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

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.