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A Jacobi polynomial based approximation for free vibration analysis of axially functionally graded material beams

机译：基于Jacobi多项式的轴向功能梯度材料梁的自由振动分析近似

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摘要

The paper presents an effective approximation for free vibration analysis of axially functionally graded material (AFGM) beams based on the Jacobi polynomial theory. An arbitrary-order derivative of the Jacobi polynomial is expressed as the expression of low-order components, whereas its boundary values are fully defined by the polynomial parameters. The particular feature is used to derive the generalized eigenvalue equation for free vibration analysis of AFGM beams in conjunction with the Euler-Bernoulli, the Timoshenko and the nonlocal strain gradient beam theories. Several numerical examples in the literature are presented to demonstrate potential applications of the Jacobi polynomial approach. A fast convergence of the approximation error for natural frequency results has confirmed high accuracy of the proposed approach. The Legendre and the Chebyshev polynomials are special cases of the Jacobi basis function. This guarantees the flexibility of the presented method for free vibration analysis of AFGM beams with nonuniform geometries and axially varying material properties.
机译：本文介绍了基于Jacobi多项式理论的轴向功能渐变材料（AFGM）光束的自由振动分析的有效近似。 Jacobi多项式的任意阶数表示为低阶分量的表达式，而其边界值完全由多项式参数完全定义。该特定特征用于与Euler-Bernoulli，TimosheNKO和非本地应变梯度波束理论一起导出AFGM光束的自由振动分析的广义特征值方程。提出了文献中的几个数值例子以证明雅各比多项式方法的潜在应用。自然频率结果的近似误差的快速收敛已经确认了所提出的方法的高精度。 Legendre和Chebyshev多项式是Jacobi基本功能的特殊情况。这保证了具有非均匀几何形状的AFGM光束的自由振动分析方法和轴向不同的材料特性的灵活性。

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