旋转Timoshenko输流管道的固有频率和稳定性分析

杨佳丽，杨虹，李伟; Yang Jiali; Yang Hong; Li Wei

引 en

旋转Timoshenko输流管道的固有频率和稳定性分析^*

杨佳丽
机构：
北京信息科技大学理学院,北京 100192
×
，杨虹
机构：
北京信息科技大学理学院,北京 100192
×
，李伟
机构：
北京信息科技大学理学院,北京 100192
×

北京信息科技大学理学院,北京 100192；

Natural Frequency and Stability Analysis of Rotating Timoshenko Pipe Conveying Fluid^*

Yang Jiali
Affiliation：
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192 , China
×
，Yang Hong
Affiliation：
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192 , China
×
，Li Wei
Affiliation：
School of Applied Science, Beijing Information Science and Technology University, Beijing 100192 , China
×

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192 , China；

通讯作者:

李伟,E-mail∶liwei_0217@qq.com

中图分类号:O321

文献标识码:A

文章编号:1672-6553-2023-21(2)-058-008

DOI:10.6052/1672-6553-2021-072

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参考文献
出版信息

参考文献 1

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目录contents

摘要 Abstract
关键词 Keywords
1 振动方程推导
2 离散求解
3 参数性能分析
4 总结
参考文献

摘要

基于Timoshenko梁模型,本文研究了旋转输流管道在自由振动状态下的流固耦合振动特性.考虑流体压力、重力、初始轴应力作用,基于Hamilton原理和欧拉角转换,推导得到了旋转Timoshenko输流管道的偏微分方程.根据Galerkin截断法将运动方程进行离散,通过求解系统的特征方程即得到输流管一阶复频率的实部和虚部,实部代表固有频率,虚部代表能量变化.在流速较高时,研究发现必须考虑4阶及以上Galerkin截断,才能得到稳定的结果.通过与Euler-Bernoulli梁模型对比,验证了本文结果的正确性.研究发现针对短粗型管道,Timoshenko梁模型更加精确.此外研究了多种参数对旋转Timoshenko输流管道固有频率和振动稳定性的影响.研究结果表明质量比、流速、剪切系数对Timoshenko输流管道流固耦合振动的稳定性影响显著,而转动惯量、重力、流体压力和初始轴应力在一定程度上也会影响管道振动的频率和稳定性.转速的出现将管道频率分为两个量值,但转速并不影响系统能量变化.

Abstract

Based on the Timoshenko beam model, the fluid structure coupling vibration characteristics of rotating pipe conveying fluid under free vibration is studied. Considering fluid pressure, gravity and initial axial stress, the partial differential equation of rotating Timoshenko pipe conveying fluid is derived by using Hamilton principle and Euler angle transformation. The partial differential equations of motion are discretized according to Galerkin truncation method. By solving the characteristic equation of the system, the real part and the imaginary part of the first order complex frequency of the pipe conveying fluid are obtained, the real part represents the natural frequency and the imaginary part denotes the energy change. When the fluid velocity is high, it is found that the fourth order and above Galerkin truncation must be considered in order to obtain stable results. By comparing with the Euler-Bernoulli beam model, the correctness of the results in this paper is verified, and it is found that the Timoshenko beam model is more accurate in the study of short and thick pipes. In addition, the effects of various parameters on the natural frequency and vibration stability of the rotating Timoshenko pipe conveying fluid are studied. The results show that the mass ratio, flow velocity and shear coefficient have significant effects on the stability of fluid structure coupling vibration of Timoshenko pipe conveying fluid. To a certain extent, the moment of inertia, gravity, fluid pressure and initial axial stress also affect the frequency and stability of the pipe conveying fluid. The frequency of the pipe is divided into two values when the rotating speed appears, and the rotating speed does not affect the energy change of the system.

关键词

Timoshenko梁模型；输流管道； Galerkin截断法；固有频率；稳定性分析

Keywords

Timoshenko beam model ； pipe conveying fluid ； Galerkin truncation method ； natural frequency ； stability analysis

