Abstract
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 (γ) and elastic foundation parameters (κ) as well as the effects of different boundary conditions on the vibrations are discussed.
Highlights
Due to the rapid improvement in the nano-mechanics, nanobeams have become one of the most important structures used extensively in nanotechnology, such as those in sensors and actuators
The nonlocal elasticity theory which was formally initiated by the papers of Eringen [ ] on nonlocal elasticity can be used for nanotechnology applications due to the small length scale in nanoapplications of the beam
The dimensionless linear elastic foundation parameter effect on the nonlinear natural frequencies of the nanobeam with γ = . and both boundary conditions are investigated in Figure that plot the nonlinear natural frequency variation versus amplitude with κ =, and. It can be seen in the same figures that an increase in dimensionless linear elastic stiffness and a fixed in nonlocal parameter increase in nonlinear frequency value occur regardless of the type of boundary condition
Summary
Due to the rapid improvement in the nano-mechanics, nanobeams have become one of the most important structures used extensively in nanotechnology, such as those in sensors and actuators. The nonlinear free vibration of the nanotube with damping effect was studied by using nonlocal elasticity theory [ ]. To the best of the knowledge of the author, there is no published work on a nonlinear free vibration of nanobeam resting on elastic foundation with the effect of damping and forcing terms.
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