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首页> 外文期刊>Composite Structures >A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory
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A novel approach for free vibration of axially functionally graded beams with non-uniform cross-section based on Chebyshev polynomials theory

机译：基于切比雪夫多项式理论的截面不均匀的轴向功能梯度梁自由振动的新方法

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摘要

A new approach based on Chebyshev polynomials theory is introduced to analyze free vibration of axially functionally graded Euler-Bernoulli and Timoshenko beams with non-uniform cross-sections. By using high-order Chebyshev expansions to approximate deflections of a beam, its potential energy and kinetic energy, both of which can be considered as weighted inner products of functions, can be expressed in matrix form. All variable geometric and material properties, such as cross-sectional area, area moment of inertia, mass density, Young's modulus, and shear modulus, are treated as weight functions. In this way, the discrete governing equation can be obtained directly by applying Lagrange's equation. Natural frequencies and mode shapes can be easily determined by solving the eigenvalue equation. Several numerical examples are carried out to verify the competency of the proposed method. All results are seen to be in excellent agreement with those presented in literature. The overall convergence is approximately exponential, and a highly accurate solution can be gained by using a small number of polynomials.(C) 2017 Elsevier Ltd. All rights reserved.
机译：引入了一种基于切比雪夫多项式理论的新方法来分析截面不均匀的轴向功能梯度Euler-Bernoulli和Timoshenko梁的自由振动。通过使用高阶切比雪夫展开近似束的偏转，可以将其势能和动能（都可以视为函数的加权内积）以矩阵形式表示。所有可变的几何和材料特性（例如横截面积，面积惯性矩，质量密度，杨氏模量和剪切模量）均视为重量函数。这样，可以通过应用拉格朗日方程直接获得离散控制方程。通过求解特征值方程，可以轻松确定固有频率和振型。进行了几个数值算例验证了所提出方法的能力。所有结果都被认为与文献中的结果非常吻合。总体收敛是近似指数的，并且可以使用少量多项式来获得高度精确的解决方案。（C）2017 Elsevier Ltd.保留所有权利。

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