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

s

x

q

s

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

s

l+d

x

l+d

s

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

hard

double well

soft

driven system
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
steady state solution
Steady State Solution
  • Try a solution, match terms

trig identities

amplitude dependence
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
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
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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