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Unformatted text preview: amical system constains hundreds of rows
and columns, it is still possible to use symmetries in the box spring to simplify
the calculations, and make the problem of determining the “natural frequencies
of vibration” of the mattress into a manageable problem.
To illustrate how the symmetries of a problem can make it much easier to
solve, let us consider a somewhat simpler problem, a system of n carts containing
identical weights of mass m and connected by identical springs of spring constant
k , moving along a circular frictionfree track as sketched in Figure 2.6.
We choose a positive direction along the track and let xi denote the displacement of the ith cart out of equilibrium position in the positive direction,
for 1 ≤ i ≤ n. The potential energy stored in the springs is
V (x1 , . . . , xn ) = 1
=− k
2 x1 1
1
1
1
k (xn −x1 )2 + k (x2 −x1 )2 + k (x3 −x2 )2 +. . .+ k (xn −xn−1 )2
2
2
2
2 −2 1
0 ···
1
x1 1 −2 1 · · ·
0 x2 0
1 −2 · · ·
0 x3 .
x2 x3 · xn ·
·
·
···...
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 Winter '14
 Equations

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