STUDY ON NATURAL CHARACTERISTICS OF FLUID-CONVEYING PIPES WITH ELASTIC SUPPORTS AT BOTH ENDS
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摘要: 输流管道广泛应用于航天航空、石油化工、海洋等重要的工程领域, 其振动特性尤其是系统固有特性一直是国内外学者研究的热点问题. 本文研究了两端弹性支承输流管道横向振动的固有特性, 尤其是在非对称弹性支承下的系统固有特性. 使用哈密顿原理得到了输流管道的控制方程及边界条件, 通过复模态法得到了静态管道的模态函数, 以其作为伽辽金法的势函数和权函数对线性派生系统控制方程进行截断处理. 分析了两端对称支承刚度、两端非对称支承刚度、管道长度以及流体质量比对系统固有频率的影响规律, 重点讨论了管道两端可能形成的非对称支承条件下固有频率的变化规律. 结果表明, 较大的对称支承刚度下管道的第一阶固有频率下降较快; 当管道两端支承刚度变化时, 管道的各阶固有频率在两端支承刚度相等时取得最值; 对于两端非对称支承的管道而言, 两端支承刚度越接近, 第一阶固有频率下降的越快, 而且相应的临界流速越小; 流体的流速越大, 其对两端非对称弹簧支承的管道固有频率的影响更为明显.Abstract: Fluid-conveying pipes have been widely used in aerospace, petrochemical, offshore and other important engineering fields. The vibration characteristics of the fluid-conveying pipes, especially the natural characteristics of the system, have been an important issue in the research of scholars around the world. This study investigates the natural characteristics of transverse vibration of a fluid-conveying pipe with elastic supports at both ends. In particular, the natural characteristics of the fluid-conveying pipe with asymmetric elastic supports at both ends are discussed. The governing equation and boundary conditions of the fluid-conveying pipe system are derived by the Hamilton’s principle. The modal functions of the static pipe are obtained by the complex modal method, and then they are used as the potential function and weight function for the Galerkin method to truncate the control equation of the linear derived system. The effects of symmetrical support stiffness at both ends, asymmetric support stiffness at both ends, pipe length and fluid mass ratio on the natural frequencies of the system are discussed. The discussion focuses on the variation of natural frequencies under the condition of asymmetric supports that may happen at both ends of the pipe. Results show that a fast decrease in the first natural frequency for large symmetrical support stiffness. When the support stiffness at both ends of the pipe changes, the natural frequencies of each order of the pipe obtain the maximum or minimum value when the support stiffness at both ends is equal. For the pipe with asymmetric supports at both ends, the closer the support stiffness at both ends, the faster the first natural frequency decreases, and the smaller the corresponding critical flow velocity. The greater the flow velocity of the fluid, the more significant is the effect on the natural frequency of the pipe supported by asymmetric supports at both ends.
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引言
在现代工业生产中, 管道被广泛应用于水、石油、天然气以及化工原料等各类流态物料的运输[1-4]. 一般而言, 当流体在管道内部发生流动时, 管道会发生振动, 如果严重的话会直接损坏管道系统. 由于管道的动力学特性会受到流体流速、两端支承条件以及其他因素的影响, 因此展开了对输液管道相关方面的研究.
管道动力学的研究主要集中在理想边界. 例如, 早在1950年, Ashley和Haviland[5]首次研究了含有流动流体管道的弯曲振动. Paidoussis等[6-8]通过牛顿第二定律推导了水平悬臂输液管道横向振动的线性方程, 计算了系统前四阶模态的复频率并通过实验进行了验证. 杨晓东和金基铎[9]使用复模态法和伽辽金法对两端铰支的输液管道的固有特性进行计算, 发现伽辽金法只对其截断阶数的前几阶较低固有频率的分析具有较好的精确性. 周永兆等[10]研究了两种铰支输液管道非线性模型下系统的固有频率, 得出了偏微分控制方程结果相对于积分-偏微分控制方程结果而言具有较强的非线性. 易浩然等[11]通过研究含集中质量的悬臂输液管道, 得到了管道的振动模态会随着集中质量的位置而发生转迁. 刘昌领等[12]对一端固定一端简支输液管道的流固耦合振动进行了分析, 发现管道的长度对固有频率的影响最大, 其次是流体速度, 液体压力的影响最小. 郭梓龙等[13]研究了悬臂输流管一端加上接地惯容式减震器后的稳定性和动态响应的影响. 张挺等[14]基于广义有限差分法对不同支承下输流直管的振动响应特性进行了研究, 得出结论: 两端简支时输流直管中点处的振幅最大, 振动频率最小; 两端固支时输流直管中点处的振幅最小, 振动频率最大; 且在端部约束限制条件不对称时, 其振动幅值最大值出现位置会向弱约束端偏移. 金基铎等[15]研究了具有弹性支承和运动约束作用的悬臂输液管道系统的运动分岔现象和混沌运动. 匡华军等[16]研究了温度对两端固支管道振动特性的影响, 为管道系统的安全性和设计提供了理论依据. 邢静忠和柳春图[17]研究了土壤中悬跨管道的弯曲和振动特性, 得出了当土壤刚度系数较大时, 管道可以近似为两端固支模型; 而只有在土壤刚度系数较小时的几个参数点上, 管道才可以近似为两端简支模型. 随岁寒和李成[18]使用有限元分析方法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. Ye等[19]研究了简支边界下输送超临界流体微曲管道的非平凡平衡位形和固有频率, 得出了曲管的固有频率会随着管长的增加而增加, 但不一定以单调的方式增加.
