运动薄膜的速度对非线性强迫振动的影响研究
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摘要
研究了运动薄膜的速度对非线性强迫振动的影响。基于Von Karman薄板理论推导出轴向运动薄膜大挠度振动方程,应用Galerkin方法对振动偏微分方程组进行离散,得到系统的状态方程,采用4阶Runge-Kutta法对系统状态方程进行数值求解,利用分岔图分析了薄膜非线性振动特性与速度的关系,得到了薄膜产生混沌的区间和稳定工作区间。通过时程图、相图、Poincare截面图和功率谱分析系统的周期运动和混沌运动。
The effect of the velocity of moving membrane on the nonlinear forced vibration is investigated in this paper. Large deflection vibration equation of an axially moving membrane is deduced by using the Von Karman thin plate theory. Galerkin method is applied to discretize vibration differential equations of the membrane,and then the state equation of the system is obtained. The state equation of the system is numerically solved by the fourth order Runge-Kutta method,and the relationship between the nonlinear vibration characteristics and the velocity of the membrane is analyzed by using the bifurcation diagram. The chaotic region and stable working region of the moving membrane are obtained. The periodic motion and chaos motion of the system are analyzed by time histories,phaseplane portraits,Poincare maps and power spectrum.
The effect of the velocity of moving membrane on the nonlinear forced vibration is investigated in this paper. Large deflection vibration equation of an axially moving membrane is deduced by using the Von Karman thin plate theory. Galerkin method is applied to discretize vibration differential equations of the membrane,and then the state equation of the system is obtained. The state equation of the system is numerically solved by the fourth order Runge-Kutta method,and the relationship between the nonlinear vibration characteristics and the velocity of the membrane is analyzed by using the bifurcation diagram. The chaotic region and stable working region of the moving membrane are obtained. The periodic motion and chaos motion of the system are analyzed by time histories,phaseplane portraits,Poincare maps and power spectrum.
引文
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[15] Banichuk N,Jeronen J,Kurki M,et al. On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates[J]. International Journal of Solids and Structures, 2011,48(13):2015-2025
[16] Wu J M,Lei W J,Wu Q M,et al. Transverse vibration characteristics and stability of a moving membrane with elastic supports[J]. Journal of Low Frequency Noise,Vibration and Active Control,2014,33(1):65-77
[2] Chen L Q,Yang X D. Steady-state response of axially moving viscoelastic beams with pulsating speed:comparison of two nonlinear models[J]. International Journal of Solids and Structures,2005,42(1):37-50
[3] Yang X D,Chen L Q. Bifurcation and chaos of an axially accelerating viscoelastic beam[J]. Chaos,Solitons&Fractals,2005,23(1):249-258
[4] Ding H,Tan X,Zhang G C,et al. Equilibrium bifurcation of high-speed axially moving Timoshenko beams[J].Acta Mechanica,2016,227(10):3001-3014
[5] Yang X D,Zhang W,Chen L Q,et al. Dynamical analysis of axially moving plate by finite difference method[J]. Nonlinear Dynamics, 2012,67(2):997-1006
[6] Ghayesh M H,Amabili M,Pa6doussis M P. Nonlinear dynamics of axially moving plates[J]. Journal of Sound and Vibration,2013,332(2):391-406
[7]刘金堂,杨晓东,闻邦椿.轴向变速运动大挠度薄板的非线性动力学行为[J].振动与冲击,2012,31(11):11-15Liu J T, Yang X D, Wen B C. Nonliear dynamic behaviors of an axially accelerating large deflection thin plate[J]. Journal of Vibration and Shock,2012,31(11):11-15(in Chinese)
[8]刘金堂,杨晓东,张宇飞.轴向运动大挠度板的非线性动力学行为[J].工程力学,2011,28(10):58-64Liu J T,Yang X D,Zhang Y F. Nonlinear dynamical behaviors of axially moving large deflection plates[J].Engineering Mechanics, 2011,28(10):58-64(in Chinese)
[9] Tang Y Q,Zhang D B,Rui M H,et al. Dynamic stability of axially accelerating viscoelastic plates with longitudinally varying tensions[J]. Applied Mathematics and Mechanics,2016,37(12):1647-1668
[10]高阳,王三民,刘晓宁.一种改进的增量谐波平衡法及其在非线性振动中的应用[J].机械科学与技术,2005,24(6):663-665Gao Y,Wang S M,Liu X N. An improved IHB method and its application to non-linear vibration[J].Mechanical Science and Technology,2005,24(6):663-665(in Chinese)
[11]鲍和云,朱如鹏,靳广虎,等.基于增量谐波平衡法的星型齿轮传动非线性动力学分析[J].机械科学与技术,2008,27(8):1038-1042Bao H Y,Zhu R P,Jin G H,et al. Nonlinear dynamic analysis of a 2-stage star gear train based on incremental harmonic balance method[J]. Mechanical Science and Technology for Aerospace Engineering,2008,27(8):1038-1042(in Chinese)
[12] Lin C C,Mote C D. Equilibrium displacement and stress distribution in a two-dimensional,axially moving web under transverse loading[J]. Journal of Applied Mechanics,1995,62(3):772-779
[13] Marynowski K. Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension[J].Chaos,Solitons&Fractals,2004,21(2):481-490
[14]赵凤群,王忠民.运动矩形薄膜的非线性振动分析[J].机械科学与技术,2010,29(6):768-771Zhao F Q,Wang Z M. Nonlinear vibration analysis of a moving rectangular membrane[J]. Mechanical Science and Technology for Aerospace Engineering,2010,29(6):768-771(in Chinese)
[15] Banichuk N,Jeronen J,Kurki M,et al. On the limit velocity and buckling phenomena of axially moving orthotropic membranes and plates[J]. International Journal of Solids and Structures, 2011,48(13):2015-2025
[16] Wu J M,Lei W J,Wu Q M,et al. Transverse vibration characteristics and stability of a moving membrane with elastic supports[J]. Journal of Low Frequency Noise,Vibration and Active Control,2014,33(1):65-77