输液管模型及其非线性动力学近期研究进展

RECENT DEVELOPMENT ON MODELS AND NONLINEAR DYNAMICS OF PIPES CONVEYING FLUID

综述了输液管系统的各类物理模型及其相应的数学模型,在流体满足基本假设条件下,对于管道内径远远小于管道长度的直管和曲管,详细叙述了梁模型管动力学数学模型的建模过程以及建模方法,针对在水动压力作用下以及管道短而且薄的情形,综述了壳模型的输液管道的动力学方程。在此基础上,概述了近几年来输液管道的非线性振动、稳定性、分岔与混沌、特别是管道控制的研究现状,并对今后的发展趋势作了分析和预测。综观非线性动力学理论的发展历程可以发现选取研究对象和典型的数学模型是至关重要的。对于低维的非线性系统,常常选用Van der Pol、Duffing、Mathieu、Lorenz等典型系统来进行研究工作的。通过本文可以看出,对于研究高维非线性系统动力学,流诱发输液管的动力学问题是非常典型的模型之一,它有着容易理解的工程背景、包含了梁和壳的振动问题,并且它的数学模型相对简单,然而却能包含非常复杂的非线性动力学现象,同时容易解释数学方法得到的结果易对应到工程中的实际现象。本文希望通过对输液管动力学模型及其非线性动力学和控制研究现状的综述,建立高维非线性动力学的分析模型,以便发展高维非线性动力学的分岔与混沌理论,同时建立相应的控制理论基础。

In this paper, various physical and mathematical models of pipes conveying fluid are summarized. Under some assumptions for the fluid in the pipe, the mathematical models of the beam-pipe are proposed to govern the dynamics of straight or curve pipes with internal radii less than their length. The proposed models are discussed in details. The governing equations of shell-pipe are given for short and thin pipes subjected to dynamic pressure in the pipes. Then, the recent research development is reviewed on nonlinear vibration, stability bifurcation and control for pipes conveying fluid. The future trends and advances are proposed. From the history of nonlinear dynamics, one may see that it is very important to choose some typical systems as models, such as van der Pol, Duffing, Mathieu, Lorenz and so on, for studying low dimensional systems. The motivation of this paper is to represent fluid-reduced vibrations as one of typical nonlinear problems. It may be considered as a model to investigate nonlinear dynamics in high dimensional systems since its engineering background is intuitive and easy to be understood, vibrations of shell and beam are included, its mathematical model is simple but may present rather rich dynamics, and the results of the mathematical analysis is easy to be explained and applied. It is expected to establish models of analysis for high-dimensional systems and to develop nonlinear dynamics in terms of discussions in this paper. In addition, it may also provide some guidance for establishing the control theory.