SHIP MOTIONS 



153 



rigid body but has an additional dosi'^c of freedom in 

 movements of the rudder. This subject will be further 

 discussed in Section 2.32. 



2.1 Ship Motions in the Plane of Symmetry: 2.11 

 Linear theory of coupled pitching ond heaving. Ship 

 motions in the plane of synnnetry involve surging, 

 heaving, and pitching. The oscillatory surging motion, 

 along the .c-axis, does not appear to be of direct interest 

 in defining the seagoing qualities of ships. Previous in- 

 vestigations of the stability of airjilanes and seaplanes 

 have also indicated that surging motion has no effect im- 

 portant in practice on the heaving and pitching motions 

 of free ships.- Two mutually interconnected, or 

 "coupled," differential equations of motion, expressing 

 Newton's second law for heaving and pitching, are usually 

 sufficient. These equations can be written (on the ha- 

 bitual assumption of infinitely small motions) as: 



mz = Z . 



Je = .1/ ^ ' 



where ??i and J are the mass and mass moment of inertia 

 of a ship, 2 is the displacement along the vertical or Z- 

 axis, and 6 is the angular displacement in pitching. Z is 

 the component of the total hydrodynamic force in the 

 vertical direction and M the total hydrodynamic pitch- 

 ing moment. By "hydrodynamic force" is usually 

 meant the resultant of those water pressin-es acting on the 

 hull which are connected with relative water velocities 

 and accelerations. However the force Z and moment 

 M in equation (1) include changes in the buoyancy force 

 resulting from shi]) oscillations. Only the velocities and 

 accelerations connected with waves and the oscillatory 

 body motions are considered. Water pressures resulting 

 from the steady forward speed of a ship and the hydro- 

 static pressures in normal flotation are not considered. 

 These are balanced by the propeller thrust and the weight 

 of a ship. The oscillatory fluctuation of the propeller 

 thrust in waves and the moments connected with it are 

 insignificant in comparison with other oscillatory forces 

 acting on a normal surface ship. They would have to be 

 taken into account in a planing-type craft, but this is out- 

 side the scope of this monf)grapli. 



The hydrodynamic force Z and moment M can be eval- 

 uated by several methods which will be discussed later. 

 The results of this e\'aluation take the form of a poly- 

 nomial with various terms proportional to displacements, 

 z, d, 1) (wave elevation), velocities, i, 6, ij and acceler- 

 ations, 2, 6, 7j. The terms in z and 9 and their derivatives 

 represent the forces connected with a body oscillating 

 in smooth water. The terms in r; and its derivatives cor- 

 respond to wave forces acting on a ship restrained from 

 heaving or pitching. When this second group is left on 

 the right-hand side and all other terms are transferred to 

 the left-hand side of equations (1), these equations take 

 the expanded form: 



^ It has been shown theoretically and experimentally (Keiss, 

 1956; Sibul, 195t)) that .surging motion oceasionaliy becomes 

 important in towing tanks because of synchronism with the inotlel 

 towing system. By the same token it may be important in towed 

 and in moored craft. References for chapters 2 and 3 ajjpear at 

 the end of this chapter. 



az + l>i + cz + (Id + cd + gd = Fe''^' 

 Ad+ Be + Ce + Dz + Ez + Gz = Me'"' 



(2) 



The first three terms on the left-hand sides of equation 

 (2) are identical with ecjuation (1) of Chapter 2 repre- 

 senting a simple oscillator. 



The last three terms on the left-hand sides are known as 

 "cross-coupling" terms and express the influence of pitch- 

 ing on heaving motion and the influence of heaving on 

 pitching. The coefficients e and E contain simple (dis- 

 sipative) damping terms arising from the fore-and-aft 

 asymmetry of the hull, and also contain inertial con- 

 tributions indicating the transfer of energy from one 

 mode of motion to another. Havelock (1955) referred to 

 these contributions as "dynamic damping." This is in 

 agreement with the fact, well known in the study of vi- 

 brations of coupled mechanical systems, that oscillations 

 can be controlled by a certain disjiosition of masses and 

 springs, without introducing dissipative damping. 



The forces caused by waves on the right-hand sides of 

 e(iuations (2) could have been presented in the same 

 form as the first three terms on the left-hand sides; 

 i.e., as forces proportional to acceleration, velocity and 

 displacement. In a simple harmonic wa\'e, however, the 

 relationships among these terms are rigidly defined, and 

 the solution of equation (2) is simiilificd by a compact 

 complex notation, with the understanding that real parts 

 are to be taken. These terms can be written as 



ReFc'"' = Re{Foe''')e'"' = Focos(iot + a) 

 ReMt'^'' = Re{Moe")e'"' = il/ocos(co< + r) 



(3) 



F and M are the "complex amplitudes"; i.e., the 

 quantities defining both the amplitudes of force and 

 moment (Fu and Mo) and the phase lag angles (o- and r). 

 These latter can be expressed with reference to an arbi- 

 trary origin, but the same convention must be retained 

 for both the force and the moment. The symbol oj is used 

 in the foregoing eciuations to represent the frec(uency of 

 the wave encounter, cof, but the value of F and i\[ also will 

 depend on the wave length; i.e., on the wave's own fre- 

 quency. 



The solution of the coupled differential equations of 

 motion (2) is given for the steady-state oscillations 

 (Korvin-Kroukovsky and Lewis, 1955; Korvin-Krou- 

 kovsky and Jacobs, 1957) as 



AIQ - FS 

 QR - PS 



FR - MP 

 QR - PS 



(i) 



where P, Q, R, and S represent the groupings of the coef- 

 ficients of equations (2) as follows: 



P = —aw- + ihw + c 

 Q = —do- -|- iew -\- g 

 R = -Dw~ + iEw -f G 

 S = -^0)2 -I- iBw + C 



(5) 



The symbols Z and d in eciuations (4) are the complex 



