This paper studies the vibration behaviors of bidirectional functionally graded (BDFG) Timoshenko beams based on the Chebyshev collocation method. The material properties of the beam are assumed to vary simultaneously in the beam length and thickness directions. The Chebyshev differentiation matrices are used to reduce the ordinary differential equations into a set of algebraic equations to form the eigenvalue problem for free vibration analysis. To validate the accuracy of the proposed model, some calculated results are compared with those obtained by other investigators. Good agreement has been achieved. Then the effects of slenderness ratios, material distribution types, gradient indexes, and restraint types on the natural frequency of BDFG beams are examined. Through the parametric study, the influences of the various geometric and material parameters on the vibration characteristics of BDFG beams are evaluated.

中文翻译：

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使用切比雪夫配置法的双向功能梯度 Timoshenko 梁的振动分析

本文基于切比雪夫配置方法研究了双向功能梯度（BDFG）Timoshenko梁的振动行为。假设梁的材料属性在梁长度和厚度方向上同时变化。Chebyshev微分矩阵用于将常微分方程简化为一组代数方程，形成自由振动分析的特征值问题。为了验证所提出模型的准确性，将一些计算结果与其他研究人员获得的结果进行了比较。达成了良好的协议。然后研究了长细比、材料分布类型、梯度指数和约束类型对 BDFG 梁的固有频率的影响。通过参数研究，