Abstract An efficient approach to obtain accurate solutions for free vibration of functionally graded beams (FGB) with variable cross-sections and resting on Pasternak elastic foundations is presented. The general expressions of displacements and stresses which thoroughly result from the dynamic differential equations and boundary conditions for FGB with arbitrarily variable cross-sections are obtained by using the separate variable method and Laplace transform on the basis of 2-D elastic theory. Meanwhile, the frequency equations for free vibration of FGB with variable cross-sections are derived by applying Fourier series expansion along the lower and upper boundary conditions of the FGB. Validity of the developed approach as well as its effectiveness and accuracy are verified through analysing several typical FGB examples, and this provides a potential alternative approach for analysing vibration of functionally graded components with variable cross-sections in case of ultra-high precision requirement in modern mechanical systems. The effect of geometric and mechanical parameters on vibration frequency and mode shapes of FGB with variable cross-sections resting on Pasternak elastic foundations is further analysed.

中文翻译：

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帕斯捷尔纳克弹性基础上具有可变截面的功能梯度梁振动的解析解

摘要 提出了一种有效的方法来获得具有可变横截面的功能梯度梁 (FGB) 自由振动的精确解并位于 Pasternak 弹性基础上。在二维弹性理论的基础上，利用分离变量法和拉普拉斯变换，得到了具有任意变截面FGB的动态微分方程和边界条件的位移和应力的一般表达式。同时，通过沿FGB的上下边界条件应用傅里叶级数展开，推导出了变截面FGB自由振动的频率方程。通过分析几个典型的 FGB 示例，验证了所开发方法的有效性及其有效性和准确性，这为分析具有可变截面的功能梯度部件的振动提供了一种潜在的替代方法，以应对现代机械系统中的超高精度要求。进一步分析了几何和机械参数对基于 Pasternak 弹性基础的变截面 FGB 振动频率和振型的影响。