然而, 当管道受到环境温度、湿度、振动疲劳等因素的影响时, 管道两端的约束形式不能简化为理想边界, 可能是较为复杂的弹性边界. 因此, 大批学者开展了弹性支承下输液管道的振动研究. 例如, 江建祥和黄幼玲[20]通过对上游铰支加扭转支承, 下游铰支的模型进行了计算和实验, 得到了管道各阶固有频率随流速的增加而降低. 倪樵等[21]使用微分求积法分析了具有弹性支承输液管道的临界流速, 并证明了该方法具有较高的精度. 李琳等[22]研究了一端弹性支承一端固定支承输液管道的稳定性,得到了静态失稳的临界流速不随质量比变化, 而动态失稳的临界流速随质量比的增加先上升后下降. Qian等[23]研究了埋在非线性弹性地基中的输液管道平面运动的非线性响应. 包日东等[24-25]分析了含有弹性支承输液管道的固有特性、动力稳定性和非线性动力学特性. 刘俊卿和王克林[26]提出了一种求解输液管道临界流速的新方法, 可用于任意弹性支承的输液管道, 支承位置可在两端也可在中间. 吴男和金基铎[27]研究了弹性地基两端铰支输液管道的模态和固有频率, 并讨论了弹性地基、质量比以及轴向力对固有频率与流速之间关系的影响. Ni等[28-29]研究了非线性弹性地基上输流弯管的内外共振及其分岔与混沌运动. 许锋等[30]研究了两端弹性支承输液管道在分布随从力作用下的稳定性, 得到了支承刚度的变化和分布随从力的大小及方向会影响输液管道的失稳类型. Li等[31]研究了两端受竖直弹簧和扭转弹簧约束的输液管道在脉动流激励下的非线性参数振动问题. Ding等[32]在输液管道两端加上以一定的方式组合起来的三个线性弹簧作为非线性隔振器以衰减基础激励引起的输液管道横向振动. 以上关于弹性支承的研究均是基于对称情况下的, 对于两端非对称弹性支承情况下的相关研究较少. 但在工程实际中, 例如海底悬跨管道, 其两端边界处多为管-土作用的非对称约束, 因此对于两端非对称弹性支承管道的研究是必不可少的.
本文以两端弹性支承输液管道为研究对象, 通过复模态法和伽辽金截断法求解静态管道和受流体流动影响的管道固有频率和模态函数, 并分析了支承刚度、管道长度、流体质量比等系统参数对管道固有频率的影响规律, 重点讨论了工程实际中由于管道受到环境温度、湿度、振动疲劳以及局部松动等情况下所出现的管道两端可能形成的非对称支承的情况. 本文中所得到的非对称支承下管道固有频率的变化规律可应用于设计一些满足特殊需求的管道.
1. 输液管道横向自由振动控制方程及伽辽金截断
图1所示为两端弹性支承输液管道模型. 当考虑Kelvin-Voigt黏弹性管材、管道几何非线性, 没有管截面轴向力以及管内流体压力作用, 基于欧拉梁模型的管道控制方程为[33]
$$ \begin{split} & \left( {{\rho _{\rm{p}}}{A_{\rm{p}}} + {\rho _{\rm{f}}}{A_{\rm{f}}}} \right)W{,_{tt}}{\rm{ + 2}}{\rho _{\rm{f}}}{A_{\rm{f}}}\varGamma W{,_{xt}}{\rm{ + }}E{I_{\rm{b}}}W{,_{xxxx}}{\rm{ + }}\\ &\qquad \mu {I_{\rm{b}}}W{,_{xxxxt}}{\rm{ + }}\left( {{\rho _{\rm{f}}}{A_{\rm{f}}}{\varGamma ^2} - \frac{{E{A_{\rm{p}}}}}{{2L}}\int_0^L {W_{,x}^2{\rm{d}}x} - } \right.\\ &\qquad \left. {\frac{{\mu {A_{\rm{p}}}}}{L}\int_0^L {{W_{,x}}{W_{,xt}}{\rm{d}}x} } \right)W{,_{xx}} = 0 \end{split}$$ (1) $$ \left.\begin{gathered} W{,_{xx}}\left( {0,t} \right) = 0,{\kern 1pt} {\text{ }}W{,_{xx}}\left( {L,t} \right) = 0 \hfill \\ {\kern 1pt} {k_{\text{L}}}W\left( {0,t} \right) + E{I_{\text{b}}}W{,_{xxx}}\left( {0,t} \right) = 0{\kern 1pt} \hfill \\ {k_{\text{R}}}W\left( {L,t} \right) - E{I_{\text{b}}}W{,_{xxx}}\left( {L,t} \right) = 0{\kern 1pt} \hfill \end{gathered} \right\}$$ (2) 其中, W(x)为管道横向位移; E为管道长度; Ib为截面转动惯量; μ为管道黏弹性系数; L为管道长度; kL, kR分别是两端线性支承弹簧的刚度; ρp为管道密度; D为管道横截面外径; d为管道内径; ρf为管道内流体密度; Af为流体截面面积; Γ为流体流速. 各参数的值如下表1所示.
