张拉平面膜结构非线性振动频率计算分析
Computation and Analysis for the Frequency of Nonlinear Vibration of Tensioned Plane Membrane Structure

作者: 黄从兵 * , 宦洪彬 :国内贸易工程设计研究院,北京; 刘衍华 :中国建筑第二工程局有限公司西南分公司,重庆; 王 琦 :基准方中建筑设计有限公司,四川 成都;

关键词: 膜结构非线性振动正交异性摄动法Membrane Structure Nonlinear Vibration Orthotropic Perturbation Method

摘要:
利用冯∙卡门薄膜大挠度理论,结合达朗贝尔原理,建立正交异性张拉平面膜结构非线性自由振动的控制方程。然后利用伽辽金法对其进行简化,并采用同伦摄动法进行求解,得到振动频率的近似解析解。通过算例,计算了结构的非线性振动频率,并将本文结果与精确解进行比较分析。分析表明:本文所求得的近似解析解与精确解之间的最大误差小于4%。因此本文的近似解析解与精确解非常接近,且本文所得解形式更为简单,计算也更方便,有利于在工程中进行推广应用。

Abstract: The nonlinear free vibration governing differential equations for the orthotropic tensioned plane membrane structure are established by Von Kármán’s membrane large deflection theory and D’Alembert’s principle. Then the governing differential equations are simplified by Bubnov-Ga- lerkin method and solved by the homotopy perturbation method (HPM), and obtained the ap-proximate analytical solution of the vibration frequency. In the computational example, the non-linear vibration frequency of the structure is computed, and the results of this paper are analyzed and compared with the exact solution. The analysis shows that the approximate analytical solution obtained in this paper is very close to the exact solution (the maximum error is less than 4%), and the approximate analytical solution obtained in this paper is more simple and convenient. This is favorable for the popularization and application in engineering.

文章引用: 黄从兵 , 宦洪彬 , 刘衍华 , 王 琦 (2015) 张拉平面膜结构非线性振动频率计算分析。 声学与振动, 3, 7-16. doi: 10.12677/OJAV.2015.32002

参考文献

[1] 陈务军 (2004) 膜结构工程设计. 中国建筑工业出版社, 北京.

[2] 张其林 (2002) 索和膜结构. 同济大学出版社, 上海.

[3] Vega, D.A., Vera, S.A. and Laura, P.A.A. (1999) Fundamental frequency of vibration of rectangular membranes with an internal oblique support. Journal of Sound and Vibration, 224, 780-783.
http://dx.doi.org/10.1006/jsvi.1999.2219

[4] Kang, S.W. and Lee, J.M. (2002) Free vibration analysis of com-posite rectangular membranes with an oblique interface. Journal of Sound and Vibration, 251, 505-517.
http://dx.doi.org/10.1006/jsvi.2001.4015

[5] Kang, S.W. (2004) Free vibration analysis of composite rectangular membranes with a bent interface. Journal of Sound and Vibration, 272, 39-53.
http://dx.doi.org/10.1016/S0022-460X(03)00305-5

[6] Reutskiy, S.Yu. (2009) Vibration analysis of arbitrarily shaped membranes. CMES—Computer Modeling in Engineering & Sciences, 51, 115-142.

[7] 钱国祯 (1982) 二向受力不等的平面薄膜自由振动问题解. 应用数学和力学, 6, 817-824.

[8] 张亿果, 袁驷 (1993) 有限元线法求解非线性模型问题—Ⅲ. 薄膜的固有振动. 工程力学, 3, 1-8.

[9] 罗吉, 罗亮生 (2010) 圆环膜自由振动的数学模型及其若干声学特性. 数学杂志, 1, 168-172.

[10] 林文静, 陈树辉 (2010) 平面薄膜自由振动的有限元分析. 动力学与控制学报, 3, 202-206.

[11] Liu, C.J., Zheng, Z.L., Yang, X.Y., et al. (2014) Nonlinear damped vibration of pre-stressed orthotropic membrane structure under impact loading. International Journal of Structural Stability and Dynamics, 14, Article ID: 1350055.

[12] He, J.H. (2003) Homotopy perturbation method: A new nonlinear analytical technique. Applied Mathematics and Com- putation, 135, 73-79.
http://dx.doi.org/10.1016/S0096-3003(01)00312-5

[13] Zheng, Z.L., Liu, C.J., He, X.T., et al. (2009) Free vibra-tion analysis of rectangular orthotropic membranes in large deflection. Mathematical Problems in Engineering, 2009, Article ID: 634362.

[14] Liu, C.J., Zheng, Z.L., He, X.T., et al. (2010) L-P perturbation solution of nonlinear free vi-bration of prestressed orthotropic membrane in large amplitude. Mathematical Problems in Engineering, 2010, Article ID: 561364.

分享
Top