引言
输流管道广泛应用于工业领域中，是一种典型的流固耦合系统.流体在管道中运动时管道不可避免地会发生振动，管道的振动又会改变流体的运动状态，当管道振动的幅值超过一定的范围就会对管道造成危害.自从1939年Bourrières首次比较系统地推导了输流管的运动方程以后，众多科研人员对输流管道的稳定性问题进行了研究^[1].目前关于输流管振动的研究方向主要是振动模型的建立、求解方法的探索和振动特性分析三部分.近年来输流管建模主要采用梁模型，Dagli和Ergut^[2]利用Rayleigh梁理论研究了非经典边界条件对管道固有频率的影响.Liang^[3]等人利用伽辽金法推导了旋转Euler-Bernoulli梁模型输流管道的运动方程，基于流固耦合理论研究了输流管振动和动态失稳的机理.Li^[4]和Zhou^[5]等人研究了考虑阻尼影响的Timoshenko梁模型的振动问题，并用数值法进行验证.Tan^[6]等人运用多尺度法、有限差分法和Galerkin截断法研究了超临界输流管道的自由振动，确定了Timoshenko输流管道模型在超临界领域内的必要性.输送流体的管道种类繁多，对于长细比（L/D）比较大的管道，Euler梁模型就可以提供较高的精确度.然而对于短粗型输流管道，Euler-Bernoulli梁模型可能就不是很适合，具体差别还有待研究，这时需要考虑管道横向剪切力的影响.
以上对输流管的研究大部分都是对静态管道的振动研究，但输流管在实际应用中可能会伴随着旋转运动^[7]，比如石油钻井中运输泥浆的钻柱管道、月球着陆器自动采样机构、航空发动机的空心主轴等.旋转引起的离心力和附加科氏力会使这类旋转问题的处理变得更加复杂，对比尚缺乏深入的理论解释.Gulyayev^[8]等人模拟钻柱的静态和动态临界状态，将钻柱的旋转效应视为周期性激励.Huo和Wang^[9]通过数值算例法研究无量纲参数对旋转双锥形梁固有频率的影响，观察到有趣的频率和模态运动现象.Yang^[10]等人探究了旋转梁自由振动时的陀螺项、静离心力和动力离心力项对固有频率的影响，利用数值算例法仿真了旋转梁的实际振动轮廓.Abdollahi^[11]等人对内外流体同时作用的柔性旋转管的稳定性进行分析，发现对于外部流体中的旋转管道，其稳定性程度取决于半径比和质量比.Zhang^[12]等人通过欧拉角建立Euler和Lagrange坐标系下的运动方程，利用模态分析法分析离心力项对颤振稳定的影响.
Wiggert^[13]等人利用特征值法对弯型输流管道的压力振动问题进行了实验，结果表现为没有完全约束的边界条件的弯管存在明显的瞬态压力波动，而固支边界条件的弯管没有引起压力波动.Mehdi等人^[14]利用随机平均法研究流速的随机性对输流管道稳定性的影响.文献^[15]研究了末端有附加质量的梁模型，给出了不稳定模式下振幅的演化并进行了详细的线性稳定性分析.Lu^[16]等人研究了管道在超临界流速时的内共振和应力分布，并表示在不同流速下存在1:3内共振时，应修改适当的轴向预压力来提高管道疲劳寿命.
本文利用Hamilton原理、欧拉角转换及Galerkin截断法，研究了旋转Timoshenko输流管道模型的振动特性，详细讨论了关键参数对振动固有频率和系统稳定性的影响.
1 振动方程推导
旋转Timoshenko输流管道的模型图参考文献^[3].假设管道绕长轴以定常速度Ω旋转，同时管道内以流速U输送流体.
对于管道系统，基于欧拉角转换，旋转Timoshenko输流管的速度矢量v_p可以表示为^[3，12]:
$v_{p} = (- φ \dot{φ} θ z - y \dot{φ} θ^{2} + z \dot{θ} - y \dot{φ}) i_{x} + (\dot{v} -$

\begin{matrix} w Ω - z Ω + z \dot{φ} θ + z φ \dot{θ} - z Ω φ^{2} - φ y Ω θ - \\ φ \dot{φ} y) i_{y} + (\dot{w} + v Ω + y Ω - y \dot{φ} θ - z θ \dot{θ} + \\ y Ω θ^{2} + φ z Ω θ + θ y \dot{φ}) i_{z} \end{matrix}

(1)

其中w（x，t）和v（x，t）分别为旋转Timoshenko输流管振动时沿两个横向方向的弯曲位移.θ（x，t）和φ（x，t）是弯矩产生的两个方向上的转角.矢量（i_x，i_y，i_z）表示坐标系x-y-z的单位矢量.
管内流体的速度矢量v_f可以表示为:

\begin{matrix} v_{f} = v_{p} + U (v^{'} i_{y} + w^{'} i_{z}) i_{y} = \\ (- φ \dot{φ} θ z - y \dot{φ} θ^{2} + z \dot{θ} - y \dot{φ}) i_{x} + (\dot{v} - w Ω - \\ z Ω + z \dot{φ} θ + z φ \dot{θ} - z Ω φ^{2} - φ y Ω θ - φ \dot{φ} y + \\ U v^{'}) i_{y} + (\dot{w} + v Ω + y Ω - y \dot{φ} θ - z θ \dot{θ} + \\ y Ω θ^{2} + φ z Ω θ + θ y \dot{φ} + U w^{'}) i_{z} \end{matrix}

