124 



THEORY OF SEAKEEPING 



frequency may occm- in a following sea. This condition 

 is not critical for heaving and pitching motions, but may 

 become of interest in an analysis of six-component mo- 

 tion in C|uartering seas. The interest in this case is in- 

 direct and is connected with the effect of bow submersion 

 on yawing and yaw-induced rolling. However, the 

 frecjuencies of encounter in quartering seas of significant 

 wa\'e lengths are usually a fraction of the resonant fre- 

 quenc.y of heaving and pitching. Large errors in damp- 

 ing estimates, therefore, have little influence on the mo- 

 tion estimates. 



The peculiar behavior of the damping in pitch at low 

 frequencies was determined analytically by Havelock 

 (195S) and was recorded by Golovato (1958) in tests on 

 a parabolic model. Golovato's results are shown on Fig. 

 19. 



3.3 Integration With Respect to Length. An individ- 

 ual ship section of length d^, at a distance ^ from the 

 origin, has a simple vertical motion lioth in space and in 

 relation to the rising and falling sea surface. It is sub- 

 jected to the four kinds of forces generated by water 

 pressures, three of which are caused by ship oscillation 

 (as in smooth water) and the fourth by wave action. 

 The first three are : 



Inertial (proportional to acceleration). 



Damping (proportional to vertical velocity). 



Displacement (proportional to changes of displaced 

 water volume). 



The wave-caused forces also can be considered as com- 

 posed of inertial, damping and displacement compo- 

 nents. In the classical wave theory, howe\er, the iner- 

 tial and displacement forces are interrelated and are 

 more conveniently combined in a single force. 



In order to analyze a ship's motion (considered as that 

 of a rigid body) in waves, the sectional forces must be 

 integrated over the ship's length. If the sum of the 

 sectional forces (exclusive of wave forces) is designated 

 as dF/d^, the integrations have the general form (with 

 reference to Fig. 1) : 



Heaving force 



Pitching moment 





All integrations are carried out over the ship's length. 

 Since dF/d^ consists of three components, the integration 

 yields the three coefficients a, b, and c of equation (1). 

 Likewise integration of the wave force yields the coeffi- 

 cient Fo of equation (1). Two separate equations for 

 heaving and pitching of the form of ecjuation (1), each 

 containing four parameters, are referred to as "uncoupled 

 equations." In reality heaving motion also causes cer- 

 tain pitching moments and pitching motion causes cer- 

 tain heaving forces. These cross-coupling effects are 

 represented Idj' coefficients resulting from an integration 

 of the form : 



0.8 

 a)(B/g)5 



Fig. 19 Damping in pitch versus frequency in region of low 

 frequencies (from Golovato, 1958) 



A more detailed exposition of these coefficients and of 

 their use in ship-motion analysis will be found in Korvin- 

 Kroukovsky and Jacobs (1957). It will also be reviewed 

 briefly in Chapter 3. The derivation of sectional forces 

 and their integration with respect to ship's length will be 

 found in Appendix C. 



Evaluation in a closed form of the integrals just shown 

 is possible only in special cases of mathematically ex- 

 pressed ship lines. An example of such a procedure can 

 be found in Weinblum and St. Denis (1950). For nor- 

 mal ship forms, these integrations can be carried out 

 either graphically or numerically by methods familiar to 

 naval architects. 



(23) 4 Forces in Lateral Motion 



Cross coupling force or moment 



/f -'* '^ 



:4) 



In the present discussion the term "lateral motion" 

 refers to yawing and side sway. (The rolling motion is 

 to be discussed separately.) As in heaving and pitch- 

 ing, the forces invoh'ed in these motions can either be 

 estimated for a body as a whole by comparison with 

 ellipsoids, obtained experimentall.v for the body as a 

 whole, or computed theoreticall.y by means of the strip 

 theory. In \'iew of the extreme scarcitj' of the available 

 data, all of these will be treated together in this section. 



Data on the force coefficients for the entire ship 

 were given for six models in the work bj' Da^'idson and 

 Schiff (1946) as part of a theoretical and experimental 

 investigation of ship stability on course in smooth water. 

 These data have been deduced partly from measure- 

 ments made on the rotating arm and partly from obser- 

 vation of the turning radii of self-powered, free-running 



