耳石器官广义分数阶粘弹性动力学模型

摘要

After introducing the generalized strain energy function with fractional order, the constitutive equation of biological visco-elastic material was obtained by means of the fractional calculus. Considering the gel layer and the endolymph of otolith organs as visco-elastic solid and fluid respectively, the Grant′s elastic model was modified. The Bode figures of the otoconial layer frequency response were colser to the recent experimental data than Grant′s Kelvin-Voight viscoelastic model, and revealed the low-frequency phase lead and accompanying amplitude reduction, which Grant′s models could not explained. As a special case, the results given by Grant et al were included in this paper, and in order to solve the obtained governing equations with fractional order, the generalized Mittag-Leffler function was used. The numerical simulations showed that the results given by our paper are consistent with the physiological phenomena.%对生物粘弹性材料，引入广义分数阶应变能函数以后，应用分数阶微积分分别给出了粘弹性固体和粘弹性流体的本构关系式。将耳石器官的胶质层和内淋巴液分别作为粘弹性固体和流体处理，修改了Grant等人所提出的模型，通过频率分析，给出了耳石器官物性参数以及粘弹特性对于系统的不同控制作用。作为特例，本文所得结果包括了Grant等人的结果，反映出Grant粘弹性模型无法描述的低频段相角滞后和减幅现象。对于所得到的分数阶微分方程组，本文应用广义Mittag-Leffler函数给出了较为简单的解析解，避免了应用Fox函数(H函数)所引起的繁琐留数计算。数值模拟表明，本文所给出的结果与生理现象是一致的。