本文應用Reissner混合變分原理(RMVT)非局部Timoshenko梁理論(TBT)分析嵌入在彈性介質中的多層奈米碳管(MWCNT)之非線性自然振動。文中，考量嵌入式MWCNT的四種不同邊界條件，以及構成每對管壁之間的凡德瓦爾互制力的兩種不同模式，並且將MWCNT與周圍介質之間的互制力以Pasternak型基礎模擬之，亦將von Kármán幾何非線性效應納入各層控制方程和相關邊界條件之推衍。文中應用Eringen非局部彈性理論探討微小尺度效應，透過微分擬合法結合直接迭代法取得嵌入式MWCNT在最大模態出現撓曲時的基本自然振動頻率參數變化。數值範例結果顯示RMVT非局部TBT所得到頻率參數解收斂快速，其收斂解與文獻中之虛位移原理非局部Timoshenko梁理論的解析解和數值解相當吻合。

Based on the Reissner mixed variational theorem (RMVT), we present a nonlocal Timoshenko beam theory (TBT) for the nonlinear vibration analysis of multi-walled carbon nanotubes (MWCNT) embedded in an elastic medium. In this formulation, four different edge conditions of the embedded MWCNT are considered, two different models with regard to the van der Waals interaction between each pair of walls constituting the MWCNT are considered, and the interaction between the MWCNT and its surrounding medium is simulated using the Pasternak-type foundation. The governing equations of an individual wall and the associated boundary conditions are derived, in which the von Karman geometrical nonlinearity is considered. Eringen’s nonlocal elasticity theory is used to account for the effects of the small length scale. Variations of the lowest frequency parameters with the maximum modal deflection of the embedded MWCNT are obtained using the differential quadrature method in conjunction with a direct iterative approach. In the numerical examples, it is shown that solutions of the frequency parameters obtained using the current RMVT-based nonlocal TBT converge rapidly.

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