(2)

旋转Timoshenko输流管道的总动能T是管道T_p和流体T_f的动能之和，如下所示:

\begin{matrix} T = T_{p} + T_{f} = \\ \frac{1}{2} \int_{0}^{L} \int_{A_{p}} ρ_{p} v_{p} \cdot v_{p} d A_{p} d x + \\ \frac{1}{2} \int_{0}^{L} \int_{A f} ρ_{f} v_{f} \cdot v_{f} d A_{f} d x = \frac{(m + M)}{2} \\ \int_{0}^{L} [({\dot{w}}^{2} + {\dot{v}}^{2}) + Ω^{2} (w^{2} + v^{2}) + 2 Ω (v \dot{w} - \\ w \dot{v})] d x + \frac{j_{p} + j_{f}}{2} \int_{0}^{L} [({\dot{φ}}^{2} + {\dot{θ}}^{2}) + 2 Ω^{2} (θ^{2} + φ^{2}) - \\ 2 Ω (φ \dot{θ} + θ \dot{φ})] d x + 2 L j Ω^{2} + \frac{M}{2} \int_{0}^{L} (2 \dot{v} v^{'} U - 2 w Ω U v^{'} + \\ U^{2} v^{' 2} + 2 U w^{'} \dot{w} + 2 v Ω U w^{'} + U^{2} w^{' 2}) d x \end{matrix}

(3)

其中M，m，A_f和A_p分别表示流体单位长度质量，管道单位长度质量，流体截面积和管道截面积.下文公式中，上标点和撇分别表示对变量t和x的偏导数，下标“p”和“f”分别表示管道和流体.转动惯量j定义如下:

j = \int_{A} ρ y^{2} d x = \int_{A} ρ z^{2} d x = ρ I

(4)

其中ρ=ρ_p+ρ_f为密度，I为惯性矩.
输流管受到的总势能表示为:

V = V_{r} + V_{t} + V_{p} + V_{g} .

(5)

V_r是管道回复力引起的弹性应变能:

\begin{matrix} V_{r} = \frac{1}{2} \int_{0}^{L} \{E I [{(θ^{'})}^{2} + {(φ^{'})}^{2}] + \\ k A_{p} G [{(v^{'} - θ)}^{2} + {(w^{'} - φ)}^{2}]\} d x \end{matrix}

(6)

其中A_p和G分别表示管道截面积和剪切模量.k是Timoshenko梁的截面形状因子，

k = \frac{6 (1 + ν) {[1 + (d / D)^{2}]}^{2}}{(7 + 6 ν) {[1 + (d / D)^{2}]}^{2} + (20 + 12 ν) (d / D)^{2}}

(7)

其中ν为泊松比.D和d分别为管道直径和管壁厚度.V_t是初始轴应力引起的弹性应变能:

\begin{matrix} V_{t} = \frac{F}{2} \int_{0}^{L} \{A_{p} [{(v^{'})}^{2} + {(w^{'})}^{2}] + \\ I_{p} [{(θ^{'})}^{2} + {(φ^{'})}^{2}]\} d x \end{matrix}

(8)

其中F表示初始轴应力.V_p是流体压力引起的弹性应变能:

\begin{matrix} V_{p} = - \frac{P}{2} (1 - 2 ν δ) \int_{0}^{L} \{A_{f} [{(v^{'})}^{2} + {(w^{'})}^{2}] + \\ \frac{A_{f}}{A_{p}} I_{p} [{(θ^{'})}^{2} + {(φ^{'})}^{2}]\} d x \end{matrix}

(9)

其中P表示流体压力，管道两端夹紧时δ=1，不受约束δ=0.V_g是流体和管道单元重力引起的弹性应变能:

\begin{matrix} V_{g} = \frac{(m + M) g}{2} \int_{0}^{L} \{(L - x) [(2 v^{'} θ - θ^{2}) + \\ (2 w^{'} φ - φ^{2})]\} d x \end{matrix}

(10)

