Purpose

Goal for the present research is investigating the effect of scale effect on free vibration of piezoelectric sandwich nanobeams on Winkler foundation. For this purpose, the effects of nonlocal parameters and strain gradient parameters on the free vibration of the model are studied.

Methods

Based on the nonlocal strain gradient theory and Timoshenko beam theory, the nonlinear vibration of piezoelectric sandwich nanobeams on Winkler foundation is investigated. The nonlinear governing equations and boundary conditions are derived using the Hamilton's principle. The partial differential equation is transformed into ordinary differential equation by Galerkin's method, and then the nonlinear vibration of piezoelectric nanobeam is numerically analyzed using the Runge-Kutta method.

Results and Conclusions

The results show that the nonlinear frequency ratio decreases with the increase of length-to-thickness ratio. When the nonlocal parameter is not less than the strain gradient length scale parameter, the piezoelectric nanobeam exhibits stiffness softening effect. When the nonlocal parameter is not greater than the strain gradient length scale parameter, the piezoelectric nanobeam exhibits stiffness hardening effect. It is also observed that both large length-to-thickness ratios and shear deformation can attenuate the nonlocal strain gradient effect. In addition, changes in the external applied voltage have a significant effect on the natural frequency of the piezoelectric nanobeams and increasing the thickness of the piezoelectric layer can enhance the structural stiffness.

中文翻译：

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温克勒基础上压电夹层纳米梁非线性振动的尺度效应

目的

本研究的目的是研究尺度效应对Winkler基座上的压电夹层纳米束自由振动的影响。为此，研究了非局部参数和应变梯度参数对模型自由振动的影响。

方法

基于非局部应变梯度理论和Timoshenko束理论，研究了Winkler基础上压电夹层纳米梁的非线性振动。非线性控制方程和边界条件是根据汉密尔顿原理导出的。用Galerkin方法将偏微分方程转化为常微分方程，然后用Runge-Kutta方法对压电纳米束的非线性振动进行数值分析。

结果与结论

结果表明，非线性频率比随长厚比的增加而减小。当非局部参数不小于应变梯度长度尺度参数时，压电纳米束表现出刚度软化效果。当非局部参数不大于应变梯度长度尺度参数时，压电纳米束表现出刚度硬化效果。还观察到，大的长厚比和剪切变形都可以减弱非局部应变梯度效应。另外，外部施加电压的变化对压电纳米束的固有频率具有显着影响，并且增加压电层的厚度可以增强结构刚度。