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热环境中超临界黏弹性输流管道自由振动分析*

  • 高思禹

    机 构：

    上海大学 力学与工程科学学院,上海 200444

    ×
  • ，毛晓晔

    机 构：

    上海大学 力学与工程科学学院,上海 200444

    ×
  • ，丁虎

    机 构：

    上海大学 力学与工程科学学院,上海 200444

    ×
  • ，陈立群

    机 构：

    上海大学 力学与工程科学学院,上海 200444

    ×

上海大学 力学与工程科学学院,上海 200444；

Free Vibration Analysis on Supercritical Viscoelastic Pipes Conveying Fluid in Thermal Environment*

  • Gao Siyu

    Affiliation：

    School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444 ,China

    ×
  • ，Mao Xiaoye

    Affiliation：

    School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444 ,China

    ×
  • ，Ding Hu

    Affiliation：

    School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444 ,China

    ×
  • ，Chen Liqun

    Affiliation：

    School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444 ,China

    ×

School of Mechanics and Engineering Science,Shanghai University,Shanghai 200444 ,China；

通讯作者:

毛晓晔,E-mail:xmao3@shu.edu.cn

中图分类号:O322

文献标识码:A

文章编号:1672-6553-2023-21(6)-039-008

DOI:10.6052/1672-6553-2023-078

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目录contents

    摘要

    基于Euler-Bernoulli梁模型,本文研究了热环境中输流管道在超临界范围内流固耦合自由振动特性.考虑温度增量以及初始轴向拉力作用,在两端简支边界条件下,利用广义Hamilton原理建立输流管道横向振动偏微分-积分控制方程.通过解析方法得到输流管道非平凡静平衡位形及临界流速精确表达式,与微分求积单元法(DQEM)数值结果吻合较好.基于复模态法,结合伽辽金(Galerkin)法离散系统偏微分-积分控制方程,得到热环境下超临界输流管道的模态函数和固有频率.结果表明,温度增量越大,临界流速越小,此时的管道越容易屈曲,但相同流速下超临界管道固有频率越大；初始拉力越大,临界流速越大,相同流速下超临界固有频率越小.该研究可以为热环境中超临界状态下的管道系统振动设计提供理论指导.

    Abstract

    Based on the Euler-Bernoulli beam theory, the current work investigates the free vibration of a supercritical pipe conveying fluid in the thermal environment. Considering the temperature increment and the initial axial force, the partial-differential-integral governing equation for the transverse vibration of the simply supported pipe is established by the generalized Hamilton’s principle. The non-trivial equilibrium configurations and corresponding critical flow velocities are obtained analytically, and verified by the differential quadrature element method (DQEM). Based on the complex mode method and the Galerkin method, the natural frequencies of the supercritical pipe conveying fluid in the thermal environment are obtained. The results show that the larger the temperature increment is, the smaller the critical flow velocity will be, which also makes the pipe buckled more easily. But the supercritical natural frequencies become bigger at the same fluid velocity. The discussion also finds that the larger the initial tension is, the larger the critical flow velocity will be. Under this condition, the natural frequencies in the super critical region become smaller at the same flow velocity. This study provides guidance for the vibration design of pipes conveying fluid in the supercritical region under the thermal environment.

  • 引言

  • 输流管道流致振动的研究有广泛的应用前景，其研究成果可直接运用于海洋和土木工程、石油和化学加工工业、核电和电力工业、船舶和飞机，以及日常生活中.管道内部流固耦合作用和外部激励引起的振动，会不断地传递到整个系统，引发系统大幅振动[1]，从而导致管道出现疲劳损伤，甚至在极端振动条件下会因管道结构破裂，导致液体泄漏而产生不可估量的后果[2].当管道中流体流速超过一定值时，管道将发生屈曲失稳，并在新的平衡位形附近振动，这被称为输流管道系统的超临界状态[3]，其特征为管道基频归零.环境温度会显著改变系统共振特性，因此，将环境温度考虑进输流管道的研究是具有现实意义的；环境温度条件下的输流管道振动特性分析也是对已有理论的补充和发展，具有一定学术价值.

