第24卷第5期蘇州大學(xué)學(xué)報(bào)工科版Vol 24 No, 52004 410 A JOURNAL OF SOOCHOW UNIVERSITY ENGINEERING SCIENCE EDITION)Oct.2004文章編號(hào):1000-19992004)5-0023-04軸向運(yùn)動(dòng)紗線非線性動(dòng)力學(xué)馮志華朱曉東蘭向軍蘇州大學(xué)機(jī)電工程學(xué)院江蘇蘇州215021)摘要以兩端約束的軸向運(yùn)動(dòng)紗線為分析對(duì)象建立了相應(yīng)的線性波動(dòng)方程。采用 Galerkin截?cái)喾ǐ@得了描述軸向運(yùn)動(dòng)紗線旳橫向振動(dòng)一階近似常微分方程?；诙喑叨确ㄖ攸c(diǎn)研究了系統(tǒng)主參激共振時(shí)系統(tǒng)的穩(wěn)定性問題獲得了穩(wěn)定性臨界曲線的近似解析表達(dá)形式。研究結(jié)果表明穩(wěn)定性區(qū)域與軸向運(yùn)動(dòng)平均速度V及波動(dòng)速度幅值V1密切相關(guān),V與V1越大不穩(wěn)定區(qū)間越寬。關(guān)鍵詞玅線κ galerkin截?cái)噢柖喑叨确ㄇf參激共振穩(wěn)定性中圖分類號(hào)313文獻(xiàn)標(biāo)識(shí)碼0前言在紡織工程的紡、絡(luò)、并、捻、槳等工序中紗紗線大多以軸向運(yùn)動(dòng)為主體完成相應(yīng)的工序隊(duì)從力學(xué)模型看此類狀況的紗線可近似作為一定張力下的軸向運(yùn)動(dòng)弦處理此時(shí)紗線的運(yùn)動(dòng)或動(dòng)力行為有可能影響其工藝乃至最終產(chǎn)品的性能與質(zhì)量。本文以前述紗線為分析對(duì)象通過研究期望了解并掌握相關(guān)非線性動(dòng)力行為〔特別是動(dòng)力穩(wěn)定性躊性為有關(guān)工程技術(shù)提供理論指導(dǎo)。近年來許多學(xué)者已對(duì)軸向運(yùn)動(dòng)弦的振動(dòng)進(jìn)行了研究取得了一定的成果。 Huang等涉及運(yùn)動(dòng)弦三維方向振動(dòng)的穩(wěn)定性問題1] Mochensturr等導(dǎo)出了弦振動(dòng)時(shí)的非平凡響應(yīng)極限環(huán)的幅值及其穩(wěn)定性的解析表達(dá)形式3 Pakdemirli、Usoy采用rloqμet理論數(shù)值分析了弦振動(dòng)響應(yīng)的穩(wěn)定性問題3并確定了軸向加速弦的穩(wěn)定性邊界4 Suweken等構(gòu)造傳送帶振動(dòng)的漸近近似解以顯現(xiàn)系統(tǒng)復(fù)雜的動(dòng)力行為51,agat46、Chen等7用數(shù)值或解析分析了靜止或運(yùn)動(dòng)弦的分岔及混沌現(xiàn)象。本文在導(dǎo)岀軸向運(yùn)動(dòng)紗線的橫向振動(dòng)方程的基礎(chǔ)上采用 Galerkin截?cái)喾ú⒔Y(jié)合多尺度法重點(diǎn)研究了系統(tǒng)主參激共振時(shí)的穩(wěn)定性特性動(dòng)力學(xué)模型考慮圖1所示軸向運(yùn)動(dòng)紗線模型紗線模型化為弦)在兩個(gè)固定羅拉間以恒定的初始張力沿x方向以速度Ⅸt)運(yùn)動(dòng)現(xiàn)只考慮其在α方向存在橫向振動(dòng)且其振動(dòng)的位移為τ(x)根據(jù)文獻(xiàn)5相應(yīng)的動(dòng)力學(xué)方程為wn t 2V0式中c=√T/p為紗線的波速,Io及ρ為相應(yīng)的恒定初下標(biāo)x、t分別代表對(duì)位置及時(shí)間的導(dǎo)數(shù)。假定L為兩羅拉在x=0及x=L處無橫向振動(dòng)即邊界條件為m YHg中國(guó)煤化oCNMHG圖1軸向運(yùn)動(dòng)紗線模型收稿日期204-08-01擇看翻單月起學(xué)禽班默接坐要計(jì)列方向?yàn)榉蔷€性動(dòng)力學(xué)與控制24蘇州大學(xué)學(xué)報(bào)工科版)第5期(0t)=0;(L(2)在下述分析中紗線的軸向運(yùn)動(dòng)速度Vt)設(shè)定為v(t)=Vo+ VIcodnt(3)式中V為平均速度,V1、a分別為速度間諧波動(dòng)的幅值及頻率。將上式代入式1)有wm,+(vc2 )Wrr 2vowzt+2Vo VIco nt wmr-nVIsir( Q2t )w+ vicos( ot )u=0(4)2穩(wěn)定性分析假定式4)滿足邊界條件式(2)的解為下列級(jí)數(shù)的展開形式ki.xx,t)=(5)式中(1)及()分別為紗線第k階廣義位移及振型函數(shù)現(xiàn)著重分析紗線橫向振動(dòng)的一階振型近似展開。將式5)代入式4)并采用 Galerkin截?cái)喾ㄓ衠1+ic-v6 )q1-212 Vo VIco at l L2 icos( nt a=0(6)并引入無量綱變量T式中ε為一小參數(shù)滿足0<≤1將式7)代入式6)得(1-y22y cod or )q1-Ety2a cos( ar )a在紡織工程中紗線的軸向運(yùn)動(dòng)速度V(t)通常要遠(yuǎn)小于其波速c即y≤1若y>1一種不太實(shí)際的情況)式8)所述系統(tǒng)將失穩(wěn)。設(shè)a=1-y2并代入式8)得622y2acod(y1=0(9)采用多尺度法對(duì)式9)進(jìn)行一次近似展開設(shè)q t e)=Eqd To ,T2)+equ To, T2)(10)式中To=τ,T2=ε2r。將式10)代入式9)并令方程兩邊有關(guān)ε同階次的系數(shù)相同得D5q10+o6q10=0Doqu +woqu =-2DoD2910+2y acod ato x(12)式中D、D2分別表示對(duì)T0T2的導(dǎo)數(shù)。