Oscillations and Resonances
PHYS 5306
Instructor : Charles Myles
Lee, EunMo
Outline
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of the talk
The Harmonic Oscillator
Nonlinear Oscillations
Nonlinear Resonance
Parametric Resonance
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The Harmonic Oscillator
(1). Basic equations of motion and solutions
d 2
2
 0  0
2
dt
Solution
 (t )  Ceiw t  C *eiw t
0
0
 (t )  A sin w0 t  B cos w0 t
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(2). Damping
The equation of motions has an additional term which
comes from the damping force:
d 2
d
2

 0  0
2
dt
dt
Soultion

2
2
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   
The underdamped case:
0
2

2
 (t )   max (
t
2


2
0
) sin(  t  )
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2
The critically damped case:  
0
2
 (t )  ( At  B) exp( 

The overdamped case:
2
t
2

)

2
0
 (t )  A exp( t )  A exp( t )
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(3). Resonance
Equation of motion of a damped and driven harmonic oscillator
d 2
d
2

  0   a cos 2ft
2
dt
dt
Solution
 (t )  A sin( 2ft)  B sin( 2ft)
Where A 

2fa
  (2f ) )

  (2f )  a
B
(  (2f )   (2f ) )
(   (2f )
2
0
2 2
2 2
2
0
2
0
2
2 2
2
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The amplitude of oscillations depend on the driving frequency.
It has its maximum when the driving frequency matches the eigenfrequency.
This phenomenon is called resonance
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In the underdamped case
 max 

a
2
0
 (2t )
  (2f ) )
2 2
2
2f
  tan ( 2
)
2
.
0  (2t )
1


The width of resonance line is proportional to 
In the critically damped and overdamped case the
resonance line disappears
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2. Nonlinear Oscillations
the total energy
d 2
2

sin   0

2
0
dt
1 d 2
2
E
  0 cos 
2
2 dt
E   0 cos  max
2
d
2
  2( E   0 cos  )
dt
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d
2( E   0 cos  )
2
  dt
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The canonical form of the complete elliptic integral
of the first kind K
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3. Nonlinear Resonance
Nonlinear resonance seems not to be so much different
from the (linear) resonance of a harmonic oscillator. But
both, the dependency of the eigenfrequency of a
nonlinear oscillator on the amplitude and the
nonharmoniticity of the oscillation lead to a behavior
that is impossible in harmonic oscillators, namely, the
foldover effect and superharmonic resonance,
respectively. Both effects are especially important in the
case of weak damping.
The foldover effect got its name from the bending of the
resonance peak in a amplitude versus frequency plot. This
bending is due to the frequency-amplitude relation which is
typical for nonlinear oscillators.
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Nonlinear oscillators do not oscillate sinusoidal. Their oscillation
is a sum of harmonic (i.e., sinusoidal) oscillations with
frequencies which are integer multiples of the fundamental
frequency (i.e., the inverse of the period of the nonlinear
oscillation). This is the well-known theorem of Jean Baptiste
Joseph Fourier (1768-1830) which says that periodic functions
can be written as (infinite) sums (so-called Fourier series) of sine
and cosine functions.
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(1) The foldover effect
g  9.81m / sec 2 , l  1m,   0.4 sec 1
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(2). Superharmonic Resonance
g  9.81m / sec 2 , l  1m,   0.1sec 1
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4. Parametric Resonance
Parametric resonance is a resonance phenomenon different from
normal resonance and superharmonic resonance because it is an
instability phenomenon.
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•1. The instability
The onset of first-order parametric resonance can be
approximated analytically very well by the ansatz:
Mathieu equation
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parametric resonance condition
This instability threshold has a minimum just at the parametric
resonance condition f  0

The minimum reads ac  2f
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2. Parametrically excited oscillations
g  9.81m / sec 2 , l  1m,   0.1sec 1 , A  0.07m
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