Duffing Oscillator

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Duffing Oscillator. Two Springs. A mass is held between two springs. Spring constant k Natural length l Springs are on a horizontal surface. Frictionless No gravity. l. k. m. k. l. Transverse Displacement. The force for a displacement is due to both springs.

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Duffing Oscillator

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Duffing Oscillator

Two Springs
• A mass is held between two springs.
• Spring constant k
• Natural length l
• Springs are on a horizontal surface.
• Frictionless
• No gravity

l

k

m

k

l

Transverse Displacement
• The force for a displacement is due to both springs.
• Only transverse component
• Looks like its harmonic

s

x

q

s

The force can be expanded as a power series near equilibrium.

Expand in x/l

The lowest order term is non-linear.

F(0) = F’(0) = F’’(0) = 0

F’’’(0) = 3

Quartic potential

Not just a perturbation

Purely Nonlinear
Mixed Potential
• Typical springs are not at natural length.
• Approximation includes a linear term

s

l+d

x

l+d

s

Quartic Potentials
• The sign of the forces influence the shape of the potential.

hard

double well

soft

Assume a more complete, realistic system.

Damping term

Driving force

Rescale the problem:

Set t such that w02 = k/m = 1

Set x such that b = ka/m = 1

This is the Duffing equation

Driven System
• Try a solution, match terms

trig identities

Find the amplitude-frequency relationship.

Reduces to forced harmonic oscillator for A  0

Find the case for minimal damping and driving force.

f, g both near zero

Defines resonance condition

Amplitude Dependence
Resonant Frequency
• The resonant frequency of a linear oscillator is independent of amplitude.
• The resonant frequency of a Duffing oscillator increases with amplitude.

A

Linear oscillator

Duffing oscillator

w

w = 1

Hysteresis
• A Duffing oscillator behaves differently for increasing and decreasing frequencies.
• Increasing frequency has a jump in amplitude at w2
• Decreasing frequency has a jump in amplitude at w1
• This is hysteresis.

A

w1

w2

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