本文基於Hamilton原理和Timoshenko梁理論發展一套混合有限元素法用以分析材料性梯度梁在簡支承和固定端兩種組合邊界條件束制下之非線性自由振動行為。功能性梯度梁是由兩相材料組成，其材料性質之變化根據組成材料體積分率的冪次函數分佈，為一材料性質沿厚度方向平滑變化之梁，且功能性梯度梁之有效材料性質是以複合材料力學中兩相材料混合原理評估。此混合有限元素法之弱形式數學方程式是從能量泛函透過變分法推導而來，其中能量泛函有將馮卡門幾何非線性效應(von Kármán geometrical nonlinearity, VKGN)考慮在內。文中數值範例的振幅-頻率關係是以前述之混合有限元素法經過迭代運算求得。本混合有限元素解與文獻中之精確解相比較，其彼此之結果相當吻合，且在迭代求解過程中此法可以快速地達到收斂。在此篇文章中，作者進一步使用了多層感知倒傳式類神經網路來預測功能性材料梁之幾何非線性振動行為。經過適當的訓練後，類神經網路所預測之振幅-頻率關係與混合有限元素法比較有相當不錯的精確度，而其運算時間比需要經過迭代的混合有限元素法大幅節省。

Based on the Hamilton principle combined with the Timoshenko beam theory, the authors develop a mixed finite element (FE) method for the nonlinear free vibration analysis of functionally graded (FG) beams with combinations of simply-supported and clamped edge conditions. The FG beam is composed of a two-phase material, the material properties of which gradually and smoothly vary through the thickness direction according to the power-law distributions of the volume fractions of the constituents, and the effective material properties of the FG beam are estimated using the rule of mixtures. A weak-form formulation for the mixed finite element analysis is derived using the variational approach, in which the von Kármán geometrical nonlinearity is considered. The finite element solutions of the amplitude-frequency relations of the FG beam are obtained using an iterative process. Implementation of the mixed finite element method shows its solutions converge rapidly and that the convergent solutions closely agree with the accurate solutions available in the literature. A multilayer perceptron (MP) back propagation neural network (BPNN) is also developed to predict the nonlinear free vibration behavior of the FG beam. After appropriate training, the prediction of the amplitude-frequency relations of the MP BPNN is quite accurate with those obtained using the mixed FE method, and its computer execution time is less time consuming than that of the mixed FE method.

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