In this paper, the effect of different profile variations on vibrational properties of non-uniform beams made of graded porous materials is studied. Timoshenko beam theory is used to present the mathematical formulation of the problem including shear deformation, rotary inertia, non-uniformity of the cross-section, and graded porosity of the beam material. Three different variations of porosities through the thickness direction are introduced. The beam is assumed with the clamped condition at both ends. To obtain a numerical solution, finite element formulations of the governing equations are presented. The non-uniform beam is approximated by another beam consisting of n elements with piecewise constant thickness to keep the volume and hence the total mass unchanged for each element. The beam response has been calculated for the first three modes of vibration. In each case, the results for different types of thickness variation and porosity distribution are compared with those obtained for a beam with uniform thickness. The effects of non-uniformity, taper parameters, and porosity distribution on the frequencies and mode shapes are investigated. It is observed that a considerable change in frequencies and mode shapes can be achieved by selection of different thickness variation and porosity distribution.
Optimization of a rectangular plate piecewise constant thickness when bending, prepared from the composite material.
