具有非对称横截面的悬臂输流管道的非线性振动特征

郭勇; Guo Yong

引 en

具有非对称横截面的悬臂输流管道的非线性振动特征^*

郭勇
机构：
安顺学院电子与信息工程学院, 安顺 561000
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安顺学院电子与信息工程学院, 安顺 561000；

Nonlinear Vibration Characteristics of Cantilevered Fluid-Conveying Pipe with Asymmetric Cross-Section^*

Guo Yong
Affiliation：
School of electronic and Information Engineering, Anshun University, Anshun 561000 , China
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School of electronic and Information Engineering, Anshun University, Anshun 561000 , China；

通讯作者:

郭勇,E-mail:gy-gates@163.com

中图分类号:O322；O326

文献标识码:A

文章编号:1672-6553-2023-21(11)-081-014

DOI:10.6052/1672-6553-2023-087

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 力学模型与运动方程
2 Galerkin方法与离散化方程
3 数值仿真
4 结论
参考文献

摘要

在形心主惯性轴坐标系下推导了横截面具有“大”对称破缺的悬臂输流管道的非线性空间弯曲振动方程.运用Galerkin方法将振动方程离散成常微分方程组.基于振动方程的6模态Galerkin离散化方程,通过数值计算考察了输流管道在两个主方向上的失稳情况.结果表明：当两个主方向的无量纲弯曲刚度的差较大时,管道在两个主方向不会同时失稳；当两个无量纲弯曲刚度的差不超过一定值时,管道在特定的质量比处可同时在两个主方向上发生失稳.若“同时失稳”发生在两条临界流速曲线的非滞后段,则流速增加时管道颤振：或做稳定的环面运动,或在较大刚度方向做稳定的周期运动,或在较小刚度方向做稳定的周期运动.若“同时失稳”发生在两条临界流速曲线的滞后段与非滞后段的交点处,则不管流速增加还是减少,管道均会颤振：流速增加时若在较大(较小)刚度方向做周期运动,则流速减少时将在较小(较大)刚度方向做周期运动.

Abstract

The nonlinear spatial bending vibration equations of cantilevered fluid-conveying pipe with “large” symmetry breaking on the cross section were derived under the coordinate system of centroid principal axes of inertia. The Galerkin method was applied to discretize the vibration equations into a system of ordinary differential equations. The loss of stabilities of the fluid-conveying pipe in two principal transverse directions were investigated through numerical calculations based on the 6-mode Galerkin discretization equations of the original vibrations. The results show that, when the difference between the dimensionless bending stiffnesses in the two principal transverse directions is relatively significant, the pipe would not lose its stabilities in both principal directions simultaneously; when the difference of the two dimensionless bending stiffnesses does not exceed a certain value, the pipe may lose its stabilities in two principal directions simultaneously for specific mass ratio. For “simultaneous loss of instabilities” occurring in the non-hysteresis part of the two critical flow velocity curves, the pipe would flutter as the flow velocity increases, performing stable torus motion, stable periodic motion in the direction of greater stiffness, or stable periodic motion in the direction of smaller stiffness. If the “simultaneous loss of stabilities” occurs at the intersection of the hysteresis and non-hysteresis parts of the two critical flow velocity curves, the pipe would flutter no matter what happened to the flow velocity (increase or decrease), and then, if the pipe performs periodic motion in the direction of greater (smaller) stiffness when the flow velocity increases, the periodic motion would occur in the direction of smaller (greater) stiffnesses as the flow velocity decreases.

关键词

输流管道；周期运动；临界流速；对称；滞后

Keywords

fluid-conveying pipe ； periodic motion ； critical flow velocity ； symmetry ； hysteresis

引言
输流管道是一种重要的工程结构，对其动力学行为的研究最早可追溯到上世纪50年代^[1].Benjamin^[2]给出了适用于悬臂输流管道的Hamilton原理.Gregory和Païdoussis^[3]用力平衡法推导了悬臂输流管道的线性振动方程，并分别运用模态分析法和Galerkin方法研究了系统的特征值随流速的变化以及管道的临界流速、临界频率.Chen^[4]研究了输流管道的线性强迫振动.Ginsberg^[5]分析了两端简支管的线性参数振动，计算了不稳定边界.Holmes最早研究了输流管道的非线性振动^[6-8].Rousselet和Herrmann ^[9]用Krylov-Bogoliubov法研究了悬臂输流管道与管内液体的耦合非线性自由振动，分析了管道的周期运动振幅随质量比的变化规律.此后，两端简支管的非线性自由振动^[10]和参数振动^[11-16]，悬臂管的非线性参数振动^[17-19]，受到约束的悬臂管^[20-25]和简支管^[26]的非线性动力学行为与控制，复杂边界条件管^[27]、复杂构形管^[28]、以及微纳尺度管^[29-33]的非线性振动特征等相继得到研究.以上文献均只考虑了管道的平面振动.
Lundgren等^[34]最早考虑了输流管道的三维动力学，用力平衡法建立了悬臂管的空间弯曲振动方程.Bajaj等^[35]研究了悬臂输流管的两种运动——平面周期运动和空间周期运动的分布，Bajaj和Sethna ^[36]进一步考虑了“小”对称破缺对这两种运动的影响.Wadham-Gagnon等^[37]运用Hamilton原理推导了包含诸多因素的悬臂输流管的空间振动方程.基于文献^[37]中的方程，Modarres-Sadeghi等^[38]考察了流速增加时管道的空间运动和平面运动的切换；Païdoussis等^[39，40]分析中间支撑弹簧对管道的动力学行为的影响；Modarres-Sadeghi等^[41，42]发现，对自由端具有集中质量的管道，在做计算时需取较多的模态截断数才能产生收敛的结果；Ghayesh等^[43]综合研究了中间支撑弹簧、自由端集中质量以及外激励对悬臂管的周期运动、混沌运动及相应振幅的影响；Chang等^[44]通过调整外周期激励的振幅和频率，使得悬臂输流管道的平面运动和空间运动按预定的方式发生切换.Guo等^[45]的研究表明，小尺度效应对悬臂输流管的平面周期运动和空间周期运动的分布有重要影响.两端支撑管的空间振动问题也得到了相关学者的关注^[46-49].在上述关于管的空间运动的文献中，管的横截面均具有O（2）对称性.
因加工误差的存在或某些特殊的使用目的，管的横截面将不再具有O（2）对称性.对这类输流管的研究，文献相对少.Javadi等^[50]研究了横截面上具有“小缺口”的输流管道的振动，发现该“小缺口”可以通过测量管道的超谐响应振幅来进行检测.Heshmati ^[51]研究了输流管道在横截面的外圆周和内圆周存在偏心时的振动特征，重点考察了“偏心程度”对管的固有频率的影响.早年Bajaj和Sethna ^[36]研究了“小”对称破缺对宏观悬臂管的平面周期运动和空间周期运动的影响.最近Guo^[52]推进了Bajaj和Sethna ^[36]的研究，在将力学模型扩大到微尺度悬臂管的同时，在更全面的不稳定域上考察了“小”对称破缺的影响.值得一提的是，文献^[50，51]仅考虑了对称破缺管的平面运动，而文献^[36，52]虽然研究了对称破缺管的空间运动，但仅是在“小”对称破缺的前提下做的考察.
综上可知，对于横截面具有“大”对称破缺的输流管道，目前尚没有相关文献对其动力学行为做过讨论.鉴于此，本文对这一问题做初步的探讨：首先推导了横截面不具有对称性的悬臂输流管的空间弯曲振动方程；其次以悬臂梁的模态函数为基函数，运用Galerkin方法对振动方程进行离散，得到一组常微分方程；最后基于模态截断数为6的Galerkin离散方程，考察了管道在两个主横向上失稳及失稳后的振动形态.
1 力学模型与运动方程
本文研究图1所示的系统.如图1（a）所示，直管的长度为L，单位长度的质量为m，密度为ρ，管内的不可压缩流体的速度大小为V，单位长度的质量为M；如图1（d）~（f）所示，管具有一般形状横截面，其形状和面积A_p沿轴向不变；管做空间弯曲振动[图1（b）]；为便于描述管的运动，拉格朗日坐标系（X，Y，Z）和欧拉坐标系（x，y，z）[图1（c）]取为重合^[37].值得一提的是，图1（a）~（c）和文献^[52]中的Fig.1（a）~（c）相同，即悬臂输流管、空间弯曲振动、坐标系；但与文献^[52]中的Fig.1（d）所呈现的“小”对称破缺不相同的是，本文所考虑的输流管其横截面具有“大”对称破缺[图1（d）~（f）].
图1（a）悬臂输流管道;（b）三维弯曲振动;（c）坐标系;（d）~（f）横截面的O（2）对称性具有大的破缺
Fig.1 (a) Schematic of a cantilevered pipe conveying fluid; (b) Three-dimensional flexural vibration; (c) Coordinate systems; (d) ~ (f) Cross-section of the pipe with large breaking of O (2) symmetry
文献^[52]已初步给出了横截面不具有对称性的悬臂输流管道的空间弯曲振动方程[式（1）]和边界条件[式（2）]，但其仅仅在“小”对称破缺情形下对方程做了分析，此时得到的最终方程其非线性项相对简单.本文将在“大”对称破缺情形下对方程（1）进行处理，得到非线性项相对复杂的最终方程，其适用于具有一般形状横截面的管道的动力学研究.
$\begin{matrix} (m + M) \ddot{w} + E I_{Y} w^{(4)} + E I_{Y Z} v^{(4)} + 2 M V {\dot{w}}^{'} + M V^{2} w^{''} + \\ 2 M V [w^{' 2} {\dot{w}}^{'} + w^{'} v^{'} {\dot{w}}^{'} - w^{''} \int_{s}^{L} (w^{'} {\dot{w}}^{'} + v^{'} {\dot{v}}^{'}) d s] + \\ M V^{2} [w^{' 2} w^{''} + w^{'} v^{'} v^{''} - w^{''} \int_{s}^{L} (w^{'} w^{''} + v^{'} v^{''}) d s] + \\ E (I_{Y} w^{' 2} w^{(4)} + 4 I_{Y} w^{'} w^{''} w^{'''} + I_{Y} w^{'' 3} + \\ \frac{2 I_{Y} + I_{Z}}{3} w^{'} v^{'} v^{(4)} + [\frac{4 I_{Y}}{3} + \frac{1}{4} (I_{Y} + \frac{I_{Z}}{3}) + \end{matrix}$

