Exact solution for large amplitude flexural vibration of nanobeams using nonlocal Euler-Bernoulli theory

In this paper, nonlinear free vibration of nanobeams with various end conditions is studied
using the nonlocal elasticity within the frame work of Euler-Bernoulli theory with
von K´arm´an nonlinearity. The equation of motion is obtained and the exact solution is established
using elliptic integrals. Two comparison studies are carried out to demonstrate
accuracy and applicability of the elliptic integrals method for nonlocal nonlinear free vibration
analysis of nanobeams. It is observed that the phase plane diagrams of nanobeams in
the presence of the small scale effect are symmetric ellipses, and consideration the small
scale effect decreases the area of the diagram.