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Duffing

Duffing. 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

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Duffing

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  1. Duffing

  2. 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

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

  4. 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

  5. Mixed Potential • Typical springs are not at natural length. • Approximation includes a linear term s l+d x l+d s

  6. Quartic Potentials • The sign of the forces influence the shape of the potential. hard double well soft

  7. 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

  8. Steady State Solution • Try a solution, match terms trig identities

  9. 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

  10. 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

  11. 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 next

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