本文應用改良Pagano方法探討具四邊簡支承，三維多層疊合與具功能性電磁彈性材料板之自由振動分析，該板具有三種不同的側向表面條件。文中板之上、下表面之自由電位勢/磁位勢為已知或自由電通量/磁通量為已知。功能性電磁彈性材料板之材料性質，係由兩相材料沿厚度方向依組成的體積比例，遵循相同的冪次法則分佈。本改良Pagano方法應用混合理論架構推衍，遇系統方程式之線性獨立解為一對共軛複數解時，則根據Euler公式轉換成一對具實數型之線性獨立解。配合傳遞矩陣解法與連續近似分析，本改良Pagano方法可適用於多層疊合或功能性電磁彈性材料板之自由振動分析。數值範例中，則探討冪級數之幂次對功能性電磁彈性材料板的自然振動頻率和其相對應磁場、電場和彈性場之模態變數之影響。

A modified Pagano method is developed for the three-dimensional (3D) dynamic responses of simply-supported, multilayered and functionally graded (FG) magneto-electro-elastic plates with three different lateral surface conditions. Either the free electric/magnetic potential or the free electric/magnetic flux conditions is prescribed on each lateral surface. The material properties of FG plates are regarded as heterogeneous through the thickness coordinate and then specified to obey a power-law distribution of the volume fractions of the constituents. The Pagano method is modified for the present analysis in that a displacement-based formulation is replaced by a mixed formulation, the complex-valued solutions of the system equations are transferred to the real-valued solutions, a method of propagator matrix is developed for the present method, and a successive approximation (SA) method is used to make the present method feasible for the coupled analysis of FG plates. In the illustrative examples, the influence of the power-law exponent on the natural frequencies of the FG magneto-electro-elastic plates and their corresponding modal variables in the magnetic, electric and elastic fields through the thickness of the plates is studied. The present modified Pagano method combined with the SA method is capable of efficiently approximating the available 3D solutions of FG plates to any desired accuracy.

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