Spring 2016
Ch4. Coupled Oscillations
1.
2.
3.
4.
Two coupled pendulums
Normal Modes and Normal Coordinates
Linear Algebra and Cramer’s Rule
More Examples: Stiffness Coupling, Mass
Coupling
5. Complexity: Many Coupled Oscillators,
Asymmetry
6. Waves Motion as Coupled Oscillations
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Coupled Pendulums Without Damping
Two pendulums oscillate
with same frequency?
Complete energy transfer?
k
x0
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x1
Coupled Pendulums Without Damping
Two pendulums oscillate
with same frequency?
Complete energy transfer?
k
x0
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x1
Coupled Pendulums Without Damping
FIXED END
x
t
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The In-phase Vibration Mode
q2  0(x0  x1 )
q1   q  0
2
1 1
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The Out-of-phase Vibration Mode
q1  0 (x0   x1 )
 2 2k 
q2   1   q2  0
m

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An Initial Condition Problem
1
1
q

q

a
cos

t

cos

t
x

 1 2 
 q1  q2   a  cos1t  cos2t 
1
2 
0
2
2
2  1  t
1  2  t
2  1  t
1  2  t




 2acos
cos
 2asin
sin
2
2
2
2
x1 
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Coupled Pendulums Without Damping
Equation of motion
x1
mx1  mg  k ( x1  x0 )
l
x0
mx0  mg  k ( x0  x1 )
l
k
x0
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x1
SHM
term
Coupling
term
Normal Co-ordinates
x1  x0  q1
x1  x0  q2
Normal modes
q1   q  0
2
0 1
 2 2k 
q2    0  q2  0
m

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q 1  x1  x0  q10 cos 1t  1 
q 2  x1  x0  q20 cos 2t  2 
Amplitudes : q10 and
q20
1  0
1/ 2
 2 2k 
2   1  
m

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The General Method for Coupled Oscillations
x1
mx1  mg  k ( x1  x0 )
l
x0
mx0  mg  k ( x0  x1 )
l
Normal modes
x0 (t )  Ce i (t   )  Dei (t   )
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x1 (t )  Ce
i (  t   )
 De
i (  t   )
Linear Algebra and Cramer’s Rule
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Normal Co-ordinates
x1  x0  q1
x1  x0  q2
Normal modes
q1   q  0
2
0 1
 2 2k 
q2    0  q2  0
m

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Energy of Coupled Oscillations
Importance of Normal modes and their coordinates
If we modify our normal coordinates to read
m
q1 
( x0  x1 )
2
m
q2 
( x0  x1 )
2
The total energy of the system
1 2 2
1 2 2
2
E  [1 q1  q1 ]  [2 q2  q 22 ]
2
2
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They are entirely independent of each other. The
energy associated with a normal mode is never
exchanged with another mode.
Degrees of Freedom and Normal Modes
1. Normal coordinates are coordinates in which the equations of motion
take the form of a set of linear differential equations with only one
dependent variable in each of them.
2. A vibration involving only one dependent variable X (or Y) is called a
normal mode of vibration and has its own normal frequency. In such a
normal mode all components of the system oscillate with the same
normal frequency.
3. The importance of the normal modes of vibration is that they are
entirely independent of each other. The energy associated with a normal
mode is never exchanged with another mode.
4. Each independent way by which a system may acquire energy is
called a degree of freedom to which is assigned its own particular
normal coordinate. This determines the number of the normal modes.
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