\begin{matrix} (I_{Y} + \frac{I_{Z}}{3})] w^{'} v^{''} v^{'''} + \frac{3}{4} (I_{Y} + \frac{I_{Z}}{3}) w^{''} v^{'} v^{'''} + I_{y} w^{''} v^{'' 2}) + \\ w^{'} (m + M) \int_{0}^{s} ({\dot{w}}^{' 2} + w^{'} {\ddot{w}}^{'} + {\dot{v}}^{' 2} + v^{'} {\ddot{v}}^{'}) d s - \\ w^{''} (m + M) \int_{s}^{L} \int_{0}^{s} ({\dot{w}}^{' 2} + w^{'} {\ddot{w}}^{'} + {\dot{v}}^{' 2} + v^{'} {\ddot{v}}^{'}) d s + \\ \frac{1}{2} A_{p} l^{2} G (2 w^{(4)} + 4 w^{'} w^{''} w^{'''} + w^{' / 3} + w^{' 2} w^{(4)}) + \\ \frac{1}{2} A_{p} l^{2} G (w^{''} v^{'' 2} + w^{''} v^{'} v^{'''} + 3 w^{'} v^{''} v^{'''} + w^{'} v^{'} v^{(4)}) = 0 \end{matrix}

(1a)

$\begin{matrix} (m + M) \ddot{v} + E I_{Z} v^{(4)} + E I_{Y Z} w^{(4)} + 2 M V {\ddot{v}}^{'} + M V^{2} v^{''} + \\ 2 M V [v^{' 2} {\dot{v}}^{'} + w^{'} v^{'} {\dot{w}}^{'} - v^{''} \int_{s}^{L} (w^{'} {\dot{w}}^{'} + v^{'} {\dot{v}}^{'}) d s] + \\ M V^{2} [v^{' 2} v^{''} + w^{'} v^{'} w^{''} - v^{''} \int_{s}^{L} (w^{'} w^{''} + v^{'} v^{''}) d s] + \\ E \{I_{Z} v^{' 2} v^{(4)} + 4 I_{Z} v^{'} v^{''} v^{'''} + I_{Z} v^{' / 3} + \\ (\frac{2 I_{Z} + I_{Y}}{3}) w^{'} v^{'} w^{(4)} + \frac{3}{4} (I_{Z} + \frac{I_{Y}}{3}) {v'}^{''} w^{'} w^{'''} + [\frac{4 I_{Z}}{3} + \\ \frac{1}{4} (I_{Z} + \frac{I_{Y}}{3}) + (I_{Z} + \frac{I_{Y}}{3})] v^{'} w^{''} w^{'''} + I_{Z} w^{'' 2} v^{''}\} + \\ v^{'} (m + M) \int_{0}^{s} ({\dot{w}}^{' 2} + w^{'} {\ddot{w}}^{'} + {\dot{v}}^{' 2} + v^{'} {\ddot{w}}^{'}) d s - \\ v^{''} (m + M) \int_{s}^{L} \int_{0}^{s} ({\dot{w}}^{' 2} + w^{'} {\ddot{w}}^{'} + {\dot{v}}^{' 2} + v^{'} {\ddot{v}}^{'}) d s + \end{matrix}$

\begin{matrix} \frac{1}{2} A_{p} l^{2} G (2 v^{(4)} + 4 v^{'} v^{''} v^{'''} + v^{'' 3} + v^{' 2} v^{(4)}) + \\ \frac{1}{2} A_{p} l^{2} G (v^{''} w^{'' 2} + v^{''} w^{'} w^{'''} + 3 v^{'} w^{''} w^{'''} + w^{'} v^{'} w^{(4)}) = 0 \end{matrix}

(1b)

\{\begin{matrix} w (0, t) = w^{'} (0, t) = w^{''} (L, t) = w^{'''} (L, t) = 0 \\ v (0, t) = v^{'} (0, t) = v^{''} (L, t) = v^{'''} (L, t) = 0 \end{matrix}

(2)