表 1 输液管道的物理参数Table 1. Physical parameters of the pipelineName Notation Value outer diameter D/m 0.02 inner diameter d/m 0.016 Young’s modulus E/GPa 72 density of fluid ρf/(kg·m−3) 1000 density of pipe ρp/(kg·m−3) 2700 length of pipe L/m 1 viscoelasticity coefficient μ/(MN·s·m−2) 5 support stiffness on the left kL/(kN·m−1) 10 support stiffness on the right kR/(kN·m−1) 10 忽略流体流动、阻尼、非线性项的影响, 得到静态管的线性控制方程和边界条件为
$$ \left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right)W{,_{tt}}{\text{ + }}E{I_{\text{b}}}W{,_{xxxx}} = 0 $$ (3) $$ \left.\begin{gathered} W{,_{xx}}\left( {0,t} \right) = 0,{\kern 1pt} {\text{ }}W{,_{xx}}\left( {L,t} \right) = 0\hfill \\ {\kern 1pt} {k_{\text{L}}}W\left( {0,t} \right) + E{I_{\text{b}}}W{,_{xxx}}\left( {0,t} \right){\text{ = 0}}{\kern 1pt} {\text{ }} \hfill \\ {k_{\text{R}}}W\left( {L,t} \right) - E{I_{\text{b}}}W{,_{xxx}}\left( {L,t} \right) = 0{\kern 1pt} \hfill \end{gathered}\right\} $$ (4) 假设静态管道横向自由振动的位移解为
$$ \tilde W\left( {x,t} \right) = \phi \left( x \right)q\left( t \right) $$ (5) 其中, ϕ(x)和q(t)分别为静态管道的模态函数和相应的广义坐标, 假设管道横向振动的模态函数通解形式为
$$ \phi \left( x \right) = {C_1}\cos (\beta x) + {C_2}\sin (\beta x) + {C_3}{\text{cosh}}(\beta x) + {C_4}{\text{sinh}}(\beta x) $$ (6) 其中C1, C2, C3, C4分别为模态系数. 将解(5)代入边界条件(4), 并联立式(6)可以得到
$$ \left[ {\begin{array}{*{20}{c}} { - 1}&0&1&0 \\ { - \cos (\beta L)}&{ - \sin (\beta L)}&{{\text{cosh}}(\beta L)}&{{\text{sinh}}(\beta L)} \\ {{k_{\text{L}}}}&{ - E{I_{\text{b}}}{\beta ^3}}&{{k_{\text{L}}}}&{E{I_{\text{b}}}{\beta ^3}} \\ {{k_{\text{R}}}\cos (\beta L) - E{I_{\text{b}}}{\beta ^3}\sin (\beta L)}&{{k_{\text{R}}}\sin (\beta L) + E{I_{\text{b}}}{\beta ^3}\cos (\beta L)}&{{k_{\text{R}}}\cosh (\beta L) - E{I_{\text{b}}}{\beta ^3}{\text{sinh}}(\beta L)}&{{k_{\text{R}}}\sinh (\beta L) - E{I_{\text{b}}}{\beta ^3}{\text{cosh}}(\beta L)} \end{array}} \right]\left\{ {\begin{array}{*{20}{c}} {{C_1}} \\ {{C_2}} \\ {{C_3}} \\ {{C_4}} \end{array}} \right\} = {\boldsymbol{0}} $$ (7) 根据式(7)可以得到静态管道的模态函数各系数的关系
$$\left. \begin{gathered} {C_1} = {C_3} \\ {C_2} = \frac{{I_{\text{b}}^2{E^2}{\beta ^6}\sin \left( {\beta L} \right) + I_{\text{b}}^2{E^2}{\beta ^6}\sinh \left( {\beta L} \right) - 2E{I_{\text{b}}}{k_{\text{L}}}{\beta ^3}\cosh \left( {\beta L} \right) - {k_{\text{R}}}E{I_{\text{b}}}{\beta ^3}\cos \left( {\beta L} \right) - {k_{\text{R}}}E{I_{\text{b}}}{\beta ^3}\cosh \left( {\beta L} \right) + 2{k_{\text{L}}}{k_{\text{R}}}\sinh \left( {\beta L} \right)}}{{E{I_{\text{b}}}{\beta ^3}\left[ {{I_{\text{b}}}E{\beta ^3}\cos \left( {\beta L} \right) - {I_{\text{b}}}E{\beta ^3}\cosh \left( {\beta L} \right) + {k_{\text{R}}}\sin \left( {\beta L} \right) + {k_{\text{R}}}\sinh \left( {\beta L} \right)} \right]}} \\ {C_4} = \frac{{I_{\text{b}}^2{E^2}{\beta ^6}\sin \left( {\beta L} \right) + I_{\text{b}}^2{E^2}{\beta ^6}\sinh \left( {\beta L} \right) - 2E{I_{\text{b}}}{k_{\text{L}}}{\beta ^3}\cos \left( {\beta L} \right) - {k_{\text{R}}}E{I_{\text{b}}}{\beta ^3}\cos \left( {\beta L} \right) - {k_{\text{R}}}E{I_{\text{b}}}{\beta ^3}\cosh \left( {\beta L} \right) - 2{k_{\text{L}}}{k_{\text{R}}}\sinh \left( {\beta L} \right)}}{{E{I_{\text{b}}}{\beta ^3}\left[ {{I_{\text{b}}}E{\beta ^3}\cos \left( {\beta L} \right) - {I_{\text{b}}}E{\beta ^3}\cosh \left( {\beta L} \right) + {k_{\text{R}}}\sin \left( {\beta L} \right) + {k_{\text{R}}}\sinh \left( {\beta L} \right)} \right]}} \\ \end{gathered} \right\}$$ (8) 图2所示为所得到的静态管道的前四阶模态函数, 从中可以看出, 在本文给定的参数条件下, 管道的第一阶和第三阶模态关于管道中点对称, 而第二阶和第四阶模态关于管道中点反对称.
在接下来的分析中, 为了求解输液管道的固有频率、横向振动模态, 采用线性静态管的模态函数作为试函数和权重函数, 应用伽辽金截断法.
