Nonlocal and surface effects on nonlinear vibration response of a graded Timoshenko nanobeam

Abstract The free and forced vibration of a graded geometrically nonlinear Timoshenko nanobeam supported by on foundation is considered in this paper. main contribution study to propose new formulation for the dynamic response beam combining nonlocal surface elasticity addition employing physical neutral axis method which eliminates quadratic nonlinearity from equation motion. Using principle virtual work, fourth-order partial differential formulated Galerkin technique employed yield ordinary with cubic temporal domain. multiple scales obtain analytical expression frequency its curve primary resonance analysis. To assess accuracy solution, it compared numerical solution obtained using quadrature method. results are successfully validated particular cases problem published other authors. effects elasticity, nonlocality, axis, aspect ratio, power-law index elastic coefficients thoroughly investigated different types boundary conditions .