Title: Oscillations and Resonances
1Oscillations and Resonances
- PHYS 5306
- Instructor Charles Myles
- Lee, EunMo
2Outline of the talk
- The Harmonic Oscillator
- Nonlinear Oscillations
- Nonlinear Resonance
- Parametric Resonance
3The Harmonic Oscillator
(1). Basic equations of motion and solutions
Solution
4(2). Damping
The equation of motions has an additional term
which comes from the damping force
Soultion
The underdamped case
5The critically damped case
The overdamped case
6(3). Resonance
Equation of motion of a damped and driven
harmonic oscillator
Solution
Where
7- 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
8In the underdamped case
.
The width of resonance line is proportional to
In the critically damped and overdamped case the
resonance line disappears
92. Nonlinear Oscillations
the total energy
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11The canonical form of the complete elliptic
integral of the first kind K
123. 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.
13Nonlinear 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.
14(1) The foldover effect
15(2). Superharmonic Resonance
164. Parametric Resonance
Parametric resonance is a resonance phenomenon
different from normal resonance and superharmonic
resonance because it is an instability
phenomenon.
17The onset of first-order parametric resonance can
be approximated analytically very well by the
ansatz
Mathieu equation
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19parametric resonance condition
This instability threshold has a minimum just at
the parametric resonance condition
The minimum reads
202. Parametrically excited oscillations