  • 目前已有的研究主要聚焦于输流管道的振动特征，这些研究包含输流管道振动特性对管道寿命影响、流固耦合作用、建立精确的输流管道动力学模型及管道大幅度振动时的非线性动力学等[4].杨晓东和金基铎发现因为假设的模态函数会使Galekin截断法得到的固有频率产生误差，于是通过复模态法与其比较，结果表明截断阶数较低时，两种方法得到的固有频率拟合较好[5].Ding等针对非线性连续系统，建立了与两个非线性隔振器耦合的输流管道非线性受迫振动动力学模型[6].周永兆等人研究了两端铰支输流管道的横向非线性自由振动，并且对偏微分方程做准静态假设得到偏微分-积分方程，研究表明偏微分控制方程有较强的非线性[7].颜雄和魏莎等研究了两端非对称弹性支承输流管道横向振动的固有特性，发现流体的流速越大，其对两端非对称弹性支承的管道固有频率的影响越明显[8].

  • 当流速超过临界值时，管道会发生静态屈曲，即管道发生屈曲失稳，系统会重新稳定在曲线平衡位形[9].Ding等分别使用伽辽金截断法计算了轴向运动梁的固有频率且通过数值方法计算了轴向运动梁在超临界运动状态下的平衡位移[10-13].随岁寒和李成利用虚功原理建立了输流管道弯曲和振动的有限元方程，给出了两种边界条件下管道自由振动的前三阶固有频率与流体流速的关系，结果表明，前三阶固有频率随流速增大而减小[14].Zhou等人研究了四种不同初始形状的悬臂式输送流体管道的非线性静力和动力特性.结果证明了管道由流动引起的管道静态变形也可能非常大，管道颤振失稳的临界流速主要取决于刚刚达到临界流速时的静力平衡构型[15].缪旭和金基铎针对超临界状态下的输流管道的扰动方程，通过伽辽金法得到了超临界状态下系统非线性振动第一阶固有频率的解析表达式[16].Mao等研究了输流管道内共振为3∶1时的受迫振动响应，研究发现当流体的速度超过临界值时，直线平衡构型会发生弯曲，原来的单稳态系统转化为双稳态系统[17].Tan等人首次用Timoshenko梁理论研究了超临界范围内输流管道的振动特性，并且与Euler-Bernoulli管道相比，Timoshenko管道的平衡构型曲率更大，两种管道模型之间的相对差异受流速的影响[18].Lu等人研究了输流管道在超临界绕流曲线处的内部共振和应力分布，以改善管道的机械疲劳性能[19].Ye等研究了超临界范围内微弯曲输送流体管道的振动特性，研究发现弯曲管道的固有频率可能会随着管道长度的增加而增加，但不一定是单调的[20].

  • 尽管近年来超临界高速输流管道的振动问题一直受到广泛关注，但目前的研究工作主要集中在理想环境的基础上.在实际工程应用中，环境温度的改变也会诱发管道发生振动，降低管道寿命.根据文献，Qian等人通过数值方法分析了温度和流速对输流管道的固有特性的影响.结果表明，动力学特性由温度和流速共同决定，温度升高也会导致管道刚度的降低，即使内部没有流体也会导致管道发散不稳定[21].Ebrahimi-Mamaghani等人研究了考虑热效应的轴向瑞利管的稳定性，发现温度越高时管道越容易以发散和颤振的形式失去稳定性[22].Babaei分析了热环境中功能梯度微曲管道在均匀侧压力作用下的非线性响应，结果发现温度升高提高了微曲管道的强度，同时也提高了微曲管道的上限载荷和下限载荷[23].但是以上工作没有涉及对热环境中输流管道超临界振动的研究.

  • 基于上述文献调研，本文建立了具有线性热弹性关系的Euler-Bernoulli输流管道在超临界状态下的运动控制方程；对管道热环境下随流速分岔的位形进行了数值验证；利用复模态法和伽辽金法得到管道的模态函数并分析温度增量和初始轴力对管道超临界固有频率的影响.

  • 1 运动方程及非平凡静平衡位形

  • 图1为两端简支输流管道的物理模型简图.管道材料密度为ρp，长度为L，外径为D，内径为d，P0为初始轴力，弹性模量E，截面绕中性轴转动惯量Ip.考虑Kelvin黏弹性本构关系，黏弹性系数为α.管道内流体假设为不可压的黏性流体，密度为ρf，流速Г恒定.管道均布F·cos（Ωt）的简谐横向激励力，只考虑激励平面内的横向位移v（x，t），其中t代表时间坐标，x表示沿管道轴向的空间坐标.此外，考虑系统处于均匀温度场中，以室温为基础，温度增量为ΔT.

