Size-dependent free vibration of functionally graded beams with a tip nanoparticle and a nonlinear spring constraint via strain gradient theory

Abstract

In this article, the free vibration of a functionally graded Euler–Bernoulli beam with a tip nanoparticle and a nonlinear/linear spring constraint is presented for the first time using strain gradient theory. It is assumed that there is a nonlinear/linear spring and a nanoparticle at one end of the beam while the other is clamped. The influence of different parameters, such as the material length scale parameter, the mass of the nanoparticle, and the nonlinear/linear spring constant on the natural frequencies, is discussed in detail. The presented benchmark tables can be used as an excellent database to verify approximate or analytical solutions in future works. The presented results can be useful for those designing micro/nano electromechanical devices.