其中g表示流体重力.
根据Hamilton原理，将方程（3）~（10）代入公式 $\int_{t 1}^{t 2} δ （ T - V ） d t = 0$ ，可得输流管运动控制方程为:

\begin{matrix} (m + M) \ddot{v} - (m + M) Ω^{2} v - 2 Ω \dot{w} (m + M) - \\ 2 M U Ω w^{'} + 2 M U {\dot{v}}^{'} - k A_{p} G (v^{''} - θ^{'}) + \\ M U^{2} v^{''} - F A_{p} v^{''} + P A_{f} (1 - 2 ν δ) v^{''} - \\ (m + M) (L - x) g θ^{'} + (m + M) g θ = 0, \\ (m + M) \ddot{w} - (m + M) Ω^{2} w + 2 Ω \dot{v} (m + M) + \\ 2 M U Ω v^{'} + 2 M U {\dot{r}}^{'} - k A_{p} G (w^{''} - φ^{'}) + \\ M U^{2} w^{''} - F A_{p} w^{''} + P A_{f} (1 - 2 ν δ) w^{''} - \\ (m + M) (L - x) g φ^{'} + (m + M) g θ^{'} = 0, \\ (j_{p} + j_{f}) \ddot{θ} - 2 (j_{p} + j_{f}) Ω^{2} θ - E I θ^{''} - \\ k A_{p} G (v^{'} - θ) + P I_{p} \frac{A_{f}}{A_{p}} (1 - 2 ν δ) θ^{''} F I_{p} θ^{''} = 0, \\ (j_{p} + j_{f}) \ddot{φ} - 2 (j_{p} + j_{f}) Ω^{2} φ - E I φ^{''} - + \\ k A_{p} G (w^{'} - φ) P I_{p} \frac{A_{f}}{A_{p}} (1 - 2 ν δ) φ^{''} - F I_{p} φ^{''} = 0 . \end{matrix}

(11)

引入如下无量纲参数:
$\begin{matrix} \bar{v} = \frac{v}{L}, \bar{w} = \frac{w}{L}, \bar{x} = \frac{x}{L}, \bar{θ} = θ, \bar{φ} = φ, \\ Ω = L^{2} Ω \sqrt{\frac{m + M}{E I}}, \bar{g} = \frac{L^{3} (m + M)}{E I} g, \end{matrix}$

\begin{matrix} M_{r} = \sqrt{\frac{M}{m + M}}, \bar{U} = U L \sqrt{\frac{M}{E I}}, \\ k_{1} = \frac{k A_{p} G L^{2}}{E I}, k_{2} = \frac{j_{p} + j_{f}}{L^{2} (m + M)}, \\ k_{3} = \frac{L^{2} [P A_{f} (1 - 2 ν δ) - F A_{p}]}{E I}, \\ k_{4} = \frac{[P A_{f} (1 - 2 ν δ) - F]}{E A_{p}} . \end{matrix}

(12)

其中无量纲参数k₁、k₂、k₃、k₄、g分别表示剪切变形系数、转动惯量系数、压力和应力对管道刚度影响系数、压力和应力对管道转角影响系数、重力系数.
将式（12）代入方程（11）中，得到旋转Timoshenko管道无量纲运动方程.为了方便书写，运动方程中无量纲参数省略了上横线.

\begin{matrix} \ddot{v} - Ω^{2} v - 2 Ω \dot{w} - 2 M_{r} U Ω w^{'} + 2 M_{r} {\dot{u}}^{'} + g θ - \\ (U^{2} - k_{1} + k_{3}) v^{''} + (k_{1} - g + g x) θ^{'} = 0, \\ \ddot{w} - Ω^{2} w + 2 Ω \dot{v} + 2 M_{r} U Ω v^{'} + 2 M_{r} {\dot{w}}^{'} + g φ - \\ (U^{2} - k_{1} + k_{3}) w^{''} + (k_{1} - g + g x) φ^{'} = 0, \\ k_{2} \ddot{θ} - 2 k_{2} Ω^{2} θ - (1 - k_{4}) θ^{''} - k_{1} (v^{'} - θ) = 0, \\ k_{2} \ddot{φ} - 2 k_{2} Ω^{2} φ - (1 - k_{4}) φ^{''} - k_{1} (w^{'} - φ) = 0 \end{matrix}

(13)