  • 图1 输流管道物理模型

  • Fig.1 Physical model of the pipe conveying fluid

  • 根据能量法建立运动方程，Euler-Bernoulli输流管道横向运动的动能Te为

  • Te=12∫0L ρpApv,t2dx+12∫0L ρfAfv,t+Γsin⁡v,x2dx+12∫0L ρfAfΓ2cos⁡v,x2dx
    (1)

  • 式中第一项表示输流管道横向振动的动能，第二项及第三项表示流体的动能.

  • 系统内力及外力做功W为

  • W=-∭V σx-σTεxdV+Up+Uf
    (2)

  • 式中V为输液管道的体积，σx（x，t）和εx（x，t）分别为正应力和正应变，σT表示热应力，Up表示轴力P0做功，Uf表示激励力做功，相应的表达式

  • Up=∫0L P0vxvdxUf=∫0L Fcos⁡(Ωt)vdx
    (3)

  • 式中管道材料应力和应变本构关系可表示为

  • σx=Eεx+αεx,t
    (4)

  • 当温度场发生变化时，管道会产生应变，沿x轴的热应变可表示为[24]

  • εT=ζΔT
    (5)

  • 其中，ζ是热膨胀系数.因此，热应力可以表示为

  • σT=EεT
    (6)

  • 假设由振动变形引起的附加应力沿管道的轴向变化很小，可以将应力取作沿输流管道在长度上的平均值

  • εx(t)=12L∫0L v,x2dx
    (7)

  • 根据广义Hamilton原理

  • δ∫t1t2 Tedt+∫t1t2 δWdt=0
    (8)

  • 从而得到热环境中黏弹性横向振动输流管道的偏微分-积分模型

  • ρpApv,tt+ρfAfv,tt+2Γvxt+Γ2v,xx-P0-EApζΔTv,xx+EIpv,xxxx+αIpv,xxxxt=EAp2Lv,xx∫0L v,x2dx+ApαLv,xx∫0L v,xv,xtdx+Apα2Lv,xxt∫0L v,x2dx+Fcos⁡(Ωt)
    (9)

  • 以及两端简支条件

  • v(0,t)=v,xx(0,t)=0,v(L,t)=v,xx(L,t)=0
    (10)

  • 表1给出了本文研究的输流管道的物理参数值及几何参数值.

  • 表1 输流管道物理参数

  • Table1 Physical parameters of the pipe conveying fluid

  • 2 热环境中输流管道非平凡静平衡位形

  • 2.1 非平凡位形解析解

  • 为求解输流管道静态屈曲构型，忽略式（9）中显含时间的项，得到只与空间坐标有关的位移函数

  • ρfAfΓ2-P0+EApζΔTv^,xx+EIpv^,xxxx=EAp2Lv^,xx∫0L v^,x2dx
    (11)

  • 式（11）中，v^即表示非平凡静平衡位形.为求得解析解，先将定积分项假设为常数ξ，该方程可以化简为

  • ρfAfΓ2-P0+EApζΔTv^,xx+EIpv^,xxxx-ξEAp2Lv^,xx=0
    (12)

  • 按照求导阶数，方程（12）可以改写为

  • v^,xxxx+λ2v^,xx=0
    (13)

  • 其中λ2=ρfAfΓ2-P0+EApζΔT-ξEAp2L/EIp，方程（13）是一个四阶常微分方程，通解为

  • v^=C1+C2x+C3cos⁡(λx)+C4sin⁡(λx)
    (14)

  • 其中Ci （i=1，2，3，4）是常数.

  • 将边界条件（10）代入常微分方程（13）得到特征方程sinλ=0或λ=kπ，k=1，2，3···，即可得到输流管的非平凡静平衡位形[25]

  • v^=Csin⁡(kπx)
    (15)

  • 将表达式（15）代入式（14）可得到屈曲位形系数

  • C=±2kπLρfAfΓ2-P0+EApζΔT-k2π2EIpEAp
    (16)

  • 从式（16）可以导出，只有当流速Γ>Γcr⁡（k）=k2π2EIp+P0-EApζΔT/ρfAf时，产生一对对称平衡曲线位形，Γcr（k）表示第k阶临界流速.由于超临界流速具有复杂多变性，输流管道分析难度加大，因此，重点关注超临界流速范围在Γcr（1）<Γ<Γcr（2）下，研究热环境中简支输流管道围绕正平衡位形v^+（x）的横向振动的非线性动力学特性.