为研究具有线性弹性边界的输液管道的自由振动特性, 根据式(1)对应的线性派生系统为
$$ \begin{split} & \left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right)W{,_{tt}}{\text{ + 2}}{\rho _{\text{f}}}{A_{\text{f}}}\varGamma W{,_{xt}} +\hfill \\ &\qquad {\text{ }}{\rho _{\text{f}}}{A_{\text{f}}}{\varGamma ^2}W{,_{xx}} + E{I_{\text{b}}}W{,_{xxxx}} = 0 \end{split} $$ (9) $$ \left.\begin{gathered} W{,_{xx}}\left( {0,t} \right) = 0,{\kern 1pt} {\text{ }}W{,_{xx}}\left( {L,t} \right) = 0 \hfill \\ {\kern 1pt} {k_{\text{L}}}W\left( {0,t} \right) + E{I_{\text{b}}}W{,_{xxx}}\left( {0,t} \right){\text{ = 0}} \hfill \\ {k_{\text{R}}}W\left( {L,t} \right) - E{I_{\text{b}}}W{,_{xxx}}\left( {L,t} \right) = 0{\kern 1pt} \hfill \end{gathered}\right\} $$ (10) 考虑流体流动作用下输液管道的自由振动位移解为
$$ W\left( {x,t} \right) = \sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}\left( x \right)} $$ (11) 其中n为伽辽金计算的截断项, 可以得到
$$ \begin{split} & - {\omega ^2}\left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right)\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}\left( x \right)}+ \hfill \\ &\qquad {\text{ 2i}}{\rho _{\text{f}}}{A_{\text{f}}}\varGamma \omega \sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_x}\left( x \right)} + \hfill \\ &\qquad {\rho _{\text{f}}}{A_{\text{f}}}{\varGamma ^2}\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_{xx}}\left( x \right)} + \hfill \\ &\qquad E{I_{\text{b}}}\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_{xxxx}}\left( x \right)} = 0 \end{split} $$ (12) 选取静态管道的模态函数为权函数, 可得
$$ \begin{split} & - {\omega ^2}\left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right)\int_0^L {{\phi _m}\left( x \right)\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}\left( x \right)} {\text{d}}x}+ \hfill \\ &\qquad {\text{ 2i}}{\rho _{\text{f}}}{A_{\text{f}}}\varGamma \omega \int_0^L {{\phi _m}\left( x \right)\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_x}\left( x \right)} {\text{d}}x} + \hfill \\ &\qquad {\text{ }}{\rho _{\text{f}}}{A_{\text{f}}}{\varGamma ^2}\int_0^L {{\phi _m}\left( x \right)\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_{xx}}\left( x \right)} {\text{d}}x} + \hfill \\ &\qquad E{I_{\text{b}}}\int_0^L {{\phi _m}\left( x \right)\sum\limits_{k = 1}^n {{Q_k}{{\text{e}}^{{\text{i}}\omega t}}{\phi _k}{,_{xxxx}}\left( x \right)} {\text{d}}x} = 0 \hfill \\ &\qquad{m = 1,2, \cdots ,n} \end{split} $$ (13) 则式(13)可化简为
$$ \left( { - {\omega ^2}{\boldsymbol{M}} + {\rm{i}}\omega {\boldsymbol{G}} + {\boldsymbol{K}}} \right){Q_k} ={\boldsymbol{ 0}} $$ (14) 其中
$$ \left.\begin{array}{l} {M_{km}} = \left\{ {\begin{array}{l} 0,\qquad{k \ne m} \\ 1,\qquad{k = m} \end{array}} \right.\\ {G_{km}} = \dfrac{{2{\rho _{\text{f}}}{A_{\text{f}}}\varGamma {N_{{\text{1}}km}}}}{{\left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right){N_{{\text{0}}m}}}} \hfill \\ {K_{km}} = \left\{ {\begin{array}{l} {0,}\qquad{k \ne m} \\ \dfrac{{E{I_{\text{b}}}{N_{{\text{4}}m}}}}{{\left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right){N_{{\text{0}}m}}}} +\\ \qquad \dfrac{{{\rho _{\text{f}}}{A_{\text{f}}}{\varGamma ^2}{N_{{\text{2}}m}}}}{{\left( {{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}} \right){N_{{\text{0}}m}}}},\qquad{k = m} \end{array}} \right. \hfill \\ \\ {N_{{\text{0}}m}} = \displaystyle\int_0^L {{{\left[ {{\phi _m}\left( x \right)} \right]}^2}{\text{d}}x} \\ {N_{{\text{1}}km}} = \displaystyle\int_0^L {{\phi _k}{,_x}\left( x \right){\phi _m}\left( x \right){\text{d}}x} \hfill \\ {N_{{\text{2}}m}} = \displaystyle\int_0^L {{\phi _m}{,_{xx}}\left( x \right){\phi _m}\left( x \right){\text{d}}x} \\ {N_{{\text{4}}m}} = \displaystyle\int_0^L {{\phi _m}{,_{xxxx}}\left( x \right){\phi _m}\left( x \right){\text{d}}x} \end{array}\right\} $$ (15) 固有频率ω可以通过以下式子求得
$$ \left| { - {\omega ^2}{\boldsymbol{M }}+ {\rm{i}}\omega {\boldsymbol{G}} + {\boldsymbol{K}}} \right| = 0 $$ (16) 对于给定的特征值, 可以确定复特征向量元素Qk, 则含有流体流动的自由横向振动输液管道的模态函数可以表示为
$$ \varphi \left( x \right) = \sum\nolimits_{k = {\text{1}}}^n {{\rm{Im}} \left[ {{Q_k}} \right]} {\phi _k}\left( x \right) $$ (17) 其中Im[Qk]表示Qk的虚部.
