Chapter 12 Coupled Oscillations

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Chapter 12Coupled Oscillations
Claude A Pruneau Wayne State University
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• 12.1 Introduction
• Coupled equations
• Normal coordinates
• Normal modes
• n degrees of freedom (n-coupled 1-d oscillators
  or n/3-coupled 3-d oscillators) leads to n normal
  modes (in general)
• some of the modes may be identical.

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12.2 Two coupled harmonic oscillators. Example
In a solid, atoms interact by elastic forces and
oscillate about their equilibrium
positions. Lets consider the following simpler
system
m1M
m2M
?1?
?2?
?12
x2
x1
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Consider a solution of the form
Frequency, ?, to be determined, and amplitudes
may be complex.
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A solution to these Eqs exist if the 2x2
determinant is null.
This yields
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There are two characteristic frequencies
The general solution is thus
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The amplitudes are not all independent given that
they must satisfy.
The solutions may thus be written
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There are four arbitrary constants - as expected
given one has two equations of second order.
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add subtract
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Weak Coupling
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X1(t)
t
X2(t)
t
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In general, we then find
where
By construction
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In summary
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finite
mij and Ajk express the coupling between the
various coordinates.
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Euler-Lagrange Eq.
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We get
A non trivial solution to this equation exists
only if
A secular Eq. of degree n in ?2. Implies n roots
for ?2
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