SHIP MOTIONS 



157 



shows the coupling-caused transfer of oscillatory motion 

 energy from pitching to heaving. 



The most conspicuous deviation of the series 60 form 

 from the assumptions of the linearized theory is in the 

 large inclination of ship sides in the stern region. In this 

 connection it is interesting to note that out of eight models 

 used by Korvin-Kroukovsky and Jacobs (1957), the 

 two, for which calculations were found to be completely 

 invalid, were models of sailing yachts. The lines of these 

 models show a large inclination of ship sides throughout 

 the model length. The need of investigating the added 

 mas.ses and tlamping of such forms was emphasized in 

 Chapter 2. 



To summarize : Reasonably reliable calculations of the 

 coupled heaving and pitching motions of conventional 

 ships can be made on the basis of the linearized theory 

 using the strip method described in Appendix C for evalu- 

 ation of coefficients. Further research on the supporting 

 material for damping is, however, needed. Various proj- 

 ects, directed to this objective, were listed in Sect inn 

 2-9. 



2.13 Variable coefficients and nonlinearities. At the 

 end of the previous section, attention was called to de- 

 ficiencies of the theoretical damping coefficients, the 

 values of which are assumed to be constant throughout 

 the motion cycle. Discrepancies between theoretically 

 computed and experimental ship motions may also be a 

 result of the latter assumption. In the linear theory, the 

 ecjuations are solved for very small wave heights and very 

 .small motions, in which case the coefficients are es.sen- 

 tially constant. In the actual fairly large motions of 

 models, however, the coefficients are not constant but are 

 functions of instantaneous ship position and, therefore, 

 functions of time. A simple explicit solution can be ob- 

 tained only when the coefficients are assumed to he con- 

 stant and independent of time. The coefficients are 

 evaluated in this case with ship sections submerged to the 

 normal still \\'aterline (for instance as in Appendix C, 

 equation 42). These coefficients can be evaluated, 

 however, for any instantaneous position of the water- 

 line at a ship section, and thus can be expressed as func- 

 tions of displacement from a normal position. The 

 solution of equations (2) can be obtained in this case 

 only numerically by means of step-by-step integration. 



An important de\'iation from the assumptions of the 

 linear theory occurs in the restoring-force coefficients c 

 and C which depend directly on the width of the water- 

 line. For sloping sides, usually found in stern sections 

 of ships, these coefficients have a higher value than mean 

 at a deeper submergence and a lower \'alue at a shallower 

 submergence. There is also an important nonlinearity 

 in the case of the damping coefficients b and B because of 

 a triple effect: (a) Changes in waterline width with sub- 

 mergence, (b) changes of mean draft, (c) a certain de- 

 pendence of damping on the square of the vertical ve- 

 locity in addition to the first power of velocity i.e., 

 b = b{z). 



The significance of item (b) in a ship's motion was 

 demonstrated in an exaggerated form by a towing-tank 



test of Akita and Ochi (1955), as shown in Fig. 4. The 

 .solid line with black dots indicates the phase angle (des- 

 ignated here as 5^-^) between pitching and heaving 

 motions. The model was 19.7 ft long, had a flat bottom 

 and vertical sides, and was tested at a very shallow draft 

 of 9.5 in. in waves 7 in. high. Because of the vertical 

 sides and fore-and-aft symmetry, the model behavior 

 should have been linear in all respects except item {b). 

 It is clear from the discussion of damping forces, ex- 

 pressions 2-(21 and 22), that the damping coefficient 

 is sensitive to draft and increases rapidly with decrease of 

 draft. In the present case, relati\'e changes of draft and 

 damping force with model pitching are large because of a 

 small mean draft. Furthermore, since in pitching a de- 

 crease of draft at one end is accompanied by an increase 

 at the other, the cross-coupling coefficients e and E of 

 equation (2) should be affected. Fig. 4 shows that at 

 zero forward speed, the heave lags behind pitch by nearly 

 90 deg, as it would in a simple noncoupled system. At 

 the speed near 1.4 m per sec, in the vicinity of synchro- 

 nism, the heave-pitch lag, however, is reduced to about 

 20 deg. Apparently the only cau.se of this behavior 

 is the nonlinearity of damping, which caused large 

 cyclically fluctuating cross-coupling terms. It is im- 

 portant to realize that nonlineui'ity of terms of the ecjua- 

 tions of motion becomes particularly important because 

 it brings about cyclically varying changes in cross-cou- 

 pling terms. The effect of nonlinear clamping is unusually 

 strong in the Akita and Ochi model because of its shallow 

 draft. It will occur, however, in cases of all ships but to a 

 smaller degree. 



Demonstrated in the foregoing for damping, the ef- 

 fects noted occur in the displacement cross-coupling terms 

 (g and G) in ships having inclined sides. This effect was 

 discus.sed by Radosavljevic (19571)). 



The nonlinear behavior of the coefficients of e<iua- 

 tions (2) can be evaluated without difficulty by applying 

 the expressions given in Appendix C to instantaneous 

 section drafts. While formal solution of equation (2) be- 

 comes impossible, numerical .solutions can be made easily 

 with the help of electronic computing machines. A re- 

 search project to carry out such computations is recom- 

 mended because of the possibility of a large effect of non- 

 linearities on motions for many ship forms. In par- 

 ticular, phase relationships appear to be strongly af- 

 fected. 



So far computation of nonlinear motion has been car- 

 ried out only by Hazen and Xims (1940) using a semi- 

 mechanical process. The nonlinearity was limited to re- 

 storing moment coefficients, C. This attempt was es- 

 sentially premature in that it was applied to an un- 

 coupled pitching motion and based on displacement com- 

 puted according to the Froude-Kriloff hypothesis. The 

 damping coefficient was crudely estimated and was taken 

 as constant. Hazen and Nim's work showed that even in 

 harmonic waves the motions deviated considerably from 

 the harmonic. However, while these deviations distorted 

 the sinu.soidal oscillatory trajectory, it has not been tiem- 

 onstrated to what extent the amplitudes were af- 



