微尺度悬臂管颤振的研究

RESEARCH ON FLUTTER OF MICRO-SCALE CANTILEVER PIPE

本文基于修正的偶应力理论并考虑Lagrange应变张量所给出的几何非线性,运用Hamilton原理建立了微悬臂管的平面振动微分方程.通过对线性方程的特征值分析,得到了微管的前四阶复频率及临界流速—质量比曲线(临界流速曲线)对材料长度尺寸参数的依赖关系;并且发现宏观管和微尺度管(或者具有不同材料长度尺寸参数的微管)的临界流速曲线可能会相交.运用基于中心流形—范式理论的投影法,计算了临界流速处系统的第一李雅普诺夫系数和退化特征值关于流速的变化率,以此为基础论证了分岔的超临界性质;并对临界流速曲线上的滞后部分及不同尺度管的该曲线的交点处的动力学性质作了探讨,发现了不同的分岔方向.

cantilever Lagrange Based on a modified couple stress theory, the governing equation for the motion of the micro-scale pipe is derived by using Hamilton＇s principle, where the geometric nonlinearity arising from the strain tensors is taken into account. An eigenvalue analysis for linear equation is performed to examine the effect of internal material length scale parameter on the graph of critical flow velocity versus mass ratio and the complex frequencies of the four lowest modes of the micro-pipe. It is found that these curves of critical flow veloci- ty versus mass ratio for the micro-scale pipes with different material length scale parameters may intersect each other. At each critical velocity, the first Lyapunov＇s coefficient and the derivation of the degenerate eigenvalue with respect to flow velocity are calculated by employing the projection methods based on the center manifold the- ory and normal form method, which demonstrates that the bifurcation is supercritical. The dynamics at the hyster- esis and intersection points of the curve of critical flow velocity versus mass ratio are also investigated, and the bifurcation diagrams inwards different directions are then detected.