在方程（1）中，E是拉伸弹性模量，G 是剪切弹性模量，l是材料长度尺寸参数^[53]（对宏观管，其等于零；对微尺度管，其不等于零）.方程（1）是在一般的坐标系[图1（c）]下建立的方程，w和v分别表示t时刻管轴线上一点（X，0，0）在z和y方向上的位移.因为悬臂管无初始轴向力，所以管轴线通常被认为是没有伸缩的^[37]，从而在管道运动过程中其轴线上一点（X，0，0）到坐标原点的弧长（记为s）是不变的，均等于初始距离X，即s=X（图2），因此管轴线上一点（X，0，0）也可表示成（s，0，0）.方程（1）中的“点”表示相应的量对时间t求导，即（˙）= $\partial / \partial t$ ，“撇”表示相应的量对弧长s求导，即（）′= $\partial / \partial t$ .I_Y、I_Z分别表示管的横截面对Y轴和Z轴的惯性矩，I_XY为惯性积，根据定义，三者分别为：
图2 轴线不可伸缩:（a）变形前;（b）变形后
Fig.2 Centroid line of pipe is not stretchable: (a) before deformation; (b) after deformation
图3 一般坐标轴和形心主惯性轴
Fig.3 General coordinate axes and centroid principal axes of inertia
$\begin{matrix} \int_{A_{p}} Z^{2} d Y d Z = I_{Y}, \int_{A_{p}} Y Z d Y d Z = I_{Y Z}, \\ \int_{A_{p}} Y^{2} d Y d Z = I_{Z} \end{matrix}$
若Y轴和Z轴用红色的形心主惯性轴（图3）代替，则I_Y和I_Z将分别由I₁和I₂代替，同时I_XY将等于0.
记

I = \frac{I_{1} + I_{2}}{2}, Δ I = \frac{I_{1} - I_{2}}{2}

(3)

引入下面的无量纲系数：

\begin{matrix} ξ = \frac{s}{L}, η = \frac{w}{L}, ζ = \frac{v}{L}, τ = {(\frac{E I}{m + M})}^{\frac{1}{2}} \frac{t}{L^{2}}, \\ ν = {(\frac{M}{E I})}^{\frac{1}{2}} V L, β = \frac{M}{m + M}, l_{0} = \frac{A_{p} l^{2} G}{2 E I}, δ = \frac{Δ I}{I} \end{matrix}

(4)

与文献^[52]中的“ $ε δ = \frac{Δ I}{I}$ ”不同的是，这里的 $\frac{Δ I}{I}$ 已不是小量，所以“δ”前无需乘以小参数“ε”.在本文中，不失一般性，可设I₁＞I₂，即δ＞0，从而管道在η方向的振动对应较大刚度，在ζ方向的振动对应较小刚度.根据方程（1）、方程（2）和式（4）可以写成：

\begin{matrix} \ddot{η} + (1 + 2 l_{0} + δ) η^{(4)} + 2 ν \sqrt{β} {\dot{η}}^{'} + ν^{2} η^{''} + \\ 2 ν \sqrt{β} [η^{' 2} {\dot{η}}^{'} + η^{'} ζ^{'} {\dot{ζ}}^{'} - η^{''} \int_{ξ}^{1} (η^{'} {\dot{η}}^{'} + ζ^{'} {\dot{ζ}}^{'}) d ξ] + \\ ν^{2} [η^{' 2} η^{''} + η^{'} ζ^{'} ζ^{''} - η^{''} \int_{ξ}^{1} (η^{'} η^{''} + ζ^{'} ζ^{''}) d ξ] + \\ (\frac{I_{1}}{I} η^{' 2} η^{(4)} + 4 \frac{I_{1}}{I} η^{'} η^{''} η^{'''} + \frac{I_{1}}{I} η^{'' 3} + \frac{2 I_{1} + I_{2}}{3 I} η^{'} ζ^{'} ζ^{(4)} + \\ \frac{31 I_{1} + 5 I_{2}}{12 I} η^{'} ζ^{''} ζ^{'''} + \frac{3 I_{1} + I_{2}}{4 I} η^{''} ζ^{'} ζ^{'''} + \frac{I_{1}}{I} η^{''} ζ^{'' 2}) + \\ η^{'} \int_{0}^{ξ} ({\dot{η}}^{' 2} + η^{'} {\ddot{η}}^{'} + {\dot{ζ}}^{' 2} + ζ^{'} {\ddot{ζ}}^{'}) d ξ - η^{''} \int_{ξ}^{1} \int_{0}^{ξ} ({\dot{η}}^{2} + \\ η^{'} {\ddot{η}}^{'} + {\dot{ζ}}^{' 2} + ζ^{'} {\ddot{ζ}}^{'}) d ξ + l_{0} (4 η^{'} η^{''} η^{'''} + η^{'' 3} + \\ η^{' 2} η^{(4)} + η^{''} ζ^{'' 2} + η^{''} ζ^{'} ζ^{'''} + 3 η^{'} ζ^{''} ζ^{'''} + \\ η^{'} ζ^{'} ζ^{(4)}) = 0 \end{matrix}

(5a)

\begin{matrix} \ddot{ζ} + (1 + 2 l_{0} - δ) ζ^{(4)} + 2 ν \sqrt{β} {\dot{ζ}}^{'} + ν^{2} ζ^{''} + \\ 2 ν \sqrt{β} [ζ^{' 2} {\dot{ζ}}^{'} + η^{'} ζ^{'} {\dot{η}}^{'} - ζ^{''} \int_{ξ}^{1} (η^{'} {\dot{η}}^{'} + ζ^{'} {\dot{ζ}}^{'}) d ξ] + \\ ν^{2} [ζ^{' 2} ζ^{''} + η^{'} ζ^{'} η^{''} - ζ^{''} \int_{ξ}^{1} (η^{'} η^{''} + ζ^{'} ζ^{''}) d ξ] + \\ (\frac{I_{2}}{I} ζ^{' 2} ζ^{(4)} + 4 \frac{I_{2}}{I} ζ^{'} ζ^{''} ζ^{'''} + \frac{I_{2}}{I} ζ^{'' 3} + \\ \frac{2 I_{2} + I_{1}}{3 I} η^{'} ζ^{'} η^{(4)} + \frac{31 I_{2} + 5 I_{1}}{12 I} ζ^{'} η^{''} η^{'''} + \\ \frac{3 I_{2} + I_{1}}{4 I} ζ^{''} η^{'} η^{'''} + \frac{I_{2}}{I} ζ^{''} η^{'' 2}) + ζ^{'} \int_{0}^{ξ} ({\dot{η}}^{' 2} + η^{'} {\ddot{η}}^{'} + \\ {\dot{ζ}}^{' 2} + ζ^{'} {\ddot{ζ}}^{'}) d ξ - ζ^{''} \int_{ξ}^{1} \int_{0}^{ξ} ({\dot{η}}^{' 2} + η^{'} {\ddot{η}}^{'} + {\dot{ζ}}^{' 2} + \\ ζ^{'} {\ddot{ζ}}^{'}) d ξ + l_{0} (4 ζ^{'} ζ^{''} ζ^{'''} + ζ^{'' 3} + ζ^{' 2} ζ^{(4)} + ζ^{''} η^{'' 2} + \\ ζ^{''} η^{'} η^{'''} + 3 ζ^{'} η^{''} η^{'''} + η^{'} ζ^{'} η^{(4)}) = 0 \end{matrix}

(5b)

\{\begin{matrix} η (0, τ) = η^{'} (0, τ) = η^{''} (1, τ) = η^{'''} (1, τ) = 0 \\ ζ (0, τ) = ζ^{'} (0, τ) = ζ^{''} (1, τ) = ζ^{'''} (1, τ) = 0 \end{matrix}

