分数导数型本构关系描述粘弹性梁的振动分析

本文研究粘弹性梁在周期激励作用下的受迫振动问题。梁的材料满足Kelvin-Volgt分数导数型本构关系。基于动力学方程、本构关系和应变－位移关系建立了小变形粘弹性梁的振动方程。采用分离变量法分析粘弹性梁的自由振动，导出模态坐标满足的常微分－积分方程和模态函数满足的常微分方程，对于两端简支的截面梁给出了固有频率和模态函数。对于简谐激励作用下粘弹性梁的受迫振动，利用模态叠加得到了稳态响应。最后给出数值算例说明本文方法的应用。

Vibration Analysis of Viscoelastic Beams with Fractional Derivative Constitute Relation

The forced vibrations of viscoelastic beams under periodic excitations were investigated in the paper. The viscoelastic material of the beams is assumed to obey the fractional derivative constitutive relation. Based on the dynamical equation, the constitutive relation and the strain-displacement relation, the vibration equation of small deflection beams was derived. Free vibrations of viscoelastic beams were treated by the separation of variables. The integro-ordinary-differential equation defining modal coordinates and the ordinary differential equation defining modal functions were obtained. In the case of two simply-supported ends, the natural frequencies and the modal functions were presented. Steady-state responses of simply harmonically forced viscoelastic beams were given by mode superposition. The application of the method in the paper were demonstrated by a numerical example.