图3给出了伽辽金方法选取不同截断项数时的系统模态函数, 这里考虑了管内流体流速的作用. 图4则给出了对应情况下的系统固有频率. 从图3和图4的结果可以看出, 与截断项数取为6的结果相比, 当截断项数取为4时, 前四阶模态函数及固有频率均已具有较好的收敛性. 因此, 后续计算中伽辽金方法的截断项数均取为4. 此外, 与图2结果相比, 图3(a)和图3(c)中的结果(第一阶和第三阶模态函数)不是关于中点对称; 图3(b)和图3(d)中的结果(第二阶和第四阶模态函数)不是关于中点反对称. 这主要是由于流体流速作用导致系统模态函数出现了不对称.
2. 固有特性分析
2.1 对称线性支承刚度对固有频率的影响
图5为管道两端对称线性弹簧的刚度分别为1 × 103 N/m, 1 × 104 N/m, 1 × 105 N/m时, 输液管道前四阶固有频率随着流体流速的变化情况. 图5(a)表示在亚临界状态下, 管道的第一阶固有频率随着管道内流体流速的增大而下降, 两端线性刚度越大, 下降的趋势越快, 且管道的临界流速越小. 此结论与文献[25]中所得结论一致, 证明了计算结果的正确性. 图5(b)中结果表明: 在亚临界状态下, 当两端支承刚度较小时(kL = kR = 1 × 103 N/m), 管道第二阶固有频率随着流速的增加而增大. 当支承刚度较大时(kL = kR = 1 × 105 N/m), 第二阶固有频率随着流速的增大而减小, 这与经典的两端简支输液管道下的结论相同[18]. 图5(c)和图5(d)表示管道在不同对称支承刚度情况下的第三和第四阶固有频率随着流速的增大而减小.
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图 5 不同流速及对称线性支承刚度下的系统固有频率Figure 5. The natural frequencies with different flow velocity and the stiffness of symmetrical linear spring表2和表3分别给出了当支承刚度很大和很小时系统的前四阶特征值. 表中也给出了刘延柱等[34]的两端简支和自由边界条件的系统特征值结果. 表2中结果表明: 当支承刚度很大时, 其结果与刘延柱等[34]计算的两端简支边界条件下的系统特征值吻合较好. 表3中结果表明: 当支承刚度很小时, 其结果与刘延柱等[34]计算的两端自由边界条件下的系统特征值吻合较好. 因此, 当两端支承刚度很大或很小时, 该模型可以分别退化为两端简支或自由边界条件, 这也说明了本文计算结果的正确性.
表 2 当支承刚度较大时的系统前四阶特征值与简支边界条件下的计算结果Table 2. The results of the first four orders eigenvalues of the system with large support stiffness and simply supported boundary conditionsBoundary conditions First-order eigen-value Second-order eigen-value Third-order eigen-value Fourth-order eigen-value kL = kR =
1 × 1012 N/m3.141588 6.283177 9.424590 12.566140 simply supported boundary [34]
3.141592
6.283185
9.424777
12.566370表 3 当支承刚度较小时的系统前四阶特征值与自由边界条件下的计算结果Table 3. The results of the first four orders eigenvalues of the system with small support stiffness and free boundary ConditionsBoundary conditions First-order eigen-value Second-order eigen-value Third-order eigen-value Fourth-order eigen-value kL = kR =
1 × 10−5 N/m4.730041 7.853204 10.995607 14.137165 free boundary [34]
4.722389
7.853982
10.995574
14.1371672.2 非对称线性支承刚度对固有频率的影响
图6表示的是管道左右两端支承刚度的总和为20 kN/m不变, 流体流速为0, 左右两端支承刚度的比值从1:10递增到10:1时, 管道固有频率的变化情况. 从中可以看出, 管道的第一阶和第二阶固有频率随着比值的增加而先增加后减小, 在左端等于右端时达到最大值. 而第三阶和第四阶固有频率随着比值的增大先减小后增大, 在左端等于右端时达到最小值, 而且通过计算结果可以得出, 互为倒数的比值对应的固有频率值相等.
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图 6 不同非对称线性支承刚度下的系统固有频率Figure 6. The natural frequencies with different stiffness of asymmetrical linear spring图7表示的是管道左右两端支承刚度的总和为20 kN/m不变, 左端支承刚度与右端支承刚度值的比分别为1:3, 1:1, 3:1时管道的固有频率随流体流速的变化情况. 通过观察可以发现, 当流速Γ = 0时, 各阶的固有频率关系满足图6中所对应的规律. 而当流速增加时, 互为倒数的刚度比值所对应的固有频率间的差别增大, 但第一阶固有频率几乎不受影响.
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图 7 不同流速及非对称线性支承刚度下的系统固有频率Figure 7. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring图8表示的是当两端非对称线性弹簧的刚度总和为2 kN/m不变, 左端支承刚度与右端支承刚度值的比分别为1:6, 1:4, 1:2时管道的固有频率随流体流速的变化情况. 可以看出, 随着流速的增加, 管道的各阶固有频率减小. 而且对于管道的第一阶固有频率而言, 两端弹簧的刚度越接近, 其下降的越快, 而且相应的临界流速越小. 第二阶固有频率随着两端支承刚度比值的增大而增大, 但第三阶和第四阶固有频率随着两端支承刚度比值的增大而减小.
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8 不同流速及非对称线性支承刚度下的系统固有频率8. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring![]()
8 不同流速及非对称线性支承刚度下的系统固有频率 (续)8. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring (continued)2.3 管道长度对固有频率的影响
图9为管道长度分别为L = 0.8 m, L = 1 m以及L = 1.2 m时, 管道前四阶固有频率的变化规律(除管长以外其他参数均与表1一致). 从图9(a)可以看出, 其他条件都一样的情况下, 越长的管道其第一阶固有频率和临界流速越小. 而从图9(b)中可以看出, 管道的长度不会改变第二阶固有频率一开始随流体流速增加而增大的变化规律. 但是, 当管道的长度增加时, 第二阶固有频率开始下降的位置会向左发生平移. 图9(c)和图9 (d)表示管道的长度不会改变第三和第四阶固有频率随着流速的增大而减小的变化规律.
