Nonlinear dynamics of Timoshenko beams on nonlinear fractional viscoelastic Pasternak foundation under a moving mass

Abstract

In this paper, the nonlinear dynamics analysis of beams resting on a fractional Pasternak visco-elastic foundation with cubic nonlinearity, under a mass traveling with a constant speed is explored. Timoshenko beams with a large amplitude and two characteristics fractional derivative orders are considered. A comprehensive mathematical procedure has been developed to derive the governing equation including the Caputo fractional derivative model. An arbitrary number of eigenmodes can be used and the associated nonlinear fractional differential system is explicitly given. A well-adapted implicit numerical method is elaborated based on the Newmark scheme coupled with a fractional derivative numerical scheme for linear and nonlinear cases. For a comparison, an industrial FE code (ANSYS FE) is also used. The parametric study investigates the effects of the fractional derivative orders ( α , λ ) on the beam’s response. The findings suggest that as α and λ increase from 0.2 to 0.7, the maximum vertical displacement decreases. In particular, the fractional viscous damping coefficient μ has a significant effect on the transverse deflection compared to the fractional rocking damping. The analysis shows that when the orders of the fractional derivatives decrease, the transverse deflection and the maximum mid-span shear response increase. Additionally, the study demonstrates that the effect of fractional orders on the transverse shear response is more significant for the viscous and rocking damping coefficients. These findings are discussed in detail, with specific numerical data provided for various parameters such as the shear deformation coefficient, velocity of the moving mass, and foundation characteristics. Furthermore, a remarkable outcome of α characterizing the fractional viscous damping is observed comparing to λ on the time response of the beam.

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