Nonlinear Vibration of a Nonlocal Functionally Graded Beam on Fractional Visco-Pasternak Foundation

This paper investigates the nonlinear dynamic behavior of a nonlocal functionally graded Euler–Bernoulli beam resting on a fractional visco-Pasternak foundation and subjected to harmonic loads. The proposed model captures both, nonlocal parameter considering the elastic stress gradient field and a material length scale parameter considering the strain gradient stress field. Additionally, the von Karman strain-displacement relation is used to describe the nonlinear geometrical beam behavior. The power-law model is utilized to represent the material variations across the thickness direction of the functionally graded beam. The following steps are conducted in this research study. At first, the governing equation of motion is derived using Hamilton's principle and then reduced to the nonlinear fractional order differential equation through the single-mode Galerkin approximation. The methodology to determine steady-state amplitude-frequency responses via incremental harmonic balance method and continuation technique is presented. The obtained periodic solutions are verified against the perturbation multiple scales method for the weakly nonlinear case and numerical integration Newmark method in the case of strong nonlinearity. It has been shown that the application of the incremental harmonic balance method in the analysis of nonlocal strain gradient theory-based structures, can lead to more reliable studies for strongly nonlinear systems. In the parametric study is shown that, on one hand, parameters of the visco-Pasternak foundation and power-law index remarkable affect the response amplitudes. On the contrary, the nonlocal and the length scale parameters are having a small influence on the amplitude-frequency response. Finally, the effects of the fractional derivative order on the system's damping are displayed at time response diagrams and subsequently discussed.