Wang Jing, Shen Huoming
School of Mechanical and Electrical Engineering, Southwest Petroleum University, Chengdu 610500, People's Republic of China.
J Phys Condens Matter. 2019 Dec 4;31(48):485403. doi: 10.1088/1361-648X/ab3bf7. Epub 2019 Aug 19.
The lateral nonlinear vibration of an axially moving simply supported viscoelastic nanobeam is analysed based on nonlocal strain gradient theory. The proposed model includes the nonlocal parameters and material characteristic length parameters, investigating the two kinds of size effects of micro-nano beam structures. Firstly, the steady-state amplitude-frequency response of the subharmonic parametric resonance is analysed by a direct multiscale method, and the stability of the (non-) zero equilibrium solution determined by the Routh-Hurwitz criterion. Subsequently, the nonlinear frequencies of the nanobeams are calculated. Finally, several numerical examples are used to illustrate the influence of the scale parameters on the nonlinear vibration characteristics of nanobeams. The results show that when subharmonic parametric resonance occurs in the system, the (non-) zero equilibrium solution and the boundary of the instability region are markedly affected by the scale parameters. In addition, the nonlocal parameters soften the system, the material characteristic length parameters harden the system, and these softening and hardening effects are strengthened (or weakened) to varying degrees in the presence of nonlinearity.
基于非局部应变梯度理论，分析了轴向运动简支粘弹性纳米梁的横向非线性振动。所提出的模型包含非局部参数和材料特征长度参数，研究了微纳梁结构的两种尺寸效应。首先，采用直接多尺度方法分析了亚谐参数共振的稳态幅频响应，并通过劳斯 - 胡尔维茨判据确定了（非）零平衡解的稳定性。随后，计算了纳米梁的非线性频率。最后，通过几个数值算例说明了尺度参数对纳米梁非线性振动特性的影响。结果表明，当系统发生亚谐参数共振时，尺度参数对（非）零平衡解和不稳定区域边界有显著影响。此外，非局部参数使系统软化，材料特征长度参数使系统硬化，并且在非线性存在的情况下，这些软化和硬化效应会不同程度地增强（或减弱）。