不考虑方程（13）后两式与前两式中的转角，方程即为旋转Euler-Bernoulli梁模型输流管道; 不考虑转速Ω，方程即退化为两个方程，即为Timoshenko梁模型输流管道.方程中前两式的第二、三、四项和后两式中的第二项为输流管旋转引起的附加科氏力和离心力（第二项）.
旋转Timoshenko输流管两端简支的边界条件为:

\begin{matrix} v (0, t) = v (1, t) = 0, w (0, t) = w (1, t) = 0 \\ θ^{'} (0, t) = θ^{'} (1, t) = 0, φ^{'} (0, t) = φ^{'} (1, t) = 0 \end{matrix}

(14)

2 离散求解
根据Galerkin截断法将得到的偏微分运动方程化为常微分方程，设:

\begin{matrix} v (x, t) = \sum_{n = 1}^{N} ϕ_{n} (x) q_{v_{n}} (t), \\ w (x, t) = \sum_{n = 1}^{N} ϕ_{n} (x) q_{w_{n}} (t), \\ θ (x, t) = \sum_{n = 1}^{N} ψ_{n} (x) q_{θ_{n}} (t), \\ φ (x, t) = \sum_{n = 1}^{N} ψ_{n} (x) q_{φ_{n}} (t) \end{matrix}

(15)

式中 $q_{v_{n}} ， q_{w_{n}} ， q_{θ_{n}} ， q_{φ_{n}}$ 分别为两个横向方向上的位移和转角，N为所取的模态函数个数， $ϕ_{n} （ x ）$ 和 $φ_{n} （ x ）$ 为满足边界条件的模态函数为:

ϕ_{n} (x) = \sqrt{2} s i n (n π x), φ_{n} (x) = \sqrt{2} c o s (n π x)

(16)

将式（15）代入式（13）进行截断，得到的常微分方程如下:

M \ddot{q} + G \dot{q} + K = 0

(17)

式中M，G，K分别表示质量、陀螺和刚度矩阵.
设解的形式为

\{\begin{matrix} q_{v_{n}} = C_{n} e^{i ω t} \\ q_{w_{n}} = C_{2 n} e^{i ω t} \\ q_{θ_{n}} = C_{3 n} e^{i ω t} \\ q_{φ_{n}} = C_{4 n} e^{i ω t} \end{matrix}\} n = 1,2, \dots, N

(18)

将式（18）代入式（17）可得到:

(- ω^{2} M + i ω G + K) C = 0

(19)

由于C为常数矩阵，式（19）存在的条件为

d e t (- ω^{2} M + i ω G + K) = 0

(20)

