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3 2017

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A homogenization procedure and a physical discrete model for geometrically nonlinear transverse vibrations of a clamped beam made of a functionally graded material

100%

Rahmouni A. , Benamar R.

tom Vol. 18, No. 3

15--20

Functionally graded materials are used in aircrafts, space vehicles and defence industries because of their good thermal resistance. Geometrically nonlinear free vibration of a functionally graded beam with clamped ends (FGCB) is modeled here by an N-dof discrete system presenting an equivalent isotropic beam, with effective bending and axial stiffness parameters obtained via a homogenization procedure. The discrete model is made of N masses placed at the ends of solid bars connected by rotational springs, presenting the flexural rigidity. Transverse displacements of the bar ends induce a variation in their lengths giving rise to axial forces modeled by longitudinal springs. The nonlinear semi-analytical model previously developed is used to reduce the vibration problem, via application of Hamilton’s principle and spectral analysis, to a nonlinear algebraic system involving the mass and rigidity tensors mij and kij and the nonlinearity tensor bijkl. The material properties of the (FGCB) examined is assumed to be graded according to a power rule of mixture in the thickness direction. The fundamental nonlinear frequency parameters found for the (FGCB) are in a good agreement with previously published results showing the validity of the present equivalent discrete model and its availability for further applications to non-uniform beam.

Linear and geometrically non-linear frequencies and mode shapes of beams carrying a point mass at various locations. An analytical approch and a parametric study

100%

Adri A. , Benamar R.

tom Vol. 18, No. 2

13--21

In the present paper, the frequencies and mode shapes of a clamped beam carrying a point mass, located at different positions, are investigated analytically and a parametric study is performed. The dynamic equation is written at two intervals of the beam span with the appropriate end and continuity conditions. After the necessary algebraic transformations, the generalised transcendental frequency equation is solved iteratively using the Newton Raphson method. Once the corresponding program is implemented, investigations are made of the changes in the beam frequencies and mode shapes for many values of the mass and mass location. Numerical results and plots are given for the clamped beam first and second frequencies and mode shapes corresponding to various added mass positions. The effect of the geometrical non-linearity is then examined using a single mode approach in order to obtain the corresponding backbone curves giving the amplitude dependent non-linear frequencies.

A discrete model for nonlinear vibrations of a simply supported cracked beams resting on elastic foundations

100%

Khnaijar A. , Benamar R.

tom Vol. 18, No. 3

39--46

The present paper introduces a discrete physical model to approach the problem of nonlinear vibrations of cracked beams resting on elastic foundations. It consists of a beam made of several small bars, evenly spaced, connected by spiral springs, presenting the beam bending stiffness. The crack is modeled by a spiral spring with a reduced stiffness and the Winkler soil stiffness is modeled using linear vertical springs. Concentrated masses, presenting the inertia of the beam, are located at the bar ends. The nonlinear effect, due to the axial forces in the bars resulting from the change in their length, is presented by longitudinal springs. This model has the advantage of simplifying parametric studies, because of its discrete nature, allowing any modification in the mass and the stiffness matrices, and in the nonlinearity tensor, to be made separately. After establishing the model, various practical applications are performed without the need of going through all the formulation again. Numerical linear and nonlinear results are given, corresponding to a cracked simply supported beam.