Duffing Oscillator Norther Illinois UniversityDuffing振子北伊利诺斯大学

1、duffing oscillatortwo springsa mass is held between two springs.spring constant natural length klmkltransverse displacementthe force for a displacement is due to both springs.qsxs22xls22222xlxlxlkf221112lxkxfqsin222lxlkfpurely nonlinearthe force can be expanded as a power series near equilibrium.exp

2、and in /221112lxlxklf3lxklf424xlkvmixed potentialtypical springs are not at natural length.sxsl+dl+d332xldlkxlkdf4324xldlkxlkdvquartic potentialsthe sign of the forces influence the shape of the potential.4242xkxkv4242xkxkv4242xkxkvhardsoftdouble welldriven systemassume a more complete, realistic sy

3、stem.damping termdriving forcetfxkkxxbxmcos23 tfxxxmbxcos2320 tfxxxxcos23 steady state solutiontry a solution, match terms)(cos)()(qtatxtftatataqqqcos)(cos)sin(2)cos()1 (332tfxxxxcos23 trig identities)(3cos)cos()(cos41433qqqttt)sin(sin)cos(coscosqqqqtftftf0)(3cos)sin(sin2)cos(cos)1 (3412432qqqtattfa

4、ttfaa0)(3cos2sin)1 (cos3412432qtaatfaatfamplitude dependencefind the amplitude-frequency relationship.reduces to forced harmonic oscillator for a 04)1(4sin)1 (cos2222432222222222432222aafatfaatf)2()1(22222 af) 1()(100)1(02342432224322aaaaresonant frequencythe resonant frequency of a linear oscillator is independent of amplitude.the resonant frequency of a duffing oscillator increases with amplitude.aduffing oscillatorlinear oscillator) 1(234a 1hysteresisa duffing oscillator behaves differently for increasing and decreasing frequencies.increasing frequency has a jump in amplitude a