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Chebyshev Formulation for In-Plane Vibration Analysis of Arbitrary Laminated Polygonal Plates
AIAA Journal ( IF 2.5 ) Pub Date : 2021-06-01 , DOI: 10.2514/1.j060366
Tao Liu 1 , Ailun Wang 1 , Qingshan Wang 1 , Bin Qin 1 , Fei Xie 1
Affiliation  

This paper proposes a unified solution for in-plane vibration analysis of laminated arbitrary polygonal plates based on the Chebyshev polynomial method. As an innovative point of present work, the arbitrarily shaped polygonal plate is composed of two arbitrarily shaped quadrilateral plates rigidly coupled for simulation. The quadrilateral plate is uniformly mapped into a unit square plate through the four-node coordinate transformation. In the present approach, whatever the shape of the plate and the type of boundary conditions, the in-plane displacement components of the plate after transformation are expanded as Chebyshev polynomial of the first kind. The artificial spring technique is adopted here to simulate the general boundary restraints and the rigid coupling of coupled plates. Based on the Rayleigh–Ritz method, the accurate solution can be obtained through the energy function of the laminated polygonal plate. The in-plane vibration characteristics including the natural frequency and the corresponding mode shapes of the polygonal plate can be easily obtained by solving the eigenvalue problem. A large number of numerical results of the laminated polygonal plate with different shapes and boundary conditions are presented, it is found the present method shows rapid convergence, high computing efficiency, and high accuracy by comparing with those solutions calculated from the finite element method and the results in the experimentation. Besides, comprehensive studies on the effects of geometrical properties and material parameters of rhombic plates, pentagonal plates, and hexagonal plates on their in-plane vibration characteristics are also presented.



中文翻译:

任意叠层多边形板面内振动分析的切比雪夫公式

本文提出了一种基于切比雪夫多项式方法的层压任意多边形板面内振动分析的统一解决方案。作为目前工作的创新点,任意形状的多边形板由两个任意形状的四边形板刚性耦合组成以进行仿真。将四边形板通过四节点坐标变换均匀映射为一个单位正方形板。在本方法中,无论板的形状和边界条件的类型如何,变换后板的面内位移分量都扩展为第一类切比雪夫多项式。这里采用人工弹簧技术来模拟一般边界约束和耦合板的刚性耦合。基于瑞利-里兹方法,通过叠层多边形板的能量函数可以得到精确解。通过求解特征值问题,可以很容易地获得包括多边形板的固有频率和相应的振型在内的面内振动特性。给出了大量不同形状和边界条件的叠层多边形板的数值计算结果,通过与有限元法计算得到的解进行比较,发现该方法收敛速度快,计算效率高,精度高。实验结果。此外,还综合研究了菱形板、五边形板和六边形板的几何特性和材料参数对其面内振动特性的影响。

更新日期:2021-06-02
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