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Nonlinear vibration of a beam resting on a nonlinear viscoelastic foundation traversed by a moving mass: a homotopy analysis

Pourseifi Mehdi, Monfared Mojtaba Mahmoudi

Engineering Transactions

2022

Vol. 70, iss. 4

355--371

In this study, the dynamic response of an Euler-Bernoulli beam resting on the nonlinear viscoelastic foundation under the action of a moving mass by considering the stretching effect of the beam’s neutral axis is investigated. A Dirac-delta function is applied to model the location of the moving mass along the beam as well as its inertial effects. The Galerkin decomposition method is used to transform a partial dimensionless nonlinear differential equation of dynamic motion into an ordinary nonlinear differential equation. Subsequently, the well-known homotopy analysis method (HAM) is employed to obtain an approximate analytical solution of this equation. The validity and accuracy of the solution are examined numerically using the fourth-order Runge-Kutta method. Finally, several examples are provided to show the effects of parameters such as linear and nonlinear stiffness coefficients of a viscoelastic foundation, velocity of the moving mass as well as Coriolis force, centrifugal force and inertia force of the moving mass on the dynamic deflection of the beam.

An analytical study on the nonlinear forced vibration of functionally graded carbon nanotube-reinforced composite beams on nonlinear viscoelastic foundation

Shafiei H., Setoodeh A. R.

Archives of Mechanics

2020

Vol. 72, nr 2

81--107

This paper deals with the nonlinear forced vibration of nanocomposite beams resting on a nonlinear viscoelastic foundation and subjected to a transverse periodic excitation. It is considered that the functionally graded carbon nanotubereinforced composite (FG-CNTRC) beam is made of an isotropic matrix reinforced by either aligned- or randomly oriented-straight single-walled carbon nanotubes (SWCNTs) with four types of distributions through the thickness direction of the beam. Both the Eshelby–Mori–Tanaka approach and extended rule of mixtures are used to predict the effective material properties of the FG-CNTRC beams. The mathematical model of the beam is developed based on the Euler–Bernoulli beam theory together with von Kármán assumptions. Subsequently, the accurate analytical solutions of the governing equation are obtained through applying the variational iteration method (VIM). Several examples are verified to have higher accuracy than those available in the literature. In addition, a comprehensive investigation into the effect of carbon nanotubes (CNTs) distribution, CNTs volume fraction, end supports, vibration amplitude, and foundation coefficients on the vibrational characteristics of the nanocomposite beam is performed and some new results are presented.