The stochastic eigenvalue problem for free vibrations of functionally graded beams with a random elastic modulus is investigated. The effective material properties and beam cross section are assumed to vary continuously in different directions according to the exponential law. The governing equations for the natural frequency of the functionally graded beams are derived from Hamilton’s principle. In the stochastic finite-element method, the random process was discretized by averaging random variables in each element to increase the accuracy of the natural frequency found. A solution of the stochastic eigenvalue problem formulated was obtained using the perturbation technique in conjunction with the finite-element method. The spectral representation was used to generate a random process to employ the Monte Carlo simulation. A good agreement was obtained between the results of the first-order perturbation technique and the Monte Carlo simulation.

中文翻译：

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具有不确定弹性模量的功能梯度梁自由振动的基于随机微扰的有限元

研究了具有随机弹性模量的功能梯度梁自由振动的随机特征值问题。根据指数定律，假设有效材料属性和梁横截面在不同方向上连续变化。功能梯度梁的固有频率的控制方程是从哈密顿原理推导出来的。在随机有限元方法中，通过对每个元素中的随机变量进行平均来将随机过程离散化，以提高所发现的固有频率的准确性。使用微扰技术结合有限元方法获得了随机特征值问题的解决方案。频谱表示用于生成随机过程以采用蒙特卡罗模拟。