将参数数值代入 $|- ω^{2} M + i ω G + K| = 0$ ，即得到不同参数对应振动频率的复频率，并可分析对应的振动模态.
3 参数性能分析
本文用数值算例对旋转Timoshenko输流管道的参数进行性能分析，输流管管长L=0.25m，管径D=0.02m，管厚d=0.001m，管道密度ρ_p=920kg/m³，流体密度ρ_f=1000 kg/ m³，杨氏模量E=900 MPa，初始应力F=100 Pa·m²，流体压力P=100 kPa.采用复模态分析法，观察频率随流速的变化情况.复频率的实部反映了Timoshenko管道振动的频率变化，虚部反映能量的变化.为了方便直观地研究复频率，本文在以下分析中只对管道的一阶固有频率进行分析.图1~图5中分别有两个图，第一个表示的是复频率的实部，第二个表示的是复频率的虚部.
图1 模态个数对复频率的影响
Fig.1 Influence of the number of modes on complex frequency
图2 剪切系数对复频率的影响
Fig.2 Influence of shear coefficient on complex frequency
当质量比M_r=0.8，剪切系数k₁=250时，图1描述了Timoshenko输流管道振动方程取不同试函数模态个数N对管道一阶复频率的影响.从图中可以看出，在二阶临界流速U=5.8之前，振动频率幅值呈现逐渐减小到零后稳定的趋势，此时试函数模态个数变化对频率的影响差别比较小，二阶Galerkin离散即可满足方程的求解.当流速较大时（U＞5.8），实部频率的幅值和虚部振动能量随着模态个数的增加变化明显.对于管内流体流速高的输流管，必须采用更高阶模态个数才能满足系统求解的正确性.从文中可看出当N＞4以后频率和能量几乎稳定不变，因此本文研究旋转Timoshenko输流管道时取N=4.
图3 质量比对复频率的影响
Fig.3 Influence of mass ratio on complex frequency
图2描述了质量比M_r=0.8时Timoshenko管道模型中剪切系数k₁对管道一阶复频率的影响.从频率和能量图中可以看出，当流速较小时剪切系数改变对自由振动的影响比较小.当流速较大时（U＞5），随着剪切系数的增加，Timoshenko管道的二阶临界流速、三阶临界流速会增大，同时管道模态耦合颤振的频率值也会提升.同时图2给出了相同管道参数下的Euler-Bernoulli管道模型的计算结果，当剪切系数为4000时两种模型计算出来的结果符合性较好，这表明了本方法的正确性.剪切系数与管道长度成正比，在管道比较长时两种模型计算结果相差不大，管道比较短时剪切力的影响较大，不能忽略，这时输流管就要考虑Timoshenko梁管道模型.
图4 转速对复频率的影响
Fig.4 Influence of rotational speed on complex frequency
随着无量纲流速的变化，质量比对Timoshenko输流管道一阶复频率的影响如图3所示.图3显示在低流速时，频率变化不大，而在高流速时，频率会发生不规则的变化过程.从图中可看到管道会经历“稳定→一阶屈曲→模态耦合颤振→二阶屈曲”的过程，质量比在流速超过第二临界流速时对稳定性的影响显著.因此研究高流速流固耦合颤振时则必须考虑质量比变化的影响.
图5 流速对复频率的影响
Fig.5 Influence of flow velocity on complex frequency
为使研究更具对比性，图4~图6质量比和剪切系数分别取M_r=0.8和k₁=250.图4描述了无量纲流速转速的变化对Timoshenko输流管道一阶复频率的影响.结合频率图和能量图可以看出管道会经历一个“稳定→一阶屈曲→模态耦合颤振→二阶屈曲→稳定的过程”.在转速Ω=0时，管道在每个阶段两个横向方向的一阶固有频率都完全重合一起颤振.考虑转速作用后，管道在两个稳定阶段和模态耦合颤振阶段两个方向以不同频率各自颤振.随着转速的增大，系统频率差值变大，振动能量完全重合没有演变.结果表明转速变化只改变管道的频率，对系统的能量变化几乎没有影响.
随着无量纲转速的变化，流速对Timoshenko输流管道一阶复频率的影响如图5所示.从图中可以看出，一阶频率随转速是线性变化的.U=4时两个横向方向颤振频率重合，随着Ω的增加频率值逐渐升高.U=6和U=8时两个横向振动方向以不同频率耦合颤振，随着Ω的增加一个方向振动频率逐渐升高，另一个方向振动频率先降低至零后再上升.这种振动现象与图3中U=4，6，8的现象也是相符的.
随着无量纲流速的变化，图6分别研究了转动惯量、重力、流动压力和初始轴应力对Timoshenko输流管道一阶复频率实部的影响.从图（a）可以看出，转动惯量对一阶频率的影响较小.从图中还可以看到，管道会经历“稳定→一阶屈曲→模态耦合颤振→二阶屈曲→稳定”的过程，频率在稳定和模态耦合颤振阶段对转动惯量变化比较敏感，振动能量在二阶屈曲时对转动惯量变化较敏感.从图（b）可以看出，管道振动频率和能量对重力参数g不敏感，但重力会对系统临界流速产生影响.图（c）和图（d）分别取转速Ω=10和Ω=0时，管道在稳定阶段振动频率幅值随着流体压力和初始轴应力的增加呈现微弱下降趋势，在模态耦合颤振阶段振动频率幅值逐渐升高.压力和初始应力对Timoshenko输流管道的稳定性有一定的影响，在流体压力变化较大时需要对此进行分析以提高计算准确度.
图6 复频率的实部
Fig.6 The real part of the complex frequency
4 总结
本文基于Hamilton原理和欧拉角转换，对旋转Timoshenko输流管道的自由振动进行了分析，研究了多种参数对系统复频率的影响.研究发现在高流速下取4阶Galerkin离散结果则更加精确，并且验证了本模型的正确性.在流速较高时，研究发现Timoshenko输流管道振动时会经历“稳定→一阶屈曲→模态耦合颤振→二阶屈曲→稳定”的过程，与Euler-Bernoulli模型相比多了一个稳定阶段，说明了在短粗型输流管道时采用Timoshenko输流管道分析的必要性.同时研究得到剪切系数、质量比对稳定性影响很大，而转动惯量、重力、流动压力、初始轴应力对稳定性也有一定的影响，转速只影响振动频率的幅值，不影响其稳定性，考虑多种参数会使计算结果精度更高.
参考文献
- [1] 刘燕,张伟,龚涛涛.轴向可伸缩复合材料悬臂梁的非线性振动研究 [J].动力学与控制学报,2020,18(4)∶ 19-25.LIU Y,ZHANG W,GONG T T.Nonlinear vibration of an axial telescopic composite cantilever [J].Journal of Dynamics and Control,2020,18(4)∶ 19-25.(in Chiniese)