(6)

其中的“点”表示相应的量对无量纲时间τ求导，即（˙）= $\partial / \partial τ$ ，“撇”表示相应的量对无量纲弧长ξ求导，即（）′= $\partial / \partial ξ$ ，下文中的“点”和“撇”的含义均是如此.
根据文献^[18，54]中的摄动技巧，可以将方程（5）中的非线性惯性项写成：

\{\begin{matrix} \int_{0}^{ξ} η^{'} {\ddot{η}}^{'} d ξ = - \int_{0}^{ξ} η^{'} (1 + 2 l_{0} + δ) d η^{(4)} - \int_{0}^{ξ} η^{'} 2 ν \sqrt{β} {\dot{η}}^{''} d ξ - \int_{0}^{ξ} η^{'} ν^{2} η^{'''} d ξ \\ = (1 + 2 l_{0} + δ) (η^{'} η^{(4)} - \int_{0}^{ξ} η^{(4)} η^{''} d ξ) - 2 ν \sqrt{β} \int_{0}^{ξ} η^{'} {\dot{η}}^{''} d ξ - ν^{2} \int_{0}^{ξ} η^{'} η^{'''} d ξ \\ \int_{0}^{ξ} ζ^{'} {\ddot{ζ}}^{'} d ξ = - \int_{0}^{ξ} ζ^{'} (1 + 2 l_{0} - δ) d ζ^{(4)} - \int_{0}^{ξ} ζ^{'} 2 ν \sqrt{β} {\dot{ζ}}^{''} d ξ - \int_{0}^{ξ} ζ^{'} ν^{2} ζ^{'''} d ξ \\ = (1 + 2 l_{0} - δ) (ζ^{'} ζ^{(4)} - \int_{0}^{ξ} ζ^{(4)} ζ^{''} d ξ) - 2 ν \sqrt{β} \int_{0}^{ξ} ζ^{'} {\dot{ζ}}^{''} d ξ - ν^{2} \int_{0}^{ξ} ζ^{'} ζ^{'''} d ξ \end{matrix}

(7)

将方程（7）代入方程（5），整理可得：

\begin{matrix} \ddot{η} + (1 + 2 l_{0} + δ) η^{(4)} + 2 ν \sqrt{β} {\dot{η}}^{'} + ν^{2} η^{''} + [(4 \frac{I_{1}}{I} - 1 - δ) η^{'} η^{''} η^{'''} + (\frac{I_{1}}{I} + \frac{1}{2} + \frac{1}{2} δ) η^{''} + \frac{4}{3} δ η^{'} ζ^{'} ζ^{(4)} + \\ \frac{31 I_{1} + 5 I_{2}}{12 I} η^{'} ζ^{''} ζ^{'''} + \frac{3}{2} δ η^{''} ζ^{'} ζ^{'''} + (\frac{I_{1}}{I} + \frac{1}{2} - \frac{1}{2} δ) η^{''} ζ^{'' 2}] + η^{'} \int_{0}^{ξ} [{\dot{η}}^{' 2} + {\dot{ζ}}^{' 2} + (1 + δ) η^{(4)} η^{''} + \\ (1 - δ) ζ^{(4)} ζ^{''} + ν^{2} (η^{'' 2} + ζ^{'' 2}) + 2 ν \sqrt{β} (η^{''} {\dot{η}}^{'} + ζ^{''} ζ^{'})] d ξ - η^{''} \int_{ξ}^{1} \int_{0}^{ξ} [{\dot{η}}^{' 2} + {\dot{ζ}}^{' 2} + (1 + δ) η^{(4)} η^{''} + \\ (1 - δ) ζ^{(4)} ζ^{''} + ν^{2} (η^{'' 2} + ζ^{'' 2}) + 2 ν \sqrt{β} (η^{''} {\dot{η}}^{'} + ζ^{''} ζ^{'})] d ξ d ξ + l_{0} [2 η^{'} η^{''} η^{'''} + 2 η^{'' 3} - η^{' 2} η^{(4)} + 2 η^{''} ζ^{'' 2} - \\ η^{''} ζ^{'} ζ^{'''} + 3 η^{'} ζ^{''} ζ^{'''} - η^{'} ζ^{'} ζ^{(4)} + 2 η^{'} \int_{0}^{ξ} (η^{(4)} η^{''} + ζ^{(4)} ζ^{''}) d ξ - 2 η^{''} \int_{ξ}^{1} \int_{0}^{ξ} (η^{(4)} η^{''} + ζ^{(4)} ζ^{''}) d ξ d ξ] = 0 (8 a) \end{matrix}

(8a)

\begin{matrix} \ddot{ζ} + (1 + 2 l_{0} - δ) ζ^{(4)} + 2 ν \sqrt{β} {\dot{ζ}}^{'} + ν^{2} ζ^{''} + [(4 \frac{I_{2}}{I} - 1 + δ) ζ^{'} ζ^{''} ζ^{'''} + (\frac{I_{2}}{I} + \frac{1}{2} - \frac{1}{2} δ) ζ^{'' 3} - \frac{4}{3} δ η^{'} ζ^{'} η^{(4)} + \\ \frac{31 I_{2} + 5 I_{1}}{12 I} ζ^{'} η^{''} η^{'''} - \frac{3}{2} δ ζ^{''} η^{'} η^{'''} + (\frac{I_{2}}{I} + \frac{1}{2} + \frac{1}{2} δ) ζ^{''} η^{'' 2}] + ζ^{'} \int_{0}^{ξ} [{\dot{η}}^{' 2} + {\dot{ζ}}^{2} + (1 + δ) η^{(4)} η^{''} + \\ (1 - δ) ζ^{(4)} ζ^{''} + 2 ν \sqrt{β} (η^{''} {\dot{η}}^{'} + ζ^{''} {\dot{ζ}}^{'}) + ν^{2} (η^{'' 2} + ζ^{'' 2})] d ξ - ζ^{''} \int_{ξ}^{1} \int_{0}^{ξ} [{\dot{η}}^{' 2} + {\dot{ζ}}^{' 2} + (1 + δ) η^{(4)} η^{''} + \\ (1 - δ) ζ^{(4)} ζ^{''} + 2 ν \sqrt{β} (η^{''} {\dot{η}}^{'} + ζ^{''} ζ^{'}) + ν^{2} (η^{'' 2} + ζ^{'' 2})] d ξ d ξ + l_{0} [2 ζ^{'} ζ^{''} ζ^{'''} + 2 ζ^{''} - ζ^{' 2} ζ^{(4)} + 2 ζ^{''} η^{'' 2} - \\ ζ^{''} η^{'} η^{'''} + 3 ζ^{'} η^{''} η^{'''} - η^{'} ζ^{'} η^{(4)} + 2 ζ^{'} \int_{0}^{ξ} (ζ^{(4)} ζ^{''} + η^{(4)} η^{''}) d ξ - 2 ζ^{''} \int_{ξ}^{1} \int_{0}^{ξ} (ζ^{(4)} ζ^{''} + η^{(4)} η^{''}) d ξ d ξ] = 0 (8 b) \end{matrix}

(8b)

