This article investigates size-dependent nonlinear dynamic modeling and vibration analysis of nanomechanical resonators equipped with piezoelectric layers employing perturbation methods. For nonlinear dynamic modeling, the nanomechanical beam resonator is modeled based on the nonlocal elasticity theory and von-kármán type of nonlinear formulation. Then, the Hamilton’s principle is employed to derive the nonlinear vibration equation of the piezoelectric laminated beam resonator. The Galerkin separation approach is employed to develop the governing equation as a time-dependent nonlinear differential one which is solved by the multiple scales perturbation method. Obtaining an analytical formulation, a detailed study is conducted to investigate the size effects on nonlinear free vibration of the beam at several modes of vibration. The obtained results indicate that the nonlocal parameter causes a reduction effect on the nonlinear frequencies particularly for its higher values and higher modes of vibration. Also, it is observed that the nonlocal effect on the natural frequencies diminishes with an increase in the vibration amplitude.

中文翻译：

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压电层合纳米机械谐振器基于尺寸的非线性动力学建模和振动分析

本文研究了尺寸微分的非线性动力学建模和纳米机械共振器的振动分析，压电共振器配备了采用扰动方法的压电层。对于非线性动力学建模，基于非局部弹性理论和von-kármán类型的非线性公式对纳米机械梁谐振器进行建模。然后，采用汉密尔顿原理导出压电叠层梁谐振器的非线性振动方程。Galerkin分离方法被用来开发控制方程，它是一个与时间有关的非线性微分方程，可以通过多尺度摄动法求解。获得分析公式，进行了详细的研究，以研究在几种振动模式下，梁对非线性自由振动的尺寸影响。获得的结果表明，非局部参数会导致非线性频率的降低，特别是对于其较高的值和较高的振动模式。另外，观察到随着振动幅度的增加，对固有频率的非局部影响减小。