Nonlinear vibrations of axially moving simply supported viscoelastic nanobeams based on nonlocal strain gradient theory

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.