7' Coupled Oscillators

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7. Coupled Oscillators
L
L
a
b
k
m
m
D
coupled equations of motion
x
0
2
Normal Modes motions where all bodies move at
the same frequency.
Sum

Normal Coordinate 1
Normal Frequency 1
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(No Transcript)
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Difference
Normal Coordinate 2
Normal Frequency 2
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Normal Mode Solutions
Abnormal motion
Initial Conditions
D
Ao
x
0
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similarly
Switch back to xa and xb
xb also beats
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m
m
k
k
k
x
0
x1
x2
Equations of motion
Look for Normal Modes!
Plug in.
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Simplify and group by As
Matrix representation
True if the Determinant is 0
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Normal Mode frequencies
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Each Normal Mode puts a condition on the
amplitude ratio
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Larger N www.falstad.com/coupled
Oscillators which exchange energy are coupled,
and their equations of motion are coupled. Their
complex motion can be described simply in terms
of normal modes in which all bodies oscillate
with the same frequency.