Analytical modeling of the linear and nonlinear dynamic characteristics of the non-uniform axially functionally graded cylindrical beam based on the strain gradient theory

In this research, the nonlinear vibration behavior of axially functionally graded (AFG) nano-scale truncated conical tubes (with non-uniform cross-section), including the porosity, is presented for the first time based on the Euler-Bernoulli beam theory and the von-Karman's nonlinear strain-displacement and nonlinear nonlocal boundary conditions. The effect of being at the tapered nanosize model is supposedly based on the nonlocal strain gradient theory using the Hamilton principle to extract the equation of motions and nonlinear time-dependent nonlocal boundary conditions. The material properties and thickness of nano-tube are varying along the length of tubes. The generalized differential quadrature method (GDQM) is utilized with the iteration method to solve the complicated nonlinear equations. The numerical results in tabular and graphical styles present for various nonlinear amplitudes, the AFG power indexes, the rate of the cross-section change, the porosity parameter, and the nonlocal gradient strain parameter for both simply supported, fully clamped, and simply supported-clamped boundary conditions in detail.