方程（8）和方程（6）就是最终的振动方程和边界条件.在方程（8a）和（8b）中，有无量纲参数l₀，δ，ν，β，以及和参数I₁，I₂，I相关的表达式.结合式（3）和式（4）分析可知，和参数I₁，I₂，I相关的表达式可通过无量纲参数δ来刻画，具体如下：
$\begin{matrix} \frac{I_{1}}{I} = 1 + δ, \frac{I_{2}}{I} = 1 - δ \\ \frac{31 I_{1} + 5 I_{2}}{12 I} = \frac{31 (1 + δ) + 5 (1 - δ)}{12} \\ \frac{31 I_{2} + 5 I_{1}}{12 I} = \frac{31 (1 - δ) + 5 (1 + δ)}{12} \end{matrix}$
因此，只要给定l₀，δ，ν，β的值，即可进行相应的数值模拟.
2 Galerkin方法与离散化方程
根据Galerkin方法，设方程（8）的解为
$η (ξ, τ) = \sum_{i = 1}^{n} ϕ_{i} (ξ) q_{i} (τ)$

ζ (ξ, τ) = \sum_{i = 1}^{n} ψ_{i} (ξ) p_{i} (τ)

(9)

其中φ_i（ξ）、ψ_i（ξ）是悬臂梁的第i阶模态函数，q_i（τ）、p_i（τ）是管道在两个横向上振动的广义坐标，n为模态截断数.φ_i（ξ）、ψ_i（ξ）的表达式为：

\{\begin{matrix} ϕ_{i} (ξ) = ψ_{i} (ξ) = c o s h λ_{i} ξ - c o s λ_{i} ξ - \\ σ_{i} (s i n h λ_{i} ξ - s i n λ_{i} ξ), \\ σ_{i} = (s i n h λ_{i} - s i n λ_{i}) / (c o s h λ_{i} + \\ \cos λ_{i}), (i = 1,2, \dots, n) \end{matrix}

(10)

λ_i是悬臂梁的第i阶频率.
将式（9）代入方程（8），用φ_i（ξ）、ψ_i（ξ）分别乘以方程（8a）和方程（8b）两边，并从0到1积分可得：
$\begin{matrix} {\ddot{q}}_{i} + c_{i j} {\dot{q}}_{j} + k_{i j}^{1} q_{j} + A_{i j k l}^{1} q_{j} q_{k} q_{l} + B_{i j k l} q_{j} q_{k} {\dot{q}}_{l} + \\ C_{i j k l} q_{j} {\dot{q}}_{k} {\dot{q}}_{l} + L_{i j k l}^{1} q_{j} p_{k} p_{l} + B_{i j k l} q_{j} p_{k} {\dot{p}}_{l} + \\ C_{i j k l} q_{j} {\dot{p}}_{k} {\dot{p}}_{l} = 0 \end{matrix}$

{\ddot{p}}_{i} + c_{i j} {\dot{p}}_{j} + k_{i j}^{2} p_{j} + A_{i j k l}^{2} p_{j} p_{k} p_{l} + B_{i j k l} p_{j} p_{k} {\dot{p}}_{l} + C_{i j k l} p_{j} {\dot{p}}_{k} {\dot{p}}_{l} + L_{i j k l}^{2} p_{j} q_{k} q_{l} + B_{i j k l} p_{j} q_{k} {\dot{q}}_{l} + C_{i j k l} p_{j} {\dot{q}}_{k} {\dot{q}}_{l} = 0

(11)

其中
$\{\begin{matrix} \begin{matrix} c_{i j} = 2 ν \sqrt{β} \int_{0}^{1} ϕ_{i} ϕ_{j}^{'} d ξ \\ k_{i j}^{1} = (1 + 2 l_{0} + δ) \int_{0}^{1} ϕ_{j}^{(4)} ϕ_{i} d ξ + ν^{2} \int_{0}^{1} ϕ_{i} ϕ_{j}^{''} d ξ, k_{i j}^{2} = (1 + 2 l_{0} - δ) \int_{0}^{1} ϕ_{j}^{(4)} ϕ_{i} d ξ + ν^{2} \int_{0}^{1} ϕ_{i} ϕ_{j}^{''} d ξ \\ A_{i j k l}^{1} = \int_{0}^{1} ϕ_{i} \cdot \{l_{0} (2 ϕ_{j}^{'} ϕ_{k}^{''} ϕ^{'''}_{l} + 2 ϕ_{j}^{''} ϕ_{k}^{''} ϕ_{l}^{''} - ϕ_{j}^{'} ϕ_{k}^{'} ϕ_{l}^{(4)} + 2 ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ - 2 ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ d ξ) + \\ ν^{2} (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{''} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{''} d ξ d ξ) + [(4 \frac{I_{1}}{I} - 1 - δ) ϕ_{j}^{'} ϕ_{k}^{''} ϕ_{l}^{''} + (\frac{I_{1}}{I} + \frac{1}{2} + \frac{1}{2} δ) ϕ_{j}^{''} ϕ_{k}^{''} ϕ_{l}^{''}] + \\ (1 + δ) (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ d ξ)\} d ξ \end{matrix} \\ \begin{matrix} A_{i j k l}^{2} = \int_{0}^{1} ϕ_{i} \cdot \{l_{0} (2 ϕ_{j}^{'} ϕ_{k}^{''} ϕ_{l}^{'''} + 2 ϕ_{j}^{''} ϕ_{k}^{''} ϕ_{l}^{''} - ϕ_{j}^{'} ϕ_{k}^{'} ϕ_{l}^{(4)} + 2 ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ - 2 ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ d ξ) + \\ ν^{2} (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{''} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{''} d ξ d ξ) + [(4 \frac{I_{2}}{I} - 1 + δ) ϕ_{j}^{'} ϕ_{k}^{''} ϕ_{l}^{''} + (\frac{I_{2}}{I} + \frac{1}{2} - \frac{1}{2} δ) ϕ_{j}^{''} ϕ_{k}^{''} ϕ_{l}^{''}] + \\ (1 - δ) (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{(4)} d ξ d ξ)\} d ξ \\ B_{i j k l} = \int_{0}^{1} ϕ_{i} \cdot 2 ν \sqrt{β} (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{'} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{''} ϕ_{l}^{'} d ξ d ξ) d ξ \\ C_{i j k l} = \int_{0}^{1} ϕ_{i} \cdot (ϕ_{j}^{'} \int_{0}^{ξ} ϕ_{k}^{'} ϕ_{l}^{'} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ϕ_{k}^{'} ϕ_{l}^{'} d ξ d ξ) d ξ \\ L_{i j k l}^{1} = \int_{0}^{1} ϕ_{i} \cdot \{l_{0} (2 ϕ_{j}^{''} ψ_{k}^{''} ψ_{l}^{''} - ϕ_{j}^{''} ψ_{k}^{'} ψ_{l}^{'''} + 3 ϕ_{j}^{'} ψ_{k}^{''} ψ_{l}^{''} - ϕ_{j}^{'} ψ_{k}^{'} ψ_{l}^{(4)} + \end{matrix} \\ \begin{matrix} 2 ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ - 2 ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ d ξ) + ν^{2} (ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{''} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{''} d ξ d ξ) + \\ [(\frac{I_{1}}{I} + \frac{1}{2} - \frac{1}{2} δ) ϕ_{j}^{''} ψ_{k}^{''} ψ_{l}^{''} + \frac{31 I_{1} + 5 I_{2}}{12 I} ϕ_{j}^{'} ψ_{k}^{''} ψ_{l}^{''}] + (1 - δ) (ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ - \\ ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ d ξ) + δ (\frac{4}{3} ϕ_{j}^{'} ψ_{k}^{'} ψ_{l}^{(4)} + \frac{3}{2} ϕ_{j}^{''} ψ_{k}^{'} ψ_{l}^{''})\} d ξ \\ L_{i j k l}^{2} = \int_{0}^{1} ϕ_{i} \cdot \{l_{0} (2 ϕ_{j}^{''} ψ_{k}^{''} ψ_{l}^{''} - ϕ_{j}^{''} ψ_{k}^{'} ψ_{l}^{'''} + 3 ϕ_{j}^{'} ψ_{k}^{''} ψ_{l}^{'''} - ϕ_{j}^{'} ψ_{k}^{'} ψ_{l}^{(4)} + 2 ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ - \\ 2 ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ d ξ) + ν^{2} (ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{''} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{''} d ξ d ξ) + [(\frac{I_{2}}{I} + \frac{1}{2} + \frac{1}{2} δ) ϕ_{j}^{''} ψ_{k}^{''} ψ_{l}^{''} + \\ \frac{31 I_{2} + 5 I_{1}}{12 I} ϕ_{j}^{'} ψ_{k}^{''} ψ_{l}^{'''}] + (1 + δ) (ϕ_{j}^{'} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ - ϕ_{j}^{''} \int_{ξ}^{1} \int_{0}^{ξ} ψ_{k}^{''} ψ_{l}^{(4)} d ξ d ξ) - \\ δ (\frac{4}{3} ϕ_{j}^{'} ψ_{k}^{'} ψ_{l}^{(4)} + \frac{3}{2} ϕ_{j}^{''} ψ_{k}^{'} ψ_{l}^{'''})\} d ξ \end{matrix} \end{matrix}$
i，j，k和l的值取遍1到n.
令

