Oscillations and Resonances
PHYS 5306 Instructor : Charles Myles
Lee, EunMo
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Outline 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 02
02
2
dtd
tiwtiw eCCet 00 *)( Solution
tBtAt ww 00cossin)(
4
2
02
Soultion
(2). Damping
The equation of motions has an additional term which comes from the damping force:
02
02
2
dtd
dtd
The underdamped case:
2
0
2
42
)sin()2
()( max ttt
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The critically damped case:
The overdamped case:
2
02
)2
exp()()( tt BAt
2
02
)exp()exp()( tAtAt
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(3). Resonance
Solution
Equation of motion of a damped and driven harmonic oscillator
ftadtd
dtd 2cos2
02
2
)2sin()2sin()( ftBftAt
))2()2((2
22220
ff
afA
Where
))2()2((
)2(2222
0
2220
ffafB
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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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))2(2(tan 22
0
1
tf
))2()2( 22220
max
ft
a
.
The width of resonance line is proportional to
In the critically damped and overdamped case the resonance line disappears
In the underdamped case
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the total energy
0sin2
02
2
dtd
cos21 2
02
2
dtdE
max2
0cosE
)cos(2 2
0 E
dtd
2. Nonlinear Oscillations
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dtE
d
)cos(2 2
0
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The canonical form of the complete elliptic integral of the first kind K
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3. Nonlinear Resonance
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.
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.
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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 effect12 sec4.0,1,sec/81.9 mlmg
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(2). Superharmonic Resonance12 sec1.0,1,sec/81.9 mlmg
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4. Parametric ResonanceParametric 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
Mathieu equation
The onset of first-order parametric resonance can be approximated analytically very well by the ansatz:
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parametric resonance condition
This instability threshold has a minimum just at the parametric resonance condition
0f
The minimum reads
fac 2
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2. Parametrically excited oscillations
mAmlmg 07.0,sec1.0,1,sec/81.9 12