  • 2.2 非平凡位形仿真解

  • 微分单元求积法（DQEM）在微分求积法（DQM）的基础上，将管道两端的旋转角度作为广义变量引入.因此，它有N个节点基础上的（N+2）个变量.基于（N-2）个内点方程和4个边界条件，这（N+2）个变量可以被求解出来.因此，DQEM可以准确地满足四阶边界值问题.由于篇幅所限，DQEM对偏微分方程插值离散公式见[26].

  • 已有文献表明当节点采用非均匀分布时，计算结果精度及收敛性更高[27]，因此采用Chebyshev-Gauss-Lobatto非均匀分布，即

  • xi=121-cos⁡(i-1)πN-1L,i=1,2,⋯,N
    (17)

  • 本文选择11个节点（N=11）进行模拟，以获得更准确的结果.

  • 图2 热环境中输流管道非平凡位形

  • Fig.2 Non-trivial equilibrium configuration of the pipe in thermal environment

  • 图2所示为热环境中输流管道的非平凡静平衡位形数值解和解析解的对比.其中图2（a）表示DQEM中（N-2）个内部节点的自由衰减情况.当它们趋于稳定时，输流管道静止于势能最小位置，该位置即屈曲平衡位形.表格表示各节点对应的坐标值.图2（b）点表示微分求积单元法计算得到的数值解，实线表示理论推导得到的解析解.从图中可以看出解析结果和数值结果吻合得很好，从而证明了非平凡静平衡位形解析解的正确性.图3为不同温度增量下非平凡静平衡位形图，结果表明温度增量越大，非平凡静平衡位形越大，管道越容易屈曲.

  • 图3 不同温度增量下的非平凡静平衡位形

  • Fig.3 Non-trivial equilibrium configurations with different temperature increments

  • 3 热环境中超临界管道固有频率

  • 将管道的非平凡静平衡位形代入式（11），v（x，t）→v（x，t）+v^（x），得到超临界输流管道非平凡平衡位形基础上的振动控制方程

  • ρpAp+ρfAfv,t+2ρfAfΓv,xt+ρfAfΓ2v,xx-P0+EApζΔTv,xx+EIpϑ,xxxx-EAp2Lv,xx∫0L v^,x+2dx-EApLv^,xx+∫0L v,xv^,x+dx=0
    (18)

  • 方程（18）的解可以写作

  • v(x,t)=φn(x)eiωnt
    (19)

  • 式中ωn为系统第n阶固有频率，φn（x）为第n阶模态函数.将式（19）代入局部线性化方程（18）可得

  • ρpAp+ρfAfφn(x)ωn2+2iρfAfΓωnφn'(x)+ρfAfΓ2-P0+EApζΔTφn''(x)+EIpφn(4)(x)-EAp2Lφn''(x)∫0L v^,x+2dx-EApLv^,xx+∫0L φn'(x)v^,x+dx=0
    (20)

  • 选用静态梁模态函数为基函数，假设

  • φn(x)=∑j=1M Dnjsin⁡(jπx)
    (21)

  • 其中，Dnj为待定系数，形式解（21）满足简支边界条件（10）.将模态函数（21）代入方程（18），在方程两边同乘sin（iπx），i=1，2，3，···，M，并在[0，L]上对x进行积分，可得到关于ωn和dnj（j=1，2，3，···，M）的超静定齐次线性代数方程组[12]

  		•
    (22)

  • 其中，dnj =Dnj/Dn1（Dn1≠0），M为质量矩阵，G为陀螺矩阵，K为刚度矩阵.

