Analytical Solutions for Vibration of Simply Supported Nonlocal Nanobeams With an Axial Force

This paper presents exact, analytical solutions for the transverse vibration of simply supported nanobeams subjected to an initial axial force based on nonlocal elasticity theory. Classical continuum theory is inherently size independent while nonlocal elasticity exhibits size dependence. The latter has significant effects on bending moment, which results in a conceptually different definition of a new effective nonlocal bending moment with respect to the corresponding classical bending moment. A sixth-order partial differential governing equation is subsequently obtained. The effects of nonlocal nanoscale on the vibration frequencies and mode shapes are considered and analytical solutions are solved. Effects of the nonlocal nanoscale and dimensionless axial force including axial tension and axial compression on the first three mode frequencies are presented and discussed. It is found that the nonlocal nanoscale induces higher natural frequencies and stiffness of the nano structures.