q_{i} = x_{i}, {\dot{q}}_{i} = x_{i + n}, p_{i} = x_{i + 2 n}, {\dot{p}}_{i} = x_{i + 3 n}

(12)

将方程（11）化成一阶常微分方程组：

\dot{X} = L X + N (X)

(13)

其中

L = [\begin{matrix} 0 & I & 0 & 0 \\ - K^{1} & - C & 0 & 0 \\ 0 & 0 & 0 & I \\ 0 & 0 & - K^{2} & - C \end{matrix}]

(14)

$X = [\underset{n}{\underset{⏟}{q_{1}, \dots q_{n}}}, \underset{n}{\underset{⏟}{{\dot{q}}_{1}, \dots {\dot{q}}_{n}}}, \underset{n}{\underset{⏟}{p_{1}, \dots p_{n}}}, \underset{n}{\underset{⏟}{{\dot{p}}_{1}, \dots {\dot{p}}_{n}}}]^{T} =$

[\underset{n}{\underset{⏟}{x_{1}, \dots x_{n}}}, \underset{n}{\underset{⏟}{x_{n + 1} \dots, x_{2 n}}}, \underset{n}{\underset{⏟}{x_{2 n + 1}, \dots x_{3 n}}}, \underset{n}{\underset{⏟}{x_{3 n + 1} \dots, x_{4 n}}}]^{T}

(15)

K¹的元素为k¹_ij，K²的元素为k²_ij，C的元素为c_ij，I为n阶单位矩阵，0为n阶零矩阵（除另有说明外，其含义均是如此）.N（X）表示非线性项：

\begin{matrix} N (X) = [\underset{1 - n}{\underset{⏟}{0, \dots 0}}, N_{1} (X), \dots, N_{n} (X), \underset{1 - n}{\underset{⏟}{0, \dots 0}}, \\ {N_{1 + n} (X), \dots, N_{2 n} (X)]}^{T} \end{matrix}

(16)

其中N_i（X）、N_i₊_n（X）分别为：

\begin{matrix} N_{i} (X) = - A_{i j k l}^{1} x_{j} x_{k} x_{l} - B_{i j k l} x_{j} x_{k} x_{l + n} - \\ C_{i j k l} x_{j} x_{k + n} x_{l + n} - L_{i j k l}^{1} x_{j} x_{k + 2 n} x_{l + 2 n} - \\ B_{i j k l} x_{j} x_{k + 2 n} x_{l + 3 n} - C_{i j k l} x_{j} x_{k + 3 n} x_{l + 3 n} \end{matrix}

(17a)

\begin{matrix} N_{i + n} (X) = - A_{i j k l}^{2} x_{j + 2 n} x_{k + 2 n} x_{l + 2 n} - \\ B_{i j k l} x_{j + 2 n} x_{k + 2 n} x_{l + 3 n} - C_{i j k l} x_{j + 2 n} x_{k + 3 n} x_{l + 3 n} - \\ L_{i j k l}^{2} x_{j + 2 n} x_{k} x_{l} - B_{i j k l} x_{j + 2 n} x_{k} x_{l + n} - \\ C_{i j k l} x_{j + 2 n} x_{k + n} x_{l + n} \end{matrix}

(17b)

对方程（13）的线性部分：

\dot{X} = L X

(18)

将其写成：

[\begin{matrix} {\dot{X}}_{1} \\ {\dot{X}}_{2} \end{matrix}] = [\begin{matrix} L_{1} & 0 \\ 0 & L_{2} \end{matrix}] [\begin{matrix} X_{1} \\ X_{2} \end{matrix}]

(19)

式（19）中的0为2n阶零矩阵，L₁、L₂，X₁、X₂分别为：

L_{1} = [\begin{matrix} 0 & I \\ - K^{1} & - C \end{matrix}], L_{2} = [\begin{matrix} 0 & I \\ - K^{2} & - C \end{matrix}]

(20)

\begin{matrix} X_{1} = [\underset{n}{\underset{⏟}{q_{1}, \dots q_{n}}}, \underset{n}{\underset{⏟}{{\dot{q}}_{1}, \dots {\dot{q}}_{n}}}]^{T} = {[x_{1}, \dots, x_{2 n}]}^{T}, \\ X_{2} = [\underset{n}{\underset{⏟}{p_{1}, \dots p_{n}}}, \underset{n}{\underset{⏟}{{\dot{p}}_{1}, \dots {\dot{p}}_{n}}}]^{T} \\ = {[x_{2 n + 1}, \dots, x_{4 n}]}^{T} \end{matrix}

(21)

