On The Dynamic Stability of Functionally Graded Material Beams Under Parametric Excitation

The dynamic stability of functionally graded material (FGM) beams subjected to parametric excitation is studied using finite element method. First order shear deformation theory (Timoshenko beam theory) is used for the analysis of the beams. The shape functions for the beam element are established from the differential equation of static equilibrium. Floquet’s theory is used to establish the stability boundaries. A steel-alumina functionally graded ordinary (FGO) beam with steel-rich bottom is considered for the analysis. For the analysis of functionally graded sandwich (FGSW) beam, alumina and steel are chosen as top and bottom skin respectively and the core is FGM with steel and alumina as constituent phases. The material properties in the direction of thickness of FGM are assumed to vary as per power law and exponential law. The effect of property distribution laws on critical buckling load, natural frequencies and parametric instability of the beams is investigated. Also, the effect of variation of power law index on the critical buckling load, natural frequencies and dynamic stability of beams is determined. It is found that the property variation as per exponential law ensures better dynamic stability than property variation as per power law. Increase in the value of power law index is found to have detrimental effect on the dynamic stability of the beams. Influence of the elastic foundations on the dynamic stability of the beams is studied. Pasternak elastic foundation is found to have more enhancing effect on the dynamic stability of the beam than Winkler elastic foundation. The dynamic stability of FGO and FGSW beams used in high temperature environment is investigated. It is observed that increase in environmental temperature has an enhancing effect on the instability of the beams. The effect of beam geometry, rotary inertia, hub radius and rotational speed on natural frequencies as well as on the parametric instability of rotating FGO and FGSW cantilever beams is studied. It is observed that increase in rotational speed enhances the dynamic stability of the beams. Parametric instability of a pre-twisted FGO cantilever beam is investigated. The effect of property distribution laws and pre-twist angle on critical buckling load, natural frequencies and parametric instability of the beam is studied. The increase in the value of power law index is found to have enhancing effect on the parametric instability of the beam. The increase in pre-twisting of the beam reduces the chance of parametric instability of the beam with respect to the first principal instability region. But the increase in pre-twist angle has a detrimental effect on the stability of the beam for second principal instability region