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