微尺度悬臂管的空间弯曲振动-非线性运动方程及尺度效应

Three-dimensional flexural vibration of a micro-scale cantilever pipe-nonlinear equations of motion and scale effect

具有圆环形横截面的微尺度悬臂输液管可以同等地向空间各方向产生弯曲振动。按照欧拉-伯努利梁理论,在分析管道上点的位移及相关几何关系的基础上,考虑Lagrange应变张量所给出的几何非线性,基于修正的偶应力理论计算了管的应变能,运用Hamilton原理建立了微尺度悬臂输液管的空间弯曲振动的非线性动力学方程。研究了无量纲材料长度尺寸参数对系统动力学性质的影响,结果表明,尺度效应增大管道的临界流速,并使得稳定的平面周期运动(空间周期运动)在整个质量比区间上占的比例越大(小)。

Flexural vibration of a micro-scale cantilever fluid-conveying pipe with annulus cross section can occur in each direction in the three-dimensional space. According to the Euler-Bernoulli beam theory,the displace components of the pipe and the relevant geometrical relations could be analyzed. The geometric nonlinearity,arising from the Lagrange strain tensor,was taken into account. Based on a modified couple stress theory,the strain energy in the pipe was calculated. The nonlinear dynamical equations of three-dimensional flexural vibration for a micro-scale cantilever fluidconveying pipe were derived by using the Hamilton principle. The effect of the dimensionless material length scale parameter on the dynamics of the system was investigated. It is found that the scale effect increases the critical flow velocity of the pipe and that the larger the dimensionless material length scale parameter is,the wider（ narrower） the region of stable planar（ spatial） periodic motion is.