- [2] DAGLI B Y,ERGUT A.Dynamics of fluid conveying pipes using Rayleigh theory under nonclassical boundary conditions [J].European Journ al of Mechanics/B Fluids,2019,77∶125-134.
- [3] LIANG F,YANG X D,QIAN Y J,et al.Transverse free vibration and stability analysis of spinning pipes conveying fluid [J].International Journal of Mechanical Sciences,2018,137∶195-204.
- [4] LI X Y,ZHAO X,LI Y H.Green's functions of the forced vibration of Timoshenko beams with damping effect [J].Journal of Sound and Vibration,2014,333(6)1781-1795.
- [5] 周远,唐有绮,刘星光.黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究[J].力学学报,2019,51(6)∶ 1897-1904.ZHOU Y,TANG Y Q,LIU X G.Research on vibration characteristics of axial moving Timoshenko beam under viscoelastic damping [J].Chinese Journal of Theoretical and Applied Mechanics,2019,51(6)∶ 1897-1904.(in Chiniese)
- [6] TAN X,DING H,CHEN L Q.Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model [J].Journal of Sound and Vibration,2019,455∶241-255.
- [7] FEYZOLLAHZADEH M,BAMDAD M.Vibration analysis of rotating beam with variable cross section using Riccati transfer matrix method [J].Structural Engineering and Mechanics,2019,70(2)∶199-207.
- [8] GULYAYEV V I,GAIDAICHUK V V,SOLOVJOV I L,et al.The buckling of elongated rotating drill strings [J].Journal of Petroleum Science and Engineering,2009,67(3-4)∶140-148.
- [9] HUO Y L,WANG Z M.Dynamic analysis of a rotating double-tapered cantilever Timoshenko beam [J].Archive of Applied Mechanics,2016,86(6)∶1147-1161.
- [10] YANG X D,LI Z,ZHANG W,et al.On the gyroscopic and centrifugal effects in the free vibration of rotating beams [J].Journal of Vibration and Control,2019,25(1)∶219-227.
- [11] ABDOLLAHI R,DEHGHANI F R,RAHMANIAN M.On the stability of rotating pipes conveying fluid in annular liquid medium [J].Journal of Sound and Vibration,2021,494∶115891.
- [12] 张永旺,杨晓东,梁峰,等.Rayleigh旋转梁的动力学建模与动态稳定性分析 [J].应用力学学报,2018,35(2)∶ 248-253,447.ZHANG Y W,YANG X D,LIANG F,et al.Dynamic modeling and dynamic stability analysis of a Rayleigh rotating beam [J].Chinese Journal of Applied Mechanics,2018,35(2)∶ 248-253,447.(in Chiniese)
- [13] WIGGERT D C,HATFIELD F J,STUCKEN B S.Analysis of liquid and structural transients in piping by the method of characteristics [J].Journal of Fluids Engineering,1987,109(2)∶161-165.
- [14] MEHDI R,HAMID R M,SAREH G.Divergence instability of pipes conveying fluid with uncertain flow velocity [J].Physica A∶ Statistical Mechanics and its Applications,2018,491∶650-665.
- [15] YAMASHITA K,NISHIYAMA N,KATSURA K,et al.Hopf-Hopf interactions in a spring supported pipe conveying fluid [J].Mechanical Systems and Signal Processing,2021,152∶107390.
- [16] LU Z Q,ZHANG K K,DING H,et al.Internal resonance and stress distribution of pipes conveying fluid in supercritical regime [J].International Journal of Mechanical Sciences,2020,186∶105900.

