DuffingTwo SpringsTransverse DisplacementPurely NonlinearMixed PotentialQuartic PotentialsDriven SystemSteady State SolutionAmplitude DependenceResonant FrequencyHysteresisDuffingDuffingTwo SpringsTwo SpringsA mass is held between two A mass is held between two springs.springs.•Spring constant Spring constant kk•Natural length Natural length llSprings are on a horizontal Springs are on a horizontal surface.surface.•FrictionlessFrictionless•No gravityNo gravityklmk lTransverse DisplacementTransverse DisplacementThe force for a displacement The force for a displacement is due to both springs.is due to both springs.•Only transverse componentOnly transverse component•Looks like its harmonicLooks like its harmonicsxs22xls 22222xlxlxlkF221112lxkxFsin222lxlkF Purely NonlinearPurely NonlinearThe force can be expanded The force can be expanded as a power series near as a power series near equilibrium.equilibrium.•Expand in Expand in xx//llThe lowest order term is non-The lowest order term is non-linear.linear.•FF(0) = (0) = FF’(0) = ’(0) = FF’’(0) = 0’’(0) = 0•FF’’’(0) = 3’’’(0) = 3Quartic potentialQuartic potential•Not just a perturbationNot just a perturbation221112lxlxklF3lxklF424xlkVMixed PotentialMixed PotentialTypical springs are not at Typical springs are not at natural length.natural length.•Approximation includes a Approximation includes a linear termlinear termsxsl+dl+d 332xldlkxlkdF 4324xldlkxlkdVQuartic PotentialsQuartic PotentialsThe sign of the forces influence the shape of the The sign of the forces influence the shape of the potential.potential.4242xkxkV4242xkxkV4242xkxkVhardsoftdouble wellDriven SystemDriven SystemAssume a more complete, Assume a more complete, realistic system.realistic system.•Damping termDamping term•Driving forceDriving forceRescale the problem:Rescale the problem:•Set Set tt such that such that 0022 = k = k//m = m = 11•Set Set xx such that such that  = k = k//m = m = 11This is the Duffing equationThis is the Duffing equationtfxkkxxbxmcos23tfxxxmbxcos2320tfxxxxcos23Steady State SolutionSteady State SolutionTry a solution, match termsTry a solution, match terms)](cos[)()( tAtxtftAtAtAcos)(cos)sin(2)cos()1(332tfxxxxcos23trig identities)(3cos)cos()(cos41433 ttt)sin(sin)cos(coscos tftftf0)(3cos)sin(]sin2[)cos(]cos)1([3412432tAttfAttfAA0)(3cos2sin)1(cos3412432tAAtfAAtfAmplitude DependenceAmplitude DependenceFind the amplitude-Find the amplitude-frequency relationship.frequency relationship.•Reduces to forced harmonic Reduces to forced harmonic oscillator for A oscillator for A  0 0Find the case for minimal Find the case for minimal damping and driving force.damping and driving force.•f, f,  both near zero both near zero•Defines resonance conditionDefines resonance condition]4)1[(4sin)1(cos2222432222222222432222AAfAtfAAtf])2()1[(22222Af)1()(10]0)1[(02342432224322AAAAResonant FrequencyResonant FrequencyThe resonant frequency of a The resonant frequency of a linear oscillator is linear oscillator is independent of amplitude.independent of amplitude.The resonant frequency of a The resonant frequency of a Duffing oscillator increases Duffing oscillator increases with amplitude.with amplitude.ADuffing oscillatorLinear oscillator)1(234AHysteresisHysteresisA Duffing oscillator behaves A Duffing oscillator behaves differently for increasing and differently for increasing and decreasing frequencies.decreasing frequencies.•Increasing frequency has a Increasing frequency has a jump in amplitude at jump in amplitude at 22•Decreasing frequency has a Decreasing frequency has a jump in amplitude at jump in amplitude at 11This is hysteresis.This is