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