具有环状运动约束的悬臂输流管道的非线性振动特征

Nonlinear Vibration Characteristics of Cantilevered Fluid-Conveying Pipe with Circular Motion Constraint

本文对具有环状运动约束的悬臂输流管道的空间弯曲振动进行研究,目的在于考察约束刚度系数、约束放置位置对管道的两类周期运动(包括平面周期运动和空间周期运动)及其稳定性的影响规律.首先,在已有文献的基础上,将运动约束对管道的作用模拟成非线性立方弹簧模,得出振动方程.其次,运用Galerkin方法将振动方程离散成常微分方程组,结合基于中心流形—范式理论的投影法与平均法,给出了决定系统定性动力学性质的相关系数(包括临界特征值随流速的变化率及非线性共振项),取模态截断数为6,在几组约束刚度值和约束位置处计算了上述系数,据此考察了运动约束对管道的周期运动的影响,总结出了如下结论:在约束位置取定时增加约束刚度,或在约束刚度取定时增大约束位置至管的固定端的距离,均会使得管道的稳定平面周期运动对应的质量比区间减小,稳定空间周期运动对应的质量比区间增大;约束位置距管的固定端越远,约束刚度的变化对管道动力学行为的影响越明显.最后,对上述通过投影法和平均法得出的结论,本文在特定的质量比处数值求解了原振动方程的6模态Galerkin离散化方程,绘制了位形图、相图和Poincaré映射图,计算了频率,从而验证了相关的分析.

In this paper,the spatial bending vibration of a cantilevered fluid-conveying pipe with circular motion constraints was studied to explore the influence of constraint stiffness coefficient and constraint placement position on the two kinds of periodic motion of pipeline(including planar periodic motion and spatial periodic motion)and their stabilities.Firstly,the vibration equation was obtained by simulating the action of the motion constraint on the pipeline as a nonlinear cubic spring mode based on the existing literature.Secondly,the vibration equation was discretized into a system of ordinary differential equa-tions by the Galerkin method.The relevant coefficients(including the rate of change of critical eigenval-ue with velocity and the nonlinear resonant term)that determine the qualitative dynamical properties of the system were given in combination with the projection method based on the center manifold-normal form theory and averaging method.By setting the truncated mode numbers to 6,the aforementioned co-efficients were calculated at several sets of constrained stiffness values and constrained positions.And then,the influence of motion constraints on the periodic motion of the pipeline was studied.The following conclusions were drawn:Increasing the constraint stiffness at a fixed constraint position or increasing the distance from the constraint position to the fixed end of the pipeline while keeping the constraint stiffness constant,both will reduce the mass ratio interval corresponding to the stable planar periodic motion of the pipeline,and increase the mass ratio interval corresponding to the stable spatial periodic motion;the farther the constraint position from the fixed end of the pipeline,the more significant the in-fluence of the changes in the constraint stiffness on the dynamic behavior of the pipeline.Finally,the 6-mode Galerkin discretization equation of the original vibration equation was numerically solved at some specific mass ratios to calculating the oscillation frequencies and generating the configuration diagrams,phase diagrams,and Poincarémapping diagrams,which validates the relevant conclusions obtained by projection and averaging methods.