devices. The force and moment exerted by the rudiler 

 are generally proportional to the rudder angle of attack 

 and therefore are not only functions of the rudder angle 

 5 but also of the ship yawing velocity /•. They are, fiu'- 

 thermore, affected to some extent by the pitching atti- 

 tude 6 and velocity 6 ^ q. The rudder angle 5 is, there- 

 fore, a time-dependent \'ariab!e (not known in advance) of 

 the same significance as the ship-displacement \-ariables, 

 X, y, z, d, 4>, and xp. Seven degrees of freedom are in- 

 volved here. 



The six e([uations (12), must l)e supplemented liy an 

 etiuation deHning the rudder angle 6 as a function of the 

 ship yawing angle, 5 = 5(4/), in whicl\ the integral and 

 first and second derivatives of ^ may be included. Pught- 

 hand sides of the first six equations will have terms de- 

 pendent on S of greater or lesser importance. Right-hand 

 sides of the .second and sixtli e(|uations will haN'e the 

 important added terms (dF, d5)c/5 and (dXdd)rl5. Addi- 

 tion of only these two terms will suffice in linear stability 

 analysis based on infinitesimal ship deviations from 

 steady rectilinear motion. In a step-by-step integration 

 for large deviation from steady conditions, and in the 

 analysis of a ship's turning, the foregoing terms will he 

 modified by the obliqueness of the water flow at the rud- 

 der. These derivatives become functions of the j^awing 

 velocity (dl'/dS) = Y6=Yi{r) and (dA'/dS) ^Ng = Xsd-). 

 The effect of a ship's trim on the rudder-force deriva- 

 tive also may be taken into account; i.e., I'j = Yiid) 

 and Ns = Ni(6). This effect of ship's trim 6 becomes par- 

 ticularly important during a j^art of the oscillating cycle 



'^ The importance of (he side-swaying force in a ship's resjionse 

 to rudder motions is brought out by Davidson and Schitf's ( M)4ti ) 

 theory of dynamic stability of shi|)s on course and of a ship's 

 turning. 



