Muvthy 



We shall now consider the problem of an ACV that has been 

 operating at sea for a long time under a constant propulsive thrust. 

 This problem is simpler and more of practical interest than that 

 where the ACV starts from rest and moves over water under the 

 action of given forces, such as these due to wind, control setting, 

 etc. , in an arbitrary seaway. After the lapse of a sufficiently long 

 time, all the transients would have disappeared, and if the propulsi- 

 ve thrust is the only force acting on the ACV, it would be moving at 

 a steady speed of translation. The linear displacement of the centre 

 of gravity of the ACV from its equilibrium position of steady transla- 

 tion may be represented by components along three axes fixed in the 

 craft (this system of axes will be described presently), namely, sur- 

 ge, sway and heave. Similarly, the angular displacements of the 

 craft may be represented by components along these axes, namely, 

 roll, pitch and yaw. It is expected that each component of displace- 

 ment will consist of two terms, one independent of time and repre- 

 senting the steady displacement that would exist during motion with 

 uniform velocity in calm water and the other an oscillatory term sim- 

 ple harmonic in the time due to the excitation by the incident waves 

 coming from infinity. If the ACV is symmetrical about a longitudinal 

 axis, it may be expected that the motion in calm water will produce 

 non-zero displacements only in pitch and heave (and, possibly, in 

 surge, which is trivial, since the steady surge displacement can be 

 absorbed in the forward motion). The complete solution of our linea- 

 rized problem depends then on the determination of the forward speed 

 for a given thrust (or thrust required for a given forward speed), the 

 steady pitch and heave displacements (usually known as trim and sin- 

 kage) and the six oscillatory components of displacement. 



An irregular, but long crested, seaway may be assumed to be 

 composed of a system of simple harmonic progressive waves, each 

 of a given frequency. Any irregular wave train may therefore be ex- 

 panded as a Fourier series with respect to time. In the linearized 

 theory, we may assume that the ACV responds to each wave compo- 

 nent as though it existed independently of the others. By the theory 

 of linear superposition the motion of the ACV will be composed of the 

 same Fourier components. Similarly, in the case of forced oscillation 

 in calm water, any arbitrary type of oscillation may be represented 

 by a Fourier series with respect to time. It is therefore only neces- 

 sary to consider a single sinusoidal component for our solution. The 

 results can then be generalised by spectral analysis. 



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