微尺度悬臂管颤振的有限维研究

Research on the Flutter of Micro-Scale Cantilever Pipes——A Finite-Dimensional Analysis

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原理建立了微尺度悬臂管平面振动的积分-微分方程.通过Galerkin方法将原积分-微分方程离散成常微分方程组,研究了临界流速-质量比曲线的不同阶Galerkin近似解与精确解的符合程度以及它们对材料长度尺寸参数的依赖性.对不同的模态截断数,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Lyapunov(李雅谱诺夫)系数和临界特征值关于流速的变化率,以此为基础分析了系统的分岔模式,探讨了模态截断数对系统动力学性质的影响.临界流速-质量比曲线的滞后部分及交点处的动力学性质表明,系统存在不同的分岔方向,用6个模态的Galerkin离散化方程作分岔图对此进行了验证,并通过理论分析及数值方法分别计算了颤振的固有频率.

Based on the modified couple stress theory,the integro-differential equations of motion for microscale cantilever pipes were derived by means of Hamilton＇s principle. The geometric nonlinearity,arising from the Lagrangian strain tensor,was taken into account. The integro-differential equations were transformed into ordinary differential equations with the Galerkin method. With different numbers of modes in the Galerkin discretization,the diagrams of critical flow velocity vs. mass ratio were given. The difference between the Galerkin approximation results and the exact solutions to the 2-point boundary problem was investigated and the effect of the internal material length scale parameter on the graphs of critical flow velocity vs. mass ratio was studied. For different numbers of modes,the first Lyapunov＇s coefficient was calculated and the critical eigenvalue with respect to the flow velocity was derived with the projection method based on the center manifold theory and the normal form method,therefrom,the bifurcation model was analyzed and the effect of the number of modes on the dynamical behaviors was examined.The dynamics of hysteresis and intersection points of the curves of critical flow velocity vs. mass ratio was also investigated and then bifurcation diagrams in different directions were found. Finally,the 6-mode ordinary differential equations of the Galerkin discretization were employed to construct the bifurcation diagrams and verify the relevant results obtained,and the natural frequencies of flutter were calculated through the theoretical analysis and with the numerical method,respectively.