Nonlocal free and forced vibration of a graded Timoshenko nanobeam resting on a nonlinear elastic foundation

The free and forced vibration of a nonlocal Timoshenko graded nanobeam resting on a nonlinear elastic foundation is investigated in this paper. The Timoshenko beam theory along with the von Kármán geometric nonlinearity is formulated while accounting for Eringen's nonlocal elasticity differential model. A power-law distribution is used to model the material distribution along the beam thickness. The equations of motion are derived using Hamilton's principle and then solved analytically using the Method of Multiple Scale (MMS) and numerically using the Differential Quadrature Method (DQM) and the Harmonic Quadrature Method (HQM). The considered boundary conditions include both Hinged-Hinged and Clamped-Clamped. The obtained nonlocal nonlinear frequencies of the nanobeam are first validated based on published analytical results that use linear mode shapes. A frequency response analysis is also conducted utilizing both MMS and DQM. The time discretization in DQM solution is performed using Spectral Method (SPM) and HQM. The primary objective of this study is to investigate the effects of the nonlocal parameter, power-law index, linear and nonlinear stiffnesses of the elastic foundation as well as the boundary conditions on the dynamic response of the nanobeam.