基本信息

中图分类号: O321
文献标识码: A
DOI: 10.6052/1672-6553-2021-072
文章编号: 1672-6553-2023-21(2)-058-008

基金信息

北京信息科技大学校科研基金资助项目(2021XJJ65)；
国家自然科学基金资助项目(11772063, 11972050)；
Supported by the Beijing Information Science and Technology University (2021XJJ65) ；
National Natural Science Foundation of China (11772063, 11972050)；

引用信息

稿件历史

收稿日期: 2021-08-15

参考文献

[1] 刘燕,张伟,龚涛涛.轴向可伸缩复合材料悬臂梁的非线性振动研究 [J].动力学与控制学报,2020,18(4)∶ 19-25.LIU Y,ZHANG W,GONG T T.Nonlinear vibration of an axial telescopic composite cantilever [J].Journal of Dynamics and Control,2020,18(4)∶ 19-25.(in Chiniese)
[2] DAGLI B Y,ERGUT A.Dynamics of fluid conveying pipes using Rayleigh theory under nonclassical boundary conditions [J].European Journ al of Mechanics/B Fluids,2019,77∶125-134.
[3] LIANG F,YANG X D,QIAN Y J,et al.Transverse free vibration and stability analysis of spinning pipes conveying fluid [J].International Journal of Mechanical Sciences,2018,137∶195-204.
[4] LI X Y,ZHAO X,LI Y H.Green's functions of the forced vibration of Timoshenko beams with damping effect [J].Journal of Sound and Vibration,2014,333(6)1781-1795.
[5] 周远,唐有绮,刘星光.黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究[J].力学学报,2019,51(6)∶ 1897-1904.ZHOU Y,TANG Y Q,LIU X G.Research on vibration characteristics of axial moving Timoshenko beam under viscoelastic damping [J].Chinese Journal of Theoretical and Applied Mechanics,2019,51(6)∶ 1897-1904.(in Chiniese)
[6] TAN X,DING H,CHEN L Q.Nonlinear frequencies and forced responses of pipes conveying fluid via a coupled Timoshenko model [J].Journal of Sound and Vibration,2019,455∶241-255.
[7] FEYZOLLAHZADEH M,BAMDAD M.Vibration analysis of rotating beam with variable cross section using Riccati transfer matrix method [J].Structural Engineering and Mechanics,2019,70(2)∶199-207.
[8] GULYAYEV V I,GAIDAICHUK V V,SOLOVJOV I L,et al.The buckling of elongated rotating drill strings [J].Journal of Petroleum Science and Engineering,2009,67(3-4)∶140-148.
[9] HUO Y L,WANG Z M.Dynamic analysis of a rotating double-tapered cantilever Timoshenko beam [J].Archive of Applied Mechanics,2016,86(6)∶1147-1161.
[10] YANG X D,LI Z,ZHANG W,et al.On the gyroscopic and centrifugal effects in the free vibration of rotating beams [J].Journal of Vibration and Control,2019,25(1)∶219-227.
[11] ABDOLLAHI R,DEHGHANI F R,RAHMANIAN M.On the stability of rotating pipes conveying fluid in annular liquid medium [J].Journal of Sound and Vibration,2021,494∶115891.
[12] 张永旺,杨晓东,梁峰,等.Rayleigh旋转梁的动力学建模与动态稳定性分析 [J].应用力学学报,2018,35(2)∶ 248-253,447.ZHANG Y W,YANG X D,LIANG F,et al.Dynamic modeling and dynamic stability analysis of a Rayleigh rotating beam [J].Chinese Journal of Applied Mechanics,2018,35(2)∶ 248-253,447.(in Chiniese)
[13] WIGGERT D C,HATFIELD F J,STUCKEN B S.Analysis of liquid and structural transients in piping by the method of characteristics [J].Journal of Fluids Engineering,1987,109(2)∶161-165.
[14] MEHDI R,HAMID R M,SAREH G.Divergence instability of pipes conveying fluid with uncertain flow velocity [J].Physica A∶ Statistical Mechanics and its Applications,2018,491∶650-665.
[15] YAMASHITA K,NISHIYAMA N,KATSURA K,et al.Hopf-Hopf interactions in a spring supported pipe conveying fluid [J].Mechanical Systems and Signal Processing,2021,152∶107390.
[16] LU Z Q,ZHANG K K,DING H,et al.Internal resonance and stress distribution of pipes conveying fluid in supercritical regime [J].International Journal of Mechanical Sciences,2020,186∶105900.

旋转Timoshenko输流管道的固有频率和稳定性分析^*

Natural Frequency and Stability Analysis of Rotating Timoshenko Pipe Conveying Fluid^*

摘要

Abstract

关键词

Keywords

1 振动方程推导

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

2 离散求解

(15)

(16)

(17)

(18)

(19)

(20)

3 参数性能分析

4 总结

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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旋转Timoshenko输流管道的固有频率和稳定性分析*

Natural Frequency and Stability Analysis of Rotating Timoshenko Pipe Conveying Fluid*

摘要

Abstract

关键词

Keywords

1 振动方程推导

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

2 离散求解

(15)

(16)

(17)

(18)

(19)

(20)

3 参数性能分析

4 总结

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

旋转Timoshenko输流管道的固有频率和稳定性分析^*

Natural Frequency and Stability Analysis of Rotating Timoshenko Pipe Conveying Fluid^*