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

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.