  • 式（22）存在非零解的条件为方程组的系数行列式为零

  • -ωn2M+iωnG+K=0
    (23)

  • 基于此，可以得到不同离散阶数系统的固有频率.图4描述了用不同Galerkin截断阶数计算得到的前两阶固有频率随超临界流速变化的趋势，从图中可以看出，随着流速的增大，超临界固有频率也增大.从误差分析可以发现在四阶和六阶截断情况下，系统第一阶固有频率曲线拟合良好，第二阶固有频率的误差不大于0.1%.因此认为四阶Galerkin截断达到了收敛，后续研究都将采用四阶Galerkin截断法对控制方程进行截断降维处理.其中，截断阶数误差表达式为

  • RG=ωn(6- Galerkin )-ωn(4- Galerkin )ωn(6- Galerkin )
    (24)

  • 图4 超临界输流管道前两阶固有频率随流速的变化及截断误差分析

  • Fig.4 Accuracy of the first-two natural frequencies changing with fluid velocity in the supercritical region

  • 4 参数对热环境中固有频率的影响

  • 4.1 温度增量对超临界固有频率的影响

  • 基于收敛性结论，图5对比了在室温基础下，不同温度增量对超临界输流管道固有频率的影响.根据非平凡静平衡位形（15），当环境温度升高时，管道的屈曲位形增大.由于引入温度增量，刚度矩阵增大，此时的弯曲刚度增强.文献[21，28]研究表明，亚临界状态下简支输流管道在温度升高时，由于热膨胀导致管道产生轴向压力，从而温度升高会使管道固有频率降低，临界流速随温度升高而降低.图5（a）显示了在给定的超临界流速（Г=1.2Гcr）下，管道固有频率随温度变化的增大而增大，与亚临界情况相反.图5（b）表明，超临界状态下流速越大，固有频率越大，与亚临界情况相反；温度增量越大，输流管道的临界流速越小，与亚临界结果一致.图5（c）表明当流速较大时，温度对管道固有频率的影响较小且温度升高对管道第二阶固有频率影响不明显.

  • 图5 温度增量对超临界固有频率的影响

  • Fig.5 Effect of the temperature increment to natural frequencies in the supercritical region

  • 4.2 轴力对热环境中超临界固有频率的影响

  • 图6为初始轴力分别为500 N、1000 N、2000 N时，热环境中超临界输流管道前两阶固有频率随流速的变化.图6（a）显示了在给定的超临界流速（Г=1.2Гcr）下，管道固有频率随轴力的增大而减小.图6（b）、（c）的结果表明，初始轴力越大，输流管道的临界流速越大.也可得，在超临界情况下，温度增量与初始轴力对输流管道固有频率的影响相反.

  • 图6 初始轴力对超临界固有频率的影响

  • Fig.6 Effect of the initial axial force to natural frequencies in the supercritical region

  • 5 结论

  • 本文以热环境中超临界自由振动的输流管道为研究对象，通过哈密顿原理建立了两端简支受温度影响的输流管道的控制方程.得到非平凡静平衡位形及临界流速，基于DQEM方法，进行数值验证.利用复模态法进行模态分析，将模态函数作为权函数，使用四阶伽辽金截断法得到流固耦合作用下管道的固有频率及不同超临界流速下的模态函数，并分析了温度增量及初始轴力对管道固有频率的影响.经过研究发现，非平凡静平衡位形近似解析解与数值解拟合良好，验证了其正确性.当管道微幅振动时，超临界固有频率和模态函数不受非线性系数影响，随着超临界流速增加，高阶固有频率增大.有趣的是，在亚临界状态下，温度增量越大或轴力越小时输流管道的固有频率越低，临界流速越低.超临界状态下，温度增量越大或初始轴力越小，固有频率反而越大，此时临界流速越小，且温度增量和初始轴力的改变对管道第二阶固有频率影响较小.此外，亚临界状态下，温度增量和流速的增大都会导致挠度增大，而轴力的增加会抑制最大挠度.与本文温度增量、流速及轴力对静平衡位形的影响一致，验证了本文结果的正确性.

  • 参考文献

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  • 基本信息

    中图分类号: O322
    文献标识码: A
    DOI: 10.6052/1672-6553-2023-078
    文章编号: 1672-6553-2023-21(6)-039-008

    基金信息

    国家自然科学基金青年基金项目(12002195)；
    国家自然科学基金杰出青年基金(12025204)；
    上海市教育委员会科研创新计划项目(2019-01-07-00-09-E00018)；
    上海市科学技术委员会浦江人才计划（A类20PJ1404000）；
    National Natural Science Foundation of China (12002195)；
    the National Science Fund for Distinguished Young Scholars (12025204)；
    the Program of Shanghai Municipal Education Commission (2019-01-07-00-09-E00018)；
    the Pujiang Project of Shanghai Science and Technology Commission (20PJ1404000)；

    引用信息

    稿件历史

    收稿日期: 2023-03-09

  • 参考文献

    • [1] LI S,KARNEY B W,LIU G.FSI research in pipeline systems-A review of the literature [J].Journal of Fluids and Structures,2015,57:277-297.

