Nonlinear vibration of nanobeam with attached mass at the free end via nonlocal elasticity theory

In the present study, nonlinear free and forced vibration of Euler–Bernoulli nanobeam with attached nanoparticle at the free end is investigated based on nonlocal elasticity theory. The effects of the different nonlocal parameters (γ) and mass ratios (α) as well as effects of fixed-free boundary conditions on the vibrations are determined. To obtain the equation of motion of the system, the Hamilton’s principle is employed. The stretching of neutral axis which introduces cubic nonlinearity is included into the equation for deriving nonlinear equation. And also effects of damping and forcing are included into the equations. The approximate solutions of the equations are derived by using the multiple scale method. Fundamental frequencies, frequency shift and mode shapes for the linear problem are estimated for a nonlocal Euler–Bernoulli nanobeam with an attached nanoparticle and graphically represented the frequency shift and mode shapes. Nonlinear frequencies are derived depending on amplitude and phase modulation. Frequency–response curves are drawn for different nonlocal parameters and different modes.