Three SpringCoupled Masses
Consider a generalized version of the mechanical system discussed in Section
4.1
that consists of three identical
masses
which slide over a frictionless horizontal surface, and are connected by identical light horizontal
springs of spring constant
. As before, the outermost masses are attached to immovable walls by springs of
spring constant
. The instantaneous configuration of the system is specified by the horizontal displacements of
the three masses from their equilibrium positions: namely,
,
, and
. This is manifestly a three
degree of freedom system. We, therefore, expect it to possess three independent normal modes of oscillation.
Equations (
149
)(
150
) generalize to
(202)
(203)
(204)
These equations can be rewritten
(205)
(206)
(207)
where
. Let us search for a normal mode solution of the form
(208)
(209)
(210)
205
)(
210
) can be combined to give the
homogeneous matrix equation
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where
. The normal frequencies are determined by setting the determinant of the matrix to zero:
that is,
(212)
or
(213)
Thus, the normal frequencies are
,
, and
. According
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 Fall '08
 Staff
 Friction, Mass, Waves And Optics, Light, Normal mode

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