Understanding and Prediction of Ship Motions 



(amplitude of ship motions) x (waterplane area); (e) motion-generated fluid 

 force = (amplitude of motion- generated waves) x (waterplane area). We can now 

 add to our catalog of orders of magnitude: 



excitation by incident waves /S^ 



hydrostatic restoring force fi^ 



ship inertial reactions /S^ 



added mass and damping force /3^ 



At resonance, the restoring force and inertial reaction add up to zero, and there 

 are no second order forces to counter the excitation force. Therefore the re- 

 sponse amplitudes are unbounded to this order of approximation. This violates 

 the assumption that motions are of order /3, but it would not be proper in the 

 perturbation analysis to try to modify the resonance prediction through use of 

 higher forces, simply because they are of higher order and therefore small by 

 comparison. 



Peters and Stoker criticized Haskind for assuming a priori the orders of 

 magnitude of the various kinds of forces, but it can be seen a fortiori that 

 Peters and Stoker have done essentially the same thing, for they also assumed 

 the orders of magnitude of certain quantities (not the forces) and arrived at un- 

 tenable conclusions. 



These authors recognized and noted this anomaly, and they suggested sev- 

 eral escapes from this predicament. For example, they discussed the "flat 

 ship" linearization. However, such an approach simply shifts the same diffi- 

 culty to the lateral motion modes. They also considered a "yacht-type" ship, 

 which would avoid the trouble in all modes except surge. Such a mathematical 

 model is quite artificial, but it might produce successful results if it could be 

 worked out. 



However, the basic difficulty with thin- ship theory may be looked at in an- 

 other way which suggests a totally different method. It was assumed that ship 

 beam, ship motions, and incident waves were all small, of the same order of 

 magnitude. However, it was found that motions near resonance could be very 

 large — to an extent that invalidated the assumptions. Newman (1960) proposed 

 that there should be more than one small parameter in the statement of the 

 problem, and he worked out a development in terms of three parameters. He 

 retained ,3, the beam/length ratio, and he added a parameter y which indicates 

 the order of magnitude of the unsteady motions and another parameter s which 

 indicates the order of magnitude of the incident waves. Such a triple expansion 

 allows for consideration of two important points: 1) There is no reason at all to 

 assume that ship beam is related in size to the amplitude of the incident waves; 

 2) it is not necessary to make any a priori assumptions about the magnitude of 

 ship motions relative to the magnitude of ship beam or incident waves. With 

 regard to point 1), we note that ship beam and incident wave amplitude remain 

 as independent parameters throughout the problem, whereas, with regard to 

 point 2), we expect that the solution of the problem will provide us with infor- 

 mation about the actual amplitudes of motion. 



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