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9 不同流速及管道长度下的的系统固有频率9. The natural frequencies with different flow velocity and pipe length2.4 流体质量比对固有频率的影响
图10为流体质量比Mr=0.3, Mr=0.4和Mr = 0.5时, 管道前四阶固有频率随流速的变化规律
${M_{\text{r}}} = {{{\rho _{\text{f}}}{A_{\text{f}}}}}/({{{\rho _{\text{p}}}{A_{\text{p}}} + {\rho _{\text{f}}}{A_{\text{f}}}}})$ . 从中可以看出, 流体质量比越大, 管道的第一阶、三阶和第四阶固有频率越小. 但对于管道第二阶固有频率而言, 在一开始它是随着流速的增大而增大. 在一定的范围内, 流体质量比越大, 管道第二阶固有频率越小, 但当流速超过这一临界点后, 流体质量比越大, 管道第二阶固有频率越大.![]()
10 不同流速及流体质量比下的系统固有频率10. The natural frequencies with different flow velocity and fluid mass ratio3. 结论
本文研究了两端弹性支承输液管道的固有特性, 通过哈密顿原理建立了两端弹性支承输液管道的控制方程, 使用复模态法得到了静态管道的前四阶模态函数, 以其作为权函数, 使用四阶伽辽金截断法得到考虑流体流动影响下的管道真实固有频率和模态函数, 并分析了流体流速、两端对称线性支承刚度、两端非对称线性支承刚度、管道长度以及流体质量比对管道固有频率的影响, 得出以下结论.
(1) 当输液管道两端为对称支承时, 支承刚度越大, 管道的一阶固有频率随着流体流速的增大下降的越快, 而且管道的临界流速越小. 第二阶固有频率在两端支承刚度较小时, 随着流速的增加一开始是增大的, 当支承刚度足够大时, 频率随着流速的增大而减小.
(2) 当输液管道两端为非对称支承, 而左右两端支承刚度总和不变时, 考虑流体流速为零的情况下, 管道的各阶固有频率在两端支承刚度相等时取得最值, 而且左右两端支承刚度的比值互为相反数所对应的固有频率相等. 而当流体流速不为零时, 流速越大, 固有频率变化的越明显.
(3) 其他参数相同的条件下, 管道越长, 其固有频率和临界流速越小; 流体质量比越大, 其一阶、三阶和四阶固有频率越小. 但对于二阶固有频率而言, 在一定的范围内, 流体质量比越大, 管道第二阶固有频率越小, 但当流速超过临界点后, 流体质量比越大, 管道第二阶固有频率越大.
(4) 文中所得到的两端对称支承与非对称支承情况下固有频率的变化规律结果可以用于设计一些满足特殊需求的输液管道.
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图 5 不同流速及对称线性支承刚度下的系统固有频率
Figure 5. The natural frequencies with different flow velocity and the stiffness of symmetrical linear spring
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图 6 不同非对称线性支承刚度下的系统固有频率
Figure 6. The natural frequencies with different stiffness of asymmetrical linear spring
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图 7 不同流速及非对称线性支承刚度下的系统固有频率
Figure 7. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring
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8 不同流速及非对称线性支承刚度下的系统固有频率
8. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring
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8 不同流速及非对称线性支承刚度下的系统固有频率 (续)
8. The natural frequencies with different flow velocity and the stiffness of asymmetric linear spring (continued)
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9 不同流速及管道长度下的的系统固有频率
9. The natural frequencies with different flow velocity and pipe length
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10 不同流速及流体质量比下的系统固有频率
10. The natural frequencies with different flow velocity and fluid mass ratio
表 1 输液管道的物理参数
Table 1 Physical parameters of the pipeline
Name Notation Value outer diameter D/m 0.02 inner diameter d/m 0.016 Young’s modulus E/GPa 72 density of fluid ρf/(kg·m−3) 1000 density of pipe ρp/(kg·m−3) 2700 length of pipe L/m 1 viscoelasticity coefficient μ/(MN·s·m−2) 5 support stiffness on the left kL/(kN·m−1) 10 support stiffness on the right kR/(kN·m−1) 10 
下载: 导出CSV
表 2 当支承刚度较大时的系统前四阶特征值与简支边界条件下的计算结果
Table 2 The results of the first four orders eigenvalues of the system with large support stiffness and simply supported boundary conditions
Boundary conditions First-order eigen-value Second-order eigen-value Third-order eigen-value Fourth-order eigen-value kL = kR =
1 × 1012 N/m3.141588 6.283177 9.424590 12.566140 simply supported boundary [34]
3.141592
6.283185
9.424777
12.566370
下载: 导出CSV
表 3 当支承刚度较小时的系统前四阶特征值与自由边界条件下的计算结果
Table 3 The results of the first four orders eigenvalues of the system with small support stiffness and free boundary Conditions
Boundary conditions First-order eigen-value Second-order eigen-value Third-order eigen-value Fourth-order eigen-value kL = kR =
1 × 10−5 N/m4.730041 7.853204 10.995607 14.137165 free boundary [34]
4.722389
7.853982
10.995574
14.137167
下载: 导出CSV
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[1] Chatjigeorgiou IK. On the effect of internal flow on vibrating catenary risers in three dimensions. Engineering Structures, 2010, 32(10): 3313-3329 doi: 10.1016/j.engstruct.2010.07.004
[2] Huang BW, Chen GS, Yu CT. Parametric resonance of a fluctuation fluid flow heat exchanger system. International Journal of Mechanical Sciences, 2018, 146: 386-395
