Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end
Non-linear dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end
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DOI：
10.1016/s0020-7462(97)00002-4
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发表时间：
1998-01-01
影响因子：
3.2
通讯作者：
Semler, C
中科院分区：
文献类型：
--
作者：
Paidoussis, MP;Semler, C
本文从理论和实验两方面对自由端附有小质量（简称“端部质量”）的输液悬臂管的平面动力学进行了研究。采用输送水的弹性体管道以及由黄铜、铝或塑料制成的端部质量进行了实验研究。主要目的是扩展科普兰（Copeland）和穆恩（Moon）的研究工作，采用一种改进的结构：运动被限制在平面内而非三维空间，并且将管道建模为具有不可忽略的抗弯刚度的梁，而非在重力作用下悬挂的细绳。与先前的研究一样，证明了对于无端部质量的系统，至少对于所考虑的参数，仅存在一种稳定的周期解。另一方面，在存在小端部质量的情况下，动力学行为要丰富得多，并且发现存在不同类型的周期解。在实验中观察到跳跃现象以及混沌振荡，从而揭示了即使是很小的质量对动力学的重要性。同时，进行了理论/数值研究。对垂直管道平面运动的非线性方程进行了修改，以考虑自由端的小集中质量。所得的离散化方程包含非线性惯性项，并使用两种专门为此类情况开发的方法进行积分：一种基于豪博尔特（Houbolt）格式的有限差分法（FDM），它产生一组非线性代数方程，用牛顿 - 拉夫森（Newton - Raphson）方法求解；以及一种增量谐波平衡法（IHB），它能够构建周期解的分岔图并确定其稳定性。对于恒定（非零）的端部质量和不断增加的流速，结果表明在第一次霍普夫（Hopf）分岔之后，系统经历一系列分岔，再次导致动力学行为的广泛多样性。与实验中一样，检测到两种不同的周期解；还针对不同的端部质量发现了跳跃现象、准周期和混沌振荡，并进行了详细研究。特别关注新解的出现，说明了为什么对该系统进行线性分析不是很有用。尽管理论和实验都有一定的局限性，但从定性和定量的角度来看，两者之间的一致性相当好。这证实了（i）当前模型的有效性，（ii）在分析中考虑系统即使很小的修改的必要性，以及（iii）从动力学角度来看悬臂输液管系统的丰富性。（C）1997年爱思唯尔科学有限公司
In this paper, the planar dynamics of a fluid-conveying cantilevered pipe with a small mass attached at the free end ('end-mass', for short) are examined theoretically and experimentally. An experimental study is undertaken with elastomer pipes conveying water and with end-masses made of brass, aluminum or plastic. The main purpose is to extend the work of Copeland and Moon on a modified configuration the motion is constrained to be planar instead of three-dimensional and the pipe is modelled as a beam having a non-negligible flexural rigidity instead of a string hanging under gravity. As in previous studies, it is demonstrated that for the system with no end-mass, only one stable periodic solution exists, at least for the parameters considered. On the other hand, in the presence of a small end-mass, the dynamics are much richer and different types of periodic solutions are found to exist. Jump phenomena as well as chaotic oscillations are observed in the experiments, revealing therefore the importance of even a small mass on the dynamics. In parallel, a theoretical/numerical investigation is undertaken. The non-linear equations for planar motions of a vertical pipe are modified to take into account the small lumped mass at the free end. The resultant discretized equations contain non-linear inertial terms and are integrated using two methods developed specifically to treat such a case: a Finite Difference Method based on Houbolt's scheme (FDM), which leads to a set of non-linear algebraic equations that is solved with a Newton-Raphson approach; and an Incremental Harmonic Balance method (IHB), which enables the construction of bifurcation diagrams of periodic solutions and the determination of their stability. For a constant (non-zero) end-mass and an increasing flow velocity, it is shown that after the first Hopf bifurcation, the system undergoes a series of bifurcations leading again to a wide diversity of dynamical behaviour. As in the experiments, two different periodic solutions are detected; also jump phenomena, quasiperiodic and chaotic oscillations are found for different end-masses and are investigated in detail. Particular attention is paid to the emergence of new solutions, showing why a linear analysis for this system is not very useful. Even though both theory and experiment have certain limitations, the agreement between the two is rather good, from both qualitative and quantitative points of view. This confirms (i) the validity of the present model, (ii) the necessity of taking account in the analysis of even small modifications to the system, and (iii) the richness of the system of a cantilevered fluid-conveying pipe from a dynamical point of view. (C) 1997 Elsevier Science Ltd.