Simultaneous effects of material and geometric nonlinearities on nonlinear vibration of nanobeam with surface energy effects

This study explores the nonlinear free vibration of a nanobeam within the framework of nonlocal elasticity theory. It incorporates the materials nonlinear behavior, von Kármán strains, and surface elasticity theory. The stress–strain relationship in this study includes the quadratic material nonlinearity, which is typically ignored in previous research. The governing equations are derived through the application of Hamiltons principle. Using Galerkins method on the partial differential equations, the nonlinear differential equation governing the system is derived. The cubic nonlinearity in this equation arises from geometrical effects, while the quantic nonlinearity is attributed to material nonlinearity. The derived nonlinear differential equation is addressed utilizing the modified Homotopy Perturbation method. This approach yields the nonlinear time response and nonlinear frequency of the nanobeam, taking into account the effects of material nonlinearity and surface phenomena. The findings demonstrated the combined impact of surface effects and nonlinear material behavior on the nonlinear time response and frequency of the nanobeam. The natural frequency of the nanobeam was analyzed using the Elman neural network. Various inputs were fed into the network, and its output was compared with the exact solution for the natural frequency to assess accuracy. Additionally, the influence of material nonlinearity and surface effects on the phase trajectories of the nanobeam is examined. For validation purposes, the results are compared with those obtained using the fourth-order Runge–Kutta numerical method and previous studies.