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The effect of added point masses on the geometrically non-linear vibrations of SCSC rectangular plates

Hamdani Mustapha, El Kadiri Mounia, Benamar Rhali

Diagnostyka

2022

Vol. 23, No. 2

art. no. 2022206

A point mass added to a plate may have a significant effect on its linear and nonlinear dynamics, including frequencies, mode shapes and the forced response to external loading. In the present paper, a simply supported clamped simply supported clamped rectangular plate (SCSCRP) carrying a point mass is examined. The expressions for the kinetic, linear and non-linear strain energies are derived by taking into account the effect of the added mass on the kinetic energy and the effect of the membrane forces induced by the non-linearity on the strain energy. The discretization of these expressions makes the mass tensor, the linear and non-linear rigidity tensors appear in a non-linear algebraic multimode amplitude equation, the iterative solution of which permit to obtain, in the neighborhood of the first non-linear mode, the basic SCSCRP function amplitude dependent contribution coefficients. Nonlinear frequency response functions have been obtained for the first time, based on an iterative numerical solution in each case of the associated complete set of nonlinear algebraic equations. Such new results are useful for a better qualitative understanding allowing an optimal dynamic design of the rectangular plates with added masses.

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.