Stability and Nonlinear Vibration Analysis of an Axially Loaded Nanobeam Based on Nonlocal Strain Gradient Theory

Abstract

The size-dependent stability mechanism and nonlinear dynamics of a nano-scale beam under an axial time-dependent load are theoretically studied based on nonlocal strain gradient theory. First, with the consideration of geometric nonlinearity of the nanobeam, the equation of motion is derived with the aid of Hamilton’s principle. Then, by using both Floquet technology and Bolotin’s method, a linear analysis is employed to investigate the size-dependent stability mechanism of the nanobeam for pinned-pinned boundary conditions. It is demonstrated that the results obtained based on the Floquet technology and Bolotin’s method agree well with each other in most cases. Furthermore, the analytical expressions for stability boundaries of the nanobeam system are derived. It is found that when the size-dependent nanobeam system is reduced to a classical one, the obtained expressions for stability boundaries are in good agreement with previous experimental results. Finally, the nonlinear responses of the nanobeam are presented in the form of time traces, bifurcation diagrams and phase portraits. It is shown that the axially excited nanobeam always undergoes a limit cycle oscillation. For all results regarding stability boundaries and nonlinear responses, the size-dependent effects are found to be significant.