正交异性薄膜非线性振动分析

PDF(1040 KB)

振动与冲击 ›› 2018, Vol. 37 ›› Issue (12) : 252-259.

论文

正交异性薄膜非线性振动分析

何泽青1,2,张冬辉1,2,宋林1,2,栗颖思2,王生2

作者信息 +

Nonlinear vibration analysis of orthotropic membrane

HE Zeqing1,2, ZHANG Donghui1,2 , SONG Lin1,2, LI Yingsi2, WANG Sheng2

Author information +

文章历史 +

摘要

根据大挠度理论建立了正交异性薄膜的非线性动力学方程并对其展开分析。首先根据薄膜动力学特性建立薄膜振动问题的控制方程组；其次根据物理和边界条件对方程组进行简化和解算，得出其线性解析解和非线性解析解。再次，利用伽辽金算法得出薄膜非线性振动频率的近似解。最后，通过算例分析验证了近似解的准确性和精度，并对其误差进行了分析。分析结果表明近似解结构简单，且具有较高的精度和较大的适用范围，能够为膜结构工程设计提供理论计算依据。

Abstract

A nonlinear kinetic equation of orthotropic membrane was established and analyzed according to the large deflection theory. Firstly, the control equations of the vibration of the membrane were established in line with its dynamic characteristics. Secondly, the equations were simplified based on the physical and boundary conditions, and then, its linear solution and nonlinear solution were obtained. Thirdly, the approximate frequency values of othotropic membrane were obtained by using the Galerkin algorithm. Finally, the approximate solutions were verified by the analysis of an example, and at the same time, the deviation was analyzed. The analysis result show that the approximate solution is simple, and has high precision and large range application. The result could provide theoretical basis for the design of membrane structure.

Key words

Nonlinear vibration;Duffing equation / Galerkin algorithm

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用

何泽青1,2,张冬辉1,2,宋林1,2,栗颖思2,王生2. 正交异性薄膜非线性振动分析[J]. 振动与冲击, 2018, 37(12): 252-259

HE Zeqing1,2, ZHANG Donghui1,2,SONG Lin1,2, LI Yingsi2, WANG Sheng2. Nonlinear vibration analysis of orthotropic membrane[J]. Journal of Vibration and Shock, 2018, 37(12): 252-259

参考文献

[1]陈务军.膜结构工程设计[M].北京:中国建筑工业出版社,2004,134-139.
CHEN Wu-Jun. Design of membrane structure engineering[M]. Beijing, China architecture & building press, 2004,134-139.
[2] S Timoshenko, S H Young, W Weaver. Vibration Problems in Engineering (4th ed.)[M]. John Wiley &Sons, 1974, 327-334.
[3]林文静,陈树辉,李森.圆形薄膜自由振动的理论解[J].振动与冲击, 2009,28(5):84-86.
LIN Wen-jing, CHEN Shu-hui, LI Sen. Analytical solution of the free vibration of circular membrane[J].Journal of vibration and shock, 2009,28(5):84-86.
[4] QIAN Guo-zhen. Solution for free vibration problem of membrane with unequal tension in two directions[J].Applied mathematics and mechanics(English Edition,Vol.3,No.6),HUST
Press,1982
[5]林文静,陈树辉.平面薄膜自由振动的有限元分析[J].动力学与控制学报,2010,8(3):202-206.
LIN Wen-jing, CHEN Shu-hui. Free vibration analysis of plane membranes by finite element method[J]. Journal of dynamics and control, 2010,8(3):202-206.
[6] 刘充,李玉宇,保宏,等.边界几何参数对空间平面张拉膜结构固有频率影响研究[J].振动与冲击,2015, 34(20):198-202.
LIU Chong, LI Yu-yu, BAO Hong, et al. Natural frequencies of pre-tensioned membrane structure with different boundary geometrical parameters [J]. Journal of vibration and shock, 2015,34(20):198-202.
[7]张俊生.薄膜二维振动数理方程的推导与求解[J].榆林学院学报,2006,16(6):29-31.
ZHANG Jun-sheng. Inferential reasoning solution of thin film 2-D vibration M&P equation [J]. Journal of YuLin University,2006,16(6):29-31.
[8]周一峰. 强非线性系统周期解的能量法[J].力学季刊,2002,23(4):514-520.
ZHOU Yi-feng. Energy iteration method for analytic periodic solutions of full strongly nonlinear vibration systems[J]. Chinese quarterly mechanics, 2002, 23(4):514-520.
[9]张琪昌,郝淑英,陈予恕. 用范式理论研究强非线性振动问题[J]．振动工程学报, 2000, 13( 3):481－486．
ZHANG Qi-chang, HAO Shu-ying, CHEN Yu-shu. Study on strongly non-linear vibration systems by normal form theory [J]. Journal of vibration engineering, 2000,13( 3):481－486.
[10]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, 2015,34(20): 57-60.
WU Ji-mei, CHEN Yuan, WANG Yan, et al. Dynamic stability of printing moving membrane based on differential quadrature method [J]. Journal of vibration and shock, 2015, 34(20): 57-60.
[11]徐兆,詹杰民．强非线性振子的受迫振动[J]．中山大学学报:自然科学版,1995,34(2):1-6．
XU Zhao, ZHAN Jie-min. Forced oscillations of strongly nonlinear oscillators [J].Journal of Sun Yat-sen university: Natural Science, 1995,34(2):1-6．
[12]余志祥,赵雷.张拉膜结构自振特性研究[J].西南交通大学学报,2004,39(6):734-739.
YU Zhi-xiang, ZHAO Lei. Research on free vibration properties of membrane structure [J]. Journalofsouthwestjiaotonguniversity, 2004,39(6):734-739.
[13] CHEN Shan-lin, ZHENG Zhou-lian. Large deformation of circular membrane under the concentrated force [J]. Applied Mathematics and Mechanics (English edition), 2003, 24(1):
28-31.
[14] WANG Jing. Nonlinear free vibration of the circular plate with large deflection [J]. Journal of South ChinaUniversity of Technology, 2001, 29(8): 4-6.
[15] LIUChang-jiang,ZHENGZhou-lian, YANGXiao-yan, et al. Nonlinear damped vibration of pre-stressed orthotropic membranestructure under impact loading [J]. InternationalJournal of Structural Stability and Dynamics, 2014, Article ID: 1350055.
[16] 乔磊,谭峰,杨庆山.薄膜结构的动力反应分析[J].振动与冲击, 2011,30(6): 109-113.
QIAOLei, TAN Feng, YANG Qing-shan. Dynamic analysis of membrane structures [J].Journal of vibration and shock, 2011, 30(6): 109-113.
[17]徐芝纶.弹性力学[M].第四版.北京,高等教育出版社,2006,32-36.
XU Zhi-lun. Elasticity[M]. Beijing: Higher education press, 2006,32-36.
[18] ZHENG Zhou-lian, LIU Chang-jiang, HE Xiao-ting, et al. Free vibration analysis of rectangular orthotropic membranes in large deflection [J]. Mathematical Problems in Engineering, 2009, Article ID: 634362.
[19] 周纪卿,朱因远.非线性振动[M].西安：西安交通大学出版社,2001,140-144.
ZHOU Ji-qing, ZHU Yin-yuan. Nonlinear Vibrations[M]．Xi’an: Xi’an jiaotong university press, 2001,140-144.
[20]刘延柱,陈立群．非线性振动[M]．北京:高等教育出版社,2001,59-60．
LIU Yan-zhu, CHEN Li-qun. Nonlinear Vibrations[M]．Beijing: Higher education press, 2001, 59-60.