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Re: Scaling and stability: Was [Nav-L] 7/8 SCALE SEXTANTS MORE
From: Joel Jacobs
Date: 2004 Aug 2, 18:41 -0400
From: Joel Jacobs
Date: 2004 Aug 2, 18:41 -0400
Thank you George, Joel ----- Original Message ----- From: "George Huxtable"To: Sent: Monday, August 02, 2004 6:04 PM Subject: Scaling and stability: Was [Nav-L] 7/8 SCALE SEXTANTS MORE > Joel Jacobs wrote- > > >I do > >have "Stability and Trim for the Ship's Officer" by La Dadge and Van Gemert > >handy. > > > >On page 35, they move beyond the standard formula Henry presents to the > >following: > > > >Saying "moment of inertia is a difficult term to define simply." and "This > >moment which resists motion of an infinite number of moments which are > >composed of the product of each elementary area and the square of the > >distance from the axis." (simple?) > > > >I = L x B3 / 12 for a rectangular waterplane > > > >For non-rectangular waterplanes I = L x B3 x k > > > >k is a constant that relates to a waterplane coefficient > > > >They conclude that the moment of inertia "is almost wholly dependent on > >breadth of the vessel." > > ================= > > This is a bit misleading. They are referring to the moment of inertia of a > thin flat waterplane, about a fore-and-aft axis, such as a raft (if it can > be said to have a fore-and-aft axis). They assume a constant "surface > density": that is, assume that the weight per surface are of the raft stays > constant when other dimensions are changed. > > Robert Gainer was describing a rather different situation, in which the > dimensions of a vessel (which could well be a raft) are ALL scaled up in > proportion. So, as the breadth of a raft is increased by a certain ratio, > so is its length increased by that same ratio, and so is the depth > (vertical thickness) of the raft. And so, therefore , is its weight per > surface area. > > Now the moment of inertia of the raft about a fore-and-aft axis is > proportional to length x (breadth)cubed x depth. So if you double all the > dimensions of that raft, you increase the moment of inertia, about its > fore-and-aft axis, by 2x2x2x2x2, the fifth power of 2, which is a factor of > 32. > > Note that before you specify a moment of inertia, you have to specify the > axis about which the rotation being considered is going to occur. The > moments of inertia about different axes will be very different. But in all > cases, when the dimensions of a vessel are all multiplied up by the same > factor (so that its shape doesn't change), the value of any moment of > inertia will change by that factor to the fifth power (i.e. multiplied by > itself 5 times). That's what Robert Gainer was pointing out. > > George. > > ================================================================ > contact George Huxtable by email at george@huxtable.u-net.com, by phone at > 01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy > Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. > ================================================================