﻿ 基于欧拉梁的管路吸振器振动特性研究
 舰船科学技术  2020, Vol. 42 Issue (8): 48-54    DOI: 10.3404/j.issn.1672-7649.2020.08.009 PDF

Research on vibrational characteristics of pipe DVA system based on Euler beam
YAO Wu-ping, PENG Xu, TANG Wen-bing, ZHANG Zhenli, LU Min-yue
Wuhan Second Ship Design and Research Institute, Wuhan 430064, China
Abstract: To control vibration transmission across pipes with dynamic vibration absorber, a continuous pipe-DVA vibration model under transverse excitation is established using Euler beam. The model is testified by finite element methods. Influence of relative location of excitation and DVA, control object and stiffness of support, is investigated. Results show that relative location of excitation and DVA has a great impact on DVA’s effect, modal-objected control differs from excitation-objected control. Stiffness of pipe support can affect pipe’s first modal frequency within a specific range.
Key words: pipe     DVA     Euler beam     elastic support
0 引　言

1 管路吸振器系统振动模型 1.1 管路—吸振器系统振动模型

 图 1 管路—吸振器振动模型 Fig. 1 Pipe-DVA vibration model

 $\rho \frac{{\partial }^{2}y}{\partial {t}^{2}}+EI\frac{{\partial }^{4}y}{\partial {x}^{4}}={d}_{j}\delta \left(x-{x}_{j}\right)+{f}_{i}\delta \left(x-{x}_{i}\right){\text{，}}$ (1)
 ${f}_{i}=c\left(\frac{\partial {y}_{id}}{\partial t}-\frac{\partial {y}_{i}}{\partial t}\right)+k({y}_{id}-{y}_{i}) {\text{，}}$ (2)
 ${f}_{i}=-m\frac{{\partial }^{2}{y}_{id}}{\partial {t}^{2}} {\text{。}}$ (3)

 ${y}_{i}\left(x,t\right)=\sum\nolimits_{n=1}^{\infty }{W}_{n}\left(x\right){q}_{n}\left(t\right) {\text{，}}$ (4)

 $EI\frac{{d}^{4}{W}_{n}\left(x\right)}{d{x}^{4}}-{\omega }_{n}^{2}\rho A{W}_{n}\left(x\right)=0 {\text{。}}$ (5)

 $\begin{split} &\sum\nolimits_{n=1}^{\infty }{W}_{n}\left(x\right)\frac{{d}^{2}{q}_{n}\left(t\right)}{d{t}^{2}}+\sum\nolimits_{n=1}^{\infty }{\omega }_{n}^{2}{{q}_{n}\left(t\right)W}_{n}\left(x\right)=\\& \frac{{d}_{j}\delta \left(x-{x}_{j}\right)+{f}_{i}\delta \left(x-{x}_{i}\right)}{\rho A}{\text{，}} \end{split}$ (6)

 $\frac{{{\rm{d}}}^{2}{q}_{n}\left(t\right)}{{\rm{d}}{t}^{2}}+{\omega }_{n}^{2}{q}_{n}\left(t\right)=\frac{{{W}_{n}\left({x}_{j}\right)d}_{j}+{W}_{n}\left({x}_{i}\right){f}_{i}}{\rho A} {\text{。}}$ (7)

 $-{\omega }^{2}{q}_{n}\left(\omega \right)+{\omega }_{n}^{2}{q}_{n}\left(\omega \right)=\frac{{{W}_{n}\left({x}_{j}\right)d}_{j}\left(\omega \right)+{W}_{n}\left({x}_{i}\right){f}_{i}\left(\omega \right)}{\rho A} {\text{。}}$ (8)

 ${q}_{n}\left(\omega \right)=\frac{{{W}_{n}\left({x}_{j}\right)d}_{j}\left(\omega \right)+{W}_{n}\left({x}_{i}\right){f}_{i}\left(\omega \right)}{\rho A\left({\omega }_{n}^{2}-{\omega }^{2}\right)} {\text{。}}$ (9)

 ${y}_{i}\left(\omega \right)=\sum\nolimits_{n=1}^{\infty }{W}_{n}\left({x}_{i}\right){q}_{n}\left(\omega \right) {\text{，}}$ (10)
 $jc\omega \left[{y}_{id}\left(\omega \right)-{y}_{i}\left(\omega \right)\right]+k\left[{y}_{id}\left(\omega \right)-{y}_{i}\left(\omega \right)\right]=m{\omega }^{2}{y}_{id}\left(\omega \right) {\text{。}}$ (11)

 ${y}_{k}\left(\omega \right)=C{d}_{j}\left(\omega \right)+\frac{AD}{1-B}{d}_{j}\left(\omega \right) {\text{，}}$ (12)

 $\begin{split} &A=\sum\limits_{n=1}^{\infty }{W}_{n}\left({x}_{i}\right)\frac{{W}_{n}\left({x}_{j}\right)}{\rho A\left({\omega }_{n}^{2}-{\omega }^{2}\right)} \text{，}\\&B=\frac{m{\omega }^{2}\left(jc\omega +k\right)}{jc\omega +k-m{\omega }^{2}}\sum\limits_{n=1}^{\infty }\frac{{W}_{n}^{2}\left({x}_{i}\right)}{\rho A\left({\omega }_{n}^{2}-{\omega }^{2}\right)}{\text{，}} \end{split}$
 $\begin{split} &C=\sum\nolimits_{n=1}^{\infty }{W}_{n}\left({x}_{k}\right)\frac{{W}_{n}\left({x}_{j}\right)}{\rho A\left({\omega }_{n}^{2}-{\omega }^{2}\right)} {\text{，}}\\&{{D}}=\frac{m{\omega }^{2}\left(jc\omega +k\right)}{jc\omega +k-m{\omega }^{2}}\bullet \sum\nolimits_{n=1}^{\infty }\frac{{W}_{n}\left({x}_{k}\right){W}_{n}\left({x}_{i}\right)}{\rho A\left({\omega }_{n}^{2}-{\omega }^{2}\right)}{\text{。}}\end{split}$

