Nonplanar free and forced vibrations of an imperfect nanobeam employing nonlocal strain gradient theory

Abstract

In this paper, the free and forced vibrations of a three-dimensional nonplanar nanobeam with initial geometric imperfection are investigated. The Kelvin–Voigt model and nonlocal strain gradient theory (NSGT) are applied to the nanobeam with viscoelastic structural damping and size-dependent effect. A mechanical model of the nanobeam is established by applying Hamilton’s principle, and an initial displacement is used to describe the initial geometric imperfection. The differential quadrature method (DQM) is used to discretize the complex partial differential equation and simulate the mid-span vibration amplitude of the nonplanar nanobeam. Comparisons between the proposed model and published results show good agreement and further demonstrate the validity of the model. Numerical simulations are performed to illustrate how the geometric imperfection, the viscoelasticity, the nonlocal parameter, the strain gradient parameter and lateral motion influence the buckling phenomena, natural frequencies and vibrational responses. It is found that the nonlocal effect can reduce the natural frequencies and enhance the complicated nonlinear phenomena. However, the contrary phenomena are induced by the strain gradient effect. The geometric imperfection causes buckling phenomena which significantly affect the natural frequencies of the nanobeam. Moreover, the viscoelastic effect, imperfect effect and lateral motion can lead to more complicated and rich vibrational responses due to the greater contribution of nonlinearity.