The presented paper investigates the nonlinear vibration of a nanobeam with a piezoelectric layer bounded to its top surface considering the nonlocal piezoelectricity theory. To do this, Hamilton’s principle is implemented to derive the governing nonlinear vibration equations of the nanobeam by assumption of nonlocal piezoelectricity and a nonlinear strain–displacement relation. Then, the Galerkin separation method is applied to transform and simplify the partial differential equation of the nonlinear oscillation to an ordinary one with quadratic and cubic nonlinearities in the time domain. By implementing the multiple-scale perturbation method, an analytical relation for the nonlinear natural frequencies is obtained as a function of the oscillation amplitude and the nonlocal size scale parameter. Then, the nonlinear vibration characteristics of the nanobeam are investigated at higher modes of vibration and the size scale effects are reviewed comprehensively. It is observed that the nonlocal parameter decreases the nonlinear natural frequencies and becomes noticeable at higher modes of vibration. Moreover, by increasing the amplitude ratio, the nonlocal effects are decreased and the nonlocal nonlinear frequency approaches the local one. Also, the amplitude ratio has increasing effects on the nonlinear frequencies.

中文翻译：

---

基于非局域压电理论的高模压电层压纳米梁的非线性振动

所提出的论文研究了考虑非局域压电理论的纳米梁的非线性振动，其中压电层绑定到其顶面。为此，通过假设非局部压电性和非线性应变-位移关系，实施哈密顿原理以推导出纳米梁的控制非线性振动方程。然后，应用伽辽金分离法，将非线性振荡的偏微分方程化简为时域上具有二次和三次非线性的普通偏微分方程。通过实施多尺度微扰方法，非线性固有频率的解析关系作为振荡幅度和非局部尺寸尺度参数的函数获得。然后，研究了纳米梁在更高振动模式下的非线性振动特性，并全面审查了尺寸尺度效应。据观察，非局部参数降低了非线性固有频率，并在更高的振动模式下变得明显。此外，通过增加幅度比，非局部效应降低，非局部非线性频率接近局部非线性频率。此外，幅度比对非线性频率的影响越来越大。非局部效应减少，非局部非线性频率接近局部非线性频率。此外，幅度比对非线性频率的影响越来越大。非局部效应减少，非局部非线性频率接近局部非线性频率。此外，幅度比对非线性频率的影响越来越大。