[3] Gao PX, Zhai JY, Qu FZ, et al. Vibration and damping analysis of aerospace pipeline conveying fluid with constrained layer damping treatment. Journal of Aerospace Engineering, 2018, 232(8): 1529-1541 doi: 10.1177/0954410017692367
[4] Lu HF, Asce SM, Huang K, et al. Vibration and stress analyses of positive displacement pump pipeline systems in oil transportation stations. Journal of Pipeline Systems Engineering and Practice, 2016, 7(1): 0000205 doi: https://doi.org/10.1061/(ASCE)PS.1949-1204.0000205
[5] Ashley H, Haviland G. Bending vibration of a pipe line containing flowing fluid. Journal of Applied Mechanics, 1950, 17(3): 229-232 doi: 10.1115/1.4010122
[6] Gregory RW, Paidoussis MP. Unstable oscillation of tubular cantilevers conveying fluid Ⅰ. Theory. Proceedings of the Royal Society of London, 1966, 293(1435): 512-527
[7] Gregory RW, Paidoussis MP. Unstable oscillation of tubular cantilevers conveying fluid Ⅱ. Experiments. Proceedings of the Royal Society of London, 1966, 293(1435): 528-542
[8] Paidoussis MP, Issid N. Dynamic stability of pipes conveying fluid. Journal of Sound and Vibration, 1974, 33(3): 267-294 doi: 10.1016/S0022-460X(74)80002-7
[9] 杨晓东, 金基铎. 输流管道流-固耦合振动的固有频率分析. 振动与冲击, 2008, 27(3): 80-81, 86 (Yang Xiaodong, Jin Jiduo. Comparison of Galerkin method and complex mode method in natural frequency analysis of tube conveying fluid. Journal of Vibration and Shock, 2008, 27(3): 80-81, 86 (in Chinese) doi: 10.3969/j.issn.1000-3835.2008.03.020 [10] 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. 振动、测试与诊断, 2012, 32: 66-68, 150 (Zhou Yongzhao, Yang Xiaodong, Jin Jiduo. Natural frequency analysis of nonlinear transverse vibration for pipes conveying fluid. Journal of Vibration,Measurement &Diagnosis, 2012, 32: 66-68, 150 (in Chinese) [11] 易浩然, 周坤, 代胡亮等. 含集中质量悬臂输流管的稳定性与模态演化特性研究. 力学学报, 2020, 52(6): 1800-1810 (Yi Haoran, Zhou Kun, Dai Huliang, et al. Study on stability and modal evolution characteristics of the cantilevered fluid-conveying pipe attached with the lumped mass. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(6): 1800-1810 (in Chinese) doi: 10.6052/0459-1879-20-280 [12] 刘昌领, 罗晓兰, 叶道辉等. 一端固定一端简支输流管道流固耦合振动分析. 机械与电子, 2013, 7: 19-23 (Liu Changling, Luo Xiaolan, Ye Daohui, et al. Vibration analysis of liquid-filled pipes with fixed-hinged under fluid structure interaction. Machinery & Electronics, 2013, 7: 19-23 (in Chinese) [13] 郭梓龙, 王琳, 倪樵等. 接地惯容式减振器对悬臂输流管稳定性和动态响应的影响研究. 力学学报, 2021, 53(6): 1769-1780 (Guo Zilong, Wang Lin, Ni Qiao, et al. Research on the influence of grounded inerter-based absorber on the stability and dynamic response of cantilevered pipe conveying fluid. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(6): 1769-1780 (in Chinese) doi: 10.6052/0459-1879-21-105 [14] 张挺, 林震寰, 郭晓梅等. 基于广义有限差分法的输流直管振动响应特性研究. 振动与冲击, 2019, 38(24): 165-171 (Zhang Ting, Lin Zhenhuan, Guo Xiaomei, et al. Numerical simulation of vibration response of pipe conveying fluid based on a generalized finite difference method. Journal of Vibration and Shock, 2019, 38(24): 165-171 (in Chinese) [15] 金基铎, 邹光胜, 张宇飞. 悬臂输流管道的运动分岔现象和混沌运动. 力学学报, 2002, 34(6): 863-873 (Jin Jiduo, Zou Guangsheng, Zhang Yufei. Bifurcations and chaotic motions of a cantilevered pipe conveying fluid. Chinese Journal of Theoretical and Applied Mechanics, 2002, 34(6): 863-873 (in Chinese) doi: 10.3321/j.issn:0459-1879.2002.06.004 [16] 匡华军, 刘永寿, 岳珠峰. 温度对两端固支管道振动特性影响分析. 机械设计与制造, 2010, 5: 207-208 (Kuang Huajun, Liu Yongshou, Yue Zhufeng. Analysis the temperature to the vibration characteristics of pipeline with both ends clamped lmpact. Machinery Design & Manufacture, 2010, 5: 207-208 (in Chinese) doi: 10.3969/j.issn.1001-3997.2010.03.087 [17] 邢静忠, 柳春图. 线弹性土壤中埋设悬跨管道的弯曲和振动特性. 海洋工程, 2008, 2: 78-82 (Xing Jingzhong, Liu Chuntu. Bending and vibration characteristics of spanning pipe buried in linear elastic soil. The Ocean Engineering, 2008, 2: 78-82 (in Chinese) doi: 10.3969/j.issn.1005-9865.2008.02.010 [18] 随岁寒, 李成. 输流管道弯曲和振动的有限元分析. 动力学与控制学报, 2021, 出版中, https://kns.cnki.net/kcms/detail/43.1409.O3.20211111.1552.002.html Sui Suihan, Li Cheng. The finite element analysis on bending and vibration of the fluid-conveying pipes. Journal of Dynamics and Control, 2021, in press, https://kns.cnki.net/kcms/detail/43.1409.O3.20211111.1552.002.html (in Chinese))
