Free and forced vibration nonlinear analysis of a microbeam using finite strain and velocity gradients theory

A nonlinear finite strain and velocity gradient framework is formulated for the Euler–Bernoulli beam theory. This formulation includes finite strain and the strain gradient within the strain energy generalization as well as velocity and its gradient within the kinetic energy generalization. Consequently, static and kinetic internal length scales are developed to capture size effects. The governing equation with initial and boundary conditions is obtained using the variational approach. Free and forced vibration of a simply supported nanobeam is studied for different values of static and kinetic length scales using the method of multiple scales.