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 frictionfree 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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This document was uploaded on 01/12/2014.
 Winter '14
 Equations

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