Nonlocal beam theory for nonlinear vibrations of a nanobeam resting on elastic foundation

In the present study, nonlinear vibrations of an Euler-Bernoulli nanobeam resting on an elastic foundation is studied using nonlocal elasticity theory. Hamilton's principle is employed to derive the governing equations and boundary conditions. The nonlinear equation of motion is obtained by including stretching of the neutral axis that introduces cubic nonlinearity into the equations. Forcing and damping effects are included in the equations of the motion. The multiple scale method, a perturbation technique for deriving the approximate solutions of the equations, is applied to the nonlinear systems. Natural frequencies and mode shapes for the linear problem are found and also nonlinear frequencies are found for a nonlocal Euler-Bernoulli nanobeam resting on an elastic foundation. In the numerical calculation, frequency-response curves are drawn for various parameters like nonlocal parameters, elastic foundation, and boundary conditions. The effects of the different nonlocal parameters (gamma) and elastic foundation parameters (.) as well as the effects of different boundary conditions on the vibrations are discussed.