//SYS21///INTEGRA/BST/VOL2/REVISES 31-7-2001/BSTC12.3D ± 482 ± [473±538/66] 30.7.2001 3:50PM
482 Basic ship theory The solution to this di erential equation is cos !t
"
where tan "
2k 1 2 1
tuning factor !=!0 ; magnification factor 1=f(1 2 )2 4k2 2 g2 : Plots of the phase angle " and magni®cation factor are presented in Fig. 12.5. It will be appreciated that these expressions are similar to those met with in the study of vibrations. The e ect of damping is to cause the free oscillation to die out in time and to modify the amplitude of the forced oscillation. In an ideal regular sea, the ship would oscillate after a while only in the period of the waves. In practice, the maximum forced roll amplitudes occur close to the natural frequency of the ship, leading to a ship at sea rolling predominantly at frequencies close to its natural frequency. Pitching and heaving in waves In this case, attention is focused on head seas. In view of the relative lengths of ship and wave, it is not reasonable to assume, as was done in rolling, that the wave surface can be represented by a straight line. The principle, however, remains unchanged in that there is a forcing function on the right-hand side of the equation and the motions theoretically exhibit a natural and forced oscillation. Because the response curve is less peaked than that for roll the pitch and heave motions are mainly in the frequency of encounter, i.e. the frequency with which the ship meets successive wave crests. Another way of viewing the pitching and heaving motion is to regard the ship/sea system as a mass/spring system. Consider pitching. If the ship moved extremely slowly relative to the wave surface it would, at each point, take up an equilibrium position on the wave. This may be regarded as the static response of the ship to the wave and it will exhibit a maximum angle of trim which will approach the maximum wave slope as the length of the wave becomes very large relative to the ship length. In practice, the ship hasn't time to respond in this way, and the resultant pitch amplitude will be the `static' angle multiplied by a magni®cation factor depending upon the ratio of the frequencies of the wave and the ship and the amount of damping present. This is the standard magni®cation curve used in the study of vibrations. Provided the damping and natural ship period are known, the pitching amplitude can be obtained from a drawing board study in which the ship is balanced at various points along the wave pro®le. Having discussed the basic theory of ship motions, it is necessary to consider in what form the information is presented to the naval architect before proceeding to discuss motions in an irregular wave system.