CHAPTER 3 



Ship Motions 



1 Introduction 



It was shown in Chapter 1 that observed sea waves can 

 be classified by various statistical methods without re- 

 gard to the weather conditions causing them. Like- 

 wise, ship motions, observed visually or with the help of 

 various instruments, can be recorded without reference 

 to sea waves which caused them. Such records can be 

 useful, for instance, in the design of equipment for which 

 a certain operational range must be specified. The sta- 

 tistical methods of handling ship-motion data of this 

 type are identical with those already described for sea 

 waves in Section 1-7.' A short reference to ship-motion 

 records of this type will be made in Section 5.2, but the 

 subject will not be developed further. 



The plan oi this monograph is to trace the cjuantitative 

 effect of wa\-es on ship motions and on ship stresses. 

 Attention is concentrated, therefore, on formulation of the 

 functional relationships on the basis of which ship mo- 

 tions and stresses can be predicted once the sea waves are 

 defined. The definition of sea waves as a function of 

 weather conditions was the subject of Chapter 1. The 

 waves cause ship motions, reduce a ship's speed and cause 

 bending stress in a ship's hull. These effects will be 

 covered respectively in Chapters 3, 4 and 5. An outline 

 of the hydrodynamic information needed in this con- 

 nection was given in Chapter 2. 



The ultimate aim of the activity to be surveyed in 

 Chapter 3 is to predict ciuantitatively all motions of a 

 sliip in a natural (always irregular) sea with various di- 

 rections of wave propagation. This broad and difficult 

 problem can be solved only by dividing it into a series of 

 sub-problems, each of which is sufificiently simple to be 

 tractable. The major subdivisions, established only 

 within the last few years, are (a) ship motions in regular 

 long-crested waves, and (6) motions in an irregular sea 

 described by its spectrum. The first of these sub-prolj- 

 lems is attacked by means of hydrodynamics and the 

 dynamics of rigid bodies, and the second mostly by the 

 methods of mathematical statistics. This second sub- 

 problem consists of mathematical operations on the re- 

 sults of the first one. All physical characteristics of 

 waves and ships are considered only in the first sub- 

 problem. The development of these two subdivisions 

 will be outlined in Sections 2 and 3 of this chapter. 



' Reference to sections, equations, figures and bibliography date 

 in preceding chapters will be designated by chapter number and 

 section, equation, reference or figure number; reference above, for 

 example, is to chapter 1, section 7. 



2 Ship Motions in Long-Crested Harmonic Waves 



A ship traveling obliciuely to the direction of wave 

 crests will experience a complicated series of translational 

 and rotational oscillations. In the analysis, these mo- 

 tions are considered as the summation of six components, 

 three translational, and three rotational. The trans- 

 lational motions are surging along the .r-axis, side-sway- 

 ing along the lateral or y-a,xis and heaving along the ver- 

 tical or i-axis. The rotations about these axes are roll- 

 ing, pitching, and yawing. In the mathematical analy- 

 sis of such a motion, a differential equation is written for 

 each mode: i.e., six simultaneous equations are formed. 



Generally motion in any one of these six modes brings 

 into play forces and moments affecting all other modes, 

 so that the analysis becomes rather complicated. 

 The general principles ha\-e been stated and analysis has 

 been made of the much simpler problem of the motions 

 of airplanes and airships, but has not yet been accom- 

 plished for surface ships. Two special simplified cases, 

 however, have been investigated : 



(a) A ship traveling in a direction normal to the 

 wave crests and experiencing only motions in the plane of 

 symmetry; i.e., surging, heaving, and pitching. 



(6) A ship at zero speed in beam seas experiencing 

 only rolling, side-swaying, and heaving. 



The two cases have been listed as physically and math- 

 ematically consistent. Additional simplifications, to be 

 outlined later, are obtained by an arbitrary disregard 

 of certain interactions between modes. These are not 

 justifiable a prion but can be accepted either on the basis 

 of experimental confirmation or because they permit 

 evaluation of certain broad trends. 



The two consistent simplified cases will be considered 

 first in Sections 2.1 and 2.2. The complete six-com- 

 ponent motion will be discussed in Section 2.3. 



The motion composed of the six components just de- 

 scribed is often referred to as one in six degrees of free- 

 dom. The motion of a rigid body is completely described 

 by its six degrees of freedom, but in a practical appHca- 

 tion this description is adequate only when the stability 

 of a body with fixed controls is to be investigated. It is 

 not adequate for investigation of the path or trajectory 

 of a controlled body's motion. Uncontrolled submerged 

 bodies eventually orient themselves broadside to the di- 

 rection of motion and an uncontrolled and unpropelled 

 ship will orient itself parallel to the wave crests. Main- 

 tenance of a useful trajectory or path of a ship requires 

 use of the rudder. A practical ship is, therefore, not a 



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