分数阶Ver Del Pol-Duffing系统的非线性动力学行为分析
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摘要
利用分数导数本构模型模拟系统的阻尼特性,构造了分数阶Ver Del Pol-Duffing系统,探讨了系统的动力特性随特征参数的变化规律。分析发现:该非线性振子具有与经典Ver Del Pol系统相似的自激振动特性,但其非线性强弱受分数导数阶值以及阻尼系数和非线性大位移系数的影响;在简谐荷载作用下,随着外荷载幅值的增大或阻尼系数的减小,系统由拟周期振动变为周期三振动最后发展为单周期振动;在地震荷载作用下,分数导数阶值的变化能改变系统的输出能量。
A fractional-order Ver Del Pol-Duffing system,a fractional derivative model was introduced to simulate the system's damping characteristics,was proposed.The numerical computation scheme of fractional derivative was deduced.The dynamic characteristics influenced by the feature parameters under different external loads were investigated.The results showed that this nonlinear system has the same self-excited vibration characteristics as those of the classic Ver Del Pol system,the q-order derivative has a large impact upon the system dynamic behavior;on the other hand,as expected,the higher the values of damping coefficient and the nonlinear displacement coefficient,the more nonlinear the system becomes;under harmonic load,with increase in amplitude or decrease in damping coefficient,the system motion changes from quasi periodic motion to period-three vibration until to single period vibration;under seismic load,the q-order derivative has a large impact upon the output energy.
A fractional-order Ver Del Pol-Duffing system,a fractional derivative model was introduced to simulate the system's damping characteristics,was proposed.The numerical computation scheme of fractional derivative was deduced.The dynamic characteristics influenced by the feature parameters under different external loads were investigated.The results showed that this nonlinear system has the same self-excited vibration characteristics as those of the classic Ver Del Pol system,the q-order derivative has a large impact upon the system dynamic behavior;on the other hand,as expected,the higher the values of damping coefficient and the nonlinear displacement coefficient,the more nonlinear the system becomes;under harmonic load,with increase in amplitude or decrease in damping coefficient,the system motion changes from quasi periodic motion to period-three vibration until to single period vibration;under seismic load,the q-order derivative has a large impact upon the output energy.
引文
[1]Ge Z M,Ou C Y,Chaos in a fractional order modified duffingsystem[J].Chaos Solitons&Fractals,2007,34(2):262-291.
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[3]Gao X,Yu J B.Chaos in the fractional order periodicallyforced complex Duffing s oscillators[J].Chaos Solitons&Fractals,2005.24(4):1097-1104.
[4]Cao J,Xie H,Jiang Z.Nonlinear dynamics of duffing systemwith fractional order damping[J].Hsi-An Chiao Tung TaHsueh/Journal of Xi an Jiaotong University,2009,43(3):50-54.
[5]Barbosa R S,Machado J A T,Ferreira I M.Dynamics of thefractional-order van der pol Oscillator[C].IEEE Transactionson Industrial Electronics:373-378.
[6]张卫,徐华,清水信行.分数算子描述的粘弹性体力学问题数值方法[J].力学学报,2004,36(5):617-622.
[2]李根国,朱正佑,程昌钧.非线性粘弹性Timoshenko梁动力学行为的分析[J].力学季刊,2001.22(3):346-352.
[3]Gao X,Yu J B.Chaos in the fractional order periodicallyforced complex Duffing s oscillators[J].Chaos Solitons&Fractals,2005.24(4):1097-1104.
[4]Cao J,Xie H,Jiang Z.Nonlinear dynamics of duffing systemwith fractional order damping[J].Hsi-An Chiao Tung TaHsueh/Journal of Xi an Jiaotong University,2009,43(3):50-54.
[5]Barbosa R S,Machado J A T,Ferreira I M.Dynamics of thefractional-order van der pol Oscillator[C].IEEE Transactionson Industrial Electronics:373-378.
[6]张卫,徐华,清水信行.分数算子描述的粘弹性体力学问题数值方法[J].力学学报,2004,36(5):617-622.
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