3 数值仿真
本文中，在用Galerkin方法对原振动方程（8）进行离散时，式（9）中的模态截断数取为6^[40]，即式（11）是12个自由度的二阶常微分方程组，式（13）是24维的一阶常微分方程组.进一步地，本文仅考虑横截面不具有对称性的宏观管，即方程（1）中的l=0或式（4）和式（5）中的l₀=0.在给定的质量比β[其定义见式（4）]处，管内流体的流速变化时，矩阵L₁、L₂的特征值会有相应的改变：使得L₁具有零实部特征值的流速称为η方向的临界流速，记为v¹_c，流速在v¹_c附近变化时管道将可能在η方向失稳；使得L₂具有零实部特征值的流速称为ζ方向的临界流速，记为v²_c，流速在v²_c附近变化时管道将可能在ζ方向失稳.
本文着重探讨如下情形：在特定的质量比处，v¹_c和v²_c可能相等，此时矩阵L具有两对纯虚数特征值，分别对应管道在η、ζ方向上振动的固有频率，流速的变化可能引起η、ζ两个方向上振动的耦合，从而出现复杂的动力学现象.v¹_c和v²_c是否可以相等及在何处相等，这可以通过研究二者随质量比β的变化曲线（统称为“临界流速曲线”，分别简记为v¹_c-β曲线和v²_c-β曲线）的交点来加以确定.计算表明：δ[其定义见式（4）]越大，v¹_c-β曲线越高，v²_c-β曲线越低，δ=0时，v¹_c-β曲线与v²_c-β曲线重合，如图4所示.
图4 δ取不同值时的临界流速随质量比β的变化曲线：（a）v¹_c-β曲线;（b）v²_c-β曲线
Fig.4 Critical flow velocity as functions of mass ratio β with different values of δ: (a) curve of v¹_c-β; (b) curve of v²_c-β
对每一条临界流速曲线，有必要将其区分为“滞后段”（即临界流速随质量比的增加而减小的段）和“非滞后段”（即临界流速随质量比的增加而增大的段），v¹_c-β曲线与v²_c-β曲线的交点发生在不同的段时，对管道的失稳行为会有影响：当交点属于两条曲线的非滞后处时，流速增加则管道颤振，流速减少则管道稳定；当交点属于某条曲线的非滞后处和另一条曲线的滞后处时，不管流速增加还是减少，管道均会发生颤振，但运动形式有所不同.
鉴于本文考虑“大”对称破缺，因此不妨将δ的取值限制在0.15以上，则η、ζ两个方向上的无量纲刚度差[其为（1+δ）-（1-δ）=2δ]不小于30%.又从图4可知，随δ值增大，v¹_c-β曲线上升，v²_c-β曲线下降，据此可推断，当δ超过一定值δ₁时，η、ζ两个方向上的临界流速曲线无交点.因此，要符合“大”对称破缺且在特定的质量比处v¹_c =v²_c的要求，这里将δ的范围限制为0.15≤δ≤δ₁.
计算可知δ₁=0.2309，δ＞δ₁时（例如δ=0.25时）v¹_c-β曲线和v²_c-β曲线无交点[如图5（a）所示]；δ刚好等于δ₁时v¹_c-β曲线和v²_c-β曲线有一个交点[如图5（b）中的黑点所示]；0.15≤δ＜δ₁时v¹_c-β曲线和v²_c-β曲线的交点数可为两个、三个及四个，具体将在下文中分析.
图5 临界流速v¹_c和v²_c随质量比β的变化曲线:（a）δ=0.25;（b）δ=0.2309
Fig.5 Critical flow velocity v¹_c and v²_c as functions of mass ratio β: (a) δ=0.25; (b) δ=0.2309
图6（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.6 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
图5（b）中的黑点属于两条曲线的非滞后段，其对应的质量比和临界流速分别为：β=0.8874，v_c=16.2814，管道在此处通过两对纯虚数特征值失稳.为探究失稳后管道的运动形式，可令流速有一个微小增量，不妨取为0.2，此时管道振动的位形图和相图如图6所示.需要说明的是，在图（6、8、9、11~15、17、18）中，彩色的线或点的含义均分别为：蓝色对应瞬态解，红色对应稳态解，品红色表示管道的轴线；其子图（a）~图（d）的含义均分别为：（a）管道自由端在两个横向上的位移关系，（b）整个管道做稳态振动时的位形图，（c）η方向上的位移-速度关系，（d）为ζ方向上的速度-位移关系.图6表明，此时管道做稳定的环面运动.
图7 δ=0.2214时临界流速v¹_c和v²_c随质量比β的变化曲线
Fig.7 Critical flow velocity v¹_c and v²_c as functions of mass ratio β for δ=0.2214
图8（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.8 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
δ=0.2214时，v¹_c-β曲线和v²_c-β曲线的交点个数为2，如图7中的黑点所示（均属于两条曲线的非滞后段），左边黑点对应的质量比和临界流速分别为：β=0.8580，v_c=15.7951，右边黑点对应的质量比和临界流速分别为：β=0.9082，v_c=16.7373.管道在这两处均通过两对纯虚数特征值失稳：左边黑点处对应的两对纯虚数特征值为±52.8012i，±72.9908i，即此时在η方向振动的固有频率为52.8012，在ζ方向振动的固有频率为72.9908；右边黑点处对应的两对纯虚数特征值为±56.6092i，±79.6542i，即此时在η方向振动的固有频率为56.6092，在ζ方向振动的固有频率为79.6542.当流速有一个微小增量（不妨均取为0.2）时：管道从左边黑点处失稳后振动的位形图和相图如图8所示，可以看出，此时管道沿η方向（即较大刚度方向）做稳定的周期运动想，自振频率为53.7484（接近52.8012）；管道从右边黑点处失稳后振动的位形图和相图如图9所示，显然，此时管道沿ζ方向（即较小刚度方向）做稳定的周期运动，自振频率为81.3884（接近79.6542）.
图9（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.9 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
δ=0.1687时，v¹_c-β曲线和v²_c-β曲线的交点个数为3，如图10中的黑点所示.左边黑点对应的质量比和临界流速分别为：β=0.6619，v_c=11.3875，其属于两条曲线的非滞后段；中间黑点对应的质量比和临界流速分别为：β=0.8540，v_c=15.4022，其属于v¹_c-β曲线的滞后段和v²_c-β曲线的非滞后段；右边黑点对应的质量比和临界流速分别为：β=0.9021，v_c=17.1847，其属于v¹_c-β曲线的非滞后段和v²_c-β曲线的滞后段.管道在这三处均通过两对纯虚数特征值失稳：左边黑点处对应的两对纯虚数特征值为±28.8399i，±38.1262i，即此时在η方向振动的固有频率为28.8399，在ζ方向振动的固有频率为38.1262；中间黑点处对应的两对纯虚数特征值为±51.5286i，±64.1231i，即此时在η方向振动的固有频率为51.5286，在ζ方向振动的固有频率为64.1231；右边黑点处对应的两对纯虚数特征值为±61.9368i，±81.6980i，即此时在η方向振动的固有频率为61.9368，在ζ方向振动的固有频率为81.6980.当流速有一个微小增量（不妨取为0.2）时，管道从左边黑点处失稳后振动的位形图和相图如图11所示，管道沿ζ方向（即较小刚度方向）做稳定的周期运动，自振频率为39.8174（接近38.1262）.虽然该交点和图5（b）中的交点类似，即都在临界流速曲线的非滞后段，但是管道在这两处失稳后所对应的稳态运动形式（图6和图11）却是不一样的.
图10 δ=0.1687时临界流速v¹_c和v²_c随质量比β的变化曲线
Fig.10 Critical flow velocity v¹_c and v²_c as functions of mass ratio β for δ=0.1687
图11（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.11 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
图12（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.12 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
图13（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.13 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
当管道从中间黑点处失稳时，流速增量的“正”或“负”影响管道最终的颤振形式.若流速增量为正（取为0.2），管道失稳后振动的位形图和相图如图12所示，即管道沿η方向（即较大刚度方向）做稳定的周期运动，自振频率为52.4473（接近51.5286）；若流速增量为负（取为-0.2），管道失稳后振动的位形图和相图如图13所示，显然，此时管道沿ζ方向（即较小刚度方向）做稳定的周期运动，自振频率为61.1205（接近64.1231）.当管道从右边黑点处失稳时，流速增量的“正”或“负”也会影响管道最终的颤振形式，但其和中间黑点处的失稳方式刚好相反：流速增加时管道沿ζ方向（即较小刚度方向）做稳定的周期运动（图14），自振频率为83.4420（接近81.6980），流速减少时管道沿η方向（即较大刚度方向）做稳定的周期运动（图15），自振频率为60.0687（接近61.9368）.
图14（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.14 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
图15（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.15 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
图16 δ=0.1628时临界流速和随质量比β的变化曲线
Fig.16 Critical flow velocity and as functions of mass ratio β for δ=0.1628
图17（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.17 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
δ=0.1628时，v¹_c-β曲线和v²_c-β曲线的交点个数为4，如图16中的黑点所示.从左至右，交点对应的质量比和临界流速依次为：β=0.6506，v_c=11.2353；β=0.6806，v_c=11.6118；β=0.8566，v_c=15.3945；β=0.8998，v_c=17.2081.左边两交点属于两条曲线的非滞后段（与图7类似），右边两交点所属段与图10中对应点的相同.管道在这四点均通过两对纯虚数特征值失稳，其中：左一黑点处对应的两对纯虚数特征值为±28.6433i，±36.2085i，即此时在η方向振动的固有频率为28.6433，在ζ方向振动的固有频率为36.2085；左二黑点处对应的两对纯虚数特征值为±29.1992i，±39.6618i，即此时在η方向振动的固有频率为29.1992，在ζ方向振动的固有频率为39.6618.当流速有一个微小增量（不妨取为0.2）时：管道从左边第一个黑点处失稳后振动的位形图和相图如图17所示，管道沿η方向（即较大刚度方向）做稳定的周期运动，自振频率为29.1023（接近28.6433）；管道从左边第二个黑点处失稳后振动的位形图和相图如图18所示，管道沿ζ方向（即较小刚度方向）做稳定的周期运动，自振频率为41.0397（接近39.6618）.通过与图（7）~图（9）的对比可看出，不仅交点所属的曲线段类似，而且失稳后管道的运动形式也类似.
管道在图16中右边两黑点处的失稳方式和图10中对应点处的相同.
图18（a）管道自由端在两个方向上的位移关系；（b）整个管道的稳态振动；（c）管道自由端在η方向上的位移-速度关系；（d）管道自由端在ζ方向上的位移-速度关系
Fig.18 (a) the position relationship of the free end of the pipe in two directions; (b) the steady-state vibration of the whole pipe; (c) the velocity-displacement relationship diagram of the free end of the pipe in η direction; (d) the velocity-displacement relationship diagram of the free end of the pipe in ζ direction
4 结论
本文在形心主惯性轴坐标系下推导了横截面具有“大”对称破缺的悬臂输流管道的空间弯曲振动方程，用Galerkin方法将其离散成常微分方程组，并取模态截断数为6，对离散化方程进行了数值分析.结果表明，当两个主方向上的无量纲刚度的差大于2δ₁=2×0.2309=0.4618时，管道在这两个方向上的临界流速曲线不会相交，即管道在这两个方向上不会同时失稳.当两个主方向上的无量纲刚度的差小于等于2δ₁=2×0.2309=0.4618时，管道在这两个横向上的临界流速曲线存在交点，随两个横向上无量纲刚度的差距的缩小，交点的个数增加.当交点属于两条临界流速曲线的非滞后段时，流速增加时管道失稳，失稳后的运动可为环面运动、较大刚度方向的周期运动、较小刚度方向的周期运动.当交点属于一条临界流速曲线的滞后段和另一条临界流速曲线的非滞后段时，不管流速增加还是减少，管道均会失稳而做周期运动，流速增加时若在较大（较小）刚度方向做周期运动，则流速减少时将在较小（较大）刚度方向做周期运动.
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基本信息

