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Unformatted text preview: retation of M to solve this problem, but here it is,
for those interested. Suppose we rotate the mass distribution around a line passing through the
120 center of gravity in the direction q ∈ R2 that lies in the plane where the mass distribution is, at
angular rate ω . Then the total kinetic energy is (ω 2 /2)q T M q .)
The goal is to choose the density ρ, subject to 0 ≤ ρ(z ) ≤ ρmax for all z ∈ R, and a ﬁxed total
mass m = mgiven , in order to maximize λmin (M ).
To solve this problem numerically, we will discretize R into N pixels each of area a, with pixel
i having constant density ρi and location (say, of its center) zi ∈ R2 . We will assume that the
integrands above don’t vary too much over the pixels, and from now on use instead the expressions
N ρi , m=a
i=1 a
c=
m N N ρi zi , M =a
i=1 i=1 ρi (zi − c)(zi − c)T . The problem below refers to these discretized expressions.
(a) Explain how to solve the problem using convex (or quasiconvex) optimization.
(b) Carry out your method on the problem in...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.
 Fall '13
 F.Borrelli
 The Aeneid

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