A size-dependent meshless model for free vibration analysis of 2D-functionally graded multiple nanobeam system

In this study, the free vibration of two-directional functionally graded (2D-FG) multiple nanobeam system (MNBS) are studied by considering Winkler elastic medium between each nanobeam. Effects of small scale are considered using the nonlocal elasticity theory. The material properties of the FG nanobeams are considered to vary over the length and thickness of the nanobeams. The equations of motion are derived using Hamilton's principle and the first-order shear deformation beam theory (FSDBT), and a meshless formulation is developed to discretize the governing equations. Numerical results are obtained for both cases of free-chain and clamped-chain multiple nanobeam system. In order to validate the accuracy of the meshless formulation, numerical results for free vibration of 1D-FG single nanobeam are compared with the available predictions of various beam theories and solution approaches. Also, free vibration of homogeneous double nanobeam system is analyzed and good agreement is observed while comparing these results with analytical solutions. In the numerical results, the effects of nonlocal parameter, slenderness ratio, FG power index, elastic medium stiffness, number of nanobeams, boundary conditions and concentrated mass on the free vibration of 1D- and 2D-FG single and multiple nanobeam system are investigated.