Functionally graded bars with enhanced dynamic performance

This paper presents analytical solutions for enhancing the dynamic performance of functionally graded material bars in axial motion. Optimized designs with maximized natural frequencies under mass equality constraint are given and discussed. The composition of the construction material is optimized by defining the spatial distribution of volume fractions of the material constituents using either continuous or discrete variations along the bar length. Three cases of boundary conditions have been examined: fixed-fixed, fixed-free and free-free bars. The major aim is to tailor the mass and stiffness distributions in the axial direction so as to maximize the frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration without the penalty of increasing total structural mass. The resulting optimization problem has been formulated as a nonlinear mathematical programming problem solved by invoking the Matlab optimization toolbox routines, which implement the method of feasible directions interacting with the associated eigenvalue problem routines. The proposed mathematical models have shown that the use of material grading can be promising in optimizing natural frequencies without mass penalty.