    • [2] GUO Y,ZHU B,YANG B,et al.Flow-induced buckling and post-buckling vibration characteristics of composite pipes in thermal environment [J].Ocean Engineering,2022,243:110267.

    • [3] 张凯凯,谭霞,丁虎,等.超临界输流管道3∶1内共振下参激振动响应 [J].应用数学和力学,2018,39(11):1227-1235.ZHANG K K,TAN X,DING H,et al.Parametric vibration responses of supercritical fluid conveying pipes in 3:1 internal resonance [J].Applied Mematics and Mechanics,2018,39(11):1227-1235.(in Chinese)

    • [4] 丁虎.输流管道振动的被动控制研究 [J].自然杂志,2022,44(4):316-322.DING H.Passive control of vibration in pipes conveying fluid [J].Chinese Journal of Nature,2022,44(4):316-322.(in Chinese)

    • [5] 杨晓东,金基铎.输流管道流-固耦合振动的固有频率分析 [J].振动与冲击,2008,(3):80-81+86+181.YANG X D,JIN J D.Analysis of the inherent frequency of flow-solid coupling vibration in flow transfer pipes [J].Journal of Vibration and Shock,2008,(3):80-81+86+181.(in Chinese)

    • [6] DING H,JI J C,CHEN L Q.Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics [J].Mechanical Systems and Signal Processing,2019,121:675-688.

    • [7] 周永兆,杨晓东,金基铎.输流管道非线性横向振动固有频率分析 [J].振动、测试与诊断,2012,32(S1):66-68,150.ZHOU Y Z,YANG X D,JIN J D.Natural frequency analysis of nonlinear transverse vibrations for pipes conveying fluid [J].Journal of Vibration,Measurement & Diagnosis,2012,32(S1):66-68,150.(in Chinese)

    • [8] 颜雄,魏莎,毛晓晔,等.两端弹性支承输流管道固有特性研究 [J].力学学报,2022,54(5):1341-1352.YAN X,WEI S,MAO X Y,et al.Study on natural characterstics of fluid-conveying pipes with elastic suppports at both ends [J].Chinese Journal of Theoretical and Applied Mechanics,2022,54(5):1341-1352.(in Chinese)

    • [9] 黄慧春,张艳雷,陈立群.受迫振动的超临界输液管Galerkin数值模拟 [J].应用数学和力学,2014,35(10):1100-1106.HUANG H C,ZHANG Y L,CHEN L Q.A Galerkin numerical method for the pipe conveying supercritical fluid under forced vibration [J].Applied Mematics and Mechanics,2014,35(10):1100-1106.(in Chinese)

    • [10] DING H,CHEN L Q.Galerkin methods for natural frequencies of high-speed axially moving beams [J].Journal of Sound and Vibration,2010,329(17):3484-3494.

    • [11] DING H,ZHANG G C,CHEN L Q.Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions [J].Mechanics Research Communications,2011,38(1):52-56.

    • [12] 张国策,丁虎,陈立群.复模态分析超临界轴向运动梁横向非线性振动 [J].动力学与控制学报,2015,13(4):283-287.ZHANG G C,DING H,CHEN L Q.Complex modal analysis of transversally non-linear vibration fo supercritically axially moving beams [J].Journal of Dynamics and Contorl,2015,13(4):283-287.(in Chinese)

    • [13] DING H,ZHANG G C,CHEN L Q,et al.Forced vibrations of supercritically transporting viscoelastic beams [J].Journal of Vibration and Acoustics,2012,134(5):051007.

    • [14] 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 pipe [J].Journal of Dynamics and Contorl,2022,20(4):83-90.(in Chinese)

    • [15] ZHOU K,NI Q,CHEN W,et al.Static equilibrium configuration and nonlinear dynamics of slightly curved cantilevered pipe conveying fluid [J].Journal of Sound and Vibration,2021,490:115711.

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