中图分类号: O322；O326
文献标识码: A
DOI: 10.6052/1672-6553-2023-087
文章编号: 1672-6553-2023-21(11)-081-014

基金信息

国家自然科学基金资助项目(12002096)；
安顺学院2022年度博士基金资助项目(asxybsjj202201)；
the National Natural Science Foundation of China (12002096)；
the 2022 Doctoral Foundation of Anshun University (asxybsjj202201)；

引用信息

稿件历史

收稿日期: 2023-06-07

参考文献

[1] HANDELMAN G H.A note on the transverse vibration of a tube containing flowing fluid [J].Quarterly of Applied Mathematics,1955,13(3):326-330.
[2] BENJAMIN T B.Dynamics of a system of articulated pipes conveying fluid-Ⅰ.Theory [J].Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences,1961,261(1307):457-486.
[3] GREGORY R W,PAÏDOUSSIS M P.Unstable oscillation of tubular cantilevers conveying fluid I.Theory [J].Proceedings of the Royal Society of London Series A Mathematical and Physical Sciences,1966,293(1435):512-527.
[4] CHEN S S.Forced vibration of a cantilevered tube conveying fluid [J].The Journal of the Acoustical Society of America,1970,48(3B):773-775.
[5] GINSBERG J H.The dynamic stability of a pipe conveying a pulsatile flow [J].International Journal of Engineering Science,1973,11(9):1013-1024.
[6] HOLMES P J.Bifurcations to divergence and flutter in flow-induced oscillations:a finite dimensional analysis [J].Journal of Sound and Vibration,1977,53(4):471-503.
[7] HOLMES P J.Pipes supported at both ends cannot flutter [J].Journal of Applied Mechanics,1978,45(3):619-622.
[8] HOLMES P,MARSDEN J.Bifurcation to divergence and flutter in flow-induced oscillations:an infinite dimensional analysis [J].Automatica,1978,14(4):367-384.
[9] ROUSSELET J,HERRMANN G.Dynamic behavior of continuous cantilevered pipes conveying fluid near critical velocities [J].Journal of Applied Mechanics,1981,48(4):943-947.
[10] 随岁寒,李成.输流管道弯曲和振动的有限元分析 [J].动力学与控制学报,2022,20(4):83-90.SUI S H,LI C.The finite element analysis on bending and vibration of the fluid-conveying pipes [J].Journal of Dynamics and Control,2022,20(4):83-90.(in Chinese)
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[12] SRI NAMCHCHIVAYA N,TIEN W M.Non-linear dynamics of supported pipe conveying pulsating fluid-Ⅱ.Combination resonance [J].International Journal of Non-Linear Mechanics,1989,24(3):197-208.
[13] JAYARAMAN K,NARAYANAN S.Chaotic oscillations in pipes conveying pulsating fluid [J].Nonlinear Dynamics,1996,10(4):333-357.
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[24] WANG L,LIU Z Y,ABDELKEFI A,et al.Nonlinear dynamics of cantilevered pipes conveying fluid:towards a further understanding of the effect of loose constraints [J].International Journal of Non-Linear Mechanics,2017,95:19-29.
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具有非对称横截面的悬臂输流管道的非线性振动特征*

Nonlinear Vibration Characteristics of Cantilevered Fluid-Conveying Pipe with Asymmetric Cross-Section*

摘要

Abstract

关键词

Keywords

1 力学模型与运动方程

(1a)

(1b)

(2)

(3)

(4)

(5a)

(5b)

(6)

(7)

(8a)

(8b)

2 Galerkin方法与离散化方程

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17a)

(17b)

(18)

(19)

(20)

(21)

3 数值仿真

4 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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具有非对称横截面的悬臂输流管道的非线性振动特征^*

Nonlinear Vibration Characteristics of Cantilevered Fluid-Conveying Pipe with Asymmetric Cross-Section^*