從式11)可得910= AC T2 xp iwo To )+cc(13)式中c代表前項(xiàng)共軛下同將式13)代入式12)得Doqu+0qu=-2iwo A exd iwo To)+ya fAex (o+wo To ]+Aex (o-wo )To 1)+ cc( 14)本文只限研究系統(tǒng)的主參激共振由此設(shè)頻率調(diào)諧參數(shù)滿足(15)并代入式14)消除qn中的永年項(xiàng)得中國(guó)煤化工2iwgA'-y2aAex( icCNMHG(16)引入笛卡爾坐標(biāo)變換A(T2)TT2)=iT2)]x(2并代入式摒離實(shí)、虛部得穩(wěn)態(tài)相應(yīng)的調(diào)制方程組為第24卷馮志華朱曉東蘭向軍軸向運(yùn)動(dòng)紗線非線性動(dòng)力學(xué)(18)因此從式18)中可方便地獲得系統(tǒng)穩(wěn)定性的臨界曲線的解析表達(dá)式為并代入式15)有設(shè)=最終式20)成為(21)0.10y=0.10.100.10y=0.2y=03unstablea0.050050.05stablestable0.000.002.00.100.10y=0.50.1010.050.000.001.52.0圖2系統(tǒng)隨參數(shù)β變化的穩(wěn)定區(qū)域圖B=0050.6B=0.1N0.30.0中國(guó)煤化工CNMHG圖3系統(tǒng)隨參數(shù)γ變化的穩(wěn)定區(qū)域圖蘇州大學(xué)學(xué)報(bào)工科版)第5期圖2顯示了系統(tǒng)在不同的參數(shù)x(無量綱平均速度Vo)下隨參數(shù)無量綱速度幅值v1)變化的穩(wěn)定區(qū)域變化圖類似地圖3顯示了系統(tǒng)在不同的參數(shù)β下隨參數(shù)γ變化的穩(wěn)定性區(qū)域變化圖。從圖中可發(fā)現(xiàn)系統(tǒng)不穩(wěn)定區(qū)域隨參數(shù)β及γ的增大而增大換言之減小VaV有利于軸向運(yùn)動(dòng)紗線的穩(wěn)定性。3結(jié)論本文在所建立的軸向運(yùn)動(dòng)紗線橫向振動(dòng)運(yùn)動(dòng)方程基礎(chǔ)上基于 Galerkin截?cái)喾ú⒔Y(jié)合多尺度法詳細(xì)分析了系統(tǒng)一階近似情況下的主參激共振的穩(wěn)定性問題獲得了穩(wěn)定性旳臨界解析表達(dá)形式。結(jié)果表明不穩(wěn)定性區(qū)域與Va、V1緊密相關(guān)隨著V與V1的增加不穩(wěn)定區(qū)域變寬。參考文獻(xiàn)1] HUANG JS FUNG R F IN C H Dynamic stability of a moving string unergoing three-dimensional vibration J International Journal of Mechanical Science 1995 3x 2): 145-160[2] Mochensturm E M Perkins N C Ulsoy A G Stability and limit eyeles of parametrically exited axially moving string[ J ] ASME Journal of Vi-bration and Acoustics 1996, 1163)346-351[ 3] Parkdemirli M soy A G Ceranoglu A. Transverse vibration of an axially accelerating string[ J ] Journal of Sound and Vibration 1994, 1692)179-196[ 4] Pakdemirli M lsoy A G Stability analysis of an axially accelerating string J I Jourmal of Sound and Vibration 1997 2035)815-532[5] Suweken G, Van Horssen W T On the transversal vibrations of a conveyor belt with a low and time- varying veloctiy[ J I Part 1 :the string-likecase. Journal of Sound and Vibration 2003 264(2): 117-133[6] Tagata G. Parametric oscillations of a non-linear string J ] Journal of Sound and Vibration 1995, 181)51-78[7] Chen L Q Zhang N H u J W. Bifurcation and chaos of an axially moving viscoelastic string J I Mechanics Research Communications 2002 29(1)81-90Nonlinear Dynamics of an Axially Moving YarnFENG Zhi-hua . HU Xiao-dong , LAN XiCollege of Mechanical and Electronic Engineering of Sorchomwe University Suzhou 215021 ChinaAbstract The initial-boundary-value problems of a linear wave equation of an axially moving yarn are consideredThe ordinary differential equation of the first order approximate motion is derived based on the galerkin truncation to deseribe the transversal vibration of the yarn and the principal parametric resonance of the system is thenanalyzed using the method of multiple scales. The equations of approximate transition curves in the plane of thedimensionless frequency and excitation parameter that separate stable from unstable solution are derived Resultsshow that the unstalbe region is closely related to the average velocity Vo and the small speed amplitude of theharmonic motion VI of the yarn ,that is with the increase of both Vo and VI the unstable region become widerKey Words yarn Galerkin truncation ,the method of multiple scales principal parametric resonance ' stability中國(guó)煤化工CNMHG