浸液轴向变速运动黏弹性板的组合参数共振
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摘要
针对浸没于液体中的轴向运动黏弹性板,考虑其速度发生扰动变化,根据经典薄板理论以及达朗贝尔原理,得到该系统的横向振动控制微分方程。假定液体为无黏、无旋、不可压缩的理想流体,流体对板的动压力由板的附加质量来描述。采用多尺度法,分析系统的偏微分方程及边界条件。根据可解性条件及Routh-Hurwitz判据,确定系统和式组合共振与差式组合共振的失稳区域,并讨论不同参数对系统两种组合共振失稳区间的影响。
An axially moving viscoelastic plate immersed in the liquid, having the variable speed, was considered. According to the classical thin plate theory and the d'Alembert's principle, the governing equation of the transverse vibration of the system was derived. The liquid was assumed as ideal fluid and thus was inviscid, irrotational, and incompressible. The dynamic pressure of fluid on the plate could be described by added plate mass. Then, using the method of multiple scales, we analyzed the partial differential equations and boundary conditions of the system. Based on the solvability conditions and the Routh-Hurwitz criterion, the instability regions for sum-type and difference-type combination resonances of the system were determined. Finally, the effects of different parameters on the instability regions of the two kinds of combination resonance were discussed.
An axially moving viscoelastic plate immersed in the liquid, having the variable speed, was considered. According to the classical thin plate theory and the d'Alembert's principle, the governing equation of the transverse vibration of the system was derived. The liquid was assumed as ideal fluid and thus was inviscid, irrotational, and incompressible. The dynamic pressure of fluid on the plate could be described by added plate mass. Then, using the method of multiple scales, we analyzed the partial differential equations and boundary conditions of the system. Based on the solvability conditions and the Routh-Hurwitz criterion, the instability regions for sum-type and difference-type combination resonances of the system were determined. Finally, the effects of different parameters on the instability regions of the two kinds of combination resonance were discussed.
引文
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[2]冯志华,胡海岩.内共振条件下直线运动梁的动力稳定性[J].力学学报,2002,34(3):389-400.FENG Zhihua,HU Haiyan.Dynamic stability of a slender beam with internal resonance under a large linear motion[J].Chinese Journal of Theoretical and Applied Mechanics,2002,34(3):389-400.
[3]WANG Y,ZU J W.Nonlinear oscillations of sigmoid functionally graded material plates moving in longitudinal direction[J].Applied Mathematics and Mechanics,2017,38(11):1533-1550.
[4]CHEN L Q,ZU J W.Solvability condition in multi-scale analysis of gyroscopic continua[J].Journal of Sound and Vibration,2008,309(1):338-342.
[5]ZHANG W,GAO M,YAO M.Global analysis and chaotic dynamics of six-dimensional nonlinear system for an axially moving viscoelastic belt[J].International Journal of Modern Physics B,2011,25(17):2299-2322.
[6]ZHANG W,SONG C.Higher-dimensional periodic and chaotic oscillations for viscoelastic moving belt with multiple internal resonances[J].International Journal of Bifurcation and Chaos,2007,17(5):1637-1660.
[7]张宇飞,王延庆,闻邦椿.轴向运动层合薄壁圆柱壳内共振的数值分析[J].振动与冲击,2015,34(22):82-86.ZHANG Yufei,WANG Yanqing,WEN Bangchun.Internal resonance of axially moving laminated thin cylindrical shells[J].Journal of Vibration and Shock,2015,34(22):82-86.
[8]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.
[9]DING H,CHEN L Q,YANG S P.Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load[J].Journal of Sound and Vibration,2012,331(10):2426-2442.
[10]HUANG J L,SU R K L,LI W H,et al.Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances[J].Journal of Sound and Vibration,2011,330(3):471-485.
[11]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.
[12]GHAYESH M H,AMABILI M.Nonlinear stability and bifurcations of an axially moving beam in thermal environment[J].Journal of Vibration and Control,2015,21(15):2981-2994.
[13]张伟,温洪波,姚明辉.黏弹性传动带1∶3内共振时的周期和混沌运动[J].力学学报,2004,36(4):443-454.ZHANG Wei,WEN Hongbo,YAO Minghui.Periodic and chaotic oscillation of a parametrically excited viscoelastic moving belt with 1∶3 internal resonance[J].Chinese Journal of Theoretical and Applied Mechanics,2004,36(4):443-454.
[14]YANG X D,ZHANG W.Nonlinear dynamics of axially moving beam with coupled longitudinal-transversal vibrations[J].Nonlinear Dynamics,2014,78(4):2547-2556.
[15]WANG L,NI Q.Vibration and stability of an axially moving beam immersed in fluid[J].International Journal of Solids and Structures,2008,45(5):1445-1457.
[16]WANG Y Q,XUE S W,HUANG X B,et al.Vibrations of axially moving vertical rectangular plates in contact with fluid[J].International Journal of Structural Stability and Dynamics,2016,16(2):1450092.
[17]WANG Y Q,HUANG X B,LI J.Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process[J].International Journal of Mechanical Sciences,2016,110:201-216.
[18]张宇飞,王延庆,闻邦椿.浸液轴向运动板的非线性自由振动和内共振分析[J].振动与冲击,2017,36(18):36-42.ZHANG Yufei,WANG Yanqing,WEN Bangchun.Analysis on the nonlinear free vibration and internal resonance of axially moving plates immersed in liquid[J].Journal of Vibration and Shock,2017,36(18):36-42.