[19] Ye SQ, Ding H, Wei S, et al. Non-trivial equilibriums and natural frequencies of a slightly curved pipe conveying supercritical fluid. Ocean Engineering, 2021, 227: 108899 doi: 10.1016/j.oceaneng.2021.108899
[20] 江建祥, 黄幼玲. 弹性支承输流管道固有频率计算. 振动工程学报, 1992, 5(4): 396-402 (Jiang Jianxiang, Huang Youling. Calculation of the eigenfrequency of the fluid-conveying pipes on spring-support. Journal of Vibration Engineering, 1992, 5(4): 396-402 (in Chinese) [21] 倪樵, 黄玉盈, 陈贻平. 微分求积法分析具有弹性支承输液管的临界流速. 计算力学学报, 2001, 18(2): 146-149 (Ni Qiao, Huang Yuying, Chen Yiping. Differential quadrature method for analyzing critical flow velocity of the pipe conveying fluid with spring support. Journal of Computational Mechanics, 2001, 18(2): 146-149 (in Chinese) doi: 10.3969/j.issn.1007-4708.2001.02.004 [22] 李琳, 唐冶, 任正义等. 具有弹性支承输流管道的稳定性和临界流速分析. 哈尔滨工程大学学报, 2010, 31(11): 1509-1513 (Li Lin, Tang Ye, Ren Zhengyi, et al. Stability and critical flow velocity analysis on a fluid-conveying pipeline with elastic support. Journal of Harbin Engineering University, 2010, 31(11): 1509-1513 (in Chinese) [23] Qian Q, Wang L, Ni Q. Nonlinear responses of a fluid-conveying pipe embedded in nonlinear elastic foundations. Acta Mechanica Solida Sinica, 2008, 21(2): 170-176 doi: 10.1007/s10338-008-0820-7
[24] 包日东, 金志浩, 闻邦椿. 端部约束悬臂输流管道的动力学特性. 工程力学, 2009, 26(1): 209-215 (Bao Ridong, Jin Zhihao, Wen Bangchun. Dynamic behaviors of restrained cantilever pipe conveying fluid. Engineering Mechanics, 2009, 26(1): 209-215 (in Chinese) [25] 包日东, 冯颖, 毕文军. 弹性支承输流管道的动力学特性. 机械设计与制造, 2010, 3(3): 129-131 (Bao Ridong, Feng Ying, Bi Wenjun. Dynamic characteristic of pipeline conveying fluid with elastic supports. Machinery Design & Manufacture, 2010, 3(3): 129-131 (in Chinese) doi: 10.3969/j.issn.1001-3997.2010.03.055 [26] 刘俊卿, 王克林. 弹性支承输液管道的临界流速. 应用力学学报, 2003, 20(1): 133-135, 168 (Liu Junqing, Wang Kelin. Critical flow velocity of pipes coneying fluid with elastic supports. Chinese Journal of Applied Mechanics, 2003, 20(1): 133-135, 168 (in Chinese) doi: 10.3969/j.issn.1000-4939.2003.01.030 [27] 吴男, 金基铎. 弹性地基两端铰支输流管的模态和固有频率. 沈阳航空工业学院学报, 2008, 1: 42-46 (Wu Nan, Jin Jiduo. The mode and nature frequency of pinned-pinned pipes conveying fluid with elastic foundations. Journal of Shengyang Institute of Aeronautical Engineering, 2008, 1: 42-46 (in Chinese) [28] Ni Q, Tang M, Luo YY, et al. Internal-external resonance of a curved pipe conveying fluid resting on a nonlinear elastic foundation. Nonlinear Dynamics, 2013, 76(1): 867-886
[29] Ni Q, Wang L, Qian Q. Bifurcations and chaotic motions of a curved pipe conveying fluid with nonlinear constraints. Computers and Structures, 2006, 84(10-11): 708-717 doi: 10.1016/j.compstruc.2005.11.006
[30] 许锋, 郭长青, 黄建红. 弹性支承输流弹性支承输液管道在分布随从力作用下的稳定性. 工程力学, 2014, 31(7): 234-238, 256 (Xu Feng, Guo Changqing, Huang Jianhong. Stability of elastically supported pipes conveying fluid with distributed follower force. Engineering Mechanics, 2014, 31(7): 234-238, 256 (in Chinese) doi: 10.6052/j.issn.1000-4750.2012.10.0763 [31] Li Q, Liu W, Lu K, et al. Nonlinear parametric vibration of a fluid-conveying pipe flexibly restrained at the ends. Acta Mechanica Solida Sinica, 2019, 33(3): 327-346
[32] Ding H, Ji JC, Chen LQ. Nonlinear vibration isolation for fluid-conveying pipes using quasi-zero stiffness characteristics. Mechanical Systems and Signal Processing, 2019, 121: 675-688 doi: 10.1016/j.ymssp.2018.11.057
[33] Semler C, Li GX, Paidoussis MP. The non-linear equations of motion of pipes conveying fluid. Journal of Sound and Vibration, 1994, 169(5): 577-599 doi: 10.1006/jsvi.1994.1035
[34] 刘延柱, 陈立群, 陈文良. 振动力学, 第3版. 北京: 高等教育出版社, 2019: 206-207 Liu Yanzhu, Chen Liqun, Chen Wenliang. Mechanics of Vibrations, 3rd edition. Beijing: Higher Education Press, 2019: 206-207 (in Chinese)
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