61 chapter 3 fourier series 31 fourier series the

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Unformatted text preview: corresponds to a constant speed motion of the carts around the track. If n is large, sin(π/n) . π/n k1 1 ω1 . = = ⇒ = , π π n 2π mn so if we were to set k/m = n2 , the lowest nonzero frequency of oscillation would approach one as n → ∞. Exercise: 2.5.1. Find the eigenvalues of the matrix 010 0 0 1 0 0 0 T = · · · 0 0 0 100 58 ··· ··· ··· ··· ··· ··· 0 0 0 · 1 0 Figure 2.8: A linear array of carts and springs. by expanding the determinant −λ 0 0 · 0 1 2.6 1 −λ 0 · 0 0 0 1 −λ · 0 0 ··· ··· ··· ··· ··· ··· 0 0 0 . · 1 −λ A linear array of weights and springs* Suppose more generally that a system of n − 1 carts containing identical weights of mass m, and connected by identical springs of spring constant k , are moving along a friction-free track, as shown in Figure 2.7. Just as in the preceding section, we can show that the carts will move in accordance with the linear system of differential equations d2 x k = Ax = P x, 2 dt m where P = −2 1 0 · 0 1 −2 1...
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