Lecture 16: Oscillator chains (9 Oct 09)
Hour exam: take home, hand out 16 Oct, due 19 Oct. at 5 PM.
A. Review: coupled oscillator problem
1. Coupled pendulums: the eigenvalue problem gives the normal mode
frequencies. Use the eigenvectors to expand the general solution and
satisfy initial conditions using amplitudes and phases of the normal
modes.
2. Planar equilateral triangle normal modes: made the kinetic energy di
agonal using Jacobi coordinates to separate cm and internal degrees of
freedom
r
,
s
. Can then get to “standard” form by scaling one of the
internal vectors (
s
) so that the nominal mass parameter is the same for
both.
B. linear chain
1. FW Sec.24
2. Longitudinal oscillations of 1D oscillator chain (without specifying the
ends..)
L
=
T

V
=
m
2
X
j
˙
η
2
j

K
2
X
j
(
η
j
+1

η
j
)
2
m
¨
η
j
=

K
[2
η
j

η
j
+1

η
j

1
]
3. The coupled equations are the same for mass points connected by a
string of tension
τ
and small amplitude
y
j
of transverse vibration.
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 Fall '09
 BRUCH
 Boundary value problem, 2k, Normal mode, 2 j

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