220 this equation is easily solved via eulers formula

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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 friction-free track as sketched in Figure 2.6. We choose a positive direction along the track and let xi denote the displacement of the i-th 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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