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Unformatted text preview: ucture. A tensile structure is modeled as a set of n masses in R2 ,
some of which are ﬁxed, connected by a set of N springs. The masses are in equilibrium, with spring
forces, connection forces for the ﬁxed masses, and gravity balanced. (This equilibrium occurs when
the position of the masses minimizes the total energy, deﬁned below.)
We let (xi , yi ) ∈ R2 denote the position of mass i, and mi > 0 its mass value. The ﬁrst p masses
ﬁxed
are ﬁxed, which means that xi = xﬁxed and yi = yi , for i = 1, . . . , p. The gravitational potential
i
energy of mass i is gmi yi , where g ≈ 9.8 is the gravitational acceleration.
Suppose spring j connects masses r and s. Its elastic potential energy is
(1/2)kj (xr − xs )2 + (yr − ys )2 ,
where kj ≥ 0 is the stiﬀness of spring j . To describe the topology, i.e., which springs connect which masses, we will use the incidence matrx
A ∈ Rn×N , deﬁned as 1 head of spring j connects to mass i
−1 tail of spring j connects to mass i
Aij = 0 otherwise. Here we arbitrarily choose a head and t...
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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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