旋转分数导数粘弹性矩形板横向自由振动分析

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摘要研究了用分数导数型本构关系描述的旋转粘弹性矩形板的横向自由振动问题。从分数导数Kelvin-Voigt三维本构方程出发，基于板的平面问题，得到了分数导数Kelvin-Voigt二维本构关系，运用Hamilton原理建立了旋转分数导数粘弹性矩形板的运动微分方程。采用微分求积法离散运动微分方程与边界条件，得到系统的复特征方程，分析了分数导数阶数、宽长比、径长比以及厚长比对系统无量纲复频率虚部的影响。结果表明，随着旋转角速度的增大，前三阶无量纲复频率虚部（固有频率）增大；随着分数导数阶数的增大，无量纲复频率虚部减小; 第三阶复频率虚部受到各参数的影响比第一、二阶较大。

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	孙羽键1
	王忠民1
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关键词 ：旋转矩形板, 分数导数型粘弹体, 横向振动, 微分求积法

Abstract：The transverse free vibration of a rotating viscoelastic rectangular plate described by fractional derivative constitutive relation is studied. Based on the plane problem of the plate, the Kelvin-Voigt two-dimensional constitutive relation with fractional derivative is obtained from the Kelvin-Voigt three-dimensional constitutive equation with fractional derivative. The differential equation of motion for rotating rectangular plate made of viscoelastic material with fractional derivative is established with Hamilton principle. Differential quadrature method is used to discretize the differential equations of motion and boundary conditions, and the complex eigen-equation of the system is obtained. The effects of fractional derivative order, width to length ratio, radius to length ratio and thickness to length ratio on the imaginary part of dimensionless complex frequency of the system are analyzed. The results show that with the increase of the rotational angular speed, the imaginary part (natural frequency) of the first three order dimensionless complex frequencies increases; with the increase of the fractional derivative order, the imaginary part of dimensionless complex frequency decreases; and the effect of each parameter on the third order imaginary part of the complex frequency is greater than the first and second order.

Key words： rotating rectangular plate viscoelasticity with fractional derivative transverse vibration differential quadrature method

收稿日期: 2022-03-15 出版日期: 2023-04-28

引用本文:

孙羽键1，王忠民1,2. 旋转分数导数粘弹性矩形板横向自由振动分析[J]. 振动与冲击, 2023, 42(8): 126-133.
SUN Yujian1，WANG Zhongmin1,2. Transverse free vibration analysis of a rotating rectangular plate made of viscoelastic material with fractional derivative. JOURNAL OF VIBRATION AND SHOCK, 2023, 42(8): 126-133.

链接本文:

http://jvs.sjtu.edu.cn/CN/ 或 http://jvs.sjtu.edu.cn/CN/Y2023/V42/I8/126

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