I know rwg has a practical engineering concern here, viz http://www.flickr.com/photos/thane/113588315/ On 4/25/07, Eugene Salamin <gene_salamin@yahoo.com> wrote:
Let rho = density, g = gravity, sigma[i,j] = stress tensor, f[i] = body force per unit volume, F[i] = surface force per unit area, n[i] = unit surface normal, k = length scale factor.
The stress obeys the following two equations (plus one more homogeneous equation that I do not need here).
sum(diff(sigma[i j], x[j]), ,j) = f[i] within the volume,
sum(sigma[i j] n[j], j) = F[i] on the bounding surface.
Suppose the system must look the same under length scale change. Then sigma[i j] must be invariant. The body force is gravity, so f = rho g. But the LHS scales as 1/k. Hence rho g must be scaled as 1/k. The LHS side of the second equation is independent of k. On the RHS, the reaction force scales as rho g k^3 and the contact area scales as k^2, so F scales as rho g k, and this is then independent of k. So your final sentence is correct.
Gene
----- Original Message ---- From: R. William Gosper <rwg@osots.com> To: math-fun@mailman.xmission.com Sent: Tuesday, April 24, 2007 11:26:01 AM Subject: [math-fun] mechanical engineering sanity check
I have a 1/k scale model of a weight on a tripod (http://gosper.org/legbox3.jpg). Alan Adler tells me beam strength scales with the width and the square of the thickness, so k^3. Likewise the load, but there is also a factor of k snapping moment. So for large k, the legs would need to thicken like k^(4/3). (DaVinci's analysis of elephants, giraffes, tree-trunks, ...?) But my k is only about 2.5, and the model is fairly robust, and I want to see how the full-scale would load without the extra leg thickening. So I just need to scale the *density* by k, right? I.e., on a planet with 2g, all the animals could be half scale. --rwg
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