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

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.