变密度印刷纸带的非线性自由振动特性分析
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摘要
关于变密度纸带振动的研究多数局限于小挠度线性问题的讨论,对于变密度印刷纸带大挠度非线性振动问题的研究很少。然而,在实际中,由于印刷图文的分布不同纸带的密度是变化的。研究变密度运动印刷纸带的非线性振动特性。基于von Karman薄板理论推导出轴向运动纸带大挠度振动方程,应用Bubnov-Galerkin方法对振动偏微分方程组进行离散,利用椭圆积分法对微分方程进行求解,得出纸带非线性振动的频率表达式。分析了不同初始条件下,密度系数,无量纲运动速度和长宽比对运动纸带大挠度振动复频率的影响。该研究为提高印刷设备的工作稳定性提供理论依据。
Most studies on the vibration of variable density paper web are limited to the discussion of small deflection linear problems. There are few researches on nonlinear vibration problems with large deflection in variable density printing web.However,in practice,the density of the web is varied due to the difference in the distribution of the printed image. In this paper,the nonlinear vibration characteristics of moving printing web with variable density are studied. Based on the von Karman plate theory,the vibration equation of the large deflection motion of the axially moving paper is deduced. The differential equation is divided by the Bubnov-Galerkin method,and the differential equation is solved by elliptic integral method. The frequency expression of the nonlinear vibration is obtained. The influence of density coefficient,dimensionless velocity and aspect ratio on the complex frequency of large deflection vibration of moving paper under different initial conditions is analyzed. This study provides a theoretical basis for improving the stability of printing equipment.
Most studies on the vibration of variable density paper web are limited to the discussion of small deflection linear problems. There are few researches on nonlinear vibration problems with large deflection in variable density printing web.However,in practice,the density of the web is varied due to the difference in the distribution of the printed image. In this paper,the nonlinear vibration characteristics of moving printing web with variable density are studied. Based on the von Karman plate theory,the vibration equation of the large deflection motion of the axially moving paper is deduced. The differential equation is divided by the Bubnov-Galerkin method,and the differential equation is solved by elliptic integral method. The frequency expression of the nonlinear vibration is obtained. The influence of density coefficient,dimensionless velocity and aspect ratio on the complex frequency of large deflection vibration of moving paper under different initial conditions is analyzed. This study provides a theoretical basis for improving the stability of printing equipment.
引文
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[17]武吉梅,陈媛,王砚,等.基于微分求积法的印刷运动薄膜动力稳定性分析[J].振动与冲击,2015(20):57-60.WU Ji Mei,CHEN Yuan,WANG Yan,et al.Dynamic stability of printing moving membrance based on differential quadrature method[J].Journal of Vibration and Shock,2015,34(20):57-60(In Chinese).
[18]徐芝伦.弹性力学(下)[M].第四版.北京:高等教育出版社,2006:142-144.XU Zhi Lun.Elastic mechanics(Part II)[M].Fourth edition.Beijing:Higher Education Press,2006:142-144(In Chinese).
[2]Jiang W A,Chen L Q,Ding H.Internal resonance in axially loaded beam energy harvesters with an oscillator to enhance the bandwidth[J].Nonlinear Dynamics,2016:1-14.
[3]Tang Y Q,Chen L Q.Internal-external combination resonance of nonlinear vibration of in-plane translating viscoelastic plates[J].Applied Mathematics and Mechanics,2013,34(5):480-487.
[4]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.
[5]刘金堂,杨晓东,闻邦椿等.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):11-15.LIU Jin Tang,YANG Xiao Dong,WEN BangChun,et al.Transverse vibration of an axially moving plate based on differential quadrature method[J].Journal of Vibration and Shock,2009,28(3):178-181(In Chinese).
[6]李银山,陈予恕,吴志强.正交各向异性圆板非线性振动的亚谐分岔[J].机械强度,2001,23(2):148-151.LI YinShan,CHEN ShuYu,WU Zhi Qiang.Subharmonic bifurcation of the nonlinear forced flexural vibration of elastic circular plates exhibiting rectilinear orthotropy[J].Journal of Mechanical Strength,2001,23(2):148-151(In Chinese).
[7]Soares R M,Gon9alves P B.Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane[J].International Journal of Solids&Structures,2012,49(3-4):514-526.
[8]李晓军,陈立群.轴向运动简支-固支梁的横向振动和稳定性[J].机械强度,2006,28(5):654-657.LI Xiao Jun,CHEN Li Qun.Transverse vibration and stability of an axially moving beam with pinned and fixed ends[J].Journal of Mechanical Strength,2006,28(5):654-657(In Chinese).
[9]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.
[10]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):768-771.ZHAO FengQun,WANG ZhongMin.Nonlinear vibration analysis of a moving rectangular membrane[J].Mechanical Sciense and Technology for Aerospace Engineering,2010,29(6):768-771(In Chinese).
[11]戈新生,李德双.轴向运动带的横向与纵向非线性振动[J].机械强度,2012,34(1):20-24.GE XinSheng,LI De Shuang.Nonlinear vibration of axially moving strip with the transverse and longitudinal[J].Journal of Mechanical Strength,2012,34(1):20-24(In Chinese).
[12]Jabareen M,Eisenberger M.Free vibrations of non-homogeneous circular and annular membranes[J].Journal of Sound and Vibration,2001,240(3):409-429.
[13]Ma L E,Wu J M,Mei X S,et al.Active vibration control of moving web with varying density[J].Journal of Low Frequency Noise,Vibration and Active Control,2013,32(4):323-334.
[14]吴晓.变密度环形薄膜的轴对称振动修正摄动解[J].振动与冲击,2002,21(4):59-60.WU Xiao.Natural vibration analysis of films with middle supports[J].Journal of Vibration and Shock,2006,25(1):140-141.(In Chinese)
[15]李世荣.非均匀变密度圆形和环形薄膜的轴对称振动[J].兰州理工大学学报,2000,26(2):98-102.LI ShiRong.Axisymmetric vibrations of circular and annular membranes with non-uniformly varying density,Journal of Lanzhou University of Technology[J].2000,26(2):98-102.(In Chinese)
[16]李世荣.连续变密度圆形和环形薄膜的轴对称振动[J].振动与冲击,2000,19(2):52-53.LI ShiRong.Axisymmetric vibrations of circular and annular membranes with continuously varying density[J].Journal of Vibration and Shock,2000,19(2):52-53.(In Chinese)
[17]武吉梅,陈媛,王砚,等.基于微分求积法的印刷运动薄膜动力稳定性分析[J].振动与冲击,2015(20):57-60.WU Ji Mei,CHEN Yuan,WANG Yan,et al.Dynamic stability of printing moving membrance based on differential quadrature method[J].Journal of Vibration and Shock,2015,34(20):57-60(In Chinese).
[18]徐芝伦.弹性力学(下)[M].第四版.北京:高等教育出版社,2006:142-144.XU Zhi Lun.Elastic mechanics(Part II)[M].Fourth edition.Beijing:Higher Education Press,2006:142-144(In Chinese).