Closed-form Solution of Dynamic Displacement for SLGS Under Moving the Nanoparticle on Visco-Pasternak Foundation

In this paper, forced vibration analysis of a single-layered graphene sheet (SLGS) under moving a nanoparticle is carried out using the non-local elasticity theory of orthotropic plate. The SLGS under moving the nanoparticle is placed in the elastic and viscoelastic foundation which are simulated as a Pasternak and Visco-Pasternak medium, respectively. Movement of the nanoparticle is considered as a linear movement with constant velocity from an edge to another edge of graphene sheet. Using the non-linear Von Kármán strain-displacement relations and Hamilton’s principle, the governing differential equations of motion are derived. The differential equation of motion for all edges simply supported boundary condition is solved by an analytical method and therefore, the dynamic displacement of SLGS is presented as a closed-form solution of that. The influences of medium stiffness (Winkler, Pasternak and damper modulus parameter), nonlocal parameter, aspect ratio, mechanical properties of graphene sheet, time and velocity parameter on dimensionless displacement (dynamic displacement to static displacement of SLGS) are studied. The results indicate that, as the values of stiffness modulus parameter increase, the maximum dynamic displacement of SLGS decreases. Therefore, the results are in good agreement with the previous researches.