Forced Transverse Nonlinear Vibration of an Axially Moving Graphene Sheet Over an Adjacent Substrate

Background: Squeeze film damping effect has been found to have a significant effect on the dynamic performance of the nano-sheet. Yet, it was not considered in previous works for axially moving nanosheets. Objectives: The objectives of this research are to study the squeez thin film effects on the dynamic behavior of an axially moving graphene sheets over a close parallel substrate and under the effect of a uniformly distributed harmonic load. The study includes frequency response, time domain response, and chaotic behavior. Methods: Within the context of the modified couple stress theory of nano-scale Euler–Bernoulli beams, the Galerkin method is utilized to develop a mathematical model that describes the dynamic behavior of the axially moving simply supported nano-beam, resembling a graphene sheet. Results: Frequency response curves are obtained to study the forced vibration of the axially moving small-size beam and understand the influence of squeeze film damping, nano-induced nonlinear damping, width-to-gap ratio, the axial motion of the nano-beam, and applied external force. Bifurcation diagrams are also studied to investigate transitions and chaotic behavior. Phase portrait and Poincaré section are studied for selected frequencies. Conclusions: Results show that the squeeze film damping has a significant impact on amplitude reduction as it damps the system and avoids failure at resonant frequencies. It is also shown that when the small size effect is considered, the natural frequency becomes higher than those of the classical beam theory. Increasing the vibration amplitude leads to a nonlinear hardening behavior in the system. Significant increasing the axial velocity of the nano-beam is found to alter frequency response, while no change in the frequency response is observed when changing the axial acceleration of the beam.