NavList:

...What's parametric rolling?

Some sentences(with my editing)from 'Ships and Water' by JAH Paffett..' ...consider a ship in a following sea where the wavelength is comparable with the ship's length. The waves themselves , with crests square to the ship's axis, will have no heeling action. If the ship is slow, the waves will pass her from stern to bow, but a fast ship(particularly in shallow water) may find herself keeping station on a single wave for some time. If the trough is amidships and the crest at bow and stern there is not much to worry about, but if the crest is amidships there could possibly be some trouble. This is because the ship's sides are usually more or less vertical amidships, but flared towards the ends. The fall in water levels at the ends reduces the waterplane widths there, while the rise amidships produces no compensating increase. The quantity 'I', the waterplane moment of inertia, is therefore diminished, leading to a reduction in GM(a measure of the ships initial statical stability). Thus a ship perched on a following wave can lose part of her effective GM over a period of time. In these circumstances a moderate heeling moment,as from wind, can produce an unexpected and even an alarming angle of heel. A related phenomenon is the so-called 'parametric rolling'. If a ship is acted upon by a constant heeling moment m (due perhaps to a beam wind or asymmetric loading) she will heel to an angle(�) given by m = W(dispacement).GZ(righting lever)= W.GM.� if � is small. Here W and m are both constant, so that GM.� has to be constant too. It follows that � will vary if GM varies. Thus, a ship in these circumstances will experience a varying �- that is, a rolling motion-even when the seas encountered are coming from dead ahead or astern. The amplitude could be troublesome near resonance, which occurs when the wave encounter period is half the natural roll period of the ship.

An online limited preview of 'Ship Stability for Masters and Mates' by D.R.Derrett 6th Edition 2006 has expanations/definitions of GM, GZ etc p.45,46 and some diagrams re parametric rolling of a container ship p367.

It seems that the lengthy expanation given to we seamen can be reconciled with your more concise 'Tennis racket theorem'.

Regards,

Kevin

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