三种分形和分数阶导数阻尼振动模型的比较研究

标准的整数阶导数方程不能准确描述粘弹性材料的记忆性参考文献1]和阻尼的分数次幂频率依赖2],因此分形导数、分数阶导数及正定分数阶导数被用于描述粘弹性介质中的阻尼振动.该文通过分析模型和数值模拟,比较了三种模型描述的振动过程.结果显示,当p小于约O.75或大于约1.9时(p为非整数阶导数的阶数),分形导数模型衰减最快;当P大于约0.75且小于约1.9时,正定分数阶导数模型衰减最快,衰减最慢的分别为分数阶导数模型(p<1)和分形导数模型(p>1).且正定分数阶导数模型衰减快于分数阶导数模型,当p接近2时,两种模型较为相近.

COMPARISON OF THREE FRACTAL AND FRACTIONAL DERIVATIVE DAMPED OSCILLATION MODELS

The standard integer order differential equation can't be used to describe the memory mechanics behavior and the power-law frequency-denpendent damping of the viscoelastic materials,so the fractal derivative, the fractional derivative and the positive fractional derivative is employed to depict the damped oscillation in the viscoelastic medium. This study compared the oscillation by these three models by analyzing models and numerical simulating motion of the oscillator. Results show that the damped oscillation by the fractal derivative model decays most fast when p (p is the order of the non-integer order derivative) is smaller than about 0.75 or bigger than about 1.9;the positive fractional derivative model decays most fast when p is bigger than about 0.75 and smaller than about 1.9, and the slowest one is the fractional derivative model (p<1) or the fractal derivative model (p>1). Moreover,the damped oscillation modeled by the positive fractional derivative model gains more dissipation than the fractional derivative model, which are very similar with p close to 2.