1.2 弹性支撑边界下欧拉梁的振动方程

 $W\left(x\right)={c}_{1}{e}^{\beta x}+{c}_{2}{e}^{-\beta x}+{c}_{3}{e}^{i\beta x}+{c}_{4}{e}^{-i\beta x} {\text{，}}$ (13)

x=0处， ${{\partial }^{2}W}/{\partial {x}^{2}}=0$ ${\partial \left({EI\partial }^{2}W\right)}/{\partial {x}^{3}}= -{K}_{1}W$ ，即

 ${c}_{1}+{c}_{2}-{c}_{3}-{c}_{4}=0 {\text{，}}$ (14)
 $EI{\beta }^{3}\left({c}_{1}-{c}_{2}-{ic}_{3}+i{c}_{4}\right)+{K}_{1}\left({c}_{1}+{c}_{2}+{c}_{3}+{c}_{4}\right)=0 {\text{。}}$ (15)

x=L处， ${{\partial }^{2}W}/{\partial {x}^{2}}=0$ ${\partial \left({EI\partial }^{2}W\right)}/{\partial {x}^{3}}={K}_{2}W$ ，即

 ${c}_{1}{e}^{\beta L}+{c}_{2}{e}^{-\beta L}-{c}_{3}{e}^{i\beta L}-{c}_{4}{e}^{-i\beta L}=0 {\text{，}}$ (16)
 $\begin{split}& EI{\beta }^{3}\left({c}_{1}{e}^{\beta L}-{c}_{2}{e}^{-\beta L}-{ic}_{3}{e}^{i\beta L}+i{c}_{4}{e}^{-i\beta L}\right)-\\&{K}_{2}\left({c}_{1}{e}^{\beta L}+{c}_{2}{e}^{-\beta L}+{c}_{3}{e}^{i\beta L}+{c}_{4}{e}^{-i\beta L}\right)=0{\text{。}} \end{split}$ (17)

 $\begin{split}&\left[\left(1-{T}_{1}-{T}_{2}\right)cos\beta L+\left({T}_{1}+{T}_{2}-2{T}_{1}{T}_{2}\right)sin\beta L\right]{e}^{\beta L}+\\&\left[\left(1+{T}_{1}+{T}_{2}\right)cos\beta L+\left({T}_{1}+{T}_{2}+2{T}_{1}{T}_{2}\right)sin\beta L\right]{e}^{-\beta L}-2=0{\text{。}} \end{split}$ (18)

2 计算模型数值验证

 图 2 刚性支撑条件下管路振型图 Fig. 2 Pipe model shape under rigid support

 图 4 弹性支撑条件下管路振型图 Fig. 4 Pipe model shape under elastic support

 图 3 自由状态下管路振型图 Fig. 3 Pipe model shape under free boundary

3 管路吸振器控制效果分析 3.1 管路吸振器参数设计

 图 5 中间加装动力吸振器前后安装部位振动响应对比（两端刚性支撑） Fig. 5 Vibration response before and after DVA install (Both ends fixed)

 图 6 中间加装动力吸振器前后安装部位振动响应对比（两端弹性支撑） Fig. 6 Vibration response before and after DVA install (Both ends with elastic support)

3.2 吸振器控制效果影响因素分析

a）吸振器与激励力位置关系

 图 7 激励及吸振器位置对特定模态控制效果的影响 Fig. 7 Influence of DVA position on control effect aimed at specific modal

b）非共振频率控制效果

 图 8 激励及吸振器位置对特定激励控制效果的影响 Fig. 8 Influence of DVA position on control effect aimed at specific excitation

3）管路支撑刚度

 图 9 管路一阶固有频率随支撑刚度的变化曲线 Fig. 9 First characteristic frequency vs support stiffness

4 结　语

1）本文提出的一种管路吸振器的振动特性计算方法，在刚性支撑、自由支撑和弹性支撑等不同类型的边界条件下，管路的前4阶固有频率与有限元计算结果偏差在1.6%以内，振型的计算结果基本一致。

2）针对特定模态频率，如果激励的位置不在振动极值点，那么吸振器离激励位置越远，控制效果越差，甚至没有控制效果；吸振器及激励的位置对管路刚性支撑条件的影响要小于弹性支撑。针对特定频率的激励，吸振器与激励位置重合时有良好的控制效果，如果吸振器的安装位置与激励位置不重合，不论管路两端是刚性还是弹性支撑，那么随着二者之间距离的增大，控制效果越来越差，在某些条件下管路振动甚至放大超过3倍。因此，在应用管路吸振器时，应区分控制对象为特定模态频率还是特定激励；以特定激励为管路系统振动控制对象时，高度重视吸振器的布置位置，尽力使其靠近激励。

3）若弹性影响因素特定区间内，管路1阶固有频率对支撑刚度十分敏感。当 $T$ 大于10时，管路两端类似刚性支撑，支撑刚度的增大对1阶固有频率的影响在10%以内；当 $T$ 小于0.1时，管路两端类似自由状态，支撑刚度的减小对1阶固有频率的影响在1%以内。因此，以管路振动模态频率为控制对象设计吸振器时，需要重视管路两端支撑刚度的影响。

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