Study of Primary and Secondary Nonlinear Resonances of Nanobeam Based on Nonlocal Strain Gradient Theory

In this paper, the nonlinear forced vibrations of nonlocal Euler-Bernoulli nanobeam that is utilized in nanoelectromechanical systems are studied using the analytical method of multiple time scales. Based on non-linear strain gradient elasticity theory, governing equation of Euler-Bernoulli nanobeam with von-karman geometric nonlinearity is derived using Hamilton principle. In the next step, using the Galerkin method, the partial differential governing equations for simply supported boundary conditions are reduced to time variable ordinary nonlinear differential equation. Subsequently, the nonlinear forced vibration equation is solved using a multiple time scalar method. After solving the nonlinear excited equation, primary and secondary resonances of nonlocal nanobeam are studied. The region of acceptable sub-harmonic responses is identified and for different values of nonlocal parameter, the frequency response curves and response amplitudes versus excitation amplitude is plotted in all resonances as primary, super-harmonic and sub-harmonic. These results show that using nonlocal strain gradient theory is a fundamental necessity for analyzing nonlinear vibration of nanobeams. The results of this paper can be used to improve the design and optimization of nanoelectromechanical systems.